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Qualitative Influence Lines using the Müller Breslau Principle
The Müller Breslau Principle is another alternative available to qualitatively develop the influence lines for different functions. The Müller Breslau Principle states that the ordinate value of an influence line for any function on any structure is proportional to the ordinates of the deflected shape that is obtained by removing the restraint corresponding to the function from the structure and introducing a force that causes a unit displacement in the positive direction.
Figure 1 - Beam structure to analyze
For example, to obtain the influence line for the support reaction at A for the beam shown in Figure 1, above, remove the support corresponding to the reaction and apply a force in the positive direction that will cause a unit displacement in the direction of YA. The resulting deflected shape will be proportional to the true influence line for this reaction. i.e., for the support reaction at A. The deflected shape due to a unit displacement at A is shown below. Notice that the deflected shape is linear, i.e., the beam rotates as a rigid body without any curvature. This is true only for statically determinate systems.
Figure 2 - Support removed, unit load applied, and resulting influence line for support reaction at A
Similarly, to construct the influence line for the support reaction YB, remove the support at B and apply a vertical force that induces a unit displacement at B. The resulting deflected shape is the qualitative influence line for the support reaction YB.
Figure 3 - Support removed, unit load applied, and resulting influence line for support reaction at B
Once again, notice that the influence line is linear, since the structure is statically determinate.
This principle will be now be extended to develop the influence lines for other functions.
To determine the qualitative influence line for the shear at s, remove the shear resistance of the beam at this section by inserting a roller guide, i.e., a support that does not resist shear, but maintains axial force and bending moment resistance.
Figure 4 - Structure with shear capacity removed at s
Removing the shear resistance will then allow the ends on each side of the section
to move perpendicular to the beam axis of the structure at this section.
Next, apply a shear force, i.e., Vs-R and Vs-L that will
result in the relative vertical displacement between the two ends to equal unity.
The magnitude of these forces are proportional to the location of the section and
the span of the beam. In this case,
Vs-L = 1/16 x 10 = 10/16 = 5/8
Vs-R = 1/16 x 6 = 6/16 = 3/8
The final influence line for Vs is shown below.
Figure 5 - Influence line for shear at s
The shear just to the left side of support B can be constructed using the ideas explained above. Simply imagine that section s in the previous example is moved just to the left of B. By doing this, the magnitude of the positive shear decreases until it reaches zero, while the negative shear increases to 1.
Figure 6 - Influence line for shear just to the left of B
To plot the influence line for the shear just to the right side of support B, Vb-R, release the shear just to the right of the support by introducing the type of roller shown in Fig. 7, below. The resulting deflected shape represents the influence line for Vb-R. Notice that no deflection occurs between A and B, since neither of those supports were removed and hence the deflections at A and B must remain zero. The deflected shape between B and C is a straight line that represents the motion of a rigid body.
Figure 7 - Structure with shear capacity removed at just to the right of B and the resulting influence line
To obtain a qualitative influence line for the bending moment at a section, remove the moment restraint at the section, but maintain axial and shear force resistance. The moment resistance is eliminated by inserting a hinge in the structure at the section location. Apply equal and opposite moments respectively on the right and left sides of the hinge that will introduce a unit relative rotation between the two tangents of the deflected shape at the hinge. The corresponding elastic curve for the beam, under these conditions, is the influence line for the bending moment at the section. The resulting influence line is shown below.
Figure 8 - Structure with moment capacity removed at s and the resulting influence line
The values of the moments shown in Figure 8, above, are calculated as follows:
The qualitative influence line for the bending moment at B is obtained by introducing a hinge at support B and applying a moment that introduces a unit relative rotation. Notice that no deflection occurs between supports A and B since neither of the supports were removed. Therefore, the only portion that will rotate is part BC as shown in Fig. 9, below.
Figure 9 - Structure with moment capacity removed at B and the resulting influence line
The shear and moment envelopes are graphs which show the variation in the minimum and maximum values for the function along the structure due to the application of all possible loading conditions. The diagrams are obtained by superimposing the individual diagrams for the function based on each loading condition. The resulting diagram that shows the upper and lower bounds for the function along the structure due to the loading conditions is called the envelope.
The loading conditions, also referred to as load cases, are determined by examining the influence lines and interpreting where loads must be placed to result in the maximum values. To calculate the maximum positive and negative values of a function, the dead load must be applied over the entire beam, while the live load is placed over either the respective positive or negative portions of the influence line. The value for the function will be equal to the magnitude of the uniform load, multiplied by the area under the influence line diagram between the beginning and ending points of the uniform load.
For example, to develop the shear and moment envelopes for the beam shown in Figure 1, first sketch the influence lines for the shear and moment at various locations. The influence lines for Va-R, Vb-L, Vb-R, Mb, Vs, and Ms are shown in Fig. 10.
Figure 10 - Influence lines
These influence lines are used to determine where to place the uniform live load to yield the maximum positive and negative values for the different functions. For example;
Figure 11 - Support removed, unit load applied, and resulting influence line for support reaction at A
The maximum value for the positive reaction at A, assuming no partial loading, will occur when the uniform load is applied on the beam from A to B (load case 1)
Figure 12 - Load case 1
The maximum negative value for the reaction at A will occur if a uniform load is placed on the beam from B to C (load case 2)
Figure 13 - Load case 2
Load case 1 is also used for:
- maximum positive value of the shear at the right of support A
- maximum positive moment Ms
Load case 2 is also used for:
- maximum positive value of the shear at the right of support B
- maximum negative moments at support B and Ms
Load case 3 is required for:
- maximum positive reaction at B
- maximum negative shear on the left side of B
Figure 14 - Load case 3
Load case 4 is required for the maximum positive shear force at section s
Figure 15 - Load case 4
Load case 5 is required for the maximum negative shear force at section s
Figure 16 - Load case 5
To develop the shear and moment envelopes, construct the shear and moment diagrams for each load case. The envelope is the area that is enclosed by superimposing all of these diagrams. The maximum positive and negative values can then be determined by looking at the maximum and minimum values of the envelope at each point.
Individual shear diagrams for each load case;
Superimpose all of these diagrams together to determine the final shear envelope.
Figure 18 - Resulting superimposed shear envelope
Individual moment diagrams for each load case;
Superimpose all of these diagrams together to determine the final moment envelope.
Figure 20 - Resulting superimposed moment envelope
Contact Dr. Fouad Fanous for more information.