Determinacy, Indeterminacy & Stability Notes

Last reviewed 16 Sept 2026 · 7 min read

Why classify structures

Before analysing a structure, decide how it can be analysed:

Statically determinate — all reactions and member forces follow from the equations of equilibrium alone.
Statically indeterminate — there are more unknowns than equilibrium equations; compatibility of deformations is also needed (force method, slope-deflection, moment distribution, matrix methods).
Unstable — the structure cannot resist some loads at all; it is a mechanism.

The number of extra unknowns is the degree of static indeterminacy ( $D_{s}$ ), also called redundancy. The number of unknown joint displacements is the degree of kinematic indeterminacy ( $D_{k}$ ), or degrees of freedom.

Equations of equilibrium available

Structure	Equations per rigid body / joint
Plane structure (whole or any free body)	3: $\sum F_{x} = 0$ , $\sum F_{y} = 0$ , $\sum M = 0$
Space structure	6: three forces, three moments
Pin joint of a plane truss	2: $\sum F_{x} = 0$ , $\sum F_{y} = 0$
Pin joint of a space truss	3

Support reactions

Support	Plane reactions	Space reactions
Roller / link	1	1
Hinge (pin)	2	3 (ball-and-socket)
Fixed	3	6
Guided roller (slider allowing translation, no rotation)	2	—

Static indeterminacy

External and internal

External indeterminacy $D_{se}$ = number of reactions $r$ minus equilibrium equations available for the whole structure (3 for plane) minus extra equations from releases.
Internal indeterminacy $D_{s i}$ = redundancy within the structure itself (closed loops, extra members).
$D_{s} = D_{se} + D_{s i}$ .

Formulas

FormulaDegree of static indeterminacy

Beams and plane rigid frames:

D_{s} = (3 m + r) - 3 j - c

$m$ = members, $r$ = reaction components, $j$ = rigid joints (including supports and free ends), $c$ = number of equations of condition from internal releases.

Alternative for frames (closed-loop method): $D_{s} = 3 C - R$ , where $C$ = number of closed loops when the supports are joined through the ground, and $R$ = number of releases (hinge support 1, roller support 2, fixed support 0, internal hinge between two members 1). Example: a portal frame with fixed feet forms one loop with the ground → $D_{s} = 3$ .

Plane truss (pin-jointed):

D_{s} = m + r - 2 j

Space truss: $D_{s} = m + r - 3 j$

Space rigid frame: $D_{s} = 6 m + r - 6 j - c$

Interpretation:

$D_{s} = 0$ and the structure is stable → determinate.
$D_{s} > 0$ → indeterminate to that degree.
$D_{s} < 0$ → unstable (mechanism).

A zero or positive $D_{s}$ is necessary but not sufficient for stability — the arrangement must also be geometrically stable (see below).

Internal releases (hinges)

An internal hinge makes the moment zero at that point, giving one extra equation of condition.

A hinge joining two members of a frame: $c = 1$ .
A hinge at a joint where $n$ members meet (all pinned together): $c = n - 1$ .
An internal roller (shear release plus moment release) in a beam: $c = 2$ .

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Determinacy, Indeterminacy & Stability