Advanced Methods of Structural Analysis

4.5 Special Types of Arches 99

The total vertical reactions may be defined as follows

RADR^0 ACZsin ̨D2:857C11:4680:08304D3:809kN;

RBDR^0 BZsin ̨D7:14311:4680:08304D6:191kN: (4.16)

Bending moment at section k:

MkDMk^0 HyD3:809 6 11:4283:5D17:144kN: (4.17)

4.5.1.2 Influence Lines for Thrust and Bending MomentMk.

ThrustSinceH D



MC^0 =H


cos ̨, then equation of influence line for thrust
becomes

IL.H /D

cos ̨

h

IL



MC^0



: (4.18)

The maximum ordinate of influence line occurs at crownCand equals

cos ̨

h



aCbC

l

D

0:9965

5:979



24  18

24 C 18

D1:71428:

Bending momentSinceMk D Mk^0 Hyk, then equation of influence line for
bending moment at sectionkbecomes

IL.Mk/DIL



Mk^0



ykIL.H / : (4.19)

Influence line may be easily constructed using the nil point method. Equation of the
lineAkis

yD

3:5

6

xD0:5833x: (d)

Equation of the lineBCis

yyCDm.xxC/!y 8 D

4:5

18

.x24/!yD 14 0:25x; (e)

wheremis a slope of the lineBC.
The nil point NP.Mk/of influence line forMkis point of intersection of linesAk
andBC. Solution of equations (d) and (e) isx 0 D16:8m. Influence lines forHand
Mkare presented in Fig.4.15. Maximum positive and negative bending moment at
sectionkoccurs if loadPis located at sectionkand hingeC, respectively. If loadP
is located within portionx 0 , then extended fibers at the sectionkare located below
the neutral line of the arch.