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The utilization of the principles of natural and sustainable design in the development of a building system

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THE COOPER UNION
ALBERT NERKEN SCHOOL OF ENGINEERING
TITLE
The Utilization of the Principles of Natural and Sustainable
Design in the Development of a Building System
by
Matthew
Waxman
A thesis submitted in partial fulfillment
of the requirements for the degree of
Master of Engineering
DATE
May 11. 2010
Cosmas Tzavelis, PhD, P.E.
Advisor
UMI Number: 1484975
All rights reserved
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UMI 1484975
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The Cooper Union for the Advancement of Science and Art
THE COOPER UNION FOR THE ADVANCEMENT OF SCIENCE AND ART
ALBERT NERKEN SCHOOL OF ENGINEERING
This thesis was prepared under the direction of the Candidate's Thesis Advisor and
has received approval. It was submitted to the Dean of the School of Engineering
and the full Faculty, and was approved as partial fulfillment of the requirements for the
degree of Master of Engineering.
S , StyvAtyw ^ e v - r W '
S-tl-iQ
Dean, School of Engineering - May 11, 2010
Professor (JosTnas^T/dvclisTPhD, P.E.
Candidate's Thesis Advisor
Acknowledgment
I would like to thank all those that in any way assisted with the completion of this
Master's Thesis. In particular, I would like to express my gratitude to Professor Cosmas
Tzavelis, who was immensely helpful in guiding my pursuit of this project. Furthermore,
I'd like to thank my classmates who provided advice ranging from the most minute
details to the more significant issues encountered. I would also like to acknowledge the
Cooper Union Library and New York University's Bobst Library, my research would
have been nearly impossible without them. The University of Maine's AEWC Advanced
Structures and Composites Center's Bridge-in-a-Backpack™ project must be mentioned
as it was a tremendously innovative, insightful and inspirational example of the potential
of fiber reinforced composites.
I would also like to thank all the staff and faculty of the Cooper Union for the
Advancement of Science and Art that have indirectly contributed to this accomplishment
at some point during my undergraduate and graduate years.
i
Abstract
The Green Design movement has been growing in momentum in recent years
spurring technological and architectural innovation in the building industry. The
contributions of the structural engineering community, however, have been limited. Yet,
there are many systems seen in nature that could have applications in structural
engineering that may be of a beneficial nature if correctly applied. Thus, a search of the
natural world was conducted for specific examples of remarkable structural aspects as
witnessed in plants and animals. The termite mound, bee hive, spider web, and the use of
fluids structurally were identified as cases of interest. The potential of fiber reinforced
composites, as observed in the Bridge-in-a-Backpack™, was subsequently discovered.
Utilizing a combination of the various principles identified in the aforementioned
points of research, a unique building system was developed, which was intended to
reduce the amount of structural material, as well as transportation and construction
demands. A feasibility study was conducted to quantify the material quantity savings
over a conventional design, if any. Therefore, two computer models were constructed and
analyzed. The results of this feasibility demonstrated that the conceptual design is
potentially beneficial, although further investigation is necessary.
Moving forward with the proposed building system requires further evaluation.
The computer model must be refined and made more encompassing. Eventually,
laboratory testing would be necessary to evaluate and further develop the proposed use of
fiber reinforced composites. There is also room for expansion of the role of fiber
reinforced composites in the building system. A more precise method of estimating cost
as well as assessing environmental impact must be developed.
ii
TABLE OF CONTENTS
1
INTRODUCTION
1.1
1
STATEMENT OF PROBLEM
2
9
RESEARCH
11
2.1
TERMITE MOUND
12
2.2
BEEHIVE
14
2.3
SPIDER WEB
16
2.4
FLUIDS
18
2.5
FIBER REINFORCED POLYMERS
19
2.5.1
2.6
3
BRIDGE-IN-A-BACKPACK™
26
CONSEQUENCES OF RESEARCH: ADDRESSING PRACTICALITY AND INSPIRATION
CONCEPT
29
34
3.1
INITIAL CONCEPT
35
3.2
REVISED CONCEPT
39
3.3
FINAL DESIGN
44
4
FEASIBILITY STUDY
47
4.1
PRECEDENT
48
4.2
COMPUTER MODELS
51
4.3
RESULTS
55
5
CONCLUSION AND FUTURE WORK
58
6
WORKS CITED
61
7
APPENDIX
64
7.1
BRIDGE-IN-A-BACKPACK™
7.1.1
65
CARBON FOOTPRINT ANALYSIS
68
7.2
ARCHES
71
7.3
BICYCLE WHEEL
76
7.4
FLAT SLAB
93
7.5
DOMED BUILDING CONCEPT
102
7.6
HEXAGONAL MESH AND FIBER REINFORCED BEAM
104
7.7
FIBER REINFORCED POLYMER DROP PANEL
108
7.8
CONCEPTUAL BUILDING DESIGN
111
7.9
COMPARISON OF CONCEPTUAL AND CONVENTIONAL DESIGN FRAME FORCES
115
in
7.10
SPECIFICATIONS FOR STRUCTURAL STEEL BUILDINGS
IV
129
LIST OF TABLES AND FIGURES
Figure 1.1: Green Classification
5
Figure 2.1: Magnetic Termite Mounds
13
Figure 2.2: Rhombic Decahedron
14
Table 2.1: Comparison of Natural Silks to High Tensile Steel
16
Figure 2.3: Overview of the Creation Process of Fiber Reinforced Composites
24
Table 2.2: Properties of Several Reinforcing Fibers
25
Table 2.3: Select Properties of Four Fiber Reinforced Polymers
25
Figure 2.4: The Neal Bridge
28
Figure 2.5: Elongated Rhombic Decahedron as a Skyscraper
31
Figure 2.6: Use of Fluid as a Structural Column
33
Figure 3.1: Illustration of the Flattening of Arches
36
Figure 3.2: Domed Structure
37
Figure 3.3: Steel Framing System Following a Hexagonal Mesh
39
Figure 3.4: Typical Concrete Design
43
Figure 4.1: The Lily Pads of the Johnson Wax Administration Building
49
Figure 4.2: Typical Suspend Floor Building Configuration
50
Table 4.1: Load Combinations per AISC-LRFD 99/ASCE 7-05
52
Figure 7.1: Proposed Lateral Bracing for Domed Building
102
Figure 7.2: Aerial View of Proposed Lateral Bracing
103
Figure 7.3: Overhead View of Hexagonal Mesh
104
Figure 7.4: Elevation View of Hexagonal Mesh
105
Figure 7.5: Isometric View of Hexagonal Mesh
106
Figure 7.6: Select Views of Concrete-Filled Fiber Reinforced Polymer Beam with Prestressing Tendon
107
Figure 7.7: Elevation View of FRP Drop Panel
108
Figure 7.8: Aerial View of FRP Drop Panel
109
Figure 7.9: Underside View of FRP Drop Panel
110
Figure 7.10: Plan View of Final Concept
111
Figure 7.11: Elevation View of Final Concept
112
Figure 7.12: View of Transfer Truss- Final Concept
113
Figure 7.13: Section View of Concrete-Filled FRP Drop Panel-Steel Column-ConcreteFilled Steel Deck Details-Final Concept
114
Figure 7.14: Reference of Frame locations
116
Table 7.1 Conceptual Model [1.2DL+1.6 LL (50 psf)]
117
Table 7.2: Conventional Model [1.2DL+1.6 LL (50 psf)]
119
Table 7.3: Conceptual Model - Conventional Model [1.2DL + 1.6 LL (50 psf)]
121
Table 7.4: Conceptual Model [1.2DL + 1.6 LL (100 psf)]
123
Table 7.5: Conventional Model [1.2DL + 1.6 LL (100 psf)]
125
v
Table 7.6: Conceptual Model - Conventional Model [1.2DL + 1.6 LL (50 psf)]
VI
127
1
INTRODUCTION
For as long as mankind can remember the flight of birds has been a natural
phenomenon of immense interest. Thus, the Wright Brothers' tireless, and in many ways
dangerous, research and development of the airplane was no surprise. Just as the ability
of flight preoccupied the thoughts of our ancestors, the mastery of flight, along with
advancements in science and technology, has turned our attention to other mysteries of
the natural world. Furthermore, the recent fears of climate change and the ensuing
environmental consciousness have shifted, to differing extents, the mindset of progress.
That is, no longer is the sole motivation of technological advancement convenience, but
rather "Green Design" has entered the fray.
While Green Design, or Sustainability has become all encompassing in recent
years, "urgent attempts by architects now to produce a green architecture in both
blueprint and in reality have in their ancestry a sequence of ideas and buildings" (Farmer,
5). That is, with respect to the design of buildings, the idea of environmentally friendly
structures is not specific to recent years. Furthermore, the achievements of modern
architects and engineers in Green Design, while perhaps being assisted by modern
technology, are not entirely novel, but likely represent the progression of previously
developed concepts; there are numerous examples of existing structures, designed and
built by past architects and engineers, that stand as a testament to ecologically-driven
design. In this brief discussion of the genesis of Green Design, it is evident that a
definition of Green Design is necessary.
John Farmer, author of Green Shift: Changing Attitudes in Architecture to the
Natural World, states that:
1
For some designers... [Green Design] means
retreating from mass production and high technology in
favour [sic] of a sophisticated knowing primitivism, while
for others the mass industrial and scientific techniques that,
used rampantly, have contributed to our various ecological
crises are seen as crucial to solving them. Some of this
ecologically concerned effort is constructional, some is
aesthetic—symbolic of green values, paralleling such
expressions in the other arts. With only a few exceptions all
are unconnected to the mass of the current design, and
without connection either to the history of modern
architecture as presently perceived—except to one aspect
or other of the organic tradition. (Farmer, vii)
Within Farmer's "framework" of Green Design, there is a definite divergence in
approach. One approach, "sophisticated knowing primitivism," as Farmer calls it,
embraces minimalism—taking a holistic approach: stripping structures down to their
simplest form by eliminating superfluous elements and/or luxuries. It is through this
simplicity—inevitably reducing electrical and water usage and raw material
consumption—that sustainability is obtained. The second approach that Farmer
enumerates focuses on the integration of technological innovation, e.g. photovoltaic cells,
gray water systems, etc., as a means of reducing environmental impact. These two
approaches are not mutually exclusive; an individual can design a building, which
combines a simplified form with cutting-edge technology. However, it is important to
note that at their essence, these two approaches represent different cultural ideologies: the
"sophisticated knowing primitivism" approach being decidedly counter-cultural at its
extreme, whereas the technological innovation approach could be viewed as a means of
abating, but more so perpetuating the negative environmental impacts inherent in our
society as presently constituted. In other words, an individual in support of "primitivism"
2
might favor town planning that removes the need for automobile transportation, while an
individual who supports using innovations might instead favor leaving an area unchanged
and instituting the use of hybrid cars. In addition to these different approaches, Farmer
notes two different potential applications.
One purpose of Green Design is "constructional," to borrow Farmer's term, which
is fairly straightforward. A constructional design aims to impart environmental
advantages, regardless of form, through the use of technology, improved methodology, or
the like. In contrast, the second application is in aesthetics—meaning form driven
design—perhaps best being described as biomorphic or by Organic Design, which could
or could not be viewed as "belonging to Green Design." In the following quotation,
Farmer elaborates on the distinction between Organic and Green design:
The word organic, misused often in hopelessly
fuzzy ways, has come to stand for that strand in
architecture which has empathy in one way or another with
natural forms. The narrower definition of the idea of the
organic which Giedion, and then Bruno Zevi set down
formed an honoured [sic] but marginal adjunct to the
mainstream of modern architecture; it was always a loose
one and has become more confused since. In contrast to
this conception of the organic we have the idea of a green
architecture, of buildings which use lightly the earth's
resources, and which are expressive also of a way of living
which thinks in terms of partnership with nature. It may be
organic in those senses, but equally may be 'inorganic,'
relating to the physical rather than biological aspects of
nature. (Farmer, 5-6)
Farmer's characterization of organic versus green centers on the idea of natural forms
versus tangible environmental benefits, respectively. Farmer mentions that even his brief
explanation of organic is a sort of consensus, implying that it could be viewed differently.
3
As example, iconic architect Frank Lloyd Wright's own idea of organic design differs,
instead meaning "an architecture that develops from within outward in harmony with the
conditions of its being [i.e. the specific environment in which it is built]" (quoted in
Farmer, 131), which though vague, does suggests some degree of integration of Organic
and Green Design. To even further complicate this idea of classifying Green Design is
the idea of Natural Design or construction: "a rational form-finding process following
natural laws... [that] is also part of a larger vision directed at a peaceful and free society
in harmony with itself and nature" (Nerdinger, 15), such as practiced by Frei Otto. Before
further discussing natural design, however, some sort of classification system for the
different types of sustainable design should be settled.
The following classification system is not authoritative, but simply an attempt to
clarify for the purposes of this discussion within the scope of this paper; see Figure 1.1
below for a visual representation of the classification. Sustainable Design and Green
Design are not identical terms; Sustainable Design is solely motivated by the reduction in
negative environmental impacts whereas Green Design encompasses anything that is in
anyway motivated by nature, whether in terms of appearance or functionality. However,
these two terms: Sustainable and Green Design, are for the purposes of this discussion
interchangeable. Green Design can be carried-out in a constructional manner or
aesthetically. In turn, the constructional application can be approached technologically or
holistically (knowing sophisticated primitivism). From the aesthetic approach comes the
idea of organic or biomorphic design, which as mentioned above could stand apart from
Green Design and furthermore, could in the spirit of Frank Lloyd Wright's quotation, just
4
as easily include constructional elements. Lastly, Natural Design is a sort of culmination
of everything mentioned above.
Sustainable/Green Design _
Constructional
Technological Innovation
Aesthetic
1
1
Organic/Biomorphic Design
Sophisticated Knowing Primitivism
Natural Design
- ^
Figure 1.1: Green Classification- A Clarification of the different sustainable practices
The idea of Natural Design or so-called "Biomimicry" focuses on the fact that
flora and fauna perform functions with efficiency, but more so, an ecological affect that,
on the whole, modern science envies. An example of science versus nature is the
production of high tensile steel versus spiders' web. Consider that "the strength of
[Araneus spiders' (a genus of spiders)] MA silk [strands originating from the major
ampulatte gland] is very high for a biological material, and comparable with, although
less than, that of high tensile steel" {Animal, 57). Furthermore, it is important to keep in
mind that high tensile steel requires an immense amount of energy to produce—mining
and transportation of materials combined with very high temperature refinement—
compared to a naturally occurring material such as spider silk; such is the motivation of
Natural Design, although with respect to buildings. That is, utilizing processes and/or
designs—constructions—found in the natural world in order to simplify the construction
of structures, resulting in an overall improved efficiency. The following Frei Otto
5
quotation provides further insight into possible avenues of Natural Design and
constructions:
When we speak of natural constructions... we are
not concerned so much with any specific examples of the
infinite variety of possible objects, but rather with that
which is typical. We are concerned with those
constructions which demonstrate with particular clarity the
physical, biological, and technical processes which objects
give rise to. We are concerned with the fundamental—to
which we even refer as the 'classical.' Even if technology,
too, is a tool of the natural objects we call human beings, a
means which humans, as a species, employ to prevail
against nature—even if technology is used against nature
and is therefore unnatural—we nonetheless view it as a
product of human beings and, as such, ultimately a part of
nature. Human beings control nature. They have put nature
to their own uses. But they are at last beginning to
recognize that, in doing so, they are harming and disrupting
and destroying nature. Increasingly, therefore, they are
searching for ways to preserve nature and to be a part of it
rather than fighting against it. And nature-oriented
technology is the tool they use to achieve this. This is the
technology we mean when we speak of natural
constructions, (quoted in Barthel, 17)
To clarify the above remarks, Otto's personal interest in the area of Natural Design
chiefly concerns that of natural constructions of a specific variety, namely that of
autonomous formation processes. As Rainer Barthel explains, by using autonomous
formation, "a large proportion of the form-finding process is subject to laws which are
outside the designer's control" (Barthel, 18). That is, by studying autonomous formation
processes the designer may be able to drastically increase the efficiency and reduce the
construction expenditures of a particular design by utilizing shapes which have a natural
propensity for some situation. An example of such a formation is the use of a cable,
6
which "stretched between two points will find its own shape when it is placed under
load" (Barthel, 18). The form that the cable takes under a given load illustrates the ideal
form for a tensile element; therefore, when this shape—that is to say, not necessarily
involving the use of a cable—is applied correctly, it can be exploited to increase the
efficiency of a given design. More so, in the case of the cable, since the cable does not
have to be molded or manipulated into the form—simply fixing the cable at its ends and
applying a load will induce the form—the material and construction demands of this
element are diminished (Barthel, 18). Thus, there are varying degrees of autonomy. The
simplest form of autonomous formation being an element's shape which has been derived
from an investigation rooted in the laws of physics, while the highest level of
autonomous design involves self-assembling elements as in the case of the cable.
Therefore, Otto's statement refers to the fact that his pursuit of autonomous
formations is not really focused on gaining ideas from one case, but in identifying and
understanding universal concepts that can be applied with practicality. Otto continues to
state that anything man-made should be considered part of nature and therefore, may
serve a hand in Natural Design. The Natural Design of Frei Otto is only one example of
the "harnessing" of nature.
The numerous ways in which green design has influenced modern times is a clear
indication of its practicality. Even when green design is applied only aesthetically,
evoking the principles of organic design, yielding no or limited environmental benefits,
the resulting structure can be quite spectacular. For those owners unconcerned with
ecologically driven design the advantages of green design can be economical as well, not
necessarily from an initial cost, but more likely a long-term cost stand point due to
7
potential electric and water use savings. Thus, green design, in its numerous forms
appears to be part of the convention of the near future.
1.1
STATEMENT OF PROBLEM
While it is easy to discuss Green Design, from a structural stand-point it is hard to
achieve. Conventional materials, steel and concrete, as well as standard practices
continue to have a stronghold on the way that buildings are designed and built. Despite
the concepts previously discussed, the large majority of sustainable practices and/or
technologies applied to buildings are completely independent of the structural system.
That is, today's "green buildings" are for the most part sustainable due to improved
mechanical, electrical, and plumbing (MEP) systems and devices. Indeed, the
contribution of architects has largely been visual and spatial, not directly addressing
energy and material consumption. Meanwhile, the approach of structural engineers has
been relatively stagnant.
Innovations in structural materials such as fiber reinforced polymers, fly ash
cement, ground granulated blast furnace slag cement, and alternative aggregate concretes
have not been widely utilized. The only innovation embraced has been the use of
computers for analysis and design. Typically, the use of software improves member
efficiency in contrast to manual methods, but the degree of which is arguable—rather, the
savings is more so in terms of time and in turn, money. Part of the problem is that
engineers are restricted by their role, usually being subordinate to the architect, and must
realize a design that they have very little input into.
Thus, this project seeks to produce a building, designed exclusively by a structural
engineer, and therefore, yield a unique structural system. By bypassing the architect, the
hope is to create a purely form-follows-function structure, by eliminating grandiose
architectural notions in favor of performance. However, the role of the architect: "to
9
reconcile... expression with the functional and social duties of a building" (Farmer, 6),
should not be forgotten; it has been minimized. It is with the Green Design concepts
previously discussed in mind, specifically those of Natural Design, that the following
thesis has been pursued—integrating elements of, or inspiration from nature in structural
engineering in such a way as to achieve something potentially beautiful, but more so
something which is more efficient than conventional building methods from a material
and constructability stand-point.
10
2
RESEARCH
The initial phase of research for this thesis involved an exploration of the natural
world. That is, a wide ranging search for animal buildings of a substantial nature took
place. In addition, although to a lesser extent, the diverse forms and functions of plants
was also considered. A few structures of note were identified from this initial research
including the Anthill/Termite Mound, the Bee Hive, and the Spider Web. Furthermore,
the concept of using a fluid as a structural element was encountered. Stilt Roots, as
observed in tropical trees, are another item of note. It is worth noting at this point that
there are myriad "natural designs" worth further investigation for potential integration
into many different fields of study, however, this thesis is limited to those that may
contribute to the field of Structural Engineering and fit within the proposed scope of this
thesis. Of the concepts identified above, which will henceforth be referred to as a
"structural system," each one has specific elements of interest.
In addition to the systems enumerated above, the use of newfangled materials was
considered. Of particular interest was the use of fiber reinforced plastics (FPRs), such as
fiberglass and carbon fiber. These materials are on the forefront of many different
industries, including construction, offering high strength to density ratios. A significantly
important use of FPRs encountered, which has greatly influenced the evolution of this
thesis and probably will be farther reaching, is the University of Maine's Bridge-in-aBackpack™ Method. A brief background of these structural systems follows.
11
2.1
TERMITE MOUND
The Termite Mound is a structural system that has attracted much attention from
the scientific community due to a number of key features. First and foremost, from a
structural engineering perspective, the Termite Mound represents a tremendous vertical
accomplishment; the building height to building occupant ratio may be as high as 800
(Built, 93). Applying said ratio to an average human height of 5 feet 6 inches equates to a
building 4,400 feet or 5/6th of a mile tall. Although Termite Mounds do not have very
high aspect ratios (height divided by footprint), having relatively large footprints, for an
insect to achieve such heights is remarkable. Termites construct these relatively massive
structures mostly through the use of a soil/fecal cement composite (Built, 42). The
Termite Mound's sheer size is not the only feature of note.
Termite Mounds possess natural ventilation systems, which are driven by either
"temperature or pressure differences" (Built, 94). Much research has been devoted to
studying the "mechanics" behind these natural ventilation systems; there are some
variations from one termite species to the other. In addition, the Magnetic Termite
Mounds of Australia—called so because of the flat surface of the Eastern and Western
faces and the North to South orientation of the long axes, see Figure 2.1 below—were
believed to demonstrate passive heating/cooling, however, later research has shown that
this shape may actually be to allow rapid drying during the wet season (Built, 44-45).
This is to say, the shape maximizes the surface area to volume ratio. Nonetheless, the
Magnetic Termite Mound is clearly indicative of the ingenuity and adaptability of
Termite Mounds in general.
12
Figure 2.1: Magnetic Termite Mounds- As seen in the Cape York Peninsula of
Queensland and the Northern Territories of Australia
Martin Harvey/NHPA
(Built, 44)
Termite Mounds also demonstrate the concept of symbiotic co-existence. That is,
the Macotermes species utilize fungi housed in special horticultural areas of the mound to
help digestion (Built, 42). The potential longevity of Termite Mounds is perhaps the most
extraordinary feature of this structural system. Consider that an age study has carbon
dated the material from the core of a mound from 4,000 to 5,500 years old (Built, 31).
13
2.2
BEEHIVE
Like the Termite Mound, the Beehive has also been consistently studied
particularly for its geometry. A beehive consists of many honeycombs, which is itself a
mesh of hexagonal wax cells; mesh is a term to describe the partitioning of a surface
using a sequence of one or many shapes. Furthermore, the wax cells of the honeycombs
are oriented "such that two opposing honeycomb layers nest into each other, with each
facet of the closed ends being shared by opposing cells, and with the open ends facing
outward" (Sarcone). Of particular interest are the wax cells, which are rhombic
decahedrons—that is, a hexagonal prism with a modified closed-end consisting of three
rhombi, see Figure 2.2. The wax cells demonstrate remarkable isoperimetric properties.
In short, the wax cell has essentially the largest known volume to surface area ratio.
(Sarcone)
Figure 2.2: Rhombic Decahedron- Consisting
of a hexagonal prism with three rhombi
forming the closed end
Gianni Sarcone (Archimedes Laboratory)
The modifier "essentially" was used because Laszlo Fejes Toth devised a slightly
more efficient cell, less than 0.1% more efficient, in 1965. The nominal improvement of
the Toth structure, however, does not reflect real conditions as proven by Denis Weaire
14
and Robert Phelan. The experiments of Weaire and Phelan, which were performed using
foam, thus, approximating walls of infinitesimal thickness, revealed that once the
thickness of the walls of the cell is taken into account, i.e. when liquid was added to the
foam generating a finite thickness, the geometry of the wax cell (rhombic decahedron)
dominates. (Sarcone)
The pattern of the honeycomb itself warrants discussion, as the hexagonal mesh is
best. That is to say, the hexagonal mesh provides the greatest amount of volume per
construction output. As a result, the beehive exhibits desirable structural performance as
well: "one kilogram of wax turned into comb can support 22 kilograms of honey"
(Hersey, 65), and because it is a space frame the amount of supports necessary is at a
minimum. It should be noted that not all beehives utilize a hexagonal mesh. (Hersey, 65)
15
2.3
SPIDER WEB
One could argue that spider webs are more aptly described as traps rather than
shelter and are therefore impractical in terms of the problem statement of this thesis, and
while this may hold true, the mechanics behind the spider web are worthy of discussion,
especially since this structural system was contemplated.
As previously mentioned, the relative strength of spider silk to steel is remarkable
considering its biological nature. Consider that "the strength of [Araneus spiders' (a
genus of spiders)] MA silk [strands originating from the major ampulatte gland] is very
high for a biological material, and comparable with, although less than, that of high
tensile steel" {Animal, 57). Furthermore, it is important to keep in mind that high tensile
steel requires an immense amount of energy to produce—mining and transportation of
materials combined with very high temperature refinement—compared to a naturally
occurring material such as spider silk. Table 2.1 below lists several mechanical
properties of the silk of different organisms with that of steel.
Table 2.1: Comparison of Natural Silks to High Tensile Steel
Organism
Araneus
Araneus
Araneus
NephUa
Bombyx
Steel
Silk type
MA
FLfvisrid spiral}
Ml
MA
Cocoon
High tensile
Stiffness. Strengths
E„,(Gto)
o^(GPa)
10
0.003
22
7
200
1.1
0.5
1.2
1.3
0.6
15
Extensibility Toughness Hysteresis
e**
(MMr3)
(%)
0.27
2.70
0.40
0.12
0.18
0 008
160
150
80
70
6
Data from Kohler and Vollrath (1995) and Gosline et al. (1999).
{Animal, 59)
16
65
65
56
The mechanical stresses of the spider web are almost exclusively tensile, that is, the
web is constructed entirely of "cable" elements. The fact that spider webs are constructed
of cables should come as no surprise, however, the fact that materials such as steel are
strongest in tension, see Section 7.10, which provides an excerpt from the AISC Steel
Construction Manual, means that the spider web represents the most desired mode of
loading. Clearly, there is a caveat that goes along with this; the loads imposed on spider
webs must clearly be transmitted to the ground in some way—without levitation
something must be in compression—in the case of spider webs the compression element
is whatever the web is attached to: a tree, a window frame, etc. However, the spider web
avoids the complication of bending moments, which are the most difficult design
condition. Spider silk possesses a great deal of strength, but lacks stiffness (Built, 168).
While the strength of spider silk had been lauded previously in comparison with
steel it should be noted that the stiffness falls well short of steel. As seen in Table 2.1, the
MA silk of the Araneus spider is twenty times less stiff than steel, and has more than
twenty times more extensibility (Animal, 59). The low stiffness and high extensibility of
spider silk plus the property of hysteresis, however, serves well for its purpose of
catching "fast-moving prey" (Built, 168); hysteresis means that the silk does not behave
the same during loading and unloading. More specifically, during the unloading phase the
stress required for a certain strain is less than it was during the loading phase; the cycle is
not perfectly elastic. Thus, the web does not rebound at the same rate as the initial
deflection, preventing prey from being launched off of the web. (Built, 168-170)
17
2.4
FLUIDS
The utilization of fluid as a load-bearing element may seem strange or implausible,
but there are various devices in daily life where such is the case; take the air tires of an
automobile as an example (French, 73). Clearly, the use of a fluid as a structural element
is intended only for certain instances: "while tension members must be solids, a
compression member can be fluid" (French, 141). Indeed, many organisms such as
"earthworms, slugs, [and] sea-anemones... rely on liquids for the compressive part of
their structure" (French, 141). As further example, flowers wilt when not watered,
because their stems consist of flexible fibers that rely on water to provide pre-stressing,
which stabilizes the stem (141). Discussion of the earthworm offers further insight into
the function of fluids as structural elements in organisms.
An earthworm is essentially a tube filled with fluids and organs; the tube itself is
comprised of muscles acting along the circumference and those running along its length,
while the internal fluids are for all intents and purposes incompressible. Utilizing the
incompressible nature of the fluids, by contracting its circumferential muscles the
earthworm undergoes a forcible elongation. This act of elongation allows the earthworm
to penetrate the ground. Alternatively, by contracting its longitudinal muscles the
earthworm can bend sideways. This sideways motion requires the development of an
internal bending moment; the tensile component of this moment is supplied by the
longitudinal muscles while the reactive, compressive component, is the worm's liquid
contents. (French, 141-142)
18
2.5
FIBER REINFORCED POLYMERS
The number of applications, types, and variations of fiber reinforced polymers
precludes an all-encompassing discussion. In truth, the study of FRPs could alone
comprise an entire thesis. Therefore, a brief history of the origins and explanation of the
composition and production, as well as the physical properties of FRPs will be presented.
At this time it should be noted that the term polymer and plastic are interchangeable.
Furthermore, a FRP is but one type of fiber reinforced composite. The distinction
between the term polymer and composite will be addressed in greater depth.
The first commercially successful FRP was Glass reinforced polymer (GRP aka
Fiber-Glass), originating in the 1940s (Smallman, 366). The original application was for
the radomes of airplanes (366). Subsequently, the types and applications of fiber
reinforced composites have grown. Today, fiber reinforced composites are used in almost
every industry imaginable including: "aircraft, space, automotive, sporting goods, and
marine engineering" (Mallick, 6). The topic to be discussed in most depth is the
characterization and properties of a fiber reinforced composite.
P.K. Mallick, author of Fiber-reinforced Composites: Materials, Manufacturing,
and Design, offers an excellent comprehensive explanation of fiber reinforced
composites:
Fiber-reinforced composite materials consist of
fibers of high strength and modulus embedded in or bonded
to a matrix with distinct interfaces (boundary) between
them. In this form, both fibers and matrix retain their
physical and chemical identities, yet they produce a
combination of properties that cannot be achieved with
either of the constituents acting alone. In general, fibers are
the principle load-carrying members, while the surrounding
matrix keep them in the desired location and orientation,
19
act as a load transfer medium between them, and protects
them from environmental damages due to elevated
temperatures and humidity, for example. Thus, even though
the fibers provide reinforcement for the matrix, the latter
also serves a number of useful functions in a fiberreinforced composite material. (Mallick, 1)
Thus, it can be seen that a fiber reinforced composite could be any combination of
materials, provided said combination can be bonded together. The typical reinforcing
fibers used include glass, carbon, Kevlar 49, boron, silicon carbide, and aluminum oxide,
while metals, ceramics, and polymers are the typical matrix materials used (Mallick, 1); a
composite using a metal, ceramic, or polymer matrix is referred to as a metal-matrix
composite (MMC), ceramic-matrix composite (CMC), or polymer-matrix composite
(PMC), respectively. The terms polymer-matrix composite and fiber-reinforced polymer
are synonymous. The distinction between a fiber reinforced composite and a fiber
reinforced polymer should now be clear—a FRP is a specific type of fiber reinforced
composite and stands in contrast to one consisting of a metal or ceramic matrix.
Furthermore, although this discussion is focused on FPRs/PMCs, MMCs and CMCs are
included for completeness/continuity. In addition to the matrix and fibers, fiber
reinforced composites contain coupling agents, coatings, and fillers; coupling agents and
coatings improve bonding between the fibers and the matrix, thereby improving
performance, while fillers serve to chiefly control cost and enhance dimensional stability
(15). A description of the effects of the chosen materials, fibers and matrix, follows.
The specific gravity, tensile strength and modulus, compressive strength and
modulus, fatigue strength and fatigue failure mechanisms, electrical and thermal
conductivities, and cost of a fiber reinforced composite is largely determined by the type
20
of fiber, its quantity and orientation (Mallick, 17); the variations and consequences of
fiber orientation will be touched upon later. The matrix, on the other hand, mostly
controls the interlaminar and planar shear characteristics (40). The matrix also provides a
certain amount of compressive strength in that it acts as a lateral support, reducing the
unbraced length, thereby increasing the critical buckling stress (40). A description of the
physical form of fiber reinforced composites follows.
Fiber reinforced composites intended for structural applications are typically
produced as laminates, especially FRPs (Mallick, 1). That is, fiber reinforced composites
consist of multiple layers, each layer a fiber-matrix composite itself, stacked on top of
each other and then fused together to a specified thickness (1). Furthermore, differing
layers can be stacked with varying sequence and fiber orientation to produce desired
physical and mechanical properties (1-2). A composite containing differing layers, in
terms of fiber material and not matrix material, is known as an interply hybrid laminate
(17). Intraply hybrid laminate refers to a composite in which there is two or more fiber
materials present in a single layer; once again there is only one matrix material present
(17). While interply and intraply hybrid laminates involve the use of a blend of fibers the
matrix is consistent; the term hybrid laminate, however, describes a composite containing
different matrices (17). Hybrid laminates typically consist of FRPs and aluminum sheets
with an intermediate adhesive layer (17). Besides the types of fiber(s) utilized there are
several different orientations and forms in which the fibers can be employed.
As far as the variations of fiber orientation mentioned earlier there includes
unidirectional, bidirectional, multidirectional and random (Mallick, 2, 17). Furthermore,
the fibers may be of a continuous or discontinuous form (1). Continuous fibers are
21
typically oriented in a uni- or bidirectional pattern—a typical bidirectional orientation has
the two directions perpendicular to each other (15). Discontinuous fibers are usually
oriented in a unidirectional or random fashion (17). See Figure 2.3 below for a graphical
depiction of the process of creating a fiber reinforced composite. The reason for choosing
one orientation or form (continuous vs. discontinuous) over the other is primarily a
function of application. Before fiber orientation and form can be addressed directly,
however, the general properties of fiber reinforced composites must be explained.
Fiber reinforced composites "offer a combination of strength and modulus that are
either comparable to or better than many traditional metallic materials... [and because] of
their low specific gravities, the strength-weight ratios and modulus-weight ratios of these
composite materials are markedly superior to those of metallic materials" (Mallick, 2).
Thus, it can be seen why use of these materials is in general advantageous, but despite
these desirable properties there is an inherent limitation: fiber reinforced composites
exhibit anisotropic properties (2). That is to say, as would be expected, fiber reinforced
composites demonstrate their greatest strength when stressed parallel to the fiber
orientation while the material is weakest when stressed perpendicular to the fiber
orientation (2). As a result, fiber reinforced composites that are expected to experience
principal stresses in two directions would be produced using bidirectional layers or
unidirectional layers could be stacked such that the fiber orientations are not parallel, for
example; multidirectional layers are an extension of this principal. Alternatively, the
utilization of discontinuous fibers implemented in a random orientation results in a
virtually isotropic material (17). Hence, fiber reinforced composites are essentially
custom designable materials (5). It is very important to realize that in order to avoid
22
complete anisotropy the material's strength is sacrificed to varying extents. That is, a
unidirectional fiber reinforced composite exhibits the greatest strength, albeit when
measured parallel to the fiber orientation, compared to the strength of a fiber reinforced
composite with a random orientation, although its strength is constant no matter the
direction of measurement. Generally, fiber reinforced composites consisting of
continuous fibers are stronger than those with discontinuous fibers, regardless of
orientation. Thus, the implications of fiber form and orientation should be understood.
To better understand the strength of the reinforcing fibers and fiber reinforced composites
see Table 2.2 and Table 2.3, respectively, below.
As can be seen in Table 2.2 the fiber with the greatest tensile strength is R glass, a
type of glass fiber, with a tensile strength of 3,600 MPa. In comparison, Table 2.1 lists
the tensile strength of high tensile strength steel at 1.5 GPa or 1,500 MPa. Therefore, R
glass is more than two times stronger than high tensile strength steel. In terms of
stiffness, however, it can be seen from Table 2.1 and Table 2.2 that the Modulus of
Elasticity of high tensile strength steel, 200 GPa or 200,000 MPa, is more than two times
greater than that of R glass, 86,000 MPa. As seen in Table 2.2, HM fibers, a type of
carbon fiber, on the other hand, exhibit a Modulus of Elasticity of 500,000 MPa—two
and a half times greater than that of steel—and HM fibers have a greater tensile strength
too. Furthermore, the densest material listed in Table 2.2 is Boron fibers; having a
density of 2.63 g/cm3. Compared with the density of steel, approximately?.85 g/cm3,
Boron is nearly three times less dense. The strengths of the composite materials listed in
Table 2.3 demonstrate more modest properties than the fibers themselves, but are still
beneficial over steel when considering density.
23
Figure 2.3: Overview of the Creation Process of Fiber Reinforced CompositesThe typical constituents, fiber forms and orientation, and stacking process of fiber
reinforced composites
CONSTITUENTS
FIBERS
+ MATRIX
+ C0UPLIN6 AGENTS
OR COATINGS
+ FILLERS
J
LAHIHA
(PLY, LAYER)
(a) UNIDIRECTIONAL CONTINUOUS
(b) BIDIRECTIONAL CONTINUOUS
m
m
(c) UNIDIRECTIONAL DISCONTINUOUS
(4) RANDOM DISCONTINUOUS
PH
<.) LMUIHTE
£>!*<&
*r
(Mallick, 16)
24
Table 2.2: Properties of Several Reinforcing Fibers
Type of fiber
Density
Modulus of elasticity
(MP*)
(gW)
Tensile strength
(MP«)
E glass
R glass
S glass
2.52
2.55
2.50
2,400
3,600
3,400
73,000
86,000
88,000
HM fibers
HS fibers
Aramid fibers
(Kevlar49)
Boron fibers
1.90
1.75
1.45
2,000
2,500
3,200
500,000
240,000
133,000
2.63
3,200
420,000
Melting point
CQ
700
800
840
Temperature
limit 230
(Erhard, 96)
Table 2.3: Select Properties of Four Fiber Reinforced Polymers-The fiber material
of these composites is carbon fiber. T800, IM7 and HTA are variations of carbon
fiber, while the numbers after the slash represents the matrix material.
Material
Elastic modulus TensUe strength Failure strain Compression strength
GPa
%
GPa
GPa
T800/5245 94(3)
1.69(0.1)
0.88(0.1)
1.67(0.9)
T80Q/924 92(8)
1.49(0.1)
0.90(0.09)
1.42 (0.09)
IM7/977
90(11)
1.52(0.15)
0.90(0.07)
1.43 (0.07)
1.73(0.1)
0.97(0.08)
HTA/913 70(4)
1.27(0.05)
(Durability,
75)
25
2.5.1
BRIDGE-IN-A-BACKPACK
A particular application of fiber reinforced composites of immense interest was
first encountered in a New York Times article, "Building a Bridge Of (and to) the Future."
The article outlines the construction of the Neal Bridge, a small overpass on Route 100
South, in Pittsfield, Maine (Fountain, Dl). Of note was the employment of the Bridge-ina-Backpack™ method, a revolutionary system developed by the University of Maine's
AEWC Advanced Structures and Composites Center in association with Advanced
Infrastructure Technologies, LLC, the Maine Department of Transportation, and the U.S.
Army Natick Soldier RD and E Center (Bridge).
The Bridge-in-a-Backpack™, currently intended for short to medium spans,
centers around the utilization of FRP arches (AEWC). These arches are filled with
concrete and serve as both formwork and reinforcement—there is no need for wood
formwork or steel reinforcing bars (AEWC). As in the case of the Neal Bridge, the
concept is that the soon-to-be FRP arches are transported simply as pliable, lightweight
carbon- and glass-fiber fabric tubes, and thus, could essentially be carried to the site in a
backpack (Fountain, Dl). Once on site the fabric tubes are inflated, bent into the desired
shape, infused with a plastic resin, forming a stiff FRP, and then filled with concrete
(Dl). A composite decking is then mounted on top of these FRP arches followed by
compacted soil, gravel, and an asphalt topping (Dl). See Figure 2.4 below, for an
illustration of the construction of the Neal Bridge.
The advantage of the Bridge-in-a-Backpack™ method is clearly in material and
construction savings. As mentioned previously, there is no need for wood formwork and
steel re-bars, while transportation and constructions demands are reduced (Fountain, D4).
26
AEWC Advanced Structures and Composites Center's life cycle analysis indicates
tangible environmental benefits; see Section 7.1.1 In addition, there are apparent cost
benefits. The cost of the Neal Bridge was supposedly $170,000 less than that of a
conventional precast concrete bridge (D4). It is also anticipated that the design will
reduce maintenance costs; the FRP tubes protect concrete from chemical-induced
deterioration (D4). As far as the span limitation, Dr. Dagher, director of the AEWC,
believes that the ability to produce 300 foot-spans is only a matter of time (D4).
Furthermore, addressing the fact that fiber reinforced composites have not been more
impactful in structural engineering, despite a long history of use in other industries, "is
that engineers and contractors have little experience with the materials, and full standards
guiding their use... have not been developed" (D4).
27
Seurc* Univaa&y of Mairm
The arches are tubes of
carbon-fiber and glass-fiber
fabric that are covered with
decking and filled with
concrete at the site. The
decking is then covered with a
thin layer of concrete, followed
by soil, gravel and asphalt.
A SEMES OF TUBES
coNcnemcone
A technology that uses fiber-reinforced
plastic arches filled with concrete may be
a solution for replacing some of the
nation's deteriorating, bridges
Totally Tubular
GRAVEL
FILL
The tubes are inflated, bent
to shape and Infused with
resin to make them rigid.
BUUXNO SEQUENCE
COMPOSITE
DECKING
28
Once in position, they are
fitted with concrete that
expands slightly as it cures.
•Hi
Fill Is placed on top and
compacted, followed by gravel
and the road surface.
' Jra*«i.» Source: University of Maine
;* ^ J s Mika Grondahl/The New York Times
% * i (Fountain, Dl)
The hollow, light tubes can
be placed without heavy-duty
cranes or other equipment.
'
Figure 2.4: The Neal
Bridge- Demonstrates
the Bridge-In-ATM,
Backpack1™ Method
2.6
CONSEQUENCES OF RESEARCH: ADDRESSING PRACTICALITY AND INSPIRATION
The purpose of this section is to serve as a recapitulation of the different areas of
research discussed in the previous sections and to more precisely describe the ways in
which this thesis was influenced by these structural systems. Thus, different concepts that
were not followed through with shall be presented including discussing the inspiration
behind them and their practicality, and application. The concept that was finally settled
on to be further developed and analyzed will be discussed in more length in later sections.
The termite mound was a big factor in the conceptual design.
The termite mound is not very relevant from a structural engineering standpoint
because of its composition, fecal matter and soil, and low aspect ratio. That is, due to
economics, owners generally want the aspect ratios of their building to be large, since
land is expensive while the air space is for the most part copious. There are many ideas
that can be taken away from the termite mound, however; such as the integration of
natural ventilation, passive heating and cooling, symbiotic co-existence, and
expandability. An idea such as symbiotic co-existence could be applied in the form of a
green roof, which is quite a popular feature of modern buildings. The green roof is
primarily intended to reduce the heat island effect—that is, the increase in ambient
temperature of urban settings relative to natural weather patterns due to the influence of
man-made materials and environments—although it also adds visually to the roof, if
accessible, as a terrace, improves a building's insulation, reduces storm water runoff, and
the hosting of plants could be considered a carbon offset. Furthermore, the longevity of
the termite mound is something to strive for in buildings. The shape of the termite mound
is of all these features the most directly applicable; the general shape of the mound—
29
obviously this varies from species to species and region to region—is approaching a
dome. Mathematics and engineering tells us that the dome or arch 2-dimensionally, is the
most efficient form for compression; in the spirit of autonomous formation processes, the
form of the dome is rooted in the laws of nature. It is the idea of a "dome building" that
was investigated further in combination with other principles; this will be discussed in the
coming sections.
The hexagonal mesh of the bee hive has vast scientific and mathematical
applications. The utilization of meshes is germane to structural engineering in general,
especially when using Finite Element Analysis. The idea of approaching the layout of a
building's structural system in a hexagonal mesh, rather than the typical rectangular grid,
was contemplated since, as was shown the hexagonal mesh is the most efficient method
of dividing an area. The use of a hexagonal mesh was considered in combination with
other ideas, such as the dome, which contributed to the final concept and will, therefore,
be discussed later. The wax cell itself also represents a viable design.
The shape of the wax cell, a rhombic decahedron, as was mentioned presents the
optimum volume to surface area ratio. Given these isoperimetric properties the shape
would serve well as a building—having the largest volume to surface area translates to
reducing facade/enclosure and framing material consumption compared to a rectangular
building of the same volume. A reduction in surface area would also indicate less thermal
conductivity. That is, the building's HVAC system would function more efficiently, as it
would be less susceptible to ambient temperatures due to the reduction in surface area.
Figure 2.5 clarifies the idea of a rhombic decahedron as a building, standing on its
hexagonal face. A drawback of this design, however, is that the hexagonal shape leads to
30
unused area when considering rectangular lots of land. The concept was never really
developed, but has potential. Further investigation would be necessary to establish the
actual material and energy savings over conventional design.
Figure 2.5: Elongated Rhombic Decahedron as a Skyscraper
(Hersey, 70)
The inspiration taken away from the spider web is the idea of utilizing steel rods
as the predominant structural element of the floors. The rods would be used in
conjunction with a steel or concrete compression ring around the perimeter, consider a
horizontal bicycle wheel. The bicycle wheel design has been utilized in the past, but has
served as a roof and not as an accessible floor, thus loading conditions are reduced. While
there is some precedent for this sort of bicycle wheel design it was not incorporated into
31
the final building concept as the required depth of this assembly may be prohibitive. See
Section 7.3 for further information regarding the bicycle wheel.
The use of a fluid as a structural element is clearly advantageous due to the
availability of water versus steel, for example. The issue is how to utilize a fluid such as
water in place of steel or concrete. A very crude design was conceptualized, which used
a hydraulic system of columns. The idea is that a steel assembly, essentially a steel tank
with a plunger-like extension at one end and an opening at the other that would house the
next story's plunger, would be filled with pressurized water, see Figure 2.6 below. The
principle is to capitalize on the very low compressibility of water to buoy the plunger of
the succeeding fitting. The steel would only act to contain the water, maintaining and
withstanding the pressure. The use of water versus steel could result in weight reduction
(unit weight of steel and water: 490 lb/ft3 and 62.4 lb/ft3, respectively) and should lead to
energy reduction when considering embodied energy. Obviously the connections would
have to be air-tight and a system would have to be installed to monitor and maintain
water levels; safety precautions would have to be devised. There would have to be MEP
systems associated with this design, which is in part why this concept was not further
pursued. It should be noted that the detail of connecting this "fluid column" assembly to
beams and slabs was not considered and would likely necessitate custom designs. In
addition, computer modeling of such a system would be difficult if not impossible, while
laboratory testing would be limited too. Therefore, this design was eliminated from
consideration, although the design may be worth further investigation.
Fiber reinforced composites are already used sparingly in structural engineering,
but have not precipitated a radical departure from steel and concrete construction despite
32
their useful attributes. Part of the problem in the past has been cost, although currently
fiber reinforced composites are very cost competitive. More so, as mentioned before, is
the lack of familiarity. The potential of these composite materials, as demonstrated by the
Bridge-in-a-Backpack™ method, compelled the inclusion of these materials into the final
concept. Therefore, the use of FRPs will be discussed in the next section.
Figure 2.6: Use of Fluid as a Structural ColumnSection view of conceptual design.
"Plunger
Tank
Pressure Gauge
Blue indicates water
Plunger and Tank are steel
33
3
CONCEPT
The concept that was finally chosen to be developed went through several variations.
The seminal principle, however, is the use of strong-cell exterior columns running along
the entire height of the structure, i.e. multiple arches forming a dome as was alluded to in
the previous section; the term strong-cell is meant to denote relatively large crosssectional elements and may also be referred to as super-columns. These strong-cell
columns are intended to be the principal load carrying elements of the building.
Furthermore, a concrete core, housing elevators and stairways, is intended in all of the
alternatives. As mentioned in the previous section, the inspiration of this design was in
part the form of the termite mound, as such, it was intended that the design also include
natural ventilation of the sort demonstrated in termite mounds. The integration of a
natural ventilation system clearly has implications on the structural design of a building;
however, designing the system itself is outside the realm of structural engineering and
therefore, was not considered. The initial concept will be discussed in full detail in the
next section.
34
3.1
INITIAL CONCEPT
Before directly discussing the design it is necessary that the principles and nuances
of arches and domes be mentioned briefly because their use is of such a pivotal nature.
When considering only two dimensions an arch is considered the most efficient way of
withstanding compression loads. Arches, when loaded uniformly avoid the complication
of bending due to their shape and symmetry. Removing the presence of bending greatly
increases the available strength of a given cross-section. Although it is impossible for
loading conditions to be entirely symmetrical, meaning bending forces will develop, the
arch is still beneficial, in that its form minimizes bending. Therefore, an arch is a
desirable shape for an element of a structure that will be in compression. Extending the
discussion to three dimensions, a dome is essentially an arch revolved 360 degrees, and
therefore exhibits the same favorable qualities. Furthermore, arches and domes perform
better the more vertical their shape. That is, as an arch is "flattened," the form approaches
a horizontal span with two vertical elements; the horizontal element provides less and
less structural support and becomes more dead weight for the vertical elements to
support; Figure 3.1 provides a graphical depiction of what is meant by flattening.
Therefore, the dome lends itself well to a high-rise building or any structure with a high
aspect ratio. The failure mechanism of the arch and dome, neglecting stress-related
failure, is that the base has the propensity to slide-out and must be sufficiently restrained.
Now that the technical aspects of arches have been touched upon a discussion of the
concept can proceed. See Section 7.2 for more information regarding arches.
35
Figure 3.1: Illustration of the Flattening of Arches
"Vertical Arch"
"Flattened Arch"
Given the known abilities of arches it is clear that their utilization can be
advantageous. Furthermore, the termite mound, arguably the most impressive animalbuilt structure besides or including those of humans, depending on your point of view,
serves as an example that domes are effective. Therefore, the use of a domed building,
not just having a dome adorn the apex like the Capitol Building, but literally the entire
building taking the shape of a dome was one of the main features of the initial design.
36
The building would consist of several rotated arches, these arches acting as the strongcell exterior columns, which are the primary compressive elements, transporting loads to
the ground, see Figure 3.2 below. Besides functionality, the domed-shape coincidently
provides a touch of aesthetics; see Section 7.5 for additional figures. Furthermore, the
intention is that the design be used for a high-rise building to fully take advantage of the
properties of an arch; a building of approximately 40 stories was considered, although
theoretically the taller the better. In combination with the general idea of arches, the use
of bicycle wheel floors, as inspired by spider webs, mentioned earlier was considered.
Figure 3.2: Domed Structure
(A) Overhead View (B) Elevation View
37
The use of these steel rod-supported floors was intended to reduce material
consumption since they would be acting primarily in tension, which has been established
is steel's greatest strength, see Section 7.10 for details regarding basic steel design
procedure. The presence of a steel or concrete compression ring around the floors is
necessary to provide resistance to the tension developed in the rods, but could also be
attached to the arches to help restrain them from sliding out. It should be pointed out that
the use of rods is intended to completely substitute the use of steel I-beams. The
practicality of using a rod-compression ring system is questionable, however.
The
required depth of the entire steel rod construction is not apparent. In the end, the use of
the rod-compression ring system was abandoned, not because it is an implausible design,
but in favor of a different approach. Once again, see Section 7.3 for further details on the
bicycle wheel roof.
38
3.2
REVISED CONCEPT
The thought of using a rod-compression ring floor was replaced with the idea of a
more conventional girder-joist-slab system with the unique aspect of the framing layout
being organized using a mesh. That is, as the hexagonal mesh is the most efficient way of
dividing a surface, based on research regarding bee hives, partitioning of the framing grid
of a building in a similar arrangement should improve efficiency; optimizing span
lengths. Figure 3.3 is representative of the hexagonal mesh applied to an elliptical
footprint domed building. There was still, however, further room for innovation in the
form of fiber reinforced polymers, see Section 7.6 for illustrative figures.
Figure 3.3: Steel Framing System Following a Hexagonal Mesh
Elevation View
39
Integrating fiber reinforced polymers into the design was the next phase. The
thought was that replacing steel beams with a concrete-filled fiber reinforced polymer, in
the spirit of the Bridge-in-a-Backpack™ method, would lead to energy savings—in terms
of materials, construction, and transportation—proportional to those observed in the
University of Maine's project, see Section 7.1.1 regarding the life cycle analysis. These
FRP beams were conceptualized as pre-stressed; having a steel tendon, see Section 7.6
for illustrative figures.
As the project evolved the FRP beams were discarded as well as the meshing
approach; it should be noted that this approach, however, appears to have potential and
could be studied in the future. Instead, a sort of flat slab, drop panel approach was
adopted. Flat slab, or alternatively flat plate, is a term used to describe a constant depth
steel-reinforced concrete element that acts only in combination with columns in
providing structural support. In contrast, in the case of concrete construction, a concrete
slab could be of a varying depth, due to the presence of concrete beam elements spanning
in one or two directions, see Figure 3.4. For steel construction, the concrete floor is
poured on top of a metal decking, which is supported by a system of joists and girders,
and columns. Flat slab design has become quite popular because it saves space due to the
lack of the additional depth of beams. Part of the flat slab design is the use of droppanels. These drop panels are an addition of concrete to the slab at columns connections
to resist punching shear. Punching shear is the tendency of the column to pierce through
the concrete slab due to the large loads on the column versus its small cross-section. See
Section 7.4 for a description of flat plate and drop panel design. It should be noted that an
alternative to drop panels is the use of shear heads. A shear head is a steel plate
40
embedded in the concrete that serves the same purpose as a drop panel. The use of a shear
head allows the slab to be completely homogenous with respect to thickness.
In particular, a concrete filled FRP will serve as a drop panel, while a steel deck
filled with concrete will rest on top of these drop panels, forming the floor slab. The FRP
will serve as the tensile reinforcement and formwork; see Section 7.7 for illustrative
figures. Furthermore, the interior columns, those supporting the drop panels and in turn
the floor slab, will be tied into the strong-cell arches, such that they will go into tension
when a live load is present. The load will be transferred through the slab to the drop panel
to the columns to the arches at the building's apex and finally to the ground. The
intended behavior is that the interior columns will withstand compression under
construction and dead loads, but will go into tension under live loads. In having the
columns acting in tension the steel will have a greater available strength, reducing the
cross-section and in turn the total tonnage of steel necessary, see Section 7.9 regarding
steel design specifications. This configuration is a sort of suspended floor design, and
could potentially utilize high tensile strength steel cables in place of columns. However,
the process of hanging floors from the top down rather than building from the bottom up,
would likely increase construction difficulty and in turn energy. Furthermore, the need
for redundant design makes compression capable elements more attractive than cables
elements that can only act in tension.
The interior columns will act as pinned elements, supporting for the most part
only axial loads. Thus, the concrete slab, metal deck, and drop panel assemblage will
resist the generated bending moments and shear stresses. The load will be transferred to
the arches in a relatively uniform manner, since the geometry will be specified by the
41
design, meaning that bending moments will be greatly reduced, see Section 7.2 regarding
arches. When considering the overall structure this should result in material savings since
the columns and arches will be virtually free of bending and shear. Construction savings
should follow as well due to the simplicity of the floor framing: steel beams do not have
to be maneuvered and connected, and in the case of concrete construction, no formwork
is necessary and the light-weight nature of the FRP limits the need of heavy construction
equipment. Transportation savings should follow using the same logic.
42
Sparafevi fettffl
(optional)
Bwnmn
J
M
Spandttf
•nttrtor
column
(6)
Figure 3.4: Typical Concrete Design
(a)Two-way flat plate (b) Two-way slab on beams (c) Waffle Slab
(Nawy, 450)
43
3.3
FINAL DESIGN
The concept presented in the previous section is what should be considered the
ideal design. Once economics are considered in combination with rectangular lots, a
circular or elliptical footprint becomes prohibitive; following the same logic discussed in
the drawbacks of utilizing a rhombic decahedron design (hexagonal footprint). Not fully
using the rectangular lot means less square footage on a per floor basis, this translates
into less rentable space and less revenue, or additional floors to make up for the
discrepancy in square footage, which presents a greater initial cost. Thus, the domed
design would have to yield significant cost savings to offset loss in revenue due to not
fully utilizing available land to persuade a developer to use this design. As a result, the
design was altered to reflect a potentially more economically attractive design. The fullheight arches were removed in favor of a dome at only the building's top and the
buildings footprint was changed to rectangular. The dome was eventually removed
completely and replaced with a transfer truss that would serve the same purpose. The
transfer truss connects to the exterior columns, but also bears on the concrete core since
the core has significant strength. Section 7.8 contains figures of this final design. Note
that the strong-cell exterior columns in combination with the transfer truss should be
viewed as essentially an independently stable frame. That is, since it is intended that the
floors behave as if they are suspended from this assembly, the transfer truss cannot bear
on the interior columns, but must provide resistance to their deflection. Therefore, in
terms of the construction methodology, the strong-cell exterior columns and transfer
truss, as well as the concrete core must be completed before the interior columns, drop
panels, and floors are put in place.
44
More specifically, the strong-cell exterior columns, including lateral bracing,
would be constructed in unison with, although independent of, the concrete core. At the
building's top, however, since the transfer truss will be tied into the core and connected
with the exterior columns, the construction must be coordinated. That is, to embed the
truss into the concrete core would require that the connections be set-up prior to forming
the concrete. Subsequently, the trusses would be maneuvered into place fully assembled,
and partially assembled for those intersecting, and simultaneously connected at both
ends.
The construction of the interior—floors, drop panels, and columns—could take
place at the same time as the exterior columns and the core. The columns would first be
placed and then the FRP drop panels would be fitted on top and filled with concrete. The
FRP drop panels could be transported already shaped, or could be shipped as a fabric and
then shaped and stiffened on the construction site. Steel decking would then be placed on
top of the drop panels and filled with concrete forming the floor slab. The process would
be identical for each successive floor. Obviously, the typical 28-day curing period for
concrete applies for each concrete element and would affect construction time.
In terms of the connections, a method/detail must be designed to effectively
interface the steel deck and FRP drop panels. Furthermore, a connection must be
designed between the FRP drop panels and interior columns such that the drop panel is
provided support by the columns locally, but bypassed globally. To clarify, it is assumed
that the drop panel cannot take the tension that is expected to be developed, since
concrete is the primary component, although the FRP reinforcing could potentially
provide enough tensile strength, nor the concrete slab, although the steel deck could also
45
potentially provide enough tensile strength. Therefore, the interior columns must be
somehow continuously connected while providing bearing for the floors and drop panels.
In order to gauge the actual benefits, if any, of this unconventional design, a
feasibility study was proposed. A description of the approach of this feasibility study and
the results are available in the next section.
46
4
FEASIBILITY STUDY
The purpose of performing this feasibility study is to determine if the theory behind
the conceptual design stands up under real world conditions. In order to understand if this
design yields any benefits two analyses had to be performed: one utilizing a conventional
design approach and the other following the conceptual design that has been discussed.
Obviously, the two models must be as similar as possible, having the same height and
footprint. A hypothetical location for the building was chosen to be 41 Cooper Square,
the Cooper Union's New Academic Building. The only ramifications of the choice of this
site is in the footprint of the building, although the footprint was altered to make an
axisymmetric situation for simplicity, as geological and geotechnical conditions were not
considered. Thus, the proposed location is of very little significance, its longest
dimension, 180 feet, was simply used as the length of each side of the building model.
The computer analysis was performed using SAP2000 Advanced Version 12.0, a finite
element analysis software released by Computers and Structures, Inc.
There is some precedent in the use of drop panels in a similar manner as well as
suspended floors. These two examples, which illustrate that the concepts discussed have
merit, will be touched upon in the next section.
47
4.1
PRECEDENT
In determining the feasibility of this design, the fact that this design integrates
different principles that have been actually used is critical. In particular, the Johnson Wax
Building, designed by Frank Lloyd Wright, offers an example of drop panels being used
as the sole form of supporting a floor. More general, is the existence of suspended floor
buildings.
Discovered after the concept was proposed, pointed out by Master's student,
Jeffrey Tan, the Johnson Wax Administration Building is often stated as one of the most
innovative designs of the twentieth century. The feature of this building that is pertinent
to this thesis is Wright's use of "lily pad" columns. These lily pad columns, consisting of
hollow reinforced concrete, serve as both column and roofing element (Etlin, 60-61). As
seen in Figure 4.1 below, the form of these columns is essentially the same as the FRP
drop panel-column assembly. The difference between these two similar designs, besides
the materials used, is that Wright's lily pad partly served as a structural element and the
roof itself, whereas the concept presented here serves to support a floor and is not itself
part of the floor. Furthermore, as seen in Figure 4.1, Wright's lily pads are very closely
spaced, much more so than has been proposed for this project. Coincidently, the FRP
drop panel-column assembly was also referred to as the lily pad prior to any knowledge
of the existence of this Frank Lloyd Wright structure. The existence and utilization of
such a design is a clear indication of the practicality of the conceptual design.
The principle of having floors acting in suspension rather than compression is a
notion that has been adopted as an alternate method of building construction. So called
"suspended floor buildings" employ a system in which the floors are essentially
48
cantilevered from a concrete core. The edges of the floor are further supported by cables
that are anchored by a framing system at the building's top (Masted, 60); see Figure 4.2.
One of the quintessential suspended floor structures is the West Coast Building, in
Vancouver, built in 1958. There is no indication, one way or the other, that either the
suspended floor building, nor Wright's system in the Johnson Wax Administration
Building results in a more efficient design.
Figure 4.1: The Lily Pads of the Johnson Wax Administration Building
Source: http://www.greatbuildings. com
49
Figure 4.2: Typical Suspend Floor Building Configuration
{Masted, 6)
50
4.2
COMPUTER MODELS
The two models have identical footprints of 180 feet by 180 feet and are 40 stories
tall, with each story being 11 feet tall for an apex of 440 feet. Both models are configured
for 30 foot spans. Dead, live, and wind loads are applied while snow and seismic loads
were not considered. The exclusion of these loads is for the sake of simplicity; dead, live,
and wind loads are a good indication of typical load conditions. Nonetheless, both models
are loaded with only these three cases such that there is no inequity. In terms of the loads
themselves, the dead loads are a function of the materials and member cross-sections,
thus, these loads are program determined. However, for the conceptual design, the dead
load of the exterior columns and the transfer truss were ignored to achieve the intended
loading mechanism. That is, as mentioned previously, the truss should not bear on the
interior columns; the discrepancy caused by this simplification should be insignificant
since these loads are nominal in comparison to the combination of the dead load of the
interior columns and concrete slabs and the uniformly distributed live loads. The live
loads, on the other hand, are user defined. A uniformly distributed live load of 100 psf
was used for all the floors, with the exception of the roof where a 40 psf load was applied
since the roof is assumed to be non-accessible. The wind load is automatically generated
by the software following the method outlined by the American Society of Civil
Engineer's/Structural Engineering Institute's Minimum Design Loads for Buildings and
Other Structures, 2005 edition (ASCE/SEI 7-05). The load combinations are also
automatically generated following the American Institute of Steel Construction's Load
and Resistance Factor Design, 1999 edition (AISC-LRFD 99) as specified by ASCE/SEI
7-05, see Table 4.1 for the load combinations that were used. The steel design was
51
performed in accordance with the American Institute of Steel Construction's Steel
Construction Manual ultimate strength design criteria (LRFD), see Section 7.10 for select
sections from the AISC Specifications for Structural Steel Buildings. Note that the design
was specified to only satisfy strength, i.e. serviceability was not considered. Therefore,
deflection limits may not be satisfied, however, this is true of both models, so there is no
inconsistency.
Table 4.1: Load Combinations per AISC-LRFD 99/ASCE 7-05
Load Combination Dead Load Factor Live Load Factor Wind Load Factor
DSTL1
0
0
1.4
1.2
0
DSTL2
1.6
1
1.6
DSTL3
1.2
DSTL4
1
-1.6
1.2
0.8
DSTL5
0
1.2
-0.8
DSTL6
0
1.2
DSTL7
0.9
0
1.6
DSTL8
0.9
0
-1.6
The limitations of analyzing the conceptual design that is the focal point of this
thesis must be noted. First and foremost, the use of FRP is a key feature of the concept,
but could not be faithfully included in computer model due to a lack of knowledge and
experience, and the software's inability to integrate the use of FRPs in the desired
capacity. In addition, the virtually endless variations of fiber reinforced composites, with
varying mechanical and physical properties makes the selection of a particular form of
fiber reinforced composite very difficult. The selection of a fiber reinforced composite
would likely require a significant amount of laboratory testing and research in order to
arrive at a material well suited for the required stress demands. Furthermore, as was
52
discussed earlier, the connections require unique details, these details were never
developed, and modeling the required behavior of these connections would be tedious.
The exclusion of the FRP drop panels simplifies the connections, however; in any case,
all connections are idealized. For both models the exterior columns are momentconnections while for the conventional model the interior columns are also momentconnections. The conceptual model utilizes pin-connections for the interior columns. For
the conventional model all joists and girders are pin-connected. For the conceptual model
the transfer truss is moment-connected for the primary axes and pin-connected
everywhere else. A truss is defined as being pin-connected, but in order to satisfy some
stability issues moment-connections are required at select points.
The ability to use shear studs to achieve composite action between the concrete
slab and the steel beams is not specifically incorporated into the conventional model. It is
not known how to properly model this sort of connection; the assigned connections may
actually achieve this behavior to some extent. However, it should be noted that the use of
shear studs, which facilitates an interface between the slab and beams, effectively
reducing the beams unbraced length, thereby reducing the beam's required cross-section,
would result in some amount of steel savings.
The necessary thicknesses of the concrete slabs were assumed for simplicity. That
is, the concrete floors were assumed rigid and capable of carrying the imposed loads. The
thicknesses were based on known typical depths. Therefore, the conventional model was
assigned 6 inch thick concrete slabs with a concrete strength of 4,000 psi while the
conceptual model was assigned 11 inch thick concrete slabs also with a concrete strength
of 4,000 psi. This discrepancy in slab depth is to account for the lack of beam supports in
53
the conceptual model, i.e. the slab itself will have to withstand the majority of the
generated bending moments and shear forces. It should be noted that a depth of 11 inches
is likely on the conservative side. In addition, the amount of reinforcing bars/grid and the
use of a steel decking were ignored as these slabs were treated as rigid, homogenous
elements. Furthermore, as was stated earlier, soil conditions were ignored; the models'
are assumed to be built on grade, located on rock of infinite strength.
54
4.3
RESULTS
The computer models yielded results, which suggest that the conceptual model
offers benefits. In terms of total steel tonnage, the conventional model requires 6,500 tons
while the conceptual model requires 4,500 tons of structural steel. Thus, the conventional
model uses 1.44 times more steel than the conceptual model. In terms of total volume of
concrete, the conventional model requires 24,300 cubic yards while the conceptual model
requires 43,500 cubic yards of 4,000 psi concrete. Thus, the conceptual model uses 1.8
times more concrete than the conventional model. The difference in concrete
consumption is actually built into the models based on the assumed slab thicknesses, as
mentioned earlier. The necessary slab thickness for the conceptual model is likely 2-3
inches less than the 11 inches used for the model. The conceptual model does not include
the concrete needed for the drop panels; however, this volume of concrete should be
insignificant in comparison to the slabs. To further compare these two models the
embodied energy of the construction materials was considered.
Applying ratios of 3,180 MJ/m3 for 30 MPa (-4,000 psi) concrete and 32 MJ/kg for
steel (Canadian Architect) to the total material consumption of the two models yields a
total embodied energy of 247,500 GJ for the conventional model and 183,200 JG for the
conceptual model. Thus, the conventional model has 1.35 times more embodied energy
than the conceptual model. While a comparison of steel and concrete usage from an
embodied energy standpoint demonstrates that the conceptual model offers benefits, a
cost estimate based only on materials, demonstrates a cost detriment. That is, a
rudimentary cost analysis, utilizing 20-city average material index values for May 2010,
obtained from the Engineering News Record ($102.49/ton of concrete and $44.27/100 lbs
55
of steel [Grogan]), yielded a total material cost of $13,000,000 for the conceptual model
and $11,000,000 for the conventional model. Thus, the conceptual model is 1.18 times
more expensive than the conventional model. These costs do not take construction
effort—hours of labor and necessary equipment—maintenance, or design life into
account, however. Typically concrete has a longer life span and requires less maintenance
than steel. Since the conceptual model uses more concrete than the conventional one, its
maintenance costs should be less while its life span should be longer. Furthermore, FRPs
typically require even less maintenance than concrete while having longer life spans.
Thus, the use of FRPs should further reduce maintenance cost while increasing life span.
It should be noted that the most efficient cross-section per element was chosen,
although typically for economics (economy of scale) and construction simplicity crosssections are grouped. That is to say, that there might only be one beam cross-section per
floor used in the actual construction, despite the fact that the most efficient design might
utilize four different cross-sections per floor, for example. Therefore, in practice, the total
quantity of steel for both models would increase to reflect this grouping of elements.
Since the most efficient cross-section was selected for every element in both models,
however, there should be no inconsistency.
A second method of comparing the two models was tried. This second method was
performed to try and rectify the fact that in order to exhibit the desired behavior of the
conceptual model the dead weight of the exterior columns and transfer truss were
ignored. In order to account for these ignored dead loads this second method of analysis
has one model in which the interior columns and truss are unconnected, such that the
exterior-cells and truss act as an independently stable frame as well as the interior
56
columns and slabs; only the dead loads are analyzed for this case. A second model in
which the truss and interior columns are connected was then analyzed for only live loads.
The stresses from these two models were then combined to reflect the total effective load.
The only change to this model is that the thickness of the concrete slab is reduced to 9
inches to reflect more realistic parameters. Furthermore, the models were analyzed for a
50 psf and 100 psf uniformly distributed live load for the sake of comparison.
For brevity, only the columns of the first floors were compared, see Section 7.9 for
tables comparing the stresses. The results of this comparison indicate that as a whole, the
forces exerted on the conceptual model are greater in magnitude than those observed in
the conventional model. It appears, however, that as the live load increases the difference
in axial loads decreases. These results may indicate that the benefits of the conceptual
model are largely due to the large cross-sections of the columns rather than the load
distribution mechanism. At this point it is unclear, additional data is necessary to fully
understand the trend. Nonetheless, the conceptual design has proven itself to be the more
efficient system based on embodied energy.
57
5
CONCLUSION AND FUTURE WORK
The results of the feasibility study demonstrate that the conceptual design has
potential benefits. The. crude nature of this feasibility study, however, definitely
necessitates further investigation and refinement. For example, as the second method of
analysis indicates the actual benefits of the conceptual design are likely more modest.
That is, the actual benefits of the conceptual design are somewhere between the results of
the two analysis approaches. Furthermore, an elementary cost estimate demonstrates that
this conceptual design is more expensive than a conventional building, although as was
mentioned previously, this estimate does not factor in maintenance or construction costs.
At this point, the conceptual design must be labeled inconclusive. More
importantly, however, the research and concepts that have been discussed in this thesis
demonstrate that there is indeed the ability to approach building design in alternative
methods. The particular building system that has been proposed and analyzed may be
subsequently definitively identified as actually being less efficient than the typical design,
but the other ideas that were considered and abandoned, see Sections 2.6 and 3, could be
rehashed and may prove advantageous. Furthermore, there are no doubt other sources of
inspiration that can be found in nature—and following Frei Otto's definition, the
creations of man are natural, thus, technological innovations are included as part of
nature—that can aid in the design of a building that were unintentionally overlooked.
This is to say, as an evaluation of the legitimacy of green and natural design in regards to
structural engineering, this thesis was tremendously successful. It is now obvious that
there is the ability to improve conventional building design; even if ultimately, only small
scale modifications prove to be the most efficient design.
58
In terms of the future work, the next step in evaluating the conceptual system is
creating a more comprehensive, more refined computer model. In doing so, it may be
necessary to identify software that can better handle the parameters of the system,
especially with regards to the integration of fiber reinforced polymers. Considering
seismic loading is definitely a necessity of any future analysis and evaluation as well.
Furthermore, a typical concrete frame building should be considered as another source of
comparison. A more in-depth cost analysis must also be performed to account for the
non-standard cross-sections associated with the strong-cell exterior columns and to reflect
the construction methodology and use of FRPs. A more in-depth environmental
assessment must be performed to fully gauge the ecological benefits.
Once it has been sufficiently demonstrated that this conceptual design is a viable
option through computer modeling lab testing would be necessary. In particular, as
mentioned earlier, determining the required mechanical properties and identifying
possible fiber reinforced composites, including fiber type, form, orientation, and lamina
stacking sequence would require some amount of laboratory testing. In general, further
developing and evaluating the FRP drop panel design would also require laboratory
testing. Furthermore, connections for the FRP drop panel to steel deck and column, need
to be designed and may require laboratory testing. Laboratory testing would also be
needed to fully understand the behavior of this structural system.
Expanding the use of FRPs is a logical next step. The integration of FRPs, even to
conventional design, applied to columns and beams, has immense potential. The benefits
of the high strength-weight ratios of FRPs are undeniable. Furthermore, the carbon
analysis of the Bridge-in-a-Backpack™ method is suggestive that the manufacturing
59
process of these awesome materials is more environmentally friendly than steel and
concrete, see Section 7.1.1. Thus, the only drawback of FRPs appears to be the lack of
familiarity, as these materials are progressively becoming more cost competitive.
In the event that the conceptual model is proven to be less efficient than conventional
design, the use of a hexagonal mesh framing system, which was one of the proposed
concepts, would be the next course of action. The building system would be conventional
with the exception of the beams and columns being organized using a hexagonal pattern.
A feasibility study similar to the one discussed in this paper would have to take place.
60
6
WORKS CITED
Barthel, Rainer. "Natural Forms—Architectural Forms." Frei Otto: Complete Works:
Lightweight Construction, Natural Design. Ed. Winfried Nerdinger.
Boston: Birkhauser, 2005. 16-31. Print.
"Bridge-in-a-Backpack™." The AEWC Advanced Structures and Composites Center.
University of Maine. Web. 1 Apr. 2010.
<http://www2.umaine.edu/aewc/content/view/185/71/>.
Erhard, G. Designing with Plastics. Cincinnati: Hanser Gardner Publications, 2006.
Google Books. Web. 1 Apr. 2010. <http://books.google.com>.
Etlin, Richard A. Frank Lloyd Wright andLe Corbusier: the Romantic Legacy.
Manchester: Manchester UP, 1994. Google Books. Web. 15 Apr. 2010.
<http://books.google.com>.
Farmer, John. Green Shift: Changing Attitudes in Architecture to the Natural World.
Ed. Kenneth Richardson. 2nd ed. Boston: Architectural, 1999. Print.
Fountain, Henry. "Building a Bridge Of (and to) the Future." The New York Times
13 Oct. 2009, Science sec: Dl, D4. Print.
French, M. J. Invention and Evolution: Design in Nature and Engineering.
New York: Cambridge UP, 1988. Print.
Grogan, Tim, and Manuela Zoninsein. "Economics." Engineering News-Record.
McGraw Hill Construction. Web. 5 May 2010. <http://www.enr.com/economics>.
Hansell, Mike. Animal Architecture. New York: Oxford UP, 2005. Print.
Hansell, Mike. Built by Animals: The Natural History ofAnimal Architecture.
New York: Oxford UP, 2007. Print.
61
Harris, Bryan. "Fatigue Behaviour of Polymer-based Composites and Life Prediction
Methods." Durability Analysis of Structural Composite Systems.
Ed. A. H. Cardon. 1st ed. Rotterdam: A.A. Balkema, 1996. 49-84. Google Books.
Web. 1 Apr. 2010. <http://books.google.com>.
Harris, James B., and Kevin Pui-K. Li. Masted Structures in Architecture.
Burlington: Architectural Press, 1996. Google Books. Web. 15 Apr. 2010.
<http://books.google.com>.
Hersey, George. The Monumental Impulse: Architecture's Biological Roots.
Cambridge, Mass.: MIT, 1999. Google Books. Web. 12 Mar. 2010.
<http://books.google.com>.
Mallick, P. K. Fiber-reinforced Composites: Materials, Manufacturing, and Design.
2nd ed. New York: M. Dekker, 1993. Google Books. Web. 26 Mar. 2010.
<http://books.google.com>.
"Measure of Sustainability: Embodied Energy." Canadian Architect. Web. 5 May 2010.
<http://www.canadianarchitect.com/asf/perspectives_sustainibility/
measures_of_sustainablity/measures_of_sustainablity_embodied.htm>.
Nawy, Edward G. Reinforced Concrete: a Fundamental Approach.
Upper Saddle River, N.J.: Pearson Prentice Hall, 2009. Print.
Nerdinger, Winfried. "Frei Otto Workin for a Better 'Earth for Mankind'"
Frei Otto: Complete Works: Lightweight Construction, Natural Design.
Ed. Winfried Nerdinger. Boston: Birkhauser, 2005. 8-15. Print.
62
Sarcone, Gianni A. "The Geometry of the Bees..." Archimedes Laboratory.
8 Mar. 2010. Web. 8 Mar. 2010.
<http://www.archimedes-lab.org/monthly_puzzles_72.html>
Smallman, R. E., and R. J. Bishop. Modern Physical Metallurgy and Materials
Engineering: Science, Process, Applications. 6th ed.
Oxford: Butterworth Heinemann, 1999. Google Books. Web. 26 Mar. 2010.
<http://books.google.com>.
63
7
APPENDIX
64
7.1
BRIDGE-IN-A-BACKPACK™
• 50+ years of truck traffic applied in 3 weeks
y JA -1. V-L -i »• A -L ~ -»-'
fffl M A TNF
DJDBB THE UNIVERSITY OF
'
AEWC Advanced Structures & Composites Center
207-581-2138
hd@umit.maine.edu
--
CONTACT:
H J Da9herDirector
Advanced SfeUCtunss
*Compo«ltaB Cantor
www.aewc.umaine.edu
WJMENC
sSaWJI. l E t A a V *
• Manufacture of 6" and 12" diameter 16, 40 and 60 ft arches of various wall thicknesses and geometries
• Tested load carrying capacity to 75,000 lbs over a 35' span
• Timeframe of less than two weeks to assemble and cast the support system for a bridge: infusion time, 20 minutes;
specimen cure, 6 hours; concrete filling, 5 minutes; concrete cure, 10 days
• U.S. Patent application 20060174549 filed 8/10/06
• Construction of a short-span bridge by Maine Dept. of Transportation using arch technology completed Nov 2008.
Milestones Achieved
• Specimens subjected to 2,000,000 cycles fatigue loading
In order to effectively carry out structural design using the arch members, AEWC researchers have developed an analysis
technique which has been validated through structural testing of arch specimens. The specimens were subjected to static
testing to failure and their load-deflection response and ultimate strengths were studied. Excellent correlation has been
seen between experimental and predicted results, providing a high level of confidence in the modeling technique.
STRUCTURAL TESTING AND MODELING
BRIDGE-IN-A-BACKPACK™
"^^TBT**-"nii ~~ •
i *. *p ^asar J
^ MAINE
_BBB| THE UNIVERSITY OF
www.aewc.umaine.edu
For mart, nforaat an almjt the AEwC
M . * i « eO Stnji tuns &«ompoate. Cmrtei * *,Ono t f
rac'»«i 10 nt.^atertKO t'OlSicrreditettiariorat.jiius •jtpertist. jot producttievujopittdd,ye U»
• Acts as sole required reinforcement - no steel rebar
needed
• Rapidly erectable, stay-in-place form
• Durability - Composite shell protects concrete from
the environment, reducing maintenance costs and
increasing the life-span of the structure.
• Simplicity of construction - formwork is simple to
place and mitigates need for heavy equipment,
cranes and large crews
• Cost competitive with alternative system
• Transportation efficiencies - compact, light-weight
bridge kits can transported to the site with a single
truck
Benefits over current state-of-the-art
njfitbfids
60' arch manufactured In AEWC laboratories it light
enough for two people to carry and place
CONTACT:
H. J. Dagher, Director
AEWC Advanced Structures & Composites Center
207 - 581 - 2138
hd@umit.maine.edu
Underside of Neal Bridge, first field
application of Bridge in a Backpack
'•) h'tK
Highway bridges in the US are quickly becoming deficient due to increasing traffic volumes, rapid deterioration, extended service life, and increasing load requirements. Repair or replacement of deficient structures is
expensive, time and labor intensive, and typically results in lengthy road closures during construction.
Researchers at the University of Maine's AEWC Advanced Structures & Composites Center have developed a lightweight, corrosion resistant system for short
to medium span bridge construction using FRP composite arch tubes that act as reinforcement and formwork for cast-in-place concrete. They are lightweight,
easily transportable, rapidly deployable and do not require the heavy equipment or large crews needed to
handle the weight of traditional construction materials.
These arches capitalize on their inherent properties to
transform vertical loads to internal axial forces, the
superiority of concrete in sustaining compression
loads, and the versatility and strength of composite
materials
Background
An innovative bridge technology using rigidified inflatable composite arches
• Simplifies construction
• Reduces life-cycle costs
• Increases the design life of bridges
• Decreases carbon footprint of bridge construction
_
.
.
_
Advanced Structures
& Composites Center
IWAEMIC
&
^^^7
Mllnt D*p»rtm»nt of Tnngportotlon
MAINEDOT
I n partnership with
Reinforcement is placed over air
bladder
Bridge Completion. The day after the concrete pour spandule headwalls and precast concrete wingwalls were installed. Granular sand backfill was placed over
the concrete and compacted on top of the arch structure. The bridge was backfilled; and road surface paved.
Neal Bridge before rehabilitation
SiWses'n-M^toeawsWpfti intlfttaAtftgiQFt&BS .
aowcHtfa wfav
* tmtngut during xw>wn
TtadmlwmwtiflfthltlsclMiqlpgyJiHtKrn ji
. wntMMIwtiw«tWU$.ftn*M^*iOW i
Bridge Monitoring. Sensors were installed for
monitoring of the bridge. The monitoring program
includes load testing with weighted vehicles, and the
installation of instrumentation to measure arch
strains and deflections. The first load test showed
significant reserve capacity. A follow-up is scheduled
for 2011.
Concrete Fill. Self-consolidating
expansive concrete (26 cy total) was
poured through individual holes in
the top of each arch; fill time = 1 hr.
Concrete was also spread on decking
which with the concrete anchors form
lateral force-resisting diaphragm.
Decking. The arches were covered with corrosion resistant FRP
corrugated decking using screws
which become concrete anchors
once the arches are filled. Decking
also assists soil retention.
The bladder is then inflated and the part is placed on a form of the desired geometry and infused with resin. Forming and infusion can be performed in a shop or in the field. AEWC has developed a telescoping, adjustable and transportable
form.
Arch Installation. All 23 arches (34' span, 12" diameter, 0.10" wall thickness) were placed in a single
day. They were lowered into place with a simple boom and placed by hand labor with no heavy equipment.
The base of the arches are encased in a concrete footing.
Composite materials selected for
geometry, cost, storage tolerance,
and resin compatibility.
The arch system was demonstrated in the replacement of the 90+ year old Neal Bridge, located on Rt. 100 in Pittsfield, ME. Twenty-three arch members were used in construction of the 34' span, 44' wide structure. The arches were installed in a single
working day, covered
with FRP23decking,
and filled with
concrete.
Next, headwalls
and wingwalls
were
bridge was
backfilled,
paved
and opened
to traffic
Fabrication
Of Arches.
arches fabricated
for bridge
reconstruction.
This process
can easily
beplaced,
done atand
thethe
construction
site;
in this first
construction,
arches
were. fabricated in the laboratory at UMaine.
Reconstruction of the Neal Bridge in Pittsfield, Maine
.
An innovative bridge technology using rigidified inflatable composite arches
BRIDGE-IN-A-BACKPACK™
7.1.1
CARBON FOOTPRINT ANALYSIS
68
Bridge-in-a-Backpack
Carbon Footprint Analysis
"A
'"••«.
i
Researchers at AEWt
_. , T M ^ _ ^ , ^ ^ ^ ^ ^ ^ _ ^
wyck eacbon footprint of tfes ^ ^ ^ ^ ^ ^ P ^ ^ ^ ^ ^ ^ ^ % w ? P " ! ^ ^ ^ » ^ ^ ^ ^ a ^ S ® B G j e t «
b*j*jfess,' a o ^ ce>fi$giasee!5-.t»foe result? c> tr.ulirinfi.il concrete .mtl stest feridge technologist
••
•• ; * * * .
• The total carbon footprint of the Bridge-in-aBackpack™ is 45.71 kg (C0 2 e/year)/sq m.
This is one third less than the carbon footprint Manufacturing (kg/yr)
of a comparable concrete bridge and one Transportation (kg/yr)
fourth less than that of a steel bridge.
• The Bridge-in-a-Backpack'^ superstructure has
half the carbon footprint of a typical concrete
bridge.
410
496
465
82
124
89
Construction (kg/yr)
245
340
413
Maintenance (kg/yr)
410
496
465
End of life Disposal (kg/yr)
28
152
-88
Total (C0 2 e/year)/sq m
45.71
69.17
59.10
• A recent report by the Federal Highway Administration concluded that 25.4% (152,316) of all bridges are
either structurally deficient, in need of repair, or functionally obsolete. If the Bridge-in-a-Backpack™ replaced
just 20% of these bridges, the equivalent amount of C 0 2 emissions reduction would equal taking 230,000 cars
off the road for a year.
Carbon Emissions Savings with the Bridge-in-a-Backpack
= 20,000 units
SAME AS
230,000
CARS
'
\
i \
SAME AS
"
" - 1,542,524
TONS OF COAL
„.
,8,201,893
\l
BARRELS OF OIL
\
ITAEWD
Adwtcod Structures & Composttus Cvntsr
AEWC
University of Maine
5 7 9 3 AEWC Building
Orono, ME 0 4 4 6 9 - 5 7 9 3
Dr. Habib Dagher, Director
Phone:(207)581-2123
Fax:
(207)581-2074
www.aewc.umaine.edu
_
THE UNIVERSITY OF
LTD MAINE
traditional concrete and steef dirMun*. l.-:irbw»lli*sitp!Hftfrresufes were normalized into equivalent carbon dioxide p<£t.
service yea$ andspes stjaare tn«rtt. i or" p.iwmem. The service lives for the three bridges compared in this stridfr *«fafcas follows: Bridge-in-a~Backp0*:'"*' "Hi vr- »rs, tuUiuun.il Luiicrete bridge -72 years, and steel bridge -58 years.
T h e c a r b o n footprint consists of the following five c o m p o n e n t s :
• Materials: This portion analyzes the production of materials that goes into each bridge, which includes the mining of
aggregate and making of cement for concrete, the manufacturing of carbon fiber reinforced polymer (CFRP) arches,
and the forming of galvanized steel. Information was obtained from Neal Bridge drawings and other similar bridge
drawings for typical material quantities and energy usage per unit of material.
• Construction: This portion includes the energy consumption of typical construction machinery operating on site
such as excavators, boom trucks, rollers, and pavers. The standard used was 22.2 lbs CO2 per gallon of diesel fuel
consumed.
• Transportation: This portion includes the energy cost of materials transport, equipment mobilization, and worker
transport. Estimates were developed on a per truck basis for materials using 4 mpg for large trucks, 7 mpg for medium
trucks, and 20 mpg for passenger vehicles (typically pick-up trucks). A two-level analysis was used on transportation of
materials: 1) from producer to distributor and 2) from distributor to site. Any earlier transport levels are factored into
the creation of materials. Distances used were estimates based on the availability of the product. For example, ready-mix
concrete has a shorter average transport distance than galvanized steel. Carbon footprint estimates for fuel were 19.4 lbs
CO2 per gallon of gasoline and 22.2 lbs of CO2 per gallon of diesel.
• Maintenance: This portion includes the energy expenditure for inspection of the bridge and periodic replacement of
asphalt wearing surface. Data for typical construction crews and equipment usage were obtained from the RS Means
Heavy Construction Guide 2008.
The analysis suggests that the Bridge-in-a-BackpacMrM technology promotes a
substantial energy savings over its design life when compared to its competitors.
7.2
ARCHES
71
V
S T LIU G T U Li IK L
A W A LY s" L S
mm
saw*
HKauMS
* »
SECTION 1-2
Classification of Structures
uimpit.'.Mon
/
I tension
I
Loading caus.es bending of truss.
which develops compression in lop
members, tension in bottom
members
given load. Fig, 1 5 . Also, a truss is constructed from long and slender
elements, which can be arranged in various ways to support a load. Most
often it is economically feasible to use a truss to cover spans ranging
from 30 ft (9 m) to 400 ft (122 m), although trusses have been used on
occasion for spans of greater lengths.
Cables and Arches. Two other forms of structures used to span long
distances are the cable and the arch. Cables are usually flexible and carry
their loads in tension. I hilike tension ties, however, the external load is not
applied along the axis of the cable, and consequently the cable takes a form
that has a defined sag. Fig. ]-6A. Cables arc commonly used to support
bridges and building roofs. When used for these purposes, the cable has an
advantage over the beam and the truss, especially for spans that are greater
than 150 ft (46 m). Because they are always in tension, cables will not
become unstable and suddenly collapse, as may happen with beams or
trusses. Furthermore, the truss will require added costs for construction
and increased depth as the span increases. Use of cables, on the other hand,
is limited only by their sag, weight, and methods of anchorage.
The arch achieves its strength in compression, since it has a reverse
curvature to that of the cable. Fig. 1-66.The arch must be rigid, however,
in order to maintain its shape, and this results in secondary loadings
involving shear and moment, which must be considered in its design.
Arches are frequently used in bridge structures, dome roofs, and for
openings in masonry walls.
\ 1,1
cables support then loads in tension
fa)
arches support their loads in compression
(b)
188
CHAPTER 5 Cables and Arches
Arches
evtrados
(or hack
spt iiiLjIinc
intrados
(or soffit)
haunch
abutment
fixed arch
(at
Like cables, arches can be used to reduce the bending moments in lonsj
span structures. Essentially, an arch acts as an inverted cable.so it leceivej
its load mainly in compression although, because of its rigidity, it miwi
also resist some bending and shear depending upon how it is loaded arJ
shaped. In particular, if the arch has a parabolic shape and it is subjected
to a uniform horizontally distributed vertical load, then from the analy$J
of cables it follows that only compressive forces will be resisted by t
arch. Under these conditions the arch shape is called a funicular nrA
because no bending or shear forces occur within the arch.
A typical arch is shown in Fig. 5 7, which specifies some of t|j|
nomenclature used to define its geometry. Depending upon t y
£cntcrlmc rise application, several types of arches can be selected to support a loading]
1
A fixed arch. Fig. 5-Ha, is often made from reinforced concrete. A Ithoum
it may require less material to construct than other types of aulies,jf
must have solid foundation abutments since it is indeterminate in ihjj
third degree and, consequently, additional stresses can be introduced into!
the arch due to relative settlement of its supports. A two-hingid arclQ
Fig. 5-Hb, is commonly made from metal or timber. It is indeleiminatei
to the first degree, and although it is not as rigid as a fixed aieh. it jjf
somewhat insensitive to settlement. We could make this siuictmcgj
statically determinate by replacing one of the hinges with a rollei Doinfj
so, however, would remove the capacity of the structure to resist IvndinSl
along its span, and as a result it would serve as a curved beam, and noil
as an arch. A three-hinged arch, Fig. 5-8c. which is also made from meti
or timber, is statically determinate. Unlike statically indeterminati
arches, it is not affected by settlement or temperature changes. Finallj|
if two- and three-hinged arches are to be constructed without the nee|
for larger foundation abutments and if clearance is not a problem. then|I
the supports can be connected with a tie rod. Fig. 5-isd. A tied arch allowjj
the structure to behave as a rigid unit, since the tie rod carries the
horizontal component of thrust at the supports. It is also unaffected by'
relative settlement of the supports.
l\vo-hinged arch
HI
three-hinged arch
(e)
tied arch
(d)
SECTION 5-5
5„5
Three-Hinged Arch
•
189
Three-Hinged Arch
To provide some insight as lo how arches transmit loads, we will now
consider the analysis of a three-hinged arch such as the one shown in Fig.
5_9«. In this case, the third hinge is located at the crown and the supports
are located at different elevations. In order to determine the reactions at
the supports, the arch is disassembled and the free-body diagram of each
member is shown in Fig. 5-9/). Here there are six unknowns for which six
equations of equilibrium are available. One method of solving this
problem is to apply the moment equilibrium equations about points A
and B. Simultaneous solution will yield the reactions C\ and C\. The
support reactions are then determined from the force equations of
equilibrium. Once obtained, the internal normal force, shear, and moment
loadings at any point along the arch can be found using the method of
sections. Here, of course, the section should be taken perpendicular to
the axis of the arch at the point considered. For example, the free-body
diagram for segment AD is shown in Fig. 5-9c.
Three-hinged arches can also take the form of two pin-connected trusses,
each of which would replace the arch ribs AC and CB in Fig. 5 9</. The
analysis of this form follows the same procedure outlined above. The
following examples numerically illustrate these concepts.
(a)
*
A,
Jl
An example of nn open-spandrel fixed-arched
bridge having a main span of 1000 ft (305 m)
!('ourtcsv of 'Bethlehem Steel Corporation.)
7.3
BICYCLE WHEEL
Source:
Salvadori, Mario, Saralinda Hooker, and Christopher Ragus. "The Hanging Sky."
Why Buildings Stand Up: the Strength of Architecture. New York: Norton, 1990.
Google Books. Web. 16 Apr. 2010. <http://books.google.com>.
76
16 The
Hanging
Sky
Domes and Dishes
Mankind has built domed structures for over two
thousand years and learned to attribute to their geometry very particular psychological properties. As a source of messages the architectural dome is related to the celestial sphere, which can be interpreted
as a dome of infinite magnitude covering the whole of humanity. The
feeling of protection given by die sky is more obvious at night when the
stars seem either to hang from a spherical ceiling or to be holes pierced
in it This feeling lingers in daytime, when the spherical quality of die
sky is less obvious and yet made present by the apparent circular path
of the sun. No other surface can give the same feeling of protection
because the sphere is the only surface that comes down equally all around
us. Moreover, a human being under a large dome-shaped roof has the
feeling of being at its center, of being the all-important point in the space
covered by the man-made sky. The dome sends a complex and ambiguous
message, composed of amazement, awe, and serenity.
If the dome as we normally think of it is 2,000 years old, only recently
has mankind perfected its opposite-the upside-down dome. Such a dish
roof must be built of steel, or another material capable of withstanding
tension, and could only be supported in large dimensions by means of
steel cables, the strongest structural material devised so far. Such cables
are made by assembling a very large number of steel wires until a sizable,
immensely strong cable of circular shape is obtained. As we have seen in
276
WHY BUILDINGS STAND UP
280
Ifci
MAPfi^ Mi>Af£
-iA<^M
Pl*W 1&>?
We now have dish roofs with diameters of over 400 feet covering
three or more acres of seats and over 20,000 spectators. Their technology
would easily allow a doubling or tripling of this number. How are they
built?
T h e Hanging Dish
Their essential components (Fig, 16.3} are a large outer compression
ring, a smaller inner tension ring, and a number of radial cables connecting the tension ring to the compression ring. The outer ring is supported
on columns or, at times, on a circular wall; the inner ring hangs from
the radial cables at the center of the space, always at a level lower than
that of the outer ring. Segmented slabs (usually of reinforced concrete)
placed athwart the cables constitute the surface of the disk Since the
cables (tensed by the weight of the slabs) pull outward on the inner
ring, that ring tends to increase in diameter and Its fibers elongate.
Elongation is always due to tension, the state of stress that pulls the
particles of a material apart. Therefore the inner ring must be built of
T H E HANGING SKY
281
steel, a material that withstands high tensile stresses. The outer ring,
conversely, is pulled in by the cables and tends to shrink. A reduction in
length is always due to compression. The outer ring is compressed and
can be built either of steel, a material that takes tension and compression
equally well, or of concrete, a less expensive material that takes compression but not tension.
The judicious choice of materials and the tremendous strength of
steel cables (no less than 1,400,000 pounds would be required to pull apart
a three-inch diameter cable) make it possible to span hundreds of feet with
a suspended dish roof. But the singular advantage of these roofs is the
economy of their construction, While the construction of a dome usually
requires the erection of a centering of great expense and complexity, the
dish roof is erected without a scaffold, The slabs forming its surface can
be fabricated at ground level, while the compression and tension rings arc
built, and then hoisted into place on cables previously strung between
the rings. The concrete used for the slabs is only a few inches thick, since
the slabs span small distances between the radial cables. Moreover, spacing the cables closely allows the slabs to be of small thickness wlwtecer
the span of the roof. (The thickness of a dome, on the other hand, must
increase with its span.)
Dish roofs are so light that a designer would ordinarily have to
worry about the danger of potential vibrations under the action of wind
gusts, but a very ingenious method has been devised to prevent these
vibrations without increasing the weight of the roof. We owe to the late
Uruguayan engineer Leonel Viera this last significant improvement in the
cable roof, which he employed in building a stadium in Montevideo,
|*,$ PtfH *ta*f ST£ik,T<s*Sr
W H Y BUILDINGS S T A N D U P
282
Uruguay, in 1957. His method aims, like all others, at stiffening the cables
and permits an economy and simplicity of construction never reached
before. Like most brilliant ideas, the principle of the Viera roof is so elementary that one wonders why it was not thought of years before, particularly since all the basic concepts and technologies required for its
realization had been available for a long time.
As usual, Viera set on the cables a concentric series of concrete slabs
of varying trapezoidal shape, more and more thin-wedged as they
approached the smaller inner ring. But then, rather than grouting and
glueing the joints between the slabs at this time, as everybody else had
done before him, Viera loaded each slab with an additional ballast of
bricks, thus increasing the weight on the cables and stretching them
further Only then did he grout, with a good cement mortar, the radial
and circumferential joints between the slabs (Fig. 16.4). When the grout
had hardened, making the slabs into a monolithic dish of concrete with
the stretched cables embedded in it, Viera removed the ballast of bricks.
At this point, the roof tended to move up under the reduced load but
could not do so since the cables, grabbed by the solid grout, were prevented from shortening. The roof became a monolithic, prestressed dish of
concrete, much more rigid than if it had been tensed by the slab weights
only, and built without costly supporting formwork. (Even the laborconsuming operations of ballast-loading and unloading have been done
away with lately by using in the grout special "expansive" cements that
increase in volume while setting, thereby posl-tensioning the cables by
their own expansive action.) Viera's stadium (Fig. 16.5) is 310 feet in
*A
4fifiirt\H4
r
*l
*7L-Afl>3 ** * £ A i - ^ r e p v j £ M
p?0f
T H E HANGING SKY
283
diameter, has slabs two inches thick, an outer concrete ring compressed
by the pull of the cables, and an inner tension ring of steel eighteen fee*
in diameter. The outer wall of the stadium supporting the concrete ring
is sixty feet high and only eight inches thick, and the inner steel ring,
initially supported on a light scaffold, hangs from the cables of the completed roof. The eight-inch outer wall even supports in addition two sets
of circular balconies!
Viera's structural masterpiece encountered an almost unsunnountable
objection when first proposed. How does one dispose of the rain water
accumulating in the dish, the weight of which may be much larger than
the weight of the roof itself? Viera let the water flow down and out of
the stadium through four inclined pipes hanging from the inner steel ring.
But this was found unacceptable from an aesthetic point of view by many
architects. It was then proposed to pump the water over the rim of the
roof, as soon as it starts accumulating at die level of the lower tension
ring. "But," critics asked, "what if the pump fails?" The obvious answer—
a back-up, fail-safe pump—was then countered by the new question,
"What if the electricity fails?" Again obviously, one would have gasoline
generators ready to activate the pumps. At long last the objections stopped
and the Viera system was adopted, in a variety of forms, all over the
world. These persistent objections encountered by Viera show the conservative bent of members of the construction guilds and the extreme
caution required for the introduction of new ideas in the field of structures.
We have seen similar difficulties in the acceptance of pneumatic membrane structures.
Another solution to the water-disposal problem was adopted in two
outstanding applications of the dish roof principle in the United States—
Madiscn Square Garden in New York (Fig. 16.6) and the Forum Sports
284
WHY BUILDINGS STAND UP
\k * lur&n *t HAP^^ jquAte4AtPen, * new f^K< c\rr
Arena in Inglewood, California (both designed by Charles Luckman
Associates). In both cases the cables support, above the dish surface, a
one-story circular building housing the air-conditioning and other
mechanical systems as well as the central lighting system for the building.
The roof of this additional one-story building is curved downward so that
the water runs radially to its perimeter, where it is channelled to the
boundary of the building.
While the circular shape is the most logical for a hanging dish, a
variety of other shapes have been used that increase the range of expression of these roofs. Elliptical dishes have been built without a center ring
by using two nets of cables at an angle to each other (Fig, 16.7). Pure
compression and tension rings can also be used with an elliptical dish,
if the inner ring is set at one of the foci* of the ellipse. While the circle is
a perfectly centered shape, ellipses can be built with any ratio of long to
short span. The greater this ratio, the more eccentric the location of the
foci. The eccentricity of the inner ring gives a totally different shape to
the dish and focuses the sight lines on a point off the center of the covered
area. A new dimension, which can be exploited for a variety of functional purposes, is thus added to the classical hanging dish: in spite of
the noncircularity of the roof, the outer ring is purely compressed and the
inner ring purely tensed, developing the same structural advantages and
material economies of a circular roof. Elliptical domes were built in the
eighteenth century to cover the crossings of some Baroque churches,
* The foci of an ellipse are two points on its longer aids, having die
property that the distances of any point on the ellipse from the foci add up to
the same length.
T H E HANGING SKY
265
but only in our time have elliptical dishes become a reality in widening
the architectural potentialities of hanging roofs.
An ingenious solution to the rainwater problem, suggested and used
by Lev Zetlin, has led to a structural variation of the dish roof (Fig. 16.8).
This has not one but two inner tension rings, interconnected and set one
above the other to form a hub. The lower ring sits, as usual, below the level
of the outer compression ring; the upper is above it. Cables are strung in
two sets from the compression ring, one connected to the upper tension
ring, the other connected to the lower tension ring. Spreader struts are set
vertically between the two sets of cables; by spreading them apart, all
the cables are put in tension. The resulting structure is nothing but a
horizontal bicycle wheel with the rim represented by the outer compression ring and the hub by the two interconnected tension rings. The
surface defined by the lower set of cables would be a dish, but since
panels are not supported on them, such a dish surface does not materialize. Panels are supported, instead, on the upper set of cables defining
a dome surface on which the rainwater flows outward to the perimeter
of the building.
A totally different cable surface can be obtained if a single inner tension ring is set higher than the compression ring. In this case the tension
ring must be supported on a center column, since it docs wot hang from
L.7 iJift^VW
e.L-.f'fV-^i- &Fff *."> K££T"*.\ifA*< Lfitdb
NET
286
W H Y BUILDINGS S T A N D U P
the cables. Although the construction procedure used to erect such a roof
is essentially the same as that used for the dish roof, the resulting shape is
A circular tent supported on a center pole (Fig. 16.9). Obviously such a
shape cannot be used for an auditorium or a sports arena, as the center
column breaks up the interior space. On the other hand, if the cables are
almost horizontal when they reach the outer compression ring, their
pull on it will be practically horizontal, and the entire vertical weight of
the roof will be carried by the center column. Thus the outer ring can be
supported on widely spaced columns and the periphery of the building
can be open to the outside. This is an essential requirement for buildings
like hangars or bus depots, and roofs of this type have been built in the
USSR with diameters of up to 500 feet.
Finally, it may be mentioned that in a dish or a tent roof the
structural function of the cables and the enclosing function of the slabs
can be combined by building the slabs out of thin steel sheet and welding
them together. This steel "membrane" becomes both cables and slabs and
presents the utmost integration of architectural and structural needs.
One may wonder what the ingenuity of humankind will invent to
satisfy the future needs of our culture. Much as we praise individualism
T H E HANGING SKY
i*.t ^htit-
^f
287
*M CrtMrfrf, j.jrf#*.r
and independence, the urge to gather in large numbers and thus increase
the emotional enjoyment of many experiences is so basic that it appears
to be a natural human trait. The Coliseum, with its hanging tent roof
protecting the spectators from the harsh Roman sun, allowed people to
enjoy spectacles which today we consider inhuman. Madison Square
Garden allows 20,000 people to enjoy the somewhat less inhuman show
of a man trying to beat another into insensibility or the edifying show of
people gathering for spiritual exaltation beyond words. One may dream
of the development of new steels of immense strength and of new plastics
as strong as steel and with optical properties varying with the light
impinging on diem. And one may dream of dish roofs, made possible
by such technological developments, under which hundreds of thousands
of people gather for the enhancement of their deepest experiences.
Perhaps the message of such man-made skies will be—as it was at
Hagia Sophia—simply the brotherhood of man.
Source:
Harris, James B., and Kevin Pui-K. Li. Masted Structures in Architecture.
Burlington, MA: Achitectural, 1996. Google Books. Web. 8 Apr. 2010.
<http://books.google.com>.
86
MASTED STRUCTURES IN ARCHITECTURE
nrfieei *truettw.'
auditorium,
Mica, Near Yon,
i960
i*j«fwotffd
neor tmtktng:
Standard 8 * >
HMdbuartws.
Jotennastiuig,
19/0-
apex, m its place there may be fm wails above roof 1.6
level, various forms of A-frames, or even siub
BICYCLE WHEEL STRUCTURES
columns set on some form of sub-structure. This
can lead to rather cumbersome structural 'hybrids' The modem form of dual-spoked, tensktned cycle
wheel was invented by James Startey of Coventry
{Bg. 1.7).
in 1874. In circular roof framevwxkswhidh take this
form, two sets of pre-stressed cables span
between an outer compression ring and an inner
tension ring or 'hub'. The prestressing enables
both sets of cables to share in supporting the dead
and live loads and the system is economic for
spans between 60m and 550m. The first architectural example of thts form was developed by the
engmear Lav Zetfin for the Municipal Auditorium at
Utica, New York, completed in 1960 (Fig. 1.8), The
two systems of unequally stressed cables span
73m and are held apart by tubular spreaders. An
equally celebrated example was the US pavibon at
the Brussels Expo of 1958 designed by Edward D.
Stone wtth a span of 100m and with its central hub
open to the sky.
1.7
SUSPENDED FLOOR BUILDINGS
This is a form ol tower block structure in which the
floors are supported centrally by a substantial core
structure, and at the perimeter by steel cables hung
from an anchorage framework at roof level. A concrete core can be conveniently constructed by 'sip
form' methods. The slender cable supports enable
column-free space, with minimal obstruction to
light and view, together with an open ground floor.
Typical examples are the West Coast Bailing in
Vancouver ol 1950. and the Standard Bank.
Johannesburg of 1970 (Fig 1,9).
1.8
AIR SUPPORTED STRUCTURES
These form a tolalry different group of tension
Source:
Comstock, Henry B. "They Built the Roof on the Ground." Popular Science
Mar. 1964: 98-101. Google Books. Web. 16 Apr. 2010.
<http://books.google.com>.
88
New York State's
Fair pavilion
is a mighty
wheel, supported
by slim towers
By Henry B. Comstock
They
Built the
Roof
on the
Ground
B
UILD a lopsided bicycle wheel so big its 2,000-ton
rim would almost girdle a football field. Assemble
it horizontally on the ground. Then snug up its 96
spokes—each a 2'&-mch cable.
Next, lift the wheel 100 feet and support its rim on
brackets projecting from 16 concrete columns. Finally,
cover it with multicolored plastic panels.
That's how they've constructed the Tent of Tomorrow
for the New York State pavilion at the coming World's
Fair, Architecturally, cable-suspension roofs like this may
be the biggest breakthrough since elevators spawned the
skyscraper. They need no interior supports, weigh only
nine pounds per square foot, vs. 80 for a conventional roof.
How they raised
Before prestressed table
roof of Tent of Tomorrow was built, 16 eoncrete columns
were
formed by the slip-mold
method. Standing more
than 100 feet high, they
encompass a 240-by320-foot oval area.
These are the sole supports for the huge steeland-plastic canopy.
Scaffolding behind them
is for adjacent observation towers.
SB POPULAR SCIENCE MARCH 1944
CONTINUED
thr roof for ihv 7 frit of
Tomorrow
"Bit-M-le-wheel" roof toi»,is!> of irinlci in -.liiipr
ni J 4 l M > y - 3 2 < M t « ) | t-llipM-. willi " h . i t l l i - . i M "
l i r , u v v t o wliu-ll i.ilili.^ nuli;tt< \ikv .sp'ikt-s h o m
OUTER RING
OF GIRDER
h u l l Ut i v h h T . It l o n k r t l likf tllis IM'IHN' plit-.tk
p.mi'K. I.<id IIVII ll« lUi-iiJili < u l i l r v n i i n p l c h d
tin nioi a m i IU.HII- it w r a t l i o r - t i y h t .
ELLIPTICAL RttG EIROCR
320 n . LONG, 240 FT. NIOC
16 SUPPORTING COLUMNS AROUND RING GIRDER
arena
MM6!
TUB
mua
c
» totraioM „
-HI If
ttCMttt
UWKO
LVNEIKMI
BUST*
mran
•~*»~».
* naiuii
!
3 I-
Thirty inches at a time, roof was mised by 32
hydraulic jacks in temporary four-sided towers,
alongside permanent concrete columns. In last
drawing, roof has been fastened at planned
Ciring the cable-*nt petition
r
100-foot height, by welding bracket section beneath it to similar section extended from column. Jacking rig, its work done, is lowered,
and temporary tower can now be removed.
roof a coat of many color*
First of 1,500 plastic panels were laid atop the roof long
Wfore it was fully raised. Working from tensioning ring,
each was Imlted to the upper cables, and all joints lapped
with weatherstripping (photo at right). On sunny days,
and at night, when 800 topside lights shine through it,
the translucent canopy will tint the exhibit area below.
Building's architect is Philip Johnson Associates.
100 POPUUR SCIENCE MARCH 1944
1
Touchy job of keeping roof constantly horizontal
was controlled from this hydraulic ner\e center.
Walkie-talkies kept the operators in continuous contact with tower-crew foremen.
JSi'tr York Stale lookouts
Two of three observation towers, taking shape beside Tent of
Tomorrow, rise above it. Their
radial roof beams will double as
supports for ring-shaped decks
suspended below. C o n c r e t e
crossarms flanking tower at
left in photo hold guides for
"capsule' elevators. Lapping
observation decks will let one
ear serve all three towers. Another ear will run express only
to the highest platform.
Had any part of hydraulic system failed, jacks
could not have dropped roof as much as an
inch. Horseshoe-shaped stops (sketch), slipped
around pistons, provided a safeguard.
trill be the highest at the Fair
7.4
FLAT SLAB
93
HANDBOOK
OF
BUILDING CONSTRUCTION
DATA FOR
ARCHITECTS, DESIGNING AND CONSTRUCTING
ENGINEERS, AND CONTRACTORS
VOLUME I
COMPILED BY A STAFF
OF FORTY-SIX SPECIALISTS
EDtTOBS-IN-CpB*
GEORGE A; IJOOL, S.B.
OOXSULTINa •NODfBBB, IfADUON, WUCOltUX
IB or
•lmucruaAL •NOINEEMNO, TB> m n i M n
or wueomm
AND
NATHAN C. JOHNSON, M.M.E.
OOOTDLTWO, KMOIWBB*. KSW Y O U OXTT
FBBT EDITION
FOUBTH IMPRESSION
McGRAW-HILL BOOK COMPANY, INC.
NEW YORK: 370 SEVENTH AVENUE
LONDON: 8 * 8 BOUVERIE ST., E. C. 4
1920
Google
fee*''f •'fr<
/
EDITORIAL STAFF
EDITORS-IN-CHIEF
George A. Hool, Consulting Engineer, Professor of Structural Engineering, The University
of Wisconsin, Madison, Wis.
Nathan C. Johnson, Consulting Engineer, New York, N. Y.
ASSOCIATE EDITORS
PART I—DESIGN AND CONSTRUCTION
James O. Betelle, of Guilbert & Betelle, Architects, Newark, N. J.
D. Knickerbacker Boyd, Architect and "Structural Standardist," Philadelphia, Pa.
John Severin Branne, Consulting Engineer, New York, N. Y.
H. J. Burt, Structural Engineer, Manager for Holabird & Roche, Architects, Chicago, 111.
Walter W. Clifford, Structural Engineer, Hyde Park, Mass.
Chas. D. Conklin, Jr., Civil and Structural Engineer, Cheltenham, Pa.
F. W. Dean, Mill Architect and Engineer, Lexington, Mass.
Henry D. Dewell, Consulting Engineer, San Francisco, Calif.
Richard G. Doerfling, Civil Engineer, San Francisco, Calif.
Wm. J. Fuller, Assistant Professor of Structural Engineering, The University of Wisconsin,
Madison, Wis.
Harry L. Gilman, Consulting Engineer, Newton Highlands, Mass.
James H. Herron, of James H. Herron Co., Consulting Engineers, Cleveland, O.
Frederick Johnck, Architect, of George C. Nimmons & Company, Architects, Chicago, 111.
LeRoy E. Kern, Structural Service Bureau, Philadelphia, Pa.
Frank R. King, Domestic Sanitary Engineer, Madison, Wis.
W. S. Kinne, Professor of Structural Engineering, The University of Wisconsin, Madison,
Wis.
W. J. Knight, Consulting Engineer, St. Louis, Mo.
Clyde T. Morris, Professor of Structural Engineering, Ohio State University, Columbus, O.
A. G. Moulton, Vice-president of Thompson-Stairett Company, Detroit, Mich.
Allan F. Owen, Structural Engineer, of George C. Nimmons & Company, Architects, Chicago,
I1L
Arthur Peabody, State Architect, Madison, Wis.
J. C. Pearson, Chief of Cement Section, United States Bureau of Standards, Washington, D. C.
Corydon T. Purdy, Civil Engineer, of Purdy 4 Henderson Company, Engineers and Contractors, New York, N. Y.
EL Ries, Professor of Dynamic and Economic Geology, Cornell University, Ithaca, N. Y.
Alfred Wheeler Roberts, Structural Engineer, of Perin and Marshall, Consulting Engineers,
New York, N. Y.
H. S. Rogers, Engineer for The Truscon Steel Company, Youngstown, Ohio.
M. Y. Seaton, Chemical Engineer, Dow Chemical Company, Midland, Mich.
W. Stuart Tait, of.Tait & Lord, Inc., Consulting Engineers, Chicago, HI.
Frank C. Thiessen, Structural Engineer, Madison, Wis.
T. Kennard Thomson, Consulting Engineer, New York, N. Y.
F. R. Watson, Professor of Experimental Physics, The University of Illinois, Urbana, 111.
Harvey Whipple, Editor of "Concrete," Detroit, Mich.
Vll
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viii
EDITORIAL
STAFF
PART H—ESTIMATING AND CONTRACTING
Arthur E. Alitis, Construction Engineer, of GiahoH Machine Company, Madison, Wis.
Daniel J. Hauer, Consulting Engineer and Construction Economist, Baltimore, Md.
Clayton W. Mayers, Chief Estimator, Aberthaw Construction Company, Boston, Mass.
Arthur Peabody, State Architect, Madison, Wis.
PART m—MECHANICAL AND ELECTRICAL EQUIPMENT
H. P. Bates, Otis Elevator Company, New York, N. Y.
G. H. Cheesman, Otis Elevator Company, New York, N. Y.
S. E. Dibble, Head of Sanitary Equipment and Installation Department, Carnegie Institute of
Technology, Schenley Park, Pittsburgh, Pa.
Ira N. Evans, Consulting Engineer, Heating and Power, Detroit, Mich.
Charles R. Goughenour, Consulting Engineer, of The United Electric Company, Canton, O.
C. M. Jansky, Professor of Electrical Engineering, The University of Wisconsin, Madison,
Wis.
Frank R. King, Domestic Sanitary Engineer, Madison, Wis.
W. G. Kirchoffer, Sanitary and Hydraulic Engineer, Madison, Wis.
W. W. Lighthipe, General Service Manager, Otis Elevator Co., New York, N. Y.
Stewart T. Smith, Architectural Engineer, of Van Rensselaer H. Greene, Refrigerating Engineer,
New York, N. Y.
^-.•M.v G o o g l e
PREFACE
These volumes have been prepared to provide the architect, engineer,
and builder with a reference work covering thoroughly the design
and construction of the principal kinds and types of modern buildings
with their mechanical and electrical equipment. Since the art of
building is now highly specialised, an unusually large number of associate editors were engaged in order to cover the field in a reliable and
comprehensive manner.
The Editors-in-Chief desire here to express their appreciation of the
spirit of cooperation shown by the Associate Editors and the Publishers. They desire also to express their indebtedness to Mr. Clifford
E. Ives for his excellent work in preparing the drawings from which
all the sine etchings were made.
G. A. H.
N. C. J.
September, 1920.
U
:7 Google
434
HANDBOOK OF BUILDING CONSTRUCTION
[Sec 8-89
FLAT SLAB CONSTRUCTION
BY W. STUART TAIT
89. In General.—Flat slab construction consists of a concrete slab of practically uniform
thickness so designed that the slab carries and transfers the load coming upon it directly to the
columns. This form of construction has become very widely used during the past ten years
until today it is used to a far greater extent for warehouse and manufacturing buildings than all
other types of concrete construction. It is also used in railroad track elevation, in bridges,
hotels, apartment buildings, and offices.
The correct method of design for this type of concrete construction has been a contentious
point among engineers for a number of years. In spite of a lot of research work, flat slab construction must still be classed as a statically indeterminate structure. The methods of design
now in general use must be considered as empirical but we have now had a sufficiently wide
experience with their application to be certain of the results to be obtained.
The Joint Committee recently adopted a ruling for the design of flat slab construction but
this ruling will not be treated here owing to the fact that it is rather too flexible to be considered
as a design method. In addition to this, it has had very little practical application and results
obtained in years of experience with other rulings do not justify the higher moments given under
the Joint Committee report. The proposed American Concrete Institute ruling agrees closely
with many building codes which have been in effect for a number of years, and which have given
highly satisfactory results. The A.C.I. ruling, however, is more complete than any city code
so far as the writer is aware and covers more completely many of the secondary features of the
design. There will be given later a number of examples of the different forms of flat slab construction, fully worked out, so that by following through and understanding the various steps,
an engineer will have no difficulty in applying any of the various city codes now in effect. It is
almost an impossibility to cover all the points of flat slab design in a handbook such as this,
which may determine the difference between a highly satisfactory structure and one which is
•imply passable. Furthermore, long experience is necessary before a designer may be able to
produce the most economical design for a given purpose. It is, therefore, desirable to have
designs of this class prepared by an engineer who has had wide experience in flat slab construction and who has proved by the satisfactory structures to his credit that he is an authority on
the subject.
A number of systems of flat slab construction have patented features which may or may not
contribute to the efficiency, economy, and strength of a design, but it is not the writer's intention to elaborate on these various systems but rather to explain and show examples of flat slab
design which can be taken as guides by practicing engineers.
In multiple-story warehouse and factory construction the flat-slab type of design shows
marked economy over other types of concrete construction. In most cases too, it offers many
physical advantages. Its execution is thoroughly understood by the greater proportion of concrete contractors and, owing to its simplicity, good construction and accurate adherence to the
designs are easily obtained. Designing engineers would do well to give this method of construction very careful consideration before deciding upon the type of design to be used for any
building, particularly where the structure has large floor areas with fairly regularly spaced
columns. It has also been found that in many hotels, offices, and apartment buildings where
regular column spacing can be obtained and in which spans of about 18 ft. or less can be used,
that the type of flat slab construction, in which large columns without any projecting capitals
are used, offers economy and some advantages.
90. American Concrete Institute Ruling.—The diagram, Fig. 128, together with the following notes, i8 a summary of this proposed ruling.1 It is inserted so that designers may easily
follow the examples worked out later. The general notation is given in Appendix A.
i While the following matter wu in the hand* of the printers, aome alight modification* to this proposed ruling
war* made at the 1910 convention. These proposed modification* are not shown in this chapter.
Google
1
Sec 9r-90)
STRUCTURAL
435
DATA
Slab Thickness.—t shall not be less than 0.02Ly/w + 1 in., nor leas than L/32 forfloorsand
L/40 for roofs.
Design Momenta.—Numerical sum of positive and negative moments shall not be less than
0.09 wh (lt — qc)*. The report allows a slight variation in the distribution of this total moment. A reasonable division of this moment in percentage is shown in Fig. 128. Note that a
slightly different distribution applies in the case of drop construction from that in cap construction. Corresponding moments shall be figured at right angles to those shown in Fig. 128.
The moments shown in Fig. 128 are calculated for a value of the cap diameter c — 0.225L, and
are for interior panels.
For exterior panels the negative moment at the first row of interior columns and the positive
moments at the center of the exterior panels on sections parallel to the wall shall be increased
n
Ccgfn/ctifip—
tout Cap or Drop
*=*•
Construction
*»g}£C0p
wTbhldaodmdfnt/oad
ptr aauarw toor.
t "ft ff™**i capitals.
Drop
Constructed
Construction
OmcfUsnArA
L -$£%}*
narkafMm
oSftmmth* longtr
Way,
Ajjmcrntirtcoafficftnlsan
fdcton oftfitt
Fio. 128.
20% over those specified for interior panels. The negative moment at the exterior column
parallel to the wall shall not be less than 50 % of that for the interior panel.
Shear.—The shearing stress which is used as a measure of diagonal tension stress is calcu0 25W
lated on a width equal to L/2, and the formula used in this calculation is r — -'..,- for cap con0.30TT,
for drop construction. Punching shear at the edge of the drop and
bjd
at the column cap is calculated by multiplying the total panel load occurring outside the area
under consideration by 1.25 and dividing this load by the perimeter of the cap (or drop as the
case may be) and by d.
Columns.—Both interior and exterior columns shall be designed for bending. The moment
in a column shall not be less than 0.022wih (It — qc)* where wx is the designed live load. In the
ease of exterior columns, the total dead and live load (w) whould be used in the above formula
instead of wt. For top story columns, this amount is all applied at one section of the column.
For columns continuous through the story above, the moment is to be divided between the
struction, and v •=
•Google
436
HANDBOOK OF BUILDING CONSTRUCTION
[S«c »-91
upper and lower column in proportion to their stiffness. Stress used in calculations for direct
load and bending may exceed the direct load stresses allowed by 50 %.
Stream*.—In the examples worked out, the stresses recommended by the Joint Committee
based on 1-2-4 gravel concrete (see Appendix J) are used as follows: /«for positive moment = 650 lb. per sq. in., / , for negative moment .«• 750 lb. per sq. in., /, = 16,000 lb. per sq.
in., shear as a measure of diagonal tension — 40 lb. per sq. in. on plain concrete. Punching
shear — 120 lb. per sq. in.
• 1 . Example of Design—Drop Construction, Four-way Arrangement—Take a panel 20
ft. square for a live load of 300 lb. per sq. ft., with cement finish laid with the slab.
Lire load Dead lewd w C M 0K-ta-
300 lb.
118 lb.
415 lb.
abb. Fireprooftng 1 in.
t - 0.02L\^i + 1 - (0.02)(20)(V7l5) + 1 - 0.15 in.
< not lew than L/32 - 7.6 in.
d for outer Motion - 9.25 - 1.26 - 8.00 in. (one layer
of steel)
d for inner section — 9.25 — 1.50 - 7.75 in. (two layer*
of Heel)
Column capital - 0.225L - 4 ft. 6 in.
M - column head section - 0.0336wli X It* - (0.0386)(416)(20)(20)<(12)
M - KM*, b - 0.3L - 8 ft. 0 in. K - 134 (see Sect. 2, Art 31a).
d*
(134H6K12)
(in.-lb.)
189 in., or d - 11.8 in.
- 11.8 + 1.00 + 1.00 (4 layers steel) - 13.8 in. Use 14 in. Slab - 9>i in. Drop - 14 - 9>i - 4Ji X « ft.
0 in. X 6 ft. 0 in. Note with 14-in. thickness, d at column becomes 12 in. (see later increase).
ou
* ,
0.3 IT
(0.3) (415) (20) (20)
....
Shear at column - - ^
- 401b.
(
«. u..
* J
,A
(415)(400 - 36X1.25)
„ „
Punchini shear at edge of drop - * 14) (72) (7 7-5)
"
_
..
.
. , ' ,
...
(415)(400 - 16) >1.25)
„,.
Punching shear at edge of capital - *
U54U12)
99 lb.
From this, it is noted that both punching shear and diagonal tention stress are within the limits prescribed.
M - column head section (see above) - 1,340,000 in.-lb.
.
1,340.000
'
(b^6Tu6ToooT(i2)
A
.
.
8 IZ q
-
.
' -' •
11 - mid. section - 0.0065 teli X It*
- (0.0066)(418)(20)(20«)(12) - 258.000 i n . * .
A
W
« " (0.86)(^£g(8.00) - " * • * ^ " 1 » ~ « - i n - «"**
M — at outer section-0.0118wti X It*
- (0.0118) (415) (20) (20)*(12) - 470.000 in.-lb.
A
-' «.m™:™Hs.oo) - * • » • * * • - » - * * • r o u a d b*~
d» required - M/Kb - T ^ C T J ^ J - 86.2 in.
d - 6 in., where we have 8.00 in.
It — at inner section - 0.0129 uli X It*
In this design we are using the four-way arrangement of steel, and consequently each bar in each diagonal band
cuts the inner section line at 45 deg. The A. C. I. ruling specifies that the sectional area of bars, crossing any
section at an angle multiplied by the sine of the angle between these bars and the section may be considered as
effective. Now we have two diagonal bands of rods, so the effective area of steel to resist the moment at the inner
section — 0.7 X 2 bands of rods — 1.4 bands. Therefore
A. each diagonal band - ^ 6 ) ( J ! o o 5 ? 8 . 0 ) ( l . 4 ) " " 8 " 1 8 ~ **>- r o t t n d b " *
We, therefore, have the following reinforcing for the interior panels:
Direct bands
22—H-in. rounds - 4.30 sq. in.
Diagonal bands
18—H-in. rounds - 3.53 sq. in.
Across direct bands.. 12—H-i». rounds - 2.35 sq. in.
If general practice is followed, and we bend up all bars at the column, we have 4.3 + (1.4K3.53) - 9.24 sq. in.
effective, and we found above that 8.12 sq. in. were required at the column head section.
Bxitrior Panel.—In case the exterior panel is the same sise as the interior, for which the design above it shownthe moment at the first interior column would be increased by 20% and becomes 1,340,000 X 1.2 - 1,610.000
in.-lb. To resist this increased moment the depth of the drop or the width must be increased. For the sake
of uniformity, it is good practice to make all drops the same sise and to let all other interior drops be governed
by the sise of the first interior. If 6 is kept 6 ft. 0 in.,
d* - liij$j£ - »»
* " « w *» - •* » m-
• Google
Sec S-91]
437
STRUCTURAL DATA
The drop then become* 13 + 2 — 9\i - 5J4 in. This increase in the interior eolumn drop tbiokneaa would permit
lea* steel to be used at the eolumn section, but it is better practice to allow the number of rods given above to remain,
aa abort ban should be avoided.
Now A. at tat int. column - — M ^ —
- 9.0 „ . In.
A, direct band normal to wall becomes - (1.2)(4.3) - 5.1 sq. in. - 26 - Jf-in. round ban.
A, «w»r-»-i band in the exterior panels - (1.2)(3.46) - 4 . 1 4 - 21 - H-in. round rods.
If we therefore band up all bare to top of slab at the let interior eolumn from M exterior span, wa have
&1 + (1.4K4.1) - 10.8 sq. in. whieh is satisfactory- Sinee the moment is the same on eaeh aide of the eolumn
the extra bars in the exterior panel must eontinue past the first interior oolumn to the quarter point of the
next span. This is shown In diagram, Fig. 129.
Fio. 120.
The A. C. I. ruling specifies a bending moment at the column head section parallel to the wall at the exterior
«WJM-»" of SO % of the interior oolumn head section moment, i.e., O.OlOSts/iIt* -- 870,000 In.-lb. The area of the
ATA ftflO
steel whieh must be provided to resist the moment aeroas thia section — (nBgwift'ooomai ~ *"®* *q* *n- W e nave
available, 5.1 + (1.4)(4.1) — 10.8 sq. in. which is more than is required. It is good practice to allow about onehalf the bars in the exterior direct band to pass through in the bottom of the slab, and the other half to be bent up
to top of slab. The moment which this steel is resisting occurs at the edge of the eolumn capital and the distance
from tt is point to the end of the bars is usually ample to develop 18,000 lb. in the steel in bond. It is, however,
good practice to bend the ends of some of the bars down into the eolumn or beam.
Now the moment in a direction normal to thia is one-half the moment of an interior eolumn head section, since
there exists but one-half a section along the wall. Therefore, the cap and drop construction will be similar to that
used at an interior column. The steel required at this section - 4.06 sq. in. We have available 2.15 -f (0.7)(4.1)
» 5.02 aq. in. So aa to provide sufficient imbedment to develop the bars in the exterior diagonal bands, it is generally advisable to eontinue the ends of the bars along the wall a short distance. The steel arrangement for thia
design is shown in Fig. 129. Note that one-half of the bars in each band only are broken at each eolumn. Thia is
a recommendation of the A. C. I.
Column MomtnU.—The ruling specifies a bending moment of 0.022wi!i(fo — ee)< for interior columns. In this
ease M - 0.0158wiP - (0.0158)(300)(20)«(12) - 455,000 in.-lb. Top-etory interior columns should be designed
for this moment combined with the direct load. The lower-etory columns, if of equal sise above and below the
floor considered, should be designed for half of this moment. If of different sixes, the moment should be divided
directly aa the stiffness of the columns, i.e., in proportion to the value r for each column, where I is the moment
of Inertia and h the height of the column. Similarly the exterior column moment
it - O.O168W0 - 0U)158)(416K%)*(12) - 620,000 in.-lb.
Co ogle
438
HANDBOOK OF BUILDING CONSTRUCTION
[Sec. *-92
must be provided for. Note particularly that the A. C. I. ruling allow* an extreme fiber street combining direct
load and bending SO % greater than the direct street allowed for columns. While no ruling or ordinance is distinct on this point, it is the writer's opinion that in designing columns for direct load and bending, the entire concrete
section may be considered. His reason for this is the fact that we are not required to deduct any portion of the
concrete in a beam in designing for negative bending and the lower aide of a beam at the supports hi just as liable t o
damage from fire as is the oolumn it rests upon.
92. Example of Design—Cap Construction, Four-way Arrangement—In the previous
example the design was accompanied by many explanations but in this case these will be
eliminated, as they would simply be repetition. Take a panel 20 X 22 ft. for a live load of
100 lb. per sq.ft. with a maplefloorfinishlaid on sleepers with a cinder concrete fill between.
Live load - 100
Floor finish - 2 0
Dead load - 112
t - 0 . 0 2 L \ / V + 1 - (0.02)(21)( V232) + 1 - 7.4 in.
I not less than L/32 - 7.85 in.
Column cap - (0.225)(21) - 4 ft. 9 in.
w - 232 lb.
it - column head section - 0.0286Wi X Pi - (0.0286)(232)(20)(22)«(12)
.,
(0.0286)(232)(20)(22)'(12)
._ .
.
d
OSKOOHZOKTZ)— "
47 6
--
(in.-lb.)
. „ .
* - wo • .
i required •• 6.9 + 1 + 1 (4 layers of steel).
Use 9-in. slab.
d at column head section — 9 — 2
- 7 in.
d at mid-eeotion and outer section - 9 — 1.25 - 7.75 is.
d a t inner section — 9 — 1.5 — 7.5 .in.
Mai. shear at column - 9 g £ - ^ g ^ 2 ^ ? ) . 85 lb. per sq. in.
Punching shear at column - < * 2 2 > l ± g ^ p ^ - 98 lb. per sq. in.
From this we find that 9-in. slab satisfies the shear requirements.
. . . . .
,
(0.0286) (232) (20) (22)'(12) T_ „ _
.
t.
A. at column head eection acroa. span /,
(0.86) (16,000) t7)
" * "* ^
. . . . . . .
,
(0.0286) (232) (22) (20)'(12)
_ ..
A. at column head section across span lj - i
(0 86)116 000) (7)
"
"*
. . 4
^.
,
(0.0142)(232)(20)(22)«(12)
„.
.
,„
,. .
.
.
A, at outer section across spaa Ii (0 86j (16 000) (7 75)
"
•Q- M. - 1* - M-w. round rods.
A * .
i
(0.0142) (232) (22) (20)*(12)
„„
.
,_
„ .
.
.
A. at outer section across span li - -—.p-ge) ( i s dool (7~7fi)
™
*q'
™ ~
"""^ * ° ^
As each diagonial band - - . - ; " 2.52 — 13 — X-in. round rods.
1.4
If all bars are bent up to top of slab at oolumn, the steel we have available across span ft — 3.6 + 3.53 - 7.13
sq. in. The steel required - 7.95 sq. in. We must therefore provide 7.95 - 7.13 - 0.82 sq. in. or 4 — H-in.
round bars extra. The sted available across Ii — 3.27 + 3.53 — 6.8 sq. in., and we require 7.22 sq. in. We must
therefore provide in this direction 0.42 sq. in. For the sake of uniformity we will add 4 — H-in. round bars in each
direction and these bars will be made 11 ft. 0 in. in length.
The exterior panels and the bending moments in the oolumn will be found and treated in a manner similar to
the case where drop construction was used. It must be borne in mind, however, that the bending moment at the
first row of interior columns will have to be increased across the section parallel to the wall. Since we must maintain the same thickness of slab, namely, 9 in., it will be necessary to introduce compression sted in this direction
provided the exterior span is the same as the interior It is convenient where the layout permits, to slightly reduce
the exterior span so that the moment at the first interior column bead section is the same as the others. This has
been done in Fig. 129 which is a plan of this design.
9S. Example of Design Where Neither Drop nor Cap Are Used.—It will have been noted
that a smaller percentage of the total bending moment was used at the column head section
in the case of cap construction than in the case of drop construction. This is on account of the
tact that drop construction is slightly stiffer than cap construction at the supports. Now
if in addition we eliminate the capital, we have still a smaller amount of stiffness at the column
section. Accordingly, a slightly smaller percentage of the total moment may be used at the
column head section. A satisfactory distribution of moments for the four-way arrangement
is shown in Fig. 128. Square columns are generally used in this design, as partitions fit up to
them better than other shapes. The writer has found that a square column having a aise of
O.llL usually proves economical and satisfactory. The bending moment coefficients shown in
fOO^I c
439
STRUCTURAL DATA
Sec S-93]
Fig. 128 for this class of design are based on this value. Designers will find it economical to
maintain the same sue of columns for a number of stories in this design. In most cases of
designs of this class, the writer has maintained one sise of column throughout the structure,
simply varying the mix and steel for the increased loads. Take a panel 16 ft. square for a live
load of 50 lb. per sq.ft., a partition load of 25 lb., plaster ceiling, and cementfinish\)4'w. thick.
live load
Partition*
Plaster online
Cement finish
Dead load (7-in. slab)
L/32 - 8 in.
0 . 0 3 £ \ / w + 1 - 5.36 in.
- SO
- 25
8
- 18
- 88
180 lb.
Minimum column aiae - (0.11X10) - 20 in. square.
sue in necessary on seoount of punching shear.
Fireproofiag below steel
FSreproofiag abore steel
d at oolumn head section
a* at outer section
d at mid-eeotion
d at inner section
M - oolumn head section
We will UM 22 in. square a* it will be found later that this
- 1 in.
— H in. (Note cement finish abore the structural abb).
- 7-H-H
- 675 ».
- 7-1-Jfe
- 6.80 in.
- 7-H~M«
- 6.30 in.
- 7-1-H
- 5.02 in.
- 0.0302wfi X It*
- (0.0302)'189)(10)(16)»(12) - 278.000 in.4b.
Had thia design been for oap construction, the diameter of the cap would have been about 0.226X, or 8.6 ft., and b
used in the reeistingmoment at the oolumn head section would have been L/% or 8 ft. In thia ease we hare a
oolumn 1.8 ft. in width and we should therefore use a width of beam - 8.00 - (3.6 - 1.8) - 6.2 ft., or 74 in.
Another good rule is to limit 6 to the width of the column plus St. In this ease we would have 22 + 56 — 78
278 000
in. In this ease we w _ use the smaller value, namely, 74 in. At the column head section, then d« — 7T34H74)
- 28.0 in. and d - 5.28 in., so the 7-in. slab assumed above ia satisfactory.
fth_.r *• ™l„ m n - _ _ ? - (0-26)(18O)(16)(16)
Shear at oolumn- - —
_ _ _ _ _ _ _ _ S3 lb. per sq. In.
_
..
.
(189)(256 - 8)(1.25)
,,„..
.
Punching shear
"««>. peraq. in.
(4)(2g)(ft.76)
The 7-in. slab ia satisfactory for shear.
A. at column head section -
(0 8 6 )
^;^)(6
A. at midsection - < $ g g « S >
76)
- 3.52 sq. in.
" *» ~
*
" * " «*" ™
d
A. at o u t - s**on - ^ g S S i f
" ™ ~ *• " * " « - ~ - —
A -* ««..«. _ - t ^ n - (0182)(189)(16)'(12)
A. at inner Motion- - - — — — — — - _ 2.18 sq. in.
2.18
A. each diagonal band - —— — 1.66 aq. in. - 14 — X-in. round rods.
« ~
—•
We have available at the column head section, 2.12 + 2.18 — 4.30 sq. in., where we require 3.52 sq. in. provided
all the bars are raised to the top of the slab at the column. It would be good policy, however, to run the excess
steel, i.e., 4.30 — 3.52 — 0.78 sq. in. or 7 — H-i— round bars in the direct band, through in the bottom of the slab
and raise the remaining 13 ban at the column.
The exterior panel may be designed in a manner similar to that given under drop construction. In this case,
as in the ease of eap construction, it is desirable where the layout permits to decrease the sise of the exterior panel.
Unless this can be done it will usually be necessary to determine the slab thickness by using the bending moment
for the column bead section which applies to the first interior columns. Where relatively thin slabs are used, as in
thia case, compressive reinforcement to provide the increased resisting moment necessary at the first row of interior
columns is very inefficient. In the above ease, the moment at the first tier of interior columns is such that a depth
of 5.8 in. is required. We have practically this depth available due to the fact that punching shear governed the
slab thickness. In eases where the increased moment warrants the addition of compression steel or increased slab
thickness at the first interior tier, increasing the slab will usually be found to be the most economical. Fig. 120 ia a
plan of this design.
Other arrangements of steel, such as the two-way or combinations of two and four-way, should be treated in the
same manner as the above .Some of the systems of reinforcing prefer slightly different distributions of the total
bending momenta from those shown in Fig. 128. The distribution shown will, however, give satisfactory results
• OO^IC
440
HANDBOOK OP BUILDING CONSTRUCTION
[Sec. 3-94
04. Construction in Which Brick Bearing Walls are Used Instead of Exterior Columns.—
This class of support for flat slab construction should be avoided by engineers wherever possible.
In the case of relatively short spans it can usually be relied upon to give satisfactory results.
Engineers who have not had extensive experience in the design of flat slab construction would,
however, be wise to avoid its design. Neither the Joint Committee nor the American Concrete
Institute make any recommendations covering the design of these exterior panels. Flat slab
construction relies to a marked extent upon the stiffness of the exterior columns to prevent
undue deflection in the exterior panels. If brick walls are used for the exterior support, this
restraining action is practically eliminated and the slab itself must therefore be stiffened up.
The Chicago Code specifies that the positive moments in these wall panels shall be increased to
50% in excess of those used for interior panels—it does not, however, specify that the slab
thickness must be increased. The Chicago Code formula for minimum slab thickness is
I — 0.023L\/w- The minimum slab thickness used for wall bearing construction should be
0.025Lv,w + 1, where L is in feet and the result in inches. It would also be well to increase
the minimum thickness to about L/28 for both floors and roofs.
Pilasters with substantial corbels on line with the interior columns should be used in the wsll. The tots! pilaster
and wall thickness should be at least equal to the minimum sise of column permitted (i/12) plus 4 in. The width
of pilaster should be at least equal to the thickness of the wall and pilaster. The corbel should hare a vertical
depth of at least two courses before the offsets begin. The corbel projection should be determined in the same way
as that of a column cap for the same length of span. It will be found that the brick wall will be subjected to some
bending in a similar manner to a concrete column. The amount of this bending will probably be less than that
occurring in a concrete column. It is well, however, to make an investigation of the stresses occurring in the
pilaster by combining the direct load with the bending moment given previously for exterior columns. The
pilaster sise used should be such that little or no tension is found upon combining the direct load and bending, and
also that the maximum compression is within that allowable upon the kind of brickwork used.
95. Rectangular Panels.—Flat slab construction proves most economical in panels which
are approximately square and engineers should endeavor to make their layouts accordingly.
Most codes and rulings provide that the methods of analysis given are limited to panels in
which the long side is not greater than 1.33 times the short side. It has been the writer's
practice in cases where this proportion was exceeded to a slight extent, to increase the length of
the short side for design purposes only so that this proportion of spans was maintained. Thus,
a panel 20 X 28 ft. would be treated in the design as if it were a panel 21 X 28 ft. It will be
noted by referring to the total bending moment formula given previously that the moment
in any band is a function of the total panel load times the first power of its span. This form of
moment equation is recommended by both the Joint Committee and the American Concrete
Institute. In some of the older building codes the bending moment in a band is a function of
the load per square foot times the cube of the span of the band. It will be found upon examining this method that the moment in a band running in the long direction of the panel is exactly
the same as that in a square panel of the same span. In the 21 X 28-ft. panel referred to
above, under this method of analysis, we would have the same moment in the band running
in the 28-ft. direction that we would have in a panel 28 X 28 ft. This is obviously incorrect
for the width of the panel would only be 21 ft. and not 28 ft. In order to insure that sufficient
reinforcement is introduced in the short span direct band, the code usually further requires
that the steel in that direction shall not be less than 75 % of the steel in the long span direction.
Most codes and rulings allow panels, in which the long side is not more than 1.05 times the short side, to be
treated as square panels having a span equal to the mean of the length of the two panel sides.
The drop panel in rectangular panels should be made rectangular since with this arrangement we tend to stiffen
up the slab on the long span. Thus, in a panel 21 X 28 ft., the drop panel would be about 6 ft. 6 in. X 8 ft. 8 in.,
the width and length both being directly proportionate to the width and length of the panel.
96. Unequal Adjoining Spans.—In fiat slab construction, as in any form of design where
we have continuity over a number of unequal spans, the correct bending moments must be
obtained by applying the Theorem of Three Moments. In flat slab construction since the
moments used are empirical we cannot apply the theorem directly but must increase or decrease the bending moment coefficients used for equal spans by applying certain factors to
GOOQIC
C '
STRUCTURAL
Sec. S-96]
441
DATA
these moments. The following is a method of applying the Theorem of Three Moments and
obtaining the factors referred to for the case shown in Fig. 130.
In this layout we have a tenet of panels 20 ft. in length and varying from 16 to 25 ft in width. While the
arrangements somewhat irregular, it will be noted that the length of the panels is in no case greater than 1.S3 times
t h e breadth. We will assume that drop construction is to be used and that the column caps are 0.226X in diameter.
T h e banding moment coefficient for uniform spans, then, will be as shown in Fig. 128. As explained previously
Mi— yi it for interior column head section
- 0.0188«* X li» - (0.0l68)(w)(X)(r,)«
or per foot of width - O.OlSwTV
(1)
3
L-h,_^
1
.. r p
V-
|i
1#eu*r
ikfouhr
Steffon
i
N*ff
1?
^r raw. -2i>.
s.
5 _ u_
•i\
- F
•
_
1
hi
S.
• — s
r
i
,*J
(
L
L
&
1, *>
M
•r
n.
*>
T
'
A
T
'
,',
»u
7
'
»'
t&
16'
t
I.
.
a.
7
*
1
T
*
Fio. 130.
Now applying Clapeyron's theorem, we have
-w(Ti» + TV>
4
MtTt + 2M»(Tt + Ti) + M,Tt- -»<JV + TV)
4
-w(Ti» + T**)
MtTt + 2M*(T, + T«) + JCiT. 4
M1T1+ 2Jfi(T, + Ti) + M*Tt -
MAT, + 2it,{Tt
+ Ti) + U»Tt - -w(r««
+ r.«>
.(3)
(4)
.(5)
Now substituting in these equations for Ti, Tu etc., we have values per foot width in the direction of span X as
follows:
Equation (1) becomes Jfi
- 4.3w
(2) becomes 82Jfi + 25 Jfi
- - 4861 0w
(3) becomes 251ft + 82 Jfi + 16Ji 4 - - 4930.2w
(4) becomes 16Jfi + 72Jf4 + 20Jf » - - 3024 w
(5) becomes 20Jf < + 88Af 1
- - 3024 «e
Solving these simultaneous equations:
A f 4 - - 24.Ow
Jf - 4.3w
J f i - - 2 8 . 6 w - J»i
Aft - - 46.8w
Jfi - - 41w
Now And the positive momenta Nt, Nt, etc., as follows:
AT, - - 0.95w
JVi - X ~ HiMl + Mt)
Ni - 6.47u>
A T « - 23.26w
Nt - 34.23w
A T i - 3.40v
The quantities Jfi, Jff , etc., and Nt, JVi, etc., are the bending moments per foot of width at their respective points
ihown in the diagram and are for one-way construction. Now, if we obtain a value for Nt similarly to the above
but for a series of spans equal to Ti, and also a value for Nt for a series of spans Tt, etc., and designate these values
by Qt, 0i, etc.. we can by dividing Nt by Qt, Nt by Qt, etc., obtain factors Ct, Ct, etc.. which are the coefficients
• Google
442
HANDBOOK OF BUILDING CONSTRUCTION
JSec 3-96
giving the influence of the adjoining unequal spans upon the bending momenta for equal spans. By solving equations (1) to (5) for equal spans and writing Qi, Qj, etc., for Ni, Nt, etc., we find
Q, - 0.066is7i< - O.OMw X 10* - 16.9t»
Qi - 0.035»ri> - 0.035» X 25> - 21.8w
Qi - 0.(H3wTi* - 0.043a. X 15' - 11.Ow
Qt - 0.041«r<< - 0.041*. X 20' - 16.4*>
Qi - 0.042ieTi» - 0.042m X IS1 - 10.8tt
-
tft
34.23ic
Qi
_
tfi
C* •" 7T •
Qi
,
„
21.8u>
-OtSe
...
,. ^
- - 0.09
11.0*0
•-S-1S —
*-$-§£-•>•»•
By means of these coefficients we may determine the correct moments across the spans Ti, Tt, etc., for the inner and
outer sections. Take the outer section in span Tu By referring to Fig. 128 we find that on this section
M - 0.01l8ioWi«, which in this case
- 0.0U8u>X7\» X 1.2 for an exterior panel with equal spans adjoining.
- 0.0118t»X7\« X C> X 1.2 for unequal panels.
- (0.0118)(t»)(20)(16)*(0.384)(1.2) M this case.
(fk-lb.)
Similarly across T» we find in this ease
i f - (0.0118)(w)(20)(25)*(1.57)
(ft.-lb.)
and across T» we find
M - (0.0118)(w)(20)(16)«(- 0.09)
(ft.-lb.)
Note particularly that the coefficient C% is negative and that in consequence we have a negative moment at the inner
and outer sections acroM J**.
The Theorem of Three Moments assumes knife edge supports at the columns. While this is not strictly correct
the assumption will give slightly higher moments in the slab on the side of the column adjoining the short span than
actually occur. The bending moment occurring in the column will be taken up later.
We previously found numerical values for Jf i, Mi, etc. for the negative momenta, considering one-way oonrtrurtion for the arrangement of spans shown. By solving equations (1) to (5) we may also obtain numerical value for
these momenta for a series of equal spans. For these negative momenta we will write Pi, Pi, etc. Then by dividing
Mi by Pi, Mt by Pi, etc., we will obtain coefficients Ci, Ct, etc., whien are measures of the influence of the unequal
spans upon the negative momenta.
In obtaining the numerical values of Aft and Pi, it is immaterial whether we use the span length of Ti or Tt,
provided in our calculations for the momenta occurring in the construction, we use the same value for It in the
equation M - 0.0336wfi/t>.
The bast method is to use in all calculations a span equal to the mean of the spans adjoining the column at which
the negative moment is being calculated.
The moment Mi m not effected by the unequal span arrangement and in consequenoe Ci is unity.
Tt + Ti
Tt + Tt
Solving equations (1) to (6) for a series of spans of
g
for Pi,
for Pi, etc., we have
Pi - - 0.0168u>7V - ( - 0.0168) (w) (18)* - - 4.3to
Pi - - 0.101» ( r ' 2 " " ' ) ' - ( " 0.101)(w)(20.5)» - - 42.5w
Pi - - 0.07»w ( - ' *
r
* ) ' - ( - 0.079) (u>) (20.5)» - - 33.3w
Pt - - 0.085U- ( — l - ^ - ' ) * - (~ C.085)(w)(18)» - r
Pt - - 0.083» ( * *
C, - ^! - =-±¥
J
27f»
1
' ' ) - - (0.083)(ic)(18)« - - 26.9*
- 1
C. - 1.23
Pi
— 4.3w
_
Mt
- 46.8«
Ci - 0.9
C
* " "Pi " - 42.5«r " X 1
Cf - 1 . 0 6
By means of these coefficients we may determine the correct momenta at the column head and mid-sections across
the series of spans Ti, Tt, etc. Take the column head section between spans Ti and Tt. By referring to Fig. 138
we find that on this section
M - 0.0336wtilt' which in this case
- (0.0336)(v)(X) ( T * ^ r ' ) ' X 1.2 for an exterior panel with spans ( ' ^ *)
adjoining.
- (0.0336) (w)(X) (J-'t T~y x 1>2 x c » f c T u»«9«»»l »P» ns
- (0.0336) (w) (20) (20.5) >(12) (1.1) in this case.
(ft.-lb.)
-Google
Sec. *>97]
STRUCTURAL DATA
443
Similarly at column head section between span Tt and T», we have in this oaae
M - 10.0336) (w) (20) (20.5)'(1.23)
(ft--lb.)
Proceeding aa abore, the momenta occurring in the slab at all sections across the span Ti, Tt, Tt, etc., may be determined. The moments at right angles to these will be entirely unaffected by the inequality of the spans Ti,
T%, etc. and may be obtained in the usual manner. The design may then be treated in the usual way. For the sake
of uniformity in the construction, the maximum slab and drop thickness should be determined for the worst cases
of bending moment and panel sise. and these thirknasses allowed to govern in all cases. The dimensions of the
drops will be laid out from the column center lines in each direction and the projection from these center lines made
the same proportion of the span in which eaeh part of the drop occurs.
In this analysis it will probably be found that the moment at the inner section across the spans T\, Tt, etc., is
not the same as that found across the span X. In two-way construction, then, the steel in these two sections will
vary. In four-way construction the steel used in each diagonal band in a rectangular panel should be the same.
The 4rsigTVftT will, therefore, take the mean of the two bending moments obtained across the inner sections in calculating the steel required in eaeh diagonal band.
By following the methods of examples given above, all of the moments across the spans T\, Tt, etc., can be
found without doubt arising in the designer's mind. For the sake of entire clearness, a few examples of the method
of obtaining the moments across the spans X will be given. Take the moment Si as indicated in Fig. 130. This
moment is made up of the moments in two half outer sections, in one case the panel width being 26 ft. and in the
other 16 ft.—the span in both cases being 20 ft. Assuming the same column capital proportion shown in Fig. 128,
we have for drop construction it for half outer section — O.OMheWt. In this case
at - (0.059)(w)(r,)(X)» + (0.05»)(»)(ri)(X)*
- 0.059wX'(?t + Tt)
_ 0.U8-X. ( ^ ^ )
Similarly 5s - 0.118«X»(—^-j—'). Si - O.lMvTtX*. and St - OATOwTtX*. The negative moments at the
column head and mid-eectiona may be found in the same manner.
The analysis given above assumes knife edge supports aa stated previously. This means that if we have uniform
loading throughout the structure, there will be no bending in the columns. This is not strictly true, but the departure of the moments obtained under this method of analysis from the precise moments is not sufficient to warrant
the application of the extremely laborious calculation necessary if the method or slopes and deflections were to be
applied. The bending occurring in any interior column, then, will be that due to any entire panel being unloaded
while the adjacent panel is loaded fully. For equal adjoining spans the A. C. L recommends the use of the formula
It - a022wili(fi - «c)»
where wi is the live load per square foot. Where unequal adjoining panels occur, the dead load moments at the
column do not cancel eaeh other. In this ease, therefore, the moment in the column between spans Tt and Tt would
become
It - 0.022wX(r, - oe)* - 0.O22DX(Ti - qe)H
where D is the dead load per square foot of the structure.
The moment in the exterior eolumn will be found in the usual way.
97. Openings.—Providing for openings is one of the places where the designer's judgment
and experience come into play. In general, small openings may be placed in the slab in the
regions of the outer and inner sections without varying the design in any way. The same opening, however, could not in most cases be introduced within the area occupied by the drop panel
without making special provisions in the design. Circular openings of reasonable diameter
may be carried through the column capital at the location of sprinkler risers, downspouts, etc.,
without damage. No definite rules about relatively small openings can be laid down; it is all
a matter of judgment and experience.
98. Use of Beams.—Where large openings occur in aflat-slabfloor,beams must be used.
In some oases also where heavy concentrated loads occur it is advisable to introduce beams.
Beams around openings must be designed to carry the loads coming upon them and in addition
a portion of floor adjoining. The width of thefloorstrip to be used cannot be governed by
definite rules. The engineer's judgment and experience must be relied upon in this. Whenever possible, it is desirous to use broadflatbeams of depth equal to the depth of the slab and
drop panel. Some city codes require that spandrel beams be designed to carry a portion of
the floor load in addition to the weight of the brick spandrel. Since the spandrel beam is
usually much stiffer than the adjoining slab, it is true that a portion of thefloorload will be
carried by the beam. The only live load falling upon the spandrel beam is that coming from
the narrow strip of floor which is carried by the beam. This is generally a small proportion
of the total load on the beam. Since the load is practically all dead and is uniformly distri-
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444
HANDBOOK OP BUILDING CONSTRUCTION
fSec 8-09
WL
buted, the positive moment for an interior beam approaches -757- We are bound by code,
WL
however, to figure for -rz-' which is almost twice as much. Under these conditions, then, it is
satisfactory to design spandrel beams to carry the wall load only.
99. Capitals at Exterior Columns.—Exterior columns are usually square or rectangular
and, in place of the usual half capital, a bracket, the width of the column, is used. This is
good and safe practice where reasonably large spandrel beams are used. Where no spandrel
beams are used, or where they are relatively thin or shallow, the full half column capital should
be used.
100. Drop at Exterior Column.—A half drop panel should be used at the exterior column
in practically all cases where drop construction is used at the interior columns.
101. Omission of Spandrel Beams.—In particular cases as in cold storage buildings, where
the enclosing walls are built as independent structures, no spanderl beams need be used. In
some other cases the beam may be upturned above the ceiling either forming a concrete spandrel
or carrying a brick spandrel. In factory construction it will usually be found that a beam
having a depth equal to the depth of the slab plus the drop will be sufficient, and in other cases
where cap construction is used the slab alone may be of sufficient depth.
108. Narrow Buildings.—The bending moments given in the A.C.I. proposed ruling are
for structures over three spans in width. For structures of less width the moments should be
increased by factors obtained by comparing the actual negative and positive moments applying
in one-way construction with those occurring in an interior span.
108. Minimum Column Size.—Neither the Joint Committee nor the A.C.I. rulings provide
a minimum column size as a function of the span. It will be usually found that if the bending
moments specified for columns are provided for that a column having a diameter or least size
of L/12 is required. It is good practice in any case, however, to limit the minimum column
sise to L/12.
104. Width of Bands.—In four-way construction the widths of the bands of steel are
usually made 0.4L. In rectangular panels where the width IB much less than the length of the
panel, the band widths should be made proportionate to the width of the panel and not a
proportion of the span of the band. Thus in a panel 20 X 24 ft. the bands spanning the 24
direction should be 8 ft. wide. In two-way construction, the bands are made 0.5L in width
with the same provision as above for rectangular panels.
108. Kinds of Bars to Use.—Either deformed or plain bars may be used but the use of
square twisted bars should be entirely avoided. A round bar is better than a square for the
reason that it packs better at the column and also that the concrete will flow round the intersecting bars more completely.
106. Construction Notes.
106a. Pouring Columns and Slabs.—If it is convenient, it is well to pour the
columns including the capital up to the underside of the drop or slab before placing the slab
steel. If the columns are to be poured after the slab steel is in place, they should be filled up
to the top of the capital and allowed to set for about two hours before the slab above is placed.
106b. Construction Joints.—Construction joints should be made at the center
of the span in all cases. Bulkheads should be set up to form vertical joints in these locations
and any concrete which has passed under the bulkhead running out to a feather edge should
be carefully removed before pouring the next section.
106c Supporting and Securing Steel.—At the center of the span the steel should
be held securely in place at the correct distance above the forms by means of one of the many
devices of this nature now on the market. The device used should be in one piece for each
band so that the bars may be securely held to an accurate spacing. Two of these spacing bars
should be used on each band of steel in the region of the mid span. At the column head,
spacing bars are not necessary but substantial supporting bars should be used. The bars must
be supported at the correct distance above the form work and while many metal devices for this
purpose have been placed on the market, a concrete block about 3 in. square serves this purpose in the most satisfactory manner. The supporting bars are placed just outside the drop
:-.,•.<•..<:. G
O O Q
Ic
STRUCTURAL
S e c 8-1064]
445
DATA
panel and are carried on three of these blocks. The blocks should have the wires imbedded in
t h e top to wire down the supporting bars. For supporting bars, jHs-in. rounds may be used
for spans up to 20 ft., %-ia.. bars for spans between 20 and 25 ft., and %-in. bars for spans
between 25 and 30 ft. One more block about %-in. higher than those under the supporting
b a n should be used at the middle of the other sides of the drop panel allowing the slab steel to
rest directly upon it. The steel in the mid-section should be securely supported in s manner
similar to the steel at the column head section.
10W. Placing Steel.—The A.G.I, gives a formula for the distance from the
column center line to the point of inflection as H (ft — V) + Mv f° r c a P construction and a
distance equal to Y^X\ — ge) + Hqc for drop construction. In a Bquare panel in which the
diameter of column cap is 0.225L, these distances become 0.25^ and 0.3b respectively. The
steel should bend down to the bottom of the slab in approximately these locations. It is essential for good construction that the negative reinforcement be securely supported with the minimum cover allowed in the upper part of the slab and be carried out parallel to the top of the
slab to approximately the line of inflexion. Arrangements of steel in which the reinforcement
droops away from the top of the slab and is some distanoe below the top in the region of the
line of inflexion will lead to unsatisfactory results.
106*. Floor Finish.—Satisfactory results from a structural point of view can
be obtained by either applying the floor finish »ith the slab or applying it after the main slab has
set. In general, however, the best and most economical results can be obtained by finishing
the structural slab with a mixture of the same mortar proportions as used in the slab, before
the slab has set.
106/. Future Extensions.—Future extensions can be provided for by introducing
a spandrel beam along the side to be extended and leaving in the upper part of the beam a seat
about 6 in. in width to receive the new slab. Sufficient steel should be left projecting in the top
of the slab to satisfy the moments at the column head and mid-sections. This steel should be
structural grade material. It should project beyond the edge of the slab about 80 diameters
for bond. After the concrete in the first portion of the building has set, the steel may be bent
up and enclosed in the spandrel wall. The usual eolumn capital should be built on the columns
projecting out for the future extension and these capitals should be reinforced with bracket
bars. This is not an entirely satisfactory method of providing for future extensions. A far
better method is to build the foundation only to allow for future extension and construct new
independent columns to support the extension later allowing the existing columns to remain
supporting the original structure.
FLAT-SLAB FLOORS—AMERICAN CONCRETE INSTITUTE RULING
/, — 10,000; ft for positive moment = 650; /«for negative moment - 750. (See Fig. 128 for
distribution of moments)
Interior panel—Superimposed load 100 lb. per sq. ft.
Panel MM
(feet)
Capital
diameter
16^16
17X17
18X18
10X19
30X20
21X31
33X33
33X23
34X34
36X35
36X36
3'6"
8'9"
4'0"
4'3"
4'6"
4'9"
B'0"
5'3"
6'6"
5'9"
6'0"
Steel in each band
Concrete
Depth Depth
Steel in.
Site of drop
in cubic
of alab of drop
Ib.per
eq.
Across
panel
(.inches) (inches) feet per aq. Direct Diagonal
ft.
direct
ft.
(inches) (inehea)
(inches)
4'10"X4*10"
6' 3"X6' 2"
6' 6"X6' 6"
«' 8"X6' 8"
6' 0"X6' 0"
6' 4"X6' 4"
6' 8"X6' 8"
7' 0"X7' 0"
V 4"X7' 4"
7' 6"X7' 6"
7'10" x 7'10"
6
6H
6«
1H
1H
8
8H
»H
9
9H
OH
2tf
2tf
3H
2H
3«
2H
3H
3H
3K
3H
3«
0.02
0.56
0.68
0.63
0.66
0.69
0.71
0.76
0.78
0.83
0.84
H-H*
16-H*
18-K*
21-H*
13-H*
16-M*
16-H*
18-H*
23-M*
24-K*
11-X*
13-X*
16-H*
16-H*
11-X+
12-H*
13-><>
14-H*
16-H*
18-H*
19-M*
9-H*
10-K*
11-X*
7-X>
8-H*
»-X*
10-H+
U-H+
12-H*
13-«#
1.88
3.00
3.12
3.33
3.43
3.68
3.67
2.77
2 05
3.14
3 27
Google
446
HANDBOOK OP BUILDING CONSTRUCTION
(Sec. *-10tf
Interior panel—Superimposed load 150 lb. per sq. ft.
Panel rise
(feet)
10X10
17X17
18X18
10X10
30X30
31X21
22X23
23X23
24X24
25X25
26X20
Capital
diameter
3'8"
3*0"
4'0"
4'3"
4'0"
4'9"
5'0"
5'3"
5'0"
5'0"
O'O"
Steel in each band
Concrete
Depth Depth
•teal in
in cubic
S u e of drop
of slab of drop
feet per aq. Direct Diagonal Across Ib.peraq.
panel
(inches) (inches)
direct
ft.
ft.
(inehea) (inches)
(inches)
4'10"X4'10"
5' 2"X5' 2"
5' 0"X5' 0"
5' 8"X5' 8"
0' 0"XO' 0"
0' 4"X0' 4"
0' 8"X0' 8"
V 0"X7' 0"
V 4"X7' 4"
7' O ' ^ r 0"
7'10"X7'10"
0
OH
o«
7H
7H
8
8>i
8%
•
OH
o«
2*S
2«
3H
3H
3H
4
4
4H
4H
4H
0.62
0.50
0.50
0.03
0.05
0.60
0.72
0.70
0.78
0.83
0.85
18-*i>
20-*»»
17-Xa#
10-Hs*
17-H*
18-H»
20-H+
22-H*
24-H*
27-H*
30-H*
15-H*
16-H*
14-Hs*
15-H«*
13-H*
15-H*
17-H*
18-M*
20-H*
23-H*
24-H*
10-H»
U-H +
10-Hs*
10-Hs*
9-H*
10-X*
11-H*
12-H*
14-X*
15-H*
16-H*
2.42
2.46
2.70
2.76
2.08
3.16
3.36
3.43
3.63
3.84
4.05
Interior panel—Superimposed load 200 lb. per sq. ft.
Panel sue
(feet)
10X10
17X17
18X18
10X10
20X20
21X21
22X22
23X23
24X24
25X25
20X20
Capitol
diameter
Steel in each band
Concrete
Depth Depth
Steel in
in cubic
Siae of drop
of slab of drop
panel
feet per aq. Direct Diagonal Aeroas lb.per sq.
(inches) (inehea)
direct
ft.
ft.
(inches) (inches)
(inches)
3'fl"
4'10"X4'10"
OH
ar
5' 2"X5' 2"
5' 0"X5' 0"
5' 8"X5' 8"
0' 0"X6' 0"
6' 4"X0' 4"
6' 8"X0' 8"
V 0"X7' 0"
7' 4"X7' 4"
T 0"X7' 0"
7'10"X7'10"
o«
4'0"
4'3"
4'0"
4'0"
O'O"
5'3"
5'0"
O'O"
O'O"
7H
7H
8
SH
8«
9H
OH
10
10H
3
3H
3*«
4
4
4H
4H
4H
5
5
«H
0.57
0.50
0.63
0.66
0.70
0.72
0.76
0.81
0.83
0.87
0.02
20-K*
24-H +
10-Hs*
22-H«+
io-H»
21-X*
23-H*
25-H*
28-H*
30-H*
83-H*
17-H*
10-H*
10-H.
18-H«#
15-H*
17-H*
10-H*
20-H#
23-H*
24-H*
27-H*
U-H*
13-H*
ll.Jfs*
12-Hs»
10-H*
12-H*
13-H*
14-H*
15-H*
17-H*
18-H*
2.70
2.00
3.03
3.32
3.36
3.60
3.79
3.88
4.15
4.22
4.50
Interior panel—Superimposed load 250 lb. per sq. ft.
Panel sise
(feet)
Capitol
diameter
16X16
17X17
18X18
10X10
20X20
21X21
22X22
23X23
24X24
25X25
36X20
3'0"
3'9"
4'0"
4'3"
4'6"
4'9"
6'0"
6'3"
5'6"
5'0"
6'0"
Steel in each band
Concrete
Depth Depth
Steel in
in cubic
Size of drop
of slab of drop
lb.
per aq.
Across
feet per sq. Direct Diagonal
panel
(inches) (inches)
ft.
direct
ft.
(inches) (inches)
(inches)
7
4'10"X4'10"
5' 2"X5' 2"
7H
5' 6"X5' 0"
7«
5' 8"X5' 8"
8H
8' 0"X6' 0"
8H
6' 4"X0' 4"
0
0' 8"X8' 8"
OH
T 0"X7' 0" 10
7' 4"X7' 4" 10H
V 0"X7' 6" 11
7'I0"X7'10"
iw
3H
3H
4
*H
4H
4H
*H
5
5
Mi
0.01
0.05
0.08
0 72
0.74
0.70
0.83
0.87
0.01
0.00
0.08
22-H»
18-H.*
21-Hs*
18-H*
21-H*
23-H*
25-H*
27-H*
30-H*
33-H*
36-H*
18-H*
12-*0
15-Hs* 10-Ks*
17-Hs* 12-Hs*
15-H*
10-H*
17-H
11-H*
18-H*
12-H*
20-H*
14-H*
22H#
1S-H*
24-H* 17-H*
28-H*
18-H*
20-H* 20-H*
Gooolc
2 85
3 04
3 28
3.48
3.75
3.84
4 04
4.20
4.40
4 80
4 90
S e c 8-107]
STRUCTURAL DATA
447
Interior panel—Superimposed load 300 lb. per sq. ft.
Panel eise
(feet)
16X18
17X17
18X18
19X19
20X20
21X21
• 22X22
23X23
24X24
26X26
26X26
Capital
diameter
3'6"
3'9"
4'0"
4'3"
4'6"
4'9"
6'0"
6'3"
6*6"
6'9"
6'0"
Steel in each band
Concrete
Depth Depth
Steel in
Siae of drop
in cubic
ofelab of drop
Acroaa Ib.per aq.
panel
feet
per
aq.
Direct Diagonal
(inches) (inehea)
ft.
direct
ft.
(inehea) (inehea)
(inehea)
4'10"X4'10"
6' 2"X6' 2"
6' 6"X6' 6"
8' 8"X6' 8"
6' 0*'X6* 0"
6* 4"X6' 4"
6' 8"X6' 8"
7H
7H
8«
8«
»«
9%
10
T vxr o" 10H
7' 4"X7' 4" 11
V 6"X7' 6" 11H
7'10"X7'10" 12
m
6
7
0.66
0.68
0.72
0.77
0.81
0.86
0.88
0.92
0.96
1.01
1.05
18-J<«*
!»->.•
18-X*
K-H*
22-H+
24-H*
27-H*
30-H*
33-H*
35-M*
39-J4*
13-H*
14-H»
16-H*
18-K*
23-H+
26-M*
29-X*
31-H»
10-Ha*
9-H*
io-«*
n-H*
12-K*
18-H*
15-H*
16-K*
18-H*
20-H*
22-H*
3.15
3.48
3.58
3.76
3.97
4.12
4.40
4.50
4.76
6.00
5.24
Interior panel—Superimposed load 350 lb. per sq. ft.
Panel aiae
(feet)
Capital
diameter
16X16
17X17
18X18
19X19
20X20
21X21
22X22
23X23
24X24
26X25
26X26
3*6"
8*9"
4'0"
4'3"
4'6"
4'9"
6'0"
5'3"
6'6"
5'9"
6'0"
Steel in each band
Concrete
Dopth Depth
Steel in
Siae of drop
in cubie
of slab of drop
Acroaa Ib.per aq.
feet
per
aq.
panel
Direct Diagonal
(inehea) (inehea)
ft.
direct
ft.
(inehea) (inehea)
(inches)
4' 10" X 4' 10"
5' 2"X5' 2"
5'6" X6' 6"
5'8" X5' 8"
6' 0"X6' 0"
V 4"X6' 4"
6' 8"X6' 8"
V 0"X7' 0"
V 4"X7' 4"
T 6"X7' 6"
7'10"X7'10"
7«
8K
8«
9H
9H
10>i
10K
11H
11«
12tf
12H
6
5H
5H
6
6M
6«
7>i
7«
8H
8H
0.68
0.73
0.77
0.82
0.86
0.90
0.95
0.99
1.04
1.00
1.13
15-H*
17-H*
19-H*
21-X*
23-X*
26-H*
28-H»
31-M»
34-H*
37-H»
41-W*
13-H*
14-H*
15-K*
17-H*
19-H*
21-H*
23-H*
25-H»
28-H>
30-K*
33-H>
6-H+
9-H>
10-H*
11-H*
13-««
14-H*
16-K*
18-H»
19-H>
21-H*
22-H*
3.56
3.68
3.80
3.97
4.13
4.40
4.58
4.80
5.05
5.22
6.63
In these panels the ateel ia lapped to develop the strength of the bar by bond. The ateel is considered to be in
approximately 2 panel lengths. The necessary supporting bare are included in the ateel weights. The concrete
in the elab and in the drop panel are included in the concrete quantities.
FLOOR SURFACES
BT ALLAN F. OWEN
107. Wood Floor Surface*.
107a. Softwood Flooring.—Soft pine is not used for flooring except some northem pine for very cheap work. It is called 1 X 6-in. matched and dressed, but comes xHt X Bhi
in. It is apt to have sap in it and be subject to warping and twisting.
Hard pine, or yellow pine, comes flat sawed and quarter sawed (see Figs. 131 and 132).
Theflat-sawedflooringshould never be used as it splinters badly with use. The quarter-sawed
or edge-grain flooring is good flooring and can be used for residences, factories, and warehouses,
although it will not wear as well as hard wood. The best yellow pine flooring is cut from logs
having the largest number of circular rings per inch of diameter and with the largest proportion
of hard summer wood in the rings and the smallest proportion of soft spring growth. Longleaf yellow pine generally has more than 8 rings per inch, and short leaf and loblolly pine gen-
Gocwle
7.5
DOMED BUILDING CONCEPT
Figure 7.1: Proposed Lateral Bracing for Domed Building- Front View,
Three-Quarters View, Side View
102
Figure 7.2: Aerial View of Proposed Lateral Bracing
103
7.6
HEXAGONAL MESH AND FIBER REINFORCED BEAM
'Steel Column
.Concrete-Filled Fiber Reinforced Polymer Beam
Figure 7.3: Overhead View of Hexagonal Mesh
104
Figure 7.4: Elevation View of Hexagonal Mesh
105
Figure 7.5: Isometric View of Hexagonal Mesh
106
Side View
Three-Quarters View
Bottom View
Pre-stressing Tendon
Isometric View
Figure 7.6: Select Views of Concrete-Filled Fiber Reinforced Polymer Beam with Prestressing Tendon
107
7.7
FIBER REINFORCED POLYMER DROP PANEL
Steel Column
•
Figure 7.7: Elevation View of FRP Drop Panel
108
UP
Figure 7.8: Aerial View of FRP Drop Panel
109
Figure 7.9: Underside View of FRP Drop Panel
110
7.8
CONCEPTUAL BUILDING DESIGN
H~
"Ur
4
w
Ur
ir
J_
-I
1
!
!-
TST
13
—T"
i
i
i
fS-
- — • ! — ( - *
i
-4
. ^ |
B-
^
•
B-
^
^
^
-e
e-l
4-._4-
s_
;
+---
JlL
JJL
*
•
-m•a-
S,
J5L
JSL
Figure 7.10: Plan View of Final Concept
111
JSL
M.
"
;
1
,
—
_
•1
§1
I
i
1
PiH
1n
H
j
1
%
"J l
'
'
Figure 7.11: Elevation View of Final Concept
112
i i
K
I\
\i MM
I
\i i\ i • i i i i\ 11 \i • /i i /i i * * i
»••!!.• L ! ? . j j i i
liiiii
n
HfsPBB
Figure 7.12: View of Transfer Truss- Final Concept
113
i
3
Figure 7.13: Section View of Concrete-Filled FRP Drop Panel-Steel
Column-Concrete-Filled Steel Deck Details-Final Concept
114
7.9
COMPARISON OF CONCEPTUAL AND CONVENTIONAL DESIGN FRAME FORCES
115
116
Table 7.1:
Frame
Text
5
6
7
Station
8
9
10
11
12
11.000
11.000
11.000
11.000
11.000
13
14
15
16
17
18
19
20
21
22
23
24
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
25
26
27
28
33
34
35
36
37
38
39
40
41
42
43
44
ft
11.000
11.000
11.000
0.000
0.000
0.000
0.000
Conceptual Mode [ [1.2DL + 1.6 LL (50 psf)]
P
Kip
V2
Kip
V3
Kip
-1987.850
-3792.902
3.499
-0.219
-3.496
-9.004
-3768.219
-3760.781
0.310
0.001
-0.308
0.222
-3.497
-9.004
-8.463
-8.504
-8.463
-9.005
-3.499
0.217
-8.462
-8.503
-8.463
-9.004
-3.498
0.218
-0.310
-0.001
0.308
-0.215
3.496
9.003
8.461
8.502
8.462
9.004
-0.310
-0.002
0.308
-0.215
3.496
9.002
8.461
8.502
8.463
9.003
3.498
-0.220
0.309
0.000
-0.308
0.215
0.000
-3768.058
-3795.232
-2002.099
-3792.268
-3766.761
-3760.121
-3767.160
-3789.749
-1988.387
-3793.101
-3768.429
-3761.034
-3767.467
-3791.597
-1990.278
-3789.360
-3766.674
-3759.796
-3766.525
-3789.291
-7772.350
-8397.825
-8361.762
-8397.954
-7771.003
-8400.335
-4590.620
-1873.467
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
-4590.186
-8401.094
-8367.970
0.000
0.000
0.000
-1876.446
0.000
117
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
T
Kip-ft
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
M2
Kip-ft
38.461
99.037
M3
Kip-ft
-38.486
2.410
93.092
93.543
93.096
99.056
38.480
-2.389
3.417
0.017
-3.379
2.374
-38.453
-99.018
-93.072
-93.528
-93.091
-99.035
-38.474
2.422
-3.405
-0.006
3.390
-2.449
38.471
-3.400
-0.001
3.391
-2.359
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
99.035
93.084
93.528
93.090
99.044
38.485
-2.407
3.410
0.012
-3.383
2.368
-38.457
-99.034
-93.071
-93.517
-93.081
-99.039
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Table 7.1: Conceptual Mode
p
Frame Station
V2
Text
ft
Kip
Kip
0.000 -1879.162 0.000
45
46
0.000 -8370.394 0.000
47
0.000 -8399.845 0.000
48
0.000 -4589.800 0.000
0.000 -1878.680 0.000
49
50
0.000 -4589.349 0.000
51
0.000 -8399.468 0.000
52
0.000 -7770.639 0.000
0.000 -8398.460 0.000
53
54
0.000 -8367.763 0.000
0.000 -8397.958 0.000
55
56
0.000 -7770.953 0.000
118
[1.2DL + 1.6 LL (50 psf)]
M2
V3
T
M3
Kip-ft Kip-ft
Kip
Kip-ft
0.000
0.000 0.000
0.000
0.000
0.000 0.000
0.000
0.000
0.000 0.000
0.000
0.000 0.000
0.000
0.000
0.000
0.000 0.000
0.000
0.000 0.000
0.000
0.000
0.000 0.000
0.000
0.000
0.000
0.000 0.000
0.000
0.000
0.000 0.000
0.000
0.000
0.000 0.000
0.000
0.000
0.000 0.000
0.000
0.000
0.000 0.000
0.000
T a b l e 7.2: C o n v e n t i o n a l M o d e l [1.2DL + 1 . 6
Frame Station
P
V2
T
V3
Text
ft
Kip
Kip
Kip
Kip-ft
5
11.000 -1,142.719 1.169 -1.385 0.000
6
11.000 -2,553.311 0.115 -4.425 0.000
7
11.000 -2,422.474 0.074 -4.284 0.000
8
11.000 -2,422.110 0.000 -4.271 0.000
9
11.000 -2,422.284 -0.074 -4.284 0.000
10
11.000 -2,553.187 -0.115 -4.426 0.000
11
11.000 -1,143.091 -1.169 -1.385 0.000
12
11.000 -2,650.750 -4.158 0.288 0.000
13
11.000 -2,496.359 -3.778 -0.077 0.000
14
11.000 -2,509.665 -3.808 0.000 0.000
15
11.000 -2,496.160 -3.778 0.078 0.000
16
11.000 -2,650.504 -4.158 -0.288 0.000
17
11.000 -1,142.466 -1.169 1.386 0.000
18
11.000 -2,552.605 -0.115 4.426 0.000
19
11.000 -2,421.840 -0.074 4.284 0.000
20
11.000 -2,421.909 0.000 4.272 0.000
21
11.000 -2,421.948 0.074 4.285 0.000
22
11.000 -2,552.907 0.115 4.426 0.000
23
11.000 -1,142.469 1.169
1.386 0.000
24
11.000 -2,650.772 4.158 -0.288 0.000
25
11.000 -2,496.169 3.778 0.078 0.000
26
11.000 -2,509.422 3.807 0.000 0.000
27
11.000 -2,496.247 3.778 -0.077 0.000
28
11.000 -2,650.645 4.158 0.289 0.000
33
11.000 -5,892.471 0.188 0.711 0.000
34
11.000 -5,668.537 -0.124 -0.288 0.000
35
11.000 -5,810.916 -0.207 0.000 0.000
36
11.000 -5,669.252 -0.124 0.290 0.000
37
11.000 -5,892.336 0.189 -0.710 0.000
38
11.000 -5,729.695 -0.302 0.466 0.000
39
11.000 -4,279.655 0.350 0.739 0.000
40
11.000 -2,951.924 1.811 0.005 0.000
41
11.000 -4,281.538 0.352 -0.733 0.000
42
11.000 -5,729.861 -0.301 -0.465 0.000
43
11.000 -5,836.653 0.000 0.312 0.000
44
11.000 -3,095.207 0.002 1.806 0.000
45
46
11.000
11.000
-3,095.112
-5,837.104
0.001
0.000
119
-1.799
-0.309
0.000
0.000
L L (50 psf)l
M3
M2
Kip-ft
Kip-ft
-12.864
15.239
48.680
-1.262
47.121
-0.815
46.983
0.000
47.121
0.815
48.681
1.261
15.240 12.863
-3.173 45.737
0.851 41.562
-0.004 41.884
-0.857 41.559
3.165 45.736
-15.243 12.863
-48.689
1.262
-47.129
0.816
-46.991
0.001
-47.131
-0.815
-48.690
-1.263
-15.243 -12.864
3.165 -45.736
-0.857 -41.558
-0.004 -41.880
0.849 -41.558
-3.174 -45.736
-7.823
-2.073
3.173
1.361
-0.005
2.278
-3.195
1.361
-2.077
7.805
-5.128
-8.127
-0.055
8.058
5.112
-3.434
3.317
-3.846
-19.918
-3.876
3.315
0.001
-19.862
19.791
-0.018
-0.017
3.403
-0.001
Table 7.2: Conventional Model
p
Frame Station
V2
Text
ft
Kip
Kip
47
11.000 -5,729.542 0.301
48
49
50
51
52
53
54
55
56
11.000
11.000
11.000
11.000
-4,279.205
-2,955.805
-4,281.540
-5,730.234
-0.352
-1.807
11.000
11.000
11.000
11.000
11.000
-5,892.509
-5,668.630
-5,811.332
-5,669.804
-0.189
0.123
0.206
0.124
-5,890.163
-0.189
-0.353
0.302
120
fl.2DL + 1.6 LL (50 psf)]
T
M2
V3
M3
Kip
Kip-ft
Kip-ft
Kip-ft
-3.315
0.466 0.000
-5.130
-8.121
3.869
0.738 0.000
0.005
-0.731
-0.465
0.711
-0.288
0.001
0.291
-0.709
0.000
0.000
0.000
-0.050
8.039
5.110
19.878
3.884
-3.318
0.000
0.000
0.000
0.000
0.000
-7.826
3.169
-0.007
2.075
-1.352
-3.198
7.802
-2.263
-1.360
2.074
Table 7.3: Conceptual Model - Conventional Model | .2DL + 1.6 LL (50 psf)]
M2
P
V2
V3
T
Frame
M3
Kip
Kip
Text
Kip
Kip-ft
Kip-ft
Kip-ft
5
845.131
2.330
2.111
23.222
25.623
0.000
0.104
6
1,239.591
4.579
50.357
0.000
1.148
7
1,345.745
0.236
4.179
45.971
0.000
2.590
1,338.671
8
0.001
4.233
0.000
46.560
0.006
0.234
45.974
2.574
9
1,345.774
4.179
0.000
10
0.107
4.579
50.375
1,242.045
0.000
1.188
11
2.328
2.114
23.241
25.607
859.008
0.000
12
4.846
0.071
0.784
1,141.518
0.000
53.298
4.684
13
1,270.402
0.233
2.566
0.000
51.523
51.644
14
1,250.456
4.695
0.001
0.013
0.000
15
1,271.000
4.685
0.230
0.000
2.522
51.531
16
1,139.245
4.846
0.073
0.000
0.791
53.308
17
845.921
2.329
2.110
0.000
23.210
25.622
18
1,240.496
0.103
4.576
0.000
50.329
1.146
19
1,346.589
0.236
4.177
45.943
2.594
0.000
20
1,339.125
0.001
4.230
0.000
46.537
0.011
0.234
21
1,345.519
4.178
45.960
2.567
0.000
22
1,238.690
0.100
4.577
50.345
0.000
1.105
23
2.327
2.112
23.231
847.809
0.000
25.593
24
1,138.588
0.068
0.000
0.742
4.845
53.298
0.231
25
1,270.505
4.683
0.000
2.543
51.514
26
1,250.374
0.000
4.695
0.000
0.003
51.637
27
4.684
1,270.278
0.231
0.000
2.542
51.523
28
1,138.646
4.846
0.074
0.815
0.000
53.302
0.711
33
1,879.879
0.000
7.823
0.188
2.073
34
0.124
2,729.288
0.288
0.000
3.173
1.361
35
2,550.846
0.207
0.000
0.000
0.005
2.278
0.124
36
2,728.702
0.290
0.000
3.195
1.361
37
1,878.667
0.710
0.000
7.805
2.077
0.189
38
0.466
5.128
2,670.640
0.302
0.000
3.317
8.127
39
310.965
0.350
0.739
0.000
3.846
40
1,078.457
0.005
0.000
0.055
19.918
1.811
41
308.648
0.352
0.733
8.058
0.000
3.876
5.112
42
2,671.233
0.465
0.000
3.315
0.301
0.312
3.434
43
2,531.317
0.000
0.000
0.001
44
45
46
1,218.761
1,215.950
2,533.290
0.002
1.806
0.001
0.000
1.799
0.309
121
0.000
0.000
19.862
19.791
0.018
0.017
0.000
3.403
0.001
Table 7.3: Conceptual Model - Conventional Model [1.2DL + 1.6 LL (50 psf)]
Frame
P
V2
V3
T
M2
M3
Kip
Kip
Kip-ft
Text
Kip
Kip-ft
Kip-ft
47
2,670.303
0.301
0.466
0.000
5.130
3.315
48
310.595
0.352
0.738
0.000
8.121
3.869
49
1.807
0.005
0.000
0.050
19.878
1,077.125
50
307.809
0.353
0.731
0.000
8.039
3.884
2,669.234
51
0.302
0.465
0.000
5.110
3.318
52
1,878.130
0.711
0.000
0.189
7.826
2.075
53
2,729.830
0.000
1.352
0.123
0.288
3.169
54
2,556.431
0.000
0.007
0.206
0.001
2.263
55
2,728.154
0.124
0.000
0.291
3.198
1.360
56
1,880.790
0.000
7.802
2.074
0.189
0.709
Summation
64,495.584
553.107
50.281
39.536
0.000
434.883
Red numbers indicate negative values, i.e. the
conceptual model exhibits lower magnitude loads.
122
Frame
Text
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
33
34
35
36
37
38
39
40
41
42
43
Table 7.4: Conceptual Mode [1.2DL + 1.6 L L
p
T
Station
V2
V3
Kip
Kip
Kip
Kip-ft
ft
4.752
11.000
-2,667.434
-4.748 0.000
11.000
-5,143.847
-0.291 -12.370 0.000
11.000
-5,099.109
0.435 -11.571 0.000
11.000
-5,096.535
0.001 -11.626 0.000
11.000
-5,098.804
-0.433 -11.572 0.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
11.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
44
45
0.000
0.000
46
0.000
-5,145.896
-2,682.304
-5,142.496
-5,098.359
-5,096.655
-5,098.977
-5,140.191
-2,667.971
-5,144.028
-5,099.304
-5,096.784
-5,098.292
-5,142.298
-2,670.593
-5,139.502
-5,098.234
-5,096.315
-5,098.254
-5,139.672
-10,657.883
-11,458.079
-11,443.533
-11,458.278
-10,655.831
-11,461.288
-6,304.638
-2,591.275
-6,304.115
-11,462.296
-11,450.818
-2,595.648
-2,598.885
-11,453.573
0.296
-4.751
-12.370
-11.570
-11.624
-11.571
-12.373
-4.753
0.288
-0.436
-0.002
0.432
-12.371
-4.752
0.290
-0.436
-0.002
0.432
-0.286
4.748
12.369
11.568
11.623
11.570
12.370
0.000
0.000
0.000
0.000
0.000
0.000
0.000
-0.286
4.747
12.367
11.569
11.624
11.571
0.000
0.000
0.000
0.000
0.000
0.000
0.000
123
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
12.370 0.000
4.751 0.000
-0.292 0.000
0.435 0.000
0.000 0.000
-0.433 0.000
0.284 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
(100 psf
M2
Kip-ft
52.235
136.071
127.287
127.886
127.293
136.098
52.282
-3.178
4.805
0.025
-4.749
3.149
-52.222
-136.041
-127.259
-127.864
-127.285
-136.065
-52.254
3.216
-4.781
-0.002
4.767
-3.125
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
n
M3
Kip-ft
-52.270
3.200
-4.786
-0.010
4.765
-3.255
52.264
136.067
127.273
127.865
127.284
136.081
52.271
-3.193
4.796
0.020
-4.752
3.142
-52.227
-136.061
-127.255
-127.847
-127.270
-136.071
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Table 7.4: Conceptual Mode
Frame Station
p
V2
ft
Kip
Text
Kip
47
0.000 -11,460.571
0.000
48
49
50
51
52
53
54
55
56
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
-6,303.585
-2,598.247
-6,303.074
-11,460.226
-10,655.374
-11,459.016
-11,450.574
-11,458.298
-10,655.445
[1.2DL + 1.6 LL (100 psf)l
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
124
V3
Kip
T
0.000
0.000
0.000
0.000
0.000
Kip-ft
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
M2
Kip-ft
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
M3
Kip-ft
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Table 7.5: Conventional Model
P
V2
Frame Station
Kip
Text
ft
Kip
-1,948.665 2.182
5 11.000
-4,472.332 0.027
6 11.000
7 11.000
-4,267.059 0.170
-4,265.124 0.000
8 11.000
-4,266.616 -0.169
9 11.000
-4,470.482 -0.026
10 11.000
-1,949.143 -2.181
11 11.000
-4,616.803 -6.752
12 11.000
-4,343.238 -6.359
13 11.000
14 11.000
-4,360.530 -6.368
-4,343.108 -6.359
15 11.000
-4,616.617 -6.751
16 11.000
-1,949.576 -2.181
17 11.000
-4,465.336 -0.028
18 11.000
-4,270.109 -0.159
19 11.000
-4,265.353 0.000
20 11.000
21 11.000
-4,465.331 0.160
-1,949.488 0.029
22 11.000
-4,616.437 2.181
23 11.000
-4,616.437 6.753
24 11.000
-4,342.902 6.360
25 11.000
-4,360.300 6.368
26 11.000
-4,343.012 6.360
27 11.000
-4,616.533 6.753
28 11.000
33 11.000 -10,199.461 -0.321
-9,811.694 -0.650
34 11.000
35 11.000 -10,043.441 -0.774
-9,812.507 -0.650
36 11.000
-10,199.802
37 11.000
-0.323
-9,943.054 -0.472
38 11.000
39
43
44
11.000
11.000
11.000
11.000
11.000
11.000
45
46
11.000
11.000
40
41
42
-7,793.725 0.826
-5,725.129 2.786
-7,797.696 0.827
-9,940.361 -0.452
-10,106.363 0.001
-5,942.629 0.003
-5,946.077 0.003
-10,106.323 0.002
125
[1.2DL + 1.6 L L (100 psf)l
T
V3
M2
M3
Kip
Kip-ft
Kip-ft
Kip-ft
-2.785 0.000 30.631 -23.997
-7.086
-7.035
-7.073
0.000
0.000
0.000
0.000
-7.035
-7.086 0.000
-2.785 0.000
0.453 0.000
-0.133 0.000
0.000 0.000
0.133 0.000
-0.453 0.000
2.785 0.000
7.087 0.000
7.016 0.000
6.856 0.000
7.017 0.000
7.088 0.000
2.785 0.000
-0.453 0.000
0.133 0.000
0.000 0.000
-0.133 0.000
0.454 0.000
0.563 0.000
-0.188 0.000
0.000 0.000
0.189 0.000
-0.564 0.000
0.397 0.000
77.942
77.381
77.804
77.382
77.945
30.632
-4.988
1.465
-0.001
-1.466
4.986
-30.631
-77.962
-77.180
-75.411
-77.183
-77.965
-30.632
4.985
-1.467
-0.002
1.463
-4.991
-6.196
2.072
0.003
-2.078
6.201
-4.368
-13.819
1.256
0.010
-1.255
-0.377
0.000
0.000
0.000
0.000
0.370
0.000
0.000
-30.981
0.000
0.000
31.059
2.380
2.816
-2.824
-0.216
-0.300
-1.866
-0.006
1.855
0.286
23.992
74.270
69.951
70.046
69.947
74.265
23.988
0.305
1.748
-0.004
-1.759
-0.321
-23.993
-74.280
-69.957
-70.051
-69.956
-74.280
3.528
7.147
8.516
7.148
3.550
5.197
-9.081
-0.108 -30.649
-9.099
13.801
4.148
4.968
-4.070
-0.014
-0.032
-0.033
-0.017
Table 7.5: Conventional Model [1.2DL
p
V2
Frame Station
V3
ft
Kip
Kip
Kip
Text
47 11.000
-9,942.256 0.475 0.397
-7,793.592 -0.827 1.255
48 11.000
49 11.000
-5,731.826 -2.780 0.009
50 11.000
-7,797.752 -0.826 -1.252
51
52
53
54
55
56
11.000
11.000
11.000
11.000
11.000
11.000
-9,940.024
0.455
-10,198.008
-9,812.104
0.323
0.651
0.775
0.653
0.326
-10,043.951
-9,812.687
-10,199.736
126
-0.377
0.564
-0.188
0.000
0.189
-0.563
+ 1.6 L L (100 psf)l
T
M2
M3
Kip-ft
Kip-ft
Kip-ft
-4.371
0.000
-5.223
0.000
0.000
0.000
0.000
-13.808
-0.100
13.767
4.144
9.097
30.577
9.085
-5.004
0.000
0.000
0.000
0.000
0.000
-6.202
2.066
-0.002
-2.082
6.196
-3.554
-7.163
-8.522
-7.178
-3.586
Table 7.6: Conceptual Model - Conventiona Model [1.2DL + 1.6 LL (50 psf)]
Frame
P
V2
V3
T
M2
M3
Text
Kip
Kip
Kip-ft
Kip-ft
Kip
Kip-ft
5
0.000
21.604
28.274
718.769
2.570
1.963
6
0.264
5.284
671.515
0.000
58.129
2.900
7
2.921
832.050
0.265
4.536
0.000
49.906
8
831.411
0.000
0.000
50.082
0.004
4.553
9
0.264
4.537
0.000
832.188
49.911
2.911
10
675.414
0.270
5.287
0.000
2.970
58.153
11
28.272
733.161
2.570
1.968
0.000
21.650
12
61.797
525.693
5.618
0.165
0.000
1.810
57.322
13
755.121
5.211
0.303
0.000
3.340
14
736.125
5.256
0.002
0.000
0.024
57.819
15
5.212
0.000
755.869
0.299
3.282
57.338
16
523.574
5.620
0.167
0.000
61.816
1.838
17
2.571
1.962
0.000
28.284
718.395
21.591
18
678.692
0.262
5.280
0.000
2.888
58.079
19
0.277
829.195
4.553
0.000
50.079
3.048
20
0.001
0.000
0.016
831.431
4.768
52.453
21
0.272
4.554
2.994
632.961
0.000
50.102
22
0.257
2.822
3,192.810
5.282
0.000
58.101
23
2.567
28.234
1,945.844
1.966
0.000
21.622
24
5.616
523.065
0.161
0.000
1.769
61.781
25
755.332
5.208
0.302
0.000
3.314
57.299
26
736.015
5.255
0.000
0.000
57.796
0.000
27
5.210
0.000
57.314
755.242
0.300
3.304
28
5.617
523.139
0.170
0.000
61.791
1.865
33
458.422
0.321
0.000
0.563
6.196
3.528
34
0.000
7.147
1,646.385
0.650
0.188
2.072
35
1,400.092
0.774
0.000
8.516
0.000
0.003
36
1,645.771
0.650
0.000
0.189
2.078
7.148
37
0.323
0.564
0.000
3.550
456.029
6.201
38
0.472
5.197
1,518.234
0.397
0.000
4.368
39
40
41
42
43
44
45
46
1,489.087
3,133.854
1,493.581
1,521.935
1,344.455
3,346.981
3,347.192
1,347.250
0.826
2.786
0.827
0.452
0.001
0.003
0.003
0.002
1.256
0.010
1.255
0.377
0.370
0.000
0.000
0.000
0.000
0.000
13.819
0.108
13.801
4.148
4.070
2.816
2.824
0.000
0.000
0.216
0.000
30.981
31.059
2.380
127
9.081
30.649
9.099
4.968
0.014
0.032
0.033
0.017
Table 7.6: Conceptual Model - Conventional Model [1.2DL + 1.6 LL (50 psf)l
V2
Frame
P
V3
T
M2
M3
Kip
Text
Kip
Kip
Kip-ft
Kip-ft
Kip-ft
0.397
47
1,518.315
0.475
0.000
4.371
5.223
48
1,490.007
0.827
1.255
0.000
13.808
9.097
49
2.780
0.009
30.577
3,133.579
0.000
0.100
1.252
50
1,494.678
0.826
0.000
13.767
9.085
0.377
51
1,520.202
0.455
0.000
4.144
5.004
52
0.564
6.202
3.554
457.366
0.323
0.000
1,646.912
0.188
53
0.651
0.000
2.066
7.163
54
0.000
0.002
8.522
1,406.623
0.775
0.000
55
1,645.611
0.189
2.082
0.653
0.000
7.178
56
455.709
0.326
0.563
6.196
0.000
3.586
Summation
41.217
17,881.675 50.053
453.425
550.640
0.000
Red numbers indicate negative values, i.e. the
conceptual model exhibits lower magnitude loads.
128
7.10
SPECIFICATIONS FOR STRUCTURAL STEEL BUILDINGS
129
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
One East Wacker Drive, Suite 700
Chicago, Illinois 60601-1802
Approved by the AISC Committee on Specifications and issued by the
AISC Board of Directors
Supersedes the Load and Resistance Factor Design Specification for Structural Steel
Buildings dated December 27,1999, the Specification for Structural Steel Buldings—
Allowable Stress Design and Plastic Design dated June 1, 1989, including Supplement
No. 1, the Specification for Allowable Stress Design of Single-Angle Members
dated June 1, 1989, the Load and Resistance Factor Design Specification for SingleAngle Members dated November 10, 2000, and the Load and Resistance Factor
Design Specification for the Design of Steel Hollow Structural Sections
dated November 10, 2000, and all previous versions of these specifications.
March 9, 2005
Specification
for Structural Steel Buildings
ANSI/AISC 360-05
An American National Standard
AMERICAN INSTITUTE OF STEEI. CONSTRUCTION, INC.
Specification for Structural Steel Buildings, March 9, 2005
Printed in the United States of America
Second Printing: July 2006
Third Printing: April 2007
Caution must be exercised when relying upon other specifications and codes developed by other bodies and incorporated by reference herein since such material
may be modified or amended from time to time subsequent to the printing of this
edition. The Institute bears no responsibility for such material other than to refer
to it and incorporate it by reference at the time of the initial publication of this
edition.
The information presented in this publication has been prepared in accordance with
recognized engineering principles and is for general information only. While it is
believed to be accurate, this information should not be used or relied upon for any
specific application without competent professional examination and verification
of its accuracy, suitability, and applicability by a licensed professional engineer, designer, or architect The publication of the material contained herein is not intended
as a representation or warranty on the part of the American Institute of Steel Construction or of any other person named herein, that this information is suitable for
any general or particular use or of freedom from infringement of any patent or
patents. Anyone making use of this information assumes all liability arising from
such use.
AII rights reserved. This book or any part thereof
must not be reproduced in any form without the
written permission of the publisher.
American Institute of Steel Construction, Inc.
by
Copyright © 2005
xliii
GLOSSARY
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Specification for Structural Steel Huildings, March 9. 2005
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10
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10
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11
B. DESIGN REQUIREMENTS
Bl. General Provisions
B2. Loads and Load Combinations
B3. Design Basis
1. Required Strength
2. Limit States
3. Design for Strength Using Load and Resistance Factor Design
(LRFD)
4. Design for Strength Using Allowable Strength Design (ASD)
5. Design for Stability
6. Design of Connections
6a. Simple Connections
6b. Moment Connections
7. Design for Serviceability
8. Design for Ponding
9. Design for Fatigue
10. Design for Fire Conditions
11. Design for Corrosion Effects
II
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12
12
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12
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1
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2
2
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A. GENERAL PROVISIONS
A1. Scope
1. Low-Seismic Applications
2. High-Seismic Applications
3. Nuclear Applications
A2. Referenced Specifications, Codes and Standards
A3. Material
1. Structural Steel Materials
la. ASTM Designations
lb. Unidentified Steel
1 c. Rolled Heavy Shapes
Id. BuiltUp Heavy Shapes
2. Steel Castings and Forgings
3. Bolts, Washers and Nuts
4. Anchor Rods and Threaded Rods
5. Filler Metal and Flux for Welding
6. Stud Shear Connectors
A4. Structural Design Drawings and Specifications
SPECIFICATION
xxix
SYMBOLS
TABLE OF CONTENTS
16.1
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DESIGN OF MEMBERS FOR COMPRESSION
El. General Provisions
E2. Slendemess Limitations and Effective Length
E3. Compressive Strength for Flexural Buckling of Members without
Slender Elements
Specification for Structural Steel Buildings. March 9, 2(X).'S
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DESIGN OF MEMBERS FOR TENSION
DI. Slendemess Limitations
D2. Tensile Strength
D3. Area Determination
1. Gross Area
2. Net Area
3. Effective Net Area
D4. Built-Up Members
D5. Pin-Connected Members
1. Tensile Strength
2. Dimensional Requirements
D6. Eyebars
1. Tensile Strength
2. Dimensional Requirements
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12. Design Wall Thickness for HSS
13. Gross and Net Area Determination
Classification of Sections for Local Buckling
1. Unstiffened Elements
2. Stiffened Elements
Fabrication, Erection and Quality Control
Evaluation of Existing Structures
TABLE OF CONTENTS
STABILITY ANALYSIS AND DESIGN
CI. Stability Design Requirements
1. General Requirements
2. Member Stability Design Requirements
3. System Stability Design Requirements
3a. Braced-Frame and Shear-Wall Systems
3b. Moment-Frame Systems
3c. Gravity Framing Systems
3d. Combined Systems
C2. Calculation of Required Strengths
1. Methods of Second-Order Analysis
la. General Second-Order Elastic Analysis
1 b. Second-Order Analysis by Amplified First-Order Elastic
Analysis
2. Design Requirements
2a. Design by Second-Order Analysis
2b. Design by First-Order Analysis
B5.
B6.
B4.
x
F.
Compressive Strength for Torsional and Flexural-Torsional
Buckling of Members without Slender Elements
Single Angle Compression Members
Buiit-Up Members
1. Compressive Strength
2. Dimensional Requirements
Members with Slender Elements
1. Slender Unstiffened Elements, Qs
2. Slender Stiffened Elements, g„
Specification for Structural Steel Buildings, March 9. 2005
AMERICAN INSTITLTK OF STEEL CONSTRUCTION, INC.
H. DESIGN OF MEMBERS FOR COMBINED FORCES
AND TORSION
HI. Doubly and Singly Symmetric Members Subject to Flexure and
Axial Force
1. Doubly and Singly Symmetric Members in Flexure
and Compression
2. Doubly and Singly Symmetric Members in Flexure
and Tension
3. Doubly Symmetric Members in Single Axis Flexure
and Compression
H2. Unsymmetric and Other Members Subject to Flexure and Axial Force. . .
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G. DESIGN OF MEMBERS FOR SHEAR
Gl. General Provisions
G2. Members with Unstiffened or Stiffened Webs
1. Nominal Shear Strength
2. Transverse Stiffeners
G3. Tension Field Action
1. Limits on the Use of Tension Field Action
2. Nominal Shear Strength with Tension Field Action
3. Transverse Stiffeners
G4. Single Angles
G5. Rectangular HSS and Box Members
G6. Round HSS
G7. Weak Axis Shear in Singly and Doubly Symmetric Shapes
G8. Beams and Girders with Web Openings
F13.
Fl 2.
Fl 1.
FIO.
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TABLE OF CONTENTS
2. Lateral-Torsional Buckling
3. Flange Local Buckling of Tees
Single Angles
1. Yielding
2. Lateral-Torsional Buckling
3. Leg Local Buckling
Rectangular Bars and Rounds
1. Yielding
2. Lateral-Torsional Buckling
Unsymmetrical Shapes
1. Yielding
2. Lateral-Torsional Buckling
3. Local Buckling
Proportions of Beams and Girders
1. Hole Reductions
2. Proportioning Limits for I-Shaped Members
3. Cover Plates
4. Built-Up Beams
16.1-xii
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16.1-xi
Specification for Structural Steel Buildings, March 9, 2(X)5
DESIGN OF MEMBERS FOR FLEXURE
Fl. General Provisions
F2. Doubly Symmetric Compact I-Shaped Members and Channels Bent
about their Major Axis
1. Yielding
2. Lateral-Torsional Buckling
F3. Doubly Symmetric I-Shaped Members with Compact Webs
and Noncompact or Slender Flanges Bent about Their Major
Axis
1. Lateral-Torsional Buckling
2. Compression Flange Local Buckling
F4. Other I-Shaped Members with Compact or Noncompact Webs,
Bent about Their Major Axis
1. Compression Flange Yielding
2. Lateral-Torsional Buckling
3. Compression Flange Local Buckling
4. Tension Flange Yielding
F5. Doubly Symmetric and Singly Symmetric I-Shaped Members with
Slender Webs Bent about Their Major Axis
1. Compression Flange Yielding
2. Lateral-Torsional Buckling
3. Compression Flange Local Buckling
4. Tension Flange Yielding
F6. LShaped Members and Channels Bent about Their Minor Axis
1. Yielding
2. Flange Local Buckling
F7. Square and Rectangular HSS and Box-Shaped Members
1. Yielding
2. Flange Local Buckling
3. Web Local Buckling
F8. Round HSS
1. Yielding
2. Local Buckling
F9. Tees and Double Angles Loaded in the Plane of Symmetry
I. Yielding
E7.
E5.
E6.
E4.
TABLE OF CONTENTS
I.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
DESIGN OF CONNECTIONS
Jl. General Provisions
1. Design Basis
2. Simple Connections
3. Moment Connections
4. Compression Members with Bearing Joints
5. Splices in Heavy Sections
6. Beam Copes and Weld Access Holes
7. Placement of Welds and Bolts
8. Bolts in Combination with Welds
9. High-Strength Bolts in Combination with Rivets
10. Limitations on Bolted and Welded Connections
J2. Welds
1. Groove Welds
1 a. Effective Area
lb. Limitations
2. Fillet Welds
2a. Effective Area
2b. Limitations
3. Plug and Slot Welds
3a. Effective Area
3b. Limitations
4. Strength
5. Combination of Welds
6. Filler Metal Requirements
7. Mixed Weld Metal
J3. Bolts and Threaded Parts
1. High-Strength Bolts
2. Size and Use of Holes
3. Minimum Spacing
4. Minimum Edge Distance
5. Maximum Spacing and Edge Distance
6. Tension and Shear Strength of Bolts and Threaded Parts
7. Combined Tension and Shear in Bearing-Type Connections
8. High-Strength Bolts in Slip-Critical Connections
9. Combined Tension and Shear in Slip-Critical Connections
10. Bearing Strength at Bolt Holes
11. Special Fasteners
12. Tension Fasteners
J4. Affected Elements of Members and Connecting Elements
1. Strength of Elements in Tension
2. Strength of Elements in Shear
3. Block Shear Strength
4. Strength of Elements in Compression
TABLE OF CONTENTS
Specification for Structural Steel Buildings, March 9, 2005
J.
16.1-xiv
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16.1-xiii
Specification for Structural Steel Buildings, March 9, 2005
DESIGN OF COMPOSITE MEMBERS
11. General Provisions
1. Nominal Strength of Composite Sections
la. Plastic Stress Distribution Method
lb. Strain-Compatibility Method
2. Material Limitations
3. Shear Connectors
12. Axial Members
1. Encased Composite Columns
1 a. Limitations
I b. Compressive Strength
1c. Tensile Strength
Id. Shear Strength
le. Load Transfer
If. Detailing Requirements
1 g. Strength of Stud Shear Connectors
2. Filled Composite Columns
2a. Limitations
2b. Compressive Strength
2c. Tensile Strength
2d. Shear Strength
2e. Load Transfer
2f. Detailing Requirements
13. Flexural Members
1. General
la. Effective Width
lb. Shear Strength
lc. Strength During Construction
2. Strength of Composite Beams with Shear Connectors
2a. Positive Flexural Strength
2b. Negative Flexural Strength
2c. Strength of Composite Beams with Formed Steel Deck
2d. Shear Connectors
3. Flexural Strength of Concrete-Encased and Filled Members
14. Combined Axial Force and Flexure
15. Special Cases
H3. Members Under Torsion and Combined Torsion, Flexure, Shear
and/or Axial Force
1. Torsional Strength of Round and Rectangular HSS
2. HSS Subject to Combined Torsion, Shear, Flexure and
Axial Force
3. Strength of Non-HSS Members under Torsion and
Combined Stress
TABLE OF CONTENTS
90
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91
91
91
92
92
92
93
93
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93
93
95
95
95
95
97
97
97
98
101
102
102
102
102
105
106
106
106
108
109
109
110
Ill
Ill
Ill
Ill
112
112
112
113
Fillers
Splices
Bearing Strength
Column Bases and Bearing on Concrete
Anchor Rods and Embedments
Flanges and Webs with Concentrated Forces
1. Flange Local Bending
2. Web Local Yielding
3. Web Crippling
4. Web Sidesway Buckling
5. Web Compression Buckling
6. Web Panel Zone Shear
7. Unframed Ends of Beams and Girders
8. Additional Stiffener Requirements for Concentrated Forces
9. Additional Doubler Plate Requirements for Concentrated Forces..
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
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M. FABRICATION, ERECTION AND QUALITY CONTROL
Ml. Shop and Erection Drawings
M2. Fabrication
1. Cambering, Curving and Straightening
2. Thermal Cutting
3. Planing of Edges
4. Welded Construction
5. Bolted Construction
6. Compression Joints
7. Dimensional Tolerances
8. Finish of Column Bases
9. Holes for Anchor Rods
10. Drain Holes
11. Requirements forGalvanized Members
M3. Shop Painting
I. General Requirements
142
141
140
138
139
140
138
137
135
136
136
137
143
143
143
144
144
144
144
144
144
HSS-to-HSS Moment Connections
1. Definitions of Parameters
2. Criteria for Round HSS
2a. Limits of Applicability
2b. Branches with In-Plane Bending Moments in T-, Y- and
Cross-Connections
2c. Branches with Out-of-Plane Bending Moments in T-, Y- and
Cross-Connections
2d. Branches with Combined Bending Moment and Axial Force in
T-, Y- and Cross-Connections
3. Criteria for Rectangular HSS
3a. Limits of Applicability
3b. Branches with In-Plane Bending Moments in T- and
Cross-Connections
3c. Branches with Out-of-Plane Bending Moments in T- and
Cross-Connections
3d. Branches with Combined Bending Moment and Axial Force in
T- and Cross-Connections
TABLE OF CONTENTS
L. DESIGN FOR SERVICEABILITY
LI. General Provisions
L2. Camber
L3. Deflections
L4. Drift
L5. Vibration
L6. Wind-Induced Motion
L7. Expansion and Contraction
L8. Connection Slip
K3.
16.1-xvi
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122
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123
123
IB
114
114
134
115
115
116
116
117
117
119
119
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16.
Specification for Structural Steel Buildings, March 9, 2005
K. DESIGN OF HSS AND BOX MEMBER CONNECTIONS
Kl. Concentrated Forces on HSS
1. Definitions of Parameters
2. Limits of Applicability
3. Concentrated Force Distributed Transversely
3a. Criterion for Round HSS
3b. Criteria for Rectangular HSS
4. Concentrated Force Distributed Longitudinally at the Center
of the HSS Diameter or Width, and Acting Perpendicular to
the HSS Axis
4a. Criterion for Round HSS
4b. Criterion for Rectangular HSS
5. Concentrated Force Distributed Longitudinally at the Center
of the HSS Width, and Acting Parallel to the HSS Axis
6. Concentrated Axial Force on the End of a Rectangular HSS
with a Cap Plate
K2. HSS-to-HSS Truss Connections
1. Definitions of Parameters
2. Criteria for Round HSS
2a. Limits of Applicability
2b. Branches with Axial Loads in T-, Y- and Cross-Connections
2c. Branches with Axial Loads in K-Connections
3. Criteria for Rectangular HSS
3a. Limits of Applicability
3b. Branches with Axial Loads in T-, Y- and Cross-Connections
3c. Branches with Axial Loads in Gapped K-Connections
3d. Branches with Axial Loads in Overlapped K-Connections
3e. Welds to Branches
J5.
J6.
J7.
J8.
19.
J10.
TABLE OF CONTENTS
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
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179
APPENDIX 4. STRUCTURAL DESIGN FOR FIRE CONDITIONS
4.1. General Provisions
4.1.1. Performance Objective
Specification for Structural Steel Buildings, March 9. 2(X)5
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159
160
160
162
163
APPENDIX 3. DESIGN FOR FATIGUE
3.1. General
3.2. Calculation of Maximum Stresses and Stress Ranges
3.3. Design Stress Range
3.4. Bolts and Threaded Parts
3.5. Special Fabrication and Erection Requirements
AMERICAN INSTITUTK OF STEKI. CONSTRUCTION. INC.
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APPENDIX 2. DESIGN FOR PONDING
2.1. Simplified Design for Ponding
2.2. Improved Design for Ponding
187
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187
188
188
188
188
188
188
189
189
189
189
190
190
APPENDIX 5. EVALUATION OF EXISTING STRUCTURES
5.1. General Provisions
5.2. Material Properties
1. Determination of Required Tests
2. Tensile Properties
3. Chemical Composition
4. Base Metal Notch Toughness
5. Weld Metal
6. Bolts and Rivets
5.3. Evaluation by Structural Analysis
1. Dimensional Data
2. Strength Evaluation
3. Serviceability Evaluation
5.4. Evaluation by Load Tests
1. Determination of Load Rating by Testing
2. Serviceability Evaluation
5.5. Evaluation Report
180
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TABLE OF CONTENTS
4.1.2. Design by Engineering Analysis
4.1.3. Design by Qualification Testing
4.1.4. Load Combinations and Required Strength
4.2. Structural Design for Fire Conditions by Analysis
4.2.1. Design-Basis Fire
4.2.1.1. Localized Fire
4.2.1.2. Post-Flashover Compartment Fires
4.2.1.3. Exterior Fires
4.2.1.4. Fire Duration
4.2.1.5. Active Fire Protection Systems
4.2.2. Temperatures in Structural Systems under Fire
Conditions
4.2.3. Material Strengths at Elevated Temperatures
4.2.3.1. Thermal Elongation
4.2.3.2. Mechanical Properties at Elevated Temperatures
4.2.4. Structural Design Requirements
4.2.4.1. General Structural Integrity
4.2.4.2. Strength Requirements and Deformation Limits
4.2.4.3. Methods of Analysis
4.2.4.3a. Advanced Methods of Analysis
4.2.4.3b. Simple Methods of Analysis
4.2.4.4. Design Strength
4.3. Design by Qualification Testing
4.3.1. Qualification Standards
4.3.2. Restrained Construction
4.3.3. Unrestrained Construction
16.1-xviii
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148
148
148
148
148
148
149
149
149
149
149
149
150
150
150
150
150
16.1-xvii
APPENDIX 1. INELASTIC ANALYSIS AND DESIGN
1.1. Genera! Provisions
1.2. Materials
1.3. Moment Redistribution
1.4. Local Buckling
1.5. Stability and Second-Order Effects
1. Braced Frames
2. Moment Frames
1.6. Columns and Other Compression Members
1.7. Beams and Other Flexural Members
1.8. Members under Combined Forces
1.9. Connections
2. Inaccessible Surfaces
3. Contact Surfaces
4. Finished Surfaces
5. Surfaces Adjacent to Field Welds
M4. Erection
1. Alignment of Column Bases
2. Bracing
3. Alignment
4. Fit of Column Compression Joints and Base Plates
5. Field Welding
6. Field Painting
7. Field Connections
M5. Quality Control
1. Cooperation
2. Rejections
3. Inspection of Welding
4. Inspection of Slip-Critical High-Strength Bolted Connections . . . .
5. Identification of Steel
TABLE OF CONTENTS
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200
203
203
204
204
204
204
207
208
208
208
208
209
COMMENTARY
INTRODUCTION
COMMENTARY GLOSSARY
A. GENERAL PROVISIONS
Al. Scope
A2. Referenced Specifications, Codes and Standards
A3. Material
1. Structural Steel Materials
la. ASTM Designations
1c. Rolled Heavy Shapes
2. Steel Castings and Forgings
3. Bolts, Washers and Nuts
4. Anchor Rods and Threaded Rods
5. Filler Metal and Flux for Welding
A4. Structural Design Drawings and Specifications
249
249
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250
250
250
250
252
253
254
254
254
255
255
256
256
258
DESIGN OF MEMBERS FOR TENSION
Dl. Slenderness Limitations
D2. Tensile Strength
D3. Area Determination
1. Gross Area
2. Net Area
3. Effective Net Area
D4. Built-Up Members
D5. Pin-Connected Members
1. Tensile Strength
2. Dimensional Requirements
D6. Eyebars
1. Tensile Strength
2. Dimensional Requirements
DESIGN OF MEMBERS FOR COMPRESSION
El. General Provisions
E2. Slenderness Limitations and Effective Length
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STABILITY ANALYSIS AND DESIGN
CI. Stability Design Requirements
1. General Requirements
2. Member Stability Design Requirements
3. System Stability Design Requirements
3a. Braced-Frame and Shear-Wall Systems
3b. Moment-Frame Systems
3c. Gravity Framing Systems
3d. Combined Systems
C2. Calculation of Required Strengths
1. Methods of Second-Order Analysis
la. General Second-Order Elastic Analysis
1 b. Second-Order Analysis by Amplified First-Order Elastic
Analysis
2. Design Requirements
2a. Design by Second-Order Analysis
2b. Design by First-Order Analysis
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TABLE OF CONTENTS
8. Design for Ponding
9. Design for Fatigue
10. Design for Fire Conditions
11. Design for Corrosion Effects
12. Design Wall Thickness for HSS
B4. Classification of Sections for Local Buckling
B5. Fabrication, Erection and Quality Control
B6. Evaluation of Existing Structures
xx
Specification for Structural Steel Buildings, March 9, 2005
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216
217
218
222
210
210
211
213
213
213
196
196
196
196
APPENDIX 7. DIRECT ANALYSIS METHOD
7.1. General Requirements
7.2. Notional Loads
7.3. Design-Analysis Constraints
B. DESIGN REQUIREMENTS
Bl. General Provisions
B2. Loads and Load Combinations
B3. Design Basis
1. Required Strength
2. Limit States
3. Design for Strength Using Load and Resistance Factor
Design (LRFD)
4. Design for Strength Using Allowable Strength Design (ASD)
5. Design for Stability
6. Design of Connections
7. Design for Serviceability
191
191
191
192
192
193
193
193
193
194
194
195
16.1-xix
APPENDIX 6. STABILITY BRACING FOR COLUMNS AND BEAMS
6.1. General Provisions
6.2. Columns
1. Relative Bracing
2. Nodal Bracing
6.3. Beams
1. Lateral Bracing
la. Relative Bracing
lb. Nodal Bracing
2. Torsional Bracing
2a. Nodal Bracing
2b. Continuous Torsional Bracing
TABLE OF CONTENTS
286
286
286
286
288
288
288
289
289
DESIGN OF MEMBERS FOR SHEAR
Gl. General Provisions
G2. Members with Unstiffened or Stiffened Webs
1. Nominal Shear Strength
2. Transverse Stiffeners
G3. Tension Field Action
1. Limits on the Use of Tension Field Action
2. Nominal Shear Strength with Tension Field Action
3. Transverse Stiffeners
Single Angles
Rectangular HSS and Box Members
Round HSS
Weak Axis Shear in Singly and Doubly Symmetric Shapes
Beams and Girders with Web Openings
TABLE OF CONTENTS
289
290
290
290
291
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
DESIGN OF COMPOSITE MEMBERS
11. General Provisions
1. Nominal Strength of Composite Sections
1 a. Plastic Stress Distribution Method
lb. Strain-Compatibility Approach
2. Material Limitations
3. Shear Connectors
12. Axial Members
1. Encased Composite Columns
1 a. Limitations
1 b. Compressive Strength
lc. Tensile Strength
Id. Shear Strength
1 e. Load Transfer
2. Filled Composite Columns
2a. Limitations
2b. Compressive Strength
2c. Tensile Strength
2d. Shear Strength
2e. Load Transfer
2f. Detailing Requirements
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306
306
308
H. DESIGN OF MEMBERS FOR COMBINED FORCES
AND TORSION
292
HI. Doubly and Singly Symmetric Members Subject to Flexure
and Axial Force
292
1. Doubly and Singly Symmetric Members in Flexure and
Compression
292
2. Doubly and Singly Symmetric Members in Flexure and Tension .. 295
3. Doubly Symmetric Members in Single Axis Flexure
and Compression
296
H2. Unsymmetric and Other Members Subject to Flexure and
Axial Force
297
H3. Members Under Torsion and Combined Torsion, Flexure, Shear
and/or Axial Force
298
1. Torsional Strength of Round and Rectangular HSS
298
2. HSS Subject to Combined Torsion, Shear, Flexure
and Axial Force
300
3. Strength of Non-HSS Members under Torsion and
Combined Stress
300
G4.
G5.
G6.
G7.
G8.
16.1-xxii
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268
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263
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275
275
275
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277
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279
279
283
283
283
284
284
285
Compressive Strength for Flexural Buckling of Members without
Slender Elements
Compressive Strength for Torsional and Flexural-Torsional
Buckling of Members without Slender Elements
Single-Angle Compression Members
Built-Up Members
1. Compressive Strength
2. Dimensional Requirements
Members with Slender Elements
1. Slender Unstiffened Elements, Qs
2. Slender Stiffened Elements, Qa
16.1 -xxi
DESIGN OF MEMBERS FOR FLEXURE
Fl. General Provisions
F2. Doubly Symmetric Compact I-Shaped Members and
Channels Bent About Their Major Axis
F3. Doubly Symmeteric I-Shaped Members with Compact Webs
and Noncompact or Slender Flanges Bent about Their Major A x i s . . . .
F4. Other I-Shaped Members with Compact or Noncompact
Webs Bent about Their Major Axis
F5. Doubly Symmetric and Singly Symmetric I-Shaped Members
with Slender Webs Bent about Their Major Axis
F6. I-Shaped Members and Channels Bent about Their Minor Axis
F7. Square and Rectangular HSS and Box-Shaped Members
F8. Round HSS
F9. Tees and Double Angles Loaded in the Plane of Symmetry
F10. Single Angles
1. Yielding
2. Lateral-Torsional Buckling
3. Leg Local Buckling
Fl 1. Rectangular Bars and Rounds
F12. Unsymmetrical Shapes
Fl 3. Proportions of Beams and Girders
1. Hole Reductions
2. Proportioning Limits for I-Shaped Members
E7.
E5.
E6.
E4.
E3.
TABLE OF CONTENTS
Flexural Members
1. General
la. Effective Width
lb. Shear Strength
lc. Strength during Construction
2. Strength of Composite Beams with Shear Connectors
2a. Positive Flexural Strength
2b. Negative Flexural Strength
2c. Strength of Composite Beams with Formed Steel Deck
2d. Shear Connectors
3. Flexural Strength of Concrete-Encased and Filled Members
Combined Axial Force and Flexure
Special Cases
TABLE OF CONTENTS
343
344
345
346
349
349
350
350
350
351
351
353
353
353
353
353
353
354
356
356
357
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Specification for Structural Steel Buildings, March 9, 2(X)5
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372
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379
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367
365
365
365
365
366
Additional Doubler Plate Requirements for Concentrated Forces .. 364
K. DESIGN OF HSS AND BOX MEMBER CONNECTIONS
Kl. Concentrated Forces on HSS
1. Definitions of Parameters
2. Limits of Applicability
3. Concentrated Force Distributed Transversely
4. Concentrated Force Distributed Longitudinally at the
Center of the HSS Diameter or Width, and Acting Perpendicular
to the HSS Axis
5. Concentrated Force Distributed Longitudinally at the Center
of the HSS Width, and Acting Parallel to the HSS Axis
6. Concentrated Axial Force on the End of a Rectangular HSS
with a Cap Plate
K2. HSS-to-HSS Truss Connections
1. Definitions of Parameters
2. Criteria for Round HSS
3. Criteria for Rectangular HSS
K3. HSS-to-HSS Moment Connections
9.
5. Maximum Spacing and Edge Distance
6. Tension and Shear Strength of Bolts and Threaded Parts
7. Combined Tension and Shear in Bearing-Type Connections
8. High-Strength Bolts in Slip-Critical Connections
9. Combined Tension and Shear in Slip-Critical Connections
10. Bearing Strength at Bolt Holes
12. Tension Fasteners
J4. Affected Elements of Members and Connecting Elements
1. Strength of Elements in Tension
2. Strength of Elements in Shear
3. Block Shear Strength
4. Strength of Elements in Compression
J5. Fillers
J6. Splices
J7. Bearing Strength
J8. Column Bases and Bearing on Concrete
J9. Anchor Rods and Embedments
J10. Flanges and Webs with Concentrated Forces
1. Flange Local Bending
2. Web Local Yielding
3. Web Crippling
4. Web Sidesway Buckling
5. Web Compression Buckling
6. Web Panel-Zone Shear
7. Unframed Ends of Beams and Girders
8. Additional Stiffener Requirements for Concentrated Forces
16.1-xxiv
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330
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331
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337
337
337
341
342
342
342
342
343
343
343
308
308
311
312
312
313
313
313
313
314
319
319
323
16.1-xxiii
Specification for Structural Steel Buildings, March 9, 2005
DESIGN OF CONNECTIONS
Jl. General Provisions
1. Design Basis
2. Simple Connections
3. Moment Connections
4. Compression Members with Bearing Joints
5. Splices in Heavy Sections
6. Beam Copes and Weld Access Holes
7. Placement of Welds and Bolts
8. Bolts in Combination with Welds
9. High-Strength Bolts in Combination with Rivets
10. Limitations on Bolted and Welded Connections
J2. Welds
1. Groove Welds
1 a. Effective Area
1 b. Limitations
2. Fillet Welds
2a. Effective Area
2b. Limitations
3. Plug and Slot Welds
3a. Effective Area
3b. Limitations
4. Strength
5. Combination of Welds
6. Filler Metal Requirements
7. Mixed Weld Metal
J3. Bolts and Threaded Parts
1. High-Strength Bolts
2. Size and Use of Holes
3. Minimum Spacing
4. Minimum Edge Distance
14.
15.
13.
TABLE OF CONTENTS
397
APPENDIX 2.
Specification for Structural Steel Buildings, March 9. 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
APPENDIX 5. EVALUATION OF EXISTING STRUCTURES
5.1. General Provisions
5.2. Material Properties
1. Determination of Required Tests
2. Tensile Properties
4. Base Metal Notch Toughness
5. Weld Metal
6. Bolts and Rivets
5.3. Evaluation by Structural Analysis
2. Strength Evaluation
5.4. Evaluation by Load Tests
1. Determination of Live Load Rating by Testing
2. Serviceability Evaluation
5.5. Evaluation Report
405
405
405
406
407
407
407
407
408
408
408
APPENDIX 4. STRUCTURAL DESIGN FOR FIRE CONDITIONS
4.1. General Provisions
4.1.1.
Performance Objective
4.1.4.
Load Combinations and Required Strength
4.2. Structural Design for Fire Conditions by Analysis
4.2.1.
Design-Basis Fire
4.2.1.1. Localized Fire
4.2.1.2. Post-FIashover Compartment Fires
4.2.1.3. Exterior Fires
4.2.1.4. Fire Duration
4.2.1.5. Active Fire Protection Systems
4.2.2.
Temperatures in Structural Systems under
Fire Conditions
4.2.3.
Material Strengths at Elevated Temperatures
4.2.4.
Structural Design Requirements
4.2.4.1. General Structural Integrity
4.2.4.2. Strength Requirements and Deformation Limits
4.2.4.3. Methods of Analysis
4.2.4.3a. Advanced Methods of Analysis
4.2.4.3b. Simple Methods of Analysis
4.2.4.4. Design Strength
4.3. Design by Qualification Testing
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417
418
418
418
418
418
419
419
419
420
408
412
412
412
413
413
413
413
413
413
400
400
400
401
402
403
TABLE OF CONTENTS
APPENDIX 3. DESIGN FOR FATIGUE
3.1. General
3.2. Calculation of Maximum Stresses and Stress Ranges
3.3. Design Stress Range
3.4. Bolts and Threaded Parts
3.5. Special Fabrication and Erection Requirements
16.1-xxvi
Specification for Structural Steel Buildings, March 9, 2005
DESIGN FOR PONDING
393
393
393
393
393
394
395
395
395
395
396
396
APPENDIX 1. INELASTIC ANALYSIS AND DESIGN
1.1. General Provisions
1.2. Materials
1.3. Moment Redistribution
1.4. Local Buckling
1.5. Stability and Second-Order Effects
1. Braced Frames
2. Moment Frames
1.6. Columns and Other Compression Members
1.7. Beams and Other Flexural Members
1.8. Members Under Combined Forces
1.9. Connections
381
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382
382
383
385
385
386
386
388
388
388
388
388
389
389
389
390
391
391
391
391
391
391
391
392
392
392
DESIGN FOR SERVICEABILITY
LI. General Provisions
L2. Camber
L3. Deflections
L4. Drift
L5. Vibration
L6. Wind-Induced Motion
L7. Expansion and Contraction
L8. Connection Slip
16.1
M. FABRICATION, ERECTION AND QUALITY CONTROL
Ml. Shop and Erection Drawings
M2. Fabrication
1. Cambering, Curving and Straightening
2. Thermal Cutting
4. Welded Construction
5. Bolted Construction
10. Drain Holes
11. Requirements for Galvanized Members
M3. Shop Painting
1. General Requirements
3. Contact Surfaces
5. Surfaces Adjacent to Field Welds
M4. Erection
2. Bracing
4. Fit of Column Compression Joints and Base Plates
5. Field Welding
M5. Quality Control
1. Identification of Steel
L.
TABLE OF CONTENTS
Specification for Structural Steel Buildings, March 9, 2(X)5
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
439
REFERENCES
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
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APPENDIX 7. DIRECT ANALYSIS METHOD
7.1. General Requirements
7.2. Notional Loads
7.3. Design-Analysis Constraints
Specification for Stniriu ral Steel Buildings, March 9, 2(X)5
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421
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425
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426
16.1 -xs
APPENDIX 6. STABILITY BRACING FOR COLUMNS AND BEAMS
6.1. General Provisions
6.2. Columns
6.3. Beams
1. Lateral Bracing
2. Torsional Bracing
TABLE OF CONTENTS
As
ASIAsf
Asr
Asl
A,
Aw
Afr
Afg
Af„
Aft
Af,
As
Ax
Af,
Agv
A„
A„<
Am
Aph
Ar
Ae
Aeff
Symbol
A
A
AB
A&M
Ah
Ah,
Ay
Ar
Ac
AD
Specification for Structural Steel Buildings, March 9. 2005
AMERICAN INSTITUTE OF STEEI. CONSTRUCTION. INC.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Effective area of the weld, in. 2 (mm 2 )
J2.4
Effective area of weld throat of any ith weld element, in. 2 (mm 2 )
J2.4
Area of steel concentrically bearing on a concrete support, in. 2 (mm 2 )
J8
Maximum area of the portion of the supporting surface that is
2
2
geometrically similar to and concentric with the loaded area, in. ( m m ) . . . J8
B
Overall width of rectangular HSS member, measured 90 degrees
to the plane of the connection, in. (mm)
Table D3.1
B
Overall width of rectangular HSS main member, measured
90 degrees to the plane of the connection, in. (mm)
K3.I
B
Factor for lateral-torsional buckling in tees and double angles
F9.2
Bh
Overall width of rectangular HSS branch member, measured
90 degrees to the plane of the connection, in. (mm)
K3.1
Bbi
Overall branch width of the overlapping branch
K.2.3
Bhj
Overall branch width of the overlapped branch
K2.3
Bp
Width of plate, measure 90 degrees to the plane of the connection,
in. (mm)
Kl.l
Bp
Width of plate, transverse to the axis of the main member,
in. (mm)
K2.3
B\, Bz Factors used in determining Mu for combined bending and axial
forces when first-order analysis is employed
C2.1
C
HSS torsional constant
H3.1
Cb
Lateral-torsional buckling modification factor for nonuniform moment
diagrams when both ends of the unsupported segment are braced
FI
Cd
Coefficient relating relative brace stiffness and curvature
App. 6.3.1
Cf
Constant based on stress category, given in Table A-3.1
App. 3.3
Cm
Coefficient assuming no lateral translation of the frame
C2.1
Cp
Ponding flexibility coefficient for primary member in a flat roof
App. 2.1
Cr
Coefficient for web sidesway buckling
J10.4
C,
Ponding flexibility coefficient for secondary member in a flat roof.. .App. 2.1
Cv
Web shear coefficient
G2.1
Cw
Warping constant, in. 6 (mm 6 )
E4
D
Nominal dead load
App. 2.2
D
Outside diameter of round HSS member, in. (mm)
Table B4.1
D
Outside diameter, in. (mm)
E7.2
D
Outside diameter of round HSS main member, in. (mm)
K2.1
D
Chord diameter, in. (mm)
K2.2
Db
Outside diameter of round HSS branch member, in. (mm)
K2.1
Z>,
Factor used in Equation G 3 - 3 , dependent on the type of transverse
stiffeners used in a plate girder
G3.3
D„
In slip-critical connections, a multiplier that reflects the ratio of the mean
installed bolt pretension to the specified minimum bolt pretension
J3.8
E
Modulus of elasticity of steel = 29,000 ksi (200 000 MPa)
Table B4.1
£,.
Modulus of elasticity of concrete = wl^yffj., ksi
(0.043wJ5 / £ , MPa)
12.1
Aw
Awi
Ai
A2
Specification for Structural Sled Buildings, March 9. 2005
Definition
Section
Column cross-sectional area, in. 2 (mm 2 )
J10.6
2
2
Total cross-sectional area of member, in. (mm )
E7.2
2
2
Loaded area of concrete, in. (mm )
12.1
2
2
Cross-sectional area of the base metal, in. (mm )
J2.4
Nominal unthreaded body area of bolt or threaded part, in. 2 (mm 2 )
J3.6
Cross-sectional area of the overlapping branch, in.- (mm 2 )
K2.3
Cross-sectional area of the overlapped branch, in. 2 (mm 2 )
K2.3
Area of concrete, in. 2 (mm 2 )
12.1
Area of concrete slab within effective width, in. 2 (mm 2 )
13.2
Area of an upset rod based on the major thread diameter,
in. 2 (mm 2 )
Table J3.2
Effective net area, in. 2 (mm 2 )
D2
Summation of the effective areas of the cross section based on the
2
2
reduced effective width, be, in. (mm )
E7.2
Area of compression flange
G3.1
2
2
Gross tension flange area, in. (mm )
F13.1
Net tension flange area, in. 2 (mm 2 )
FI3.1
2
2
Area of tension flange, in. (mm )
G3.1
2
2
Gross area of member, in. (mm )
B3.13
2
2
Gross area of section based on design wall thickness, in. (mm )
G6
Gross area of composite member, in. 2 (mm 2 )
12.1
2
2
Chord gross area, in. (mm )
K2.2
Gross area subject to shear, in. 2 (mm 2 )
J4.3
2
2
Net area of member, in. (mm )
B3.13
2
2
Net area subject to tension, in. (mm )
J4.3
Net area subject to shear, in. 2 (mm 2 )
J4.2
2
2
Projected bearing area, in. (mm )
J7
Area of adequately developed longitudinal reinforcing steel within
2
2
the effective width of the concrete slab, in. (mm )
13.2
Area of steel cross section, in. 2 (mm 2 )
12.1
Cross-sectional area of stud shear connector, in. 2 (mm 2 )
12.1
Shear area on me failure path, in. 2 (mm 2 )
D5.1
Area of continuous reinforcing bars, in. 2 (mm 2 )
12.1
Stiffener area, in. 2 (mm 2 )
G3.3
Net tensile area, in. 2 (mm 2 )
App. 3.4
Web area, the overall depth times the web thickness, dtw, in. 2 (mm 2 )
G2.1
The section or table number in the right-hand column refers to where the symbol is first
used.
SYMBOLS
Available flexural stress at the point of consideration about the
Fm
F^.
Nominal tensile stress Fn,, or shear stress, F„v, from Table J3.2,
F„
FM,
Fwi
F„
Fum
F„
F„
Fu
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
App. 4.2
J2.4
J2.4
J3.10
Kl.l
12.1
D2
App. 3.1
J3.7
J3.7
App. 3.3
J3.7
J3.6
H3.3
x component of stress F„./, ksi (MPa)
J2.4
y component of stress Fwi, ksi (MPa)
J2.4
Specified minimum yield stress of the type of steel being used.
ksi (MPa). As used in this Specification, "yield stress" denotes either
the specified minimum yield point (for those steels that have a yield
point) or specified yield strength (for those steels that do not have a
yield point)
Table B4.1
Specified minimum yield stress of the compression flange,
ksi (MPa)
App. 1.3
Specified minimum yield stress of the column web, ksi (MPa)
JI0.6
Specified minimum yield stress of HSS member material, ksi (MPa)
Kl.l
Specified minimum yield stress of HSS main member material,
ksi (MPa)
K2.1
Specified minimum yield stress of HSS branch member material,
ksi (MPa)
K2.1
Specified minimum yield stress of the overlapping branch material,
ksi (MPa)
K2.3
Specified minimum yield stress of the overlapped branch material,
ksi (MPa)
K2.3
Specified minimum yield stress of the flange, ksi (MPa)
J10.1
Specified minimum yield stress of the type of steel being used at
elevated temperature, ksi (MPa)
App. 4.2
Specified minimum yield stress of plate, ksi (MPa)
Kl.l
Specified minimum yield stress of reinforcing bars, ksi (MPa)
12.1
Specified minimum yield stress of the stiffener material, ksi (MPa)
G3.3
Specified minimum yield stress of the web, ksi (MPa)
J 10.2
Shear modulus of elasticity of steel = 11,200 ksi (77 200 MPa)
E4
Story shear produced by the lateral forces used to compute Al{,
kips (N)
C2.1
Overall height of rectangular HSS member, measured in the
plane of the connection, in. (mm)
Table D3.1
Overall height of rectangular HSS main member, measured in
the plane of the connection, in. (mm)
K2.1
Flexural constant
E4
Overall height of rectangular HSS branch member, measured in
the plane of the connection, in. (mm)
K2.1
Overall depth of the overlapping branch
K2.3
Moment of inertia in the place of bending, in.4 (mm4)
C2.1
Moment of inertia about the axis of bending, in.4 (mm4)
App. 7.3
4
4
Moment of inertia of the concrete section, in. (mm )
12.1
Moment of inertia of the steel deck supported on secondary
4
4
members, in. (mm )
App. 2.1
Moment of inertia of primary members, in.4 (mm4)
App. 2.1
4
4
Moment of inertia of secondary members, in. (mm )
App. 2.1
Specification for Structural Steel Buildings, March 9, 2005
Nominal tensile stress modified to include the effects of shearing
stress, ksi (MPa)
Nominal shear stress from Table J3.2, ksi (MPa)
Design stress range, ksi (MPa)
Threshold fatigue stress range, m a x i m u m stress range for
indefinite design life from Table A-3.1,ksi (MPa)
Specified minimum tensile strength of the type of steel being used,
ksi (MPa)
Specified minimum tensile strength of a stud shear connector,
ksi (MPa)
Specified minimum tensile strength of the connected material,
ksi (MPa)
Specified minimum tensile strength of HSS material, ksi (MPa)
Specified minimum tensile strength of the type of steel being
used at elevated temperature, ksi (MPa)
Nominal strength of the weld metal per unit area, ksi (MPa)
Nominal stress in any j'th weld element, ksi (MPa)
F'm
Fm,
FSR
FTU
Nominal tensile stress from Table J3.2, ksi (MPa)
F„,
ksi (MPa)
Nominal torsional strength
F„
strength, ksi (MPa)
E4
E4
J2.4
E4
C1.3
E4
E4
F12.2
E3
K2.2
H2
H2
J2.4
Table B4.1
Electrode classification number, ksi (MPa)
Fnxx
A calculated stress used in the calculation of nominal flexural
Elastic flexural buckling stress about the major axis, ksi (MPa)
Fet
F/.
Elastic critical buckling stress, ksi (MPa)
FF
Elastic flexural buckling stress about the minor axis, ksi (MPa)
Critical torsional buckling stress, ksi (MPa)
Frr.
Elastic torsional buckling stress, ksi (MPa)
Critical stress about the minor axis, ksi (MPa)
Fcrv
Fez
Buckling stress for the section as determined by analysis, ksi (MPa)
Fcr
F^.
Available stress, ksi (MPa)
Critical stress, ksi (MPa)
Fcr
minor axis, ksi (MPa)
Available flexural stress at the point of consideration about the
Fe
Fhz
Nominal strength of the base metal per unit area, ksi (MPa)
F,
major axis, ksi (MPa)
Available axial stress at the point of consideration, ksi (MPa)
Eleff
E,„
E,
H2
Modulus of elasticity of concrete at elevated temperature,
ksi (MPa)
App. 4.2.3
Effective stiffness of composite section, kip-in. : (N-mm 2 )
12.1
Modulus of elasticity of steel at elevated temperature, ksi ( M P a ) . . -App. 4.2.3
Modulus of elasticity of steel = 29,000 ksi (200 000 MPa)
12.1
Ecm
L„
Lv
L,
Lp
Lpd
Lq
Lp
Le
Lt,
L,
Le
Lh
L
L
L
L
L
L
L
Kz
J
K
Kz
K\
/,
Isr
/», / v
h
lz
lyC
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Absolute value of moment at quarter point of the unbraced segment,
kip-in. (N-mm)
Fl
Ma
Required flexural strength in chord, using ASD load combinations,
kip-in. (N-mm)
K2.2
Mn
Absolute value of moment at centerline of the unbraced segment,
kip-in. (N-mm)
Fl
Mbr
Required bracing moment, kip-in. (N-mm)
App. 6.2
Mc
Absolute value of moment at three-quarter point of the unbraced
segment, kip-in. (N-mm)
Fl
M,.(x. y> Available flexural strength determined in accordance with Chapter F,
kip-in. (N-mm)
Hl.l
Mcx
Available flexural-torsional strength for strong axis flexure
determined in accordance with Chapter F, kip-in. (N-mm)
HI.3
Me
Elastic lateral-torsional buckling moment, kip-in. (N-mm)
FI0.2
Mi,
First-order moment under LRFD or ASD load combinations caused
by lateral translation of the frame only, kip-in. (N-mm)
C2.1
Mmax Absolute value of maximum moment in the unbraced segment,
kip-in. (N-mm)
Fl
M„
Nominal flexural strength, kip-in. (N-mm)
Fl
M„,
First-order moment using LRFD or ASD load combinations assuming
there is no lateral translation of the frame, kip-in, (N-mm)
C2.1
Mp
Plastic bending moment, kip-in. (N-mm)
Table B4.I
Mr
Required second-order flexural strength under LRFD or ASD load
combinations, kip-in. (N-mm)
C2.1
Mr
Required flexural strength using LRFD or ASD load combinations,
kip-in. (N-mm)
HI
Mr
Required flexural strength in chord, kip-in. (N-mm)
K2.2
Mr-ip
Required in-plane flexural strength in branch, kip-in. (N-mm)
K3.2
Mr-0p
Required out-of-plane flexural strength in branch, kip-in. (N-mm)
K3.2
M„
Required flexural strength in chord, using LRFD load combinations,
kip-in. (N-mm)
K2.2
Mv
Yield moment about the axis of bending, kip-in. (N-mm)
Table B4.1
Mi
Smaller moment, calculated from a first-order analysis, at the
ends of that portion of the member unbraced in the plane of bending
under consideration, kip-in. (N-mm)
C2.1
Mz
Larger moment, calculated from a first-order analysis, at the ends of
that portion of the member unbraced in the plane of bending under
consideration, kip-in. (N-mm)
C2.1
JV
Length of bearing (not less than k for end beam reactions), in. (mm)
J10.2
N
Bearing length of the load, measured parallel to the axis of the HSS
member, (or measured across the width of the HSS in the case of the
loaded cap plates), in. (mm)
K1.1
N
Number of stress range fluctuations in design life
App. 3.3
Nh
Number of bolts carrying the applied tension
J3.9
MA
SYMBOLS
Specification for Structural Steel Buildings. March 9. 2005
Moment of inertia of steel shape, in. 4 (mm 4 )
12.1
Moment of inertia of reinforcing bars, in. 4 (mm 4 )
12.1
Moment of inertia about the principal axes, in. 4 (mm 4 )
E4
Out-of-plane moment of inertia, in. 4 (mm 4 )
App. 6.2
Minor principal axis moment of inertia, in. 4 (mm 4 )
F30.2
Moment of inertia about y-axis referred to the compression flange,
or if reverse curvature bending referred to smaller flange,
in. 4 (mm 4 )
Fl
Torsional constant, in. 4 (mm 4 )
E4
Effective length factor determined in accordance with Chapter C
CI.2
Effective length factor for torsional buckling
E4
Effective length factor in the plane of bending, calculated based on
the assumption of no lateral translation set equal to 1.0 unless
analysis indicates that a smaller value may be used
C2.1
Effective length factor in the plane of bending, calculated based on a
sidesway buckling analysis
C2.1
Story height, in. (mm)
C2.1
Length of the member, in. (mm)
H3
Actual length of end-loaded weld, in. (mm)
J2.2
Nominal occupancy live load
App. 4.1.4
Laterally unbraced length of a member, in. (mm)
E2
Span length, in. (mm)
App. 6.2
Length of member between work points at truss chord centerlines,
in. (mm)
E5
Length between points that are either braced against lateral
displacement of compression flange or braced against twist of the
cross section, in. (mm)
F2
Distance between braces, in. (mm)
App. 6.2
Length of channel shear connector, in. (mm)
13.2
Clear distance, in the direction of the force, between the edge of the
hole and the edge of the adjacent hole or edge of the material,
in. (mm)
J3.10
Total effective weld length of groove and fillet welds to rectangular
HSS, in. (mm)
K2.3
Limiting laterally unbraced length for the limit state of yielding
in. (mm)
F2.2
Column spacing in direction of girder, ft (m)
App. 2
Limiting laterally unbraced length for plastic analysis, in. (mm)
App. 1.7
Maximum unbraced length for Mr (the required flexural strength),
in. (mm)
App. 6.2
Limiting laterally unbraced length for the limit state of inelastic
lateral-torsional buckling, in. (mm)
F2.2
Column spacing perpendicular to direction of girder, ft (m)
App. 2.1
Distance from maximum to zero shear force, in. (mm)
G6
16.1-xxxiv
S
S
S
Sc
12.1
12.1
Specification for Structural Steel Buildings. March 9, 2005
AMERICAN INSTITUTE OF STEFI. CONSTRUCTION, INC.
Up
Us
7]j
Tt,
T,~
Tn
Tr
T„
U
U
Uhs
Sx, Sv
Sy
T
Coefficient to account for group effect
13.2
Factor in Equation C2-6b dependent on type of system
C2.1
Cross-section monosymmetry parameter
Fl
Nominal strength, specified in Chapters B through K
B3.3
Nominal slip resistance, kips (N)
J3.8
Position effect factor for shear studs
13.2
Web plastification factor
F4.1
Reduction factor for reinforced or nonreinforced transverse
partial-joint-penetration (PJP) groove welds
App. 3.3
Web plastification factor corresponding to the tension flange
yielding limit state
F4.4
Required strength (LRFD)
B3.3
Total nominal strength of longitudinally loaded fillet welds, as
determined in accordance with Table J2.5
J2.4
Total nominal strength of transversely loaded fillet welds,
as determined in accordance with Table J2.5 without the alternate
in Section J2.4 (a)
J2.4
Elastic section modulus of round HSS, in. 1 (mm 1 )
F8.2
Lowest elastic section modulus relative to the axis of bending,
in. 1 (mm 1 )
F12
Spacing of secondary members, ft (m)
App. 2.1
Nominal snow load
App. 4.1.4
Chord elastic section modulus, in.-1 (mm-1)
K2.2
Elastic section modulus to the toe in compression relative to
3
1
the axis of bending, in. (mm )
F10.3
Effective section modulus about major axis, in.-1 (mm 3 )
F7.2
Elastic section modulus referred to tension and compression
1
3
flanges, respectively, in. (mm )
Table B4.1
Elastic section modulus taken about the principal axes, in. 3 ( m m 1 ) . . .F2.2, F6
For channels, taken as the minimum section modulus
F6
Nominal forces and deformations due to the design-basis fire
defined in Section 4.2.1
App. 4.1.4
Tension force due to ASD load combinations, kips (kN)
J3.9
Minimum fastener tension given in Table J3.1 or J3.1M, kips (kN)
J3.8
Available torsional strength, kip-in. (N-mm)
H3.2
Nominal torsional strength, kip-in. (N-mm)
H3.1
Required torsional strength, kip-in. (N-mm)
H3.2
Tension force due to L R F D load combinations, kips (kN)
J3.9
Shear lag factor
D3.3
Utilization ratio
K2.2
Reduction coefficient, used in calculating block shear rupture
strength
J4.3
Stress index
App. 2.2
Stress index
App. 2.2
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
App. 3.3
App. 2.2
A 1.1
B3.4
H1.2
J10.6
K3.2d
K.2.2
App. 1.4
C2.2
E7
E7.2
K2.2
12.1
E7.1
C2.2
Seff
S w , Sxc
S
S
C2.1
D2
C2.1
Rwl
R„
Rwi
Rp,
Rf,
R,„
R,„
Rn
R„
Rp
Rp,.
RPJP
C2.1
C2.1
App. 7.3
C2.2
App. 7.3
J3.8
K2.2
App. 3.4
App. 6.2
H1.1
H1.2
H1.3
16.1-
Specification for Structural Steel Buildings, March 9, 2005
Nj
Additional lateral load
Nj
Notional lateral load applied at level /, kips (N)
Ns
Number of slip planes
Ov
Overlap connection coefficient
P
Pitch, in. per thread (mm per thread)
Phr
Required brace strength, kips (N)
Pr
Available axial compressive strength, kips (N)
Pc
Available tensile strength, kips (N)
Pro
Available compressive strength out of the plane of bending, kip (N)
Pei. Pel Elastic critical buckling load for braced and unbraced frame,
respectively, kips (N)
Pei,
Euler buckling load, evaluated in the plane of bending, kips (N)
Pi{,,C) First-order axial force using LRFD or ASD load combinations
as a result of lateral translation of the frame only (tension or
compression), kips (N)
Pnu.ri First-order axial force using LRFD or ASD load combinations,
assuming there is no lateral translation of the frame (tension or
compression), kips (N)
Pn
Nominal axial strength, kips (N)
Pn
Nominal axial compressive strength without consideration of length
effects, kips (N)
Pp
Nominal bearing strength of concrete, kips (N)
Pr
Required second-order axial strength using LRFD or ASD load
combinations, kips (N)
Pr
Required axial compressive strength using LRFD or ASD load
combinations, kips (N)
Pr
Required tensile strength using LRFD or ASD load combinations,
kips (N)
Pr
Required strength, kips (N)
Pr
Required axial strength in branch, kips (N)
Pr
Required axial strength in chord, kips (N)
Ptl
Required axial strength in compression, kips (N)
Pv
Member yield strength, kips (N)
Q
Full reduction factor for slender compression elements
Q„
Reduction factor for slender stiffened compression elements
Qf
Chord-stress interaction parameter
Q„
Nominal strength of one stud shear connector, kips (N)
£>,
Reduction factor for slender unstiffened compression elements
R
Nominal load due to rainwater or snow, exclusive of the ponding
contribution, ksi (MPa)
R
Seismic response modification coefficient
Ru
Required strength (ASD)
Ryu.
Reduction factor for joints using a pair of transverse fillet welds
only
SYMBOLS
Gl
3
Plastic section modulus about the principal axes, in. ( m m )
stud (in other words, in the direction of m a x i m u m m o m e n t for a
F4.2
E7.2
neglecting the welds, in. (mm)
K2.3
B4.1
F4.2
G3.1
Effective width of the branch face welded to the overlapped brace
Flange width, in. ( m m )
Compression flange width, in. ( m m )
Width of tension flange, in. ( m m )
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
to the member axis of buckling, in. (mm)
Distance between centroids of individual components perpendicular
in. (mm)
distance between the flanges less the inside comer radius on each side,
Specification for Structural Steel Buildings, March 9. 2005
h
K2.3
Effective width of the branch face welded to the chord
used; for tees, the overall depth; for rectangular HSS, the clear
lines of fasteners or the clear distance between flanges when welds are
D5.1
force, in. ( m m )
Clear distance between flanges less the fillet or comer radius for
J10.6
J7
D5.1
J7
E7.1
B4.1
II. 1
H2
H2
13.2
K2.1
J10.6
E6.1
B4.2
K2.1
B3.13
J3.7
App. 2.2
rolled shapes; for built-up sections, the distance between adjacent
h
Gap between toes of branch members in a gapped K-connection,
E7.1
F11.2
App. 4.2
the edge of the part measured in the direction normal to the applied
Effective edge distance; the distance from the edge of the hole to
Reduced effective width, in. ( m m )
G4
J 10.6
Width of column flange, in. ( m m )
gage lines, in. (mm)
radius on each side, in. (mm)
g
Transverse center-to-center spacing (gage) between fastener
$
width b is the clear distance between the webs less the inside corner
Width of the angle leg resisting the shear force, in. (mm)
Required shear strength per unit area, ksi (MPa)
fv
adjacent lines of fasteners or lines of welds; for rectangular H S S , the
B 4 . 1 , B4.2
due to rainwater or snow exclusive of the ponding contribution),
ksi (MPa)
edge to the first row of fasteners or line of welds, or the distance between
Stress due to D + R (the nominal dead load + the nominal load
/„
nominal dimension; for plates, the width b is the distance from the free
Specified m i n i m u m compressive strength of concrete at elevated
temperatures, ksi (MPa)
f'an
members and tees, the width b is half the full-flange width, bf\ for legs
Specified m i n i m u m compressive strength of concrete, ksi (MPa)
minor axis) using L R F D or A S D load combinations, ksi (MPa)
Required flexural stress at the point of consideration (major axis,
A S D load combinations, ksi (MPa)
of angles and flanges of channels and zees, the width b is the full
f'r
E7.1
fbfw.zi
Width of unstiffened compression element; for flanges of I-shaped
Full width of longest angle leg, in. ( m m )
Outside width of leg in compression, in. ( m m )
F10.3
of major axis bending moment alone to the area of the compression
flange components
simply supported beam), in. (mm)
Required axial stress at the point of consideration using L R F D or
Ratio of two times the web area in compression due to application
fa
at mid-height of the deck rib, and in the load bearing direction of the
the thickness of the tension-loaded plate, in. ( m m )
App. 3.3
Half the length of the nonwelded root face in the direction of
em>d-ki
Distance from the edge of stud shank to the steel deck web, measured
D5.1
J3.3
B4.1
App. 3.4
branches, in. ( m m )
Eccentricity in a truss connection, positive being away from the
Nominal diameter (body or shank diameter), in. (mm)
Column depth, in. ( m m )
dc
e
db
measured parallel to direction of force, in. ( m m )
E6.1
F13.2
Beam depth, in. ( m m )
Roller diameter, in. ( m m )
Pin diameter, in. (mm)
Diameter, in. (mm)
E5
E5
App. 6.2
Shortest distance from edge of pin hole to edge of member
Distance between connectors in a built-up member, in. ( m m )
Clear distance between transverse stiffeners, in. ( m m )
F2, F6.1
db
3
d
3
K3.3
d
d
in. 3 ( m m 3 )
F7.1
F13.1
Branch plastic section m o d u l u s about the correct axis of bending,
Plastic section modulus about the axis of bending, in. ( m m )
3
Full nominal depth of tee, in. ( m m )
d
A S D load combination applied at level i, kips (N)
Hole reduction coefficient, kips (N)
Full nominal depth of section, in. ( m m )
d
Gravity load from the L R F D load combination or 1.6 times the
C2.2
Depth of rectangular bar, in. ( m m )
d
Full nominal depth of tee, in. ( m m )
d
H3.2
Full nominal depth of the section, in. ( m m )
kips (N)
d
Nominal fastener diameter, in. ( m m )
Stiffener width for one-sided stiffeners, in. ( m m )
b„
d
Shorter leg of angle, in. ( m m )
Longer leg of angle, in. ( m m )
bs
b\
Required shear strength using L R F D or A S D load combinations,
G3.3
Nominal shear strength, kips (N)
Required shear strength at the location of the stiffener, kips (N)
G3.3
12.1
Available shear strength, kips (N)
12.1
Required shear force transferred by shear connectors, kips (N)
16.1-x>
Required shear force introduced to column, kips (N)
SYMBOLS
K1.3
G2.2
J10.2
B4.2
J3.8
B4.2
F2.2
16.1-xxxix
Specification for Structural Steel Ruilding.s, March 9. 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Radius of gyration about y-axis, in. (mm)
E4
Radius of gyration for the minor principal axis, in. (mm)
E5
Longitudinal center-to-center spacing (pitch) of any two consecutive
holes, in. (mm)
B3.13
Thickness of element, in. (mm)
B4.2
Wall thickness, in. (mm)
E7.2
Angle leg thickness, in. (mm)
FI0.2
Width of rectangular bar parallel to axis of bending, in. (mm)
F l 1.2
Thickness of connected material, in. (mm)
J3.10
Thickness of plate, in. (mm)
D5.1
Design wall thickness for HSS equal to 0.93 times the nominal wall
thickness for ERW HSS and equal to the nominal wall thickness for
SAW HSS, in. (mm)
B3.12
Total thickness of fillers, in. (mm)
J5
Design wall thickness of HSS main member, in. (mm)
K2.1
Design wall thickness of HSS branch member, in. (mm)
K2.1
Thickness of the overlapping branch, in. (mm)
K2.3
Thickness of the overlapped branch, in. (mm)
K2.3
Thickness of the column flange, in. (mm)
J10.6
Thickness of the loaded flange, in. (mm)
J10.1
Flange thickness of channel shear connector, in. (mm)
13.2
Compression flange thickness, in. (mm)
F4.2
Thickness of plate, in. (mm)
K1.1
Thickness of tension loaded plate, in. (mm)
App. 3.3
Thickness of the attached transverse plate, in. (mm)
K2.3
Web stiffener thickness, in. (mm)
App. 6.2
Web thickness of channel shear connector, in. (mm)
13.2
Beam web thickness, in. (mm)
App. 6.3
Web thickness, in. (mm)
Table B4.1
Column web thickness, in. (mm)
J10.6
Thickness of element, in. (mm)
E7.1
Width of cover plate, in. (mm)
F13.3
Weld leg size, in. (mm)
J2.2
Subscript relating symbol to major principal axis bending
Plate width, in. (mm)
Table D3.1
Leg size of the reinforcing or contouring fillet, if any, in the
direction of the thickness of the tension-loaded plate, in. (mm)
App. 3.3
Weight of concrete per unit volume (90 < MY < 155 lbs/ft3 or
3
1500 < HY < 2500 kg/m )
12.1
Average width of concrete rib or haunch, in. (mm)
13.2
Subscript relating symbol to strong axis
Coordinates of the shear center with respect to the centroid, in. (mm)
E4
Connection eccentricity, in. (mm)
Table D3.1
SYMBOLS
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
H>
x
xn, y0
3f
H>
t
t
tt,
thi
fhj
tcf
tf
tf
tfc
tp
tp
tp
r,
tw
tw
tw
tw
rM.
w
w
w
w
w
/
t
/
t
t
t
t
rv
rz
5
16.1-xl
Specification for Structural Steel Buildings, March 9, 2005
Coefficient for slender unstiffened elements, in. (mm)
Table B4.1
Slip-critical combined tension and shear coefficient
J3.9
Web plate buckling coefficient
G2.1
Largest laterally unbraced length along either flange at the point of
load, in. (mm)
J10.4
Length of bearing, in. (mm)
J7
Length of connection in the direction of loading, in. (mm)
Table D3.1
Number of nodal braced points within the span
App. 6.2
Threads per inch (per mm)
App. 3.4
Ratio of element i deformation to its deformation at maximum stress — J2.4
Projected length of the overlapping branch on the chord
K.2.2
Overlap length measured along the connecting face of the chord
beneath the two branches
K.2.2
Governing radius of gyration, in. (mm)
E2
Distance from instantaneous center of rotation to weld element with
minimum A„/r, ratio, in. (mm)
J2.4
Minimum radius of gyration of individual component in a built-up
member, in. (mm)
E6.1
Radius of gyration of individual component relative to its centroidal
axis parallel to member axis of buckling, in. (mm)
E6.1
Polar radius of gyration about the shear center, in. (mm)
E4
Radius of gyration of the flange components in flexural compression
plus one-third of the web area in compression due to application of
major axis bending moment alone
F4.2
Effective radius of gyration used in the determination of Lr for the
lateral-torsional buckling limit state for major axis bending of doubly
symmetric compact 1-shaped members and channels
F2.2
Radius of gyration about geometric axis parallel to connected leg,
in. (mm)
E5
1.5/ i f unknown, in. (mm)
Twice the distance from the centroid to the following: the inside face
of the compression flange less the fillet or corner radius, for rolled
shapes; the nearest line of fasteners at the compression flange
or the inside faces of the compression flange when welds are used,
for built-up sections, in. (mm)
Distance between flange centroids, in. (mm)
Twice the distance from the plastic neutral axis to the nearest line
of fasteners at the compression flange or the inside face of the
compression flange when welds are used, in. (mm)
Hole factor
Factor defined by Equation G2-6 for minimum moment of inertia
for a transverse stiffener
Distance from outer face of flange to the web toe of fillet, in. (mm)
Outside corner radius of the HSS, which is permitted to be taken as
SYMBOLS
k
kp
kPf
Xpy,.
kr
krf
km
u.
T|
t,
7
Am
A„
A
An
A,-
$»
$eop
p,W(.
fW
^eff
3r
(3
3
Specification for Structural Steel Buildings. March 9. 2(X)5
AMERICAN INSTITUTE OF STKKI. CONSTRUCTION, INC.
AMERICAN INSTITIJTF. OF STEEL CONSTRUCTION. INC.
Resistance factor, specified in Chapters B through K
B3.3
Resistance factor for bearing on concrete
12.1
Resistance factor for
flexure
Fl
Resistance factor for compression
El
Resistance factor for axially loaded composite columns
12.1 b
Resistance factor for shear on the failure path
D5.1
Resistance factor for torsion
H3.1
Resistance factor for tension
D2
Resistance factor for shear
G1
Safety factor
B3.4
Safety factor for bearing on concrete
12.1
Safety factor for
flexure
F1
Safety factor for compression
El
Safety factor for axially loaded composite columns
12.1 b
Safety factor for shear on the failure path
D5.1
Safety factor for torsion
H3.1
Safety factor for tension
D2
Safety factor for shear
Gl
Minimum reinforcement ratio for longitudinal reinforcing
12.1
Angle of loading measured from the weld longitudinal axis, degrees
J2.4
Acute angle between the branch and chord, degrees
K2.1
Strain corresponding to compressive strength, f'c
App. 4.2
Parameter for reduced flexural stiffness using the direct analysis
method
App. 7.3
Specification for Structural Steel Buildings, March 9, 2005
Separation ratio for built-up compression members = - —
E6.1
2rih
Reduction factor given by Equation J2-1
J2.2
Width ratio; the ratio of branch diameter to chord diameter for round
HSS; the ratio of overall branch width to chord width for rectangular
HSS
K2.1
Brace stiffness requirement excluding web distortion, kip-in./radian
(N-mm/radian)
App. 6.2
Required brace stiffness
App. 6.2
Effective width ratio; the sum of the perimeters of the two branch
members in a K-connection divided by eight times the chord width
K2.1
Effective outside punching parameter
K2.3
Web distortional stiffness, including the effect of web transverse
stiffeners, if any, kip-in./radian (N-mm/radian)
App. 6.2
Section property for unequal leg angles, positive for short legs in
compression and negative for long legs in compression
F10.2
First-order interstory drift due to the design loads, in. (mm)
C2.2
First-order interstory drift due to lateral forces, in. (mm)
C2.1
Deformation of weld elements at intermediate stress levels, linearly
proportioned to the critical deformation based on distance from the
instantaneous center of rotation, rt, in. (mm)
J2.4
Deformation of weld element at maximum stress, in. (mm)
J2.4
Deformation of weld element at ultimate stress (fracture), usually in
element furthest from instantaneous center of rotation, in. (mm)
J2.4
Chord slenderness ratio; the ratio of one-half the diameter to the wall
thickness for round HSS; the ratio of one-half the width to wall
thickness for rectangular HSS
K2.1
Gap ratio; the ratio of the gap between the branches of a gapped
K-connection to the width of the chord for rectangular HSS
K2.1
Load length parameter, applicable only to rectangular HSS; the ratio
of the length of contact of the branch with the chord in the plane of
the connection to the chord width
K2.1
Slenderness parameter
F3
Limiting slenderness parameter for compact element
B4
Limiting slenderness parameter for compact
flange
F3
Limiting slenderness parameter for compact web
F4
Limiting slenderness parameter for noncompact element
B4
Limiting slenderness parameter for noncompact
flange
F3
Limiting slenderness parameter for noncompact web
F4
Mean slip coefficient for class A or B surfaces, as applicable, or as
established by tests
J3.8
a
C2.1
Subscript relating symbol to weak axis
Subscript relating symbol to minor principal axis bending
Factor used in B2 equation
v
z
a
SYMBOLS
GLOSSARY
Composite. Condition in which steel and concrete elements and members work as a
unit in the distribution of internal forces.
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
Specification for Structural Steel Buildings. March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Complete-joint-penetration groove weld (CJP). Groove weld in which weld metal
extends through the joint thickness, except as permitted for HSS connections.
Compact section. Section capable of developing a fully plastic stress distribution and possessing a rotation capacity of approximately three before the onset of local buckling.
Combined system. Structure comprised of two or more lateral load-resisting systems of
different type.
Column. Structural member that has the primary function of resisting axial force.
Cold-formed steel structural member]. Shape manufactured by press-braking blanks
sheared from sheets, cut lengths of coils or plates, or by roll forming cold- or hotrolled coils or sheets; both forming operations being performed at ambient room
temperature, that is, without manifest addition of heat such as would be required for
hot forming.
Cladding. Exterior covering of structure.
Chord member. For HSS, primary member that extends through a truss connection.
Charpy V-Notch impact test. Standard dynamic test measuring notch toughness of a
specimen.
Camber. Curvature fabricated into a beam or truss so as to compensate for deflection
induced by loads.
Built-up member, cross-section, section, shape. Member, cross-section, section or shape
fabricated from structural steel elements that are welded or bolted together.
Buckling strength. Nominal strength for buckling or instability limit states.
Buckling. Limit state of sudden change in the geometry of a structure or any of its
elements under a critical loading condition.
Branch member. For HSS connections, member that terminates at a chord member or
main member.
Branch face. Wall of HSS branch member.
Braced frame]. An essentially vertical truss system that provides resistance to lateral
forces and provides stability for the structural system.
Block shear rupture. In a connection, limit state of tension fracture along one path and
shear yielding or shear fracture along another path.
Bearing-type connection. Bolted connection where shear forces are transmitted by the
bolt bearing against the connection elements.
Bearing (local compressive yielding). Limit state of local compressive yielding due to
the action of a member bearing against another member or surface.
16.1-xliv
Bearing. In a bolted connection, limit state of shear forces transmitted by the bolt to the
connection elements.
Beam-column. Structural member that resists both axial force and bending moment.
Beam. Structural member that has the primary function of resisting bending moments.
Batten plate. Plate rigidly connected to two parallel components of a built-up column or
beam designed to transmit shear between the components.
Average rib width. Average width of the rib of a corrugation in a formed steel deck.
Available stress*. Design stress or allowable stress, as appropriate.
Available strength*]. Design strength or allowable strength, as appropriate.
Authority having jurisdiction. Organization, political subdivision, office or individual
charged with the responsibility of administering and enforcing the provisions of the
applicable building code.
ASD load combination]. Load combination in the applicable building code intended for
allowable strength design (allowable stress design).
ASD (Allowable Strength Design)]. Method of proportioning structural components such
that the allowable strength equals or exceeds the required strength of the component
under the action of the ASD load combinations.
Applicable building code]. Building code under which the structure is designed.
Amplification factor. Multiplier of the results of first-order analysis to reflect secondorder effects.
Allowable stress. Allowable strength divided by the appropriate section property, such
as section modulus or cross-section area.
Allowable strength* ]. Nominal strength divided by the safety factor, R„/Q.
(1) Terms designated with t are common AISI-AISC terms that are coordinated between
the two standards developers.
(2) Terms designated with * are usually qualified by the type of load effect, for example,
nominal tensile strength, available compressive strength, design flexural strength.
(3) Terms designated with ** are usually qualified by the type of component, for example, web local buckling, flange local bending.
Notes:
Terms that appear in this Glossary are italicized throughout the Specification, where they
first appear within a sub-section.
GLOSSARY
16.1-xliii
Specification for Structural Steel Ruildin%s. March 9. 2005
AMERICAN INSTITUTE OF STEEI. CONSTRUCTION. INC.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Eyebar. Pin-connected tension member of uniform thickness, with forged or thermally
cut head of greater width than the body, proportioned to provide approximately equal
strength in the head and body.
Expansion roller. Round steel bar on which a member bears that can roll to accommodate
expansion.
Expansion rocker. Support with curved surface on which a member bears that can tilt
to accommodate expansion.
Engineer of record. Licensed professional responsible for sealing the contract documents.
End return. Length of fillet weld that continues around a corner in the same plane.
End panel. Web panel with an adjacent panel on one side only.
Encased composite column. Composite column consisting of a structural concrete
column and one or more embedded steel shapes.
Elastic analysis. Structural analysis based on the assumption that the structure returns
to its original geometry on removal of the load.
Effective width. Reduced width of a plate or slab with an assumed uniform stress
distribution which produces the same effect on the behavior of a structural member
as the actual plate or slab width with its nonuniform stress distribution.
Effective section modulus. Section modulus reduced to account for buckling of slender
compression elements.
Effective net area. Net area modified to account for the effect of shear lag.
Effective length. Length of an otherwise identical column with the same strength when
analyzed with pinned end conditions.
Effective length factor, K. Ratio between the effective length and the unbraced length of
the member.
Drift. Lateral deflection of structure.
Doubter. Plate added to, and parallel with, a beam or column web to increase resistance
to concentrated forces.
Double-concentrated forces. Two equal and opposite forces that form a couple on the
same side of the loaded member.
Double curvature. Deformed shape of a beam with one or more inflection points within
the span.
Distortional stiffness. Out-of-plane flexural stiffness of web.
Specification for Structural Steel Buildings, March 9, 2005
Direct analysis method. Design method for stability that captures the effects of residual
stresses and initial out-of-plumbness of frames by reducing stiffness and applying
notional loads in a second-order analysis.
Diaphragm}. Roof, floor or other membrane or bracing system that transfers in-plane
forces to the lateral force resisting system.
Diaphragm plate. Plate possessing in-plane shear stiffness and strength, used to transfer
forces to the supporting elements.
Diagonal stiffener. Web stiffener at column panel zone oriented diagonally to the
flanges, on one or both sides of the web.
Diagonal bracing. Inclined structural member carrying primarily axial force in a braced
frame.
Design wall thickness. HSS wall thickness assumed in the determination of section
properties.
Design stress*. Design strength divided by the appropriate section property, such as
.section modulus or cross section area.
Design stress range. Magnitude of change in stress due to the repeated application and
removal of service live loads. For locations subject to stress reversal it is the algebraic
difference of the peak stresses.
Design strength*}. Resistance factor multiplied by the nominal strength, $/?„.
Design load*\. Applied load determined in accordance with either LRFD load
combinations or ASD had combinations, whichever is applicable.
Cross connection. HSS connection in which forces in branch members or connecting
elements transverse to the main member are primarily equilibrated by forces in other
branch members or connecting elements on the opposite side of the main member.
Cover plate. Plate welded or bolted to the flange of a member to increase cross-sectional
area, section modulus or moment of inertia.
Cope. Cutout made in a structural member to remove a flange and conform to the shape
of an intersecting member.
Connection}. Combination of structural elements and joints used to transmit forces
between two or more members.
Concrete-encased beam. Beam totally encased in concrete cast integrally with the slab.
Distortional failure. Limit state of an HSS truss connection based on distortion of a
rectangular HSS chord member into a rhomboidal shape.
GLOSSARY
Concrete haunch. Section of solid concrete that results from stopping the deck on
each side of the girder in a composite floor system constructed using & formed steel
deck.
16.1-xlvi
Direct bond interaction. Mechanism by which force is transferred between steel and
concrete in a composite section by bond stress.
16.1-xlv
Concrete crushing. Limit state of compressive failure in concrete having reached the
ultimate strain.
GLOSSARY
16.1-xlvii
GLOSSARY
Horizontal shear. Force at the interface between steel and concrete surfaces in a
composite beam.
Flat width. Nominal width of rectangular HSS minus twice the outside corner radius.
In absence of knowledge of the corner radius, the flat width may be taken as the total
section width minus three times the thickness.
Joint eccentricity. For HSS truss connection, perpendicular distance from chord member
center of gravity to intersection of branch member work points.
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Specification for Structural Steel Buildings, March 9. 2005
AMERICAN INSTITUTE OF STEEL CONSTRCCTION, INC.
instability. Limit state reached in the loading of a structural component, frame or
structure in which a slight disturbance in the loads or geometry produces large
displacements.
In-plane instability. Limit state of a beam-column bent about its major axis while lateral
buckling or lateral-torsional buckling is prevented by lateral bracing.
Inelastic analysis. Structural analysis that takes into account inelastic material behavior,
including plastic analysis.
User Note: A pipe can be designed using the same design rules for round HSS
sections as long as it conforms to ASTM AS3 Class B and the appropriate parameters
are used in the design.
Gap connection. HSS truss connection with a gap or space on the chord face between
intersecting branch members.
Gage. Transverse center-to-center spacing of fasteners.
Fully restrained moment connection. Connection capable of transferring moment with
negligible rotation between connected members.
Formed steel deck. In composite construction, steel cold formed into a decking profile
used as a permanent concrete form.
Formed section. See cold-formed steel structural member.
Force. Resultant of distribution of stress over a prescribed area.
Flexural-torsional buckling]. Buckling mode in which a compression member bends
and twists simultaneously without change in cross-sectional shape.
HSS. Square, rectangular or round hollow structural steel section produced in accordance
with a pipe or tubing product specification.
Gusset plate. Plate element connecting truss members or a strut or brace to a beam or
column.
Flexural buckling. Buckling mode in which a compression member deflects laterally
without twist or change in cross-sectional shape.
Groove weld. Weld in a groove between connection elements. See also AWS Dl. I.
Flare V-groove weld. Weld in a groove formed by two members with curved surfaces.
Grip (of bolt). Thickness of material through which a bolt passes.
Gravity load. Load, such as that produced by dead and live loads, acting in the downward
direction.
Gravity frame. Portion of the framing system not included in the lateral load resisting
system.
Gravity axis. Axis through the center of gravity of a member along its length.
Flare bevel groove weld. Weld in a groove formed by a member with a curved surface
in contact with a planar member.
Fitted bearing stiffener. Stiffener used at a support or concentrated load that fits tightly
against one or both flanges of a beam so as to transmit load through bearing.
First-order analysis. Structural analysis in which equilibrium conditions are formulated
on the undeformed structure; second-order effects are neglected.
Fillet weld. Weld of generally triangular cross section made between intersecting
surfaces of elements.
Fillet weld reinforcement. Fillet welds added to groove welds.
Gouge. Relatively smooth surface groove or cavity resulting from plastic deformation
or removal of material.
Girt]. Horizontal structural member that supports wall panels and is primarily subjected
to bending under horizontal loads, such as wind load.
Filler. Plate used to build up the thickness of one component.
Girder. See Beam.
Filler metal. Metal or alloy to be added in making a welded joint.
Girder filler. Narrow piece of sheet steel used as a fill between the edge of a deck sheet
and the flange of a girder in a composite floor system constructed using a formed steel
deck.
Geometric axis. Axis parallel to web, flange or angle leg.
General collapse. Limit state of chord plastification of opposing sides of a round HSS
chord member at a cross-connection.
16.1-xlviii
Filled composite column. Composite column consisting of a shell of HSS or pipe filled
with structural concrete.
Faying surface- Contact surface of connection elements transmitting a shear force.
Fatigue. Limit state of crack initiation and growth resulting from repeated application
of live loads.
Fastener. Generic term for bolts, rivets, or other connecting devices.
Factored \oad\. Product of a load factor and the nominal load.
GLOSSARY
16.1-xlix
GLOSSARY
Moment connection. Connection that transmits bending moment between connected
members.
Lateral-torsional buckling. Buckling mode of a flexurai member involving deflection
normal to the plane of bending occurring simultaneously with twist about the shear
center of the cross-section.
Nondestructive testing. Inspection procedure wherein no material is destroyed and
integrity of the material or component is not affected.
Notch toughness. Energy absorbed at a specified temperature as measured in the Charpy
V-Notch test.
Specification for Structural Steel Buildings, March 9. 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Local crippling**. Limit state of local failure of web plate in the immediate vicinity of
a concentrated load or reaction.
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
Noncompact section. Section that can develop the yield stress in its compression elements
before local buckling occurs, but cannot develop a rotation capacity of three.
Nominal strength*]. Strength of a structure or component (without the resistance factor
or safety factor applied) to resist load effects, as determined in accordance with this
Specification.
Nominal rib height. Height of formed steel deck measured from the underside of the
lowest point to the top of the highest point.
Nominal load]. Magnitude of the load specified by the applicable building code.
Nominal dimension. Designated or theoretical dimension, as in the tables of section
properties.
Nodal brace. Brace that prevents lateral movement or twist independently of other
braces at adjacent brace points {see relative brace).
Net area. Gross area reduced to account for removed material.
Local buckling * *. Limit state of buckling of a compression element within a cross section.
Local bending**. Limit state of large deformation of a flange under a concentrated
tensile force.
Load factor]. Factor that accounts for deviations of the nominal load from the actual
load, for uncertainties in the analysis that transforms the load into a load effect and
for the probability that more than one extreme load will occur simultaneously.
Load effect]. Forces, stresses and deformations produced in a structural component by
the applied loads.
Load]. Force or other action that results from the weight of building materials, occupants
and their possessions, environmental effects, differential movement, or restrained
dimensional changes.
Limit state. Condition in which a structure or component becomes unfit for service and
is judged either to be no longer useful for its intended function (serviceability limit
state) or to have reached its ultimate load-carrying capacity (strength limit state).
Length effects. Consideration of the reduction in strength of a member based on its
unbraced length.
Moment frame]. Framing system that provides resistance to lateral loads and provides
stability to the structural system, primarily by shear and flexure of the framing
members and their connections.
Milled surface. Surface that has been machined flat by a mechanically guided tool to a
flat, smooth condition.
Leaning column. Column designed to carry gravity loads only, with connections that
are not intended to provide resistance to lateral loads.
Mill scale. Oxide surface coating on steel formed by the hot rolling process.
Lateral load. Load, such as that produced by wind or earthquake effects, acting in a
lateral direction.
Mechanism. Structural system that includes a sufficient number of real hinges,
plastic hinges or both, so as to be able to articulate in one or more rigid body
modes.
Main member. For HSS connections, chord member, column or other HSS member to
which branch members or other connecting elements are attached.
LRFD load combination]. Load combination in the applicable building code intended
for strength design (load and resistance factor design).
LRFD (Load and Resistance Factor Design)]. Method of proportioning structural
components such that the design strength equals or exceeds the required strength of
the component under the action of the LRFD load combinations.
Local yielding**. Yielding that occurs in a local area of an element.
16.1-1
Lateral load resisting system. Structural system designed to resist lateral loads and
provide stability for the structure as a whole.
Lateral bracing. Diagonal bracing, shear walls or equivalent means for providing
in-plane lateral stability.
Lap joint. Joint between two overlapping connection elements in parallel planes.
Lacing. Plate, angle or other steel shape, in a lattice configuration, that connects two
steel shapes together.
/(•connection. HSS connection in which forces in branch members or connecting
elements transverse to the main member are primarily equilibriated by forces in other
branch members or connecting elements on the same side of the main member.
Joint]. Area where two or more ends, surfaces, or edges are attached. Categorized by
type of fastener or weld used and method of force transfer.
GLOSSARY
16.1-Ii
GLOSSARY
Root of joint. Portion of a joint to be welded where the members are closest to each other.
Specification for Structural Steel Buildings. March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Reverse curvature. See double curvature.
Resistance factor, 4>t. Factor that accounts for unavoidable deviations of the nominal
strength from the actual strength and for the manner and consequences of failure.
Required strength*]. Forces, stresses and deformations acting on the structural
component, determined by either structural analysis, for the LRFD or ASD load
combinations, as appropriate, or as specified by this Specification or Standard.
Relative brace. Brace that controls the relative movement of two adjacent brace points
along the length of a beam or column or the relative lateral displacement of two
stories in a frame (see nodal brace).
Reentrant. In a cope or weld access hole, a cut at an abrupt change in direction in which
the exposed surface is concave.
Rational engineering analysis]. Analysis based on theory that is appropriate for the
situation, relevant test data if available, and sound engineering judgment.
Quality control. System of shop andfieldcontrols implemented by the fabricator and erector to ensure that contract and company fabrication and erection requirements are met.
Quality assurance. System of shop and field activities and controls implemented by the
owner or his/her designated representative to provide confidence to the owner and
the building authority that quality requirements are implemented.
P-A effect. Effect of loads acting on the displaced location of joints or nodes in a
structure. In tiered building structures, this is the effect of loads acting on the laterally
displaced location of floors and roofs.
P-b effect. Effect of loads acting on the deflected shape of a member between joints or
nodes.
Purlin]. Horizontal structural member that supports roof deck and is primarily subjected
to bending under vertical loads such as snow, wind or dead loads.
Punching load. Component of branch member force perpendicular to a chord.
Prying action. Amplification of the tension force in a bolt caused by leverage between
the point of applied load, the bolt and the reaction of the connected elements.
Properly developed Reinforcing bars detailed to yield in a ductile manner before
crushing of the concrete occurs. Bars meeting the provisions of ACI 318 insofar as
development length, spacing and cover shall be deemed to be properly developed.
Pretensioned joint. Joint with high-strength bolts tightened to the specified minimum
pretension.
Post-buckling strength. Load or force that can be carried by an element, member, or
frame after initial buckling has occurred.
Ponding. Retention of water due solely to the deflection of flat roof framing.
16.1-IH
Plug weld. Weld made in a circular hole in one element of a joint fusing that element
to another element.
Plate girder. Built-up beam.
Plastification. In an HSS connection, limit state based on an out-of-plane flexural yield
line mechanism in the chord at a branch member connection.
Plastic stress distribution method Method for determining the stresses in a composite
member assuming that the steel section and the concrete in the cross section are fully
plastic.
Plastic moment. Theoretical resisting moment developed within a fully yielded cross
section.
Plastic hinge. Yielded zone that forms in a structural member when the plastic moment
is attained. The member is assumed to rotate further as if hinged, except that such
rotation is restrained by the plastic moment.
Plastic analysis. Structural analysis based on the assumption of rigid-plastic behavior,
in other words, that equilibrium is satisfied throughout the structure and the stress is
at or below the yield stress.
Pitch. Longitudinal center-to-center spacing of fasteners. Center-to-center spacing of
bolt threads along axis of bolt.
Pipe. See HSS.
Permanent ioad\. Load in which variations over time are rare or of small magnitude.
All other loads are variable loads.
Percent elongation. Measure of ductility, determined in a tensile test as the maximum
elongation of the gage length divided by the original gage length.
Partially restrained moment connection. Connection capable of transferring moment
with rotation between connected members that is not negligible.
Partial-joint-penetration groove weld (PJP). Groove weld in which the penetration is
intentionally less than the complete thickness of the connected element.
Panel zone. Web area of beam-to-column connection delineated by the extension of
beam and column flanges through the connection, transmitting moment through a
shear panel.
Overlap connection. HSS truss connection in which intersecting branch members
overlap.
Out-of-plane buckling. Limit state of a beam-column bent about its major axis
while lateral buckling or lateral-torsional buckling is not prevented by lateral
bracing.
Notional load. Virtual load applied in a structural analysis to account for destabilizing
effects that are not otherwise accounted for in the design provisions.
GLOSSARY
16.1-liii
GLOSSARY
Stiffness. Resistance to deformation of a member or structure, measured by the ratio of
the applied force (or moment) to the corresponding displacement (or rotation).
Specification for Structural Steel Buildings. March 9, 200.S
AMERICAN INSTITUTE OF STKKL CONSTRUCTION, INC.
Sheet steel. In a composite floor system, steel used for closure plates or miscellaneous
trimming in a formed steel deck.
Specification for Structural Steel Buildings. March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Stiffener. Structural element, usually an angle or plate, attached to a member to distribute
load, transfer shear or prevent buckling.
Stiffened element. Flat compression element with adjoining out-of-plane elements along
both edges parallel to the direction of loading.
Stability. Condition reached in the loading of a structural component, frame or structure
in which a slight disturbance in the loads or geometry does not produce large
displacements.
Splice. Connection between two structural elements joined at their ends to form a single,
longer element.
Specified minimum yield stress]. Lower limit of yield stress specified for a material as
defined by ASTM.
Specified minimum tensile strength. Lower limit of tensile strength specified for a
material as defined by ASTM.
Snug-tightened joint. Joint with the connected plies in firm contact as specified in
Chapter J.
Slot weld. Weld made in an elongated hole fusing an element to another element.
Slip-critical connection. Bolted connection designed to resist movement by friction on
the faying surface of the connection under the clamping forces of the bolts.
Slip. In a bolted connection, limit state of relative motion of connected parts prior to
the attainment of the available strength of the connection.
Slender-element section. Cross section possessing plate components of sufficient
slenderness such that local buckling in the elastic range will occur.
Single curvature. Deformed shape of a beam with no inflection point within the span.
Single-concentrated force. Tensile or compressive force applied normal to the flange of
a member.
Simple connection. Connection that transmits negligible bending moment between
connected members.
Sidewall crushing. Limit state based on bearing strength of chord member sidewall in
HSS truss connection.
Sidewall crippling. Limit state of web crippling of the sidewalls of a chord member at
a HSS truss connection.
Sidesway buckling. Limit state of lateral buckling of the tension flange opposite the
location of a concentrated compression force.
Shim. Thin layer of material used to fill a space between faying or bearing surfaces.
16.1-Iiv
Shear yielding (punching). In an HSS connection, limit state based on out-of-plane
shear strength of the chord wall to which branch members are attached.
Shear yielding. Yielding that occurs due to shear.
Shear wall]. Wall that provides resistance to lateral loads in the plane of the wall and
provides stability for the structural system.
Shear rupture. Limit state of rupture (fracture) due to shear.
Shear connector strength. Limit state of reaching the strength of a shear connector, as
governed by the connector bearing against the concrete in the slab or by the tensile
strength of the connector.
Shear connector. Headed stud, channel, plate or other shape welded to a steel member
and embedded in concrete of a composite member to transmit shear forces at the
interface between the two materials.
Shear buckling. Buckling mode in which a plate element, such as the web of a beam,
deforms under pure shear applied in the plane of the plate.
Serviceability limit state. Limiting condition affecting the ability of a structure to
preserve its appearance, maintainability, durability or the comfort of its occupants or
function of machinery, under normal usage.
Service load]. Load under which serviceability limit states are evaluated.
Service load combination. Load combination under which serviceability limit states are
evaluated.
Seismic response modification coefficient. Factor that reduces seismic load effects to
strength level.
Second-order effect. Effect of loads acting on the deformed configuration of a structure;
includes P-b effect and P-A effect.
Second-order analysis. Structural analysis in which equilibrium conditions are formulated on the deformed structure; second-order effects (both P-8 and P-A, unless
specified otherwise) are included.
Safety factor, S2f. Factor that accounts for deviations of the actual strength from the
nominal strength, deviations of the actual load from the nominal had, uncertainties
in the analysis that transforms the load into a had effect, and for the manner and
consequences of failure.
Rupture strength. In a connection, strength limited by tension or shear rupture.
Rotation capacity. Incremental angular rotation that a given shape can accept prior to
excessive load shedding, defined as the ratio of the inelastic rotation attained to the
idealized elastic rotation at first yield.
GLOSSARY
compatibility
method.
Method for determining the stresses in a composite
!6.1-lv
limit state. Limiting condition affecting the safety of the structure, in which
concentration.
Localized stress considerably higher than average (even in
analysis].
Determination of load effects on m e m b e r s and connections
system.
Bridges.
HSS connection
in which the branch member
or connecting element is
tension.
M a x i m u m tension force that a m e m b e r is capable of
Yielding that occurs due to
(of member).
Plate element used to join two parallel components of a built-up
Tie plate.
column,
in a
GLOSSARY
buckling.
Steel reinforcement in the form of closed ties or welded
Yielding that occurs due to torsion.
reinforcement.
column.
mode in which a compression m e m b e r twists about its
column.
method. Procedure whereby the specified pretension in high-strength bolts
oriented perpendicular to the flanges, attached to the
load distribution.
In an HSS connection,
end. The end of a m e m b e r not restrained against rotation by stiffeners or
element.
system.
System of shear
nails,
braced frames
load.
steel. High-strength, low-alloy steel that, with suitable precautions, can be
Limit state of out-of-plane compression buckling of the w e b
buckling. Limit state of lateral buckling of the tension flange opposite the
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEF.L CONSTRUCTION. INC.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
location of a concentrated compression force.
Web sidesway
due to a concentrated compression force.
buckling.
Limit state of lateral instability of a web.
Web compression
Web buckling.
used in normal atmospheric exposures (not marine) without protective paint coating.
Weathering
Weak axis. Minor principal centroidal axis of a cross section.
through one or m o r e floors of a building.
bracing
Variable load]. Load not classified as permanent
Vertical
or both, extending
Flat compression element with an adjoining out-of-plane element
along one edge parallel to the direction of loading.
Unstiffened
connection elements.
Unframed
readily determined.
distributed through the cross section of connected elements in a m a n n e r that can be
Uneven
condition in which the load is not
length. Distance between braced points of a m e m b e r , measured between the
centers of gravity of the bracing m e m b e r s .
Unbraced
bolt has been snug tightened.
is controlled by rotating the fastener c o m p o n e n t a predetermined amount after the
Turn-of-nut
composite
stiffener. W e b stiffener
Tubing. See HSS.
web.
Transverse
in an encased concrete
wire fabric providing confinement for the concrete surrounding the steel shape core
Transverse
Torsional yielding.
Buckling
Bracing resisting twist of a beam or
shear center axis.
Torsional
Torsional bracing.
section fillet.
Toe of fillet. Junction of a fillet weld face and base metal. Tangent point of a rolled
16.1-lvi
Specification for Structural Steel Buildings, March 9, 2005
shear between them.
girder or strut rigidly connected to the parallel components and designed to transmit
cut. Cut with gas, plasma or laser.
Thermally
m a n n e r similar to a Pratt truss.
stiffeners
Behavior of a panel under shear in which diagonal tensile forces
develop in the web and compressive forces develop in the transverse
Tension field action.
tension and shear force.
Tension and shear rupture. In a bolt, limit state of rupture (fracture) due to simultaneous
Tensile yielding.
sustaining.
Tensile strength
Tensile strength (of material)].
M a x i m u m tensile stress that a material is capable of
sustaining as defined by A S T M .
Tensile rupture. Limit state of rupture (fracture) due to tension.
are primarily equilibriated by shear in the m a i n member.
perpendicular to the main member and in which forces transverse to the main m e m b e r
T-cotwection.
Standard
based
An assemblage of load-carrying c o m p o n e n t s that are joined together
and
to provide interaction or interdependence.
Structural
Practice for Steel Buildings
steel. Steel elements as defined in Section 2.1 of the A I S C Code of
Structural
Member, connector, connecting element or assemblage.
component].
Structural
on principles of structural mechanics.
Structural
Strong axis. Major principal centroidal axis of a cross section.
geometry or localized loading.
uniformly loaded cross sections of uniform thickness) due to abrupt changes in
Stress
Stress. Force per unit area caused by axial force, m o m e n t , shear or torsion.
the ultimate load-carrying capacity is reached.
Strength
with respect to the neutral axis of the cross section.
m e m b e r considering the stress-strain relationships of each materia! and its location
Strain
GLOSSARY
t6.t-lvii
Specification for Structural Steel Buildings. March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Yielding (yield moment). Yielding at the extreme fiber on the cross section of a member
when the bending moment reaches the yield moment.
Yielding (plastic moment). Yielding throughout the cross section of a member as the
bending moment reaches the plastic moment.
Yielding. Limit state of inelastic deformation that occurs after the yield stress is reached.
Yield stress}. Generic term to denote either yield point or yield strength, as appropriate
for the material.
Yield strength}. Stress at which a material exhibits a specified limiting deviation from
the proportionality of stress to strain as defined by ASTM.
Yield point}. First stress in a material at which an increase in strain occurs without an
increase in stress as defined by ASTM.
Yield moment. In a member subjected to bending, the moment at which the extreme
outer fiber first attains the yield stress.
Y-connection. HSS connection in which the branch member or connecting element is
not perpendicular to the main member and in which forces transverse to the main
member are primarily equilibriated by shear in the main member.
Weld root. See root of joint.
Weld metal. Portion of a fusion weld that has been completely melted during welding.
Weld metal has elements offillermetal and base metal melted in the weld thermal cycle.
GLOSSARY
16.1-10
CHAPTER B
DESIGN REQUIREMENTS
The general requirements for the analysis and design of steel structures that are applicable to all chapters of the specification are given in this chapter.
The chapter is organized as follows:
B1.
B2.
B3.
B4.
B5.
B6.
Bl.
General Provisions
Loads and Load Combinations
Design Basis
Classification of Sections for Local Buckling
Fabrication, Erection and Quality Control
Evaluation of Existing Structures
GENERAL PROVISIONS
The design of members and connections shall be consistent with the intended
behavior of the framing system and the assumptions made in the structural analysis. Unless restricted by the applicable building code, lateral load resistance and
stability may be provided by any combination of members and connections.
B2.
LOADS AND LOAD COMBINATIONS
The loads and load combinations shall be as stipulated by the applicable building
code. In the absence of a building code, the loads and load combinations shall be
those stipulated in SEI/ASCE 7. For design purposes, the nominal loads shall be
taken as the loads stipulated by the applicable building code.
User Note: ForLRFD designs, the load combinations in SEI/ASCE 7, Section
2.3 apply. For ASD designs, the load combinations in SEI/ASCE 7, Section
2.4 apply.
B3.
DESIGN BASIS
Designs shall be made according to the provisions for Load and Resistance Factor
Design (LRFD) or to the provisions for Allowable Strength Design (ASD).
1.
Required Strength
The required strength of structural members and connections shall be determined
by structural analysis for the appropriate load combinations as stipulated in Section B2.
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
DESIGN BASIS
16.1-11
required strength (LRFD)
nominal strength, specified in Chapters B through K
resistance factor, specified in Chapters B through K
design strength
R,< 5 $Rn
(B3-1)
Ra
=
R„ =
£2 =
R„/Q =
where
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
The overall structure and the individual members, connections, and connectors
shall be checked for serviceability. Performance requirements for serviceability
design are given in Chapter L.
Design for Serviceability
(a) Fully-Restrained (FR) Moment Connections
A fully-restrained (FR) moment connection transfers moment with a negligible
rotation between the connected members. In the analysis of the structure, the
connection may be assumed to allow no relative rotation. An FR connection
shall have sufficient strength and stiffness to maintain the angle between the
connected members at the strength limit states.
(b) Partially-Restrained (PR) Moment Connections
Partially-restrained (PR) moment connections transfer moments, but the rotation between connected members is not negligible. In the analysis of the structure, the force-deformation response characteristics of the connection shall be
included. The response characteristics of a PR connection shall be documented
in the technical literature or established by analytical or experimental means.
The component elements of a PR connection shall have sufficient strength,
stiffness, and deformation capacity at the strength limit states.
A moment connection transmits moment across the connection. Two types of
moment connections, FR and PR, are permitted, as specified below.
Moment Connections
A simple connection transmits a negligible moment across the connection. In the
analysis of the structure, simple connections may be assumed to allow unrestrained
relative rotation between the framing elements being connected. A simple connection shall have sufficient rotation capacity to accommodate the required rotation
determined by the analysis of the structure. Inelastic rotation of the connection is
permitted.
Simple Connections
User Note: Section 3.1.2 of the Code of Standard Practice addresses communication of necessary information for the design of connections.
Connection elements shall be designed in accordance with the provisions of Chapters J and K. The forces and deformations used in design shall be consistent with
the intended performance of the connection and the assumptions used in the structural analysis.
Design of Connections
Specification for Structural Steel Buildings. March 9, 2005
7.
Design for Stability
[Sect. B3
Stability of the structure and its elements shall be determined in accordance with
Chapter C.
DESIGN BASIS
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
(B3-2)
6b.
6a.
6.
5.
16.1-12
Specification for Structural Steel Buildings, March 9, 200.*i
required strength (ASD)
nominal strength, specified in Chapters B through K
safety factor, specified in Chapters B through K
allowable strength
Ra<R„/Q
Design shall be performed in accordance with Equation B3-2:
Design according to the provisions for Allowable Strength Design (ASD) satisfies
the requirements of this Specification when the allowable strength of each structural component equals or exceeds the required strength determined on the basis
of the ASD load combinations. All provisions of this Specification, except those
of Section B3.3, shall apply.
Design for Strength Using Allowable Strength Design (ASD)
R„ =
R„ =
0 =
<)>/?„ =
where
Design shall be performed in accordance with Equation B3-1:
Design according to the provisions for Load and Resistance Factor Design (LRFD)
satisfies the requirements of this Specification when the design strength of each
structural component equals or exceeds the required strength determined on the
basis of the LRFD load combinations. All provisions of this Specification, except
for those in Section B3.4, shall apply.
Design for Strength Using Load and Resistance Factor Design
(LRFD)
Design shall be based on the principle that no applicable strength or serviceability
limit state shall be exceeded when the structure is subjected to all appropriate load
combinations.
Limit States
Design by elastic, inelastic or plastic analysis is permitted. Provisions for inelastic
and plastic analysis are as stipulated in Appendix 1, Inelastic Analysis and Design.
The provisions for moment redistribution in continuous beams in Appendix 1,
Section 1.3 are permitted for elastic analysis only.
B3.]
DESIGN BASIS
16.1-13
[Seel. B3.
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEI. CONSTRUCTION, INC.
Sections are classified as compact, noncompact, or slender-element sections. For
a section to qualify as compact its flanges must be continuously connected to
the web or webs and the width-thickness ratios of its compression elements must
not exceed the limiting width-thickness ratios kp from Table B4.1. If the widththickness ratio of one or more compression elements exceeds kp, but does not
exceed K from Table B4.1, the section is noncompact. If the width-thickness ratio
of any element exceeds Ar, the section is referred to as a slender-element section.
CLASSIFICATION OF SECTIONS FOR LOCAL BUCKLING
User Note: Section J4.1(b) limits A„ to a maximum of 0.85Ag for splice
plates with holes.
Tn determining the net area across plug or slot welds, the weld metal shall not
be considered as adding to the net area.
For slotted HSS welded to a gusset plate, the net area, A„, is the gross area minus the product of the thickness and the total width of material that is removed
to form the slot.
For angles, the gage for holes in opposite adjacent legs shall be the sum of the
gages from the back of the angles less the thickness.
s = longitudinal center-to-center spacing (pitch) of any two consecutive
holes, in. (mm)
g = transverse center-to-center spacing (gage) between fastener gage lines,
in. (mm)
where
For a chain of holes extending across a part in any diagonal or zigzag line,
the net width of the part shall be obtained by deducting from the gross width
the sum of the diameters or slot dimensions as provided in Section J3.2, of all
holes in the chain, and adding, for each gage space in the chain, the quantity
s2/4g
In computing net area for tension and shear, the width of a bolt hole shall be
taken as l /i6 in. (2 mm) greater than the nominal dimension of the hole.
The net area, A„, of a member is the sum of the products of the thickness and
the net width of each element computed as follows:
b. Net Area
The gross area, AK, of a member is the total cross-sectional area.
a. Gross Area
Gross and Net Area Determination
DESIGN BASIS
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
B4.
13.
16.1-14
Specification for Structural Steel Buildings, March 9, 2005
The design wall thickness, t, shall be used in calculations involving the wall
thickness of hollow structural sections (HSS). The design wall thickness, t, shall
be taken equal to 0.93 times the nominal wall thickness for electric-resistancewelded (ERW) HSS and equal to the nominal thickness for submerged-arc-welded
(SAW) HSS.
Design Wall Thickness for HSS
Where corrosion may impair the strength or serviceability of a structure, structural
components shall be designed to tolerate corrosion or shall be protected against
corrosion.
Design for Corrosion Effects
User Note: Design by qualification testing is the prescriptive method specified
in most building codes. Traditionally, on most projects where the architect is
the prime professional, the architect has been the responsible party to specify
and coordinate fire protection requirements. Design by Engineering Analysis
is a new engineering approach to fire protection. Designation of the person(s)
responsible for designing for fire conditions is a contractual matter to be addressed on each project.
Nothing in this section is intended to create or imply a contractual requirement for
the engineer of record responsible for the structural design or any other member
of the design team.
Two methods of design for fire conditions are provided in Appendix 4, Structural Design for Fire Conditions: Qualification Testing and Engineering Analysis.
Compliance with the fire protection requirements in the applicable building code
shall be deemed to satisfy the requirements of this section and Appendix 4.
Design for Fire Conditions
Fatigue shall be considered in accordance with Appendix 3, Design for Fatigue,
for members and their connections subject to repeated loading. Fatigue need not
be considered for seismic effects or for the effects of wind loading on normal
building lateral load resisting systems and building enclosure components.
Design for Fatigue
See Appendix 2, Design for Ponding, for methods of checking ponding.
The roof system shall be investigated through structural analysis to assure adequate strength and stability under ponding conditions, unless the roof surface
is provided with a slope of 'A in. per ft (20 mm per meter) or greater toward
points of free drainage or an adequate system of drainage is provided to prevent
the accumulation of water.
Design for Ponding
B3.|
CLASSIFICATION OF SECTIONS FOR LOCAL BUCKLING
16.1-15
Description of
Element
0.45 y/Wry
0MjkeE/FrW
o.sey?/^:
0.95 A ; £ / F t "
\r
(noncompact)
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
0.36/EJT^
(compact)
Specification for Structural Steel Buildings, March 9, 2005
b/t
b/t
b/t
Limiting WldlhThlckness Ratios
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Uniform
compression in
flanges of rolled
I-shaped sections,
plates projecting
from rolled I-shaped
sections:
outstanding legs of
pairs of angles in
continuous contact
and flanges of
channels
Uniform
compression in
flanges of built-up
I-shaped sections
and plates or angle
legs projecting from
built-up I-shaped
sections
Uniform
compression in legs
of single angles,
legs of double
angles with
separators, and all
other unstiffened
elements
Flexure in legs of
single angles
Flexure in flanges of
doubly and singly
symmetric I-shaped
built-up sections
Flexure in flanges of
rolled I-shaped
sections and
channels
Width
Thickness
Ratio
TABLE B4.1
Limiting Width-Thickness Ratios for
Compression Elements
CLASSIFICATION OF SECTIONS FOR LOCAL BUCKLING
Specification for Structural Steel Buildings. March 9,2005
For tapered flanges of rolled sections, the thickness is the nominal value halfway
between the free edge and the corresponding face of the web.
User Note: Refer to Table B4.1 for the graphic representation of stiffened element dimensions.
(a) For webs of rolled or formed sections, h is the clear distance between flanges
less the fillet or corner radius at each flange; hc is twice the distance from the
centroid to the inside face of the compression flange less the fillet or corner
radius.
(b) For webs of built-up sections, h is the distance between adjacent lines of
fasteners or the clear distance between flanges when welds are used, and hr
is twice the distance from the centroid to the nearest line of fasteners at the
compression flange or the inside face of the compression flange when welds
are used; hp is twice the distance from the plastic neutral axis to the nearest
line of fasteners at the compression flange or the inside face of the compression
flange when welds are used.
(c) For flange or diaphragm plates in built-up sections, the width b is the distance
between adjacent lines of fasteners or lines of welds.
(d) For flanges of rectangular hollow structural sections (HSS), the width b is the
clear distance between webs less the inside comer radius on each side. For
webs of rectangular HSS, h is the clear distance between the flanges less the
inside corner radius on each side. If the corner radius is not known, b and b
shall be taken as the corresponding outside dimension minus three times the
thickness. The thickness, t, shall be taken as the design wall thicbiess, per
Section B3.12.
For stiffened elements supported along two edges parallel to the direction of the
compressi on force, the width shall be taken as follows:
Stiffened Elements
User Note: Refer to Table B4.1 for the graphic representation of unstiffened
element dimensions.
(a) Forflangesof I-shaped members and tees, the width b is one-half the full-flange
width, bf.
(b) For legs of angles and flanges of channels and zees, the width b is the full
nominal dimension.
(c) For plates, the width b is the distance from the free edge to the first row of
fasteners or line of welds.
(d) For stems of tees, d is taken as the full nominal depth of the section.
For unstiffened elements supported along only one edge parallel to the direction
of the compression force, the width shall be taken as follows:
Unstiffened Elements
B4]
0.11E/r>
0.31 E/Fy
NA
A,
(noncompact)
0.07 E/Fy
A,
(compact)
Example
B6.
Evaluation of Existing Structures.
Provisions for the evaluation of existing structures are presented in Appendix 5,
EVALUATION OF EXISTING STRUCTURES
Control.
meet the requirements stipulated in Chapter M, Fabrication, Erection, and Quality
Shop drawings, fabrication, shop painting, erection, and quality control shall
FABRICATION, ERECTION AND QUALITY CONTROL
M kc = - » - . but shall not be taken less than 0.35 nor greater than 0.76forcalculation purposes. (See
Cases 2 and 4)
M
FL = 0.7Fyforminor-axis bending, major axis bending of slender-web built-up I-shaped members, and
major axis bending of compact and noncompact web built-up I-shaped members with S^/S,e > 0.7; FL =
Fy$,tlS,c > 0.5FJ,formajor-axis bending of compact and noncompact web built-up I-shaped members with
&rt/S,c < 0.7. (See Case 2)
Circular hollow
sections
In uniform
compression
In flexure
Description ol
Element
Uniform
compression in all
other stiffened
elements
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
ywjEiFy
W/f/7&W)7J7ff/fA
T/J/&/S//»Af/S//&\' \
JfcsJ
Example
Llmltlng WidthThickness Ratios
Specification for Structural Steel Buildings, March 9. 2005
i.i2yE7*>
(0.54^-0.09)
5.70jE]Fy
0.75^EjFy
A,
(noncompact)
m/EjFy
Width
Thlck-
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
h/tw
d/t
(compact)
Limiting WidthThickness Ratios
TABLE B4.1 (cont.)
Limiting Width-Thickness Ratios for
Compression Elements
FABRICATION. ERECTION AND QUALITY CONTROL
Sperificationfor Structural Steel Buildinf-x, March 9, 2005
Flexure in webs of
rectangular HSS
Uniform
compression in
flanges of
rectangular box and
hollow structural
sections of uniform
thickness subject to
bending or
compression; flange
cover plates and
diaphragm plates
between lines of
fasteners or welds
Flexure in webs of
singly-symmetric
I-shaped sections
Uniform
compression in
webs of doubly
symmetric I-shaped
sections
Flexure in webs of
doubly symmetric
I-shaped sections
and channels
Uniform
compression in
stems of tees
Width
ThickDescription of
ness
Element
Ratio
Flexure in flanges of bit
tees
TABLE B4.1 (cont.)
Limiting Width-Thickness Ratios for
Compression Elements
CLASSIFICATION OF SECTIONS FOR LOCAL BUCKLING
2.
General Requirements
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Where elements are designed to function as braces to define the unbraced length
of columns and beams, the bracing system shall have sufficient stiffness and
strength to control member movement at the braced points. Methods of satisfying
User Note: Local buckling of cross section components can be avoided by the
use of compact sections defined in Section B4.
Individual member stability is provided by satisfying the provisions of Chapters
E, F, G, H and I.
Member Stability Design Requirements
In structures designed by inelastic analysis, the provisions of Appendix 1, Inelastic
Analysis and Design, shall be satisfied.
(1) Calculation of the required strengths for members, connections and other elements using one of the methods specified in Section C2.2, and
(2) Satisfaction of the member and connection design requirements in this specification based upon those required strengths.
In structures designed by elastic analysis, individual member stability and stability
of the structure as a whole are provided jointly by:
Stability shall be provided for the structure as a whole and for each of its elements.
Any method that considers the influence of second-order effects (including P-A
and P-h effects), flexural, shear and axial deformations, geometric imperfections,
and member stiffness reduction due to residual stresses on the stability of the
structure and its elements is permitted. The methods prescribed in this chapter and
Appendix 7, Direct Analysis Method, satisfy these requirements. All component
and connection deformations that contribute to the lateral displacements shall be
considered in the stability analysis.
STABILITY DESIGN REQUIREMENTS
1.
Stability Design Requirements
Calculation of Required Strengths
CI.
CI.
C2.
The chapter is organized as follows:
This chapter addresses general requirements for the stability analysis and design of members and frames.
STABILITY ANALYSIS AND DESIGN
CHAPTER C
20
[Sect. CI.
Specification for Structural Sleet Buildings, March 9, 2005
AMERICAN iNsrrrtTE OF STEEL CONSTRUCTION, INC.
Except as permitted in Section C2.2b, required strengths shall be determined using
a second-order analysis as specified in Section C2.1. Design by either secondorder or first-order analysis shall meet the requirements specified in Section C2.2.
CALCULATION OF REQUIRED STRENGTHS
Combined Systems
The analysis and design of members, connections and other elements in combined
systems of moment frames, braced frames, and/or shear walls and gravity frames
shall meet the requirements of their respective systems.
Gravity Framing Systems
Columns in gravity framing systems shall be designed based on their actual length
(AT = 1.0) unless analysis shows that a smaller value may be used. The lateral stability of gravity framing systems shall be provided by moment frames, braced
frames, shear walls, and/or other equivalent lateral load resisting systems. P-A
effects due to load on the gravity columns shall be transferred to the lateral load resisting systems and shall be considered in the calculation of the required strengths
of the lateral load resisting systems.
In frames where lateral stability is provided by the flexural stiffness of connected
beams and columns, the effective length factor K or elastic critical buckling stress,
Fe, for columns and beam-columns shall be determined as specified in Section C2.
Moment-Frame Systems
User Note: Knee-braced frames function as moment-frame systems and should
be treated as indicated in Section CI.3b. Eccentrically braced frame systems
function as combined systems and should be treated as indicated in Section
CI.3d.
In structures where lateral stability is provided solely by diagonal bracing, shear
walls, or equivalent means, the effective length factor, K, for compression members shall be taken as 1.0, unless structural analysis indicates that a smaller value
is appropriate. In braced-frame systems, it is permitted to design the columns,
beams, and diagonal members as a vertically cantilevered, simply connected truss.
Braced-Frame and Shear-Wall Systems
Lateral stability shall be provided by moment frames, braced frames, shear walls,
and/or other equivalent lateral load resisting systems. The overturning effects of
drift and the destabilizing influence of gravity loads shall be considered. Force
transfer and load sharing between elements of the framing systems shall be considered. Braced-frame and shear-wall systems, moment frames, gravity framing
systems, and combined systems shall satisfy the following specific requirements:
System Stability Design Requirements
this requirement are provided in Appendix 6, Stability Bracing for Columns and
Beams.
STABILITY DESIGN REQUIREMENTS
CALCULATION OF REQUIRED STRENGTHS
16.1
5, =
Pr = Pnl + B2P„
(C2-lb)
and
(Sect. (
= elastic critical buckling resistance of the member in the plane of bending,
calculated based on the assumption of zero sidesway, kips (N)
•jj-EI
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
SP e , = RM^1
A//
(C2-6b)
_
(C2-6a)
(K2L)2
For all types of lateral load resisting systems, it is permitted to use
E/>, _ E
For moment frames, where sidesway buckling effective length factors
K2 are determined for the columns, it is permitted to calculate the elastic
story sidesway buckling resistance as
ft, = - ^
(C2-5)
(KXL?
T,pe2 = elastic critical buckling resistance for the story determined by sidesway
buckling analysis, kips (N)
Pei
(C2-4)
where M\ and M2, calculated from a first-order analysis, are the
smaller and larger moments, respectively, at the ends of that portion
of the member unbraced in the plane of bending under consideration. M\IM2 is positive when the member is bent in reverse curvature,
negative when bent in single curvature,
(ii) For beam-columns subjected to transverse loading between supports,
the value of Cm shall be determined either by analysis or conservatively taken as 1.0 for all cases.
Cm = 0.6 - 0.4(M, JM2)
(i) For beam-columns not subject to transverse loading between supports
in the plane of bending,
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
a = 1.60 (ASD)
CALCULATION OF REQUIRED STRENGTHS
= required second-order flexural strength using LRFD or ASD load combinations, kip-in. (N-mm)
M„, = first-order moment using LRFD or ASD load combinations, assuming
there is no lateral translation of the frame, kip-in. (N-mm)
Mi, = first-order moment using LRFD or ASD load combinations caused by
lateral translation of the frame only, kip-in. (N-mm)
Pr = required second-order axial strength using LRFD or ASD load combinations, kips (N)
P„, = first-order axial force using LRFD or ASD load combinations, assuming
there is no lateral translation of the frame, kips (N)
£/>„, = total vertical load supported by the story using LRFD or ASD load combinations, including gravity column loads, kips (N)
Pi, = first-order axial force using LRFD or ASD load combinations caused by
lateral translation of the frame only, kips (N)
Cm = a coefficient assuming no lateral translation of the frame whose value
shall be taken as follows:
Mr
•2
Specification for Structural Steel Buildings, March 9, 2005
a = 1.00 (LRFD)
User Note: Note that the B2 amplifier (Equation C2-3) can be estimated in
preliminary design by using a maximum lateral drift limit corresponding to the
story shear HH in Equation C2-6b.
HPe2
%-— > 1
(C2-2)
]-aPr/Pel
For members subjected to axial compression, B\ may be calculated based on the
first-order estimate PF = P„, + Pi,.
User Note: Si is an amplifier to account for second order effects caused by
displacements between brace points (P-5) and B2 is an amplifier to account
for second order effects caused by displacements of braced points (P-A).
For members in which Bx < 1.05, it is conservative to amplify the sum of the
non-sway and sway moments (as obtained, for instance, by a first-order elastic
analysis) by the B2 amplifier, in other words, Mr = B2(M„, + Mtt).
where
The following is an approximate second-order analysis procedure for calculating
the required flexural and axial strengths in members of lateral had resisting
systems. The required second-order flexural strength, Mr, and axial strength, Pr,
shall be determined as follows:
Mr = B,Mnt + B2Mn
(C2-la)
User Note: A method is provided in this section to account for second-order
effects in frames by amplifying die axial forces and moments in members and
connections from a first-order analysis.
Second-Order Analysis by Amplified First-Order Elastic Analysis
The Amplified First-Order Elastic Analysis Method defined in Section C2.1b is
an accepted method for second-order elastic analysis of braced, moment, and
combined framing systems.
Any second-order elastic analysis method that considers both P- A and P-S effects
may be used.
General Second-Order Elastic Analysis
Second-order analysis shall conform to the requirements in this Section.
Methods of Second-Order Analysis
12.]
2a.
2.
CALCULATION OF REQUIRED STRENGTHS
[Sect. C2
aPr < 0.5P,
(C2-7)
Specification for Structural Steel Buildings, March 9. 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
N, = 2.1(A/L)r( > 0.0042^
(C2-8)
Y, = gravity load from the LRFD load combination or 1.6 times the ASD load
combination applied at level i, kips (N)
(2) All load combinations include an additional lateral load, M, applied in combination with other loads at each level of the structure, where
a = 1.0 (LRFD) a = 1.6 (ASD)
Pr = required axial compressive strength under LRFD or ASD load combinations, kips (N)
Pv = member yield strength (= AFy), kips (N)
where
(1) The required compressive strengths of all members whose flexural stiffnesses
are considered to contribute to the lateral stability of the structure satisfy the
following limitation:
Required strengths are permitted to be determined by a first-order analysis, with
all members designed using K = 1.0, provided that
Design by First-Order Analysis
(4) Where the ratio of second-order drift to first-order drift is less than or equal to
1.1, members are permitted to be designed using K — 1.0. Otherwise, columns
and beam-columns in moment frames shall be designed using a K factor or
column buckling stress, Fe, determined from a sidesway buckling analysis
of the structure. Stiffness reduction adjustment due to column inelasticity is
permitted in the determination of the K factor. For braced frames, K for
compression members shall be taken as 1.0, unless structural analysis indicates
a smaller value may be used.
User Note: The minimum lateral load of 0.002*1, in conjunction with the
other design-analysis constraints listed in this section, limits the error that
would otherwise be caused by neglecting initial out-of-plurnbness and member
stiffness reduction due to residual stresses in the analysis.
(3) All gravity-only load combinations shall include a minimum lateral load applied at each level of the structure of 0.002 F(, where K, is the design gravity
load applied at level /, kips (N). This minimum lateral load shall be considered
independently in two orthogonal directions.
User Note: The amplified first order analysis method of Section C2.1b incorporates the 1.6 multiplier directly in the #1 and B2 amplifiers, such that no
other modification is needed.
(2) For design by ASD, analyses shall be carried out under 1.6 times the ASD load
combinations and the results shall be divided by 1.6 to obtain the required
strengths.
CALCULATION OF REQUIRED STRENGTHS
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
24
Specification for Structural Steel Buildings. March 9, 2005
Where required strengths are determined by a second-order analysis:
(1) The provisions of Section C2.1 shall be satisfied.
Design by Second-Order Analysis
For the methods specified in Sections 2.2a or 2.2b:
(1) Analyses shall be conducted according to the design and loading requirements
specified in either Section B3.3 (LRFD) or Section B3.4 (ASD).
(2) The structure shall be analyzed using the nominal geometry and the nominal
elastic stiffness for all elements.
User Note: The ratio of second-order drift to first-order drift can be represented by B2, as calculated using Equation C2-3. Alternatively, the ratio can
be calculated by comparing the results of a second-order analysis to the results
of a first-order analysis, where the analyses are conducted either under LRFD
load combinations directly or under ASD load combinations with a 1.6 factor
applied to the ASD gravity loads.
These requirements apply to all types of braced, moment, and combined framing
systems. Where the ratio of second-order drift to first-order drift is equal to or less
than 1.5, the required strengths of members, connections and other elements shall
be determined by one of the methods specified in Sections C2.2a or C2.2b, or by
the Direct Analysis Method of Appendix 7. Where the ratio of second-order drift
to first-order drift is greater than 1.5, the required strengths shall be determined
by the Direct Analysis Method of Appendix 7.
Design Requirements
A H = first-order interstory drift due to lateral forces, in. (mm). Where An varies
over the plan area of the structure, A« shall be the average drift weighted
in proportion to vertical load or, alternatively, the maximum drift.
£ / / = story shear produced by the lateral forces used to compute A#, kips (N)
User Note: Methods for calculation of K2 are discussed in the Commentary.
E = modulus of elasticity of steel = 29,000 ksi (200 000 MPa)
RM = 1.0 for braced-frame systems;
= 0.85 for moment-frame and combined systems, unless a larger value is
justified by analysis
/ = moment of inertia in the plane of bending, in.4 (mm4)
L — story height, in. (mm)
K\ = effective length factor in the plane of bending, calculated based on the
assumption of no lateral translation, set equal to 1.0 unless analysis indicates that a smaller value may be used
K2 = effective length factor in the plane of bending, calculated based on a
sidesway buckling analysis
where
Sect. C2.1
16.1-25
User Note: The drift A is calculated under LRFD load combinations directly
or under ASD load combinations with a 1.6 factor applied to the ASD gravity
loads.
A/L = the maximum ratio of A to L for all stories in the structure
A
= first-order interstory drift due to the design loads, in. (mm). Where A
varies over the plan area of the structure, A shall be the average drift
weighted in proportion to vertical load or, alternatively, the maximum
drift.
L
= story height, in. (mm)
CALCULATION OF REQUIRED STRENGTHS
DESIGN OF MEMBERS FOR TENSION
CHAPTER D
Slenderness Limitations
Tensile Strength
Area Determination
Built-Up Members
Pin-Connected Members
Eyebars
SLENDERNESS LIMITATIONS
AMERICAN INSTITUTE OP STEEL CONSTRUCTION, INC.
Q, = 1.67 (ASD)
Specification for Structural Steel Buildings, March 9, 2005
to = 0.90 (LRFD)
P„ = F},Ag
(a) For tensile yielding in the gross section:
(D2-1)
The design tensile strength, §,Pn, and the allowable tensile strength, P„fClt, of
tension members, shall be the lower value obtained according to the limit states
of tensile yielding in the gross section and tensile rupture in the net section.
TENSILE STRENGTH
User Note: For members designed on the basis of tension, the slenderness ratio
L/r preferably should not exceed 300. This suggestion does not apply to rods
or hangers in tension.
There is no maximum slenderness limit for design of members in tension.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
D2.
Dl.
User Note: For cases not included in this chapter the following sections apply:
• B3.9
Members subject to fatigue.
•Chapter H
Members subject to combined axial tension and flexure.
•J3.
Threaded rods.
• J4.1
Connecting elements in tension.
• J4.3
Block shear rupture strength at end connections of tension
members.
Dl.
D2.
D3.
D4.
D5.
D6.
The chapter is organized as follows:
This chapter applies to members subject to axial tension caused by static forces acting
through the centroidal axis.
16.1-26
Specification for Structural Steel Buildings, March 9, 2005
(3) The non-sway amplification of beam-column moments is considered by applying the B\ amplifier of Section C2.1 to the total member moments.
This additional lateral load shall be considered independently in two orthogonal
directions.
Seel. C2.J
AREA DETERMINATION
Gross Area
|Scci.D3.
(D3-1)
AMERICAN INSTITUTE OF STEEI, CONSTRUCTION. INC.
J2, = 2.00 (ASD)
Specification for Structural Steel Buildings. March 9. 2005
4, = 0.75 (LRFD)
Pn =2tbeffFu
(a) For tensile rupture on the net effective area:
(D5-1)
The design tensile strength, ty,P„, and the allowable tensile strength, P„/^lt, of
pin-connected members, shall be the lower value obtained according to the limit
states of tensile rupture, shear rupture, bearing, and yielding.
Tensile Strength
PIN-CONNECTED MEMBERS
User Note: The longitudinal spacing of connectors between components should
preferably limit the slenderness ratio in any component between the connectors
to 300.
Either perforated cover plates or tie plates without lacing are permitted to be used
on the open sides of built-up tension members. Tie plates shall have a length not
less than two-thirds the distance between the lines of welds orfasteners connecting
them to the components of the member. The thickness of such tie plates shall not be
less than one-fiftieth of the distance between these lines. The longitudinal spacing
of intermittent welds or fasteners at tie plates shall not exceed 6 in. (150 mm).
For limitations on the longitudinal spacing of connectors between elements
in continuous contact consisting of a plate and a shape or two plates, see
Section J3.5.
BUILT-UP MEMBERS
Members such as single angles, double angles and WT sections shall have connections proportioned such that U is equal to or greater than 0.60. Alternatively, a
lesser value of U is permitted if these tension members are designed for the effect
of eccentricity in accordance with H1.2 or H2.
where U, the shear lag factor, is determined as shown in Table D3.1.
Ae = A„U
The effective area of tension members shall be determined as follows:
Effective Net Area
User Note: Section J4.1(b) limits A„ to a maximum of 0.85 Ag for splice plates
with holes.
In determining the net area across plug or slot welds, the weld metal shall not be
considered as adding to the net area.
AREA DETERMINATION
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
;
Specification for Structural Steel Buildings, March 9, 2005
For slotted HSS welded to a gusset plate, the net area, A„, is the gross area minus
the product of the thickness and the total width of material that is removed to form
the slot.
For angles, the gage for holes in opposite adjacent legs shall be the sum of the
gages from the back of the angles less the thickness.
s = longitudinal center-to-center spacing (pitch) of any two consecutive holes,
in. (mm)
g = transverse center-to-center spacing (gage) between fastener gage lines, in.
(mm)
where
For a chain of holes extending across a part in any diagonal or zigzag line, the net
width of the part shall be obtained by deducting from the gross width the sum of
the diameters or slot dimensions as provided in Section J3.2, of all holes in the
chain, and adding, for each gage space in the chain, the quantity s2/4g
In computing net area for tension and shear, the width of a bolt hole shall be taken
as Vie in. (2 mm) greater than the nominal dimension of the hole.
The net area, A„, of a member is the sum of the products of the thickness and the
net width of each element computed as follows:
Net Area
The gross area, AK, of a member is the total cross-sectional area.
AREA DETERMINATION
1.
When members without holes are fully connected by welds, the effective net area
used in Equation D2-2 shall be as defined in Section D3. When holes are present
in a member with welded end connections, or at the welded connection in the
case of plug or slot welds, the effective net area through the holes shall be used in
Equation D2-2.
D3.
2.
f2, = 2.00 (ASD)
(D2-2)
16.1-27
Ae = effective net area, in.2 (mm2)
Ag = gross area of member, in.2 (mm2)
Fy = specified minimum yield stress of the type of steel being used, ksi (MPa)
F„ — specified minimum tensile strength of the type of steel being used, ksi
(MPa)
where
4>, = 0.75 (LRFD)
Pn = F„Ae
(b) For tensile rupture in the net section:
Sect.D3.]
„
with two side gusset / >
plates
with a single concentric gusset plate
H...U : 1 - * / ,
B 2 +2BH
I >-\.3D...U=-\.0
0</<1.3D...U=1-*//
• 3
Example
AMERICAN INSTITUTE O,F STEEL CONSTRUCTION, INC.
Specification for Structural Steel Buildings, March 9, 2005
with flange con- bi > 2/3rf... U = 0.90
nected with 3 CM bt < 2 / 3 r f . . . U = u.85
more fasteners per
line in direction
loading
with web connected
with 4 or more fasteners per line in the
direction of loading
Single angles
with 4 or more fas{If U is calculated teners per line in diper Case 2, the rection of loading
larger value is per- with 2 or 3 fasteners
mitted to be used)
per line in the direction of loading
/ = length of connection, in. (mm); w= plate width, in. (mm); H- connection eccentricity, in. (mm); B- overall
width of rectangular HSS member, measured 90 degrees to the plane of the connection, in. (mm); H = overall
height ot rectangular HSS member, measured in the plane of the connection, in. (mm)
W, M, S or HP
Shapes or Tees cut
from these shapes.
(If U is calculated
per Case 2, the
larger value is permitted to be used)
Rectangular HSS
Round HSS with a single concentric gusset plate
and
An = area of the directly
connected elements
Plates where the tension load is transmit- / > 2 w . . . U = 1 . 0
ted by longitudinal welds only.
2 w > / > 1 . 5 w . . . ( 7 = 0.87
1 . 5 w > / > w . . . t y = 0.75
Shear Lag Factor, U
TABLE D3.1
Shear Lag Factors for Connections
to Tension Members
PIN-CONNECTED MEMBERS
Description of Element
All tension members where the tension
load is transmitted directly to each of
cross-sectional elements by fasteners or
welds, (except as in Cases 3,4, 5 and 6)
All tension members, except plates and
HSS, where the tension load is transmitted to some but not all of the crosssectional elements by fasteners or longitudinal welds (Alternatively, for W, M, S and
HP, Case 7 may be used.)
All tension members where the tension
load is transmitted by transverse welds
to some but not all of the cross-sectional
elements.
| Case I
App.DS]
30
Qsf = 2.00 (ASD)
(D5-2)
[Sect. D.V
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
Specification for Structural Steel Buildings, March 9. 200S
Eyebars shall be of uniform thickness, without reinforcement at the pin holes, and
have circular heads with the periphery concentric with the pin hole.
Dimensional Requirements
For calculation purposes, the width of the body of the eyebars shall not exceed
eight times its thickness.
The available tensile strength of eyebars shall be determined in accordance with
Section D2, with A^ taken as the cross-sectional area of the body.
Tensile Strength
EYEBARS
The corners beyond the pin hole are permitted to be cut at 45° to the axis of the
member, provided the net area beyond the pin hole, on a plane perpendicular to
the cut, is not less than that required beyond the pin hole parallel to the axis of the
member.
The width of the plate at the pin hole shall not be less than 2beg + d and the
minimum extension, a, beyond the bearing end of the pin hole, parallel to the axis
of the member, shall not be less than 1.33 x beg.
The pin hole shall be located midway between the edges of the member in the
direction normal to the applied force. When the pin is expected to provide for
relative movement between connected parts while under full load, the diameter
of the pin hole shall not be more than lhi in. (1 mm) greater than the diameter of
the pin.
Dimensional Requirements
(c) For bearing on the projected area of the pin, see Section J7.
(d) For yielding on the gross section, use Equation D2-1.
where
Asf = 2t(a + d/2), in. : (mnr)
a = shortest distance from edge of the pin hole to the edge of the member
measured parallel to the direction of the force, in. (mm)
beff= 2t +0.63, in. ( = It + 16, mm) but not more than the actual distance
from the edge of the hole to the edge of the part measured in the direction
normal to the applied force
d = pin diameter, in. (mm)
t = thickness of plate, in. (mm)
$sf = 0.75 (LRFD)
P„ = Q.6F„Asf
(b) For shear rupture on the effective area:
PIN-CONNECTED MEMBERS
General Provisions
Slenderness Limitations and Effective Length
Compressive Strength for Flexural Buckling of Members without Slender
Elements
Compressive Strength for Torsional and Flexural-Torsional Buckling of
Members without Slender Elements
Single Angle Compression Members
Built-Up Members
Members with Slender Elements
GENERAL PROVISIONS
S2C = 1.67 (ASD)
Specification for Structural Steel Buildings. March 9. 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
The effective length factor, K, for calculation of column slenderness, KL/r, shall
be determined in accordance with Chapter C,
SLENDERNESS LIMITATIONS AND EFFECTIVE LENGTH
fc. = 0.90 (LRFD)
(a) For doubly symmetric and singly symmetric members the limit state of flexural
buckling is applicable.
(b) For singly symmetric and unsymmetric members, and certain doubly symmetric members, such as cruciform or built-up columns, the limit states of torsional
or flexural-torsional buckling are also applicable.
The nominal compressive strength, P„, shall be the lowest value obtained according to the limit states of flexural buckling, torsional buckling mdflexural-torsional
buckling.
The design compressive strength, $CP„, and the allowable compressive strength,
P„f£lc* are determined as follows:
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
E2.
El.
User Note: For members not included in this chapter the following sections apply:
• HI. - H3. Members subject to combined axial compression and flexure.
• H4.
Members subject to axial compression and torsion.
•J4.4
Compressive strength of connecting elements.
•12.
Composite axial members.
E5.
E6.
E7.
E4.
E1.
E2.
E3.
The chapter is organized as follows:
This chapter addresses members subject to axial compression through the centroidal
axis.
Specification for Structural Steel Buildings, March 9, 2005
A thickness of less than l/i in. (13 mm) is permissible only if external nuts are
provided to tighten pin plates and filler plates into snug contact. The width from
the hole edge to the plate edge perpendicular to the direction of applied load
shall be greater than two-thirds and, for the purpose of calculation, not more than
three-fourths times the eyebar body width.
For steels having F v greater than 70 ksi (485 MPa), the hole diameter shall not
exceed five times the plate thickness, and the width of the eyebar body shall be
reduced accordingly.
DESIGN OF MEMBERS FOR COMPRESSION
The pin diameter shall not be less than seven-eighths times the eyebar body width,
and the pin hole diameter shall not be more than l/n in. (1 mm) greater than the
pin diameter.
16.1-31
CHAPTER E
EYEBARS
The radius of transition between the circular head and the eyebar body shall not
be less than the head diameter.
Sect. D6.1
E3.
COMPRESSIVE STRENGTH FOR FLEXURAL BUCKLING
16.1-33
fv
(or Fe < 0.44Fv)
r = [O.fi6 5 8 *
(or Fe > 0.44FV)
(E3-3)
(E3-2)
(E3-1)
AF„FmH
(Fm + F„
(E4-2)
(E4-1)
(E4-3)
•n-EC„,
]zh
rr + GJ
4F„F,zH
(F„ + Fel
(E4-5)
(E4-4)
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
=0
Specification for Structural Steel Buildings, March 9. 2005
-F;(F,-F„)(y\
(F, - F„)(F, - F„)(F, - F„) - F;(.Fe - F,y)
(y\
(E4-6)
(iii) For unsymmetric members, Fe is the lowest root of the cubic equation:
i*m
(ii) For singly symmetric members where v is the axis of symmetry:
F,--
(i) For doubly symmetric members:
(b) For all other cases, Fcr shall be determined according to Equation E3-2 or E3-3,
using the torsional or flexural-torsional elastic buckling stress, Fe, determined
as follows:
GJ
A„7~
where Fcry is taken as Fcr from Equation E3-2 or E3-3, for flexural buckling
KL KL
about the y-axis of symmetry and — = — , and
r
r.
-(*#*)[-
(a) For double-angle and tee-shaped compression members;
P„ = F„A,
The nominal compressive strength, P„, shall be determined based on the limit
states of flexural'-torsional and torsional buckling, as follows:
This section applies to singly symmetric and unsymmetric members, and certain
doubly symmetric members, such as cruciform or built-up columns with compact
and noncompact sections, as defined in Section B4 for uniformly compressed
elements. These provisions are not required for single angles, which are covered
in Section E5.
COMPRESSIVE STRENGTH FOR TORSIONAL AND
FLEXURAL-TORSIONAL BUCKLING OF MEMBERS
WITHOUT SLENDER ELEMENTS
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
E4.
COMPRESSIVE STRENGTH FOR TORS. AND FLEX.-TORS. BUCKLING
Specification for Structural Steel Buildings, March 9,2005
User Note: The two equations for calculating die limits and applicability of
Sections E3(a) and E3(b), one based on KUr and one based on Fe, provide the
same result.
•n E
2
Fe = elastic critical buckling stress determined according to Equation E3-4,
Section E4, or the provisions of Section C2, as applicable, ksi (MPa)
KL
E
(b) When — > 4.71 / —
r
V Fv
KL
[~E
(a) When — < 4.71 / —
r
y Fy
The flexural buckling stress, Fcr, is determined as follows:
Pn = FcrAg
The nominal compressive strength, P„, shall be determined based on the limit state
of flexural buckling.
User Note: When the torsional unbraced length is larger than the lateral
unbraced length, this section may control the design of wideflangeand similarly
shaped columns.
This section applies to compression members with compact and noncompact sections, as defined in Section B4, for uniformly compressed elements.
COMPRESSIVE STRENGTH FOR FLEXURAL BUCKLING
OF MEMBERS WITHOUT SLENDER ELEMENTS
User Note: For members designed on the basis of compression, the slenderness ratio KUr preferably should not exceed 200.
where
L = laterally unbraced length of the member, in. (mm)
r = governing radius of gyration, in. (mm)
K = the effective length factor determined in accordance with Section C2
Seci. E3.]
E5.
= ("
•)*J
." + GJI — ^
ffl
m
Tt e
2
x; + .y;
A.
= gross area of member, in.2 (mm2)
= warping constant, in.6 (mm6)
,
/ , + /,
(E4-11)
(E4-10)
(E4-9)
(E4-8)
SINGLE ANGLE COMPRESSION MEMBERS
[Sect. 1
— > 80:
rx
— =72 + 0.75r
rx
0 < — < 80:
rx
(E5-1)
KL
L
— = 6 0 + 0.8—
(E5-3)
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
AMERICAN INSTITUTE OF STEEI, CONSTRUCTION, INC.
L = length of member between work points at truss chord centerlines,
in. (mm)
bi = longer leg of angle, in. (mm)
bs = shorter leg of angle, in. (mm)
r, = radius of gyration about geometric axis parallel to connected leg,
in. (mm)
rz = radius of gyration for the minor principal axis, in. (mm)
(c) Single angle members with different end conditions from those described in
Section E5(a) or (b), with leg length ratios greater than 1.7, or with transverse
loading shall be evaluated for combined axial load and flexure using the provisions of Chapter H. End connection to different legs on each end or to both
where
— > 75:
rx
rx
L
0 < — < 75:
KL
L
— = 45 + — < 200
(E5-4)
r
rx
For unequal-leg angles with leg length ratios less than 1.7 and connected
through the shorter leg, KUr from Equations E5-3 and E5-4 shall be increased by adding 6[(bi/b,)2 - \], but KUr of the member shall not be less
than 0.82/Vr;,
(ii) When
(i)When
(b) For equal-leg angles or unequal-leg angles connected through the longer leg
that are web members of box or space trusses with adjacent web members
attached to the same side of the gusset plate or chord:
KL
L
— = 32 + 1.25— < 200
(E5-2)
r
rx
For unequal-leg angles with leg length ratios less than 1.7 and connected
through the shorter leg, KUr from Equations E5-1 and E5-2 shall be increased
by adding 4[(fc//fr,)2 - 1], but KUr of the members shall not be less than
0.95L/r : .
(ii) When
(i) When
(a) For equal-leg angles or unequal-leg angles connected through the longer leg
that are individual members or are web members of planar trusses with adjacent
web members attached to the same side of the gusset plate or chord:
*6
Specification for Structural Steel Buildings, March 9, 2005
The effects of eccentricity on single angle members are permitted to be neglected
when the members are evaluated as axially loaded compression members using
one of the effective slenderness ratios specified below, provided that: (1) members
are loaded at the ends in compression through the same one leg; (2) members are
attached by welding or by minimum two-bolt connections; and (3) there are no
intermediate transverse loads.
The nominal compressive strength, Pn> of single angle members shall be determined in accordance with Section E3 or Section E7, as appropriate, for axially
loaded members, as well as those subject to the slenderness modification of Section
E5(a) or E5(b), provided the members meet the criteria imposed.
SINGLE ANGLE COMPRESSION MEMBERS
User Note: For doubly symmetric I-shaped sections, Cw may be taken as
lyhgIA, where h0 is the distance between flange centroids, in lieu of a more
precise analysis. For tees and double angles, omit term with Cw when computing Fez a n d tate x0 as 0.
= shear modulus of elasticity of steel = 11,200 ksi
(77 200 MPa)
Ix, / ¥ = moment of inertia about the principal axes, in.4 (mm4)
J
= torsional constant, in.4 (mm4)
Kz
— effective length factor for torsional buckling
x„, y0 ~ coordinates of shear center with respect to the centroid, in. (mm)
r0
= polar radius of gyration about the shear center, in. (mm)
rv
= radius of gyration about y-axis, in. (mm)
G
F„
Af.
Cw
SINGLE ANGLE COMPRESSION MEMBERS
BUILT-UP MEMBERS
16.1-37
Compressive Strength
(E6-1)
)m
—columnslendernessofbuilt-upmemberactingasaunitin thebuckling direction being considered
= distance between connectors, in. (mm)
= minimum radius of gyration of individual component, in. (mm)
= radius of gyration of individual component relative to its centroidal
axis parallel to member axis of buckling, in. (mm)
= separation ratio = h/2ru,
= distance between centroids of individual components perpendicular
to the member axis of buckling, in. (mm)
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Specification for Structural Steel Buildings, March 9, 2005
Individual components of compression members composed of two or more shapes
shall be connected to one another at intervals, a, such that the effective slenderness
Dimensional Requirements
(b) The nominal compressive strength of buiit-up members composed of two or
more shapes or plates with at least one open side interconnected by perforated
cover plates or lacing with tie plates shall be determined in accordance with
Sections E3, E4, or E7 subject to the modification given in Section E6.!(a).
a
h
I— J
"
a
r;
rih
r
— | = modified column slenderness of built-up member
where
(-W(T)/^(0
(ii) For intermediate connectors that are welded or pretensioned bolted:
(").-/(")>©-
(i) For intermediate connectors that are snug-tight bolted:
(a) The nominal compressive strength of built-up members composed of two or
more shapes that are interconnected by bolts or welds shall be determined in
accordance with Sections E3, E4, or E7 .subject to the following modification.
In lieu of more accurate analysis, if the buckling mode involves relative deformations that produce shear forces in the connectors between individual shapes,
KUr is replaced by {KUr),„ determined as follows:
BUILT-UP MEMBERS
1.
legs, the use of single bolts or the attachment of adjacent web members to opposite sides of the gusset plate or chord shall constitute different end conditions
requiring the use of Chapter H provisions.
E6.
Sect. E6.1
16.1-38
(Seel. E6.
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
User Note: It is conservative to use the limiting width/thickness ratio for
Case 14 in Table B4.1 with the width, b, taken as the transverse distance
between the nearest lines of fasteners. The net area of the plate is taken at the
widest hole. In lieu of this approach, the limiting width thickness ratio may be
determined through analysis.
Open sides of compression members built up from plates or shapes shall be provided with continuous cover plates perforated with a succession of access holes.
The unsupported width of such plates at access holes, as defined in Section B4,
is assumed to contribute to the available strength provided the following requirements are met:
(1) The width-thickness ratio shall conform to the limitations of Section B4.
Along the length of built-up compression members between the end connections
required above, longitudinal spacing for intermittent welds or bolts shall be adequate to provide for the transfer of the required forces. For limitations on the
longitudinal spacing of fasteners between elements in continuous contact consisting of a plate and a shape or two plates, see Section J3.5. Where a component of
a built-up compression member consists of an outside plate, the maximum spacing shall not exceed the thickness of the thinner outside plate times 0.75 v/E/Fv,
nor 12 in. (305 mm), when intermittent welds are provided along the edges of
the components or when fasteners are provided on all gage lines at each section.
When fasteners are staggered, the maximum spacing on each gage line shall not
exceed the thickness of the thinner outside plate times \.\2y/E/Fv nor 18 in.
(460 mm).
At the ends of built-up compression members bearing on base plates or milled
surfaces, all components in contact with one another shall be connected by a weld
having a length not less than the maximum width of the member or by bolts spaced
longitudinally not more than four diameters apart for a distance equal to 1'/; times
the maximum width of the member.
User Note: It is acceptable to design a bolted end connection of a built-up
compression member for the full compressive load with bolts in shear and bolt
values based on bearing values; however, the bolts must be pretensioned. The
requirement forClass A or B faying surfaces is not intended for the resistance of
the axial force in the built-up member, but rather to prevent relative movement
between the components at the end as the built-up member takes a curved
shape.
ratio Kalr, of each of the component shapes, between \\\z fasteners, does notexceed
three-fourths times the governing slenderness ratio of the buiit-up member. The
least radius of gyration, rt, shall be used in computing the slenderness ratio of
each component part. The end connection shall be welded or pretensioned bolted
with Class A or B faying surfaces.
BUILT-UP MEMBERS
E7.
MEMBERS WITH SLENDER ELEMENTS
16.1-39
(E7-J)
(E7-3)
(E7-2)
(iii) When b/l>l
mjE/F,
^
8. = 1.415- 0.74 ( T ) / f
1.03JE/K
Q, = 1.0
(ii) WhenQ.56jEjF~y < bft <
b
\~E
( i ) W h e n - < 0.56 / —
t
y FT
(E7-6)
(E7-4)
(a) For flanges, angles, and plates projecting from rolled columns or other compression members:
The reduction factor Qs for slender unstiffened elements is defined as follows:
Slender Unstiffened Elements, Qt
User Note: For cross sections composed of only unstiffened slender elements,
Q = Qs {Qa — 1-0). For cross sections composed of only stiffened slender
elements, Q = Qa (Qs ~ 1.0). For cross sections composed of both stiffened
and unstiffened slender elements, Q = Qs Qa.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
buckling.
Fcr = 0.877 Fe
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
P„ = FrrAg
flexural-torsional
Fv
Specification for Structural Steel Buildings, March 9, 2005
and
ft
(or Fc < 0MQFv)
Frr-Q\0.658
(or F, > 0MQFy)
Fe = elastic critical buckling stress, calculated using Equations E3-4 and
E4-4 for doubly symmetric members. Equations E3-4 and E4-5 for singly
symmetric members, and Equation E4-6 for unsymmetric members, except for single angles where Fe is calculated using Equation E3-4.
Q = 1.0 for members with compact and noncompact sections, as defined in
Section B4, for uniformly compressed elements
= Qs Qa for members with slender-element sections, as defined in Section
B4, for uniformly compressed elements.
(b)When — > 4.71
KL
(a) W h e n — < 4.71 ,
MEMBERS WITH SLENDER ELEMENTS
Specification for Structural Steel Buildings, March 9, 2005
states of ftexural, torsional
The nominal compressive strength, P„, shall be determined based on the limit
This section applies to compression members with slender sections, as defined in
Section B4 for uniformly compressed elements.
MEMBERS WITH SLENDER ELEMENTS
For additional spacing requirements, see Section J3.5.
User Note: The inclination of lacing bars to the axis of the member shall
preferably be not less than 60° for single lacing and 45° for double lacing.
When the distance between the lines of welds or fasteners in the flanges is
more than 15 in. (380 mm), the lacing shall preferably be double or be made
of angles.
Lacing, including flat bars, angles, channels, or other shapes employed as lacing,
shall be so spaced that the Llr ratio of the flange included between their connections shall not exceed three-fourths times the governing slenderness ratio for the
member as a whole. Lacing shall be proportioned to provide a shearing strength
normal to the axis of the member equal to 2 percent of the available compressive
strength of the member. The Llr ratio for lacing bars arranged in single systems
shall not exceed 140. For double lacing this ratio shall not exceed 200. Double
lacing bars shall be joined at the intersections. For lacing bars in compression, / is
permitted to be taken as the unsupported length of the lacing bar between welds
or fasteners connecting it to the components of the built-up member for single
lacing, and 70 percent of that distance for double lacing.
As an alternative to perforated cover plates, lacing with tie plates is permitted at
each end and at intermediate points if the lacing is interrupted. Tie plates shall
be as near the ends as practicable. In members providing available strength, the
end tie plates shall have a length of not less than the distance between the lines of
fasteners or welds connecting them to the components of the member. Intermediate
tie plates shall have a length not less than one-half of this distance. The thickness
of tie plates shall be not less than one-fiftieth of the distance between lines of
welds or fasteners connecting them to the segments of the members. In welded
construction, the welding on each line connecting a tie plate shall total not less
than one-third the length of the plate. In bolted construction, the spacing in the
direction of stress in tie plates shall be not more than six diameters and the tie
plates shall be connected to each segment by at least three fasteners.
(2) The ratio of length (in direction of stress) to width of hole shall not exceed two.
(3) The clear distance between holes in the direction of stress shall be not less than
the transverse distance between nearest lines of connecting fasteners or welds.
(4) The periphery of the holes at all points shall have a minimum radius of 1V: in.
(38 mm).
Seer. E7.1
Q, = 10
Q,-
•©"
\t)\Ek,
(E7-9)
(E7-8)
(E7-7)
Q, = 1.0
''©'
e. = -
0.9l/Ej%
ei ., 3 4-0,6(^)y5
< b/t < 0.91 /EjF\.
Slender Stiffened Elements, Qa
A
(E7-16)
fr
(E7-17)
except
0.38
/£
"(»/f)V/
AMERICAN INSTITUTE OF STEKL CONSTRUCT ION, INC.
•fr
(b) For flanges of square and rectangular slender-element sections of uniform thick.. b
/ is taken as Fcr with Fcr calculated based on Q -
where
"(fr/oy/.
0.34 / ¥
flanges of square and rectangular sections of uniform thickness:
(a) For uniformly compressed slender elements, with - > 1.49,
The reduced effective width, be, is determined as follows:
A = total cross-sectiona! area of member, in.2 (mm2)
Aeff= summation of the effective areas of the cross section based on the
reduced effective width, bet in.- (mm2)
where
*
Specification for Structural Steel Buildings, March 9, 2005
(E7-13)
(E7-15)
The reduction factor, Q„, for slender stiffened elements is defined as follows:
AMERICAN INSTITUTE OF STEE1. CONSTRUCTION. INC.
Q, = 1 0
(E7-12)
(E7-1I)
(E7-10)
0.69E
(E7-14)
b = width of unstiffened compression element, as defined in Section B4,
in. (mm)
d = the full nominal depth of tee, in. (mm)
t = thickness of element, in. (mm)
where
e, =
Si
rf/,<1.03/-
Q, = 1.908- 1.22
:
(iii) When d/t > 1.03 i
(ii) When 0.75 l
MEMBERS WITH SLENDER ELEMENTS
Specification for Structural Steel Buildings, March 9, 2(105
d
[E~
( i ) W h e n - < 0.75 / —
(d) For stems of tees
^ = full width of longest angle leg, in. (mm)
where
(iii) When bit >
(ii) When 0.45 JE/K
b
j~E~
( i ) W h e n - < 0.45 / —
I
V Fv
(c) For single angles
calculation purposes
•Jhju,
kc = — : = = , and shall not be taken less than 0.35 nor greater than 0.76 for
(iii) When ft/' > 1-17.
1k~c
Q, = 1.415
Ek~
Ekr
(ii) When 0.64 —1 < b/t < 1.17 / — -
b
\Ekc
( i ) W h e n - < 0.64 — t
V Fv
(b) For flanges, angles, and plates projecting from built-up columns or other compression members:
MEMBERS WITH SLENDER ELEMENTS
16.1
2
»=™v7j + 3
General Provisions
Doubly Symmetric Compact I-Shaped Members and Channels Bent about
Their Major Axis
Doubly Symmetric I-Shaped Members with Compact Webs and Noncompact or Slender Flanges Bent about Their Major Axis
Other I-Shaped Members with Compact or Noncompact Webs Bent about
Their Major Axis
Doubly Symmetric and Singly Symmetric I-Shaped Members with
Slender Webs Bent about Their Major Axis
I-Shaped Members and Channels Bent about Their Minor Axis
Square and Rectangular HSS and Box-Shaped Members
Round HSS
Tees and Double Angles Loaded in the Plane of Symmetry
Single Angles
Rectangular Bars and Rounds
Unsymmetrical Shapes
Proportions of Beams and Girders
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STFEI. CONSTRUCTION, INC.
For guidance in determining the appropriate sections of this chapter to apply, Table
User Note Fl.l may be used.
Members subject to biaxial flexure or to combined flexure and
axial force.
• H4.
Members subject toflexureand torsion.
• Appendix 3. Members subject to fatigue.
• Chapter G.
Design provisions for shear.
• H1-H3.
User Note: For members not included in this chapter the following sections apply:
F6.
F7.
F8.
F9.
F10.
Fl 1.
F12.
F13.
F5.
F4.
F3.
Fl.
F2.
The chapter is organized as follows:
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
"
(E7 19>
This chapter applies to members subject to simple bending about one principal axis. For
simple bending, the member is loaded in a plane parallel to a principal axis that passes
through the shear center or is restrained against twisting at load points and supports.
DESIGN OF MEMBERS FOR FLEXURE
CHAPTER F
Specification for Structural Steel Buildings, March 9, 2005
where
D = outside diameter, in. (mm)
t = wall thickness, in. (mm)
G= e
0.038E
User Note: In lieu of calculating / = PJA^r, which requires iteration, /
may be taken equal to Fy. This will result in a slightly conservative estimate
of column capacity.
where
MEMBERS WITH SLENDER ELEMENTS
(c) For axially-loaded circular sections:
E D
E
When 0.11— < - < 0.45—
F,
t
Fy
E7.]
\u
•i
t__ /
1 -
~TT~T
ex
I I I
1
I •
-
_.. —
-
13-
Cross Section
i
•
N/A
N/A
N/A
N/A
N/A
N/A
CNC
N/A
s
CNC
c
C, NC, S
N/A
C.NC.S
CNC.S
C.NC.S
C.NCS
NC, S
Flange
Web
Slenderness Slenderness
0
c
LTB, FLB
Limit
States
Y, LTB
16.1-45
Y.LTB
Y, LTB, LLB
Y, LTB, FLB
Y, LB
Y, FLB, WLB
Y.FLB
Y, LTB, FLB, TFY
Y, LTB, FLB, TFY
TABLE User Note F1.1
Selection Table for the Application
of Chapter F Sections
GENERAL PROVISIONS
GENERAL PROVISIONS
[Sect. Ft.
Qh = 1.67 (ASD)
1 2 ^ 5 ^
2.5Mmax + 3A/,,+4A/ f l +3A/c
<
curvature
bending
= moment of inertia about the principal y-axis, in.4 (mm4)
— moment of inertia about y-axis referred to the compression flange,
or if reverse curvature bending, referred to the smaller flange, in.4
(mm4)
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
In singly symmetric members subjected to reverse curvature bending, the lateraltorsionai buckling strength shall be checked for both flanges. The available flexural strength shall be greater than or equal to the maximum required moment
causing compression within the flange under consideration.
Iy
Iyc
= 0.5 + 2 I -^- \ , singly symmetric members subjected to reverse
Mmax = absolute value of maximum moment in the unbraced segment,
kip-in. (N-mm)
MA — absolute value of moment at quarter point of the unbraced segment,
kip-in. (N-mm)
MB = absolute value of moment at centerline of the unbraced segment,
kip-in. (N-mm)
Mc = absolute value of moment at three-quarter point of the unbraced
segment, kip-in. (N-mm)
Rm = cross-section monosymmetry parameter
= 1.0, doubly symmetric members
= 1.0, singly symmetric members subjected to single curvature
bending
where
=
Ch = lateral-torsionai buckling modification factor for nonuniform moment
diagrams when both ends of the unsupported segment are braced
The following terms are common to the equations in this chapter except where
noted:
and the nominal flexural strength, M„, shall be determined according to Sections F2 through F12.
(2) The provisions in this chapter are based on the assumption that points of
support for beams and girders are restrained against rotation about their longitudinal axis.
<t»h = 0.90 (LRFD)
(1) For all provisions in this chapter
The design flexural strength, <tv,M„, and the allowable flexural strength, M„/Qh,
shall be determined as follows:
GENERAL PROVISIONS
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
Fl.
16.1-46
Specification for Structural Steel Buildings, March 9, 2005
Unsymmetrlcal shapes
F12
N/A
N/A
All limit states
Y = yielding, LTB = lateral-torsionai budding, FLB = flange local buckling, WLB = web local buckling,
TFY = tensnnflangeyielding, LL6 = leg local budding, LB = local budding, C = compact, NC = noncompact,
S = slender
F11
F10
F9
F8
F7
F6
F5
F4
F3
Section
In
Chapter F
F2
Sect. F 1.1
2.
1.
F2.
Sect. F2.
16.1-47
M„ = M P =FyZx
(F2-1)
M„ = F„SX < Mp
~ M"
(F2-3)
(F2-2)
Fcr =
Sxho
+0.078^- ( ^ ) "
Specification for Structural Steel Buildings. March 9. 2(X)5
AMERICAN INSTITCTK OF STEEL CONSTRtCTION, INC.
m
^£JX
(F2-4)
Lh = length between points that are either braced against lateral displacement
of compression flange or braced against twist of the cross section, in.
(mm)
where
(c) When Lb > Lr
Mn = Ch \MP - (Mp - 0.7FySx) (LLbZLL")]
(a) When Lh < Lp, the limit state of lateral-torsional buckling does not apply.
(b) When Lp < Lh < Lr
Lateral-Torsional Buckling
Fy = specified minimum yield stress of the type of steel being used, ksi (MPa)
Z v = plastic section modulus about the x-axis, in.1 (mm1)
where
Yielding
The nominal flexural strength, Mn> shall be the lower value obtained according
to the limit states of yielding (plastic moment) and lateral-torsional buckling.
User Note: All current ASTM A6 W, S, M, C and MC shapes except W21 x48,
Wt4x99, W14x90, W12x65, W10xl2, W8x31, W8xl0, W6xl5, W6x9,
W6x8.5, and M4x6 have compact flanges for Fy = 50 ksi (345 MPa); all
current ASTM A6 W, S, M, HP, C and MC shapes have compact webs at
Fy < 65 ksi (450 MPa).
This section applies to doubly symmetric I-shaped members and channels bent
about their major axis, having compact webs and compact flanges as defined in
Section B4.
DOUBLY SYMMETRIC COMPACT I-SHAPED MEMBERS
AND CHANNELS BENT ABOUT THEIR MAJOR AXIS
User Note: For doubly symmetric members with no transverse loading between brace points, Equation Fl-1 reduces to 2.27 for the case of equal end
moments of opposite sign and to 1.67 when one end moment equals zero.
Cj, is permitted to be conservatively taken as 1.0 tor all cases. For cantilevers or
overhangs where the free end is unbraced, Cb = 1.0.
DOUBLY SYMMETRIC COMPACT I-SHAPED MEMBERS AND CHANNELS
[Sect. F2.
I + . / 1 + 6.76
ho H7
c= — / ~
V E
/ 0 . 7 F y,
'
Sxh0\,
Jc }
(F2-8b)
(F2-8a)
(F2-7)
(F2-6)
(F2-5)
2SX
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
»Hfe)
r,t may be approximated accurately and conservatively as the radius of gyration
of the compression flange plus one-sixth of the web:
bf
"
2 h^_
/ h
For doubly symmetric I-shapes
with rectangular flanges, Cw *= - — and thus
I-sl
Equation F2-7 becomes
Note that this approximation can be extremely conservative.
User Note: If the square root term in Equation F2-4 is conservatively taken
equal to 1, Equation F2-6 becomes
h0 = distance between the flange centroids, in.. (mm)
where
For a channel:
For a doubly symmetric I-shape: c = 1
and
^•«»mj&i
The limiting lengths Lp and LT are determined as follows:
User Note: The square root term in Equation F2-4 may be conservatively
taken equal to 1.0.
E = modulus of elasticity of steel = 29,000 ksi (200 000 MPa)
/ = torsional constant, in.4 (mm4)
Sx = elastic section modulus taken about the x-axis, in.3 (mm3)
DOUBLY SYMMETRIC COMPACT I-SHAPED MEMBERS AND CHANNELS
2.
1.
F3.
[Sec). F4.
=
™*Ml
(F3-2)
(F3-1)
(F4-1)
- F,.Srr)
Myc = FySxe
FL = F,^->0.5F,
FL = Q.1F,
^
t^M^
R
(F4-6b)
(F4-6a)
'""
(F4-4)
(F4-7)
-^ -=~
(m
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
1 + . / 1 + 6.76
Specification for Structural Steel Buildings, March 9, 2005
J
AMERICAN INSTITUTE OF STEFX CONSTRUCTION, INC.
E
L, = 1.95r,—./
(F4-8)
The limiting unbraced length for the limit state of inelastic lateral-torsional buckling. Lr, is
LP=l-lr,J§-
"2^
(F4-3)
(F4
The limiting laterally unbraced length for the limit state of yielding, Lp, is
O'i) For j - < 0.7
(i) For — > 0.7
The stress. FL, is determined as follows:
For -^- < 0.23, J shall be taken as zero.
( * ) •
( ^ l j ^ ) l
M„ = FerSxr < RpcMye
~ {RpcMyc
' — ^ / ' • ™ , * 3Sx,; ( £ ) "
where
(c) When Lb > Lr
M„ = Ch lltpcMvr
(a) When Lh < Lp, the limit state of lateral-torsional buckling does not apply.
(b)WhenLp < Lb < Lr
Specification for Structural Steel Buildings, March 9, 2005
User Note: I-shaped members for which this section is applicable may be
designed conservatively using Section F5.
This section applies to: (a) doubly symmetric I-shaped members bent about their
major axis with noncompact webs; and (b) singly symmetric I-shaped members
with webs attached to the mid-width of the flanges, bent about their major axis,
with compact or noncompact webs, as defined in Section B4.
OTHER I-SHAPED MEMBERS WITH COMPACT OR NONCOMPACT
WEBS BENT ABOUT THEIR MAJOR AXIS
2'/
XPf = Xp is the limiting slenderness for a compact flange. Table B4.1
Xrf = Xr is the limiting slenderness for a noncompact flange, Table B4.1
4
kc =
and shall not be taken less than 0.35 nor greater than 0.76 for
s/h/t*.
calculation purposes
where
„.
\Ar/-Ap//J
[M,-<M,-O.IF,S,)(±^)]
(b) For sections with slender flanges
M. =
(a) For sections with noncompact flanges
Compression Flange Local Buckling
For lateral-torsional buckling, the provisions of Section F2.2 shall apply.
Lateral-Torsional Buckling
The nominal flexural strength, Mn, shall be the lower value obtained according
to the limit states of lateral-torsional buckling and compression flange local
buckling.
User Note: The following shapes have noncompact flanges for Fy — 50 ksi
(345 MPa): W21x48, W14x99, W14x90, W12x65, W10xl2, W8x31,
W8x 10, W6x 15, W6x9, W6 x 8.5, and M4x6. All other ASTM A6 W, S, M,
and HP shapes have compact flanges for Fy < 50 ksi (345 MPa).
Lateral-Torsional Buckling
M„ = Rpc Myc = Rp,. Fy Sxc
Compression Flange Yielding
OTHER I-SHAPED MEMBERS WITH COMPACT OR NONCOMPACT WEBS
This section applies to doubly symmetric I-shaped members bent about their major axis having compact webs and noncompact or siender flanges as defined in
Section B4.
16.1
The nominal flexural strength, M„, shall be the lowest value obtained according
to the limit states of compression flange yielding, lateral-torsional buckling,
compression flange local buckling and tension flange yielding.
OTHER T-SHAPED MEMBERS WITH COMPACT OR NONCOMPACT WEBS
DOUBLY SYMMETRIC I-SHAPED MEMBERS WITH COMPACT
WEBS AND NONCOMPACT OR SLENDER FLANGES BENT
ABOUT THEIR MAJOR AXIS
Sect. F4.|
OTHER I-SHAPED MEMBERS WITH COMPACT OR NONCOMPACT WEBS
=
M;
™
(F4-9a)
16.1-51
tW
= A.p, the limiting slenderness for a compact web. Table B4.1
= A.r, the limiting slenderness for a noncompact web, Table B4.1
hctw
(F4-11)
(F4-10)
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Specification for Structural Sleel Buildings, March 9, 2005
(ii) For I-shapes with channel caps or cover plates attached to the compression
flange:
r, = radius of gyration of the flange components in flexural compression
plus one-third of the web area in compression due to application of
major axis bending moment alone, in. (mm)
aw = the ratio of two times the web area in compression due to application
of major axis bending moment alone to the area of the compression
flange components
bfc = compression flange width, in. (mm)
tfc = compression flange thickness, in. (mm)
_
where
(i) For I-shapes with a rectangular compression Mange:
The effective radius of gyration for lateral-torsional buckling, r,, is determined
as follows:
A-/WXrw
Mp
= ZxFy < ].6SxcFy
SXCi $xt = elastic section modulus referred to tension and compression flanges,
respectively, in.1 (mm3)
where
*
R
M£-(£-06££)]*£
(ii) For - i > Xpw
(i) For — < lpw
The web plastification factor, Rpr, is determined as follows:
Sect. F4]
[Sect. F4.
- (RpcMyc - FfSxc) ( ^ ~ _ X f ) ]
(F4-13)
(F4-12)
Mn = RplMvl
(F4-14)
RP, = ^
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
Specification for Structural Steel Buildings. March 9. 2005
(i) For — < kpw
(F4-15a)
The web plastification factor corresponding to the tension flange yielding
limit state, Rpl, is determined as follows:
My, = FySXI
where
(a) When Sxl > Sxc, the limit state of tension flange yielding does not apply.
(b) When S.(, < Sxe
Tension Flange Yielding
2//r
XPf = kp, the limiting slenderness for a compact flange. Table B4.1
Xrf = Xr, the limiting slenderness for a noncompact flange. Table B4.1
calculation purposes
Fr is defined in Equations F4-6a and F4-6b
Rpc is the web plastification factor, determined by Equations F4-9
4
kr = -•'""••••• and shall not be taken less than 0.35 nor greater than 0.76 for
(c) For sections with slender flanges
0.9EkrSxr
M„ = \RPCM„
(a) For sections with compact flanges, the limit state of local buckling does not
apply.
(b) For sections with noncompact flanges
Compression Flange Local Buckling
K ,+ W
User Note: For I-shapes with a rectangular compression flange, r, may be
approximated accurately and conservatively as the radius of gyration of the
compression flange plus one-third of the compression portion of the web; in
other words,
OTHER I-SHAPED MEMBERS WITH COMPACT OR NONCOMPACT WEBS
™
(F5-2)
(F5-1)
CuTt-F
(F5-7)
J^
calculation purposes
y/fi/tw
4
= —*=== and shall not be taken less than 0.35 nor greater than 0.76 for
(F5-9)
-
(F5 8)
\f„ = Mp = FyZy < 1.6FVSV
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
Yielding
(F6-1)
The nominal fiexural strength, M„, shall be the lower value obtained according
to the limit states of yielding (plastic moment) and flange local buckling.
This section applies to I-shaped members and channels bent about their minor
axis.
I-SHAPED MEMBERS AND CHANNELS BENT ABOUT
THEIR MINOR AXIS
(a) When Sx, > Sxc, the limit state of tension flange yielding does not apply.
(b) When Sxl < Sxc
Mn = FySxl
(F5-10)
Tension Flange Yielding
kpf — kp, the limiting slenderness for a compact flange. Table B4.1
krf — kr, the limiting slenderness for a noncompact flange. Table B4.1
k
kc
where
m
(c) For sections with slender flange sections
Q.9Ekc
Hf'-(a3F-<^)]
AMERICAN INSTITUTE OF STEEI. CONSTRUCTION, INC.
1.
F6.
(F5-6)
(a) For sections with compact flanges, the limit state of compression flange local
buckling does not apply.
(b) For sections with noncompact flanges
M„ = R„F„SXC
Compression Flange Local Buckling
Specification for Structural Steel Buildings. March 9, 2005
-
(F55)
(F5-4)
(F5-3)
4.
?1
( — - 5 . 7 / — ] < 1.0
I200 + 300aB, \tw
y FyJ ~
(Sect. FS.
aw is defined by Equation F4-11 but shall not exceed 10
and
r, is the effective radius of gyration for lateral buckling as defined in
Section F4.
n
Rpt = i
RPg is the bending strength reduction factor:
DOUBLY SYMMETRIC AND SINGLY SYMMETRIC I-SHAPED MEMBERS
Specification for Structural Steel Buildings, March 9. 2005
^"•S
Lp is defined by Equation F4-7
where
(c) When Lh > Lr
Frr = Ch \Fy - (0.3F,) ( ^ I ^ ) j < Fy
(a) When Lb < Lp, the limit state of lateral-torsional buckling does not apply.
(b)WhenLp < Lb < Lr
M„ = RpxFcrS.yr
Lateral-Torsional Buckling
M„ = Rpf,FySxc
Compression Flange Yielding
The nominal fiexural strength, M„, shall be the lowest value obtained according
to the limit states of compression flange yielding, lateral-torsional buckling,
compression flange local buckling and tension flange yielding.
This section applies to doubly symmetric and singly symmetric I-shaped members with slender webs attached to the mid-width of the flanges, bent about their
major axis, as defined in Section B4.
DOUBLY SYMMETRIC AND SINGLY SYMMETRIC I-SHAPED
MEMBERS WITH SLENDER WEBS BENT ABOUT
THEIR MAJOR AXIS
kpw = kp, the limiting slenderness for a compact web, defined in Table B4.1
krn, = kr, the limiting slenderness for a noncompact web, defined in
Table B4.1
where
^-[£-(g-)(^:)]-g
fc
(ii) For — > kpw
DOUBLY SYMMETRIC AND SINGLY SYMMETRIC I-SHAPED MEMBERS
Sccl.F7.
16.1-55
;
m
=r
0.69£
(F6-2)
(F6-4)
(F6-3)
M„ =MP = F,Z
(F7-1)
B#-4
(F7-3)
[Seel. F7
0.38 E
b/t V Fy
<b
(F7-4)
(Mp - FySx) ( 0 . 3 0 5 — J ^ - 0.738 ] < Mp
(VIS)
M„ = Mp = FyZ
(F8-1)
AMERICAN INSTITUTE OF STEEL CONSTRUCTION. INC.
= elastic section modulus, in.3 (mm3)
t
D
_ 0.33E
M„ = FrrS
t
Specification for Structural Steel Buildings, March 9, 2005
S
Frr-
where
(c) For sections with slender walls
(F8-4)
(F8-3)
(a) For compact sections, the limit state of flange local buckling does not apply.
(b) For noncompact sections
Local Buckling
Yielding
The nominal flexural strength, M„, shall be the lower value obtained according
to the limit states of yielding (plastic moment) and local buckling.
0.45E
This section applies to round HSS having D/t ratios of less than —=—.
M„=Mp-
(a) For compact sections, the limit state of web local buckling does not apply.
(b) For sections with noncompact webs
Web Local Buckling
fc«l.Mr/£
Seff is the effective section modulus determined with the effective width of the
compression flange taken as:
where
M„ = FySeff
(c) For sections with slender flanges
SQUARE AND RECTANGULAR HSS AND BOX-SHAPED MEMBERS
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
56
Specification for Structural Steel Buildings, March 9, 2005
M„ = Mp - (Mp - FyS) ( 3 . 5 7 - . / ^ - - 4.0 ) < Mp
(F7-2)
(a) For compact sections, the limit state of flange local buckling does not apply.
(b) For sections with noncompact flanges
Flange Local Buckling
Z = plastic section modulus about the axis of bending, in.3 (mm1)
where
Yielding
The nominal flexural strength, M„, shall be the lowest value obtained according
to the limit states of yielding (plastic moment), flange local buckling and web
local buckling under pure flexure.
This section applies to square and rectangular HSS, and doubly symmetric boxshaped members bent about either axis, having compact or noncompact webs
and compact, noncompact or slender flanges as defined in Section B4.
SQUARE AND RECTANGULAR HSS AND BOX-SHAPED MEMBERS
XPf = Xp, the limiting slendemess for a compact flange. Table B4.1
Xrf = kr, the limiting slendemess for a noncompact flange, Table B4.1
Sy for a channel shall be taken as the minimum section modulus
x
F„ =
where
M„ = FcrS>
(c) For sections with slender flanges
M, = U p - (Mp - OJFyS,) (j—rf~)\
(b) For sections with noncompact flanges
User Note: All current ASTM A6 W, S, M, C and MC shapes except
W21x48, W14x99, W14x90, W12x65, W10xl2, W8x31, W8xl0,
W6xl5, W6x9, W6x8.5, and M4x6 have compact flanges at F, = 50
ksi (345 MPa).
(a) For sections with compact flanges the limit state of yielding shall apply.
Flange Local Buckling
SQUARE AND RECTANGULAR HSS AND BOX-SHAPED MEMBERS
F10.
3.
2.
1.
F9.
SINGLE ANGLES
16.1-57
M„=Mp
(£)y?
-
<F95)
(F9-4)
(F9-2)
(F9-3)
(F9-!)
(F9-6)
m
(F9-8)
,FM,
[Sect. F10.
Mn = 1.5A/y
(F10-1)
M,'„
= M.92 - lAlJ—^jMy < \.5My
M„ = ( o . 9 2 - ^ ) ^
(F10-3)
(F10-2)
—
+ 1
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
1+0.78
/t.
Specification for Structural Steel Buildings, March 9. 2005
Q.66Eb*tCf
>e
(b) With maximum tension at the toe
^.aoraW/
(F10-4b)
(FKMa)
(i) For bending about one of the geometric axes of an equal-leg angle with no
lateral-torsional restraint
(a) With maximum compression at the toe
Me, the elastic lateral-torsional buckling moment, is determined as follows:
where
(b) When Me > My
(a) When Me < Mv
For single angles without continuous lateral-torsional restraint along the length
Lateral-Torsional Buckling
My = yield moment about the axis of bending, kip-in. (N-mm)
where
Yielding
The nominal flexural strength, M„, shall be the lowest value obtained according
to the limit states of yielding (plastic moment), lateral-torsional buckling and leg
local buckling.
User Note: For geometric axis design, use section properties computed about
the x- and y-axis of the angle, parallel and perpendicular to the legs. For
principal axis design use section properties computed about the major and
minor principal axes of the angle.
Single angles with continuous lateral-torsional restraint along the length shall be
permitted to be designed on the basis of geometric axis (x, y) bending. Single
angles without continuous lateral-torsional restraint along the length shall be designed using the provisions for principal axis bending except where the provision
for bending about a geometric axis is permitted.
SINGLE ANGLES
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
2.
1.
16.1-58
Specificationfor Structural Steel Buildings, March 9, 2005
This section applies to single angles with and without continuous lateral restraint
along their length.
SINGLE ANGLES
(c) For slender sections
.P.(,.»-0.«,(i)vE)
(a) For compact sections, the limit state of flange local buckling does not apply.
(b) For noncompact sections
Sxc is the elastic section modulus referred to the compression flange.
Fir is determined as follows:
M„ = F„SXC
Flange Local Buckling or Tees
The plus sign for B applies when the stem is in tension and the minus sign applies
when the stem is in compression. If the tip of the stem is in compression anywhere
along the unbraced length, the negative value of B shall be used.
fi=±2j
where
M„ = Mcr
TijEfv GJ r
,
n
= —~\B + y/l + B2\
Lateral-Torsional Buckling
Mp = FyZ_x < 1.6MV for stems in tension
< My for stems in compression
where
Yielding
The nominal flexural strength, Mn, shall be the lowest value obtained according
to the limit states of yielding (plastic moment), lateral-torsional buckling and
flange local buckling.
This section applies to tees and double angles loaded in the plane of symmetry.
TEES AND DOUBLE ANGLES LOADED IN THE PLANE
OF SYMMETRY
Sect. F10.1
(F10-6)
-^-1 My < Mp
(Fll-2)
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
F „ = i ^
(F1.-4)
Lhd
t2
t — width of rectangular bar parallel to axis of bending, in. (mm)
d = depth of rectangular bar, in. (mm)
Lh = length between points that are either braced against lateral displacement of the compression region or braced against twist of the cross
section, in. (mm)
Lbd
1.9E
(b) For rectangular bars with —z- >
bent about their major axis:
tFy
M„ = FcrSx <Mp
(Fll-3)
where
M„ = Cb Tl.52 - 0.274 (~f\
, u
.u 0.08E
Lhd
1.9E u
,.
w•
•
(a) For rectangular bars with —-— < —— < ——— bent about their major
axis:
Lateral-Torsional Buckling
. L Lhd
0.08E , . , _ , _ . . .
For rectangular bars with -—- <
bent about their major axis, rectangular
t~
Fy
bars bent about their minor axis, and rounds:
M„ = Mp = FyZ< l.6Mv
(Fll-1)
The nominal fiexural strength, M„, shall be the lower value obtained according
to the limit states of yielding (plastic moment) and lateral-torsional buckling, as
required.
Yielding
Specification for Structural Steel Buildings, March 9, 20()5
(F10-7)
2.
1.
(F10-9)
(F10-8)
This section applies to rectangular bars bent about either geometric axis and
rounds.
Fll. RECTANGULAR BARS AND ROUNDS
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
FySc (2.43 - 1.72 ( - ) J — j
e)
0.71£
M„ = FcrSc
ISect. F
b = outside width of leg in compression, in. (mm)
Sc = elastic section modulus to the toe in compression relative to the axis
of bending, in.3 (mm3). For bending about one of the geometric axes
of an equal-leg angle with no lateral-torsional restraint, Sc shall be
0.80 of the geometric axis section modulus.
where
(c) For sections with slender legs
SINGLE ANGLES
Specification for Structural Steel Buildings, March 9. 2005
M„
(a) For compact sections, the limit state of leg local buckling does not apply.
(b) For sections with noncompact legs
The limit state of leg local buckling applies when the toe of the leg is in compression.
Leg Local Buckling
User Note: The equation for % and values for common angle sizes are listed
in the Commentary.
Ch is computed using Equation Fl-1 with a maximum value of 1.5.
L = laterally unbraced length of a member, in. (mm)
Iz = minor principal axis moment of inertia, in.4 (mm4)
rz = radius of gyration for the minor principal axis, in. (mm)
t — angle leg thickness, in. (mm)
0,„ = a section property for unequal leg angles, positive for short legs in
compression and negative for long legs in compression. If the long leg
is in compression anywhere along the unbraced length of the member,
the negative value of (3,,. shall be used.
^(f^M^)
(iv) For bending about the major principal axis of unequal-leg angles:
My shall be taken as the yield moment calculated using the geometric section
modulus.
(iii) For bending about the major principal axis of equal-leg angles:
0A6Eb2t2Ch
Me =
(F10-5)
Me shall be taken as 1.25 times Me computed using Equation F10-4a or
F10-4b.
(ii) For bending about one of the geometric axes of an equal-leg angle with
lateral-torsional restraint at the point of maximum moment only
User Note: M„ may be taken as My for single angles with their vertical leg
toe in compression, and having a span-to-depth ratio less than or equal to
My shall be taken as 0.80 times the yield moment calculated using the geometric section modulus.
SINGLE ANGLES
PROPORTIONS OF BEAMS AND GIRDERS
M„ = F„S
(F12-1)
Lateral-Torsional Buckling
2.
F„ = Fcr < Fy
F„ = Fy
(F12-3)
(F12-2)
F„ = Fcr < Fy
(F12-4)
Hole Reductions
1.
[Seci.F13.
=
fg
A
^JlSx
(H3-1)
(F13-2)
®
a
0.42E
Fy
(F13-4)
(F13-3)
Specification for Structural Steel Buildings, March 9. 2005
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
High-strength bolts or welds connecting flange to web, or cover plate to flange,
shall be proportioned to resist the total horizontal shear resulting from the bending
forces on the girder. The longitudinal distribution of these bolts or intermittent
welds shall be in proportion to the intensity of the shear.
The total cross-sectional area of cover plates of bolted girders shall not exceed
70 percent of the total flange area.
Flanges of welded beams or girders may be varied in thickness or width by
splicing a series of plates or by the use of cover plates.
Cover Plates
where
a = clear distance between transverse stiffeners, in. (mm)
In unstiffened girders h/tw shall not exceed 260. The ratio of the web area to the
compression flange area shall not exceed 10.
( b ) F o r - > 1.5
h
h
(a) For?- < 1.5
I-shaped members with slender webs shall also satisfy the following limits:
0 . l < — <0.9
Singly symmetric I-shaped members shall satisfy the following limit:
Proportioning Limits for I-Shaped Members
where
Afg = gross tension flange area, calculated in accordance with the provisions
of Section D3.1, in.2 (mm2)
Afi, = net tension flange area, calculated in accordance with the provisions of
Section D3.2, in.2 (mm2)
Y, = 1 . 0 for Fy/FH < 0 . 8
= 1.1 otherwise
Mn
(b) For FuAfi, < Y, FyAfg, the nominal flexural strength, A/„, at the location of the
holes in the tension flange shall not be taken greater than:
PROPORTIONS OF BEAMS AND GIRDERS
AMERICAN INSTITUTE OF STEEL CONSTRUCTION, INC.
3.
2.
16.1-62
Specification for Structural Steel Buildings. March 9, 2005
(a) For F„Af„ > Y,FyAfg, the limit state of tensile rupture does not apply.
In addition to the limit states specified in other sections of this Chapter, the
nominal flexural strength, M„, shall be limited according to the limit state of
tensile rupture of the tension flange.
This section applies to rolled or built-up shapes, and cover-plated beams with
holes, proportioned on the basis of flexural strength of the gross section.
PROPORTIONS OF BEAMS AND GIRDERS
F13.
Frr = buckling stress for the section as determined by analysis, ksi (MPa)
where
Local Buckling
User Note: In the case of Z-shaped members, it is recommended that Fcr be
taken as 0.5 Fcr of a channel with the same flange and web properties.
Fcr = buckling stress for the section as determined by analysis, ksi (MPa)
where
Yielding
1.
S = lowest elastic section modulus relative to the axis of bending, in.3 (mm3)
where
The nominal flexural strength, M„, shall be the lowest value obtained according
to the limit states of yielding (yield moment), lateral-torsional buckling and local
buckling where
This section applies to all unsymmetrical shapes, except single angles.
F12. UNSYMMETRICAL SHAPES
3.
16.1-
(c) For rounds and rectangular bars bent about their minor axis, the limit state of
lateml-torsiona! buckling need not be considered.
Sect. F13.I
PROPORTIONS OF BEAMS AND GIRDERS
Sect.FB.]
16.1-63
However, the longitudinal spacing shall not exceed the maximum permitted for
compression or tension members in Section E6 or D4, respectively. Bolts or welds
connecting flange to web shall also be proportioned to transmit to the web any
loads applied directly to the flange, unless provision is made to transmit such
loads by direct bearing.
Partial-length cover plates shall be extended beyond the theoretical cutoff point
and the extended portion shall be attached to the beam or girder by high-strength
bolts in a slip-critical connection orfilletwelds. The attachment shall be adequate,
at the applicable strength given in Sections J2.2, J3.8, or B3.9 to develop the cover
plate's portion of the flexural strength in the beam or girder at the theoretical cutoff
point.
For welded cover plates, the welds connecting the cover plate termination to the
beam or girder shall have continuous welds along both edges of the cover plate
in the length a', defined below, and shall be adequate to develop the cover plate's
portion of the strength of the beam or girder at the distance a' from the end of
the cover plate.
(a) When there is a continuous weld equal to or larger than three-fourths of the
plate thickness across the end of the plate
a =w
(F13-5)
where
w = width of cover plate, in. (mm)
(b) When there is a continuous weld smaller than three-fourths of the plate thickness across the end of the plate
a' = i.5w
(F13-6)
(c) When there is no weld across the end of the plate
a' = 2w
4.
(F13-7)
Built-Up Beams
Where two or more beams or channels are used side-by-side to form a flexural
member, they shall be connected together in compliance with Section E6.2. When
concentrated loads are carried from one beam to another, or distributed between
the beams, diaphragms having sufficient stiffness to distribute the load shall be
welded or bolted between the beams.
Specification for Structural Steel Buildings, March 9, 2005
AMERICAN INSTITUTE OE STEEL CONSTRUCTION, INC.
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