Influence Line Calculator
Plot the influence line for a reaction, shear, or moment on a simply supported beam as a unit load moves across the span.
📈 What is an Influence Line?
An influence line is a graph that shows how one specific structural response, a support reaction, or the internal shear or bending moment at one fixed section, changes as a single unit load travels across the entire span of a beam. It is the mirror image of a familiar bending moment diagram: a moment diagram fixes the load in place and shows the response everywhere along the beam, while an influence line fixes the point of interest and shows the response for every possible load position.
Bridge and crane rail engineers rely on influence lines constantly, since their structures are governed by moving loads (vehicles crossing a bridge, a trolley moving along a crane girder) rather than static ones. A bridge engineer uses the moment influence line at a critical section to find where a truck axle should sit to produce the worst-case bending moment there. A crane designer uses the reaction influence lines to find how much of the load each rail support must carry as the trolley moves. Even textbook problems on vehicle loading, train axle spacing, and pedestrian bridge design lean directly on influence line concepts.
A common point of confusion is mixing up an influence line with a bending moment diagram, they look similar (both are plotted against a position along the beam) but answer completely different questions. A moment diagram answers "what is the moment everywhere, for this one load case?" An influence line answers "what is the response at this one section, for every possible load position?" Getting this backwards leads to using the wrong shape entirely when checking a moving-load scenario.
This calculator plots the exact influence line for a simply supported beam in all four of the classic response cases, the reaction at each support, the shear at a chosen section, and the moment at a chosen section, and reports the live ordinate as you move the unit load with a slider.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Reaction at A, Load Near Midspan
L = 10 m, unit load at x = 3 m
Example 2 — Reaction at B, Same Load Position
L = 10 m, unit load at x = 3 m
Example 3 — Shear at a Section, Load Past the Section
L = 10 m, section at a = 4 m, unit load at x = 6 m
Example 4 — Moment at a Section, Load Past the Section
L = 10 m, section at a = 4 m, unit load at x = 6 m
Example 5 — Moment Influence Line at Its Own Peak
L = 10 m, section at a = 4 m, unit load at x = 4 m (directly at the section)
❓ Frequently Asked Questions
🔗 Related Calculators
What is an influence line in structural engineering?
An influence line is a graph showing how a single response quantity (a support reaction, or the shear or moment at one fixed section) varies as a unit load moves across the entire span of a beam. It is the reverse of a bending moment diagram, which fixes the load and shows the response along the whole beam.
What is the influence line formula for the reaction at support A?
For a simply supported beam of span L, the influence line ordinate for the reaction at A, when a unit load sits at position x, is (L minus x) divided by L. It equals 1 when the load is directly over A (x=0) and 0 when the load is directly over B (x=L).
What is the influence line formula for the reaction at support B?
The influence line ordinate for the reaction at B is x divided by L, where x is the unit load's position measured from support A. It equals 0 at A (x=0) and 1 at B (x=L), the mirror image of the reaction-at-A influence line.
Why does the shear influence line have a jump?
The shear influence line has a jump discontinuity of exactly 1.0 at the section itself, because moving the unit load from just left of the section to just right of it instantly changes which side of the section carries the full unit load, flipping the internal shear by that same amount.
Where is the moment influence line at its maximum?
The moment influence line for a section at distance a from support A peaks exactly at x = a (the unit load sitting right at the section), with peak ordinate a(L-a)/L, and is zero when the load sits directly over either support.
How is an influence line different from a shear force or bending moment diagram?
A shear force or bending moment diagram shows the response at every point along the beam for one fixed load position. An influence line does the opposite: it fixes the point being examined (a support or a section) and shows how that one response value changes as the load position moves across the beam.
How do you use an influence line to find the maximum reaction or moment?
Multiply each individual load in a moving load system by the influence-line ordinate at that load's position, then sum the products, this superposition gives the total response for any load arrangement without re-solving equilibrium from scratch, and the worst-case position is found by testing where this sum is largest.
What is the influence line ordinate range for reactions on a simply supported beam?
Both reaction influence lines range from 0 to 1.0 across the span, since a single unit load is always shared between the two supports in proportion to how close the load is to each one, and the two ordinates at any position always sum to exactly 1.0.
Does this calculator work for cantilever or continuous beams?
No, this calculator is scoped specifically to a simply supported beam (pin at one end, roller at the other, two supports total). Cantilever and continuous (multi-span) beams have different, more complex influence line shapes that this tool does not compute.
Why is the section position (a) needed for shear and moment modes but not for the reaction modes?
The reaction influence lines depend only on the overall span L, since a support's reaction responds to the load's position relative to the whole beam. Shear and moment influence lines describe the internal force at one specific interior section, so their shape depends on exactly where that section (a) sits along the span.