Title:
Empirical Modeling of Cracking in Reinforced Concrete
Author(s):
Evan C. Bentz
Publication:
Structural Journal
Volume:
116
Issue:
3
Appears on pages(s):
233-242
Keywords:
cracking; design equations; direct tension; modulus of rupture; size effect; split cylinder; web-shear cracking
DOI:
10.14359/51714476
Date:
5/1/2019
Abstract:
A source of confusion in design codes is that multiple equations are often provided for concrete tensile strength—for example, one for modulus of rupture and another for direct tension strength. This paper proposes that these differences in tensile strength result from a size effect that is based on the volume of concrete in tension. Volumes larger than approximately 30 L (1 ft3) do not show this size effect and crack at the stress recommended by ACI for diagonal web-shear cracking. For smaller volumes, the tensile stress at cracking can be up to three or more times larger than this value. When the volume of the specimen is accounted for by the presented empirical size effect equation, the different test methods show consistent results. Using a set of 511 tension tests on plain or reinforced specimens of different types and sizes, the proposed unified tension model is justified. Finally, a simple design equation for flexural cracking is presented.
Related References:
1. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14),” American Concrete Institute, Farmington Hills MI, 2014, 519 pp.
2. Branson, D. E., “Instantaneous and Time-Dependent Deflections of Simple and Continuous Reinforced Concrete Beams,” HPR Report No. 7, Part 1, Alabama Highway Department, Bureau of Public Roads, Montgomery, AL, 1965.
3. Bischoff, P. H., “Rational Model for Calculating Deflection of Reinforced Concrete Beams and Slabs,” Canadian Journal of Civil Engineering, V. 34, No. 8, 2007, pp. 992-1002. doi: 10.1139/l07-020
4. ASTM C78/C78M-18, “Standard Test Method for Flexural Strength of Concrete (Using Simple Beam with Third-Point Loading),” ASTM International, West Conshohocken, PA, 2018, 5 pp.
5. Carrasquillo, R. L.; Nilson, A. H.; and Slate, F. O., “Properties of High-Strength Concrete Subjected to Short-Term Loads,” ACI Journal Proceedings, V. 78, No. 3, May 1981, pp. 171-178.
6. Zheng, W.; Kwan, A. K. H.; and Lee, P. K. K., “Direct Tension Test of Concrete,” ACI Structural Journal, V. 98, No. 1, Jan.-Feb. 2001, pp. 63-71.
7. Rossi, P.; Wu, X.; Le Maou, F.; and Belloc, A., “Scale Effect on Concrete in Tension,” Materials and Structures, V. 27, No. 8, 1994, pp. 437-444. doi: 10.1007/BF02473447
8. Davies, J. D., and Bose, D. K., “Stress Distribution in Splitting Tests,” ACI Journal Proceedings, V. 65, No. 8, Aug. 1968, pp. 662-669.
9. Kupfer, H.; Hilsdorf, H. K.; and Rusch, H., “Behavior of Concrete under Biaxial Stresses,” ACI Journal Proceedings, V. 66, No. 8, Aug. 1969, pp. 656-666.
10. Japaridze, L., “Stress-Deformed State of Cylindrical Specimens during Indirect Tensile Strength Testing,” Journal of Rock Engineering and Geotechnical Engineering, V. 7, 2015, pp. 509-518.
11. Masukawa, J., “Degradation of Shear Performance of Beams due to Bond Deterioration and Longitudinal Bar Cutoffs,” PhD thesis, University of Toronto, Toronto, ON, Canada, 2012, 175 pp.
12. Xie, L., “The Influence of Axial Load and Prestress on the Shear Strength of Web-Shear Critical Reinforced Concrete Elements,” PhD thesis, University of Toronto, Toronto, ON, Canada, 2009, 344 pp.
13. Gopalaratnam, V. S., and Shah, S. P., “Softening Response of Plain Concrete in Direct Tension,” ACI Journal Proceedings, V. 82, No. 3, May-June 1985, pp. 310-323.
14. Navalurkar, R. K., and Ansari, F., “Tensile Properties of High-Performance Concrete,” International Workshop on High-Performance Concrete, SP-159, American Concrete Institute, Farmington Hills, MI, 1996, pp. 283-293.
15. Vecchio, F. J., and Collins, M. P., “The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear,” ACI Journal Proceedings, V. 83, No. 2, Mar.-Apr. 1986, pp. 219-231.
16. Vecchio, F. J.; Collins, M. P.; and Aspiotis, J., “High-Strength Concrete Elements Subjected to Shear,” ACI Structural Journal, V. 91, No. 4, July-Aug. 1994, pp. 423-433.
17. Ruggiero, D. M. V., “The Behaviour of Reinforced Concrete Subjected to Reversed Cyclic Loads,” PhD dissertation, University of Toronto, Toronto, ON, Canada, 2015, 433 pp.
18. Proestos, G. T., “Influence of High-Strength Reinforcing Bars on the Behaviour of Reinforced Concrete Nuclear Containment Structures Subjected to Shear,” MASc thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 2014, 155 pp.
19. Kuchma, D., “The Influence of T-Headed Bars on the Strength and Ductility of Reinforced Concrete Walls.” PhD thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 1997, 269 pp.
20. Biederman, J. D., “The Design of Reinforced Concrete Shell Elements: An Analytical and Experimental Study,” MASc thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 1987, 59 pp.
21. Porasz, A., “An Investigation of the Stress-Strain Characteristics of High Strength Concrete in Shear,” MASc thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 1989, 171 pp.
22. Bentz, E. C., “Sectional Analysis of Reinforced Concrete,” PhD thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 2000, 316 pp.
23. Meyboom, J., “Experimental Investigation of Partially Prestressed, Orthogonally Reinforced Concrete Elements Subjected to Membrane Shear,” MASc thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 1987, 180 pp.
24. Kirschner, U., and Collins, M. P., “Investigating the Behaviour of Reinforced Concrete Shell Elements,” Publication No 86-09, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, Sept. 1986, 209 pp.
25. Belarbi, A., and Hsu, T. T. C., “Constitutive Laws of Softened Concrete in Biaxial Tension-Compression,” ACI Structural Journal, V. 92, No. 5, Sept-Oct. 1995, pp. 562-572.
26. Pang, X., and Hsu, T. T. C., “Constitutive Laws for Reinforced Concrete in Shear,” Research Report UHCEE 92-1, Department of Civil and Environmental Engineering, University of Houston, Houston, TX, Dec. 1992, 180 pp.
27. Zhang, L.-X., “Constitutive Laws of Reinforced Concrete Membrane Elements with High Strength Concrete,” PhD thesis, University of Houston, Houston, TX, Aug. 1995, 303 pp.
28. European Committee for Standardization, “CEN EN 1992-1-1:2004 Eurocode 2: Design of Concrete Structures- Part 1-1: General Rules and Rules for Buildings,” Brussels, Belgium, 2004, 225 pp.
29. Shioya, T.; Iguro, M.; Nojiri, Y.; Akiyama, H.; and Okada, T., “Shear Strength of Large Reinforced Concrete Beams,” Fracture Mechanics: Application to Concrete, SP-118, American Concrete Institute, Farmington Hills, MI, 1989, pp. 259-279.
30. Battista, D. D., “Minimum Reinforcement Requirements for High-Strength Concrete Slabs,” MASc thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 1992, 172 pp.
31. Rocco, C.; Guinea, G. V.; Planas, J.; and Elices, M., “The Effect of Boundary Conditions on the Cylinder Splitting Strength,” Proceedings of FRAMCOS-2, 1995, pp. 75-84.
32. Mirza, M. S., “An Investigation of Combined Stresses in Reinforced Concrete Beams,” PhD thesis, Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, QC, Canada, 1967, 234 pp.
33. Wright, P. J. F., and Garwood, F., “The Effect of the Method of Test on the Flexural Strength of Concrete,” Magazine of Concrete Research, V. 4, No. 11, 1952, pp. 67-76. doi: 10.1680/macr.1952.4.11.67
34. Podgorniak-Stanik, B. A., “The Influence of Concrete Strength, Distribution of Longitudinal Reinforcement Amount of Transverse Reinforcement and Member Size on Shear Strength of Reinforced Concrete Members,” MASc thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 1998, 369 pp.
35. Lubell, A. S., “Shear in Wide Reinforced Concrete Members,” PhD thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 2006, 475 pp.
36. Khavaran, A., “Investigation of Shear-Critical One-Way Concrete slabs Internally Reinforced with GFRP Bars,” PhD thesis, Department of Civil and Mineral Engineering, University of Toronto, Toronto, ON, Canada, 2018, 452 pp.
37. Mihaylov, B. I., “Behavior of Deep Reinforced Concrete Beams under Monotonic and Reversed Cyclic Load,” PhD thesis, European School for Advanced Studies in Reduction of Seismic Risk, Pavia, Italy, 2008, 379 pp.
38. Bentz, E. C., and Collins, M. P., “The Toronto Size Effect Series,” Shear in Structural Concrete, SP-328, American Concrete Institute, Farmington Hills, MI, 2018, pp. 1-12.
39. Lash, S. D., “Ultimate Strength and Cracking Resistance of Lightly Reinforced Beams,” ACI Journal Proceedings, V. 50, No. 6, Feb. 1953, pp. 573-582.
40. Hunter, M. D., “Towards Stochastic Finite Element Analysis of Reinforced Concrete Structures,” MASc thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 2016, 281 pp.
41. Bentz, E. C., and Buckley, S., “Repeating a Classic Set of Experiments on Size Effect in Shear of Members Without Stirrups,” ACI Structural Journal, V. 102, No. 6, Nov.-Dec. 2005, pp. 841-847.
42. Angelakos, D., “The Influence of Concrete Strength and Longitudinal Reinforcement Ratio on the Shear Strength of Large-Size Reinforced Concrete Beams with, and without, Transverse Reinforcement,” MASc thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 1999, 193 pp.
43. Sherwood, E., G., “One-Way Shear Behaviour of Large Lightly-Reinforced Concrete Beams and Slabs,” PhD thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 2008, 567 pp.
44. Yoshida, Y., “Shear Reinforcement for Large, Lightly-Reinforced Concrete Members,” MASc thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 2000, 161 pp.
45. Cao, S., “Size Effect and the Influence of Longitudinal Reinforcement on Shear Response of Large Reinforced Concrete Members,” MASc thesis, Department of Civil Engineering, University of Toronto, Toronto, ON, Canada, 2001, 193 pp.
46. Collins, M. P.; Bentz, E. C.; Quach, P. T.; and Proestos, G. T., “The Challenge of Predicting the Shear Strength of Very Thick Slabs,” Concrete International, V. 37, No. 11, Nov. 2015, pp. 29-37.
47. Mancuso, C., and Bartlett, F. M., “ACI 318-14 Criteria for Computing Instantaneous Deflections,” ACI Structural Journal, V. 114, No. 5, Sept.-Oct. 2017, pp. 1299-1310. doi: 10.14359/51689726