8.2 Canonical Equations of Displacement Method 277
introduced constraints. The canonical equations of the displacement method will be
written as follows:
r 11 Z 1 Cr 12 Z 2 CCr1nZnCR1PD 0
r 21 Z 1 Cr 22 Z 2 CCr2nZnCR2PD 0
(8.4)
rn1Z 1 Crn2Z 2 CCrnnZnCRnPD0:The number of canonical equations is equal to number of primary unknowns of the
displacement method.
Interpretation of the canonical equations Coefficientrikrepresents the unit re-
action, i.e., the reaction (force or moment), which arises in theith introduced
constraint (first letter in subscript) caused byunitdisplacementZkD 1 ofkth intro-
duced constraint (second letter in subscript). The termrikZkrepresents the reaction,
whicharisesintheith introduced constraint due to the action ofrealunknown dis-
placementZk.FreetermRiPis the reaction in theith introduced constraint due to
the action of the applied loads. Thus, the left part of theith equation represents a
total reaction, which arises in theith introduced constraint due to the actions of all
real unknownsZas well as the applied load.
The total reaction in each introduced constraint in the primary system caused by
all primary unknowns (the linear and angulardisplacements of the introduced con-
straints) and the applied loads is equal to zero. In this case, the difference between
the given structure and the primary systemvanishes, or in other words, the behavior
of the given and the primary systems is the same.
8.2.2 Calculation of Unit Reactions................................
The frame presented in Fig.8.5a allows angular displacement of the rigid joint and
horizontal displacement of the crossbar. So the structure is twice kinematically inde-
terminate. The primary system of the displacement method is presented in Fig.8.5b.
Constraints 1 and 2 are additional introduced constraints that prevent angular and
linear displacements. In constraint 1, which prevents angular displacement, only the
reactive moment arises; in constraint 2,which prevents only linear displacement,
only the reactive force arises. Thecorresponding canonical equations are
r 11 Z 1 Cr 12 Z 2 CR1PD0;
r 21 Z 1 Cr 22 Z 2 CR2PD0:To determine coefficientsrikof these equations, we consider two states. State 1
presents the primary system subjected to unit angular displacementZ 1 D 1. State 2
presents the primary system subjected to unit horizontal displacementZ 2 D 1 .For
both states, we will show the bending moment diagrams. These diagrams caused by
the unit displacements of introduced constraintsZ 1 D 1 andZ 2 D 1 are shown in
Fig.8.5c, d, respectively. The elastic curves are shown by dashed lines; the asterisk