Define stiffness coefficient

Stiffness coefficient $k_{ij}$ is the force (or moment) required at coordinate $i$ to produce a unit displacement (or rotation) at coordinate $j$, while the displacement at every other coordinate is held at zero. It is the basic quantity of the stiffness (displacement) method; its unit is force per unit displacement (kN/m for a translation, kNm/rad for a rotation).

Explanation

• Each column of the structure stiffness matrix $[K]$ is the set of forces needed to hold the structure in the shape produced by a unit displacement at one coordinate. Thus $k_{ij}$ is the force at $i$ when coordinate $j$ is displaced by one unit.
• The equation $\{P\}=[K]\{\Delta\}$ follows by superposition.
• By Betti's law $k_{ij}=k_{ji}$, and $k_{ii}>0$.
• Examples for a prismatic member of length $L$ and rigidity $EI$:
  • unit rotation at one end, with the far end fixed and no translation: $k=4EI/L$ (moment at the same end) and $2EI/L$ (moment carried to the far end)
  • unit relative transverse translation of the ends, with both ends held against rotation: $k=12EI/L^3$ (shear) and $6EI/L^2$ (end moment)
  • unit axial extension: $k=AE/L$.