Structural idealization; stability condition of truss

Structural idealization

It is the replacement of an actual structure by a simple analytical model (line members, ideal supports, pinned or rigid joints, idealized loads and linear elastic material) that can be analysed and still represents the real behaviour with acceptable accuracy. A roof truss is idealized as pin-jointed bars with loads at the joints.

Stability of a plane truss

A truss is stable when it keeps its shape and position under any load. Two conditions are checked.

1. Necessary condition (count)

• Internally, the members must satisfy $m \ge 2j - 3$ (for a stable, internally rigid truss).
• Including supports, $m + r \ge 2j$.

If $m + r < 2j$ the truss is unstable (a mechanism). If $m + r = 2j$ it is determinate if stable, and if $m + r > 2j$ it is indeterminate.

2. Sufficient condition (arrangement) The count is not enough, so the arrangement must also be right:

• Each part must be a rigid unit, normally a chain of triangles. A panel without a diagonal is a mechanism.
• Reactions must be at least 3 and must not be all parallel or all concurrent at a point, otherwise the truss can move as a rigid body.
• No joint may be held by only two collinear members, and no part may form a collapsible chain of links; such arrangements are geometrically unstable even when the count is satisfied.

A truss is stable when $m + r \ge 2j$ and it is built from properly arranged triangles with supports preventing rigid-body motion. Example: a triangle with hinge and roller has $j=3,\ m=3,\ r=3$, and $m + r = 6 = 2j$, so it is stable and determinate.