Non-Linear Rupture Surface and Response of Retaining Wall under Static Loading Condition during Active State of Equilibrium

(1)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 6, June 2013)

509

Non-Linear Rupture Surface and Response of Retaining Wall

under Static Loading Condition during Active State of

Equilibrium

Sima Ghosh

1

, Arijit Saha

2

1_{Asst. Prof. Dept. of Civil Engg. NIT Agartala} 2_{PG Student NIT Agartala}

Abstract— In this paper, an effort has been made to determine the active earth pressure on rigid retaining wall considering non-linear failure surface. The backfill of c-Φ nature has been considered for the purpose. Analytical solutions have been obtained considering non-linearity of the rupture surface using Horizontal Slicing Method. The non-linear model is supposed to provide more general solution compared to linear nature of failure surface. The effects of variation of all the parameters have been studied and it is seen that the shape of the rupture surface may be hogging or sagging in nature.

Keywords—Active earth pressure, horizontal slicing method, c-Φ backfill, rigid retaining wall, wall inclination, curvilinear rupture surface.

I. INTRODUCTION

Research works on earth pressure coefficients for retaining walls started from the analyses given by Rankine (1857) and Coulomb (1776). They have given the solutions for lateral earth pressure behind retaining walls supporting Ф backfill under the consideration of linear failure surface. Culmann‟s (1865) has given the graphical solution for the determination of Coulomb lateral earth pressure. All these analyses have been done considering the Φ nature of backfill with linear failure surface. Whereas, in practical conditions, many of the retaining walls constructed, now a days, are subjected to c-Φ nature of backfill material. Thus the analysis for Φ nature of backfill is not sufficient for design of these walls. We have the Bell (1915) equation to find the effects of cohesion (c) on the failure wedge. Terzaghi (1943) has considered a log spiral sliding surface for the analysis of lateral earth pressure to show the nonlinearity of failure surface. The analysis of Ghanbari and Ahmadabadi (2010) has suggested Horizontal Slicing Method (HSM) for better results considering linear failure surface. Hereunder, an effort has been made to determine the active earth pressure coefficients for retaining walls using horizontal slice method.

Also, the effects of cohesive resistance of soil mass and adhesive resistance of wall surface have been taken into consideration to optimize the values of lateral earth pressure coefficients.

II. METHODOLOGY

A retaining wall of height, H inclined at an angle, α with the vertical as shown in the Fig.1 has been considered having the curvilinear failure surface. The failure surface makes an angle, θn at the bottom and another angle, θ1 with

the vertical at the bottom of top slice. The failure wedge is split into „n‟ number of thin slices of thickness, ΔH. The rate of change of inclination of failure surface (θ1 and θn)

has been assumed as θr={(θ1 ~ θn)/(n-1)}.

Fig.1 Battered-faced retaining wall (Active Case)

Ѳn

α

Ø

R

W

c

a

c

δ

P

a

H

Ѳ1+ѲR

Ѳ1

LAYER -1

LAYER -2

(2)

International Journal of Emerging Technology and Advanced Engineering

510

Free Body Diagram of retaining wall-backfill system under active state of equilibrium is shown in Fig.2.

Fig.2 Shows various components of the retaining wall along with slices (Active case)

The pressures acting on the wall has been calculated by considering the following parameters:

Wi=Weight of the failure wedge of i th

slice.

Vi-1, Vi=Vertical load (UDL) on top and bottom of ith slice.

(Hs)i-1, (Hs)i=Horizontal shear acting on the top and bottom

of the ith slice.

Φ=The angle of internal friction of soil. δ=The angle of wall friction.

c=Cohesion acting on the failure surface. ca=Adhesion acting on the wall surface.

Pai=Active earth pressure on ith slice.

Ri=The reaction of the retained soil on ith slice.

Applying the force equilibrium conditions for ith slice, we can solve the equations in the following pattern:

0 



H











cos sin ) ) 1 ( cos( ) ) 1 ( sin( ) ( ) ) 1 ( cos( ) cos( 1 1 1                a R R i S R i i c i i c i R P (1) 0 



V

















cos cos ) ) 1 ( cos( ) ) 1 ( cos( )} ) 1 ( tan( {tan ) 2 1 ( ) ) 1 ( sin( ) sin( 1 1 1 2 1                     a R R R R i i c i i c i i i R P (2)

Solving the Eqn (1) and (2) we get,

) ) 1 ( sin( ) ) 1 ( sin( ) ( 2 cos ) ) 1 ( cos( ) ) 1 ( cos( cos ) ) 1 ( cos( )) ) 1 ( tan( )(tan 1 2 {( ) ( 2 1 1 1 1 1 1 2 r r i s r s r s r r i a i i i M i N i i i P                                                                

(3)

Where, i s i s i

s H H

H ) ( ) ( )

(  1







































 

)

tan

)

1 (

(tan(

)}

tan

)

1 (

)(tan(

1 (

tan

)

(

)

(tan(

{

tan

1 1 1 1 2

















R s R n i m R

i

N

i

n

m

Where,

,

0 )

tan(

1







  n i m R

m



When, m=n

V

1

(H

s

)

1

Ø

δ

c

1

R

1

P

1

α

c

a1 W1 ΔH

Ѳ

1

V

n-1

_(H

s

)

n-1

Ø

δ

c

n

R

n

P

n

α

c

an Wn Ѳn

V

i-1

V

i

(H

s

)

i-1

(H

s

)

i

Ø

δ

c

i

R

i

P

i

α

c

ai Wi

Ѳ1+(i-1)ѲR

LAYER - 1

LAYER - n LAYER - i

ΔH

H

(3)

International Journal of Emerging Technology and Advanced Engineering

511







C

N

_S

2







Ca

M_S 2

The active earth pressure coefficient can be simplified as,

2 1

2 







n i

ai a

P K

(4)

Optimisation of the active earth pressure coefficient Ka

is done for the variables θ1 and θn satisfying the

optimization criteria. The optimum value of Ka which are

denoted as Kae are given in Table.1.

Table1

Active earth pressure coefficients

Φ δ Mc

Nc=0.1 Nc=0.2

α=-20 α=0 α=+20 α=-20 α=0 α=+20

10 0

0 0.543 0.634 0.785 0.453 0.582 0.754

Nc/2 0.497 0.607 0.767 0.372 0.539 0.737

Nc 0.453 0.582 0.754 0.297 0.502 0.729

Φ/2

0 0.499 0.603 0.769 0.418 0.556 0.743

Nc/2 0.457 0.578 0.754 0.343 0.516 0.73

Nc 0.418 0.556 0.743 0.275 0.482 0.724

Φ

0 0.466 0.582 0.767 0.391 0.539 0.744

Nc/2 0.428 0.559 0.754 0.322 0.502 0.732

Nc 0.391 0.539 0.744 0.256 0.469 0.728

20 0

0 0.296 0.429 0.612 0.217 0.38 0.576

Nc/2 0.255 0.403 0.592 0.144 0.339 0.554

Nc 0.217 0.380 0.576 0.079 0.305 0.542

Φ/2

0 0.253 0.395 0.615 0.182 0.35 0.578

Nc/2 0.217 0.372 0.593 0.117 0.313 0.556

Nc 0.182 0.350 0.577 0.056 0.28 0.544

Φ

0 0.23 0.385 0.666 0.163 0.342 0.622

Nc/2 0.196 0.363 0.644 0.101 0.305 0.597

Nc 0.163 0.342 0.621 0.043 0.272 0.583

30 0

0 0.143 0.281 0.481 0.076 0.24 0.446

Nc/2 0.108 0.26 0.461 0.016 0.207 0.425

Nc 0.076 0.24 0.445 -0.037 0.179 0.414

Φ/2

0 0.113 0.26 0.554 0.052 0.22 0.513

Nc/2 0.082 0.239 0.534 -0.003 0.185 0.473

Nc 0.052 0.22 0.512 -0.055 0.157 0.454

Φ

0 0.104 0.277 0.769 0.043 0.239 0.757

Nc/2 0.073 0.259 0.762 -0.012 0.201 0.739

Nc 0.043 0.239 0.758 -0.065 0.167 0.723

40 0

0 0.049 0.176 0.357 -0.004 0.143 0.326

Nc/2 0.021 0.158 0.348 -0.052 0.117 0.328

Nc -0.004 0.143 0.326 -0.095 0.098 0.324

Φ/2

0 0.029 0.17 0.642 -0.02 0.136 0.618

Nc/2 0.004 0.153 0.629 -0.069 0.106 0.601

Nc -0.02 0.136 0.619 -0.113 0.08 0.573

Φ

0 0.034 0.346 -- -0.022 0.269 --

Nc/2 0.005 0.306 -- -0.076 0.225 --

Nc -0.022 0.272 -- -0.127 0.185 --

III. DISCUSSION ON RESULTS

A detailed Parametric study has been conducted to find the of variations of static active earth pressure with various other parameters like soil friction angle (Φ), wall inclination (α), wall friction angle (δ), cohesion (c), adhesion (ca). For Φ=100, 200, 300, 400; δ=0, Φ/2, Φ; Nc=0,

0.1, 0.2; Ms=0, Nc/2, Nc and α=+200, 0, -20⁰ the details of

these studies are presented below:

Where,

N

_C



(



H

/

H

)

N

_S,

S C

H

M



(



/

)

Fig.3. Shows the variation of active earth pressure coefficient with respect to soil friction angle (Ф) at different Wall inclination angles

(4)

International Journal of Emerging Technology and Advanced Engineering

512 A. Effect of Inclination of the Wall (α)

Fig.3 shows the variation of active earth pressure coefficient (Kae) with soil friction angle (Φ) for different

values of α. It is observed that the value of Kae increases

with the increase of wall inclination angle (α). For example, at Nc=0.1, Mc= Nc, δ=Φ/2, α=-20°, the values of

Kae are 0.418, 0.182 and 0.052 for Φ=10°, 20°, 30°

respectively; whereas, the values of Kae increases up to

0.743, 0.577, 0.512 for α=+20° for all other parameters remaining unchanged.

Fig.4. Shows the variations of active earth pressure coefficient with respect to soil friction angle (Ф) at different Wall friction angles (δ= 0,

Ф/2, Ф), Nc=0.1, Mc=Nc for α=00.

B. Effect of Wall Friction Angle (δ)

Fig.4. shows the variation of active earth pressure coefficient (Kae) with soil friction angle (Φ) for different

values of δ. It is observed that the value of Kae decreases

with the increase of wall friction angle (δ) for lower values of soil friction angles, but at higher value of Φ, with increase in the value of wall friction angles, there is an abrupt change in the value of Kae. This may be due to

change in orientation in the direction of Pae. For example, at

Nc=0.1, Mc=Nc, Φ=10°, α=0° the values of Kae are 0.582,

0.556 and 0.539 for δ=0, Φ/2 and Φ respectively. Whereas; the value of Kae increases with the increase of δ from Φ/2

to Φ at Φ=25° onward. The orientation of the active earth depends not only on the angle of wall friction but also on wall inclination. Depending upon the angle of wall friction and wall inclination it may happen that there is an abrupt change in the value of seismic active earth pressure and due to that at higher value of angle of wall friction, the nature of the active earth pressure co-efficient do not follow the same trend as in the lower value of angle of wall friction.

Fig.5. Shows the variations of active earth pressure coefficient with respect to soil friction angle (Ф) at Mc=0, Nc/2, Nc for Nc=0.1, δ=Ф/2

and α=200_.

Fig.6. Shows the variations of active earth pressure coefficient with respect to soil friction angle (Ф) at Nc=0, 0.1, 0.2 for Mc=Nc, δ=Ф/2

and α=200_.

C. Effect of Cohesion and Adhesion

Fig.5 and 6 shows the variations in active earth pressure (Kae) with respect to soil friction angle (Φ). At Mc=0, Nc/2,

Nc for Nc=0.1, δ=Ф/2 and α=20°,the value of Kae gradually

decreases with the increase of Mc value. For example, at

Φ=30°, Mc=Nc/2, δ=Ф/2 and α=20°, the values of Kae are

0.534, and 0.473 for Nc=0.1 and 0.2 respectively. Whereas;

the value decreases upto 0.512, and 0.454 for Mc=Nc

(5)

International Journal of Emerging Technology and Advanced Engineering

513

Again, it is also found that the value of Kae decreases

with the increase of Nc value. For example, at Φ=20°,

δ=Ф/2 and α=20°,the values of Kae are0.625, 0.566 and

0.544 for Mc=0, Nc/2 and Nc respectively for Nc=0.1.

Whereas; the value decreases upto 0.578, 0.556 and 0.544 for Nc=0.2 with all other conditions remaining unchanged.

Fig.7. Shows the variations of active earth pressure coefficient with respect to soil friction angle (Ф) for different heights of retaining wall

(H=5m, 7.5m, 10m ), Nc=0.1, Mc=Nc for α=200, δ=Ф/2.

D. Effect of Height (H)

Fig.7 shows the variations of active earth pressure coefficient with respect to soil friction angle (Ф) for different heights of retaining wall at Nc=0.1, Mc= Nc,

α=20° and δ=Ф/2. It is found that the coefficient of active earth pressure increases with the increase in height. As the height increases, the increment in active earth pressure also increases gradually. The value of Nc=2c/γH and Mc=2ca/γH

decreases with increase in height (H) of the retaining wall. So, the coefficient of active earth pressure also increases.

E. Wall Inclination and Nonlinearity of Failure Surface

Fig.8 shows the nonlinearity of failure surface of backfill (active case) for different values of wall inclinations like α=-200_{, 0, +20}0_{. It is found that, at α=20°,}

i.e., wall inclines away from backfill; the failure surface is sagging in nature. Again, at α=0°, i.e., when the wall is vertical, the failure surface is slightly hogging in nature.

With a decrease of wall inclination upto α=-20°, i.e., wall inclines towards the backfill then the failure surface is hogging in nature. Also Fig.9 and 10 show that the failure wedge is linear in case of Ghosh and Sengupta (2012) analysis, whereas, the curvilinear surface is found in the present analysis. The comparison shows that the value of failure angle is 28º in case of Ghosh and Sengupta (2012) for wall inclination α=20°. And for this case, on the basis of present study, the failure angle at top and bottom are -43° and 68° respectively. So, failure wedge consumes more soil mass in the present study in comparison to Ghosh and Sengupta (2012) analysis.

Fig.8. Shows the nonlinearity of failure surface of backfill (active case) for different values of wall inclinations at Φ=30º, δ= Φ/2, Nc=0.1,

Mc=Nc.

Fig.9. Shows the comparison between failure surface of backfill (active case) for wall inclination angle, α=+200 _at_{Φ=30º, δ= Φ/2,}

Nc=0.1, Mc=Nc.

Wall face (α=+20°) Wall face (α=0°) Wall face (α=-20°)

Failure surface (α=+20°) Failure surface (α=0°)

(6)

International Journal of Emerging Technology and Advanced Engineering

514

Fig.10. Shows the comparison between failure surface of backfill (active case) for wall inclination angle, α =-20º at Φ=30º, δ= Φ/2,

Nc=0.1, Mc=Nc.

F. Comparison of Results

Fig.11 shows the variations of active earth pressure coefficient with respect to soil friction angle (Φ) at δ= 2Φ/3, Nc=0.1, Mc=Nc for α=20

0

. Kae increases uniformly

with the increase in the value of soil friction angle (Φ). The values obtained in present study of nonlinear analysis are higher than the values of Kae for Sharma and Ghosh (2010)

analysis. Fig.12 shows the comparison of results obtained from present study with the experimental results obtained from Ghosh and Sharma (2013). The present values are 0.4 to 9% greater than the experimental values.

Fig.11. Shows the comparison of active earth pressure coefficient with respect to soil friction angle (Φ) with previous analytical analysis at

wall friction angles δ= Φ/2, Nc=0.1, Mc=Nc for α=200.

Fig.12. Shows the comparison of active earth pressure coefficient with respect to soil friction angle (Φ) with experimental analysis at wall

friction angles δ= Φ/2, Nc=0.1, Mc=Nc for α=200.

Considering the curvilinear rupture surface and using the horizontal slices method, an analysis has been developed for the determination of active earth pressure on the back of a retaining wall supporting c-Φ backfill. Very interestingly, it is seen that the shape of the rupture surface may be sagging or hogging in nature. Generally, the values of active earth pressure coefficients obtained using this method is 10-15% higher in comparison to Sharma and Ghosh (2010). This fact suggests the acceptability of the model which may be extended for the analysis of retaining wall under seismic loading condition.

REFERENCES

[1] Bell, A. L. (1915), “Lateral Pressure and Resistance of Clay and the Supporting Power of Clay Foundations, Minutes, Proceedings of the Institution of Civil Engineers, London, Vol. 199”.

[2] Coulomb, C.A. (1776), “Essai Sur Une Application Des Maximis et Minimis a Queques problems Des Statique Relatifsa1‟Architecture”, Nem. Div. Sav.Acad, Sci.Vol.7.

[3] Culmann, K. (1866), “Die Graphische Statik, Mayer and Zeller, Zurich.

[4] Ghanbari, A. and Ahmadabadi, M. (2010), “Pseudo-Dynamic Active Earth Pressure Analysis of Inclined Retaining Walls Using Horizontal Slices Method” Transaction A: Civil Engineering Vol. 17, No. 2, pp. 118-130, © Sharif University of Technology. [5] Ghosh, S. and Sengupta, S. (2012), “Extension of Mononobe-Okabe

(7)

International Journal of Emerging Technology and Advanced Engineering

515

[6] Ghosh, S. and Sharma, R. P. (2013), “Experimental Study and Pseudo-Dynamic Solution for Seismic Active Earth Pressure on Model Retaining Wall Supporting c- Φ Backfill”1st_annual international conference, pp324-333.

[7] Rankine, W. J. M. (1857), “On the Stability of Loose Earth”, Phil. Tras. Royal Society (London).

[8] Sharma, R. P. and Ghosh. S. (2010), “Pseudostatic Seismic Active Response of Retaining Wall Supporting c-Φ Backfill” EJGE Vol. 15(2010), Bund. A.