An Exact Elasticity Solution for Monoclinic Functionally Graded Beams

Abstract

In this study, an elasticity solution is presented for monoclinic functionally graded beams subject to a transverse pressure distributed sinusoidally. Monoclinic material properties are assumed to vary exponentially throughout the thickness of the beam’s layers. An analytical formulation based on the classical Euler–Bernoulli beam theory is also derived for comparison purposes of simply supported monoclinic functionally graded beams. In benchmark examples, the numerical results of normal stresses, transverse shear stress, as well as axial and vertical displacements are presented. The effect of material grading, fiber angle, and beam length to thickness ratio on the stress and displacement distributions is comprehensively investigated. The proposed elasticity-based analytical solution and presented numerical results can be used for verification or comparison purposes of numerical procedures.

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Appendices

Appendix A: Details of some expressions

Expressions of \(m_{j}\), \(k_{j}\) and \(L_{j}\) appearing in Eqs. (12) and (13) are given below

$$ m_{j} = - \frac{{\overline{C}_{55}^{0} (k_{j} + n_{j} )\beta + (\overline{C}_{13}^{0} - \overline{C}_{33}^{0} k_{j} n_{j} )(n_{j} \beta + \gamma )}}{{\overline{C}_{45}^{0} n_{j} \beta + \overline{C}_{36}^{0} (n_{j} \beta + \gamma )}} $$
(A1)
$$ \begin{aligned} k_{j} & = \left[ {\overline{C}_{16}^{0} \beta ((\overline{C}_{36}^{0} + \overline{C}_{45}^{0} )n_{j} \beta + \overline{C}_{36}^{0} \gamma ) + \overline{C}_{13}^{0} (n_{j} \beta + \gamma )( - \overline{C}_{66}^{0} \beta + \overline{C}_{44}^{0} n_{j} (n_{j} \beta + \gamma ))} \right. \\ & \quad + \left. {n_{j} ( - \overline{C}_{45}^{0} (n_{j} \beta + \gamma )((\overline{C}_{36}^{0} + \overline{C}_{45}^{0} )n_{j} \beta + \overline{C}_{36}^{0} \gamma ) + \overline{C}_{55}^{0} \xi ( - \overline{C}_{66}^{0} \beta + \overline{C}_{44}^{0} n_{j} (n_{j} \beta + \gamma )))} \right]/ \\ & \quad \left[ {((\overline{C}_{36}^{0} + \overline{C}_{45}^{0} )^{2} n_{j}^{2} + (\overline{C}_{55}^{0} - \overline{C}_{33}^{0} n_{j}^{2} )(\overline{C}_{66}^{0} - \overline{C}_{44}^{0} n_{j}^{2} ))\beta^{2} + n_{j} ((\overline{C}_{36}^{0} + \overline{C}_{45}^{0} )^{2} - \overline{C}_{44}^{0} \overline{C}_{45}^{0} - \overline{C}_{33}^{0} \overline{C}_{66}^{0} + } \right. \\ & \quad + \left. {2\overline{C}_{33}^{0} \overline{C}_{44}^{0} n_{j}^{2} )\beta \gamma + (\overline{C}_{36}^{0} \overline{C}_{45}^{0} + \overline{C}_{33}^{0} \overline{C}_{44}^{0} n_{j}^{2} )\gamma^{2} } \right] \\ \end{aligned} $$
(A2)
$$ L_{1} = 3\gamma /\beta $$
(A3)
$$ \begin{aligned} L_{2} & = (\beta^{3} \left[ { - (\overline{C}_{13}^{0} )^{2} \overline{C}_{44}^{0} + \overline{C}_{11}^{0} \overline{C}_{33}^{0} \overline{C}_{44}^{0} - 2\overline{C}_{16}^{0} \overline{C}_{33}^{0} \overline{C}_{45}^{0} - (\overline{C}_{36}^{0} )^{2} \overline{C}_{55}^{0} + 2\overline{C}_{13}^{0} (\overline{C}_{36}^{0} \overline{C}_{45}^{0} + (\overline{C}_{45}^{0} )^{2} } \right. - \\ & \quad \left. {\overline{C}_{44}^{0} \overline{C}_{55}^{0} ) + \overline{C}_{33}^{0} \overline{C}_{55}^{0} \overline{C}_{66}^{0} } \right] + \gamma^{2} \beta \left[ {3\overline{C}_{33}^{0} + (\overline{C}_{45}^{0} )^{2} - 3\overline{C}_{33}^{0} \overline{C}_{44}^{0} \overline{C}_{55}^{0} } \right])/\left( {\beta^{3} \overline{C}_{33}^{0} \left[ {(\overline{C}_{45}^{0} )^{2} - \overline{C}_{44}^{0} \overline{C}_{55}^{0} } \right]} \right) \\ & \quad \left. {\overline{C}_{44}^{0} \overline{C}_{55}^{0} ) + \overline{C}_{33}^{0} \overline{C}_{55}^{0} \overline{C}_{66}^{0} } \right] + \gamma^{2} \beta \left[ {3\overline{C}_{33}^{0} + (\overline{C}_{45}^{0} )^{2} - 3\overline{C}_{33}^{0} \overline{C}_{44}^{0} \overline{C}_{55}^{0} } \right])/\left( {\beta^{3} \overline{C}_{33}^{0} \left[ {(\overline{C}_{45}^{0} )^{2} - \overline{C}_{44}^{0} \overline{C}_{55}^{0} } \right]} \right) \\ \end{aligned} $$
(A4)
$$ \begin{aligned} L_{3} & = (2\beta^{2} \gamma \left[ { - (\overline{C}_{13}^{0} )^{2} \overline{C}_{44}^{0} + \overline{C}_{11}^{0} \overline{C}_{33}^{0} \overline{C}_{44}^{0} - 2\overline{C}_{16}^{0} \overline{C}_{33}^{0} \overline{C}_{45}^{0} - } \right.(\overline{C}_{36}^{0} )^{2} \overline{C}_{55}^{0} + 2\overline{C}_{13}^{0} (\overline{C}_{36}^{0} \overline{C}_{45}^{0} + (\overline{C}_{45}^{0} )^{2} \\ & \quad - \left. {\overline{C}_{44}^{0} \overline{C}_{55}^{0} ) + \overline{C}_{33}^{0} \overline{C}_{55}^{0} \overline{C}_{66}^{0} } \right] + \gamma^{3} \overline{C}_{33}^{0} \left[ {(\overline{C}_{45}^{0} )^{2} - \overline{C}_{44}^{0} \overline{C}_{55}^{0} } \right])/\left( {\beta^{3} \overline{C}_{33}^{0} \left[ {(\overline{C}_{45}^{0} )^{2} - \overline{C}_{44}^{0} \overline{C}_{55}^{0} } \right]} \right) \\ \end{aligned} $$
(A5)
$$ \begin{aligned} L_{4} & = (\beta^{3} \left[ {(\overline{C}_{16}^{0} )^{2} \overline{C}_{33}^{0} } \right. - 2\overline{C}_{16}^{0} (\overline{C}_{13}^{0} (\overline{C}_{36}^{0} + \overline{C}_{45}^{0} ) + \overline{C}_{36}^{0} \overline{C}_{55}^{0} ) + \overline{C}_{13}^{0} (\overline{C}_{13}^{0} + 2\overline{C}_{55}^{0} )\overline{C}_{66}^{0} + \overline{C}_{11}^{0} ((\overline{C}_{36}^{0} )^{2} + \\ & \quad \left. { + 2\overline{C}_{36}^{0} \overline{C}_{45}^{0} + (\overline{C}_{45}^{0} )^{2} - \overline{C}_{44}^{0} \overline{C}_{55}^{0} - \overline{C}_{33}^{0} \overline{C}_{66}^{0} )} \right] + \gamma^{2} \beta \left[ { - (\overline{C}_{13}^{0} )^{2} } \right.\overline{C}_{44}^{0} + \overline{C}_{11}^{0} \overline{C}_{33}^{0} \overline{C}_{44}^{0} - 2\overline{C}_{16}^{0} \overline{C}_{33}^{0} \overline{C}_{45}^{0} \\ & \quad - \left. {(\overline{C}_{36}^{0} )^{2} \overline{C}_{55}^{0} + \overline{C}_{13}^{0} (2\overline{C}_{36}^{0} \overline{C}_{45}^{0} + 3(\overline{C}_{45}^{0} )^{2} - 3\overline{C}_{44}^{0} \overline{C}_{55}^{0} ) + \overline{C}_{33}^{0} \overline{C}_{55}^{0} \overline{C}_{66}^{0} } \right])/\left( {\beta^{3} \overline{C}_{33}^{0} \left[ {(\overline{C}_{45}^{0} )^{2} - \overline{C}_{44}^{0} \overline{C}_{55}^{0} } \right]} \right) \\ \end{aligned} $$
(A6)
$$ \begin{aligned} L_{5} & = (\gamma \beta^{2} \left[ {(\overline{C}_{16}^{0} )^{2} \overline{C}_{33}^{0} } \right. - 2\overline{C}_{16}^{0} (\overline{C}_{13}^{0} (\overline{C}_{36}^{0} + \overline{C}_{45}^{0} ) + \overline{C}_{36}^{0} \overline{C}_{55}^{0} ) + \overline{C}_{13}^{0} (\overline{C}_{13}^{0} + 2\overline{C}_{55}^{0} )\overline{C}_{66}^{0} + \overline{C}_{11}^{0} ((\overline{C}_{36}^{0} \\ & \quad + \left. {\overline{C}_{45}^{0} )^{2} - \overline{C}_{44}^{0} \overline{C}_{55}^{0} - \overline{C}_{33}^{0} \overline{C}_{66}^{0} )} \right] + \gamma^{3} \overline{C}_{13}^{0} \left[ {(\overline{C}_{45}^{0} )^{2} - \overline{C}_{44}^{0} \overline{C}_{55}^{0} } \right])/\left( {\beta^{3} \overline{C}_{33}^{0} \left[ {(\overline{C}_{45}^{0} )^{2} - \overline{C}_{44}^{0} \overline{C}_{55}^{0} } \right]} \right) \\ \end{aligned} $$
(A7)
$$ L_{6} = \beta^{3} \left[ { - (\overline{C}_{16}^{0} )^{2} + \overline{C}_{11}^{0} \overline{C}_{16}^{0} } \right] + \gamma^{2} \beta \left[ { - \overline{C}_{13}^{0} \overline{C}_{16}^{0} \overline{C}_{45}^{0} + \overline{C}_{11}^{0} \overline{C}_{36}^{0} \overline{C}_{45}^{0} - \overline{C}_{16}^{0} \overline{C}_{36}^{0} \overline{C}_{55}^{0} + \overline{C}_{13}^{0} \overline{C}_{55}^{0} \overline{C}_{66}^{0} } \right] $$
(A8)

Appendix B: Figures of displacement components along the beam axis

See Figs. 13 and 14

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Çömez, İ., Aribas, U.N., Kutlu, A. et al. An Exact Elasticity Solution for Monoclinic Functionally Graded Beams. Arab J Sci Eng (2021). https://doi.org/10.1007/s13369-021-05434-9

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Keywords

  • Functionally graded beam
  • Heterogeneous beam
  • Monoclinic material
  • Elasticity solution
  • Exact solution