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

Static analysis of composite beams on variable stiffness elastic foundations by the homotopy analysis method

New analytical solutions for the static deflection of anisotropic composite beams resting on variable stiffness elastic foundations are obtained by the means of the Homotopy Analysis Method (HAM). The method provides a closed-form series solution for the problem described by a non-homogeneous system of coupled ordinary differential equations with constant coefficients and one variable coefficient reflecting variable stiffness elastic foundation. Analytical solutions are obtained based on two different algorithms, namely conventional HAM and iterative HAM (iHAM). To investigate the computational efficiency and convergence of HAM solutions, the preliminary studies are performed for a composite beam without elastic foundation under the action of transverse uniformly distributed loads considering three different types of stacking sequence which provide different levels and types of anisotropy. It is shown that applying the iterative approach results in better convergence of the solution compared with conventional HAM for the same level of accuracy. Then, analytical solutions are developed for composite beams on elastic foundations. New analytical results based on HAM are presented for the static deflection of composite beams resting on variable stiffness elastic foundations. Results are compared to those reported in the literature and those obtained by the Chebyshev Collocation Method in order to verify the validity and accuracy of the method. Numerical experiments reveal the accuracy and efficiency of the Homotopy Analysis Method in static beam problem

Closed form solutions for an anisotropic composite beam on a two-parameter elastic foundation

Beams resting on elastic foundations are widely used in engineering design such as railroad tracks, pipelines, bridge decks, and automobile frames. Laminated composite beams can be tailored for specific design requirements and offer a desirable design framework for beams resting on elastic foundations. Therefore, the analysis of flexural behaviour of laminated composite beams on elastic foundations is of important consequence. Exact solutions for flexural deflection of composite beams with coupling terms between stretching, shearing, bending and twisting, resting on two-parameter elastic foundations for various types of loading and boundary conditions, are presented for the first time. The proposed new formulation is based on Euler–Bernoulli beam theory having four degrees of freedom, namely bending in two principal directions, axial elongation and twist. Governing equations and boundary conditions are derived from the principle of virtual work and expressed in a compact matrix–vector form. By decoupling bending in both principal directions from twist and axial elongation, the fourth-order differential equation for bending is derived and transformed into a system of first-order differential equations. An exact solution of this system of equations is obtained using a fundamental matrix approach. Fundamental matrices for different configurations of elastic foundation are provided. The ability of the presented mathematical model in predicting flexural behaviour of beams on elastic foundations is verified numerically by comparison with results available in the literature. In addition, the deflection of anisotropic beams is analysed for different types of stacking sequences, boundary and loading conditions. The effect of elastic foundation coefficients on the flexural behaviour is also investigated and discussed

Closed-form solutions for the coupled deflection of anisotropic euler–bernoulli composite beams with arbitrary boundary conditions

The fully anisotropic response of composite beams is an important consideration in diverse applications including aeroelastic responses of helicopter rotor and wind turbine blades. Our goal is to present exact analytical solutions for the first time for coupled deflection of Euler–Bernoulli composite beams. Towards this goal, two approaches are proposed: (1) obtaining the exact analytical solutions directly from the governing equations of Euler–Bernoulli composite beams and (2) extraction of the solutions from Timoshenko composite beam solutions. For the direct solution approach, based on Euler–Bernoulli theory, new variationally-consistent field equations are obtained, in which four degrees of freedom, i.e. in-plane bending, out-of-plane bending, twist and axial elongation are fully coupled. By expressing the coupled system of differential equations in a compact matrix form, a novel expression for the eccentricity of neutral axes from the midplane, as well as the shift in shear centre from the centre of beam, is obtained. This eccentricity matrix serves to decouple the bending in the two principal directions from in-plane and twist deformations. Then, the general closed-form analytical solutions for the decoupled system are derived simply using direct integration. Additionally, the analogous closed-form analytical solutions are retrieved from the previously obtained Timoshenko composite beam solution and it is proven that they are identical to those obtained from the current direct approach for conditions where Euler–Bernoulli beam theory apply. To study the effects of anisotropy, numerical results are obtained for a number of examples with different composite stacking sequences showing various coupled behaviours. The results are compared against the Chebyshev collocation method as well as against less comprehensive analytical solutions available in the literature, noting that excellent agreement is observed, where expected. The present exact solutions can serve as benchmark problems for assessing the accuracy and convergence of various analytical and numerical method