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)

186 Reinforced concrete beams-the serviceability limit states
From the condition of equilibrium,
I' - (lsAs) + UsAs)es e
Je2 - A 1 s (5.5-10)
(5.5-11)
where es is the eccentricity of As from the centroid of the transformed
section and A the cross-sectional area. From the compatibility condition,
E _ let = Is (5.5-12)
es Ee Es
Eliminating lc1 from eqns (5.5-11) and (5.5-12), we have
(5.5-13)
where ae = EsiEe, {3 =AsiA and e = 1/A (k is the radius of gyration).
Substituting eqn (5.5-13) into eqns (5.5-10) and (5.5-11), we have
,, ~ E't -iJ[(I + ~)- -;;:J
1 + aef3 1 + ~
f.,~ E,t e!][l + ~]
1 + aef3 1 + J.
From eqns (5.5-7) to (5.5-9),
1 E2 - Et let - le2
res= d = Eed
Using eqns (5.5-14) and (5.5-15),
}~ ~ (a,t e!] [:~ l
1 + aef3 1 + J.
(5.5-14)
(5.5-15)
(5.5-16)
As will be explained later in Example 9.4-1, the quantity aef3(1 + e~/ k 2 ) is
small compared with unit. Hence, eqn (5.5-16) becomes
EesaeSs
=-!-
where S 5 = Ases and I = Ak2.