Engineering Structures 27 (2005) 1373–1384 www.elsevier.com/locate/engstruct A numerical method for static or dynamic stiffness matrix of non-uniform members resting on variable elastic foundations Z. Canan Girgin a,∗ , Konuralp Girgin b a Yapı Merkezi Construction Inc., Çamlıca, 34676, Istanbul, Turkey b Department ofCivil Engineering, Istanbul Technical University, Maslak, 34469, Istanbul, Turkey Received 24 November 2004; received in revised form 16 March 2005; accepted 5 April 2005 Available online 2 June 2005 Abstract This paper presents a generalized numerical method which is based on the well-known Mohr method. Static or dynamic stiffness matrices, as well as nodal load vectors for the static case, of non-uniform members are derived for several effects. The method focuses on the effects of resting on variable one- or two-parameter elastic foundations or supported by no foundation; a variable iterative algorithm is developed for computer application of the method. The algorithm enables the non-uniform member to be regarded as a sub-structure. This provides an important advantage to encompass all the variable effects in the stiffness matrix of this sub-structure. Stability and free-vibration analyses of the sub-structure can also be carried out through this method. Parametric and numerical examples are given to verify the accuracy and efficiency of the submitted method. © 2005 Elsevier Ltd. All rights reserved. Keywords: Non-uniform member; Arbitrarily variable; Two-parameter elastic foundation; Geometric non-linearity; Stiffness matrix; Stability and free- vibration analysis 1. Introduction Members including cross-sectional variations partially or along the whole length have been widely used in many fields to improve the strength of some regions, to reduce the dimensions of cross-sections for more economical solutions, etc. Generally, analyses of non-uniform members receive attention due to their relevance to structural, mechanical and aeronautical engineering. The exact dynamic stiffness matrices for beams of arbitrarily varying cross-sections were derived by Banerjee and Williams [1] and Mou et al. [2]. These derivations were performed by means of Bessel’s functions [1] for integer powers of area and moment of inertia, and by means of complicated algebraic manipulations [2] for arbitrary real-number powers as well. Many numerical or approximate methods have been conducted for the vibration analysis of non-uniform ∗ Corresponding author. Tel.: +90 216 3219000; fax: +90 216 3219013. E-mail address: canan.girgin@gmail.com (Z.C. Girgin). 0141-0296/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2005.04.005 members; for example, the Galerkin method [3], the step- reduction method [4] and more recently, the Rayleigh–Ritz method [5]. Static and dynamic analyses of beams or beam- columns resting on elastic foundations play an important role in many problems related to soil-structure interaction (railway tracks, piles, pipelines, highway pavements, strip and ring foundations, retaining walls, etc.). The most used model in the solution of these problems is the Winkler hypothesis. Uniform beam and beam-columns resting on this type of elastic foundation were early studied by Hetenyi [6], Timoshenko and Gere [7]. In this model, the elastic foundation acts as if it consisted of infinitely many closely spaced linear springs. In order to improve this model, several two-parameter foundation models were developed. These models account for the interactions between springs by means of a second parameter. Ref. [8] provides detailed and actual knowledge about available models. In order to obtain the exact static stiffness matrices and nodal load vectors of uniform beams resting on a two-parameter elastic foundation with constant coefficient, several methods using exact