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Civil
Shear influence ordinate at a chosen section
Shear at section a of a simply supported span per unit downward force at z.
- Ref
- CIV-29
- Rev
- v1
- Field
- Civil
- Result
- ηV
The shear at a section equals the left reaction minus the loads strictly to the left of the cut. A unit load at z to the left of the section gives −z/L. A unit load to the right gives (L − z)/L. The line therefore jumps by one unit at the section, from −a/L to (L − a)/L. At z = a the page returns the value for a load just to the right.
Shear at section a of a simply supported span per unit downward force at z. Section shear is the left reaction less all forces strictly to the left of the cut. The line jumps by one unit at the section; at z = a the page returns the value for a load just to the right. Dimensionless.
Calculator
- Value just left of the section, −a/L−0.375
- Value just right of the section, (L − a)/L0.625
- Jump at the section1
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.
- The ordinate of 0.5 means a 10 kN load here would cause 5 kN of shear at the section, positive.
- Because of the one-unit jump, the worst positive shear needs the load just right of the section, and the worst negative shear needs it just left.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Section position | 0 to 8 m | −0.5 | 0.5 | 1 | +10% | 0 % |
| z Unit load position | 0 to 8 m | −0.3667 | 0.6 | 0.9667 | +10% | −10 % |
| L Span | 4 to 40 m | 0 | 0.9 | 0.9 | +10% | +9.09 % |
Sweep of Unit load position
| Unit load position (m) | Shear influence ordinate |
|---|---|
| 0 | 0 |
| 1 | −0.125 |
| 2 | −0.25 |
| 3 | 0.625 |
| 4 | 0.5 |
| 5 | 0.375 |
| 6 | 0.25 |
| 7 | 0.125 |
| 8 | 0 |
Sweep one variable
Variables and how each one contributes
Shear 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. The ordinate falls (becomes more negative) as the load moves right to the section, then drops back to positive after the jump. Raising it by 10 % (to 4.4) moves the result by −10 %, from 0.5 to 0.45.
Section position
The position of the section, in m. Moves the jump and changes the two end values −a/L and (L − a)/L. Raising it by 10 % (to 3.3) moves the result by +0 %, from 0.5 to 0.5.
Span
The span, in m. Scales the slope of the line; the jump of one unit stays the same. Raising it by 10 % (to 8.8) moves the result by +9.09 %, from 0.5 to 0.5455.
Worked example with the values above
- The load is at or right of the section, so η = (L − z)/L.
- (8 − 4) / 8.
- η_V = 0.5.
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