Advanced Methods of Structural Analysis

94 4 Three-Hinged Arches

4.4 Nil Point Method for Construction of Influence Lines

Each influence line shown in Figs.4.8–4.10has the specified points labeled as./.
These points are called nil (or neutral) point of corresponding influence line. Such
points of influence lines indicate a position of the concentrated load on the arch, so
internal forcesM,Q,andNin the given sectionkwould be zero. Nil points may be
used for simple procedure for construction of influence lines for internal forces and
checking the influencelines which were constructed by analytical approach. This
procedure for symmetrical three-hinged arch of spanlis discussed below.

4.4.1 Bending Moment.............................................

Step 1.Find nil point (NP) of influence lineMk.IfloadPis located on the left half
of the arch, then reaction of the supportBpass through crownC. Bend-
ing moment at sectionkequals zero, if reaction of supportApass through
the pointk. Therefore, NP.Mk/is point of intersection of lineBCandAk
(theorem about three concurrent forces). The nil point./is always located
between the crownCand sectionk(Fig.4.11).
Step 2.Lay off along the vertical passing through the supportA, the abscissa of
sectionk, i.e.,xk.
Step 3.Connect this ordinate with nil point and continue this line till a vertical
passing through crownCand then connect this point with supportB.
Step 4.Take into account indirect load application; connecting line between joints
2 and 3 is not shown.

Inf. line Mk

xk

+

uM * −

A

C

B

1

3 5

2

k

7

6

xk

P= 1

NP(Mk)

l 2

yk f

l

RA RB

Fig. 4.11 Construction of Influence lineMkusing nil point method