Structural
Influence Lines
How a single response changes as a load rolls across — the moving-load engineer's map
An influence line is a diagram of how one specific response — a support reaction, the internal shear at a section, or the bending moment at a section — varies as a single unit load moves across a structure. Its ordinate at position x is the value of that response when the unit load sits at x, so multiplying the ordinate under a real load by that load's magnitude gives the load's contribution. Influence lines pin down the worst-case position of moving traffic on bridges, build the maximum-effect envelope that girders are designed against, and are drawn in seconds with the Müller-Breslau principle — the influence line has the shape of the deflected form obtained by releasing the response and imposing a unit displacement. For a simply supported span of length L, the moment influence line at midspan is a triangle peaking at L/4, and the AASHTO HL-93 truck plus 9.3 kN/m lane load are positioned by exactly this reasoning.
- PlotsOne response vs. moving-load position
- OrdinateResponse when unit load sits at x
- ConstructionMüller-Breslau deflected shape
- Determinate ILPiecewise straight lines
- Midspan M peakL/4 for a simple span
- Design useMoment & shear envelopes (AASHTO HL-93)
Interactive visualization
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Watch the 60-second explainer
A condensed visual walkthrough — narrated, captioned, under a minute.
Why influence lines matter
Almost every structure that carries traffic is loaded by things that move. A truck rolls across a girder bridge, a locomotive tracks along a plate girder, an overhead crane trolley traverses a runway beam, a forklift crosses a warehouse slab. The internal forces at any given section are not fixed — they rise and fall as the load changes position. A single static analysis of one load case cannot prove it found the worst case, because a load that is harmless at one location may be devastating at another. The influence line answers the question a fixed-load diagram cannot: for this one section, how does the force here change as the load moves, and where is it worst?
- Bridges. Highway and railway spans are sized by moving-load envelopes built from influence lines — trucks, trains, and lane loads positioned for maximum moment and shear.
- Crane runways. Wheel loads from bridge and gantry cranes traverse the beam; the peak reaction and moment come from placing wheels on influence-line vertices.
- Continuous structures. In multi-span bridges, loading alternate spans maximizes interior-support moments — a pattern the influence line makes obvious and static intuition hides.
- Fatigue. The stress range a detail sees per truck passage is the swing between the maximum and minimum influence-line effects, the input to fatigue-life checks.
- Rating and assessment. Load-rating an existing bridge uses influence lines to place the rating vehicle for the governing effect at each critical section.
How an influence line works, step by step
Fix your attention on one response quantity R — say the bending moment at a section C, or the left reaction A. Now imagine a downward unit load (1 kN, or dimensionless 1) placed at position x. Solve the structure and record the value of R. Move the unit load to a new x, solve again, record again. Plot R against x. That plot is the influence line for R.
- Choose the response. One number: a reaction, a shear at a cut, or a moment at a cut. The influence line is specific to that one quantity.
- Walk the unit load. Conceptually slide a load of magnitude 1 from one end of the structure to the other.
- Record the ordinate. At each load position x, the value of R is the influence-line ordinate at that x. Positive ordinates mean the response has its defined positive sign for a load there.
- Read it for real loads. For a concentrated load P at position a, its contribution to R is P · y(a), where y(a) is the influence-line ordinate. For a distributed load w over a region, its contribution is w times the area under the influence line in that region.
- Superpose. Sum contributions from every load — the influence line linearizes the moving-load problem into a weighted read-off.
The governing read-off equation is deceptively simple. If a set of concentrated loads Pi sit at positions with influence-line ordinates yi, and a distributed load of intensity w covers regions of net influence-line area Aw, then the response is:
R = Σ Pi · yi + w · Aw
- R — the response of interest (reaction in kN, shear in kN, or moment in kN·m).
- Pi — the i-th concentrated (axle/wheel) load, in kN.
- yi — influence-line ordinate under Pi; dimensionless for a reaction or shear IL, and in units of length (m) for a moment IL, so that P·y carries the right units.
- w — distributed (lane) load intensity, in kN/m.
- Aw — the area under the influence line over the loaded region; for maximum effect, load only regions where y has the desired sign.
The Müller-Breslau shortcut
You rarely re-solve the structure at hundreds of load positions. Heinrich Müller-Breslau's principle (1886) gives the influence line's shape directly from a single deflected form. To get the influence line for a force quantity, release the restraint that quantity represents and impose a unit displacement in its positive direction; the deflected shape is the influence line.
- Reaction. Remove the support and lift the released point by a unit. The rigid-body deflected line is the reaction influence line — for a simple span it goes from 1 at the support to 0 at the far end, linearly.
- Shear at a cut. Insert a shear release at the section (the two faces can slide vertically but not rotate relative to each other), and impose a unit relative vertical slip. The influence line for a simple span at fraction a·L jumps across the cut, running from −a on one side to +(1−a) on the other.
- Moment at a cut. Insert a hinge at the section and impose a unit relative rotation. For a simple span the moment IL at a distance a from the left support (span L, b = L − a) is a triangle with peak ordinate a·b/L.
For statically determinate structures the released shape is composed of straight rigid segments, so the influence line is piecewise linear. For statically indeterminate structures — continuous beams, frames, arches — the released structure bends elastically, so the influence line is smoothly curved, with alternating positive and negative lobes over successive spans.
Worked example: simply supported span
Take a simple beam of span L = 20 m, pinned at A (left) and roller at B (right). We want the influence lines for the reaction at A, and for the shear and moment at a section C located a = 8 m from A (so b = 12 m).
| Response at section C (a = 8 m, b = 12 m, L = 20 m) | Influence-line shape | Key ordinate |
|---|---|---|
| Reaction RA | Straight line, 1 at A to 0 at B | 1.0 at A (dimensionless) |
| Shear VC | Two parallel lines, jump of 1 at C | −a/L = −0.40 just left of C; +b/L = +0.60 just right of C |
| Moment MC | Triangle, apex under C | a·b/L = 8·12/20 = +4.8 m |
Single concentrated load. Place a 100 kN wheel at the moment-IL peak, i.e. directly over C. The moment at C is R = P · ypeak = 100 kN · 4.8 m = 480 kN·m. No other load position produces a larger MC for a single wheel — the influence line proves it.
Uniform lane load. Apply w = 9.3 kN/m (the AASHTO lane load) over the full 20 m span. The moment IL is a triangle with base L = 20 m and height 4.8 m, so its area is Aw = ½ · 20 · 4.8 = 48 m². The moment at C is R = w · Aw = 9.3 kN/m · 48 m² = 446 kN·m. Because the moment IL is entirely positive, loading the whole span is the worst case here.
Two-axle vehicle. Put a 145 kN drive axle over C (ordinate 4.8 m) and a 35 kN steer axle 4.3 m to the left, where the IL ordinate is 4.8 · (8 − 4.3)/8 = 2.22 m. Then MC = 145 · 4.8 + 35 · 2.22 = 696 + 78 = 774 kN·m. Sliding the truck and re-reading ordinates is how the governing axle arrangement is found — the maximum falls where a heavy axle sits on the peak and the increase-decrease criterion no longer improves the sum.
Building the envelope
Design does not use one section — it uses all of them. For each of many sections along the span, form that section's own moment influence line, position the loads for the maximum, and record the result. Plotting those section-by-section maxima against position gives the moment envelope; repeating for shear gives the shear envelope. These envelopes bound every possible live-load effect and are the curves the girder flanges, web, and reinforcement are proportioned against. For a simple span under a moving concentrated load, the absolute-maximum moment does not occur at midspan for a load train — it occurs under a specific wheel positioned by the classic rule that the section and the resultant of the loads are equidistant from midspan.
Common misconceptions and failure modes
- Confusing it with a moment diagram. An influence line fixes the section and moves the load; a bending-moment diagram fixes the load and moves the section. Same axis units, opposite meaning.
- Reading ordinates as stress. The ordinate is the response for a unit load, not a stress or a final force. You must multiply by real loads and superpose.
- Loading the whole span always. For a UDL, load only regions where the ordinate has the sign you want. Loading negative lobes reduces the peak — a costly mistake in continuous spans.
- Ignoring sign reversal of shear. The shear IL jumps across the cut, so shear at a section can flip sign as a load crosses it; the design shear is the larger of the positive and negative envelopes.
- Forgetting alternate-span loading. In continuous beams, interior-support hogging moment is maximized by loading alternate spans, not all spans — the influence line's negative lobes reveal this.
- Treating indeterminate ILs as straight. Only determinate influence lines are piecewise linear; continuous and framed structures give curved ILs that require an elastic solution or Müller-Breslau shape.
- Dropping dynamic effects. Moving loads bounce; codes add a dynamic load allowance (impact factor, e.g. 33% on the AASHTO truck) on top of the static influence-line result.
Influence line vs. load diagram — the distinction that trips everyone
| Aspect | Influence line | Shear / moment (load) diagram |
|---|---|---|
| What is fixed | The response section (point of interest) | The load (a fixed load case) |
| What varies (x-axis) | Position of a moving unit load | Position along the member |
| Ordinate (y-axis) | Value of one response for unit load at x | Internal shear or moment at section x |
| Answers | Where is the load worst for this section? | What is the force everywhere for this loading? |
| Determinate shape | Piecewise straight lines | Piecewise linear (V) / parabolic under UDL (M) |
| Primary use | Moving loads, worst-case position, envelopes | A single static load case, member sizing |
Frequently asked questions
What is an influence line?
An influence line is a diagram showing how one specific response — a support reaction, the shear at a section, or the bending moment at a section — changes as a single unit load moves across the structure. The horizontal axis is the position of the moving load; the vertical ordinate is the value of that response when the unit load stands at that position. Reading it: the influence-line ordinate under a real load, multiplied by that load, gives its contribution to the response. It is a per-response, moving-load tool, not a picture of stress along the beam.
How is an influence line different from a bending-moment or shear diagram?
A shear or bending-moment diagram fixes the loads and plots the internal force at every section — the horizontal axis is position along the member. An influence line fixes the section (the response point) and moves a unit load — the horizontal axis is the position of the load. One diagram (shear/moment) answers 'what is the force everywhere for this loading?'; the influence line answers 'how does the force here change as the load moves?'. They have the same units on the axes but opposite roles, and confusing them is the most common student error.
What is the Müller-Breslau principle?
The Müller-Breslau principle states that the influence line for any force quantity (a reaction, shear, or moment) has the same shape as the deflected form the structure takes when you remove the restraint corresponding to that quantity and impose a unit displacement in its direction. For a reaction, remove the support and lift it a unit; for shear at a cut, introduce a shear release and slide the two faces a unit apart while keeping slopes equal; for moment at a cut, insert a hinge and rotate a unit relative angle. The resulting deflected shape, scaled by the flexibility, is the influence line — for statically determinate structures it is made of straight lines; for indeterminate structures it is curved.
How do you find the worst-case position of a moving load?
For a single concentrated load, place it at the peak (maximum ordinate) of the influence line. For a uniformly distributed lane load, load only the parts of the span where the ordinate has the sign you want — positive regions for maximum positive response — because a UDL contributes the load intensity times the area under the loaded portion of the influence line. For a train of wheel loads (a truck or locomotive), slide the group so the resultant sits near the peak and check candidate positions where a wheel is over a break point; the maximum occurs when a heavy axle is at a vertex, checked by an increase-decrease criterion.
What is a maximum-effect envelope?
An envelope is the curve of the largest response of a given type at every section, taken over all admissible load positions. You compute the maximum bending moment (or shear) at many sections along the span using each section's own influence line, then plot those maxima against position. The resulting moment envelope and shear envelope bound all live-load effects and are what bridge girders and reinforcement are designed against — for example AASHTO LRFD combines a design truck (HL-93, a 325 kN tractor plus trailer axles) with a 9.3 kN/m lane load, both positioned by influence lines.
Do influence lines work for indeterminate structures?
Yes. For statically determinate beams the influence line is piecewise linear and can be found from equilibrium alone. For statically indeterminate structures — continuous bridges, portal frames, arches — the influence line is curved and is obtained from the Müller-Breslau deflected shape, from a flexibility or stiffness analysis, or by unit-load solutions at many positions. Continuous-beam influence lines have alternating positive and negative lobes over successive spans, which is why alternate-span loading governs interior-support moments and why lane loads are patterned span by span.
Why do influence lines matter for real bridges?
Bridge loads move: trucks, trains, and crane trolleys occupy every position along the deck during service. Static analysis of one fixed load case cannot guarantee it caught the worst case. Influence lines make the worst position provable — you multiply ordinates by axle loads and read the maximum — and they drive the envelopes used to size girders, place reinforcement, and check fatigue. They also reveal counter-intuitive effects, such as a load on one span reducing the moment in the next, or shear that reverses sign as a load crosses a section, which fixed-load diagrams hide.