Asymptotic Prediction of Energetic-Statistical Size Effect from Deterministic Finite-Element Solutions
Publication: Journal of Engineering Mechanics
Volume 133, Issue 2
Abstract
An improved form of a recently derived energetic-statistical formula for size effect on the strength of quasibrittle structures failing at crack initiation is presented and exploited to perform stochastic structural analysis without the burden of stochastic nonlinear finite-element simulations. The characteristic length for the statistical term in this formula is deduced by considering the limiting case of the energetic part of size effect for a vanishing thickness of the boundary layer of cracking. A simple elastic analysis of stress field provides the large-size asymptotic deterministic strength, and also allows evaluating the Weibull probability integral which yields the mean strength according to the purely statistical Weibull theory. A deterministic plastic limit analysis of an elastic body with a through-crack imagined to be filled by a perfectly plastic “glue” is used to obtain the small-size asymptote of size effect. Deterministic nonlinear fracture simulations of several scaled structures with commercial code ATENA (based on the crack band model) suffice to calibrate the deterministic part of size effect. On this basis, one can calibrate the energetic-statistical size effect formula, giving the mean strength for any size of geometrically scaled structures. Stochastic two-dimensional nonlinear simulations of the failure of Malpasset Dam demonstrate good agreement with the calibrated formula and the need to consider in dam design both the deterministic and statistical aspects of size effect. The mean tolerable displacement of the abutment of this arch dam is shown to have been approximately one half of the value indicated by the classical deterministic local analysis based on material strength.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
Bažant’s work, and Vořechovský and Novák’s visiting appointments at Northwestern University, were partly supported by U.S. National Science Foundation under Grant No. NSFCMS-9713944 to Northwestern University. Vořechovský’s visiting appointment was further supported by a grant from the Fulbright Foundation. Vořechovský and Novák’s work at the Brno University of Technology was also supported by the Czech Ministry of Education under Project Clutch No. UNSPECIFIED1K-04-110 and Project VITESPO No. UNSPECIFIED1ET409870411, from the Academy of Sciences of Czech Republic.
References
Ang, A. H.-S., and Tang, W. H. (1984). Probability concepts in engineering planning and design. Decision, risk and reliability, Vol. II, Wiley, New York.
Bartle, A., ed. (1985). “Four major dam failures re-examined.” Int. Water Power Dam Constr., 37(11), 33–36, 41–46.
Bažant, Z. P. (1995). “Scaling theories for quasibrittle fracture: Recent advances and new directions.” Fracture Mechanics of Concrete Structures, Proc., 2nd Int. Conf. on Fracture Mechanics of Concrete and Concrete Structures, FraMCoS-2, ETH, Zürich, F. H. Wittmann, ed., Aedificatio Publishers, Freiburg, Germany, 515–534.
Bažant, Z. P. (1997). “Scaling of quasibrittle fracture: Asymptotic analysis.” Int. J. Fract., 83(1), 19–40.
Bažant, Z. P. (2001). “Probabilistic modeling of quasibrittle fracture and size effect.” Proc., 8th Int. Conf. on Structural Safety and Reliability (ICOSSAR), Newport Beach, Calif., R. B. Corotis, G. I. Schueller, and M. Shinozuka, eds., Swets and Zeitinger, Balkema, 1–23.
Bažant, Z. P. (2002). Scaling of structural strength. Hermes Penton Science (Kogan Page), London, 2nd ed., Elsevier 2005 (French transl. with updates, Hermes Science Publ., Paris 2004).
Bažant, Z. P. (2004a). “Scaling theory for quasibrittle structural failure.” Proc. Natl. Acad. Sci. U.S.A., 101(37), 13397–13399.
Bažant, Z. P. (2004b). “Probability distribution of energetic-statistical size effect in quasibrittle fracture.” Probab. Eng. Mech., 19(4), 307–319.
Bažant, Z. P., and Chen, E.-P. (1997). “Scaling of structural failure.” Appl. Mech. Rev., 50(10), 593–627.
Bažant, Z. P., and Novák, D. (2000a). “Probabilistic nonlocal theory for quasibrittle fracture initiation and size effect. I. Theory.” J. Eng. Mech., 126(2), 166–174.
Bažant, Z. P., and Novák, D. (2000b). “Energetic-statistical size effect in quasi-brittle failure at crack initiation.” ACI Mater. J., 97(3), 381–392.
Bažant, Z. P., and Novák, D. (2001). “Proposal for standard test of modulus of rupture of concrete with its size dependence.” ACI Mater. J., 98(1), 79–87.
Bažant, Z. P., and Novák, D. (2003). “Stochastic models for deformation and failure of quasibrittle structures: Recent advances and new directions.” Computational Modeling of Concrete Structures, Proc., EURO-C Conf., St. Johann im Pongau, Austria, N. Bićanić, R. de Borst, H. Mang, and G. Meschke, eds., A. A. Balkema Publ., Lisse, The Netherlands, 583–598.
Bažant, Z. P., and Oh, B. H. (1983). “Crack band theory for fracture of concrete.” Mater. Constr. (Paris), 16, 155–177.
Bažant, Z. P., and Pang, S.-D. (2005a). “Effect of size on safety factors and strength of quasibrittle structures: Beckoning reform of reliability concepts.” Proc., Structural Engineering Convention (SEC 2005), J. M. Chandra Kishen and D. Roy, eds., Indian Institute of Science, Bangalore, 2–20.
Bažant, Z. P., and Pang, S. D. (2005b). “Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture.” Struct. Eng. Rep. No. 05-07/C441a, Northwestern Univ., also J. Mech. Phys. Solids, 54, (2006), in press.
Bažant, Z. P., and Pang, S.-D. (2006). “Statistical mechanics of failure risk for quasibrittle fracture.” Proc. Natl. Acad. Sci. U.S.A., 103(25), 9434–9439.
Bažant, Z. P., Pang, S.-D., Vořechovský, M., and Novák, D. (2007). “Energetic-statistical size effect simulated by SFEM with stratified sampling and crack band model.” Int. J. Numer. Analytical Methods in Engrg., (in press).
Bažant, Z. P., and Planas, J. (1998). Fracture and size effect in concrete and other quasibrittle materials, CRC Press, Boca Raton, Fla., Chap. 12.
Bažant, Z. P., and Xi, Y. (1991). “Statistical size effect in quasibrittle structures: II. Nonlocal theory.” J. Eng. Mech., 117(11), 2623–2640.
Bergmeister, K., Novák, D., and Pukl, R. (2004). “An advanced engineering tool for reliability assessment of concrete structures.” Progress in Structural Engineering, Mechanics and Computation, Proc., 2nd Int. Conf., Cape Town, South Africa, Zingoni, ed., Taylor and Francis Group, London, 899–904, ISBN 90-5809-568-1.
Bouchaud, J.-P., and Potters, M. (2000). Theory of Financial Risks: From Statistical Physics to Risk Management, Cambridge Univ., Cambridge, U.K.
Caner, F. C., and Bažant, Z. P. (2000). “Microplane model M4 for concrete: II. Algorithm and calibration.” J. Eng. Mech., 126(9), 954–961.
Carmeliet, J. (1994). “On stochastic descriptions for damage evolution in quasibrittle materials.” DIANA computational mechanics, G. M. A. Kusters and M. A. N. Hendriks, eds., Delft, The Netherlands.
Carmeliet, J., and Hens, H. (1994). “Probabilistic nonlocal damage model for continua with random field properties.” J. Eng. Mech., 120(10), 2013–2027.
Červenka, J., Bažant, Z. P., and Wierer, M. (2005). “Equivalent localization element for crack band approach to mesh-sensitivity in microplane model.” Int. J. Numer. Methods Eng., 62(5), 700–726.
Červenka, V., and Pukl, R. (1994). “SBETA analysis of size effect in concrete structures.” Size effect of concrete structures, H. Mihashi, H. Okamura, and Z. P. Bažant, eds., E & FN Spon, London, 323–333.
Červenka, V., and Pukl, R. (2006). ATENA—Computer program for nonlinear finite element analysis of reinforced concrete structures, Program documentation. Červenka Consulting, Prague.
de Borst, R., and Carmeliet, J. (1996). “Stochastic approaches for damage evaluation in local and non-local continua.” Size-scale effects in the failure mechanism of materials and structures, A. Carpinteri, ed., E & FN Spon, London, 242–257.
Fisher, R. A., and Tippett, L. H. C. (1928). “Limiting forms of the frequency distribution of the largest and smallest member of a sample.” Proc. Cambridge Philos. Soc., 24, 180–190.
Fréchet, M. (1927). “Sur la loi de probabilité de l’écart maximum.” Ann. Soc. Polon. Math., Cracow, 6, 93.
Gumbel, E. J. (1958). Statistics of extremes. Columbia Univ., New York.
Gutiérrez, M. A., and de Borst, R. (1999). “Deterministic and stochastic analysis of size effects and damage evaluation in quasibrittle materials.” Arch. Appl. Mech., 69, 655–676.
Gutiérrez, M. A., and de Borst, R. (2001). “Deterministic and probabilistic length-scales and their role in size-effect phenomena.” ICOSSAR 2001, Proc., 8th Int. Conf. on Structural Safety and Reliability, Calif., R. B. Corotis et al., eds., Swets and Zeitinger, Lisse, The Netherlands, 1–8.
Hartford, D. N. D., and Baecher, G. B. (2004). Risk and uncertainty in dam safety. Thomas Telford, London.
Levy, M., and Salvadori, M. (1992). Why buildings fall down? W. W. Norton, New York.
Novák, D., Bažant, Z. P., and Vořechovský, M. (2003b). “Computational modeling of statistical size effect in quasibrittle structures.” Proc., ICASP 9, Int. Conf. on Applications of Statistics and Probability in Civil Engineering, San Francisco, Der Kiureghian, A., et al., eds., Millpress, Rotterdam, 621–628.
Novák, D., Vořechovský, M., Lehký, D., Rusina, R., Pukl, R., and Červenka, V. (2005). “Stochastic nonlinear fracture mechanics finite element analysis of concrete structures.” ICOSSAR’05 Proc., 9th Int. Conf. on Structural Safety and Reliability, Augusti, G., Schuëller, G. I., Ciampoli, M. eds., Millpress, Rotterdam, The Netherlands, Rome, Italy, pp. 781–788.
Novák, D., Vořechovský, M., Pukl, R., and Červenka, V., (2001). “Statistical nonlinear analysis—Size effect of concrete beams.” Fracture Mechanics of Concrete and Concrete Structures, Proc., 4th Int. Conf. on Fracture Mechanics of Concrete Structures, FraMCoS-4, Cachan, France, de Borst R. et al., eds., Swets and Zeitlinger, Lisse, 823–830.
Novák, D., Vořechovský, M., and Rusina, R. (2003a). “Small-sample probabilistic assessment—Software FREET.” 9th Int. Conf. on Applications of Statistics and Probability in Civil Engineering-ICASP 9, San Francisco, 91–96.
Novák, D., Vořechovský, M., and Rusina, R. (2006). FREET—Feasible reliability engineering efficient tool, user’s and theory guides. Brno/Červenka Consulting, Czech Republic.
Pattison, K. (1998). “Why did the dam burst?” Invention and Technology, 14(1), 22–31.
Pukl, R., Červenka, V., Strauss, A., Bergmeister, K., and Novák, D. (2003). “An advanced engineering software for probabilistic-based assessment of concrete structure using nonlinear fracture mechanics.” Proc., ICASP 9, Int. Conf. on Applications of Statistics and Probability in Civil Engineering, A. Der Kiureghian, et al., eds. San Francisco, Millpress, Rotterdam, 1165–1171.
Pukl, R., Eligehausen, R., and Červenka, V. (1992). “Computer simulation of pullout test of headed anchors in a state of plane-stress.” Concrete design based on fracture mechanics, W. Gerstle and Z. P. Bažant, eds., ACI SP-134, Am. Concrete Inst., Detroit, 79–100.
Pukl, R., Novák, D., and Eichinger, E. M. (2002). “Stochastic nonlinear fracture mechanics analysis.” First Int. Conf. on Bridge Maintenance, Safety and Management, IABMAS 2002, Joan R. Casas, Dan M. Frangopol, and Andrzej S. Nowak, eds. Barcelona, Spain, 218–219.
RILEM Committee QFS (2004). “Quasibrittle fracture scaling and size effect.” Mater. Struct., 37(272), 547–586.
Vořechovský, M. (2004). “Stochastic fracture mechanics and size effect,” Ph.D. thesis, Brno Univ. of Technology, Brno, Czech Republic. ISBN 80-214-2695-0.
Vořechovský, M. (2006). “Interplay of size effects in concrete specimens under tension studied via computational stochastic fracture mechanics.” Int. J. Solids Struct., in press.
Vořechovský, M., and Chudoba, R. (2006). “Stochastic modeling of multi-filament yarns. II: Random properties over the length and size effect.” Int. J. Solids Struct., 43(3–4), 435–458.
Vořechovský, M., and Matesová, D. (2006). “Size effect in concrete specimens under tension: Interplay of sources.” EURO-C 2006 Computational Modelling of Concrete Structures, Meschke, de Borst, Mang, Bičanič, eds., Taylor & Francis Group, London, pp. 905–914.
Vořechovský, M., and Novák, D. (2004). “Modeling statistical size effect in concrete by the extreme value theory.” 5th Int. Ph.D. Symp. in Civil Engineering, Vol. 2, Walraven, J. et al., eds., A. A. Balkema, Delft, The Netherlands, pp. 867–875.
Weibull, W. (1939). “The phenomenon of rupture in solids.” Proc., Royal Swedish Institute of Engineering Research (Ingenioersvetenskaps Akad. Handl.), 153, 1–55.
Zech, B., and Wittmann, F. H. (1977). “A complex study on the reliability assessment of the containment of a PWR. Part II. Probabilistic approach to describe the behavior of materials.” Trans., 4th Int. Conf. on Structural Mechanics in Reactor Technology (held in San Francisco), H. T. A. Jaeger and B. A. Boley, eds., European Communities, Brussels, Vol. J, 1–11.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 133 • Issue 2 • February 2007
Pages: 153 - 162
Copyright
© 2007 ASCE.
History
Received: Jan 24, 2005
Accepted: Jan 31, 2006
Published online: Feb 1, 2007
Published in print: Feb 2007
Notes
Note. Associate Editor: Arvid Naess
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.