A SIMPLIFIED APPROACH TO ESTIMATING THE SOIL STRESS DISTRIBUTION DUE TO A SINGLE PILE Keywords analytical solution; soil stress distribution; skin friction; end bearing Abstract This paper reports a simplified analytical solution for estimating the soil stress distribution due to pile dependence on the pile dimensions. The exponentially increasing ultimate skin friction along the pile shaft is derived by means of an equilibrium analysis of the soil element around the pile. The soil stress distribution due to the exponentially increasing skin friction and uniformly distributed end bearing of the piles is proposed. The estimated soil stresses are compared using the original Geddes solution, which validates the derivation and formulae obtained. 1 INTRODUCTION The Boussinesq solution [1] is normally used to estimate the soil stress distribution due to a vertical point load applied to the ground surface. It proposes that the ground is a semi-infinite, elastic, isotropic and homogeneous medium that obeys Hookes law. Through integration, the stress distribution induced by any type of external vertical load can be obtained. Thus, the ground surface settlement can be calculated using a layer-wise summation. However, there are many cases in which the vertical loads are applied not to the ground surface, but to the interior of a semi-infinite medium, i.e., the stress condition during excavation and in deep foundations. Problems based on these conditions cannot be solved using the Boussinesq method. Mindlin [2] carried out some theoretical analyses based on Boussinesqs assumption to estimate the stress induced by an internal point load applied to positions in an isotropic, homogeneous, elastic half-space. Because the stress distribution due to an internal point load can be obtained using this approach, the stress due to any type of internal loading (i.e., increments varying linearly, uniform with circular and rectangular distributions), can also be calculated by integration. Based on an ideal elastic soil mass assumption, the Mindlin solution has been widely used to analyze the behavior of pile foundations, such as Geddes [3], Seed and Reese [4], D' Appolonia and Romualdi [5], Coyle and Reese [6], Poulos and Davis [7], Ramiah and Chickanagappa [8], Clancy and Randolph [9], Horikoshi and Randolph [10], and Kitiyodom and Matsumoto [11]. Based on the Mindlin solution, some load-transfer mechanisms of inclusion were reported by Luk&Keer [12] and Selvadurai & Rajapakse [13]. Suriyamongkal et al. [14] investigated the stress distribution in soil induced by piles under a vertical applied load in order to develop the radial ring forces and vertical forces acting on the interior of a half space. The soil stress for pile Ping Li Hohai University, 210098, Nanjing, PR China Tao Yu (corresponding author) Hohai University, 210098, Nanjing, PR China E-mail: yutaohhu@163.com Dongdong Zhang (corresponding author) PLA University of Science and Technology, Nanjing 210007, PR China E-mail: zhangdodo_163@163.com Acta Geotechnica Slovenica, 2017/2 19. P. Li et al.: A simplified approach to estimating the soil stress distribution due to a single pile foundations has been analyzed by Geddes [3] based on the Mindlin solution, in which the end bearing (EB) of the pile foundation is assumed to be a point load on the center of the cross-section of the pile toe, and the skin friction (SF) is modeled as a line load distributed along the vertical axis of the pile shaft. Due to the complexity of the integral equation and the limited computer availability, the solution was used in practice up until approximately 1966. Geddes [15] reported tabulated values (m and n) of the stress coefficients that were defined as the relevant stress value simply multiplied by H2/P to determine one of the stress components at a point, where H is the pile length and P is the applied load, and they were widely used in practice. However, Wang & Pan reported [15,16] that the Geddes solution does not account for the cross-sectional geometry and dimensions because the distance between the calculated position and the point load does not vary with these factors. This simplification causes some errors, particularly for non-circular piles by Lv et al [17] and Zhang et al. [18] and for circular piles with large cross-sectional diameters. In addition, the skin friction is ideally simplified as a uniform and/or linear variation along the pile shaft, as reported by Wang et al. [19]. This paper modifies the assumptions of the Geddes solution so that the skin friction is distributed on the external surface of the pile shaft and the end bearing is uniformly distributed on the pile toe. The skin friction is derived via equivalent analyses. Analytical solutions are derived by considering the effects of the pile dimensions, and are calculated using MATLAB software. 2 BASIC ASSUMPTIONS There are four basic assumptions: (1) the construction effects on piles are not considered; (2) the ground is assumed to be a semi-infinite, isotropic and homogeneous medium; (3) the skin friction is distributed on the external surface of the pile shafts and the end bearing is uniformly distributed on the pile toes; and (4) the ultimate skin friction is exponentially increased with the pile depth. The technique of this analytical solution is illustrated by Figure 1. The soil stress at the point (rj, aj) is governed by the distance between the position of (rj,(Xj) to the position of the skin friction and the end bearing. In the Geddes solution, this distance is r, which is constant for piles with different diameters. In this analytical solution, this distance is r, which varies with (r;,a;), i.e., the pile dimension. The stress distributions due to the skin friction and the end bearing can be analyzed using the normalized stress coefficient by multiplying the relevant stress and H2/P. r Figure 1. Graphical representation of the technique for the exponentially increased skin friction and the uniformly distributed end bearings. 3 DERIVATION PROCEDURES Under ideal conditions, the effective vertical stress of the semi-infinite ground (crj) at depth z is o'v = yz, where y is the effective unit weight of the soil. However, due to the existence of piles, the ground is influenced by vertical shearing effects at the pile-soil interfaces. The free bodies of a soil element adjacent to the pile shafts are equivalently analyzed, as shown in Figure 2. According to the theory of the vertical shearing mechanism, the skin friction on the pile surface, rs, is not entirely equal to the shear stress on the free body of Figure 2. Equilibrium analysis. 72. Acta Geotechnica Slovenica, 2017/2 P. Li et al.: A simplified approach to estimating the soil stress distribution due to a single pile the soil element. For a circular pile with a diameter of rp, the influencing zone of the skin friction is arp. The ratio of the shear stress, Ars, to the skin friction is A. The equilibrium equation of the free body was derived from (j'v , r5, and Ats as follows: Aa[ _ 2i]K(j[ tant» aX-1 " = (2) (1) where k denotes the coefficient of earth pressure at rest. In this analysis, the interactions between the adjacent free bodies are taken as the internal force of a soil element. The top and the bottom areas of the free body are Aop = Aottom = (a2 ~ l)'p2d0 / 2 The inner surface area of the free body is (3) A^=rpd0dz (4) The outer surface area of the free body is Am« =arpd0dz (5) It is assumed that the local shaft friction at failure, ts , is governed by the Coulomb equation according to Lehane et al. [20]: ts = c + Kctv tan 5 (6) where 8 is the friction angle of the pile-soil interface. The effective cohesion (c) is not considered in general. The integration of Eq. 1 and the incorporation of the boundary condition yields f 2 t]K tan S_ ^ C>, =- V 2r]K tan S -1 (7) The parameters K and S remain constant along the pile length at the ultimate stress state. Therefore, the ultimate skin friction on the pile surfaces, rs, is obtained by t = Koita&S = 2r¡ f Ir/KtaaS _ \ e ^ -1 V (8) / After the integration of Eq. (8) on the pile surface, the total force due to an external skin friction is obtained by p = ESF nr2/' 2rjK tant? _ -1 2 tjK tan § (9) The skin friction on the pile surfaces over a normalized depth ôh is SPESF = J rSds5h. The normalized soil stress H2/PEsf at any position due to the exponentially increased skin friction is given by IESF H1 j" j xscj_dsdh (10) where IEgp is the normalized soil stress due to the exponentially increased skin friction; 1 denotes the pile perimeter; PESF is the total force due to the external skin friction, which is determined by Eq. (9); az is the stress at any given position M(x, y, z) due to an internal point load P(0, 0, h), which is obtained by Mindlin [2] (1-2n){z-h) (\-2n)(z-h) | 3(z-hf , 30fe(z + /z)3 8^(1 -ju) 3z(3 - 4/¿)(z + h f - 3h{z + h)(5z - h) (ii) r: where n is the Poisson ratio, Rx = i/x2 + y2 +(z-h)2 and R2 =yjx2 +y2 +{z + hf The end bearing over the normalized cross-sectional area