Available online at www.sciencedirect.com Procedia Engineering ProcediaProcedia Engineering 00 (2011) Engineering 28 000–000 (2012) 489 – 493 www.elsevier.com/locate/procedia 2012 International Conference on Modern Hydraulic Engineering Determination of Local Damage Probability in Concrete Structure Gu Peiyinga, Deng Changa, Tang Leia,a* a Nanjing Hydraulic Research Institute, Nanjing 210029, P R China Abstract Applying renormalization group theory to evaluate the safety of overall structure, local damage probability must be obtained at first According to the results of unit detection test and numerical simulation, the methods how to determine local damage probability was presented in the paper For small unit, meaning the unit size is far less than the maximum primitive cell or the structure size, if the unit is damaged by detection test, the local damage probability of it is defined as Otherwise it is defined as For large unit, the local damage probability is expressed by the ratio of volume summation of all damages to volume of the unit The process of local damage probability to be obtained was also introduced Based on the results of numerical simulation, the local damage probability how to be obtained was mainly expounded Concrete strength was assumed to obey two parameter Weibull distribution Therefore, damage probability based on stress of concrete structure may be developed to obey it The steps of obtaining local damage probability are stated Furthermore, determination of parameters was introduced in similar structure © 2012 Published by Elsevier Ltd Selection and/or peer-review under responsibility of Society for Resources, © 2011 Published by Elsevier Ltd Environment and Engineering Keywords: Local damage; probability; detection; numerical simulation; Weibull distribution; renormalization group theory; safety evaluation Introduction With rapid development of science and technology, hydraulic and civil engineerings usually play roles of lifeline engineerings nowadays The safety of hydraulic concrete structure is also very important Therefore, it is necessary to study the safety of overall structure There are two processes to determine damage of concrete structure now First, visual damage of the structure is inspected Second, detailed * Corresponding author Tel.: +86-25-8582-9647 E-mail address: pygu@nhri.cn 1877-7058 © 2012 Published by Elsevier Ltd Selection and/or peer-review under responsibility of Society for Resources, Environment and Engineering doi:10.1016/j.proeng.2012.01.756 490 Gu Peiying al / Procedia Engineering 28 (2012) 489 – 493 Author name /etProcedia Engineering 00 (2011) 000–000 damage is determined through the special means of NDT or detection with local damage But, it is difficulty that the safety evaluation of overall structure is determined now A new method to evaluate the safety of overall structure is advanced by author [1] The method is based on renormalization group theory The safety evaluation of overall structure involves the probability of it First of all, local damage probability of unit must be studied According to the composition of concrete, mechanical property of aggregate, interface and cement mortar is related to the factors of source, proportioning, mixing process and so on The mechanics parameters obey the rule of random distribution respectively Weibull distribution and logarithmic normal distribution are two basic forms of random distribution Therefore, the values of tensile strength and elastic modulus are determined by the way of random number generation The statistic of the mechanics parameters was assumed, furthermore, the parameters of Weibull distribution and histogram of logarithmic normal distribution were compared [2] The Weibull probability density function was applied to describing mechanical characteristics of concrete components by Van, Zhang, et al [3-5] The statistical characteristics of load and basic variable were provided by An, Guan, et al [6,7] The load included dead load, hydrostatic pressure and earthquake action The basic variable included the strength of steel bar and concrete, physical dimension of element According to meso-mechanics analysis of fiberreinforced composites by Coleman [8], it was assumed that Weibull distribution was applied to the distribution of fiber strength Then the statistical model of it was established This paper concerned with the method which determined local damage probability from two aspects In other words, according to the results of unit detection test and numerical simulation, local damage probability was determined The method based on the results of unit detection test According to renormalization group theory, a level primitive cell consists of units A level primitive cell consists of level primitive cells, and so on It is assumed that every unit or primitive cell is either broken or intact The damage probability of level primitive cell is obtained based on ones of unit That of level primitive cell is obtained based on ones of level primitive cell, and so on Furthermore, precondition of renormalization group theory is that the unit size is far less than the maximum primitive cell or the structure size Therefore, according to the unit size, two aspects are discussed about local damage probability based on the results of unit detection test In other words, the two aspects included the unit size satisfying and dissatisfying the precondition They are called small unit and large unit for short 2.1 Small unit When the unit size is so small that it is far less than the maximum primitive cell or the structure size, the assumption of renormalization group theory is supported Every unit is either broken or intact According to the results of unit detection test, if the unit is damage, it is considered broken Correspondingly, the local damage probability of it is defined as Otherwise it is defined as Generally, the units to be detected can’t be very small in actual engineering because of technology, cost and time So the precondition isn’t supported It is necessary to study how to determine the local damage probability of large unit based on the results of detection test 2.2 Large unit Same damages usually exist in concrete structure, such as crack, hole, loose concrete, falling block at 491 Gu Author Peiyingname et al // Procedia Procedia Engineering Engineering 00 28 (2011) (2012) 000–000 489 – 493 concrete surface, and damage in interface The volume of large unit is assumed to be V The local damage probability of it is expressed as p0 V (1) d V where, p0 is the damage probability of large unit; V is the volume of large unit; summation of all damages Vd may be expressed as V V V V V V where, V , V , V , V , and V are d c c h h l l s d is the volume (2) i i s V the volume summation of crack, hole, loose concrete, falling block at concrete surface, and damage in interface, respectively Based on large unit detection test, the process of local damage probability to be obtained is introduced as follows First, the units to be detected are divided according to some principles Second, special detection test is carried, and testing data is analyzed Third, the units are redivided into smaller ones when the discreteness of damages is large, and the second step is repeated Fourth, the damage probability is calculated Fifth, according to the results of test and criteria, the degree of unit local damage is classified To simplify the units of approximately equal damage probability are classified into the same group Then, the units of same group are given one damage probability which probability of occurrence is high The method based on the results of numerical simulation If testing data is incomplete, local damage probability is obtained by statistical analysis of similar engineerings or numerical simulation The method of statistical analysis is easy to understand The method of numerical simulation will be mainly expounded According to the existing probabilistic model of concrete, mechanics parameters such as strength, elastic modulus obey Weibull distribution Therefore, local damage probability may be assumed to obey Weibull distribution Weibull distribution is continuous It is usually applied in strength theory of brittle material And, it has been extensively applied in reliability field It has the advantage in small samples applications For concrete strength, the probability density function of three parameter Weibull distribution is expressed as f ( ) m m1 ( ( ) e 0 0 0 )m (3) where, is concrete strength; m is shape parameter, indicating dispersion degree of random quantity ; is the parameter related to average value, average value for short, sometime it is called scale parameter; is location parameter is the minimum value of If , the structure is safer Therefore, the formula (3) is simplified to two parameter Weibull distribution f ( ) m m1 ( ) ( ) e m 0 0 The probability distribution function of two parameter Weibull distribution is expressed as (4) 492 Gu Peiying al / Procedia Engineering 28 (2012) 489 – 493 Author name /etProcedia Engineering 00 (2011) 000–000 F ( ) e ( m ) 0 (5) Two parameter Weibull distribution is suited to brittle material, such as concrete, glass and ceramics Concrete strength obeys two parameter Weibull distribution in formula (5) The damage state of concrete structure can be indicated by stress state Along with the increase of stress, damage probability will increase Therefore, damage probability based on stress of concrete structure may be developed to obey formula (5), where, concrete strength is replaced by concrete stress Applying the formula (5), it is discussed how to obtain local damage probability by numerical simulation in this section The steps are as follows First, element stress is obtained by calculation Second, the parameters m and are determined Third, substituting , m and into the formula (5), the probability F ( ) of elements is obtained The probability F ( ) is also local damage probability defined in the paper According to the steps stated above, determination of parameters m and is vital The parameter m is applied to evaluating dispersity of materials strength The material is more uniform, the parameter m is greater The parameter is the strength value which failure probability is 63.21%, while average strength is the strength value which failure probability is 50% [9] So properly speaking, the parameter is greater than average strength Through fitting experimental data, the parameters m and may be determined Usual fitting methods include maximum likelihood estimation, linear fitting, and moment method [10] The experimental study of cement mortar was discussed in reference [11] Most of the parameter m varied in the range from 1.873 to 3.185 Only one result of it was greater because of applying quality control Based on the results of triaxial test, the parameters m and of rock were determined in reference [12] The parameter m was relevant to confining pressure It varied in the range from 2.945 to 5.584 However, the parameter m of plain concrete was 27.1 in reference [10] Conclusion According to the results of unit detection test and numerical simulation, the methods of determining the local damage probability was expounded in the paper For small unit, if the unit is damaged by detection test, the local damage probability of it is defined as Otherwise it is defined as For large unit, the local damage probability is expressed by the ratio of volume summation of all damages to volume of the unit Based on the results of numerical simulation, the local damage probability to be obtained is mainly presented The steps of obtaining local damage probability are stated The present results are the precondition of the new method to evaluate the safety of overall structure The method is based on renormalization group theory Therefore, the study how to determine the local damage probability of unit is the fundamental research of the method Acknowledgements The research is financially supported by national natural science foundation of china under grant no.51179107 and no.51079087 References [1] Gu Peiying, Deng Chang, Tang Lei Damaged probability of concrete for hydraulic structure based on renormalization group GuAuthor Peiyingname et al./ /Procedia ProcediaEngineering Engineering00 28(2011) (2012)000–000 489 – 493 theory [J] Hydro-scince and Engineering, 2010, ( ) : 1-5 [2] Li Jianbo, Lin Gao, Chen Jianyun, et al Mesoscopic numerical influence analysis of random mechanical parameter of concrete damage evolution [J] Journal of Architecture and Civil Engineering, 2007, 24 ( ) : 7-12 [3] Tang Xinwei, Zhang Chuhan Meso-scale modeling of concrete: effects of heterogeneity [J] Journal of Hydroelectric Engineering, 2009, 28 ( ) : 56-62 [4] Van Mier, Van Vliet, Wang T K Fracture mechanisms in particle composites: statistical aspects in lattice type analysis[J] Mechanics of Materials, 2002, 34 ( 11 ) : 705-724 [5] 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probability distribution [J] Journal of Materials Science & Engineering, 2007, 25 ( ) : 830-832 [11] Chen Yuhai Study on cumulative failure probability of brittle material [J] Journal of Hydraulic Engineering, 1996, ( ) : 45-48, 44 [12] Cao Wengui, Zhao Minghua, Liu Chengxue Study on the model and its modifying method for rock softening and damage based on Weibull random distribution [J] Chinese Journal of Rock Mechanics and Engineering, 2004, 23 ( 19 ) : 3226-3231 493 ... evaluation of overall structure involves the probability of it First of all, local damage probability of unit must be studied According to the composition of concrete, mechanical property of aggregate,... the methods of determining the local damage probability was expounded in the paper For small unit, if the unit is damaged by detection test, the local damage probability of it is defined as Otherwise... determine the local damage probability of large unit based on the results of detection test 2.2 Large unit Same damages usually exist in concrete structure, such as crack, hole, loose concrete,