Effect of Arching on Passive Earth Pressure Coefficient.

Transcription

1 The 12 th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India Effect of Arching on Passive Earth Pressure Coefficient. Rupa Sunil Dalvi Department of Civil Engineering, Geotechnical Engineering Division, PIET s College of Engineering, Pune , Maharashtra, India. Pise P.J. Department of Civil Engineering, Geotechnical Engineering Division, PIET s College of Engineering, Pune , Maharashtra, India. (Former Professor, IIT Kharagpur ) Keywords: Arching, Passive earth pressure, Sandy soil, Wall friction, Retaining Wall ABSTRACT : Analysis has been proposed by Paik & Salgado (2003) for the non linear distribution of active earth pressure on a translating retaining wall considering arching effect. In the present paper, formulation is proposed for calculating passive earth pressure on a rigid retaining wall undergoing horizontal translation based on his approach (Paik & Salgado, 2003). It takes into account arching effect in the backfill. The proposed formulation is compared with Coulomb s results. The comparison between proposed and Coulomb s values shows that the proposed equation predict values of earth pressure much less than those of Coulomb s values. In order to facilitate calculation of passive earth pressures, using the proposed equation, a modified passive pressure coefficient is provided. It is a function of soil friction angle φ and soil - wall friction angle, δ 1 Introduction Arching is universal phenomena that involves transfer of pressure from the yielding part of soil to the adjoining part. The soil is said to be arch over the yielding part of support. The state of stress within the zone of arching depends upon the amount of yield. As yielding increases, arching effect is gradually reduced. However,the effect of arching is permanent in character as the shear strength property of soil.arching effect is less if shear strength is less. Though arching phenomenon occurs in number of geotechnical engineering problems, it has not received much attention. Several researchers have attempted to estimate the active earth pressures exerted against rigid retaining walls considering arching effect in the retained soil mass. Janssen (1895) set up the differential equation for pressures in the silos by considering force equilibrium for any differential flat element in the silo. Based on Janssen s arching theory Spranglar & Handy (1984) and Wang (2000)suggested equations to estimate the non linear distribution of active pressure on retaining walls. Handy (1985) and Harrop- Williams (1989) also proposed lateral active earth pressure coefficients K aw and equation for calculating non-linear active earth pressures. Dalvi et.al.(2005,2007) have used approach similar to that of Handy and derived coefficient of passive earth pressure (K pw ). Estimation of passive earth pressure acting on a rigid retaining wall is very important in the design of many geotechnical engineering structures, particularly retaining walls. Civil engineers have traditionally calculated the passive earth pressure against rigid wall using either Coulomb s or Rankine s formulation. Both assume that the distribution of the passive earth pressure exerted against the wall is triangular. However, many experimental results Naran et.al.(1969) and Fang & Ishibashi (1986) shows that the distribution of passive earth pressure on the face of the rough wall depends on the wall movement (rotation about top, rotation about bottom, and horizontal translation)and is non-linear. This is different from the assumption made by both Coulomb and Rankine. 1.1 Scope of Study In this study the effect of arching on passive earth pressure in the non-cohesive backfill is considered. The backfill is assumed to rise upward in a circular arc form due to arching. The vertical and lateral stress acting at arbitrary point along a differential flat element in backfill is calculated. These stresses are the function of internal friction angle φ and soil to wall interface friction angle δ. The ratio of lateral to vertical stress is denoted by K pwn. A modified passive earth 236

2 pressure coefficient K pwn considering arching effect has been derived. 2 Proposed Method of Analysis The approach is similar to that of Paik and Salgado(2003) for active earth pressure and Dalvi.et.al. (2005,2007) for passive earth pressure. Following assumptions have been made in the analysis. 1. The soil is cohesionless, semi infinite, homogenous, isotropic and backfill is horizontal. 2. The problem is a plane strain problem i.e. two- dimensional. 3. The soil mass is bounded between two parallel, un-yielding rough vertical walls. The walls are assumed to translate towards the soil mass creating passive case. 4. The sliding surfaces are vertical and pass through the outer edge of yielding wall. 5. The soil mass moves up in curved path taken as arc of circle. 6. Full shear strength s is mobilized on these vertical surface and it is expressed by Coulomb s empirical law s = c + σ tanφ. 7. The major and minor principal stresses have been considered to be constant along the length of the arc. 8. The ratio of horizontal to vertical pressure σ h to σ v is considered constant K = σ h /σ v Let us assume that two parallel, rigid vertical walls retain granular soil as shown in Fig.1. When the translation of wall takes place towards the soil mass, passive state is created and the soil moves in upward direction. The frictional resistance at walls causes changes in the direction of principal stresses acting on the differential element. The minor principal stresses on the differential flat element are applied normal to the upward arc, which is denoted by dotted lines. While the direction of major principal stresses is tangential to the direction of upward arc becoming horizontal at the centre of the element. Rigid retaining wall with rough face moves towards the soil horizontally. The direction of element in Fig.2 changed due to frictional resistance of a wall. The major principal stresses σ 1 on the differential flat element behind the wall act along the convex arc as shown in Fig.2 Where minor principal stresses, σ 3 are perpendicular to convex arc. The shape of convex arc is assumed in the form of arc of a circle. Considering that the slip lines of soil make angle of + (45 -φ /2) with the horizontal and the angle between the slip plane and major principal stress must be (45 -φ /2). Figure 1. Trajectory of major principal stresses. The width of the differential flat element at a depth z can be expressed as follows B z = Rcosθ (1) where R is radius of major principal stress trajectory and θ is the angle of major principal plane with respect to the 237

3 horizontal at the wall. The passive lateral stress on the wall σ phw can be calculated by considering the horizontal force equilibrium in the triangular element at the left edge of the convex arc as shown in Fig. 2 The lateral stress on the wall σ phw = σ 1 sin 2 θ + σ 3 cos 2 θ (2) Similarly the lateral stress at point D of the convex arch which was originally located at point B is σ phw = σ 1 sin 2 Ψ+ σ 3 cos 2 Ψ (3) where Ψ is the angle between tangent to the arch at D and vertical. Dividing equation 3 by σ 3 and put σ 1/ σ 3 = 1/N for the soil in the passive condition. Where,N is the ratio of minor to major principal stresses, σ 3 / σ 1 = tan 2 (45-φ/2) σ phw / σ 3 = (1/ N )sin 2 Ψ+ cos 2 Ψ (4) As σ 1 σ Phw = σ v σ 3 Substitution for σ phw gives σ v / σ 3 = sin 2 Ψ+ 1/N cos 2 Ψ (5) Therefore the vertical and lateral stresses at arbitrary points along a differential flat element can be obtained by substitution of Ψ= θ in equation 4 and Determination of θ When wall friction angle δ θ. The rotational angle θ of the principal stress for the wall can be obtained using Mohr s circle as shown in Figure 3. from the two triangles OAB and ABC, we can write τ w = σ phw tanδ = (σ 1 σ phw ) tan θ w (6) σ phw / σ 1 tan θ w = tanδ (7) 1- σ phw / σ 1 Dividing equation (2) by σ 1 substitute equation (8) in equation (7) gives second order equation σ phw / σ 1 = sin 2 θ + N cos 2 θ (8) 238

4 Figure 2. Stresses on differential flat element In backfill tan 2 θ w +N tan θ w = tanδ (1- N) Solving this equation for θ the expression for θ is, (1-N)+ [ (1-N) 2-4N tanδ 2 ] θ = tan (9) 2 tan δ From the two values of θ given by equation (9), the maximum value corresponds to the condition on a retaining wall. 239

5 Figure 3(a) Mohr circle for stress at wall Passive Lateral Stress Ratio Figure 3 (b) Detail at C Dalvi et.al( 2007) has determined the lateral stress ratio K pw at the wall using the average vertical stress across a given differential flat element. They(2007) have given the following equation for K pw, for the values of φ in the range of 100 to400 and for δ = φ condition. 240

6 σ phw K pw = = 0.862(cos 2 θ + 1/Nsin 2 θ) (10) σ v In above equation σ phw is passive lateral stress ratio at the wall and σ v is average vertical stress across the soil element. In Equation (10) if we put φ = 0 which gives K pw = Although, the passive stress ratio for φ = 0 must be equal to 1. Therefore there is an error in values of K pw calculated using equation (10). This error decreases with increasing φ and δ. We shall now derive a new relationship for K pw that reflects the variation of σ v with φ and δ. The differential vertical force dv on the shaded element at pint B in Fig.2 can be expressed as dv =daσ v = σ 3 ( 1/N cos 2 Ψ + sin 2 Ψ) (Rd Ψ sinψ)) (11) In which da is the width of the shaded element at point B. The average vertical stress σv across the differential flat element shown in Fig. 2 can be obtained by dividing the vertical force V acting on the differential element by width of the element, B z = Rcos θ σ v V 1 π/2 = --- = ---- dv B z B z θ π/2 σ v = σ 3 (sin 2 Ψ + 1/N cos 2 Ψ) (Rd Ψ sinψ) θ Integration of this equation yields N -1 σ v = σ cos 2 θ (12) 3N Dividing equation (2) by Eq.(12) we get a new ratio K pwn of the passive lateral stress at the wall to the average vertical stress over the differential flat element. σ phw σ 1 sin 2 θ + σ 3 cos 2 θ K pwn = = (13) σ v σ 3 ( 1- [(N-1)/3N ] cos 2 θ 3(sin 2 θ +Ncos 2 θ) K pwn = (14) 3N - (N-1) cos 2 θ The above equation is for modified passive pressure coefficient. This is equal to 1 for φ = 0 and matches the values of Rankine s passive stress ratio for δ = Illustrative Example To illustrate effect of arching in passive case and Coulomb s pressure an example for the soil wall data given below has been solved. Wall height 2 m, φ =32and δ= 0.2φ and δ= 0.6φand γ = 18 KN/m 3 The results have been compared in Fig.4 for pressure distribution considering arching effect as well as Coulomb s results. In both cases the earth pressure on the wall increases linearly with height. At any depth Coulomb s pressure is more than that predicted by considering arching effects. The difference 241

7 depends on δ -values. It increases with increase in δ -values For δ=0.2φ, the pressure predicted by present method is 0.77 times Coulomb s lateral pressure at base. However, for δ=0.6φ, it is 0.34 times Coulomb s lateral pressure at base. For δ=0.2φ the pressure predicted by Dalvi et al(2007) is times Coulomb s lateral pressure at base. However, for δ=0.6φ, it is 0.35 times Coulomb s pressure at base 0 Lateral earth pressure(kn/m2) Depth (m) Coulomb s pressure for delta=0.6phi Coulomb s pressure for delta=0.2phi Present method for delta=0.2phi Dalvi et.al.(2007) for delta=0.2phi Present method for delta=0.6phi Dalvi et.al.(2007) for delta=0.65phi Figure 4. Lateral earth pressure distribution with depth and comparison with coulomb theory. 3 Conclusion The estimation of passive earth pressure acting on retaining wall is very important in geotechnical design. Soil arching in passive state is shown by trajectory of major principal stress considering arc of a circle. Due to rotation of principal stresses at the rough wall,the lateral and vertical stresses are modified. The ratio of lateral to average vertical stress, K pwn, a modified passive lateral stress ratio has been derived. Lateral earth pressure on the retaining wall due to arching is less than the pressure predicted by Coulomb s analysis Limitations For simplicity the failure plane is assumed to be the planer. In practice the failure plane may be a curved one. The failure surface may not be a plane and the deformations assumed in arc of a circle during arching may not be possible in actual situation. These are the limitations of the present study. 242

8 4 References Dalvi,R.S.,Bhosale,S.S.,and Pise,P.J Analysis for passive earth pressure-catenary arch in soil. Indian Geotechnical Journal, 35, No 4, Dalvi,R.S.,Bhosale,S.S.,and Pise,P.J Analysis of arching in soil. Communicated to International Journal of Geo-mechanics, ASCE, under re-review Fang, Y. and Ishibhishi,I. (1986). Static earth pressure with various wall movements, Journal of Geotechnical Engineering. ASCE, 112, No3, Handy, R. L The arch in soil arching. Journal of Geotechnical Engineering. ASCE, III, No3, Harrop-Williams, K Geostatic wall pressures. Journal of Geotechnical Engineering. ASCE, 115(9), Jagdish Narian,Swami Saran and P. Nandkumaran. (1969). Model study of passive pressure in sand. Journal of soil mechanics and foundation engineering. ASCE, 95 (4), Janssen, h.a versche uber Getreidedruck in silozellen, Z.Ver.dut. Ingr, 39, 1895 pp1045(partial English translation in proceedings of the Institute of Civil Engineers, London,England 1896, pp 553 Paik, K.H. and Salgado, R Estimation of active earth pressure against rigid retaining wall considering arching effects. Geotechnique, 53,No7,pp Quinlan, J.F Discussion of arch in soil arching. By R.L.Handy. Journal of Geotechnical Engineering. Div. ASCE 113(3), Spanglar,M.G. and Handy,R.L Soil Engineering,4th Ed.Harper and Row,New york. Wang, Y.Z. (2000). Distribution of earth pressure on a retaining wall. Geotechnique, 50, No.1, pp