 # Chris Fox's Engineering Section

## Plane Trusses

A plane truss is a framework consisting of long, thin members joined at their ends to form a rigid structure. For maximum rigidity, the members form an array of triangles. Trusses are used in such structures as bridges, roof supports, cranes and pylons.

The joints can be welded, bolted or riveted, and may have reinforcing plates attached.

For our purposes, deformation is assumed to be negligible, in which case the geometry does not change. In reality, however, some deformation does happen and must be taken into account. For example, one end of a truss is usually mounted on rollers to allow for thermal expansion (expansion due to heat).

Those members in compression must be strong enough not to buckle. The critical buckling load PCRIT=π²EI/l², where E=Young's modulus, I=second moment of area and l=length. Young's modulus and the second moment of area will be covered later.

Those members in tension must be strong enough not to fail either by yield - where a material undergoes plastic (irreversible) deformation - or by fracture. Some of those members can be cables.

In order to simplify a problem, some idealisation of trusses is necessary. The joints are represented by pin joints - joints of negligible size - and loads are considered to be applied only at the pin joints. The joints are considered frictionless, so that members are in simple tension or compression. And the members are two-force systems; as they are in equilibrium, they are either in tension at both ends or in compression at both ends, the forces in each member having equal magnitude.

To analyse the forces in a truss, start by using equilibrium equations to calculate the external reactions. Then identify and disregard any members in which the force is 0N. To take an example, consider a joint where two members that are mutually in line meet only one other member and where there are no applied forces or reactions; as one component of the force in the third member would be all that acted perpendicular to the other two members, the value of that force must be 0N.

The forces in each member can be calculated using either of two methods. The method of joints can be used to find the forces acting in a lot of members. Draw a free body diagram of each joint, showing the name and direction of each force. Assume all members are in tension, and show the forces in them acting away from each joint; if a member is in compression, the value of the force in that member will then be negative. Denote each joint by a letter; the force in each member can then be labelled in terms of the joints at the ends of that member, for example AB. Then use equilibrium equations to calculate the forces.

If one only needs to find the forces acting in a few members, the method of sections can be used. Cut the truss in two through the members of interest and draw a free body diagram of either part of the structure. Then calculate the forces as above.