Analysis of the Mechanical and Thermal Buckling of Laminated Beams by New Refined Shear Deformation Theory

Abstract

Thermo-mechanical buckling analysis of symmetric and antisymmetric laminated composite beams is performed based on a refined simple nth higher-order shear deformation theory. The theory accounts for the parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The governing equations and corresponding simply boundary conditions are obtained with the aid of minimum total potential energy principle. The effects of temperatures on non-dimensional critical buckling loads are investigated. Numerical results due to present theory are compared with data available in the literature to show the accuracy and simplicity of the proposed theory in analyzing the thermo-mechanical buckling of laminated composite beams.

Appendix

(1) Consider a laminate beam made of n plies. Each ply has a thickness of $t_{k}^{{}}$. Then, the thickness of the laminate h is

$$h = \sum\limits_{k = 1}^{n} {t_{k}^{{}} }$$

Then, the location of the mid-plane is h/2 from the top or the bottom surface of the laminate. The z-coordinate of each ply k surface (top and bottom) is given by

Ply 1:

$$\begin{aligned} &h_{0}^{{}} = - \frac{h}{2}\quad \left( {{\text{top}}\;{\text{surface}}} \right) \hfill \\& h_{1}^{{}} = - \frac{h}{2} + t_{1}^{{}} \quad \left( {{\text{bottom}}\;{\text{surface}}} \right) \hfill \\ \end{aligned}$$

Ply k: $\left( {k = 2,3, \ldots n - 2,n - 1} \right)$

$$\begin{aligned} & h_{k - 1}^{{}} = - \frac{h}{2} + \sum\limits_{ = 1}^{k - 1} {t\quad \left( {{\text{top}}{\kern 1pt} \;{\text{surface}}} \right)} \\ & h_{k}^{{}} = - \frac{h}{2} + \sum\limits_{ = 1}^{k} {t{\kern 1pt} \quad {\kern 1pt} \left( {{\text{bottom}}\;{\text{surface}}} \right)} \\ \end{aligned}$$

(2) Find the value of the reduced stiffness matrix $[Q]$ for each ply using its six elastic moduli, $E_{1}^{{}} ,E_{2}^{{}} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} G_{12}^{{}} ,G_{13}^{{}} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} G_{23}^{{}} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} \nu_{12}^{{}}$ in constants $Q_{11} ,Q_{12} ,Q_{22} ,Q_{66} ,{\kern 1pt} Q_{44} {\kern 1pt}$ and $Q_{55}$.

(3) Find the value of the transformed reduced stiffness matrix for each ply using the $[\bar{Q}]$ matrix calculated in step 2, and the angle of the ply and transformed coefficient of thermal expansion can be referred to any standard texts such as (Reddy (1997)).

(4) Knowing the thickness, $t_{k}^{{}}$, of each ply, find the coordinate of the top and bottom surface, $h_{i}^{{}} ,\quad i = 1 \ldots ,n,$ of each ply, using the following equation:

Ply n:

$$\begin{aligned} & h_{n - 1}^{{}} = \frac{h}{2} - t_{n}^{{}} \left( {{\text{top}}{\kern 1pt} \;{\text{surface}}} \right) \\ & h_{n}^{{}} = \frac{h}{2}\left( {{\text{bottom}}\;{\text{surface}}} \right) \\ \end{aligned}$$

(5) Use the $[\bar{Q}]$ matrices from step 3 and the location of each ply from step 4 to find the six beam stiffness $(A_{11} ,{\kern 1pt} {\kern 1pt} B_{11} ,D_{11} ,B_{11}^{s} ,D_{11}^{s} ,H_{11}^{s}$ and $A_{55}^{s}$) from Eq. (7).

(6) Substitute the stiffness matrix values found in step 5 and the applied forces and moments in Eq. (6).

(7) Solve the three simultaneous Eqs. (14a–14c). Closed-form solutions are obtained using the Navier solution for simply supported laminated composite beams Eqs. (15a–15c), and the eigenvalue problem is solved to get the corresponding eigenvalues for buckling load equation with the effect temperature reduces the critical buckling load (18) and the critical temperature Eq. (19).