INFLUENCE LINES
Influence lines have important application for the design of structures that
resist large live loads. In this topic, we will discuss how to draw the influence
line for a statically determinate structure. The theory is applied to structures
subjected to a distributed load or a series of concentrated forces, and
specific applications to floor girders and bridge trusses are given.
If a structure is subjected to a live or moving load, the variation of the shear
and bending moment in the member is best described using the influence
line. An influence line represents the variation of either the reaction, shear,
or moment at a specific point in a member as a concentrated force
moves over the member. Once this line is constructed, one can tell at a
glance where the moving load should be placed on the structure so that
it creates the greatest influence at the specified point.
PROCEDURE FOR ANALYSIS
Either of the following two procedures can be used to construct the
influence line at a specific point P in a member for any function (reaction,
shear, or moment).
TABULATE VALUES
• Place a unit load at various locations, x, along the member, and at each
location use statics to determine the value of the function at the specified
point.
• If the influence line for a vertical force reaction at a point on a beam is
to be constructed, consider the reaction to be positive at the point when
it acts upward on the beam.
• If a shear or moment influence line is to be drawn for a point, take the
shear or moment at the point as positive according to the same sign
convention used for drawing shear and moment diagrams.
• All statically determinate beams will have influence lines that consist of
straight line segments. After some practice one should be able to minimize
computations and locate the unit load only at points representing the end
points of each line segment.
INFLUENCE-LINE EQUATIONS
• The influence line can also be constructed by placing the unit load at a
variable position x on the member and then computing the value of R, V,
or M at the point as a function of x. In this manner, the equations of the
various line segments composing the influence line can be determined
and plotted.
Loadings. Once the influence line for a function (reaction, shear, or
moment) has been constructed, it will then be possible to position the live
loads on the beam which will produce the maximum value of the function.
Two types of loadings will now be considered.
Concentrated Force. For any concentrated force F acting on the
beam at any position x, the value of the function can be found by
multiplying the ordinate of the influence line at the position x by the
magnitude of F.
Uniform Load. In general, the value of a function caused by a
uniform distributed load is simply the area under the influence line
for the function multiplied by the intensity of the uniform load.
Illustration
Draw the influence line for the reaction at A for the beam shown.
A
B
10 m
Solution
Procedure 1: TABULATE VALUES
Procedure 2: EQUATIONS
Illustration
Draw the influence line for the shear at bending moment at C for the beam
shown.
Sample Problem #1:
Given the compound beam as shown below, construct the influence lines
for RA, RC, RE, VB and MB.
C
A
E
D
2m
2m
3m
3m
Sample Problem #2:
The figure shows a compound beam. Draw influence for RA, RB, MB, VD, MD.
VE and ME.
D
E
B
C
2m
2m
2m
Müller-Breslau Principle
Influence Lines for the Reaction at D :
The basis of the Müller-Breslau Principle is that we can find the influence
line for a determinate beam by:
1) Removing the restraint caused by the parameter that we want to
find the influence line for
2) Then, displace or rotate the resulting structure by one unit.
If the restraint that is removed is an internal shear or a vertical or horizontal
support, then the second step is to displace the structure at that same
location. If the restraint that is removed is an internal moment or a
rotational restraint, then the second step is to rotate the structure at that
same location.
Influence Lines for the Reaction at B :
Influence Lines for the Reaction at F:
Influence Lines for the moment reaction at point F :
Influence Lines for the internal moment at C’ :
Influence Lines for shear at point C' which is 6m to the right of point C :
Influence Lines for shear in the beam at point D’ which is 2m to the right of
point D :
Influence Lines for the internal moment at D’
Trusses are often used as primary load-carrying elements for bridges.
Hence, for design it is important to be able to construct the influence lines
for each of its members. As shown in the figure below, the loading on the
bridge deck is transmitted to stringers, which in turn transmit the loading to
floor beams and then to the joints along the bottom cord of the truss.
Sample Problem #3:
Draw the influence line for the force in member BC of the Warren truss.
Indicate numerical values for the peaks. All members have the same
length.
Sample Problem #4:
For the truss shown, construct the influence lines for the bar for each of the
lettered bars.
Sample Problem #5:
Given a simple beam 24 ft long, construct the influence lines for the shear
and bending moment at a section 8 ft from the left end, and obtain the
maximum shear and maximum bending moment for the section resulting
from a moving uniform load of 3 kips/ft and a movable concentrated load
of 50 kips.
Sample Problem #6:
A simple beam 45ft long carries moving loads of 10 kips, 10 kips and 5 kips
spaced 5ft apart. Calculate (a) the maximum left reaction and (b) the
maximum shear and bending moment at a section 15ft from the left end.