Statical determinacy

Statically determinate or not?

In order to perform any calculations by hand to get the forces in each member of a truss structure, there needs to be checked if the structure is statically determinate. A truss can fall in 3 different categories. The truss can be either statically determinate, statically indeterminate or a mechanism. To check if the truss is statically determinate the following 3 equations belonging to the 3 types of trusses can be used:

Where $N_{m}$ is equal to the number of members, $N_{r}$ to the number of reaction forces and $N_{j}$ to the number of joints.

The structure has equation 1 if it is statically determinate, equation 2 if it is statically indeterminate and equation 3 if it is a mechanism.

Example

In figure 2 the structure of figure 1 is given in a free body diagram, exposing the reaction forces acting on the truss. The weight is divided over the 2 vertical reaction forces so the magnitude for those is 0.5W. And the horizontal reaction force in point A is 0, since that is the only horizontal reaction force and the structure is in equilibrium.

For this structure $N_m = 7$, $N_r = 3$ and $N_j = 5$. Following equation 1, this structure is statically determinate so we can calculate the forces of each member by hand.

Figure 1. Schematic visualisation of a truss of a small bridge

Figure 2. Free body diagram of the truss from figure 1

Now is shown how to check whether a truss structure is statically determinate or not and combined with the equations for static equilibrium, it is possible to do calculations on forces acting in a truss structure, by hand. In the next articles the methods for doing that are explained in further detail beginning with the method of joints.