Compatibility conditions; Maxwell's theorem

Compatibility conditions

Compatibility conditions are the geometric requirements that the deformed structure must satisfy: displacements must be continuous inside members and at the joints, and must agree with the support conditions (for example zero deflection at a rigid support, zero rotation at a fixed end, equal rotation of connected members at a rigid joint). They are independent of the material and loading.

In an indeterminate structure, equilibrium alone cannot give all the forces. The extra equations come from compatibility. In the force method, the redundants $X_j$ are found from

$$\Delta_{i0} + \sum_j f_{ij}X_j = \delta_i$$

Example: for a propped cantilever with prop reaction $R_B$, the compatibility condition is that the deflection at the prop is zero, $\Delta_{B0} - f_{BB}R_B = 0$, so $R_B = \dfrac{wL^4/8EI}{L^3/3EI} = \dfrac{3wL}{8}$.

Maxwell's reciprocal theorem

In a linearly elastic structure, the displacement at A due to a unit load at B equals the displacement at B due to a unit load at A:

$$\delta_{AB} = \delta_{BA}, \qquad f_{ij} = f_{ji}$$

It follows from Betti's law (or from the independence of strain energy from the order of loading). It makes the flexibility and stiffness matrices symmetric, which reduces the number of coefficients to be found in the force method and gives a check on the computation. Example: for a cantilever of length $L$, the deflection at mid-span due to a unit tip load is $\dfrac{5L^3}{48EI}$, which equals the tip deflection due to a unit load at mid-span.