Statically indeterminate

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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.

Based on Newton's second law, the equilibrium equations available for a two-dimensional body are

• : 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;

• : the sum of the moments (about an arbitrary point) of all forces equals zero.

In the beam construction on the right, the four unknown reactions are V_A, V_B, V_C and H_A. The equillibrium equations are:

Σ V = 0:

V_A − F_v + V_B + V_C = 0

Σ H = 0:

H_A − F_{h = 0}

_{Σ MA = 0:}

F_v · a − V_B · (a + b) - V_C · (a + b + c) = 0.

Since there are four unknowns forces (or variables) (V_A, V_B, V_C and H_A) but only three equillibrium equations, this system of simultaneous equations cannot be solved. The structure is therefore classified statically indeterminate. Considerations in the material properties and compatibility in deformations are taken to solve statically indeterminate systems or structures.

[edit] Statically determinate

If the support at B or C is removed, the reactions V_B or V_C cannot occur, and the system becomes statically determinate. If the support at A is designed as a roller support, the number of reactions are reduced to three (without H_A), but then the system becomes unstable.

[edit] Static indeterminacy

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 static indeterminacy of structural systems depends on the internal structure as well as on the external supports.

The degree of static 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.

[edit] See also

• flexibility method
• Hardy Cross
• Otto Mohr
• structural engineering