Module 4: Moving Loads and Influence Lines - Theory & Concepts

Module 4: Moving Loads and Influence Lines

While dead loads are static and fixed in position, live loads—such as vehicles on a bridge or overhead cranes in a factory—move across a structure. The internal forces (shear, bending moment) and reactions caused by these moving loads vary depending on their position. This module introduces the concept of influence lines, a powerful graphical tool used to determine the critical placement of moving loads that produces the maximum structural response.

Moving Loads

Loads that change their position on a structure over time. They represent transient forces exerted by traffic (vehicles, pedestrians), traveling equipment (cranes, gantries), or even movable machinery.

Concept of Moving Loads

In the design of structures like highway bridges, railway bridges, and some industrial buildings, it is essential to determine the position of the live load that produces the maximum stress, deflection, or other critical structural response. This critical position is often not obvious, especially for complex structures or multiple axle loads.

Checklist

Vehicular Loads: Trucks, trains, or cars that travel across a bridge deck. They are often modeled as a series of concentrated wheel loads with fixed spacing, or a uniform lane load. Building codes (e.g., AASHTO for bridges) specify standard vehicle models.
Moving Equipment: Overhead cranes or gantries that travel along rails. They induce moving concentrated loads on the supporting girders, which fluctuate as the crane trolley moves transversely.

Key Takeaways

Moving loads create varying responses (shear, moment, reactions) in a structure, requiring specialized analysis techniques to find the absolute maximum values.

Influence Lines

An influence line is a graph that plots the variation of a specific structural response (e.g., support reaction, shear force, bending moment, or deflection at a particular point in a structure) as a unit load moves across the entire structure.

Properties of Influence Lines

The abscissa (horizontal axis) represents the position of the moving unit load.
The ordinate (vertical axis) represents the magnitude of the specific structural response being plotted, caused by the unit load at that position.
For statically determinate structures, the influence line for any reaction, shear, or moment consists solely of straight-line segments. For statically indeterminate structures, the influence line is composed of curved segments.

Müller-Breslau Principle

The Müller-Breslau Principle is a powerful method for rapidly sketching the shape of an influence line. It states that the influence line for a specific structural function (reaction, shear, moment) is to the same scale as the deflected shape of the structure when the structure is acted upon by a unit displacement (or rotation) corresponding to that function.

For a Reaction: Remove the support and apply a unit vertical displacement. The resulting deflected shape is the influence line for that reaction.
For Shear: Cut the beam at the point of interest and apply a unit relative transverse displacement (sliding without rotation). The deflected shape is the influence line for shear.
For Moment: Insert an internal hinge at the point of interest and apply a unit relative rotation (bending). The deflected shape is the influence line for moment.

Influence Line Simulation

Use this interactive tool to visualize how the influence lines for reactions, shear, and moment change as you select different points along a simply supported beam.

Truss Influence Line Simulator

Select Member

Position of Unit Load (x)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

Position of Unit Load (x)5.0 m

Reactions

A_y = 0.50 kN

B_y = 0.50 kN

Horizontal Thrust

H = 0.50 kN

Key Takeaways

An influence line graphs the variation of a specific structural response at a single, fixed point as a unit load traverses the structure. This is distinct from shear and moment diagrams, which plot responses along the entire structure for a fixed set of loads.
The Müller-Breslau principle allows rapid sketching of influence lines based on the structure's deflected shape under a unit displacement corresponding to the response of interest.

Influence Line vs. Shear/Moment Diagram

It is absolutely critical to distinguish between these two tools:

Shear/Moment Diagram: Shows the internal shear or moment across the entire length of the beam for a fixed set of loads. The x-axis represents the position along the beam.
Influence Line: Shows the internal shear or moment at a single, specific point on the beam as a unit load moves across the structure. The x-axis represents the position of the moving load.

Absolute Maximum Moment Procedure

When a series of concentrated wheel loads (like a truck) moves across a simple beam, the absolute maximum bending moment under the entire span occurs under a specific wheel load. The procedure to find it:

Calculate the magnitude and location of the Resultant Force ( $R$ ) of the entire wheel load system.
Position the load system on the beam such that the centerline of the beam span perfectly bisects the distance between the Resultant ( $R$ ) and the closest heavy wheel load ( $P_i$ ). Note that all loads must remain on the span.
The absolute maximum moment will occur directly underneath that specific wheel load ( $P_i$ ). You may need to check adjacent heavy loads if the exact critical wheel is unclear.

Maximum Response from Moving Loads

Once an influence line is drawn, it can be used to determine the maximum value of a structural response caused by a given moving load or series of moving loads.

Checklist

Concentrated Load: Multiply the magnitude of the concentrated load by the corresponding ordinate of the influence line directly beneath the load's position. To maximize the response, place the load at the peak (highest ordinate) of the influence line.
Uniformly Distributed Load (UDL): Multiply the intensity of the uniform load by the area under the influence line over the length where the load is applied. To maximize a positive response, place the UDL only over the positive areas of the influence line. To maximize a negative response, place the UDL only over the negative areas.

Force Envelopes (Maximum Effect Diagrams)

While an influence line shows the effect at a single point as a load moves, a force envelope (or maximum shear/moment diagram) shows the absolute maximum and minimum values of shear or moment that can occur at every point along the entire structure due to all possible positions of the moving load.

The envelope is created by superimposing the peak values from many individual shear/moment diagrams (each drawn for a different load position).
Designers use the force envelope to size the member, ensuring it is strong enough at every section to resist the worst-case scenario.

Absolute Maximum Shear and Moment

When a series of concentrated loads (like a truck's wheel axles) moves across a simple beam, we must determine the absolute maximum shear and absolute maximum moment that occur anywhere in the beam.

Absolute Maximum Shear

For a simple beam, the absolute maximum shear will always occur immediately adjacent to one of the supports when one of the loads (usually the heaviest or the first/last in the series) is placed directly over that support.

Absolute Maximum Moment

For a simple beam subjected to a series of moving concentrated loads, the absolute maximum bending moment occurs under one of the loads when that specific load and the resultant of the entire load series are positioned equidistant from the centerline of the beam span.

Procedure

STEP-BY-STEP

Find the magnitude and location of the resultant force ( $R$ ) of the entire moving load series.
Assume one of the loads (usually the heaviest one near the resultant) will cause the absolute maximum moment. Let's call this load $P_i$ .
Calculate the distance ( $d$ ) between $P_i$ and the resultant $R$ .
Position the load series on the beam such that the centerline of the beam exactly bisects the distance ( $d$ ) between $P_i$ and the resultant $R$ .
Calculate the bending moment under the load $P_i$ at this position. This is a strong candidate for the absolute maximum moment.
Repeat the process for other heavy loads near the resultant to ensure the true absolute maximum is found.

Key Takeaways

To find the maximum positive response from a uniform live load, the load must be placed only over the positive area(s) of the corresponding influence line. For concentrated loads, place the heaviest load at the peak ordinate.
The absolute maximum moment under a series of moving loads occurs when the centerline of the beam bisects the distance between a specific load and the resultant of the load series.

PreviousModule 3: Analysis of Statically Determinate Structures - Examples

NextModule 4: Moving Loads and Influence Lines - Examples