316 9 Mixed Method
9.2 Canonical Equations of the Mixed Method...........................
Compatibility conditions for structure in Fig.9.2may be presented in canoni-
cal form
ı 11 X 1 Cı^012 Z 2 C1PD0;
r^021 X 1 Cr 22 Z 2 CR2PD0:
(9.1)Thus the mixed method for this particular design diagram reduces the number of
unknowns to two.
The first equation means thatdisplacementin the direction of theeliminated
constraint 1, due to reaction of this constraint (primary unknownX 1 of the force
method), rotation of the introduced joint 2 (primary unknownZ 2 of the displace-
ment method), and applied load must be zero.
The second equation means thatreactionin theintroducedjoint 2 due to reaction
X 1 of eliminated constraint 1, rotationZ 2 of introduced joint 2 and applied load
must be zero.
9.2.1 The Matter of Unit Coefficients and Canonical Equations
These equations contain the coefficients, which belong to four different groups, i.e.:
ı 11 represents adisplacementdue to the unitforce
ı^012 represents adisplacementdue to the unitdisplacement
r 210 represents areactiondue to the unitforce
r 22 represents areactiondue to the unitdisplacement
So, the coefficientsı 11 andr 22 are unit coefficients of the classical force and dis-
placement method, respectively. The essence of the unit coefficientsı 120 andr 210
is different from the coefficientsı 11 andr 22 : coefficientı^012 is unitdisplacement
caused by the unknown of thedisplacementmethod, and coefficientr 210 is unitre-
actioncaused by the unknown of theforcemethod.
Each term of (9.1) has the following meaning:
First equation:
ı^011 X 1 Ddisplacementin the direction of the eliminated constraint (point A,
vertical direction) caused by the unknownforceX 1
ı^012 Z 2 Ddisplacementof pointAin the same direction caused by the unknown
angle of rotationZ 2
1P Ddisplacementof pointAin the same direction caused by theapplied
load.
Second equation:
r 210 X 1 Dreactionof the introduced constraint 2 due to the unknownforceX 1
r 22 Z 2 Dreactionof the same constraint 2 due to the unknownangle of rota-
tionZ 2
R2PDreactionof the same constraint 2 due to theapplied loads