Numerical investigation of the ultimate lateral resistance of piles in soft clay

Abstract

The paper presents a numerical study on the undrained lateral response of a single, free-head, reinforced concrete pile in soft clays. Soil conditions simulating normally consolidated clays are examined—undrained shear strength increasing with depth—and the pile-soil interaction under static lateral loading is analyzed. The nonlinear p‒y curves proposed in literature for soft clays are imported into a beam-on-nonlinear-Winkler-foundation simulation in order to predict the pile head lateral load—displacement curve and the distribution of the horizontal displacement and bending moment along the pile. The striking differences among these methods require further investigation via 3D finite element analyses. The determination of the ultimate soil resistance p_ult from the results of the finite element analyses aims at providing the estimation of a range of values for the ultimate soil resistance coefficient N_p with depth and the comparison of the derived values to the corresponding ones proposed by existing methodologies.

Keywords

laterally loaded / pile / soft clay / p‒y curves / finite element method (FEM)

Introduction

p‒y curves are nowadays a common practice for the calculation of bending moment and horizontal displacement along laterally loaded piles. They are applicable as nonlinear springs along a beam-on-nonlinear-Winkler-foundation (BNWF) simulation of the laterally loaded piles. p‒y curves for static pile head lateral loading are investigated in the current paper, due to their importance for the formulation of p‒y curves for cyclic or dynamic loading. It is important to start with a brief reference of existing p-y curves methodologies for soft clays.

Numerous methodologies of p‒y curves are applicable to clayey soils [1–4] taking into account the undrained shear strength c_u and the characteristic strain at half the maximum compressive strength ϵ₅₀ of the soil, with the ones of Georgiadis and Georgiadis [5] being the most recent. However, only Matlock [1] and Det Norske Veritas [2] clearly claim that their p-y curves are applicable to soft clays.

Therefore, it is necessary to evaluate the aforementioned methodologies and test whether other unified methods are applicable to soft clays, namely normally consolidated clays with linear distribution of the undrained shear strength with depth. p‒y curve formulations mainly consist of three parts: the initial small-strain stiffness, the yielding section of the curve and the ultimate lateral resistance. The scope of the present paper is the determination of the last part of the p‒y curves for normally consolidated clays.

p-y curves for soft clays

There are various methodologies for the design of laterally loaded piles in cohesive soils using p‒y curves. The striking differences between the p‒y curves computed by different methodologies for soft clays at a specific depth are presented in Fig. 1.

The existence of such a variety of p‒y curves for a specific soil type leads to the computation of different deformational values and internal forces along the pile for each separate case. Therefore, a comparison between the p‒y curves for soft clays is very important. The present study compares the values of horizontal displacements, bending moments and shear forces along the pile, computed by the application of different p‒y curve formulations.

Simulation parameters

Soil conditions

The investigation of the pile response is carried out for the case of undrained loading conditions, since this is considered critical. In this paper, the ultimate lateral soil resistance of soft, normally consolidated clays is simulated and investigated. The undrained shear strength of such clays is considered to increase linearly with depth (z), according to the following Equation:

(1)c_{\mathrm{u}}(z)=c_{\mathrm{u}0}+A\cdot {{\sigma }^{\prime }}_{\mathrm{v}\mathrm{o}}(z),

where σ′_vo(z) stands for the vertical effective stress of the soil at the specific depth. For soft to medium-stiff clays, A ranges from 0.15 to 0.35 [6]. To simulate natural states of a soft clay, the present study considered c_u0 = 10kPa at ground surface, A = 0.15‒0.35, buoyant soil unit weight γ′ = 10 kN/m³, ground water table at ground surface and coefficient of horizontal geostatic stress k_o = 0.60. Finally, the ratio of the undrained Young’s modulus to the in situ undrained shear strength E_u/c_u of the present study is taken into account according to Table 1 [7].

Pile lateral loading

The results presented herein for p-y curves in soft clays are derived from numerical lateral loading tests of a free-head, reinforced concrete pile with length L = 20 m and diameter D = 1 m (Fig. 4). The pile is loaded with a concentrated lateral load H at its head, applied in specific load increments imposed by the convergence demands of the solution. Structural serviceability limit state design demands the elastic behavior of the pile. Therefore, the pile is considered elastic with Young’s modulus E_p = 25 GPa and Poisson’s ratio v = 0.20.

2D simulation of a single laterally loaded pile

A 2D finite element code developed in NTUA Geotechnical Department is utilized in order to acquire y₀‒H, y‒z and M‒z graphs (where y₀ = pile head horizontal displacement, H = pile head lateral load, y = horizontal displacement, z = soil depth, M = bending moment) of the laterally loaded pile described in the previous paragraph. The BNWF code incorporates p‒y curves for different soil depths, pile properties and pile head loading conditions and calculates the displacement, rotation, bending moment, shear force and soil lateral pressure along the pile for each load increment applied.

The procedure aims at an initial estimation of the differentiation of the aforementioned results, when simulating the soft clay with p‒y curves proposed by different methodologies. The p‒y curves are derived every 0.5m along the pile for each case of undrained shear strength distribution (A = 0.15, 0.25, 0.35). A total concentrated lateral load H = 3000 kN is applied at the pile head in load increments of 100 kN. The analysis is terminated when convergence of the solution is judged unlikely, according to a specific maximum tolerance limit defined for the nodal displacement and rotation increments computed along the pile. To discern the impact of the p-y curves differentiation on the displacements and the internal forces along the pile, the comparative graphs of Fig. 2 are presented. The corresponding results refer to A = 0.25 coefficient of undrained shear strength c_u(z) distribution with depth. However, similar trends appear for the other two coefficient values referring to normally consolidated clays (A = 0.15, 0.35). Therefore, the graphs of Fig. 2 lead to general quantitative conclusions on the existing p-y curves methodologies for soft clays.

It is obvious that different p-y curves methodologies compute a wide range of displacements, moments and shear forces along the pile, especially for high lateral loads. For H = 1100 kN in Fig. 2, the divergence of pile head lateral displacements y₀ from an average value is 40%, while the same measurement for the maximum bending moment along the pile is 50%. Therefore, the ultimate lateral resistance of normally consolidated clays is investigated in the following paragraphs. Furthermore, the suitability of ultimate lateral soft clay resistance derived for constant undrained shear strength with depth for the simulation of normally consolidated clays is investigated.

Ultimate lateral soil resistance of soft clays

The ultimate lateral resistance of clays is quantified by the non-dimensional coefficient N_p (Eq. (2)).

(2)N_{\mathrm{p}}={\displaystyle \frac{p_{\mathrm{u}\mathrm{l}\mathrm{t}}}{c_{\mathrm{u}}D}}.

A variety of methods referring to the ultimate lateral resistance of clays is located in the literature [1-4,8-10] (Hansen, 1961, Broms, 1964, Georgiadis K. & M., 2012). An attempt is made to group these methods in a single diagram representative of a soft, normally consolidated clay with undrained shear strength increasing linearly with depth. The aforementioned methods are presented in Fig. 3, depicting the variation of N_p for the specific soil type, especially at small depths (z < 6D).

The existence of such a miscellaneous diagram for the coefficient of ultimate lateral soil resistance N_p imposes the numerical simulation of the problem. To clarify the typical range of N_p for soft, normally consolidated clays, 3D finite element analyses are carried out and their results are presented in the following paragraphs.

3D simulation of a single laterally loaded pile

The 3D finite element model

A 3D finite element model is designed in the commercial code ABAQUS [11] in order to simulate the single laterally loaded pile, as presented by Tzivakos [12]. Half the cross-section of the pile and the surrounding soil block is simulated for symmetry reasons (Fig. 4), with a concetrated lateral load H applied on the pile head. Solid, 8-node, full integration elements are used to model the soft clay, while 2-node beam elements acting in the three-dimensional space simulate the pile. The Drucker-Prager constitutive model is assigned to the soil elements-widely applicable for the simulation of cohesive soils in various geotechnical problems (e.g., Ref. [13])- and total stress analyses are carried out. The material of the pile is considered elastic.

The surface interaction between the pile and the surrounding soil in the shear direction is simulated according to the Coulomb friction law, allowing for relative slippage of the pile against the soil if the ultimate shear strength of the interface is reached. Referring to the normal stress interaction between the pile and the surrounding soil, a proper simulation of pile-soil interface is carried out, allowing for the formulation of a gap behind the pile (Fig. 6)- as long as the undrained shear strength c_u of the soil suffices for it to stand vertical. The accurate contact pressure simulation on the pile-soil interface follows the exponential law depicted in Fig. 5, in order to avoid potential numerical instabilities caused by a corresponding hard contact simulation. According to this law, contact pressure starts to build up between the two interacting surfaces at infinitesimal clearance c_o. The contact pressure transmitted between the surfaces then increases exponentially as the clearance continues to diminish. For the FEA of the current study, clearance c_o = 10^-6 m and pressure p_o = 1 kPa were assigned to the prescribed exponential law.

The calculation of the ultimate lateral soil resistance p_ult through FEA is complicated. Initially, the assumption is made that the lateral soil pressure p is calculated according to the beam-on-elastic foundation solution:

(3)p=E_{\mathrm{p}}{I}_{\mathrm{p}}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{4}}y}{\mathrm{d}{z}^{\mathrm{4}}}}={\displaystyle \frac{\mathrm{d}Q}{\mathrm{d}z}}.

A clarification remark is important here, namely that the ultimate lateral soil resistance in the present study is treated as the asymptote of the hyperbolic p-y curve in very large horizontal displacements. To acquire this ultimate value, a transformation of the hyperbolic p-y equation to an equivalent linear equation is necessary:

(4)p={\displaystyle \frac{y}{{\displaystyle \frac{\mathrm{1}}{k}}+{\displaystyle \frac{y}{p_{\mathrm{u}\mathrm{l}\mathrm{t}}}}}}\to {\displaystyle \frac{y}{p}}={\displaystyle \frac{\mathrm{1}}{p_{\mathrm{u}\mathrm{l}\mathrm{t}}}}\cdot y+{\displaystyle \frac{\mathrm{1}}{k}}.

Thus, the results of the FEA are depicted in y‒y/p diagrams and via linear regression-similarly to the method used by Zhu et al. [14] – the ultimate lateral soil resistance for each depth is determined (Fig. 7).

The simulation of the physical problem leads the surrounding clay to yield up to a specific depth range. Moreover, the upper part of the pile-soil interaction (z/D<6) is crucial in static lateral loading. Thereafter, the coefficient of ultimate lateral resistance N_p remains constant with depth.

Parametric investigation of the simulation variables

Two main variables of the problem are studied parametrically. The coefficient of horizontal geostatic stress k_o and the pile-clay adhesion factor α, which receives values 0 ÷ 1 for a smooth or rough pile-soil interaction respectively. The analyses of Table 2 are carried out for this verification.

The results of the specific study showed that for all undrained shear strength distributions (A = 0.15, 0.25, 0.35) the divergence between the p_ult values obtained by altering k_o is less than 5% for depth to diameter ratios z/D<8. On the contrary, the adhesion factor α is of significant importance for the computation of p_ult, with the difference of the specific results ranging between 10 and 20% for the same z/D ratios.

Estimation of the ultimate lateral soft clay resistance coefficient N_p

The coefficient of ultimate lateral soil resistance N_p that resulted from the 3D finite element analyses is presented in the same graph with the corresponding proposals of the existing methodologies for comparative reasons (Fig. 9). In addition, the three cases of undrained shear strength distribution with depth were simulated by two layers of constant c_u for each half length of the pile. The specific constant values were derived according to the assumption of Fig. 8 for the soft clay strength distribution with depth.

Three additional FEA were carried out for the rough soil-pile interaction case (α = 1), in order to ascertain the effects of the aforementioned simulation on the determination of the ultimate lateral soil resistance of soft clays with depth. The corresponding output is marked in the diagrams with the “c_u_AVG” index.

It is clearly distinguished that the calculated N_p values are higher than almost every other existing methodology. Both the ultimate lateral soil resistance for pile-clay adhesion α = 0 as for α = 1 receive a N_p value between 3 ÷ 4.5 at ground level. For depth to diameter ratios z/D>4, a stabilization of N_p occurs. For α = 0, N_p = 11 ÷ 12 for z/D>4, while for α = 1 the corresponding value is N_p = 12 ÷ 13. The values of N_p in the midrange are closer to the ones proposed by Stevens-Audibert [8] or the upper bound of Murff-Hamilton [10], Randolph-Houlsby [9] and Georgiadis and Georgiadis [5], i.e., for α = 1. Attention needs to be drawn to the assumption of the present study for the ultimate lateral soil resistance; p_ult is considered the soil pressure value of the asymptote of the p‒y curve at high lateral displacements of the soil.

Finally, referring to the two-layered simulation of the linear undrained shear strength distribution with depth, the following observation is derived: for A = 0.15, no alteration of the calculated N_p coefficient is noticed, while for A = 0.25‒0.35 the effect of the aforementioned simulation becomes more significant with the increase of A.

Therefore, the derived ultimate lateral soil resistance for z/D>4 and A>0.25 is to be utilized with caution, since it is calculated according to the specific assumption.

Conclusions

A thorough study has been carried out on the ultimate lateral resistance p_ult of soft, NC clays under undrained loading conditions. The existing methodologies seem to underestimate the actual lateral resistance of such soils. Three-dimensional finite element analyses in the comercial code ABAQUS [11] demonstrate closer association of the caclulated ultimate lateral soft clay resistance to the proposals of Stevens and Audibert [8] and the upper bound of Murff and Hamilton [10], Randolph and Houlsby [9] and Georgiadis [5] for soil depth less than 4 pile diameters. Ultimate soil resistance values acquired for greater depth are higher than almost all the existing methodologies. The coefficient of ultimate lateral soil resistance for the latter case is calculated N_p = 11 ÷ 12 and N_p = 12 ÷ 13 for a smooth and rough pile-soil interaction respectively. In addition, the assumption of a constant undrained shear strength—instead of a linear c_u distribution with depth – leads to overestimation of the ultimate lateral soft clay resistance for A>0.25 and z/D>4.

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