Stress and strain of uncracked sections - Concrete Structures: Stresses and Deformations - page 66

Strain and stress at time t 0
At completion of the installation of the precast elements the concrete
and steel act as separate sections; the concrete section is subjected to the
prestressing force 2200 kip at the centroid and the steel section is sub-
jected to a bending moment

168 000 kip-in. The area of the trans-
formed concrete section composed of A c and

=

α ns (excluding the area of

4535 in 2 ; with

ducts) 6.51. The strain and stress distributions

at this stage are shown in Fig. 2.16(b).

= A = α ns =

Strain and stress at time t
After connection of concrete and steel at time t 0 , the section becomes
composite. Select the reference point O at the centroid of the structural
steel and follow the four analysis steps outlined in Section 2.6:

Step 1 The instantaneous strain and stress at t 0 have been determined
in Fig. 2.16(b).

Step 2 If creep and shrinkage were free to occur, the change in strain
between t 0 and t would be:

( ∆ ε O ) free =− 220×10 −6 − 2.2(112.8 × 10 −6 )

=− 468.2 × 10 −6 ; (∆ ψ ) free = 0.

Step 3 T he age-adjusted elasticity modulus of concrete (Equation
(1.31) ), E c =

1558 ksi.
The stress necessary to restrain creep and shrinkage (Equation
(2.45) ) is:

468.2 × 10 6 )

( σ c ) restraint =− 1558 ( − = 0.729 ksi.

Step 4 The area properties of concrete are:

A c = 4471 in 2 ; B c =− 216 800 in 3 ; I c = 10.57 × 10 6 in 4 .

The forces necessary to restrain creep and shrinkage (Equation
(2.18) ) are:

( ∆ N ) creep + shrinkage = 3261 kip; (∆ M ) creep + shrinkage

=−

158.2 × 10 3 kip-in.