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In order to study the performance of base force element method (BFEM), a new 2-node degenerated plane element model based on the BFEM of complementary energy principle is presented for geometrically nonlinear problem, respectively, in this chapter. The plane truss element model can easily be obtained by degenerating from a 4-mid-node plane element of the BFEM based on complementary energy principle for geometrically nonlinear problem. The compliance matrix of the rod element model is the same as the compliance matrix of the 4-mid-node plane element of the BFEM. According to the characteristics of analysis for truss structure, the nodal equilibrium conditions, and the displacement coordination conditions of each rod at the nodes have been considered when the compliance matrix of the structure is integrated. In order to verify the feasibility of the degenerated plane element model, several truss problems are analyzed and their solutions are compared to the analytical solutions and the numerical results, which are calculated with the traditional displacement FEM. It is found that the model has shown high precision and good performance.

Base force element method (BFEM) is a new method of finite element method which takes the base forces as fundamental unknown variables. In recent years, base force element method has great development and wide applications. In this chapter, a new 3D model of base force element method (BFEM) on complementary energy principle is presented using the concept of base force which is a new physical quantity to describe the stress state by Gao. First, a three-dimensional model of compliance matrix for space truss is derived through complementary energy principle which is described using base forces by dyadic vectors. Then, the explicit control equations of base force element method on complementary energy principle are derived using Lagrange multiplier method. Thereafter, the forming method of the overall compliance matrix of the BFEM on complimentary energy principle is proposed according to the stress and deformation characteristics of space truss. And the procedure of base force element method for space truss is developed. Finally, several examples are analyzed to illustrate the reliability and accuracy of the formulation and the procedure.

The base force element method (BFEM) is a new finite element method. It describes the stress state of a point by “base force”. Based on Chaps. 10, 12 and 13, this chapter will study the application of the BFEM to the elastic–plastic analysis of plane truss structure and space truss structure. Using the idea of incremental method and elastoplastic model, the BFEM programs suitable for material nonlinear analysis are developed for plane truss problems and space truss problems, respectively, and a lot of truss examples are calculated.

A new finite element method—the base force element method (BFEM) has been developed in recent years for the analysis of solid mechanics problems. In this chapter, the 4-mid-node plane model of base force element method (BFEM) on complementary energy principle in Chap. 4 is used to analyze the rock mechanics problems. First, the chapter studies the problem of applying gravity load of rock mass on a 4-mid-node element according to the principle of virtual work. Second, the problem of exerting different material properties in the calculation of BFEM was studied. Then, the method to simulate the crack propagation using the BFEM on complementary energy principle is proposed. And the calculation method of safety factor for rock mass stability was presented for the BFEM on complementary energy principle. Finally, some typical numerical example is used to verify the validity and reliability of the above studies. The performance of the BFEM on complementary energy principle and the large-scale computing problems of the BFEM are tested. The numerical researches show that the results of the BFEM are consistent with the results of the conventional quadrilateral isoparametric element (Q4 model) and quadrilateral reduced integration element (Q4R model). The new nonlinear model of base force element method on complementary energy principle can be applied to rock mechanics and engineering.

Combined with the modified complementary energy principle and force interpolation function method, a planar beam model of the base force element method (BFEM) is proposed, and its compliance matrix, control equation and the node displacement expression are also given. Using the concept of base forces and base moments as state variables, a new function for the beam element complementary energy functional is constructed. In order to verify the applicability of this model, the model is programmed into a MATLAB program, combined with calculating some typical calculation examples, the results are compared with the theoretical solutions and the structural mechanics solver solutions. Its good agreement proved that this element can be used to analysis the static problem of the two-dimensional framed structures.

The complementary energy principle is the foundation of the base forces element method. In the references (Gao in Science in China: Series G Physics, Mechanics and Astronomy 49:341–356, 2006), Gao put forward a complementary energy principle for the problem of large elastic deformation. The form of the complementary energy principle is the same as that of the small deformation case. In recent years, some scholars have studied Gao’s complementary energy principle and put forward some generalized complementary energy principles. In this chapter, we will introduce Gao's complementary energy principle.

In this chapter, the elastic–plastic compliance matrix of an element expressed by the base forces is derived from the material nonlinear model. In this chapter, the elastic–plastic compliance matrix of an element expressed by the base forces is derived from the material nonlinear model. The equilibrium conditions are released by the Lagrange multiplier method, and a modified elastic–plastic complementary energy principle described by the base forces is used. A nonlinear procedure of the BFEM on complementary energy principle is developed using MATLAB language. The numerical results of examples show that this nonlinear model of the BFEM for the material nonlinear problem is feasible.

In this chapter, the model of the base force element method (BFEM) on the convex polygonal mesh is studied based on the complementary energy principle for two-dimensional geometrically nonlinear problems. The performances of 2D BFEM for geometrically nonlinear problems are researched. A number of highly geometrically nonlinear problems are solved using the BFEM. The results of BFEM are compared with the traditional 4-node quadrilateral isoparametric element (Q4 model) and 4-node quadrilateral reduced integration element (Q4R model). The numerical results of examples show that the BFEM has good performance. The two-dimensional formulation of BFEM with complementary energy principle provides reliable predictions for convex polygonal mesh problems in geometrically nonlinear analysis.

In the references (Gao in Foundation of solid mechanics. Railway Publisher of China, Beijing, 1999, Gao in Archive of Applied Mechanics 73:171–183, 2003; Gao et al. in Applied Mechanics Reviews 61:1–16, 2008), Gao introduced the concept of “base forces” in detail, and formed the elastic theory based on the concept of base forces. In this chapter, we will introduce the concept of base forces and the theory of base forces. This theory is the basis of the research work in this book.

Using the concept of base forces given by Gao as fundamental variables to describe the stress state of an elastic system, this chapter presents an explicit expression of two-dimensional element compliance matrix on the complementary energy principle with concave polygonal meshes. The detailed 2D formulations of governing equations for the new finite element method—the base force element method (BFEM) are written in terms of the base forces concept using the Lagrange multiplier method. The explicit displacements expression of nodes is given. To verify the model proposed in this paper, a program on the 2D BFEM using MATLAB language is made and a number of examples on concave polygonal meshes and aberrant meshes are provided to illustrate the BFEM. Good agreement has been obtained comparing numerical results using the proposed BFEM to available theoretical results. The concave polygonal element model can be used efficiently for the elasticity analysis with increasing element aspect ratios and distortion meshes compared with the traditional finite element method (FEM).

Based on the concept of the base forces by Gao, a new finite element method—the base force element method (BFEM) on complementary energy principle for two-dimensional geometrically nonlinear problems is presented. A 4-mid-node plane element model of the BFEM for the geometrically nonlinear problem is derived by assuming that the stress is uniformly distributed on each side of a plane element. The explicit formulations of the control equations for the BFEM are derived using the modified complementary energy principle. The BFEM is naturally universal for small displacement and large displacement problems. A number of example problems are solved using the BFEM and the results are compared with corresponding analytical solutions and those obtained from the standard displacement finite element method. A good agreement of the results, and better performance of the BFEM, compared to the displacement model, in the large displacement and large rotation calculations, is observed.

Using the base forces as fundamental variables to describe the stress state and the displacement gradients that are the conjugate variables of the base forces to describe the deformation state for the two-dimensional elasticity problems, a 4-mid-node plane model of base force element method (BFEM) based on complementary energy principle is proposed. In this chapter, the complementary energy of an element of the BFEM is constructed using the base forces. The equilibrium conditions are released by the Lagrange multiplier method, and a modified complementary energy principle described by the base forces is obtained. The formulation of the 4-mid-node plane element of the BFEM is derived by assuming that the stress is uniformly distributed on each side of the plane element. A procedure of the BFEM on complementary energy principle is developed using MATLAB language. The numerical results of examples show that this model of the BFEM has high precision and is free from mesh sensitivity. This model shows good performances.