Structural Concrete - Hassoun

550 Chapter 15 Design for Torsion
b. Max. s = P h /8 = 91/8 = 11.4 in. Use s = 6 in., as calculated.
c. A vt /s = 50b w /f yt = 50(14)/60,000 = 0.017 in. 2 /in., which is less than the 0.032 in. 2 /in. used. Use
no. 4 closed stirrups spaced at 6 in.
6. Find the distribution of longitudinal bars. Total A l is 1.28 in. 2 Use one-third, or 0.43 in. 2 ,atthe
top, at the bottom, and at middepth.
a. Total top bars = 0.88 + 0.43 = 1.31 in. 2 ; use three no. 6 bars (1.32 in. 2 ).
b. Total bottom bars = 4.0 + 0.43 = 4.43 in. 2 ; use five no. 9 bars (5.0 in. 2 ). Total A l used = (1.32
− 0.88) + (5 − 4) = 1.44 in. 2
c. Use two no. 4 bars at middepth (0.40 in. 2 ). Reinforcement details are shown in Fig. 15.16.
Spacing of longitudinal bars is at 7.5 in. < 12 in. The diameter of no. 4 bars used is 0.5 in.,
which is greater than no. 3 or stirrup spacing, s∕24 = 6 = 0.25in. Add no. 4 longitudinal bars
24
on all corners of closed stirrups in beam web and flange.
SUMMARY
Sections 15.1–15.7
1. Torsional stresses develop in a beam when a moment acts on the beam section parallel to its
surface.
2. In most practical cases, a structural member may be subjected to combined shear and torsional
moments.
3. The design methods for torsion rely generally on two basic theories: the skew bending theory
and the space truss theory. The ACI Code adopted the space truss theory.
Sections 15.8 and 15.9
A summary of the relative equations in U.S. customary units and SI units is given here.
Note that (1.0 √ f ′ c) in psi is equivalent to (0.08 √ f ′ c) in MPa N/mm 2 ,1in.≈25 mm, and
f yt ≤ 400 MPa.
Equation U.S. Customary Units SI Units
15.16 T c = 2φ √ ∑
f c ′ x 2 y T c = 0.17φ √ ∑
f c ′ x 2 y
15.17 φT n ≥ T u Same
15.19 T cr = 4λ √ ( )
A
2
f c
′ cp
T
P cr =(λ √ f c ′ ∕3)(A 2 cp ∕P cp )
cp
15.20 T u = φλ √ ( )
A
2
f c
′ cp
T
P u ≤ φλ( √ f c ′ ∕12)(A 2 cp ∕P cp )
cp 15.21
√ ( ) ( )
V 2 2 [( )
u Tu P
+ h
Vc
≤ φ + 8 √ ]
f
b w d 1.7A 2 c
′ (U.S.)
b
0h
w d
√ ( ) ( )
V 2 2 [( )
u Tu P
+ h
Vc
≤ φ +(2 √ ]
f
b w d 1.7A 2 c ′ ∕3) (SI)
b
0h
w d