Structural Concrete - Hassoun

15.8 Torsion in Reinforced Concrete Members (ACI Code Procedure) 535
The reinforcement required for torsion must be added to that required for shear, bending
moment, and axial forces. The reinforcement required for torsion must be provided such that the
torsional moment strength of the section φT n is equal to or exceeds the applied factored torsional
moment T u computed from factored loads:
φT n ≥ T u (15.17)
When torsional reinforcement is required, the torsional moment strength φT n must be calculated
assuming that all the applied torque, T u , is to be resisted by stirrups and longitudinal bars with
concrete torsional strength, T c = 0. At the same time, the shear resisted by concrete, v c , is assumed
to remain unchanged by the presence of torsion.
15.8.2 Torsional Geometric Parameters
In the ACI Code, Section 22.7, the design for torsion is based on the space truss analogy, as shown
in Fig. 15.8. After torsional cracking occurs, the torque is resisted by closed stirrups, longitudinal
bars, and concrete compression diagonals. The concrete shell outside the stirrups becomes relatively
ineffective and is normally neglected in design. The area enclosed by the centerline of the outermost
closed stirrups is denoted by A 0h , the shaded area in Fig. 15.11. Because other terms are used in
the design equations, they are introduced here first to make the equation easier to comprehend.
Referring to Fig. 15.11, the given terms are defined as follows:
A cp = area enclosed by outside perimeter of concrete section, in. 2
P cp = perimeter of concrete gross area, A cp ,in.
A 0 h = area enclosed by centerline of outermost closed transverse torsional
reinforcement, in. 2 (shaded area in Fig. 15.11)
A 0 = gross area enclosed by shear flow path and may be taken equal to 0.85A 0h
(A 0 may also be determined from analysis [18,19]).
P h = perimeter of concrete of outermost closed transverse torsional reinforcement
Θ = angle of compression diagonals between 30 ∘ and 60 ∘ (may be taken equal to 45 ∘
for reinforced concrete members)
In T- and L-sections, the effective overhang width of the flange on one side is limited to the
projection of the beam above or below the slab, whichever is greater, but not greater than four times
the slab thickness (ACI Code, Sections 9.2.4.4).
15.8.3 Cracking Torsional Moment, T cr
The cracking moment under pure torsion, T cr , may be derived by replacing the actual section, prior
to cracking, with an equivalent thin-walled tube, t = 0.75 A cp /P cp , and an area enclosed by the wall
centerline, A 0 = 2 A cp /3. When the maximum tensile stress (principal stress) reaches 4λ √ f c, ′ cracks
start to occur and the torque T in general is equal to
T = 2A 0 τt (15.18)
where τ is the torsional shear stress, which is 4λ √ f c ′ for torsional cracking.
Replacing τ by 4λ √ f c,
′
T cr = 4λ √ ( )
A
2
f c
′ cp
= T
P n and T u = φT cr (15.19)
cp