Uplift Capacity of Pile by Finite Element Method
Assistant Professor, Civil Engineering Group
Former students, Civil Engineering Group,
In this paper a single pile and group of piles of varying cross-section have been analysed by nonlinear finite element method under plane strain condition. Each row of group of piles in the transverse direction has been converted into equivalent strip of volume equal to the total number of piles. The top of the piles have been considered to be connected with a rigid pile cap such that when under uplift load each of the piles undergo the same vertical displacement. The cap, pile and soil have been discretized into four nodded isoparametric elements. The soil has been modeled as elastoplastic medium by Drucker-Prager yield criterion. The load deflection curves have been provided for an individual pile and a single pile taken from the group of piles. The effect of varying cross-section on load deflection curve has also been analysed. The load carrying capacity of pile of varying cross-section is more than that of the straight shaft pile of same volume of concrete. The interaction between the piles has been found more at closer spacing and least at higher spacing resulting into more deflection of a pile in a group. The load carrying capacity of an individual pile has been found more than that of a pile in a group in case of piles under uplift load and of varying cross-section. The load carrying capacity of group of piles has been found more than an individual pile.
Keywords: Individual pile, Varying cross-section, Analysis, Deflection, Load carrying capacity, Group of piles.
The piles under vertical load are mostly used for heavy structures where the lateral load acting on the pile is considered to be negligible. For the structures where lateral load has got significance the application of pile is a must where the plies are under uplift load. These piles are very useful for the stability and safety of the structure. These piles carry load through skin friction. The piles can be provided as an individual unit or it can be provided as a group depending upon the uplift resistance to be carried by the piles for the stability of the structure. The present paper aims to analyse a single and group of piles of varying cross-section by finite element method to find its load carrying capacity and to understand its behaviour.
Figure 1. Pile with straight shaft and of varying cross-section
Some of the important literatures related to this paper are Potts and Martins (1982), Chow (1987), Polo and Clemente (1988), Chow (1991), Trochanis (1991), Well and Naggar (1998), Castelli and Maugeri (2002).
Potts and Martins (1982) reported mobilisation of shear stress along a rough pile shaft in normally consolidated clay.
Chow (1987) presented a numerical method for the analysis of general three dimensional pile groups. Polo and Clemente (1988) presented a method of predicting pile-group settlements for cases in which piles are held together by means of rigid pile cap. Chow (1991) presented a numerical analysis to study the behaviour of vertically loaded pile groups embedded in a non homogeneous soil with the pile cap in contact with the ground. Trochanis (1991) studied the effect of nonlinear soil behaviour on the axial and lateral response of piles due to monotonic and cyclic loading, Well and Naggar (1998) discussed the experimental investigation of the tapered pile, Castelli and Maugeri (2002) proposed an approximate approach for the analysis of nonlinear response of vertically loaded pile groups
FINITE ELEMENT FORMULATION
The element stiffness matrix, element load vector, the assembly of element stiffness matrix and load vector, the constitutive model considered, derivation of elastoplastic matrix, the iterative method for solving the nonlinear finite element equation and the validation of the two dimensional finite element model considered are same as reported by Author, Maharaj (2003-Ppr-338)
FINITE ELEMENT ANALYSIS
Figure 2 shows the finite element discretization considered for the analysis of pile under uplift load. The pile and soil have been discretized into four noded isoparametric finite elements. The soil has been modeled as elastoplastic medium by Drucker-Prager Yield Criterion. The Nonlinear Finite Element equation has been solved by Newton-Rapshon Iterative procedure. A zone of soil equal to 28 meter length and 15 meter depth has been considered in the analysis such that the boundary of the soil on either end is at a distance of 10 meter from the center of the end piles. The center to center spacing between the piles has been kept equal to 4 meter. The bottom boundary of the domain of soil considered has been considered to have no degree of freedom while the edge boundary of soil has been considered to have only the vertical degree of freedom.
Figure 2. Finite element discretization for group of piles with varying cross-section
The ranges of the material properties and other and parameters used were as follows.
Pile Length = 12 m
Pile width = 1.0, 2.0, 3.0 m which has been varied at each 4.0 meter depth.
Pile width for uniform pile = 2.0 m
Modulus of cap and pile material = 2 x 107 kN/m2.
Poissonís ratio of cap and pile material = 0.30
Young's Modulus of soil = 3200, 76000 kN/m2
Poissonís Ratio of soil = 0.45
Cohesion of Soil = 29.10 and 63.4 kN/m2
RESULTS AND DISCUSSIONS
Figure 3 shows the load deflection curve for a single pile with varying cross-section for soil modulus of 76000 kN./m2. The load-deflection curve is initially linear and then becomenonlinear. For a known deflection. of the pile the load carrying capacity of pile under uplift load can be found. Similarly when the load is known the deflection of the pile can be obtained. One must consider the permissible deflection to find the allowable uplift capacity of the pile.
Figure 3. Load-displacement curve for pile with Es = 76000 kPa
Figure 4 shows the comparison between the load deflection curve of pile with straight shaft and that with varying cross-section keeping the total volume of concrete same. The effect of varying cross-section is to improve the load carrying capacity of the pile. This improvement has been seen significant at higher deflection of the pile.
Figure 4. Comparison of Load-displacement curves for piles with pile section geometry
Figure 5 shows the comparison of load deflection curve for an individual pile and a pile taken from the group of two piles. It can be seen that the load carried by an individual pile is more than that of a single unit in the group of piles. This shows that the piles interact with each other in a group of piles and hence its capacity reduces.
Figure 6 shows the comparison of load deflection curve for an individual pile and a pile taken from the group of three piles ( width wise). It can be seen that the load carried by an individual pile is more than that of a single pile in group of piles. This is due to the interaction between the piles which reduces its capacity. When compared with a group of two piles it can be observed that the load carrying capacity of a pile in a group has further reduced. This is due to the fact that when the spacing between the piles has reduced the interaction effect between the piles is more and hence the capacity of the pile has reduced.
Figure 7 shows the comparison of deflection for an individual pile and a pile in a group of two piles and a group of three piles. It is clear from the figure that a pile in a group of three piles undergoes more deflection than in a group of two piles. An individual pile undergoes least deflection. This shows that closer are the piles less is the capacity of the pile. Hence it can be said that the efficiency of group of piles is less than one in clayey soil.
Figure 5. Comparison of Load-displacement curves for single piles and pile groups
Figure 6. Comparison of Load-displacement curves for single piles and a pile in a group
Figure 7. Comparison of Load-displacement curves for single piles and a pile in a group
The uplift capacity of pile improves significantly by providing pile of varying cross-section keeping the volume of concrete same. In other words the load carrying capacity of pile of varying cross-section is more than that of the straight shaft pile of same volume of concrete. The interaction between the piles is more at closer spacing and least at higher spacing resulting into more deflection of a pile in a group. The load carrying capacity of a single pile is always more than that of a pile in a group in case of piles under uplift load and of varying cross-section. The load carrying capacity of group of piles has been found more than a single pile but the load carrying capacity for group of piles is less than the capacity of same number of individual piles. The efficiency of group of piles is less than one.
The Authors acknowledge Professor L. K .Maheshwari, Director, Birla Institute of Technology and Science for providing all facilities. The authors also acknowledge Professor A. K. Sarkar (Dean Instruction Division, Dean Faculty Division-I and Unit Chief of Community Welfare Division). The first author acknowledges staffs of all groups specially his Civil Engineering Group. The first author cannot remain without acknowledging his wife and his loving sons Ashish and Manish for their full support in this paper.
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