Advanced Methods of Structural Analysis

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/: