Exact solution and stability of postbuckling configurations of beams |
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Authors: | Ali H Nayfeh Samir A Emam |
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Institution: | (1) Department of Engineering Science and Mechanics, MC 0219, Blacksburg, VA 24061, USA;(2) Department of Mechanical Design and Production, Faculty of Engineering, Zagazig University, Zagazig, 44519, Egypt |
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Abstract: | We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged
boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the
governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution
for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode
shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the
dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium
position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are
unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first
three buckled configurations. The results show that many internal resonances might be activated among the vibration modes
around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed
beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration
and the first vibration mode around the second buckled configuration. |
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Keywords: | Exact solution Stability Free vibration Buckled beams |
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