8.1 Fundamental Idea of the Displacement Method 273
ab513 5(^24)
j 2 j 2 j 4
1
3
2 4
Hinged
scheme
1
3 5
2 4
Δ 1 Δ 2 Δ 4
1
3 5
2 4
Hinged
scheme
Fig. 8.1 (a,b) Design diagrams of frames, deflected shapes and hinged schemes
Hinged scheme
Design
diagram
A
D
C
B
E
C*
ab
cd
Fig. 8.2 Calculation of independent linear joints displacements
Figure8.1b presents the same frame, but with amovablesupport 1. The number
of unknown angles of rotation of rigid joints is, as before,nrD 2 ; angular displace-
ments' 2 and' 4 are not shown. In the case of rolled support 1 the structure has the
linear displacementsof joints 1, 2, and 4 as well. According to conditions 1 and 2
of the assumptions, these linear displacementsare the same for all joints. So the
number of independent linear displacements isndD 1. Indeed, the hinged structure
presents a geometricallychangeable system because support 1 is movable. To ob-
tain a geometrically unchangeable structure, only one additional member needs to
be introduced into this system (as shown by line 3-4), thusndD 1. Therefore, the
total degree of kinematical indeterminacy equals three. The unknowns of the dis-
placement method are angular displacements' 2 and' 4 and lateral displacement.
Some statically indeterminate frames and their hinged schemes are presented in
Fig.8.2. In case (a), only one additional member (shown by the dashed line) trans-
forms the hinged system into a geometrically unchangeable structure, sondD 1 ;the