Keywords: boundary element method, orthotropy, Saint-Venant, shear stresses, torsional rigidity, shear center.

The increasing use of composites in construction calls for a thorough analysis of their anistotropic behaviour. Beams manufactured by an assembly of longitudinal sheets can be modeled, after homogeneization, by an orthotropic behaviour which can deviate notably from the transversally isotropic response. This paper deals with the flexure and torsion problem for an orthotropic prismatic beam.

The analysis of the orthotropic Saint-Venant problem was investigated by means of analytical procedures based on series expansions by Lekhnitskii [1] and Tolf [2]. These approaches are inevitably limited to simple cross-sections. Numerical models for stress analysis of beams have been developed using both the finite and the boundary element approach. The latter has some appealing features (easy discretization process, computational efficiency) which have attracted many researchers in last years. Jaswon and Ponter [3] analysed the torsion problem in the isotropic case, using a boundary element model with constant interpolation. Gracia and Doblare [4] used a boundary element discretization for an optimization problem of an elastic orthotropic shaft under torsion. Chou [5] employed boundary elements to compute the shear center location of isotropic sections. Friedman and Kosmatka [6] solved the isotropic torsion and flexure problem with a refined boundary analysis, using a three-node isoparametric boundary element.

In this paper the flexure and torsion of an orthotropic beam is treated by constructing a boundary element model. To this aim, the equations governing the tangential stress field are converted, by a coordinate tranformation, into three Neumann problems which form the starting point of a boundary integral formulation using the same fundamental solution as the isotropic case. The numerical solution of these problems requires a unique assembling and factorization of the system matrix, solving for three known terms defined by different boundary conditions.

The model aims to furnish an accurate evaluation of the stress field at low computational cost. Numerical efficiency is achieved by refining the boundary interpolation and the integration process. The boundary mechanical quantities are described by means of macroelements on which a quadratic B-spline approximation ensures a continuity using few control points [7]. All the integrals involved in the formulation are transferred onto the boundary and analytically computed using hierarchically structured formulae, suitable for computer implementation. It should be noted that the exact evaluation of integrals is decisive in order to describe the inner stress field at points very close to the boundary.

The numerical procedure furnishes the shear stress field, through the evaluation of the section properties and the shear center location. Numerical results allow an analysis of the performance of the model and an investigation of the effects due to orthotropy. It is worth noting that orthotropy significantly affects stress field due to torsion and the location of the shear center, whereas stresses due to shear have a distribution close to the isotropic case.

1

S.G. Lekhnitskii, "Theory of Elasticity of an Anisotropic Body", Mir publishers, Moscow, 1981.

2

G. Tolf, "Saint-Venant bending of an orthotropic beam", Composite Structures, 4, 1-14, 1985. doi:10.1016/0263-8223(85)90017-0

3

M.A. Jaswon, A.R.S. Ponter, "An integral equation solution of the torsion problem", Proceedings of the Royal Society of London, 282(A), 237-246, 1963. doi:10.1098/rspa.1963.0085

4

L. Gracia, M. Doblare, "Shape optimization of elastic orthotropic shafts under torsion by using boundary elements", Computers & Structures, 30(6), 1281-1291, 1988. doi:10.1016/0045-7949(88)90193-9

5

S.I. Chou, "Determination of centers of flexure using the boundary element method", Engineering Analysis with Boundary Elements, 12, 321-324, 1993. doi:10.1016/0955-7997(93)90059-T

6

Z. Friedman, J.B. Kosmatka, "Torsion and flexure of a prismatic isotropic beam using the boundary element method", Computers & Structures, 74, 479-494, 2000. doi:10.1016/S0045-7949(99)00045-0

7

M. Aristodemo, E. Turco, "Boundary element discretization of plane elasticity and plate bending problems", International Journal for Numerical Methods in Engineering, 37, 965-987, 1994. doi:10.1002/nme.1620370606

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 £135 +P&P)