New analytical and semi-analytical solutions for static deflection of composite beams

Anisotropic composite structures are widely used in aerospace, marine, civil, and biomedical areas of engineering due to their advantages, including excellent specific strength, resistance to fatigue and damage tolerance behaviour. Multiple crucial slender structural components of aircraft, automobiles, buildings designed to withstand various loads are modelled as composite beams, thus it is very important to understand the structural behaviour of composite beams and to investigate the mechanism that causes their static deflection. In this thesis mathematical models describing static deflection of composite beams and composite beams resting on elastic foundations are investigated using both analytical and semi-analytical methods based on Euler-Bernoulli and Timoshenko beam theories. These models for the static deflection of composite beams, presented by a system of coupled ordinary differential equations with corresponding boundary conditions, are rigorously derived. The nature of the governing equations depends on the particular problem. For example, the homogeneity of equations is affected by the type of applied loads, while the coefficients of the governing equations are determined by constant or variable stiffness properties of the beam and elastic foundation. In order to obtain closed-form analytical solutions for the problem, coupled governing equations are rewritten in a compact matrix form enabling direct integration to uncouple unknown variables. Closed-form solutions are presented by formulae computationally more efficient compared to commonly used numerical methods such as finite difference or finite element methods, providing deep insight into the mechanism and physics of the static displacement of beams, and quantifying the role and importance of model parameters. Subsequently, semi-analytical techniques, namely the variational iteration method and the homotopy analysis method, are used to predict the static behaviour of composite beams. The presented analytical models are fast and computationally efficient which can be utilised during the preliminary design stages. The derived results can be utilised as benchmark solutions to assess the accuracy and convergence of various analytical and numerical methods.