Bending Moment Calculations for Piles Based on the Finite Element Method
Bending Moment Calculations for Piles Based on the Finite Element Method
Yu-xin Jie, Hui-na Yuan, Hou-de Zhou, and Yu-zhen Yu
State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China
Received 17 March 2013; Accepted 1 June 2013
Academic Editor: Fayun Liang
Copyright © 2013 Yu-xin Jie et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Using the finite element analysis program ABAQUS, a series of calculations on a cantilever beam, pile, and sheet pile wall were made to investigate the bending moment computational methods. The analyses demonstrated that the shear locking is not significant for the passive pile embedded in soil. Therefore, higher-order elements are not always necessary in the computation. The number of grids across the pile section is important for bending moment calculated with stress and less significant for that calculated with displacement. Although computing bending moment with displacement requires fewer grid numbers across the pile section, it sometimes results in variation of the results. For displacement calculation, a pile row can be suitably represented by an equivalent sheet pile wall, whereas the resulting bending moments may be different. Calculated results of bending moment may differ greatly with different grid partitions and computational methods. Therefore, a comparison of results is necessary when performing the analysis.
1. Introduction
As the finite element method (FEM) develops, pile foundations are increasingly being analyzed using FEM [1–8]. Solid elements are used to simulate soil or rock in geotechnical engineering. Other structures embedded in soil such as piles, cut-off walls, and concrete panels are also often simulated with solid elements. However, internal force and bending moment are generally used for engineering design. So it is necessary to calculate the bending moment with stress and displacement obtained using FEM.
Theoretically, the following two methods are both appropriate.
(a) Calculating Bending Moment with Stress
The bending moment is directly calculated by summing the total moments of the elements across the specified pile section. When using this method, sufficient grids are necessary to partition the pile section.
(b) Calculating Bending Moment with Displacement
The bending moment is indirectly calculated using the quadratic differential of deflection (lateral displacement) of the pile. This method uses fewer grids, but the differential process will result in reduced accuracy.
The bending moment can also be obtained by integrating the area of the shear force diagram [9] which is a complex process and is not considered in this paper.
As we know, shear locking occurs in first-order (linear) fully integrated elements that are subjected to bending, while second-order reduced-integration elements can yield more reasonable results in this case and are often used in the analysis of piles subjected to lateral pressure [1–4, 10]. However, calculating second-order elements is time consuming and increases complexity and computational effort, particularly when the problem involves contact conditions. So we consider that the linear element method with appropriate meshing is still useful for the analysis of piles.
A row of piles can be simplified as a plane strain wall (sheet pile wall) and modeled using 2D plane strain elements [11–13]. This simplification can greatly reduce computational effort. However, the influence of bending moment on the computational results merits further research.
In this paper, a series of calculations on cantilever beam, pile, and sheet pile wall examples were conducted to study the abovementioned problems. The main aim of the work was to investigate the computational methods for bending moment and the influences of element type and mesh partition. Hence, no interface element was introduced, that is, the pile was assumed to be fully attached to the soil, and the soil and pile were both assumed to have linear elastic behavior.
2. Cantilever Beam Example2.1. Analytical Solution
The cantilever beam example is shown in Figure 1. The width of the square beam is 1 m. The length is 30 m. A distributed load kPa is applied to the beam. The analytical solution equations arewhere bending moment, the position coordinate, , the Young’s modulus, the moment of inertia, is the deflection of the beam, and displacement in the direction.
Figure 1: Cantilever beam.
The beam parameters are taken as Young’s modulus MPa and Poisson’s ratio in the computation. The element used in the FEM is the 4-node first-order plane stress element (CPS4). The following two methods were used to calculate the bending moment.
(a) Calculating Bending Moment with Stress
The bending moment was directly computed wit (...truncated)