Statically indeterminate

In statics, a structure is statically indeterminate when the static equilibrium equations are not sufficient for determining the internal forces and reactions on that structure.

The equilibrium equations available for a two-dimensional body are

$\sum {\vec {F}}=0$ : the vectorial sum of the forces acting on the body equals zero. This translates to

Σ H = 0: the sum of the horizontal components of the forces equals zero;

Σ V = 0: the sum of the vertical components of forces equals zero;

Σ M = 0: the sum of the moments (about an arbitrary point) of all forces equals zero.

File:Staticallyindeterminate.jpg

In the beam construction on the right, the four unknown reactions are V_A, V_B, H_A and H_B. The vertical reactions can be found as follows:

Σ V = 0:

V_A − F_v + V_B = 0, or V_A + V_B = F_v

Σ M_A = 0:

F_v · a − V_B · (a + b) = 0, or V_B = F_v · a / (a + b)

hence V_A = F_v · b / (a + b).

Since only the horizontal equilibrium equation remains, the unknown reactions H_A and H_B cannot be solved, and the structure is classified statically indeterminate. This conclusion can be reached merely by observing that the three equilibrium equations available for the body are insufficient for determining the four unknown forces.

Statically determinate

If the support at B is designed as a roller support, the reaction H_B cannot occur, and the system becomes statically determinate. All three support reactions can then be calculated using the three equilibrium equations. For the remaining reaction H_A:

Σ H = 0:

H_A − F_h = 0, or H_A = F_h

Statical indeterminacy

File:Determinacy.jpg

A system can be statically indeterminate even though its reactions are determinate as shown in Fig.(a) on the right. On the other hand, the system in Fig.(b) has indeterminate reactions, and yet, the system is determinate because its member forces, and subsequently the reactions, can be found by statics. Thus, in general, the statical indeterminacy of structural systems depends on the internal structure as well as on the external supports.

The degree of statical indeterminacy of a system is M-N where

M is the number of unknown member forces, and optionally, reactions in the system;
N is the number of independent, non-trivial equilibrium equations available for determining these M unknown forces.

If M includes reaction components, then N must include equilibrium equations along these reaction components, one for one. Thus, we may, in fact, choose to exclude reactions from the above relation.

Statically determinate

Statical indeterminacy

See also