Static and Kinematic Indeterminacy

Before analysing a structure, find out whether equilibrium alone can solve it. The degree of static indeterminacy (Ds) counts the extra unknowns; the degree of kinematic indeterminacy (Dk) counts the unknown joint displacements.

Static indeterminacy

Ds = (number of unknown forces) − (number of independent equilibrium equations).

Plane frames (rigid joints):

Ds = 3m + r − 3j − (releases)

m = members, r = support reactions, j = joints. Each internal hinge where n members meet gives (n − 1) releases.

Plane trusses (pin joints): Ds = m + r − 2j.

Space frames: Ds = 6m + r − 6j; space trusses: m + r − 3j.

• Ds = 0: statically determinate. Ds > 0: indeterminate. Ds < 0: a mechanism (unstable).
• External indeterminacy = r − 3 (for a plane structure); internal = Ds − external.
• A closed ring (one cell of a rigid frame) is internally indeterminate to degree 3.

Supports

Kinematic indeterminacy

Dk is the number of unknown independent joint displacements:

• Each free rigid joint of a plane frame has 3 (two translations, one rotation). A roller support has 2, a hinge 1, a fixed support 0.
• Dk = 3j − r for plane frames (minus 1 per member if members are treated as axially rigid, a common assumption).
• For plane trusses: Dk = 2j − r.

Force methods are efficient when Ds is small; displacement methods when Dk is small.

Stability

A structure can satisfy the counting rule but still be geometrically unstable: all reactions parallel, or all reactions passing through one point. Always check the arrangement, not just the numbers.

Worked example

A propped cantilever (fixed at A, roller at B) has r = 4 and is a single member: Ds = 4 − 3 = 1. A fixed–fixed beam has r = 6: Ds = 3 (or 2 if axial forces are ignored).

Practice