







A new book and software for modeling of shells based on the KirchhoffLove theory
Available with source codes and intended for scientific and engineering research



Shallow Shell Example from the QUADSHELL Book



To illustrate the use of the QUADSHELL
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 onefifth
of the shell inplane 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:
 for the deflection w
 for the normal stress resultant force acting in the direction of the xaxis
 for the bending moment about the xaxis computed at points located along
the middle line x=a/2 of the elliptic paraboloid
Each diagram displays the numerical solution obtained using the
QUADSHELL 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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