Advanced Methods of Structural Analysis

92 4 Three-Hinged Arches

Step 4.Influence line between joints 2 and 3 presents a straight line; this connected

line is shown by solid line. Final influence line IL.Nk/is shown in

Fig.4.10.

4.3.2.4 Properties of the Influence Lines for Internal Forces

1.Influence line for bending moment has significantly less ordinates than for ref-
erence beam. This influence line containsthe positive and negative ordinates. It
means that at sectionkextended fibers can be located below or above the neutral
line depending on where the load is placed.
2.Influence line for shear, as in case of reference beam, has two portions with
positive and negative ordinates; all ordinates are significantly less than in the
reference beam. Influence line for axial force has only negative ordinates. So in
case of arbitrary load the axial forces in arch are always compressed.

4.3.3 Application of Influence Lines...............................

Assume that arch is subjected to fixed loads as shown in Fig.4.5. The reactions of

supports and internal forces caused by fixed load may be calculated by formula

ZD

X

PiyiC

X

qjj;

whereZis any force, for which the influence line is constructed;ythe ordinate of

influence line under the concentrated loadP;andis the area of influence line

graph under the distributed loadq.

Reactions of Supports

Ordinates of influence line forRAat the points of application the loadsP 1 andP 2

are 0.75 and 0.125, respectively. The area of the influence line under the uniformly

distributed load is

D

0:5C0:25

2

 8 D3:0:

Therefore, the reactionRADP 1 0:75Cq 3 CP 2 0:125D14:5kN

The thrustHof the arch, using influence line equals

HDP 1 0:5Cq

1 C0:5

2

 8 CP 2 0:25D 19 kN:

These values of reactions coincide with those computed previously in Sect. 4.2.1.