Combined Torsion and Bending in Reinforced and Prestressed Concrete Beams Using Simplified Method for Combined Stress-Resultants. Paper by Khaldoun N. Rahal/AUTHOR’S CLOSURE
(ProQuest: … denotes formula omitted.)
Discussion by Himat Solanki
Professional Engineer, Building Dept., Sarasota County Government, Sarasota, FL.
The author has presented an interesting concept on a simplified model for design and analysis of reinforced and partially- and fully-prestressed concrete beams subjected to combined torsional and bending moments. The discusser would like to offer the following:
1. Design and analysis of membrane elements require uniformly distributed reinforcement in the x- and y-directions, either skewed or orthogonal.25 The application in beams subjected to combined torsional and bending moments appears to be questionable even though the author?s Table 3 indicates it, because the beam subjected to combined torsional and bending moments does not always have uniformly distributed reinforcement. Normally, it has top and bottom bars with stirrup reinforcement. Therefore, the discusser believes that any theory in beams subjected to combined torsional and bending moments is as good as author?s proposed method;
2. It is unclear how Fig. 3 was developed. Neither Eq. (18), (19), nor (20) indicates any degree of freedom;
3. To calculate shear stress v in a beam subjected to combined torsion and bending, there is some influence of shear reinforcement that cannot be ignored, and it can expressed as
4. The author has converted a solid section in to an arbitrary hollow section, but the solid section cracked and failed at higher loads than the hollow section. The smaller the ratio of torsion to bending, the larger the differences in failure load between the hollow and solid sections26;
5. In Appendix B, ETH Zurich Beam TBU has unsymmetrical longitudinal reinforcement (No. 4 bars at the top of No. 3 bars in the vertical elements and No. 8 bars at the bottom). Also from Fig. 8(c)) at Point TBU3, M [asymptotically =] 326 kN-m (240 kip-ft) and corresponding T [asymptotically =] 219 kN-m (162 kip-ft) and, at this point, the author?s SMCS curve predicts T [asymptotically =] 173, which is approximately 82% of Beam TBU3?s value. Therefore, the discusser believes that any theory in beams subjected to combined torsional and bending moments is as good as the author?s proposed method;
6. Although the author has simplified the t^sub d^ value in Eq. (11) based the study of numerous hollow beams subjected to torsion; torsion and bending; and torsion, bending, and shear, the wall thickness could be approximated to 3 to 5 times the concrete cover in the beams; and
7. The discusser has studied beams as specified in References 18, 20, 22, 23, and 26 through 35, and the degree of freedom was found to be from 0.05 to 0.47; therefore, the degree of freedom can not be ignored in either Fig. 3 or Eq. (1), (2), (18), (19), and (21).
25. Kuyt, B., “Zur Frage der Netzbewehrung von Fl?chentragwerken,” Beton und Stahlbetionbau, V. 59, July 1964, pp. 158-163.
26. Alnunaimi, A. S.; Al-Jabri, K. S.; and Hago, A., “Comparison between Solid and Hollow Reinforced Concrete Beams,” Materials and Structures, V. 41, No. 2, Mar. 2008, pp. 269-286.
27. Alnuaimi, A. S., and Bhatt, P., “Direct Design of Hollow Reinforced Concrete Beams: Part II, Experimental Investigation,” Structural Concrete, V. 5, No. 4, 2004, pp. 147-160.
28. McGee, D., and Zia, P., “Prestressed Concrete under Torsion, Shear and Bending,” ACI JOURNAL, Proceedings V. 73, No. 1, Jan. 1976, pp. 26-32.
29. Wu, C.-W., “Behavior of Reinforced Concrete Beams Subjected to Torsion and Bending,” MSc thesis, Civil Engineering Department, Chung Yuan Christian University, 2000. (in Chinese)
30. Jackson, N., and Esta?ero, R. A., “The Plastic Flow Law for Reinforced Concrete Beams under Combined Flexure and Torsion,” Magazine of Concrete Research, V. 23, No. 77, Dec. 1971, pp. 169-180, and discussion, V. 24, No. 81, Dec. 1982, pp. 242-246.
31. Evans, R. H., and Sarkar, S., “A Method of Ultimate Design of Reinforced Concrete Beams in Combined Bending and Torsion,” The Structural Engineer, V. 43, No. 10, Oct. 1965, pp. 337-344.
32. Ashour, A. S.; Shihata, S. A.; Akhtaruzaman, A. A.; and Wafa, F. F., “Prestressed High-Strength Concrete Beams under Torsional Bending,” Canadian Journal of Civil Engineering, V. 26, No. 2, Apr. 1999, pp. 197-207.
33. Semple, W. J., “Torsional Strength of Prestressed Concrete Hollow Beams Subjected to Combined Torsion and Bending,” MEngSc thesis, Department of Civil Engineering, University of Queensland, Brisbane, Queensland, Australia, 1971.
34. Swamy, N., “The Behavior and Ultimate Strength of Prestressed Concrete Hollow Beams under Combined Bending and Torsion,” Magazine of Concrete Research, V. 14, No. 40, Mar. 1962, pp. 13-24.
35. Nakayama, T., “An Experimental Study of Reinforced Concrete Beams Subjected to Flexural, Shear, and Torsion-Part 2: Beams with Stirrups,” Transactions of Architectural Institute of Japan, No. 282, Tokyo, 1979, pp. 23-35. (in Japanese)
The author thanks the discusser for his interest in the paper. The author provides the following clarifications:
1. The author disagrees with the discusser that analysis of membrane elements requires uniformly distributed orthogonal reinforcement. For example, Vecchio36 successfully modeled reinforced concrete beams subjected to shear and bending as a series of finite elements. The webs in these beams did not have any longitudinal reinforcement in the web and were modeled as membrane elements. Moreover, torsion design provisions in codes such as the ACI Building Code1 and the AASHTO LRFD Specifications2 assume that sections such as those shown in Fig. 5 are adequately reinforced for torsion in the longitudinal direction. The longitudinal reinforcement in these sections can hardly be described as uniformly distributed, but the thin walls are considered capable of resisting shear stresses. The same is true for sections subjected to shear. The discusser?s statement that “any theory in beams subjected to combined torsional and bending moments is as good as the author?s proposed method” has no basis, and he is invited to present such methods.
2. The discusser found the description of how Fig. 3 was developed not clear. In fact, the paper provided only a brief description of the development of the graphs to allow providing adequate details of how the application of the method is extended to the case of combined torsion and bending. The section on p. 402 describing the development of Fig. 3 (SMCS for pure shear), however, clearly cited Reference 9 for full details, and the discusser is directed to this reference for clarification. The author is not clear what the discusser means by the statement that Eq. (18) to (21) do not have any degrees of freedom.
3. The shear stress indeed depends on the amount of reinforcement, and that is reflected in the method. The shear stress due to torsion is obtained from Fig. 3 based on the reinforcement indexes in the x- and y-directions calculated using Eq. (18) and (19). The equation the discusser presented is dimensionally incorrect because the left-hand side is a force while the right-hand side is a force per unit distance.
4. Experimental results15,37 confirmed that solid cross sections crack at torsional moments larger than those of a solid but otherwise similar section, but found insignificant difference in the ultimate strength of these sections. At relatively lower torque-to-moment ratio, the difference between the strength of hollow and solid sections is slightly larger as stated by the discusser, but this did not bias the results of SMCS, as shown in Fig. 7 and 8.
5. The TBU specimens were Toronto tests4 and not ETH Zurich23 tests, as stated by the discusser. The ratio of the SMCS to observed strength of Beam TBU3 was 0.85, which led the discusser to conclude that any theory is as good as the author?s theory. This statement has no scientific basis. In fact, the ratio of the ACI-to-observed-strength obtained from the same graph the discusser used is 0.55, which proves the conclusion of the discusser is not correct.
6. The author did not provide reference to the studies that showed that the thickness t^sub d^ ranges from three to five times the concrete cover. On the other hand, there are numerous models to calculate this thickness based on equilibrium equations13,15,38 or more simplified models.11,13,39 The suggestion that td is related to the thickness of the cover does not make it an undebatable fact, and the author is entitled to propose an alternative model.
7. It is not clear what the discusser means by “degree of freedom.”
36. Vecchio, F. J., “Nonlinear Finite Element Analysis of Reinforced Concrete Membranes,” ACI Structural Journal, V. 86, No. 1, Jan.-Feb. 1989, pp. 26-35.
37. Hsu, T. T. C., “Torsion of Structural Concrete-Behavior of Reinforced Concrete Rectangular Members,” Torsion of Structural Concrete, SP-18, American Concrete Institute, Farmington Hills, MI, 1968, pp. 261-306.
38. Hsu, T. T. C., and Mo, Y. L., “Softening of Concrete in Torsional Members-Theory and Tests,” ACI JOURNAL, Proceedings V. 82, No. 3, May-June 1985, pp. 290-303.
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