Materials Science, Vol. 46, No. 1, 2010 REFINED SOLUTION OF THE TIMOSHENKO PROBLEM FOR AN ORTHOTROPIC BEAM ON A RIGID BASE V. I. Shvab’yuk, 1, 2 Ya. M. Pasternak, 1 and S. V. Rotko 1 UDC 539.3 We have obtained the solution of the problem of a composite (orthotropic) beam partly supported on a perfectly rigid base under a uniformly distributed load. For calculations, we use the refined model of beams that takes into account the strains of transverse shear and compression. We have derived an equa- tion for evaluating the size of the contact area and relations for calculating the contact pressure of the rigid base on the external surface of the beam. Numerical results obtained by the refined model for iso- tropic and orthotropic materials are compared with those corresponding to the classical theory, the shear model of beams, and the plane problem of the theory of elasticity. Keywords: refined model of beams, transverse shear and compression, contact pressure, contact area, rigid base. The problem of contact of a beam partly supported on a rigid base was solved by Timoshenko [1] in the statement of the Kirchhoff classical theory of beams. Based on analysis of the results of investigations, Gu- dramovich and Mossakovskii [2] also applied a similar method in studying the interaction of an elastic ring stiff- ening a shell with a perfectly rigid base. For refining the solution obtained by Timoshenko and Feodos’ev [3], the theory of beams (with regard for shear strain) was used; Keer and Silva [4] considered this problem as a mixed problem of the theory of elasticity for a half-strip. In the present work, we apply the refined model of beams [5] that, in addition to the strain of transverse shear, takes into account transverse compression. We compare the results obtained by this model with the nu- merical solution additionally found by the boundary element method (BEM). A similar problem for the case where the beam is glued to the base was solved in [6] with the help of the Timoshenko theory of plates with ad- ditional regard for compression. Statement of the Problem Consider the symmetric bending of a beam of length 2l under the action of its own weight q = 2 P /2l and forces P 1 applied to the right and left ends and raising them. The middle part of the beam lies on a rigid base and is subjected to the action of its own weight q and contact pressure p( x ) . We assume that each of the forces P 1 has to be smaller than half of the weight of the right or left part of the beam – P = ql . It is neces- sary to determine the pressure distribution p( x ) in the contact zone and the size of contact area 2a between the beam and rigid base where each of the forces is P 1 = P /3 . In view of the symmetry of this problem, we con- sider the equilibrium of only the right half (Fig. 1). 1 Luts’k State Technical University, Luts’k, Ukraine. 2 Corresponding author; e-mail: shvabyuk@lutsk-ntu.com.ua. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 46, No. 1, pp. 51–56, January–February, 2010. Original article submitted October 18, 2008. 56 1068-820X/10/4601–0056 © 2010 Springer Science+Business Media, Inc.