Engineering Structures 31 (2009) 208–215 Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct An approximate method for simultaneous modification of natural frequencies and buckling loads of thin rectangular isotropic plates R. Mirzaeifar a , S. Shahab a , H. Bahai b,∗ a Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran b School of Engineering and Design, Brunel University, Uxbridge, UB8 3PH, UK article info Article history: Received 26 March 2008 Received in revised form 4 July 2008 Accepted 5 August 2008 Available online 20 September 2008 Keywords: Natural frequency Buckling load Eigenvalue Eigenvector Sensitivity analysis abstract In this paper, the first and second order derivatives of natural frequencies and buckling loads with respect to an arbitrary geometrical or physical property of a plate structure are calculated. Based on these eigenderivatives, and by using first and second order Taylor expansions, an approximate method is presented for simultaneous modification of natural frequencies and buckling loads. In the proposed method, the structural modification is considered as an inverse eigenvalue problem and calculated eigenderivatives are used for transforming this inverse problem to solving a system of algebraic equations. By solving this set of equations, the necessary changes in design parameters are calculated, in order to achieve predefined simultaneous shifts in natural frequencies and buckling loads. A plate with two boundary conditions is considered as a case study. The plate is divided into eight regions and the plate thickness in each region is considered as a design parameter. By considering several case studies, the required modification in design variables is calculated for achieving a predefined change in natural frequencies, buckling loads or simultaneous change of both. In each case, the calculated changes in design variables are implemented on the initial structure, and the changes in eigenvalues are computed using FE method. It is shown that the results from presented method compare very well with those obtained from a direct optimization procedure. © 2008 Published by Elsevier Ltd 1. Introduction Both the free vibration and linear buckling behaviour of structures are formulated as standard eigenvalue problems. Derivatives of natural frequencies and mode shapes (as the eigenvalues and eigenvectors of the free vibration equation) with respect to geometrical or physical properties of structure play an important role in structural design. Consequently, the calculation of the free vibration eigenderivatives has been an active field of research since the early work of Fox and Kapoor [1]. However, no attention has been paid to the problem of derivatives of buckling loads and buckling modes of a structure (as the eigenvalues and eigenvectors of the stability equation). Since the work of Fox and Kapoor, which was based on a modal method, several methods have been developed for calculating the eigenvalue and eigenvector derivatives of symmetric un- damped systems, including the finite-difference method [2], the iterative method [3], the Nelson’s method [4] and the modified modal method [5]. Among these methods, the modal method, which expresses the derivatives of each eigenvector as a linear combination of all eigenvectors, is the most common approach in use. ∗ Corresponding author. E-mail address: hamid.bahai@brunel.ac.uk (H. Bahai). In design applications, calculating the derivatives of eigenval- ues and eigenvectors with respect to arbitrary parameters for a structural system is considered to have very useful implications. Based on a Taylor expansion, the eigenderivatives can be used to approximately formulate the direct and inverse eigenvalue prob- lems. Direct approximation for eigenvalue problem allows the cal- culation of the change of each eigenvalue due to an arbitrary change in design parameters. In the inverse approximate eigen- value problem formulation, the required structural modifications to achieve predefined shifts in eigenvalues and eigenvectors are computed. Based on the first and second-order derivatives of the natural frequencies and modal shapes, many studies have been carried out by the authors of the present paper for developing approximate solutions of direct and inverse eigenvalue problems. Djoudi and Bahai [6] established a relationship between geometric and material properties of pin-jointed truss structures and their eigenvalues in order to relocate natural frequencies of structures to a specified level. Bahai and Aryana [7,8] used the first and second order approximations of a Taylor expansion with local modification for performing sensitivity analysis on two- dimensional plane structures. Mirzaeifar et al. [9] used the first and second derivatives of the eigenvalues for formulating the direct and inverse approximate eigenvalue problem for laminated composite plates. Aryana et al. [10] proposed a similar approach 0141-0296/$ – see front matter © 2008 Published by Elsevier Ltd doi:10.1016/j.engstruct.2008.08.006