Stability analysis of a sandwich composite magnetostrictive nanoplate coupled with FG porous facesheets

Abstract

The main aim of this study is to examine the buckling behavior of a composite material that has both magnetostrictive capabilities and functionally graded facesheets. The effective material parameters of the functionally graded layer are determined using the power-law model. Eringen’s nonlocal theory has been used for the quantification of the small-scale parameter. In contrast, the suggested system is based on the theoretical framework established by Winkler and Pasternak, which incorporates the analysis of an elastic medium. The use of higher-order sinusoidal shear deformation theory has been employed to derive the governing equation. This governing equation is then solved analytically using the Galerkin solution method, considering various boundary conditions. In order to assess the precision and effectiveness of the ongoing inquiry, the findings are juxtaposed with the existing literature articles. Furthermore, this study examines the impact of many factors, including aspect ratio, velocity feedback gain, and foundation, on the critical buckling load. The findings of the present research indicate that there is a positive correlation between the porosity volume parameter and the buckling load of the structure. The current study aims to provide engineers and designers with a better understanding and predictive capability about buckling response. This knowledge may be advantageous in the design of nanoscale systems, including highly sought-after technologies like as sensors and actuators.

Appendix 1

$$K_{11} = A_{11} \left( {\mathop \int \limits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + A_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$K_{12} = A_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + A_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$\begin{aligned} K_{13} = & - B_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} X_{m} Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) - B_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - B_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$K_{14} = S_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$K_{15} = S_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$K_{21} = A_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) + A_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$K_{22} = A_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) + A_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$\begin{aligned} K_{23} = & - B_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) - B_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} Y_{n} X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - 2B_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$K_{24} = S_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$K_{25} = S_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$\begin{aligned} K_{31} = & B_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + B_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & + 2B_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$\begin{aligned} K_{32} = & B_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + B_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & + 2B_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$\begin{aligned} K_{33} = & - D_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} X_{m} Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) - 2D_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - D_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} Y_{n} X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}} \right){\text{d}}x\,{\text{d}}y} \right) - 4D_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$\begin{aligned} K_{34} = & M_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + M_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & + 2M_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{4} X_{m} }}{{\partial x^{4} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$\begin{aligned} K_{35} = & M_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + M_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{4} Y_{n} }}{{\partial y^{4} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & + 2M_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$K_{41} = S_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$K_{42} = S_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$\begin{aligned} K_{43} = & - M_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} X_{m} Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) - M_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - 2M_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$\begin{aligned} K_{44} = & O_{11} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{3} X_{m} }}{{\partial x^{3} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) + O_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - L_{55} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}Y_{n} } \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$K_{45} = O_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right) + O_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{2} Y_{n} }}{{\partial y^{2} }}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$K_{51} = S_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$K_{52} = S_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) + S_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$\begin{aligned} K_{53} = & - M_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) - M_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} Y_{n} X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - 2M_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$K_{54} = O_{12} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) + O_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}Y_{n} \frac{{\partial X_{m} }}{\partial x}\frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right)$$

$$\begin{aligned} K_{55} = & O_{22} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial^{3} Y_{n} }}{{\partial y^{3} }}} \right){\text{d}}x\,{\text{d}}y} \right) + O_{66} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {\frac{{\partial^{2} X_{m} }}{{\partial x^{2} }}\frac{{\partial Y_{n} }}{\partial y}X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) \\ & - L_{44} \left( {\mathop \int \nolimits_{0}^{b} \mathop \int \nolimits_{0}^{a} \left( {X_{m} \frac{{\partial Y_{n} }}{\partial y}} \right){\text{d}}x\,{\text{d}}y} \right) \\ \end{aligned}$$

$$C_{11} = C_{12} = C_{14} = C_{15} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{13} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{31} \left( \alpha \right)K_{c} C\left( t \right){\text{d}}z$$

$$C_{21} = C_{22} = C_{24} = C_{25} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{23} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{32} \left( \beta \right)K_{c} C\left( t \right){\text{d}}z$$

$$C_{31} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{31} \left( \alpha \right)K_{c} C\left( t \right)dz\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{32} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{31} \left( \beta \right)K_{c} C\left( t \right){\text{d}}z$$

$$C_{33} = c_{d} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{34} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{31} \left( \alpha \right)K_{c} C\left( t \right){\text{d}}z$$

$$C_{35} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{32} \left( \beta \right)K_{c} C\left( t \right){\text{d}}z$$

$$C_{41} = C_{42} = C_{44} = C_{45} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{43} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{31} \left( \alpha \right)K_{c} C\left( t \right){\text{d}}z$$

$$C_{51} = C_{52} = C_{54} = C_{55} = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C_{53} = \mathop \int \nolimits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{32} \left( \beta \right)K_{c} C\left( t \right){\text{d}}z$$

$$A_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{11}^{f} {\text{d}}z$$

$$A_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{12}^{f} {\text{d}}z$$

$$A_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{22}^{f} {\text{d}}z$$

$$A_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{44}^{f} {\text{d}}z$$

$$A_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{55}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{55}^{f} {\text{d}}z$$

$$A_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{66}^{f} {\text{d}}z$$

$$B_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{11}^{f} {\text{d}}z$$

$$B_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{12}^{f} {\text{d}}z$$

$$B_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{22}^{f} {\text{d}}z$$

$$B_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{44}^{f} {\text{d}}z$$

$$B_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{55}^{f} {\text{d}}z$$

$$B_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{66}^{f} {\text{d}}z$$

$$D_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{11}^{f} {\text{d}}z$$

$$D_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{12}^{f} {\text{d}}z$$

$$D_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{22}^{f} {\text{d}}z$$

$$D_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{44}^{f} {\text{d}}z$$

$$D_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{55}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{55}^{f} {\text{d}}z$$

$$D_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{66}^{f} {\text{d}}z$$

$$S_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{11}^{f} {\text{d}}z$$

$$S_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{12}^{f} {\text{d}}z$$

$$S_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{22}^{f} {\text{d}}z$$

$$S_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{44}^{f} {\text{d}}z$$

$$S_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{55}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{55}^{f} {\text{d}}z$$

$$S_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)C_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)C_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)C_{66}^{f} {\text{d}}z$$

$$M_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{11}^{f} {\text{d}}z$$

$$M_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{12}^{f} {\text{d}}z$$

$$M_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{22}^{f} {\text{d}}z$$

$$M_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{44}^{f} {\text{d}}z$$

$$M_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{55}^{f} {\text{d}}z$$

$$M_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)zC_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)zC_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)zC_{66}^{f} {\text{d}}z$$

$$O_{11} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f(z)^{2} C_{11}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{2} C_{11}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{2} C_{11}^{f} {\text{d}}z$$

$$O_{12} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{2} C_{12}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f(z)^{2} C_{12}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{2} C_{12}^{f} {\text{d}}z$$

$$O_{22} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{2} C_{22}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{2} C_{22}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{2} C_{22}^{f} {\text{d}}z$$

$$O_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{2} C_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{2} C_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f(z)^{2} C_{44}^{f} {\text{d}}z$$

$$O_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{2} C_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f(z)^{2} C_{55}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f(z)^{2} C_{55}^{f} {\text{d}}z$$

$$O_{66} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f(z)^{2} C_{66}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{2} C_{66}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{2} C_{66}^{f} {\text{d}}z$$

$$L_{44} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{\prime 2} C_{44}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{\prime 2} C_{44}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{\prime 2} C_{44}^{f} {\text{d}}z$$

$$L_{55} = \mathop \int \nolimits_{{ - h_{c} /2}}^{{h_{c} /2}} f\left( z \right)^{\prime 2} C_{55}^{c} {\text{d}}z + \mathop \int \nolimits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} f\left( z \right)^{\prime 2} C_{55}^{f} {\text{d}}z + \mathop \int \nolimits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} f\left( z \right)^{\prime 2} C_{55}^{f} {\text{d}}z$$