Numerical Modeling of Initial Stress Stiffening Effect on the Dynamic Behavior of Axisymmetric Shells

This paper presents a numerical model to simulate the initial stress stiffening effect, induced by radial pressure and/or axial load on the dynamic behavior of axisymmetric shells. This effect is particularly important for thin shells since their bending stiffness is very small compared to membrane stiffness. The theoretical formulation is based on a combination of the finite element method and classical shell theory. For a perfect geometrical consistency, two semi-analytical finite elements, conical and cylindrical, are used to model axisymmetric shells. The displacement functions are derived from exact solutions of Sanders' shell equilibrium equations. The results obtained using this approach are remarkably accurate. The potential energy is calculated to estimate the initial stiffening effect using direct membrane forces per unit width and rotations about the orthogonal axes. The final stiffness matrix of each finite element is composed of the regular stiffness matrix and the added stiffness matrix generated by membrane loads. The frequencies of vibration are compared with those obtained in other experimental and theoretical research works and very good agreement is observed.

A Weak Form Implementation of Nonlinear Axisymmetric Shell Equations With Examples

We present a weak form implementation of the nonlinear axisymmetric shell equations. This implementation is suitable to study the nonlinear deformations of axisymmetric shells, with the capability of considering a general mid-surface shape, non-homogeneous (axisymmetric) mechanical properties and thickness variations. Moreover, given that the weak balance equations are arrived to naturally, any external load that can be expressed in terms of an energy potential can, therefore, be easily included and modeled. We validate our approach with existing results from the literature, in a variety of settings, including buckling of imperfect spherical shells, indentation of spherical and ellipsoidal shells, and geometry-induced rigidity (GIR) of pressurized ellipsoidal shells. Whereas the fundamental basis of our approach is classic and well established, from a methodological view point, we hope that this brief note will be of both technical and pedagogical value to the growing and dynamic community that is revisiting these canonical but still challenging class of problems in shell mechanics.

Stressed State of the Axisymmetric Adhesive Joint of Two Cylindrical Shells under Axial Tension

The research of the deflected mode of the construction, composed of two coaxially-glued cylindrical pipes, is done. Pipes are considered as thin-walled axisymmetric shells, which are joined by adhesive layer of a certain thickness. The shearing stresses in the glue are considered to be constant over the thickness of the adhesive layer, and normal stresses are linearly dependent on the radial coordinate. The shearing stresses in the adhesive layer are considered to be proportional to the difference in the longitudinal displacements of the shell sides that are faced to the adhesive layer. Normal stresses are proportional to the difference in radial displacement of the shells. It is supposed that the change in the adhesive layer thickness under deformation does not affect the stress, that is, the linear model is considered. The problem of the joint deflected mode finding is reduced to the system of four ordinary differential equations relative to the radial and longitudinal displacements of the layers. The system is solved by the matrix method. Displacements of layers outside of the adherent area can be found by the classical theory of axisymmetric shells. Satisfaction of boundary conditions and conjugation conditions leads to a system of twenty two linear equations with twenty two unknown coefficients. The model problem is solved; the results are compared with the computation made by the finite element method. The tangential and normal stresses in the glue reach the maximum values at the edges of the adhesive line. It is shown that the proposed model describes the stressed state of the joint with high accuracy, and this joint has an influx of glue residues at the ends of the adhesive line but can not be applied in the absence of adhesive influxes. Because in this case, the tangential stresses due to the parity rule reach maximum values not on the edge, but at some distance from the edge of the line. As a result, the distribution of normal stresses at the edge of the line also substantially changes. Thus, the proposed model with certain restrictions has sufficient accuracy for engineering problems and can be used to solve design problems.