Load-Deflection Response of Laterally Loaded
Single Pile by Nonlinear Finite Element Analysis

 

Dilip Kumar Maharaj

Assistant Professor, Civil Engineering Group
Birla Institute of Technology and Science, Pilani, Rajasthan, India
e-mail:dilip_maharaj@yahoo.com or dkm@bits-pilani.ac.in

ABSTRACT

In this paper a square pile with square cap embedded in ground and under lateral load has been analysed by three dimensional nonlinear finite element method. The soil has been modelled as an elastoplastic medium by Drucker-Prager Yield Criteria. The cap, pile and soil have been discretized into eight noded isoparametric brick finite elements. Parametric study has been done for load-deflection response of pile under lateral load by varying parameters pile cap thickness, pile cap size and pile length. Based on nonlinear finite element analysis results have been produced for load-deflection.The increase in thickness of pile cap and size of pile cap have been found to increase the lateral load carrying capacity of pile significantly. This is more effective for shorter pile than for longer pile. With increase in length of pile the lateral load carrying capacity of pile cap system as a whole increases. This increase is upto the critical length of pile. The results obtained from the finite element model considered compares well with that reported in literature.

Keywords: Analysis, Nonlinear, Deflection, Cap, Pile

INTRODUCTION

The analytical approaches developed for single pile and pile group under lateral load are the subgrade reaction approach, the elastic approach and the finite element approach. The main disadvantage of the subgrade reaction approach is that the continuum nature of soil is ignored, whereas the elastic approach assumes the soil to be an ideal elastic continuum. This approach does not take into account the soil yielding and it is therefore only suitable for prediction of load-deflection response of laterally loaded piles at small strain levels. Due to the limitation of the above mentioned methods three dimensional nonlinear finite element analysis has been considered to predict the actual load-deflection behaviour of pile under lateral load. Figure 1 shows a pile with cap which is to be analysed under lateral load. The side of square cap, side of square pile, the thickness of pile cap and the length of pile have been represented by B, d, T and L respectively and all these dimensions are considered in metre.

 



Figure 1. Single pile with cap

LITERATURE REVIEW

The important literature reported on single pile and pile groups under lateral loads are Matlock and Reese(1960), Banerjee and Davies (1978), Poulos (1971), Pise (1983), Pise (1984), Budhu and Davis (1988), Alizadeh and Lalvani (2000) and Mokwa and Duncan (2001). Matlock and Reese (1960) provided generalised solution for elastic and rigid pile under lateral load. Poulos (1971) and Banerjee and Davies (1978) reported the elastic solution for laterally loaded pile. Pise (1983) presented theoretical results for fixed head piles while Pise (1984) presented theoretical results for free head piles. Budhu and Davies (1988) reported elasto-plastic analysis of laterally loaded pile based on boundary-element method. Alizadeh and Lalvani(2000) provided useful results of full-scale, field lateral load tests on four instrumented single piles installed in sand. Mokwa and Duncan(2001) reported the cap resistance of single pile and pile group based on field tests.

Based on this literature review it has been found that there is need for understanding the three dimensional nonlinear behaviour of pile under lateral load especially when the cap is embedded in ground. In the present analysis single square pile with fixed square cap has been analysed by nonlinear finite element analysis in order to understand the lateral load-deflection response of pile till failure. This load-deflection response has been studied for single pile without cap and single pile with cap.

FINITE ELEMENT FORMULATION

The cap, pile and the soil have been discretized into eight noded isoparametric brick elements. The cap and pile have been considered as linear elastic material. The soil has been modelled as Drucker-Prager (1952) elastoplastic medium. The stiffness matrix and force vector for an element has been obtained from energy principle. The stiffness matrix and load vetor for the complete pile cap soil system has been obtained by assembling the stiffness matrix and load vectors for all the elements. The nonlinear finite element equation obtained for the complete structure has been solved by Full Newton-Raphson Iterative Procedure. The stiffness matrix, load vector for the eight noded brick element and its assembly, the derivation of elastoplastic constitutive matrix and the Full Newton-Raphson Iterative Procedure considered in this analysis are same as discussed by the author [Maharaj (2003), EJGE Paper 0334] and hence the description is not given here.

VALIDATION OF THE 3D FE MODEL

Figure 2 shows the comparison of the results obtained from the present analysis and that reported in the literature for vertical load under identical conditions.The two results are in excellent agreement.

 



Figure 2. Validation of 3D finite element model

FINITE ELEMENT ANALYSES

Figure 3 shows the finite element discretization considered in three dimensional nonlinear finite element analysis. Only half of the pile, cap and the surrounding soil have been considered in the finite element analysis. The soil domain considered from the center of pile is 30 times the side of pile both in x and y directions. The depth of soil considered below the cap is 70 times the side of pile. The domain of soil considered has been found very much suitable for the analysis of the laterally loaded pile (without cap and with cap) as when loaded till failure the soil elements at and near boundary do not experience any deflection. Also the soil elements at and near to the bottom boundary do not experience any deflection. The boundary conditions considered are shown in Figure 3.



Figure 3. Finite element discretization for a single pile with cap

 

Material Properties and Parameters Varied

Modulus of elasticity for pile (Ep) = 20 GN/m2 (20 GPa)

Poisson’s ratio of pile (np) = 0.30

Modulus of elasticity for soil (Es) = 32000 kN/m2 (32 MN/m2 or 32 MPa)

Cohesion of soil (c) = 29.10 kN/m2

Poisson’s ratio for soil (ns) = 0.45

Thickness of pile cap (T) = 0.5, 1.0, 2.0

Size of pile cap (B x B) = 1.6 m ´ 1.6 m and 4.0 m ´ 4.0 m

Length of pile (L) = 8, 12, 16 m.

Size of pile (d ´ d) = 0.4 m ´ 0.4 m

RESULTS AND DISCUSSIONS

Figure 4 shows the load-deflection curve for laterally loaded piles of different lengths. The load carrying capacity of smallest pile is least while the longest pile takes maximum load. It can be seen that load-deflection curve for each of the three piles are same upto 1000 kN load beyond which the piles of length 12 metres and 16 metres follow the same load deflection curve up to load of 2400 kN. Beyond this load the pile of length 16 metres carries more load than the pile of length 12 metres. The smallest pile reaches to its ultimate lateral load carrying capacity first followed by 12-metre long pile and then the 16-metre long pile.

 



Figure 4. Load-deflection curves (piles only)

Figure 5 shows the effect of cap thickness on load-deflection curves of laterally loaded piles without cap and with cap. It can be seen that the load carrying capacity of pile increases significanty due to the presence of cap. This is due to the resistance offered by the pile cap which increases the pile capacity significantly. This increase in the lateral resistance of pile with cap can be seen significant even at 0.5 metre thickness of pile cap beyond which further increase in thickness of the cap does not show significant contribution in increasing the lateral load carrying capacity of pile.

 



Figure 5. Load-deflection curves (L = 8 m, cap size B = 1.6 m square)

Figure 6 shows the load-deflection curves for pile of same length but larger cap size. It can be seen that the lateral resistance offered by the pile cap multiplies the lateral load carrying capacity of pile by nearly four times at 0.5 metre thickness of pile cap. Increase in thickness of pile cap from 0.5 metre does not show significant contribution in increasing the lateral load carrying capacity of the pile.

 



Figure 6. Load-deflection curves (L = 8 m, cap size B = 4.0 m square)

Figure 7 shows the load-deflection curves for 12 metre length of pile with and without cap. The lateral resistance of pile cap increases the load carrying capacity of pile significantly though this increase is comparatively less than that of the shorter pile (Figure 4). This is due to the fact that the lateral resistance of longer pile is more than that of the shorter pile as seen in Figure 4 and hence the contribution of equal cap resistance doesnot show as much improvement as in case of shorter pile though the overall load carrying capacity of longer pile cap system is more than that of the shorter pile cap system. The increase in thickness of pile cap from 0.5 metre does not show significant increase in the load carrying capacity of pile.

 



Figure 7. Load-deflection curves (L = 12, cap size B = 1.6 square)

Figure 8 shows the load-deflection curves for a pile of 12 metre length and increased pile cap size. The lateral resistance of pile cap is significant in this case than the previous case (Figure 7) due to the increase in size of pile cap. Increase in thickness from 0.5 metre does not show significant increase in the lateral resistance of pile cap and hence the pile cap system.

Figure 9 shows the effect of length of pile on the load-deflection curves of pile cap system. It can be seen that upto load of 2400 kN the load-deflection curves for all the three piles are same and beyond this the load carried by the longer pile [12 metre] cap system is more than that of the shorter pile [8 metre] cap system. Further increase in length of pile does not increase the load carrying capacity. The load-deflection curves of 12 metre and 16 metre pile lengths overlap each other. This clearly shows that the pile has reached to its critical length.

 



Figure 8. Load-deflection curves (L = 12 m, cap size B = 4.0 m square)

 


Figure 9. Load-deflection curves (T = 1.0 m, cap size B = 1.6 m square)

Figure 10 shows the effect of pile cap size on the lateral load-deflection curves of pile. With increase in cap size the lateral load carrying capacity of pile increases. For same deflection the load carried by pile with larger cap size is more than the pile with smaller cap size. The increase in cap size is very effective in increasing the load carrying capacity of pile. In other words the lateral resistance of pile cap increases with increase in size of cap. Hence while designing pile under lateral load cap resistance must be considered which makes the design economical.

 



Figure 10. Load-deflection curves (L = 8 m, T = 0.5 m)

Figure 11 shows the effect of pile cap size on the lateral load-deflection curves of pile of length 12 metre. The effect of increase in cap size is to increase the laod carrying capacity of pile. The effect of increase in cap size is seen less significant for longer pile than that of the shorter pile. This is due to the fact that cap resistance remaining the same , the lateral resistance of longer pile without pile cap is more than that of the shorter pile without pile cap.

 



Figure 11. Load-deflection curves (L = 12 m, T = 0.5 m)

 

CONCLUSIONS

The lateral load carrying capacity of longer pile has been found more than that of the shorter pile. The addition of a cap to a pile and taking advantage of the lateral load resistnace of pile cap have been found to increase the lateral load carrying capacity of pile significantly. The lateral load resistance of the pile cap can be increased by increasing the thickness of the pile cap and the size of the cap. The increase in size of pile cap has been found to increase the lateral load carrying capacity of pile significantly. The effect of increase in thickness of pile cap is to increase the lateral load carrying capacity of pile. At 0.5 metre thickness of pile cap significant increase in lateral load carrying capacity of pile has been found though further increase in thickness doesnot show significant increase in load carrying capacity of pile. For the same cap resistance, this increase is seen more effective for shorter pile than the longer pile as the lateral load carrying capacity of longer pile (without cap) is more than that of the shorter pile (without cap). The lateral load carrying capacity of pile cap system as a single unit has been found more for longer pile than that of the shorter pile. When pile reaches to its critical length, further increase in length of pile is not recommended as it will not increase the load carrying capacity. The results obtained from the finite element model compares well with that reported in literature.

ACKNOWLEDGEMENTS

The author wishes to thank Birla Institute of Technology and Science, Pilani, Rajasthan for providing computing facility. The author thanks his wife and loving sons Ashish and Manish for their sincere effort in preparing this paper. The author also thanks all the groups specially his Civil Engineering Group for their cooperation.

references

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