Module 4: Moving Loads and Influence Lines

Learning Objectives

  • Distinguish an influence line from a shear-force or bending-moment diagram.
  • Construct influence lines for reactions, shear, and bending moment in statically determinate structures.
  • Apply the Müller-Breslau principle to predict the qualitative shape and sign of an influence line.
  • Calculate a structural response caused by moving concentrated and distributed loads using influence ordinates.
  • Position moving loads to maximize or minimize a selected response quantity.
  • Interpret response envelopes and verify moving-load results using sign, units, and boundary conditions.

Influence Line

An influence line shows how one selected structural response quantity at a fixed location—such as a reaction, section shear, section moment, or member force—varies as a unit load moves across the structure.

Influence Line versus Internal-Force Diagram

A shear-force or bending-moment diagram shows how response varies along the structure for one fixed loading case. An influence line instead holds the response location fixed and moves a unit load across the structure. Confusing these two plots leads to incorrect moving-load calculations.

Influence-Line Ordinates

The ordinate at a load position is the value of the selected response caused by a unit load placed there. Multiplying that ordinate by an actual concentrated load gives that load's contribution to the selected response. The ordinate units depend on the response quantity: reaction and shear influence ordinates are commonly dimensionless, while moment influence ordinates have units of length when the moving load is represented as a unit force.

Response from Moving Concentrated Loads

Superposition of concentrated moving loads using the influence ordinate at each load location.

R=∑i=1nPi y(xi)R=\sum_{i=1}^{n} P_i\,y(x_i)

Variables

SymbolDescriptionUnit
RRSelected structural response-
PiP_iMagnitude of concentrated load i-
y(xi)y(x_i)Influence-line ordinate at the position of load i-

Response from a Distributed Moving Load

The response equals the integral of load intensity times influence ordinate over the loaded region.

R=∫abw(x) y(x) dxR=\int_a^b w(x)\,y(x)\,dx

Variables

SymbolDescriptionUnit
RRSelected structural response-
w(x)w(x)Distributed load intensity-
y(x)y(x)Influence-line ordinate-
a,ba,bLimits of the loaded region on the structure-

Müller-Breslau Principle

For a linear elastic structure, the qualitative influence line for a response quantity has the shape of the compatible displaced configuration obtained by releasing the restraint associated with that response and imposing a unit displacement or rotation in its positive direction.

Using Müller-Breslau Correctly

For a support reaction, release that support component and impose a unit displacement in the positive reaction direction. For a section moment, introduce a moment release at the section and impose a unit relative rotation consistent with the positive moment convention. For a section shear, release the corresponding shear transfer and impose a compatible relative translation. The method is especially powerful for identifying the sign pattern and where moving loads should be placed.

Qualitative Shape Is Not Automatically a Numeric Scale

The Müller-Breslau principle gives the influence-line shape directly. For a statically determinate structure, geometry and equilibrium often provide exact ordinates easily. For an indeterminate structure, a quantitative influence line generally requires an elastic analysis after the release/kinematic interpretation.

Interactive Exploration

Use the influence-line visualizers to move a load across the span and observe how a fixed reaction, shear, moment, or truss-member response changes. Track the response value and the influence ordinate together rather than treating the graph as decoration.

Truss Influence Line Simulator

5.0 m

Move the load across the bottom chord of the Pratt truss to see how the force in the selected member changes. Negative values indicate compression, and positive values indicate tension.

Three-Hinged Arch: Influence Line for Thrust

5.0 m
Reactions

Ay = 0.50 kN

By = 0.50 kN

Horizontal Thrust

H = 0.50 kN

Maximizing a Response

For independent downward concentrated loads, place the largest loads on the influence-line ordinates with the largest favorable values when seeking a maximum positive response. Avoid unfavorable-sign regions unless the load arrangement or spacing forces loads to occupy them. For a continuous distributed load, load the positive region for maximum positive response and the negative region for maximum negative response, subject to the actual loading rules.

Response Envelopes

A moving-load envelope records the maximum and minimum value of a response quantity at each location as the load pattern traverses the structure. Unlike a single shear or moment diagram, an envelope summarizes many load positions and is useful for identifying governing design demands.

Moving-Load and Influence-Line Workflow

Use the workflow for each target response separately. A vehicle position that maximizes support reaction is generally not the same position that maximizes moment at a particular section, so the response quantity must be selected before positioning the load.

Moving-Load and Influence-Line WorkflowChoose the response quantity and fixed location → Construct or obtain the influence line with sign convention; Construct or obtain the influence line with sign convention → Are exact numeric ordinates required?; Are exact numeric ordinates required? — Yes → Determine ordinates by equilibrium or elastic analysis; Are exact numeric ordinates required? — No → Use Müller-Breslau shape for qualitative positioning; Determine ordinates by equilibrium or elastic analysis → Position moving concentrated/distributed loads on favorable ordinates; Use Müller-Breslau shape for qualitative positioning → Position moving concentrated/distributed loads on favorable ordinates; Position moving concentrated/distributed loads on favorable ordinates → Compute response from P_i y_i and/or integral w y dx; Compute response from P_i y_i and/or integral w y dx → Is a full maximum/minimum envelope required?; Is a full maximum/minimum envelope required? — Yes → Sweep admissible load positions and retain extrema; Is a full maximum/minimum envelope required? — No → Verify units, signs, zero ordinates, and equilibrium checks; Sweep admissible load positions and retain extrema → Verify units, signs, zero ordinates, and equilibrium checks; Verify units, signs, zero ordinates, and equilibrium checks → Governing moving-load response

Choose the response quantity and fixed location → Construct or obtain the influence line with sign convention; Construct or obtain the influence line with sign convention → Are exact numeric ordinates required?; Are exact numeric ordinates required? — Yes → Determine ordinates by equilibrium or elastic analysis; Are exact numeric ordinates required? — No → Use Müller-Breslau shape for qualitative positioning; Determine ordinates by equilibrium or elastic analysis → Position moving concentrated/distributed loads on favorable ordinates; Use Müller-Breslau shape for qualitative positioning → Position moving concentrated/distributed loads on favorable ordinates; Position moving concentrated/distributed loads on favorable ordinates → Compute response from P_i y_i and/or integral w y dx; Compute response from P_i y_i and/or integral w y dx → Is a full maximum/minimum envelope required?; Is a full maximum/minimum envelope required? — Yes → Sweep admissible load positions and retain extrema; Is a full maximum/minimum envelope required? — No → Verify units, signs, zero ordinates, and equilibrium checks; Sweep admissible load positions and retain extrema → Verify units, signs, zero ordinates, and equilibrium checks; Verify units, signs, zero ordinates, and equilibrium checks → Governing moving-load response

  • Choose the response quantity and fixed location: terminator
  • Construct or obtain the influence line with sign convention: process
  • Are exact numeric ordinates required?: decision
  • Determine ordinates by equilibrium or elastic analysis: process
  • Use Müller-Breslau shape for qualitative positioning: process
  • Position moving concentrated/distributed loads on favorable ordinates: process
  • Compute response from P_i y_i and/or integral w y dx: process
  • Is a full maximum/minimum envelope required?: decision
  • Sweep admissible load positions and retain extrema: process
  • Verify units, signs, zero ordinates, and equilibrium checks: process
  • Governing moving-load response: terminator

Verification Rules

A load outside the structural domain has zero influence ordinate. A simple-support reaction influence line must recover the expected support values at the ends. A section-moment influence line is zero where a unit load cannot create moment at that section under the adopted idealization. These boundary checks catch many sign and interpolation errors.

Key Takeaways
  • An influence line varies load position while holding one response location fixed.
  • Concentrated-load effects are summed as PiyiP_i y_i; distributed-load effects are obtained from the area/integral of w(x)y(x)w(x)y(x).
  • Müller-Breslau gives the physically meaningful influence-line shape by releasing the target response and imposing its positive generalized displacement.
  • Maximum response depends on the selected response quantity and the admissible moving-load arrangement.
  • Response envelopes retain the governing extrema from many load positions.
  • Influence-line graphs must be read with explicit signs, units, boundaries, and response definitions.

References