How is a diagram for a linear support calculated?

 Tisk

There are several ways in which results on a linear support can be displayed:  as Values, Vectors, Diagrams, Uniform diagrams or Torsors.

A model representing a wall is proposed  as an example,  in order to illustrate how the diagram display of the reaction forces over a linear support is obtained.

The model shown in Figure 1 consists of shell elements with no self-weight, having a linear fixed support at the base. The dimensions of the wall are 10 x 5 m. A uniformly distributed linear load of 5 kN/m is placed at the top of the wall. Another point load is placed at the top of the wall, at midspan. The mesh is made out of rectangular elements, having a size of 0.5 x 0.5 m. There is only one combination defined: 1*Point Load + 1*Linear load.

Figure 1 - Model Description

The reactions on the support - Fz (Figure 2), displayed as vectors - have the following values:

Figure 2 - Fz Reaction forces over the linear support - Vectors display

The reactions on the support - Fz (Figure 3), displayed as a diagram - have the following shape:

Figure 3 - Fz Reaction forces over the linear support - Diagram display

The diagram shape and values are calculated using the vectors.

Firstly, the vectors are transformed into a histogram - Figure 4 - by distributing the vectors on a length equal to half the length of the mesh element to the left side of the node and half the length of the mesh element to the right side. The first and last vector are distributed on half of the length of the adjacent mesh element.

For example, for the first vector, the value of the distributed reaction is calculated as: 10.74/0.25 = 42.96 kN/m.

For the second vector, it is computed as: 12.03/0.5=24.06 kN.

Figure 4 – Histogram of the reaction forces - Fz

Secondly, the diagram of the reaction forces along the support - Figure 5 - is constructed by connecting the midpoints of the histogram bars. For the bars at the supports edges, the corner points are used.

Figure 5 - Diagram - constructed from the histogram

Due to the discretization of the diagram, some discrepancies may occur between the total reaction computed as the sum of the vectors and the one computed as the total area under the diagram. Therefore, the diagram is normalized by a factor, β, computed as follows:

Finally, the values of the reaction forces defining the diagram are multiplied by the β factor and the final diagram of the reaction forces variation over the support is obtained (Figure 6).
For the proposed example, the β factor is calculated as:

Figure 6 - Fz Reaction forces over the linear - Diagram display

Note: For the proposed example, the β factor is negligible; hence, the normalized diagram is similar to the initial one. However, there could be cases where the effect of this factor becomes more significant. E.g.: elements with a coarse mesh, supports with a sudden change in the reaction force (due to applied point loads).