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 167
lTD
h d
r 1ss 1
b
')"L ..
feu = 40 N/mm 2
fy = 460 N/mm 2
A 5 = 2 -size20
Fig. 5.2-9
36 kNm. Using the assumptions appropriate to a partially cracked section,
determine:
(a) the long-term curvature of the beam under the permanent load, if fct
(see Fig. 5.2-8(d)) has the specified value of 0.55 N/mm 2 appropriate
to long-term loading;
(b) the instantaneous curvatures under the total load and the
permanent load, if fct = 1 N/mm 2 for short-term loading; and
(c) the difference between the instantaneous curvatures under the total
and permanent loads.
Given: E, = 200 kN/mm 2 , Ec from Table 2.5-6, Ec (long term) = Ecf
(1 + cp) where the creep coefficient cp may in this example be taken as 2.5.
SOLUTION
(a) Long-term curvature llr1p due to MP. From Table 2.5-6, Ec =
28 kN/mm 2 . Therefore
_28_ 28- 2
Ec (long term) - 1 + cp - 1 + 2 _5 - 8 kN/mm
ac = 200/8 = 25
628
ae(J = (25) (185)(340) = 0.25
From Figs. 5.2-3 and 5.2-4,
xld = 0.505 /cfbd 3 = 0.105
Therefore
x = 0.505d = 171 mm
Ic = 0.105bd 3 = (763)(10 6 ) mm 4
From eqn (5.2-20),
- 6 1{185(375 - 171) 3 }
Mp(net) - (36)(10) Nmm - 3 ( 340 _ 171 ) (0.55)
= (36)(10 6 ) Nmm - (2)(10 6 ) Nmm
= (34)(10 6 ) Nmm
Using eqn (5.2-23), the long-term curvature is