Carry over factor and distribution factor

Distribution factor (DF)

The distribution factor of a member at a joint is the fraction of the unbalanced (external) moment at that joint that is resisted by that member:

$$DF_{jk}=\frac{K_{jk}}{\sum K_j},\qquad K=\frac{4EI}{L}\ \text{(far end fixed)},\quad K=\frac{3EI}{L}\ \text{(far end hinged)}$$

The sum of the DFs of all the members meeting at a joint is 1. For a fixed support $DF=0$ and for a roller or hinge at the end of a single member $DF=1$. Because only ratios are needed, $EI/L$ may be used in place of $4EI/L$ for all members of a joint with fixed far ends.

Example: at a joint with two members of $K=2$ and $K=3$, the DFs are $2/5=0.4$ and $3/5=0.6$.

Carry-over factor (COF)

The carry-over factor is the ratio of the moment induced at the far end of a member to the moment applied at the near end when the near end is rotated and the far end is held fixed:

$$COF=\frac{M_{far}}{M_{near}}=\frac{2EI/L}{4EI/L}=\frac{1}{2}$$

If the far end is hinged or on a roller, no moment can be developed there, so $COF=0$ (the modified stiffness $3EI/L$ is then used). The carried-over moment has the same sign as the applied moment (clockwise near-end moment gives a clockwise far-end moment).