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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:

• Tabular Method (Analytical)
• Müller-Breslau Principle (Qualitative)

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:

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:

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.

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