Structures composed of Timonshenko beams resting on two parameter elastic foundation.

The objective of this dissertation is to present an algorithm and fundamental formulae, for the analysis of linear skeletal structures composed of Timoshenko beam elements resting on two parameter elastic foundations. The formulae are derived from first principles of structural mechanics, in which case, an energy equation for a finite beam element is formulated by considering the kinematics of its deformation. Differential equations, together with the corresponding generalised and natural boundary conditions, am derived applying variation calculus principles. In this case the first variation of the energy functional for a system in equilibrium is equated to zero, the main purpose being to minimize the total potential energy. Exact finite element formulae, which constitute the elements of transport matrix, also called transfer matrix, are obtained by solving the differential equations. This matrix enables one to give complete information of displacements and internal forces at any point along the neutral beam axis in terms of the initial values of displacements and forces at the left end of the beam element. Further, formulae for stiffness matrix coefficients and nodal actions are derived for both generalised and particular beam elements. These formulae are essential for the displacement method of structural analysis. Numerical examples are presented to demonstrate the applicability of the developed algorithm. A parametric study is carried out to investigate the differences which may occur, as a result of different hypotheses adopted for problems of beams on elastic foundation. The results are presented graphically. It has been found that both the effects of transverse shear deformation of the beam element and the second parameter of elastic foundation can have a significant influence on the nodal actions as well as the displacements and internal forces. In view of the fact that relatively powerful computers are available, it is important to ensure that all beams are properly analysed, with the help of the formulae derived, it is hoped that, exact static solutions for problems of linear skeletal structures consisting of beams resting on elastic foundation can now be obtained.