Stiffness, flexibility and reciprocity theorem

Equations of motion

A system of three masses $m_1, m_2, m_3$ in a line, connected by springs $k_1$ (wall to $m_1$), $k_2$ and $k_3$ in series, has

$$\begin{bmatrix} m_1 & 0 & 0 \\ 0 & m_2 & 0 \\ 0 & 0 & m_3 \end{bmatrix}\begin{Bmatrix} \ddot{x}_1 \\ \ddot{x}_2 \\ \ddot{x}_3 \end{Bmatrix} + \begin{bmatrix} k_1+k_2 & -k_2 & 0 \\ -k_2 & k_2+k_3 & -k_3 \\ 0 & -k_3 & k_3 \end{bmatrix}\begin{Bmatrix} x_1 \\ x_2 \\ x_3 \end{Bmatrix} = \begin{Bmatrix} F_1 \\ F_2 \\ F_3 \end{Bmatrix}$$

i.e. $[M]\{\ddot{x}\} + [K]\{x\} = \{F\}$.

Influence coefficients

• Stiffness influence coefficient $k_{ij}$: the force at coordinate $i$ required to produce a unit displacement at coordinate $j$, with all other displacements held at zero.
• Flexibility influence coefficient $a_{ij}$: the displacement at coordinate $i$ due to a unit force at coordinate $j$ (other forces zero).

Therefore $\{F\} = [k]\{x\}$ and $\{x\} = [a]\{F\}$.

Relation between them

Substituting $\{x\} = [a]\{F\}$ in $\{F\} = [k]\{x\}$ gives $\{F\} = [k][a]\{F\}$ for every $\{F\}$, so

$$[k][a] = [I] \;\Rightarrow\; [a] = [k]^{-1}$$

The flexibility matrix exists only if the system is restrained (no rigid-body motion). For an unrestrained system $[k]$ is singular.

Maxwell's reciprocity theorem

Statement: The displacement at point $i$ due to a unit load at point $j$ equals the displacement at $j$ due to a unit load at $i$: $a_{ij} = a_{ji}$ (so $[a]$ and $[k]$ are symmetric).

Proof: Apply load $P_i$ first and then $P_j$. The work done (strain energy stored) is

$$U_1 = \tfrac12P_i(a_{ii}P_i) + \tfrac12P_j(a_{jj}P_j) + P_i(a_{ij}P_j)$$

(The last term: while $P_j$ is being applied, the full load $P_i$, already in place, moves through the extra deflection $a_{ij}P_j$ at $i$ caused by $P_j$.) Now apply the loads in the reverse order:

$$U_2 = \tfrac12P_j(a_{jj}P_j) + \tfrac12P_i(a_{ii}P_i) + P_j(a_{ji}P_i)$$

The final state is the same, and the energy stored in a linear elastic system does not depend on the order of loading, so $U_1 = U_2$, giving $a_{ij} = a_{ji}$.