Keywords: beam theory, bending, stress components, composite beam, warping function, finite element.

In this paper, two formulations for determining the distributions of the stress components in the cross-section of a composite beam subject to two-axial bending are studied theoretically and numerically. These formulations can be regarded as improvements of the classical formulas for determining stresses in a cross-section of a beam caused by the corresponding stress resultants. They are based on improved kinematic assumptions for the displacements. In the first formulation (f1) the classical beam theory expression for the axial displacement is augmented with the help of two warping functions. In the second formulation (f2) the classical expressions for the two transverse displacements are further augmented using three contraction functions each.

The theoretical part of the study considers determining the warping and contraction functions and obtaining the stress components with the help of these. Simple two-dimensional boundary value problems, which are defined in the cross-section, are derived for both the warping and contraction functions. These boundary value problems are of the Poisson- and plane elasticity-type for the warping and contraction functions, respectively. Because these problems have only Neumann boundary conditions, special attention is paid to setting constraints, which are required for obtaining single valued solutions to these problems. It is further shown, that in connection with homogeneous and isotropic material f2 reduces to a theory, which has been considered in connection with a cantilever beam with a terminal load in classical textbooks of the theory of elasticity [1], and first solved numerically in [2]. Compared to the classical formulas of beam theory, f1 gives an identical result to the axial normal stress, but improves the two typical shear stress components so, that the axial equilibrium equation and the corresponding boundary condition are satisfied, and f2 improves all the stress components so, that all the equilibrium equations and the corresponding boundary conditions are satisfied.

In the numerical part of this study, the finite element results of the two formulations are compared to those of the finite prism method. Biquadratic iso-parametric finite elements with 3x3 Gauss quadrature are used. Compared to the numerical results obtained with finite prism method, both formulations give results, which describe the typical beam stresses quite accurately. The results f2 for the additional stress components are also satisfactory, but slightly lower than those of the finite prism method.

1

S.P. Timoshenko, J.N. Goodier, "Theory of Elasticity", McGraw-Hill, 354-358, 1970.

2

W.E. Mason, L.R. Herrmann, "Elastic shear analysis of general prismatic bars", Journal of the Engineering Mechanics Division ASCE, No. EM4, 965-983, 1968.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £65 +P&P)