Structural Concrete - Hassoun

540 Chapter 15 Design for Torsion
when the principal stress in concrete is about 4λ √ f ′ c. When T u > T cr , a torque equal
to T cr Eq. 15.19, may be assumed to occur at the critical sections near the faces of the
supports.
The ACI Code limits the design torque to the smaller of T u from factored loads or φT cr from Eq.
15.19.
15.8.5 Limitation of Torsional Moment Strength
The ACI Code, Section 22.7.7, limits the size of the cross-sectional dimension by the following
two equations:
1. For solid sections,
) ( )
√ 2 2 [( )
√(
Vu T
+ u P h
Vc
≤ φ + 8 √ ]
f
b w d 1.7 A 2 c
′ b
oh
w d
(15.21)
2. For hollow sections,
) ( )
T
+ u P h
b w d 1.7 A 2 oh
(
Vu
[( )
Vc
≤ φ + 8 √ ]
f c
′ b w d
(15.22)
where V c equals 2λ √ f cb ′ w d, which is shear strength for normal-weight concrete. All other
terms were defined in Section 15.8.2.
This limitation is based on the fact that the sum of the stresses due to shear and torsion
(on the left-hand side) may not exceed the cracking stress plus 8 √ f ′ c. The same condition was
applied to the design of shear without torsion in Chapter 5. The limitation is needed to reduce
cracking and to prevent crushing of the concrete surface due to inclined shear and torsion
stresses.
15.8.6 Hollow Sections
Combined shear and torsional stresses in a hollow section are shown in Fig. 15.6, where the
wall thickness is assumed constant. In some hollow sections, the wall thickness may vary around
the perimeter. In this case, Eq. 15.22 should be evaluated at the location where the left-hand
side is maximum. Note that at the top and bottom flanges, the shear stresses are usually negligible.
In general, if the wall thickness of a hollow section t is less than A oh /P h , then Eq. 15.22
becomes
(ACI Code, Section 22.7.7).
15.8.7 Web Reinforcement
V u
b w d +
T u
1.7A 0h t ≤ φ [(
Vc
b w d
)
+ 8 √ f ′ c
]
(15.23)
As was explained earlier, the ACI Code approach for the design of the members due to torsion is
based on the space truss analogy in Fig. 15.8. After torsional cracking, two types of reinforcement
are required to resist the applied torque, T u : transverse reinforcement, A t , in the form of