STRESS AND STRAIN RECOVERY FOR THE IN-PLANE DEFORMATION OF AN ISOTROPIC TAPERED STRIP-BEAM

The variational-asymptotic method was recently applied to create a beam theory for a thin strip-beam with a width that varies linearly with respect to the axial coordinate. For any arbitrary section, ratios of the cross-sectional stiffness coefficients to their customary values for a uniform beam depend on the rate of taper. This is because for a tapered beam the outward-directed normal to a lateral surface is not perpendicular to the longitudinal axis. This changes the lateral-surface boundary conditions for the cross-sectional analysis, in turn producing different formulae for the cross-sectional elastic constants as well as for recovery of stress, strain and displacement over a cross-section. The beam theory is specialized for the linear case and solutions are compared with those from plane-stress elasticity for stress, strain and displacement. The comparison demonstrates that for beam theory to yield such excellent agreement with elasticity theory, one must not only use cross-sectional elastic constants that are corrected for taper but also the corrected recovery formulae, which are in turn based on cross-sectional in-and out-of-plane warping corrected for taper.