Effect of Arching on Passive Earth Pressure for Rigid Retaining Walls

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1 Effect of Arching on Passive Earth Pressure for Rigid Retaining Walls R.S. Dalvi & R.S. Kulkarni Department College of Engineering Pune k.sharine@gmail.com Abstract Arching involves stress transfer from yielding part of soil to unyielding part of soil. Many authors considered arching action for active earth pressure. In this paper arching action is considered for passive earth pressure in non-cohesive backfill. The backfill is assumed downward in form of parabolic shape due to arching. The value of θ (the angle of major principal plane) is calculated from different values of δ and φ. An illustrative example has been solved to show the effect on the earth pressure distribution on retaining wall considering arching for different wall friction angle. The results obtained show non-linear distribution of passive earth pressure behind retaining wall which is same as previous researchers. Keywords Arching; passive earth pressure; wall friction; retaining wall. I. INTRODUCTION Arching involves stress transfer from yielding part of the soil to the unyielding part of the soil. It is first described by Terazaghi in It depends on shear strength of soil and yielding of soil. Many researchers have studied active earth pressure against the rigid retaining wall considering arching effect. Janseen (1895) proposed differential equation for pressure in silos. He provides theoretical basis for understanding the effect of arching. Spanglar and Handy (1982) and Wang(2000) suggested procedures to estimate the non linear pressure distribution of active earth pressure on the back of the wall based on Janseen theory (1985), Fang and Ishibhishi (1986) show that lateral earth pressure behind retaining wall depends on the mode of wall movement or rotation of the wall at the top or bottom and the pressure distribution is non linear. This non linearity is attributed due to arching effect by Handy(1985). II. REVIEW OF PREVIOUS WORK A. Review Stage Handy(1985) considered two stages of arching for active earth pressure when the back of the wall is rough. He assumed the shape of arch as caternary. Stage I arching is due to change of direction of principal stresses. Considering Stage I arching the earth pressure estimated is more than the predicted by classical theory. As the wall movement is continuous semi arch is formed between the wall and the boundary of the slip surfaces separating mobile and immobile soil. Due to this minor principal stress becomes horizontal. This is termed as second stage of arching. The second stage of arching reduces vertical and horizontal pressure particularly near the base of the wall. Due to this effect pressure distribution becomes rounded at the base. The equation proposed by Handy (1985) and Horrop- Williams (1989) have some limitations. They have not considered the dependence of vertical stress σv and horizontal stress σh due to soil friction angle, φ and soil to wall friction angle, δ. Paik and Salgado(2004) have proposed an equation considering the effect of φ and δ on σv and σh. They compared the existing test results with the values calculated from the equation given by other authors. They have also presented design charts based on their formulation. Dalvi, Bhosale and Pise (2005) considered the effect of arching on the passive earth pressure in non cohesive soil. They have considered stage I arching for different heights of retaining wall. Dalvi and Pise(2012) calculated effect of arching on passive earth pressure and compared with results of previous work. B. Final Stage In this paper effect of arching on passive earth pressure for rigid retaining wall is considering translation mode and formulation is carried out. The analytical results are compared with experimental work. III. ASSUMPTION Following assumptions have been made in the analysis; 1) The soil is cohesion less, semi infinite, homogenous, isotropic and the backfill is horizontal. 1

2) The problem is a plane strain problem i.e. twodimensional. 3) The soil mass is bounded between two parallel, un-yielding rough vertical walls.

6) The soil mass move up in a curved path which is considered as parabolic arch. 7) Wall friction angle δ is less than an equal to soil friction angle φ. Fig(1)Representation Of Soil Arching IV.

Let us assume that two parallel, rigid vertical wall retain granular soil as shown in fig(1).

Consider a small strip of a soil mass having thickness dh at a depth of h below the ground surface in the soil mass. V is the vertical upward force acting on the strip.

2 2) The problem is a plane strain problem i.e. twodimensional. 3) The soil mass is bounded between two parallel, un-yielding rough vertical walls. 4) The walls are assumed to rotate towards the soil mass creating passive case. 5) The sliding surfaces are vertical and pass through the outer edge of the yielding wall. 6) The soil mass move up in a curved path which is considered as parabolic arch. 7) Wall friction angle δ is less than an equal to soil friction angle φ. Fig(1)Representation Of Soil Arching IV. ANALYSIS Fig(2) Stress on differential flat element on backfill Fig(1) shows the model considered for representing arching in the soils. Let us assume that two parallel, rigid vertical wall retain granular soil as shown in fig(1). The distance between walls is B when rotation of the wall taken place towards soil mass, passive state is developed and the soil moves in upward direction as shown. Consider a small strip of a soil mass having thickness dh at a depth of h below the ground surface in the soil mass. V is the vertical upward force acting on the strip. The weight of the strip is γbdh. The force acting on the strip is ( as shown in fig 1.) lateral force and static friction at the end of wall. The frictional resistance is acting in the downward direction, this frictional resistance is equal to lateral force times the coefficient of friction i. e. μ=tanδ, where δ is soil wall friction angle. V. SHAPE OF SLIP SURFACE IN RETAINED SOIL MASS A rigid retaining wall may yield either by tilting or translating towards the backfill. The slope of the slip surface, which occurs in the retained soil after sufficient yielding of the wall depends not only wall frictional angle,δ, but also on the yielding mode(terazaghi 1943). A. Failure surface In Coulomb s theory (1776) the slip surface is assumed to the plane is used in practice. In this paper the simplifying assumption made by Coulomb 1776) where the slip surface for the translating rigid wall is plane and makes at an angle of 45-φ/2 to the horizontal is used to calculate the passive earth pressure on the wall. Considering the arching effect in the backfill Shape of failure surface is considered as planar failure surface with parabolic arch. Rotation of B. Principal stress The major and minor principal planes are as shown in fig(3). The vertical and horizontal stress at the wall is σ v and σ p. Inside the soil mass the trajectory of major principal stress (σ 1 ) given continuous tension arch in the upward direction. Due to parabolic arch considered direction of the principal stress (σ 1 ) makes an angle θ with the wall as shown in fig(3). This slip line makes angle θ= 45-φ/2. The minor principal stress (σ 3 ) acts along the convex arch. VI. STRESS IN ARCH Fig(2) shows stress acting on AC. Considered a small element of width da at a depth Z from the top of wall. dz is thickness of element.then σph is lateral stress acting on the wall and δw is shear stress developed. Then (1) Fig(3)Mohr s circle for stress at wall Similarly σp is lateral stress act a point D on parabolic arch is given by, 2

3 (2) Where ψ is angle between the tangent to the arch at a point D and its vertical. Dividing eqution (2) by σ1 and substituting σ3 / σ1 = 1/N for passive condition we get, σ p /σ 1 = cos 2 ψ + 1/N sin 2 ψ (3) Where N= ratio of major principal stress N = tan2(45-φ/2) Using property of stress variants, σ 1 + σ 3 =σ v + σ p (4) Substituting σp from (3) gives σ v /σ 1 = sin 2 ψ + 1/N cos 2 ψ (5) For triangle OAB and triangle ABC in fig(2) ζ w = σ ph tan δ = (σ 3 σ ph ) tan θ (6) Thus tan θ = σ ph /σ 3 1 σ ph /σ 3 tan δ (7) Dividing equation(3) by σ 3 σ ph /σ 3 = N cos 2 θ + sin 2 θ (8) By substituting Equation (8) in (7 We get second order equation tan θ = N+tan 2 θ 1 N 3 tan δ (9) Solution of this equation is θ = tan 1 N 1 ± N 1 4N2 tan 2 δ 2tanδ (10) Equation (10) gives two values of θ out of that maximum value is considered. A. Planar Failure surface with parabolic arch In this paper, it is assumed that trajectory of the minor principal stress takes the form of parabola. Considering that slip planes in the soil makes an angle α= 45-φ/2 with horizontal and that the angle between the slip plane and minor principal stress must be 45-φ/2, it follows that at the right edge of the differential flat element as shown in fig(4) the minor principal stress must be horizontal. The parabolic arch is expressed in polar form and general point is expressed as (r,). The coordinate at the edge of the wall is (r0, θ). The general equation of parabolic arch is expressed in equation (11) r = (2β)/(1 cos ψ) (11) Where, β= parameter define parabolic shape of arch. Width Bz of differential flat element is, B z = r 0 cos θ (12) Where r0 = radius of minor principal stress trajectory at the wall. θ = angle of minor stress plane with respect to horizontal at the wall. The differential vertical force dv on element at point B in Fig(4) can be expressed as; dv = σ v da = σ 1 (sin 2 ψ+1/n cos 2 ψ)(γdψsinψ) (13) Where da= width of shaded element at point B. The average vertical stress σ v across the differential flat element as shown in fig(3) can be obtained by dividing vertical force V acting on the differential element by the width of the element at B. B z = r 0 cos θ σ v = V B z = and r 0 = 2β/(1 cos ψ) π/2 θ Integrating above equation we get; σ 1 sin 2 ψ+1/n cos 2 ψ 2βsinψ dψ 1 cosψ 2βcosθ 1 cosθ } σ v = σ 1 1 cosθ cosθ 1/N cosθ+ln1 cosθcosθ} (15) (14) Dividing Eq(2) by (15) gives new lateral stress ratio Kpw VIII. ESTIMATION OF EARTH PRESSURE BEHIND WALL (16) Fig(5) shows the stress acting on the differential flat element behind a rigid wall. As the slip surface in the backfill forms an angle of 45-φ/2 with horizontal, the major and minor principal stresses at the right edge of the differential element must be applied in vertical and horizontal direction respectively. Accordingly, the shear stress in vertical and horizontal planes at right edge of the differential flat element is zero, however at the left edge of the differential flat element is non-zero along the wall. At the left edge of differential element shear stress along the wall δw is given as; ζ w = σ ph tan δ = σ v K pw tan δ (17) 3

The rectangular differential element with thickness dz is subjected to stress is as shown in fig(5). The element is in equilibrium.

Summation of all vertical forces acting on differential element is; (18) Kpw = new passive lateral stress ratio from eq(16) and γ= unit weight of backfill.

Applying boundary conditions at z=0 to (19)We get (20) Multiplying equation (20) by (21) Fig(5) Free Body Diagram Of Differential Flat Element IX.

4 The rectangular differential element with thickness dz is subjected to stress is as shown in fig(5). The element is in equilibrium. The average vertical stress ; the shear stress δw acting on the interface of wall; and the self weight of element. Summation of all vertical forces acting on differential element is; (18) Kpw = new passive lateral stress ratio from eq(16) and γ= unit weight of backfill. Substituting general solution of differential equation is Where H= Height of retaining wall and Z= depth from surface of the backfill. C= constant of integration. Applying boundary conditions at z=0 to (19)We get (20) Multiplying equation (20) by (21) Fig(5) Free Body Diagram Of Differential Flat Element IX. MAGNITUDE AND HEIGHT OF APPLICATION OF LATERAL PASSIVE FORCE The lateral passive force P ph on the rigid retaining wall can be obtained by integrating (21) with respect to z and substituting the expression for σ ph as derived above. (22) In the case of nonlinear distributed passive earth pressure, the height of application of the total passive force, which coincides with the point of application of the lateral force, can be obtained by dividing the moment of the passive earth pressure about the wall base by the lateral passive force. The moment, M, of the lateral active earth pressure about the wall can be obtained by integration and substitution of σph, yielding the expression for moment as follows: (23) Now dividing (23) by (22) gives the height of application of the lateral passive force, h Fig(4) Trajectory Of Minor Principal Stress Behind Retaining Wall (24) For δ =0 the height of application of lateral passive force is one third of the wall height. This is consistent with the results obtained from Rankine s theory. 4

5 X. ILLUSTRATIVE EXAMPLE Fig.(6) variation of passive Lateral stress with wall height for different δ/φ nature of variation of lateral passive earth pressure for varying δ /φ ratios. This is consistent with [Terzaghi, 1943] prediction of nonlinear variation of earth pressure along the wall height. It may be observed that when δ /φ =0, the earth pressure variation becomes triangular as suggested by [Rankine, 1857]. Again as δ /φ increase the distribution of passive earth pressure changes from triangular to non linear. It is noted that the earth pressure attains a maximum value at 0.7 to 0.9 times the wall height. In order to study the variation of lateral passive pressure with wall height as a function of internal friction angle of soil, the nondimensionalized parameter (σph/γh) has been plotted with respect to z/h for varying values of φ for the case δ /φ =0.5. From Figure (7), it is noted that as the internal frictional angle of backfill increases, the lateral earth pressure on the wall decreases except for φ =0. It may also be worth noting that the maximum lateral earth pressure occurs at times the height of wall beyond which it decreases and achieves a minimum value of zero at the wall base. XI. HEIGHT OF APPLICATION OF EARTH PRESSURE The height at which the resultant earth pressure centered from the base of the wall is determined. The results are presented in fig(8). It is observed that for φ=0, the normalized height of application of lateral passive force is at 0.33 regardless of δ/φ. This is consistent with the prediction made by Rankine s (1857). It is also observed that the highest normalized height of application of lateral passive force is 0.45, which is in agreement with prediction made by Handy (1985). It is also observed that δ/φ increases, for any given φ, the height at which lateral passive force centered is also increased. Fig.(7) variation of passive lateral stress with wall height for different φ To check the present analysis, an example is taken having 1m height of retaining wall. Lateral stress is calculated for different values of φ = 10,20,30,40,50 and for different δ/φ=0.25,0.5,0.75 conditions. Results are as shown in fig(6) and (7). Fig(6) shows variation of lateral earth pressure along the depth of retaining wall for various δ/φ ratios. The lateral passive earth pressure has been nondimensionalized with respect to wall height (H) and the unit weight of the backfill soil (γ). The depth coordinate along the wall height (z) has been nondimensionalized with respect to the wall height (H).These diagrams clearly bring out the curvilinear Fig(8) Variation Of Height of application(h) For Different δ/φ 5

6 Fig(9) Variation of force with δ XII. COMPARISON WITH OTHER THEORY To facilitate a clear understanding of the effect on the lateral passive earth force due to various parameters like shape of critical failure surface, shape of an arch and mobilization of the soil wall friction angle, a parametric study has been conducted. The results from the study are then compared with the proposed by Rankine and Coulomb theories. The following parameters for retaining wall and backfill were taken; height of wall (H)= 2m; unit weight of the backfill (γ)=18kn/m3 ; angle of internal friction of backfill (φ)= 400. The results are as shown in fig(9). It is observed that passive earth force is depends on coefficient of earth pressure and it has minimum value at δ= 100. For present analysis the passive force is very less as compared to the Coulomb s theory. XIII. COMPARISON WITH MODEL TEST RESULTS In order to check the applicability of proposed formulations, the prediction from the equation are compared with field test results of Fang(2002) and Fang (1997), who measured passive earth pressure behind retaining wall. 1) Model test results of Fang, Chen and Wu(1994) Fang, Chen and Wu(1994) conducted an experiment on retaining wall. The model wall is 1000mm wide, 550mm high and 120mm thick solid plate and is made of steel. The effective wall height H is only 500mm. The properties of soil are γ= 15.5 kn/m3; φ=30.90;δ=19.20.fig(10) shows the comparison between predicted values of lateral passive stress by present theory and measured values of lateral stress for different height of wall from surface. It is observed that predicted values are in reasonable agreement with the experimental values. At the top of wall it is same as height increases ; the predicted values 30% less than measured. 2) Comparison with Fang, Chen and Ho(2002) Fang, Chen and Ho (2002) conducted an experiment on retaining wall for critical state concept. The model was 1000mm wide, 550mm high, and 120mm thick solid plate, is made of steel. The effective wall height H is only 500mm. The backfill is taken as loose sand, medium sand and dense sand. The parameters of loose sand, medium and dense sand are; Table1 Parameters for loose, medium and dense backfill Backfill condition Unit weight (γ) kn/m3 Internal friction angle φ Loose Wall friction angle δ Medium Dense Fig (11), (12) and (13) shows the comparison between predicted values of lateral passive stress by present theory and measured values of lateral stress for different height of wall from surface. It is observed that predicted values are in reasonable agreement with the experimental values. At the top of wall it is same as height increases the lateral stress increases. The predicted values more close to experimental for medium and dense backfill however for loose backfill it nearly 30% more than predicted. The shape of lateral stress is non linear as given by Handy (1985). XIV.CONCLUSION This study shows the change of passive earth pressure distribution for rigid retaining walls considering the arching effect and also the shape of the critical failure surface. A new formulation was proposed for passive earth pressure on retaining wall considering arching effect. In this planar failure surface with parabolic arch is considered. The present analysis gives 25% less result as that of Coulomb s theory. Charts for modified passive earth pressure coefficient and height of application of lateral force have also been suggested. The present study gives 70% results same as that of experimental results. It was found from the above study that planar failure surface with parabolic arch shape predicts closest to the experimental values. The distribution of earth pressure behind retaining wall is Non-linear. This is due to arching effect. The distribution of passive earth pressure is depending more 6

7 on mode of translation rather than wall friction angle between backfill and soil. Fig(13) Comparison with experimental Fang 2002 for Dense sand Fig(10)Comparison Experimental Fang 1994 Fig(11) Comparison with Experimental Fan 2002 for loose sand. Fig(12) Comparison with Experimental Fang 2002 for Medium sand XV. REFERENCE [1] Dalvi,R.S.,Bhosale,S.S.,and Pise,P.J. (2005): Analysis for passive earth pressure-catenary arch in soil. Indian Geotechnical Journal, Vol.35, No 4, [2] Fang, Y. and Ishibhishi,I. (1986): Static earth pressure with various wall movements, Journal of Geotechnical Engineering. ASCE, 112, No3, [3] Etezer, Z. Komovnik, A. and Mazurik, A.( 1968): Model study on arching above buried structures. Journal of Geotechnical Engineering. ASCE, 1968, [4] Handy, R. L.(1985) : The arch in soil arching. Journal of Geotechnical Engineering. ASCE, Vol. III, No3, [5] Harrop-Williams, K.(1989): Geostatic wall pressures. Journal of Geotechnical Engineering. ASCE, 115(9), [6] Marston, A. and Anderson,A.O. (1913); The theory of loads on pipes in ditches and tests of cement and clay drain tile and sewer pipe, Iowa Engineering experiment station bulletin, Iowa State Callege, Ames, Ipwa, No31,1913, 181pages. [7] Janssen, h.a.(1895): versche uber Getreidedruck in silozellen, Z.Ver.dut. Ingr, Volume 39, 1895 pp1045(partial English translation in proceedings of the Institute of Civil Engineers, London,England 1896, pp 553 [8] Paik, K.H. and Slagado, R. (2003): Estimation of active earth pressure against rigid retaining wall 7

8 considering arching effects. 53,No7,pp Geotechnique, [9] Quinlan, J.F.(1987): Discussion of arch in soil arching. By R.L.Handy. Journal of Geotechnical Engineering. Div. ASCE 113(3), [10] Spanglar,M.G. and Handy,R.L.(1982) : Soil Engineering,4 th Ed.Harper and Row,New york. [11] Sherif,M.A.,Ishibhashi,I. and Lee,C.D.(1982): Earth Pressure Against Rigid Retaining Wall, Journal of the Geotechnical Engineering Division,ASCE,Vol.108,No GT5, [12] Terzaghi, K.(1947):. Theoretical Soil Mechanics. London: Chapman & Hall Ltd. IV edition, pp 66-76, [13] Tsagareli,Z.V.(1965): Experimental investigation of the pressure of a loose medium retaining wall with a vertical back face and horizontal backfill surface, Journal of soil mechanics and foundation engineering. ASCE, 91 (4), [14] Jagdish Naran,Swami Saran and P. Nandkumaran. (1969): Model study of passive pressure is sand. Journal of soil mechanics and foundation engineering. ASCE, 95 (4), [15] Wang,W.L.,andYen,B.C.(1974): Soil Arching in Slopes. Journal of the Geotechnical Engineering Division,ASCE,Vol.100,No GT1, [16] Yung-Show, Tsang-Jiang Chen and Bin-Fern Wu(Augest 1994): Passive Earth Pressure with Various Wall Moments Journal of the Geotechnical Engineering Division,ASCE,Vol.120. [17] Yung-Show, Tsang-Jiang Chen and Ying-Chien Ho(Augest 2002): Passive Earth Pressure with Critical State concept Journal of the Geotechnical Engineering Division,ASCE,Vol.128 [18] Ching Hung Ting; Sanjay Kumar Shukla; and Nagaratnam Sivakuga(January/ February 2011): Arching in Soils Applied to Inclined Mine Stopes International Journal Of Geomechanics [19] S. Singh1; N. Sivakugan, M.ASCE2; and S. K. Shukla3 (2010): Can Soil Arching Be Insensitive to?, International Journal Of Geomechanics [20] A. Shelke1 and N. R. Patra2 (2008): Effect of Arching on Uplift Capacity of Pile Groups in Sand, International journal of geomechanics. Spanglar,M.G. and Handy,R.L Soil Engineering,4th Ed.Harper and Row,New york 8

Active Earth Pressure on Retaining Wall Rotating About Top

INTERNATIONAL JOURNAL OF GEOLOGY Volume 9, 05 Active Earth Pressure on Retaining Wall Rotating About Top Ahad Ouria and Sajjad Sepehr Abstract Traditional methods for calculation of lateral earth pressure