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Teaching Nonlinear Mechanics at the Undergraduate and Graduate Level—Two Examples

R. KIENZLER, P. SCHNEIDER

Abstract


In undergraduate courses and often also in graduate courses on strength-ofmaterials, the post-buckling behavior of structures are not treated. Usually it is indicated that the subject is much too complicated and might be attacked only numerically. In the paper, we give two examples in which the analytical treatment is feasible. The results admit extensive and instructive physical interpretations. As a first example (undergraduate level) we treat a rigid vertical bar elastically clamped (by a torsional spring) at the lower end and loaded by a compressive vertical force V and a horizontal force H at the upper end. It is shown that the post-buckling deformation and the bifurcation point can be easily calculated for arbitrarily large angels φ. Additionally, the existence of a uniquely determined pair (V, H) is shown for any given angle φ. that gives rise to bifurcation of equilibrium. As a second example (graduate level) we revisit the problem of Euler’s elastica, i.e., a simple supported elastic beam under a compressive normal force P. The buckling load is well known but it is almost fallen in oblivion that a closed form solution is available for the post-buckling state involving elliptic integrals. The bifurcation diagram and the load-point displacement diagram are governed by remarkable simple formulae, which might easily be plotted and interpreted in physical terms.

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