Advanced Methods of Structural Analysis

7.6 Temperature Changes 253

2.Construct the unit bending moment and axial force diagrams

3.Calculate the unit displacements. In case of bending structures take into account

only bending moments

4.Calculate the free terms of canonical equations. For this:

(a)Calculate the average temperature and temperature gradient for each member

of a structure

(b)Apply formula (7.18a)

5.Solve the canonical equation (7.17) with respect to primary unknownsXi.

6.Construct the internal force diagrams.

7.Calculate the reactions of supports and provide their verifications.

Let us consider fixed-rolled uniform beam (the height of the cross section of the
beam ish), subjected to following change of temperature: the temperature of the
above beam is increased byt 1 ı, while below of the beam increased by thet 2 ı,t 1 >t 2
(Fig.7.19a).

1 2

t 2

t 1

EI

h M

1

1

M 1

b X 1 Mt

c

d

a

Fig. 7.19 (a) Design diagram, (b) Primary system; (c) Unit bending moment diagram; (d)Final
bending moment diagram

The primary system is the simply supported beam and the primary unknown is
the moment at the fixed support (Fig.7.19b). Bending moment diagram for unit state
in primary system is shown in Fig.7.19c.
The canonical equation isı 11 X 1 C1tD^0. Bending moment diagram due to
unit primary unknown is shown in Fig.7.19c. The unit displacement is

ı 11 D

MN 1 MN 1

EI

D

1

2

l 1 

2

3



1

EI

D

l

3 EI

:

The axial forceNN 1 in primary system due to the primary unknown is zero, so
the first term of (R 7.18a) is zero. The second term of (7.18a) contains the expression
MN 1 ds, which is the area of the bending moment diagram in the unit state. In our
case the free term may be presented as

1tD ̨

t

h

Z

MN 1 dsDat^1 t^2

h



1

2

 1 l

„ ƒ‚ ...

areaMN 1

D

al

2h

.t 1 t 2 /: