Library / Civil / CIV-28
Civil
Moment influence ordinate at a chosen beam section
Bending moment at section a of a simply supported span, per unit downward force at position z, both measured from the left support.
- Ref
- CIV-28
- Rev
- v1
- Field
- Civil
- Result
- ηM [m]
The moment at section a caused by a unit downward load at z is the influence ordinate. For a load left of the section the right reaction is z/L and the moment is z(L − a)/L; for a load to the right it is a(L − z)/L. The ordinates form a triangle with its peak a(L − a)/L under the section. The unit is length because the load is a unit force.
Bending moment at section a of a simply supported span, per unit downward force at position z, both measured from the left support. The influence line is a triangle with its peak a(L − a)/L at the section. The result is sagging moment per unit force, so its unit is length.
Calculator
- Peak ordinate (load at the section), a(L − a)/L1.875 m
- Load position as a fraction of the span, z/L0.5
- Moment from a 10 kN load here15 kN·m
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.
- A unit force here produces 1.5 m of moment, 80 % of the peak value 1.875 m, which is reached with the load directly over the section.
- The ordinate is zero at both supports, so a load there causes no moment at the section.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Span | 4 to 40 m | 0 | 2.7 | 2.7 | +10% | +9.09 % |
| a Section position | 0 to 8 m | 0 | 2 | 2 | +10% | +10 % |
| z Unit load position | 0 to 8 m | 0 | 1.833 | 1.833 | +10% | −10 % |
Sweep of Unit load position
| Unit load position (m) | Moment influence ordinate (m) |
|---|---|
| 0 | 0 |
| 1 | 0.625 |
| 2 | 1.25 |
| 3 | 1.875 |
| 4 | 1.5 |
| 5 | 1.125 |
| 6 | 0.75 |
| 7 | 0.375 |
| 8 | 0 |
Sweep one variable
Variables and how each one contributes
Moment influence ordinate
What the equation gives you. Everything else feeds it.
Unit load position
The position of the unit load from the left support, in m. Moving the load toward the section raises the ordinate linearly to its peak. Raising it by 10 % (to 4.4) moves the result by −10 %, from 1.5 to 1.35 m.
Section position
The position of the section from the left support, in m. Moves the peak and raises it as the section approaches midspan. Raising it by 10 % (to 3.3) moves the result by +10 %, from 1.5 to 1.65 m.
Span
The span, in m. A longer span scales all the ordinates up. Raising it by 10 % (to 8.8) moves the result by +9.09 %, from 1.5 to 1.636 m.
Worked example with the values above
- The load is right of the section, so η = a(L − z)/L.
- a(L − z) = 3 × 4 = 12, divided by L = 8.
- η_M = 1.5 m.
Assumptions and limits
- A simply supported span with superposition valid, and static loads placed at their stated positions.
- Loads outside the span contribute nothing.
Common mistakes
- Putting the unit load on the wrong side of the section for shear, where the value jumps by one.
- Applying the closed-form maximum outside its stated spacing range.
- Using an influence line for moment to find shear.
Statistics and uncertainty
Influence-line results are exact for a linear elastic simply supported span, but they only give the structural response per unit load. The uncertainty lies in the moving load: its magnitude, spacing, and dynamic effects, which design codes cover with amplification factors of 10 % to 30 %.
Where it is used
- Finding worst positions of vehicles and crane wheels on bridges and runways.
- Building live-load envelopes for floors and decks.
Maths you need
- Influence line: response at one section as a unit load moves along the span
- Superposition: responses to separate loads add
- Resultant: the single force equivalent to a set of loads