Torsion

Code References:

ACI 318-14, PCI Handbook 8^th 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).

Calculation Methods:

Three calculation methods are available in the analysis: ACI 318, Zia/Hsu, and slender spandrel. In all methods, note that only the non-composite section is used for the torsion resistance.

The ACI 318 method requires the use of closed stirrups, and generally requires more reinforcement than  other methods, especially the longitudinal steel.  This method does not take prestressing into account, which is probably why there is a higher longitudinal steel requirement.  There does not appear to be any requirement to check the end region in this method, nor is there any language that deals with checking ledge hanger steel (if any) against combined torsion and shear steel.

The calculations steps for the ACI 318 method can be summarized as follows:

1. Calculate factored loading at a section

2. Check if torsion may be neglected (threshold torsion), 22.7.1.1

3. Calculate the area enclosed by the centerline of the outermost closed transverse torsional reinforcement, A_oh (see discussion in R22.7.6.1.1)

4. Determine required area of stirrups for torsion, (A_t,reqd, per face), 22.7.6.1

5. Calculate required area of stirrups for shear (A_v,reqd, total)

6. Determine combined shear and torsion stirrup requirements (A_v/2 + A_t, per face)

7. Check maximum stirrup spacing for both torsion (9.7.6.3.3) and shear, use the smaller

8. Check minimum stirrup area (A_v + 2A_t, total), 9.6.4.2

9. Check for crushing of the concrete compression struts (combined effects of torsion and shear) and maximum shear resistance (V_c + max V_s), 22.7.7.1

10. Calculate longitudinal torsion reinforcement (A_l, total), 22.7.6.1

11. Check minimum longitudinal torsion reinforcement (A_l, min) 9.6.4.3

12. Check the end regions for plate bending (PCI Handbook 5.4.3)

13. 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/b_t) 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:

1. Calculate factored loading at a section

2. Calculate the shear and torsional constant (X²Y)).  The quantity X²Y term is based on the sum of the component rectangles in the member, where X is the shorter dimension and Y is the longer dimension of each component rectangle.  The designer should select the components such that the computed sum of X²Y terms is maximized.  Also note that the overhanging flange width used in this calculation should not be greater than three times the flange thickness.

3. Check if torsion may be neglected (threshold torsion)

4. Check the maximum allowable torsional moment and maximum allowable shear

5. Calculate the nominal torsional moment strength provided by the concrete under pure torsion (T’_c)

6. Calculate the nominal shear strength provided by the concrete without torsion (V’_c)

7. Calculate the nominal torsional moment under combined loading (T_c)

8. Calculate the nominal shear strength under combined loading (V_c)

9. Compute transverse reinforcement required for torsion, A_t,reqd (per face).  Note that the x₁ and y₁ terms used in this equation are the centerline dimensions of the closed stirrups used, and not necessarily equivalent to the X and Y terms previously used in the torsional constant.

10. Compute transverse reinforcement required for shear, A_v,reqd (total)

11. Calculate the combined transverse reinforcement, A_v/2 + A_t (per face)

12. Check minimum amount of web reinforcement required for ductility, A_v + 2A_t (total)

13. Calculate the maximum allowable stirrup spacing (use ACI requirements here)

14. Calculate the longitudinal torsional reinforcement, A_l (total) (does not appear to have a minimum required for this reinforcement)

15. Check the end regions for plate bending (PCI Handbook 5.4.3)

16. 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 (defined as the ratio of the height / width of the section resisting torsion) 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:

1. Calculate the factored loading at a section

2. Divide the length of the beam into the following regions:

   1. End region (face of bearing to H)

   2. Transition region (H to 2*H)

   3. Flexure region (remainder of beam)

3. Check the maximum allowable torsion in the end region

4. Calculate the required longitudinal reinforcement for flexural resistance

5. Calculate the required transverse reinforcement for one-way shear

6. Calculate the required vertical reinforcement required to resist plate bending in the end and transition regions

7. 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

8. 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

9. Provide sufficient reinforcement on the outer web face (basically Av/2)

10. 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.

Exceptions and notes:

1. Torsion resistance element(s) 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. 

2. The slender spandrel method is not a torsional analysis per se, it is simply a recognition that the stem of a ‘slender spandrel’ will fail in plate bending before the spandrel fails in torsion.

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, 8^th  Ed., PCI, Chicago, 2017.