---
title: "Carry over factor and distribution factor"
subject: "Theory of Structures II"
chapter: 3
key: "semester_5/theory_of_structures_ii:define-carry-over-distribution-factor"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/define-carry-over-distribution-factor
site: BCT Engineering
---

# Carry over factor and distribution factor

Theory of Structures II · Chapter 3: Displacement method

- Asked in: 2070 Chaitra (old course), 4 marks
- Video: [Lect67-Distribution factor,Carry over factor in M.D.M.](https://www.youtube.com/watch?v=y_bPdy9KIDs) — Dr. Ashish Agrawal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/define-carry-over-distribution-factor
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-3

## Question

Define carry over factor and distribution factor.

## Short answer

Distribution factor is the fraction of an unbalanced joint moment taken by a member, DF = K/ΣK with K = 4EI/L (far end fixed) or 3EI/L (hinged). Carry-over factor is the ratio of the far-end moment to the applied near-end moment when the far end is fixed, COF = 1/2; if hinged, COF = 0.

## Answer

#### 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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The distribution factor of a member at a joint is:
   A. the moment carried to its far end
   B. the fraction of the unbalanced joint moment resisted by that member, K / sum K
   C. the ratio of its length to the total length
   D. the fixed-end moment divided by the span
   Answer: B. DF = K of the member divided by the sum of K of all members at the joint, so the DFs at a joint add up to 1. (Key point: Meaning of distribution factor)

2. Which stiffness K is used when computing the DF of a member whose far end is hinged?
   A. 4EI/L
   B. 3EI/L
   C. 2EI/L
   D. 6EI/L^2
   Answer: B. A hinged far end gives modified stiffness 3EI/L; 4EI/L applies when the far end is fixed. (Key point: Stiffness for far end condition)

3. What is the carry-over factor when the far end is fixed, and when it is hinged?
   A. 1/2 for fixed and 0 for hinged
   B. 1 for fixed and 1/2 for hinged
   C. 1/2 for both
   D. 0 for fixed and 1/2 for hinged
   Answer: A. COF = (2EI/L)/(4EI/L) = 1/2 for a fixed far end; a hinge or roller cannot develop moment so COF = 0. (Key point: Carry-over factor value)

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