Advanced Methods of Structural Analysis

28 2 General Theory of Influence Lines

Uniformly Distributed Load

Value of any functionZdue to action of uniformly distributed loadqis determined
by formulaZD ̇q!, with!the area of influence line graph for functionZwithin
the portion where loadqis applied. If the influence line within the load limits has
different signs then the areas must be taken with appropriate signs. The sign of the
area coincides with sign of ordinates of influence line.

Couple

If a structure is loaded by coupleM, then functionZ, due to this moment is
ZD ̇Mtan ̨,where ̨is the angle between the base line and the portion of in-
fluence line for functionZwithin whichM is applied. If couple tends to rotate
influence line toward base line through an angle less than 90 then sign is positive; if
angle is greater then 90 then sign is negative.

Summary

Influence line for any function may be used for calculation of this function due to
arbitraryfixed loads. In a general case, any functionZas a result of application of
a several concentrated loadsPi, uniform loads intensityqj, and couplesMkshould
be calculated as follows:

ZD

X

PiyiC

X

qj!jC

X

Mktan ̨k; (2.9)

whereyiis the ordinates of corresponding influence line, these ordinates are mea-
sured at all the load points;!jthe area bounded by corresponding influence line,
thex-axis, and vertical lines passing through the load limits; and ̨kis the angle of
inclination of corresponding influence line to thex-axis.
The formula (2.9) reflects the superposition principle and may be applied for any
type of statically determinate and indeterminate structures. Illustration of this for-
mula is shown below. Figure2.11presents a design diagram of a simply supported
beam and influence line for reactionRA.

Fig. 2.11 Design diagram of
the beam

RA

q

P M

l/2 l/2 RB

+

y

(^1) α
Inf. line RA