Functionally Graded Beams as Surrogate Structural Models: Shear Beam with Exponentially Graded Rigidity

Abstract

An exact analytical solution, in terms of Bessel functions, was derived for the dynamic response of a shear beam with uniform cross section and mass density, and shear modulus that varies exponentially in the axial direction. The solution was generalized to a chain of such beam elements. The model may be appropriate for frame structures, which deform predominantly in shear, with gradually increasing rigidity toward the base. The results of a parametric study are presented for the frequencies of vibration and their ratios, mode shapes, transfer functions, and impulse response functions. The gradation affected the ratios between the frequencies of the higher modes of vibration and that of the fundamental mode, which were smaller than the corresponding ratios for a homogeneous shear beam and were progressively smaller for larger gradation constants. The mode shapes and the ratios of the amplitudes of the pulses in impulse response functions had amplitude amplification toward the top, which was larger for larger gradation constants, and suggests trapping of energy in the upper part of the structure. This effect may be responsible for observed damages and failure of structures in parts far from the base. The gradation also affected the symmetry and shape of the pulses in impulse response functions. The results of the parametric study may be useful for interpretation of the seismic response of full-scale structures. The model is intended for use as a simple surrogate model in structural system identification and health monitoring of frame structures.