Widad Ibraheem Majeed*
| Ibtehal Abbas Sadiq
© 2025 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
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This paper presents a combination of hyperbolic and polynomial four variable refined plate theory for first time to analyze buckling rectangular laminated plates with all simply supported edges. Parabolic variation of transverse shear stress over the thickness is presented to satisfy zero traction on the top and bottom surfaces of the plate. The governing equations solved for simply supported boundary conditions using Navier’s functions are formulated and based on the total potential energy. Changing design effects (aspect, thickness, orthotropic) ratio and layers scheme on the buckling load of laminated plates under uniaxial and biaxial loading conditions are investigated in detail. This theory gives good results when compared other theories for buckling of both thick and thin plates but there were changing in mode number for some cases.
laminated plate, new shear deformation theory, refined theory, uniaxial and biaxial buckling
Mechanical behavior of laminated plates is very important to allow safe structural design so that many researchers have searched using different methods such as theoretical (three and two) dimension elasticity theory, Numerical (finite element) solution with experimental methods. Akavci and Tanrikulu [1] used hyperbolic displacement models to investigate the buckling and frequency for simply supported [0/90] plate, the models used give good results when compared with other displacement. Aydogdu [2] improved a new higher order theory using three dimensions’ elasticity bending solutions to study buckling of plate, this theory gives results accurate to three-dimension elasticity solutions. Djedid et al. [3] developed a new refined plate theory to obtain the buckling of functionally graded simply supported plates, Wankhade and Niyogi [4] used Reddy higher order (HOST) as a refined plate theory to analyze buckling of composite plates, Schreiber and Mittelstedt [5] improved the analytical stability analysis of antisymmetric laminated structures, using two methods, first method based on (classical theory with first and third deformation theory). The second method based on reduced bending method. Tounsi et al. [6] presented a novel hyperbolic higher-order shear deformation theory (HSDT) for buckling analysis of functionally graded plates, Belbachir et al. [7] employed a refined plate theory based on a hyperbolic function to obtain buckling of cross-ply composite plates, Srivastava et al. [8] developed for the initial buckling response of two-directional functionally graded material (TDFGM) plate using energy principle and discretized of radial basis function (RBF) based on considering the higher theory, Kettaf et al. [9] used different theories of thick plates to analyze mechanical and thermal buckling of laminated. Nguyen et al. [10] used Airy's stress function to obtain the buckling load of functionally graded composite plates and solved using (ABAQUS) for buckling response of laminated plate combined with geometric nonlinearity, Di Sciuva and Sorrenti [11] presented zigzag theory to analysis buckling of functionally graded plate of carbon nanotube reinforcement. Sorrenti et al. [12] investigate buckling of angle-ply multilayered and sandwich plates using the enhanced Refined Zigzag Theory (en RZT), Majeed and Abed [13] used Rayleigh-Ritz solution depending on classical laminated plate theory to investigate buckling a laminated thin plate for different boundary conditions, Hashim and Sadiq [14] used a polynomial refined plate theory (RPT) to obtain the thermal buckling analysis behavior of laminated simply supported plates, Yahea and Majeed [15] investigated vibration of laminated plates under thermal load using refined theory. Majeed and Sadiq [16] analyzed buckling and fundamental frequencies of [0/90] composite plates using new higher order theory. Singh and Chakrabarti [17] carried out buckling analysis of laminated composite plates using finite element method based on zigzag theory. Nguyen-Van et al. [18] investigated buckling load of composite plate and shell by developing the flat element.
For present work, a new RPT plate theory was employed to address the mechanical buckling analysis of simply supported plates modelled with four unknowns and no need to use correction factor. The present theory considers a parabolic distribution of transverse shear stress in the thickness, which gives exactly the boundary conditions on the free surfaces of the plate. The derived equations used in this work based on Hamilton Principle of total potential energy. Next, theoretical analysis of the buckling plates subjected to biaxial and uniaxial loading conditions has been found using the Navier’s solution. The obtained results computed by the present model for critical buckling load are verified by comparing them with other results.
With the same mathematical model for displacement components of the plate, in present work, the shape function is chosen as a combination of hyperbolic and polynomial functions to satisfy the zero strain on the inner and outer surfaces of the plate is taken as [19]:
$\mathrm{u}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{u}(\mathrm{x}, \mathrm{y})-\mathrm{z}\left(\frac{\partial \mathrm{w}_{\mathrm{b}}}{\partial \mathrm{x}}\right)-\mathrm{F}(\mathrm{z})\left(\frac{\partial \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{x}}\right)$ (1)
$\mathrm{v}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{v}(\mathrm{x}, \mathrm{y})-\mathrm{z}\left(\frac{\partial \mathrm{w}_{\mathrm{b}}}{\partial \mathrm{x}}\right)-\mathrm{F}(\mathrm{z})\left(\frac{\partial \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{x}}\right)$ (2)
$\mathrm{w}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\mathrm{w}_{\mathrm{b}}(\mathrm{x}, \mathrm{y})+\mathrm{w}_{\mathrm{s}}(\mathrm{x}, \mathrm{y})$ (3)
where,
$\mathrm{F}(\mathrm{z})=\mathrm{z}-\mathrm{h} * \sinh \left(\frac{\mathrm{z}}{\mathrm{h}}\right)-\left(\frac{4}{3} * \frac{\mathrm{z}^3}{\mathrm{~h}^2}\right) \cosh (0.5)$ (4)
And $u, v, w_b, w_s$ the four unknown functions.
The strain-displacement relations are:
$\varepsilon_{\mathrm{xx}}=\frac{\partial \mathrm{u}}{\partial \mathrm{x}}$ (5)
$\varepsilon_{\mathrm{y y}}=\frac{\partial \mathrm{v}}{\partial \mathrm{y}}$ (6)
$\varepsilon_{\mathrm{zz}}=\frac{\partial \mathrm{w}}{\partial \mathrm{z}}$ (7)
$\varepsilon_{\mathrm{xy}}=\frac{1}{2} \gamma_{\mathrm{xy}}=\frac{1}{2}\left(\frac{\partial \mathrm{u}}{\partial \mathrm{y}}+\frac{\partial \mathrm{v}}{\partial \mathrm{x}}\right)$ (8)
$\varepsilon_{\mathrm{xz}}=\frac{1}{2} \gamma_{\mathrm{xz}}=\frac{1}{2}\left(\frac{\partial \mathrm{u}}{\partial \mathrm{z}}+\frac{\partial \mathrm{w}}{\partial \mathrm{x}}\right)$ (9)
$\varepsilon_{\mathrm{yz}}=\frac{1}{2}\left(\frac{\partial \mathrm{v}}{\partial \mathrm{z}}+\frac{\partial \mathrm{w}}{\partial \mathrm{y}}\right)=\frac{1}{2} \gamma_{\mathrm{yz}}$ (10)
Substituting Eqs. (1)-(3) into Eqs. (5)-(10) to give:
$\varepsilon_{\mathrm{xx}}=\varepsilon_{\mathrm{xx}}^0-\mathrm{z} \varepsilon_{\mathrm{xx}}^1-\mathrm{F}(\mathrm{z}) \varepsilon_{\mathrm{xx}}^2$ (11)
$\varepsilon_{\mathrm{yy}}=\varepsilon_{\mathrm{yy}}^0-\mathrm{z} \varepsilon_{\mathrm{yy}}^1-\mathrm{F}(\mathrm{z}) \varepsilon_{\mathrm{yy}}^2$ (12)
$\varepsilon_{\mathrm{xy}}=\varepsilon_{\mathrm{xy}}^0-\mathrm{z} \varepsilon_{\mathrm{xy}}^1-\mathrm{F}(\mathrm{z}) \varepsilon_{\mathrm{xy}}^2$ (13)
$\gamma_{\mathrm{xz}}=\varepsilon_{\mathrm{xz}}^0-\mathrm{g}(\mathrm{z}) \varepsilon_{\mathrm{xz}}^3$ (14)
$\gamma_{\mathrm{yz}}=\varepsilon_{\mathrm{yz}}^0-\mathrm{g}(\mathrm{z}) \varepsilon_{\mathrm{yz}}^3$ (15)
where,
$\left\{\begin{array}{c}\varepsilon_{\mathrm{xx}}^0 \\ \varepsilon_{\mathrm{yy}}^0 \\ \gamma_{\mathrm{xy}}^0\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial \mathrm{u}}{\partial \mathrm{x}} \\ \frac{\partial \mathrm{u}}{\partial \mathrm{x}} \\ \frac{\partial \mathrm{u}}{\partial \mathrm{y}}+\frac{\partial \mathrm{v}}{\partial \mathrm{x}}\end{array}\right\}$ (16)
$\left\{\begin{array}{c}\varepsilon_{\mathrm{xx}}^1 \\ \varepsilon_{\mathrm{yy}}^1 \\ \gamma_{\mathrm{xy}}^1\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial^2 \mathrm{w}_{\mathrm{b}}}{\partial \mathrm{x}^2} \\ \frac{\partial^2 \mathrm{w}_{\mathrm{b}}}{\partial \mathrm{y}^2} \\ 2 \frac{\partial^2 \mathrm{w}_{\mathrm{b}}}{\partial \mathrm{x} \partial \mathrm{y}}\end{array}\right\}$ (17)
$\left\{\begin{array}{c}\varepsilon_{\mathrm{xx}}^2 \\ \varepsilon_{\mathrm{yy}}^2 \\ \gamma_{\mathrm{xy}}^2\end{array}\right\}=\left\{\begin{array}{c}\frac{\partial^2 \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{x}^2} \\ \frac{\partial^2 \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{y}^2} \\ 2 \frac{\partial^2 \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{x} \partial \mathrm{y}}\end{array}\right\}$ (18)
$\left\{\begin{array}{c}\gamma_{\mathrm{xz}}^0 \\ \gamma_{\mathrm{yz}}^0\end{array}\right\}=\left\{\begin{array}{l}\frac{\delta \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{x}} \\ \frac{\delta \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{y}}\end{array}\right\}$ (19)
$g(z)=1-F^{\prime}(z)$ （20)
Using refined theory to derive equations of motion with the principle of virtual displacements [20]:
$0=\int_0^{\mathrm{v}}(\delta \mathrm{U}+\delta \mathrm{V}) \mathrm{dv}$ (21)
where:
δU: virtual strain energy and
δV: virtual work done by applied forces.
$\begin{gathered}\delta \mathrm{U}=\left[\int_{\frac{-\mathrm{h}}{2}}^{\frac{\mathrm{h}}{2}}\left\{\int_{\Omega}^{\mathrm{k}} \sigma_{\mathrm{xx}} \delta \varepsilon_{\mathrm{xx}}^{\mathrm{k}}+\sigma_{\mathrm{yy}} \delta \varepsilon_{\mathrm{yy}}^{\mathrm{k}}+\sigma_{\mathrm{xy}} \delta \varepsilon_{\mathrm{xy}}^{\mathrm{k}}+\sigma_{\mathrm{yz}} \delta \varepsilon_{\mathrm{yz}}^{\mathrm{k}}\right.\right. \\ \left.\left.\left.+\sigma_{\mathrm{xy}} \delta \varepsilon_{\mathrm{xy}}^{\mathrm{k}}\right] \mathrm{dxdy}\right\} \mathrm{dxdy}\right]=0\end{gathered}$ (22)
$\begin{aligned} \delta \mathrm{U}=\int\left(\mathrm{N}_1 \delta \varepsilon_{\mathrm{xx}}^0+\right. & \mathrm{M}_1^{\mathrm{b}} \delta \varepsilon_{\mathrm{xx}}^1+\mathrm{M}_1^{\mathrm{s}} \delta \varepsilon_{\mathrm{xx}}^2+\mathrm{N}_2 \delta \varepsilon_{\mathrm{yy}}^0 \\ & +\mathrm{M}_2^{\mathrm{b}} \delta \varepsilon_{\mathrm{yy}}^1+\mathrm{M}_2^{\mathrm{s}} \delta \varepsilon_{\mathrm{yy}}^2+\mathrm{N}_6 \delta \varepsilon_{\mathrm{xy}}^0 \\ & +\mathrm{M}_6^{\mathrm{b}} \delta \varepsilon_{\mathrm{xy}}^1+\mathrm{M}_6^{\mathrm{s}} \delta \varepsilon_{\mathrm{xy}}^2+\mathrm{Q}_5 \delta \varepsilon_{\mathrm{yz}}^0 \\ & \left.+\mathrm{Q}_4 \delta \varepsilon_{\mathrm{xz}}^0\right) \mathrm{dxdy}\end{aligned}$ (23)
where,
$(\mathrm{Ni}, \mathrm{Mi}, \mathrm{Pi})=\sum_{\mathrm{k}=1}^{\mathrm{N}} \int_{\mathrm{z}^{\mathrm{k}-1}}^{\mathrm{z}^{\mathrm{k}}} \sigma_{\mathrm{i}}^{\mathrm{k}}(1, \mathrm{z}, \mathrm{F}(\mathrm{z})) \mathrm{dz}, \quad \mathrm{i}=(1,2,6)$
$(\mathrm{Qi})=\sum_{\mathrm{k}=1}^{\mathrm{N}} \int_{\mathrm{z}^{\mathrm{k}-1}}^{\mathrm{z}^{\mathrm{k}}} \sigma_{\mathrm{i}}^{\mathrm{k}}\left(\mathrm{g}^2\right) \mathrm{dz} \quad(\mathrm{i}=4,5)$
$\begin{aligned} \delta \mathrm{V}=\int_{\Omega}\left[\mathrm{N}_{\mathrm{xx}} \delta\left(\frac{\partial^2 \mathrm{w}_{\mathrm{b}}}{\partial \mathrm{x}^2}\right.\right. & \left.+\frac{\partial^2 \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{x}^2}\right) \left.\quad+\mathrm{N}_{\mathrm{yy}} \delta\left(\frac{\partial^2 \mathrm{w}_{\mathrm{b}}}{\partial \mathrm{y}^2}+\frac{\partial^2 \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{y}^2}\right)\right] \mathrm{dx} \mathrm{dy}\end{aligned}$ (24)
Substituting Eqs. (11)-(15) into Eqs. (23) and (24) and integrating by parts to get energy equation in form of displacement components and resultant forces.
Substituting Eqs. (23) and (24) in Eq. (21) and to give four equations of motion as follows:
$\delta \mathrm{u}: \frac{\partial \mathrm{N}_1}{\partial \mathrm{x}}+\frac{\partial \mathrm{N}_6}{\partial \mathrm{y}}=0$ (25)
$\delta \mathrm{v}: \frac{\partial \mathrm{N}_2}{\partial \mathrm{y}}+\frac{\partial \mathrm{N}_6}{\partial \mathrm{x}}=0$ (26)
$\begin{aligned} \delta \mathrm{wb}: \frac{\partial^2 \mathrm{M}_1^{\mathrm{b}}}{\partial \mathrm{x}^2}+\frac{\partial^2 \mathrm{M}_2^{\mathrm{b}}}{\partial \mathrm{y}^2} & +2 \frac{\partial^2 \mathrm{M}_6^{\mathrm{b}}}{\partial \mathrm{x} \partial \mathrm{y}} +\left(\mathrm{N}_{\mathrm{xx}} \frac{\partial^2 \mathrm{w}_{\mathrm{b}}}{\partial \mathrm{x}^2}+\mathrm{N}_{\mathrm{yy}} \frac{\partial^2 \mathrm{w}_{\mathrm{b}}}{\partial \mathrm{y}^2}\right)=0\end{aligned}$ (27)
$\begin{aligned} \delta \mathrm{ws}: \frac{\partial^2 \mathrm{M}_1^{\mathrm{s}}}{\partial \mathrm{x}^2}+ & \frac{\partial^2 \mathrm{M}_2^{\mathrm{s}}}{\partial \mathrm{y}^2}+2 \frac{\partial^2 \mathrm{M}_6^{\mathrm{s}}}{\partial \mathrm{x} \partial \mathrm{y}} +\left(\mathrm{N}_{\mathrm{xx}} \frac{\partial^2 \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{x}^2}+\mathrm{N}_{\mathrm{yy}} \frac{\partial^2 \mathrm{w}_{\mathrm{s}}}{\partial \mathrm{y}^2}\right)=0\end{aligned}$ (28)
The result forces are given by:
$\left\{\begin{array}{l}\mathrm{N}_1 \\ \mathrm{~N}_2 \\ \mathrm{~N}_6\end{array}\right\}=\sum_{\mathrm{k}=1}^{\mathrm{N}} \int_{\mathrm{z}^{\mathrm{k}}}^{\mathrm{z}^{\mathrm{k}+1}}\left\{\begin{array}{l}\sigma_1 \\ \sigma_2 \\ \sigma_6\end{array}\right\} \mathrm{dz}$ (29)
$\left\{\begin{array}{l}\mathrm{M}_1^{\mathrm{b}} \\ \mathrm{M}_2^{\mathrm{b}} \\ \mathrm{M}_6^{\mathrm{b}}\end{array}\right\}=\sum_{\mathrm{k}=1}^{\mathrm{N}} \int_{\mathrm{z}^{\mathrm{k}}}^{\mathrm{z}^{\mathrm{k}+1}}\left\{\begin{array}{l}\sigma_1 \\ \sigma_2 \\ \sigma_6\end{array}\right\} \mathrm{zdz}$, (30)
$\left\{\begin{array}{l}\mathrm{M}_1^{\mathrm{s}} \\ \mathrm{M}_2^{\mathrm{s}} \\ \mathrm{M}_6^{\mathrm{s}}\end{array}\right\} \sum_{\mathrm{k}=1}^{\mathrm{N}} \int_{\mathrm{z}^{\mathrm{k}}}^{\mathrm{z}^{\mathrm{k}+1}}\left\{\begin{array}{l}\sigma_1 \\ \sigma_2 \\ \sigma_6\end{array}\right\} \mathrm{F}(\mathrm{z}) \mathrm{dz}$ (31)
$\left\{\begin{array}{l}\mathrm{Q}_4 \\ \mathrm{Q}_5\end{array}\right\}=\sum_{\mathrm{k}=1}^{\mathrm{n}}\left\{\begin{array}{l}\sigma_5 \\ \sigma_4\end{array}\right\}\left(\mathrm{g}^2\right) \mathrm{dz}$ (32)
The plane stress reduced stiffnes Qij is:
$\begin{gathered} \mathrm{Q}_{11} = \frac{\mathrm{E}_1}{1 - \mathrm{v}_{12} \mathrm{v}_{21}}, \quad \mathrm{Q}_{12} = \frac{\mathrm{v}_{12} \mathrm{E}_2}{1 - \mathrm{v}_{12} \mathrm{v}_{21}} \\ \mathrm{Q}_{22} = \frac{\mathrm{E}_2}{1 - \mathrm{v}_{12} \mathrm{v}_{21}}, \quad \mathrm{Q}_{66} = \mathrm{G}_{12}, \quad \mathrm{Q}_{44} = \mathrm{G}_{23} \\ \mathrm{Q}_{55} = \mathrm{G}_{13} \end{gathered}$ (33)
where: G (shear modulus), E (Young’s modulus) and υ (poison’s ratio) of plate.
The transformed stress-strain relation is:
$\begin{aligned}\left\{\begin{array}{l}\sigma_{\mathrm{xx}} \\ \sigma_{\mathrm{yy}} \\ \sigma_{\mathrm{xy}}\end{array}\right\}= & {\left[\begin{array}{l}\mathrm{Q}_{11} \mathrm{Q}_{12} \mathrm{Q}_{16} \\ \mathrm{Q}_{12} \mathrm{Q}_{22} \mathrm{Q}_{26} \\ \mathrm{Q}_{16} \mathrm{Q}_{26} \mathrm{Q}_{66}\end{array}\right]\left\{\begin{array}{l}\varepsilon_{\mathrm{xx}} \\ \varepsilon_{\mathrm{yy}} \\ \gamma_{\mathrm{xy}}\end{array}\right\}, } \\ & \left\{\begin{array}{l}\sigma_{\mathrm{yz}} \\ \sigma_{\mathrm{xz}}\end{array}\right\}=\left[\begin{array}{ll}\mathrm{Q}_{44} & \mathrm{Q}_{45} \\ \mathrm{Q}_{45} & \mathrm{Q}_{55}\end{array}\right]\left\{\begin{array}{l}\gamma_{\mathrm{yz}} \\ \gamma_{\mathrm{xz}}\end{array}\right\}\end{aligned}$ (34)
The force results are:
$\begin{aligned}\left\{\begin{array}{l}\mathrm{N}_1 \\ \mathrm{~N}_2 \\ \mathrm{~N}_6\end{array}\right\} & =\left[\begin{array}{l}\mathrm{A}_{11} \mathrm{~A}_{12} \mathrm{~A}_{16} \\ \mathrm{~A}_{12} \mathrm{~A}_{22} \mathrm{~A}_{26} \\ \mathrm{~A}_{16} \mathrm{~A}_{26} \mathrm{~A}_{66}\end{array}\right]\left\{\begin{array}{l}\varepsilon_{x x}^0 \\ \varepsilon_{y y}^0 \\ \gamma_{x y}^0\end{array}\right\} \\ & +\left[\begin{array}{l}\mathrm{B}_{11} \mathrm{~B}_{12} \mathrm{~B}_{16} \\ \mathrm{~B}_{12} \mathrm{~B}_{22} \mathrm{~B}_{26} \\ \mathrm{~B}_{16} \mathrm{~B}_{26} \mathrm{~B}_{66}\end{array}\right]\left\{\begin{array}{l}\varepsilon_{x x}^1 \\ \varepsilon_{y y}^1 \\ \gamma_{x y}^1\end{array}\right\} \\ & +\left[\begin{array}{l}\mathrm{E}_{11} \mathrm{E}_{12} \mathrm{E}_{16} \\ \mathrm{E}_{12} \mathrm{E}_{22} \mathrm{E}_{26} \\ \mathrm{E}_{16} \mathrm{E}_{26} \mathrm{E}_{66}\end{array}\right]\left\{\begin{array}{l}\varepsilon_{x x}^2 \\ \varepsilon_{y y}^2 \\ \gamma_{x y}^2\end{array}\right\}\end{aligned}$ (35)
$\begin{gathered}\left\{\begin{array}{c}\mathrm{M}_1^{\mathrm{b}} \\ \mathrm{M}_2^{\mathrm{b}} \\ \mathrm{M}_6^{\mathrm{b}}\end{array}\right\}=\left[\begin{array}{l}\mathrm{B}_{11} \mathrm{~B}_{12} \mathrm{~B}_{16} \\ \mathrm{~B}_{12} \mathrm{~B}_{22} \mathrm{~B}_{26} \\ \mathrm{~B}_{16} \mathrm{~B}_{26} \mathrm{~B}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{\mathrm{xx}}^0 \\ \varepsilon_{\mathrm{yy}}^0 \\ \gamma_{\mathrm{xy}}^0\end{array}\right\}+\left[\begin{array}{l}\mathrm{D}_{11} \mathrm{D}_{12} \mathrm{D}_{16} \\ \mathrm{D}_{12} \mathrm{D}_{22} \mathrm{D}_{26} \\ \mathrm{D}_{16} \mathrm{D}_{26} \mathrm{D}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{\mathrm{xx}}^1 \\ \varepsilon_{\mathrm{yy}}^1 \\ \gamma_{\mathrm{xy}}^1\end{array}\right\} \\ +\left[\begin{array}{l}\mathrm{F}_{11} \mathrm{~F}_{12} \mathrm{~F}_{16} \\ \mathrm{~F}_{12} \mathrm{~F}_{22} \mathrm{~F}_{26} \\ \mathrm{~F}_{16} \mathrm{~F}_{26} \mathrm{~F}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{\mathrm{xx}}^2 \\ \varepsilon_{\mathrm{yy}}^2 \\ \gamma_{\mathrm{xy}}^2\end{array}\right\}\end{gathered}$ (36)
$\begin{gathered}\left\{\begin{array}{c}\mathrm{M}_1^{\mathrm{s}} \\ \mathrm{M}_2^{\mathrm{s}} \\ \mathrm{M}_6^{\mathrm{s}}\end{array}\right\}=\left[\begin{array}{l}\mathrm{E}_{11} \mathrm{E}_{12} \mathrm{E}_{16} \\ \mathrm{E}_{12} \mathrm{E}_{22} \mathrm{E}_{26} \\ \mathrm{E}_{16} \mathrm{E}_{26} \mathrm{E}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{\mathrm{xx}}^0 \\ \varepsilon_{\mathrm{yy}}^0 \\ \gamma_{\mathrm{xy}}^0\end{array}\right\}+\left[\begin{array}{l}\mathrm{F}_{11} \mathrm{~F}_{12} \mathrm{~F}_{16} \\ \mathrm{~F}_{12} \mathrm{~F}_{22} \mathrm{~F}_{26} \\ \mathrm{~F}_{16} \mathrm{~F}_{26} \mathrm{~F}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{\mathrm{xx}}^1 \\ \varepsilon_{\mathrm{yy}}^1 \\ \gamma_{\mathrm{xy}}^1\end{array}\right\} \\ +\left[\begin{array}{l}\mathrm{H}_{11} \mathrm{H}_{12} \mathrm{H}_{16} \\ \mathrm{H}_{12} \mathrm{H}_{22} \mathrm{H}_{26} \\ \mathrm{H}_{16} \mathrm{H}_{26} \mathrm{H}_{66}\end{array}\right]\left\{\begin{array}{c}\varepsilon_{\mathrm{xx}}^2 \\ \varepsilon_{\mathrm{yy}}^2 \\ \gamma_{\mathrm{xy}}^2\end{array}\right\}\end{gathered}$ (37)
$\left\{\begin{array}{l}\mathrm{Q}_4 \\ \mathrm{Q}_5\end{array}\right\}=\left[\begin{array}{ll}\mathrm{L}_{44} & \mathrm{~L}_{45} \\ \mathrm{~L}_{45} & \mathrm{~L}_{55}\end{array}\right]\left\{\begin{array}{c}\gamma_{\mathrm{yz}}^0 \\ \gamma_{\mathrm{xz}}^0\end{array}\right\}$ (38)
where,
$A_{\mathrm{ij}}=\int_{\frac{-h}{2}}^{\frac{h}{2}} Q_{\mathrm{ij}} \, \mathrm{d}z \quad \mathrm{i}=(1,2,4,5,6)$ (39)
$\begin{aligned} \mathrm{B}_{\mathrm{ij}}, \mathrm{D}_{\mathrm{ij}}, \mathrm{E}_{\mathrm{ij}}, \mathrm{F}_{\mathrm{ij}}, \mathrm{H}_{\mathrm{ij}} & =\int_{\frac{-\mathrm{h}}{2}}^{\frac{\mathrm{h}}{2}} \mathrm{Q}_{\mathrm{ij}}\left(\mathrm{z}, \mathrm{z}^2, \mathrm{~F}(\mathrm{z}), \mathrm{z}\right. \\ & \left.* \mathrm{~F}(\mathrm{z}),(F(z))^2\right) \mathrm{dz} \quad \mathrm{i}=(1,2,6)\end{aligned}$ (40)
$\mathrm{L}_{\mathrm{ij}}=\int_{\frac{-\mathrm{h}}{2}}^{\frac{\mathrm{h}}{2}} \mathrm{Q}_{\mathrm{ij}}\left(\mathrm{g}^2\right) \mathrm{dz} \quad \mathrm{i}, \mathrm{j}=(4,5)$ (41)
Substituting Navier's equations which satisfied simply supported boundary conditions [20], for (cross – angle) ply with the force and moment resultants from Eqs. (29)-(32) into equations of motion Eqs. (25)-(28), the following eigenvalue equation is obtained:
$\left[\begin{array}{cccc}\mathrm{C}_{11} & \mathrm{C}_{12} & \mathrm{C}_{13} & \mathrm{C}_{14} \\ - & \mathrm{C}_{22} & \mathrm{C}_{23} & \mathrm{C}_{24} \\ - & - & \mathrm{C}_{33}-\left(\alpha^2 * \mathrm{~N}_{\mathrm{x}}+\mathrm{k} * \beta^2 * \mathrm{~N}_{\mathrm{y}}\right) & \mathrm{C}_{34} \\ - & - & - & \mathrm{C}_{44}-\left(\alpha^2 * \mathrm{~N}_{\mathrm{x}}+\mathrm{k} * \beta^2 * \mathrm{~N}_{\mathrm{y}}\right)\end{array}\right]\left\{d_{i j}\right\}=0$
where,
$\left\{\mathrm{d}_{\mathrm{ij}}\right\}=\left\{\mathrm{U}_{\mathrm{mn}}, \mathrm{V}_{\mathrm{mn}}, \mathrm{W}_{\mathrm{bmn}}, \mathrm{W}_{\mathrm{smn}}\right\}$
Cij= stiffness element which given in appendix.
A comparison of buckling load for laminated plates using a combination of hyperbolic and polynomial displacement function [19], with other plate theories and solving techniques are investigated also plates with different thickness (a/h) ratio, orthotropic ratio, number of plies, loading conditions and aspect (a/b) ratio are studied and solved by Matlab22 program.
Different theories and solution methods used by other researchers are compared to present theory which takes little efforts than other analytical or numerical methods based on five variables refined or third order plate theories as shown in Table 1, which present a comparison between present four variable refined theory, TSDT and finite element method for [0/90] square plate for different (a/h), the results give good agreement and mode number not changed.
Table 1. Effect of thickness ratios (a/h) on nondimentional critical uni-axial buckling loads (Ncr) for simply supported [0/90] square plate
|
Source |
(a/h) |
|||
|
(10) |
(20) |
(50) |
(100) |
|
|
Our Work |
11.126 |
12.452 |
12.884 |
12.948 |
|
Ref. [16] |
11.616 |
12.602 |
12.910 |
12.955 |
|
Ref. [17] |
11.310 |
12.427 |
12.800 |
12.873 |
|
Ref. [18] |
11.360 |
12.551 |
12.906 |
13.039 |
|
Ref. [21] |
11.349 |
12.510 |
12.879 |
12.934 |
|
Ref. [22] |
11.563 |
12.577 |
12.895 |
12.942 |
Notes: Using material 1. Mode for all: (q=s=1) accept when written.
Table 2. Effect of thickness ratios (a/h) on normalized critical uni-axial buckling loads (Ncr) for angle-ply square plate, using material 2. Mode for all: (q=s=1) accept when written
|
a/h |
Source |
Layers Type |
||
|
[5 /-5]6 |
[30/ -30]6 |
[45/ -45]6 |
||
|
5 |
Ref. [20] |
11.082 |
13.546 |
12.169 |
|
Present work |
13.572 q=2, s=1 |
13.357 q=2, s =1 |
12.7635 q=3, s =1 |
|
|
10 |
Ref. [20] |
22.592 |
33.701 |
32.405 |
|
Present work |
26.399 |
35.806 |
34.680 q=2, s =1 |
|
|
20 |
Ref. [20] |
31.577 |
47.643 |
53.198 |
|
Present work |
33.628 |
50.572 |
56.166 |
|
|
50 |
Ref. [20] |
35.657 |
53.951 |
60.760 |
|
Present work |
36.434 |
57.227 |
64.496 |
|
Notes: Using material 2. Mode for all: (q=s=1) accept when written.
Table 2 presents buckling load for different angle plates and give good agreement with those obtained by other researchers used TSDT, but with changing mode number for some thick and moderately thick plates. Table 3 shows the effect of (b/a) for the laminated plate on buckling load which give the same behavior to those obtained by other researchers, also changing orthotropic ratio (E1/E2) shown in Table 4, but under biaxial in plane loading and give results close to those obtained by studies [22].
Table 3. Nondimentional uni-axial buckling loads (Ncr) for different (a/h) and (a/b) for [0/90]s square plate
|
a/b |
Source |
a/h |
||||
|
0.5 |
Ref. [16] |
5 |
10 |
20 |
50 |
100 |
|
8.848 |
18.488 |
25.856 |
29.151 |
29.693 |
||
|
Ref. [17] |
8.739 |
18.347 |
25.746 |
29.087 |
29.657 |
|
|
Present work |
9.113 |
18.931 |
26.092 |
29.200 |
29.706 |
|
|
1 |
Ref. [16] |
12.029 |
23.394 |
31.716 |
35.400 |
36.005 |
|
Ref. [17] |
11.858 |
23.134 |
31.517 |
35.278 |
35.923 |
|
|
Present work |
11.614 |
23.519 |
31.888 |
35.442 |
36.016 |
|
|
2 |
Ref. [16] |
16.681 (q=3, s=1) |
48.119 (q=2, s=1) |
93.579 (q=2, s=1) |
113.21 (q=1, s=1) |
115.335 (q=1, s=1) |
|
Ref. [17] |
15.000 |
47.368 |
92.847 |
112.81 |
115.029 |
|
|
Present work |
15.631 q=3, s=1 |
46.458 q=2, s=1 |
92.572 |
111.52 |
114.891 |
|
Notes: Using material 1. Mode for all: (q=s=1) accept when written.
Table 4. Normalized critical biaxial buckling load for different thickness and orthotropic ratios for different angle-lamination square plate
|
a/h |
Source |
E1/E2=10 |
|
|
[45/ -45] |
[45/ -45]4 |
||
|
10 |
Ref. [22] |
3.923 |
6.771 |
|
Present work |
4.043 |
6.994 |
|
|
100 |
Ref. [22] |
4.526 |
8.792 |
|
Present work |
4.542 |
8.844 |
|
|
a/h |
Source |
E1/E2=25 |
|
|
[45/ -45] |
[45/ -45]4 |
||
|
10 |
Ref. [22] |
6.115 |
12.067 |
|
Present work |
6.468 |
12.715 |
|
|
100 |
Ref. [22] |
7.717 |
20.437 |
|
Present work |
7.735 |
20.502 |
|
Notes: Using material 1. Mode for all: (q=s=1) accept when written.
Buckling load for symmetric cross and angle ply plate with antisymmetric ones are listed in Table 5, from which it is noted that symmetric plies have larger buckling load than antisymmetric and angle plied plate has larger buckling load than cross plied plate same behavior given by other theories. As expected, critical buckling load for laminated plate under uniaxial load are larger than those under biaxial load as shown in Table 6.
Material1: the first one used in present work is: E1/E2 = 40, G12 = G13 = 0.6 E2 (Gpa), G23 = 0.2 E2 (Gpa), $v$12 = $v$13= 0.25.
Material 2: the second one used in present work is: E1/E2 =40, G12=G13=0.6E2 (Gpa), G23=0.5E2 (Gpa), $v$12= $v$13=0.25 and Ncr = (N˟a2/E2˟H3).
Table 5. Comparison of nondimentional uni-axial buckling loads (Ncr) with different orthotropic ratio for different cross and angle-ply square plate
|
Layers |
E1/E2 |
||||
|
5 |
10 |
20 |
30 |
40 |
|
|
[0/ 90]s |
6.887 |
10.256 |
16.267 |
21.459 |
25.992 |
|
[0/ 90]2 |
6.481 |
9.267 |
14.284 |
18.693 |
22.602 |
|
[45 /-45]s |
9.100 |
15.269 |
25.232 |
31.102 q=2, s=1 |
35.086 q=2, s=1 |
|
[45/ -45]2 |
8.387 |
13.565 |
22.065 |
27.894 q=2, s=1 |
31.584 q=2, s=1 |
Notes: Using material 2. Mode for all: (q=s=1) accept when written.
Table 6. Comparison of nondimentional critical biaxial buckling loads (Ncr) with various orthotropic ratio for different cross and angle-ply square plate, (a/h = 10)
|
Layers |
E1/E2 |
|||
|
Uniaxial buckling |
Biaxial buckling |
|||
|
20 |
40 |
20 |
40 |
|
|
[0 /90]s |
16.267 |
25.992 |
8.133 |
12.996 |
|
[0/ 90]2 |
14.284 |
22.602 |
7.142 |
11.301 |
|
[30/ -30]s |
21.201 |
27.856 q=2, s=1 |
10.600 |
15.839 |
|
[30/ -30]2 |
18.783 |
25.575 q=2, s=1 |
9.391 |
14.061 |
Notes: Using material 1. Mode for all: (q=s=1) accept when written.
Buckling analysis of cross and angle laminated simply supported plates is studied by using refined hyperbolic shear theory for first time, under two types of mechanical loadings. The displacement field of the proposed theory contains four unknowns, and involves a hyperbolic shape function to account for more acceptable distribution of the transverse shear strains through the thickness; with no need for shear correction factor. The equations are derived by using the Hamilton’s principle and the analytical solutions are obtained using the Navier’s solution method. The reliability of the present approach is checked by comparing it with various shear deformation theories. The numerical results show that the proposed refined plate theory is in excellent agreement with respect to other higher-order shear deformation theories for the evaluation of critical buckling of laminated plates but changing mode number.
|
a |
Plate dimension in x-direction (m) |
|
b |
Plate dimension in y-direction (m) |
|
h |
Plate thickness |
|
E1, E2, E3 |
Elastic modulus components (GPa) |
|
$\begin{aligned} |
Extension, bending, extension coupling (N/m) |
|
k=0 or 1 |
In plane load factor |
|
n |
Total number of plate layers |
|
$\begin{aligned} |
In-plan force per unit length (N/m) |
|
$\begin{aligned} & |
Bending moment result per unit length (N.m/m) |
|
$\mathrm{M} |
Force per unit length due to shear moment (N/m) |
|
$\mathrm{Q} |
Transverse shear force (N) |
|
TSDT |
Third shear deformation theory |
|
x, y, z |
Cartesian Coordinate system |
|
Wb, Ws |
Displacement in, bending and shear respectively |
|
us, vs |
Displacement in x ans y direction due to shear respectively |
Greek symbols
|
$\varepsilon_{\mathrm{x}}, \varepsilon_{\mathrm{y}}, \varepsilon_{\mathrm{z}}$ |
Strain components (m/m) |
|
$\gamma_{\mathrm{xz}}, \gamma_{\mathrm{yz}}$ |
Transverse shear strain (m/m) |
|
$\sigma_{\mathrm{x}} \sigma_{\mathrm{y}} \sigma_{\mathrm{xy}} \sigma_{\mathrm{yz}} \sigma_{\mathrm{xz}}$ |
Stress components (Gpa) |
|
$\mathrm{v}_{12} \mathrm{v}_{21}$ |
Poisson's ratio |
C11 = - A11 * (α2) - A66 * (β2);
C12 = - A12 * α * β - A66 * α * β;
C13 = (B11 * α3) + (B12 * α * β2) + (2 * B66 * α * β2);
C14 = (E12 * α * β2) + (E11* α3) + (2 * E66 * α * β2);
C22 = (-A66 * α2) - (A22 * β2);
C23= (2 * B66 * β * α2) + (B12 * β * α2) + (B22 * β3);
C24 = (E12 + (2 * E66)) *(β * α2) + (β3) * (E22);
C33 = (-D22 * β4) - (D11 * α4) - (2 * D12 * (α2) * (β2)) - (4 * D66 * (β2) * (α2));
C34 = (α4) * (-F11) + ((α2) * β2) * (-2 * F12 – 4 * F66) + (β4) * (-F22);
C44 = (-H11) * (α4) + ((α2) * β2) * (-2 * H12 – 4 * H66) + (β4 * (-H22) -(L11) * (α2) - (L22) * (β2).
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