Stress and strain of uncracked sections - Concrete Structures: Stresses and Deformations - page 27

2.4 Strain and stress due to non-linear
temperature variation
Analysis of the change in stresses due to creep, shrinkage of concrete and
relaxation of prestressed steel in concrete structures can be done in the same
way as the analysis of stresses due to temperature (as will be shown in Sec-
tions 2.5, 5.4 to 5.6 and 10.7). For this reason, we shall consider here the
strain and stress in a cross-section subjected to a temperature rise of magni-
tude T ( y ), which varies over the depth of the section in an arbitrary fashion
(Fig. 2.2).
In a statically determinate frame, uniform or linearly varying temperature
over the depth of the cross-section of a member produces no stresses. When
the temperature variation is non-linear (Fig. 2.2), stresses are produced
because each

bres is not free to acquire the
full expansion due to temperature. The stresses produced in this way in an
individual cross-section must be self-equilibrating; in other words the tem-
perature stress in a statically determinate structure corresponds to no change
in the stress resultants (the internal forces). We shall discuss below the analy-
sis of the stresses produced by a rise of temperature which varies non-linearly
over the depth of a member of a statically determinate framed structure.
The self-equilibrating stresses caused by non-linear temperature variation
over the cross-sections of statically determinate frame are sometimes referred
to as the eigenstresses. If the structure is statically indeterminate , the elonga-
tions and/or the rotations of the joints of the members are restrained or
prevented, resulting in a statically indeterminate set of reactions which are
also self-equilibrating, but these will produce statically indeterminate internal
forces and corresponding stresses. Statically indeterminate forces produced
by temperature will be discussed in Section 10.8. The present section is con-
cerned with the axial strain, the curvature and the self-equilibrating stresses
in a cross-section of a statically determinate structure subjected to a rise of
temperature which varies non-linearly over the depth of the section (Fig. 2.2).
The hypothetical strain that would occur at any

fi bre being attached to adjacent fi

fi bre if it were free is:

ε f = α t T (2.21)

where T = T ( y ), the temperature rise at any fi bre at a distance y below a

reference point O and α t = coe cient of thermal expansion.

If this strain is arti fi cially prevented, the stress in the restrained condition

will be

σ restrained =− E ε f (2.22)

where E is the modulus of elasticity, which is considered, for simplicity, to be
constant over the whole depth of the section.