Quotient space (linear algebra)

In statics, a structure is statically indeterminate (or hyperstatic) when the static equilibrium equations are insufficient 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.

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

Σ V = 0:

V_A − F_v + V_B + V_C = 0

Σ H = 0:

H_A − F_h = 0

Σ M_A = 0:

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

Since there are four unknown forces (or variables) (V_A, V_B, V_C and H_A) but only three equilibrium equations, this system of simultaneous equations does not have a unique solution. The structure is therefore classified as statically indeterminate. Considerations in the material properties and compatibility in deformations are taken to solve statically indeterminate systems or structures.

Statically determinate

If the support at B is removed, the reaction V_B cannot occur, and the system becomes statically determinate (or isostatic). Note that the system is completely constrained here. The system becomes an exact constraint kinematic coupling. The solution to the problem is

$H_{A}=F_{h}$ ,

$V_{C}={\frac {F_{v}\cdot a}{a+b+c}}$ ,

$V_{A}=F_{v}-V_{C}$ .

If, in addition, the support at A is changed to a roller support, the number of reactions are reduced to three (without H_A), but the beam can now be moved horizontally; the system becomes unstable or partially constrained -- a mechanism rather than a structure. In order to distinguish between this and the situation when a system under equilibrium is perturbed and becomes unstable, it is preferable to use the phrase partially constrained here. In this case, the 2 unknowns V_A and V_C 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 is not possible to satisfy the horizontal force equation unless $F_{h}=0$.

Statical indeterminacy is the existence of a non-trivial (non-zero) solution to the homogeneous system of equilibrium equations. It indicates the possibility of self-stress (stress in the absence of an external load) that may be induced by mechanical or thermal action.

See also

• Christian Otto Mohr
• Flexibility method
• Hardy Cross
• Moment distribution method
• Structural engineering
• overconstrained mechanism

External links

• Beam calculation online (Statically indeterminate)

References

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