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.

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Abbreviations

a :

Length of plate

b :

Width of plate

h :

Number of series terms in the GITT

\(k_{s} ,k_{x}\) :

Inelastic buckling coefficients

\(k_{s}^{e} ,k_{x}^{e}\) :

Elastic buckling coefficients

m, n, r, s :

Positive integers

q :

Shape parameter to describe the curvature of stress–strain curve in the Ramberg–Osgood representation

\(\overline{q}\) :

Integer of corresponding q in the boundary of linear and bilinear approximations (\(R = 0.999\))

\(s_{ij} ,c_{i}\) :

Fundamental parameters to find \(S_{1}\), \(S_{2}\) and \(C\) (i, j = 1, 2)

t :

Thickness of plate

z :

Distance from the middle surface of plate

\(C\) :

Intercept of the second line in bilinear approximation of \(k_{s} - \xi\) curve

\(D_{ij}\) :

Arrays of stiffness matrix (i, j = 1, 2, 3)

\(E\) :

Young’s modulus (or the slop of stress–stain curve at zero stress)

\(E_{{\rm sec}}\) :

Secant modulus

\(E_{{\rm tan}}\) :

Tangent modulus

\(M_{mn}^{rs}\) :

Arrays of coefficient matrix (m, n, r, s = 1, 2, …, h)

\(N_{x}\), \(N_{y}\), \(N_{xy}\) :

In-plane loads in the x-, y- and xy-directions per unit length

\(R\) :

Correlation coefficient of linear approximation in linear least squares

\(S_{1}\), \(S_{2}\) :

Slope of the first and the second line for approximation of \(k_{s} - \xi\) curve

\(X_{m} \left( x \right)\), \(Y_{n} \left( y \right)\) :

Kernels of double integral transform in x- and y-direction (m, n = 1, 2, …, h)

\(\alpha_{m}\), \(\beta_{n}\) :

Roots of transcendental beam frequency equations in x- and y- directions (m, n = 1, 2, …, h)

\(\gamma\) :

Shear strain

\(\delta w\left( {x,y} \right)\) :

Variation of out of plane displacements in z- direction

\(\delta w_{mn}\) :

Variation of transformed out of plane displacements (m, n = 1, 2, …, h)

\(\delta M_{x}\), \(\delta M_{y}\) :

Variation of bending moments in the x- and y-directions per unit length

\(\delta M_{xy}\) :

Variation of twisting moment per unit length

\(\delta \gamma_{0}\) :

Variation of middle surface shear strain

\(\delta \varepsilon_{0x}\), \(\delta \varepsilon_{0y}\) :

Variation of middle surface strains in x- and y-directions

\(\delta \kappa_{x}\), \(\delta \kappa_{y}\) :

Variation of curvatures in x- and y-directions

\(\delta \kappa_{xy}\) :

Variation of twist

\(\delta \sigma_{x}\), \(\delta \sigma_{y}\) :

Variation of stresses in x- and y-directions

\(\delta \tau\) :

Variation of shear stress

\(\varepsilon_{x}\), \(\varepsilon_{y}\) :

Strain in x- and y-directions

\(\xi\) :

Secant modulus-to-Young’s modulus ratio

\(\eta\) :

Tangent modulus-to-Secant modulus ratio

\(\lambda\) :

Thickness ratio of plate

\(\nu\) :

Poisson’s ratio

\(\nu_{\rm e}\) :

Elastic Poisson’s ratio

\(\sigma_{.7E}\) :

Stress corresponding to intersection of the stress–strain curve and a secant of 0.7E in Ramberg–Osgood representation

\(\sigma_{i}\) :

Stress intensity

\(\sigma_{x}\), \(\sigma_{y}\) :

Stresses in x- and y-directions

\(\tau\) :

Shear stress

\(\sigma_{{x,{\text{cr}}}}\), \(\tau_{{{{\rm cr}}}}\) :

Critical stresses

\(\phi\) :

Aspect ratio of plate

\(\psi_{x}\), \(\psi_{y}\), \(\overline{\psi }_{y}\), \(\overline{\psi }_{xy}\) :

Load ratios

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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
figure8

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

Fig. 9
figure9

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

Fig. 10
figure10

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

Fig. 11
figure11

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

Fig. 12
figure12

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

Fig. 13
figure13

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

Fig. 14
figure14

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

Fig. 15
figure15

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

Fig. 16
figure16

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

Fig. 17
figure17

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

Fig. 18
figure18

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

Fig. 19
figure19

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 2023 and 2427 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
figure20

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

Fig. 21
figure21

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

Fig. 22
figure22

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

Fig. 23
figure23

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

Fig. 24
figure24

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

Fig. 25
figure25

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

Fig. 26
figure26

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

Fig. 27
figure27

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

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Jahanpour, A., Kouhia, R. An explicit solution for inelastic buckling of rectangular plates subjected to combined biaxial and shear loads. Acta Mech (2021). https://doi.org/10.1007/s00707-020-02926-x

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