## Geometric Instability

33 The stability of a structure is determined not only by the number of reactions but also by their arrangement.

34 Geometric instability will occur if:

1. All reactions are parallel and a non-parallel load is applied to the structure.

2. All reactions are concurrent, Fig. 12.4.

Figure 12.4: Geometric Instability Caused by Concurrent Reactions

3. The number of reactions is smaller than the number of equations of equilibrium, that is a mechanism is present in the structure.

35 Mathematically, this can be shown if the determinant of the equations of equilibrium is equal to zero (or the equations are inter-dependent).

12.2.6 Examples

Example 12-1: Simply Supported Beam

Determine the reactions of the simply supported beam shown below.

 V 1 \ 1 1 \ 1 / The beam has 3 reactions, we have 3 equations of static equilibrium, hence it is statically determinate. Ray + Rdy - 60 k - (4) k/ft(12) ft =0 12Ray - 6Rdy - (60)(6) =0 or through matrix inversion (on your calculator) ay dy ay dy Alternatively we could have used another set of equations: (+^)£Mzd = 0; (Ray)(18) - (60)(12) - (48)(6)= 0 ^ Ray =56 kt Check:
+1 0

### Responses

• Olo Puddifoot
What is geometric instability?
8 years ago