Compwcrs & Smctvrer Vol. 23, No. 3. pp. 3514% 1986 004s7949/8653.00 + .a, Pdmed in GreatBtiuia. Q 1986 Pergmon JoumrlrLtd. zyxwvutsrqp ANALYSIS OF A BEAM COLUMN ON ELASTIC FOUNDATION Davm Z. YXWELEVSKY and MOSHE EISENBERGER Faculty of Civil Engineering. Technion-Israel Institute of Technology, Haifa 32000, Israel zyxwvutsrqponmlkjihgfedcb (Reteived 24 April 1985) Abstract-The analysis of beams on elastic Winkler foundation is very common in engineering. In many app~~~ons, transverse as weh as axial forces exist. An zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH exact analytical solution of a finite element beam column resting on a Winkler foundation is performed from which the exact stiffness terms are determined. The stiffness matrix is incorporated into a common beam program. Nodes are required only at points of discontinuity in stiffness, loading, or supports. Comparisons are made with case results appearing in literature. INTRODCCTIOS The analysis of beams on a Winkler elastic foun- dation is very common in engineering. The Winkler foundation consists of an infinite number of closely spaced unconnected linear springs and it is defined by the foundation modulus k. Closed form solutions to relatively simple problems of beams on elastic foundation have been proposed by numerous au- thors(3,6, IO, 161. Improved models assume inter- actions between the springs that may represent shear transfer in the foundation bed and be ex- pressed as a two parameter elastic foundation[5. 7. 8, 14, 171. A common problem is that in which trans- verse load as well as axial load act simultaneously on the beam, like in analysis of end bearing piles. frames resting on or buried in soil, etc. Some ana- lytical solutions of beam columns on elastic foun- dations are found in Refs. 19 and 151. In recent years finite element solutions have been proposed to the problem [l. 2. 11, 12. 181. in most of which the simple Winkler model is used. The shape functions used are those of the free beam which yield approximate solutions to our problem. Recently an exact stiffness matrix for a beam on elastic foundation has beeen formulated for the Winkler foundationl4] and for the two parameter foundation[f9]. thus leading to exact solutions with only a small number of elements. In this work a formulation of a beam column on an elastic foundation is presented and from the so- lution of finite beam to unit end rotation and dis- placement. a direct determination of the stiffness terms is obtained. Fig. 1 leads to the equation dstxf - = k*?‘(x) - q(x)* dv in which k = the Winkler’s foundation modulus. The equilibrium of moments yields dM(x) - N.!x - S(x) .r dx . The moment-curvature equation of elementary beam theory in that coordinate system is EI d2yW - = M(x). dx* (3) The norrmal shear S,(x) which is pt,pendicular to the deflection line at each section equals to (41 Differentiating eqn (3) and substituting eqn 13) yields the expression for the vertical shear force Differentiation of eqn (5) and using eqn (1) yields the governing differential equation of the deflection line in the present problem: E, d4.M d% u) N f dx4 d? + XJO) = 9(X). (6) BASICEQUATIONS A beam on elastic foundation which is subjected Using the parameter A as the characteristic of the both to vertical loading and a pair of horizontal ten- beam sile forces N acting at the center of gravity of the end cross sections is considered. Force equilibrium in 2’direction on an infinitesimal element shown in A= (7) 351