Mathematical Problems in Engineering

Volume 2017, Article ID 6492081, 33 pages

https://doi.org/10.1155/2017/6492081

## A Refined Higher-Order Hybrid Stress Quadrilateral Element for Free Vibration and Buckling Analyses of Reissner-Mindlin Plates

^{1}School of Science, Yanshan University, Qinhuangdao 066004, China^{2}Key Laboratory of Liaoning Province for Composite Structural Analysis of Aerocraft and Simulation, Shenyang Aerospace University, Shenyang 110136, China

Correspondence should be addressed to Xu Ma; nc.ude.usy@428uxam

Received 3 July 2017; Revised 20 September 2017; Accepted 6 November 2017; Published 27 November 2017

Academic Editor: Giovanni Garcea

Copyright © 2017 Tan Li 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

This paper is devoted to develop a new 8-node higher-order hybrid stress element (QH8) for free vibration and buckling analysis based on the Mindlin/Reissner plate theory. In particular, a simple explicit expression of a refine method with an adjustable constant is introduced to improve the accuracy of the analysis. A combined mass matrix for natural frequency analysis and a combined geometric stiffness matrix for buckling analysis are obtained using the refined method. It is noted that numerical examples are presented to show the validity and efficiency of the present element for free vibration and buckling analysis of plates. Furthermore, satisfactory accuracy for thin and moderately thick plates is obtained and it is free from shear locking for thin plate analysis and can pass the nonzero shear stress patch test.

#### 1. Introduction

It is well-known that the finite element method for free vibration and buckling analysis of plates is highly significant in civil, mechanical, and aerospace engineering applications. The patch test [1], which can be used to examine the convergence of the element and construct a convergence element, has been seen as a criterion for assessing the convergence of the finite element for a long time.

The potential energy function of Mindlin plate element contains the displacements and the first-order derivatives of the displacements. According to the continuity condition, it is quite easy to establish interpolation functions of deflection and rotation. Historically, the displacement-based approach was the first attempt in the formulation of effective Mindlin plate bending elements [2]. However, as we all know, the original displacement element tends to cause the shear-locking phenomenon which induces overstiffness as the plate becomes progressively thinner for low-order interpolation polynomials in the Mindlin elements. During this time, the convergence characteristics of the Mindlin plate elements were performed by means of numerical computation of pure bending and pure torsion [3, 4]. In order to avoid shear locking, various numerical techniques and effective modifications have been proposed and tested, such as the reduced integration and selective reduced integration schemes proposed by Zienkiewicz et al. [5], Pugh et al. [6], and Hughes et al. [7]; assumed natural strain method introduced by Hughes and Tezduyar [8]. It is acknowledged that the methods of reduced integration and selective integration are efficient approaches to prevent the appearance of the shear-locking phenomenon. However, it is found that such elements often exhibit extra zero-energy modes and also produce oscillatory results for some problems. Moreover, these solutions are not applicable to very thin plate; the thickness/span ratio of the plate is about restricted to .

Later, Belytschko et al. [9] proposed the stabilization procedure to remove the zero-energy modes by perturbing the stiffness. In addition, several efficient 9-DOF triangular elements based on the discrete Kirchhoff constraint and the equilibrium conditions were developed by Batoz et al. [10, 11]. These elements can eliminate locking phenomenon and converge towards the discrete Kirchhoff plate bending elements when the thickness of the plate is very thin. On the other hand, no element is free of shear locking in theory. Bathe et al. [12] introduced the MITC element and proposed strain energy patch test function to evaluate the convergence. Based on Timoshenko beam function, Soh et al. [13] proposed a triangular 9-DOF plate bending element which can be employed to analyze very thin plate (the thickness/span ratio of the plate is about ). Soon, Soh et al. [14] introduced a quadrilateral 12-DOF plate bending element. At this time, the progressively thinner plate which has the thickness/span ratio can be calculated. Wanji and Cheung [15] proposed the zero shear stress patch test functions. It is apparent that this patch test is more rigorous than the patch test using numerical computation of pure bending and pure torsion of a small-scale plate. Then, elements such as RDKQM [15], RDKTM [16], AC-MQ4 [17], and QC-P4 [18] that can pass the above patch test functions were proposed, indicating that the shear-locking problem is solved. All these elements can be used to solve the extremely thin plate problem (the thickness/span ratio of the plate can reach to ). In other words, these elements can accurately converge to thin plate finite element solution.

Chen proposed the enhanced patch test [19] and presented the zero shear deformation patch test and nonzero constant shear deformation test functions of Mindlin plate [20]. Current patch test for Mindlin plate only satisfies the zero shear deformation condition. The patch test of nonzero constant shear for Mindlin plate problem cannot be performed. The convergence test should be performed during the process of developing finite element method. Only passing the rigorous nonzero constant shear stress patch test, the convergence can be completely guaranteed. The programs of this commercial software have no proof of convergence. The enhanced patch test is stronger than the original test; the original constant stress patch test is just a special case of it. This paper is devoted to establish Mindlin plate element which can pass the strict constant shear patch test.

Different from the classical Timoshenko beam function, Jelenić and Papa [21] proposed a new arbitrary-order Timoshenko beam function in 2011. So far, it is the only function which can be used to construct the functions of nonzero constant shear patch test for thick beam element. Since beam function can be regarded as a function on the boundary, the adopted hybrid stress method just requires the boundary function rather compared to the domain function. Because this beam function is arbitrary order, thus it has high enough order to perform the nonzero constant shear stress patch test. Since a complete cubic polynomial for the element function to pass the constant shear stress patch test is required, it was used to develop the higher-order hybrid stress triangular Mindlin plate bending element named TH6 [22] and quadrilateral Mindlin plate bending element named QH8 [23]. The results of static analysis have proved that the TH6 element and QH8 element can pass the rigorous nonzero constant shear stress patch test and its accuracy is quite high. Only passing the rigorous nonzero constant shear stress patch test, the convergence can be completely guaranteed.

The purpose of this paper is to develop an 8-node Mindlin plate bending finite element for free vibration analysis and buckling analysis within an assumed stress formulation, whose main feature is that passing the rigorous nonzero constant shear stress enhanced patch test. To achieve this objective, the following steps have been taken. The first step concerns the choice of the variational framework with the adoption of complementary energy principle. Then boundary displacement interpolation function is established based on the new arbitrary-order Timoshenko beam function. Since the choice of the stress approximation is a crucial issue in developing reliable hybrid finite element, selecting a suitable stress approximation which satisfies the plate equilibrium equations is not trivial. In order to improve the performance of the constructed element, a refined mass matrix for calculation of the natural frequency and a refined geometric stiffness matrix for buckling analysis are developed by using refined element method [24–26].

#### 2. A Brief Introduction of the Higher-Order Hybrid Stress Quadrilateral Mindlin Plate Bending Element

##### 2.1. Fundamental Equations of Mindlin Plate

Consider a plate referred to as a Cartesian coordinate frame , with the origin on the mid-surface and the -axis in the thickness direction, , where is the plate thickness. Let be the boundary of . The Reissner-Mindlin theory, that is, the first-order shear deformable theory, is employed. Thus it is assumed that where , , are displacements along the -, -, and -axes, respectively, , are the rotations of the transverse normal about the - and -axes, and is the transverse displacement field.

The geometric equations can be written as follows:where , , and are, respectively, the rotations, the curvatures, and the shear strains: and operators , and are given by the following:

The equilibrium equations can be obtained from the strain energy in the following form:where vectors and are, respectively, the moment and shear resultants:

The boundary forces can be written as follows:where is the angle between the normal of edge and the local -axis of element.

For a linearly elastic material, the constitutive equations can be written as follows:where and are the elasticity matrices of bending and transverse shear moduli. In the isotropic case, the elasticity matrices specialize aswhere is Young’s modulus, the shear modulus, Poisson’s ration, and a correction factor to account for nonuniform distribution of shear stresses through the thickness.

##### 2.2. Hybrid Stress Formulation

Based on a modified complementary energy principle, the assumed stress hybrid formulation pioneered by Pian [28, 29] can be used to avoid the difficulty of forming the displacement field interpolation functions, in particular, after the work of Malkus and Hughes [30] on the equivalence between reduced integration displacements and mixed/hybrid stress models. This kind of approach based on hybrid stress element method became very useful in recent years [31–36]. A good number of effective elements which are free from shear locking have been developed by authors such as Tong [37], Bathe and Dvorkin [38], Ayad et al. [39], Brasile [40], and Li et al. [22, 23]. The higher-order hybrid stress quadrilateral Mindlin plate bending element QH8 is based on complementary energy principle. The complementary energy principle can be written aswhere is the stress vector, is the elasticity matrices, is the vector of boundary force, is the boundary displacement vector, is the transverse displacement, and , are the rotations of the transverse normal about the - and -axes.

The approximation for stress and boundary displacements can now be incorporated in the functional. The stress field is described in the interior of the element as follows:where is matrix of stress interpolation functions and is the unknown stress parameters.

The boundary force can be represented as follows:where is the combination of direction cosine for the boundary normal.

The boundary displacement field is described bywhere are interpolation functions and is nodal displacement parameters.

Substituting the stress equation (11), boundary force equation (12), and displacement approximations equation (13) into the functional (10),whereThe form of (15) is directly amenable to numerical integration (i.e., Gauss quadrature).

Then the internal strain energy can be expressed as follows:

By means of , we obtainedConsequently,

Substitution of in (16), the internal strain energy reduces toCompared with , the element stiffness matrix can be taken as

The solution of the system yields the unknown nodal displacement . After is determined, element stress or internal forces can be recovered by use of (18) and (11). Thus

##### 2.3. The Displacement Interpolation Function of QH8 Element

Euler-Bernoulli beam function has been successfully employed in the construction of refined thin plate elements. It is well-known that when constructing Mindlin plate element, both thick and thin plates should be taken into account, and it is necessary to eliminate shear-locking phenomenon. To seek out such element displacement function is definitely very difficult. Note that a closed form solution for both thick and thin beams exists in the form of the Timoshenko beam function, and it is possible to use it to derive more efficient Mindlin plate elements [41]. However, the use of Timoshenko beam function is capable of solving the problem of shear locking; it cannot solve the problem of passing the nonzero constant shear patch test. This problem has not resolved for many years. In 2011, Jelenić and Papa [21] presented a new arbitrary-order Timoshenko beam function as follows:where is the beam length, and are the values of the displacements and the rotations at the nodes equidistantly spaced between the beam ends, are the standard Lagrange polynomials of order , and for and otherwise, in which is the coordinate along the beam.

An 8-node quadrilateral element was designed as given in Figure 1. If any quadrilateral side is taken as a beam element, take the 1-2 boundary as example and the deflection and rotations , can be derived as follows:where , , , , , , is the length of 1-2 boundary and is the coordinate along the 1-2 edge.