Performance Based Design System for Concrete Mixture with Multi

PERFORMANCE BASED DESIGN SYSTEM FOR CONCRETE MIXTURE WITH MULTI-OPTIMIZING GENETIC ALGORITHM Takafumi Noguchi1, Ippei Maruyama 1 and Manabu Kanematsu 1 1

Department of Architecture, University of Tokyo, Tokyo, Japan. E-mail: [email protected]

ABSTRACT This paper presents a method for optimizing concrete mixture proportions according to the required performance and also presents several case studies. Since various qualities are usually required of concrete, we characterize the proportioning problems as multi-criteria optimization problems. We dealt with the notion of Pareto optimality to derive the optimum solution and applied it to a genetic algorithm. In this contribution, two proportioning problems are solved by the genetic algorithm: a request of cost-performance at 60 MPa in strength as a simple problem, and a concrete mix for mass concrete in cold weather at the coast. 1. INTRODUCTION Concrete is required to exhibit performance corresponding with applied environment. And nowadays, thanks to technological progress, it is possible to make the concrete meet those requirements. However, there has been no established method. Only a few attempts [1][2] have so far been made in relation to these problems. The main reason for few outcomes is that a wide variety of mixture proportions are possible and there is no way to optimize mathematically the problem under many criteria, which are represented by objective functions. This contribution describes the method of optimizing mixture proportions of concrete by application of the Genetic Algorithm (GA) and several engineering models representing the relationship between the properties of concrete and mix proportions. Proportioning problems cannot be solved by the usual methods that consist of searching for the best solution with a single objective function, such as linear programming problems and nonlinear programming problems. The reason for this impossibility is that various performances are required of concrete corresponding to the environments in which the concrete is used, and it is impossible to express the plural requests in a single objective function. It follows from the features of the proportioning problem mentioned above that a proportioning problem is considered to be a multicriteria optimization problem. 2. HOW TO APPLY GENETIC ALGORITHM TO MULTI-CRITERIA PROBLEMS 2.1 Genetic Algorithms Genetic algorithms are optimizing, learning and searching algorithms based on the mechanism of natural selection and natural genetics. What is important in GA is that it is unnecessary to develop a method to solve the target problem. Because GA can solve the problem with the relative evaluation of the non-inferior set and is easier to formulate rules of evaluation than any other existing method, GA is widely applied to the engineering field, especially to combination problems. Since proportioning is a type of combination problem, there is a good reason to apply GA. Proceedings of the 11th International Congress on the Chemistry of Cement (ICCC) ‘Cement’s Contribution to the Development in the 21st Century’ Hosted by: The Cement and Concrete Institute of South Africa CD-ROM produced by: Document Transformation Technologies 1921

11 - 16 May 2003, Durban, South Africa ISBN Number: 0-9584085-8-0 Editors: Dr G. Grieve and G. Owens Congress Organised by: Event Dynamics

In GA “genotype” is the parameter set, which means a vector in the set of optimization problem, coded as a finite-length string with binary digits. The simple string is called “individual”. To apply GA to a proportioning problem, the genotype is designed to represent various components and various mixture proportions (see Fig.1). The genotype consists of two parts. One is a coded binary string and the other is a database containing components such as cement, aggregate, admixtures, and so on. As demonstrated in Figure 1, the string has linkage parts and volumetric ratio parts. The linkage parts have a connection to the database, which is used when the fitness is calculated with volumetric ratios. The volumetric ratio parts show the volumes of components in concrete as component/water ratios. In reality, the binary string is fixed as 256 bytes whereas it is not shown in Figure 1. The algorithms make a “phenotype”, which demonstrates characteristic form and quality in the designed system, from each individual according to its genotype. Each designed genotype has its phenotype. In our system, the term phenotype means the concrete properties and performance that are estimated from mixture proportions coded in the genotype. The kinds of concrete properties and performance are also shown in Figure 1. Prediction formulas are formulated statistically for each property.

Figure 1. Schematic of GA applied for proportioning problem “Fitness value” gives the numerical evaluation of each individual and each phenotype in the designed system. According to the fitness values, which can be plural in Pareto optimality problem, an individual in the “population” representing the set of individuals will be reproduced with crossover and mutation from generation to generation. Thus the individuals in the set become fitted under applied environment. In the developed algorithm, suitable functions are designed to meet the way of performance requesting and those are shown in Figure 2 together with the equations. U is the parameter representing the required performance of concrete mixture and T is the parameters 1922

that determine shape of function. These two parameters express the allowable range of requirements for performance.

Figure 2. Fitness function and its shapes 2.2 Genetic Algorithms Applied to Multi-criteria Problem In multi-criteria optimization, the notion of optimality is not obvious at all. For example, it is difficult to compare strength with fluidity. There does not exist the way to evaluate and compare different things. But the concept of Pareto optimality helps the evaluation of multiple criteria in a rational way. The Pareto optimal set is defined as stated below. When a vector x is partially less than y and all criteria are minimizing criteria, the mathematical expression of the vector x in the Pareto optimal set is: (x < py ) = (∀i )( xi < yi ) ∧ (∃i )( xi < yi )

(1)

Under the condition of equation (1), point x “dominates” point y. If the point is not dominated by any other, that point is “non-inferior”. The set of these non-inferior points is what we call a Pareto optimal set. According to this definition, if there is a point, which is not less than any other points by all criteria, only the best point will get a good evaluation. If there is no such a point, a set of non-inferior points, which trade off one of the set of points for another, will be evaluated as good. To apply genetic algorithms to a multi-criteria problem, an algorithm that can derive the Pareto optimal set is developed. Goldberg [3] had designed similar algorithms. Improvement on Goldberg’s algorithms and application to the proportioning problem are conducted. The developed program applied to concrete mixture problem is named MixGA. The program-flow of MixGA is detailed as follows. 1. 2. 3.

4.

Assume that there exist P criteria and N individuals Make N individuals randomly. Select criterion No. 1 and determine the fitness value of each individual's genotype. Choose parents A and B with the roulette method in which the probability of its selection is in proportion to fitness value. Crossover A and B, and reproduce child C and child D. Repeat steps (2) and (3) from criteria No. 1 to No. P. 1923

5.

After getting N individuals in the new population (child generation) and N individuals in the old population (parent generation). 6. Produce a temporary generation with the child and parent generations. 7. Mutate genes by reversing the number at certain loci arbitrarily with a constant probability of 1%. (Loci is the plural word of locus. Locus means the position of the gene.) 8. Select Pareto individuals from the temporary generation and make the next generation that consists of N individuals. Reduction rule of population is dominated by two roles below. (i) If the number of Pareto individuals is less than N, then preserve all the Pareto individuals. Up to the number of all individuals becomes N, select individuals one by one from the rest with the criteria No. 1 to No. P. The probability of selection is in proportion to their fitness value. This method intends that the good genes in the remainder should be carried on in the next generation. (ii) On the other hand, if the number of Pareto individuals is N or more, select individuals from the Pareto individuals according to criteria No. 1 to No. P. The probability of selection is in proportion to their fitness in order. 9. Iterate steps (2) to (8) until the given number of times. 10. Conduct final selection. The algorithm detailed above is characterized by the evolution of population with highly evaluated genes and Pareto optimal conditions. The highly evaluated individual has a good feature in a part of the binary string. And these parts of a binary string in individuals are inherited to next generation explicitly. An individual in the next generation inherits several good parts of a binary string that can be highly evaluated through phenotype. This process of evolution makes it possible to find out the optimal concrete mixture among an almost infinite number of combinations of materials and proportions; otherwise the process can be a random search and that will fail. But it should be noted, however, that a Pareto optimum set is a set of non-inferior points. This definition implies it is possible that the evolved population has individuals, which have an outstanding performance by a certain performance criterion in exchange for bad fitness values of the others. The Pareto set conditions do not provide a single exact solution but help to search for an optimal set widely. Because of this undesirable aspect of Pareto optimality, the concept of “metaproperty” is used. In MixGA the average of fitness values of target properties is used as one of properties and this kind of property of individual is called meta-property. With the meta-property concept it is possible to inherit another possibility in gene that shows balancing in target properties of concrete to the next generation. Besides a final process of evaluation, a final selection is conducted using the fitness values and eliminating the individuals that do not meet the required performance. 3. TRIAL OF MIXGA The followings are two case studies, in which the mixture proportion of concrete is optimized under required conditions by using MixGA. 3.1 Case study 1 In case study 1, simple cost performance problem of strength property is examined. In detail, the concrete is required to have 60 MPa, Slump 18 cm and lower cost. Other properties are less significant. 3.1.1 Prediction of Strength of Concrete As is mentioned above, predicting functions that enable us to calculate the properties of concrete with any mix proportion are formulated. Regarding the strength properties of concrete, the

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formulating is partially based on the theoretical approach and partially based on the statistical approach. The formulas used are shown below. Strength of mortar: Fm = (a ( B / W ) +b) ⋅ K

(2) where Fm denotes the strength of mortar, a, b and K denote material parameter depending on cement, W denotes the water content per unit weight [kg/m3], and B denotes the summation of cement and admix content per unit weight [kg/m3]. In this contribution 4 kinds of cement are formulated; namely Ordinary Portland Cement (OPC), Low Heat Cement (LHC), Moderate Heat Cement (MHC) and High Early Strength Cement (HESC). Effect of coarse aggregate on strength:    1  V  rg = 1 −  d  ⋅   × (1 − ( e ⋅ B / W + f ) ⋅ ( LogX − LogA ) )    W / B + 1  1000  

(3)

where rg denotes the effect of coarse aggregate on strength, d, e and f denote material parameter depending on aggregate, V denotes the volume of coarse aggregate per unit weight, X denotes the maximum size of coarse aggregate and A denotes the minimum size of aggregate that is able to affect the loss of strength. Effect of type of coarse aggregate focused on the paste interface: rg 2 = k

(4)

where rg2 denotes the effect of coarse aggregate type and k denotes the material parameter of aggregate type depending on crushed aggregate (CR) or gravel (GR). Effect of air content: rair = 1 − k2 ⋅Vair

(5)

where rair denotes the effect of air content on strength, k2 denotes the parameter depending on water to binder ratio and type of coarse aggregate, and Vair denotes the volume of air per unit volume of concrete. Effect of addition of Ground Granulated Blast furnace Slag (GGBS):   X  1  X < 0.3 : rmix = 1 +  ( g ( X 2 − h)3/ 2 ) ⋅   + ( lX 2 + m )  ×  X1 + 1    0.3 X ≥ 0.3 : r0.3 × ( a1 X 2 + b1 ) ×

X − 0.3 0.4

(6)

(7)

where rmix denotes the effect of admixture on strength, g, h, l, m, a1, b1 denote the material parameter, X denotes the replacement ratio, X1 denotes the water to binder ratio, X2 denotes specific surface area of slag and r0.3 denotes the value calculated by equation (6) with X of 0.3.

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Effect of addition of Fly-Ash (FA): X < 0.45 : rmix = 1 − ( a2 X 1 − b2 ) × X

(8)

X ≥ 0.45 rmix = 1.0

(9)

where rmix denotes the effect of admix on strength, a2 and b2 denote the material parameter, X denotes the replacement ratio and X1 denotes the water to binder ratio. Effect of addition of Silica Fume (SF): X > 0.2 rmix = 1 + X

(10)

X ≥ 0.2 rmix = 1.2

(11)

where rmix denotes the effect of admix on strength, X denotes the replacement ratio. Strength of Concrete: Fc = Fm × rg × rg 2 × rair × rmix

(12)

where Fc denotes the strength of concrete. 3.1.2 Prediction of cone slump Function for predicting of the value of cone slump is developed statistically with rheological approach. Kikukawa [4] proposed equation (13), which expresses the rheological parameters of cement paste at 20 oC that are plastic viscosity and yield value, as a function of solid content of cement and volumetric density of cement paste. This expression was on the basis of study of Roscoe [5]. Plastic viscosity of cement paste:  V η p = ηw 1 −   c

− ( K1V + K 2 )

(13)

where η p denotes plastic viscosity of cement paste [Pa s], ηw denotes plastic viscosity of water, V denotes volumetric density of cement paste, c denotes solid content of cement, K1,K2 denote constants affected by the properties of the cement paste. –17.5 and 12.0 are used respectively. Yield value : τ p = aη p + b

(14)

where τ p denotes yield value of cement paste [Pa], a and b denote constant parameters. 15.505 and 1.244 are used respectively

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Temperature effect on the rheological parameters:

η pT = 0.00387η p 20 ⋅ T + η p 20

(15)

τ pT = 3.03τ p 20 ⋅ T + τ p 20

(16)

where η p 20 denotes plastic viscosity of cement paste at 20 oC, τ p 20 denotes yield value of cement paste at 20 oC, T denotes temperature, η pT denotes plastic viscosity of cement paste at T oC and

τ pT

denotes yield value of cement paste at T oC.

Effect of admixtures and additions: Collecting the result of experiments, the effects of the contents on the rheological parameters are integrated into the formulas statistically. radmix ,η = fi ( X )

(17)

radmix ,τ = Gi ( X )

(18)

where radmix ,η and radmix ,τ are the coefficient of viscosity and yield value depending on the admixtures and X is the admixture to cement ratio. In this contribution, 4 type of admixtures are formulated; namely High Range Water Reducer (HRWR), AE agent (AE), Accerelator (AC) and Retarder (RE) radd ,η = f j ( X )

(19)

radd ,τ = G j ( X )

(20) where radd ,η and radd ,τ are the coefficient of viscosity and yield value depending on the addition and X denotes the replacement ratio. Solid content of aggregate: Ooi [6] investigated the complex nature of packing particles with random shape and wide grading, and developed mathematical model predicting the compacted bulk density of aggregate. This model is expressed as a second order polynomial of the sieve residuals as follows: n

n

i

j

Z = ∑∑ Aij X i X j

(21)

where Z denotes solid content of combined fine aggregate and coarse aggregate, Aij is coefficient representing the packing performance of combined aggregates of two different diameter, Xi and Xj express volumetric ratios of the aggregates at the representative sieve sizes. Rheological parameters of fresh concrete: Recently Oh [7] found a phenomenon that the rheological parameters of fresh concrete have good correlation to a relative thickness of excess paste on the basis of excess paste theory developed by Kennedy [8]. The relation between the relative thickness of excess paste and the rheological parameters of fresh concrete is formulated as follows:

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Pe = 1 − Va / Z Γ=

(22)

Pe

(23)

n

∑n s D i i

pi

i

where Pe denotes volume of excess paste, Va is volume of aggregate, Γ denotes relative thickness of excess paste, ni is number of aggregate of size i, si denotes surface area of each aggregate of size i, Dpi is diameter of aggregate of size i. Using relative thickness of excess paste and rheological parameters of cement paste, the viscosity and yield value of concrete can be calculated.

ηc = η p × (1 + 0.0705Γ −1.69 )

(23)

τ c = τ p × (1 + 0.0705Γ −2.22 )

(24)

where ηc denotes plastic viscosity of fresh concrete and τ c denotes yield value of fresh concrete Cone slump: Using the experimental data of slump predicted rheological values, the regression curve is formulated as the function of yield value and parameter of mix proportion as below: S = a3 × Ln(τ c ) ×

1 + b3 1+W / B

(25)

where S denotes the value of slump, a3 and b3 are coefficients. 3.1.3 Costs In this problem, the function of cost is simply the summation of material cost. 3.2 Case study 2 In case 2, assuming that construction site is in cool temperate zone and average temperature is 10 o C, concrete is required 15 cm of slump, 40 MPa of strength, 6.0x10-4 of drying shrinkage, 35oC of adiabatic temperature rise at 7 day for mass concrete. Additionally 100 of durability factor, which means relative dynamic modulus of elasticity after 300 cycles of freezing and thawing actions, and 5.0x10-8 of Chlorides-diffusion-coefficient are required strictly. The other parameters are less important. 3.2.1 Adiabatic temperature rise Adiabatic temperature rise is modelled as follows: Q = Q∞ (1 − Exp( −γ t ))

(26)

Q∞ = a4 B + b4

(27)

γ = a5 B + b5

(28)

where Q denotes the adiabatic temperature rise at t day, a4, b4, a5 and b5 denote the material parameters and B denotes the weight of binder per unit volume. 1928

3.3 Results The required parameters of concrete in both cases are summarised in the same Table 1. Table 1. Required performances 　

Chlorides Carbonation Adiabatic Durability Young's Drying diffusion speed temperatur Strength factor modulus shrinkage coefficient coefficient e rise

Initial setting

Final setting

Slump

Cost

Unit

　

cm2/year

cm/year0.5

MPa

GPa

　10-6

K

hour

hour

cm

Yen

Case 1

0

1.00E-06

0.27

60

26

800

40

5

8

18

5000

Case 2

100

5.00E-08

0.25

40

25

600

35

5

8

15

10000

Case 1: After running the MixGA with 120 individuals and 100 generations, mix proportions shown in Tables 2 are derived regarding with case 1. As final selection, elimination of the individuals, which do not meet required performance of strength and slump simultaneously, is conducted. Additionally in the final selection, as an index of cost performance, a value of strength divided by cost is used. Three mix proportions listed in Table 2 are top-to-third in strength-cost index. In left of Figure 3 shows the degree of conformity of three mix proportions listed in Table 2. Among three proportions, mix A has the best strength-cost index. It should be noted that, as is shown in left graph of Figure 3, mix C is much cheaper than mix A. The value of strength of mix A is higher than that of mix C. But in MixGA both mix A and Mix C is evaluated equally and they survive selections from generation to generation. Table 2. Pareto optimal mixture proportions derived by MixGA W/B

Water

Cement (type)

Addition (type)

Fine aggregate (type)

Coarse aggregate (type)

Admixture 1 (type)

Admixture 2 (type)

A

0.42

138

275 (OPC)

52 (GGBS)

936 (CR)

987 (GR)

0.000

0.000

B

0.43

145

264 (HESC)

76 (GGBS)

967 (CR)

925 (GR)

0.232 (AC)

0.006 (AE)

　 Case 1

Case 2

C

0.41

145

301 (MHC)

52 (GGBS)

962 (CR)

921 (CR)

0.000

0.048 (AE)

D

0.47

132

280 (LHC)

0

1035 (CR)

898 (CR)

0.115 (HRWR)

0.512 (AE)

E

0.47

132

281 (LHC)

0

1029 (CR)

893 (CR)

0.005 (HRWR)

0.526 (AE)

F

0.47

131

281 (LHC)

0

1028 (CR)

906 (CR)

0.195 (HRWR)

0.512 (AE)

G

0.48

132

273 (MHC)

0

1034 (CR)

898 (CR)

0.0215 (HRWR)

0.526 (AE)

Figure 3. Ratio between phenotype and required performance of concrete mixtures derived by MixGA 1929

Case 2: After running the MixGA with 140 individuals and 200 generations with meta-property concept, mix proportions shown in Tables 2 are derived regarding with case 2. As final selection, elimination of the individuals, which do not meet required performance of strength, slump, drying shrinkage, chlorides diffusion coefficient, durability factor and adiabatic temperature simultaneously, is conducted. In right of Figure 3 shows the degree of conformity of four mix proportions listed in Table 2. As shown in Figure 3, almost all the mixtures meet the required performance. It should be noted here that mix E has low value in slump and this mix is evaluated equally from the Pareto optimal point of view. If only the Pareto optimization is used, it is possible that such undesirable individual that do not have good value in all several properties is able to survive the selection. The genetic algorithm with the notion of Pareto optimality has a good potential in initial convergence tendency but it has a risk of undesirable acceleration in a few properties as well. With meta property concept, which disappreciate the prominence and unbalancing it is possible to derive the sufficient set of mixture proportions. Using the suitable fitness function and evaluation method is important to get sufficient set. 4. CONCLUSIONS The results of this contribution are summarized as follows: 1. 2.

3.

A genetic algorithm system integrating the concept of Pareto optimality, which is named MixGA, was developed for solving multicriteria optimization problems in concrete mix proportioning. As shown in the examples presented in this study, MixGA can derive the appropriate mix proportions from the vast combinations of sorts of content and proportions of mixture to explore. This system is maintained by suitable fitness evaluation, reasonable reproduction and correct prediction formulas. The genetic algorithm with the notion of Pareto optimality has a good potential in initial convergence tendency but it has a risk of undesirable acceleration in a few properties as well. Using meta property concept, it is possible to compensate for this risk of the notion of Pareto optimality.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8]

Marks, W. and Potrzebowski, J., Multicriteria optimization of structual concretemixes, Architecture and Civil Engineering 38(4), 1992, pp.77-01 Piasta, Z. and Czarneski, L., Analysis of material efficiency of resin concrete. in Brittle Matrix Composite, Elsevier Apllied Science London and New York, 1989, pp.593-602 Goldberg, A. E., Genetic Algorithms in Search Optimization & Machine Learning, Addison Wesley, 1989, pp.192-208 Kikukawa, H., Studies on viscosity equation of Portland cement paste, Journal of Material, Concrete Structure and Pavements, V-2, 354, 1985, pp.109-118 (In Japanese) Roscoe, R., The Viscosity of Suspension of Rigid Spheres, British Jouranl of Applied Physics, Vol.3, 1952, pp.267-269 Ooi, T., Comparted bulk density of aggregate with random shape and widely ranged particle size distribution, Journal of Structure and Construction Engineering. AIJ, No.423, 1991, pp.11-16 (In Japanese) Oh, S. G.., Noguchi, T., and Tomosawa, F., Toward Mix Design for Rheology of Self-Compacting Concrete, 1st International RILEM Symposium on Self-Compacting Concrete, Stockholm, 1999, pp.361-372 Kennedy, C.T., The Design of Concrete Mixes, Proceedings of the American Concrete Institute, Vol.36, 1940, pp373-400

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