Abstract

TRIC is a facet triangular shell element, which is based on the natural mode method. It has been shown that the TRIC shell element satisfies the individual element test and in the framework of the nonconsistent formulation the convergence requirements are fulfilled, while it has been proved to be very efficient in linear and nonlinear static problems. Moreover, another major advantage in the formulation of this element is the incorporation of the transverse shear deformations in a way that defies the shear-locking phenomenon. In this work the derivation of the consistent and lumped mass matrices of the TRIC element is presented so that it can be used in linear and nonlinear dynamic problems. Both translational and rotational inertia are included in the consistent mass matrix, which is conceived, using kinematical and geometrical arguments consistent with the assumed natural rigid body and straining modes of the element. All the kinematical and geometrical arguments that are invoked for the derivation of the consistent mass matrix are briefly presented. Moreover, two formulations of the lumped mass matrix of TRIC are derived. The first formulation is based entirely on geometrical considerations whereas the second is based on lumping the consistent mass matrix of TRIC. Finally, the element’s robustness and accuracy will be shown by applying it to properly selected benchmark examples of nonlinear shell dynamics, while its computational efficiency will be demonstrated by comparing the CPU performance of the element with the other available shell elements.

Subject-index terms: 21; 28; 67; 80

Article Outline

1. Introduction

2. The stiffness matrix of TRIC––A brief description

2.1. Kinematics of the element

2.2. Natural modes and generalized forces and moments

2.3. Axial and symmetric bending stiffness terms

2.4. Antisymmetric bending and shearing and azimuth stiffness terms

2.5. The geometric stiffness

3. Mass matrix of TRIC––consistent and lumped formulations

3.1. The principal of virtual works in dynamics

3.2. The consistent mass matrix

3.3. The lumped mass matrix

4. Time integration methods for the solution of nonlinear equations in structural dynamics

4.1. Explicit integration

4.2. Implicit integration

5. Numerical examples

5.1. Spherical cap under uniform impulse loading

5.2. Cylindrical shell with dynamic snap through

6. Conclusions

Appendix A

Appendix B. Computation of the consistent mass matrix components

References