Abstract
This paper presents the development of a model of homogeneous, moderately thick shells for elastodynamic problems. The model is obtained by adapting and modifying SAM-H model (stress approach model of homogeneous shells) developed by Domínguez Alvarado and Díaz in (2018) for static problems. In the dynamic version of SAM-H presented herein, displacements and stresses are approximated by polynomials of the out-of-plane coordinate. The stress approximation coincides with the static version of SAM-H when dynamic effects are neglected. The generalized forces and displacements appearing in the approximations are the same as those involved in a classical, moderately thick shell model (CS model) but the stress approximation adopted herein is more complex: the 3D motion equations and the stress boundary conditions at the faces of the shell are verified. The generalized motion and constitutive equations of dynamic SAM-H model are obtained by applying a variant of Euler–Lagrange equation which includes pertinently Hellinger–Reissner functional. In the constitutive equations, Poisson’s effect of out-of-plane normal stresses on in-plane strains is not ignored; this is one important feature of SAM-H. To test the accuracy of dynamic SAM-H model, the following structures were considered: a hollow sphere and a catenoid. In each case, eigenfrequencies are first calculated and then a frequency analysis is performed applying a harmonic load. The results are compared to those of a CS model, MITC6 (mixed interpolation of tensorial components with 6 nodes per element) shell element calculations, and solid finite element computations. In the two problems, CS, MITC6, and dynamic SAM-H models yield accurate eigenfrequencies and eigenmodes. Nevertheless, the frequency analysis performed in each case showed that dynamic SAM-H provides much more accurate amplitudes of stresses and displacements than the CS model and the MITC6 shell finite element technique.
Highlights
Shell-like structures are a common solution for light and innovative designs; their structural analysis may be performed by making use of finite element calculations. e application of 3D solid elements requires high computational resources
As to validate the SAM-H model, its results are compared to those obtained with solid finite element calculations, the classical shell model (CS) model developed in [5], and MITC6 shell finite elements [25] included in COMSOL Multiphysics
For an analytical resolution of the SAM-H dynamic equations, a hollow sphere subjected to a harmonic internal pressure is considered; the external face of the sphere is stress free. e radius of the middle surface and the thickness-to-radius ratio are denoted by R and η, respectively. e material is isotropic and characterized by its Young’s modulus E, Poisson’s ratio ], and density ρ. e internal pressure p is given by p(t) p0 sin(2πft), where p0 and f are the amplitude of the internal pressure and the excitation frequency, respectively
Summary
Shell-like structures are a common solution for light and innovative designs; their structural analysis may be performed by making use of finite element calculations. e application of 3D solid elements requires high computational resources. In the three theories above, the applied loads at the inner and outer faces of the shell do not appear explicitly in the motion equations Other inconvenience of these models is the neglection of Poisson’s effect of out-ofplane normal stresses on in-plane strains; this can lead to an inaccurate displacement evaluation, for thick and moderately thick shells [1]. E generalized motion and constitutive equations dynamic SAM-H involve explicitly the applied normal and shear stresses at the faces of the shell: this is a second originality of the model. When dealing with plates or shells, mixed-approach models provide an approximation of the 3D stress and displacement fields In this approximation, polynomials of the coordinate across the thickness are usually applied. E generalized motion equations and force edge conditions are obtained from equation (8). e generalized constitutive equations and displacement edge conditions are given by equation (9)
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