Advanced Methods of Structural Analysis

6.4 Displacement Due to Settlement of Supports and Errors ofFabrication 171

According to this principle, the elementary work done by all the active forces on any

virtual displacement, which is compatible with constraints, is zero. This principle

may be applicable for structure with finite displacements. The forceRshould be

considered as active one and (6.16) becomes

XkCRD0:

SinceXD 1 ,then

kDR: (6.17)

The formula (6.17) may be generalized for case of displacements caused by settle-

ments ofseveralsupports

kdD

X

R; (6.18)

wherekdis the displacement inkth direction due to settlement of supports,is

the given settlement of support, andRis the reaction in the support which is settled;

this reaction caused by unit load whichcorresponds to the required displacement.

Summation covers all supports.

Procedure for computation of displacement caused by the settlement of support

is as follows:

1.At the pointKwhere displacement should be determined we need to apply unit
generalized forceXD 1 corresponding to the required displacement
2.To show reactionsRat the settled support, caused by unit generalized force
XD 1 and compute these reactions
3.Calculate the work done by these reactions on the displacements of the support;
multiply this result by negative unity
Discussion.Equation (6.18) reflects akinematicalcharacter of problem; it means
that displacements of any point of statically determinate (SD) structure are deter-
mined by the geometrical parameters of a structure without taking into account the
deformations of its elements. Any settlement of support of SD structure leads to
displacement of its separate parts as rigid discs. Displacements of any point of SD
structure caused by settlement of supports do not depend on the stiffness of the
structure.
Let us consider a procedure (6.18) for portal frame in more detail. The support
Bis moved onas shown in Fig.6.15a; it is necessary to calculate the angular
and linear displacements of the jointK(Fig.6.15c). Procedure (6.18)ispresented
in Table6.4. The first line of this table presents unit generalized forceXiD 1 which
corresponds to required displacements. Each unit force may be shown in arbitrary
direction. The second line of the table shows the reaction, which arise at the settled
support; each of these reactions is caused by unit generalized forceXiD 1.

Example 6.11.Two structures are hinged together atCas shown in Fig.6.16a.

Compute the horizontaldisplacement of pointKdue to the following settlements of

the supportA: 1 D0:02m, 2 D0:03m,D0:01rad (scales for displacements

 1 , 2 ,andand for dimensions of the structure are different).