Advanced Methods of Structural Analysis

282 8 The Displacement Method

fibers, and the bending moment diagram caused by the unit primary unknown are
shown in Fig.8.6b. The bending moment at the fixed support for a fixed-pinned
beam is 3 EI=l(TableA.3). The free-body diagram of joint 1 from diagramMN 1 is
shown in Fig.8.6c. According to the elastic curve, the extended fibers in the vicinity
of joint 1 are located above the neutral line to the left of point 1 and below the neutral
line to the right of point 1. The moments0:375EIand0:3EIare shown according
to the location of the extended fibers. Unit reactive momentr 11 is shown assuming
its positive direction (clockwise). Equilibrium condition

P
M D 0 leads to unit
reactionr 11 D0:675EI.kN m=rad/.
To calculate free termR1Pof the canonical equation, we need to construct the
bending moment diagram in the primary system caused by the given load. This
diagram is shown in Fig.8.6d; each element is considered as a separate beam; the
location of the extended fibers is shown by the dashed line. The extended fibers in
the vicinity of joint 1 are located above the neutral line to the left and right of joint

1. The bending moment at the fixed support for the left span subjected to uniformly
   distributed loadq, according to TableA.3, equals

M1A^0 D

ql 12

8

D

q 82

8

D16 .kN m/:

The bending moment at the specified points for the right span subjected to con-
centrated forceP, according to TableA.3, equals

M1B^0 D

Pl 2

2





1 ^2



D

12  10

2

0:4



1 0:4^2



D20:16 .kN m/;

Mk^0 D

Pl 2

2

u^2 .3u/D

12  10

2

0:6^2 0:4 .30:4/D20:736 .kN m/:

The free-body diagram of joint 1 from diagramMP^0 is shown in Fig.8.6e. Ac-
cording to the location of the extended fibers, the moment of 16 kN m is shown
to be counterclockwise and moment of 20.16 kN m is clockwise. Reactive moment
RP1Pis assumed to have a positive direction,i.e., clockwise. Equilibrium condition
MD 0 leads toR1PD4:16kN m.
Canonical equation (a) becomes0:675EIZ 1 4:16D 0. The root of this equa-
tion, i.e., the primary unknown is

Z 1 D

6:163

EI

.rad/: (b)

The bending moments at the specified points can be calculated by the following
formula

MPDMN 1 Z 1 CMP^0 : (c)