Determinacy and stability of frame

Frame

A frame is a rigid-jointed structure of beams and columns in which members carry axial force, shear force and bending moment.

Stability and external determinacy

Let r = number of unknown support reactions, and the number of equations of static equilibrium = 3 for a plane structure.

• If r < 3: the structure is unstable (a mechanism).
• If r = 3 and the reactions are not all parallel and do not meet at one point: stable and statically determinate.
• If r > 3: statically indeterminate externally to degree r − 3 (if stable).
• Add one extra equation for each internal hinge (M = 0 at the hinge). So the number of available equations is 3 + h, where h is the number of internal hinges (releases).

Geometric instability: even if r ≥ 3, the structure is unstable if the reactions are all parallel or all pass through one point.

Internal determinacy (closed loops)

For a plane frame with m members, j joints and r reactions, with rigid joints: Degree of indeterminacy = (3m + r) − 3j − (extra equations of condition, e.g. hinges)

• = 0 determinate, > 0 indeterminate, < 0 unstable. Each closed loop of rigid members adds 3 to the degree of indeterminacy.

Examples

• Portal frame with a hinge and a roller (r = 3, no loop): determinate and stable. Reactions are found by ΣFx = ΣFy = ΣM = 0.
• Portal frame fixed at both bases (r = 6): indeterminate to the 3rd degree.
• Three-hinged portal frame (hinge at each base and one at the crown): r = 4 and h = 1, so the degree = 4 − (3 + 1) = 0, determinate and stable.
• Cantilever frame with fixed support: r = 3, determinate.
• A frame resting on three rollers (r = 3, but all three reactions vertical and parallel) is geometrically unstable: it cannot resist any horizontal load.