The free vibration analysis of laminated composite cylindrical, spherical, hypar, saddle, and elliptical shells with cutouts and concentrated mass is presented by developing a $C^0$ finite element formulation based on third-order shear deformation theory (TSDT). The proposed model satisfies the zero transverse shear stress conditions at shell top and bottom without using a shear correction factor. This is the first finite element implementation of the TSDT to solve the vibration problem of laminated composite shells with cutouts and concentrated mass incorporating cross curvature effects in shells. Parametric vibration study has been performed for different shells with cutouts and concentrated mass.

This article presents a novel mathematical model for moderately thick and deep laminated composite conoidal shell. The zero transverse shear stress at top and bottom of conoidal shell conditions is applied. Novelty in the present formulation is the inclusion of curvature effect in displacement field and cross curvature effect in strain field. This present model is suitable for deep and moderately thick conoidal shell. The peculiarity in the conoidal shell is that due to its complex geometry, its peak value of transverse deflection is not at its center like other shells. The C¹ continuity requirement associated with the present model has been suitably circumvented. A nine-node curved quadratic isoparametric element with seven nodal unknowns per node is used in finite element formulation of the proposed mathematical model. The present model results are compared with experimental, elasticity, and numerical results available in the literature. This is the first effort to solve the problem of moderately thick and deep laminated composite conoidal shell using parabolic transverse shear strain deformation across the thickness of conoidal shell. Many new numerical problems are solved for the static study of moderately thick and deep laminated composite conoidal shell considering 10 different practical boundary conditions, four types of loadings, six different hl/hh (minimum rise/maximum rise) ratios, and four different laminations.

The present paper is the first study of free vibration analysis of laminated composite rhombic (skew) elliptic paraboloids with mass variation via multiple cutouts and concentrated mass. To solve the vibration problem of a laminated composite skew elliptic paraboloid with multiple cutouts and concentrated mass, cubic variation in the displacement field and cross-curvature effects of the shell are considered. The parabolic variation of transverse shear strain is considered; hence, no shear correction factor is required. The zero transverse shear stress conditions at the top and bottom of the shell are imposed in the mathematical model. The finite-element implementation of the present realistic mathematical model is done using a nine-noded curved isoparametric element with seven unknowns at each node. The ${\mathrm{C}}^0$ finite-element (FE) implementation of the present mathematical model has been done and coded in FORTRAN. The present model results are compared and found in good agreement with experimental, analytical, and other solutions published in the literature. A free vibration study has been done for skew elliptic paraboloids with multiple cutouts and concentrated mass having different curvatures, ply orientations, and thickness ratios.

In the presented paper, a study of bi-axial buckling of the laminated composite plate with mass variation through the cutout and additional mass is carried out using the improved shear deformation theory (ISDT). The ISDT mathematical model employs a cubic variation of thickness co-ordinates in the displacement field. A realistic parabolic distribution of transverse shear strains through the plate thickness is assumed and the use of shear correction factor is avoided. A C° finite element formulation of the mathematical model is developed to analyze the buckling behavior of laminated composite plate with cutout and additional mass. As no results based on ISDT for the considered problem of bi-axial buckling of the laminated composite plate with mass variation are available in the literature, the obtained results are validated with the data available for a laminated composite plate without cutout and additional mass. Novel results are obtained by varying geometry, boundary conditions and ply orientations.