Elsevier

Computers & Structures

Volume 147, 15 January 2015, Pages 236-243
Computers & Structures

Proof of convergence for a set of admissible functions for the Rayleigh–Ritz analysis of beams and plates and shells of rectangular planform

https://doi.org/10.1016/j.compstruc.2014.09.008Get rights and content

Abstract

This work presents a discussion on the characteristics of sets of admissible functions to be used in the Rayleigh–Ritz method (RRM). Of particular interest are sets that can lead to converged results when penalty terms are added to model constraints and interconnection of elements in vibration and buckling problems of beams, as well as plates and shells of rectangular planform. The discussion includes the use of polynomials, trigonometric functions and a combination of both. In the past, several sets of admissible functions that have a limit on the number of terms that can be included in the solution without producing ill-conditioning were used. On the other hand, a combination of trigonometric and low order polynomials have been found to produce accurate results without ill-conditioning for any number of terms and any number of penalty parameters that can be accommodated by the computer memory.

Introduction

In [1] Meirovitch states that the classical Rayleigh–Ritz method consists of selecting N comparison functions ui to be included in the Rayleigh quotient. These functions must satisfy natural and geometric boundary conditions and be differentiable 2p times (p being the order of the highest differential operator in the functional used) to construct the linear combinationwn=i=1Naiuiwhere ai are unknown coefficients. However, it was noted that a set of admissible functions ϕi, which have to satisfy only geometric boundary conditions and be only p times differentiable can be used instead. Furthermore, Meirovitch [1] also states that orthogonal and normalized functions such as Bessel functions and Legendre polynomials, as well as the Gram-Schmidt orthogonalization process have been often used aiming to reduce computational work, although these operations add computational cost. It is also worth noting that comparison functions are a subset of the admissible functions.

The motivation for this work is to find a set of admissible functions that allow solution to vibration and buckling problems of structural elements such as beams, plates and shells, in which the constraints are modelled by penalty parameters (artificial springs). Sets of functions built by simple polynomials have shown to be prone to ill-conditioning, whilst trigonometric functions alone introduce additional geometric constraints (sine functions requiring zero displacements and cosine functions requiring zero slope). A good strategy then is to combine simple polynomials and trigonometric functions to build a function that does not have a limitation on the number of terms due to ill-conditioning and converges fast with respect to the number of terms used in the set. The set of functions in this work is the simplest set of admissible functions that can be built using simple polynomials and transcendental functions, which has proved to compute very accurate results for vibration and buckling problems of beams, plates and shells [2], [3]. This paper presents a proof of convergence of the selected set of admissible functions.

Simple polynomials have a severe limitation on the number of terms that can be included in the solution before an ill-conditioning problem arises. Other sets of admissible functions built by orthogonal polynomials using the Gram-Schmidt process presented by Bhat in [4] have been proven to give excellent results for plates involving free edges, as shown in a publication by Yuan and Dickinson [5]. This procedure has been used to build sets of admissible functions by many researchers, even though Brown and Stone raised some criticism of this work in [6], where it is stated that the convergence of a vibration problem is independent of the selection of the set of admissible functions (no need for orthogonal polynomials) and that it depends only on the degree of the polynomial represented in the set. In the same work Brown and Stone stated that for plate problems, orthogonality of the functions should be targeted only on the second derivative of the functions, although they also recognized that special polynomials are only needed if higher order polynomials are included in the set of admissible functions. This is to make the set of functions more stable with respect to inversion and the extraction of eigenvalues of the resulting stiffness and mass matrices, although in [7] Li reported that even when orthogonal polynomials are used in the RRM, the higher order polynomials become numerically unstable due to round-off errors.

Transcendental functions also have some disadvantages. For instance, Li and Daniels [8] show that certain sets of admissible functions built by trigonometric functions have limitations converging when penalty parameters are included in the solution. Sets of functions using trigonometric and hyperbolic functions are very complex and are likely to become numerically unstable when several terms are used in the solution. This was noticed by Blevins [9] who recommends using a high degree of precision when higher modes are included, as well as Jaworski and Dowell [10] who used trigonometric and hyperbolic functions to solve vibration problems of beams with multiple steps using a set of functions for clamped-free beams. Jaworski and Dowell reported that numerical problems arise due to the difference between the values of the hyperbolic functions. In [10] the set of admissible functions built by trigonometric and hyperbolic functions was substituted by an approximation in higher modes with a combination of sine, cosine and exponential functions previously used by Dowell [11].

More recently Dozio [12] published a comprehensive study on the use of a set of trigonometric functions, originally proposed by Beslin and Nicolas [13], used to solve vibration problems of rectangular orthotropic plates. The method by Dozio offers the same advantages as the proposed set of functions of the present work, but the matrices of the system are built with more complex terms, and even though many terms of the matrices using the set of admissible functions by Beslin and Nicolas [13] become zero, the matrices of the present method are even more sparse.

In contrast with all the previous options to build sets of admissible functions, several publications including the works by Li [7], [14] and Zhou [15] have shown that when polynomials and trigonometric functions are used to build sets of admissible functions, the solutions have a fast convergence rate and results are also accurate for higher modes. Although it is now known that only the sum of the series of the functions should satisfy the geometric boundary conditions, many researchers have proposed to build sets of functions starting with a series containing trigonometric and polynomial functions, but enforcing boundary conditions for each term. This approach was used in [7], [14], [15]. In [7], [14] Li built a series of admissible functions by mixing polynomials and trigonometric (cosine) functions. Li stated that the polynomials are introduced to take all the relevant discontinuities with the original displacement and its derivatives at the boundaries. More recently Dal and Morgül [16] presented a similar approach to those presented by Li [7], [14] and Zhou [15]. Dal and Morgul used sine functions as in the work by Zhou [15] and also enforced geometric boundary conditions for each term. Polynomials in the publications by Li [7], [14] were of order 4, while in the approaches of Zhou [15] and Dal and Morgül [16] the maximum order of a polynomial was 3. Filipich and Rosales [17] presented another interesting set of admissible functions that uses trigonometric functions and a bidimensional function combining a constant term, a linear term and a sine term that satisfy the geometric boundary conditions as a whole. The procedure in [17] introduced the so-called “whole element method” that consists of the definition of a proper functional and the introduction of an extremizing sequence that defines the set of admissible functions.

It is important to remember that high order polynomials are the cause of numerical instabilities and ill-conditioning. Thus to keep the solution as simple as possible and free of numerical problems the minimum number of polynomial functions with the lowest order possible are included in the proposed set of admissible functions presented in this work.

Section snippets

Building a set of admissible functions

As mentioned earlier, this work presents a set of functions that can be used to model beams, plates and shells; converges fast and allows the use of a large number of functions without causing ill-conditioning. In addition, the selected set of admissible functions models an unconstrained structure and complex geometric boundary conditions can be modelled adding as many constraints as necessary using penalty functions.

In the past some researchers gave guidelines to develop sets of admissible

Examples

To show the versatility and stability of the set of admissible functions presented in Eq. (2a), (2b), (2c), (2d) the first six natural frequencies of thin plates with free (F), simply supported (S), sliding (G) and clamped edges (C) are presented using 40 terms in each direction. Consider a rectangular plate as shown in Fig. 4, with dimensions a and b along directions x and y, thickness h and flexural rigidity D defined asD=Eh3121-ν2,where ν is Poisson’s ratio and E is Young’s modulus.

The

Results

Results of the frequency parameters of the 55 cases of rectangular plates with simply supported, clamped, sliding and free conditions were obtained by assigning appropriate penalty parameters kˆ. In a publication by Williams [31] two important characteristics of the classical RRM (without penalty parameters) are mentioned. The first characteristic is that the lower modes converge first. The second characteristic is that when the RRM is used to solve vibration problems, the natural frequencies

Conclusions

In this work a discussion on set of admissible functions to be used in the RRM is presented and it has been shown how a set built by cosine functions and a linear and square terms can be used to model beams, plates and shells in fully unconstrained condition. A proof of the convergence of the set is also presented in this work. The penalty method can be used to obtain results for cases including constraints. Since the set of functions presented here does not impose a limit in the number of

Acknowledgment

The second author wishes to acknowledge the financial support from the Marsden Fund (http://www.royalsociety.org.nz/programmes/funds/marsden/).

References (38)

Cited by (51)

View all citing articles on Scopus
View full text