Advanced Methods of Structural Analysis

8.5 Special Types of Exposures 301

The bending moment diagram in unit state is presented in Fig.8.12c. The loaded
state (Fig.8.12d) presents-displacement of jointB. The memberABis not sub-
jected to bending; corresponding elastic curve is shown by a dashed line. The
momentR 1 eis a reactive moment at the introduced constraint caused by displace-
ment.TableA.3presents the reactions caused by theunitdisplacement, therefore
the specified ordinate of the bending moment diagram is



3 EI=h^2



.

It is obvious that

r 11 D

4 EI

l

C

3 EI

h

;R 1 eD

3 EI

h^2

:

So the angle of rotation of jointBbecomes

Z 1 D

3



4h^2

l C3h

D

3 0:008m

4 2:8

2

5 .m/C^3 2:8 .m/

D0:001635rad:

The resulting bending moments at the specified points are

MBAD

4 EI

l

Z 1 D

4  19979 kN m^2

5:0m

0:001635radD26:1kN mI

MABD0:5MBAD13:05kN mI

MBCD

3 EI

h

Z 1 

3 EI

h^2

D

3  19979 kN m^2

2:8m

0:001635rad



3  19979 kN m^2

2:8^2 m^2

0:008mD26:1kN m:

The corresponding bending moment diagram is presented in Fig.8.12e.
Construction of shear and axial forces and computation of reactions of supports
should be performed as usual.

Discussion. We can see that an insignificant error of fabrication leads to significant
internal forces in the structure. This fact lays the basis of the inverse problem, which
can be formulated as follows: find the initial displacement of specified points of a
structure for obtaining the required distribution of internal forces. For example, let
us consider the two-span continuous beam subjected to a uniformly distributed load
shown in Fig.8.13.

ll

q

D

Fig. 8.13 Controlling of the stresses in the beam by initial displacement of middle support