212 7 The Force Method
system (principal or released structure)is such structure, which is obtained from
the given one by eliminating redundant constraints and replacing them by primary
unknowns.
Let us consider some statically indeterminate structures, the versions of primary
systems, and the corresponding primary unknowns. A two-span beam is presented
in Fig.7.1a. The total number of constraints, and as result, the number of unknown
reactions, is four. For determination of reactions of this planar set of forces, only
three equilibrium equations may be written.Therefore, the degree of redundancy is
nD 4 3 D 1 , where four is a total number of reactions, while three is a number
of equilibrium equations for given structure. In other words, this structure has one
redundant constraint or statical indeterminacy of the first degree.
X 1cX 1a bd e fHARA RB RCABCX 1X (^1) X
1 Wrong primary system
Fig. 7.1 (a) Design diagram of a beam; (b–e) The different versions of the primary system;
(f) Wrong primary system
Four versions of the primary system and corresponding primary unknowns are
showninFig.7.1b–e. The primary unknownX 1 in cases (b) and (c) are reaction
of supportBandC, respectively. The primary unknown in cases (d) and (e) are
bending moments. In case (d), the primary unknown is the bending moment at any
point in the span, while in case (e), the primary unknown is the bending moment at
the supportB. Each of the primary systems is geometrically unchangeable and stat-
ically determinate; the structure in Fig.7.1d is a Gerber–Semikolenov beam; in case
(e), the primary system is a set of simply supported statically determinate beams.
The constraint which prevents thehorizontal displacement at supportAcannot be
considered as redundant one. Its elimination leads to the beam on the three parallel
constrains, i.e., to the geometrically changeable system. So the structure in Fig.7.1f
cannot be considered as primary system. The first condition for the primary system –
geometrically unchangeable – is the necessary condition. The second condition –
statical determinacy – is not a necessary demand; however, in this book we will
consider only statically determinate primary systems.
The statically indeterminate frame is presented in Fig.7.2a. The degree of redun-
dancy isnD 4 3 D 1. The structure in Fig.7.2b presents a possible version of
the primary system. Indeed, the constraintwhich prevents horizontal displacement
at the right support is not a necessary one inorder to provide geometrical unchange-
ability of a structure (i.e., it is a redundant constraint) and it may be eliminated,
so the primary unknown presents the horizontal reaction of support. Other version
of the primary system is shown in Fig.7.2c; in this case the primary unknown
presents the bending moment at the rigid joint.