Advanced Methods of Structural Analysis

288 8 The Displacement Method

The canonical equations of the displacement method are

r 11 Z 1 Cr 12 Z 2 CR1PD0;

r 21 Z 1 Cr 22 Z 2 CR2PD0:

To calculate unit reactionsrik, it is necessary to construct in primary system the
unit bending moment diagramsMN 1 ,MN 2. To construct unit bending moment diagram
MN 1 , it is necessary torotateintroduced constraint 1 by angleZ 1 D 1 clockwise. To
construct bending moment diagramMN 2 , it is necessary toshiftintroduced constraint
2 to the right by distanceZ 2 D 1. The bending moment diagrams for states 1 and 2
as well as the free-body diagrams for the calculation of unit reactionsrikare shown
in Fig.8.8d,e. The positive reactions are shown by dashed arrows.
To calculate free termsRiP, it is necessary to construct in primary system the
diagramMP^0 caused by the applied load (loaded state). This state and corresponding
bending moment diagram-is shown in Fig.8.8f.
The ordinates of the bending moment diagrams for standard uniform elements
due to different loads are presented in TablesA.3–A.6. The ordinates of theMP^0 dia-
gram at the specified sections according to TableA.4(for element 1-3) and TableA.3
(for element 6-8) are

M 1 DM 3 D

ql 12 - 3

12

D4:1667kN mI

M 2 D

ql 12 - 3

24

D2:0833kN mI

M 6 D

Pl

2





1 ^2



D

8  10

2

0:6



1 0:6^2



D15:36kN mI

M 7 D

Pl

2

u^2 .3u/D

8  10

2

0:4^2 0:6 .30:4/D9:984kN m:

All the bending moment diagrams are plotted on the extended fibers of the frame
(Fig.8.8d–f). Elastic curves are shown by dashed lines. The asterisks ()onthe
elastic curves show the points of inflection.
To calculate reactive momentsr 11 ,r 12 ,andR1P, it is necessary to consider the
free-body diagrams of joint 1 usingMN 1 ,MN 2 ,andMP^0 diagrams. The ordinates of
the bending moments infinitely close to the joint are taken from the corresponding
diagrams. The direction of each of these moments must correspond to the location of
the extended fibers in each diagram.The calculations of unit reactionsr 11 ,r 12 ,and
R1Pare presented in Table8.1. Positive reactive moments are directed clockwise;
locations of extended fibers are shown by dashed lines.
To calculate reactiveforces r 21 ,r 22 ,andR2P, it is necessary to consider the
equilibrium of horizontal member 6-8. For this we need to cut off element 6-8 from
diagramsMN 1 ,MN 2 ,andMP^0 by sections infinitely close to joint 1 from above and
below. The calculations of unit reactionsr 21 ,r 22 , and loaded reactionR2Pare pre-
sented in Table8.2.