INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng. 44, 657–696 (1999) BIFURCATION AND STABILITY OF A THREE-HINGED ROD UNDER A CONSERVATIVE LOAD S. RAJENDRAN 1 , A. Y. T. LEUNG 2; ∗ , A. G. STARR 2 AND J. K. W. CHAN 3 1 Singapore Productivity and Standards Board, 1 Science Park Drive, Singapore 118221 2 School of Engineering, The University of Manchester, Manchester M13 9PL, U.K. 3 Department of Civil and Structural Engineering, The University of Hong Kong, Hong Kong SUMMARY The bifurcation solutions and their stability of a three-hinged rod under conservative compressive force are investigated. The equations for the system are non-linear, and possess some symmetry properties. The symmerty group concepts are employed to exploit these symmetry properties. The symbolic computer software, Mathematica, is used for the analytical and numerical solutions. The loci of codimension-one singularity are plotted on a two-dimensional control parameter space. These curves partition the parameter space into regions of qualitatively similar bifurcation diagrams. The bifurcation solutions and their stability at typical points in the parameter diagram, and the perturbation of codimension-one singularities are discussed. Copyright ? 1999 John Wiley & Sons, Ltd. KEY WORDS: bifurcation; group theoretic; three-hinged rod; singularity 1. INTRODUCTION Buckling analysis forms an important stage in the design of compressively loaded slender structures such as columns, plates, shells and frames. In this paper, we study the post-buckling behaviour of a three-hinged rod, with one end xed, spring restraints at the joints, and a conservative compressive load at the other end. Such a system forms a preliminary model of a robot arm. The problem is also a natural extension of the two-hinged rod problem which has been studied extensively in the literature. The governing equations for buckling problems may be generalized as f (x; ; \)=0 (1) where f is a smooth non-linear mapping, f : ℜ n ×ℜ×ℜ m →ℜ n , x is the vector of state variables,  is the bifurcation parameter or distinguished parameter, and \ is the vector of auxiliary pa- rameters or control parameters. The reason for making one parameter ‘distinguished’ from the others is that it is often easy to vary one parameter independent of the others in experiments. In buckling problems the load is usually considered as the distinguished parameter. ∗ Correspondence to: A. Y. T. Leung, School of Engineering, The University of Manchester, Manchester M13 9PL, U.K. E-mail: andrew levung@man.ac.uk CCC 0029–5981/99/050657–40$17.50 Received 2 January 1997 Copyright ? 1999 John Wiley & Sons, Ltd. Revised 14 May 1998