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Structural Mechanics Simulator

Influence Line Simulator — Moving Load and the Response Quantity

Draws the "influence line" in real time, showing how the reaction, shear force, and bending moment change as a unit load moves across the beam. Learn the basics of bridge and crane girder design intuitively.

Parameters
Response quantity
Span length L
m
Section of interest x/L
Load position a/L
Load magnitude P
kN

"Move load" makes the unit load travel automatically along the beam.

Results
—
Influence ordinate η(a)
—
Actual response η·P
—
Peak ordinate η_max
—
Position a of the peak
Beam and Moving Unit Load

Triangle = pin support A / Circle = roller support B / Red dashed = section of interest x

Influence Line Diagram (ordinate η vs load position a)

Horizontal axis = position a of the unit load / Vertical axis = influence ordinate η of the response quantity

Theory & Key Formulas

Formulas for the influence ordinate η(a) when a unit load moves at position a along a simply supported beam of span L.

Reactions (independent of the section of interest x):

$$\eta_{R_A}(a) = \frac{L-a}{L}, \qquad \eta_{R_B}(a) = \frac{a}{L}$$

Shear force at section c (negative to the left of c, positive to the right):

$$\eta_V(a) = -\frac{a}{L}\ \ (a \lt c), \qquad \eta_V(a) = \frac{L-a}{L}\ \ (a \gt c)$$

Bending moment at section c (peaks at a = c):

$$\eta_M(a) = \frac{a(L-c)}{L}\ \ (a \le c), \qquad \eta_M(a) = \frac{c(L-a)}{L}\ \ (a \ge c)$$

When a real load P sits at position a, the response quantity is η(a)·P. The peak bending moment ordinate is c(L−c)/L at a = c, which equals L/4 at midspan c = L/2.

What is the Influence Line Simulator

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What is the difference between an "influence line" and a bending moment diagram? They both look like beam graphs to me…
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Good question. Roughly speaking, the meaning of the horizontal axis is different. A bending moment diagram is a graph where "the load is fixed and the section position moves." An influence line diagram is the opposite: "the section is fixed and the load position moves." Try moving the "Load position a/L" slider in the simulator above. The yellow load travels along the beam, and the point on the influence line follows it, right?
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It really does! The value changes as I move the load. But why do we need to move the load in the first place?
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Think of a bridge or a crane girder. Vehicles and trolleys move along the structure. What the designer wants to know is "when the load is at which position is this section most dangerous." If you place the load where the influence-line ordinate is maximum, you find the worst case for that section. In practice, finding the "most unfavorable load arrangement" is the main purpose of influence lines.
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I see! When I set the "response quantity" to bending moment M and the section of interest x/L to 0.5, the influence line becomes a triangle. Does that mean something?
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That is the Müller-Breslau principle. The influence line for bending moment has the same shape as the deformation of the beam when a "unit rotation" is applied at the section. At midspan it is a left-right symmetric bent line, that is, a triangle. The height of the peak is $c(L-c)/L$, and at midspan $c = L/2$ it is $L/4$. Multiply by the real load $P$ and you get the familiar $PL/4$. Compare it with the formula in the theory box.

Frequently Asked Questions

In the influence line for shear force, the ordinate becomes negative when the unit load is to the left of the section of interest. This means that for that load arrangement, the shear force at the section acts in the negative direction (the direction that shifts the beam counterclockwise). The sign indicates that both the "maximum positive shear" and the "maximum negative shear" must be considered in design.
A reaction is a quantity of support A itself, so it does not depend on the section of interest x somewhere along the beam. The influence line $\eta_{R_A}(a)=(L-a)/L$ is a function of the load position a only. The section of interest x/L becomes meaningful when you choose shear force V or bending moment M. It is always shown as a red dashed line on the beam diagram.
The response quantity for a distributed load w is "the area enclosed by the influence line over the loaded span × w." Applying the distributed load only over the regions with positive ordinates gives the maximum positive response, and over negative regions gives the maximum negative response. When concentrated and distributed loads are mixed, the two are summed.
The principle is the same, but influence lines for statically indeterminate structures are curves rather than straight lines. This simulator deals with the statically determinate simply supported beam. For indeterminate cases, grasp the shape with the Müller-Breslau principle and compute exact ordinates with the slope-deflection method or the matrix displacement method. A characteristic of influence lines for continuous beams is that they become smooth curves with continuous curvature over the supports.

Real-World Applications

Bridge design: On road bridges and railway bridges, vehicles and trains move along the bridge. Designers draw the influence line for each section of the main girders and deck slabs, and find the bending moment and shear force when the live load is placed in the most unfavorable position. For railway bridges, a train load with multiple axles is moved along the influence line to find the axle arrangement that produces the maximum response.

Crane girders and overhead cranes: In factory overhead cranes, the trolley (hoisting device) moves along the girder. The trolley position that maximizes the bending moment of the girder is immediately apparent from the influence line and is used to determine the girder cross section.

Continuous beams and rigid-frame bridges: In multi-span continuous bridges, placing a live load on one span also affects the member forces in adjacent spans. With influence lines, you can efficiently judge "which span the load should be placed on to make the section of interest worst," which is essentially important in the design of multi-span structures.

Structural health monitoring: In load tests of existing bridges, a test vehicle of known weight is moved along the bridge and the response of strain gauges is measured. By comparing the measured response curve with the theoretical influence line, damage such as a loss of bridge stiffness or a seized bearing can be detected.

Common Misconceptions and Cautions

The most common misconception is to confuse an influence line diagram with a bending moment diagram. The horizontal axis of an influence line is "the position of the load," not "the position of the section." For example, the point where the ordinate is maximum in a bending moment influence line means "placing the load there maximizes the moment of the section of interest," not "the moment is maximum at that point." Keep the section of interest x/L fixed in the simulator and move only the load position a/L until the meaning of the horizontal axis becomes second nature.

Next is the mistake of thinking the influence-line ordinate itself is the response value. An influence line is the response to a unit load (magnitude 1), so it is dimensionless (reaction, shear) or has a length dimension (bending moment). The actual response is always calculated as "ordinate × real load P." That is why this tool provides a separate "Actual response η·P" card. Note that the bending moment influence ordinate is in metres, and only after multiplying by P does it become kN·m.

Finally, the pitfall of judging the load position that gives the maximum from a single point. Because the shear force influence line jumps discontinuously at the section of interest, both the positive and negative maxima are needed in design. In real structures, multiple concentrated loads and distributed loads act simultaneously, so you need to slide the whole load group along the influence line to find the worst combination, not just the maximum position of a single load. Bridge design codes specify the "method of applying live loads" in detail in order to capture this worst arrangement without omission.

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