Golden Ratio in the Crack Pattern of Reinforced Concrete Structures
Publication: Journal of Engineering Mechanics
Volume 139, Issue 9
Abstract
Among all the proportions, the golden ratio has been taken into consideration for its geometrical and morphological properties, which can be found in a huge number of natural patterns, and therefore has been always considered as a model of beauty. Nevertheless, as discussed for the first time in the present paper, the cracking phenomenon of quasi-brittle materials also brings the golden ratio into play. In particular, such an irrational number appears when the average crack spacing of RC ties and beams is evaluated at different scales. This conjecture is corroborated by the results of a tension-stiffening model capable of predicting both the crack width and the crack spacing measured by the tests. In other words, it can be argued that the centrality of the golden ratio in the crack pattern of concrete members has profound physical meanings and reveals the existence of a size-effect law of crack spacing. The practical interest of this law lies in the possibility of predicting the crack pattern of large structures without knowing the material performances but by testing prototypes of the lower dimensions.
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References
Al-Fayadh, S., Engström, B., and Magnusson J. (2001). “Cracking behaviour of reinforced concrete tensile members.” RILEM TC 147-FMB Rep., Rilem, Bagneux, France, 1–121.
Bažant, Z. P. (1984). “Size effect in blunt fracture: Concrete, rock, metal.” J. Engrg. Mech. Div., 110(4), 518–535.
Beeby, A. W. (2004). “The influence of the parameter on crack widths.” Struct. Concr., 5(2), 71–83.
Beeby, A. W., et al. (2005). “Discussion: The influence of the parameter on crack widths.” Struct. Concr., 6(4), 155–165.
Borosnyói, A., and Balázs, G. L. (2005). “Models for flexural cracking in concrete: The state of the art.” Struct. Concr., 6(2), 53–62.
Burtscher, S., et al. (2004). “RILEM TC QFS quasibrittle fracture scaling and size effect—Final report.” Mater. Struct., 37(8), 547–568.
Carpinteri, A., Chiaia, B., and Ferro, G. (1995). “Size effects on nominal tensile strength of concrete structures: Multifractality of material ligaments and dimensional transition from order to disorder.” Mater. Struct., 28(6), 311–317.
Ciampi, V., Eligehausen, R., Bertero, V. V., and Popov, E. P. (1982). “Analytical model for concrete anchorages of reinforcing bars under generalized excitations.” UCB/EERC Rep. 82/23, Univ. of California at Berkeley, Berkeley, CA.
Fantilli, A. P., Ferretti, D., Iori, I., and Vallini, P. (1998). “Flexural deformability of reinforced concrete beams.” J. Struct. Eng., 124(9), 1041–1049.
Fédération Internationale du Béton (fib). (2000). “Bond of reinforcement in concrete.” Bulletin 10, fib, Lausanne, Switzerland.
Fédération Internationale du Béton (fib). (2010). “Model Code 2010—First complete draft—Volume 1.” Bulletin 55, fib, Lausanne, Switzerland.
Hejazi, M. (2005). “Geometry in nature and Persian architecture.” Build. Environ., 40(10), 1413–1427.
Lackner, R., and Mang, H. A. (2003a). “Scale transition in steel-concrete interaction. I: Model.” J. Eng. Mech., 129(4), 393–402.
Lackner, R., and Mang, H. A. (2003b). “Scale Transition in steel-concrete interaction. II: Applications.” J. Eng. Mech., 129(4), 403–413.
Livio, M. (2002). The golden ratio: The story of phi, the world’s most astonishing number, Broadway Books, New York.
Minelli, F., Tiberti, G., and Plizzari, G. A. (2011). “Durability and crack control in FRC RC elements: An experimental study.” Proc., Int. RILEM Conf. on Advances in Construction Materials through Science and Engineering, RILEM, Bagneux, France, 435–443.
Mitchell, D., Abrishami, H. H., and Mindess, S. (1996). “The effect of steel fibers and epoxy-coated reinforcement on tension stiffening and cracking of reinforced concrete.” ACI Mater. J., 93(1), 61–68.
Moorman, C. M., and Goff, J. E. (2007). “Golden ratio in a coupled-oscillator problem.” Eur. J. Phys., 28(5), 897–902.
Swamy, R. N., and Qureshi, S. A. (1971). “Strength, cracking and deformation similitude in reinforced T-beams under bending and shear.” ACI J. Proc., 68(3), 187–195.
Weibull, W. (1939). A statistical theory for the strength of materials, Swedish Royal Institute for Engineering Research, Stockholm, Sweden.
Zhi-Jian, Y., Shi-Jie, W., and Xiang-Jian, W. (1997). “Precise solutions of elastic-plastic crack line fields for cracked plate loaded by antiplane point forces.” Eng. Fract. Mech., 57(1), 75–83.
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Information
Published In
Journal of Engineering Mechanics
Volume 139 • Issue 9 • September 2013
Pages: 1178 - 1184
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Feb 10, 2012
Accepted: Sep 13, 2012
Published online: Sep 17, 2012
Published in print: Sep 1, 2013
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