Advanced Methods of Structural Analysis

10.4 Kinematical Method for Construction of Influence Lines 359

1.Indicate the constraint or section, in which factorXarises.
2.Show the new system by eliminating constraint where factorXarises.
3.Apply theXD 1 instead of eliminated constraint.
4.Show the elastic curve due toXD 1 in new system. This curve is a model for
influence line of factorX.
Figure10.16presents elimination of constraintkwhere factorXarises and re-
placing this constraint by corresponding forceX; case (a) should be used for
construction of influence line for reaction at any support of continuous beam; the
cases (b) and (c) for construction of influence lines for bending moment and shear
at any sectionk, respectively.

k

X=Rk=1

a

k

k

X=M=1

b

X=Q=1

X=Q

k

k

c

Fig. 10.16 Elimination of constraint and replacing it by corresponding forceXD 1

In case (a) a support is eliminated and unit reaction is applied. In case (b) we

introduce hingek. Now both parts of the beam can rotate at sectionkrespect to

each other, but relative horizontal and relative vertical displacements are absent. In

case (c) we introduce special device thatallows for the displacements of each part

in vertical direction, but the relative horizontal and relative angular displacements

are absent.

Figure10.17a presents two-span uniform continuous beam. The required bending

moment at support 1 is considered as a primary unknownX 1. The primary system

presents two separate simply supported beams (Fig.10.17b). Equation of influence

line forX 1 is described by (10.24).

P= 1

1345268910711

ll

a

P= (^1) X 1 Primary system
d 1 P d 1 P
P= 1
Elastic curve due to P=1
b M=X 1
X 1 =1
dP (^1) Elastic curve due to X dP 1
1 =1
c X
1 =1
dP 1
x
y
R=1/l Left span
d
Fig. 10.17 Design diagram, primary system and illustration of reciprocal displacement theorem