Influence Lines and Moving Loads

Introduction and Context

Structural analysis usually evaluates the response of a structure under loads fixed at known positions. Bridges, crane girders, floor beams, gantries, and construction platforms introduce a different problem: loads move. A vehicle crossing a bridge, a crane trolley travelling along a runway, or construction materials being shifted across a temporary deck can produce different reactions, shears, moments, and deflections at every position.

An influence line is a graphical or analytical representation of how one response quantity at one specified location varies as a unit load moves across a structure. It is not the same as a bending-moment diagram. A bending-moment diagram gives bending moment at many sections for one fixed loading arrangement; an influence line gives one selected response at one section for many load positions.

Influence-line methods are especially efficient for statically determinate structures. They also provide the conceptual basis for more advanced moving-load analysis of indeterminate structures, where compatibility, stiffness methods, or structural software are required.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Distinguish influence lines from shear-force and bending-moment diagrams.
2. Construct influence lines for reactions, shear, and moment using equilibrium.
3. Apply the Müller–Breslau principle to identify the qualitative shape of influence lines.
4. Determine maximum response caused by moving point loads and distributed loads.
5. Analyze practical bridge and construction-system loading scenarios.
6. Recognize assumptions, limitations, and professional checks required in moving-load analysis.

1. Fundamental Concepts

1.1 Moving load and response function

Let a unit load move along a structure, and let its position be represented by coordinate x. If the response quantity selected is a reaction, shear, or moment Q at a fixed location, its influence ordinate is

The influence line is the graph of I_Q(x) against x. The unit load is dimensionless in the idealized definition, so the ordinate has the units of the response:

If an actual point load P is placed at position xₚ, its effect is

For several point loads Pᵢ at positions xᵢ,

These expressions apply to a linearly elastic structure when superposition is valid, and to a statically determinate structure when the influence line has been obtained from equilibrium.

For a distributed load w(x), with units of force per unit length,

The integral is taken over the portion of the structure carrying the moving load. If w is constant, the response equals w multiplied by the signed area under the relevant part of the influence line.

1.2 Influence line versus response diagram

A response diagram, such as a bending-moment diagram, is constructed after fixing the loads and moving from section to section. An influence line fixes the response location and moves a load across the structure. This distinction has practical consequences:

• A bending-moment diagram commonly has units of force × length.
• A moment influence line for a unit force has units of length.
• A response diagram may be curved for a distributed load, whereas the influence line for a determinate prismatic beam is often piecewise linear.
• Influence lines identify critical load positions directly.

1.3 Sign convention

A sign convention must be stated before constructing an influence line. A common convention is upward reaction positive, positive shear producing a chosen face rotation, and sagging bending moment positive. Changing the convention reverses signs but does not change the physical magnitude of the response.

2. Influence Lines for a Simply Supported Beam

Consider a simply supported beam AB of span L. Let a unit downward load be at distance x from A. The reactions are obtained from equilibrium:

where R_A and R_B are upward reactions. Thus the influence line for R_A is a straight line from ordinate 1 at A to 0 at B. The influence line for R_B is a straight line from 0 at A to 1 at B.

At a section C, distance a from A and b from B, where L = a + b, the influence line for bending moment at C is triangular. When the unit load is left of C,

When the unit load is right of C,

At the section itself, the maximum ordinate is

The moment influence line is zero at both supports and reaches its peak at C. The ordinate has units of length because it represents moment due to a unit force.

For shear at C, the influence line has a jump at C. With the convention that positive shear is the algebraic sum of forces on the left face, the ordinates are commonly written as

The difference between the right-hand and left-hand ordinates is 1, equal to the magnitude of the moving unit load. The exact signs depend on the adopted shear convention; the jump magnitude is invariant.

Worked Example 1: Moment caused by a moving point load

A simply supported beam has span 12 m. Determine the bending moment at a section 5 m from the left support when a 40 kN point load is positioned 3 m from the left support.

Here, L = 12 m, a = 5 m, and b = 7 m. Since the load is left of the section, the moment influence ordinate is

Therefore,

The result is positive sagging moment under the stated convention. The calculation is valid for a statically determinate beam and a point load treated as acting on the beam centreline.

3. Constructing Influence Lines by Equilibrium

The equilibrium method is direct and reliable for statically determinate structures.

General procedure

1. Select the response quantity and its fixed location.
2. Place a unit load at a general position x.
3. Draw a free-body diagram of the complete structure or a suitable part.
4. Apply ∑F = 0 and ∑M = 0 to express the response in terms of x.
5. Evaluate the expression at key positions, including supports, hinges, section locations, and discontinuities.
6. Plot the ordinates and join them according to the response function.
7. Check the result using limiting positions and equilibrium.

For a determinate truss, the same principle applies. A unit load is moved along the loaded chord, and member force is obtained using the method of joints, method of sections, or a suitable equilibrium equation. The resulting member-force influence line may contain positive and negative regions.

Discontinuities and ordinates

A reaction influence line is generally continuous for a beam with ordinary supports. A shear influence line has a unit jump at the selected section because the moving force crosses the cut. A moment influence line is continuous at an ordinary section, although it may change slope. Internal hinges and changes in structural arrangement can create additional changes in slope or discontinuity in selected response quantities.

4. The Müller–Breslau Principle

The Müller–Breslau principle provides a rapid method for determining the qualitative shape of an influence line. It states:

The influence line for a response quantity is proportional to the deflected shape of the structure obtained by releasing the restraint associated with that response and imposing a unit displacement or rotation in the positive direction of the response.

The principle is based on virtual work and applies most directly to linear elastic structures. It gives the shape and sign pattern, but not always the numerical scale. Numerical ordinates should be obtained from equilibrium, geometry, or a structural-analysis model.

Typical releases

For a simply supported beam, releasing the moment at an interior section produces a triangular mechanism with a kink at that section. This agrees with the triangular moment influence line. Releasing shear produces a vertical discontinuity, agreeing with the jump in the shear influence line.

The principle is particularly useful for complex beams, frames, and indeterminate systems because it reveals where positive and negative load regions occur before numerical analysis. It also helps engineers detect incorrect computer-generated influence lines.

5. Moving Point Loads

5.1 Single point load

For a point load P, the response is obtained by multiplying the influence ordinate by P:

The maximum positive response occurs where the load occupies the largest positive ordinate. The maximum negative response occurs where it occupies the most negative ordinate.

For a simply supported beam and a single downward load, maximum moment at a selected section occurs when the load is directly over that section, because the moment influence line has its maximum ordinate there.

5.2 A train of point loads

For a set of loads with fixed spacing, the response is

where s is the position of the load train and xᵢ(s) is the position of load i. The train should be moved through the structure, and candidate positions should be examined. Critical positions often occur when a heavy axle is near a peak of the influence line, but the exact position depends on all axle weights and spacings.

For a simply supported beam under a moving system of concentrated loads, maximum moment at a particular section is not necessarily obtained by placing the heaviest axle exactly at the section. A nearby arrangement may place several substantial axles on favorable ordinates and produce a larger total.

5.3 Absolute maximum moment

A distinction must be made between:

• maximum moment at a specified section, and
• absolute maximum moment anywhere in the span.

The first uses one influence line for a fixed section. The second requires considering possible critical sections and load positions. For symmetrical loading and a symmetrical beam, symmetry can reduce the number of positions examined, but it should not replace a check of the actual load arrangement.

6. Moving Distributed Loads

For a uniformly distributed load w occupying a length ℓ, the response is the load intensity multiplied by the signed area of the influence line covered by the load:

where A_I is the signed influence-line area under the loaded region. This expression is valid when w is constant and the load is applied continuously along the relevant structural line.

To maximize a positive response, place the distributed load over the positive region of the influence line. To maximize a negative response, place it over the negative region. If the load length is limited, slide the load to the position producing the greatest signed area.

For a simply supported beam, the moment influence line at an interior section is entirely positive. A full-span uniform load therefore produces the largest moment at that section when it covers the full span. For shear, the influence line has positive and negative regions separated by a jump; a partial distributed load may be more critical than a full-span load because loading an unfavorable region can reduce the desired shear.

Worked Example 2: Uniform load using an influence line

A 10 m simply supported beam is subjected to a moving uniform load of 8 kN/m over the entire span. Determine the moment at the midspan section.

The midspan moment influence line is a triangle with base 10 m and height L/4 = 2.5 m. Its area is

Therefore,

The units are (kN/m) × m² = kN·m. This equals the familiar static result wL²/8, but the influence-line method remains useful when the load covers only part of the span or when several load patterns must be compared.

7. Influence Lines for Bridges and Construction Systems

7.1 Bridge decks

Bridge analysis commonly includes permanent dead load, lane loads, axle loads, impact or dynamic allowances, braking forces, wind, temperature, and construction-stage loads. Influence lines are used to identify critical positions of wheel loads and lane loading for reactions, deck moments, shears, cross-girder forces, and support effects.

The idealized beam model must represent the intended load path. A wheel load may first distribute through a slab, transverse beam, and longitudinal girder before reaching a support. Applying the wheel directly to an inappropriate one-dimensional member can produce misleading results.

7.2 Floor beams and crane runways

In industrial buildings, a crane trolley can generate moving wheel loads on runway beams. Influence lines locate the positions producing maximum vertical reaction, bending moment, shear, and sometimes support uplift. Horizontal acceleration and braking effects require separate influence lines for longitudinal forces.

7.3 Temporary works and construction platforms

Construction materials, mobile equipment, and temporary props may move across partially completed structures. The structure can have a different support condition at each construction stage. Influence lines should therefore be constructed for the actual stage being checked, not automatically for the final structure.

8. Illustrative Applied Case: Two-Axle Maintenance Vehicle

Background

Consider an illustrative simply supported bridge beam of span 18 m. A maintenance vehicle has two downward axle loads: 80 kN and 120 kN, spaced 4 m apart. The vehicle travels from left to right. The design team wishes to estimate the largest bending moment at midspan using a moving-load idealization.

Problem and assumptions

The beam is assumed prismatic and statically determinate. Dynamic effects, transverse distribution, impact, braking, and load amplification are excluded from this preliminary calculation. The 120 kN axle is taken as the leading axle during the critical trial position.

At midspan, the moment influence line is triangular with base 18 m and maximum ordinate

Let the 120 kN axle be at midspan, x₁ = 9 m. The 80 kN axle is then at x₂ = 5 m. For a unit load at position x on the left half, the midspan ordinate is

Thus,

The resulting moment is

Alternative trial position

Place the 80 kN axle at midspan and the 120 kN axle 4 m to its right, at x = 13 m. For the right half, the ordinate is

Therefore,

and

The first trial gives 740 kN·m, which is larger than the second. However, a professional assessment should slide the axle train through additional candidate positions or use a systematic influence-line search. It should also check shear, reactions, fatigue, dynamic effects, and all applicable bridge-loading regulations.

Learning points from the case

1. The response is the sum of each axle load multiplied by its influence ordinate.
2. The largest axle need not be positioned at the response location in every load arrangement.
3. Influence lines support rapid screening, but final bridge design requires code-prescribed load models and combinations.
4. A preliminary static moving-load result is not automatically a design value.

9. Indeterminate Structures and Computational Practice

For statically indeterminate structures, equilibrium alone cannot determine reactions and internal forces. Influence lines can still be obtained by the Müller–Breslau principle, force method, displacement method, virtual work, unit-load analysis, or matrix structural analysis.

In a computer model, a common procedure is:

1. Define the structural model, supports, releases, and stiffness properties.
2. Apply a unit load at successive positions.
3. Record the selected response at the fixed location.
4. Plot response versus load position.
5. Apply actual axle or distributed loads by superposition or direct moving-load analysis.
6. Check mesh spacing, sign conventions, boundary conditions, and convergence.

For a continuous beam or frame, the influence line may be curved rather than piecewise linear, and it may contain positive and negative regions. The Müller–Breslau shape remains a valuable qualitative check. Software output should never be accepted without checking units, load direction, response extraction point, and support idealization.

10. Limitations, Dynamic Effects, and Best Practice

10.1 Principal assumptions

Elementary influence-line analysis commonly assumes:

• small displacements;
• linear elastic behavior and superposition;
• idealized supports and load paths;
• loads acting along a defined structural line;
• negligible or separately treated dynamic effects;
• a response quantity that is clearly defined by sign convention.

Plastic behavior, cracking, large displacement, contact changes, and nonlinear restraints may invalidate direct superposition.

10.2 Static versus dynamic response

A moving vehicle can excite vibration. Road roughness, speed, suspension properties, structural natural frequencies, and damping affect the dynamic response. A static influence line gives the response to a load position under quasi-static assumptions; it does not itself provide an impact factor. Dynamic amplification must be evaluated using applicable standards or a validated dynamic model.

10.3 Regulatory and professional responsibilities

Bridge and construction-system design must comply with the governing structural code, including prescribed vehicle models, lane factors, impact provisions, fatigue requirements, load combinations, and safety factors. Engineers must document assumptions and obtain appropriate review. In temporary works, responsibilities include checking construction sequence, access control, equipment limits, inspection, and the possibility of accidental eccentric loading.

10.4 Best-practice checklist

• Define the response location and sign convention.
• Draw the influence line before applying moving loads.
• Check zero ordinates, jumps, slopes, and symmetry.
• Use signed influence areas, not absolute areas, unless explicitly seeking an absolute effect.
• Test both point-load and distributed-load arrangements.
• Check reactions and global equilibrium.
• Distinguish serviceability, strength, fatigue, and construction-stage requirements.
• Compare hand calculations with software output.
• Record load positions that govern the design.

Contemporary Developments

Modern bridge-management systems combine influence lines with weigh-in-motion data, structural health monitoring, and digital twins. Measured vehicle signatures can be compared with predicted influence-line responses to identify changes in stiffness or damage. Automated moving-load envelopes can rapidly assess thousands of vehicle configurations, but the reliability of the result still depends on model calibration, sensor quality, load-path representation, and engineering judgment. Artificial intelligence can assist in pattern recognition and anomaly detection, but it does not remove the need for code compliance, transparent assumptions, and independent verification.

Chapter Summary

An influence line describes the variation of one selected structural response as a unit load moves across a structure. For point loads, the response equals the load multiplied by the influence ordinate; for distributed loads, it equals the integral of load intensity times influence ordinate, or load intensity times signed influence-line area when the load is uniform. Equilibrium provides direct influence lines for statically determinate structures. The Müller–Breslau principle gives the qualitative shape by releasing the selected restraint and imposing the corresponding unit displacement or rotation. Shear influence lines contain a jump, whereas moment influence lines are continuous at ordinary sections. Moving axle trains require summation of individual load effects, and distributed loads must be positioned over favorable positive or negative regions. Practical applications include bridges, crane runways, floor beams, and temporary construction systems. Dynamic effects, code requirements, support conditions, load paths, and model verification must be considered before a result is used for design.

Key Terms

Influence line: Graph showing the variation of a selected response as a unit load moves.

Moving load: A load whose position changes along a structure.

Influence ordinate: Response value at a specified load position on an influence line.

Unit load: Idealized load of magnitude one used to construct an influence line.

Response quantity: Selected reaction, shear, moment, axial force, or displacement.

Müller–Breslau principle: Principle relating influence-line shape to a released structural mechanism.

Moving point load: Concentrated load travelling along a structural member.

Axle train: Set of concentrated vehicle loads with specified spacing.

Distributed load: Load spread continuously over a length, commonly measured in force per unit length.

Signed area: Area under an influence line retaining positive and negative signs.

Critical position: Load location producing a governing structural response.

Shear jump: Discontinuity in a shear influence line as a unit load crosses the section.

Statically determinate structure: Structure whose reactions and internal forces follow from equilibrium alone.

Indeterminate structure: Structure requiring compatibility and stiffness relations in addition to equilibrium.

Superposition: Addition of responses from separate loads under linear conditions.

Load path: Physical route by which applied load is transferred through structural components.

Dynamic amplification: Increase in response caused by motion and vibration relative to static response.

Moving-load envelope: Maximum and minimum response obtained over relevant load positions.

Construction-stage analysis: Analysis of the structure in its temporary support and loading condition.

Review Questions

Short Answer Questions

1. What is an influence line, and how does it differ from a bending-moment diagram?
2. Why does a shear influence line have a jump at the selected section?
3. State the Müller–Breslau principle in your own words.
4. How is the response to a uniform moving load obtained from an influence line?
5. List two assumptions commonly used in elementary influence-line analysis.

Long Answer Questions

1. Derive the reaction influence lines for a simply supported beam of span L using equilibrium.
2. Explain the construction and interpretation of a bending-moment influence line at an interior section.
3. Describe a systematic method for finding the critical position of a train of moving axle loads.
4. Compare equilibrium construction and the Müller–Breslau principle, including their advantages and limitations.
5. Discuss the use of influence lines in bridge design, including dynamic and regulatory considerations.

Analytical / Application Questions

1. A 15 m simply supported beam has a section 6 m from the left support. Determine the moment at that section caused by a 50 kN load located 4 m from the left support.
2. A 12 m simply supported beam carries a 6 kN/m uniform load over the full span. Use the midspan influence line to determine the midspan moment.
3. A two-axle vehicle has loads of 70 kN and 110 kN spaced 3 m apart and crosses an 18 m simply supported beam. Explain how you would locate and verify the position producing the maximum midspan moment, including at least two trial positions and checks for shear.

Multiple Choice Questions with Answers

Question 1

What does an influence line represent?

A. The response at every section for one fixed load arrangement

B. The variation of one selected response as a unit load moves

C. The deflected shape under the complete design load

D. The distribution of material strength along a beam

Correct Answer: B. The variation of one selected response as a unit load moves

Question 2

For a simply supported beam of span L, what is the influence ordinate for the left reaction when a unit load is at distance x from the left support?

A. x/L

B. L/x

C. (L − x)/L

D. x(L − x)

Correct Answer: C. (L − x)/L

Question 3

At an interior section of a simply supported beam, the bending-moment influence line is generally:

A. Triangular with zero ordinates at the supports

B. Rectangular over the complete span

C. Discontinuous at the section

D. Constant and negative

Correct Answer: A. Triangular with zero ordinates at the supports

Question 4

What characteristic does a shear influence line have as a unit load crosses the selected section?

A. A zero ordinate at the section

B. A jump whose magnitude equals the unit load effect

C. A parabolic maximum

D. No change in slope or value

Correct Answer: B. A jump whose magnitude equals the unit load effect

Question 5

According to the Müller–Breslau principle, the qualitative influence-line shape is obtained by:

A. Doubling the applied permanent load

B. Releasing the response restraint and imposing the corresponding unit movement

C. Dividing the span into equal load intervals only

D. Integrating the bending-moment diagram

Correct Answer: B. Releasing the response restraint and imposing the corresponding unit movement

Question 6

A 10 m simply supported beam has a midspan moment influence-line maximum ordinate of 2.5 m. What is the midspan moment from a 20 kN load placed at midspan?

A. 8 kN·m

B. 25 kN·m

C. 50 kN·m

D. 200 kN·m

Correct Answer: C. 50 kN·m

Question 7

For a constant distributed load w, the response obtained from an influence line equals:

A. w divided by the influence-line area

B. w multiplied by the signed influence-line area covered by the load

C. The maximum ordinate divided by w

D. The span multiplied by the largest load only

Correct Answer: B. w multiplied by the signed influence-line area covered by the load

Question 8

Which statement about a train of moving point loads is correct?

A. Only the heaviest axle can affect the response

B. The response is the sum of each load multiplied by its influence ordinate

C. All axles always produce the same influence ordinate

D. The loads must be replaced by a single load at the span centre

Correct Answer: B. The response is the sum of each load multiplied by its influence ordinate

Question 9

Why is a static influence line insufficient by itself for determining vehicle impact effects?

A. It contains no response ordinates

B. It assumes the structure has no supports

C. It represents quasi-static load-position effects and does not model vibration

D. It can only be used for distributed loads

Correct Answer: C. It represents quasi-static load-position effects and does not model vibration

Question 10

Which additional issue is especially important when applying influence lines to temporary construction systems?

A. The final structure can always be used without modification

B. Construction-stage supports and load paths may differ from the final condition

C. Moving loads are never permitted on temporary works

D. Equilibrium equations become unnecessary

Correct Answer: B. Construction-stage supports and load paths may differ from the final condition