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COMPUTATIONAL METHODS FOR ENGINEERING SCIENCE
Edited by: B.H.V. Topping
Analytical Non-Linear Stability of a Continuous System as a Benchmark Study for Computational Formulations
M.A. Bradford and Y.-L. Pi
Centre for Infrastructure Engineering and Safety, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, Australia
M.A. Bradford, Y.-L. Pi, "Analytical Non-Linear Stability of a Continuous System as a Benchmark Study for Computational Formulations", in B.H.V. Topping, (Editor), "Computational Methods for Engineering Science", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 14, pp 351-362, 2012. doi:10.4203/csets.30.14
Keywords: arches, benchmark solutions, buckling, elasticity, non-linear, validations.
Over a significant period, many research investigators have reported sophisticated numerical solutions for the non-linear behaviour of continuous structural systems such as frameworks and plates. An important aspect of any numerical formulation is that it is able to replicate real behaviour, which may include sometimes profoundly non-linear effects, and to this end approximations or simplifications are often included in numerical formulations. When a proposed numerical procedure is to be justified, it must be able to predict so-called "exact" solutions derived on the basis of fundamental structural mechanics.
Structural arches are one of the only continuous systems that possess significant geometric non-linearities and that lend themselves to exact solutions in closed form. As such, they are attractive insofar as the solution for their behaviour is a well-posed problem mathematically. However, despite the fact that the solution for their response starts from a well-posed formulation, most analytical solutions possess a number of simplifications and so their use to benchmark numerical solutions can be fraught with inaccuracy and inconsistencies of logic. In particular, most researchers have ignored the coupling of pre-buckling deformations on the in-plane and out-of-plane elastic buckling of arches, and those researchers who have identified this phenomenon have merely mentioned it. A recent study reported by the authors  has focused on the significance of the pre-buckling equilibrium configuration on the buckling and post-buckling of circular, pin-ended arches under uniformly distributed radial loading, and it was concluded in this study that the coupling of pre and post-buckling deformations is significant in the non-linear response of shallow arches.
This paper describes and presents a solution of the non-linear in-plane buckling of arches in closed form, including the coupling of pre-buckling and post-buckling deformations. It considers a circular arch with one end being pin-supported and the other end fixed, with a central concentrated load. By invoking the principle of minimisation of the total potential with a non-linear representation of the strain-displacement relationship, it is shown that the behaviour of an arch of this type is profoundly non-linear, therefore presenting a challenge to numerical schemes that purport to be able to model its behaviour. Some discussion of numerical schemes that aim to include these non-linearities is given, including a sophisticated curved-beam element reported elsewhere [2-4] and the commercial ABAQUS software . It is shown that the solutions are very valuable for calibrating numerical schemes that aim to capture highly geometric non-linearity.
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