Mathematical Problems in Engineering

Volume 2015, Article ID 292809, 16 pages

http://dx.doi.org/10.1155/2015/292809

## Application of Base Force Element Method on Complementary Energy Principle to Rock Mechanics Problems

The Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology, Beijing 100124, China

Received 24 August 2014; Accepted 5 September 2014

Academic Editor: Song Cen

Copyright © 2015 Yijiang Peng 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

The four-mid-node plane model of base force element method (BFEM) on complementary energy principle is used to analyze the rock mechanics problems. The method to simulate the crack propagation using the BFEM is proposed. And the calculation method of safety factor for rock mass stability was presented for the BFEM on complementary energy principle. The numerical researches show that the results of the BFEM are consistent with the results of conventional quadrilateral isoparametric element and quadrilateral reduced integration element, and the nonlinear BFEM has some advantages in dealing crack propagation and calculating safety factor of stability.

#### 1. Introduction

The finite element method (FEM) has been playing a very important role in solving various problems in engineering and science. However, the conventional finite element method (FEM) based on the displacement model has some shortcomings, such as large deformation, treatment of incompressible materials, bending of thin plates, and moving boundary problems. In the past decades, numerous efforts techniques have been proposed for developing finite element models which are robust and insensitive to mesh distortion, such as the hybrid stress method [1–4], the equilibrium models [5, 6], the mixed approach [7], the integrated force method [8–11], the incompatible displacement modes [12, 13], the assumed strain method [14–17], the enhanced strain modes [18, 19], the selectively reduced integration scheme [20], the quasiconforming element method [21], the generalized conforming method [22], the Alpha finite element method [23], the new spline finite element method [24, 25], the unsymmetric method [26–29], the new natural coordinate methods [30–33], the smoothed finite element method [34], and the base force element method [35–43].

In recent years, some scholars are studying other types of numerical analysis methods, such as boundary element method [44, 45] and meshless method [46, 47]. And some scholars still adhere to explore the finite element method based on complementary energy principle [48–51]. However, these methods have not been widely applied in engineering.

In this paper, the base force element method (BFEM) on complementary energy principle is used to analyze the engineering problems of rock mechanics. The “base forces” was introduced by Gao [52], who used the concept to replace various stress tensors for the description of the stress state at a point. These base forces can be directly obtained from the strain energy. For large deformation problems, when the base forces were adopted, the derivation of basic formulae was simplified by Gao [53] and Gao et al. [54–56]. Based on the concept of the base forces, precise expressions for stiffness and compliance matrices for the FEM were obtained by Gao [52]. The applications of the stiffness matrix to the plane problems of elasticity using the plane quadrilateral element and the polygonal element were researched by Peng et al. [37]. Using the concept of base forces as state variables, a three-dimensional formulation of base force element method (BFEM) on complementary energy principle was proposed by Peng and Liu [35] for geometrically nonlinear problems. And the new finite element method based on the concept of base forces was called as the Base Force Element Method (BFEM) by Peng and Liu [35]. A three-dimensional model of base force element method (BFEM) on complementary energy principle was proposed by Liu and Peng [36] for elasticity problems. A 4-mid-node plane element model of the BFEM on complementary energy principle was proposed by Peng et al. [38] for geometrically nonlinear problem, which is derived by assuming that the stress is uniformly distributed on each edges of a plane element. In the paper [39], an arbitrary convex polygonal element model of the BFEM on complementary energy principle was proposed for geometrically nonlinear problem. In the paper [43], a 4-mid-node plane model of BFEM on complementary energy principle was researched, and its computational performance was studied. The convex polygonal element model of BFEM on complementary energy principle was given by Peng et al. [40] for arbitrary mesh problems. In the paper [41], the concave polygonal element model of BFEM on complementary energy principle was proposed for the concave polygonal mesh problems. In the paper [42], the BFEM on potential energy principle was used to analyze recycled aggregate concrete (RAC) on mesolevel, in which the model of BFEM with triangular element was derived, and the simulation results of the BFEM agree with the test results of recycled aggregate concrete. In recently, the BFEM on damage mechanics has been used to analyze the compressive strength, the size effects of compressive strength, and fracture process of concrete at mesolevel, and the analysis method is the new way for investigating fracture mechanism and numerical simulation of mechanical properties for concrete.

The purpose of this paper is to survey the base forces element method on complementary energy principle for large-scale computing problems in rock engineering problems.

#### 2. Model of the BFEM

##### 2.1. Compliance Matrix

Consider a 4-mid-node plane element as shown in Figure 1; the compliance matrix of a base force element can be obtained as [43]in which is Young’s modulus, is Poisson’s ratio, is the area of an element, is the unit tensor, and is the dot product of radius vectors and at points and .