Advanced Methods of Structural Analysis

350 10 Influence Lines Method

Table 10.5 Calculation of IL



Q^0 k



LoadPD 1 is located to the left of

sectionk,.u 0:6/

LoadPD 1 is located to the right

of sectionk.u 0:6/

Point u Q^0 kDRP Point u Qk^0 DR

1 1 IL.Qk/D

u^2

2

. 3 u/ 1 0 k; 3 0.6 IL

Q^0 k



D

u^2

2

.3u/ 0.432

2 0.8 0.296 4 0.4 0.208

k,3 0.6 0.568 5 0.2 0.056

6 0 0

LoadP D 1 in the left span Shear force at sectionkdepends on the location of

the loadPD 1 with respect to sectionk(to the left or to the right). Calculation of

shear at sectionkis presented in Table10.5.

LoadPD 1 is in the right span In this case the shear at section k does not arise,

and influence line has zeros ordinates. The final influence line for shearQkis con-

structed using the expression (10.23). This influence line is presented in Fig.10.13c.

The same result had been obtained early by the force method.

Final

Inf. line Qk

a

3 EIIL(Z

IL (Qk) = − l 2 1 )+ IL(Qk^0 )

+ Inf. line Qk 0

−

0.296 0.568

0.432

0.2080.056

−^3 EI⋅IL(Z 1 )

− l 2

+

0.0480.0840.0960.072

0.0720.0960.0840.048

0.516

0.248 0.4840.3040.128

+ 0.0720.0960.084

−

0.048

11

P = 1

13452689107

k

c

b

Fig. 10.13 (a–c) Construction of influence line for shearQk

Discussion

1.Influence lines forMk^0 andQk^0 for the force method are bounded by thestraight
lines, because the primary system is a set of staticallydeterminatesimply sup-
ported beams. The same influence lines in the displacement method are bounded
by thecurvedlines, because the primary system is a set of staticallyindetermi-
natebeams.