Special cases of uncracked sections and calculation of displacements - Concrete Structures: Stresses and Deformations - page 95

3.9 Example worked out in British units

Example 3.6 Parametric study.
The structure shown in Fig. 3.9(a) represents a 1 ft wide (305 mm) strip
of a post-tensioned, simply supported solid slab. At time t 0 , the structure
is subjected to dead load q = 0.40 kip/ft (5.8 kN/m) and an initial pre-

stressing force P 290 kip (1300 kN), which is assumed constant over

the length. The objectives of this example are to study the e

=

ects of the
presence of the non-prestressed steel on the stress distributions between
concrete and the reinforcement and on the mid-span de

ff

ection at time t
after occurrence of creep, shrinkage and relaxation. Non-prestressed
steel of equal cross-section area A ns is provided at top and bottom. The
steel ratio

fl

A ns / bh , is considered variable between zero and 1 per cent.
The modulus of elasticity of concrete E c ( t 0 )

ρ ns =

4350 ksi (30 GPa); the
change in E c with time is ignored. The modulus of elasticity of the
prestressed and the non-prestressed steel E s

=

= 29 000 ksi (200 GPa).

Other data are:

300×10 −6 ;
∆ σ pr =−

φ ( t , t 0 ) = 3.0; ε cs ( t , t 0 ) =−

9.3 ksi ( − 64 MPa).

The e ff ects of varying the values of φ and ε cs on the results will also be

discussed.
The dead load q produces a bending moment at mid-span =

1500 kip-in (169 kN-m).
Only the results of the analyses are given and discussed below. For
ease in verifying the results, the simplest cross-section is selected. Also
the variation of the initial prestressing force P because of friction is
ignored and the di erence in the cross-section area of the tendon and

the area of the prestressing duct is neglected.
Table 3.2 gives the concrete stresses at midspan at time t after occur-
rence of creep, shrinkage and relaxation. It can be seen that the stress at
the bottom

ff

fi bre varies between − 1026 and − 502 psi ( − 7.08 and

− 3.46 MPa) as the non-prestressed steel ratio, ρ ns is increased from zero

to 1%.
In other words, ignoring the non-prestressed steel substantially over-
estimates the compressive stress provided by prestressing to prevent or
to control cracking by subsequent live load; the overestimation is of the
same order of magnitude as the tensile strength of concrete. The