Influence lines are graphs that show how a specific beam response changes as a load moves across the structure. The response can be a support reaction, shear force at a section, or bending moment at a section. They matter because bridges, crane girders, and floor systems often carry moving loads rather than fixed loads.
An influence line helps engineers place those loads where they cause the largest effect.
To build an influence line, imagine a unit load moving from one end of the beam to the other and calculate the chosen response at each load position. The height of the influence line at any position tells how strongly a load at that position affects the response. For several moving loads, multiply each load by the influence-line ordinate under it and add the results.
The worst-case position occurs when the loads are arranged to maximize the total value of the reaction, shear, or moment being studied.
Key Facts
- Influence line value = structural response caused by a unit load at a specified position.
- Response from one point load: effect = P y, where P is the load and y is the influence-line ordinate under the load.
- Response from multiple point loads: total effect = Σ P_i y_i.
- For a simply supported beam of span L, the influence line for left reaction is R_A = (L - x) / L for a unit load at distance x from A.
- For a simply supported beam of span L, the influence line for right reaction is R_B = x / L.
- Worst-case loading is found by placing the largest loads where the influence-line ordinates are largest and have the desired sign.
Vocabulary
- Influence line
- A graph showing how a chosen reaction, shear, or moment changes as a unit load moves across a structure.
- Ordinate
- The vertical value of the influence line at a particular load position.
- Unit load
- A load with magnitude 1 used to calculate the influence line for a structural response.
- Moving load
- A load whose position changes along a structure, such as a truck wheel load on a bridge.
- Critical position
- The load arrangement that produces the maximum or minimum value of the response being investigated.
Common Mistakes to Avoid
- Confusing the influence line with the bending moment diagram for a fixed load. An influence line varies load position, while a moment diagram varies location along the beam for loads that are already placed.
- Using the wrong ordinate under a moving load. The ordinate must be taken directly below the current load position, not below the section where the response is being measured unless the load is there.
- Ignoring the sign of the influence line. Loads placed on negative ordinates reduce a positive response and may be critical when looking for the most negative response.
- Adding loads without multiplying by ordinates. For moving point loads, each contribution is P_i y_i, so the load magnitude alone does not give the structural effect.
Practice Questions
- 1 A simply supported beam has span L = 10 m. A unit load is located 3 m from the left support A. Find the influence-line ordinates for R_A and R_B.
- 2 For a beam response, the influence-line ordinates under three wheel loads are 0.8, 0.5, and -0.2. The loads are 40 kN, 60 kN, and 30 kN at those positions. Find the total response using Σ P_i y_i.
- 3 A positive bending moment influence line has its largest positive ordinate near midspan and negative ordinates near an overhang. Explain where heavy loads should be placed to maximize positive moment and which regions should be avoided.