Advanced Methods of Structural Analysis

10.4 Kinematical Method for Construction of Influence Lines 361

P= 1

1268910345 711

ll

a

X 1 =1

M 1

1.0

b

0.0480.0840.0960.072 0.0720.0960.0840.048

Inf. line X 1

(factor l)

c

• –

Fig. 10.18 Unit bending moment diagram and influence line for bending moment at support 1

free term of canonical equation of the force method. In Sect.10.1we calculated a
free termı1P, which is the angle of rotation at support 1, caused by moving unit
load. In Sect.10.4we calculated a free termıP1, which is the vertical displace-
ment at the point of application of the moving load caused by momentX 1 D 1 at
the middle support 1. Now it is evident that for the construction of influence line
for truss, as shown in Sect.10.3,wehavealsousedtheM ̈uller–Breslau principle;
however for computation of displacement of the joints, which belong to the loaded
contour, the elastic load method was applied.
As shown above for calculation ofıP1the precise analytical method has been ap-
plied. As a result, influence line was presented withcomputed ordinatesat specified
points using formula (10.24). Without consideration of constant factor.1=ı 11 /,
this formula allows constructing themodelof influence line. Thus, the model of in-
fluence line is constructed using only functionıP1, which presents the elastic curve.
In other words, if we apply at pointathe unit factorX, then the elastic curve
presents the model of the influence lineXat pointa. If we apply at pointathe
unit displacement, which corresponds to the required factorX, then the elastic curve
presents genuine influence lineXat pointa.

Example 10.4.Design diagram for continuous beam is presented in Fig.10.19.
Construct the models of influence lines for reactions and internal forces.

Solution.

Influence line for reactionR 1 In this case we need to eliminate a constraint, where
the vertical reactionR 1 arises and apply the positive reactionR 1. Elastic curve due
to unit reactionR 1 in the new system is a model of influence line forR 1 .Ifload
P D 1 is located above support 1, then reactionR 1 D 1 , therefore ordinate of