No abstract

The variational-asymptotic method is used to obtain an asymptotically-exact expression for the strain energy of a tapered strip-beam. The strip is assumed to be sufficiently thin to warrant the use of twodimensional elasticity. The taper is represented by a nondimensional constant of the same order as the ratio of the maximum cross-sectional width to the wavelength of the deformation along the beam, and thus its cube is negligible compared to unity. The resulting asymptotically-exact section constants, being functions of the taper parameter, are then used to find section constants for a generalized Timoshenko beam theory. These generalized Timoshenko section constants are then used in the associated one-dimensional beam equations to obtain the solution for the deformation of a linearly tapered beam subject to pure axial, pure bending, and transverse shear forces. These beam solutions are then compared with plane stress elasticity solutions, developed for extension, bending, and flexure of a linearly tapered isotropic strip. The agreement is excellent, and the results show that correction of the section constants using the taper parameter is necessary in order for beam theory to yield accurate results for a tapered beam. E I = Et (2b) 3 12 = 2Etb 3 3. Customarily, this expression remains the same regardless of whether or not the beam is uniform. For example, when b = b(x), one just replaces b with b(x); the local taper of the beam b (x) = − τ (x) does not further influence the local bending stiffness. [Boley 1963] showed that the accuracy of predictions by beam theory, performed in the described manner, worsened as τ increased.

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 crosssectional 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.A list of symbols can be found on page 975.

The variational-asymptotic method is used to obtain an asymptotically-exact expression for the strain energy of a tapered strip-beam. The strip is assumed to be sufficiently thin to warrant the use of twodimensional elasticity. The taper is represented by a nondimensional constant of the same order as the ratio of the maximum cross-sectional width to the wavelength of the deformation along the beam, and thus its cube is negligible compared to unity. The resulting asymptotically-exact section constants, being functions of the taper parameter, are then used to find section constants for a generalized Timoshenko beam theory. These generalized Timoshenko section constants are then used in the associated one-dimensional beam equations to obtain the solution for the deformation of a linearly tapered beam subject to pure axial, pure bending, and transverse shear forces. These beam solutions are then compared with plane stress elasticity solutions, developed for extension, bending, and flexure of a linearly tapered isotropic strip. The agreement is excellent, and the results show that correction of the section constants using the taper parameter is necessary in order for beam theory to yield accurate results for a tapered beam.