Advanced Methods of Structural Analysis

254 7 The Force Method

The negative sign means that bending moment diagram is plotted on the more cold
fibers.
Canonical equation becomes

l

3 EI

X 1 

al

2h

.t 1 t 2 /D0:

So, the bending moment at the clamped support isX 1 DM 1 D.3aEI=2h/.t 1 t 2 /.
Final bending moment diagramMtis constructed on the basis of (7.20)and
shown in Fig.7.19d.
Kinematical control leads to the following result
Z
MtMN 1
EI

dsC1tD

1

EI



1

2

 1 l

„ ƒ‚ ...

̋



2

3



3aEI

2h

.t 1 t 2 /

„ ƒ‚ ...

yc



al

2h

.t 1 t 2 /D0:

Discussion:Let the height of rectangular cross section of the beam is increased
byntimes. In this case, the bending moment at clamped support increases byn^2
because

M 1 D

3aE

2.nh/

b.nh/^3

12

tD

3aEI

2h

tn^2 :

Example 7.4.Design diagram of the redundant frame is same as in Example 7.2.
Stiffness for vertical and horizontal members are 1EIand 2EI, respectively. Heights
of the cross sectionhfor vertical and horizontal members are 0.60 m. The frame is
subjected to temperature changes as presented in Fig.7.20a. Construct the internal
force diagrams and calculate reactions of supports.

Solution.Let us accept a primary system as shown in Fig.7.20b.

Canonical equations of the force method are

ı 11 X 1 Cı 12 X 2 C1tD0;

ı 21 X 1 Cı 22 X 2 C2tD0; (a)

whereıikare unit displacements and1tand2tare displacements in a primary
system in the direction of primary unknownsX 1 andX 2 , respectively, due to change
of temperature.
The unit states and corresponding bending moment and axial force diagrams
caused by unit primary unknowns are presented in Fig.7.20c. All normal forces due
toXN 2 D 1 are zero, i.e.,NN 2 D 0.
The unit displacementsıikfor this primary system have been obtained in
Example 7.2; they are