Abstract

This work presents an experimental study performed to investigate the effect of the size of the specimen on the fracture response of geometrically similar concrete notched beams. To the best of the authors’ knowledge, this study is the largest of its kind and includes beams of five different sizes. Twenty-eight beams were tested using a three-point bending (TPB) test setup designed according to draft guidelines. Five depths D, namely, 75, 150, 250, 500, and 1,000 mm, were considered, and the width b was kept constant and equal to 150 mm for all specimens. A new setup is proposed for large specimens. Load responses, peak loads, failure modes, fracture energy, and size-effect analysis are presented and discussed.

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Data Availability Statement

All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

Dr. Carloni would like to acknowledge the start-up fund provided by Case Western Reserve University, which was partially used to support this research. Mr. Mattia Baldassari, who helped design the test fixtures and cast the specimens, is gratefully acknowledged. Mr. Michael Butler, who worked as Laboratory Engineer at Case Western Reserve University and assisted the authors during the tests, is also gratefully acknowledged.

References

ACI (American Concrete Institute). 2016. Guide to external curing of concrete. Farmington Hills, MI: ACI.
ASTM. 2017a. Standard test method for air content of freshly mixed concrete by the pressure method. C231/C231M-17a. West Conshohocken, PA: ASTM.
ASTM. 2017b. Standard test method for splitting tensile strength of cylindrical concrete specimens. C496/C496M-17. West Conshohocken, PA: ASTM.
ASTM. 2020. Standard test method for slump of hydraulic-cement concrete. ASTM C143/C143M-12. West Conshohocken, PA: ASTM.
ASTM. 2021. Standard test method for compressive strength of cylindrical concrete specimens. ASTM C39/C39M-21. West Conshohocken, PA: ASTM.
Baietti, G., L. Carabba, G. Quartarone, C. Carloni, S. Manzi, and M. Bignozzi. 2019. “Fracture properties of alkali activated mortars.” In Proc., Int. Conf. on Fracture Mechanics of Concrete and Concrete Structures, edited by G. Pijaudier-Cabot, P. Grassl, and C. La Borderie, 23–26. Bayonne, France: ATENA and Lafarge Holcim.
Baietti, G., G. Quartarone, L. Carabba, S. Manzi, C. Carloni, and M. C. Bignozzi. 2020. “Use of digital image analysis to determine fracture properties of alkali-activated mortars.” Eng. Fract. Mech. 240 (Mar): 107313. https://doi.org/10.1016/j.engfracmech.2020.107313.
Bažant, Z. 1998. “Size effect in tensile and compression fracture of concrete structures: Computational modeling and design.” Fract. Mech. Concr. Struct. 3 (Mar): 1905–1922.
Bažant, Z. P. 1984. “Size effect in blunt fracture: Concrete, rock, metal.” J. Eng. Mech. 110 (4): 518–535. https://doi.org/10.1061/(ASCE)0733-9399(1984)110:4(518).
Bažant, Z. P. 2002. “Concrete fracture models: Testing and practice.” Eng. Fract. Mech. 69 (2): 165–205. https://doi.org/10.1016/S0013-7944(01)00084-4.
Bažant, Z. P., and E. Becq-Giraudon. 2002. “Statistical prediction of fracture parameters of concrete and implications for choice of testing standard.” Cem. Concr. Res. 32 (4): 529–556. https://doi.org/10.1016/S0008-8846(01)00723-2.
Bažant, Z. P., and G. Lewis. 2003. “Scaling of structural strength.” Appl. Mech. Rev. 56 (5): B70–B72. https://doi.org/10.1115/1.1584419.
Bažant, Z. P., and Y.-N. Li. 1994. “Penetration fracture of sea ice plate: Simplified analysis and size effect.” J. Eng. Mech. 120 (6): 1304–1321. https://doi.org/10.1061/(ASCE)0733-9399(1994)120:6(1304).
Bažant, Z. P., and J. Planas. 1998. Fracture and size effect in concrete and other quasibrittle materials. Oxfordshire, UK: Routledge.
Çağlar, Y., and S. Şener. 2016. “Size effect tests of different notch depth specimens with support rotation measurements.” Eng. Fract. Mech. 157 (May): 43–55. https://doi.org/10.1016/j.engfracmech.2016.02.028.
Carloni, C. 2014. “Analyzing bond characteristics between composites and quasi-brittle substrates in the repair of bridges and other concrete structures.” In Advanced composites in bridge construction and repair, 61–93. Amsterdam, Netherlands: Elsevier.
Carloni, C., G. Cusatis, M. Salviato, J.-L. Le, C. G. Hoover, and Z. P. Bažant. 2019a. “Critical comparison of the boundary effect model with cohesive crack model and size effect law.” Eng. Fract. Mech. 215 (Jun): 193–210. https://doi.org/10.1016/j.engfracmech.2019.04.036.
Carloni, C., M. Santandrea, and G. Baietti. 2019b. “Influence of the width of the specimen on the fracture response of concrete notched beams.” Eng. Fract. Mech. 216 (Jul): 106465. https://doi.org/10.1016/j.engfracmech.2019.04.039.
Carpinteri, A. 1994. “Multifractal scaling law for the nominal strength variation of concrete structures.” In Proc., Fracture Mechanics of Concrete Structures, FRAMCOS-2, edited by F. H. Wittmann, 193–206. Freiburg, Germany: Aedificatio Publishers.
Carpinteri, A. 2012. Vol. 5 of Mechanical damage and crack growth in concrete: Plastic collapse to brittle fracture. New York: Springer.
Carpinteri, A., B. Chiaia, and G. Ferro. 1995. “Size effects on nominal tensile strength of concrete structures: Multifractality of material ligaments and dimensional transition from order to disorder.” Mater. Struct. 28 (Apr): 311–317. https://doi.org/10.1007/BF02473145.
Cedolin, L., and G. Cusatis. 2008. “Identification of concrete fracture parameters through size effect experiments.” Cem. Concr. Compos. 30 (9): 788–797. https://doi.org/10.1016/j.cemconcomp.2008.05.007.
CEN (European Committee for Standardization). 2010. Eurocode-2: Design of concrete structures—Part 1-1: General rules and rules for buildings. EN 1992-1-1. Brussels, Belgium: CEN.
Coleman, T. F., and Y. Li. 1994. “On the convergence of interior-reflective newton methods for nonlinear minimization subject to bounds.” Math. Program. 67 (1–3): 189–224. https://doi.org/10.1007/BF01582221.
Coleman, T. F., and Y. Li. 1996. “An interior trust region approach for nonlinear minimization subject to bounds.” SIAM J. Optim. 6 (2): 418–445. https://doi.org/10.1137/0806023.
Cooper, T. P. 2000. “Size effects (macro-and micro-scale) on the fracture behavior of high strength concrete.” M.S. thesis, Dept. of Civil Engineering, Case Western Reserve Univ.
Cusatis, G., and E. A. Schauffert. 2009. “Cohesive crack analysis of size effect.” Eng. Fract. Mech. 76 (14): 2163–2173. https://doi.org/10.1016/j.engfracmech.2009.06.008.
Czernuschka, L.-M., R. Wan-Wendner, and J. Vorel. 2018. “Investigation of fracture based on sequentially linear analysis.” Eng. Fract. Mech. 202 (Oct): 75–86. https://doi.org/10.1016/j.engfracmech.2018.08.008.
Duan, K., X. Hu, and F. H. Wittmann. 2003. “Boundary effect on concrete fracture and non-constant fracture energy distribution.” Eng. Fract. Mech. 70 (16): 2257–2268. https://doi.org/10.1016/S0013-7944(02)00223-0.
Duan, K., X. Hu, and F. H. Wittmann. 2006. “Scaling of quasi-brittle fracture: Boundary and size effect.” Mech. Mater. 38 (1–2): 128–141. https://doi.org/10.1016/j.mechmat.2005.05.016.
Elices, M., G. Guinea, and J. Planas. 1992. “Measurement of the fracture energy using three-point bend tests: Part 3—Influence of cutting the p-δ tail.” Mater. Struct. 25 (Jul): 327–334. https://doi.org/10.1007/BF02472591.
Elices, M., and G. Planas. 2002. “21 prediction of size-effect based on cohesive crack models.” In Proc., Int. Union of Theoretical and Applied Mechanics (IUTAM) Symp. on Size-Scale Effects in the Failure Mechanisms of Materials and Structures, edited by A. Carpinteri, 309. London: Alberto Carpinteri.
Elices, M., and J. Planas. 1989. “Material models.” In Fracture mechanics of concrete structures, edited by L. Elfgren, 16–66. London: Chapman and Hall.
Ferrian, F., P. Cornetti, L. Marsavina, and A. Sapora. 2022. “Finite fracture mechanics and cohesive crack model: Size effects through a unified formulation.” Frattura ed Integrità Strutturale 16 (61): 496–509. https://doi.org/10.3221/IGF-ESIS.61.33.
fib. 2010. fib model code for concrete structures 2010. Hoboken, NJ: Wiley.
Guinea, G. 1993. “Correlation between the softening and the size effect curves.” In Size effect in concrete structures, 233–244. London: E and FN Spon.
Guinea, G., M. Elices, and J. Planas. 1997. “On the initial shape of the softening function of cohesive materials.” Int. J. Fract. 87 (2): 139–149. https://doi.org/10.1023/A:1007416926604.
Guinea, G., M. Elices, and J. Planas. 2000. “Assessment of the tensile strength through size effect curves.” Eng. Fract. Mech. 65 (2–3): 189–207. https://doi.org/10.1016/S0013-7944(99)00115-0.
Guinea, G., J. Pastor, J. Planas, and M. Elices. 1998. “Stress intensity factor, compliance and CMOD for a general three-point-bend beam.” Int. J. Fract. 89 (Jan): 103–116. https://doi.org/10.1023/A:1007498132504.
Guinea, G., J. Planas, and M. Elices. 1994. “A general bilinear fit for the softening curve of concrete.” Mater. Struct. 27 (Mar): 99–105. https://doi.org/10.1007/BF02472827.
Hillerborg, A. 1985a. “Results of three comparative test series for determining the fracture energy gf of concrete.” Mater. Struct. 18 (Sep): 407–413. https://doi.org/10.1007/BF02472416.
Hillerborg, A. 1985b. “The theoretical basis of a method to determine the fracture energy of concrete.” Mater. Struct. 18 (Jul): 291–296. https://doi.org/10.1007/BF02472919.
Hillerborg, A., M. Modéer, and P.-E. Petersson. 1976. “Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements.” Cem. Concr. Res. 6 (6): 773–781. https://doi.org/10.1016/0008-8846(76)90007-7.
Hoover, C. G., and Z. P. Bažant. 2013. “Comprehensive concrete fracture tests: Size effects of types 1 & 2, crack length effect and postpeak.” Eng. Fract. Mech. 110 (Sep): 281–289. https://doi.org/10.1016/j.engfracmech.2013.08.008.
Hoover, C. G., and Z. P. Bažant. 2014a. “Cohesive crack, size effect, crack band and work-of-fracture models compared to comprehensive concrete fracture tests.” Int. J. Fract. 187 (1): 133–143. https://doi.org/10.1007/s10704-013-9926-0.
Hoover, C. G., and Z. P. Bažant. 2014b. “Universal size-shape effect law based on comprehensive concrete fracture tests.” J. Eng. Mech. 140 (3): 473–479. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000627.
Hoover, C. G., Z. P. Bažant, J. Vorel, R. Wendner, and M. H. Hubler. 2013. “Comprehensive concrete fracture tests: Description and results.” Eng. Fract. Mech. 114 (Dec): 92–103. https://doi.org/10.1016/j.engfracmech.2013.08.007.
Hu, X., and K. Duan. 2008. “Size effect and quasi-brittle fracture: The role of FPZ.” Int. J. Fract. 154 (Nov): 3–14. https://doi.org/10.1007/s10704-008-9290-7.
Hu, X., and K. Duan. 2010. “Mechanism behind the size effect phenomenon.” J. Eng. Mech. 136 (1): 60–68. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000070.
Hu, X., J. Guan, Y. Wang, A. Keating, and S. Yang. 2017. “Comparison of boundary and size effect models based on new developments.” Eng. Fract. Mech. 175 (Apr): 146–167. https://doi.org/10.1016/j.engfracmech.2017.02.005.
Hu, X., and F. Wittmann. 2000. “Size effect on toughness induced by crack close to free surface.” Eng. Fract. Mech. 65 (2–3): 209–221. https://doi.org/10.1016/S0013-7944(99)00123-X.
Hu, X., and F. H. Wittmann. 1991. “An analytical method to determine the bridging stress transferred within the fracture process zone: I, general theory.” Cem. Concr. Res. 21 (6): 1118–1128. https://doi.org/10.1016/0008-8846(91)90072-P.
Hu, X., and F. H. Wittmann. 1992. “An analytical method to determine the bridging stress transferred within the fracture process zone: II, application to mortar.” Cem. Concr. Res. 22 (4): 559–570. https://doi.org/10.1016/0008-8846(92)90006-H.
Irwin, G. 1958. Vol. 6 of Handbuch der Physik. Berlin: Springer.
Karihaloo, B. 2002. “22 physical causes for size effect in concrete structures.” In Proc., Int. Union of Theoretical and Applied Mechanics (IUTAM) Symp. on Size-Scale Effects in the Failure Mechanisms of Materials and Structures, edited by A. Carpinteri, 325. London: Taylor and Francis.
Knott, J. F. 1973. Fundamentals of fracture mechanics. London: Butterworth Group.
Mihashi, H., N. Nomura, M. Izumi, and F. Wittmann. 1991. “Size dependence of fracture energy of concrete.” In Fracture processes in concrete rocks and ceramics, edited by J. G. M. van Mier, J. G. Rots, and A. Bakker, 441–450. London: E and FN Spon.
Ozbolt, J., and R. Eligehausen. 2002. “Size effect in concrete and reinforced concrete structures.” In Proc., Int. Union of Theoretical and Applied Mechanics (IUTAM) Symp. on Size-Scale Effects in the Failure Mechanisms of Materials and Structures, edited by A. Carpinteri, 290. London: Taylor and Francis.
Papaioannou, I., and D. Straub. 2021. “Variance-based reliability sensitivity analysis and the form -factors.” Reliab. Eng. Syst. Saf. 210 (Jun): 107496. https://doi.org/10.1016/j.ress.2021.107496.
Pastor, J., G. Guinea, J. Planas, and M. Elices. 1995. “Nueva expresión del factor de intensidad de tensiones para la probeta de flexión en tres puntos.” An. Mecánica Fractura 12 (May): 85–90.
Petersson, P.-E. 1981. Crack growth and development of fracture zones in plain concrete and similar materials. Lund, Sweden: Lund Institute of Technology.
Planas, J., and M. Elices. 1989. “Conceptual and experimental problems in the determination of the fracture energy of concrete.” In Fracture toughness and fracture energy: Test methods for concrete and rock, edited by H. Mihashi, H. Takahashi, and F. H. Wittman, 165–181, Rotterdam, Netherlands: Balkema.
Planas, J., M. Elices, and G. Guinea. 1992. “Measurement of the fracture energy using three-point bend tests: Part 2—Influence of bulk energy dissipation.” Mater. Struct. 25 (Jun): 305–312. https://doi.org/10.1007/BF02472671.
Planas, J., M. Elices, and G. Guinea. 1993. “Cohesive cracks versus nonlocal models: Closing the gap.” Int. J. Fract. 63 (Sep): 173–187. https://doi.org/10.1007/BF00017284.
Planas, J., G. Guinea, and M. Elices. 1997. “Generalized size effect equation for quasibrittle materials.” Fatigue Fract. Eng. Mater. Struct. 20 (5): 671–687. https://doi.org/10.1111/j.1460-2695.1997.tb00300.x.
Rackwitz, R., and B. Flessler. 1978. “Structural reliability under combined random load sequences.” Comput. Struct. 9 (5): 489–494. https://doi.org/10.1016/0045-7949(78)90046-9.
Rice, J. R. 1968. “Mathematical analysis in the mechanics of fracture.” In Vol. 2 of Fracture—An Advanced Treatise, edited by H. Liebowitz, 191–308. New York: Academic Press.
Rots, J. G., and S. Invernizzi. 2004. “Regularized sequentially linear saw-tooth softening model.” Int. J. Numer. Anal. Methods Geomech. 28 (7–8): 821–856. https://doi.org/10.1002/nag.371.
Santandrea, M., F. Focacci, C. Mazzotti, F. Ubertini, and C. Carloni. 2020. “Determination of the interfacial cohesive material law for srg composites bonded to a masonry substrate.” Eng. Fail. Anal. 111 (Apr): 104322. https://doi.org/10.1016/j.engfailanal.2019.104322.
Shah, S., S. E. Swartz, and B. Barr. 2002. Fracture of concrete and rock: Recent developments. Boca Raton, FL: CRC Press.
Trivedi, N., R. Singh, and J. Chattopadhyay. 2015. “A comparative study on three approaches to investigate the size independent fracture energy of concrete.” Eng. Fract. Mech. 138 (Apr): 49–62. https://doi.org/10.1016/j.engfracmech.2015.03.021.
Tschegg, E., H. Kreuzer, and M. Zelezny. 2003. “Fracture in concrete under biaxial loading–numerical evaluation of wedge splitting test results.” In Fracture mechanics of concrete structures, 455–460. Boca Raton, FL: CRC Press.
Vorel, J., and P. Kabele. 2019. “Inverse analysis of traction-separation relationship based on sequentially linear approach.” Comput. Struct. 212 (Feb): 125–136. https://doi.org/10.1016/j.compstruc.2018.10.005.
Walsh, P. 1972. “Linear fracture mechanics in orthotropic materials.” Eng. Fract. Mech. 4 (3): 533–541. https://doi.org/10.1016/0013-7944(72)90064-1.
Zhu, R., S. Y. Alam, and A. Loukili. 2020. “An experimental investigation on the correlation between the aggregate size effect and the structural size effect.” Eng. Fract. Mech. 234 (Jul): 107101. https://doi.org/10.1016/j.engfracmech.2020.107101.

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Go to Journal of Engineering Mechanics
Journal of Engineering Mechanics
Volume 150Issue 2February 2024

History

Received: Feb 19, 2023
Accepted: Sep 16, 2023
Published online: Nov 16, 2023
Published in print: Feb 1, 2024
Discussion open until: Apr 16, 2024

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Zahra Ameli, S.M.ASCE [email protected]
Ph.D. Student, Dept. of Civil and Environmental Engineering, Case Western Reserve Univ., Cleveland, OH 44106. Email: [email protected]
Ph.D. Student, Dept. of Civil and Environmental Engineering, Case Western Reserve Univ., Cleveland, OH 44106. ORCID: https://orcid.org/0000-0003-4128-8787. Email: [email protected]
Associate Professor, Dept. of Civil and Environmental Engineering, Case Western Reserve Univ., Cleveland, OH 44106 (corresponding author). ORCID: https://orcid.org/0000-0003-1663-7535. Email: [email protected]

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