296 8 The Displacement Method
Shear within all vertical members isQD.M 0 CM 0 =h/DP=3. The reaction
of all supportsR 0 DQDP=3. Figure8.10f shows the final deflection curve of the
frame and inflection pointof all vertical members.
Note that a frame with an absolutely rigid crossbar has only one unknown of the
displacement method for any number of vertical members.
8.5 Special Types of Exposures............................................
The displacement method in canonical formcan be effectively applied for analy-
sis of statically indeterminate frames subjected to special types of exposures such
as settlements of supports and errors of fabrication. For both types of problems, a
primary system of the displacement method should be constructed as usual.
8.5.1 Settlements of Supports......................................
For a structure withndegrees of kinematical indeterminacy, the canonical equa-
tions are
r 11 Z 1 Cr 12 Z 2 CCr1nZnCR 1 sD 0
r 21 Z 1 Cr 22 Z 2 CCr2nZnCR 2 sD 0
(8.6)
rn1Z 1 Crn2Z 2 CCr2nZnCRnsD0;where the free termsRjs(jD1;2;:::;n) represent the reaction in thejth intro-
duced constraint in the primary system due to thesettlements of a support.These
terms are calculated using TablesA.3–A.6, taking into account theactualdisplace-
ments of the support. Unit reactionsrikshould be calculated as before. The final
bending moment diagram is constructed by the formula
MsDMN 1 Z 1 CMN 2 Z 2 CCMN 2 Z 2 CMs^0 ; (8.7)whereMs^0 is the bending moment diagram in the primary system caused by the
givensettlements of the support.
Example 8.3.The redundant frame in Fig.8.11is subjected to the following set-
tlement of fixed supportA:aD 2 cm,bD 1 cm, and'D0:01radD 3403000.
Construct the bending moment diagram.