Advanced Methods of Structural Analysis

276 8 The Displacement Method

moment arises and the second numeral isthe label of the constraint, which is
rotated (in this example, they are the same).
Sign rule. If constraint 1 is rotated clockwise andafter that it is released,then
this constraint tends to rotate back counterclockwise. Therefore, the reactive mo-
ment will act in an opposite direction, i.e., clockwise. Thus, the positive reaction
coincideswith the positive displacement of the introduced constraint. Figure8.4d
shows the positive directions for displacementZ 1 and unit reactionr 11.
4.The total reaction caused by rotation of the introduced constraint and the given
load isr 11 Z 1 CR1P, where the first termr 11 Z 1 is the reactive moment in con-
straint 1 due to the real angle of rotationZ 1. The second term represents the
reactive moment in constraint 1 due to the actual load. Lowercase letterrmeans
that this reaction is caused by theunitrotation while capital letterRmeans that
this reaction is caused by therealexternal load. If this total reactive moment
(due to both the given load and the angular displacement of the included con-
straint) is equal to zero, then the behavior of the given and primary structures is
identical. This statement may be written in the following form:

r 11 Z 1 CR1PD0: (8.2)

Equation (8.2) presents thedisplacement method in canonical formfor a structure

of the first degree of kinematical indeterminacy.

The primary unknown is obtained asZ 1 DR1P=r 11. Knowing the primary

unknownZ 1 , we can consider each element of the frame to be a standard beam due

to the action of found angleZ 1 and the external load applied to this particular ele-

ment. Moments of supports for these standard beams are found in TablesA.3–A.6.

Using the principle of superposition, the final bending moment diagram for the

entire frame is constructed by the formula

MPDMN 1 Z 1 CMP^0 ; (8.3)

whereMN 1 is the bending moment diagram caused by the unit primary unknown;

termMN 1 Z 1 is the bending moment diagram caused by theactualprimary unknown

Z 1 ; and the bending moment diagram in the primary system caused by the given

load is denoted asMP^0.

8.2 Canonical Equationsof Displacement Method.......................

8.2.1 Compatibility Equations in General Case...................

Now we consider an arbitraryn-times kinematically indeterminate structure. The

primary unknownsZi.iD1;:::;n/are displacements (linear and/or angular) of