Nonlinear p-y analysis is the most widely used method for design of laterally loaded piles due to its simplicity; the successful application of a p-y method depends upon the availability of detailed information on a spatial distribution of soil properties which are key factors in the design of laterally loaded deep foundation. The subgrade reaction method models soils as Winkler springs and piles as beams; hence, pile geometry can be considered only indirectly. The finite element method provides a more precise tool that is capable of modelling soil continuity, soil nonlinearity, pile-soil interface behaviour, and 3-D boundary conditions. It is more rigorous in its analytical methodology than any other existing methods. Randolph [1] used linear strain triangles in the semi- analytical finite element formulation to avoid special integration techniques. From the results of parametric study, a simplified expression was developed for the response of single and group piles embedded in elastic soil. The nonlinear behaviour of pile group was studied using 3-D FEA with nonlinear elastic soil model [2] [3] . In order to include the interaction effects involving relative slip and separation, a thin layer of interface element was used. Najjar and Zaman [4] [5] studied the effects of loading sequence and soil nonlinearity on the deformation behaviour of a pile group using a nonlinear 3-D finite element technique. Ladhane and Sawant [6] presented dynamic analysis of pile group and examined the effect of different pile configurations on dynamic response.

The review of literature has revealed that behaviour of laterally loaded piles is significantly influenced by the material non-linearity. Behaviour of pile group is further influenced by the arrangements of piles in the group. A 3-D finite element program has been developed for the analysis of a pile groups in clay considering the above aspects. Soil and pile media have been discretised into 3-D isoparametric continuum elements. The pile elements have been assumed to remain in elastic state at all the time. On the other hand, the soil elements are assumed to undergo plastic yielding according to the von Mises yield criterion. This model has been selected because it is suitable for analyzing the behaviour of purely cohesive soils under undrained condition. To simulate the stress transfer between soil and pile under lateral load, interface elements have been introduced at the soil-pile interface. Normal and tangential stiffness of these elements are assumed in such a way that shearing at the soil pile interface is allowed but gapping will be restricted. The formulation and implementation of additional features required for incorporating nonlinear behaviour are presented in this paper.

Mainly, there are two types of materials involved in present study i.e. reinforced concrete and soil. The pile material is treated as linear elastic, whereas, the soil is replaced by an idealised material which behaves elastically up to some state of stress at which yielding occurs and beyond yielding, the theory of plasticity is implemented to account for the constitutive law of the soil. The constitutive relationships arising from the theory of plasticity are incremental in nature due to the stress path dependence of the material behaviour. The essential features of the plasticity theory are [7] :

・ A yield function, separates the elastic and plastic states of stresses in the body under consideration.

・ A plastic potential function defining the direction of plastic straining when yielding occurs (flow rule).

・ A hardening/softening law describing the dependence of the yield function on the plastic strains.

In the present study, von-Mises, Drucker-Prager [8] and modified Mohr-Coulomb [9] models have been considered. Viladkar et al. [7] have converted these models into convenient forms to make the computational aspect simple and presented a generalized approach to elasto-plastic finite element analysis. The associative flow rule is assumed, in which the yield function and plastic potential function are expressed by same function. The Yield criterions used to identify the onset of the yielding in the soil are explained here. Most of them are defined in terms of stress invariants. Here J₁, J₂, J₃ are first, second and third stress invariant and is second stress invariant of the deviatoric stress components.

Nayak and Zienkiewicz [11] have converted the yield criteria into convenient forms to make the computational aspect simple and presented a generalized approach to elasto-plastic finite element analysis. Further, Viladkar et al. [7] transform several yield criteria with isotropic hardening and associated flow characteristics into convenient forms and make them available for easy implementation into the finite element codes.

This formulation is due to Nayak and Zienkiewicz [11] and Viladkar et al. [7] and is achieved by expressing the yield function as,

where, q is an alternative to the third stress invariant and determined by,

Here J₁, J₂, J₃ are first, second and third stress invariant and is second stress invariant of the deviatoric stress components.

The flow vector, a, is rewritten in terms of J₁, and q in the following way using the chain rule:

Substituting from Equation (9), the flow vector can be expressed as:

where,

Calculation of C₁, C₂ and C₃:

Differentiating Equation (9) with respect to s,

Substituting in Equation (18),

Simplifying radical in third term from above equation,

Substituting in Equation (12),

From Equations (11) and (14)

Calculation of a₁, a₂ and a₃:

Following are the stress invariants required to evaluate the constants a₁, a₂ and a₃.

The stress invariants are given as,

The second deviatoric stress invariants is given as,

From Equations (11) and (16) and substituting,

The constants C₁, C₂ and C₃ can be evaluated from the expression of yield function using Equation (15).

The total load is applied in the ten equal increments and the stiffness matrix, [K] is kept constant for all the load increments in Figure 2. Following is the algorithm used for nonlinear analysis:

1) Apply incremental load and solve for incremental displacement,

2) Update the total displacements u_i for i^th increment by adding incremental displacement in

3) Set m_e = 0, where m_e indicates number of yielded elements.

4) For each element, set m_p = 0 (number of yielded points within the given element), obtain the element incremental displacement vector from global incremental displacement vector,.

5) For each Gauss point, compute the incremental stress and total stress for i^th increment from following relations,

Check for yield, if the point have not yielded then go to next Gauss point, else, set m_p = m_p + 1, and calculate extra stress over yield stress, {Δσ}_ext

Set total stress at yield level

Update the additional force vector to be applied in next iteration

Complete loop over the Gauss points of the element

6) If m_p > 0, then assemble load vector and set m_e = m_e + 1.

7) Repeat Step 4) to 6) for all the elements.

8) If m_e = 0, then not a single element have yielded, go to next load level.

9) If m_e > 0, then check for convergence using following displacement criteria,

where, e_d = displacement norm, q_i = total displacement at the i^th iteration and u_i−1 = total displacement at the iteration.

If the convergence criterion is satisfied, then apply next load increment. If a convergence criterion is not satisfied then repeat the procedure from step 1) to step 9), till the displacements are converged.

For validation, the field data reported by Gabr et al. [12] and Ismael and Klym [13] are used. A detailed geotechnical investigation of the soil at Canons Park site (England) was carried out as reported by Gabr et al. [12] . The soil profile consisted of stiff slightly gravelly fissured London clay to the depth of 4 m. The test pile, 0.17 m in diameter, was bored cast in place with an embedment length of 4.5 m. Cohesion of the soil linearly increased with depth at the rate of 10 kPa/m, with a value of 45 kPa at ground level. The EI value of the pile was 1500 KNm². Soil was assumed to follow von-Mises yield criterion. Measured versus predicted lateral-pile response is presented in Figure 3 along with 3-D FEM prediction by Dewaikar et al. [14] . A reasonably good agreement is observed at lower load levels. But the difference at higher load level suggests to adopt different yield criterion. von-Mises yield criterion is independent of hydrostatic stress which resulted discrepancy at higher load level. So in the present study, Drucker-Prager and Mohr-Coulomb yield criterion are employed.

Comparison with pile load test data reported by Ismael and Klym [13] at Ontario is presented in Figure 4. A concrete pile with diameter 60 in (1.52 m) was embedded in over-consolidated clay with cohesion 2000 lb/ft² (96.0 kPa). Total pile length was 39 ft (12.0 m) with one foot (0.30 m) length above ground-line. The flexural rigidity EI of pile was 93 × 10¹⁰ lb-in² (2.675 × 10⁶ kN∙m²). Comparison between measured and predicted pile response shows a good agreement at higher load levels.

A parametric study is carried out to examine the effect of pile spacing and pile group configuration on static nonlinear response of pile groups (pile diameter D = 1.0 m, L/D ratio = 20). Six different pile group configurations (Figure 5) are considered as 2 × 2, 3 × 3, 3PP, 3PS, 2PP, 2PS (PP―piles in parallel arrangement, PS―piles in series arrangement). The pile is assumed linear elastic throughout the analysis and soil is modeled using von Mises yield criteria. The lateral force H = 2000 kN is applied at top. The values of the various parameters assumed for pile and soil in this study are summarized in Table 1.

A finite element program has been developed to perform the static nonlinear analysis of pile group with six different configurations subjected to lateral loads. A brief description of the numerical algorithm and its features has been presented. The pile has been assumed to remain elastic all the time whereas the soil has been assumed to undergo plastic yielding following von Mises yield criterion. The formulation of elasto-plastic analysis following von Mises yield criterion has been explained. The effect of Drucker-Prager and Mohr Coulomb yield criteria on the response of pile group is also investigated. The whole analysis is based on incremental load application. The external load is applied in small increments and the stresses are initially computed assuming elastic constitutive relation. These stresses in the soil elements are then checked against the corresponding yield criteria and are adjusted to satisfy the required conditions. The excess stresses are applied back again in the form of residual forces in the next iteration. This procedure is continued until a satisfactory convergence is arrived at. Significant effect of soil nonlinearity is observed at smaller pile spacing which reduces with increase in spacing. Combined structural stiffness of pile and pile cap in square arrangement is considerably higher than that of those groups in series or parallel arrangement.

V. A.Sawant,K. B.Ladhane, (2016) Nonlinear FEA of Pile Group Subjected to Lateral Load. Open Journal of Civil Engineering,06,19-30. doi: 10.4236/ojce.2016.61003