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. 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.

Stress and strain recovery for the in-plane deformation of an isotropic tapered strip-beam

“…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.…”

“…An asymptotic beam theory for an isotropic strip-beam with linearly tapered width was presented in [Hodges et al 2008]. Section stiffnesses for this theory depend on taper in ways other than the simplistic approach noted above.…”

Planar Timoshenko-like model for multilayer non-prismatic beams

“…Furthermore, generalizing the results already discussed by Auricchio et al (2015) and Balduzzi et al (2016), the horizontal stress r x could be seen as the independent variable that completely defines the stress state on the interlayer surfaces. Finally, generalizing the results discussed by Boley (1963) and Hodges et al (2008Hodges et al ( , 2010, the shear stress jumps within the crosssection depend on the variation of the mechanical properties of the material-determining the jumps of horizontal stress-and on the slopes of the interlayer surfaces h 0 i x ð Þ. Therefore the latter seem to be crucial for the determination of the beam behavior.…”

“…To the author's knowledge, the most enhanced modeling approaches that seem capable to overcome all the so far discussed limitations have been presented by Rubin (1999), Hodges et al (2008Hodges et al ( , 2010, Auricchio et al (2015), Beltempo et al (2015a), and Balduzzi et al (2016). In greater detail, Rubin (1999), Hodges et al (2008Hodges et al ( , 2010 limit their investigations to planar tapered beams whereas Auricchio et al (2015), Beltempo et al (2015a), and Balduzzi et al (2016) consider more complex geometries.…”

“…In greater detail, Rubin (1999), Hodges et al (2008Hodges et al ( , 2010 limit their investigations to planar tapered beams whereas Auricchio et al (2015), Beltempo et al (2015a), and Balduzzi et al (2016) consider more complex geometries. On the one hand, the beam model proposed by Rubin (1999) seems to achieve the best compromise between simplicity and effectiveness.…”