Advanced Methods of Structural Analysis

7.1 Fundamental Idea of the Force Method 213

X 1

X 1

X 1

a b cd

Fig. 7.2 (a) Design diagram of a portal frame; (b–c) The different versions of the primary system;
(d) Wrong primary system

The system shown in Fig.7.2d is geometrically changeable because three remain-
ing support bars would be incapable of preventing rotation of the frame with respect
to the left pinned support. Indeed, the constraint which prevents vertical displace-
ment at the right support is anecessaryconstraint in order to provide a geometrical
unchangeability of a structure (i.e., it isnot a redundant one). It means that the sys-
tem shown in Fig.7.2d cannot be accepted as a version of primary system.
Another statically indeterminate frame is presented in Fig.7.3a. The degree of
redundancy isnD 6  3 D 3.

Closed

contour

X (^3) X X 1
3
X 2
X 1
X 1 X 2
X 3
X 2
X 1
X 2
X 3
X 2
ab
cdef
Fig. 7.3 (a) Statically indeterminate frame; (b–e) The different versions of the primary system;
(f) A concept of closed contour
One version of the primary system and corresponding primary unknowns is
showninFig.7.3b. The primary unknowns are reactions of support. The structure
shown in Fig.7.3c presents the primary system where primary unknowns are in-
ternal forces (axial forceX 1 , shearX 2 , and momentX 3 /, which appear in pairs.
Figure 7.3d presents another version of the primary system. In this case, we elim-
inate two constraints which prevent twodisplacements (horizontal and angular) at
support and one constraint which preventsmutualangular displacement, i.e., the
primary unknowns are a combination of reactionsX 1 andX 2 and internal moment
X 3. Is it obvious that three-hinged frame (Fig.7.3e) can be adopted as the primary
system.
The structure shown in Fig.7.3a can also be considered as a system with
closed contour (Fig.7.3f). One closed contour has three degrees of redundancy,
and primary system can be similar as in Fig.7.3c.