Structural rigidity is the property of a structure that it does not bend or flex under an applied force. A structure is built of rigid components, that hold their ends apart, but it may also have components that pull their ends together, called "tension elements". The components in the simplest cases are straight rods, or rods and solid triangles and rectangles. For the theory one assumes the components are perfectly rigid, or perfect tension elements. The opposite of rigidity is flexibility.

The term rigidity of frameworks refers to the rigidity problem for structures built of straight rods, and possibly also linear tension elements.

There are two essentially different kinds of rigidity. Finite or macroscopic rigidity means that the structure will not flex, fold, or bend by a positive amount. Infinitesimal rigidity means that the structure will not flex by even an amount that is too small to be detected even in theory. (Technically, that means certain differential equations have no nonzero solutions.) The importance of finite rigidity is obvious, but infinitesimal rigidity is also crucial because infinitesimal flexibility in theory corresponds to real-world minuscule flexing, and consequent deterioration of the structure.

The fundamental problem is how to predict the rigidity of a structure by theoretical analysis, without having to build it. Even in the simplest cases it is not yet always known how to do so despite the existence of considerable mathematical theory.

One of the founders of the mathematical theory of structural rigidity was the great physicist James Clerk Maxwell. The late twentieth century saw an efflorescence of the mathematical theory of rigidity, which continues in the twenty-first century.

[edit] See also

• Torsion

[edit] References

• Alfakih, Abdo Y. (2007), On dimensional rigidity of bar-and-joint frameworks. Discrete Applied Mathematics, Vol. 155, No. 10, pp. 1244-1253.

• Connelly, Robert (1980), The rigidity of certain cabled frameworks and the second-order rigidity of arbitrary triangulated convex surfaces. Advances in Mathematics, Vol. 37, pp. 272-299.

• Crapo, Henry (1979), Structural rigidity. Topologie Structurale (Structural Topology), Vol. 1, pp. 26-45.

• Maxwell, J. C. (1864), On reciprocal figures and diagrams of forces.

Philosophical Magazine (4th Series), Vol. 27, pp. 250-261.

• Rybnikov, Konstantin, and Zaslavsky, Thomas (2005), Criteria for balance in abelian gain graphs, with applications to piecewise-linear geometry. Discrete and Computational Geometry, Vol. 34, No. 2, pp. 251-268.

• Whiteley, W. (1988), "The union of matroids and the rigidity of frameworks", SIAM Journal on Discrete Mathematics 1 (2): 237–255, doi:10.1137/0401025 .