GGU-ELASTIC: General
An analytical solution is only possible for simple systems. When modelling complicated systems, numerical solutions are required. In the main
finite-difference-methods (FDM) and
finite-element-methods (FEM)
are used. Using finite methods, the whole area is subdivided into a large amount of smaller (finite) areas (elements). Generally, using FEM, triangles are selected for these areas. Within these triangles simple, generally linear, approximation functions are used. The actual, complicated, whole solution is pieced together like a mosaic from the many simple partial solutions. Equation systems result, in which the number of variables corresponds to the number of system nodes. With the finite-difference-method one generally has only the possibility of defining a discrete whole using rectangular partial-areas. In contrast to FEM therefore, FDM is a lot less flexible when it comes to adjusting to complicated boundary structures. Also, in some areas, mesh refinement is not as easy. Further, the resulting equation systems are numerically more stable for FEM. The main advantage in FDM is in the theoretically simpler mathematical relationships, which generally will be of little interest to the program user. The GGU-ELASTIC program uses the finite-element-method.
When using the application please remember that all finite-element or finite-difference-methods are approximation methods. The quality of the approximation to the actual solution increases with an increase in mesh refinement. You should pay attention to the fact that the mesh should be finer for areas in which the main forces act (e.g. point loads). The type of triangle used also exerts a certain influence. Optimum conditions are achieved with equilateral triangles. You can get an overview of the solution quality by analysing the same system with a coarser and a finer mesh subdivision, and comparing the deviation of the two results.
The following general notes to the GGU-ELASTIC program are important:
Triangular elements are used.
Hooke’s Law is valid.
Analytical solutions for the differential equation for plane and axis-symmetrical strain conditions only exist for a few special cases, so that for problems of daily design practice (with variously distributed loads, free, fixed or supported boundaries etc.) numerical methods must be relied upon.
The differential equation is solved by the program using finite-element-methods. Triangular elements are used for this. Some simple assumptions are made for these triangular elements with references to displacements. In the present case, a linear displacement assumption is used, which is described in Zienkiewicz (Carl-Hanser-Verlag, 1984, Chapters 4 and 5). The selected assumption leads to equation systems, whose number of variables correspond to twice the number of system nodes. The complete solution is put together like a mosaic from the many partial solutions of the triangular elements. It is clear that with increasing refinement of the finite-element mesh, the quality of the solution also increases.
The stresses are determined by numerical differentiation of the strains. As a linear displacement assumption was selected, the stresses are constant for each element. To compensate for this, the program follows a suggestion of Zienkiewicz’ thus:
For each element node the stresses from the neighbouring elements are added and then divided by the number of neighbouring elements. In this way, the stress course can be better presented. Naturally, the results at the boundary nodes are not quite as exact. Further, the approximation of stresses in the region of element nodes belonging to elements with different material types can be worsened by doing this. If the stresses at such boundary nodes are of great interest, an improvement can be achieved by refining the FEM mesh in these areas.
The quality of the calculated strains is, generally, excellent. If you are only interested in strains, you need not worry about the previous explanations.
Continue to remember that all finite-element-methods are approximation methods. The quality of the approximation increases with increasing mesh density.
The case of a freely supported boundary is automatically considered by the finite-element-method. Valid is, that all system boundaries or partial system boundaries, which have no force or displacement boundary conditions, are automatically freely supported.
In finite-element theory this type of boundary condition is also known as a natural boundary condition.