The Statics and Dynamics of a Transversely Isotropic Timoshenko Beam

Abstract

A partial differential equation is derived that describes the motion of a transversely isotropic Timoshenko beam under initial stress, initial displacement, and transverse loading. Suitable specializations of this equation permits one to investigate buckling behavior, vibrational be havior as affected by initial stress, and wave propagation behavior in initially stressed infinite beams. Each separate topic provides important results for designing structures with high values of the ratio of longi tudinal modulus to longitudinal-transverse shear modulus which are common among advanced composites. The buckling investigation pre dicts moderate decreases in the buckling coefficient for simply sup ported beams but predicts very large decreases for clamped beams. The vibration investigation shows that initial tension and compression have practically no effect on the thickness shear frequencies for all modes. The wave propagation investigation shows (a) the extreme sensitivity of the thickness shear cutoff frequency as a function of the ratio of in-plane modulus of elasticity to transverse shear modulus and (b) that initial tension introduces a gap in the imaginary branch of the dispersion curve in which "trapped waves" cannot exist.