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|These pages are under development and may
contain some errors.
Any questions, suggestions, or errors should be directed to Dr. Fouad Fanous
An influence line for a given function, such as a reaction, axial force, shear force, or bending moment, is a graph that shows the variation of that function at any given point on a structure due to the application of a unit load at any point on the structure.
An influence line for a function differs from a shear, axial, or bending moment diagram. Influence lines can be generated by independently applying a unit load at several points on a structure and determining the value of the function due to this load, i.e. shear, axial, and moment at the desired location. The calculated values for each function are then plotted where the load was applied and then connected together to generate the influence line for the function.
For example, the influence line for the support reaction at A of the structure shown in Figure 1, is found by applying a unit load at several points (See Figure 2) on the structure and determining what the resulting reaction will be at A. This can be done by solving the support reaction YA as a function of the position of a downward acting unit load. One such equation can be found by summing moments at Support B.
MB = 0 (Assume
counter-clockwise positive moment)
The graph of this equation is the influence line for the support reaction at A (See Figure 3). The graph illustrates that if the unit load was applied at A, the reaction at A would be equal to unity. Similarly, if the unit load was applied at B, the reaction at A would be equal to 0, and if the unit load was applied at C, the reaction at A would be equal to -e/L.
Once an understanding is gained on how these equations and the influence lines they produce are developed, some general properties of influence lines for statically determinate structures can be stated.
There are two methods that can be used to plot an influence line for any function. In the first, the approach described above, is to write an equation for the function being determined, e.g., the equation for the shear, moment, or axial force induced at a point due to the application of a unit load at any other location on the structure. The second approach, which uses the Müller Breslau Principle, can be utilized to draw qualitative influence lines, which are directly proportional to the actual influence line.
The following examples demonstrate how to determine the influence lines for reactions, shear, and bending moments of beams and frames using both methods described above.
|Influence lines for a simple beam by developing the equations|
|Qualitative influence lines using the Müller Breslau Principle|
|Qualitative influence lines for a statically determinate continuous beam|
|Calculation of maximum and minimum shear force and moments on a statically determinate continuous beam|
|Qualitative influence lines and loading patterns for a multi-span indeterminate beam|
|Qualitative influence lines and loading patterns for an indeterminate frame|
Contact Dr. Fouad Fanous for more information.