Dynamic stiffness formulation for isotropic and orthotropic plates with point nodes

Highlights

• The dynamic stiffness theory for point noded plates is developed for the first time.

• Discrete Fourier transform is applied in a unique way to develop the theory.

• Usefulness and accuracy of the theory in free vibration of plates are demonstrated.

• The new theory permitted the application of non-classical boundary conditions.

• Computational method for free vibration analysis of plate is significantly advanced.

Abstract

In this paper, the dynamic stiffness method for isotropic and orthotropic rectangular plates with point nodes is developed, making it possible to integrate the dynamic stiffness properties for plates with the dynamic stiffness properties of other elements such as bars and beams, but importantly, the advanced theory allows amalgamation of the dynamic stiffness method with the conventional finite element method for the first time. The derivation of the dynamic stiffness matrices for isotropic and orthotropic plates with point nodes has been accomplished by implementing the Fourier coefficients of the boundary values of the amplitudes of forces and displacements of the plate to form the force–displacement relationship at nodal points, including the corners. This innovative objective has been achieved by developing a new form of discrete Fourier transform technique for modified trigonometric functions. Using some carefully chosen illustrative examples, the convergence of results is ascertained by using different number of node points and their locations on the plate edges. The proposed theory has substantial advantages over conventional dynamic stiffness theories for plates, particularly when applying non-classical different boundary conditions on plate edges. The computed numerical results are discussed with significant conclusions drawn.

Introduction

For free vibration analysis of structures, the dynamic stiffness method (DSM) is well-known as a powerful alternative to the finite element method (FEM) with significantly better model accuracy. This is mainly because DSM is a differential equation model whereas FEM is based on assumed shape functions. In the early seventies, Wittrick and Williams developed the DSM for isotropic and anisotropic rectangular plates when two opposite sides of the plate are simply supported [1]. Side by side to this work, they also developed a robust solution technique [2], [3] for the DSM to extract the eigenvalues of the structure, which are generally natural frequencies in free vibration problem and critical load factors in buckling problem. Their solution technique [2], [3] was really a breakthrough and it is now known as the Wittrick-Williams algorithm which has featured in literally hundreds of papers. The DSM is an exact method which uses exact solution of the governing differential equations of the boundary value problem of a freely vibrating structure. Prior to the development of the DSM, the transverse vibration problem of an individual rectangular plate with two opposite edges simply supported was solved in an exact sense by many investigators using the so-called Levy type solution [4], [5], [6]. This is relatively an easy task because the mode shapes in one of the directions are sine functions for the simply supported plate, which made the governing bi-harmonic equation of the plate amenable to exact solution. These investigations are by and large, mostly confined to single plates rather than an assembly of plates, but of course approximate solution for plates and plate assemblies using different boundary conditions can be successfully obtained by using the FEM [7], [8]. Levy type solution for simply supported plates has been suitably extended to cover anisotropic plates [9] and there are also some refined plate theories in the published literature [10], [11], [12]. Clearly, for the case of a simply supported plate, there exists an exact solution for the free transverse vibration in the form of a trigonometric series whereas for other boundary conditions, there are approximate methods such as Rayleigh-Ritz [13], [14], [15], extended Kantorovich [16], Galerkin [17], [18], differential quadrature [19] and boundary element [20], [21] amongst a few other methods. Using the Levy type solution one can build an accurate, if not exact, analytical relationship between the amplitudes of forces and displacements at the plate boundaries which are basically the sides or edges of the plate. The relationship essentially defines the dynamic stiffness matrix of a simply supported plate element as originally reported by Wittrick and Williams [1]. Following the work of Wittrick and Williams, Boscolo and Banerjee [22], [23] applied DSM and improved the quality of the dynamic stiffness element for plates by including the first order shear deformation theory. Pagani et al. [24] developed DSM for multi-layered structures by using higher order theories to investigate their free vibration characteristics. Other notable contributors are, amongst others, Eisenberger and Deutsch [25], Casimir et al. [26] and Ghorbel et al. [27] who used DSM to solve the plate vibration problem. The application of DSM to shell structure is outside the scope of this paper, but interested readers are referred to the work of Fazzolari [28] and Chen and Ye [29].

It should be noted at this point that by assembling the dynamic stiffness matrices of individual elements in a structure, it is possible to construct the global dynamic stiffness matrix of the final structure and then the natural frequencies of the final structure can be computed in a straightforward manner by applying the Wittrick-Williams algorithm [2]. In this way, complex structures can be analyzed for their free vibration characteristics by using the DSM. There are established computer programs [30], [31] to accomplish such tasks. The area of applicability of DSM for plates was significantly narrowed in the past by the assumption of Levy type solution, based on the premises that the two opposite sides of the plate are simply supported. Overcoming this restriction continued to be a daunting task for many years. The difficulty was primarily associated with seeking an exact general solution for the bi-harmonic equation which governs the free vibration behaviour of a plate for the most general case. The much-expected breakthrough for the DSM development of plates for the general case, came in recent years when the DSM approach for free vibration analysis of rectangular plates and their assemblies with any arbitrary boundary conditions became an area of intense research activity, see for example Nefovska-Danilovich et al. [32], [33] and Kim and Lee [34]. Some background information is necessary to explain how the limitation of the classical DSM based on simply supported boundary conditions of rectangular plates was overcome. Gorman’s superposition method [35], [36], [37] although different from DSM, was a significant development which stimulated the research in the advancement of DSM for general boundary conditions of plates. By using Gorman’s superposition method [35], [36], [37], the infinite series representing the general solution of the plate can be truncated at some suitable point to achieve reasonable accuracy. However, for the computation of higher order natural frequencies, it would be necessary to increase the number of terms needed in the series which cannot be predicted easily in advance. To alleviate this problem, Banerjee et al. [38] developed DSM for a rectangular plate with any arbitrary boundary conditions by taking full advantage of the symmetry of the plate, but importantly by considering the effect of the remainders of the infinite series solution beyond the cut-off point. This was ignored by other investigators. (In this context, it is worth noting that Papkov [39] demonstrated the importance of considering the remainders of the infinite series terms in free vibration analysis, which have significant effect on the accuracy of results.) Essentially, the authors of [38] split the plate into four equal quarters and then on the enforced planes of symmetry, they applied all four possible boundary conditions which are: (i) symmetric-symmetric (SS), (ii) symmetric-anti-symmetric (SA), (iii) anti-symmetric-symmetric (AS) and (iv) anti-symmetric-anti-symmetric (AA). In this way, they constructed the dynamic stiffness matrix of each of the four components in an exact sense and the overall dynamic stiffness matrix was obtained by summing up the individual dynamic stiffness matrices for all four cases. The symmetry reduced the size of the problem, and there was no approximation involved in the theory in any way. In their formulation, Banerjee et al. [38] used an exact inversion of an infinite matrix system based on an asymptotic expansion of linear equations to derive the dynamic stiffness matrix. This is in sharp contrast to the work described in [32], [33]. One of the main differences between [38], [32], [33] is that the former publication accounts for the infinite remainders of the series solution of the plate whereas the latter publications do not make allowances of the remaining terms of the infinite series by truncating it. In this respect, a recently published paper by Kim and Lee [34] makes interesting reading. Although the notations used in [34] are different from [38], nevertheless, the authors of [34] applied quite a similar approach to that of [38], but there were some significant differences too. The solution in [34] was described as a sum of two partial solutions unlike [38] where the solution was not split in that way. One of the other differences was that [34] used exponential Fourier series representation in the solution which contrasts with trigonometric form of solution in [38]. Liu and Banerjee [40] generalized the approach presented in [38] when they investigated the free vibration behaviour of orthotropic plates. Recently, Wei et al [41] made a noteworthy contribution when they formulated the dynamic stiffness matrix for transverse and in-plane vibration of rectangular plates with arbitrary boundary conditions. Their work differs from the work described in [38], [42], [43] in that the choice of the trigonometric function to describe the series solution was somehow different, which yielded slightly different results. It should be noted that strictly speaking, for all the above approaches, the dynamic stiffness matrix in the exact formulation is an infinite matrix because the boundary values of the displacements and forces form an infinite system of equations. All the above publications dealt with dynamic stiffness formulation for plates with line nodes and none of the previous publications dealt with the dynamic stiffness formulation for plates with point nodes. Regarding this matter, the following comments are made.

The assumed shape functions utilized in FEM are generally lower order polynomials, and thus the formulation of the free vibration problem somehow leads to acceptably good results, particularly in the low and medium frequency ranges, but in the high frequency range, the results from FEM become progressively less accurate. Attempts to improve results using FEM have been made by some investigators by using higher order polynomials which enhance the shape function representation of the structural deformation, see for example Kulla [44]. On the other hand, Doyle [45], and Lee [46] developed spectral element method (SEM) using frequency-dependent trigonometric and hyperbolic functions to address the free vibration problem of plates. Interestingly, Birgersson et al [47] went a step further and proposed a spectral super element model in their endeavour to improve the accuracy of results for the plate vibration problem. In many ways, SEM is similar to DSM as it does not construct separate element mass and stiffness matrices, unlike the FEM. Thus, SEM is effectively a spectral version of the DSM and basically, the two methods differ mainly in their numerical implementation when obtaining results.

Despite significant progress made over the years, the DSM has not still been established as a sufficiently versatile tool like the FEM, even though DSM is properly recognized as a powerful and accurate alternative to FEM, but admittedly DSM is used to solve a specific minority group of vibration problems. Without an effective integration of DSM with FEM which has much greater distribution and coverage, the application of DSM, particularly for plate elements remains strictly restricted. For two-noded line elements such as bars and beams, the task of combining DSM and FEM is relatively simple and straightforward [48], but for plate elements, the problem is extremely difficult. This is mainly because the DSM research for plate elements so far has continued to use line nodes whereas the FEM understandably works very well with point nodes for structural elements. The dynamic stiffness formulation for plate elements relating boundary forces with boundary displacements at some chosen nodal points will no-doubt be a significant step forward which can be exploited to great advantage with possible amalgamation of DSM with FEM. If the versatility of FEM is combined with the uncompromising accuracy of DSM, the scope of structural analysis with hybrid computational algorithm will no-doubt be substantially enhanced. Thus, the essential purpose of this paper is to develop through the applications of the exact solution of the governing differential equation, the dynamic stiffness matrix for isotropic and orthotropic plates characterized by nodal points. The resulting dynamic stiffness matrix with point nodes developed in this paper is finally applied through the implementation of the Wittrick-William algorithm as solution technique to compute natural frequencies and mode shapes for a wide range of problems and some of the results are validated against published results. The accuracy of results on the choice of the number of selective nodes on the plate edges is also demonstrated. The paper concludes with its principal findings with the expectation that it will pave the way for further research in DSM.