218 7 The Force Method
elimination of redundant constraints andreplacing them by reactions of these con-
strains. Primary unknownsXirepresent reactions (forces or moments) in eliminated
redundant constrains.
Let us consider a simple redundant structure, such as clamped-pinned beam. The
number of redundant constraints isnD 4 3 D 1. Assume that the right rolled
support is the redundant one. Thus the reaction of this constraint,X 1 ,isaprimary
unknown. Given and primary systems are shown in Fig.7.7.
X 1 =1d 11X 1d 11 X 1
Δ 1 PP
Given
structure ABPPAB
X 1Primary
systemFig. 7.7 Simple redundant structure. The idea of the force method and the concept of unit
displacement
The compatibility condition may bewritten in the following formyBDyB.P /CyB.X 1 /D0; (7.2)whereyB.P /is displacement of pointBin primary system due to given loadP,
andyB.X 1 /is displacement of pointBin primary system due to primary unknown
RBDX 1.
DisplacementyB.X 1 /caused by unknownX 1 may be presented as
yB.X 1 /Dı 11 X 1 ; (7.2a)whereı 11 presents the displacement in direction 1 (first index) caused by the force
X 1 D 1 (second index). Coefficientı 11 is called theunit displacementsince it is
caused byunitprimary unknownX 1 D 1 .Thetermı 11 X 1 presents the displacement
in the direction of the eliminated constraint 1 caused by theactualprimary unknown
X 1. If displacement in direction 1 caused by given loadyB.P /is denoted as1P,
then (7.2) may be rewritten in the following form
ı 11 X 1 C1PD0: (7.3)Left part of equation presents a total displacement in the direction of eliminated
constraint 1 (first index) caused by primary unknownX 1 and a given load. If this
total displacement is zero, then behaviorof both structures (entire structure sub-
jected to given load and primary structure subjected to given load as well as primary
unknownX 1 ) is identical.
The compatibility equation in form (7.4) is called thecanonical equation of the
force methodfor any structure with one redundant constraint; the free term1Pis