Advanced Methods of Structural Analysis

6.4 Displacement Due to Settlement of Supports and Errors ofFabrication 173

Actual state

A¢

C B

A K

5m

3m

4m 6m 2m

Δ 1

q

Δ

Δ 2

ab

Unit state

R X=1

B

RC

HC

H¢C

RA

A

CB

MA K

HA

R′C

Fig. 6.16 (a) Hinged structure and settlements of supportA;(b) Interaction diagram and unit state

Solution.1.Apply the unit dimensionless horizontal forceXD 1 corresponding
to the required horizontal displacement at pointK.
2.Now we have to calculate reactions at the settled supportA. Since supportA
has the horizontal and vertical componentof the settlements as well as the angle
of rotation, i.e., 1 , 2 ,and, then we need to calculate reactions in these
directions, i.e.,HA,RA,andMA.
For secondary structureCBKthe following reactions arises:

RB!

X

MCD 0 W X 3 RB 6 D 0 !RBD0:5;

RCDRBD0:5I HCDPD1:

ReactionsHCandRCfrom the secondary structure are transmitted on the pri-

mary structureACas active forcesHC^0 andRC^0. Reactions, which arise at support

A,are

HAD 1 I RADR^0 CD0:5I MADR^0 C 4 CHC^0  5 D7:

All reactions are dimensionless, while theMAis measured in the unit of length;
in our caseMAD 7 m. All these reactions are shown in real direction which
corresponds to direction ofXD 1.
3.The principle of virtual displacements is

X

ıWactD0: (a)

The work done by all active forces on the displacements, which are compatible

with constraints, is zero, i.e.,

XCHA 1 RA 2 MAD0: (b)

SinceXD 1 ,then

DHA 1 CRA 2 CMA: (c)