Refined theories may be needed for vibration analysis of structures with overhang

The e}ect of shear deformation and rotary inertia terms on the free vibration of a beam with overhang was investigated[ A recently proposed modi_ed Timoshenko!type equations of motion were used to analyze the vibration of the structure[ Two di}erent sets of boundary conditions\ with either a _xed or hinged end support\ were studied[ The results were compared with those obtained for the classical Bernoulli Euler beam theory[ The comparison shows that for a hinged end beam with very long overhang\ where the span between the supports is less than one tenth of the overall beam length\ the classical theory signi_cantly overestimates the values of the fundamental natural frequencies\ even for isotropic shear rigidity[ On the other hand\ the span e}ect is reversed for the clamped end beam\ for which a relatively signi_cant di}erence between the classical theory and shear theory results may occur only for a long span[ For transversely isotropic beamsthe re_ned theory predictions of the fundamental natural frequencies may be much smaller than those obtained through the rigid shear theory\ especially for short span hinged end beams and long span clamped end beams[