Influence Lines for Statically Determinate Structures
Understanding how structural responses vary as loads move across a structure, and finding maximum effects.
What is an Influence Line?
Influence Line
A graph that shows the variation of a specific structural response (such as a reaction, shear force, or bending moment at a given point) as a unit load moves across the structure.
Purpose
Unlike a shear or moment diagram that shows the effect of fixed loads at all points along a member, an influence line shows the effect of a moving load at a single, fixed point. Influence lines are essential for determining the critical positions of moving loads (like vehicles on a bridge or crane loads) to produce the maximum possible effects on the structure.
Constructing Influence Lines
Procedure
- Define the specific response (e.g., reaction , shear , or moment ) for which the influence line is needed.
- Imagine a single concentrated unit load moving slowly from one end of the structure to the other. Let its position be denoted by .
- Determine the equation for the response as a function of the load's position using statics.
- Plot the equation. The resulting graph is the influence line. For determinate structures, the influence lines are composed of straight-line segments.
Müller-Breslau Principle
Qualitative Method
Müller-Breslau Principle:
A qualitative method to quickly draw the shape of an influence line. It states that the influence line for a particular response (reaction, shear, or moment) is given by the deflected shape of the structure when the constraint corresponding to that response is removed and a unit displacement or rotation is introduced.
Application of Influence Lines
Once constructed, influence lines can be used to determine the response due to various types of loading.
- Concentrated Load (): The response equals the magnitude of the load multiplied by the ordinate of the influence line at the point where the load is applied.
- Uniform Distributed Load (): The response equals the intensity of the load multiplied by the area under the influence line over the loaded segment. To find the maximum positive response, place the uniform load over all positive areas of the influence line.
Influence Lines for Beams and Trusses
Beams vs. Trusses
While the fundamental concept remains the same, the construction and application of influence lines differ slightly between solid beams and framed trusses.
- Beams: The load can move continuously along the span. Influence lines for shear and moment at a specific point are composed of straight lines with jumps or slope changes directly at .
- Trusses: Loads are typically applied to the floor system (stringers and floor beams), which then transmit the loads only to the truss joints (panel points). Therefore, the influence line for a truss member force is composed of straight-line segments connecting the values calculated at the panel points. The load cannot be applied directly to the members themselves between joints.
Absolute Maximum Shear and Moment
Determining the maximum possible internal forces anywhere in the structure under a moving load system.
Absolute Maximums
An influence line gives the maximum response at a specific point. However, for design, we often need the absolute maximum response that occurs anywhere in the beam as a series of concentrated loads moves across it.
- Absolute Maximum Shear: In a simply supported beam, this always occurs adjacent to one of the supports. It is found by placing the load system such that the heaviest loads are near the support.
- Absolute Maximum Moment: This does not necessarily occur at midspan. The absolute maximum bending moment under a series of concentrated loads occurs under one of the concentrated loads when that load and the resultant of the entire load system are equidistant from the centerline of the beam.
Series of Concentrated Loads
When dealing with a series of moving point loads (like a truck's axles), determining the maximum response (shear or moment) requires finding the critical position of the load train.
Procedure
- Draw the influence line for the desired response.
- Place the series of loads on the structure and move them across.
- Calculate the response for trial positions where one of the large concentrated loads is placed directly over the maximum ordinate of the influence line.
- Compare the results of the trial positions to find the absolute maximum response.
Interactive Tool: Influence Line Simulation
Observe how reactions, shears, and moments change as a point load or a distributed load moves across a simply supported or overhanging beam. Visualize the corresponding influence lines.
Influence Line Generator
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Qualitative Influence Lines
Müller-Breslau Principle for Indeterminate Structures
While the Müller-Breslau Principle produces straight-line influence lines for determinate structures, it can also be used qualitatively for indeterminate structures. The deflected shape represents the influence line, but because indeterminate structures resist deflection, the resulting influence lines are curved rather than straight.
Envelopes for Maximum Effects
Influence Line Envelopes
For moving loads on bridges, designers create an "envelope" diagram. By passing the load across the structure, the absolute maximum and absolute minimum (often negative) values of shear and moment at every point are recorded and plotted. This envelope dictates the required design capacity along the entire span.
Influence Lines for Floor Girders
Analyzing influence lines for floor systems where loads are applied to floor beams rather than directly to the main girder.
Floor Systems
In many bridge and building floor systems, the deck is supported by stringers, which transfer the load to floor beams, which in turn transfer the load to the main girders at specific connection points (panel points).
- Because the load is not applied directly to the girder, the influence line for shear or moment in the girder is modified.
- Between the panel points, the influence line consists of straight-line segments connecting the ordinate values calculated at the adjacent panel points.
- This results in a "panel shear" effect, where the shear influence line slopes between panel points rather than having a vertical jump.
Key Takeaways
- An influence line shows the variation of a specific response at a fixed point as a unit load moves across the structure.
- They are distinct from shear and moment diagrams, which show the variation of response across the structure for a fixed load.
- The Müller-Breslau principle allows for the rapid qualitative construction of influence lines.
- Influence lines are used to determine the critical placement of live loads (both point and distributed) to produce maximum structural effects.
- An influence line shows the variation of a specific reaction, shear, or moment at a specific point as a single unit load moves across the structure.
- Unlike shear and moment diagrams (which show effects along the entire beam for a fixed load), influence lines show the effect at one point for a moving load.
- The Müller-Breslau Principle states that the influence line for a function is proportional to the deflected shape of the structure when the ability to resist that function is removed.
- Influence lines are critical for determining where to place live loads to maximize internal forces.