Applied Mathematical Sciences, Vol. 9, 2015, no. 35, 1697 - 1719 HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.5182

Accurate Approximate and Analytical

Methods for Vibration and Bending

Problems of Plates: A Literature Survey

Thongperm Yoowattana1, Patiphan Chantarawichit1

Jakarin Vibooljak2 and Yos Sompornjaroensuk1,*

1 Department of Civil Engineering Mahanakorn University of Technology

Bangkok, 10530, Thailand * Correspondence author

2 Vibool Choke Ltd., Part Bangkok, 10110, Thailand

Copyright © 2015 T. Yoowattana 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

Three types of both accurate approximate and analytical solution methods, namely, the Rayleigh-Ritz method, the superposition method, and the integral transform method, that have been used with considerable success to solve many problems of plate having stress singularities either vibration or static bending problems are dealt with in this paper with emphasis on explaining the details of each method. In additions, some problems of plate that have been solved efficiently by these mentioned methods are also explored and reviewed.

Mathematics Subject Classification: 35Q74, 44A05, 65N30, 74H45, 74K20

Keywords: Analytical method, Approximate method, Integral transform method, Plate, Rayleigh-Ritz method, Static bending, Stress singularities, Superposition method, Vibration.

1698 Thongperm Yoowattana et al. 1 Introduction In solving the differential equation of plates, it is naturally desirable to obtain the exact solutions. However, the exact solutions are found only in a limited number of problems so that many research works on plate analysis have been conducted using a wide range of methods. There are two main types of solution that can be classified herein, namely, approximate solution and analytical solution which depended upon the methods used in the analysis. Analytical solutions are the most desirable, but not always easily attainable, as in the case of the plates with mixed boundary conditions (discontinuities in the edge support) [3-5,9,21,24-28,45,52-58,63] or plates with irregular boundaries [22,23,33,36-39,44,61]. This is due to the difficulty lied in the requirements to satisfy both, the governing differential equation and all of the boundary conditions exactly [59]. Approximate mathematical techniques have been developed to remedy the situation. Concerning bending problems of plates, a valuable study was made by Leissa et al.[32], to compare approximate methods.

Generally, the approximate methods can further be categorized into continuum types like the Rayleigh-Ritz method and the Galerkin method, and discrete types like the finite difference method, finite element method, boundary element method, and recently, the meshfree method. Some of them are, however, basically discussed herein depending upon whether the solutions satisfy the differential equation or the boundary conditions.

The Rayleigh-Ritz method [31] is an energy method that makes use of the energy concept by determining the maximum strain energy and maximum kinetic energy (only for the vibration problems) of the system and then, minimizing these energy terms to obtain the condition of equilibrium equations. An inherent advantage of this method is that no need to solve the governing differential equation, only the suitable series of functions are required to be chosen in satisfying exactly the prescribed geometric boundary conditions of the problem. Forced or natural boundary conditions may be ignored. In the Galerkin method, it only requires that the residual of the differential equation be orthogonal to each term of the series that satisfy the boundary conditions.

The finite element method is one of the powerful methods and has advanced with the development of computers. It has been very successful in finding the solution of various problems, particularly its applicability to problems having irregularly shaped boundaries, which involved in discretizing the numerical solution domain. However, incorrect results may be obtained if the element-type is poorly chosen to model and the number of elements is not enough to represent the overall domain properly.

The boundary element method is another numerical method which makes use of the fundamental solution of problem considered together with the reciprocal theorems. Its characteristic feature is that only the boundary of the solution domain is needed to be modeled. This leads to a relatively smaller set of resulting algebraic equations when compared to the finite element method, but the mathematical formulation of problems is more difficult than those of finite element method. Nevertheless, finite

Accurate approximate and analytical methods 1699 element method has become the first choice for many researchers, while the Rayleigh- Ritz method is still popular.

It is obvious that in the last five decades, there are many other approximate methods employed to analyze the plate problems. Most approximate solution methods appear to be limited to special problems. These solutions have a place as a check for the finite element method. However, it should not be forgotten that they are still only approximate in nature because the solutions do not satisfy exactly the governing differential equation, the all prescribed boundary conditions, or even both. With this reason, analytically oriented methods are still of importance to verify the results obtained from those approximate methods and they are good means to yield accurate numerical results since the solutions satisfy both the governing equation exactly and the boundary conditions with an arbitrary degree of exactitude. Two types of analytical method will be presented and described in details in the subsequent sections.

2 The Rayleigh-Ritz method Because the Rayleigh-Ritz method is one of the most effective approximate methods to obtain a highly accurate solution when a more enough terms of appropriate admissible functions being used, only the published papers relevant to this method that found in the scientific or technical literature will then be focused and explored in this section.

The Rayleigh method was first introduced by Lord Rayleigh in 1877 for solving the vibration problems to obtain the fundamental natural frequency of continuum systems by assuming the mode shape, and setting the maximum values of potential and kinetic energy in a cycle of motion equal to each other. In 1908, W. Ritz des- cribed his method for solving the boundary value problems, and the eigenvalue problems to determine the frequencies and their corresponding mode shapes by choosing multiple admissible displacement functions, and minimizing a functional involving both potential and kinetic energies [31]. Nevertheless, many research articles and many books have appeared to use the terms either Ritz method or Rayleigh- Ritz method in their works.

Before proceeding further to the detailed explanation of the Rayleigh-Ritz method, the governing biharmonic differential equation for the static bending problem of isotropic rectangular thin plates with uniform thickness (h) in terms of deflection function W(x,y) and subjected to a distributed load Q(x,y) can be given in form as

4 4 4

4 2 2 4

( , ) ( , ) ( , ) ( , )2

W x y W x y W x y Q x y

x x y y D

, (1)

and

3

212(1 )

EhD

, (2)

in which D is the plate rigidity, E and υ are the Young’s modulus and Poisson’s ratio

1700 Thongperm Yoowattana et al. of the plate, respectively [59].

For the strain energy (U) of the plate, it can be expressed by [59]

2 2

222 2

( , ) ( , )U ( , ) 2(1 )

2 A

D W x y W x yW x y

x y

22 ( , )W x y

dxdyx y

, (3)

whereas A is the total area of the plate and 2 is the Laplacian operator. The work done (W) that produced by the external load is

1W ( , ) ( , )

A

Q x y W x y dxdyD

. (4)

Thus, the corresponding total potential energy or energy functional (Π) can be defined by U W . (5)

In the case of free vibration problem of plates, the kinetic energy (T) has the form as given in Leissa [29],

2

2T [ ( , )]2 A

hW x y dxdy

, (6)

where ρ is the density of mass per unit area of the plate, and ω is the circular frequency of plate vibration.

Therefore, the energy functional for free vibration of the plates can be defined by the sum of expressions given in (3) and (6),

U T . (7)

Since the Rayleigh-Ritz method is applied, the deflection function W(x,y) can be chosen in terms of admissible function ψij(x,y) satisfying boundary conditions of the plate as

1 1

( , ) ( , )I J

ij iji j

W x y a x y

, (8)

where I, J are the highest integer numbers to be used in approximation of W(x,y), aij is defined to be the adjustable arbitrary coefficients, and the function ψij(x,y) may be assumed to be the algebraic polynomials or trigonometric functions satisfying essential (geometric) boundary conditions (i.e., deflection or slope) of the plate.

When the admissible functions ψij(x,y) cannot be satisfied in some points of

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Accurate approximate and analytical methods 1705

Huang et al.[22] obtained the accurate natural frequencies of rectangular plates having V-notches along their edges (see Figure 8) by using two sets of algebraic polynomials from a mathematically complete set of functions and corner functions as the admissible displacement functions. The results were demonstrated for free, square plates with a single V-notch only. Additionally, Huang et al.[23] extended the previous work [22] to obtain the results for the case of cantilevered rectangular plates with an edge V-notch as depicted in Figure 9, and further investigated the cases of V-notch angle equal to zero (slit). 3 The superposition method The present and the next sections aim to exhaustively propose the two efficient analytical solution methods for investigating the problem of plates with various boundary conditions because the plate problems have been analyzed and successfully solved using the advantages of these two methods. The first is referred to the building block superposition method [12] and the second is the integral transform method [51]. For the method of superposition, it has a little limitation on applications that the plates must have straight edges only. This is due to the solution of problems obtained by means of the Levy-type solution [59]. In the case of integral transform method, it can be applied to solve the problem of rectangular or circular plates.

Among the methods utilized for plate analysis in the past, one of them is known to be the superposition method developed by D.J. Gorman for analyzing free vibration of rectangular plates with numerous different types of boundary conditions and point supports. This powerful method gives the best values for the natural frequencies of plates having various aspect ratios when compared with other computing methods. It can, however, be noted that the method of superposition itself is not new, in fact it has long been in use for solving static plate problems for more decades.

The superposition method has been successfully applied for analysis of vibrations of plates having classical boundary conditions, such as, clamped support [10,35,46-48], free support [11,40] and simple support [34], and more complicated systems, such as elastic support [14,15], orthotropic [13,16,17,41,64], and shear deformable plates [42].

In many cases the results from the superposition method are the benchmarks for the natural frequencies because it is very efficient and accurate for a range of geometric shapes. However, it is interesting to note that the solutions obtained by this method would give an upper bound or a lower bound result depending on whether the boundary conditions of the building blocks (separate plate problems) are stiffer or more flexible than those of the actual system.

Based on the method mentioned in paragraph above, solutions always satisfy the governing differential equation and all boundary conditions. If the solutions are possible to sum infinity terms then the solutions would be exact. Therefore, the superposition method is an analytical-type method of solution because it is closed form as an infinite series [12]. Although it is of course necessary to truncate the series, it can be done at any highest desired degree of accuracy with the high speed electronic

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Accurate approximate and analytical methods 1709 point supports symmetrically placed along the diagonals were given. However, a large number of plate aspect ratios and point support locations were also demonstrated in Saliba [47] together with the experimental results. It can be observed from this work that data obtained from the testing of plates show excellent agreement between the experimental and theoretical results where the maximum deviation is seen to be of the order of 4% with a mean deviation of about 1%. 4 The integral transform method Another analytical method that will be stated is the method of integral transforms. Various types of integral transform method have been used successfully to solve many kinds of mixed boundary value problems in engineering and in mathematical physics.

By the integral transform method, most problems are usually reduced to the final equation form of integral equations of special type [49] depending on the integral transform techniques used [51] and some of their solutions have already been provided in Polyanin and Manzhirov [43]. If the closed-form solutions of integral equation are not possible to be determined due to the complexity of the kernel then the numerical treatments would be made there [1,2].

A more comprehensive presentation of mixed boundary value problems using the method of integral transforms can be found in the textbook of Sneddon [50]. However, it is remarkable that the main task for obtaining the final integral equation governing the problem considered instead of the differential equation, some certain identities involving the integral representations are needed to be employed. These may necessarily be consulted with the help of handbook of Gradshteyn and Ryzhik [19].

It is generally well-known in the plate problems that the moment singularities existed in the problem of plates with mixed boundary conditions are in the order of an inverse-square-root type at the transition points of discontinuous support [61-62]. Moreover, the necessity of singularity considerations in the static bending problems of plates with mixed boundary conditions were given by Leissa et al.[32], while the eigenvalue problems (free vibration and buckling) were discussed by Leissa [30]. In their works, it is concluded that the accurate solutions cannot be achieved without the use of appropriate singularity functions. With this conclusion, one of the advantages of the integral transform method is that it can be coped with the singularity at the points of discontinuity of the boundary conditions.

Therefore, a brief review of important works that pertained to solve the mixed boundary value problems of plates using the method of integral transforms are listed in the following paragraphs.

Yang [63] dealt with the static bending problem of a rectangular plate with an internal line support in which the plate configuration is illustrated in Figure 16. The problem was reduced to the solution of a singular integral equation governing the pressure distribution along the internal line support by means of the Hilbert transform techniques. Based on a technique proposed by Westmann and Yang [60], the problem

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1712 Thongperm Yoowattana et al. having mixed boundary conditions around all edges, was analyzed by Kiattikomol et al. [28] and Thonchangya [58]. In the recent years, Sompornjaroensuk and Kiattikomol [52,54,55] demonstrated the rigorous treatment of uniformly loaded rectangular plates with a partial internal line support. The deformations and stress resultants of the plates including some important physical quantities, namely, the strain energy and the stress intensity factors were presented in the form of graphs and tables.

In the field of contact problems of plates [20], the advancing contact between a plate and sagged supports under a uniformly distributed load was first presented by Dundurs et al.[9] using the Hankel integral transform method. It is worth noticing that the shear distribution along the line of sagged supports was singular in the order of square root type corresponding to the nature of contact problems [6-8]. In addition, the extent of contact was increased with increasing the loading and the support reaction was not proportional to the applied external load.

The receding contact at the vicinity of a right-angle corner of semi-infinite plate with unilaterally constrained was formulated and solved by Keer and Mak [24]. The finite Fourier integral transform was used through the dual-series equations and reduced to the singular integral equation. For the case of finite plates, Dempsey et al. [5] studied the receding contact problem of square plate using the method similar to that of Keer and Mak [24]. Moreover, Dempsey and Li [3,4] further investigated the problems of rectangular plate with sagged and no sag supports. The mixed boundary conditions were formulated as a coupled system of dual-series equations and then, reduced to a system of two simultaneously Cauchy singular integral equations. Recently, a similar problem with respect to the previous work that made by Dempsey et al.[5] was reexamined by Sompornjaroensuk and Kiattikomol [53] using the method of Hankel integral transforms together with introducing an inverse-square-root shear singularity at the tips of contact interval between the plate and the unilateral supports [6]. The governing equation was finally reduced to an inhomogeneous Fredholm integral equation of the second kind. Furthermore, the obtained results were in agreement with the previous work, and also well-compared to other numerical methods which are the finite element method [45] and boundary element method [21].

Additionally, it may also be expected, by observing the previous works, that the final form of equations governing to the problem of rectangular plate having mixed boundary conditions in two directions would be a coupled system of integral equations similar to those treated by Keer and Stahl [25], Keer and Mak [24], Dempsey and Li [3], and Thonchangya [58]. Since the Hankel integral transforms are selected to be applied in the static bending problem of rectangular plates, the integral equations are in the form of inhomogeneous Fredholm-type of the second kind.

1

1 11 1 12 2 1

0

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and

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( ) [ ( , ) ( ) ( , ) ( )] ( )K r r K r r dr f , (13)

where ρ is an independent variable and 0 ≤ ρ ≤ 1, r is a dummy variable, Φ1(ρ) and

Accurate approximate and analytical methods 1713 Φ2(ρ) are the unknown auxiliary functions to be determined, and introduced in the Hankel integral transform techniques which related to the unknown constants of integration of plate’s differential equation, f1(ρ) and f2(ρ) are known functions that involved to the load applied on the plate, and K11(ρ,r) through K22(ρ,r) are the kernels of integral equation.

5 Summary In closing, the efficient accurate methods which are the Rayleigh-Ritz method, the superposition method, and the integral transform method are presented and reviewed for solving the eigenvalue and boundary value problems of plate having stress singu- larities due to geometry or boundary condition. Some important relevant research works that have been treated successfully for this class of mentioned problems are also explored and then exhaustively explained in the present paper.

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