Structural Concrete - Hassoun

19.1 Prestressed Concrete 729
Adding stresses due to the dead and live loads (Fig. 19.2) gives
Top stress =−225 − 675 =−900 psi
Bottom stress =+225 + 675 =+900 psi
(compression)
(tension)
The tensile stress is higher than the modulus of rupture of concrete, f r = 7.5λ √ f ′ c = 503 psi;
hence, the beam will collapse.
2. In the case of stresses due to uniform prestress, if a compressive force P = 259.2 K is applied at
the centroid of the section, then a uniform stress is induced on any section along the beam:
σ P =
P 259.2 × 1000
= =−900 psi (compression)
area 12 × 24
Final stresses due to live and dead loads plus prestress load at the top and bottom fibers
are 1800 psi and 0, respectively (Fig. 19.2). In this case, the prestressing force has doubled
the compressive stress at the top fibers and reduced the tensile stress at the bottom
fibers to 0. The maximum compressive stress of 1800 psi is less than the allowable stress of
2050 psi.
3. For stresses due to an eccentric prestress (e = 4 in.), if the prestressing force P = 259.2 K is placed
at an eccentricity of e = 4 in. below the centroid of the section, the stresses at the top and bottom
fibers are calculated as follows. Moment due to eccentric prestress is Pe:
σ P =− P A ± (Pe)c
I
259.2 × 1000
=− ±
12 × 24
=−900 ± 900
=−1800 psi
=− P A ± 6(Pe)
bh 2
6(259.2 × 1000 × 4)
12(24) 2
at the bottom fibers and σ P = 0 at the top fibers. Consider now an increase in the live load of
L 2 = 2100 lb/ft:
2.1 ×(24)2
M LL = = 151.2K⋅ ft
8
6(151.2 × 12, 000)
σ L2
= =±1575 psi
12(24) 2
Final stresses due to the dead, live, and prestressing loads at the top and bottom fibers are
1800 psi and 0, respectively (Fig. 19.2). Note that the final stresses are exactly the same as
those of the previous case when the live load was 900 lb/ft; by applying the same prestressing
force but at an eccentricity of 4 in., the same beam can now support a greater live load (by
1200 lb/ft).
4. For stresses due to eccentric prestress with maximum eccentricity, assume that the maximum
practical eccentricity for this section is at e = 6 in., leaving a 2-in. concrete cover; then the bending
moment induced is Pe = 259.2 × 6 = 1555.2 K⋅in. = 129.6 K⋅ft. Stresses due to the prestressing
force are
259.2 × 1000 6 ×(129.6 × 12, 000)
σ P =− ±
12 × 24
12(24) 2
=−900 ± 1350 psi
=−2250 psi and + 450 psi