Keywords: stability, lateral buckling, post-buckling, timber arches, arc-length method, three-dimensional beam theory.

This article presents the stability analyses of glulam arches subjected to point loads. Glulam timber arches tend to be slender and, therefore, the stability test is even more significant. Stability analyses and load-deflection path tracing, of initially curved elements, have been studied many times. Authors mostly consider in-plane buckling of arches using two dimensional theories. Thus, the lateral buckling of the arch should be prevented by lateral support if realistic results from the in-plane analysis are to be obtained.

In this paper a numerical analysis for determining critical buckling loads of arches and their post-buckling behaviour is presented. The method used in this analysis is based on the numerical formulation of geometrically exact, three dimensional beam theory by Zupan and Saje [1], in which the strain vectors are the only unknown functions. Their formulation allows for an arbitrary initial geometry to be assumed and proves to be both very accurate and numerically efficient.

The post-critical load deflection path is traced by adapting a modified arc-length method introduced by Feng et al. [2] to the strain-based beam theory. The buckling load is determined by observing the sign of the determinant of the tangent stiffness matrix.

The fundamental load-deflection path contains an infinite number of critical points. These points are symmetric bifurcation points, where the primary path is unstable and the secondary, bifurcation path, is stable. The higher order stable paths are hence particularly convenient, as we can consider the inflection points of the out-of-plane eigenvector as lateral supports, introducing thus the positions of the lateral supports which are optimal with regard to the bearing capacity of the arch.

With an extensive parametric study we came to the conclusion, that the relative height of the arch strongly affects the lateral buckling capacity. The optimal height/span length ratio is 0.2. The initial curvature of the arch affects the higher buckling mode shapes and therefore the zeros of the buckling modes are not equidistant.

Rather than lateral buckling, the bending failure occurs, if the arch has six or more optimal supports. In this case, the allowable deflections, according to EC 5 [3], are also exceeded. In all the examples discussed, the radial stresses, obtained by [4], were not decisive for the arch failure.

1

D. Zupan, M. Saje, "Finite element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures", Comput. Methods Appl. M., 193, 5209-5248, 2003. doi:10.1016/j.cma.2003.07.008

2

Y.T. Feng, D. Peric, D.R.J. Owen, "Determination of travel directions in path-following methods", Math. Comput. Model., 21, 43-59, 1995. doi:10.1016/0895-7177(95)00030-6

3

Eurocode 5, Design of timber structures, Part 1.1, General rules and rules for buildings, 2007.

4

J.T. Oden, "Mechanics of Elastic Structures", McGraw-Hill Book Company, New York, 1967.

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