COMPARISON BETWEEN THE NORMAL AND WEIBULL DISTRIBUTIONS FOR ANALYZING THE COMPRESSIVE STRENGTH OF THE CONCRETE

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1 1 COMPARISON BETWEEN THE NORMAL AND WEIBULL DISTRIBUTIONS FOR ANALYZING THE COMPRESSIVE STRENGTH OF THE CONCRETE Jorge Michael Colan 1, Paulo Eduardo Teodoro, Matheus Piazzalunga Neivock 3, Gilson Secco Riva 4, Sidiclei Foragini 5 Recebido e 07 de arço de 014; recebido para revisão e 9 de abril de 014; aceito e 18 de aio de 014; disponível on-line e XXX de novebro de 014. Espaço para QR Code KEYWORDS:: Concrete structures; Structural design; Noral distribuition; Weibull distribuition. ABSTRACT: The design of concrete structures and their atheatical odeling is rather subjective in its nature. Therefore, it is the purpose of this study to see whether the Weibull or Noral distributions can be applied to the copressive strengths of coercially batched ready-ixed concrete. The study was conducted during the year 011 in the city of Capo Grande / MS. The copressive strength was evaluated in 189 test saples at 8 days fro different concrete constructions conducted in the city. The trials took place as prescribed by NBR 5739 (ABNT, 007). To quantify the degree to which the proposed Weibull and Noral distributions fit the experiental data were utilized three goodness of fit tests: chi-squared, Anderson Darling, and Kologorov Sirnov. Based on the present investigation, the Weibull distribution can be applied to copressive strength data for concrete. This suggests that, despite the coplex processes involved in copression failure for a quasi-brittle coposite aterial like concrete, a statistic strength odel is possible. Furtherore, in coparing the goodness-of-fit, there is large practical difference between the Weibull and Noral distributions. The inforation is an iportant experiental addition to the body of knowl- edge regarding the failure of quasi-brittle aterials. * Authors contacts: 1 e-ail : [email protected] ( J. M. Colan ) Titulação...Função... do curso de Engenharia Civil da Universidade Anhanguera-UNIDERP - Capo Grande / MS e-ail : [email protected] ( P. E. Teodoro ) Discente do curso de Engenharia Civil da Universidade Anhanguera-UNIDERP - Capo Grande / MS 3 e-ail : [email protected] ( M. P. Neivock ) Engenheiro de Materiais, Mestre e Engenharia de Materiais e Modelage Mateática, docente do curso de Engenharia Civil da Universidade Anhanguera-UNIDERP - Capo Grande / MS. 4 e-ail : [email protected] ( G. S. Riva ) Titulação...Função..., MECFOR Tecnologia e Concreto Ltda. - Capo Grande / MS 5 e-ail : [email protected] ( S. Foragini ) Titulação...Função..., Centro de Ciências Exatas e tecnologia CCET da Universidade Federal de Mato Grosso do Sul UFMS.. ISSN: REEC - Todos os direitos reservados. 1. INTRODUÇÃO The design of concrete structures and their atheatical odeling is rather subjective in its nature. Ordered increasingly with respect to the cost of input inforation acquisition, a coparison shows that deterinistic analyses are the cheapest, however, their results are of liited validity. Probabilistic analyses provide the designer with extensive inforation including the distribution of the sought quantities, however, the input data acquisition is considerably expensive and, in soe cases such as the design or calculation of residual lifetie of unique structures, its use is irrelevant due to the lack of knowledge about the input paraeters (KARPÍSEK et al., 010). According to authors, the evaluation of the safety level in concrete structures should be

2 carried out considering the stochastic behavior of the ain paraeters involved. Particularly in concrete structures, the large variability of echanical and rheological paraeters ay give rise to significant deviations fro the expected behavior if a deterinistic approach is used. On the other hand, it is well known that the probability density function and its paraeters cannot be univocally defined. Heterogeneity of ceent concrete invariably causes recording of different strengths in saples collected fro any given concrete. This eans that a statistical test on strengths is required (Gibb & Harrison, 010). This is done usually by assuing a noral distribution and the paraeters to describe this curve, naely, average strength ( population ean) and standard deviation are obtained fro a very large nuber of the test results (Rajanae et al., 01). The concrete strength is considered atheatically as a rando variable for the Noral Distribution Curve and plotted as abscissa (x-axis) on this curve. (Tuidajski et al., 006). The fit of the noral distribution to concrete strengths is purely phenoenological. There is no theoretical basis in choosing the noral distribution. There are a nuber of other distributions that have theoretical justification and ay fit the data better. One such distribution is the Weibull distribution that based on weakest link statistics for brittle aterials, strengths are soeties odeled by this distribution (Weibull, 1951). The direction application this odel for concrete copressive strengths is coplicated by a nuber of factors. First, concrete is not a true brittle aterial. Concrete is a quasi-brittle aterial, which eans there is stress redistribution and energy release prior to failure (Bazant et al. 1991; Xxxx Bazant and Novak 000 a, 000 b). Furtherore, concrete copressive strength failure is caused by the slow extension of any cracks to for a crushed zone (Yip et al. 1995; Carpinteri et al. 1999) rather than by the rapid and unstable propagation of icro- cracks, which is the echanis expected fro weakest link theory (Tuidajski et al., 006). Therefore, it is the purpose of this study to see whether the Weibull or Noral distributions can be applied to the copressive strengths of coercially batched ready-ixed concrete. The inforation is an iportant experiental addition to the body of knowl- edge regarding the failure of quasi-brittle aterials.. MATERIAL AND METHODS The study was conducted during the year 011 in the city of Capo Grande / MS, olded in situ in several works. The copressive strength was assessed in 189 test saples after 8 days fro different concrete constructions ade in the city. The data used in this work were provided by the technological control laboratory MECFOR - Concrete Technology Ltda. The concrete was ade for six etering stations installed in the city of Capo Grande, MS. The concrete saples were collected according to the procedures of NBR NM 33 (ABNT, 1998), which were shaped speciens according to NBR 5738 (ABNT, 003) in large works that perfored quality control within the tie specified by the NBR 1655 (ABNT, 006) and tested in the laboratory according to NBR 5739 (ABNT, 007). Data were analyzed using Excel spreadsheet, which were estiated ean, standard deviation and coefficient of variation of saples shown in Table 1. TABLE 1. Average values, standard deviation and coefficient of variation of the copressive strength in 189 test saples at 8 days fro different concrete constructions in the city of Capo Grande / MS. Paraeter Value Maxiu Miniu 6.95 Average Standar deviation 3.65 Coefficient of variation (%) 10.38

3 For the noral distribution of concrete strength at 8 days (σ), the probability density function, (σ ), and the cuulative distribution function, F(σ ), are, respectively, given by Equations 1 and : Where µ is the ean, S is the standard deviation, and is the error function. The probability density function, (σ), and the cuulative distribution function, F (σ), for a Weibull distribution of σ, are given, respectively, by Equations 3 and 4: To quantify the degree to which the proposed Weibull and Noral distributions fit the experiental data were utilized three goodness of fit tests: chi-squared, Anderson Darling, and Kologorov Sirnov. The chi-squared test is an area test based on the probability density function, and the Anderson Darling and Kologorov Sirnov tests are distance tests based on the cuulative distribution function. These tests identifying exactly what the best distribution fits the data evaluated. Onde: µ: S: 1 f ( ) exp 0.5. S S Eq. [1] F( ) Eq. [] S : Toda variável deve ser identificada e atribuída unidade; f ( ) 0 1 exp 0 Eq. [3] F ) S ( Eq. [4] Onde: Toda variável deve ser identificada e atribuída unidade; 3. RESULTS AND DISCUSSION In the Table were outlined the results for Goodness of fit tests, where all reveal better atch the Weibull distribution to data copression strength of the concrete. Regardless of the test, a saller value of the goodness of fit tests statistic eans that there is an increased confidence that the data were produced by the proposed distribution. Table. Goodness of fit tests for Weibull and Noral distributions. Weibull distribution Noral distribution Chi-squared 9.50ª Anderson Darling 0.68ª 1.49 Kologorov Sirnov 0.04ª 0.07 ª: Adequability to test. Fro a statistical perspective, the chisquared test is a probability distribution test. The reliability of these kinds of tests increases as the nuber of experiental points increases. Both the Anderson Darling and Kologorov Sirnov tests are cuulative distribution tests which can reliably handle saller saple sizes. Therefore, in analyzing the results of Table, there is slightly ore reliability in the chi-squared rankings. Consequently, the Weibull distribution is overall a better representation than the noral distribution of the concrete copressive strength failure data (Figure 1 and ). Tuidajski et al. (006) when they copared these distributions which best represented the copression behavior of concrete coercial lots in Canada, did not identify significant differences between these differing fro the results of this research. Elgueta et al. (007) to analyzed which the distribution (Noral or Weibull) is best suited to the design paraeters of structural design, concluded that the Weibull distribution is ore reliable, resebling these results. The constants σ 0 and are known as Weibull paraeters and these paraeters deterine the distribution of breakdown voltage. 3

4 Equation 3 can be solved, and the result is a straight line with ln[-ln(1 - F)] on the ordinate and ln σ, on the abscissa. The Weibull distribuition allowed to find the breakdown voltage of the saples, in other words 63.0% of the test saples break at the MPa, while the paraeter =11:51 indicates hoogeneity of the copressive strength values (Figure 3). 4 X X FIGURE 1: Weibull distribuition for 189 values of copressive strength. FIGURE. Noral distribuition for 189 values of copressive strength. FIGURE 3: Weibull plot for data of copressive strength.

5 4. CONCLUSIONS Based on the present investigation, the Weibull distribution can be applied to copressive strength data for concrete. This suggests that, despite the coplex processes involved in copression failure for a quasi-brittle coposite aterial like concrete, a statistic strength odel is possible. Furtherore, in coparing the goodnessof-fit, there is large practical difference between the Weibull and Noral distributions. 5. ACKNOWLEDGEMENT The copany Mecfor Tecnologia e Concreto Ltda for cooperation in conducting the tests. ELGUETA, M.; DÍAZ, G.; ZAMORANO, S.; KITTL, P. On the use of the Weibull and the noral cuulative probability odels in structural design. Materials & Design, v. 8, n. 1, p , 007. GIBB, I.; HARRISON, T. Use of control charts in the production of concrete. ERMCO, p. KARPÍSEK, Z.; STEPÁNEK, P.; JURÁK, P. weibull fuzzy probability distribution for reliability of concrete structures. Engineering MECHANICS, v. 17, n. 5/6, p , 010. RAJANAME, N. P.; NATARAJA, M. C.; GANESA, T. P. A technical look at Individual test result criterion for concrete acceptance as per IS 456:000. The Indian Concrete Journal, v. 4, n. 1, p. 6-37, 01. TUMIDAJSKI, P.J.; FIORE, L.;. KHODABOCUS, T; LACHEMI, M.; PARI, R. Coparison of Weibull and noral distributions for concrete copressive strengths. Canadian Journal of Civil Engineer. v. 33, n. 1, p , BIBLIOGRAPHICS REFERENCES Concreto - Procediento para oldage e cura de corpos-de-prova. Rio de Janeiro, : Concreto Ensaio de copressão de corpos deprova cilíndricos. Rio de Janeiro, WEIBULL, W. A statistical distribution function of wide applicability. Journal of Applied Mechanics, v. 18, n.1, p , YIP W. K.; TAM, C. T.; GARY, G. Concrete strength evaluation through the use of sall diaeter cores. Magazine of Concrete Research, v. 40, n. 143, p , Concreto Preparo, controle e recebiento. Rio de Janeiro, 006. NM 33 - Concreto - Aostrage de concreto fresco. Rio de janeiro, BAZANT, Z. P.; XI, Y.; REID, S. G. Statistical size effect in quasi-brittle structures. I. Is Weibull theory applicable? Journal of Engineering Mechanics, v. 117, n. 11, p , BAZANT, Z.P.; NOVAK, D. Energetic-statistical size effect in quasibrittle failure at crack initiation. ACI Materials Journal, v. 97, n. 3, p , 000. BAZANT, Z. P.; NOVAK, D. Probabilistic nonlocal theory for quasibrittle fracture initiation and size effect. II: application. Journal of Engineering Mechanics, v. 16, n., p , 000. CARPINTERI, A.; FERRO, G.; MONETTO, I. Scale effects in uni-axially copressed concrete speciens. Magazine of Concrete Research, n. 51, v. 3, p. 17 5, 1999.