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Kinematic Limit Analysis Approach for Seismic Active Earth
Thrust Coefficients of Cohesive-Frictional Backfill
Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved.
Jagdish Prasad Sahoo1 and R. Ganesh2
Abstract: A theoretical study has been performed for evaluating maximum resultant active force exerted by a cohesive-frictional backfill
against a rigid wall with inclined back face undergoing outward horizontal translational movement in the presence of pseudostatic seismic
loadings. With the application of a new kinematically admissible translational mechanism in the context of the upper bound limit theorem of
plasticity, seismic active thrust has been derived as a function of nondimensional seismic active earth thrust coefficients because of the contributions of soil unit weight, surcharge pressure, and cohesion of soil. The present analysis was performed by postulating a composite collapse
mechanism comprised of a central radial shearing zone enclosed between triangular blocks at either side. The kinematically admissible velocity field of the radial shear zone freely varies in between the velocity field used for defining the conventional circular to log spiral shear zones
available in the literature. The influence of seismic acceleration coefficients, location of surcharge pressure from the wall crest, backfill slope
angles, characteristics of interface between soil and wall, orientations of wall, and properties of backfill on the magnitude of active earth force
has thoroughly been examined. DOI: 10.1061/(ASCE)GM.1943-5622.0001030. © 2017 American Society of Civil Engineers.
Author keywords: Active earth thrust; Kinematically admissible; Limit analysis; Retaining wall; Seismic loadings.
Introduction
The determination of active earth pressure induced by backfill soil
mass against a rigid retaining wall is a major problem in the field of
geotechnical engineering. A vast number of research attempts have
been made to estimate the static and seismic active earth pressure
exerted by soil as backfill on a rigid retaining wall by using different
approaches. The theories developed by Coulomb (1776) and
Rankine (1857) are generally used to determine the active earth
pressure against a rigid retaining wall. For computing the seismicinduced active earth force against rigid retaining walls with frictional backfill, Okabe (1924) and Mononobe and Matsuo (1929)
have extended the static earth pressure theory of Coulomb (1776)
by taking into account the pseudostatic seismic forces in the analysis; this modified Coulomb’s theory is commonly referred to as the
Mononobe-Okabe theory. By perceiving the development of nonplanar failure surfaces in the backfill at their critical active state,
thrust coefficients were presented in the tabular form (Terzaghi
1943; Jumikis 1962; Kerisel and Absi 1990) for determining the
static lateral earth thrust based on the limit equilibrium analysis of
curved failure surfaces. Saran and Prakash (1968) have extended the
Mononobe-Okabe theory for retaining walls with general cohesivefrictional soil as backfill and reported that the presence of cohesion
in the backfill has a significant influence on the magnitude of active
earth thrust. Therefore, the contribution of soil cohesion cannot be
omitted simply in the analysis and design of retaining structures.
1
Assistant Professor, Dept. of Civil Engineering, Indian Institute of
Technology, Roorkee 247667, India (corresponding author). E-mail:
jpscivil@gmail.com
2
Research Fellow, Dept. of Civil Engineering, Indian Institute of
Technology, Roorkee 247667, India. E-mail: ravishivaganesh@gmail
.com
Note. This manuscript was submitted on January 30, 2017; approved
on July 6, 2017; published online on October 25, 2017. Discussion period
open until March 25, 2018; separate discussions must be submitted for
individual papers. This paper is part of the International Journal of
Geomechanics, © ASCE, ISSN 1532-3641.
© ASCE
Based on the upper bound theorem of limit analysis and by analyzing six different translational mechanisms, Chen and Rosenfarb
(1973) have produced rigorous upper bound solutions for evaluating maximum static active earth thrust on a retaining wall with a
general cohesive-frictional backfill. With the inclusion of pseudostatic seismic body forces and using the upper bound mechanism
presented by Chen and Rosenfarb (1973), Chen and Liu (1990)
computed seismic active earth thrust coefficients on retaining walls
with inclined back face due to sloping cohesive-frictional backfill.
Das and Puri (1996) have addressed the limitations of the modified
Mononobe-Okabe theory proposed by Saran and Prakash (1968) in
which the backfill surface was considered horizontal and the effect
of vertical seismic acceleration was neglected. Soubra and Macuh
(2002) have computed the static active earth thrust coefficients corresponding to components of surcharge and soil cohesion based on
a rotational log spiral mechanism. The studies performed by Chen
and Rosenfarb (1973), Chen and Liu (1990), Saran and Prakash
(1968), and Das and Puri (1996) were based on the independent
maximization scheme in which the seismic active earth thrust coefficients, because of components such as soil weight, surcharge
pressure, and cohesion of soil, were discovered with the help of the
superposition rule. Based on the limit equilibrium approach, the
total active earth force computed by Shukla et al. (2009) due to
the contribution of soil weight and cohesion of soil without using
the superposition rule was reported to differ from that determined
by Saran and Prakash (1968) with the principle of superposition.
By applying Coulomb’s rigid wedge planar mechanism the influence of line surcharge, uniformly distributed surcharge of infinite
and finite extends applied at a certain distance away from the wall
crest on the magnitude of active earth forces, has been addressed
(Motta 1994; Greco 2005, 2006). Nevertheless, the applications of
these studies are limited only to purely frictional backfill. No further information seems to have been presented describing the influence of proximity of surcharge pressure on the maximum active
force regarding varying wall geometry, backfill slope angle, properties of soil, and so forth, which are quite useful to better understand the design philosophy associated with a retaining wall under
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different backfill surcharges. Jarquio (1981) and Misra (1980)
developed solutions for finding lateral earth pressure on walls due
to different surcharges on the backfill based on the theory of elasticity in which the effect of the strength parameters of backfill were
not taken into account. For estimating the total active thrust on
retaining walls in the presence of live load surcharge, Kim and
Barker (2002) used a mixed approach in which the active force was
calculated by considering backfill to be simultaneously in an active
state of failure and in the elastic state due to soil weight and surcharges, respectively. The applicability of solutions computed
using the elastic theory and the mixed approach are found to be
suitable only for unyielding walls, whereas its application to yielding walls, that is, for the case of active failure, is really questionable
(Steenfelt and Hansen 1983, 1984; Motta 1994; Georgiadis and
Anagnostopoulos 1998; Greco 2005, 2006). Mylonakis et al.
(2007) have computed seismic earth thrust coefficients in sand
with the help of the lower bound theorem of plasticity. Nian and
Han (2013) examined the effects of backfill cohesion, backfill
slope angle, depth of tension crack, and pseudostatic seismic forces
by extending Rankine’s (1857) earth pressure theory. Based on the
limit-equilibrium concept considering the arching effect in backfill
soil, theoretical solutions were developed for calculating the active
pressure on rigid retaining walls (Li and Wang 2014; Rao et al.
2016; Chen et al. 2017). It is known that the solutions obtained
from these methods, such as limit equilibrium, stress characteristics, and the lower bound theorem of limit analysis, may ensure the
conditions required for static admissibility; however, the requirement of kinematic admissibility conditions may not always be satisfied (Huntington 1957; Kumar and Subba Rao 1997). Such kinematical admissibility conditions are essential to ensure the
kinematics of the problem, or the deformations produced by the
mechanism are practically feasible in accordance with flow rule of
the material.
From the literature there are number of factors, in particular,
backfill slope angle, continuous surcharge pressure on the backfill,
soil cohesion, wall inclination, and seismic accelerations, which
influence the magnitude of the active earth pressure of a retaining
wall. In most of the previous studies, the effect of a continuous surcharge acting on the backfill has been considered; in practice, however, the uniform surcharge may act at a particular distance away
from the wall on the backfill. There seems to be very little research
done to tackle these kinds of problems, which are quite relevant to
practical situations. All the reported studies are based on the
assumption of a simple rigid wedge planar mechanism behind the
wall, and its applications are limited to frictional backfill. A rigorous
solution to address the influence of a uniform surcharge applied at
some predetermined distance away from the wall on the cohesivefrictional soil is still lacking. In the present research, a new composite collapse mechanism has been developed, which consists of a radial shearing zone sandwiched between two triangular blocks. In
this collapse mechanism, the kinematically admissible velocity field
of the radial shear zone converges on the velocity field of conventional circular and log spiral shear zones when the inclination of
velocities are equal to the internal friction angle of soil and zero,
respectively, with the normal and all radial lines emerging from a
singular point. By applying the upper bound theorem of limit analysis on this new collapse mechanism to the framework of the pseudostatic approach, the analysis was performed, and rigorous solutions
were developed for determining the magnitude of total seismic
active earth force against a rigid wall translating horizontally away
from the cohesive-frictional soil backfill. The effect of the inclination of the wall back face and backfill surface, soil-wall interface
friction angle and soil-wall interface adhesion force, and the location
© ASCE
of the ground surcharge from the wall have been examined. The
influence of the superposition rule on the solutions has also been
discussed. The effectiveness of the present approach was examined by comparing the solutions obtained in this paper with the
experimental and theoretical solutions reported in the previous
studies. For a larger negative inclination of wall back face, the
selection of a simple planar mechanism without inclusion of any
shear zone has been found to be more effective compared with the
composite mechanism introduced in the present work or in the
earlier studies in obtaining the maximum magnitude of the seismic active earth force. This may be due to the imposition of more
constraints in the case of the composite mechanism than that of
the planar mechanism for satisfying the kinematic admissibility
conditions.
Statement of Problem
A rigid retaining wall with an inclined back face supporting cohesive-frictional soil as backfill is illustrated in Fig. 1. The height of
the wall is h, and l refers to the length of the back face of the wall,
which is sloped at an angle a with the vertical plane. The surface of
the backfill makes an angle of b with the horizontal plane and
loaded with a uniform surcharge pressure of magnitude q at a distance l l from the wall top, where l is a multiplier constant whose
value varies from 0 to 1. The unit weight of backfill soil mass is
denoted as g . The interface strength t int between the wall and backfill is assumed equal to cw þ s w tand , where cw = adhesion per unit
length of wall whose magnitude is c(tan d /tanf ); s w = normal
stress on the wall; d = soil-wall interface friction angle; and the parameters c and f = cohesion and internal friction angle of soil,
respectively. In the case of rough walls, the resultant active force Pa
is directed at an angle d from the normal to the wall surface, and a
tangential adhesive force Cw is assumed to be acting along the surface of wall provided the soil is not fully cohesive ( f = 0). To perform the pseudostatic analysis, the earthquake acceleration in the
horizontal direction is assumed to be distributed uniformly in the
soil domain with a magnitude equal to khg, where kh and g = peak
coefficient of horizontal earthquake acceleration and the
q
λl
α
khq
O
β
Cw
E
khγ
l
h
δ
γ
Pa
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A
Fig. 1. Problem definition
Int. J. Geomech.
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acceleration due to gravity, respectively. The aim is to determine
the magnitude of total active earth force on a wall with inclined
back surface retaining a general cohesive-frictional sloping backfill
with a surcharge placed at different positions. The soil mass is idealized as a rigid perfectly plastic material and obeys the MohrCoulomb failure criterion with an associated flow rule. It is presumed that the properties of the backfill mass remain the same at all
the points within the soil domain during the occurrence of an earthquake. The magnitude of d is assumed to be constant in the present
analysis; however, the distribution of the magnitude of d along the
length of the wall is not constant, which is largely dependent on the
movement of the wall; addressing this issue is beyond the scope of
this article.
Theoretical Analysis
The theoretical analysis was performed by using the upper bound
theorem of limit analysis based on an assumed collapse mechanism
satisfying kinematical admissibility conditions. The upper bound
analysis enables the determination of the magnitude of collapse
load in any kinematically admissible velocity fields by equating the
rate of total work done by the body and external applied forces to
the rate of total internal energy dissipation. Any velocity field can
be said to be kinematically admissible as long as it satisfies the flow
rule, velocity compatibility, and boundary conditions. The magnitude of the collapse load computed on the basis of an upper bound
limit analysis will always remain either greater or equal to the magnitude of true collapse load, and unrestricted plastic flow must
impend when a kinematically admissible velocity field exists. The
virtual energy-work balance expression in the upper bound theorem
of limit analysis can be written mathematically as follows:
ð
ð
ð
(1)
Ti Vi dS þ Xi Vi dX ¼ D_ ɛ_ ij dX
S
X
X
where Ti = traction on the loaded boundary surface S; Xi = vector of
distributed forces within the region X including horizontal and vertical body forces; Vi = kinematically
admissible velocity field along
_ ɛ_ ij ¼ s_ ij ɛ_ ij = rate of incremental
the boundary surface S; and D
internal energy dissipation per unit volume corresponding to a strain
rate ɛ_ ij in the kinematically admissible velocity field, where s_ ij =
stress tensor associated with ɛ_ ij . Thus, the right side of Eq. (1)
implies the rate of total incremental internal energy dissipation during an incipient failure of soil medium.
Collapse Mechanism and Proposed Radial Shear Zone
With the assumption of six different kinds of collapse mechanisms,
Chen and Rosenfarb (1973) have determined the active earth force
on retaining walls subjected to translational displacement, in which
the composite collapse mechanism consisting of either circular
shearing zone or logarithmic spiral zone sandwiched between triangular rigid blocks, yielded better results when compared with
others. The circular shear zone was formed by assuming the velocities or displacement rates inclined at an angle equal to the internal
friction angle of soil ( f ) with the normal to the radial lines emerging from a singular point; whereas the log spiral shear zone in which
the inclination of velocities are normal to the all radial lines emerging from a singular point. In a similar fashion, it is attempted in this
paper to introduce a radial shear zone by taking the inclination of
velocities with the normal to the radial lines emerging from the singular point as variable, which has been found to yield better upper
bound solutions. Although the computed maximum magnitude of
active earth force from different collapse geometry does not differ
© ASCE
much from those obtained from a simple planar mechanism, in the
present study an improved composite collapse mechanism, as illustrated in Fig. 2, has been used for performing the analysis, in which
the velocity field is proposed using a radial shearing zone OBC
sandwiched between the two rigid triangular blocks OAB and OCD
that have planar discontinuities. Note that zone OBC is a continuous
deforming shear zone in which the energy dissipation occurs within
the region OBC and along the discontinuous shearing boundary surface BC, whereas blocks OAB and OCD are rigid where uniform
velocity field is implied because the energy would not be dissipated
within the region of these blocks except along the planar rigid
boundary surfaces AB and CD. In Fig. 2(a) the wall OA is prescribed to translate outward horizontally with a velocity V0, which
will cause the triangular blocks OAB and OCD to slide downward
with velocities V1 and V2, respectively. The relative velocity V01 of
block OAB with respect to the wall is kept parallel to the back face
of the wall, which is similar to that of Chen and Rosenfarb (1973),
for accounting interface characteristics of the soil and wall. For the
presented mechanism to be kinematically admissible, the direction
of tangential velocity jump vector V01 between the wall and the
adjoining soil block OAB must be directed downward for the problem under consideration. Fig. 2(b) gives the complete velocity hodograph of the proposed composite mechanism, and using this velocity hodograph the relationship among the velocities V1, V2, and V01,
respectively, can easily be interpreted in terms of any reference velocity of the system, which in this paper is taken to be the velocity
of wall V0. For a material obeying an associated flow rule, the displacement rates of each block are an inclined angle f with the discontinuous rigid boundary ABCD. Unlike in the earlier studies in
which velocities are assumed to be directed at an angle either equal
to zero or f from the normal to the radii of the continuous shearing
zone, it is considered in this paper that this makes an angle u , which
is a variable whose value ranges from 0 to f .
Velocity and Radius Increments in the Radial Shear Zone
The radial shearing zone with an included angle « is constructed in
this paper by assuming that it is comprised of n number of
qL′
λl
D
khqL′
α
E
β
O
ε
μ
khW3
W3
dχ
χ
C
rχ
khWχ
Cw
V2
r0
V01
δ
Pa
φ
V0
Wχ
α +μ -θ
khW2
khW1
W2
V1
Vχ
φ
W1
V0
φ
B
V01
α
V1
ε
V2
A
(a)
(b)
Fig. 2. (a) Composite collapse mechanism and (b) velocity hodograph
associated with composite collapse mechanism; the directions marked
for angles and forces are considered as positive in the analysis (Note:
The weights of various blocks denoted as W with subscripts 1, 2, and 3
indicate, respectively, the blocks OAB, OBC, and OCD)
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and
infinitesimal triangular rigid blocks and is characterized by a constant interior angle equal to D« (« /n), as illustrated in Fig. 3(a).
These blocks share a common apex point (singular point), and their
bases and sides constitute the discontinuous rigid shearing boundary surfaces separating the stationary and moving soil masses and
the discontinuous velocity surfaces. Let the blocks Oa0a1, Oa1a2…,
Oan-2an-1, and Oan-1an be prescribed to move with velocities equal
to V0, V1…, Vn-1, and Vn, respectively. All these velocities are uniform, constant, and make an angle of u normal to the respective radial lines emerging from the singular point. To satisfy an associated
flow rule, the relative velocity of block n – 1 with respect to block n
should be directed at an angle equal to f , as depicted in Fig. 3(b),
which shows the velocity hodograph of a particular elemental block
interface. Similarly, the inclination of the rigid shearing boundary
surfaces with the block velocities should also be equal to f .
Following the approach used by Chen (1975) and based on the compatible velocity diagram given in Fig. 3(b) for an infinitesimal elemental block interface, the velocity of the nth block can be given by
the following relation:
2
3
Dɛ
6 cos f þ u þ
7
2 7
6
Vn ¼ Vn1 6
(2)
7
4
Dɛ 5
cos f þ u 2
ε
2
Vn1
3
Dɛ
6 cos f þ u þ
7
2 7
6
¼ Vn2 6
7
4
Dɛ 5
cos f þ u 2
(3)
By substituting Eq. (3) into Eq. (2), Vn can be expressed in terms
of Vn-2 as
2
32
Dɛ
6 cos f þ u þ
7
2 7
6
Vn ¼ Vn2 6
7
4
Dɛ 5
cos f þ u 2
(4)
If the previous process is repeated n times, the velocity of the nth
block could be written in terms of the initial velocity V0 as follows:
2
3n
Dɛ
6 cos f þ u þ
7
2 7
6
Vn ¼ V0 6
7
4
Dɛ 5
cos f þ u 2
(5)
Initial radial line
O
Δε
Δε
Δε
Δε
rn
rn-1
an
Vn
Θ
Θ
φ
Vn-1
an-1
φ
r1
Final radial line
Note: Θ =
Θ
r0
π
2
φ
Θ
−θ
V1
V0
φ
a1
a0
(a)
1
(π − 2θ − Δε )
2
Δε
an-2
Surfaces
a0O
a1O
.
an-1O
anO
Description
Discontinuous
velocity surfaces
that exist between
the block
boundaries
a0a1
a1a2
.
an-2an-1
an-1an
Rigid shearing
boundary
surfaces between
stationary and
moving soil
V0
V1
.
Vn-1
Vn
Respective block
velocities which
is constant within
the block
Velocity
discontinuity surface
Vn-1
Vn
Vn-1, n
δu
δv
φ
δu: Tangential velocity jump vector
δv: Normal velocity jump vector
(b)
Fig. 3. (a) Proposed velocity field; (b) velocity hodograph for an infinitesimal element in the radial shearing zone
© ASCE
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Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
After some algebraic manipulations and substituting the limits
for n tending to infinity, the following expression for the velocity of
the nth block can be obtained:
Vn ¼ V0 eɛ tan ð f þu Þ
(6)
In a similar way, the following relation between the radius of initial and final radial lines can also be presented
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rn ¼ r0 eɛ tan ðf u Þ
(7)
Similar to Eqs. (6) and (7), the length of the radial line and velocity at an angle x from the initial radial line OB, as shown in Fig. 2,
will be presented as follows in Eqs. (8a) and (8b):
r x ¼ r0 e x
V x ¼ V1 e x
tanð f u Þ
tanð f þu Þ
(8a)
(8b)
Thus, the complete geometry of the collapse mechanism or the
velocity field can be constructed by using three independent variables, such as u , m , and « , where u = angle made by velocities with
normal to the radial lines whose value ranges from 0 to f ; m =
b
b and « = BOC.
AOB;
Incremental Energy Dissipation in the Radial Shearing Zone
During plastic shearing, the dissipation of energy is obvious along
the discontinuous rigid boundary surfaces that exist between the
moving and stationary masses, and along the block interfaces,
which combined produce the radial shearing zone. Following Chen
(1975) and referring to Fig. 3(b), the incremental energy dissipation
rate per unit length along the block interface, which represents the
velocity discontinuity surface, will be computed by the following
expression:
D_ ¼ crn1 d u
(9)
From Fig. 3(b), d u can be written as d u ¼ Vn1 sin Dɛ
cos f = cos ð f þ u Dɛ=2Þ, and the Taylor series expansion of
the resulting function from Eq. (9) for Dɛ ! 0 with the omission of
higher order terms involving Dɛ gives
D_ ¼
c cos f
rn1 Vn1 Dɛ
cosð f þ u Þ
For an infinitesimal rigid triangular block with an included angle
of D« , substituting the length of the boundary surface, Ln1 ¼
rn1 = cosð f u ÞDɛ from Fig. 2(a), the following expression will
be deduced from Eq. (12):
cr0 V0 cos f eɛ x 0 1
D_ base ¼
(13)
cosð f u Þ
x0
Thus, the total incremental energy dissipation rate during plastic
flow in the radial shear zone becomes
ɛx
e 0 1
1
1
D_ radial zone ¼ cr0 V0 cos f
þ
cosð f u Þ cosð f þ u Þ
x0
(14)
It can easily be shown that Eqs. (6), (7), (11), and (13) obtained
in this analysis are equivalent to the solutions of Chen (1975) presented for circular and log spiral shear zones by substituting u = f
and u = 0, respectively. Thus, the velocity field defined using this
new radial shear zone can have an obvious advantage of varying
freely in between the velocity field used for presenting the conventional circular to the log spiral shear zone; that is, when u varies
from f to zero, the shear zone and its velocity field changes corresponding to that of the circular to log spiral shear zone. The proposed shear zone is the generalized one in which the value of u
varies from 0 to f ; thus, it has an advantage of analyzing the range
of possible critical rupture surfaces for finding the critical collapse
load.
Seismic Active Earth Force
Maximum Seismic-Induced Active Force
After application of the upper bound theorem of limit analysis based
on the virtual energy-work balance expression presented in Eq. (1),
the resultant seismic active force Pa for an inclined rough wall in
the proposed rigid translational mechanism, satisfying the kinematical admissibility conditions, can be presented in a simplified form
as
Pa ðu ; m ; ɛÞ
¼
(10)
v_ 1 þ v_ 2 þ v_ 3 þ v_ q D_ 1 D_ 2 D_ 3 chV0 U1 ðu ; m Þ
V0 U2 ðu ; m Þ
(15)
in which
If the interior angles of blocks Dɛ are taken as this small, the
integration of Eq. (10) within the limits 0 to « will provide the dissi_ sides along the interfaces of the blocks that form vepation energy D
locity discontinuity surfaces within the radial shearing zone as
follows:
ɛx
0
_ sides ¼ cr0 V0 cos f e 1
D
(11)
cosð f þ u Þ
x0
where x 0 ¼ tan ð f u Þ tan ð f þ u Þ.
Again, the energy dissipation that takes place along the bases of
triangular blocks, that is, along the discontinuous rigid boundary
surface of the radial shear zone, can be expressed as
ð
_ base ¼ c Ln1 Vn1 cos f
D
K
© ASCE
(12)
tan d
U1 ð u ; m Þ ¼
tan f
sinða þ m u Þ
tan a
cos a cosð m u Þ
and
U2 ð u ; m Þ ¼
sinða þ m u Þ
sind þ cosða þ d Þ
cosð m u Þ
The terms v_ 1 , v_ 2 , and v_ 3 refer to the rate of work done by the
horizontal and vertical body forces in the regions OAB, OBC, and
OCD. v_ q is the work rate of the horizontal and vertical tractions on
_ 2 , and D_ 3 represent the
the loaded boundary surface ED, and D_ 1 , D
rate of energy dissipation along the discontinuity surfaces AB, DC,
and in the continually shearing zone OBC. Referring to Fig. 2, the
total work done v_ 2 by the horizontal and vertical body forces in the
radial shearing OBC can be given by the following expression:
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Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
v_ 2 ¼
8
>ðɛ
g<
2>
:
Kaq g s
kh r2x V x cosða þ m þ x u Þd x
þ r2x V x sinða þ m þ x u Þd x
9
=
;
(16)
Kaq g cs ¼
0
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The total work done v_ q by the horizontal and vertical tractions
on the loaded boundary surface ED can be presented as
v_ q ¼ qL0 V2 ½kh cos ða þ m þ ɛ u Þ þ sin ða þ m þ ɛ u Þ
(17)
where L0 is the effective length of the surcharge in the critical collapse mechanism, which can be expressed as Eqs. (18a) and (18b)
h
cosð m þ f u Þ
λ
0
ɛ tanð w u Þ
e
L ¼
cos a sinða þ m þ ɛ þ f b u Þ
cos b
(18a)
cos b cosð m þ f u Þ
eɛ tanð f u Þ λ
sinða þ m þ ɛ þ f b u Þ
(18b)
1
U2 ð u ; m Þ
(
)
3
8
X
X
Cn ðu ; m ;ɛÞ þ Nq C4 ðu ; m ;ɛÞ Nc
Cn ðu ; m ;ɛÞ
n¼1
Pa ðu ; m ; ɛÞ ¼
g h2
Ka g s þ qhKaqs chKacs
2
(19)
g h2
Kaq g s chKacs
2
(20)
g h2
Kaq g cs
2
(21)
Pa ðu ; m ; ɛÞ ¼
Pa ðu ; m ; ɛÞ ¼
where Ka g s , Kaqs , and Kacs = individual seismic active earth thrust
coefficients due to the components of soil unit weight, surcharge,
and soil cohesion, respectively; Kaqg s = combined seismic active
earth thrust coefficient, which describes the effect of both the soil
unit weight and surcharge components in terms of a nondimensional
factor Nq (2q/ g h); and Kaqg cs = combined seismic active earth thrust
coefficient, which describes the effect of all components in terms of
nondimensional factors Nq (2q/ g h) and Nc (2c/ g h). The expressions for finding coefficients Ka g s , Kaqs , and Kacs will be written as
follows in Eqs. (22a)–(22c):
Ka g s ¼
3
X
1
Cn ðu ; m ; ɛÞ
U2 ðu ; m Þ n¼1
(22a)
C4 ðu ; m ; ɛÞ
U2 ð u ; m Þ
(22b)
8
X
1
Cn ðu ; m ; ɛÞ
U2 ðu ; m Þ n¼5
(22c)
Kaqs ¼
Kacs ¼
Similarly, the following expressions can be deduced for finding
the combined seismic active earth force coefficients
n¼5
(24)
The earth thrust coefficients presented in Eqs. (19)–(24) correspond to the seismic condition. By simply putting kh = 0 into Eqs.
(15)–(17), the expressions given in Eqs. (15)–(24) will be applicable for static condition. The notations used for seismic earth thrust
coefficients Ka g s , Kaqs , Kacs , and Kaq g cs will be replaced with Ka g ,
Kaq , Kac , Kaq g , and Kaq g c , respectively, for denoting earth thrust
coefficients under static conditions.
The value of Cn ðu ; m ; ɛÞ for n = 1–8 has been presented in
Appendix I. The critical resultant active force can be found by maximizing Pa given in Eqs. (19)–(21) with respect to the variables u ,
m , and « , which are used to define the shape of the chosen composite collapse mechanism. Thus, the maximum magnitude of the resultant active force can be determined with the help of the following
equality conditions in Eqs. (25a)–(25c):
The resultant seismic active force Pa given in Eq. (15) can be
expressed in the following forms:
© ASCE
(23)
0
ðɛ
when
(
)
3
X
1
¼
Cn ðu ; m ; ɛÞ þ Nq C4 ðu ; m ; ɛÞ
U2 ðu ; m Þ n¼1
∂Pa ðu ; m ; ɛÞ
¼0
∂u
(25a)
∂Pa ðu ; m ; ɛÞ
¼0
∂m
(25b)
∂Pa ðu ; m ; ɛÞ
¼0
∂ɛ
(25c)
Application of the superposition principle enables the computation of the thrust coefficients defined in Eqs. (19)–(20), in which the
thrust coefficient corresponding to one component can be determined to be independent of other component(s). Using the principle
of superposition, solutions have been obtained for earth pressure
problems in static conditions (Chen and Rosenfarb 1973; Soubra
and Macuh 2002) and in seismic conditions (Saran and Prakash
1968; Chen and Liu 1990; Das and Puri 1996). However, it was
found that for the same properties of backfill and the same magnitudes of seismic acceleration coefficients, the total active force
obtained by Shukla et al. (2009) due to cohesion and self-weight of
backfill differs largely from that computed by Saran and Prakash
(1968), although the method of analysis is the same in both studies.
In the analysis by Saran and Prakash (1968), the failure surfaces
from the effect of soil weight and cohesion were optimized separately, which is of course too far from reality; whereas the total
active force was obtained due to the combined contribution of soil
weight and cohesion by optimizing a single failure plane (Shukla et
al. (2009). In the present analysis, the difference in the solutions
obtained with and without the superposition principle depend on the
magnitudes of kh, Nq, Nc, l , and f , which is discussed later.
Theoretically, the definition of the thrust coefficient given in Eq.
(21) is exact when the factors g , q, and c are all greater than zero.
This is because of the realization of the unique failure surface from
a mathematical standpoint. However, such a definition requires two
additional unbounded factors, such as Nq and Nc, for presenting the
numerical solutions. Thus, in the present study, the computations
were performed with and without the superposition principle,
04017123-6
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Int. J. Geomech.
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depending on the dependence of the thrust coefficients on factors Nq
and Nc.
Consideration of the Influence of the Preexisting Crack in
the Backfill
The previously developed formulation has been extended to consider that the possible effect of tension crack in the cohesivefrictional backfill on the magnitude of active earth force is essential for both static and seismic conditions. It is also known that the
region up to the crack depth in the backfill does not permit any development of soil resistance against the failure of soil mass under
the active state behind a wall. Therefore, the soil mass in the cracked
region should be treated as an additional surcharge that acts on the
remaining part of backfill lying below the cracked zone while evaluating the total active force. The maximum expected depth of crack
(zc) to be developed vertically under soil tension may be readily
taken into account if available or computed alternatively using the
classical Rankine earth pressure theory
2c
p f
zc ¼ tan
þ
(26)
g
4
2
It can be noted that the analytical expression provided by
Mazindrani and Ganjali (1997) for the depth of crack is the same as
that given in Eq. (26), which is independent of backfill slope angle.
This is contradictory to the finding of Nian and Han (2013), in
which the depth of crack is a function of backfill slope angle.
Nevertheless, the present formulation addresses only the effect of
preexisting dry cracks on the seismic active earth thrust; therefore,
the depth of crack given in Eq. (26) is expressed independent of the
seismic earth thrust coefficient, backfill slope angle, and orientation
of wall. The use of this approximation for accounting the effects of
a preexisting crack in the formulation of seismic active earth thrust
has also been reported by Das and Puri (1996). Further, the present
formulation also can be used explicitly if the expected depth of
crack is known. Accordingly, the seismic active earth thrust coefficient presented in Eq. (24) will be rewritten considering the influence of preexisting crack as
9
8
3
>
>
X
>
>
>
>
2
0
>
>
ð
j
Þ
C
u
;
m
;
ɛ
þ
N
C
u
;
m
;
ɛ
1
ð
Þ
ð
Þ
>
>
n
q
4
=
<
1
0
n¼1
Kaq g cs ¼
8
X
>
U2 ð u ; m Þ >
>
>
>
>
ð1 j ÞNc Cn ðu ; m ; ɛÞ
>
>
>
>
;
:
n¼5
(27)
and from Eq. (21)
Pa ð u ; m ; ɛ Þ ¼
g h2 0
K
2 aq g cs
(28)
where j (zc/h) = normalized depth of tension crack whose value
ranges from 0 to 1. The value of C04 ðu ; m ; ɛÞ is given in Appendix.
I. Eq. (28) is therefore a generalized expression that can be used to
find the resultant static and seismic active earth force exerted by cohesive-frictional or purely frictional soil as backfill supported by a
wall while considering the influence of surcharge loadings and tension crack in the backfill.
Solutions and Discussions
The active earth thrust coefficients defined in Eqs. (22)–(27), which
are used for determining the resultant active earth thrust, have been
computed in this paper separately by using computer code in
© ASCE
MATLAB. The numerical solutions for active earth thrust coefficients obtained from the present analysis under both static and seismic conditions have been presented for two situations of surcharge
on the backfill: (1) uniform surcharge applied throughout the backfill (l = 0) and (2) uniform surcharge applied at a certain distance
away from the backfill (l > 0).
Uniform Surcharge Applied throughout the Backfill (k = 0)
Static Condition. The variation of static active earth thrust coefficients Ka g and Kaq , and Kac with b / f for different combination of
parameters, such as d /f , a, and f , are shown in Figs. 4 and 5. Fig. 4
shows that for any given value of f , the magnitude of both coefficients Ka g and Kaq increases continuously with an increase in the
values of b / f from negative to positive, and attains its maximum
values corresponding to the limiting value of b / f equal to 1
beyond which the sloping backfill of purely frictional material can
no longer be in stable condition. This is one of the consequences of
the application of the superposition rule in obtaining solutions that
are applicable for cohesive-frictional backfill in which the sloping
backfill can be stable by itself, even for b / f > 1. The computed
magnitude of coefficients Ka g and Kaq shows an increasing trend
when the value of a changes from negative to positive. This
increase in the magnitude of thrust coefficients due to the change in
orientation of the wall from negative to positive is attributed to the
involvement of a large plastic region of soil mass behind the wall at
the time of collapse under the active state. There is a marginal
decrease or increase in the magnitude of coefficients Ka g and Kaq
with respect to change in the values of the soil-wall interface friction depending on the parameters f , b / f , and a. Further, a backfill with higher internal friction angle reduces significantly the
magnitude of coefficients Ka g and Kaq . It is confirmed that the
magnitude of coefficient Kac can be related to Kaq for the case in
which b = a = 0 by using the theorem of Caquot and Kerisel
(1948), and it is expressed in the following:
1
Kaq
Kac ¼ cos d
tan f
(29)
The parameters u , m , and « , which define the critical collapse
geometry for finding coefficients Kac and Kaq , remain exactly the
same, implying that the solutions are free from errors resulting from
the superposition rule. This is realized mainly because of the adoption of the magnitude c(tan d /tan f ) as the soil-wall interface adhesion per unit length of the wall. For given values of d / f , a, and f ,
the magnitude of coefficient Kac is found to increase when the value
of b / f changes from its negative to positive values. Significant
reduction in the values of Kac occurs when there is an increase in the
values of f and a decrease in the value of d . The changes in wall
orientation from positive to negative values have been found to
reduce the magnitudes of Kac in almost all cases.
Seismic Condition. Computations revealed that the magnitude of
Kacs is independent of earthquake acceleration coefficient kh. The
seismic independency of Kacs can also be easily confirmed by looking into Eq. (22c); hence, it can be said that Kacs ¼ Kac . This seismic
independency of coefficient Kacs arises mainly due to the application of the superposition rule in obtaining solutions for soils with
g = 0. Also, the use of the coefficients Kacs for designing retaining
structures with cohesive-frictional backfill in seismic conditions
may become uneconomical unless the error between the values
computed and the true values obtained including the weight of
backfill mass becomes marginal. Despite this fact, only the variation
of Ka g s and Kaqs with kh for different combinations of parameters,
04017123-7
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
1.4
1.4
α = +20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
1
= +10°
= 0°
= -10°
= -20°
Kaγ , Kaq
Kaγ , Kaq
Kaγ , Kaq
0.6
Kaγ
Kaq
Ka γ
Kaq
0.2
-0.5
0
0.5
1
-0.5
β/φ
0
0.5
0.2
1
-0.5
= +10°
= 0°
= -10°
= -20°
α = +20°
φ = 30°, δ/φ = 0.5
= +10°
= 0°
= -10°
= -20°
0.9
Kaγ , Kaq
Ka γ
Kaq
0.5
0.5
-0.5
0
0.5
0.1
-0.5
1
0
0.5
1
-0.5
1.2
1.2
α = +20°
φ = 40°, δ/φ = 0
= +10°
= 0°
= -10°
= -20°
0.8
α = +20°
φ = 40°, δ/φ = 0.5
= +10°
= 0°
= -10°
= -20°
Kaγ , Kaq
Kaγ , Kaq
Kaγ , Kaq
0.4
Ka γ
Kaq
0
0
-0.5
0
0.5
1
-0.5
β/φ
Ka γ
Kaq
0.4
0.4
0
φ = 40°, δ/φ = 1
0.8
0.8
Ka γ
Kaq
1
(f)
1.2
= +10°
= 0°
= -10°
= -20°
0.5
β/φ
(e)
α = +20°
0
β/φ
β/φ
(d)
Ka γ
Kaq
0.5
0.1
0.1
φ = 30°, δ/φ = 1
0.9
Kaγ , Kaq
Kaγ , Kaq
Ka γ
Kaq
1
1.3
α = +20°
φ = 30°, δ/φ = 0
0.9
0.5
(c)
1.3
1.3
= +10°
= 0°
= -10°
= -20°
0
β/φ
(b)
α = +20°
Kaγ
Kaq
β /φ
(a)
φ = 20°, δ/φ = 1
1
0.6
0.2
(g)
α = +20°
φ = 20°, δ/φ = 0.5
1
0.6
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1.4
α = +20°
φ = 20°, δ/φ = 0
0
0.5
-0.5
1
(h)
0
0.5
1
β/φ
β/φ
(i)
Fig. 4. Variation of Kag and Kaq with b / f and a for (a) f = 20° and d / f = 0, (b) f = 20° and d / f = 0.5, (c) f = 20° and d / f = 1, (d) f = 30° and
d / f = 0, (e) f = 30° and d / f = 0.5, (f) f = 30° and d / f = 1, (g) f = 40° and d / f = 0, (h) f = 40° and d / f = 0.5, and (i) f = 40° and d / f = 1
© ASCE
04017123-8
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
2.2
φ = 20°, δ/φ = 0
1.7
Kac
Kac
1.4
α = +20°
0.5
1
β /φ
(a)
1.2
-0.5
0
1.7
φ = 30°, δ/φ = 0
0.5
Kac
Kac
Kac
1.3
α = +20°
0.5
1
β /φ
(d)
0.9
-0.5
0
1.5
1.2
0.5
-0.5
1
β /φ
(e)
φ = 40°, δ/φ = 0
Kac
Kac
Kac
1.1
α = +20°
0.5
α = +20°
= +10°
= 0°
= -10°
= -20°
0.7
0.6
0
φ = 40°, δ/φ = 1
= +10°
= 0°
= -10°
= -20°
0.6
1
-0.5
β /φ
1
1.5
= +10°
= 0°
= -10°
= -20°
-0.5
0.5
β /φ
1.9
φ = 40°, δ/φ = 0.5
0.9
α = +20°
0
(f)
1.2
0.9
α = +20°
= +10°
= 0°
= -10°
= -20°
0.8
0
φ = 30°, δ/φ = 1
= +10°
= 0°
= -10°
= -20°
0.8
1
1.7
= +10°
= 0°
= -10°
= -20°
-0.5
0.5
β /φ
2.1
φ = 30°, δ/φ = 0.5
1.1
α = +20°
0
(c)
1.4
1.1
1.5
-0.5
1
β /φ
(b)
1.4
(g)
= +10°
= 0°
= -10°
= -20°
1
0
α = +20°
= +10°
= 0°
= -10°
= -20°
0.9
1.7
1.6
α = +20°
= +10°
= 0°
= -10°
= -20°
-0.5
φ = 20°, δ/φ = 1
2
1.8
1.3
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2.4
φ = 20°, δ/φ = 0.5
Kac
2.1
0
0.5
-0.5
1
(h)
0
0.5
1
β /φ
β /φ
(i)
Fig. 5. Variation of Kac with b / f and a for (a) f = 20° and d / f = 0, (b) f = 20° and d / f = 0.5, (c) f = 20° and d / f = 1, (d) f = 30° and d / f = 0,
(e) f = 30° and d / f = 0.5, (f) f = 30° and d / f = 1, (g) f = 40° and d / f = 0, (h) f = 40° and d / f = 0.5, and (i) f = 40° and d / f = 1
© ASCE
04017123-9
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
such as d / f , b / f , a, and f , are shown in Figs. 6–8. According to
the concept of shear fluidization (Richards et al. 1990) and stability
criteria of slopes (Sarma 1999), the maximum value of b / f is restricted in this paper to obtaining the results satisfying the following
expression:
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jb j
tan1 ðkh Þ
1
f
f
(30)
Figs. 6–8 show that the magnitude of both the coefficients, Ka g s
and Kaqs , has been found to increase always with an increase in the
magnitude of kh irrespective of the magnitudes of d / f , b / f , a, and
f . This shows that the assumed direction for earthquake acceleration is critical; thus, the larger magnitudes of Kaqs are obtained
when the critical direction of kh is considered to act along the direction of the wall movement. Any increase in the values of f reduces
the magnitudes of coefficients Ka g s and Kaqs . On the other hand, the
changes in the values of d / f may either increase or decrease the
magnitudes of Ka g s and Kaqs depending on the parameters b / f , a,
and f . Nevertheless, this increase or decrease in the magnitudes of
Ka g s and Kaqs with respect to changes in d / f is only marginal for
negatively sloped backfill surfaces compared with that of the horizontal and positively sloped backfill surfaces. Further, some peculiar observations between the magnitudes of Ka g s and Kaqs with
changes in the values of wall orientation and backfill slope angles
are noticed. The magnitude of Kaqs in the case of negatively sloped
backfill surface is found to be (1) always greater than or equal to
Ka g s for positively inclined walls and (2) lower or higher than the
magnitude of Ka g s for negatively inclined walls depending on the
values of kh, d / f , and f .On the other hand, in the case of positively sloped backfill surface, the magnitude of Kaqs is found to be
(1) always less than or equal to Ka g s for positively inclined walls
and (2) always slightly higher than the value of Ka g s for negatively
inclined walls irrespective of the value of kh. A similar kind of trend
has also been observed for all the values of d / f and f that have
been taken into consideration in this analysis.
Uniform Surcharge Applied at a Certain Distance Away from
the Backfill (k > 0)
The usage of the superposition rule leads to the conclusion that the
coefficient Kac remains unaffected with changes in the magnitude
of l . This can also be confirmed from the expressions presented for
Kac , as given in Eq. (22c). In fact, this observation makes no sense
for retaining walls with a general cohesive-frictional backfill
because this type of presentation inherently neglects the effects of
the combined interaction that exists among the various factors. In
such situations, the accuracy of the solution, therefore, largely
depends on the absolute magnitude of factors, such as Nq and Nc.
Nevertheless, in this section with the assumption of Nc = 0, the solutions have been presented. Unlike in the case l = 0, the total active
earth resistance cannot be derived from the independent contribution of the surcharge and soil weight components based on the principle of superposition. Such an application of this principle in
obtaining solutions for l > 0 does not guarantee accuracy unless
the difference between the solutions obtained with and without the
superposition rule are negligible. Hence, the following sections
present the numerical solutions for static and seismic active earth
thrust coefficients based on the definition introduced in Eq. (23).
The necessity of this kind of definition for thrust coefficients is
explained briefly later.
Static Condition. The variations of Kaq g with l for different
combinations of Nq (2q/ g h), b / f , d / f , a, and f are presented in
Figs. 9–11. The magnitudes of Kaq g decrease continuously with an
© ASCE
increase in the value of l up to a certain critical value, which is
referred to as the critical surcharge distance ratio l cr, beyond which
the magnitudes of Kaqg become exactly equal to that of Ka g , indicating that the resultant active force is independent of the magnitudes of Nq. However, the value of critical surcharge distance ratio
l cr is found to vary largely depending on the magnitude of Nq for
any given combination of parameters, such as b / f , d / f , a, and f .
For a given orientation of wall, higher magnitudes of Kaq g are
always realized with a positively sloped backfill surface compared
with that of the horizontal backfill. A considerable decrease or
increase in the magnitudes of Kaq g is found depending on the values
of Nq and f . Generally, the value of l cr has been found to increase
with an increase in Nq, whereas the same decreases with an increase
in f . Further, it has been found that the value of l cr for retaining
walls (1) with negative inclination is always lower and (2) with positive inclination is higher compared with that of vertical walls. For
any given orientation of wall and backfill friction angle, the changes
in the magnitude of l cr may also become significant when the backfill slope angles change from positive to negative values. Generally,
the magnitude of l cr is found to reduce when the backfill slope
angles change from positive to negative values.
Seismic Condition. In Figs. 12–14, the variations of Kaq g s with kh
and l for d / f = 2/3 with different combinations of Nq, f , b / f ,
and a have been presented. Similar to the static case, a significant
reduction in the magnitudes of Kaq g s has been observed with an
increase in the value of l up to a certain critical surcharge distance
ratio l cr for any values of kh. Under any given set of parameters, the
minimum values of Kaqg s become exactly equal to that of Ka g s .
This shows that for values of l equal to or greater than l cr, the
magnitudes of Kaq g s are independent of surcharge pressure. Any
increase in the values of kh always results in (1) an increase in the
magnitude of Kaq g s and (2) a reduction in the magnitude of l cr. The
values of critical surcharge distance ratio l cr are found to be highly
dependent on the magnitudes of kh, Nq, f , b / f , and a. Generally,
the magnitude of l cr decreases when the backfill sloping surface
and orientation of the wall changes from its positive to negative values. A similar trend can also be noticed with an increase in the values of f .
Influence of Tension Cracks in the Backfill
For b / f = 0.5, d / f = 0.5, a = 0°, and Nc = 0.1, the variations of
0
Kaq
g cs with l for different combinations of Nq, f , and kh have been
shown in Fig. 15. Two cases have been considered in this paper:
(1) backfill free from any crack, i.e., j = 0 and (2) backfill possessing a crack, i.e., j > 0 in which the value of the crack depth is
calculated based on Eq. (26). The consideration of the preexisting
crack in the formulation may not always be a critical scenario,
0
that is, the magnitudes of Kaq
g cs computed for j > 0 may be either
less than or greater than that computed without considering crack
in the backfill depending on the magnitudes of Nc and Nq.
However, a clear-cut distinction between the values of critical
surcharge distance ratio l cr computed for the two cases has been
noted. The definition of l cr carries the same meaning that was
0
explained in previous sections, that is, the magnitudes of Kaq
g cs
corresponding to l cr will be free from the influence of the magnitude of backfill surcharge. For given values of kh, Nq, f , b / f ,
and a, the values of l cr always have been found to reduce in the
presence of crack in the backfill. This reduction in the values of
l cr may be due to the fact that the presence of vertical cracks
decreases the effective span of surcharge (L0 ) toward estimating
0
the magnitudes of Kaq
g cs from the critical active failure mechanisms formed in the backfill. For the same reason, depending on
the magnitudes of Nc and Nq, the potential driving forces due to
external surcharge pressure acting on the cracked zone and its
04017123-10
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
1
1
φ = 20°, δ/φ = 0, β/φ = -0.15
α = +20°
α = +20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
Kaγs, Kaqs
0.7
0.6
Ka γ s
Kaqs
0.4
0.4
0.4
0.1
kh
0.2
0.3
(a)
0
0.1
kh
0.2
1.4
φ = 20°, δ/φ = 0.5, β/φ = 0
α = +20°
α = +20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
0.9
1
0.6
Ka γ s
Kaqs
0.2
Ka γ s
Kaqs
0.3
0.1
kh
0.2
0.3
(d)
0.2
0
0.1
kh
0.2
0.3
(e)
0.9
1.2
φ = 20°, δ/φ = 1, β/φ = 0
α = +20°
α = +20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
0.9
Kaγs, Kaqs
0.9
0.6
Ka γ s
Kaqs
0.3
0.2
0.3
Ka γ s
Kaqs
0.3
kh
0.2
0.6
Ka γ s
Kaqs
0.1
kh
φ = 20°, δ/φ = 1, β/φ = +0.15
α = +20°
0.6
0
0.1
(f)
1.2
φ = 20°, δ/φ = 1, β/φ = -0.15
0
Kaγs, Kaqs
1.2
0.3
0.6
Ka γ s
Kaqs
0
0.2
Kaγs, Kaqs
Kaγs, Kaqs
0.5
kh
φ = 20°, δ/φ = 0.5, β/φ = +0.15
α = +20°
Kaγs, Kaqs
0.8
0.1
(c)
1.2
φ = 20°, δ/φ = 0.5, β/φ = -0.15
0
0.3
(b)
1.1
(g)
Kaγs
Kaqs
Kaγs
Kaqs
Kaγs, Kaqs
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1
Kaγs, Kaqs
0.8
0.6
0
φ = 20°, δ/φ = 0, β/φ = +0.15
α = +20°
Kaγs, Kaqs
0.8
1.3
φ = 20°, δ/φ = 0, β/φ = 0
0.3
0.3
0
0.1
kh
0.2
(h)
0.3
0
0.1
kh
0.2
0.3
(i)
Fig. 6. Variation of Kag s and Kaqs with kh and a for f = 20° with (a) d / f = 0 and b / f = –0.15, (b) d / f = 0 and b / f = 0, (c) d / f = 0 and b / f =
þ0.15, (d) d / f = 0.5 and b / f = –0.15, (e) d / f = 0.5 and b / f = 0, (f) d / f = 0.5 and b / f = þ0.15, (g) d / f = 1 and b / f = –0.15, (h) d / f = 1 and
b / f = 0, and (i) d / f = 1 and b / f = þ0.15
© ASCE
04017123-11
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
0.8
0.8
φ = 30°, δ/φ = 0, β/φ = -0.4
= +10°
= 0°
= -10°
= -20°
Kaγs, Kaqs
Kaγs, Kaqs
Kaγs, Kaqs
0.5
Ka γ s
Kaqs
Ka γ s
Kaqs
Ka γ s
Kaqs
0.2
0.2
0.1
kh
0.2
0.2
0
0.3
(a)
0.1
kh
0.2
0.3
(b)
0.7
= +10°
= 0°
= -10°
= -20°
0.8
0.4
1.4
φ = 30°, δ/φ = 0.5, β/φ = 0
Ka γ s
Kaqs
0.6
0.2
0.3
α = +20°
= +10°
= 0°
= -10°
= -20°
1
Kaγs, Kaqs
Kaγs, Kaqs
kh
φ = 30°, δ/φ = 0.5, β/φ = +0.4
= +10°
= 0°
= -10°
= -20°
0.4
0.3
0.1
α = +20°
Kaγs, Kaqs
0.5
0
(c)
0.8
φ = 30°, δ/φ = 0.5, β/φ = -0.4
0.6
α = +20°
= +10°
= 0°
= -10°
= -20°
0.1
0
0.1
kh
0.2
Kaγs
Kaqs
(d)
0.2
0
0.1
kh
0.2
0.3
(e)
1
1.7
φ = 30°, δ/φ = 1, β/φ = 0
α = +20°
α = +20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
1.2
Kaγs, Kaqs
0.7
0.4
Ka γ s
Kaqs
0.1
0.1
kh
0.2
0.2
0.3
0.3
0.7
Ka γ s
Kaqs
0.1
kh
φ = 30°, δ/φ = 1, β/φ = +0.4
α = +20°
0.4
0
0.1
(f)
φ = 30°, δ/φ = 1, β/φ = -0.4
0.7
0
Kaγs, Kaqs
1
(g)
Ka γ s
Kaqs
0.2
0.3
Kaγs, Kaqs
Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved.
α = +20°
= +10°
= 0°
= -10°
= -20°
0.6
0.4
0
φ = 30°, δ/φ = 0, β/φ = +0.4
α = +20°
α = +20°
0.6
1.1
φ = 30°, δ/φ = 0, β/φ = 0
Kaγs
Kaqs
0.2
0
0.1
kh
0.2
(h)
0.3
0
0.1
kh
0.2
0.3
(i)
Fig. 7. Variation of Kag s and Kaqs with kh and a for f = 30° with (a) d / f = 0 and b / f = –0.4, (b) d / f = 0 and b / f = 0, (c) d / f = 0 and b / f =
þ0.4, (d) d / f = 0.5 and b / f = –0.4, (e) d / f = 0.5 and b / f = 0, (f) d / f = 0.5 and b / f = þ0.4, (g) d / f = 1 and b / f = –0.4, (h) d / f = 1 and b / f =
0, and (i) d / f = 1 and b / f = þ0.4
© ASCE
04017123-12
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
0.7
φ = 40°, δ/φ = 0, β/φ = -0.55
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
0.8
Kaγs, Kaqs
0.3
0.4
0.3
Ka γ s
Kaqs
Kaγs
Kaqs
0.1
0
0.1
0
0.1
kh
0.2
0.3
(a)
0
0.1
kh
0.2
α = +20°
α = +20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
0.5
1
Kaγs, Kaqs
Ka γ s
Kaqs
Ka γ s
Kaqs
0.1
0
0.1
kh
0.2
0.3
(d)
0
0.1
kh
0.2
2.1
φ = 40°, δ/φ = 1, β/φ = 0
α = +20°
α = +20°
= +10°
= 0°
= -10°
= -20°
= +10°
= 0°
= -10°
= -20°
0.6
1.4
Kaγs
Kaqs
0
0
kh
Kaγs
Kaqs
0
0
0.3
0.3
0.7
Kaγs
Kaqs
0.2
0.2
Kaγs, Kaqs
0.3
0.1
kh
φ = 40°, δ/φ = 1, β/φ = +0.55
= +10°
= 0°
= -10°
= -20°
0.3
0
0.1
α = +20°
Kaγs, Kaqs
0.6
0
(f)
0.9
φ = 40°, δ/φ = 1, β/φ = -0.55
Ka γ s
Kaqs
0
0.3
(e)
0.9
0.3
0.5
0.3
0
0.2
φ = 40°, δ/φ = 0.5, β/φ = +0.55
= +10°
= 0°
= -10°
= -20°
0.23
(g)
1.5
φ = 40°, δ/φ = 0.5, β/φ = 0
α = +20°
Kaγs, Kaqs
0.46
kh
(c)
0.7
φ = 40°, δ/φ = 0.5, β/φ = -0.55
0.1
Kaγs, Kaqs
0.69
0
0.3
(b)
Kaγs, Kaqs
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α = +20°
= +10°
= 0°
= -10°
= -20°
0.5
Kaγs, Kaqs
0.5
φ = 40°, δ/φ = 0, β/φ = +0.55
α = +20°
Ka γ s
Kaqs
α = +20°
1.2
φ = 40°, δ/φ = 0, β/φ = 0
Kaγs, Kaqs
0.7
0.1
kh
0.2
(h)
0.3
0
0.1
kh
0.2
0.3
(i)
Fig. 8. Variation of Kag s and Kaqs with kh and a for f = 40° with (a) d / f = 0 and b / f = –0.55, (b) d / f = 0 and b / f = 0, (c) d / f = 0 and b / f =
þ0.55, (d) d / f = 0.5 and b / f = –0.55, (e) d / f = 0.5 and b / f = 0, (f) d / f = 0.5 and b / f = þ0.55, (g) d / f = 1 and b / f = –0.55, (h) d / f = 1 and
b / f = 0, and (i) d / f = 1 and b / f = þ0.55
© ASCE
04017123-13
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
1.2
β/φ = +0.5
β /φ = 0
β/φ = -0.5
Kaqγ
Kaqγ
φ = 20°, δ/φ = 0, α = -20°
0.4
0.8
(a)
λ
1.2
0.8
φ = 20°, δ/φ = 0, α = 0°
0.3
1.6
2
0
0.4
0.8
(b)
λ
1.2
0.4
0.8
λ
1.2
1.6
0.9
1.6
2
β/φ = +0.5
β /φ = 0
β/φ = -0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Kaqγ
Kaqγ
0
β /φ = 0
β/φ = -0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.6
2
β/φ = +0.5
β /φ = 0
β/φ = -0.5
0.8
1.6
(c)
1.2
β/φ = +0.5
φ = 20°, δ/φ = 0, α = +20°
0.4
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.2
Kaqγ
0
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.2
0.6
0.8
0.6
0.4
φ = 20°, δ/φ = 0.5, α = -20°
0
0.4
0.8
(d)
λ
1.2
φ = 20°, δ/φ = 0.5, α = 0°
0.3
1.6
2
0
0.4
0.8
(e)
λ
1.2
0.4
0.8
λ
1.2
1.6
Kaqγ
2
β /φ = 0
β/φ = -0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.9
1.6
β/φ = +0.5
β /φ = 0
β/φ = -0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.6
0
2
β/φ = +0.5
β /φ = 0
β/φ = -0.5
0.8
1.6
(f)
1.2
β/φ = +0.5
φ = 20°, δ/φ = 0.5, α = +20°
0.4
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.2
Kaqγ
0.2
Kaqγ
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Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.9
0.5
0.3
β/φ = +0.5
β /φ = 0
β/φ = -0.5
β/φ = 0
β/φ = -0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.7
1.6
β/φ = +0.5
Kaqγ
0.9
0.6
0.8
0.4
φ = 20°, δ/φ = 1, α = -20°
0.2
0
(g)
0.4
0.8
λ
1.2
φ = 20°, δ/φ = 1, α = 0°
0.3
1.6
0
2
(h)
0.4
0.8
λ
1.2
φ = 20°, δ/φ = 1, α = +20°
0.4
1.6
2
0
(i)
0.5
1
λ
1.5
2
2.5
Fig. 9. Variation of Kaqg with l , Nq, and b / f for f = 20°: (a) d / f = 0 and a = –20°; (b) d / f = 0 and a = 0°; (c) d / f = 0 and a = þ20°; (d) d / f =
0.5 and a = –20°; (e) d / f = 0.5 and a = 0°; (f) d / f = 0.5 and a = þ20°; (g) d / f = 1 and a = –20°; (h) d / f = 1 and a = 0°; (i) d / f = 1 and a = þ20°
© ASCE
04017123-14
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
β /φ = 0
β/φ = -0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.6
Kaqγ
Kaqγ
β/φ = +0.5
β/φ = 0
β/φ = -0.5
β /φ = 0
β/φ = -0.5
0.8
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.5
1.4
β/φ = +0.5
β/φ = +0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1
Kaqγ
0.7
0.3
0.6
φ = 30°, δ/φ = 0, α = -20°
0
0.2
0.4
(a)
λ
0.7
φ = 30°, δ/φ = 0, α = 0°
0.2
0.6
0.8
0
1
0.5
(b)
1
λ
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Kaqγ
Kaqγ
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.6
φ = 30°, δ/φ = 0.5, α = -20°
0.4
(d)
λ
0.6
0.8
β /φ = 0
β/φ = -0.5
0.5
(e)
1
λ
1
0
1.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Kaqγ
φ = 30°, δ/φ = 1, α = -20°
λ
0.8
1.8
β/φ = +0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.1
0.7
φ = 30°, δ/φ = 1, α = 0°
0.1
1.2
1.2
β/φ = 0
β/φ = -0.5
0.4
0.4
λ
1.5
β/φ = +0.5
0.7
0.2
0
0.6
β /φ = 0
β/φ = -0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0
φ = 30°, δ/φ = 0.5, α = +20°
0.2
(f)
β /φ = 0
β/φ = -0.5
0.4
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1
φ = 30°, δ/φ = 0.5, α = 0°
0
β/φ = +0.5
1.5
0.6
0.2
1.2
1
β/φ = +0.5
Kaqγ
0
λ
1.4
0.4
0.1
0.5
β /φ = 0
β/φ = -0.5
0.3
(g)
0
β/φ = +0.5
β /φ = 0
β/φ = -0.5
0.5
1.5
(c)
0.8
β/φ = +0.5
φ = 30°, δ/φ = 0, α = +20°
0.2
Kaqγ
0.1
Kaqγ
Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved.
0.4
0
0.5
λ
φ = 30°, δ/φ = 1, α = +20°
0.3
1
(h)
1.5
0
0.6
λ
1.2
1.8
(i)
Fig. 10. Variation of Kaqg with l , Nq, and b / f for f = 30°: (a) d / f = 0 and a = –20°; (b) d / f = 0 and a = 0°; (c) d / f = 0 and a = þ20°; (d) d / f =
0.5 and a = –20°; (e) d / f = 0.5 and a = 0°; (f) d / f = 0.5 and a = þ20°; (g) d / f = 1 and a = –20°; (h) d / f = 1 and a = 0°; (i) d / f = 1 and a = þ20°
© ASCE
04017123-15
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
0.7
β/φ = +0.5
β /φ = 0
β/φ = -0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Kaqγ
0.5
Kaqγ
0.2
β/φ = +0.5
β /φ = 0
β/φ = -0.5
β /φ = 0
β/φ = -0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.3
1.1
β/φ = +0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.8
Kaqγ
0.4
0.5
0.3
φ = 40°, δ/φ = 0, α = -20°
0
0.3
(a)
λ
0.3
0.6
φ = 40°, δ/φ = 0, α = 0°
0.1
0.9
0
0.4
(b)
Kaqγ
Kaqγ
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
φ = 40°, δ/φ = 0.5, α = -20°
0.3
0.6
0.4
(e)
0
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Kaqγ
0.5
φ = 40°, δ/φ = 1, α = -20°
λ
0.6
1
1.4
β/φ = +0.5
β/φ = +0.5
1
φ = 40°, δ/φ = 1, α = 0°
0
(h)
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.6
0.1
0.9
1.5
β /φ = 0
β/φ = -0.5
0.3
0.3
λ
β /φ = 0
β/φ = -0.5
0.1
0
0.5
(f)
0.7
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0
φ = 40°, δ/φ = 0.5, α = +20°
0.2
0.8
λ
β /φ = 0
β/φ = -0.5
(g)
0.8
φ = 40°, δ/φ = 0.5, α = 0°
0
0.9
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.5
0.1
β/φ = +0.5
0.2
β/φ = +0.5
Kaqγ
(d)
λ
1.2
β /φ = 0
β/φ = -0.5
0.3
0.3
0.8
1.1
β/φ = +0.5
0.5
0.1
0
λ
β /φ = 0
β/φ = -0.5
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0
0.4
(c)
β /φ = 0
β/φ = -0.5
0.2
0
0.8
λ
0.7
β/φ = +0.5
φ = 40°, δ/φ = 0, α = +20°
0.2
Kaqγ
0
Kaqγ
Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved.
0.1
0.4
λ
φ = 40°, δ/φ = 1, α = +20°
0.2
0
0.8
(i)
0.5
λ
1
1.5
Fig. 11. Variation of Kaqg with l , Nq, and b / f for f = 40°: (a) d / f = 0 and a = –20°; (b) d / f = 0 and a = 0°; (c) d / f = 0 and a = þ20°; (d) d / f =
0.5 and a = –20°; (e) d / f = 0.5 and a = 0°; (f) d / f = 0.5 and a = þ20°; (g) d / f = 1 and a = –20°; (h) d / f = 1 and a = 0°; (i) d / f = 1 and a = þ20°
© ASCE
04017123-16
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
1.6
kh = 0.3
kh = 0.2
kh = 0.1
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.2
φ = 20°, β/φ = -0.15, α = -20°
0.3
0
0.7
λ
1.4
φ = 20°, β/φ = -0.15, α = 0°
0
0.8
λ
1.6
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.4
0.7
φ = 20°, β/φ = 0, α = -20°
0
0.8
λ
1.6
φ = 20°, β/φ = 0, α = 0°
2.4
0
0.9
λ
1.8
2.7
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.6
φ = 20°, β/φ = +0.15, α = -20°
0.2
0
0.9
λ
1.8
2
3
kh = 0.3
kh = 0.2
kh = 0.1
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.4
φ = 20°, β/φ = +0.15, α = 0°
0.4
0
2.7
λ
2.3
1
0.8
1
3.2
kh = 0.3
kh = 0.2
kh = 0.1
Kaqγs
1.4
0
Kaqγs
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
φ = 20°, β/φ = 0, α = +20°
(f)
2.2
kh = 0.3
kh = 0.2
kh = 0.1
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.5
(e)
2
2.7
1.2
0.4
(d)
1.8
kh = 0.3
kh = 0.2
kh = 0.1
1.9
0.9
0.3
λ
2.6
kh = 0.3
kh = 0.2
kh = 0.1
Kaqγs
Kaqγs
1.1
0.9
(c)
1.9
kh = 0.3
kh = 0.2
kh = 0.1
0
2.4
Kaqγs
1.5
φ = 20°, β/φ = -0.15, α = +20°
0.4
(b)
(a)
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1
0.4
2.1
kh = 0.3
kh = 0.2
kh = 0.1
1.6
0.8
Kaqγs
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0.7
(g)
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Kaqγs
Kaqγs
1.1
2.2
kh = 0.3
kh = 0.2
kh = 0.1
Kaqγs
1.5
1.1
λ
2.2
(h)
φ = 20°, β/φ = +0.15, α = +20°
0.5
3.3
0
1.1
λ
2.2
3.3
(i)
Fig. 12. Variation of Kaqg s with l , Nq, and kh for f = 20° and d / f = 2/3: (a) b / f = –0.15 and a = –20°; (b) b / f = –0.15 and a = 0°; (c) b / f = –
0.15 and a = þ20°; (d) b / f = 0 and a = –20°; (e) b / f = 0 and a = 0°; (f) b / f = 0 and a = þ20°; (g) b / f = þ0.15 and a = –20°; (h) b / f = þ0.15
and a = 0°; (i) b / f = þ0.15 and a = þ20°
© ASCE
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Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
1.1
kh = 0.3
kh = 0.2
kh = 0.1
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.8
φ = 30°, β/φ = -0.40, α = -20°
0.1
0
0.4
λ
0.8
φ = 30°, β/φ = -0.40, α = 0°
0
0.4
λ
0.8
1.2
(b)
1
1.1
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.8
0.4
φ = 30°, β/φ = 0, α = -20°
0
0.5
λ
1
φ = 30°, β/φ = 0, α = 0°
0
1.5
0.6
λ
1.2
1.8
0
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.4
φ = 30°, β/φ = +0.40, α = -20°
0.2
0
0.7
λ
1.4
1.4
2.1
kh = 0.3
kh = 0.2
kh = 0.1
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.4
φ = 30°, β/φ = +0.40, α = 0°
0.2
0
2.1
λ
2.3
0.8
0.6
0.7
3.2
kh = 0.3
kh = 0.2
kh = 0.1
Kaqγs
1
φ = 30°, β/φ = 0, α = +20°
Kaqγs
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
(f)
2
kh = 0.3
kh = 0.2
kh = 0.1
1.5
kh = 0.3
kh = 0.2
kh = 0.1
0.3
(e)
1.4
1
0.8
0.2
(d)
λ
1.3
0.5
0.1
0.5
1.8
kh = 0.3
kh = 0.2
kh = 0.1
Kaqγs
Kaqγs
0.7
0
(c)
kh = 0.3
kh = 0.2
kh = 0.1
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
φ = 30°, β/φ = -0.40, α = +20°
0.3
Kaqγs
(a)
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.7
0.2
1.2
kh = 0.3
kh = 0.2
kh = 0.1
1.1
0.5
Kaqγs
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0.3
(g)
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Kaqγs
Kaqγs
0.5
1.5
kh = 0.3
kh = 0.2
kh = 0.1
Kaqγs
0.7
1
λ
2
(h)
φ = 30°, β/φ = +0.40, α = +20°
0.5
3
0
1.1
λ
2.2
3.3
(i)
Fig. 13. Variation of Kaqg s with l , Nq, and kh for f = 30° and d / f = 2/3: (a) b / f = –0.40 and a = –20°; (b) b / f = –0.40 and a = 0°; (c) b / f = –
0.40 and a = þ20°; (d) b / f = 0 and a = –20°; (e) b / f = 0 and a = 0°; (f) b / f = 0 and a = þ20°; (g) b / f = þ0.40 and a = –20°; (h) b / f = þ0.40
and a = 0°; (i) b / f = þ0.40 and a = þ20°
© ASCE
04017123-18
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
0.7
kh = 0.3
kh = 0.2
kh = 0.1
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.5
φ = 40°, β/φ = -0.55, α = -20°
0
0
0.2
λ
0.4
φ = 40°, β/φ = -0.55, α = 0°
0
0.3
λ
0.6
0.9
(b)
0.6
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.7
0.2
φ = 40°, β/φ = 0, α = -20°
0
0.3
λ
0.6
φ = 40°, β/φ = 0, α = 0°
0.9
0
0.4
λ
0.8
1.2
φ = 40°, β/φ = +0.55, α = -20°
0
0.6
λ
1.2
λ
1.2
1.8
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.4
φ = 40°, β/φ = +0.55, α = 0°
0.2
1.8
kh = 0.3
kh = 0.2
kh = 0.1
2.3
0.7
0.1
0.6
3.2
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
1.2
0.4
0
kh = 0.3
kh = 0.2
kh = 0.1
Kaqγs
0.7
φ = 40°, β/φ = 0, α = +20°
Kaqγs
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
(f)
1.7
kh = 0.3
kh = 0.2
kh = 0.1
0.9
kh = 0.3
kh = 0.2
kh = 0.1
0.3
(e)
1
0.6
0.7
0.1
(d)
λ
1.1
0.4
0
0.3
1.5
kh = 0.3
kh = 0.2
kh = 0.1
Kaqγs
Kaqγs
0.4
0
(c)
1
kh = 0.3
kh = 0.2
kh = 0.1
φ = 40°, β/φ = -0.55, α = +20°
0.2
Kaqγs
(a)
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
0.5
0.1
0.6
kh = 0.3
kh = 0.2
kh = 0.1
0.8
0.3
Kaqγs
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0.2
(g)
Nq = 1.0
= 0.8
= 0.6
= 0.4
= 0.2
= 0.0
Kaqγs
Kaqγs
0.4
1.1
kh = 0.3
kh = 0.2
kh = 0.1
Kaqγs
0.6
0
0.9
λ
1.8
(h)
φ = 40°, β/φ = +0.55, α = +20°
0.5
2.7
0
1
λ
2
3
(i)
Fig. 14. Variation of Kaqg s with l , Nq, and kh for f = 40° and d / f = 2/3: (a) b / f = –0.55 and a = –20°; (b) b / f = –0.55 and a = 0°; (c) b / f =
−0.55 and a = þ20°; (d) b / f = 0 and a = –20°; (e) b / f = 0 and a = 0°; (f) b / f = 0 and a = þ20°; (g) b / f = þ0.55 and a = –20°; (h) b / f = þ0.55
and a = 0°; (i) b / f = þ0.55 and a = þ20°
© ASCE
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Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
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Fig. 15. Variation of K0 aqg cs obtained with and without considering tensile crack in the backfill corresponding to l and f for Nc = 0.1, b / f = 0.5,
d / f = 0.5, and a = 0° with (a) kh = 0.0, (b) kh = 0.1, (c) kh = 0.2, and (d) kh = 0.3
self-weight can be either smaller or larger than the resisting forces
arising from internal stresses. Thus, the smaller and larger magni0
tudes of Kaq
g cs are realized for the case in which the effect of
crack in the backfill is included. Furthermore, and not surpris0
ingly, it is seen that the magnitudes of Kaq
g cs and l cr always
increase considerably with an increase in the values of kh for both
j = 0 and j > 0. The present study follows the assumption that
the cracks existing in the backfill are perfectly vertical. However,
the possibility of the existence of nonvertical cracks in the backfill can be expected, which will be a subject for future research.
Comparison of Results
The present theoretical analysis has been validated by comparing
solutions produced in this paper with the experimental results
© ASCE
reported by Fang et al. (1997) and theoretical solutions of Coulomb
(1776), Chen and Rosenfarb (1973), Kerisel and Absi (1990),
Motta (1994), Soubra and Macuh (2002), and Shukla et al., (2009).
In Fig. 16, the present values of Ka g are compared with the laboratory experimental results of Fang et al. (1997) for different backfill
slope angles ( b ). The results of Fang et al. (1997) corresponded to a
vertical retaining wall (a = 0) with a height of 0.3 m. A dry sand
backfill was used with the unit weight ( g ) and internal friction angle
( f ) of 15.5 kN/m3 and 30.9°, respectively. The friction angle of the
soil-wall interface (d ) has been reported to be 19.2°. In Fig. 16, the
solutions obtained from Coulomb’s theory for different values of d
are also presented. For d = 19.2°, both solutions from the Coulomb
theory and the present analysis are lower than that of experimental
results. However, present theoretical formulation shows improvement greater than that of Coulomb’s theory for different values of b
and d . Table 1 shows a comparison of earth thrust coefficients Ka g
04017123-20
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
0.5
obtained in this study with those reported by Chen and Rosenfarb
(1973) for an inclined wall with horizontal backfill. The values of
Ka g obtained from three different collapse mechanisms by Chen
and Rosenfarb (1973) do not differ much from that of the present
solutions. Table 1 shows that for a = –20°, the planar failure mechanism of Chen and Rosenfarb (1973) provided better solutions compared with their composite collapse mechanism and the mechanism
used in the present analysis. This is because of the requirement of
additional constraints for the satisfaction of kinematic admissible
conditions in the composite collapse mechanism due to which the
composite mechanisms of Chen and Rosenfarb (1973) and the present study cannot degenerate effectively to those of planar mechanisms. However, a very close agreement among the tabulated solutions can be noted. Fig. 17 provides a comparison of the present
values of static earth thrust coefficients Ka g and Kaq with the theoretical results of Kerisel and Absi (1990) for f = 45° and d / f = 1
in the case of (1) a vertical wall retaining an inclined backfill and (2)
an inclined wall retaining a horizontal backfill. The present upper
bound solutions are slightly lower than the limit equilibrium solutions of Kerisel and Absi (1990) for negatively sloped surfaces of
backfill and wall back; however, a close agreement for the coefficients Ka g and Kaq computed between the two solutions can be
observed. The solutions from the limit equilibrium approaches may
φ = 30.9°
Present analysis
Coulomb (1776)
Fang et al. (1997)
Kaγ
0.4
δ
0°
10°
19.2°
0.2
-30
-10
10
30
β
Fig. 16. Comparison of results from the present analysis with the solutions obtained from Coulomb’s theory and the experimental results
reported by Fang et al. (1997) for f = 30.9° and a = 0°
Table 1. Comparison of Ka g Obtained for Horizontal Backfill in the Present Study with That Reported by Chen and Rosenfarb (1973)
Chen and Rosenfarb (1973)
LSM
f (degrees)
30
40
CSM
PLM
Present study
a (degrees)
d / f = 1/2
d /f = 1
d / f = 1/2
d /f = 1
d / f = 1/2
d /f = 1
d / f = 1/2
d /f = 1
þ20
0
–20
þ20
0
–20
0.475
0.302
0.188
0.368
0.200
0.095
0.501
0.302
0.178
0.428
0.214
0.095
0.475
0.301
0.181
0.365
0.197
0.090
0.501
0.297
0.168
0.418
0.210
0.088
0.476
0.301
0.215
0.370
0.199
0.115
0.501
0.297
0.237
0.428
0.210
0.146
0.476
0.303
0.189
0.369
0.200
0.096
0.502
0.302
0.179
0.429
0.214
0.096
Note: LSM = log sandwich; CSM = circular sandwich; PLM = planar mechanism.
0.4
0.5
Present analysis
Kerisel and Absi (1990)
Present analysis
Kerisel and Absi (1990)
0.4
0.3
Kaγ , Kaq
0.3
Kaγ , Kaq
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0.3
Kaq
0.2
Kaq
0.2
Kaγ
Ka γ
0.1
0
β = 0°, φ = 45°, δ/φ = 1, λ = 0
-0.5
(a)
-0.3
-0.1
0.1
α/φ
0.3
α = 0°, φ = 45°, δ/φ = 1, λ = 0
0.1
-1
0.5
-0.7
(b)
-0.4
-0.1
β /φ
0.2
0.5
0.8
Fig. 17. Comparison of the present analysis with those reported by Kerisel and Absi (1990) for f = 45°, d / f = 1, and l = 0 for (a) vertical wall with
different backfill slope surfaces and (b) inclined wall with horizontal backfill
© ASCE
04017123-21
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
Remarks
Applicability of the Principle of Superposition
In most of the earlier studies, the solutions were obtained based
on an independent maximization scheme in which the total active
force has been expressed as the summation of the maximum force
induced due to the individual contribution of unit weight, cohesion, and surcharge; thus, this involves the inherent assumption
that the principle of superposition is valid. Generally, the use of
the superposition principle in the computation of the total active
earth force for a general cohesive-frictional backfill with surcharge loadings requires three distinct critical slip surfaces with
respect to soil weight, cohesion, and surcharge components rather
than predicting a unique slip surface that has been observed in the
field or laboratory trials. The use of the principle of superposition
in computing total active force will be on the safer side; however,
depending on the wall geometry, properties and configuration of
backfill, surcharge loadings, and so forth, the magnitude of total
active force is sometimes overestimated highly compared with
the actual values. Thus, for examining the applicability of the
superposition principle, a comparison of the normalized values of
resultant seismic active force (2Pa/ g h2) computed with and without the superposition principle is presented Tables 3 and 4 for l =
0 and l > 0, respectively, with d / f = 2/3, b / f = 1/3, and a = 0.
Table 3 shows that the computed values of 2Pa/ g h2 using the
superposition principle always remains higher than the corresponding solutions obtained without the superposition rule. For
any given value of Nq, the error involved in the computation of
the resultant active force using the superposition principle
increases when (1) the magnitudes of kh and Nc are higher and (2)
the magnitudes of f are lower. This is a consequence mainly due
to the omission of interaction effects associated with the previously mentioned factors. However, when the effect of the tension
crack is included, the error in the computation due to the superposition principle has been reduced, which is caused by the imposition of an additional surcharge in the formulation for the cracked
region. For the computation of active earth thrust with purely frictional backfill, individual coefficients obtained with the help of
the superposition principle can be applied for lower magnitudes
of kh because the error involved in the computations is practically
0.7
Present analysis
Motta (1994)
0.6
Nq = 1.0
= 0.5
= 0.25
= 0.0
0.5
Kaqγ
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present theory compared with that of Shukla et al. (2009) for lower
values of kh. For higher values of kh, there is no significant difference between the two solutions.
be more conservative than the limit analysis solutions; however, the
kinematic admissibility of the collapse mechanism cannot always
be ensured. In Fig. 18, the present static earth thrust coefficient Kaqg
has been compared with the numerical results of Motta (1994) for
f = 30° and d / f = 1 in the case of a vertical wall supporting a horizontal frictional backfill. The numerical results of Motta (1994) and
the present theoretical results are found to be exactly the same. For
different combinations of f , d / f , and b / f , Table 2 presents a
comparison of static earth thrust coefficients Ka g , Kaq , and Kac computed from the present theory with the upper bound solutions of
Soubra and Macuh (2002) based on rotational log spiral collapse
mechanisms. The present solution has been found to be practically
the same when compared with the rigorous upper bound solutions
obtained by Soubra and Macuh (2002). Nevertheless, the theory of
Soubra and Macuh (2002) could not be extended simply to compute
0
the combined thrust coefficients Kaqg s and Kaq
g cs or to address the
effect of the position of the surcharge placed on the backfill. Fig. 19
0
presents a comparison of the seismic earth thrust coefficient Kaq
g cs
obtained using the present theory with the limit equilibrium solutions of Shukla et al. (2009) for different values of Nc in the case of
a smooth vertical wall retaining horizontal backfill. There is mar0
ginal improvement in the coefficients Kaq
g cs computed from the
0.4
0.3
φ = 30°, β/φ = 0,δ/φ = 0.5, α = 0°
0.2
0
0.2
0.4
λ
0.6
0.8
1
Fig. 18. Comparison of the present analysis with those reported by
Motta (1994) for f = 30°, b / f = 0, d / f = 0.5, and a = 0°
Table 2. Comparison of Active Thrust Coefficients Obtained for Vertical Walls in the Present Study with That Reported by Soubra and Macuh (2002)
Soubra and Macuh (2002)
Ka g
f (degrees) d / f
20
30
40
© ASCE
0
1/2
1
0
1/2
1
0
1/2
1
Present analysis
Kaq
Kac
Ka g
Kaq
Kac
b / f = 0 bf = 2/3 b / f = 0 b / f = 2/3 b / f = 0 bf = 2/3 b / f = 0 b / f = 2/3 b / f = 0 b / f = 2/3 b / f = 0 b / f = 2/3
0.490
0.449
0.436
0.333
0.303
0.304
0.217
0.200
0.215
0.611
0.577
0.570
0.441
0.415
0.425
0.296
0.282
0.310
0.490
0.450
0.441
0.333
0.304
0.309
0.217
0.201
0.219
0.628
0.593
0.586
0.469
0.442
0.453
0.332
0.316
0.347
1.400
1.553
1.712
1.155
1.267
1.466
0.933
1.029
1.295
1.616
1.755
1.917
1.353
1.450
1.656
1.079
1.164
1.445
0.49
0.448
0.434
0.333
0.303
0.302
0.217
0.200
0.214
04017123-22
Int. J. Geomech., 2018, 18(1): 04017123
0.611
0.577
0.571
0.441
0.415
0.426
0.296
0.282
0.311
0.490
0.451
0.449
0.333
0.305
0.315
0.217
0.202
0.224
0.628
0.593
0.591
0.469
0.442
0.458
0.332
0.316
0.352
1.401
1.549
1.691
1.155
1.265
1.454
0.933
1.028
1.289
1.610
1.743
1.883
1.346
1.440
1.635
1.073
1.158
1.433
Int. J. Geomech.
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Fig. 19. Comparison of the present analysis with those reported by Shukla et al. (2009) for a smooth vertical wall retaining a horizontal c- f backfill
with no load acting on it with (a) Nc = 0, (b) Nc = 0.05, (c) Nc = 0.1, and (d) Nc = 0.2
Table 3. Comparison of Normalized Resultant Seismic Active Thrust (2Pa/ g h2) Computed with and without Using the Principles of Superposition for a =
0°, d / f = 2/3, b / f = 1/3, l = 0, and j = 0
f = 20°
f = 30°
Nq = 0.5
kh
0
0.3
Nc
2Paw/ g h
0.025
0.05
0.1
0.2
0.025
0.05
0.1
0.2
0.695
0.653
0.568
0.399
1.834
1.791
1.706
1.537
2
Nq = 1
2Pawo/ g h
0.694
0.649
0.562
0.389
1.772
1.668
1.46
1.068
2
2Paw/ g h
2
Nq = 0.5
2
0.943
0.901
0.816
0.646
2.462
2.419
2.334
2.165
2
Nq = 1
2Pawo/ g h
2Paw/ g h
2Pawo/ g h
0.941
0.897
0.809
0.636
2.400
2.296
2.088
1.672
0.479
0.444
0.373
0.232
1.165
1.129
1.059
0.918
0.478
0.441
0.368
0.222
1.134
1.068
0.941
0.723
2
2Paw/ g h
0.653
0.618
0.547
0.406
1.569
1.534
1.463
1.322
2
2Pawo/ g h2
0.651
0.614
0.541
0.395
1.538
1.472
1.339
1.108
Note: 2Paw/ g h2 and 2Pawo/ g h2 refer to 2Pa/ g h2 computed with and without using the superposition principle, respectively.
insignificant. Table 4 shows that the application of the superposition principle leads to poor solutions for estimating the values of
critical surcharge distance ratio l cr. The error involved in the
© ASCE
computation of 2Pa/ g h2 and l cr has been found to be over 20 and
200%, respectively, depending on the magnitudes of kh and Nq.
Therefore, it is recommended to take into account the combined
04017123-23
Int. J. Geomech., 2018, 18(1): 04017123
Int. J. Geomech.
Table 4. Comparison of Normalized Resultant Seismic Active Thrust (2Pa/ g h2) Computed with and without Using the Principles of Superposition for a =
0°, d / f = 2/3, b / f = 1/3, and Nc = 0
kh = 0
kh = 0.1
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Nq = 0.5
f
l
2Paw/ g h
20°
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
0.680
0.638
0.606
0.580
0.558
0.541
0.527
0.515
0.506
2
Nq = 1
2Pawo/ g h
0.671
0.614
0.561
0.514
0.490
0.490
0.490
0.49
0.49
2
2Paw/ g h
2
Nq = 0.5
2Pawo/ g h
0.870
0.786
0.722
0.669
0.627
0.591
0.563
0.539
0.522
2
0.857
0.750
0.658
0.578
0.508
0.490
0.490
0.490
0.490
2Paw/ g h
0.867
0.829
0.797
0.770
0.748
0.728
0.711
0.695
0.680
2
Nq = 1
2Pawo/ g h
2
2Paw/ g h
0.863
0.813
0.767
0.725
0.686
0.650
0.616
0.611
0.611
2
2Pawo/ g h2
1.124
1.047
0.983
0.930
0.885
0.846
0.812
0.781
0.749
1.118
1.023
0.941
0.868
0.802
0.740
0.687
0.635
0.611
Note: 2Paw/ g h2 and 2Pawo/ g h2 refer to 2Pa/ g h2 computed with and without using the superposition principle, respectively.
marginal decrease or increase in the magnitude of active earth
thrust has been observed with respect to the change in the values of soil-wall interface friction depending on the parameters
f , b , and a.
For given values of f , d , and b , the magnitude of earth thrust
has been found to always reduce with the change of wall orientation from positive to negative values.
The location of surcharge (l ) applied on the backfill has a
larger impact on reducing the active earth thrust up to a critical surcharge distance ratio l cr beyond which the computed
values of active earth force are independent of the magnitude
of the surcharge pressure. For the case of purely fictional
backfill, this quantity becomes exactly equal to that resulting
from the component of soil unit weight. However, the value
of l cr also increases with an increase in the values of the factors Nq and kh.
The magnitudes of earth thrust coefficient and critical surcharge distance ratio always increase with an increase in the
values of kh for the backfill with and without any tension crack.
However, the consideration of the preexisting crack in the formulation may not always form a critical scenario for the case of
seismic loadings and surcharge located at some distance from
the wall. Proper attention, therefore, must be paid in evaluating
total active earth force because it depends mainly on the magnitudes of Nc and Nq.
The composite failure mechanism with shear zone in the form
of a log spiral or circular sandwich between two triangular
blocks considered in the previous studies and the composite
mechanism introduced in the present analysis are not effective
compared with the planar failure mechanism for higher values
of negative wall inclination.
effect of the ground surcharge and soil self-weight components as
presented in Figs. (9)–(14) to determine the resultant active earth
thrust.
3.
Applicability of Present Solutions for Nonassociated
Flow Rule Material and in the Presence of Vertical
Earthquake Acceleration
For computing the magnitude of the active earth thrust based on the
nonassociated flow rule, the solutions obtained in the present study
can still be used by using the corrections suggested by Drescher and
Detournay (1993) and Michalowski and Shi (1995). The total active
earth thrust in the presence of vertical earthquake acceleration can
be computed from the solutions produced in the present analysis by
following the approach devised by Somers (2003). Thus, the earthquake acceleration in the vertical direction is not included in the
present theoretical formulation.
4.
5.
Conclusions
In this paper, extensive useful design charts for determining the
static and pseudostatic seismic resultant active earth force against
an inclined wall undergoing a horizontal translational movement
have been developed by performing a theoretical analysis. With
the application of the upper bound theorem of limit analysis and
the consideration of a composite collapse mechanism involving a
new kinematically admissible velocity field, the analysis was performed. The magnitudes of resultant active force under different
wall orientations and backfill properties can be estimated by taking into account the influence of the location of continuous surcharge pressure and the tension crack in the backfill. The specific
conclusions drawn from the present analysis are summarized as
follows:
1. From using the superposition principle, it can be said that the
active earth thrust coefficient corresponding to the cohesion
component remains independent of seismic loading. However,
considerable increase in the active earth thrust coefficients corresponding to soil self-weight and ground surcharge components has been noticed with an increase in the magnitudes of
seismic acceleration coefficients.
2. For a given wall orientation (a), the magnitude of active earth
thrust increases continuously with a decrease in the values of
f and with an increase in the values of b . However, a
© ASCE
6.
Appendix I. Analytical Functions for Finding the Seismic
Active Earth Thrust Coefficients
Various functions required to obtain the seismic active earth thrust
coefficients with the help of Eqs. (22)–(24) and (27) are presented
as follows:
C1 ðu ; m ; ɛÞ ¼
04017123-24
Int. J. Geomech., 2018, 18(1): 04017123
cosð m þ f u Þcosða þ m u Þsin m
cosa cosð f u Þcosð m u Þ
½kh þ tanða þ m u Þ
(31)
Int. J. Geomech.
8*
+9
M0 ɛ
>
>
ð
Þ
cos
k
a
þ
m
u
a
þ
m
u
þ
ɛ
1
þ
M
cos
ð
Þ
ð
Þe
>
>
0
h
=
<
M0 ɛ
2
ð
Þ
þ
M
sin
k
a
þ
m
u
a
þ
m
u
þ
ɛ
sin
ð
Þ
ð
Þe
cos ð m þ f u Þ
0
h
>
C2 ðu ; m ; ɛÞ ¼
>
>
>
;
cosa cos2 ð f u Þcosð m u Þ :
1 þ M20
C3 ðu ; m ; ɛÞ ¼
cos2 ð m þ f u Þcosða þ m þ ɛ b Þcosða þ m þ ɛ u ÞeM0 ɛ
½kh þ tanða þ m þ ɛ u Þ
cosa cosð f u Þcosð m u Þsinða þ m þ ɛ þ f b u Þ
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cosða þ m þ ɛ u Þ
C4 ðu ; m ; ɛÞ ¼
cosð m u Þ
(
cosð m þ f u ÞeM1 ɛ
λeɛ tanð f þu Þ
sinða þ m þ ɛ þ f b u Þ
cos b
(32)
(33)
)
½kh þ tanða þ m þ ɛ u Þ
(34)
8
9
>
ð1 j Þcosð m þ f u ÞeM1 ɛ ðλ j sinaÞeɛ tanð f þu Þ >
<
=
cosða þ m þ ɛ u Þ
C04 ðu ; m ; ɛÞ ¼
½kh þ tanða þ m þ ɛ u Þ
sinða þ m þ ɛ þ f b u Þ
cos b
>
>
cosð m u Þ
00
:
;
þC ðu ; m ; ɛÞ
4
(35)
"
#
cosða b Þ 2 j ð1 j Þcos b cosð m þ f u ÞeM1 ɛ
C 4 ðu ; m ; ɛ Þ ¼
þ j 2 sina eɛ tanð f þu Þ
cosa cos b Nq
sinða þ m þ ɛ þ f b u Þ
00
C5 ðu ; m ; ɛÞ ¼
sin m cos f
cosð f u Þcosð m u Þ
(37)
cosð m þ f u Þ½1 eM1 ɛ cosð f þ u Þ
C6 ðu ; m ; ɛÞ ¼
1þ
cosð f u Þ
2 sin f cosð m u Þ
C7 ðu ; m ; ɛÞ ¼
cosða þ m þ ɛ b Þcosð m þ f u Þcos f eM1 ɛ
cosð f u Þcosð m u Þsinða þ m þ ɛ þ f b u Þ
C8 ðu ; m ; ɛÞ ¼
tan d
tan f
M0 ¼ 2 tanð f u Þ þ tanð f þ u Þ
(41)
M1 ¼ tanð f u Þ þ tanð f þ u Þ
(42)
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© ASCE
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