Analysis of transverse shear strains in pre-twisted thick beams using variational asymptotic method

Maqsood M. Ameen, Dineshkumar Harursampath

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The cross-sectional stiffness matrix is derived for a pre-twisted, moderately thick beam made of transversely isotropic materials and having rectangular cross sections. An asymptotically-exact methodology is used to model the anisotropic beam from 3-D elasticity, without any further assumptions. The beam is allowed to have large displacements and rotations, but small strain is assumed. The strain energy is computed making use of the beam constitutive law and kinematical relations derived with the inclusion of geometrical nonlinearities and an initial twist. The energy functional is minimized making use of the Variational Asymptotic Method (VAM), thereby reducing the cross section to a point on the beam reference line with appropriate properties, forming a 1-D constitutive law. VAM is a mathematical technique employed in the current problem to rigorously split the 3-D analysis of beams into two: a 2-D analysis over the beam cross-sectional domain, which provides a compact semi-analytical form of the properties of the cross sections, and a nonlinear 1-D analysis of the beam ref-erence curve. In this method, as applied herein, the cross-sectional analysis is performed asymptotically by taking advantage of a material small parameter and two geometric small parameters. 3-D strain components are derived using kinematics and arranged in orders of the small parameters. Closed-form expressions are derived for the 3-D non-linear warping and stress fields. Warping functions are obtained by the minimization of strain energy subject to certain set of constraints that render the 1-D strain measures well-defined. The zeroth-order 3-D warping field thus yielded is then used to integrate the 3-D strain energy density over the cross section, resulting in the 1-D strain energy density, which in turn helps identify the corresponding cross-sectional stiffness matrix. The model is capable of predicting interlaminar and transverse shear stresses accurately up to first order.

Original languageEnglish
Title of host publicationProceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014, ICNAAM 2014
PublisherAmerican Institute of Physics
Number of pages5
ISBN (Electronic)9780735412873
Publication statusPublished - 1 Jan 2015
Event12th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2014) - Rodos Palace Hotel, Rhodes, Greece
Duration: 22 Sep 201428 Sep 2014
Conference number: 12

Publication series

NameAIP Conference Proceedings


Conference12th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2014)
Abbreviated titleICNAAM 2014
Internet address


  • Advanced Beam Theories
  • Composite Mechanics
  • Laminated Beams
  • VAM


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