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)

72 Axially /oaded reinforced concrete co/umns
Because of the bond between the reinforcement and the concrete, they will
have equal strains under load (bond will be discussed in Chapter 6).
Therefore, from the condition of compatibility,
fel Ee = J.! Es
(concrete strain = steel strain) (3.3-2)
where Ee and Es are respectively the modulus of elasticity of the concrete
(see Table 2.5-6) and the steel; the modulus of elasticity of steel is usually
taken as 200 kN/mm 2 in design.
From eqns (3.3-1) and (3.3-2),
i.e.
t. - N
e - Ae + aeAsc
ţ = aeN
s Ae + aeAse
(3.3-3)
(3.3-4)
fs = aefe (3.3-5)
where ae = Esi Ee is called the modular ratio.
In eqns (3.3-3) and (3.3-4), the quantity Ae + aeAse represents the
concrete area plus ae times the steel area, and is of ten referred to as the
are a of the transformed section or equivalent section. Therefore, when the
stresses are within the linear ranges in Figs 3.2-1(a) and 3.2-2(a), the
concrete stress fe is obtained by dividing the load N by the are a of the
transformed section, and the steel stress fs is ae times fe.
EtTects of creep and shrinkage
Equations (3.3-3) and (3.3-4) may give the false impression that the
stresses fe and fs in a given column are uniquely defined once the load N is
specified. In fact, it is almost impossible to determine these stresses
accurately; this is because of the effects of the creep and shrinkage of the
concrete. What we are quite certain of, however, is that in practice the
steel stressfs is usually much larger than that given by eqn (3.3-4) and the
concrete stressfe is much less than that given by eqn (3.3-3). First consider
creep, which was discussed in Section 2.5(b). It is clear from Fig. 2.5-3
that if concrete is subjected to a sustained stress, the total strain is the
elastic strain plus the creep strain, and this increases with time. Therefore,
unless the load N is applied only for a short time (and this is rarely
the case with columns in actual structures), the modulus Ee in eqn (3.3-2)
must be the etTective modulus of elasticity defined as the ratio of the stress
to the total strain for the particular duration of loading concerned;
simiIarly, the etTective modular ratio must be used in eqns (3.3-3) to
(3.3-5). Creep reduces the effective modulus of elasticity of the concrete
and hence increases the effective modular ratio. If the effective values of ac
for various durations of loading are inserted in eqns (3.3-3) and (3.3-4) it
will be found that for the column under a contant load, there is a gradual
but significant redistribution of stress with time: the concrete gradually
sheds off the load it carries and this is picked up by the steel. This
redistribution may continue for years untiI the effective modular ratio