STUDENT MATERIALS
Introduction to Influence Lines
Objectives of the materials covered: The student should be able to explain what
influence lines basically are, and what they are commonly used for.
Definition: An influence line is a plot of the magnitude of the resulting reaction/axial
force/shear/moment generated in a beam or structure as a unit load travels across its
length. Influence lines can be generated for any of these actions (reactions, axial forces,
shears, or moments) at any point in a structure.
A typical influence line: As an example, an influence line for the left reaction of a
simply supported beam can be generated either by moving a single concentrated unit load
across the length of the beam or structure, or by using the Maxwell-Betti principal, both
of which will be described later. Thus the “influence” of the unit load is shown by
plotting the magnitude of the resulting left reaction vs. any location of the unit load. A
typical influence line, in this case for the left reaction of a simply supported beam, is
shown below. Do not worry about how we got it. That will be explained later.
x
Figure 1
This diagram indicates that should a concentrated load be placed over the left reaction it
will cause a reaction of magnitude 1.0 times the value of the concentrated load. It also
shows that placing a concentrated load in the center of the beam will cause a left reaction
of magnitude 0.5 times the value of the load. In fact, the result of placing a load anywhere
on the beam can now be found by simply multiplying the magnitude of the load times the
height of the influence line at the location of the load, x.
Influence lines are used to determine where to place loads on a structure for maximum
results, and to compute the resulting magnitude of the reaction or other actions
(shear/moment/axial force) once the loads are placed in their critical positions.