Stretching and bending deformations due to normal and shear tractions of doubly curved shells using third-order shear and normal deformable theory

Abstract

We analyze static infinitesimal deformations of doubly curved shells using a third-order shear and normal deformable theory (TSNDT) and delineate effects of the curvilinear length/thickness ratio, R/a, a/h, radius of curvature/curvilinear length, and the ratio of the two principal radii on through-the-thickness stresses, strain energies of the in-plane and the transverse shear and normal deformations, and strain energies of stretching and bending deformations for loads that include uniform normal tractions on a major surface and equal and opposite tangential tractions on the two major surfaces. In the TSNDT the three displacement components at a point are represented as complete polynomials of degree three in the thickness coordinate. Advantages of the TSNDT include not needing a shear correction factor, allowing stresses for monolithic shells to be computed from the constitutive relation and the shell theory displacements, and considering general tractions on bounding surfaces. For laminated shells we use an equivalent single layer TSNDT and find the in-plane stresses from the constitutive relations and the transverse stresses with a onestep stress recovery scheme. The in-house developed finite element software is first verified by comparing displacements and stresses in the shell computed from it with those from either analytical or numerical solutions of the corresponding 3D problems. The strain energy of a spherical shell is found to approach that of a plate when R/a exceeds 10. For a thick clamped shell of aspect ratio 5 subjected to uniform normal traction on the outer surface, the in-plane and the transverse deformations contribute equally to the total strain energy for R/a greater than 5. However, for a cantilever shell of aspect ratio 5 subjected to equal and opposite uniform tangential tractions on the two major surfaces, the strain energy of in-plane deformations equals 95-98% of the total strain energy. Numerical results presented herein for several problems provide insights into different deformation modes, help designers decide when to consider effects of transverse deformations, and use the TSNDT for optimizing doubly curved shells.

Improved forming accuracy through controlling localized sheet metal deformation in the friction-assisted stretch bending process

“…To investigate the friction mechanism at the rubber/ sheet interface, Shah and Batra [30] used third-order shear and normal deformable theory (TSNDT) to describe the effects of normal and shear traction between two layers of the laminated shell on the strain energy during bending and stretching deformation. For a spherical shell subjected to equal and opposite tangential traction, transverse shear deformation contributed to about 20% of the total strain energy when the curvilinear length/thickness ratio was equal to 1.…”

An assessment of refined hierarchical kinematic models for the bending and free vibration analyses of laminated and functionally graded sandwich cylindrical panels

“…[54] and Shah and Batra [55] have also extended higher order theories for laminated composites with various stress recovery schemes for transverse stresses. Further, Shah and Batra [56] have studied effect of bending and stretching strain energies in doubly curved shell deformation using a third-order shear deformation theory. The behavior of shell structures is much influenced by its geometrical parameters, therefore, assessment of bending and stretching components can effectively communicate information about the generalized shell deformations.…”