348 10 Influence Lines Method
If load is located on the right spanthen angular displacement at supportBis posi-
tive, i.e., this displacement occurs clockwise.
10.2.1.2 Influence Line for Bending MomentMk
This influence line should be constructed by formula (10.18). The bending mo-
ment in primary system at sectionkdue to primary unknownZ 1 D 1 is shown
in Fig.10.11c and equals to
MkD3 EI
l0:4D1:2EI
lIthe negative sign means that extended fibers at sectionkare located above the
neutral line. Therefore
IL.Mk/D1:2EI
lIL.Z 1 /CIL
Mk^0
: (10.22)The first term1:2 .EI=l/IL.Z 1 /of (10.22) is presented in Fig.10.12a. The second
term IL
Mk^0
presents the influence line of bending moment at sectionkin primary
system. For construction of this influence line we need to consider a loadPD 1
placed within the left and right spans (Fig.10.12b).
LoadPD 1 in the left span (Fig.10.12b). Reaction of the left support isRD
u^2 =2 .3u/(TableA.3). The bending moment at sectionkdepends on the location
of the loadPD 1 with respect to sectionk(Table10.4).
Table 10.4 Calculation ofIL
Mk^0LoadPD 1 is located to the left of
sectionk.u 0:6/
LoadPD 1 is located to the right
of sectionk.u 0:6/
Point u Mk^0 DR0:4lP.0:4ll/ Point u Mk^0 DR0:4l
1 1
u^2
2
. 3 u/0:4ll.0:4/D 0 k; 3 0.6
u^2
2
. 3 u/0:4l 0:1728l
2 0.8
u^2
2
. 3 u/0:4ll.u0:6/ 0:08161l 4 0.4 0:0832l
k,3 0.6 0:1728l 5 0.2 0:0224l
6 0 0
LoadPD 1 in the right span. In this case the bending moment at sectionkdoes
not arise (because the introduced constraint), and influence line has zeros ordinates.
Influence lineMk^0 isshowninFig.10.12b.
The final influence line for bending moment at sectionkis constructed using
expression (10.18) and is shown in Fig.10.12c. The same result had been obtained
early by the force method.