Influence Lines
Walk a unit load across the span and watch the marker move with the LOAD, not along the beam — which is the distinction the topic turns on.
Skip to the animationAn influence line plots one chosen quantity against the position of a moving unit load — so its horizontal axis is where the load is, not where you are on the structure, which is the opposite of every other diagram in the subject.
The distinction the topic turns on
| Bending moment diagram | Influence line | |
|---|---|---|
| Load | Fixed in place | One unit load, moving |
| Horizontal axis | Position along the structure | Position of the load |
| Ordinate | The quantity at that section | One fixed quantity, for a load here |
| Answers | Where is the structure worst stressed? | Where should the load be to make this worst? |
The two diagrams look alike and mean opposite things, and confusing them is the single most common error in this topic. Read the horizontal axis first, every time.
Constructing one
Place a unit load at a position, compute the chosen quantity, and plot that value against the load's position. Repeat across the span. For a simply supported beam the influence line for R_A is a straight line from 1 at A to 0 at B; for the midspan moment it is a triangle peaking at L/4.
Using one
- A point load W: value =
W × ordinateat that position. - A distributed load w: value =
w × areaunder the relevant part of the line. - For the maximum: place the load where the ordinate is largest.
- For a train of wheel loads: put the heaviest wheel at the peak, then check a few positions either side.
All of these are read off the diagram with no re-analysis of the structure — which is the entire point. Bridge codes work precisely this way, running a standard vehicle across the influence line.
Müller-Breslau's principle
Release the restraint corresponding to the quantity, impose a unit displacement in its direction, and the deflected shape is the influence line. Lift the support for a reaction; insert a hinge for a bending moment; introduce a shear release for a shear force.
It follows from Maxwell's reciprocal theorem: deflection at X due to a load at Y equals deflection at Y due to a load at X — which is exactly the swap between "quantity fixed, load moving" and "load fixed, response moving".
So a qualitative influence line can be sketched in seconds with no calculation at all, which is invaluable for deciding where to place load before doing any arithmetic.
Determinate against indeterminate
For a determinate structure the quantity varies linearly with load position, so the influence line is made of straight segments — and a curved one means you have made an error.
For an indeterminate structure the lines are curved, because the load distributes according to relative stiffness, so EI enters. Müller-Breslau still applies, but the deflected shape must be computed rather than sketched.
The numbers you will be asked for
- Point load
value = W × y
y is the ordinate at the load's position.
- Distributed load
value = w × (area under the line)
Over the loaded length only.
- R_A, simple span
y = (L − x)/L
1 at A, 0 at B — a straight line.
- Midspan moment, simple span
peak = L/4
Triangular, peaking with the load at midspan.
- Müller-Breslau
influence line = deflected shape after unit release
Follows from Maxwell's reciprocal theorem.
Advantages and disadvantages
Advantages
- Answers the moving-load question once, then reused for any load pattern.
- Point and distributed loads are handled by an ordinate and an area.
- Müller-Breslau gives the shape qualitatively with no calculation.
- Extends directly to trusses, where the load moves along a loaded chord.
Disadvantages
- One diagram per quantity, so a full check needs many of them.
- Easily confused with a bending moment diagram, which it resembles and contradicts.
- Curved and much harder to construct for indeterminate structures.
- Assumes a single moving unit load; load trains need extra positioning work.
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.