Stress and strain of cracked sections - Concrete Structures: Stresses and Deformations - page 208

7.1 Introduction
Cracks occur in reinforced and partially prestressed members when the
stresses exceed the tensile strength of concrete. After cracking, the stresses in
concrete normal to the plane of the crack cannot be tensile. Thus, the internal
forces in a section at the crack location must be resisted by the reinforcement
and the uncracked part of the concrete cross-section. The part of the concrete
cross-section area which continues to be e

ective in resisting the internal
forces is subjected mainly to compression and some tension not exceeding the
tensile strength of concrete. At sections away from cracks, concrete in tension
also contributes in resisting the internal forces and hence to the sti

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ff ness of

the member.
Two extreme states are to be considered in the calculation of displacements
in a cracked member, as will be further discussed in Chapter 8. In state 1, the
full area of the concrete cross-section is considered e ective and the strains in

the concrete and the reinforcement are assumed to be compatible. In state 2,
concrete in tension is ignored; thus, the cross-section is assumed to be com-
posed of the reinforcement and concrete in compression. The cross-section in
state 2 is said to be fully cracked .
The actual elongation or curvature of a cracked member can be calculated
by interpolation between the two extreme states 1 and 2.
In Chapters 2 and 3 we analysed the stresses, axial strain and curvature in
an uncracked section, including the e

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ects of creep, shrinkage and relaxation
of prestressed steel. The section was assumed to be subjected to an axial force
and/or a bending moment. The values and the time of application of these
forces were assumed to be known. With prestressing, the initial prestress force
was assumed to be known, but the changes in the stresses in the prestressed
and non-prestressed steel due to creep, shrinkage and relaxation were deter-
mined by the analysis. The full concrete cross-section area was considered to
be e

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ective, whether the stresses were tensile or compressive.
In the present chapter, fully cracked reinforced concrete sections without
prestressing are analysed. The section is assumed to be subjected to an axial
force, N , and a bending moment, M , of known magnitudes. With the con-
crete in tension ignored, these forces are resisted by the concrete in compres-
sion and by the reinforcement. The analysis will give axial strain, curvature
and corresponding stresses immediately after application of N and M and
after a period of time in which creep and shrinkage occur.
Analysis of a partially prestressed section is also included in this chapter.
The section is assumed to remain in state 1 (uncracked) under the e

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ect of
prestress and loads of long duration, such as the dead load. After a given
period of time, during which creep, shrinkage and relaxation have occurred,
live load is assumed to be applied, producing cracking. With this assumption,
the equations of Chapter 2 can be used to determine the stress and strain
in concrete and the reinforcement at the time of prestressing and after a

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