102 4 Three-Hinged Arches
Note, that the discontinuity of the shear and normal forces at section E left and right
at the vertical member EF areNEFcos'andNEFsin', respectively.
4.5.2.2 Influence Lines for Thrust and Bending Moment at the Section k
Vertical reactions Influence lines for vertical reactionsRAandRBfor arch and for
reference simply supported beam coincide, i.e.,
IL.RA/DIL
R^0 A
I IL.RB/DIL
R^0 B
:Thrust According to expression (4.20), the equation of influence line for thrust be-
comes
IL.H /D1
ff 0IL
MC^0
: (4.22)The maximum ordinate of influence line forHat crownC
1
.ff 0 /l
4D48
4 .122/D1:2: (c)Influence line for thrust may be considered as key influence line.Bending moment According to expression (4.21) for bending moment at any sec-
tion, the equation of influence line for bending moment at sectionkbecomes
IL.Mk/DIL
Mk^0
.ykf 0 /IL.H /DIL
Mk^0
9:25IL.H / : (4.23)Influence line Mk^0 presents a triangle with maximum ordinate
.akbk/= lD.1830/=48D 11:25m at sectionk, so the ordinate at crownC
equals to 9 m. Influence line for thrustH presents the triangle with maximum
ordinate 1.2 at crownC. Ordinate of the graph.ykf 0 /IL.H /at crownC
equals.11:252/1:2D11:1m, so ordinate at sectionkequals 8.325 m. De-
tailed construction of influence lineMkisshowninFig.4.17. Since both terms in
(4.23)hasdifferentsigns, they should be plotted on theone sideon the basic line;
the final ordinates of influence line are locatedbetweentwo graphs IL
Mk^0
and
9:25IL.H /.
Maximum bending moment at sectionkoccurs if loadPis located above section
kand crownC. Bending moment at sectionkmay be positive, negative, and zero.
If loadPis located within the portionA-NP.Mk/, then extended fibers at section
kare located below neutral line of the arch.
Figure.4.17also presents the construction of influence lines for bending moment
using nil points; pay attention that construction of this point must be done on the
basis of conventional supportsA^0 andB^0.