Influence Lines

Introduction

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 Y_A as a function of the position of a downward acting unit load. One such equation can be found by summing moments at Support B.

Figure 1 - Beam structure for influence line example

Figure 2 - Beam structure showing application of unit load

M_B = 0  (Assume counter-clockwise positive moment)
-Y_A(L)+1(L-x) = 0
Y_A = (L-x)/L = 1 - (x/L)

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.

Figure 3 - Influence line for the support reaction at A

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.

1. For a statically determinate structure the influence line will consist of only straight line segments between critical ordinate values.
2. The influence line for a shear force at a given location will contain a translational discontinuity at this location. The summation of the positive and negative shear forces at this location is equal to unity.
3. Except at an internal hinge location, the slope to the shear force influence line will be the same on each side of the critical section since the bending moment is continuous at the critical section.
4. The influence line for a bending moment will contain a unit rotational discontinuity at the point where the bending moment is being evaluated.
5. To determine the location for positioning a single concentrated load to produce maximum magnitude for a particular function (reaction, shear, axial, or bending moment) place the load at the location of the maximum ordinate to the influence line. The value for the particular function will be equal to the magnitude of the concentrated load, multiplied by the ordinate value of the influence line at that point.
6. To determine the location for positioning a uniform load of constant intensity to produce the maximum magnitude for a particular function, place the load along those portions of the structure for which the ordinates to the influence line have the same algebraic sign. The value for the particular function will be equal to the magnitude of the uniform load, multiplied by the area under the influence diagram between the beginning and ending points of the uniform load.

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.

Examples