UBC Theses and Dissertations
Analysis of elastic shells of revolution with membrane and flexure stresses under arbitrary loading using trapezoidal finite elements Agrawal, Krishna Murari
Analysis of a general shell of revolution with arbitrary loading and boundary conditions using the Finite Element approach, well-suited for use with the electronic computer, is presented. The shell is approximated by an assemblage of flat, equilateral trapezoids and isosceles triangles connected to each other at the corners. The assumptions involved in transforming a piece of plate into a finite element are defined. Uncoupled plane stress and flexure stiffness matrices for the above-mentioned shapes of the finite elements are derived from considerations of (i) statics, and (ii) virtual work (energy). Statics matrices are asymmetric with the exception of the triangle plane stress stiffness matrix. However, it is important to note that irrespective of the size of the trapezoid element, in conditions of uniform stress the nodal forces satisfy Betti's reciprocal theorem. When a trapezoid reduces to a rectangle, the asymmetry of plane stress and flexure stiffness matrices disappears. Asymmetry of the Statics matrix is removed by averaging the matrix and its transpose. This process corresponds to introducing self-equilibrating nodal forces which disappear in conditions of uniform stress. Suitable direction cosine matrices are derived to transform the displacements and forces from the element coordinate system to the shell coordinate system. The accuracy of the formulation is demonstrated in several examples by comparing the finite element solution with the elasticity solution. The comparison suggests convergence of the results to the correct solution on reduction of the element size.
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