Indian Geotechnical Conference (December 18-20, 2003)

Numerical Analysis of a Pile Subjected to Lateral Loads IGC 2009, Guntur, INDIA

NUMERICAL ANALYSIS OF A PILE SUBJECTED TO LATERAL LOADS

Mohammed Younus Ahmed Graduate Student, Earthquake Engineering Research Center, IIIT Hyderabad, Gachibowli, Hyderabad–32, India. E-mail: [email protected] Satyam D. Neelima Assistant Professor, Earthquake Engineering Research Center, IIIT Hyderabad, Gachibowli, Hyderabad–32, India. E-mail: [email protected]

ABSTRACT: In the case of foundations of bridges, transmission towers, offshore structures and for other type of huge structures, piles are also subjected to lateral loads. This lateral load resistance of pile foundations is critically important in the design of structures under dynamic loading. Load carrying capacity and load deformation behavior of a single pile and group of piles subjected to lateral load is obtained using nonlinear finite element method of analysis. According to Poulos & Davis (1980) the maximum deflection of the pile is the major criterion in its design and made the initial attempts to study the lateral behavior of piles included two-dimensional finite element models in the horizontal plane. Several investigations have attempted to study the behavior of pile under lateral load using 3D finite element analysis. In this paper a detailed study on the piles subjected to lateral loads are investigated.

1. INTRODUCTION elements of a three dimensional finite element mesh are Pile foundations are used to support the heavy structure and represented by 20 noded brick elements. The dimensions of can act in the dual role of carrying the applied load to deeper, the element size are user defined given by the user. A model strong layers, and also of reinforcing the soil. In the case of mesh is shown in Figure 1. According to (Karthigeyan et al. foundations of bridges, transmission towers, offshore structures 2007), the soil mass dimension depends on the pile diameter and other type of huge structures, piles are also subjected to and length. The width of soil mass is taken as 40B, in which, lateral loads. This lateral load resistance of pile foundations B is the pile diameter or pile width. The soil mass effect on the is critically important in the design of structures under loading pile response is diminishing for the width more than 40B. The from earthquakes, soil movement, waves etc. According to height of soil mass is L+20B, in which, L is the length of pile. Poulos & Davis (1980) the maximum deflection of the pile is the major criterion on the design. Three-dimensional linear and nonlinear finite element analyses are gradually becoming commonplace for geotechnical appli-

cations. With the Available computational power continues to m c

allow for such more realistic representation of the actual involved 0 0 6

geometric scenarios. In this paper, nonlinear three dimensional 1

finite element analysis are carried out to investigate the x a

behavior of single pile subjected to laterally applied load. g n o l a

2. NUMERICAL MODEL t g n e

Finite element analyses were performed using the nonlinear L three dimensional analysis code. According to finite element method a continuum was divided into a number of (volume) elements. Each element consists of number of nodes. Each node has a number of degrees of freedom that correspond to discrete values of unknowns in the boundary value problem to be solved. Width along Y axis 600 cm Width along X axis 600 cm 2.1 Mesh Generation Fig. 1: Finite Element Model of Soil (6 m × 6 m × 16 m) In order to perform the finite element calculations, the with a Square Pile geometry have be divided into elements. A composition of finite elements is called a finite element mesh. The basic soil

157 Numerical Analysis of a Pile Subjected to Lateral Loads 2.2 Linear Elastic Model of Pile The usual definition of the equation of Mohr-Coulomb surface is given below (Smith & Griffith, 1982). This model represents Hooke's law of isotropic linear elasticity used for modeling the stress-strain relationship of  '  '  '  ' F  1 3 sin   1 3  c'cos (1) the pile material. The model involves two elastic stiffness 2 2 parameters, namely Young's modulus, E, and Poisson's ratio, υ as shown in Figure 2. 3. INTERFACE ELEMENTS

perfectly plastic Interfaces are modeled with 16-node interface elements. σ Interface elements consist of eight pairs of nodes, compatible with the 8-noded quadrilateral side of a soil element. Along perfectly elastic degenerated soil elements, interface elements are composed E = young’s modulus of six node pairs, compatible with the triangular side of the degenerated soil element. Each interface has a 'virtual 1 thickness' assigned to it which is an imaginary dimension Є used to obtain the stiffness properties of the interface. The virtual thickness is defined as the virtual thickness factor Fig. 2: Stress Strain Curve (Johnson 2006) times the average element size. The average element size is determined by the global coarseness setting for the 2D mesh 2.3 Mohr-Coulomb Soil Model generation. The default value of the virtual thickness factor that is used in this study is 0.1. The stiffness matrix for This elastic - plastic model is based on soil parameters that quadrilateral interface elements is obtained by means of are known in most practical situations. The Mohr-Coulomb Gaussian integration using 3 × 3 integration points. The model is used to compute realistic bearing capacities and position of these integration points (or stress points) is collapse loads of footings, as well as other applications in chosen such that the numerical integration is exact for linear which the failure behavior of the soil plays a dominant role. stress distributions. The 8-node quadrilateral elements provide a The model involves two main parameters, namely the second-order interpolation of displacements. Quadrilateral cohesion intercept, c and the friction angle, υ. In addition to elements have two local coordinates (ξ and η) (Fig. 4). three parameters namely Young's modulus, E, Poisson's ratio, ν, and the militancy angle, ψ need to calculate the ξ= –1.0 ξ= 0 ξ= –1.0 complete σ–ε behavior. Mohr Coulomb’s failure surface 4 7 3 criteria shown in Figure 3. According to Johnson et al. η=1.0 (2006), the failure envelope only depend on the principal stresses (σ1, σ3), and is independent of the intermediate principle stress (σ2). When mapped in three-dimensional stress space, Mohr–Coulomb criteria resolved into an irregular hexagonal pyramid. This pyramid forms the 4 3 failure/yield envelope, which is turn governs how soil will behave. The material behaves elastically if the stress point 8 6 η=0.0 lies within the failure envelope. However, if the stress 2 reaches the yield surface the material will undergo a degree 1 of the plastic deformation. In the Mohr–Coulomb model used herein, it is assumed that until the soil has linear elastic relationship until failure.

(τ) η = –1.0 Mohr coulomb failure envelope Shear 5 2 stress 1 Φ friction Fig. 4: Local Numbering and Positioning of Nodes (•) and Integration Points (×) of an 8-node Quadrilateral Element

Cohesive 3 4. FINITE ELEMENT ANALYSIS intercept σ σ 3 1 Normal stress (σ) The pile and soil are discretized into a number of 20 nodded isoparametric continuum elements. The interface between the Fig. 3: Mohr-Coulomb’s Failure Surface soil and pile is modeled using 16 noded isoparametric (Johnson et al., 2006) interface elements, having aero dimension.

158 Numerical Analysis of a Pile Subjected to Lateral Loads 4.1 Continuum Element Start Relationship between strains and nodal displacements expressed as Read input number of element nodes, {Є}е= [B]{δ}е (2) load, boundary conditions

Where {Є}е is the strain vector, {δ}е is a vector consisting of nodal displacements, and [b] represents the strain– Form and reduce the elastic stiffness matrix displacement transformation matrix. The stress strain relation is given by, For all load (displacement) increments

{σ}е = {D}{{Є}е (3) Read in applied load increment iteration loop starts here Where {σ}е is the stress vector, and [d] is the constitutive relation matrix. Add body-loads to applied loads The stiffness matrix [k] , of an element is given as, Solve equilibrium equations to give displacement increment е Set ICON=1 if displacements hardly changed from iteration T [K] е = [B] [D][B] dv (4)  For all elements 4.2 Interface Element For all gauss points The relative displacement (strains) between the surfaces of soil and structure induce stresses in the interface element and Compute elastic strain increments these relative displacements are given by, Compute elastic stress increment and add to stresses left from end of last load increment {Є}е= [B]f{δ}е (5)

Where [B]f represents the strain-displacement transformation No matrix. Failure criterion The element stiffness matrix is obtained by usual expression, exceeded?

T [k]e = [B]F [D][B] dv (6) Accumulate visco-platic stains;  Form integrals to obtain nodal forces Where [D], is the constitutive relation matrix for the interface. End all gauss points

4.3 Boundary Conditions Accumulate element nodal forces in body-loads vectors In this simulation the boundary conditions were No (i) Base of the ground domain was fixed in longitudinal (x), Convergence? transverse(y), and vertical (z) directions Is (ICON=1) (ii) Left, right, back and front were fixed in x, y directions Yes (the lateral directions) and free in the z direction. Update element stresses ready for next load step 4.4 Computer Simulation Details End all the elements Element size for all simulation was 50 cm × 50 cm × 100 cm. Details of simulation are given below. The program was End of the load increments implemented on Intel Core2Duo 3.0 GHz processor. Mesh size was taken to be 6m × 6m × 16 m. Number of 20 nodded Print displacement, stresses etc. elements in the mesh was 1728 and number of nodes associated with the mesh was 8281. Flowchart for FEM code Stop is given in Figure 5. Properties of the four different soils used in the study are given in Table 1. Fig. 5: Flowchart for Finite Element Program (Smith & Griffith, 1982) Table 1: Properties of Four Types of Solid Considered in the Study 9 2 Parameter Sand I Sand II Clay I Clay II The Modulus of elasticity of pile is 2 × 10 kN/m and Young’s Modulus (kN/m2) 14000 13000 8500 10000 Poisson ratio 0.15. Three cases were taken for Finite Element Poisson’s Ratio 0.3 0.3 0.3 0.35 Analysis, each having two layered soil and details of it are Cohesion (c ) (kN/m2) 1 1 10 5 given below in Table 2. Friction Angle φ0 45 31 21 25

159 Numerical Analysis of a Pile Subjected to Lateral Loads Table 2: Three Cases of Soil Strata Considered in the to 1m and varying the length of pile to 8m, 12 and 16m. FEM Analysis Lateral deflections were obtained along the longitudinal Cases Layer 1(7m) Layer 2 (9m) direction for the three cases considered are shown in Figure 1. Sand II Sand II 6. A lateral load of 40 k N was applied at the ground surface. 2. Clay I Clay II The results from the finite element method code were 3. Clay II Sand I compared and checked against Opensees PL (2006) software. From this research work it is observed that the slender ratio 5. RESULTS AND DISCUSSIONS of the pile increase there is a corresponding decrease in the deflection of the pile which is evident from the lateral Three slenderness ratios were taken to analyze i.e., L/d = 8, deflection of the pile. The deflection from the finite element 12 and 16. By keeping the cross-section of the pile constant code is well comparable with the Opensees PL software. 6. CONCLUSIONS The upper part of pile is the most critical part of pile in case of laterally loaded pile (Poulos & Davis, 1980) because of its greater deflection and its ability to carry higher lateral loads m c than the lower parts. It is important to study the lateral soil n i

h pressure along pile depth in order to understand which part of t p

e the pile carry large soil pressure that may cause pile d foundation collapse. From this research work on Numerical simulation of a single pile subjected to lateral loads, it is concluded that the lateral soil pressure under lateral loads changed with pile depth deflection in mm depending on the load intensity. It is also concluded that, (a) Case 1 water table depth influences the response of the lateral piles. That is dry soil condition gives higher lateral resistance than fully saturated soils. Here, the results are well compared with Opensees PL software and concluded that the slenderness ratio of the pile increases with the decrease in the deflection of the pile. m c

h t REFERENCES p e d Johnson, K., Lemcke, P., Karunasena, W. and Sivakugan, N. (2006). “Modelling the Load–Deformation Response of Deep Foundation Under Oblique Load”, Environment Modelling and Software, No. 21, 1375–1380. Karthigeyan, S., Ramakrishna, V.V.G.S.T. and Rajagopal K. deflection in mm (2006). “Influence of Vertical Load on the Lateral (b) Case 2 Response of Piles in Sand”, Computer and Geotechnics, Vol. 33, 121–131. Karthigeyan, S., Ramakrishna, V.V.G.S.T. and Rajagopal K. (2007). “Numerical Investigation of the Effect of Vertical Load on the Lateral Response of Piles”, Journal of m c Geotechnical and Geo environmental Engineering, Vol. n i

h 133, No. 5, 512–521. t p

e Mazzoni, S., McKenna, F. and Fenves, G.L. (2006). “Open d System for Earthquake Engineering Simulation User Manual,” Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA (http://opensees.berkele.edu/). deflection in mm Poulos, H.G. and Davis, E.H. (1980). “Pile Foundation Analysis (c) Case 3 and Design”, John Wiley & Sons, Inc, United States. Fig. 6: Lateral Deflections of the Pile with L/d =8,12,16 and Smith, I.M. and Griffith, D.V. (1982). “Programming the Compared with Opensees Pl (a) Case 1; (b) Case 2; Finite Element Method”, Second Edition, John Wiley & (c) Case 3 Sons, Chisester, U.K.