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Influence Lines
Qualitative Influence Lines and Loading Patterns for an Indeterminate Frame


problem statement

Using the Müller Breslau Principle, draw the influence lines for the moment and shear at the midspan of beam AB, and the moment at B in member BC. Draw the loading cases to give the maximum positive moment at the midpsan of beam AB, the maximum and minimum shear at the midspan of beam AB, and the maximum negative moment at B in member BC in the indeterminate frame below.

Frame structure to analyze
Figure 1 - Frame structure to analyze


Influence line for moment at midspan of AB, and the loading case for maximum positive moment at this location.

The influence line for beam ABCD can be constructed by following the procedure outlined in the Multi-span Indeterminate Beam example. To construct the rest of the influence line, make use of the fact that the angles between a column and a beam after deformation must be equal to that before deformation. In this example, these angles are 90°. Therefore, once the deflected shape of beam ABCD is determined, the deflected shape for the columns can be constructed by keeping the angles between the tangent of the deflect shape of the beam and the column equal to 90° (see Figure 2).

To get the maximum positive result for the moment, apply a distributed load at all locations where the value of the influence line is positive (see Figure 3).

 
Figure 2 - Influence lines for moment at midspan of AB Figure 3 - Load case for maximum positive moment at midspan of AB

Influence line for shear at the midspan of member AB, and the load case for maximum positive shear at this location.

Figure 4 - Influence lines for shear at midspan of AB
 
Figure 5 - Load case for maximum positive shear at midspan of AB   Figure 6 - Load case for maximum negative shear at midspan of AB

Influence line for moment at B in member BC, and the load case for maximum negative moment at this location.

 
Figure 7 - Influence lines for moment at B   Figure 8 - Load case for maximum positive moment at B

Contact Dr. Fouad Fanous for more information.