Strength Targeted PBMD of HSC based on One-parameter ...

Strength Targeted PBMD of HSC based on One-Parameter BayesianProbabilistic MethodHung Duc Phan, Jang-HoJay Kim Na-Hyun Yi, , Young-Jun You, Jong-Wook KimJournal of Advanced Concrete Technology, volume ( ), pp.10 2012 137-150

Compressive Stress-Strain Behavior of Small Scale Steel Fibre Reinforced High Strength Concrete CylindersPradeep Bhargava , Umesh K.Sharma Surendra. K. Kaushik, ,Journal of Advanced Concrete Technology, volume ( ), pp.4 2006 109-121

Particle Arrangements in Freshly-Mixed Mortars: Basis for ModelingSidney DiamondJournal of Advanced Concrete Technology, volume ( ), pp.5 2007 277-283

Mechanical Properties of Concrete and Reinforcement- State-of-the-art Report on HSC and HSS in Japan -Minehiro NishiyamaJournal of Advanced Concrete Technology, volume ( ), pp.7 2009 157-182

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Journal of Advanced Concrete Technology Vol. 10, 137-150, April 2012 / Copyright © 2012 Japan Concrete Institute 137

Scientific paper

Strength Targeted PBMD of HSC based on One-parameter Bayesian Probabilistic Method Hung Duc Phan1, Jang-Ho Jay Kim2, Na-Hyun Yi3, Young-Jun You4 and Jong-Wook Kim5

Received 30 June 2011, accepted 23 March 2012 doi:10.3151/jact.10.137

Abstract This paper is about application of recently proposed Performance Based Mixture Design (PBMD) for mix proportion design of high strength concrete (HSC) that can satisfy required performances. The PBMD method is a performance oriented concrete mix proportion design procedure easily applicable to any conditions and environments for a possible replacement to the current prescriptive design standards. Based on extensive experimental results obtained for various material and performance parameters of HSC, the application feasibility of the developed PBMD procedure for HSC has been verified. Also, the proposed PBMD procedure has been used to perform application examples to obtain desired target performances of HSC with optimum concrete mixture proportions using locally available materials, local envi-ronmental conditions, and available concrete production technologies. The validity and precision of HSC mix propor-tion design obtained using the PBMD method is verified with the experimental and ACI presented results to check the feasibility for actual design usage.

1. Introduction

The main goal of the paper is to present the application of Performance Based Mixture Design (PBMD) for high strength concrete (HSC) mix proportion design using Satisfaction Curves obtained from the one-parameter Bayesian method. In this PBMD, a design procedure of PBMD with detailed methodologies, mainly composed of three main stages with seven steps, was introduced to show a step-by-step mix proportion design procedure to satisfy performance requirements. Some new concepts such as: (1) inter-relationship parameters of various per-

formance levels of concrete, (2) Importance factors of parameters obtained based on the degree of each mate-rial parameter affecting a target criterion, (3) a method to combine multiple satisfaction curves, (4) Goodness value concept representing a common value for all con-crete material parameter values were introduced in the PBMD-procedure. The practical application possibility of using the PBMD procedure for normal strength con-crete (NSC) concrete mix proportion design was veri-fied (Kim et al. 2011, 2012).

Over the last few decades, the concrete usages have drastically changed from normal to high performance concrete (HPC). Particularly, applications requiring en-hanced workability, easier placement, reduced segrega-tion, improved durability properties, better early-age properties, better resistance to severe environments, etc have been implemented in constructions of longer span bridges and taller high-rise buildings (Holland 1993; Aïtcin 1998). High performance properties of HPC de-manded by its users include high-ultimate strength, high-early age strength, high modulus of elasticity, high durability, increased workability, better pumpability, easier finishability, improved placeablilty in hot and cold temperatures, and better controlled concrete hydra-tion. High strength concrete (HSC) is one type of HPCs such as self-compaction concrete (SCC), highly durable concrete (HDC), etc. HSC is commonly used to build high-rise buildings by reducing member size and super slim superstructures of long-span bridges, thereby creat-ing maximum useable spaces and reducing column sizes to resist dead loads, respectively. Currently, HSC is used to satisfy the specific needs of special construction ap-plications for enhanced durability, modulus of elasticity, and flexural strength for dams, grandstand roofs, marine foundations, parking garages, heavy duty industrial floors, etc (NRMCA 2001). ACI defines HSC as con-

1School of Civil and Environmental Engineering, Yonsei University, Seoul, South Korea. Currently, Faculty of Civil Engineering and Applied Mechanics, University of Technical Education of Ho Chi Minh City, Vietnam. 2Associate Professor, School of Civil and Environ-mental Engineering, Yonsei University, Seoul, South Korea. E-mail: [email protected] 3Doctoral Student, School of Civil and Environmental Engineering, Yonsei University, Seoul, South Korea. 4Doctoral Student, School of Civil and Environmental Engineering, Yonsei University, Seoul, South Korea. Researcher, Structure Research Department, Korea Institute of Construction Technology (KICT), Goyang-siSouth Korea. 5Doctoral Student, School of Civil and Environmental Engineering, Yonsei University, Seoul, South Korea. Research Manager, Industry Promotion Department, Technology Implementation Division, Korea Institute of Construction & Transportation Technology Evaluation and Planning (KICTEP), Anyang-si, South Korea.

H. D. Phan, J.-H. J. Kim, N.-H. Yi, Y.-J. You, and J.-W. Kim / Journal of Advanced Concrete Technology Vol. 10, 137-150, 2012 138

crete that has a specified compressive strength for de-sign of 41 MPa or greater (ACI 2008). Therefore, the HSC considered in this research was concrete with de-sign compressive strength within the range of 40-100 MPa. However, concrete with compressive strength exceeding 100 MPa (Ultra High Strength Concrete; UHSC) was excluded from the scope of this study (ACI 1999). Although there is sufficient number of study re-ports on HSC, most of the research works are centered on the individual property of HSC (Jaturapitakkul et al. 2004; Walraven 2009; Reinhardt and Rinder 2006; Mazloom 2008; Matsuda et al. 2009; Fujikake et al 2006), various special admixture materials for HSC (Shannag 2000; Hossain et al. 2008; Mazloom et al. 2004; Sakai et al. 2006; Brooks et al. 2000; Sui et al. 2004; Behnood and Ziari 2008), and structure member performance using HSC (Kimura et al. 2007; Tanimura et al. 2007; Sharma et al. 2007; Okamoto et al. 2004). However, reported study results on methods to evaluate the performance satisfactions of the performance criteria based on selected material parameters are quite limited. Therefore, in this study, the proposed design procedure of PBMD is applied to HSC mix proportion design to check the feasibility of using the proposed PBMD for HSC. Unlike NSC, various cementitious binders, chemical admixture, aggregate types in varying contents are used in mix proportion design to achieve desired high strength in HSC. However, since it is inefficient to come up with an optimum mix proportion design using various constituents of HSC by trial-and-error process, the aim of this study is to propose an efficient PBMD method to determine an optimal mix design of HSC. Based on the proposed PBMD process, a practical de-sign example was performed to explain the design proc-ess of using PBMD for HSC. Then, an analytical mix-ture design obtained from the PBMD method was veri-fied by comparing with experimental results to check its practical application possibility. Also, the analytical results were verified by comparing with the results ob-tained using the method presented in ACI 214R-02 (2002).

2. Application of PBMD to HSC

The proposed PBMD procedure is a process of coming up with an optimal concrete mixture design using Satis-faction Curve concept. The methodology of the pro-posed PBMD procedure is shown in Fig. 1 as a flow-chart with three main stages composed of (1) the pre-design stage of making assumptions based on a client’s requests, (2) the evaluation stage of initial designing and optimizations, and (3) the comparison stage for inter-mediate modifications and final determination of design. In the first stage, a design engineer must precisely un-derstand a client’s requests in the first step. Then, a de-signer makes initial decisions on material properties, design target criteria, a structure performance class, and an initial standard concrete mix proportion in the second

step. In the second stage, the performance evaluation of the initial design is performed by collecting data and developing Satisfaction Curves in the third step, and then multiple Satisfaction Curves are combined into a single Satisfaction Curve using Importance Factor and Goodness Value concepts in the fourth step. The last and third stage checks whether the standard concrete mix proportion satisfies the target performance criteria se-lected in the first step of the first stage. If the mix pro-portion does not satisfy all of performance criteria, then the mix proportion has to be modified so that the final mix proportion satisfies the target performances. The

Client’s requests1.Structure type2.Total cost3.Service life4.Special concrete : self-compacting concrete, high strength

concrete, crack controlling concrete, etc

Design enginner’s selection1.Target design strength, slump, water penetration depth,

shringkage, etc.2.Substrate material properties : cement type, aggregate

type, max coarse aggregate size, admixture type

Design enginner’s decision1.Performance class :

Excellent / Good / Moderate / Minimal2.Percentage of satisfaction P %

Selectstandard mix proportion

Data collectionUse existing results or obtain

new results when necessary

Modify mix proportion

Developsatisfaction curves for mix proportions

Combine satisfaction curvesOffer trends of satisfaction for next levels

Rangelimit of satisfaction

Rangelimit of satisfaction

Evaluatemix proportion’s

feasibility

Evaluatemix proportion’s

feasibility

Verifyclient’s

requirements

Verifyclient’s

requirements

Final Concrete Mixture Design

Set Importance Factor (I.F)Goodness Value (G.V)

Satisfactory

Unsatisfactory

Unsatisfactory

Unsatisfactory

Satisfactory

Satisfactory

Fig. 1 Main procedures of PBMD using Bayesian method.


comparison of originally selected performance criteria to actual evaluated performances, modifications and optimizations of the concrete mix proportion, and final verifications of a client’s requests are three tasks that are carried out in the fifth, sixth, and seventh steps, respec-tively, in the third stage.

The basic materials used to produce NSC (i.e., ce-ment, aggregates, water, and admixtures) are also used to produce HSC (Holland 1993; Aïtcin 1998). The most noticeable differences between NSC and HSC are in-creased cement content, reduced water content, and in-creased usage of chemical and mineral admixtures. Most HSC produced currently contain binding materials such as fly ash, silica fume, and ground granulated blast furnace slag in addition to cement to improve compres-sive strength and durability properties. At the same time, chemical admixtures such as high range water reducing agents are needed to ensure easy transportation, place-ment, and surface finishing of HSC. For HSC, a combi-nation of mineral and chemical admixtures is nearly always mandated to achieve a required strength (ACI 2008).

Since the constituent material types and proportions dictate the performance of HSC, the performance evaluations based on desired target criteria have to be performed to achieve the optimal mix design.

Therefore, in this study, the relationships between various parameters for strength criteria (e.g. compres-sive, splitting tensile, and flexural strengths) are deter-mined using parameter to performance relationships proposed in the PBMD procedure from previous studies (Kim et al. 2011, 2012). The main parameter for this study were selected as the mix proportions of super plasticizer (SP), silica fume (SC), cement (C), water (W), and aggregate contents (AC) based on the recom-mendations from ACI 211.4R-08 (2008). Also, in order to perform performance evaluations for the above men-tioned main parameters, the target strength and slump criteria are defined within the range values recom-mended in ACI 211.4R-08 (2008).

3. Experiment descriptions and satisfaction curves development

An experimental study was carried out to collect the data needed to develop the relationship between con-crete material parameters and various HSC criteria (i.e. compressive, splitting tensile, and flexural strengths) in the form of Satisfaction Curves. For the experiments, silica fume, a ACI 211.4R-08 (2008) recommended ce-mentitious binder that produces early age strength due to high pozzolanic reaction, was selected. In order to negate the negative characteristic of silica fume of re-ducing slump of concrete mix due to increase in its co-hesivity, super-plasticizer, a representative high range water reducing admixture, was used to improve worka-bility.

3.1 Experiment details 3.1.1 Materials The fine aggregates used for the experiments were river sand with a fineness modulus of 2.80 and a specific gravity of 2.60. Coarse aggregates with a maximum nominal size of 20 mm and a specific gravity of 2.65 were used in this study. Type I ordinary Portland cement (OPC) was used. A commercially available silica fume (SF) in powder form with an average of 93% silicon dioxide was used for casting of the SF concrete. The naphthalene formaldehyde sulfonated superplasticizer (SP) in solid type was used to cast the concrete. 3.1.2 Mix proportions The objective of the experiments was to produce HSC with compressive strength target of 70 MPa. According to ACI 211.4R-08 (2008), a common approach is to start with mix proportions that have been used successfully in the other projects with similar requirements. Given this starting point, trial mixtures can be cast under con-sistent environmental conditions to verify HSC per-formances of an actual HSC mix proportion used in previous projects. The present experimental research program is composed of various mix proportions to as-sess the effect of individual substrate material contribu-tions to the different performances of HSC. To deter-mine the isolated contribution of a mix design parameter on the properties of concrete, the variability of other mix design parameters should be eliminated by keeping them constant. Hence, the experimental program is composed of four sets of concrete mixes as shown in Table 1. In the first set, only SP parameter was varied using the dosage of 0%, 0.5%, 1.0% and 1.5% of the total binding materials. The purpose of this set was to choose a suitable dosage of SP to obtain a fresh concrete with over 180mm slump. Thus, 1% dosage of SP was used for the following set. In the second set, the dosage of cement replacement by SF was varied as 5%, 8%, 12% and 15% of the total binding materials to evaluate the effect of SF to the compressive strength of concrete. The third set comprised of four sub-sets was used to consider the variation of w/b parameter with the change of w and b, and the variation of sand-to-total aggregate ratio (s/a) parameter with the change of fine aggregate content (fa) and coarse aggregate content (ca). The mix-tures in the third set contained 8% SF. The fourth set was similar to the third set except that it contained 5% SF. 3.1.3 Specimens and test results For compressive and splitting tensile strength tests, 100x200 mm cylinder specimen was used. For flexural tensile strength test, 100x100x400 mm beam specimen was used. A symmetrical three-point loading setup with a beam span of 300 mm was used for the flexural test. The support was roller-hinge type. Fresh mixes were tested for workability by slump test according to KS F 2402 (2007). After casting, concrete specimens were


covered with wet burlap and polyethylene sheets and kept in the laboratory at an ambient room temperature for 24 hours. After demolding, all of the specimens were wet cured at an ambient room temperature until testing. Specimens were tested according to KS F 2405 (2005), KS F 2423 (2006), and KS F 2408 (2000) for compres-sive, splitting tensile, and flexural strength tests, respec-tively. The concrete compressive, splitting tensile, and flexural strength results for 32 mix proportions along with their measured slump values are tabulated in Table 2. The tensile strength criterion value was selected based on the split cylinder tensile strength value ( '

spf ) of 4.92 MPa calculated using ACI 363R-92 (1992) '

spf equation as shown in Eq. (1). For the flexural strength ( '

flf ) criterion value, since f’fl is generally between the

range of (1/5~1/8) of compressive strength, 'flf was

selected as 1/7 of 'cf (≒ 10.5 MPa).

' '

'0.59

(21 83 )sp c

c

f f MPaMPa f MPa=

< < (1)

where, 'cf is compressive strength, '

spf is splitting ten-sile strength.

The test results are tabulated in Table 2, which showed that the variation of dosage of SP causes a min-ute change in strength of concrete; however, high dos-age of SP causes segregation in fresh concrete resulting in a low strength of hardened concrete. As shown in Table 2, the strength test results of the trial mixes showed that SF5, SF8, 8WC2SW, 8WC28W, 8WC25C, 5WC25W, and 5WC28W exceeded the target strengths of '

cf = 70.0MPa, 'spf = 4.92MPa, and '

flf = 10.5MPa. However, among the mixes satisfying the strength criteria, only SF5 and SF8 satisfied the slump target criterion of 180 mm. In ACI 211.4R-08 (2008), the minimum slump requirement range for HSC with a high range water reducing agent such as super plasti-cizer is 177-203 mm (7-8 in). Therefore, in accordance with ACI 211.4R-08 (2008), the slump requirement of 180 mm was used in the study. Therefore, among the trial mix proportions, SF5 and SF8 can be considered as standard mix proportions. The strength of SF8 is slightly higher than that of SF5, but the cost to produce SF8 is also higher due to the higher price of SF than regular type I OPC.

Table 1 Concrete mix proportions. Mix proportion

Set Symbol w/b (%)

s/a (%)

w (kg/m3)

SP (kg/m3)

c (kg/m3) SF (kg/m3) fa

(kg/m3) ca

(kg/m3) SP0.0 0.30 0.42 143.0 0.00 476.0 0 764 1047 SP0.5 0.30 0.42 143.0 2.38 476.0 0 764 1047 SP1.0 0.30 0.42 143.0 4.76 476.0 0 764 1047 1

SP1.5 0.30 0.42 143.0 7.14 476.0 0 764 1047 SF5 0.30 0.42 143.0 4.76 452.2 23.80 764 1047 SF8 0.30 0.42 143.0 4.76 437.9 38.08 764 1047 SF12 0.30 0.42 143.0 4.76 418.9 57.12 764 1047 2

SF15 0.30 0.42 143.0 4.76 404.6 71.40 764 1047 8WC25W 0.25 0.42 119.0 4.76 437.9 38.08 764 1047 8WC28W 0.28 0.42 133.3 4.76 437.9 38.08 764 1047 8WC32W 0.32 0.42 152.3 4.76 437.9 38.08 764 1047 3-1

8WC35W 0.35 0.42 166.6 4.76 437.9 38.08 764 1047 8WC25C 0.25 0.42 143.0 5.72 526.2 45.76 764 1047 8WC28C 0.28 0.42 143.0 5.11 469.9 40.86 764 1047 8WC32C 0.32 0.42 143.0 4.47 411.1 35.75 764 1047 3-2

8WC35C 0.35 0.42 143.0 4.09 375.9 32.69 764 1047 8SA40FA 0.30 0.40 143.0 4.76 437.9 38.08 698 1047 3-3 8SA44FA 0.30 0.44 143.0 4.76 437.9 38.08 823 1047 8SA40CA 0.30 0.40 143.0 4.76 437.9 38.08 764 1146 3-4 8SA44CA 0.30 0.44 143.0 4.76 437.9 38.08 764 972 5WC25W 0.25 0.42 119.0 4.76 452.20 23.80 764 1047 5WC28W 0.28 0.42 133.3 4.76 452.20 23.80 764 1047 5WC32W 0.32 0.42 152.3 4.76 452.20 23.80 764 1047 4-1

5WC35W 0.35 0.42 166.6 4.76 452.20 23.80 764 1047 5WC25C 0.25 0.42 143.0 5.72 543.4 28.60 764 1047 5WC28C 0.28 0.42 143.0 5.11 485.2 25.54 764 1047 5WC32C 0.32 0.42 143.0 4.47 424.5 22.34 764 1047 4-2

5WC35C 0.35 0.42 143.0 4.09 388.1 20.43 764 1047 5SA40FA 0.30 0.40 143.0 4.76 452.20 23.80 698 1047 4-3 5SA44FA 0.30 0.44 143.0 4.76 452.20 23.80 823 1047 5SA40CA 0.30 0.40 143.0 4.76 452.20 23.80 764 1146 4-4 5SA44CA 0.30 0.44 143.0 4.76 452.20 23.80 764 972


3.2 Satisfaction curves development Satisfaction Curve for a particular criterion value is ob-tained by computing the conditional probabilities within

the range of that criterion value for various concrete material parameters. The conditional probability are defined as

[ | ]ik i kP P S s Y y= ≥ = (2)

where, Pik is probability of exceeding criterion level si for a given concrete material parameter yk; S is criterion random variable defined on criterion level vector S={s0,s1,…,sn}; Y is concrete material random variable.

The relationships between each concrete mixture pa-rameter (i.e., water-to-binder ratio w/b, water content w, binder content b, fine aggregate content fa, and coarse aggregate content ca) to the concrete compressive, split-ting tensile, and flexural strength performance require-ments of 70.0 MPa, 4.92 MPa, and 10.5 MPa, respec-tively, are shown in Figs. 2, 3, and 4, respectively. It is important to note that the Satisfaction Curves are devel-oped using the one-parameter Bayesian method (Kim et al. 2012).

The test results presented as Satisfaction Curves were used to evaluate the required satisfaction probability criteria of concrete parameters. The following section describes the procedures of performing PBMD using the developed Satisfaction Curves to come up with an opti-mum HSC mix proportion.

4. Design example

4.1 Assumption based on client’s request 4.1.1 Structure type A client’s request for concrete mix proportion to be used for construction of a new structure with a high safety level requirement would result in selection of HSC us-age and appropriate requirements. High safety level requirement is needed since it is more difficult to manu-facture concrete with high strength than ordinary strength, thereby requiring higher performance satisfac-tion criteria of HSC than NSC. The initial HSC com-pressive, splitting tensile, and flexural strength design targets are 70.0 MPa, 4.92 MPa and 10.5 MPa, respec-tively. The other concrete performance criteria are not considered for this example.

Table 2 Test results.

Compressive strength, '

cf (MPa) Splitting tensile strength,

'spf (MPa)

Flexural strength, 'flf (MPa) Symbol Slump

(cm) 1 2 3 1 2 3 1 2 3

SP0.0 0 59.5 58.6 56.0 4.73 4.61 4.54 9.93 9.79 9.40 SP0.5 14 60.5 57.9 56.8 4.75 4.71 4.59 10.08 9.69 9.52 SP1.0 22 63.6 59.0 56.7 4.79 4.75 4.61 10.25 9.85 9.51 SP1.5 25 52.2 49.6 45.3 4.57 4.43 4.34 9.83 9.44 8.95 SF5 21 73.0 71.8 69.3 5.02 4.96 4.89 10.94 10.70 10.41 SF8 18 74.2 72.8 69.6 5.03 5.00 4.91 11.07 10.85 10.38

SF12 16 71.4 69.6 68.2 4.98 4.91 4.86 10.59 10.34 9.77 SF15 14 68.5 66.6 65.1 4.91 4.87 4.81 10.33 9.99 9.47

8WC25W 13 75.2 74.7 72.2 5.09 5.03 4.98 11.21 11.11 10.82 8WC28W 16 74.9 73.3 70.9 5.08 5.01 4.96 11.24 11.06 10.61 8WC32W 21 70.9 67.9 65.5 4.95 4.86 4.74 10.54 10.29 9.81 8WC35W 26 66.6 61.4 55.8 4.89 4.71 4.64 8.99 8.21 8.37 8WC25C 12 78.3 77.1 74.8 5.20 5.13 5.07 11.26 11.13 10.94 8WC28C 15 76.6 74.5 72.1 5.11 5.07 4.96 11.19 10.97 10.65 8WC32C 19 71.1 69.5 66.9 4.95 4.89 4.80 10.48 10.23 9.91 8WC35C 21 68.3 63.7 60.6 4.89 4.75 4.61 9.77 9.46 8.99 8SA40FA 20 70.9 69.3 67.3 4.97 4.89 4.85 10.79 10.47 10.26 8SA44FA 17 74.9 72.1 70.5 5.06 5.01 4.93 11.07 10.89 10.63 8SA40CA 16 71.5 68.9 67.3 4.99 4.91 4.79 10.97 10.70 10.14 8SA44CA 20 75.1 72.9 70.4 5.05 4.98 4.93 11.12 10.87 10.67 5WC25W 15 75.0 72.9 71.4 5.05 5.00 4.95 11.18 10.95 10.72 5WC28W 19 73.6 72.0 70.5 5.05 4.98 4.93 11.04 10.85 10.53 5WC32W 24 68.5 66.1 62.1 4.89 4.78 4.73 10.53 10.05 9.72 5WC35W 29 59.2 56.2 52.0 4.57 4.40 4.30 8.92 8.53 8.03 5WC25C 14 76.8 74.3 72.2 5.13 5.06 4.95 11.32 11.12 10.70 5WC28C 17 74.1 73.4 70.8 5.03 4.99 4.92 11.22 10.96 10.60 5WC32C 22 69.6 65.5 63.9 4.91 4.86 4.79 10.44 10.31 10.04 5WC35C 24 61.1 59.3 57.3 4.63 4.52 4.46 9.17 8.90 8.45 5SA40FA 22 69.2 66.9 64.7 4.93 4.88 4.76 10.63 10.33 10.09 5SA44FA 19 73.8 71.7 70.2 5.03 4.97 4.91 11.00 10.72 10.53 5SA40CA 19 69.2 67.7 63.6 4.97 4.86 4.78 10.68 10.58 10.30 5SA44CA 23 73.5 71.8 70.1 5.04 4.99 4.92 11.04 10.84 10.61


4.1.2 Performance class and compatible satis-faction probability For this type of structure with a high safety level, the performance class requirement for concrete strength criteria should be within a “good performance class,” which is equivalent to the required probability of ex-ceeding this criterion of approximately 75 %. It is im-portant to note that performance satisfaction probability criterion value is selected based on the authors’ knowl-edge. However, more detailed considerations and stud-ies need to be performed in the future to determine cor-rect and precise satisfaction probability criteria. 4.1.3 Substrate material selection and standard mixture design The aim of this step is to obtain the appropriate data for Bayesian probability analysis. The data can be any type of available analytical and/or experimental results or even previous design results. Based on this underlying usage requirement and the target design criteria, the experimental results presented in the previous section are compatible for this design example. The standard mixture proportion design to achieve the required per-formance targets was SF5 or SF8. 4.2 Methodology of evaluation 4.2.1 Satisfaction curve developments For the material performance evaluations, data used to develop Satisfaction Curves must be compatible with a design engineer’s intention. The Satisfaction Curves developed in the experimental section (i.e., Figs. 2, 3, and 4) can be used for the example. 4.2.2 Satisfaction curve combination using goodness value and Importance factor Once all of Satisfaction Curves are developed from the required threshold criteria for a given mix proportion design, then a combining process of these Satisfaction Curves is performed to come up with a unified Satisfac-tion Curve for a mix proportion’s threshold criteria. Goodness value In the combining process, since all parameter values and ranges are different, they must be somehow calibrated so that the x-axis parameter value represents a common value in a combined Satisfaction Curve. In order to achieve this task, a calibrated parameter called Good-ness Value is introduced to represent and calibrate all concrete material parameter values. Using the normaliz-ing method, Goodness Value for each parameter is cal-culated as Gi as follows.

min

max min

i ii

i i

X XG

X X−

=−

(3)

where, Xi is concrete parameter; Ximin and Ximax are the minimum and maximum values of the limit range of the parameter Xi, respectively.

Using Eq. 3 with minimum and maximum values of

0

0.2

0.4

0.6

0.8

1

0.25 0.27 0.29 0.31 0.33 0.35

w/b

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

0

0.2

0.4

0.6

0.8

1

0.25 0.27 0.29 0.31 0.33 0.35

w/b

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

(a) water-to-binder ratio

(b)

0

0.2

0.4

0.6

0.8

1

110 130 150 170

w (kg/m3)Pr

obab

ility

of c

ompr

essi

ve

stre

ngth

sat

isfa

ctio

n

f 'c=70MPa

SF=8%

SF=5%

(b)

0

0.2

0.4

0.6

0.8

1

110 130 150 170

w (kg/m3)Pr

obab

ility

of c

ompr

essi

ve

stre

ngth

sat

isfa

ctio

n

f 'c=70MPa

SF=8%

SF=5%

(b) water content

(c)

0

0.2

0.4

0.6

0.8

1

400 430 460 490 520 550 580

b (kg/m3)

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

(c)

0

0.2

0.4

0.6

0.8

1

400 430 460 490 520 550 580

b (kg/m3)

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

(c) binder content

(d)

0

0.2

0.4

0.6

0.8

1

690 710 730 750 770 790 810 830

fa (kg/m3)

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

(d)

0

0.2

0.4

0.6

0.8

1

690 710 730 750 770 790 810 830

fa (kg/m3)

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

(d) fine aggregate content

(e)

0

0.2

0.4

0.6

0.8

1

970 1000 1030 1060 1090 1120 1150

ca (kg/m 3)

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f'c=70MPa

SF=8%

SF=5%

(e)

0

0.2

0.4

0.6

0.8

1

970 1000 1030 1060 1090 1120 1150

ca (kg/m 3)

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f'c=70MPa

SF=8%

SF=5%

(e) coarse aggregate content

Fig. 2 Compressive strength satisfaction curve.

f'c=70MPa

f'c=70MPa

f'c=70MPa

f'c=70MPa

f'c=70MPa


concrete material parameters described in Table 3, ma-terial parameters can be converted to Goodness Value of parameters with the range from 0 to 1. For example, Goodness value of w/b parameter is calculated as (w/b-0.25)/(0.35-0.25), where w/b has a value in the range from 0.25 to 0.35.

Importance factor The Importance Factors are determined based on the past knowledge of the parameter’s degree of importance to the mixture. The parameters that greatly influence the criterion are considered as the main parameters which would be assigned with higher values. The Importance Factor of each concrete mixture parameter is assumed in Table 4. As same as the satisfaction probability criterion selection, these values are only hypothetically selected and used for HSC criteria for this example and can be changed for other criteria and cases. Therefore, further studies and detailed considerations are required to de-termine correct and precise values of Importance Fac-tors. It is important to note that the total sum of all of Importance Factors used for a batch cannot exceed the value of 1.0. This is to prevent and to limit haphazard usage of Importance Factor where the selection of Im-portance Factor can be biased. Combination of satisfaction curves To create an absolute parameter value as a reference value in a design, minimum satisfaction requirement percentage is defined as the “reference value”. In order to give a hypothetical case as an example, all Goodness Values of material parameters of the standard mixture proportion can be equilibrated to 0.5 by shifting the Sat-isfaction Curves in x-axis direction to that reference point. Therefore, the Satisfaction Curve of each parame-ter will be shifted based on the reference of Goodness Value of 0.5 using compatible values shown in Table 3.

(a)

0

0.2

0.4

0.6

0.8

1

0.25 0.27 0.29 0.31 0.33 0.35

w/b

Prob

abili

ty o

f spl

ittin

g te

nsile

stre

ngth

sat

isfa

ctio

n

f 'sp=4.9MPa

SF=8%

SF=5%

(a)

0

0.2

0.4

0.6

0.8

1

0.25 0.27 0.29 0.31 0.33 0.35

w/b

Prob

abili

ty o

f spl

ittin

g te

nsile

stre

ngth

sat

isfa

ctio

n

f 'sp=4.9MPa

SF=8%

SF=5%


(b)

0

0.2

0.4

0.6

0.8

1

110 130 150 170

w (kg/m 3)

Prob

abili

ty o

f spl

ittin

g te

nsile

stre

ngth

sat

isfa

ctio

n

f 'sp=4.9MPa

SF=8%

SF=5%

(b)

0

0.2

0.4

0.6

0.8

1

110 130 150 170

w (kg/m 3)

Prob

abili

ty o

f spl

ittin

g te

nsile

stre

ngth

sat

isfa

ctio

n

f 'sp=4.9MPa

SF=8%

SF=5%

(b) water content

(c)

0.0

0.2

0.4

0.6

0.8

1.0

400 430 460 490 520 550 580

b (kg/m 3)

Prob

abili

ty o

f spl

ittin

g te

nsile

stre

ngth

sat

isfa

ctio

n

f 'sp=4.9MPa

SF=8%

SF=5%

(c)

0.0

0.2

0.4

0.6

0.8

1.0

400 430 460 490 520 550 580

b (kg/m 3)

Prob

abili

ty o

f spl

ittin

g te

nsile

stre

ngth

sat

isfa

ctio

n

f 'sp=4.9MPa

SF=8%

SF=5%

(c) binder content

(d)

0.0

0.2

0.4

0.6

0.8

1.0

690 710 730 750 770 790 810 830

fa (kg/m3)

Prob

abili

ty o

f spl

ittin

g te

nsile

stre

ngth

sat

isfa

ctio

n

f'sp=4.9MPa

SF=8%

SF=5%

(d)

0.0

0.2

0.4

0.6

0.8

1.0

690 710 730 750 770 790 810 830

fa (kg/m3)

Prob

abili

ty o

f spl

ittin

g te

nsile

stre

ngth

sat

isfa

ctio

n

f'sp=4.9MPa

SF=8%

SF=5%


(e)

0.0

0.2

0.4

0.6

0.8

1.0

970 1000 1030 1060 1090 1120 1150

ca (k g/m 3)

Prob

abili

ty o

f spl

ittin

g te

nsile

stre

ngth

sat

isfa

ctio

n

f 's p=4.9MPa

SF=8%

SF=5%

(e)

0.0

0.2

0.4

0.6

0.8

1.0

970 1000 1030 1060 1090 1120 1150

ca (k g/m 3)

Prob

abili

ty o

f spl

ittin

g te

nsile

stre

ngth

sat

isfa

ctio

n

f 's p=4.9MPa

SF=8%

SF=5%


Fig. 3 Splitting tensile strength satisfaction curve.

Table 3 The concrete material parameter values.

Concrete material parameter Min value

Max value

Shifted compatible

value w/b ratio

Water content (kg/m3) Binder content (kg/m3) Fine aggregate content

(kg/m3) Coarse aggregate content

(kg/m3)

0.25 119.0 408.6 698.0

972.0

0.35 133.3 572.0 822.6

1146.0

+0.00 +0.00 +0.09 - 0.03

+0.07

Table 4 The Importance factors of concrete parameters.

Parameter Importance Factor w/b ratio 0.50

Water content 0.10 Binder content 0.30

Fine aggregate content 0.05 Coarse aggregate content 0.05

… 0.00 Total 1.00

f'sp=4.9MPa

f'sp=4.9MPa

f'sp=4.9MPa

f'sp=4.9MPa

f'sp=4.9MPa


For example, w/b parameter of the standard mix propor-tion is 0.3. The shifted compatible value of Goodness Value of w/b parameter is obtained as 0.5-(0.3-0.25)/(0.35-0.25)=0.

Moreover, in material design or characteristics, the global trend is not always in the same direction. In order to combine the material parameter Satisfaction Curves into a single overall Satisfaction Curve for each level, then the overall trends have to be compatible. With re-spect to the positive trend conversion, a conversion is required for w/b, w, and ca parameters when strength criterion is applied. For example, the increase in w/b parameter results in decreased strength, then the trend can be converted to the “positive” trend by replacing Gi value with (1-Gi) while strictly maintaining the com-patible probability P%. Now, the x-axis values of the compressive strength, splitting tensile strength, and flexural strength Satisfaction Curves shown in Figs. 2, 3, and 4 can be recalculated to draw Figs. 5, 6, and 7, re-spectively.

The combination process is followed by calculating the probability at each Goodness Value as

kj i iP Pα= (4)

where, kjP is probability value of criterion j at level

k ; iP is probability of material parameters; iα are Importance Factors of material parameters in the mix-ture design for a specified criterion; the double subscript i, i means summation of i iPα values.

The purpose of the combining process is to help de-sign engineers to clearly understand the performance trend in each level of material hierarchy. Another goal is to optimize concrete mixture proportion based on the availability and quality of substrates to satisfy the type of concrete needed for a required structure type accord-ing to client’s requirements and structural usage. The final combined Satisfaction Curves, which represent an overall trend of concrete mixture satisfying compressive, splitting tensile, and flexural strength criteria of 70.0 MPa, 4.92 MPa, and 10.5 MPa, respectively, are shown in Fig. 8.

In this example, since the selected performance crite-ria are compressive, splitting tensile, and flexural strength criteria, these strength Satisfaction Curves are combined to obtain the final strength Satisfaction Curve of the HSC mix proportion shown in Fig. 9. This com-bining process is performed by using Eq. 4 and consid-ering the Importance Factors of criteria assumed in Ta-ble 5. Once, all of these one-parameter Satisfaction

(a)

0

0.2

0.4

0.6

0.8

1

0.25 0.27 0.29 0.31 0.33 0.35

w /b

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(a)

0

0.2

0.4

0.6

0.8

1

0.25 0.27 0.29 0.31 0.33 0.35

w /b

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%


(b)

0

0.2

0.4

0.6

0.8

1

110 130 150 170

w (kg/m 3)Pr

obab

ility

of f

lexu

ral

stre

ngth

sat

isfa

ctio

n

f 'fl=10.5MPa

SF=8%

SF=5%

(b)

0

0.2

0.4

0.6

0.8

1

110 130 150 170

w (kg/m 3)Pr

obab

ility

of f

lexu

ral

stre

ngth

sat

isfa

ctio

n

f 'fl=10.5MPa

SF=8%

SF=5%

(b) water content

(c)

0

0.2

0.4

0.6

0.8

1

400 430 460 490 520 550 580

b (kg/m3

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(c)

0

0.2

0.4

0.6

0.8

1

400 430 460 490 520 550 580

b (kg/m3

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(c) binder content

(d)

0

0.2

0.4

0.6

0.8

1

690 710 730 750 770 790 810 830

fa (kg/m 3)

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(d)

0

0.2

0.4

0.6

0.8

1

690 710 730 750 770 790 810 830

fa (kg/m 3)

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%


(e)

0

0.2

0.4

0.6

0.8

1

970 1000 1030 1060 1090 1120 1150

ca (kg/m3)

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(e)

0

0.2

0.4

0.6

0.8

1

970 1000 1030 1060 1090 1120 1150

ca (kg/m3)

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%


Fig. 4 Flexural strength satisfaction curve.

Table 5 The Importance factors for safety satisfaction curve.

Parameter Importance Factor Compressive strength 0.50

Splitting tensile strength 0.25 Flexural strength 0.25

… 0.00 Total 1.00

f'fl=10.5MPa

f'fl=10.5MPa

f'fl=10.5MPa

f'fl=10.5MPa

f'fl=10.5MPa

b (kg/m3)


(a)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gw/b

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

(a)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gw/b

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%


(b)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gw

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

(b) water content

( )

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gb

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

( )

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gb

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

(c) binder content

(d)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gfa

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f'c=70MPa

SF=8%

SF=5%

(d)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gfa

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f'c=70MPa

SF=8%

SF=5%


( )

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gca

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%

( )

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gca

Prob

abili

ty o

f com

pres

sive

st

reng

th s

atis

fact

ion

f 'c=70MPa

SF=8%

SF=5%


Fig. 5 Compressive strength satisfaction curve of good-ness value.

(a)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gw/b

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'fl=10.5MPa

SF=8%

SF=5%

(a)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gw/b

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'fl=10.5MPa

SF=8%

SF=5%


(b)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

GwPr

obab

ility

of f

lexu

ral

stre

ngth

sat

isfa

ctio

n

f 'f l=10.5MPa

SF=8%

SF=5%

(b)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

GwPr

obab

ility

of f

lexu

ral

stre

ngth

sat

isfa

ctio

n

f 'f l=10.5MPa

SF=8%

SF=5%

(b) water content

(c)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gb

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(c)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gb

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(c) binder content

(d)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gfa

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(d)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gfa

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%


(e)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gca

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(e)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gca

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%


Fig. 6 Splitting tensile strength satisfaction curve of goodness value.

f'sp=4.9MPa

f'sp=4.9MPa

f'sp=4.9MPa

f'sp=4.9MPa

f'sp=4.9MPa

f'c=70MPa

f'c=70MPa

f'c=70MPa

f'c=70MPa

f'c=70MPa

Pro

babi

lity

of s

plitt

ing

tens

ile

stre

ngth

sat

isfa

ctio

n Pr

obab

ility

of s

plitt

ing

tens

ile

stre

ngth

sat

isfa

ctio

n Pr

obab

ility

of s

plitt

ing

tens

ile

stre

ngth

sat

isfa

ctio

n P

roba

bilit

y of

spl

ittin

g te

nsile

st

reng

th s

atis

fact

ion

Prob

abili

ty o

f spl

ittin

g te

nsile

st

reng

th s

atis

fact

ion


Curves are combined for a given level, then the process can be repeated in the next higher level until the Satis-faction Curves of safety, constructability, durability, and damageability are developed to come up with overall satisfaction trends for a given mix proportion.

4.3 Mix proportion design verification and modification 4.3.1 Actual performance evaluation to original request comparison The standard mixture proportion is used to evaluate the material satisfaction probabilities, which must satisfy the required probability satisfaction P%. In this example, the Goodness Values of all parameters of an example standard mixture are selected to be equal to 0.5, which can be considered as a “reference point”. The corre-sponding probability satisfaction for 70 MPa compres-

(a)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gw /b

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(a)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gw /b

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%


(b)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gw

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(b)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gw

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(b) water content

(c)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gb

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(c)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gb

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(c) binder content

(d)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gfa

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(d)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gfa

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%


(e)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gca

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%

(e)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8 1.0

Gca

Prob

abili

ty o

f fle

xura

lst

reng

th s

atis

fact

ion

f 'f l=10.5MPa

SF=8%

SF=5%


Fig. 7 Flexural strength satisfaction curve of goodness value.

(a)

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Fig. 8 Combined satisfaction curve of concrete mix pro-portion.

f'c=70MPa

f'c=70MPa

f'c=70MPa

f'fl=10.5MPa

f'fl=10.5MPa

f'fl=10.5MPa

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sive strength, 4.92 MPa splitting tensile strength, and 10.5 MPa flexural strength are 82.2%, 81.1%, and 77.6% for SF8 and 76.7%, 73.3%, and 74.5% for SF5, respectively, obtained from Fig. 8. The differences be-tween the satisfaction probabilities using PBMD and the required probability equal to 75% are expressed as fol-low.

100%PBMDP PProbability Difference

P−⎛ ⎞= ⎜ ⎟

⎝ ⎠ (5)

where, P is the required criterion satisfaction probability equal to 75% and PPBMD is the required criterion satis-faction probability obtained using PBMD method for a material parameter.

Using Eq. 5, the probability differences of compres-sive, splitting tensile, and flexural strength are 9.6%, 8.1%, and 3.5% for SF8 and 2.3%, -2.3%, and -0.7% for SF5, respectively. The deviations indicate that the per-formance probabilities of the standard mix proportions SF5 and SF8 are not very far off from the required range of satisfaction. Therefore, the initially selected perform-ance class and corresponding probability satisfaction are reasonable for this mix proportion with respect to the design usage.

4.3.2 Concrete mix proportion modification Generally, an initially selected standard mix proportion can not satisfy all of required design target criteria by a single trial. This means that a mix proportion design can only satisfy some of selected criteria. In this step, the concrete material parameters are checked to see which ones do not satisfy the required performance satisfaction probability. Then, these unsatisfying parameters will be modified by using the developed Satisfaction Curves to satisfy required probability values.

The previously calculated probability differences show that all of the satisfaction probabilities of SF8 satisfy the required probability. Compared to the differ-ences in SF8 (i.e., 9.6%, 8.1% and 3.5%), the differ-ences in SF5 (i.e., 2.3%, -2.3%, and -0.7%) are smaller and acceptable. However, the negative difference values of splitting tensile strength and flexural strength satis-faction probabilities of SF5 do not satisfy the require-ment albeit the difference is small (i.e., -2.3% and -0.7%). Therefore, if the mix proportion is to satisfy the required compressive, splitting tensile, and flexural strength satisfaction probabilities, SF8 can be used or SF5 need to be modified. However, it is more important to satisfy the main criteria among all of the selected ones. In this example, the main criterion that a mix pro-portion must satisfy is compressive strength whereas splitting tensile and flexural strength can be considered less important. And if the requirements are considered based on the strength of concrete mix proportion, the satisfaction probabilities of SF8 and SF5 obtained from Fig. 9 are 80.8% and 75.3%, respectively, and both of them satisfy the requirement and SF5 can be used with-out modification. However, in order to illustrate modifi-

cation details of the concrete mix proportion to satisfy the required value, SF5 is modified.

Based on the Importance Factors of concrete parame-ters for strength criterion shown in Table 4, the main parameters of the mix proportion are w/b and b. There-fore, these two parameters are modified, and other pa-rameters are kept constant. The w/b and b parameter values of 0.3 and 476.0 (kg/m3) will be replaced by 0.295 and 484.7 (kg/m3), respectively. Goodness Value of w/b and b are calculated as (0.295-0.25)/(0.35-0.25)+0.0=0.45 and (484.7-408.6)/(572.0-408.6)+0.09= 0.55, respectively. Because the trend of Satisfaction Curve of w/b needs to be converted to the “positive” trend, the final Goodness Value of w/b is 1-0.45=0.55. The modified mix proportion’s parameter values, Good-ness Value of parameters are tabulated in Table 6.

Goodness Value of the modified mix proportion can be combined by the following equation

i iG Gα= (6)

where, iG is Goodness Value of material parameters; iα are Importance Factors of material parameters in the

mixture design for a specified criterion; the double sub-script i, i means summation of i iGα values.

Goodness Values of MSF5 is 0.54. Using Fig. 8, the final satisfaction probabilities of the modified mix pro-portion for compressive, splitting tensile, and flexural strengths are 83.2%, 79.3%, and 81.0%, respectively, which are larger than the required probabilities, indicat-ing that the mix proportion design satisfies the required satisfaction probabilities. Hence, the modified concrete mix proportion MSF5 can be considered an acceptable design.

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Fig. 9 Combined strength satisfaction curve trend of concrete mix proportion.

Table 6 The modified mix proportion MSF5.

Parameter SF5 MSF5 Goodness Value

w/b 0.3 0.295 0.55 w (kg/m3) 143.0 143.0 0.5 b (kg/m3) 476.0 484.7 0.55 fa (kg/m3) 764.0 764.0 0.5 ca (kg/m3) 1047.0 1047.0 0.5


4.3.3 Client request verification for final con-crete mix proportion design The final modified mix proportion needs to be checked whether it satisfies the client requested satisfaction probability trends. In this example, the strength Satis-faction Curve trends shown in Fig. 9 need to be checked whether they are sufficient for construction usage. The unified strength Satisfaction Curves linearly increase in the Goodness Value range of 0.25 to 0.55 and remain mostly horizontal for the range of 0.0 to 0.25 (unsatis-factory) and 0.55 to 1.0 (satisfactory). The trend of the Satisfaction Curves indicates that the change of Good-ness Value of material parameter does not cause the drastic change in satisfaction probability. If the Satisfac-tion Curve trend has a large unsatisfied range or drastic change in a narrow range of Goodness Value, one can conclude that the Satisfaction Curve trend is not suffi-ciently safe for construction usage. 5. Verification and discussion

The performance verification of SF8 (with 437.9 kg/m3 of cement, 38.08 kg/m3 of silica fume) and MSF5 (with 460.9 kg/m3 of cement, 23.8 kg/m3 of silica fume) are performed using the compressive strength, splitting ten-sile strength, and flexural strength experimental results. The experimental results are shown in Table 7. The probabilities of satisfying for 70.0 MPa compressive strength, 4.92 MPa splitting tensile strength, and 10.5 MPa flexural strength criteria calculated based on these compression test data using the method presented in ACI 214R-02 (2002) are 80%, 80.5%, and 79.2% for SF8 and 79.5%, 76.7%, and 79.8% for MSF5, respec-tively. The probability differences between ACI and PBMD method of compressive, splitting tensile, and flexural strength are 2.8%, 0.4% and -2.0% for SF8 and 4.7%, 3.4% and 1.5% for MSF5, respectively. With re-spect to ACI compressive strength distribution probabil-ity, the maximum range is approximately 95.45%. Hence, we chose 5% difference accuracy from 100-95.45 equal to approximately 5% accuracy limit. The probability differences for all three strengths are within an allowable range of 5%. The close proximity of the results proves that the mix proportions satisfy the re-

quired performance requirements. Moreover, the ana-lytical results obtained from the one-parameter method using PBMD scheme are shown to be reliable enough to be used in concrete mix proportion design as PBMD approach.

In this example, the application example of PBMD to HSC mix design used three strength performance re-quirements of compressive, splitting tensile, and flex-ural strengths. Other types of HPC may be needed for other construction projects such as a high-rise structure may require performance requirements of high modulus of elasticity, high pumpability, high workability, etc where as a long-span coastal bridge may require per-formance requirements of high fatigue resistant, high durability, and high freeze-thaw resistant, etc. The pro-posed PBMD is useful since the same procedure can be applied to any type of HPCs and concrete structures. By using this PBMD for HPC mix design, the relationships between properties of HPC can be clearly associated with required performance with a level of precision re-quired for engineers and designers.

6. Conclusions

The following conclusions can be drawn based on the investigation of the paper: 1. The advantages of the PBMD are the inter-relatable

characteristics and sensitive to change in mix pro-portions of HSC to its performance capacities.

2. The practical application possibility of the PBMD for HSC mix proportion design to satisfy concrete performance required criteria are shown and veri-fied.

3. The differences between the analytical results and the results obtained using the method presented in ACI 214R-02 (2002) are less than 5%, which is within an acceptable difference. This difference in-directly shows that the proposed method can be used to design and predict the satisfaction probabil-ity of required concrete performance criteria for a realistic range of material parameter values for HSC.

Acknowledgements This research was supported by Engineering Research Center (ERC) of the National Research Foundation of Korea(NRF) grant funded by the Korea government (MEST) (No. 2011-0030846) and supported from Na-tional Research Foundation of Korea (NRF) grant funded by the Ministry of Education, Science, and Tech-nology (No.2011-0014752). References ACI, (1992). “Report on high-strength concrete.”

Michigan: American Concrete Institute, ACI 363R-92. ACI, (2002). “Evaluation of strength test results of

concrete.” Michigan: American Concrete Institute, ACI 214-02.

Table 7 The extra tests for SF8 and MSF5.

Compressive strength (MPa)

Splitting tensile

strength (MPa)

Flexural strength (MPa)

SF8 MSF5 SF8 MSF5 SF8 MSF51 74.5 73.9 5.09 5.12 11.08 11.182 73.3 70.6 5.16 5.09 11.12 10.933 72.6 70.9 4.98 4.93 10.89 10.754 71.1 71.8 5.05 5.03 10.64 10.615 69.1 68.8 4.94 4.91 10.35 10.426 69.9 73 4.86 4.88 10.48 10.52

Mean 71.8 71.5 5.01 4.99 10.76 10.74Standard deviation 2.08 1.82 0.108 0.101 0.32 0.28


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