Advanced Methods of Structural Analysis

7.6 Temperature Changes 251

Kinematical verification The summary unit bending moment diagramMN†DMN 1 C
MN 2 is shown in Fig.7.18e. The formula (7.16) leads to the following result

MMN†

EI

C

X

is

D

2

6

(^66)
4
1
1 EI

1
2
 3  3 
2
3
2:1228
„ ƒ‚ ...
portion 4 - 5

1
2 EI

1
2
 10  10 
2
3
1:119
„ ƒ‚ ...
portion 6 - 8
3
7
(^77)
5
 10 ^3 EI
C
1
1 EI

5
6
.2 13 1:0038C 2  18 4:5418C 13 4:5418C 18 1:0038/ 10 ^3 EI
„ ƒ‚ ...
portion 1 - 3
C.0:12/
„ƒ‚...
1s
C.0:09/
„ƒ‚...
2s
D0:22860:2286D0:
After verification for bending moment diagram, we can construct the shear and axial
force diagrams and find reactions of supports. These procedures have been described
in Example 7.2.
Discussion:In the case of settlements of supports, the primary unknowns as well
as the reactions and internal forces (bending moment, shear, and normal force) de-
pends on both the relative and absolute stiffnessesEI. This is the common property
of any statically indeterminate structure subjected to settlement of supports.

7.6 Temperature Changes..................................................

If an arbitrary statically indeterminate structure is subjected to change of temper-
ature, then the internal forces arises in the members of the structure. Analysis of
such structure may be effectively performed on the basis of a canonical equation of
the force method. A primary system and the primary unknowns of the force method
are chosen as usual. The canonical equations for structure withnredundant con-
straints are

ı 11 X 1 Cı 12 X 2 CCı1nXnC1tD 0

ı 21 X 1 Cı 22 X 2 CCı2nXnC2tD 0

 (7.17)

ın1X 1 Cın2X 2 CCınnXnCntD0;

whereXnare primary unknowns andıikanditare displacements of the primary
system in the direction ofith primary unknown caused by the unit primary unknown
XkD 1 and by the change of temperature, respectively.