Advanced Methods of Structural Analysis

166 6 Deflections of Elastic Structures

section of the member ish 0. The upper and bottom fibers of the member are
subjected to temperature increaset 1 andt 2 , respectively, above some reference
temperature. Corresponding distribution of temperature (temperature profile) is pre-
sented in Fig.6.12. If the change of temperature for bottom and uppers fibers is equal
.t 1 Dt 2 /, then this case presents the uniform change of temperature; ift 1 ¤t 2 then
this case is referred as nonuniform change of temperature.

Bottom fibers

+t 1

+t 2

t 1 + t 2

2

Temperature profile

Upper fibers

dx

+t 1

+t 2

Du/2

h 0

Dθ t /2

Db /2

Dxt /2 Dxt /2

Db /2

D^ qt /2

Fig. 6.12 Distribution of temperature and displacements within the height of cross section

The expansion of the upper and bottom fibers equals touD ̨t 1 dxandbD
̨t 2 dx, respectively; these expressions contain coefficient of thermal expansion ̨
of member material. In the case of symmetrical cross section, the expansion of the
fiber at the mid-height equals to

xtD ̨

t 1 Ct 2

2

dx: (a)

The mutual angle of rotation of two plane sections, which are located apart from
each other on distance dx

tD ̨

jt 1 t 2 j

h 0

dx: (b)

Now we can substitute (a) and (b) into (6.14). Finally the displacement inkth direc-
tion due to change of temperature may be presented in the following form:

ktD

X

Zs

0

̨

t 1 Ct 2

2

NNkdsC

X

Zs

0

̨

jt 1 t 2 j

h 0

MNkds; (6.15)

whereM;N NNare bending moment and axial force due to the unit generalized force
inkth direction; this force should be corresponding to required temperature dis-
placements.
A differencet 1 t 2 is a temperature gradient; a half-sum.t 1 Ct 2 /=2is a temper-
ature at the centroid of the symmetric cross section (the axis of symmetry coincides