Mathematical Problems in Engineering

Volume 2015, Article ID 582326, 10 pages

http://dx.doi.org/10.1155/2015/582326

## Element for Beam Dynamic Analysis Based on Analytical Deflection Trial Function

^{1}College of Water Resources & Civil Engineering, China Agricultural University, Beijing 100083, China^{2}China Aerospace Construction Group Co., Ltd., Beijing 100071, China

Received 18 September 2014; Accepted 15 December 2014

Academic Editor: Chenfeng Li

Copyright © 2015 Qiongqiong Cao 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

For beam dynamic finite element analysis, according to differential equation of motion of beam with distributed mass, general analytical solution of displacement equation for the beam vibration is obtained. By applying displacement element construction principle, the general solution of displacement equation is conversed to the mode expressed by beam end displacements. And taking the mode as displacement trial function, element stiffness matrix and element mass matrix for beam flexural vibration and axial vibration are established, respectively, by applying principle of minimum potential energy. After accurate integral, explicit form of element matrix is obtained. The comparison results show that the series of relative error between the solution of analytical trial function element and theoretical solution is about and the accuracy and efficiency are superior to that of interpolation trial function element. The reason is that interpolation trial function cannot accurately simulate the displacement mode of vibrating beam. The accuracy of dynamic stiffness matrix method is almost identical with that of analytical trial function. But the application of dynamic stiffness matrix method in engineering is limited. The beam dynamic element obtained in this paper is analytical and accurate and can be applied in practice.

#### 1. Introduction

Dynamic structural analysis is essential in structure engineering design. It is especially important for large-scale structure in earthquake area, such as high-rise buildings, dam, hydropower station, oil pipelines, and gas pipelines. Dynamic problem of beam structure is common in engineering. Vibration happens at beam under earthquake, gas-liquid flows, and impact. Resonance occurs when frequency of external load is close to natural frequency of structure. It is a great threat for structure safety. Therefore, perfecting accuracy and efficiency of dynamic structural analysis is to ensure structure safety and reliability.

A large number of theories and methods for dynamic structural analysis have been suggested. The analysis methods include direct solving method, energy method, and numerical method. Clough and Penzien [1] have obtained analytical solution of displacement equation for flexural vibrating beam and presented the first three-order vibration mode and the corresponding frequency of cantilever beam and simply supported beam. Jin [2] provided analytical solution of Timoshenko beam clamped two ends and subjected to uniformly distributed load according to different jump condition. Guo et al. [3] established structure element property matrix by energy principle considering flexural and torsional deformation of T-beam, and the effect of bridge local member (such as diaphragm plate). Fang and Wang [4] attained beam natural vibration frequency by analyzing dynamic property of external prestressing beam using energy method, and the solution has a better match with numerical solution. Lou and Hong [5] derived the approximation analysis technique for dynamic characteristics of the prestressed beam by applying the mode perturbation method. Carrer et al. [6] analyzed the dynamic behavior of Timoshenko beam by using boundary element method. Wu [7] analyzed the dynamic behavior of two-dimension frame with stiffening bar randomly distributed by using elastic-rigid composite beam element. De Rosa et al. [8] studied the dynamic behavior of slender beam with concentrated mass at beam end, and numerical solution of frequency equation was obtained. Among them, numerical analysis represented by finite element is the main and efficient method for dynamic analysis (such as natural vibration analysis and forced vibration analysis). Finite element method was firstly proposed by Clough [9] in an article about plane elastic problem, and it is perfect in theory as a numerical method.

At present, the methods of constructing dynamic element for beam include dynamic stiffness matrix method, Galerkin method, Ritz method, energy variation method. Dynamic stiffness matrix can accurately solve differential equation of motion according to initial displacement field without any assumption, and then accurate results can be obtained irrespective of element number. This method was proposed by Koloušek [10]. To gain more accurate results, stiffness matrix of tensile torsion bar and Euler beam about frequency, that is, dynamic stiffness matrix, was firstly derived from analytical solution for studying vibration characteristics of plane truss. A lot of work on the research and development of dynamic stiffness matrix method was also done by Long and Bao [11], Hashemi and Richard [12], Chen et al. [13], Banerjee et al. [14], and Banerjee et al. [15]. Shavezipur and Hashemi [16] put forward an accurate finite element method. In this method, closed form solution of differential equation of beam not coupling flexural vibration torsional vibration was obtained by merging Galerkin weighted residual method and dynamic stiffness matrix (DSM). Result of dynamic stiffness matrix is more accurate, but analytical solution of differential equation cannot be derived when structure load or displacement boundary condition is too complex. Then dynamic stiffness matrix method is not suitable any more. According to principle of virtual displacement, a large number of research achievements on dynamic analysis of thin-walled open section beam, elastic foundation beam, and composite beam have been conducted by Chopra [17], Hu and Dai [18], Wang et al. [19], Hashemi and Richard [20], and Pagani et al. [21]. Nabi and Ganesan [22] put forward a finite element method based on free vibration analysis theory of composite beam with the first order shear deformation. Zhao and Chen [23] developed the dynamic analysis of a unified stochastic variational principle and the corresponding stochastic finite element method via the instantaneous minimum potential energy principle and the small parameter perturbation technique. On the basis of energy variation principle, Wang et al. [24] derived governing differential equation and natural boundary condition of dynamic response for I-shaped beam and obtained closed solution of the corresponding generalized dynamic displacement.

Accuracy and efficiency of beam element depend on beam displacement trial function by applying potential energy variation principle to constructing beam displacement element. For current beam element, cubic polynomial displacement mode is used to obtain a series of static and dynamic element widely applied to the software, such as ANSYS and NASTRAN. For dynamic analysis, vibrating beam displacement mode has a big difference from polynomial mode. Precision requirements cannot be met by taking polynomial function as vibrating beam displacement trail function. The basic analytical solution is used as the element trial function in analytical trial function method. Discrete finite element method takes advantage of analytical solution. It embodies the superiority of trial function using basic analytical solution.

This paper focuses on constructing element for beam dynamic analysis using analytical deflection trail function based on variational method of principle of minimum potential energy and displacement element construction theory. The fruits are useful to beam dynamic analysis.

#### 2. Displacement Trial Function for Beam Element

Selecting displacement trial function is one of the main contents of constructing displacement element. Appropriate displacement trial function should be in accordance with element deformation behavior and should be easy for integral of element energy functional. Element accuracy is determined by the accuracy of displacement trial function. The corresponding functional integral has a direct effect on element calculation efficiency and accuracy.

With regard to beam element, displacement mode based on interpolation function is used for element displacement trial function of all kinds of problems. Linear polynomial Lagrange interpolation function meeting the continuity condition at is used for axial displacement. Cubic polynomial Hermit interpolation function meeting the continuity condition at is used for flexural deformation.

Take flexural deformation, for example; for the static problems of uniform cross section beam subjected to uniform distributed load, is stiffness equilibrium equation about deflection* w*(*x*). Here, is beam section flexural stiffness and* q* is the uniform distributed load. The accurate solution of this equation is a cubic polynomial. Cubic polynomial Hermit interpolation function meeting the continuity condition of beam end displacement is actual displacement of element. And the corresponding potential energy functional has analytic, derivable, integrable solution. For this reason, static beam element derived from Hermit interpolation shape function is accurate element.

However, for dynamic and nonlinear straight beam and every nonlinear beam, deflection equation is not cubic polynomial due to the change of equilibrium differential equation mode. Therefore, element constructed by Hermit interpolation trial function is an approximate element. The key of constructing accurate element for all kinds of problems of beam is to search analytical trial function having functional integrability.

#### 3. Analytical Trial Function of Displacement for Vibrating Beam Element

The method to construct analytical trial function of displacement for vibrating beam element is as follows:(1)to deduce the general solution of displacement equation for beam vibration containing undetermined parameters according to differential equation of equilibrium for beam vibration;(2)to determine the undetermined parameters in displacement equation according to displacement condition of vibrating beam end;(3)to write out displacement equation for beam vibration expressed by beam end displacement, and then to obtain displacement trial function for vibrating beam.

To construct beam dynamic element, local coordinate as shown in Figure 1 is established. It is in accordance with general beam element. The positive direction of linear displacement of beam end and force is in accordance with coordinate direction, and the positive direction of rotation angle and moment is in accordance with clockwise direction.