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Influence Line

Influence lines show how internal forces or reactions at a specific location vary as a unit load moves across a structure. They are particularly useful in determining the critical positions of live loads.

Two Common Methods:

1. Tabular Method (Table-Based Analytical Influence Lines)

This method is best suited for statically determinate structures. It involves placing a unit load at key points and solving for the response of the desired function (e.g., support reaction, shear, or moment).

Steps:

  1. Select the function you want to analyze: a support reaction, shear at a section, or moment at a section.
  2. Divide the beam into key load positions — usually at supports and points of interest.
  3. Place a unit load (1.0 kN or 1.0 unit) at one position at a time.
  4. Using equilibrium, solve for the value of the function of interest at each load position.
  5. Plot or tabulate the values to create the influence line diagram.

Example Table Format:

Unit Load Position Support Reaction $R_A$ Shear at Section X Moment at Section X
At A1.0......
Midspan.........
At B0.0......
The tabular method is exact and useful for numerical influence line diagrams, especially for statically determinate systems.

2. Müller-Breslau Principle (Qualitative Influence Lines)

The Müller-Breslau Principle provides a fast, visual way to sketch the shape of an influence line, even for indeterminate structures. It states:

“The influence line for a response function (reaction, shear, or moment) has the same shape as the deflected shape of the structure when the corresponding restraint is released and given a unit displacement in the direction of the function.”

Steps:

  1. Identify the function you want to draw an influence line for (e.g., $R_A$, $V_x$, $M_x$).
  2. Modify the structure by releasing the restraint or cut at the location of the function.
  3. Apply a unit displacement in the direction of the desired function:
    • Vertical displacement for vertical reactions and shear
    • Rotation for moments
  4. Sketch the deflected shape — this is the qualitative influence line.
  5. Positive displacement direction = positive influence value. The relative height of the sketch indicates the magnitude.
The Müller-Breslau Principle is particularly useful for indeterminate structures or quick checks of influence line shapes.

After using the Müller-Breslau sketch, you can optionally apply the conjugate beam method or unit load method to compute exact ordinates if needed.

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