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Concept
Statically indeterminate
Summary
In statics and structural mechanics, a structure is statically indeterminate when the static equilibrium equations - force and moment equilibrium conditions - 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:
the vectorial sum of the forces acting on the body equals zero. This translates to:
the sum of the horizontal components of the forces equals zero;
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 equilibrium equations are:
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.
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:
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.
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Related concepts (5)
Free body diagram
In physics and engineering, a free body diagram (FBD; also called a force diagram) is a graphical illustration used to visualize the applied forces, moments, and resulting reactions on a body in a given condition. It depicts a body or connected bodies with all the applied forces and moments, and reactions, which act on the body(ies). The body may consist of multiple internal members (such as a truss), or be a compact body (such as a beam). A series of free bodies and other diagrams may be necessary to solve complex problems.
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