Solved Problems

Nonlinear structural elements are considered, for which several problems are proposed and solved by perturbation methods. The nonlinear elastostatic response of a taut string under uniform load and a taut membrane under double sinusoidal load, are evaluated. The buckling and postbuckling of a plate, uniformly compressed in one direction, is successively studied, to determine the bifurcated path. Various problems of nonlinear dynamics are addressed, namely: (a) the nonlinear free vibrations of a suspended cable; (b) the primary external resonance of an axially unrestrained beam undergoing a harmonic transverse motion at a support; (c) the primary external resonance of a plate, loaded by a transverse force; (d) the sub- and super-harmonic resonances of an axially restrained beam under transverse loads; (e) the comparison, in terms of free nonlinear oscillations, between beams with the same boundary conditions relevant to transverse displacements, but unrestrained or restrained against longitudinal displacements. The nonlinear Bolotin beam (i.e., a hinged-rolled beam loaded at the tip by a pulsating load), exhibiting the parametric excitation phenomenon, is analyzed. As an example of system undergoing dynamic bifurcation, the nonlinear Beck beam (i.e., a cantilever loaded at the tip by a follower force), is studied. In closing, the bifurcation equations relevant to all the dynamic problems tackled in the book are resumed and compared, and their structure discussed.