Improved torsional analysis of laminated box beams

Abstract

The improved torsional analysis of the laminated box beams with single- and double-celled sections subjected to a torsional moment is performed by introducing 14 displacement parameters. For this, a thin-walled laminated box beam theory considering the effects of shear and elastic couplings is presented. The governing equations and the force-displacement relations are derived from the variation of the strain energy. The system of linear algebraic equations with non-symmetric matrix is constructed by introducing the displacement parameters and by transforming the higher order simultaneous differential equations into first order ones. This numerical technique determines eigenmodes corresponding to 12 zero and 2 non-zero eigenvalues and derives displacement functions for displacement parameters based on the undetermined parameter method. Finally, the element stiffness matrix is determined using the member force-displacement relations. The theory developed by this study is validated by comparing several torsional responses from the present approach with those from the finite element beam model using the Lagrangian interpolation polynomials and three-dimensional analysis results using the shell elements of ABAQUS for coupled laminated beams with single- and double-celled sections.

Appendices

Appendix A: Stress resultants of the beam

(A.1)

where F _x and F _y are the shear forces in the x and y directions, respectively; F _z is the axial force; M _x and M _y are the bending moments about the x- and y-axes, respectively; M _ϕ is the bimoment; T and M _t are the two contributions to the total twisting moment.

Appendix B: Constitutive equations for the laminate box beam

$$ \left\{ \begin{array}{c} F_{z} \\ M_{x} \\ M_{y} \\ M_{\phi } \\ M_{t} \\ F_{x} \\ F_{y} \\ T \end{array} \right\} = \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} E_{11} & E_{12} & E_{13} & E_{14} & E_{15} & E_{16} & E_{17} & E_{18} \\ & E_{22} & E_{23} & E_{24} & E_{25} & E_{26} & E_{27} & E_{28} \\ & & E_{33} & E_{34} & E_{35} & E_{36} & E_{37} & E_{38} \\ & & & E_{44} & E_{45} & E_{46} & E_{47} & E_{48} \\ & & & & E_{55} & E_{56} & E_{57} & E_{58} \\ & & & & & E_{66} & E_{67} & E_{68} \\ & \mathrm{Symm}. & & & & & E_{77} & E_{78} \\ & & & & & & & E_{88} \end{array} \right] \left\{ \begin{array}{c} U'_{z} \\ \omega'_{x} \\ - \omega'_{y} \\ f' \\ \omega'_{p} - f \\ U'_{x} - \omega_{y} \\ U'_{y} + \omega_{x} \\ f + \omega'_{p} \end{array} \right\} $$

(A.2)

where E _ij are the laminate stiffnesses which depend on the cross-section of the laminated box beam.

Appendix C: Matrices X and E in Eqs. (38) and (42), respectively

(A.3)

where

(A.4)

and

(A.5)

(A.6)

where

(A.7)