Statically indeterminate

In statics and structural mechanics, a structure is statically indeterminate when the equilibrium equations – force and moment equilibrium conditions – are insufficient for determining the internal forces and reactions on that structure.^[1]^[2]

Mathematics

Based on Newton's laws of motion, the equilibrium equations available for a two-dimensional body are:^[2]

\sum \mathbf {F} =0:

the vectorial sum of the forces acting on the body equals zero. This translates to:

\sum \mathbf {H} =0:

the sum of the horizontal components of the forces equals zero;

\sum \mathbf {V} =0:

the sum of the vertical components of forces equals zero;

\sum \mathbf {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:^[2]

{\begin{aligned}\sum \mathbf {V} =0\quad &\implies \quad \mathbf {V} _{A}-\mathbf {F} _{v}+\mathbf {V} _{B}+\mathbf {V} _{C}=0\\\sum \mathbf {H} =0\quad &\implies \quad \mathbf {H} _{A}=0\\\sum \mathbf {M} _{A}=0\quad &\implies \quad \mathbf {F} _{v}\cdot a-\mathbf {V} _{B}\cdot (a+b)-\mathbf {V} _{C}\cdot (a+b+c)=0\end{aligned}}

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.

To solve statically indeterminate systems (determine the various moment and force reactions within it), one considers the material properties and compatibility in deformations.

Statically determinate

If the support at $B$ is removed, the reaction $V B$ cannot occur, and the system becomes statically determinate (or isostatic).^[3] Note that the system is completely constrained here. The system becomes an exact constraint kinematic coupling. The solution to the problem is:^[2]

{\begin{aligned}\mathbf {H} _{A}&=\mathbf {F} _{h}\\\mathbf {V} _{C}&={\frac {\mathbf {F} _{v}\cdot a}{a+b+c}}\\\mathbf {V} _{A}&=\mathbf {F} _{v}-\mathbf {V} _{C}\end{aligned}}

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 partly 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 partly constrained here. In this case, the two 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 $F h = 0$ .^[2]

Statical determinacy

Descriptively, a statically determinate structure can be defined as a structure where, if it is possible to find internal actions in equilibrium with external loads, those internal actions are unique. The structure has no possible states of self-stress, i.e. internal forces in equilibrium with zero external loads are not possible. Statical indeterminacy, however, 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.

Mathematically, this requires a stiffness matrix to have full rank.

A statically indeterminate structure can only be analyzed by including further information like material properties and deflections. Numerically, this can be achieved by using matrix structural analyses, finite element method (FEM) or the moment distribution method (Hardy Cross) .

Practically, a structure is called 'statically overdetermined' when it comprises more mechanical constraints – like walls, columns or bolts – than absolutely necessary for stability.

References

^ Matheson, James Adam Louis (1971). Hyperstatic structures: an introduction to the theory of statically indeterminate structures (2nd ed.). London: Butterworths. ISBN 0408701749. OCLC 257600.
^ ^a ^b ^c ^d ^e Megson, Thomas Henry Gordon (2014). "Analysis of statically indeterminate structures". Structural and stress analysis (Third ed.). Amsterdam: Elsevier. pp. 489–570. ISBN 9780080999364. OCLC 873568410.
^ Carpinteri, Alberto (1997). Structural mechanics: a unified approach (1st ed.). London: E & FN Spon. ISBN 0419191607. OCLC 36416368.

External links

Beam calculation online (Statically indeterminate)

[1] Matheson, James Adam Louis (1971). Hyperstatic structures: an introduction to the theory of statically indeterminate structures (2nd ed.). London: Butterworths. ISBN 0408701749. OCLC 257600.

[:0-2] Megson, Thomas Henry Gordon (2014). "Analysis of statically indeterminate structures". Structural and stress analysis (Third ed.). Amsterdam: Elsevier. pp. 489–570. ISBN 9780080999364. OCLC 873568410.

[3] Carpinteri, Alberto (1997). Structural mechanics: a unified approach (1st ed.). London: E & FN Spon. ISBN 0419191607. OCLC 36416368.

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