Elastic/plastic buckling analysis of skew plates under in-plane shear loading with incremental and deformation theories of plasticity by GDQ method

Abstract

The present study is concerned with the elastic/plastic buckling analysis of a skew plate under in-plane shear loading. The governing equations for moderately thick skew plates are analytically derived based on first-order shear deformation theory, whereas the incremental and deformation theories of plasticity are employed. Two types of shear loads, i.e. rectangular shear (R-shear) and skew shear (S-shear) have been investigated. The buckling coefficient values are significantly affected by the direction of stresses. Since the problem is geometrically and physically nonlinear, the generalized differential quadrature method as an accurate, simple and computationally efficient numerical tool is adopted to discretize the governing equations and the related boundary conditions. Then, a direct iterative method is employed to obtain the buckling coefficients of skew plates. To demonstrate the accuracy of the present analytical solution, a comparison is made with the published experimental and numerical results in literature. The influences of the aspect and thickness ratios, skew angle, incremental and deformation theories and various boundary conditions are examined for R-shear and S-shear buckling coefficients. Finally, some mode shapes of the skew thick plates are illustrated. The present results may serve as benchmark solutions for such plates.

Appendices

Appendix (A): The boundary conditions in this study

• (a) Clamped edge (C)

• for $\xi$ = 0 and $\xi$ = a

  $$w_{1j} = w_{{N_{\xi } j}} = 0,\quad \varphi_{1j}^{\xi } = \varphi_{{N_{\xi } j}}^{\xi } = 0,\quad \varphi_{1j}^{\eta } = \varphi_{{N_{\xi } j}}^{\eta } = 0,\quad \quad j = 1,\theta, \ldots ,N_{\eta } .$$

  (A.1)

• for $\eta$ = 0 and $\eta$ = b

  $$w_{i1} = w_{{iN_{\eta } }} = 0,\quad \varphi_{i1}^{\xi } = \varphi_{{iN_{\eta } }}^{\xi } = 0,\quad \varphi_{i1}^{\eta } = \varphi_{{iN_{\eta } }}^{\eta } = 0,\quad \quad i = 1, \ldots ,N_{\xi } .$$

  (A.2)

• (b) Simply supported edge (S)

• for $\xi$ = 0 and $\xi$ = a

  $$\begin{gathered} w_{1j} = w_{{N_{\xi } j}} = 0,\quad \varphi_{1j}^{\eta } = \varphi_{{N_{\xi } j}}^{\eta } = 0,\quad \quad i = 1, \ldots ,N_{\xi } ,\;j = 1, \ldots ,N_{\eta } . \hfill \\ (\alpha \cos^{2} (\theta ) + \beta \sin^{2} (\theta ))\sum\limits_{m = 1}^{{N_{\xi } }} {A_{im}^{\xi } } \varphi_{mj}^{\xi } - \beta \sin (\theta )\left(\sum\limits_{m = 1}^{{N_{\xi } }} {A_{im}^{\xi } } \varphi_{mj}^{\eta } + \sum\limits_{n = 1}^{{N_{\eta } }} {A_{jn}^{\eta } } \varphi_{in}^{\xi } \right) + \beta \sum\limits_{n = 1}^{{N_{\eta } }} {A_{jn}^{\eta } } \varphi_{in}^{\eta } = 0, \hfill \\ \end{gathered}$$

  (A.3)

• for $\eta$ = 0 and $\eta$ = b

  $$\begin{gathered} w_{i1} = w_{{iN_{\eta } }} = 0,\quad \varphi_{i1}^{\xi } = \varphi_{{iN_{\eta } }}^{\xi } = 0,\quad \quad i = 1, \ldots ,N_{\xi } ,\;j = 1, \ldots ,N_{\eta } . \hfill \\ (\beta \cos^{2} (\theta ) + \gamma \sin^{2} (\theta ))\sum\limits_{m = 1}^{{N_{\xi } }} {A_{im}^{\xi } } \varphi_{mj}^{\xi } - \gamma \sin (\theta )\left(\sum\limits_{m = 1}^{{N_{\xi } }} {A_{im}^{\xi } } \varphi_{mj}^{\eta } + \sum\limits_{n = 1}^{{N_{\eta } }} {A_{jn}^{\eta } } \varphi_{in}^{\xi }\right ) + \gamma \sum\limits_{n = 1}^{{N_{\eta } }} {A_{jn}^{\eta } } \varphi_{in}^{\eta } = 0. \hfill \\ \end{gathered}$$

  (A.4)

Appendix (B): The grid points employed in the computations are designed as follow: [30]

$$\xi_{i} = - \cos \left(\frac{i - 1}{{N_{\xi } - 1}}\pi \right), \quad i = 1,2,.. \ldots ,N_{\xi } .$$

(B.1)

$$\xi_{i} = \frac{1}{2}\left(1 - \cos\left (\frac{i - 1}{{N_{\xi } - 3}}\pi \right)\right), \quad i = 3, \ldots ..,N_{\xi } - 2.$$

(B.2)

$$\xi_{i} = \frac{1}{2}\left(1 - \cos \left(\frac{i - 2}{{N_{\xi } - 3}}\pi \right)\right), \quad i = 1,2, \ldots ..,N_{\xi } .$$

(B.3)

$$\xi_{i} = \frac{1}{2}\left(1 - \cos \left(\frac{2i - 1}{{2N_{\xi } }}\pi \right)\right), \quad i = 1,2,.. \ldots ,N_{\xi } .$$

(B.4)

The distributions of grid spacing of Chebyshev–Gauss–Lobatto (C-G-L) have the best convergence and highest accuracy [31, 32]. In this study, the following relation is used

$$\begin{gathered} \xi_{i} = \frac{1}{2}\left(1 - \cos \left(\frac{i - 1}{{N_{\xi } - 1}}\pi \right)\right),\,\,i = 1,2,.. \ldots ,N_{\xi } . \hfill \\ \eta_{j} = \frac{1}{2}\left(1 - \cos \left(\frac{j - 1}{{N_{\eta } - 1}}\pi \right) \right), \quad j = 1,2,.. \ldots ,N_{\eta } . \hfill \\ \end{gathered}$$

(B.5)