Effects of temperature - Concrete Structures: Stresses and Deformations - page 370

temperature rise subsequently vanishes slowly over a much longer period.
The stresses due to this temperature change may be analysed in steps by
dividing the time into intervals and considering that increments of tempera-
ture or stresses occur suddenly at the middle of the intervals. For each inter-
val, an appropriate creep coe

cient and modulus of elasticity is employed
(see Section 5.8). Considering creep in this fashion will result in substantially
di erent stresses from a calculation in which creep and change in modulus of

elasticity are ignored.
In fact, considering these time-dependent e

ff

ects may indicate that the
stresses developed at peak temperature reverse signs after a long time when
the heat of hydration is completely lost. 11 This can be seen in Example 10.2
which treats the problem using a step-by-step numerical analysis.
A general procedure for a step-by-step procedure of stress analysis of
concrete structures is discussed in Section 5.8. Consider here the application
of the method for analysis of the self-equilibrating stresses in a cross-
section of a concrete member due to a rise of temperature which varies with
time. Divide the time, during which the temperature change occurs, into a
number of intervals. The symbols t i −

ff

2 represent the age of
concrete at the beginning, middle and end of the i th interval. At the end
of any interval i , the strain due to free temperature expansion is the
summation:

2 , t i and t i +

1 1

i

α t ( ∆ T ) j (10.32)

j = 1

This strain is prevented arti ∆ σ restraint ) j

at the middle of the intervals. The combined strain caused by temperature
and these stress increments is zero. For the end of the i th interval, we can
write

fi cially by the introduction of stress (

i i

(

∆ σ restraint ) j
E c ( t j )

α t ( ∆ T ) j + [1 + φ ( t i +

2 , t j )]

= 0 (10.33)

1

j = 1 j = 1

where E c ( t j ) is the modulus of elasticity of concrete at the middle of the j th
interval;

2 , t j ) is the ratio of creep occurring between the middle of the j th
interval and the end of the i th interval to the instantaneous strain when a
stress is introduced at t j . The summation in the second term of the equation
represents the instantaneous strain plus creep caused by the stress increments
during the intervals 1, 2, . . . , i .
In a step-by-step analysis, when Equation (9.33) is applied at any interval i ,
the stress increments are known for the earlier intervals. Thus, the equation
can be solved for the stress increment in the i th interval, giving:

φ ( t i +

1