Procedia Engineering XXXXXXXXXX2758 – 2766
XXXXXXXXXX © 2012 Published by Elsevier Ltd.
doi: XXXXXXXXXX/j.proeng XXXXXXXXXX
* Co
esponding author. Tel.: XXXXXXXXXX
E-mail address: XXXXXXXXXX.
ICMOC
Study on Free Vi
ation 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
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
esults 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 alge
aic equations best
expressed in matrix notation. The finite element method of structural analysis enables the designer to
found stress, vi
ation and thermal effects during the design process and to evaluate design changes
efore 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.
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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 XXXXXXXXXX2758 – 2766
In the past some of the researchers have studied the free vi
ation analysis of plates. Tanaka
et.al.[7]studied the integral equation approach for free vi
ation 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 vi
ation 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 vi
ation analysis of a
itrary 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 vi
ation analysis of
plate structures. Moon and Choi [4] have formulated the Transfer Dynamic Stiffness Coefficient method
for Vi
ation 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 vi
ation 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 vi
ation analysis of unstiffened and stiffened co
ugated plates. Their analysis
ca
ied on the stiffened co
ugated 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 vi
ation
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 vi
ation analysis of any thin rectangular plate structure. In the case of
varying thickness of the rectangular plate structures natural frequencies parameter e
or 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 XXXXXXXXXX2758 – 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 XXXXXXXXXX2758 – 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= XXXXXXXXXX
displacement in the y direction
At the same point, the displacement in the y direction is:
V= = XXXXXXXXXX
The curvatures (rate of change of the angular displacements) of the plate are:
2.3a XXXXXXXXXX3b
and XXXXXXXXXX3c
Using the definitions of in-plane strains, the in-plane strain/ displacement equations are:
x= 2.4a y= XXXXXXXXXX4b
and xy= XXXXXXXXXX4c
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
XXXXXXXXXX2.5c
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 XXXXXXXXXX2758 – 2766
XXXXXXXXXX
Substituting strains for stresses gives:
XXXXXXXXXX
Using the curvature relationships, the moments become:
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Where is called the bending rigidity of the plate.
The governing differential equations are:
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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:
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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:
; XXXXXXXXXX2.11
Therefore the element has twelve degrees of freedom and the displacement function of the element can be
epresented by a polynomial having twelve terms as shown in (3.2)
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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
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Substituting (3.4) and (3.5) becomes (3.6)
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In deriving this result, it is simpler to use the expression Eq XXXXXXXXXXfor w and substitute for {a} after
performing the integration. A typical integral is then is the element stiffness matrix, and
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