Writing / 2026-05-09

Influence Lines — What They Are and Why They Matter

A structural engineer's guide to influence lines: how a response at one point changes as a unit load travels across a structure.

The core idea

An influence line (IL) answers a specific question: as a unit load (P = 1) moves from one end of a structure to the other, how does a particular response — a reaction force, a shear force, or a bending moment at a chosen section — change in value? The result is a graph whose horizontal axis is the load position and whose vertical axis is the magnitude of that response.

This is different from an ordinary bending-moment diagram. An ordinary BMD shows how the moment varies along the whole beam for one fixed load position. An influence line keeps the response location fixed and varies the load position. Once you have the IL, you can place a real moving load (a truck, a crane, a train) anywhere and immediately read off the worst-case response.

Key insight: the IL ordinate at any position x equals the response value when the unit load sits at x. To find the response due to a real load P at position a, just multiply: Response = P × IL(a). For a distributed load w over a length, integrate w × IL(x) dx over that length.

Simply supported beam: the three ILs

For a simply supported beam (pin at A, roller at B, span L), with a unit load at position x from A: Reaction at A = (L − x) / L, and Reaction at B = x / L. Both vary linearly from 1 to 0 (or 0 to 1) — the ILs are straight lines.

For shear at section C (distance a from A, b = L − a from B): when the load is to the right of C, V_C = (L − x)/L = positive; when the load is to the left of C, V_C = −x/L = negative. There is a jump of magnitude 1 exactly at C. The IL is bilinear with a discontinuity at the section.

For bending moment at C: M_C = x·b/L when the load is left of C, and M_C = a·(L−x)/L when right. The IL is triangular, peaking at x = a with a value of a·b/L. Note that the peak equals a·b/L, not a·b — the unit load and the span cancel out to give units of length (metres), which is the ordinate unit for moment ILs.

Müller-Breslau principle: the shape of the IL for any response equals the deflected shape of the structure when the restraint corresponding to that response is removed and a unit displacement (or rotation) is applied at that location. This is powerful for quickly sketching ILs for complex structures.

Cantilever beam

For a cantilever (fixed at the wall, free end), the vertical reaction is always 1 regardless of load position — the IL is a horizontal line. The shear IL at section C is a step function: 0 when the load is between the wall and C, and 1 when the load is between C and the free end. The moment IL at C is 0 from the wall to C, then drops linearly (negative, hogging) toward the free end.

When does an IL become curved?

For statically determinate beams, ILs are always straight-line (piecewise linear). Curved ILs only appear in statically indeterminate structures (continuous beams, frames with fixed ends). This is because, for a determinate structure, equilibrium alone gives the reactions as linear functions of the load position — and shear and moment follow by linear combination.

Video explanations

The three videos below each approach influence lines from a different angle. Watch them in sequence, or jump to whichever level matches where you are.

Video 1 — Influence lines: the concept and the simply supported beam

Video 2 — Drawing ILs for shear and moment step by step

Video 3 — Applying ILs to find maximum effects from moving loads

Practice in the lab

Once you feel comfortable with the theory, head to the Influence Line Lab to build your own beams and verify your intuition interactively.