F. K. Kong MA, MSc, PhD, CEng, FICE, FIStructE, R. H. Evans CBE, DSc, D ès Sc, DTech, PhD, CEng, FICE, FIMechE, FIStructE (auth.)-Reinforced and Prestressed Concrete-Springer US (1987)

Elastic theory: cracked, uncracked and partially cracked sections 157
and crack widths, should special circumstances warrant such calculations to
be done. Serviceability is concerned with structural behaviour under
service loading, and service loading is sufficiently low for the results of an
elastic analysis to be relevant. Therefore in the next section, we shall give
an account of the elastic theory for reinforced concrete beams, leading to
concepts and results which have applications in deflection and crack-width
calculations.
5.2 Elastic theory: cracked, uncracked and partially
cracked sections
In this section we shall describe the elastic theory for reinforced concrete
for three types of member sections: the cracked section (Case 1), the
uncracked section (Case 2) and the partially cracked section (Case 3). Case
1 is the classical elastic theory for reinforced concrete, which once occupied
a central position in design but which has little direct application today; it
is, however, still of some use in crack-width calculations (see Section 5.6).
Case 2 is important in prestressed concrete design (see Chapters 9 and 10),
and Case 3 is currently used for calculating deflections (see Section 5.5).
Case 1: The cracked section
Figure 5.2-1(a) shows the cross-section of a beam subjected to a bending
moment M. The following simplifying assumptions are made:
(a)
(b)
(c)
Plane sections remain plane after bending. In other words, the strains
vary linearly with distances from the neutral axis. (For a critical
review of the research on strain distribution, see Reference 5.)
Stresses in the steel and concrete are proportional to the strains.
The concrete is cracked up to the neutral axis, and no tensile stress
exists in the concrete below it. (For this reason, the section in Fig.
5.2-1(a) is referred to as a cracked section.)
From assumption (a), the steel strains can be expressed in terms of the
concrete strain Ec on the compression face (Fig. 5.2-1(b )):
X- d' d- X
t:' = --e · e = -- E: (5.2-1)
s X c• s X c
From assumption (b) the concrete stress lc on the compression face, the
tension steel stress Is and the compression steel stress ~~ are
~~ = Est:~ = acEct:~
Is = EsE:s = acEcE:s
(5.2-2(a))
(5.2-2(b))
(5.2-2(c))
where Es and Ec are the moduli of elasticity of the steel and concrete
respectively and ac is the modular ratio Esl Ec.
Since in Fig. 5.2-1(b) the concrete below the neutral axis is to be
ignored (assumption (c)), the effective cross-section is that of Fig.
5.2-2(a). From the condition of equilibrium of forces,