7.2 Canonical Equations of Force Method 219
called the loaded term (loaded displacement, free term). The solution of the canoni-
cal equation allows us to calculate the primary unknownX 1 ,i.e.,X 1 D1P=ı 11.
General case of canonical equations. The canonical equations of the force
method for a statically indeterminate structure withnredundant constraints are
written as follows
ı 11 X 1 Cı 12 X 2 CCı1nXnC1PD 0
ı 21 X 1 Cı 22 X 2 CCı2nXnC2PD 0 (7.4)
ın1X 1 Cın2X 2 CCınnXnCnPD^0The form of presentation of the canonical equations as shown in (7.4)isalwaysthe
same; it does not depend on the type of a structure, its peculiarities, and type of
external exposures (forces, support settlements, temperature change, fabrication er-
ror). The numbernof these equations equals to the degree of statical indeterminacy
of a given structure.
All coefficientsıikof canonical equations represent adisplacementof the pri-
mary structure due tounitprimary unknowns; these coefficients are called theunit
displacements.
Coefficientıikis the displacement along the direction of unknownXi due to
action of unit unknownXk;termıikXkpresents displacement along the direction of
unknownXidue to action of real unknownXk. Coefficientsıik, which are located
on the principal diagonal (iDk) are called the principal (main) displacements. All
other displacementsıik(i¤k) are called the secondary unit displacements.
Free termiPpresents displacement along the direction of unknownXidue to
action of actual load in primary system. DisplacementsiPcaused by applied loads
are called the loaded terms or free terms.
Physical meaning of the canonical equations. The left part of theith equation
presents the total displacement along the direction of unknownXidue to action of
all real unknownsXkas well as applied load. Total displacement of the primary
structure in directions of eliminated restrictions caused by primary unknowns and
applied load equals zero. In this case, the difference between the given and primary
structures is vanished.
7.2.2 Calculation of Coefficients and Free Terms
of Canonical Equations
Computation of coefficients and free terms of canonical equations presents signif-
icant and very important part of analysis of any statically indeterminate structure.
For their calculation, any methods can be applied. The graph multiplication method
is best suited for beams and framed structures. For this, it is necessary in primary
system to construct bending moment diagramsMN 1 ;MN 2 ;:::;MNndue to unit primary
unknownsXi,iD1;:::;nand diagramMP^0 due to given load. Unit displacements
and loaded terms are calculated by formulas