A Finite Element Method to solve for the shape function, mass matrix and natural frequency of 4-noded rectangular element with free edges using matlab
Procedia Engineering 38 ( 2012 ) 2758 – 2766 1877-7058 © 2012 Published by Elsevier Ltd. doi: 10.1016/j.proeng.2012.06.323 * Corresponding author. Tel.: +91-9692146667 E-mail address:511me109@nitrkl.ac.in. ICMOC Study on Free Vibration Analysis of Rectangular Plate Structures Using Finite Element Method Ramu Ia,*, S.C. Mohantyb a,Research scholar, Department of Medchanical Engineering, National Institute of Technology, Rourkela-769008, India b Associate Prof., Department of Mechanical Engineering, National Institute of Technology, Rourkela-769008, India Abstract The present research work aims to determine the natural frequencies of an isotropic thin plate using Finite element method. The calculated frequencies have been compared with those obtained from exact Levy type solution. Based on this Kirchhoff plate theory, the stiffness and mass matrices are calculated using Finite Element Method (FEM). This methodology is useful for obtaining the natural frequencies of the considered rectangular plate. Numerical results obtained from FEM of the simply supported rectangular plates are giving close agreement with the exact solutions results. More parameter. Key words: Rectangular plate, Kirchhoff plate theory, Finite Element Method, Natural Frequencies, Mode Shape 1. Introduction Different methods have been developed for performing static and dynamic analysis of plate like structures. In case of complicated shapes generally it is difficult to obtain an accurate analytical solution for structures with different sizes, various loads, and different material properties. Consequently, we need to apply on approximate numerical methods for obtaining appropriate solutions of static and dynamic problems. The finite element method (FEM) is widely used and powerful numerical approximate method. The finite element method involves modeling the structure using small inter connected elements called finite elements. A displacement function is associated with each finite element. Every interconnected element is linked, directly or indirectly, to every other element through common interfaces, including nodes or boundary lines. From stress/strain properties of the material making up the structure, one can determine the behaviour of a given node in terms of the properties of every other element in the structure. The total equations describing the behaviour or each node results in a series of algebraic equations best expressed in matrix notation. The finite element method of structural analysis enables the designer to found stress, vibration and thermal effects during the design process and to evaluate design changes before the construction of a possible prototype. Thus assurance in the acceptability of the prototype is improved. © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Noorul Islam Centre for Higher Education. Available online at www.sciencedirect.com Open access under CC BY-NC-ND license. Open access under CC BY-NC-ND license. 2759 I. Ramu and S.C. Mohanty / Procedia Engineering 38 ( 2012 ) 2758 – 2766 In the past some of the researchers have studied the free vibration analysis of plates. Tanaka et.al.[7]studied the integral equation approach for free vibration problems of elastic plate structures. In their approach the boundary integral equation method was used to determine the eigen frequency by means of direct search of the zero-determinant value of the system matrix. This new integral equation approach and its solution give an approximate fundamental solution to the static problem. Karunasena and Kitipornchai [1] determined the free vibration analysis of shear deformable triangular plate element. Wu and Liu [8] have been studied the new numerical solution technique called as differential cubature method for free vibration analysis of arbitrary shaped thick plates. In their approach a linear differential operation such as a continuous function, as a weighted linear sum of discrete function values chosen within the overall domain of a problem. Later researchers have been developed the new methods for free and forced vibration analysis of plate structures. Moon and Choi [4] have formulated the Transfer Dynamic Stiffness Coefficient method for Vibration Analysis of Frame Structures. They developed the concept based on the transfer of the dynamic stiffness coefficient which is related to the force and displacement vector at each node from the left end to the right end of the structure. Myung [5] has developed the Finite Element-Transfer Stiffness Coefficient Method for free vibration analysis of plate structures. His approach is based on the combination of the modelling techniques in FEM and the transfer technique of the stiffness coefficient in the transfer stiffness coefficient method. Liew et.al.[2]have investigated a mesh-free Galerkin method for free vibration analysis of unstiffened and stiffened corrugated plates. Their analysis carried on the stiffened corrugated plates, treated as composite structures of equivalent orthotropic plates and beams, and the strain energies of the plates and beams are added up by the imposition of displacement compatible conditions between the plate and the beams. The stiffness matrix of the whole structure was derived. Lu et.al.[3]have investigated the differential quadrature method for free vibration analysis of rectangular Kirchhoff plates with different boundary conditions. In their analysis the differential quadrature procedure was applied in the direction of line supports, while exact solution was sought in the transfer domain perpendicular to the line supports using the state space method. The present work deals with the Levy type exact solution in order to obtain the natural frequencies of a simply supported rectangular plate. Here, Kirchhoff plate theory is used for finite element analysis. By using the finite element method stiffness and mass matrices were determined. These matrixes are used to calculate the natural frequencies of rectangular plate by solving the eigen value problem. Numerical results of the simply supported rectangular plates showed that this present method can be successfully applied to the free vibration analysis of any thin rectangular plate structure. In the case of varying thickness of the rectangular plate structures natural frequencies parameter error varies and it is constant with increase the plate thickness. Nomenclature U displacement in x direction V displacement in y direction z, h plate thickness along the z direction w displacement is the function of x and y Rotation about x axis 2760 I. Ramu and S.C. Mohanty / Procedia Engineering 38 ( 2012 ) 2758 – 2766 Rotation about y axis 1st order Partial differential equations w. r t. x and y Rate of change of the angular displacements Bending and shear stresses Normal and shear strains G Modulus of rigidity Poisons ratio , and Bending moments acting along edge of plate in x and y direction Flexural rigidity of plate q Transverse distributed load , Transverse shear loads Rotations along x and y directions Local coordinates along x and y direction Vector constants Shape function A length of plate B width of plate Element stiffness matrices Element mass matrices Density of the material Global stiffness matrix Global mass matrix Natural frequencies 2761 I. Ramu and S.C. Mohanty / Procedia Engineering 38 ( 2012 ) 2758 – 2766 2. Methodology 2.1. Kirchhoff plate theory Kirchhoff plate theory [6] makes it easy to drive the basic equations for thin plates. The plate can be considered by planes perpendicular to the x axis as shown in the fig.1, to drive the governing equation. Based on Kirchhoff assumptions, at any point P, due to a small rotation displacement in the x direction U= 2.1 displacement in the y direction At the same point, the displacement in the y direction is: V= = 2.2 The curvatures (rate of change of the angular displacements) of the plate are: 2.3a 2.3b and 2.3c Using the definitions of in-plane strains, the in-plane strain/ displacement equations are: x= 2.4a y= 2.4b and xy= 2.4c The two points of equation 2.1&2.2 are used in beam theory. The remaining two equations are new to plate theory. According to Kirchhoff theory, the plane stress equations for an isotropic material are: 2.5a 2.5b 2.5c Where ) The in-plane normal stresses and shear stress are acting on the edges of the plate as shown in the fig.2. The stresses are varying linearly in the Z-direction from the mid surface of the plate. Although transverse shear deformation is neglected, transverse shear stresses yz and are present. Through the plate thickness, these stresses are varying quadratically. Fig 1 shear and bending stress in plane normal The bending moments acting along the edge of the plate related to the stresses as follows: 2762 I. Ramu and S.C. Mohanty / Procedia Engineering 38 ( 2012 ) 2758 – 2766 2.6 Substituting strains for stresses gives: 2.7 Using the curvature relationships, the moments become: 2.8 Where is called the bending rigidity of the plate. The governing differential equations are: 2.9 Where q is the transverse distributed loading And and are the transverse shear loads as shown in Fig.3 Fig 3 Transverse distributed load Substituting the moment/curvature expressions in the last equation 2.9 list above solving for and , and substituting the results into the equation 2.7 listed above, the final form of governing partial differential equation for isotropic thin-plate in bending is: 2.10 2763 I. Ramu and S.C. Mohanty / Procedia Engineering 38 ( 2012 ) 2758 – 2766 From Eq.(2.10),The solution of thin-plate bending is a function of the transverse displacement w. 2.2 FEA formulation for 4-noded rectangular element Rectangular four node element is having one node at each corner as shown in Fig.4. There are three degrees of freedom at each node, the displacement component along the thickness (w), and two rotations along X and Y directions in terms of the ( , ) coordinates: ; 2.11 Therefore the element has twelve degrees of freedom and the displacement function of the element can be represented by a polynomial having twelve terms as shown in (3.2) 2.12 This function is a complete cubic to which have been added two quadratic terms and which are symmetrically placed in Pascal's triangle. This will ensure that the element is geometrically invariant. Fig.4 Geometry of the rectangular element 2.13 2.14 2.15 Substituting (3.4) and (3.5) becomes (3.6) 2.16 2.17 2.18 In deriving this result, it is simpler to use the expression Eq. (2.18) for w and substitute for {a} after performing the integration. A typical integral is then is the element stiffness matrix, and 2.19 2764 I. Ramu and S.C. Mohanty / Procedia Engineering 38 ( 2012 ) 2758 – 2766 2.20