A2. What is a finite element?

The resolution of the efforts in the elements is carried out following the calculation of the node displacements. The method is specific for each type of element and depends on the software used.

However, the main principle is common to all software, it consists of "isolating" an element to calculate the forces at the Gauss points from the nodal displacements.

The position of the Gauss points is normally specified in the software documentation; in the case of a 4 node shell element as below, they could be located at a distance from the edge of the element equal to about 1/5 of its width.

Example of a 4-node element

The efforts in the center of the element are calculated as the average of the forces at the Gauss points, the efforts at the nodes are extrapolated from the Gauss points.

To summarize, the software calculates:

the efforts at the Gauss points G1 to G4 from the displacements at the nodes n1 to n4
the efforts at center C which are the average of the efforts at the Gauss points G1 to G4
the efforts at nodes n1 to n4 which are extrapolated from the efforts at the Gauss points G1 to G4

These calculations are carried out for all the elements. In the end, there are as many efforts at the nodes as there are elements connected to this node (here 4 elements E1 to E4 connected to node n1).

We can then deduce:

either the maximum effort at the node (maximum of the efforts calculated from the elements E1 to E4)
or the average effort (average of the efforts calculated from E1 to E4)

Main remarks concerning common usage

Generally, quadrangular elements will lead to better accuracy of the results than triangular elements.
Results at Gauss points are the most accurate, but they are generally not accessible to users.
The results at the center of the elements are more reliable than the ones at the nodes because they are not extrapolated.
It is up to the engineer to choose the type of result (maximum, average, smoothened, etc.) according to the behavior of the structure. There are no predefined rules.

Example of a load on a bridge slab illustrating the differences in results during a FE calculation.