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 laws of motion, 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;
  • \sum \vec M = 0 : the sum of the moments (about an arbitrary point) of all forces equals zero.
Free body diagram of a statically indeterminate beam.
Free body diagram of a statically indeterminate beam.

In the beam construction on the right, the four unknown reactions are VA, VB, VC and HA. The equillibrium equations are:

Σ V = 0:

VA − Fv + VB + VC = 0

Σ H = 0:

HA − Fh = 0

Σ MA = 0:

Fv · a − VB · (a + b) - VC · (a + b + c) = 0.

Since there are four unknown forces (or variables) (VA, VB, VC and HA) 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 is removed, the reaction VB cannot occur, and the system becomes statically determinate. Note that the system is completely constrained here. The solution to the problem is

HA = Fh ,
V_C=\frac{F_v\cdot a}{a+b+c} ,
VA = Fv − VC .

If, in addition, the support at A is changed to a roller support, the number of reactions are reduced to three (without HA), but the beam can now be moved horizontally; the system becomes unstable or partially constrained. In order to distinguish between this and the situation when a system under equilibrium is perturbed and becomes unstable, it's preferable to use the phrase partially constrained here. In this case, the 2 unknowns VA and VC can be determined by resolving the vertical force equation and the moment equation simultaneously. The solution yields the same results as previously obtained. However, it's not possible to satisfy the horizontal force equation unless Fh = 0.

[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