The effect of taper on section constants for in-plane deformation of an isotropic strip

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.

Untitled

“…This paper focuses on the recovery of the stress, strain, and displacement fields for the linearly tapered isotropic strip-beam. This aspect was not addressed in [Hodges et al 2008]. The recovery is performed by the variationalasymptotic method (VAM) and is consistent with the derivation of the stiffness constants in that paper.…”

“…Section 2 of this paper revisits the previous work of [Hodges et al 2008] and reviews the importance of including taper in the stiffness constants. Section 3 provides details of the procedure to determine the recovery relations using the VAM and presents a comparison with the corresponding elasticity solutions.…”

“…For better understanding of the results to be presented, a brief review of the variational-asymptotic method and a summary of the results from [Hodges et al 2008] is presented here. The VAM is used to perform cross-sectional analysis of beams using the principle of minimum total potential energy, exploiting the presence of small parameters.…”

“…We now proceed to outline a procedure to obtain the cross-sectional constants using the VAM for a tapered-strip beam as in Figure 1. For details, the reader is encouraged to consult [Hodges et al 2008]. The two small parameters of the system are a slenderness parameter δ = a/l and a taper parameter τ = −b (x), which are assumed to be of the same asymptotic order.…”

Serviceability Analysis of Non-Prismatic Timber Beams: Derivation and Validation of New and Effective Straightforward Formulas

“…In recent years, several non-prismatic beam models have been proposed in an attempt to overcome the so far discussed problematic [21] [22] [23] [24]. Unfortunately, the most of them suffer from severe limitations e.g., they can tackle only symmetric and linearly tapered beams, present energy inconsistency, or lead to extremely complicated equations.…”