# Progressive Failure Analysis of Laminated Composite Cylindrical Shell Roofs

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## Abstract

A progressive failure analysis of laminated composite cylindrical roofs under transverse uniformly distributed static loadings has been carried out in the present paper using the finite element method. In the finite element analysis, an eight-noded isoparametric shell elements are taken up. The first-ply failure load and the ultimate ply failure load are evaluated and compared using different stress-based failure criteria. The effect of orientation of fibers in a lamina, stacking sequence, and varying boundary conditions, on the strength of the laminate is carried out. Factors of safety and load factors are suggested based on the values of first and ultimate ply failure loads. The damage progression using the failure criterion corresponding to the first-ply failure load has been shown. The areas of damage are correlated with deflection values through explicit equations so as to get an idea about the damage extents without going into elaborate testing. The results are interpreted from practical engineering standpoint to extract meaningful conclusions.

## Keywords

Progressive failures Finite element method First-ply failure Ultimate ply failure## Notations

*a*,*b*Length and width of shell in plan

- D
Flexural rigidity matrix of the laminate

- {
*d*} Global displacement vector

*E*_{11},*E*_{22}Young’s moduli of a lamina along and transverse to the fibers, respectively

*G*_{12},*G*_{13},*G*_{23}Shear modulus of a lamina in 1–2, 1–3, and 2–3 planes corresponding to the local axes of that lamina

*h*Shell thickness

*M*_{x,}*M*_{y}Moment resultants per unit length of cylindrical shell

*M*_{xy}Torsion resultant per unit length of cylindrical shell

*N*_{i}Shape functions for first to eight nodes of an element, respectively

*N*_{x,}*N*_{y}In-plane normal force resultants per unit length in

*X*- and*Y*-direction*N*_{xy}In-plane shear force resultant per unit length of shell

*Q*_{x,}*Q*_{y}Transverse shear resultants per unit length of shell

*q*Intensity of uniformly distributed load

*u*,*v*,*w*Translational degrees of freedom along

*X*-,*Y*- and*Z*-direction, respectively*w*Transverse displacement in cm

*X*,*Y*and*Z*Global co-ordinates of the laminate

*Z*_{k},*Z*_{k−1}Top and bottom distance of the

*k*th ply from mid-plane of a laminate*α*,*β*Rotational degrees of freedom about

*Y*- and*X*-axis, respectively- \(\sigma_{x}\), \(\sigma_{y}\), \(\sigma_{z}\)
Normal stresses in

*X*-,*Y*- and*Z*-direction, respectively*τ*_{xy,}*τ*_{yz,}*τ*_{xz,}Shear stress in

*XY*-,*YZ*- and*XZ*-plane, respectively*ε*_{x,}*ε*_{y}Strain along

*X*- and*Y*-direction, respectively*ε*_{xy}Shear strain

*θ*Angle of lamination with respect to the

*X*axis of the cylindrical shell- \(\nu_{\text{ij}}\)
Poisson’s ratio which characterizes compressive strain along

*x*_{j-}direction produced by a tensile strain applied in*x*_{i}-direction- \(\varepsilon_{x}^{0} ,\varepsilon_{y}^{0} , \gamma_{xy}^{0} , \gamma_{xz}^{0} , \gamma_{yz}^{0}\)
In-plane and transverse strains of the mid-plane

- \(\kappa_{x} , \kappa_{y} , \kappa_{xy} , \kappa_{xz} , \kappa_{yz}\)
Curvature of shell

*F*_{XT,}*F*_{YT,}*F*_{ZT}Tensile strength in

*X*-,*Y*- and*Z*-direction, respectively*F*_{XC,}*F*_{YC,}*F*_{ZC}Compressive strength in

*X*-,*Y*- and*Z*-direction, respectively- \(F_{SXY}\), \(F_{SYZ} , F_{SXZ}\)
Shear strength in

*XY*-,*YZ*- and*XZ*-plane, respectively*ε*_{XT,}*ε*_{XC}Allowable normal strain strengths of a lamina along the fiber direction in tension and compression, respectively

*ε*_{YT,}*ε*_{YC}Allowable normal strain strengths of the matrix along the perpendicular to the fiber direction in tension and compression, respectively

- \(\varepsilon_{{\text{S}}_{XY}},\,\varepsilon_{{\text{S}}_{YZ}},\, \varepsilon_{{\text{S}}_{XZ}},\)
Shear strain strengths of a lamina in 2–3, 1–3 and 1–2 planes, respectively

## Notes

### Acknowledgments

The second author gratefully acknowledges the financial assistance of Technical Education Quality Improvement Programme, Phase-II (A World Bank aided project of Govt. of India) of Serial No. 1893 of Jadavpur University, Kolkata-700032, India.

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