Pseudo-dynamic passive resistance of battered-faced retaining wall

Pseudo-dynamic passive resistance of batteredfaced retaining wall supporting c-W backfill considering Rayleigh wave Arijit Saha* and Sima Ghosh The concept of seismic passive resistance is very much essential for the design of retaining walls under seismic loading conditions, as the damage of such earth structures may lead to catastrophic failure. The earlier analytical methods provide the methodology of computation of seismic passive resistance without considering the effects of Rayleigh waves though it considers about 67% of the total seismic energy. In this present analysis, a new methodology is proposed by considering the Rayleigh wave along with the shear wave velocity (Vs) and primary wave velocity (Vp). Using the limit equilibrium principle, a methodology is developed to calculate the pseudo-dynamic passive resistance on the back of battered-faced retaining walls supporting cohesive-frictional backfill. Results are presented in both tabular and graphical non-dimensional forms. Effects of a wide range of variation of parameters such as wall inclination angle (a), wall friction angle (d), soil friction angle (W), and horizontal and vertical seismic accelerations (kh, kv) have been studied on the seismic passive resistance coefficients. Keywords: Pseudo-dynamic approach, Passive resistance, Rayleigh wave, Battered-faced retaining wall, c-W backfill

Introduction The evaluation of seismic passive resistance is very much essential for the design of retaining walls under seismic loading conditions. The purpose of evaluating seismic passive resistance is to account for toe resistance, in the case of gravity retaining walls and cantilever sheet pile walls, and to account for earth pressure developed behind horizontal supports, in the case of braced walls and tieback walls. A number of research works have been carried out to predict the seismic passive resistance on a rigid retaining wall as a result of earthquake loading. The pioneering works on retaining structures for granular backfill under seismic loading conditions are Okabe (1926), Mononobe and Matsuo (1929), and Subba Rao and Choudhury (2005) using a pseudo-static method which are based on Coluomb (1776) in static condition. Ghosh and Sengupta (2012) extended the pseudo-static method for the evaluation of seismic passive resistance supporting c-W backfill for the simultaneous action of unit weight, surcharge, and cohesion. The influence of phase difference on the calculation of earth pressure on a retaining wall supporting W backfill was investigated by Steedman and Zeng (1990), Choudhury and Nimbalkar (2005), Basha and Babu (2010), and Civil Engineering Department, National Institute of Technology, Agartala 799055, India

396

Ghosh (2007). Ghosh and Sharma (2012) extended the pseudo-dynamic method for the evaluation of seismic passive resistance on the back of a non-vertical retaining wall considering the c-W nature of the backfill. In all of the above studies, it is seen that they have considered only the effect of shear wave velocity (Vs) and primary wave velocity (Vp) in the computation of seismic earth pressures. But from the study of seismic wave theory, it is observed that the surface waves contribute majorly to the seismic energy, for example the Rayleigh wave carries about 67% of the total seismic energy (Woods 1968). In 1885, the English scientist Lord Rayleigh demonstrated theoretically that waves can be propagated over the free surface of an elastic half-space. The amplitude of these surface wave decays rapidly with depth, i.e. the vibration is limited to a shallow layer of approximately one wave length below the surface. It has a significant effect on slope failure which shows the importance of such surface wave in geotechnical analysis. Here in this analysis, an attempt is made to solve this problem of pseudo-dynamic passive resistance by considering the total seismic energy that has been majorly contributed by Rayleigh waves along with primary wave and shear wave. Using the limit equilibrium principle, with a planer failure surface behind the rigid retaining wall is considered in this analysis.

Method of analysis

*Corresponding author, email [email protected]

An arbitrary rigid retaining wall of height H, inclined at an angle a with the vertical, supporting a c-W horizontal backfill

ß 2014 W. S. Maney & Son Ltd Received 31 August 2013; accepted 15 November 2013 DOI 10.1179/1939787913Y.0000000025

International Journal of Geotechnical Engineering

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Pseudo-dynamic passive resistance

1 Rigid retaining wall considering pseudo-dynamic passive resistance

is shown in Fig. 1. All the possible earth quake waves such as shear wave, primary wave, as well as Rayleigh waves are considered in this present study. In p a ffiffiffiffiffiffiffiffiffi pseudo-dynamic analysis, both shear p wave velocity Vs ~ G=r and primary ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wave velocity Vp ~ (2G(1{m))=(r(1{2m)) of the earthquake waves through backfill medium are assumed to act within the soil mass during earthquake, where G, r, and m are shear modulus, density, and Poisson’s ratio of backfill material, respectively with the Rayleigh wave progressing horizontally in the x direction with displacement u and vertically in the z direction with displacement w. Now the displacement components u and w are as follows: Displacement potential, d~+wz+y

(1)

Displacement components (a) Transverse particle displacement u~

Lw Ly { Lx Lz

(2)

(b) Normal particle displacement w~

Lw Ly z Lz Lx

(3)

Q~A e{qz ei(kr x{vt)

(4)

y~B e{sz ei(kr x{vt)

q

~kr2 {v2 =Vp2 ,

kr %

0:87z1:12m 1zm

From this equation, kr50?93 as m50?3 (Viktorov, 1967; Mechkour, 2002). The surface wave solution is a combination of coupled partial (longitudinal and shear) waves. The B/A ratio is determined by the condition that the surface is traction free. Both partial waves are evanescent, i.e. kr.ks.kp, where ks 2 ~

v2 Vs 2

and

s

~kr2 {v2 =Vs2 ,

and velocity ratio ($)~Vr =Vs Exact Rayleigh equation kr 6 {8kr 4 z8(3{2x2 )kr 2 {16(1{x2 )~0

and

v2 Vp 2 s2 ~kr 2 {ks 2

The boundary conditions require that both normal and transverse stresses be zero on the surface at z50.

tzx ~txz ~Lm 2

kp 2 ~

(5)

where 2

In a number of papers, an approximation of this root depending on Poisson’s ratio (m) is given using Bergmann’s (1954) formula

So, q2 ~kr 2 {kp 2

Potential function

and

where kr is the Rayleigh wave number and x is the ratio of shear wave velocity and primary wave velocity that depends on Poisson’s ratio (m) of backfill soil. sffiffiffiffiffiffiffiffiffiffiffiffiffi Vs 1{2m x~ ~ 2{2m Vp

(6)

L2 w L2 y L2 y z 2{ 2 2 LxLz Lx Lz

!

~(Ll z2Lm )q2 w{Ll kr2 w{2Lm iskr y ! L2 w L2 y L2 y tzx ~txz ~Lm 2 { 2 z LxLz Lx2 Lz 2 ~{2Lm iqkr w{Lm s zkr2 y


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where Lm and Ll are Lame constants of the isotropic material. At the surface, without the common term of eiðkr x{vtÞ # " Ll z2Lm s2 {Ll kr 2 {2Lm iskr 0 A ~ 2 2 0 {2Lm iqkr {Lm (s zkr ) B 2 where Ll z2Lm q {Ll kr 2 ~2Lm kr 2 { Ll z2Lm kp 2 ~ 2Lm kr 2 {Lm ks 2 ~Lm s2 zkr 2 , therefore # " 2 {2iskr s zkr 2 0 A ~ 2 2 0 B {2iqkr { s zkr

In Fig. 1 considering a thin element of thickness dz at depth z from the top of the surface and width dx at a distance x from the wall, the mass is c m(x,z)~ dx dz g

The weight of the whole wedge can be calculated as ð H ð (H{z)tan h cG 2 (tan hztan a) (12) W ~g m(x,z)~ 2 0 (z{H)tan a The total horizontal seismic inertia force acting on the wall can be written as ð H ð (H{z)tan h Qh (t)~ m(x,z)ah (x,z,t) (13)

The resulting characteristic equation 2 2 s zkr 2 {4sqkr 2 ~0

0

Qh (t)~

2

B s zkr 2iqkr ~ ~{ A 2iskr s2 zkr 2

a1 ~

b1 ~

where f5amplitude ratio (Viktorov, 1967; Rayleigh, 1885). Shear wave velocity Vs5100 m s21 and ratio of primary and shear wave velocities Vp/Vs51?87 (Das, 1993). 2p v~ T where T5time period of vibration generated by earthquake. For most geotechnical structures, T50?3 s (Prakash, 1981). From the above equations and the considered input values, the following values have been calculated q50?92, s50?91, and f51?00 Now, the normal and tangential displacement components for Rayleigh wave can be written as u~iA(kr e

{fs e

{sz

i(kr x{vt)

)e

q2 zkr 2

"

#

qf{e{qH cos vtzcos(kr H tan azvt)g 2 tan a {kr tan afe{qH sin vt{sin(kr H tan azvt)g

" # sf{e{sH cos vtzcos(kr H tan h{vt)g fs s2 zkr 2 tan2 h zkr tan hfe{sH sin vtzsin(kr H tan h{vt)g

d1 ~

" # sfe{sH cos vt{cos(kr H tan azvt)g fs s2 zkr 2 tan2 a zkr tan afe{sH sin vt{sin(kr H tan azvt)g

Qv (t)~

(z{H)tan a

cv2 A (a2 zb2 zc2 zd2 ) gkr

(16)

where a2 ~

b2 ~

L2 u ~v2 A(kr e{qz {fs e{sz )sin(kr x{vt) (9) Lt2 L2 w av (x,z,t)~ 2 ~v2 A(q e{qz {fkr e{sz ) cos(kr x{vt) (10) Lt

8

kr

#

qfe{qH cos vt{cos(kr H tan h{vt)g 2 tan h {kr tan hfe{qH sin vtzsin(kr H tan h{vt)g

c1 ~

ah (x,z,t)~

VOL

q2 zkr 2

"

Integrating twice within the limits

(8)

2014

kr

0

where u and w are displacements in the x and z directions, respectively. Now, the seismic accelerations with amplitude can be obtained from the displacement equations (7) and (8) twice w.r.t. time by considering only real parts that are expressed as


(14)

The total vertical seismic inertia force acting on the wall can be written as ð H ð (H{z)tan h Qv (t)~ m(x,z)av (x,z,t) (15)

(7)

w~{A(q e{qz {fkr e{sz )ei(kr x{vt)

398


where

By substituting the Rayleigh wave number from the characteristic equation, a simplified ratio is obtained. rffiffiffi B q ~{i ~{if or B~{iAf A s

{qz

(z{H)tan a

Integrating twice within the limits

The amplitude ratio between the vector and scalar potentials 2

(11)

c2 ~

NO

4

q q2 zkr 2 q q2 zkr 2

"

#

"

#

qfe{qH sin vtzsin(kr H tan h{vt)g 2 tan h zkr tan hfe{qH cos vt{cos(kr H tan h{vt)g qf{e{qH sin vtzsin(kr H tan azvt)g {qH 2 zk cos vt{cos(kr H tan azvt)g tan a r tan afe

" # sf{e{sH sin vt{sin(kr H tan h{vt)g fkr s2 zkr 2 tan2 h zkr tan hf{e{sH cos vtzcos(kr H tan h{vt)g

Saha and Ghosh

d2 ~

" # sfe{sH sin vt{sin(kr H tan azvt)g fkr s2 zkr 2 tan2 a zkr tan af{e{sH cos vtzcos(kr H tan azvt)g

Qh(t) and Qv(t) can be expressed in the form of kh and kv by replacing ‘‘A’’ in equations (14) and (16) with ‘‘A’’ in equations (9) and (10), respectively. Because the variation in the x direction is not very much significant in the case of retaining wall, all the values are computed for x50 (i.e. at the back face of the retaining wall) and z5H in this present study. So now Qh(t) and Qv(t) can be written as Qh (t)~ Qh (t)~

Qh (t)~


gkr (kr

e{qH {fs

cah (a1 zb1 zc1 zd1 ) e{sH ) sin(kr x{vt)

ckh (a1 zb1 zc1 zd1 ) kr (kr e{qH {fs e{sH ) sin({vt)

(17)

ckv (a2 zb2 zc2 zd2 (18) kr (q e{qH {fkr e{sH ) cos(vt)

where kv is the vertical seismic acceleration that is equal to av/g. Applying limit equilibrium principle, the summation of total horizontal forces equals to zero. Pp cos(d{a)~R cos(Q{h){Qh (t)z cH sin h ca H sin a z cos h cos a

(19)

Applying limit equilibrium principle, the summation of total vertical forces equals to zero. Pp sin(d{a)~{R sin(Q{h)zQv (t){ W{

cH cos h ca H cos a { cos h cos a

(20)

Solving equations (19) and (20) Pp ~

2 Pp ce H 2

(23)

where Kp is the coefficient of pseudo-dynamic earth pressure (passive state of equilibrium) and ce is equilibrium unit weight of soil. ce ~

QzW AB

where Q5total surcharge load5qH(tan aztan h) q5surcharge load per unit length AB5total area of backfill wedge5(H 2 =2)(tan aztan h) ce ~

where kh is the horizontal seismic acceleration that is equal to ah/g. Similarly Qv (t)~

Kp ~


2qzcH H

Results and discussion The passive resistance coefficients are calculated on optimization Kp with respect to h and t/T. The optimum values of Kp here are represented as Kpe. These passive resistance coefficients (Kpe) are presented in Tables 1 and 2. The variation of parameters is as follows: W520u, 30u, and 40u; d50, W/2, and W; a5210u, 0u, and z10u; kh50?1, 0?2, and 0?3; kv50, kh/2, and kh; Nc50, 0?1, and 0?2; and Mc50, Nc/2, and Nc. The constant parameters are kr50?93, q50?91, s50?85, f51?03, Vp5g/T5104?16 m s21, and Vs5l/T555?55 m s21 considering H/g50?16 and H/l50?3. H55 m and v52p/T where T50?3 s for most geotechnical structures.

Effect of seismic passive resistance coefficient for W Figure 2 shows the variations of passive resistance coefficient with respect to horizontal seismic acceleration (kh) at different soil friction angles (W520u, 30u, 40u) for d5W/2, a50u, Nc50?1, Mc5Nc, and kv5kh/2. It is seen that passive resistance coefficient (Kpe) increases with increase in soil friction angle (W). For example at kh50?2, seismic passive resistance coefficient (Kpe) increases about 91 and 375% for the increase in W from 20u to 30u and 20u to 40u, respectively. The reason behind this is that the selfresistance of soil increases for higher values of W.

½W cos(Q{h)zQh (t) sin(Q{h){Qv (t) cos(Q{h)zcH cos Q=cos hzca H cos(Q{a{h)=cos a sin(hza{Q{d)

(21)

W cos(Q{h)zQh (t) sin(Q{h){Qv (t) cos(Q{h)zce H 2 =2fNc cos Q=cos hzMc cos(Q{a{h)=cos ag Pp ~ sin(hza{Q{d) where Nc5cohesion factor52c/ceH and Mc5adhesion factor52ca/ceH. In equation (22), W, Qh(t), and Qv(t) are given from equations (12), (17), and (18), respectively. Again Pp ~

ce H 2 Kp 2

(22)

Effect of seismic passive resistance coefficient for d Figure 3 shows the variations of passive resistance coefficient with respect to seismic acceleration (kh) at different wall friction angles (d50, W/2, W) for W530u, a50u, Nc50?1, Mc5Nc, and kv5kh/2. It is seen that passive resistance coefficient (Kpe) increases with increase in wall friction angle (d). For example at kh50?2, seismic passive


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different wall inclination angles (a5210u, 0u, 10u) for d5W/2, W530u, Nc50?1, Mc5Nc, and kv5kh/2. It is seen that passive resistance coefficient (Kpe) decreases with increase in wall inclination angle (a). At kh50?2, seismic passive resistance coefficient (Kpe) decreases about 40 and 27% when a increases from 210u to 0u and 0u to z10u, respectively. The increase in inclination is imposing a greater amount of backfill soil on the wall; thus the passive resistance is getting reduced.

Effect of seismic passive resistance coefficient for kh and kv Table 3 shows that variation of passive resistance coefficients for different values of horizontal seismic acceleration (kh) at d5W/2, a50, kv5kh/2, Nc50?1, and Mc5Nc. It is seen that passive resistance coefficient (Kpe) decreases with increase in horizontal seismic acceleration (kh). For example at W530u, seismic passive resistance coefficient (Kpe) decreases about 8 and 15% because of increase in kh from 0?1 to 0?2 and 0?1 to 0?3, respectively. Figure 5 shows the variations of passive resistance coefficient with respect to seismic acceleration (kh) at different vertical seismic accelerations (kv50, kh/2, kh) for W530u, a50u, Nc50?1, Mc5Nc, and d5W/2. It is seen that Kpe decreases with the increase in vertical seismic acceleration (kv) also. At kh50?2, seismic passive resistance coefficient (Kpe)

2 The variations of passive resistance coefficient with respect to seismic acceleration (kh) at different soil friction angles (W520u, 30u, 40u) for d5W/2, a50u, Nc50?1, Mc5Nc, and kv5kh/2

resistance coefficient (Kpe) increases about 39 and 72% for the increase in d from 0 to W/2 and 0 to W. The friction between wall and soil is increasing the passive resistance.

Effect of seismic passive resistance coefficient for a Figure 4 shows the variations of passive resistance coefficient with respect to seismic acceleration (kh) at

Table 1 Pseudo-dynamic passive earth pressure coefficient for kh50?1 kh50?1 kv50

kv5kh

Ø

d

Nc5Mc

a5210

a50

a5z10

a5210

a50

a5z10

a5210

a50

a5z10

20u

0

0 0?1 0?2 0 0?1 0?2 0 0?1 0?2

2?388 2?931 3?467 3?156 3?862 4?569 4?736 5?767 6?796

2?086 2?512 2?948 2?61 3?126 3?642 3?426 4?103 4?78

1?948 2?29 2?633 2?321 2?725 3?129 2?823 3?317 3?81

2?286 2?819 3?351 3?073 3?787 4?502 4?735 5?755 6?773

1?941 2?356 2?77 2?393 2?904 3?415 3?21 3?896 4?582

1?809 2?153 2?495 2?097 2?494 2?891 2?58 3?07 3?559

2?264 2?796 3?229 3?064 3?779 4?493 4?757 6?793 3?631

1?839 2?253 2?663 2?267 2?778 3?289 3?068 3?753 4?439

1?673 2?015 2?355 1?938 2?334 2?731 2?389 2?878 3?367

0 0?1 0?2 0 0?1 0?2 0 0?1 0?2

3?748 4?434 5?119 7?269 8?491 9?714 – – –

3?063 3?578 4?088 4?892 5?686 6?48 10?072 11?632 13?191

2?646 3?05 3?448 3?778 4?327 4?877 6?518 7?448 8?377

3?644 4?339 5?033 7?272 8?496 9?72 – – –

2?813 3?313 3?813 4?657 5?477 6?293 9?826 11?393 12?96

2?446 2?836 3?226 3?465 4?018 4?571 6?249 7?168 8?088

4?333 5?036 7?316 8?54 9?764 – – –

2?683 3?183 3?683 4?481 5?301 6?12 9?706 11?273 12?84

2?269 2?658 3?048 3?22 3?773 4?326 6?034 6?953 7?873

0 0?1 0?2 0 0?1 0?2 0 0?1 0?2

6?615 7?532 8?487 – – – – – –

4?589 5?213 5?837 11?726 13?178 14?634 – – –

3?661 4?13 4?6 7?495 8?364 9?231 – – –

6?723 7?643 8?556 – – – – – –

4?281 4?924 5?566 11?467 12?928 14?388 – – –

3?363 3?826 4?289 7?207 8?07 8?933 – – –

6?786 7?697 8?609 – – – – – –

4?109 4?753 5?396 11?35 12?81 14?271 – – –

3?132 3?595 4?058 6?867 7?833 8?698 – – –

Ø/2

Ø

30u

0

Ø/2

Ø

40u

0

Ø/2

Ø

400

kv5kh/2


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3 The variations of passive resistance coefficient with respect to seismic acceleration (kh) at different wall friction angles (d50, W/2, W) for W530u, a50u, Nc50?1, Mc5Nc, and kv5kh/2

decreases about 9 and 16% because of increase in kv from 0 to kh/2 and 0 to kh, respectively. The modeled ground acceleration captures the effects of inertial forces acting away from the wall, reducing passive resistance.

Effect of seismic passive resistance coefficient for cohesion (c) Figure 6 shows the variations of passive resistance coefficient with respect to seismic acceleration (kh) at


4 The variations of passive resistance coefficient with respect to seismic acceleration (kh) at different wall inclination angles (a5210u, 0u, 10u) for d5W/2, W530u, Nc50?1, Mc5Nc, and kv5kh/2

different cohesion factors (Nc50, 0?1, 0?2) for W530u, a50u, kv5kh/2, Mc5Nc, and d5W/2. It is seen that passive resistance coefficient (Kpe) increases with increase in cohesion factor (Nc). For example at kh50?2, seismic passive resistance coefficient (Kpe) increases about 19 and 39% because of increase in Nc from 0 to 0?1 and 0 to 0?2, respectively. Cohesion increases the intermolecular attraction; thus, the passive resistance also increases.

Table 2 Pseudo-dynamic passive earth pressure coefficient for kh50?2 kh50?2 kv50

kv5kh/2

kv5kh

Ø

d

Nc5Mc

a5210

a50

a5z10

a5210

a50

a5z10

a5210

a50

a5z10

20u

0

0 0?1 0?2 0 0?1 0?2 0 0?1 0?2

2?303 2?839 3?376 3?035 3?742 4?448 4?632 5?67 6?702

2?04 2?477 2?91 2?532 3?048 3?564 3?323 4 4?677

1?93 2?181 2?577 2?284 2?693 3?101 2?771 3?264 3?757

2?137 2?67 3?203 2?865 3?58 4?294 4?647 5?677 6?705

1?738 2?153 2?567 2?134 2?645 3?157 2?862 3?547 4?233

1?617 1?96 2?302 1?873 2?27 2?667 2?297 2?787 3?276

2?101 2?634 3?166 2?849 3?564 4?278 4?728 5?746 6?764

1?533 1?947 2?362 1?89 2?401 2?912 2?565 3?259 3?947

1?342 1?684 2?027 1?555 1?951 2?348 1?917 2?406 2?895

0 0?1 0?2 0 0?1 0?2 0 0?1 0?2

3?649 4?335 5?021 7?171 8?394 9?617 – – –

3?028 3?539 4?042 4?79 5?584 6?378 9?977 11?541 13?094

2?64 3?05 3?456 3?729 4?278 4?827 6?492 7?423 8?353

3?47 4?164 4?858 7?209 8?431 9?653 – – –

2?563 3?063 3?563 4?242 5?062 5?082 9?453 11?019 12?586

2?211 2?602 2?994 3?127 3?68 4?234 5?858 6?887 7?855

3?446 4?149 4?852 7?314 8?538 9?762 – – –

2?303 2?803 3?303 3?862 4?701 5?53 9?213 10?779 12?346

1?864 2?256 2?647 2?637 3?19 3?744 5?056 6?085 7?113

0 0?1 0?2 0 0?1 0?2 0 0?1 0?2

6?545 7?487 8?436 – – – – – –

4?524 5?148 5?772 11?648 13?102 14?555 – – –

3?687 4?161 4?629 7?503 8?39 9?262 – – –

6?713 7?625 8?537 – – – – – –

3?962 4?605 5?249 11?117 12?578 14?038 – – –

3?071 3?532 3?993 6?664 7?645 8?627 – – –

6?82 7?731 8?643 – – – – – –

3?619 4?263 4?906 10?852 12?329 13?804 – – –

2?626 3?088 3?549 5?822 6?814 7?796 – – –

Ø/2

Ø

30u

0

Ø/2

Ø

40u

0

Ø/2

Ø


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5 The variations of passive resistance coefficient with respect to seismic acceleration (kh) at different vertical seismic accelerations (kv50, kh/2, kh) for W530u, a50u, Nc50?1, Mc5Nc, and d5W/2

6 The variations of passive resistance coefficient with respect to seismic acceleration (kh) at different cohesion factors (Nc50, 0?1, 0?2) for W530u, a50u, kv5kh/2, Mc5Nc, and d5W/2

Effect of seismic passive resistance coefficient for adhesion (ca)

reduced. So, these reductions effect toward a decrease in cohesion and adhesion and thus decrease in passive resistance of backfill material.

Figure 7 shows the variations of passive resistance coefficient with respect to seismic acceleration (kh) for different adhesion factors (Mc50, Nc/2, Nc) for W530u, a50u, kv5kh/2, Nc50?1, and d5W/2. It is seen that passive resistance coefficient (Kpe) increases with increase in adhesion factor (Mc). For example at kh50?2, seismic passive resistance coefficient (Kpe) increases around 3 and 5% when Mc increases from 0 to Nc/2 and 0 to Nc, respectively. Thus, it is observed that the increased cohesive and adhesive properties of backfill material enhance the seismic passive resistance of the retaining wall.

Effect of seismic passive resistance coefficient for q Figure 9 shows the variations of 2Ppe/H2 with respect to seismic acceleration (kh) at different surcharge loads on the backfill surface (q510, 20, 30 kN m22) for W530u, a50u, kv5kh/2 and d5W/2, Nc50?1, Mc5Nc. It is seen that passive resistance (Pae) increases with increase in surcharge load (q). For example at kh50?2, seismic earth pressure (Ppe) increases about 18 and 36% for increase in q from 10 to 20 kN m22 and 10 to 30 kN m22, respectively.

Pressure distribution

Effect of seismic passive resistance coefficient for H

The passive resistance distribution behind the wall can be determined by taking the partial derivative of Ppe(z,t) with respect to z, and is expressed as

Figure 8 shows the variations of passive resistance coefficient with respect to seismic acceleration (kh) at different heights of retaining wall (H55, 6, 7 m) for W530u, a50u, kv5kh/2, 2c/c52ca/c51, and d5W/2. It is seen that passive resistance coefficient (Kpe) decreases with increase in wall height (H). For example at kh50?2, seismic passive resistance coefficient (Kpe) decreases about 8 and 17% for increase in H from 5 to 6 m and 5 to 7 m, respectively. Owing to increase in height, the cohesion factor (Nc) and adhesion factor (Mc) are going to be

Table 3 Variation of passive earth pressure coefficients for different values of horizontal seismic acceleration (kh) at d5W/2, a50, kv5kh/2, Nc50?1, and Mc5Nc d5Ø/2, a50, kv5kh/2, Nc50?1, Mc5Nc kh

402

Ø

0

0?1

0?2

0?3

0?4

0?5

20u 30u 40u

3?083 5?668 13?074

2?904 5?477 12?928

2?645 5?062 12?578

2?385 4?648 12?228

2?066 4?121 11?72

1?806 3?706 11?37


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7 The variations of passive resistance coefficient with respect to seismic acceleration (kh) at different adhesion factors (Mc50, Nc/2, Nc) for W530u, a50u, kv5kh/2, Nc50?1, and d5W/2

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8 The variations of passive resistance coefficient with respect to seismic acceleration (kh) at different heights of retaining wall (H55, 6, 7 m) for W530u, a50u, kv5kh/2, 2c/ c52ca/c51, and d5W/2

ppe (z,t)~


9 The variations of passive resistance coefficient with respect to seismic acceleration (kh) at different surcharge loads on the backfill surface (q510, 20, 30 kN m22) for W530u, a50u, kv5kh/2, and d5W/2, Nc50?1, Mc5Nc

Lz W (z,t) cos(Q{h)zQh (z,t) sin(Q{h){Qv (z,t) cos(Q{h)zcz cos Q=cos hzca z cos(Q{a{h)=cos a Lt sin(hza{Q{d)

Lz ½W (z,t)~ce z(tan aztan h)~w Lt

(24)

d3 ~ 2

3 t t {sz cos 2p tan a sin (k z tan az2p zk )g sf{se r r 6 7 fs T T 6 7 t t 5 s2 zkr 2 tan2 a 4 zkr tan af{se{sz sin 2p {kr tan a cos (kr z tan az2p )g T T

Lz Qh (z,t)~ Lt Lz ckh (a1 zb1 zc1 zd1 ) {qH {fs e{sH ) sinð{2pt=T Þ Lt kr (kr e

qv ~

~qh

where

ckh qh ~ (a3 zb3 zc3 zd3 ) kr (kr e{qH {fs e{sH ) sinð{2pt=T Þ

ckv (a zb zc zd ) 4 4 4 4 kr (q e{qH {fkr e{sH ) cosð2pt=T Þ

Lz a2 (z,t)~a4 Lt

where a4 ~ 2

3 t t qf{qe{qz sin 2p zkr tan h cos (kr z tan h{2p )g 6 7 q T T 6 7 5 t t q2 zkr 2 tan2 h 4 zkr tan hf{qe{qz cos 2p zkr tan h sin (kr z tan h{2p )g T T

Lz a1 (z,t)~a3 Lt a3 ~ 2 n 3 t t o q {q e{qz cos 2p zkr tan h sin kr z tan h{2p 6 7 kr T T 6 7 n t t o 5 q2 zkr 2 tan2 h 4 {qz sin 2p zkr tan h cos kr z tan h{2p {kr tan h {q e T T

Lz b2 (z,t)~b4 Lt b4 ~ 2

3 t t {qz sin 2p zkr tan a cos (kr z tan az2p )g 6 qfqe 7 T T 6 7 t t 5 q2 zkr 2 tan2 a 4 zkr tan af{qe{qz cos 2p zkr tan a sin (kr z tan az2p )g T T

Lz b1 (z,t)~b3 Lt

q

b3 ~ 2

3 t t qfqe{qz cos 2p {kr tan a sin (kr z tan az2p )g 6 7 kr T T 6 7 5 t t q2 zkr 2 tan2 a 4 {kr tan af{qe{qz sin 2p {kr tan a cos (kr z tan az2p )g T T

Lz c2 (z,t)~c4 Lt c4 ~

Lz c1 (z,t)~c3 Lt 3 2 n t t o s s e{sH cos 2p {kr tan h sin kr z tan h{2p fs 7 6 T T c3 ~ 4 n

o 5 s2 zkr 2 tan2 h zk tan h {s e{sz sin 2p t zk tan h cos k z tan h{2p t r r r T T

Lz d1 (z,t)~d3 Lt

2 3 t t sfse{sz sin 2p {kr tan h cos (kr z tan h{2p )g fkr 6 7 T T 4 5 s2 zkr 2 tan2 h zk tan hfse{sz cos 2p t {k tan h sin (k z tan h{2p t )g r r r T T

Lz d2 (z,t)~d4 Lt d4 ~ 2

3 t t sf{se{sz sin 2p {kr tan a cos (kr z tan az2p )g 6 7 fkr T T 6 7 5 t t s2 zkr 2 tan2 a 4 zkr tan afse{sz cos 2p {kr tan a sin (kr z tan az2p )g T T


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12 The variation of normalized seismic passive resistance distribution for different wall inclination angles (a5210u, 0u, 10u) for W520u, d5W/2, Nc50?1, Mc5Nc, and kh50?2, kv5kh/2.

10 The variation of normalized seismic passive resistance distribution for different values of soil friction angles (W520u, 30u, 40u) for d5W/2, a510u, Nc50?1, Mc5Nc, and kh5 0?2, kv5kh/2

ppe (z,t)~

404

w cos(Q{h)zqh sin(Q{h){qv cos(Q{h)zc cos Q=cos hzca cos(Q{a{h)=cos a sin(hza{Q{d)

(25)

values of cohesion factors (Nc50, 0?1, 0?2) for W530u, d5W/2, a50u, Mc5Nc, and kh50?2, kv5kh/2. It is seen that passive resistance distribution (ppe) increases with increase in cohesion factor (Nc).

Figure 10 shows the variation of normalized seismic passive resistance distribution for different values of soil friction angles (W520u, 30u, 40u) for d5W/2, a510u, Nc50?1, Mc5Nc, and kh50?2, kv5kh/2. It is seen that passive resistance distribution (ppe) increases with increase in soil friction angle (W). Figure 11 shows the variation of normalized seismic passive resistance distribution for different values of wall friction angles (d50, W/2, W) for W530u, a50u, Nc50?1, Mc5Nc, and kh5 0?2, kv5kh/2. It is seen that passive resistance distribution (ppe) increases with increase in wall friction angle (d). Figure 12 shows the variation of normalized seismic passive resistance distribution for different values of wall inclination angles (a5210u, 0u, 10u) for W530u, d5W/2, Nc50?1, Mc5Nc, and kh5 0?2, kv5kh/2. It is seen that passive resistance distribution (pae) decreases with increase in wall inclination angle (a). Figure 13 shows the variation of normalized seismic passive resistance distribution for different

Figure 14 shows the comparison of passive resistance coefficient with respect to horizontal seismic acceleration (kh) between previous seismic analyses and present study for W530u, a50u, Nc50, Mc5Nc, kv5kh/2, and d5W/2. It is seen that the values from the present analyses are more than from the previous pseudo-static analyses such as Mononobe and Matsuo (1929) and Okabe (1926). For example, at kh50?2, the values obtained from the present study are about 17% more than Mononobe and Okabe (1929) analysis. On the other hand, the values as obtained from the present analysis are less than previous pseudo-dynamic analysis Choudhury

11 The variation of normalized seismic passive resistance distribution for different values of wall friction angles (d50, W/2, W) for W530u, a510u, Nc50?1, Mc5Nc, and kh50?2, kv5kh/2

13 The variation of normalized seismic passive resistance distribution for different values of cohesion factors (Nc50, 0?1, 0?2) for W530u, d5W/2, a50u, Mc5Nc, and kh50?2, kv5kh/2.


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H Kpe

14 The comparison of present analysis with previous seismic analysis with respect to seismic acceleration (kh) for W530u, a50u, kv5kh/2, Nc50, Mc5Nc, and d5W/2

and Nimbalkar (2005) up to kh50?25. For example at kh50?2, the values obtained from the present study are 15% less than Choudhury and Nimbalkar (2005) analysis. Rayleigh wave contributes majorly (about 67%) to the total seismic energy, so it has a significant effect on the earth pressure. As the available pseudo-dynamic methods did not consider the effect of Rayleigh wave, the present analysis gives more significant values than these earlier analyses.

Conclusion Using a pseudo-dynamic approach and applying the limit equilibrium principle, the seismic passive resistance on the back of the non-vertical retaining wall supporting c-W backfill considering total seismic energy has been developed. In this present analysis, the effect of Rayleigh wave has been considered along with primary wave and shear wave velocity. The effect of soil friction angle, wall inclination angle and wall friction angle, horizontal and vertical earthquake acceleration, and shear wave and primary wave velocity has been determined on the basis of total seismic passive resistance. It is seen that the magnitude of seismic passive resistance coefficient (Kpe) decreases with the increase in the values of wall inclination angle (a), seismic accelerations (kh and kv), and height of retaining wall (H) but increases with the increase in the values of wall friction angle (d), soil friction angle (W), and cohesion and adhesion (c and ca). Results obtained from this analysis are given in tabular form and to use these results, for intermediate parameters, linear interpolation is suggested.

Notations c ca kh kr kv t u w G

cohesion adhesion horizontal seismic acceleration Rayleigh wave number vertical seismic acceleration any time during earthquake displacement in x direction displacement in z direction shear modulus of backfill material

total height of retaining wall coefficient of pseudo-dynamic passive resistance Mc adhesion factor Nc cohesion factor Ppe total seismic passive resistance Qh horizontal inertia force Qv vertical inertia force R reaction of the retained soil on the failure wedge T time period of vibration generated by earthquake Vs shear wave velocity Vp primary wave velocity W weight of the failure wedge c unit weight of soil ce equilibrium unit weight of soil W angle of internal friction of soil d wall friction angle a wall inclination angle with the vertical h failure surface angle with vertical r density of backfill material m Poisson’s ratio of backfill material f amplitude ratio v angular frequency

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