Library / Civil / CIV-27
Civil
Moment distribution factor at a rigid joint
Fraction of an unbalanced joint moment taken by member i, equal to its rotational stiffness over the sum for all members meeting at the joint.
- Ref
- CIV-27
- Rev
- v1
- Field
- Civil
- Result
- DFi
When a joint is out of balance, each member takes a share of the unbalanced moment in proportion to its rotational stiffness K. The distribution factor is DFi = Ki/ΣKj, and the factors at a joint add to one. For a prismatic member without sway, K = 4EI/L if the far end is fixed and 3EI/L if it is hinged.
Fraction of an unbalanced joint moment taken by member i, equal to its rotational stiffness over the sum for all members meeting at the joint. For a prismatic member without sway use K = 4EI/L if the far end is fixed or 3EI/L if it is hinged. Up to four members here; leave unused ones at zero. The page returns the factor for the member selected by i.
Calculator
- Total joint stiffness, ΣK2.4×107 N·m/rad
- Largest distribution factor0.5
- Distribution factor of member 20.25
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Member 1 takes 50 % of any moment applied at the joint; the factors for all members are 0.5, 0.25, 0.25.
- If the far end of this member were hinged instead of fixed its stiffness would fall to three-quarters of its value, so its factor would shrink and the others’ would rise.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| K1 Stiffness of member 1 | 0 to 1e8 N·m/rad | 0 | 0.8929 | 0.8929 | +10% | +4.76 % |
| K2 Stiffness of member 2 | 0 to 1e8 N·m/rad | 0.1017 | 0.6667 | 0.565 | +10% | −2.44 % |
| K3 Stiffness of member 3 | 0 to 1e8 N·m/rad | 0.1017 | 0.6667 | 0.565 | +10% | −2.44 % |
| i Member to report | 1 to 4 | 0 | 0.5 | 0.5 | +1 | −50 % |
| K4 Stiffness of member 4 | 0 to 1e8 N·m/rad | 0.09677 | 0.5 | 0.4032 | +10% | 0 % |
Sweep of Stiffness of member 1
| Stiffness of member 1 (N·m/rad) | Distribution factor |
|---|---|
| 0 | 0 |
| 1.25e7 | 0.5102 |
| 2.5e7 | 0.6757 |
| 3.75e7 | 0.7576 |
| 5e7 | 0.8065 |
| 6.25e7 | 0.8389 |
| 7.5e7 | 0.8621 |
| 8.75e7 | 0.8794 |
| 1e8 | 0.8929 |
Sweep one variable
Variables and how each one contributes
Distribution factor
What the equation gives you. Everything else feeds it.
Stiffness of member 1
The rotational stiffness of member 1 at the joint, in N·m/rad (zero if absent). A stiffer member takes a larger share of the unbalanced moment. Raising it by 10 % (to 1.32×107) moves the result by +4.76 %, from 0.5 to 0.5238.
Stiffness of member 2
The rotational stiffness of member 2 at the joint, in N·m/rad (zero if absent). A stiffer member takes a larger share of the unbalanced moment. Raising it by 10 % (to 6,600,000) moves the result by −2.44 %, from 0.5 to 0.4878.
Stiffness of member 3
The rotational stiffness of member 3 at the joint, in N·m/rad (zero if absent). A stiffer member takes a larger share of the unbalanced moment. Raising it by 10 % (to 6,600,000) moves the result by −2.44 %, from 0.5 to 0.4878.
Stiffness of member 4
The rotational stiffness of member 4 at the joint, in N·m/rad (zero if absent). A stiffer member takes a larger share of the unbalanced moment. Raising it by 10 % (to 0) moves the result by +0 %, from 0.5 to 0.5.
Member to report
The member whose factor is reported, numbered 1 to 4. Selects which member’s distribution factor is shown. Raising it by 10 % (to 1.1) moves the result by +0 %, from 0.5 to 0.5.
Worked example with the values above
- Total stiffness: ΣK = 1.2×107 + 6,000,000 + 6,000,000 = 2.4×107 N·m/rad.
- Member 1: K1 = 1.2×107 N·m/rad.
- DF1 = 1.2×107 / 2.4×107 = 0.5.
Assumptions and limits
- Rigid supports with no settlement, and a prismatic beam of constant EI.
- Linear elastic behaviour, so compatibility and superposition apply.
Common mistakes
- Using 4EI/L for a member whose far end is hinged, when 3EI/L applies.
- Leaving out a member that meets the joint.
- Adding the factors from different joints as if they shared one total.
Statistics and uncertainty
Redundant reactions depend on the stiffness of the supports being as assumed. A prop that settles by only a few millimetres, or a clamp that is not fully rigid, can change the reaction by 10 % or more, so these ideal results are best read as limits on the real values.
Where it is used
- Sizing props, continuous beams and fixed-ended members.
- Setting up moment distribution analysis of frames.
Maths you need
- Compatibility: deformation at a redundant support must match the real support condition
- Fixed-end moment: the moment at a clamped end under load
- Rotational stiffness: moment needed to rotate a joint by one radian