Advanced Methods of Structural Analysis

292 8 The Displacement Method

Ta b l e 8. 4 (continued)

Meaning of

equations

To t a ldisplacementin the

direction ofeliminated

constraints caused by the action

of all primary unknowns

(forces or moments)and

applied forces is zero

To t a lreactionin the direction of

introducedconstraints caused

by the action of all primary

unknowns(linear or angular

displacements)and applied

forces is zero

Character of

canonical

equations

Kinematical:the left part of

canonical equations represents

displacements

Statical:the left part of canonical

equations representsreactions

Matrix of

coefficients

of canonical

equations

AD 2

(^66)
4
ı 11 ı 12 ı1n
ı 21 ı 22 ı2n
   
ın1ın2ınn
3
(^77)
5 ,detA>0,
whereAis theflexibility
matrix
RD 2
(^66)
4
r 11 r 12 r1n
r 21 r 22 r2n
   
rn1rn2rnn
3
(^77)
5 ,detR>0,
whereRis thestiffnessmatrix
Meaning of unit
coefficients
Unitdisplacementıikpresents
displacementin the direction
ofitheliminatedconstraints
due to primary unknown
(force)XkD 1
Unitreactionrikpresentsreaction
in theithintroduced
constraints due to primary
unknown(displacement)
ZkD 1
Meaning of free
terms
DisplacementiPpresents
displacementin the direction
ofitheliminatedconstraint
due to applied forces
ReactionRiPpresentsreactionin
theithintroducedconstraint
due to applied forces
Dimensions of
unit
coefficients
ıik– Dimension ofdisplacement
atiis divided by dimension of
action(force or moment)atk
rik– Dimension offorceatiis
divided by dimension of action
(linear or angular
displacement)atk
We are providing only the classical approach, however, an experienced reader may choose to des-
ignate the primary system of the force method as statically indeterminate, providing that he/she
has all the necessary formulas for calculating the accepted statically indeterminate primary system

8.3.1 Properties of Canonical Equations...........................

1.The main coefficients of the canonical equations of the force and displacement
methods are strictly positive:ıii>0;rii>0.
2.The matrix of coefficients of the canonical equations is symmetrical with respect
to the main diagonal:ıikDıki;rikDrki. These coefficients may be positive,
negative, or zero.
3.The coefficients of the canonical equations depend only on the type of struc-
ture, but do not depend on external loads, settlements of supports, temperature
changes, or errors of fabrication.