Stress and strain of uncracked sections - Concrete Structures: Stresses and Deformations - page 50

time of casting of the deck slab, t 2 = 60 days; the fi nal stress and strain

4100 × 10 3 N;
(920 kip); the bending moment due to self-weight of the prestress beam
(which is introduced at the same time as the prestress), M 1

are required at age t 3 = ∞ . The prestress force, P =

1400 × 10 3
N-m (12 400 kip-in); additional bending moment introduced at age t 2
(representing the e

=

ff ect of the weight of the slab plus superimposed

1850 × 10 3 N-m (16 400 kip-in). The modulus of elas-
ticity of concrete of the precast beam E c1 (3)

dead load), M 2 =

= 25 GPa (3600 ksi) and

E c1 (60) 37 GPa (5400 ksi).

Soon after hardening of the concrete, the composite action starts to
develop gradually. Here we will ignore the small composite action
occurring during the

=

fi rst three days. Consider that age t 2 = 60 days for

the precast beam corresponds to age 3 days of the deck at which time

the modulus of elasticity of the deck E c2 (3)

=

= 23 GPa (3300 ksi).

Creep and aging coe cients and the free shrinkage values to be used

are

Concrete part 1:
[ φ (60, 3)] 1 = 1.20 [ φ ( ∞ , 3)] 1 = 2.30 [ φ ( ∞ , 60)] 1 = 2.27;

[ χ (60, 3)] 1 = 0.86 [ χ ( ∞ , 60)] 1 = 0.80

[ ε cs (60, 3)] 1 =− 57×10 −6 [ ε cs ( ∞ , 60)] 1 =− 205 × 10 −6

Concrete part 2:
[ φ ( ∞ , 3)] 2 = 2.40 [ χ ( ∞ , 3)] 2 = 0.78

[ ε cs ( ∞ , 3)] 2 =− 269×10 −6

Reduced relaxation ∆ σ pr = − 85 MPa (12 ksi) of which − 15 MPa

(2.2 ksi) in the rst 57 days. Modulus of elasticity of the prestressed and

non-prestressed steels

fi

200 GPa.
The dimensions and properties of areas of the concrete and steel in
the two parts are given in Fig. 2.10.

=

(a) Stress and strain immediately after prestressing of the
precast beam
The geometric properties of the precast beam are calculated in Table
2.4, with the reference point O chosen at the centroid of concrete cross-
section and E ref 25 GPa.

The prestress force and the bending moment introduced at t 1 are
equivalent to an axial force at O plus a bending moment given by
Equation (2.31)

= E c1 (3) =