An explicit solution for inelastic buckling of rectangular plates subjected to combined biaxial and shear loads

Abstract

In this study, the inelastic buckling equation of a thin plate subjected to all in-plane loads is analytically solved and the inelastic buckling coefficient is explicitly estimated. Using the deformation theory of plasticity, a multiaxial nonlinear stress–strain curve is supposed which is described by the Ramberg–Osgood representation and the von Mises criterion. Due to buckling, the variations are applied on the secant modulus, the Poisson’s ratio and the normal and shear strains. Then, the inelastic buckling equation of a perfect thin rectangular plate subjected to combined biaxial and shear loads is completely developed. Applying the generalized integral transform technique, the equation is straightforwardly converted to an eigenvalue problem in a dimensionless form. Initially, a geometrical solution and an algorithm are presented to find the lowest inelastic buckling coefficient $\left( {k_{s} } \right)$. The solution is successfully validated by some results in the literature. Then, a semi-analytical solution is proposed to simplify the calculation of $k_{s}$. The method of linear least squares is applied in two stages on the obtained results and an approximate polynomial equation is found which is usually solved by trial and error. The obtained results show good agreement between the proposed semi-analytical and geometrical methods, so that the differences are < 12%. The semi-analytical solution is easily programmed in usual scientific calculators and can be applied for practical purposes.

Appendices

Appendix 1: Linear/bilinear approximation of ${{\varvec{k}}}_{{\varvec{s}}}={\varvec{f}}\left({\varvec{\xi}};{\varvec{\phi}},{{\varvec{\psi}}}_{{\varvec{x}}},{{\varvec{\psi}}}_{{\varvec{y}}},{\varvec{q}},{{\varvec{\nu}}}_{{\varvec{e}}}\right)$

Supposing the boundary conditions of the plate and the specific values for $0<{\nu }_{\rm e}<0.5$, $1\le \phi \le 4$, $-1\le {\psi }_{x}\le 1$, $-1\le {\psi }_{y}\le 1$ and $2\le q\le 20$, the suggested algorithm (Fig. 2) is applied and several examples may be solved to obtain the curves of ${k}_{s}-\xi$. Figures 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19 show the obtained curves for some examples in which the curves of SSSS and CCCC plates are drawn in Figs. 8, 9, 10, 11, 12 and 13 and Figs. 14, 15, 16, 17, 18 and 19, respectively. In these figures, ${\nu }_{\rm e}=0.33$, $\phi =1, 1.5, 2, 4$, ${\psi }_{x}=-0.5, 1$, ${\psi }_{y}=-1, 1$ and $q=3, 10, 20$. Initially, the method of linear least squares (LLS) is used and the correlation coefficient (R) of linear estimation is obtained for each curve as shown in Figs. 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19. If $R\ge 0.999$ the linear estimation is proposed; otherwise, the bilinear estimation (Eq. (49)) is used to improve the approximation. In Figs. 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 and 19, the linear/bilinear approximations are only plotted for $\phi =1$ (the dashed lines). Similarly approximated curves can be evidently plotted for the other aspect ratios. Supposing constant values of $q$ and $\phi$ and increasing ${\psi }_{x}$ and ${\psi }_{y}$, the linear estimations are mostly converted to the bilinear estimations. If $R=0.999$, the boundary of conversion is found for which only the integer value of the corresponding $q$ is considered ($\stackrel{-}{q}$ in Tables 8, 9). For example, if $\phi =4$ and ${\psi }_{x}=$ ${\psi }_{y}=1$, then $\stackrel{-}{q}=5$ for SSSS plates; thus, if $q=3$ or $q=10$, then $R=0.9996$ (linear estimation, Fig. 9) or $R=0.9964$ (bilinear estimation, Fig. 11) respectively.

Fig. 8

Linear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Fig. 9

Bilinear and linear approximations of the ${k}_{s}-\xi$ curves for $\phi =1, 1.5, 2$ and $\phi =4$ respectively

Fig. 10

Linear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Fig. 11

Bilinear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Fig. 12

Linear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Fig. 13

Bilinear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Fig. 14

Linear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Fig. 15

Bilinear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Fig. 16

Linear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Fig. 17

Bilinear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Fig. 18

Linear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Fig. 19

Bilinear approximations of the ${k}_{s}-\xi$ curves for all aspect ratios

Appendix 2: Semi-logarithm estimation of ${{\varvec{S}}}_{1}$, ${{\varvec{S}}}_{2}$ and ${\varvec{C}}$

In Appendix 1 and Eq. (49), a bilinear approximation is described with slopes of both lines (${S}_{1}$ and ${S}_{2}$) and intercept of the second line ($C$), while a linear approximation is only described with the slope of one line (${S}_{1}$). Reapplying the method of linear least squares (LLS) on several examples, ${S}_{1}$, ${S}_{2}$ and $C$ can be linearly estimated versus $\mathrm{ln}q$. Figures 20–23 and 24–27 show the estimations for SSSS and CCCC plates, respectively. If linear approximation is applied on the ${k}_{s}-\xi$ curves, then ${S}_{1}$ is only estimated as shown in Figs. 20 and 24 (${\psi }_{x}=-0.5$, ${\psi }_{y}=-1$); if bilinear approximation is applied, then ${S}_{1}$ (Figs. 21, 25), ${S}_{2}$ (Figs. 22, 26) and $C$ (Figs. 23, 27) are estimated (${\psi }_{x}={\psi }_{y}=1$). Equation (54) shows the semi-logarithm estimation,

$$ \left\{ {\begin{array}{l} {S_{1} = s_{11} \ln q + s_{12} } \\ {S_{2} = s_{21} \ln q + s_{22} } \\ {C = c_{1} \ln q + c_{2} } \\ \end{array} } \right. $$

(54)

where ${s}_{11}$, ${s}_{21}$ and ${c}_{1}$ are the slopes and ${s}_{12}$, ${s}_{22}$ and ${c}_{2}$ are the intercept of the ${S}_{1}$, ${S}_{2}$ and $C$ curves, respectively. For SSSS plates with $\phi =1$, ${\psi }_{x}=-0.5$ and ${\psi }_{y}=-1$, Fig. 20 shows that ${s}_{11}=-1.294$ and ${s}_{12}=117.37$. Similarly, the parameters of Eq. (54) will be obtained for the different boundary and load conditions as shown in Tables 8 and 9. The obtained correlation coefficients show that the semi-logarithm estimation is acceptable in this step.

Fig. 20

Linear approximation of ${S}_{1}-\mathrm{ln}q$ in Figs. 8, 10 and 12

Fig. 21

Linear approximation of ${S}_{1}-\mathrm{ln}q$ in Figs. 9, 11 and 13

Fig. 22

Linear approximation of ${S}_{2}-\mathrm{ln}q$ in Figs. 9, 11 and 13

Fig. 23

Linear approximation of $C-\mathrm{ln}q$ in Figs. 9, 11 and 13

Fig. 24

Linear approximation of ${S}_{1}-\mathrm{ln}q$ in Figs. 14, 16 and 18

Fig. 25

Linear approximation of ${S}_{1}-\mathrm{ln}q$ in Figs. 15, 17 and 19

Fig. 26

Linear approximation of ${S}_{2}-\mathrm{ln}q$ in Figs. 15, 17 and 19

Fig. 27

Linear approximation of $C-\mathrm{ln}q$ in Figs. 15, 17 and 19