Computational Mechanics Australia Pty Ltd
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A new book and software for modeling of shells based on the Kirchhoff-Love theory

Available with source codes and intended for scientific and engineering research
Shallow Shell Example from the QUAD-SHELL Book
To illustrate the use of the QUAD-SHELL finite element, consider deformation of a thin shallow shell.

A thin shell is called shallow if the shell radius of curvature significantly exceeds its overall linear dimensions.For practical purposes, a thin shell is considered to be shallow if the maximum rise of the shell middle surface does not exceed one-fifth of the shell in-plane dimensions.

Quite often, thin shallow shells comprise components of civil engineering structures.
The popularity of thin shallow shells is attributed to the fact that even a very shallow shell responds to loading markedly different than a plate of similar dimensions. The reason is that a shallow shell is stiffer than a geometrically similar plate, and responds well not only to bending but also to stresses uniformly distributed through the shell thickness.

The best way to improve the strength and stability of a shallow shell without increasing its weight is to increase the curvature: even a small increase in the shell curvature markedly improves the shell load bearing capacity.

As a numerical example we shall concider a thin shallow shell whose middle surface is an elliptic paraboloid.
The geometrical equation of the shell middle surface is
The edges of the shell are simply supported. The shell is subjected to a uniformly distributed load q.

A numerical solution of this problem is presented in the book "Osnovy Rascheta Uprugikh Obolochek" by N. V. Kolkunov (in Russian). Using the variational approach the solution is obtained in the form of the series.
The deflection w of the shell middle surface is
The calculations were performed for the shallow shell made of a material with the following properties: the Young modulus is 20000 MPa, the Poisson ratio is zero, D/Eh is 0.000833.
The values of the k coefficients used in the approximation are: 0.04 and 0.0278, respectively.

We present three diagrams:
  1. for the deflection w
  2. for the normal stress resultant force acting in the direction of the x-axis
  3. for the bending moment about the x-axis computed at points located along the middle line x=a/2 of the elliptic paraboloid

Each diagram displays the numerical solution obtained using the QUAD-SHELL software and the variational solution based on the theory of shallow shells.
It might be noticed that the numerical solution values of the bending moment at the edges of the surface are small but do not equal zero. The reason for that is because the edge points at which the bending moment is calculated do not coincide with the corner nodes of finite elements, so the boundary conditions are not applied at such points. Therefore the bending moment values (theoretically equal zero) at edge points can be considered as estimations of numerical discrepancies brought by the use of the finite element method.
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