Keywords: thin shallow arch, geometric nonlinearity, snap-through, truss model, coarse discretization, displacement control, step-by-step time integration, trapezoidal rule, predictor-corrector technique.

Thin shallow arches are often used in civil, mechanical and aeronautical structures. When the ratios of arch span to its height, as well as to its thickness, are high, strong geometric nonlinearities, due to large displacements, appear. And for a critical high value of the loading, the arch may be subject to snap-through. On unloading, the arch snaps-back, following a different loading path. So, the nonlinear generalized load-displacement curve exhibits an hysteresis loop [1].

The usual finite elements present difficulties in describing the geometric nonlinearity of a structure. Here, as an alternative, a truss model is used. As a result of the very simple geometry of a truss, the equilibrium equations can be easily written and the global stiffness matrix can be easily updated with respect to the deformed truss.

Usually, a refined spatial discretization is applied to a structure by a large number of finite elements. As a consequence, in dynamic analysis, very high frequencies appear, which dictate a very small time-step length for the algorithm and we have to follow many complicated very small vibrations, which are useless to the engineer. For this reason, often some rather complicated techniques are developed, the so-called reduced-order techniques, in order to suppress the high vibration modes [2].

Here, as an alternative to the above reduced-order techniques, a very coarse discretization is applied. So, the high vibration modes are suppressed, in a very simple way, from the beginning.

Two very short computer programs have been developed for the geometrically nonlinear, static analysis, using the displacement control, of a plane truss model of a structure, as well as for its dynamic analysis, by the step-by-step time integration algorithm of trapezoidal rule, combined with a predictor corrector technique. These two very short, fully documented, special purpose computer programs, compared with the often used very large general purpose programs, have the advantages of more transparency, simplicity and clarity of assumptions.

The above two computer programs are applied to the geometrically nonlinear, static and dynamic analysis of a specific thin shallow arch subject to snap-through.

1

Y.-L. Pi, M.A. Bradford, F. Tin-Loi, "Non-linear, in-plane buckling of rotationally restrained shallow arches under a central concentrated load", International Journal of Non-Linear Mechanics, 43, 1-17, 2008. doi:10.1016/j.ijnonlinmec.2007.03.013

2

S.M. Spottwood, J.J. Hollkamp, T.G. Eason, "Reduced order models for shallow curved beam under combined loading", AIAA Journal, 48(1), 47-55, 2010.

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