Exact analytical solution for static deflection of Timoshenko composite beams on two-parameter elastic foundations

New exact analytical solutions are presented for the static deflection of coupled Timoshenko composite beams resting on two-parameter elastic foundations subject to arbitrary boundary and loading conditions. Governing differential equations are obtained from the principle of virtual work in which four degrees of freedom, namely axial elongation, twist, bending and transverse shear are coupled. In order to obtain the analytical solutions describing the static deflection of coupled Timoshenko composite beams resting on elastic foundations, the governing equations are first rewritten in matrix form to enable decoupling of axial displacement and twist from bending and transverse shear. Applying a separation of variables technique, axial elongation and twist are eliminated and the matrix differential equation for bending and transverse shear is transformed into a system of first-order equations which is solved using a fundamental matrix approach. The results obtained from the analytical solutions are firstly verified by comparison with solutions from the literature for isotropic homogeneous beams on elastic foundations and then with the numerical results from the Chebyshev collocation method obtained for composite beams on elastic foundations, and found to be in excellent agreement. The influence of elastic foundation coefficients, coupling terms and length-to-thickness ratio on the static deflection of Timoshenko composite beams resting on elastic foundations are investigated and discussed.