Critical Comparison of Bresse–Timoshenko Beam Theories for Parametric Instability in the Presence of Pulsating Load

In this paper, we investigate parametric instability of Bresse–Timoshenko columns subjected to periodic pulsating compressive loads. The results are derived from three theories, namely the Bernoulli–Euler model for thin beams and two versions of the Bresse–Timoshenko model valid for thick beams: The truncated Bresse–Timoshenko model and the Bresse–Timoshenko model based on slope inertia. The truncated Bresse–Timoshenko model has been derived from asymptotic analysis, whereas the Bresse–Timoshenko model based on slope inertia is an alternative shear beam model supported by variational arguments. These models both take into account the rotary inertia and the shear effect. Simple supported boundary conditions are considered, so that the time-dependent deflection solution can be decomposed into trigonometric spatial functions. The instability domain in the load–frequency space is analytically characterized from a Meissner-type parametric equation. For small slenderness ratio, these last two Bresse–Timoshenko models coincide but for much higher slenderness ratio, the parametric instability regions in the load–frequency space shift to the left and widen them as compared to the Bernoulli–Euler model. The importance of these effects differs between the models.