Development of satisfaction curves to evaluate concrete mix design performance using a Bayesian probabilistic method Jang-Ho Jay Kim a,⇑ , Hung Duc Phan b , Byung-Yun Kim c , Jung-Wook Choi c , Tong-Seok Han a a School of Civil and Environmental Engineering, Yonsei University, Engineering Building #A, 134 Shinchon-dong, Seodaemun-gu, Seoul 120-794, South Korea b School of Civil and Environmental Engineering, Yonsei University, Concrete Structural Engineering Laboratory, A360, Engineering Building #A, 134 Shinchon-dong, Seodaemun-gu, Seoul 120-794, South Korea c Korea Concrete Institute Research Center, 635-4 Yoksamdong, Kangnam-gu, Seoul 135-703, South Korea article info Article history: Received 27 November 2010 Received in revised form 31 May 2011 Accepted 8 July 2011 Available online 8 September 2011 Keywords: Bayesian method Satisfaction curve Fragility curve Performance based design Concrete mix proportion Material parameters Performance satisfaction abstract Due to the restrictions of the current prescriptive design methods and standards, structural and material engineers sought to ﬁnd an innovative design method to implement creative and novel approaches in designing. One type of design method receiving a keen interest in civil engineering ﬁeld is performance based design (PBD) method. Presently, PBD is used in seismic design of structures and is speciﬁed in the code standards of earthquake resistant design. One speciﬁc approach in structural seismic design and damage evaluation is called fragility method developed by Shinozuka. Fragility method is developed based on Bayesian method, which is presented as the conditional probability of exceeding some limit state (minor, moderate, major, and collapse) for a given ground motion. In this study, the focus is to apply the Bayesian approach for the evaluation of concrete mixture design in assessing concrete material per- formance based on selected conditional parameters (i.e., safety, durability, etc.). The performance based evaluation of concrete mix proportion is presented as a satisfaction probability percentage (e.g., 0–100%) as a function of speciﬁed material parameter (i.e., water, water-to-cement ratio, cement content, etc.) for a speciﬁed performance criterion value (i.e., compressive strength, slump, elastic modulus, etc.). This relationship curve is titled satisfaction curve (SC). The paper presents a detailed explanation of develop- ing and drawing SC based on Bayesian method, an example of application of SC, and result comparisons between the results obtained using our SC method with ACI 214R-02 normal distribution probability method. The result veriﬁcation shows that concrete mix proportion performance evaluation using SC is reliable and the difference is within an acceptable range of accuracy of 5% difference. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Performance evaluation-oriented design methods have attracted great interest from structural and material design engineers world- wide in recent years, because of the need for a more ﬂexible and innovative design process. Design engineers could obtain additional information about the structural and material performance of their chosen materials using a performance evaluation-oriented design method, which could in turn increase cost effectiveness and result in more creative and optimal design results [1–3]. This concept has been commonly used in earthquake engineering ﬁeld to exam- ine seismic performance of designed structures and to evaluate seis- mic performance levels of structures for various design limit states (minor, moderate, major, and collapse) [4–7]. One type of seismic structural performance evaluation method is fragility method, which has been developed and utilized by Shinozuka et al. [4] and Singhal and Kiremidjian [6,7]. Fragility method has been effectively used in the seismic structural performance analysis to evaluate the probability of exceeding some structural performance limit state for a given ground motion intensity (i.e., peak ground motion (PGA, PGV, PGD)). Fragility method has been used in various applica- tions such as highway trafﬁc assessment and storm damage evalua- tion [5,6]. In this study, the same methodology has been used for an evaluation of satisfaction probability of a predeﬁned concrete mate- rial performance criterion (i.e., compressive strength, slump, carbonation depth, etc.) as a function of a speciﬁed concrete material parameter (i.e., water content, cement content, water-to-cement ratio, etc.) instead of an applied loading parameter (i.e., Peak Ground Acceleration (PGA), Peak Ground Velocity (PGV), Peak Ground Dis- placement (PGD), etc.). Since material performance satisfaction evaluation is different than damage assessment, we refer to the newly developed curve as the satisfaction curve (SC) rather than the fragility curve, which is the term used in structural seismic per- formance evaluation. Most of concrete mix proportion’s performances with respect to material parameters have been thoroughly studied in the past. 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.07.005 ⇑ Corresponding author. Tel.: +82 2 2123 5802; fax: +82 2 364 1100. E-mail address: jjhkim@yonsei.ac.kr (J.-H.J Kim). Construction and Building Materials 27 (2012) 578–584 Contents lists available at SciVerse ScienceDirect Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat