Mathematical Problems in Engineering

Volume 2015, Article ID 762539, 8 pages

http://dx.doi.org/10.1155/2015/762539

## Accurate Element of Compressive Bar considering the Effect of Displacement

^{1}College of Water Resources & Civil Engineering, China Agricultural University, Beijing 100083, China^{2}China Machinery TDI International Engineering Company Limited, Beijing 100083, China

Received 18 September 2014; Accepted 5 January 2015

Academic Editor: Chenfeng Li

Copyright © 2015 Lifeng Tang 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

By constructing the compressive bar element and developing the stiffness matrix, most issues about the compressive bar can be solved. In this paper, based on second derivative to the equilibrium differential governing equations, the displacement shape functions are got. And then the finite element formula of compressive bar element is developed by using the potential energy principle and analytical shape function. Based on the total potential energy variation principle, the static and geometrical stiffness matrices are proposed, in which the large deformation of compressive bar is considered. To verify the accurate and validity of the analytical trial function element proposed in this paper, a number of the numerical examples are presented. Comparisons show that the proposed element has high calculation efficiency and rapid speed of convergence.

#### 1. Introduction

Compressive bar element analysis is essential in structural engineering design. It can be used to calculate the static and stability problem. A large number of theories and methods for compressive bar element analysis have been suggested. The analysis methods include energy method and numerical method.

At present, constructing the compressive bar element needs to calculate the stiffness. References [1–3] all agreed that the stiffness matrix of accurate element of static problem for compressive bar could be derived easily. But for the linear stability problem of compressive bar, the accurate solution cannot be obtained based on the common structural mechanics. A more exact solution can be got though the finite element method which was firstly proposed by Clough [4] in an article about plane elastic problem and was perfect in theory as a numerical method. The finite element method includes direct stiffness method, analytical trial function method, or interpolation trial function method. Direct stiffness method can accurately solve differential equation of static but has low efficiency affected by the number of meshes [5]. So Chen [5] proposed refined direct stiffness method evolved from the generalized hybrid element, which could be used in any displacement function (consistent or inconsistent) and had high precision. Interpolation shape function [6] is one of the main methods of numerical differentiation. This method based on interpolation polynomial can be used for the numerical differential formulas derivation. In practice, the element is needed to be divided into several nodes for interpolation, so interpolation shape function has low precision. Analytical trial function method [7–9] which had solved many traditional problems had widely been used in the elastic mechanics. The boundary element method [10], the boundary collocation method [11], and the hybrid element method [12] all utilized the analytical function for analysis. Fu [13] had carried on the system discussion based on the analytical trial function and proposed a series of analytical trial function (ATF) element, which solved many traditional finite element problem successfully. Cen et al. [14] introduced the variational principles of stress function into analytical trial function element and proposed a new idea of high precision and accurate element. The characteristic differential equation [15–17] which is based on the theory of operator matrix provided the basis method for finite element method to construct the analytical trial function.

Nowadays the finite element method which performs very well only is for the case of small deformation problems and examples ranges from standard mixed elements [18–20] to enhanced strain elements [21, 22]. But for elements which seem to be ideal from a numerical and a theoretical perspective, it may fail in the large deformation range, due to the high compression states and many interesting methods [23–28] have been developed to solve the problem. However, a satisfactory analysis of finite element methodologies for large deformation problems is still not proposed imperfectly.

In the present paper, using the differential equations of equilibrium, and considering the effect of large deformation, the displacement shape function is derived. And then based on the minimum potential energy principle, the elastic stiffness matrix and geometrical stiffness matrix are got. Utilizing the stiffness equilibrium equation, we can get the accurate ultimate load value and the buckling mode of the compressive bar subjected to different constraints. Several examples are presented in order to verify the accuracy and validity of the proposed analytical trial function method. Results show that the present method has many advantages including accuracy, efficiency, and simplicity compared to the direct stiffness method and the interpolation shape function element.

#### 2. Characteristic of Large Displacement of Compressive Bar

For the static problem of compressive bar, the accurate solution can be obtained by the traditional finite element method. But for the stability issue, when compressive bar is subjected to the critical load, the large deformation is generated, which results in the interaction between the axial load and lateral displacement. Then the lateral displacement derived from the cubic polynomial Hermit interpolation trial function had a huge system error, while by using the direct Hermit interpolation trial function, the generated deformation of compressive bar element is an approximate and the error is also large.

The stability problem of compressive bar should be ranked as a geometrical nonlinear problem; in which situation, the large deformation should be considered. Thus is not a high order and small value, and the nodal lateral displacement generated by the longitudinal displacement cannot be ignored. Then the nodal lateral and longitudinal displacement are not two independent but interactive variables. Also, because of the influence of large deformation, the differential equilibrium equation between the shear force and bending moment [29] is not suitable anymore.

Based on the characteristic of large displacement of compressive bar, the main key of generating the accurate element is searching for the accurate curve expression of lateral displacement.

#### 3. Analytical Trial Function for Displacement of Compressive Bar Element

The steps of creating the displacement analytical shape function of the compressive bar element are as follows:

*Step 1. *Based on the differential equations of equilibrium, obtain the displacement equation of compressive bar, including the undetermined parameters;

*Step 2. *Based on the above equation, establish the nodal displacement equation of element.

*Step 3. *According to the equation of Step 2, calculate the undermined parameters.

*Step 4. *On the basis of the above equations, create the deformation shape function.

##### 3.1. Stiffness Balance Equation of Compressive Bar

Figure 1 shows the elastic compressive bar, whose length is , subjected to axial load , bending moment , and shear force at its two ends. The equilibrium differential governing equation [30] iswhere is elastic modulus; is the moment of inertia; denotes the coordinates; represents the corresponding displacement vector; is axial load acting on the bar and is taken as negative when it is pressure; is bending moment acting on end 1 of the compressive bar; and is shear force, as shown in Figure 1.