Advanced Methods of Structural Analysis

7.6 Temperature Changes 257

Kinematical verification The summary unit bending moment diagramMN†is shown
in Fig.7.20e.
The expression (7.21) leads to the following result:

XZ MtMN†

EI

dsC

X

itD

5

1 EI 6

18 43:99C 4 

13 C 18

2



43:99C68:37

2

C 13 68:37

!

 ̨EI



1

1 EI



1

2

 3  3 

2

3

.14:628/ ̨EI

„ ƒ‚ ...

portion 4 - 5



1

2 EI



1

2

 10  10 

2

3

82:99 ̨EI

„ ƒ‚ ...

portion 6 - 8

C4;191:7 ̨

„ƒ‚...

1t

C1;450 ̨

„ƒ‚...

2t

D5;686:32 ̨C5;685:58 ̨Š0:

Summary:Distribution of internal forces (bending moment, shear force and nor-
mal force) as well as reactions depends on both relative and absolute stiffnesses, as
well as on coefficient of thermal expansion. This is the property of any statically
indeterminate structure subjected to change of temperature.

Statically indeterminate trusses If any members of a truss are subjected to change
of temperature, then the unit displacements and free terms of canonical equations of
the force method (7.17) should be calculated by the formulas

ıikD

X

j

NNiNNkl

EA

;

itD ̨

X

j

NNitl; (7.22)

wherelis a length ofjth member of the truss;NNi;NNkare internal forces injth
member due to unit primary unknownsXiD 1 ,XkD 1 ;tis a thermal gradient;
and ̨is a coefficient of thermal expansion.
In expressions (7.22), summation is done on all members of the truss (subscripts
jare not shown).
Axial force in members of the truss using the superposition principle is deter-
mined by the formula

NDNN 1 X 1 CNN 2 X 2 CCNNnXn; (7.23)

wherenis degree of statical indeterminacy.

Example 7.5.Design diagram of the truss is presented in Fig.7.21. Axial stiffness
for diagonal and vertical members equalsEA, for upper and lower chords equal 2EA.
Determine the reaction of the middle support and internal forces in all members of
the truss caused by temperature changes ontdegrees. Consider two cases: the
temperature gradient is applied to (a) all members of the truss and (b) the lower
chord only.