6.4 Displacement Due to Settlement of Supports and Errors ofFabrication 175
However,XD 1 , and therefore the required displacement, according to (6.18)
equals
CDD
X
RdD.RA 1 RB 2 /: (b)
For giveni.iD1; 2/, the angle of rotation of barCDbecomes
CDD0:250:02C0:250:03D0:0025rad
Positive sign of the required displacement means that the adopted clockwise cou-
pleXwithin the real angular displacement produce the positive work. It is obvious,
that the horizontal displacement at supportAdoes not affect on the angle of rotation
of any bar, since horizontal reaction at pointAdue to applied unit couple is zero.
Deflections of the structural members may occur as a result of the geometric
misfit. This topic is sometimes referred to be the name geometric incompatibility.
The following procedure may be applied for this type of problems:
1.At the pointKwhere displacement should be determined we need to apply unit
generalized forceXD 1 corresponding to the required displacement
2.Compute all reactions caused by unit generalized forceXD 1
3.Calculate the work done by these reactions on the displacements
Example 6.13.The tieABof the archACBin Fig.6.18isD0:02m longer then
required length l. Find the vertical displacement at pointC,iflD 48 m,fD 6 m.
A
C
f
Actual state
C ¢
B ¢
l
B
DC
D
DC
D
A
C
Unit state X=1
H B
RA l RB
Fig. 6.18 Design diagram of the arch (error fabrication) and unit state
Solution.The actual position of the tie isAB^0 instead of projectABposition. For
computation of the vertical displacementC we have to apply unit vertical force
atC. Reactions of the three-hinged arch and thrust in tie caused by forcePD 1
equalsRADRBD0:5,HDMC^0 =fDl=4fD 2.
Application of principle of virtual displacements leads to the following
expression
XCHD0:
SinceXD 1 , then the required displacement becomes
CDCHDC0:04m .downward/: