Stress and strain of cracked sections - Concrete Structures: Stresses and Deformations - page 246

height 0.159 m above the neutral axis at time t , is ignored although it
would have been subjected to compressive stress. Because the ignored
area is close to the neutral axis, the error involved is small.

7.6 Partial prestressed sections
Consider a prestressed concrete section which is also reinforced by non-
prestressed steel. The prestress is applied at age t 0 at which time a part of the
dead load is also introduced and, shortly after, a superimposed dead load is
applied. At a much later date t , the live load comes into e ect and produces

cracking. What is the procedure of analysis to determine the strain and stress
distributions at age t after cracking? The term partial prestressing is used
throughout this topic to refer to the case when the prestressing forces are not
su

ff

cient to prevent cracking at all load stages.
We shall assume here that all the time-dependent changes due to creep and
shrinkage of concrete and relaxation of prestressed steel take place prior to
age t and that no cracking occurs up to this date. Thus, the method of
analysis presented in Section 2.5 for uncracked sections can be applied to
determine the strain and stress distributions at age t just before application of
the live load. The problem that needs to be discussed in the present section
may be stated as follows. Given the stress distribution in an uncracked section
reinforced by prestressed and non-prestressed reinforcement, what are the
instantaneous changes in stress and strain caused by the application of an
additional bending moment and axial force causing cracking?
Figure 7.10(a) shows a cross-section with several layers of prestressed and
non-prestressed reinforcement. At time t , the distribution of stress on the
section is assumed to be known and σ c ( t ), the concrete stress, is assumed to

vary linearly over the depth without producing cracking. This stress distribu-
tion may be completely de

fi ned by the stress value σ O ( t ) at an arbitrary refer-

ence point O and stress diagram slope, /d y . The additional bending

moment M and axial force N at O are applied, producing cracking of the
section. It is required to

γ ( t ) = d σ

fi nd the changes in strain and in stress due to M and

N .
Partition each of M and N in two parts, such that (see Fig. 7.10(c) and (e) ):

M = M 1 + M 2 (7.33)

N = N 1 + N 2 (7.34)

M 1 and N 1 represent the part of the internal forces that will bring the stresses
in the concrete to zero and M 2 and N 2 represent the remainder of the internal
forces. With M 1 and N 1 , the section is in state 1 (uncracked). Cracking is