Internal hinge properties; point of contraflexure

Properties of an internal hinge

An internal hinge is a pin connection inside a structure (for example in a beam or frame) that joins two parts.

1. It transmits axial force and shear force from one part to the other, but cannot transmit bending moment, so the bending moment at the hinge is zero.
2. It allows free relative rotation of the two connected parts, so the slope (rotation) is discontinuous at the hinge.
3. It provides one extra equation of condition ($\Sigma M_{hinge} = 0$ for the forces on either side). This is why a hinge reduces the degree of static indeterminacy by one (it releases one internal moment) and can make a structure with extra reactions determinate (for example, a three-hinged frame).
4. Hinges must be located and counted carefully: too many hinges turn the structure into a mechanism (unstable).
5. A hinge is a point of zero bending moment, but not necessarily a point of contraflexure, because the moment need not change sign there.

Point of contraflexure

A point of contraflexure is a point on the span of a beam where the bending moment changes sign (from sagging to hogging or the reverse), passing through zero. The curvature of the elastic curve changes direction there. It is found by setting $M(x) = 0$. Such points matter in design: the moments on either side have opposite signs, so the tension face switches from the bottom to the top, and reinforcement can be curtailed or shifted there.