Assessment on dynamic characterization of the cylindrical laminated composite shallow shell panel through experimental and numerical approach

Abstract

In the present study, the dynamic characterization of the glass fiber-reinforced polymer (GFRP)-laminated composite cylindrical open shallow shell panels is explored with the experimental and numerical approach. In the finite element modeling (FEM), the governing equilibrium equation of the cylindrical shell panel is developed with higher-order shear deformation theory by considering the nine-noded rectangular elements. The convergence and validation study of the present FEM is accomplished with the available literature. The cylindrical laminated composite shallow shell panels are fabricated with a curvature radius of 0.8 m, and the experimentation on the free vibration investigation is performed with various boundary conditions; then, the experimental outcomes are compared with the present FEM to verify the effectiveness. The detailed parametric investigation is executed to explore the impact of curvature radius (R), boundary conditions, thickness ratio, aspect ratio (L/B), and stacking sequence on the structural response of the cylindrical laminated composite shallow shell panel. Transverse vibration response of the GFRP cylindrical laminated composite shallow shell panel is also performed with the various curvature ratios at clamped at all end conditions.

Appendix A

In the present FEM, a nine-noded rectangular element is considered and the respective shape function is given as

$$\begin{aligned} N_{1} & = \frac{1}{4}\left( {1 - \xi } \right)\left( {1 - \eta } \right)\eta \xi ;\quad N_{2} = \frac{1}{4}\left( {1 + \xi } \right)\left( {1 - \eta } \right)\eta \xi ;\quad N_{3} = \frac{1}{4}\left( {1 + \xi } \right)\left( {1 + \eta } \right)\eta \xi ; \\ N_{4} & = - \frac{1}{4}\left( {1 - \xi } \right)\left( {1 + \eta } \right)\eta \xi ;\quad N_{5} = - \frac{1}{2}\left( {1 - \xi^{2} } \right)\left( {1 - \eta } \right)\eta ;\quad N_{6} = \frac{1}{2}\left( {1 + \xi } \right)\left( {1 - \eta^{2} } \right)\xi ; \\ N_{7} & = \frac{1}{2}\left( {1 - \xi^{2} } \right)\left( {1 + \eta } \right)\eta ;\quad N_{8} = - \frac{1}{2}\left( {1 - \xi } \right)\left( {1 - \eta^{2} } \right)\xi ;\quad N_{9} = \left( {1 - \xi^{2} } \right)\left( {1 - \eta^{2} } \right). \\ \end{aligned}$$

(A1)

Axial distortion strain–displacement field matrix $B_{{{{\mathrm{cs}}} }}^{1} (x,y)$ of the cylindrical shell panel

$$B_{{{{\mathrm{cs}}} }}^{1} (x,y) = \left[ {\begin{array}{*{20}c} {\frac{{\partial N_{i}^{ss} }}{\partial x}} & 0 & {N_{i}^{ss} \times ( - S_{22} )} & 0 & 0 & 0 & 0 \\ 0 & {\frac{{\partial N_{i}^{ss} }}{\partial y}} & 0 & 0 & 0 & 0 & 0 \\ {\frac{{\partial N_{i}^{ss} }}{\partial y}} & {\frac{{\partial N_{i}^{ss} }}{\partial x}} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{{\partial N_{i}^{ss} }}{\partial x}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{\partial N_{i}^{ss} }}{\partial y}} & 0 & 0 \\ 0 & 0 & 0 & {\frac{{\partial N_{i}^{ss} }}{\partial y}} & {\frac{{\partial N_{i}^{ss} }}{\partial x}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{{\partial N_{i}^{ss} }}{\partial x} \times S_{11} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{\partial N_{i}^{ss} }}{\partial y} \times S_{11} } \\ 0 & 0 & 0 & 0 & 0 & {\frac{{\partial N_{i}^{ss} }}{\partial y} \times S_{11} } & {\frac{{\partial N_{i}^{ss} }}{\partial x} \times S_{11} } \\ \end{array} } \right].$$

(A2)

Strain–displacement matrix $B_{{{{\mathrm{cs}}} }}^{2} (x,y)$ of the cylindrical shell panel in concerning with the transverse deformation.

$$B_{{{{\mathrm{cs}}} }}^{2} (x,y) = \left[ {\begin{array}{*{20}c} {N_{i}^{ss} \times S_{22} } & 0 & {\frac{{\partial N_{i}^{ss} }}{\partial x}} & {N_{i} } & 0 & 0 & 0 \\ 0 & 0 & {\frac{{\partial N_{i}^{ss} }}{\partial y}} & 0 & {N_{i} } & 0 & 0 \\ 0 & 0 & 0 & {N_{i}^{ss} \times S_{22} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {N_{i}^{ss} \times S_{33} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {N_{i}^{ss} \times S_{33} } \\ 0 & 0 & 0 & 0 & 0 & {N_{i}^{ss} \times S_{44} } & 0 \\ \end{array} } \right]\quad i = 1,2,3, \ldots ,9.$$

(A3)

For the orthotropic cylindrical laminated composite shell panel, the inertia matrix is given as

$$I_{SS} = \left[ {\begin{array}{*{20}c} {I_{SS}^{0} } & 0 & 0 & {I_{SS}^{1} } & 0 & {I_{SS}^{3} } & 0 \\ 0 & {I_{SS}^{0} } & 0 & 0 & {I_{SS}^{1} } & 0 & {I_{SS}^{3} } \\ 0 & 0 & {I_{SS}^{0} } & 0 & 0 & 0 & 0 \\ {I_{SS}^{1} } & 0 & 0 & {I_{SS}^{2} } & 0 & { - S_{8} \times I_{SS}^{4} } & 0 \\ 0 & {I_{SS}^{1} } & 0 & 0 & {I_{SS}^{2} } & 0 & { - S_{8} \times I_{SS}^{4} } \\ { - S_{8} \times I_{SS}^{3} } & 0 & 0 & { - S_{8} \times I_{SS}^{4} } & 0 & { - \left( {S_{8} } \right)^{2} \times I_{SS}^{6} } & 0 \\ 0 & { - S_{8} \times I_{SS}^{3} } & 0 & 0 & { - S_{8} \times I_{SS}^{4} } & 0 & { - \left( {S_{8} } \right)^{2} \times I_{SS}^{6} } \\ \end{array} } \right].$$

(A4)