[American Society of Civil Engineers Geo-Shanghai 2014 - Shanghai, China (May 26-28, 2014)] Advances...

Computation of Seismic Active Earth Thrust on Retaining Wall for c- Backfill

V. B. Deshmukh1, N. H. Babaria2 and D. M. Dewaikar3

1 Associate Professor, Department of Structural Engineering, V.J.T.I. Matunga, Mumbai – 400019, INDIA; [email protected] 2 Student, Department of Structural Engineering, V.J.T.I. Matunga, Mumbai – 400019, INDIA; [email protected] 3 Professor, Department of Civil Engineering, IIT Bombay Mumbai – 400019, [email protected] ABSTRACT: The concept of seismic active earth thrust is essential for the safe design of earth retaining walls in the seismic zone. In the proposed method, reaction along the planar failure surface is obtained using Kötter’s (1903) equation. Further, limit equilibrium method is effectively used to compute active earth thrust considering pseudo-static condition with both horizontal and vertical seismic co-efficient. An attempt is made to locate point of application of active thrust using moment equilibrium condition in c- backfill. This study is further extended to compute active earth thrust considering pseudo-dynamic condition which is more realistic in representing time and phase difference within the soil backfill. Effect of various parameters such as angle of internal friction, wall friction angle, angle of backfill inclination, horizontal and vertical seismic co-efficient, shear wave, compression wave etc. are discussed. It is seen that seismic active earth thrust decreases with increase in both friction angle and cohesion. Further active earth thrust increases with increase in horizontal seismic co-efficient and surcharge angle. The results obtained by trial and error method are underestimated by 2% than the results obtained from pseudo-static and pseudo-dynamic analytical approach suggested by Sharma and Ghosh (2010). Similarly, predictions made in case of static approach are in good agreement with results obtained from analytical approach suggested by Sharma and Ghosh (2010). INTRODUCTION Coulomb’s wedge theory (1776) was the first analysis to calculate lateral earth thrust under static condition. Considering earthquake induced forces, Okabe (1926) and Mononobe-Matsuo (1929) were the forerunner to calculate seismic lateral earth pressure considering pseudo-static approach in cohesionless backfill. This pseudo-static approach, which incorporates inertia forces from earthquake into Coulomb’s

525Advances in Soil Dynamics and Foundation Engineering GSP 240 © ASCE 2014

Advances in Soil Dynamics and Foundation Engineering

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static earth pressure analysis, is known as the Mononobe-Okabe method. Prakash and Basavanna (1969) determined distribution and point of application of dynamic lateral earth pressure analytically for active case on back of retaining wall with cohesionless backfill. Dewaikar and Halkude (2002) determined both active and passive thrust due to Coulomb’s sliding wedge but by considering Kötter’s (1903) equation to calculate distribution of soil reaction along failure plane. Kötter’s (1903) equation is successfully applied in uplift plate anchors (Deshmukh et.al, 2011). This method locates the point of application of thrust by using moment equilibrium equation. Shukla et. al. (2009) developed an expression for total seismic active force from c- backfill on smooth vertical retaining wall. Ghosh and Sharma (2010) provided analytical solution for active earth pressure considering pseudo-static and pseudo-dynamic approach for c- backfill. In the proposed method, an attempt is made to calculate active thrust and its point of application by using Dewaikar and Halkude’s (2002) approach of equilibrium equation which is modified for c- backfill. It takes into account the parameters such as cohesion and adhesion, angle of internal friction, angle of wall friction, inertia force considering pseudo-static and pseudo-dynamic approach. The results of proposed method are compared with available results. PROPOSED PSEUDO-STATIC APPROACH FIG. 1 shows FBD of failure wedge for active case. The forces acting on the wedge are active thrust Pa, at an angle i.e. wall friction angle, body force W. Inertia force in horizontal direction Kh.W and vertical direction Kv.W, and soil reaction R on failure plane AB. Forces due to cohesion, C and adhesion, Ca are included along with the above mentioned forces.

FIG. 1. FBD of failure wedge ABC for active case



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Distribution of reaction along failure plane Using Kötter’s (1903) equation reaction, R along failure plane AB is obtained,

2 . tan . sin( )dp d

pds ds

(1)Where, dp = differential reaction pressure along failure surface, ds = differential length of arc of failure surface, = angle of soil internal friction, dα = differential angle, α = inclination of tangent at the point of interest with the horizontal, and, γ = unit weight of soil. Eq.1 gives the distribution of reaction along curved failure surface for active state. The total reaction force, R along failure surface is expressed as,

22

2 2

1 sin ( )sin( )

2 sin ( )sinR H

(2)

By equating moment of distribution of reaction pressure about point A (FIG. 1) with the moment of resultant total reaction, R about same point, point of application of reaction, r is obtained at one-third the distance of failure plane AB from point A. Weight of sliding wedge Based on geometry of failure wedge for trial failure angle, . The weight, W of failure wedge ABC (FIG. 1) is expressed as,

22

2

1 sin ( ) sin( 90 ) sin(90 )

2 sin sin( ) sin( )W H

(3)

Cohesion and adhesion The total cohesive force, C due to cohesion, c (kN/m2) of backfill acting in the wedge along the failure plane as shown in Fig.1 is expressed as,

. .sin( )

sin( ).sin

c HC

(4)

Similarly adhesion, Ca along the back face of retaining wall is expressed as,

. .

sina

c HC

(5)

Where, ψ = Adhesion factor. Total active thrust Pa Active thrust, Pa is obtained by using horizontal and vertical force equilibrium equation, (FIG. 1) and is expressed as



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Horizontal force equilibrium,

sin .cos .cos

sinh a

a

R K W C CP

(6)

Vertical force equilibrium,

1 cos .sin sin

cosv a

a

K W R C CP

(7)

Eq. 6 and Eq. 7 gives the magnitude of unknown active thrust, Pa. For a unique value of α, these two equations give similar values for Pa which corresponds to those at failure. This unique value of α is obtained by trial and error method. A trial value of α is assumed and corresponding forces such as weight, W, cohesive force, C, adhesive force, Ca, total reaction on failure surface, R are computed. Using vertical equilibrium values of Pa are obtained. If the trial value of α is equal to its characteristic value corresponding to the failure condition, the two computed values for Pa will be same, otherwise, these will be different. Till the convergence up to specified accuracy is reach between these two values of Pa, various trials for α is carried out. Point of application of active thrust Using moment equilibrium equation, point of application of active thrust is determined by taking moments along point A (FIG. 1). On re-arranging the terms from xp is expressed as,

. . (1 ). . .cos .

.cosh w v w

pa

K W y K W x R rx

P

(8)

Point of application of active thrust from base of the retaining wall is obtained as,

.sinph x (9)

PROPOSED PSEUDO-DYNAMIC APPROACH An inclined retaining wall of height H is considered with c- backfill of unit weight γ. In FIG. 3, forces acting on retaining wall with its geometry for typical active case of equilibrium are shown. The back face of wall AD is inclined at an angle, . The additional inertia forces acting due to triangular wedge DBE are deducted from apparent failure wedge ABE to get net inertia forces on actual retaining wall with back face AD. The height of apparent retaining wall with AE extended to E is denoted by He. In this approach, shear wave velocity, Vs = (G/ρ)1/2 and primary wave velocity, Vp = [2G(1-µ)/ρ(1-2µ)]1/2 , where, ρ = density of soil media and µ= poisons ratio of backfill media are assumed to act within the soil medium due to seismic loading. The acceleration equation includes time period, T for lateral shaking and expressed in terms of angular frequency, i.e. T = 2 / .



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FIG. 3. FBD of failure wedge ABD for active case under pseudo-dynamic condition

Therefore, mass of elemental strips is expressed as,

11 1

( ) 1 1( ) . .1

tan tan

H zem z dzg

(11)

Horizontal and vertical acceleration, ah(z1,t) and av(z1,t) with their respective amplitude Kh.g and Kv.g, as a result of sinusoidal base shaking are expressed as,

11

( )( , ) . .sin e

h hs

H za z t K g t

V

(12)

11

( )( , ) . .sin e

v vp

H za z t K g t

V

(13)

The horizontal inertia force, Qh1(t) is exerted on the small element due to horizontal earthquake acceleration, ah(z1,t). It is expressed as m(z1). ah(z1,t). Therefore, the total horizontal inertia force, Qh1(t) acting on wedge ABE is presented as,

1 2

. . (tan tan )( ) 2 . .cos 2 sin 2 sin 2

4 .tan . tanh e e

h e

K H Ht t tQ t H

T T T

(14)

Similarly, the vertical inertia force, Qv1(t) is exerted on the small element due to vertical earthquake acceleration, av(z1,t). It is expressed as m(z1).av(z1,t). Therefore, the total vertical inertia force Qv1(t) acting on wedge ABE is presented as,



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1 2


4 .tan .tanv e e

v e

K H Ht t tQ t H

T T T

(15)

Computation of seismic forces Qh2(t) and Qv2(t) Similarly, the horizontal and vertical inertia forces acting on ΔDBE (refer FIG. 3), is expressed as follows,

. . (tan tan )( ) 2 . .cos 2 - sin 2 - - sin 22 24 .tan .tan

K t h t h thQ t hh T T T

(16)

2 2


4 .tan .tanv

v

K t h t h tQ t h

T T T

(17)

Net horizontal inertia force, Qh(t) is obtained from Eq. (14) and Eq. (16) as follows,

1 2( ) ( ) ( )h h hQ t Q t Q t (18)

Similarly, net vertical inertia force, Qv(t) is obtained from using Eq. (15) and Eq. (17) as follows,

1 2( ) ( ) ( )v v vQ t Q t Q t (19) Total seismic active thrust, Pad Seismic active thrust, Pad is obtained by using horizontal and vertical force equilibrium equation, (FIG.3). From horizontal force equilibrium,

sin ( ) .cos .cos

sinh a

ad

R Q t C CP

(20)

Using vertical force equilibrium,

( ) cos .sin sin

cosv a

ad

W Q t R C CP

(21)

The procedure for computing seismic active earth thrust is similar to that followed in pseudo-static approach. The difference lies in assuming value of t/T ratio along with α to compute horizontal and vertical inertia force using Eq. (18) and Eq. (19) respectively. To get maximum value of earth thrust, trial values of α in the range of 00 to 900 and t/T ratio in the range of 0 to 1 are used in combination. A Trial value for failure angle, α is assumed for a particular t/T ratio and corresponding forces; weight of failure wedge, W, cohesive force, C, adhesive force, Ca, reaction, R and inertia forces are computed. If the trial value of α is equal to its characteristic value corresponding to the failure condition, the two computed values for Pad will be same, otherwise, these will be different. Till the convergence up to specified accuracy is reach between these two values of Pad, various trials for α and t/T is carried out. For various combinations of α and t/T, same values of Pad using Eq. (20) and Eq. (21) are obtained. The maximum value of all gives seismic active earth thrust.



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Point of application of seismic active thrust Using moment equilibrium equation, point of application of seismic active thrust is determined by taking moments along point A (FIG. 3) and xp is obtained,

( ) ( ) .cos .

.cos

. . .vp

ad

w w wh t Q t R rx

P

Q y x W x

(22)

Point of application of seismic active thrust from base of the retaining wall is obtained as,

.sind ph x (23)

RESULTS AND DISCUSSION The necessity of this approach is to compute seismic active earth thrust and its point of application, study the influence of various parameters, and compare results of pseudo-static and pseudo-dynamic approaches. For the results and calculations, unit adhesion is taken as unit cohesion i.e. ψ =1. Pseudo-static approach

FIG. 4. Variation of Pa/γH2 with for different values of cohesion, c [Kh=0.1, Kv=Kh/2, = 60o, =2/3]

FIG. 5. Variation of Pa/γH2 with Kh for different values of [Kv=Kh/2, c=10 kN/m2, = 60o, =2/3]

FIG. 4 shows variation of seismic active earth thrust with soil friction angle for different values of cohesion. It is seen that seismic active earth thrust decreases with increase in soil friction angle, and cohesion, c. FIG. 5 shows variation of normalized seismic active earth thrust with horizontal seismic coefficient for different soil friction angle. It is clear that seismic active earth thrust increases with increase in horizontal seismic coefficient Kh.

Table 1 shows effect of c and on point of application of active earth thrust.



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Table 1. Influence of Cohesion and Soil Friction Angle on h/H

c (kN/m2) 20o 30o 40o

5 0.3812 0.3992 0.4142 10 0.3936 0.4096 0.4184 15 0.427 0.4366 0.4328 h/H

Table 2 shows effect of Kh on point of application of earth thrust and it is observed that h/H ratio reduces with increase in Kh.

Table 2. Influence of Horizontal Seismic Acceleration Co-Efficient on h/H

Kh

20o 30o 40o

0 0.386 0.415 0.443 0.1 0.394 0.410 0.418 0.2 0.376 0.382 0.374 0.3 0.333 0.334 0.310 0.4 0.269 0.267 0.225 0.5 0.186 0.182 0.113 h/H

Table 3. Comparison of Seismic Active Earth Thrust, Pa with Available Methods

Cohesion c (kN/m2) (deg.)

Sharma and Ghosh (2010), Pa (kN)

Proposed Method Pa (kN)

5 20 173 172.84 30 147 147.018 40 129.75 129.707

7.5 20 157.25 156.022 30 133.75 133.167 40 119 118.455

10 20 141.25 138.312 30 120.75 118.649 40 108.25 106.625

Table 3 shows the comparison of seismic active thrust obtained from proposed method with the results of method suggested by Sharma and Ghosh (2010) for different soil friction angle, . It is observed that values obtained are in very close proximity.



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Pseudo-Dynamic approach The variation of various parameters by pseudo-dynamic approach remains similar to that by pseudo-static approach. Table 4 shows comparison of seismic active earth thrust with available method suggested by Ghosh and Sharma (2010) by pseudo-dynamic approach. It is observed that the results are in close agreement. Table 5 shows comparison between pseudo-static and pseudo-dynamic approach, and it is observed that pseudo-dynamic approach under-estimates seismic active earth thrust.

Table 4 Comparison of Seismic Active Earth Thrust, Pad with Available Methods by Pseudo-Dynamic Analysis

(deg.)

(deg.)

Seismic active earth, Pad (kN) thrust Ghosh and

Sharma(2010) Proposed Method

0 20 190.075 189.035 30 155.26 153.731 40 126.56 124.111

/2 20 188.71 188.813 30 156.78 156.828 40 132.52 132.296

20 198.98 198.645 30 178.875 179.005 40 174.92 175.47

Table 5. Comparison of Pseudo-Static Approach and Pseudo-Dynamic

Approach.

(deg.) δ(deg.) Pseudo-Static Analysis Pseudo-Dynamic Analysis

Pa (kN) h/H Pad (kN) hd/H 20

2129.315 0.427 125.522 0.431

30 113.802 0.435 109.113 0.444

40 105.77 0.429 99.798 0.445

20 23

131.805 0.396 127.565 0.403

30 119.187 0.390 113.744 0.403

40 116.051 0.365 108.637 0.386

c = 15kN/m2, Kh=0.2, Kv=Kh/2, θ = 60o, 0

CONCLUSIONS A closed form solution based on semi empirical equation is obtained for retaining wall under seismic conditions using Kötter’s equation. Point of application is obtained for active case using moment equilibrium condition. Results show good agreement with available results.



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REFERENCES Coulomb, C. A. (1773). “Essai sur une application des regles des maximis et minimis

a quelque problems de statique relatifs a 1’architecture.” Memoires d’Academie Roy. Pres. Diverssavants: 7.

Deshmukh, V. B., Dewaikar, D. M. and Choudhury, D. (2011) “Uplift capacity of horizontal strip anchors in cohesionless soil”, J. Geotechnical and Geological Engineering, Springer (29): 977-988.

Dewaikar, D. M., and Halkude, S. A. (2002). “Seismic passive/active thrust on retaining wall-Point of application.” Japanese Geotechnical Society 42(1): 9-15.

Ghosh, S., and Sharma, R. P. (2010). “Pseudo-dynamic active response of non-vertical retaining wall supporting c- backfill.” Geotech. Geol. Eng., 28(5): 633–641.

Kötter’s, F. (1903). “Die Bestimmung des Drucks an gekrmmten Gleitflchen, eine Aufgaba aus der Lehre vom Erddruck.” Sitzungsberichte der Akademie der Wissenschaften, Berlin: 229-233.

Mononobe, N. and Matsuo, H. (1929). “On the determination of earth pressures during earthquakes.” Proc., World Engineering Conf., Vol. 9: 274–280.

Okabe, S. (1926). “General theory of earth pressure.” J. of the Japanese Society of Civil Engineers, Tokyo, Japan: 12(1).

Prakash, S. Basavanna, B. M. (1969). “Earth pressure distribution behind retaining wall during earthquake”, Proc. Fourth World Engrg. Conf. on Earthquake Engrg., Santiago, Chile, A 5: 133-148.

Sharma, R. P. and Ghosh, S. (2010) “Pseudo-static seismic active response of retaining wall supporting c- backfill”, EJGE, vol. 15, Bund. A, pp: 533-546.

Shukla, S. K., Gupta, S. K., and Sivakugan, N. (2009). “Active earth pressure on retaining wall for c- soil backfill under seismic loading condition.” J. Geotech. Geoenviron. Eng., 135(5): 690–696.



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[American Society of Civil Engineers Geo-Shanghai 2014 - Shanghai, China (May 26-28, 2014)] Advances...

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