Keywords: analysis, arch, curved-beam element, elasto-plastic, nonlinear, rigid body movement, slender, steel, three-dimensional.

After the occurence of lateral-torsional buckling [1], slender steel arches are subjected to torsional, bending and compressive actions. Hence, to predict the postbuckling elasto-plastic behaviour of steel arches, a finite element formulation needs to be able to deal with elasto-plastic torsional, bending and compressive responses accurately. This paper presents the formulation of a curved-beam element for the nonlinear three-dimensional elasto-plastic analysis of slender steel arches. In the development of beam elements, a rotation matrix is usually used to obtain the nonlinear strain-deformation relationships. Since there is a coupling between the nonlinear displacements, to facilitate the derivation of the nonlinear relationships, approximations are usually made to simplify the rotation matrix. Because of these approximations, the rotation matrices do not satisfy the orthogonality and unimodular conditions, and so the approximations may lead to a loss of some significant terms in the nonlinear strains. Without these terms, the rigid-space body motions cannot be separated from the real deformations, which may affect significantly the prediction of the responses, lead to an overestimate of the maximum load carrying capacity, and produce the incorrect nonlinear displacement response, particularly when torsional action is significant [2,3]. An exact rotation matrix for three dimensional deformations that satisfies the orthogonal and unimodular conditions [2] is used in this paper to account for the geometric nonlinearity.

For material nonlinearities and plasticity, standard elasto-plastic incremental stress-strain relationships will be derived based on the von Mises yield criterion, the associated flow rule, and the isotropic hardening rule. However, the standard elasto-plastic stress-strain relationships are accurate only for infinitesimal increments of stresses and strains. In the incremental-iterative numerical analysis, the increments of stresses and strains may be finite. Using finite increments of stresses and strains with the standard incremental stress-strain relationships may lead to error accumulations and subsequently to an unsafe drift from the yield surface. As a result, the quadratic asymptotic convergence rate of Newton-Raphson method is lost. To solve this problem, for rate-independent plasticity, a consistent tangent operator instead of the standard tangent operator will be used in conjunction with the consistent return mapping algorithm based on the operator split methodology [4]. A spatially curved-beam element, which integrates the effects of initial geometric imperfections, residual stresses, elastic restraints and supports, and the effects of the load and restraint positions and which can predict nonlinear three-dimensional elasto-plastic deformations of slender steel arches, is developed in this paper.

1

Y.-L. Pi, N.S. Trahair, "Three Dimensional Nonlinear Analysis of Elastic Arches", Engineering Structures, 18(1), 49-63, 1996. doi:10.1016/0141-0296(95)00039-3

2

Y.-L. Pi, M.A. Bradford, B. Uy, "A Spatially Curved-beam Element with Warping and Wagner Effects", International Journal for Numerical Methods in Engineering, 63, 1342-1639, 2005. doi:10.1002/nme.1337

3

Y.-L. Pi, M.A. Bradford, B. Uy, "A Rational Elasto-Plastic Spatially Curved Thin-Walled Beam Element", International Journal for Numerical Methods in Engineering, 70(3), 253-290, 2007. doi:10.1002/nme.1873

4

M.A. Crisfield, "Non-linear Finite Element Analysis of Solids and Structures", Wiley, Chichester, UK, 1991.

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