Advanced Methods of Structural Analysis

2.1 Analytical Method for Construction of Influence Lines 23

LoadPD 1 left at sectionk LoadPD 1 right at sectionk

Qk!

P

YrightD 0 , Qk!

P

YleftD 0 ,

QkDRB!IL.Qk/DIL.RB/ QkDRA!IL.Qk/DIL.RA/

Using these expressions, we can trace theleft-handportion of the influence line

for shear as the influence line for reactionRBwith negative sign andright-hand

portion of the influence line for shear as influence line for reactionRA(Fig.2.5).

Units of ordinate IL.Qk/are dimensionless.

Fig. 2.5 Simply supported

beam. Construction of

influence line for shear at

sectionk

P= 1

x

RA l RB

k

ab

AB

−IL(RB)

+

1

1

Inf. line Qk

−

IL(RA)

Operating range 1

−

−IL(RB)

Load P=1 left at section k

Operating range

+

1

IL(RA)

Load P=1 right at section k

In order to construct the influence line for shear at sectionkthe following proce-

dure should be applied:

1.Plot ordinateC 1 (upward) and 1 (downward) along the vertical lines passing
through the left-hand and right-hand support, respectively
2.Join each of these points with base point at the other support
3.Connect both portions at sectionk
4.Show the operating range of influence line: operating range of left-hand portion
is negative and operating range of right-hand portion is positive
The negative sign of the left-hand portion and jump at sectionkmay be explained
as follows: If the loadPD 1 is located on the left part of the beam, then shear
QkDRAP<0. When loadPis infinitely close to the sectionkto the left,
then shearQkDRB. As soon as the loadPD 1 moves over sectionk, then shear
QkDRA.
It is obvious, that the influence line for shear for the section which is infinitely
close to supportAcoincides with influence line for reactionRA,i.e.,IL.Q/D
IL.RA/. If section is infinitely close to supportB, then the influence line for
shear coincides with influence line for reactionRB with negative sign, i.e.,
IL.Q/DIL.RB/.