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*Influence Lines*

Qualitative Influence Lines and Loading Patterns for an Multi-span Indeterminate Beam

The Müller Breslau Principle, used previously to draw the influence lines for statically determinate structures, can also be extended to define the influence lines for indeterminate structures. This principle simply states that the influence line for a function is proportionally equivalent to the deflected shape of the structure when it undergoes a displacement as a result of the application of the function.

For indeterminate structures, an understanding of how complex structures deflect and react when acted upon by a force is required in order to draw accurate diagrams.

- Influence lines for reactions

To determine the influence line for the support reaction at A, the Müller
Breslau Principle requires the removal of the support restraint and the
application of a positive unit deformation at this point that corresponds
to the direction of the force. In this case, apply a unit vertical displacement
in the direction of Y_{A}.

Figure 1 - Structure with support reaction removed, unit deformation applied, and resulting influence line

The resulting deflected shape, due to the application of the unit deformation, is then proportionally equivalent to the influence line for the support reaction at A. Notice that in statically indeterminate structures, the deflected shape is not a straight line, but rather a curve. The ordinates of the deflected shape decrease as the distance increases from the point of application of the unit deformation.

Similarly, for the other support reactions, remove the support restraint
and apply a unit deformation in the direction of the removed restraint.
For example, the influence line for the support reaction at C is obtained by
removing the reaction at C and applying a unit displacement in the vertical
direction at C. The resulting deflected shape is a qualitative representation
of the influence line at R_{C} (see Figure 2).

Figure 2 - Structure with support reaction removed, unit deformation applied, and resulting influence line

Influence lines for the remaining support reactions are found in a similar manner.

- Influence lines for shears

For shear at a section, using the Müller Breslau Principle, the shear resistance at the point of interest is removed by introducing the type of support shown in Figure 3, below. Shear forces are applied on each side of the section in order to produce a relative displacement between the two sides which is equal to unity. The deflected shape of the beam under these conditions will qualitatively represent the influence line for the shear at the section. Notice that unlike the statically determinate structure, the magnitude of the shear force on the right and left can not easily be determined.

Figure 3 - Structure with shear carrying capacity removed at section S1, deformations applied, and resulting influence line

- Influence lines for moments

For the moment at a section, using the Müller Breslau Principle, the moment resistance at the point of interest is removed by introducing a hinge at the section as shown in Figure 4, below. Then a positive moment that introduces a relative unit rotation is applied at the section. The deflected shape of the beam under these conditions will qualitatively represent the influence line for the moment at the section.

Figure 4 - Structure with moment capacity removed at section S1, unit rotation applied, and resulting influence line

For the moment at a support, the moment resistance is again removed by inserting a hinge at the support. This hinge only prevents the transfer of moments, so the vertical translation remains fixed due to the support. By applying negative moments that induces a relative rotation of unity at this section, a deflected shape is generated. Again, this deflected shape qualitatively represents the influence line for the moment at a support.

Figure 5 - Structure with moment capacity removed at support B, unit rotation applied, and resulting influence line

- Loading cases for moment and shear envelopes

Using the influence lines found above, illustrate the loading cases needed to
calculate the maximum positive and negative R_{A}, R_{C},
M_{B}, V_{S1}, and M_{S1}.

The load cases are generated for the maximum positive and negative values by placing a distributed load on the spans where the algebraic signs of the influence line are the same. i.e., to get a maximum positive value for a function, place a distributed load where the influence line for the function is positive.

Figure 6 - Multi-span structure

Load case for maximum positive reaction at support A

Figure 7 - Maximum positive reaction at support A

Load case for maximum negative reaction at support A

Figure 8 - Maximum negative reaction at support A

Load case for maximum positive reaction at support C

Figure 9 - Maximum positive reaction at support C

Load case for maximum negative reaction at support C

Figure 10 - Maximum negative reaction at support C

Load case for maximum positive moment at support B

Figure 11 - Maximum positive moment at support B

Load case for maximum negative moment at support B

Figure 12 - Maximum negative moment at support B

Load case for maximum positive shear at s

Figure 13 - Maximum positive shear at s

Load case for maximum negative shear at s

Figure 14 - Maximum negative shear at s

Load case for maximum positive moment at s

Figure 16 - Maximum positive moment at s

Load case for maximum negative moment at s

Figure 17 - Maximum negative moment at s

Contact Dr. Fouad Fanous for more information.