Page 306 - Handbook of Civil Engineering Calculations, Second Edition

PRESTRESSED CONCRETE 2.91

FIGURE 58. Steel beam encased in concrete. (a) Section for positive moment; (b) section
for negative moment.

(3048 mm); 16t 11 16(4.5) 11 83 in. (2108 mm); this governs. Transformed
width 83/9 9.22 in. (234.2 mm).
Assume that the neutral axis lies within the flange, and take static moments with re-
2
spect to this axis; or /2(9.22y ) 11.77(10 y) 0; y 3.93 in. (99.8 mm).
1
4
4
3
1
Compute the moment of inertia. Slab: ( /3)9.22(3.93) 187 in (7783.5 cm ). Beam:
4
4
2
515.5 11.77 (10 3.93) 949 in (39,500.4 cm ); I 187 949 1136 in 4
3
3
4
(47,283.9 cm ); S c 1136/3.93 289.1 in (4737.5 cm ); S bs 1136/14.07 80.7 in 3
3
(1322.4 cm ).
3. Transform the composite section in the region of negative
moment to an equivalent section of steel; compute the
section moduli
Referring to Fig. 58b, we see that the transformed width 11/9 1.22 in. (31.0 mm).
2
Take static moments with respect to the neutral axis. Or, 11.77(10 y) /2(1.22y ) 0;
1
y 7.26 in. (184.4 mm). Compute the moment of inertia. Thus, slab: ( /3)1.22(7.26)
3
1
4
4
4
2
4
155.6 in (6476.6 cm ). Beam: 515.5 11.77(10 7.26) 603.9 in (25,136.2 cm ); I
3
4
4
155.6 603.9 759.5 in (31,612.8 cm ). Then S c 759.5/7.26 104.6 in (1714.1
3
3
3
cm ); S ts 759.5/10.74 70.7 in (1158.6 cm ).
4. Compute the bending stresses at midspan
The loads carried by the noncomposite member are: slab, (4.5)150(10)/12 563 lb/lin ft
(8.22 kN/m); stem, 11(15.5)150/144 178 lb/lin ft (2.6 kN/m); steel, 40 lb/lin ft (0.58
kN/m); total 563 178 40 781 lb/lin ft (11.4 kN/m). The load carried by the com-
posite member 145(10) 1450 lb/lin ft (21.2 kN/m). Then M ( /8)781(28.5) 12
n
2
1
c
2
1
951,500 in.·lb (107.5 kN·m); M ( /20)1450(28.5) 12 706,600 in.·lb (79.8 kN·m);
f c 706,600/[289.1(9)] 272 lb/sq.in. (1875 kPa), which is acceptable. Also, f bs
(951,500/64.4) (706,600/80.7) 23,530 lb/sq.in. (162.2 MPa), which is acceptable.
5. Compute the bending stresses at the support
c
Thus, M 706,600( /12) 1,177,700 in.·lb (132.9 kN·m); f c 1,177,700/[104.6(9)]
20
1251 lb/sq.in. (8.62 MPa), which is satisfactory. Also, f ts 1,177,700/70.7 16,600
lb/sq.in. (114.9 MPa), which is acceptable. The design is therefore satisfactory with
respect to flexure.
6. Investigate the composite member with respect to horizontal
shear in the concrete at the section of contraflexure
Assume that this section lies at a distance of 0.2L from the support. The shear at this sec-
c
tion is V 1450(0.3)(28.5) 12,400 lb (55.2 kN).