214 7 The Force Method
More complicated statically indeterminate frame is presented in Fig.7.4a. This
structure contains three support bars andone closed contour. All reactions of sup-
ports may be determined using only equilibrium equations, while the internal forces
in the members of the closed contour cannotbe obtained using equilibrium equa-
tions. Thus, this structure isexternallystatically determinate andinternallystatically
indeterminate. The degree of redundancy isnD 3. The primary unknowns are in-
ternal forces as shown in Fig.7.4b. They are axial forceX 1 , shearX 2 , and bending
momentX 3.
X 1 X 1
X 2 X 2X 4X 5X (^3) X 3 X 3 X 3
abX 2 c dX 2
Fig. 7.4 (a,b) Internally statically indeterminate structure and primary system and (c,d)exter-
nally and internally statically indeterminate structure and primary system
The frame in Fig.7.4c contains five support bars and one closed contour. So this
structure is externally statically indeterminate in the second degree and internally
statically indeterminate in the third degree. A total statical indeterminacy is five.
Primary unknowns may be chosen as shown in Fig.7.4d.
Degree of statical indeterminacy of structures does not depend on a load. It is
evident that inclusion of each redundant constraint increases the rigidity of a struc-
ture. So the displacements of statically indeterminate structures are less than the
displacements of corresponding structures without redundant constraints.
The different forms of the force method are considered in this chapter. However,
first of all we will consider the superposition principle, which is the fundamental
basis for the analysis of any statically indeterminate structure. The idea of analysis
of a statically indeterminate structure using the superposition principle is presented
below.
7.1.2 Compatibility Equation in Simplest Case
Two-span beam subjected to arbitrary loadPis shown in Fig.7.5. This structure
is statically indeterminate to the first degree. Two versions of primary system are
shown in Table7.1.
Assume that the middle supportBis the redundant one (Version 1 of the primary
system). Thus the reaction of this constraint, i.e.,RBDX, is a primary unknown.
For the given structure, the displacement at pointBis zero. The primary unknown