Code References:
ACI 318-14, PCI Handbook 8th Edition
Definition:
For the purposes of this discussion, torsion is defined as a moment about the centerline of a horizontal member. For design, the horizontal member is idealized as a tube, and the center portion of the solid beam can conservatively be neglected.
Critical sections:
For horizontal simply supported members, 9.4.4.3 has defined the critical section for torsion to be a distance d from the face of the support for non-prestressed members and h/2 for prestressed members. If a concentrated load is present between the critical section and the face of the support, then the face of support should become the critical section. The user must make an affirmative response to move the critical section away from the support face. The critical section for non-simply supported members is the face of support (9.4.4.2).
Only the non-composite section is used for torsion resistance.
Calculation Methods:
The ACI 318 method requires the use of closed stirrups, and generally requires more reinforcement than other methods, especially the longitudinal steel. Does not take prestressing into account, which is probably why this method requires more longitudinal steel. May not need to check the end region, as there is no language that appears to require this, but probably should anyway for good practice. Could not find any language that deals with checking hanger steel against combined steel, but it is probably good engineering practice to do so.
The calculations steps for the ACI 318 method can be summarized as follows:
Calculate factored loading at a section
Check if torsion may be neglected (threshold torsion), 22.7.1.1
Calculate the area enclosed by the centerline of the outermost closed transverse torsional reinforcement, Aoh (see discussion in R22.7.6.1.1)
Determine required area of stirrups for torsion, (At,reqd, per face), 22.7.6.1
Calculate required area of stirrups for shear (Av,reqd, total)
Determine combined shear and torsion stirrup requirements (Av/2 + At, per face)
Check maximum stirrup spacing for both torsion (9.7.6.3.3) and shear, use the smaller
Check minimum stirrup area (Av + 2At, total), 9.6.4.2
Check for crushing of the concrete compression struts (combined effects of torsion and shear) and maximum shear resistance (Vc + max Vs), 22.7.7.1
Calculate longitudinal torsion reinforcement (Al, total), 22.7.6.1
Check minimum longitudinal torsion reinforcement (Al, min) 9.6.4.3
Check the end regions for plate bending (PCI Handbook 5.4.3)
Calculate any required hanger steel and compare to the combined transverse steel
The Zia/Hsu method is recognized as alternate method in 9.5.4.6 when the aspect ratio (h/bt) of the beam is 3 or greater and is an update to the earlier Zia/Mcgee method. This method accounts for the prestressing force in the member (if present) and requires closed stirrups. The PCI Design Handbook suggests checking the end region. Hanger steel is not additive to shear and torsion steel.
The calculation steps for the Zia/Hsu method can be summarized as follows:
Calculate factored loading at a section
Calculate the shear and torsional constant (X2Y)
Check if torsion may be neglected (threshold torsion)
Check the maximum allowable torsional moment and maximum allowable shear
Calculate the nominal torsional moment strength provided by the concrete under pure torsion (T’c)
Calculate the nominal shear strength provided by the concrete without torsion (V’c)
Calculate the nominal torsional moment under combined loading (Tc)
Calculate the nominal shear strength under combined loading (Vc)
Compute transverse reinforcement required for torsion, At,reqd (per face)
Compute transverse reinforcement required for shear, Av,reqd (total)
Calculate the combined transverse reinforcement, Av/2 + At (per face)
Check minimum amount of web reinforcement required for ductility, Av + 2At (total)
Calculate the maximum allowable stirrup spacing (use ACI requirements here)
Calculate the longitudinal torsional reinforcement, Al (total) (does not appear to have a minimum required for this reinforcement)
Check the end regions for plate bending (PCI Handbook 5.4.3)
Calculate any required hanger steel and compare to the combined transverse steel
The slender spandrel method is allowed under 9.5.4.7 when the aspect ratio of the beam is greater than 4.5. This procedure recognizes that the member is acting as a plate near the ends, and not in torsion. Closed stirrups are not required in this method.
The calculation steps for the slender spandrel method can be summarized as follows:
Calculate the factored loading at a section
Divide the length of the beam into the following regions:
End region (face of bearing to H)
Transition region (H to 2*H)
Flexure region (remainder of beam)
Check the maximum allowable torsion in the end region
Calculate the required longitudinal reinforcement for flexural resistance
Calculate the required transverse reinforcement for one-way shear
Calculate the required vertical reinforcement required to resist plate bending in the end and transition regions
Check the required vertical reinforcement on the inner web face (ledge or corbel) against the required hanger steel and provide the larger of the two
Verify that the amount of reinforcement crossing a plane along a 45 degree line drawn from the lower tieback at the support and the top of the member is sufficient
Provide sufficient reinforcement on the outer web face (basically Av/2)
Calculate the required longitudinal web reinforcement for the end and transition regions to satisfy plate bending. There is no need to consider plate being in the flexure region.
May want to at least calculate a threshold torsion for the flexure region based on Zia-Hsu (and flag it?)
Combined Shear and Torsion Reinforcement:
(not sure if I need this section)
Calculation of Area of Steel based on Stirrup Input, At,prov:
For the calculation of At,prov at each section, the following steps are performed:
Start with closest stirrup, that is your base area of steel
Scan to left, divide that distance by 2 (S1/2)
Scan to right, divide that distance by 2 (S2/2)
Ar,prov = base area / (S1/2 + S2/2)
S1 and S2 are limited by end of member and Smax (0.x*h or 0.x*d, depending on what ACI 318 has for dv limits)
If a stirrup does not have another stirrup (or the end of the member) within Smax on one or both sides, then the At,prov for that entire stirrup ‘region’ (S1/2+S2/2) is equal to zero
Exceptions and notes:
Torsion resistance elements and stiffnesses. The supporting element should be the stiffest element (as measured by the torsional inertia), and one that also contains closed stirrups. When multiple elements have roughly the same torsional stiffness (inertia) and contain closed stirrups, the torsion is probably distributed based on the relative stiffnesses. However, not sure how to calculate the torsional inertia easily, as the only references I can find on Saint Venant’s torsion inertia are not promising as to ease of calculation. Might be a good routine to write someday, that is, a routine that takes in a cross section made up of polygons and calculates the torsional inertia (heck, we could probably sell that routine in a toolbox of sorts, if we ever get around to writing a toolbox, that is).
The so-called slender spandrel method is not a torsional analysis, it is simply a recognition that the stem of a ‘slender spandrel’ will fail in plate bending before the spandrel fails in torsion. I suspect that in a slender spandrel, the ledge is actually the part of the member that resists the most torsion, as it is probably much more torsionally stiff than the stem. We could probably get a better handle on when the slender spandrel method controls over a more traditional torsion analysis if we could calculate the torsional inertia of the various elements of the spandrel. While the slender spandrel method implies that you do not need to check torsion resistance in 'slender spandrels', I still think that the ledge of a slender spandrel does indeed resist torsion, but the torsion is likely reduced because of the cracking of the stem (not sure how to calculate this reduction), plus, as mentioned above, the stem will fail in plate bending before the ledge will fail in torsion.
Critical section for torsion. The critical section for torsion is based on the ACI sections noted above - however, I do not see any way loads on the beam can go directly into the support without the mechanism of beam action and its connections to the support. Based on ACI 318, we can assume that the depth used to calculate the critical section is based on the cross section used to resist torsion, and this cross section should be reasonably torsionally stiff and contain closed stirrups. If there is an applied concentrated torque between the critical section and the support, then the critical section is the face of the support.
Is display better by face instead of total for beam? Maybe, if Ash and Awv/Avi steel is included in table, perhaps per/face for members with one ledge (spandrels), and total for members with two ledges (it beams) or just let people choose for themselves
Currently end region and hanger steel checks are in separate section in Beam - combine in single table?
References:
ACI Committee 318, Building Code Requirements for Structural Concrete (ACI 318-14), American Concrete Institute, Detroit, 2014.
Zia, P. and Hsu T.T.C., “Design for Torsion and Shear in Prestressed Concrete Flexural Concrete Members,” PCI Journal, V. 49, No. 3, May-June 2004, pp. 34-42.
Lucier, G., C. Walter, S. Rizkalla, P. Zia, and G. Klein, “Development of a Rational Design Methodology for Precast Concrete Slender Spandrel Beams: Part 1, Experimental Results,” PCI Journal, V. 56, No. 2, Spring, pp. 88–112.
Lucier, G., C. Walter, S. Rizkalla, P. Zia, and G. Klein, “Development of a Rational Design Methodology for Precast Concrete Slender Spandrel Beams: Part 2, Analysis and Design Guidelines,” PCI Journal, V. 56, No. 4, Fall, pp. 106-133.
PCI Industry Handbook Committee, PCI Design Handbook, 8th Ed., PCI, Chicago, 2017.