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Technical Note
Generalized Approach for Determining the Maximum Moment
under HL-93 Loading on Simple Spans
Downloaded from ascelibrary.org by University Of Florida on 10/26/17. Copyright ASCE. For personal use only; all rights reserved.
Steven Lowinger, A.M.ASCE1
Abstract: With the institution of AASHTO HL-93 loading, calculating the maximum live-load moment on simple spans became more complex than it had been previously. In the past, to find the maximum live-load moment from the HS-20 design truck, the middle axle was offset
from midspan by 0.711 m (2 ft 4 in.), and the moment under that axle was the maximum moment on the beam. However, with the combination
of the design vehicle and the distributed design lane load, this general approach became obsolete. The purpose of this study was to develop a
new generalized approach to determine the location and value of the maximum moment under HL-93 loading on simple spans. This was
accomplished by creating moment envelopes and studying their extremum. Subsequently, equations, charts, and tables were developed that
describe the location and value of the maximum moment on a beam under HL-93 loading, consisting of design truck or design tandem loading
in combination with the design lane load for a simple span of any length. DOI: 10.1061/(ASCE)BE.1943-5592.0001160. © 2017 American
Society of Civil Engineers.
Author keywords: HL-93; AASHTO; Maximum moment.
Introduction
Method
Before the introduction of HL-93 loading, to determine the value of
the maximum moment on a simply supported beam under an HS-20
design truck, the middle axle load was placed 0.711 m (2 ft 4 in.)
from the center of the span, with the center of mass of the vehicle
located on the other half of the span. However, because HL-93 loading also contains a distributed lane load, this approach is no longer
valid. Shipman (2014) developed a method for determining the
location and value of the maximum moment of the HL-93 loading
consisting of a design truck and a distributed lane load for simply
supported spans greater than 12.192 m (40 ft).
However, the goal of this work was to propose a different and
more general formulation for finding the location and value of
the maximum moment under all AASHTO (2012) HL-93 loading
on simple spans of any length. Additionally, this work provides
simple equations, tables, and charts that display these quantities
as functions of span length for both the design truck combined
with the design lane load and for the design tandem combined
with the design lane load. This was accomplished by generating
a boundary curve, or envelope, that encompasses multiple
moment diagrams and identifying the maximum moment in the
beam.
The value and location of the maximum moment is kept in a
general form so that it can be applied for a span of any length,
any loads, and any load spacing. However, results specific to the
HL-93 loading are provided as well. The value and location of
the maximum moment are tabulated as a function of span length
for both the HL-93 design truck and HL-93 design tandem
loading.
Before showing the results, the author will introduce the method by
which the moment diagrams were analyzed. The method will be
explained using the design truck; as will be shown, the generality of the
method can easily be extended to the design tandem. To find the worstcase moment at any point on the beam, the moment diagrams are
tracked as the design vehicle moves along the beam. To better explain
the method of analysis, the author defines certain variables and their set
values under AASHTO (2012) HL-93 design truck loading:
d = distance between the left support to the rear axle of the
design vehicle;
s = spacing between the rear axle and the middle axle [HL-93:
4.267–9.144 m (14–30 ft)];
t = spacing between the middle axle and front axle [HL-93:
4.267 m (14 ft)];
‘ = span length;
P1 = axle load from the rear axle [HL-93: 1.33 (142.34 kN or
32 k)] with impact factor as per AASHTO (2012, Section 3.6.2);
P2 = axle load from the middle axle [HL-93: 1.33 (142.34 kN or
32 k)] with impact factor;
P3 = axle load from the front axle [HL-93: 1.33 (35.59 kN or 8 k)]
with impact factor; and
w = distributed lane load applied to the beam [HL-93: 9.34 kN/m
(0.64 k/ft)].
All of these variables are depicted in Fig. 1 for design truck loading combined with the design lane load. The value of s can vary
from 4.267 to 9.144 m (14–30 ft). However, in this work, s will
always be taken as 4.267 m (14 ft). Additionally, the impact factor,
IM, that will be used in this work is 1.33. To generate a general
moment diagram, the left vertical support reaction, Ay, is determined
for a generalized d. Using basic statics, Ay is represented by Eq. (1).
P1 ðdÞ þ P2 ðd þ sÞ þ P3 ðd þ s þ tÞ
w‘
Ay ¼ ðP1 þ P2 þ P3 Þ þ
‘
2
(1)
1
Assistant Engineer, WSP, 1 Penn Plaza, New York, NY 10119.
E-mail: Lowingersteven@gmail.com
Note. This manuscript was submitted on February 1, 2017; approved
on July 13, 2017; published online on October 26, 2017. Discussion period
open until March 26, 2018; separate discussions must be submitted for
individual papers. This technical note is part of the Journal of Bridge
Engineering, © ASCE, ISSN 1084-0702.
© ASCE
Therefore, using singularity functions, the moment on the beam
is represented by Eq. (2). H[x] is the Heaviside step function, also
06017008-1
J. Bridge Eng., 2018, 23(1): 06017008
J. Bridge Eng.
known as the unit step function, and x represents the location along
the beam.
(
P1 d þ P2 ðd þ sÞ þ P3 ðd þ s þ tÞ
MðxÞ ¼ ðP1 þ P2 þ P3 Þ ‘
)
w‘
þ
x P1 ðx dÞH ½x d P2 ½x ðd þ sÞH ½x ðd þ sÞ
2
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P3 ½x ðd þ s þ tÞH ½x ðd þ s þ tÞ wx2
2
(2)
In every moment diagram, the moments under the axle loads are
studied because these locations will yield the maximum moment.
The values of the moment at the locations of the axle loads are
referred to as potential maximums. These values are tracked to
understand how they change as the design truck moves along the
span.
To properly track these potential maximums, the locations of the
axles are studied with respect to d, the distance between the left support and the rear axle. The rear axle is located at x1 = d, the middle
axle at x2 = d þ s, and the front axle at x3 = d þ s þ t. The moment
diagram is therefore evaluated at these three locations, x = {x1, x2,
x3}, to generate the values of the potential maximums, {M(x1), M
(x2), M(x3)}, respectively. However, to keep the dependent variable
as x, d will be replaced with a function of x that is unique to each
axle under consideration. As an example, M(x2), where x = x2 =
d þ s, is the moment at the location x = x2. Within M(x2), all d must
be substituted with x – s. A similar process can be followed for M
(x1) and M(x3). After these substitutions are made, the values of the
moment functions under the axle loads (the potential maximums)
are functions of x, the location on the beam; ‘, the span length of the
beam; and the loading variables. Eqs. (3)–(5) are the envelope
curves and bound the moment diagrams [Fig. 2(a)].
(
Mðx1 ¼ dÞ ¼
Fig. 1. Design truck on a simply supported span with all applicable
variables displayed
ðP1 þ P2 þ P3 Þ
)
P1 x þ P2 ðx þ sÞ þ P3 ðx þ s þ tÞ
w‘
wx2
x
þ
‘
2
2
(3)
Fig. 2. Plot of multiple moment diagrams and the envelope curves corresponding to each axle of the design truck for different span lengths: (a) span
length of ‘ = 25 m (82.02 ft) with all three design truck axle loads on the span; (b) span length of ‘ = 8 m (26.25 ft) with the two rear design truck axle
loads on the span
© ASCE
06017008-2
J. Bridge Eng., 2018, 23(1): 06017008
J. Bridge Eng.
Mðx2 ¼ d þ sÞ
¼
P1 ðx sÞ þ P2 x þ P3 ðx þ tÞ
w‘
ðP1 þ P2 þ P3 Þ þ
x
‘
2
P1 s wx2
2
(4)
M ðx3 ¼ d þ s þ tÞ
Downloaded from ascelibrary.org by University Of Florida on 10/26/17. Copyright ASCE. For personal use only; all rights reserved.
¼ ðP1 þ P2 þ P3 Þ P1 ðs þ tÞ P2 t P1 ðx s tÞ þ P2 ðx tÞ þ P3 x
w‘
þ
x
‘
2
wx2
2
(5)
The maximum value of all of the envelopes corresponds to the
maximum moment on the beam under the HL-93 loading. This
method can also be applied to loading scenarios in which one or two
axles have exited the span. These circumstances will be addressed
later in the article.
Based on Fig. 2(a) and the author’s studies, the maximum
moment always occurs under the middle axle load. The function for
the envelope curve of this axle is found in Eq. (4). Therefore, finding the location and value of the maximum of this curve would provide an equation that explains the worst-case moment on the beam.
Taking the derivative of Eq. (4) with respect to x, setting it equal to
zero, and solving for x results in
x¼
‘
P1 s P3 t
þ
2 2ðP1 þ P2 þ P3 Þ þ w‘
(6)
The first term of Eq. (6) refers to the location of the midspan, and
the second term represents the offset associated with the design truck
combined with the lane load. When w = 0 kN/m, corresponding to a
loading scenario in which there is only a HS-20 design truck on the
span, but no lane load, the value of the offset coincides with the offset
of 0.711 m (2 ft 4 in.) that was used prior to HL-93 loading. Eq. (6)
represents xmax the location of the maximum moment on the beam of
a span ‘. Plugging Eq. (6) into Eq. (4) and simplifying results in
(
1
2‘ðP1 þ P2 þ P3 Þ 4ðP1 s þ P3 tÞ þ w‘2
Mmax1 ¼
8
4ðP1 s P3 tÞ2
þ
‘½2ðP1 þ P2 þ P3 Þ þ w‘
Table 1. Value and Location of the Maximum Moment under HL-93
Design Truck and HL-93 Design Tandem Loading
HL-93 design truck
Span
m (ft)
Maximum
moment
kN·m
1 (3.28)
48.5
2.094 (6.87)
104.2
3 (9.84)
152.5
6 (19.69)
326.1
7.408 (24.31)
414.7
9 (29.53)
585.7
10.115 (33.19)
710.9
12.329 (40.45) 1,000.9
15 (49.21)
1,367.5
18 (59.06)
1,800.2
21 (68.90)
2,254.6
24 (78.74)
2,730.5
27 (88.58)
3,227.6
30 (98.43)
3,746.0
35 (114.8)
4,656.9
40 (131.2)
5,626.5
45 (147.6)
6,654.6
50 (164.0)
7,741.2
55 (180.4)
8,886.2
60 (196.9)
10,089.7
Location
HL-93 design tandem
Maximum
moment
Location
k-ft
(xmax =‘)
kN·m
k-ft
(xmax =‘)
35.8
76.9
112.5
240.5
305.9
432.0
524.3
738.2
1,008.6
1,327.8
1,662.9
2,013.9
2,380.6
2,762.9
3,434.8
4,149.9
4,908.2
5,709.6
6,554.2
7,441.8
0.500
0.500
0.500
0.500
0.500
0.607
0.594
0.551
0.541
0.533
0.528
0.523
0.520
0.518
0.515
0.512
0.511
0.509
0.508
0.507
38.1
82.5
150.9
399.8
525.1
672.6
779.7
1,000.9
1,283.3
1,620.4
1,978.7
2,358.0
2,758.4
3,179.8
3,928.9
4,736.3
5,602.2
6,526.5
7,509.2
8,550.2
28.1
60.9
111.3
294.8
387.3
496.1
575.0
738.2
946.5
1,195.2
1,459.4
1,739.2
2,034.5
2,345.3
2,897.8
3,493.3
4,132.0
4,813.7
5,538.5
6,306.3
0.500
0.500
0.597
0.546
0.537
0.530
0.526
0.521
0.516
0.513
0.511
0.509
0.508
0.507
0.506
0.505
0.504
0.503
0.503
0.503
Note: The bolded values correspond to the controlling maximum moment
for that span length.
)
(7)
Eq. (7) is a general equation for the maximum moment on a simply supported beam under HL-93 design truck loading. For HL-93
design truck loading, Eqs. (6) and (7) are applicable to all spans of
‘ > 10.115 m (33.19 ft). When dealing with spans smaller than
10.115 m (33.19 ft), the loading scenario in which the front axle of
the design truck has already exited the span and only the two rear
axles are on the span leads to a larger maximum moment. The equation for the maximum moment for this loading scenario is shown
later in the article. The value 10.115 m (33.19 ft) was determined by
comparing the maximum moment under design truck loading with
three axles on the span and design truck loading with only the two
rear axles on the span.
With spans in the range of 4.267 m (14 ft) < ‘ ≤ 8.534 m (28 ft),
a maximum of two axles of the design truck can fit on the span. For
spans 7.408 m (24.31 ft) < ‘ ≤ 8.534 m (28 ft), the loading scenario
© ASCE
in which the rear two axle loads are on the span will govern.
However, for span lengths ‘ ≤ 7.408 m (24.31 ft), the loading scenario where only the one rear axle remains on the span will govern.
For these spans, the rear axle is able to reach midspan after the middle and front axles have exited the span. The magnitude of this
moment is larger than the maximum moment when the rear and
middle axles are both on the span.
Therefore, in the span range of 7.408 m (24.31 ft) < ‘ ≤
10.115 m (33.19 ft), the loading scenario of having the rear two
axle loads on the span will govern. The maximum moment can
be determined by taking Eq. (7) and making the simple substitutions of P3 = 0 kN, t = 0 m, which results in Eq. (8). Fig. 2(b)
Fig. 3. Plot representing the maximum moment under HL-93 design
truck and design tandem loading for any span ‘
06017008-3
J. Bridge Eng., 2018, 23(1): 06017008
J. Bridge Eng.
shows a moment envelope for a span of 8 m (26.25 ft), in which
the two rear axle loads are on the span.
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Mmax2 ¼
2‘ðP1 þ P2 Þ 2P1 s þ w‘2
8‘ð2P1 þ 2P2 þ w‘Þ
2
In the span range ‘ ≤ 7.408 m (24.31 ft), in which the loading scenario of having only one axle of the design truck on the span controls,
the value of the maximum moment can be expressed by Eq. (9).
(8)
Mmax3 ¼
P1 ‘ w‘2
þ
4
8
(9)
In summary, to properly express the maximum moment on a
beam under HL-93 design truck loading for any span, three span
ranges must be analyzed: ‘ > 10.115 m (33.19 ft), 7.408 m
(24.31 ft) < ‘ ≤ 10.115 m (33.19 ft), and ‘ ≤ 7.408 m (24.31 ft).
Applying Eqs. (7)–(9) to their respective ranges will result in a
function that displays the maximum moment on a span of any
length ‘ under HL-93 design truck loading. Table 1 contains the
values of this piecewise function evaluated at selected span
lengths and the corresponding location of this maximum moment.
Because of the generality of the approach, this method can very
easily be applied to the design tandem by simply using the previous
equations and making the following replacements: P3 = 0 kN, P1 =
P2 = 1.33(111.21 kN or 25 k), s = 1.22 m (4 ft), t = 0 m. As an example, Eq. (7) becomes Eq. (10).
Fig. 4. Plot representing the location of the maximum moment under
HL-93 design truck and design tandem loading for any span
‘ > 10.115 m (33.19 ft). The vertical axis corresponds to the location of
the maximum moment, xmax, divided by the span length, ‘
Mmax
2‘ðP1 þ P2 Þ 2P1 s þ w‘2
¼
8‘ð2P1 þ 2P2 þ w‘Þ
2
(10)
This equation represents the maximum moment under HL-93
design tandem loading for spans of length ‘ > 2.093 m (6.87 ft).
Additionally, the location of the maximum moment is
Table 2. Equations for the Location and Value of Maximum Moment under HL-93 Design Truck and Design Tandem Loading
Loading type
HL-93 design truck
Mmax kN·m (k-ft)
xmax m (ft)
504:91 þ 106:49‘ þ 1:168‘2 þ
19; 651:89
91:21‘ þ ‘2
156; 016:98
372:40 þ 23:94‘ þ 0:08‘ þ
299:25‘ þ ‘2
2
403:93 þ 94:66‘ þ 1:168‘2 þ
297:92 þ 21:28‘ þ 0:08‘2 þ
34; 936:59
81:08‘ þ ‘2
277; 363:52
266‘ þ ‘2
47:33‘ þ 1:168‘2
‘
0:711
þ
2 1 þ 0:011‘
1
0
7
C
B‘
3
C
B þ
@2
4 A
‘
1þ
1197
‘
1:067
þ
2 1 þ 0:012‘
1
0
7
C
B‘
2
C
B þ
@2
1 A
‘
1þ
266
‘
2
90:16 þ 73:95‘ þ 1:168‘2 þ
66:50 þ 16:63‘ þ 0:08‘2 þ
1; 740:7
63:34‘ þ ‘2
13; 819:53
207:81‘ þ ‘2
© ASCE
06017008-4
J. Bridge Eng., 2018, 23(1): 06017008
7.408 m < ‘ 10:115 m
(24.31 ft < ‘ 33:19 ft)
‘ 7:408 m
‘
:3048
þ
2 1 þ 0:016‘
‘ > 2:094 m
0
‘
@2 þ
1
1
16 A
‘
1þ
3325
‘
2
(8:31‘ þ 0:08‘2 )
(‘ > 33:19 ft)
(‘ 24:31 ft)
‘
2
36:98‘ þ 1:168‘2
‘ > 10:115 m
‘
2
(10:64‘ þ 0:08‘2 )
HL-93 design tandem
Span m (ft)
(‘ > 6:87 ft)
‘ 2:094 m
(‘ 6:87 ft)
J. Bridge Eng.
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x¼
‘
P1 s
þ
2 2ðP1 þ P2 Þ þ w‘
(11)
Similar to the earlier description, with spans ‘ ≤ 2.093 m
(6.87 ft), the loading scenario in which only one axle remains on
the span controls. Therefore, for HL-93 design tandem loading,
two span ranges must be analyzed: ‘ > 2.093 m (6.87 ft), in which
two axles on the span lead to a larger moment, and ‘ ≤ 2.093 m
(6.87 ft), in which having only one axle on the span leads to a
larger moment.
After making the aforementioned replacements to the equations
and following the same procedure, a piecewise function can also be
generated for HL-93 design tandem loading. The value and location
of maximum moment for selected span lengths can be found in
Table 1. Additionally, these piecewise functions for the value and
location of the maximum moment under all HL-93 loading can be
found in Figs. 3 and 4, respectively. To maintain clarity, Fig. 4 only
includes spans ‘ > 10.115 m (33.19 ft).
Based on Table 1, the span length after which the design truck
loading begins to govern over the design tandem loading is
12.329 m (40.45 ft). The final table and charts provide an efficient
way for engineers to determine the maximum moment under HL-93
loading for a simple span of any length. These results are produced
using the HL-93 loading variables set forth previously. If one of
these variables were modified, the results would be different, but
the method would be the same.
The results found for the value and location of the maximum
moment have been confirmed using structural analysis software.
Additionally, for the design truck loading on spans greater than
12.192 m (40 ft), the results presented match those found by
Shipman (2014). However, when comparing results to those of
Shipman, the difference in orientation of x relative to the design
vehicle must be noted. Additionally, in the beginning of his work,
Shipman (2014) rounded 14/3 to 4.67, which leads to a small difference in results. Without this rounding, the results coincide
exactly.
HL-93 Equations
The equations presented thus far have been expressed using generalized variables. The author will now present equations that result
from replacing all variables with the values from HL-93 loading (as
defined in the earlier), s = 4.267 m (14 ft), and an impact factor of
1.33. The equations for the location and value of the maximum
moment under HL-93 design truck or design tandem loading can be
found in Table 2.
© ASCE
Example
Find the location and value of the maximum moment under HL-93
loading on one lane on a 45-m (147.6-ft) simply supported beam.
• Refer to Table 1 and recognize that the HL-93 design truck
loading will govern. Then, using the same table, Fig. 3, Eq.
(7), or the applicable equation from Table 2, one can find that
the maximum moment is 6,654.6 kN·m (4,908.2 k-ft).
• Using Table 1, Fig. 4, Eq. (6), or the applicable equation from
Table 2, one can determine that xmax/‘ = 0.511 or that the location of the maximum moment is xmax = 23 m (75.44 ft) =
(0.511)(45) m. This corresponds to a 0.5-m (1.64-ft) offset
from midspan.
Conclusion
The author presented a generalized method for determining the
maximum moment on a simple span under HL-93 loading. This
method can be applied to any design code, any vehicle, or any
impact factor because the equations are all functions of variables
that can be altered. The basis of the work was to develop a boundary, or envelope, curve that encompasses the moment diagrams for
different locations of the design vehicle. Subsequently, the value
and location of the maximum of this envelope curve were determined as functions of the span length, and generalized equations
were presented. The author also developed tables and charts that
explain the value and location of the maximum moment under HL93 loading for any span length. These results will aid in the understanding of the behavior of a beam under HL-93 loading and will
significantly improve the efficiency of the beam design process.
Acknowledgments
I thank Dr. Fredy R. Zypman for reviewing this paper and educating
me on how to approach problems analytically; his courses informed
the method developed in this paper. I also thank Dr. Tom T.
Panayotidi for teaching me about all aspects of structural
engineering; his courses encouraged me to approach this problem.
References
AASHTO. (2012). AASHTO LRFD bridge design specifications, 7th Ed.,
Washington, DC.
Shipman, C. L. (2014). “Finding maximum moment: Determining Hl-93
truck position on simple spans.” J. Bridge Eng., 10.1061/(ASCE)BE
.1943-5592.0000591, 06014003.
06017008-5
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J. Bridge Eng.
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