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A New Strength Model Based on Water/Cement Ratio and Capillary Porosity

K.S. Pann

1

, Tsong Yen

2

, Chao-Wei Tang

3

, T.D. Lin

4

1 Lecturer, Civil Engineering Department, Cheng-Shiu Institute of Technology, Kaohsiung, Taiwan.2 Professor, Civil Engineering Department, National Chung-Hsing University, Taiwan.3 Assistant Professor, Civil Engineering Department, Cheng-Shiu Institute of Technology, Kaohsiung, Taiwan.4 Consultant, Lintek International, Inc., USA.Corresponding author: K.S. Pannis a lecturer of civil engineering at Cheng-Shiu Institute of Technology. He is a

Ph.D. candidate in the department of civil engineering at National Chung-Hsing University.

ACI member Tsong Yen is a professor of civil engineering at National Chung-Hsing University. He obtained his

Ph.D. from the Technical University in Berlin, Germany, in 1975. His research interests include concrete technology,

concrete structures and scaffold structures.

Chao-Wei Tangis an assistant professor of civil engineering at Cheng-Shiu Institute of Technology. He received his

Ph.D. from National Chung-Hsing University. His research interests include high-performance concrete and seismic

design of reinforced concrete.

T. D. Linworked for the Portland Cement Association as a researcher in the field of construction technology for 22

years. He is a consultant of Lintek Corporation and working with the Space Colonization Workshop Group and the

European Moon Project Team. He is also a member of Mars Society of America and ACI Fire Safety Committee,

ACI-216. And he has been frequently invited to speak on concrete applications to construction on the Moon and

Mars in the United States and abroad.

ABSTRACT

This paper presents a new mathematical model that includes the effects of w/cratios and capillary porosity that

was left out in the Abrams formula. Basically, the capillary pores are remnants of water-filled space in between the

partially hydrated cement grains, and the gel pores are formed within C-S-H gels, the primary products of hydration.

This physiochemical behavior enables us to develop a mathematical strength model to include the effect of capillary

porosities and w/c ratios. Nonlinear regression analysis has been applied to solve the equation. To check the

adequacy of the proposed model, 34 mixed-batch data taken from published literatures were used as inputs in the

analyses. The calculated results reflect a greater accuracy than that obtained from the Abrams Law. A

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back-propagation neural network was also applied to assess the compressive strengths of the same 34 data sets. The

results verify the validity of the proposed model.

Keywords:capillary pores, hydration, water-cement ratios, and compressive strengths.

INTRODUCTION

Concrete is a composite material of aggregates and cement paste that fills in the spaces between aggregate

particles and binds them together to form a rocklike solid. Under a microscopic examination, cement paste is a

non-homogeneous and anisotropic matrix composed of irregularly shaped and unevenly distributed pores attributed

to the evaporation of free water and gel pore formation in the C-S-H hydrates. The porosity greatly influences the

strength development of the cement paste. In general, a greater porosity causes a weaker strength, and the porosity

related problems should be dealt with care for the sake of the building safety.

Incidents involving collapses of freshly cast concrete elements during construction have been reported and the

investigations often found poor concrete strength the prime factor. For example, concrete made with extra high w/c

ratios and/or improper compaction develops high porosity and low strength. Concrete strength is a vital index for

concrete quality in terms of mechanical properties like elastic modulus, impermeability, and weather/wear

resistances.

Published data on compression tests and subsequent SEM examinations on samples taken from the broken

cylinders show concrete with high capillary porosity developed low strengths. The purpose ofthis study is to include

the effect of capillary porosity on concrete strength in addition to the w/cratio, the sole variable dealt in the Abrams

Law. Published data on hydration degrees and capillary porosities in conjunction with w/cratios were collected to

form a database in support of the proposed mathematical model that is basically a multivariable nonlinear equation

solvable via the regression analysis and the curve fitting method. The developed equations were refined through

repeatedly calibrated computations through the use of the published data. The calculated results were subsequently

compared against those obtained from the Abrams equation to optimize the performance of the proposed model.

RESEARCH SIGNIFICANCE

Abrams Law is simply a formula that uses water-cement ratios to predict concrete strengths at 28 days. It works

well for a properly compacted concrete made with a w/cratio within a normal range, but not for concrete containing

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large voids attributed to incomplete compaction or unusually low w/cratio. The shortcoming has signaled a need for

a better evaluation procedure with a greater accuracy. In the present research, a new mathematical model including

the effects of w/cratios and capillary porosity was successfully developed. The results provide convincing evidence

that the proposed model indeed yields a greater accuracy than the Abrams Law does.

FACTORS AFFECTING STRENGTH

Factors that affect the strength development of concrete are several including materials used, mixture procedures,

curing environment, test methods, and others. However, our discussion is limited to the water/cement ratio, degree of

cement hydration, and porosity in hydrated cement pastethat are closelyrelated tothe subject of this paper.

Water/cement ratios

That the w/cratio controls the workability of the fresh concrete and the strength of the hardened concrete was a

well-known engineering principle as early as concrete was first used more than a century ago. Duff Abrams in 1918

put together engineering rules on relation between water-cement ratios and concrete strength and earned wide

attention from the global concrete community. The published rule, are expressed as Eq. (1), was called the Abrams

Law in honoring his remarkable work.

cwcB

Af/

'=

(1)

Wherefc'stands for the compressive strength of concrete at a designated age, whileAandBare empirical constants

summarizing the effects of cement type, aggregate, admixture, curing, testing conditions, and concrete age at the

time of test. Eq. (1) can be rewritten in a linear logarithm form as Eq. (2).

)/(log)/(loglog' cwbbBcwAfc 10 +=-=

(2)

If single-size aggregates were used in the concrete mixture proportions, the goodness of fit of Eq. (1) or (2), that is,

the Abrams formula, is characterized byR2= 0.99 whereRis the correlation coefficient, as shown in Fig. 1 [1]. This

signifies a good relationship between the compressive strength and the w/cratio. Nonetheless, it is impractical if not

impossible to use uniform single-sized aggregates for concrete production. Fig. 2 shows that the correlation

coefficient between the compressive strength and w/c ratio declines (R2 =0.78), when the concrete mixture

proportions were made with various size aggregates, illustrating the fact that the w/cratio alone cannot be used to

precisely predict the compressive strength of concrete.

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Degree of cement hydration

When cement and water are mixed together, the constituent compounds of cement interact with water to form

crystals that are collectively referred to as hydration products. Basically, the principal source of strength in the

hydrate products is the Van der Waals force of attraction. The small crystals of calcium silicate hydrates (abbreviated

C-S-H) make up approximately 50-60 % of the total volume of the completely hydrated cement paste. Enormous

surface adhesive characteristics are responsible for the mechanical properties of the hardened cement paste. This

shows that concrete must have C-S-H gels to bind aggregates together to form strong solid materials.

Concrete is in fact a porous solid of which the volume of voids in the hydrated cement paste decreases

proportionally with respect to the hydration time; namely, more hydrates are formed in the course of hydration. As a

result, cement paste with less void percentage increases in strength, and the less void percentage can be achieved by

a high degree of hydration. Other than the w/c ratio, curing moisture/temperature and specimen age all have

significant influences on cement hydration as shown in Fig. 3 [4]. The same figure also reveals that the degree of

hydration rises rapidly in the w/c ratios ranging from 0.23 to 0.47 and stays relatively flat thereafter. It also

demonstrates the proportional relationship between the degree of hydration and the water-cement ratio cured under a

room temperature of 21 for same length of time.

Porosity in hydrated cement paste

The hydrated cement paste contains capillary pores and gel pores. The former is the remnants of water-filled

spaces and the latter is the micron spaces that exist among the partially hydrated cement grains, such as C-S-H

hydrates. To differentiate capillary pores from gel pores is by the rule of pore sizes. In accordance with the Powers

classification, capillary pores are those having diameters larger than 10 nm and gel pores are those smaller than 10

nm. Porosity is generally expressed as a void percentage of the hydrated cement paste and can be effectively used to

estimate the concrete strength.

Theoretically speaking, the formation of gel pores depends largely on the degree of hydration, not w/cratios,

provided with sufficient water for hydration. It has been demonstrated that the minimum w/cratio for a complete

hydration is 0.42 and capillary pores formed through the evaporation excessive water when the w/cratio of a paste

exceeds 0.42. In another word, capillary porosity,Pc, does depend on the degree of cement hydration, ,as shown in

Eq. (3).

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360-= .c

wPc

(3)

As stated, there exists an inverse relationship between porosity and strength. Equation (4) shows the relationship.

kpeSS -0= (4)

Where Sis the strength of the material of a given porosity,p; S0is the intrinsic strength of paste with zero porosity;

and kis a constant. Fig. 4 [4] shows an inverse relationship between the total porosity and concrete strength obtained

through the use of a mercury intrusion porosimetry (MIP). Eq. (5) shows the regression expression:

tPptS

9233-10978=

.. , (R2=0.887) (5)

WherePtis the total volume of mercury compressed in pores in the MIP experiment. Likewise, Fig. 5 [4] and Eq. (6)

show an inverse relationship between measured capillary porosity and the corresponding concrete strength:

cPpcS

.3204-10510= , (R2=0.926) (6)

WherePcis the volume of mercury compressed in capillary pores (cc/g) in the MIP experiment. By comparing Eq. (5)

with Eq. (6), it is not hard to visualize that the correlation coefficient of Eq. (6) is greater than that of Eq. (5) and Eq.

(6) is a better tool for the strength prediction.

NEW STRENGTH MODEL

Despite extensive research carried out in the past decades to quantitatively predict concrete strength, it seems

there is room in this specific field for improvement. This study was intended to develop a mathematical model

covering the effects of w/cratio and capillary porosity simultaneously. The experimental data needed for developing

related equations were taken from published literature [10-20]. The data were sorted in accordance with the w/c

ratios in relation to corresponding 28-day strengths of tested specimens made through different mix procedures and

cured in a same ambient temperature of 21.

Degree of cement hydration

Assessing the degree of hydration, the first step in developing a mathematic strength model, can be done through

the use of the loss-on-ignition (LOI) method. The degree of hydration increases with respect to the increase of w/c

ratio and specimen ages under a specified temperature. Fig 3 [4] shows the relationship. The nonlinear regression

equation in function of w/cratio was derived for the purpose of predicting the correlation coefficient, , as shown in

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Eq. (7).

32 890696+0531144-42621+02844-= )(.)(.)(.. cwcwcw (7)

Table 1 lists the values of w/c ratios and corresponding degrees of hydration. The correlation coefficient (R2) for Eq.

(7) is 0.998. A correlation coefficient of nearly 1.0 suggests a high reliability of the nonlinear regression analysis.

Capillary porosity

In general, cement paste of high w/cratio develops a high capillary porosity as shown in Fig. 6 [4]. The figure

demonstrates the inverse relationship between hydration degree and capillary porosity for cement pastes of various

w/cratios[4]. If the w/cratio is held constant, the capillary voids decreases as the degree of hydration increases as

shown in Fig. 6. Eq. (8)shows the relationship between capillary porosity and hydration degree [4].

BAPc -= (8)

Through the linear regression analysis, correlation coefficients (R2) of 0.84 to 0.94 were found for Eq. (8). A

correlation coefficient of less than 0.94 is somewhat undesirable and thus, after a thorough study, an alternative

approach was proposed as shown in Eq. (9).

DC

BLnAP

cwc++=

/)(

(9)

Through the multiple nonlinear analyses, using the w/c ratios and hydration degree listed in Table 2, Eq. (10) was

established and is shown expressed as the following.

6210-3070

3340+2550-= .

.

.)(.

/ cwcLnP (10)

The obtained correlation coefficient (R2) for the above equation is 0.9762 that is substantially improved in

comparison with that of Eq. (8).

New strength model

Based on aforementioned results, a new strength model like Eq. (11) was formulated in function of w/c ratio and

capillary porosity following the format of Eq. (1).

cPcwc D

C

B

Af +=

/'

(11)

Wherefc'stands for the compressive strength of concrete at 28-day;A,B, CandDare empirical constants, w/cis the

water/cement ratio, and Pcis the calculated capillary porosity from Eq. (10). The w/c depends closely on the mix

proportion while pcis intimately related to the consolidation process.

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Thirty-four sets of test data on the mixture proportions and 28-day strengths of pure cement concrete were taken

from literature [10-20] and are listed in Table 3. The data in the third and fourth columns of Table 3 were obtained

from Equations (7) and (10), respectively. Substituting the values of w/c ratio and hydration degree listed in Table 3

into Eq. (11), the corresponding compressive strengths were calculated and are listed in Column 7 of Table 3. For the

purpose of comparison, compressive strengths were calculated from Eq. (1) and the results are listed in Column 5.

The test strengthfexptaken from the reference literature and the predicted strengthsfpredcalculated from the proposed

model and the Abrams Law are shown in Fig. 7. The strength data obtained from the proposed model have less data

scattering than that of the Abram Law as shown in Eq. (1) in reference to the theoretical line. This verifies the

validity of the proposed strength model.

In addition to the values of fexp and fpred, Table 3 provides the correlation coefficient (R2) of the two analytic

models as 0.97 and 0.91 for the proposed model and Abrams' model, respectively.

The Root-Mean-Square(RMS) error, as shown in Eq. (12), was also utilized to determine the accuracy of the two

strength models.

( )

M

YT

RMS

M

p

2pp

=

(12)

Where Tp is the target value of example p, Yp is the predicted value of example p, and M is the number of

examples. Table 3 also gives the values of RMSerror for the proposed model and Abrams' model as 3.94 and 5.2

MPa, respectively. Of course, the smaller is the better. This implies that the proposed model is superior to the

Abrams model as far as the accuracy is concerned.

COMPARISON WITH BACK-PROPAGATION NEURAL NETWORK

The artificial neural network is an ideal computational tool capable of simulating the architecture of the neurons

system and information process of a human brain. It consists of a number of artificial neurons grouped into two or

more layers in a logical sequence such that they interact each other via weighted connections in the network, as

shown in Fig. 8. Most neural network applications are based on the back-propagation paradigm using the

gradient-descent method to minimize possible malfunctions. The back-propagation algorithm involves the forward

error propagation when a set of input patterns is given to the network. The backward error propagation begins at the

output layer and moves on to the intermediate layers and finally reaches the input layer. The forward and backward

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propagation continues until the error is minimized to an acceptable level. In general, the RMSerror is adopted to

provide a measure of the output network performance over the number of training iterations.

In this study, a back-propagation neural (BPN) network was established to predict the compressive strength of

cement concrete. Among the 34 sets of concrete data listed in Table 3, 24 are sampled randomly as training examples

and the remaining 10 are regarded as testing examples. The data are rearranged in such a way that water/cement ratio,

w/c, hydration degree, , and capillary porosity, Pc, are inputs while the compressive strength, fc', is the

corresponding output. Their corresponding ranges are listed in Table 5.

A viable network configuration became obvious after 10000 cycles of back-propagations. The learning processes

were repeated with different parameters. The obtained training and testing results for different configurations are

listed in Table 6. After a number of trials, the values of network parameters used in this investigation are given in the

following:

Number of Hidden Layers = 2 Number of Hidden Units in First Layer = 6 Number of Hidden Units in Second Layer = 6 Normal Input = 5 Normal Output = 1 Logarithmic Input = 5 Logarithmic Output = 1 Exponential Input = 5 Exponential Output = 1 Number of Train Examples = 34

Number of Test Examples = 34 Train Cycles = 10000 Range of Weights = 0.3 Random Seed = 0.456 Learn Rate = 1 Learn Rate Reduced Factor = 0.95 Learn Rate Minimum Bound = 0.1 Momentum Factor = 0.5 Momentum Factor Reduced Factor = 0.95 Momentum Factor Minimum Bound = 0.1

Fig. 9 shows the convergence histories of training/testing sets. It demonstrates that the values ofRMSerror decrease

with respect to the increase of learning circles. All calculatedR2values are greater than 0.96 for both training and test

sets. This indicates a good correlation between the independent variables (i.e., w/c, , and Pc) and the measured

dependent variable (fc'). In other words, the trained neural network provides sufficient information about the

compressive strength of tested concrete.

The experimental strength data of fexpcollected from the referenced literature and the values of fpredcalculated

from the proposed and BPN models are listed in Fig. 10. The comparisons suggest a nice agreement. The values of

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fexpandfpred, the correlation coefficients ofR2, andRMSerrors obtained from the three analytic models are listed in

Table 7 for the appropriate comparisons. By comparing the R2values of 0.91, 0.97, 0.98 and theRMSerrors of 5.2,

3.94, 3.38 MPas for Abrams' model, the proposed model, and the BPN model respectively, it is not hard to visualize

the BPN model yields the most accurate results and followed by the proposed model and the Abrams' model. Please

be reminded that the difference between data obtained from the proposed model and BPN model is insignificant.

CONCLUSION

This study has formulated a new model based on w/cratio and capillary porosity to predict concrete strengths and

draws the following conclusions:

1. The intimate relationship between hydration degree and w/cratio constitutes the base for predicting the degreeof hydration of cement paste cured under room temperature of 21 . The nonlinear regression equation for

predicting the hydration degree in term of w/cratio as shown in Eq. (7) offers accurate solutions.

2. Eq. (10) is an effective formula for evaluating the capillary porosity through the calculated hydration degree.The accuracy of the evaluation is exceedingly good. The proposed strength model including both w/c ratio and

capillary porosity as shown in Eq. (11) yields accurate solutions.

REFERENCE

1. L.K.A Sear, J. Dews, B.Kite, F. C. Harris and J.F. Troy, Abrams rule, air and high water-to-cement ratios,Construction and Building Materials, Vol. 10, No. 3, 1996, pp. 221-226.

2. Mindess. S., and Young, J. F., Concrete, Prentice-Hall, Inc., New Jersey, 1981.3. Ramachandran, V.S. and R.F. Feldman and J.J. Beaudoin, Concrete Science, Division of Building Research,

National Research Council, Canada(1981).

4. Ren-Yih Lin, Study on the Deduction Model of Basic Properties Between Hydration Parameters of PortlandCement, Department of Construction Engineering National Taiwan Institute of Technology Ph.D. Dissertation,

1991. (in Chinese)

5. Mehta, P.K. Concrete-Structure, Material, and Properties, Prentice-Hall, Inc., Engle wood cliffs, New Jersey,

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(1986).

6. Sergio Lai and Mauro Serra, Concrete Strength Prediction by Means of Neural Network, Construction andBuilding Materials, Vol. 11, No. 2, 1997, pp. 93-98.

7. I-Cheng Yeh, Modeling Concrete Strength With Augment-Neuron Networks, Journal of Materials in CivilEngineering, Vol. 10, No. 4, Nov. 1998, pp. 263-268.

8. I-Cheng Yeh, Modeling Of Strength Of High-Performance Concrete Using Artificial Neural Networks,Cement and Concrete research, Vol. 28, No. 12, 1998, pp.1797-1808.

9. S.Rajasekaran, M.F.Febin, and J.V.Ramasamy, Artificial Fuzzy Neural Networks in Civil Engineering,Computer & Structures, Vol. 61, No. 2, 1996, pp. 291-302

10. Francis A. Olukun, Edwin G. Burdette, and J. Harold Deatherage, Splitting Tensile Strength and CompressiveStrength Relationship at Early Ages, ACI Material Journal, March-April 1991, pp. 115121.

11. N. J. Gardner, Effect of Temperature on the EarlyAge Properties of Type I, Type III, and Type I/Fly AshConcretes, ACI Material Journal, JanuaryFebruary 1990, pp. 68-78.

12.

Arshad A. Khan, William D. Cook, and Denis Mitchell, Thermal Properties and Transient Thermal Analysis of

Structural Members during Hydration, ACI Material Journal, MayJune 1998, pp. 293303.

13. Nicholas J. Carino, H. S. Lew, and Charles K. Volz, Early Age Temperature Effects on Concrete StrengthPrediction by the Maturity Method, ACI Material Journal, MarchApril 1983, pp. 93101.

14. Rajesh, C. Tank and Nicholas J. Carino, Rate Constant Functions for Strength Development of Concrete, ACIMaterial Journal, JanuaryFebruary 1991, pp. 7483.

15. 15 Gilles Chanvillard and Laetitia DAloia, Concrete Strength Estimation at Early Ages: Modification of theMethod of Equivalent Age, ACI Material Journal, NovemberDecember 1997, pp. 520530.

16. Takashi Kuwahara and Yoshiro Koh, Computerized Thermal and Strength Simulation System for ConcreteStructures, ACI Material Journal, MarchApril 1995, pp. 117124.

17. Michel Lessard, Omar Chaalal, and Pierre-Claude Aitcin, Testing High-Strength Concrete CompressiveStrength, ACI Material Journal, July-August 1993, pp. 303308.

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18. Tarun R. Naik, and Bruce W. Ramme, Low Cement Concrete High Strength Concrete, Cement and ConcreteResearch, Vol. 17, 1987, pp. 283-294.

19. Walker, Stanton, and Delmar L., Effects of Aggregate Size on Properties of Concrete, ACI Journal,Proceedings Vol. 57, No. 9, Sept. 1960, pp. 283-298.

20. Welch, G. B., Discussion of Water Cement Ratio Versus StrengthAnother Look, by Herbert J. Gilkey, ACIJournal, Proceeding Vol. 58, Dec. 1961, pp. 1866-1868.

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LIST OF NOTATIONS

A = empirical constants

B = empirical constants

C = empirical constants

D = empirical constants

M = the number of examples

S = the strength of the material of a given porosity,p;

k = constant

= degree of cement hydration

Pc = the capillary porosity volume of mercury compressed in capillary pores (cc/g) in the MIP experiment

Pt = the total volume of mercury compressed in pores in the MIP experiment

So = the intrinsic strength of paste with zero porosity

Tp = the target value of example p

Yp = the predicted value of example p

fc = compressive strength of concrete

w/c = water-cement ratios

RMS = Root-Mean-Square error

Spc = concrete strength calculated by use of capillary porosity of mercury intrusion porosimetry

Spt = concrete strength calculated by use of total porosity of mercury intrusion porosimetry (MIP)

fexp = the experimental strength data

fpred = calculated strength from the proposed and BPN models

LIST OF TABLES

Table 1 The values of w/cratio, hydration degree in paste

Table 2 The values of w/cratio, hydration degree, and capillary porosity in paste

Table 3 Comparison of predicted compressive strength obtained by proposed model and

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Abrams' model

Table 4 Empirical constants of proposed model obtained by nonlinear regression

Table 5 Ranges of parameters in database

Table 6 Results of the BPN model

Table 7R2andRMS errors for different analytic models

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LIST OF FIGURES

Fig. 1 Relation between concrete strength and w/cratio using single-size aggregates

Fig. 2 Relation between concrete strength and w/cratio using various sizes aggregates

Fig. 3 Influence of w/cratio and moist curing age on the degree of cement hydration

Fig. 4 Relationship between total porosity and compressive strength

Fig. 5 Relationship between capillary porosity and compressive strength

Fig. 6 Relationship between hydration degree and capillary porosity for varying w/cratiosin paste

Fig.7 Comparison of actual strength and predicted strength by proposed model and

Abrams' model

Fig. 8 A typical Back-Propagation neural network

Fig. 9RMS error convergence histories for training and testing set

Fig. 10 Comparison of actual strength and predicted strength by proposed model and BPN model

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Table 1 The values of w/cratio, hydration degree in paste

(cured at temperature of 21 for cure 28 days) [4]

w/c hydration degree (%)

0.23 46.702

0.35 63.584

0.47 67.069

0.59 67.901

0.71 69.795

Table 2 The values of w/cratio, hydration degree, and capillary porosity in paste

(cure Temperature: 21 ; cure Age: 28 days) [4]

w/cHydration degree

(%)

Capillary porosity

(cc/g)w/c

Hydration degree

(%)

Capillary porosity

(cc/g)

0.23 0.368032 0.044543 0.47 0.630979 0.0644723

0.23 0.399472 0.0387913 0.47 0.631021 0.0632541

0.23 0.416462 0.0328047 0.47 0.631953 0.0630251

0.23 0.437089 0.014352 0.59 0.433681 0.280458

0.35 0.413054 0.11312 0.59 0.507186 0.191888

0.35 0.490052 0.0861891 0.59 0.574438 0.160624

0.35 0.554814 0.0350529 0.59 0.658985 0.152604

0.35 0.634297 0.0300818 0.71 0.444378 0.352652

0.47 0.424119 0.182188 0.71 0.515583 0.33189

0.47 0.507271 0.148051 0.71 0.589331 0.303863

0.47 0.57386 0.0865071 0.71 0.660971 0.245532

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Table 3 Comparison of predicted compressive strength obtained by proposed model and Abrams' model

w/c f 'c(28) hydration capillary Abram Abram's Error Predicted Error of

degree porosity Predicted

MPa cc/g Mpa % Mpa %

0.238 66.610 0.484971617 0.154381833 75.68 13.6 70.25 5.5

0.249 64.450 0.504545958 0.158121079 73.83 14.6 69.25 7.4

0.260 60.330 0.524522669 0.162664983 71.85 19.1 68.27 13.2

0.263 66.560 0.529504093 0.163909976 71.34 7.2 68.02 2.2

0.273 64.160 0.544779786 0.168001178 69.72 8.7 67.27 4.9

0.287 61.510 0.565142798 0.174085336 67.43 9.6 66.22 7.7

0.300 67.100 0.582498107 0.179828415 65.33 -2.6 65.23 -2.8

0.300 71.000 0.582498107 0.179828415 65.33 -8.0 65.23 -8.1

0.300 74.300 0.582498107 0.179828415 65.33 -12.1 65.23 -12.2

0.300 72.900 0.582498107 0.179828415 65.33 -10.4 65.23 -10.5

0.300 71.900 0.582498107 0.179828415 65.33 -9.1 65.23 -9.3

0.300 73.370 0.582498107 0.179828415 65.33 -11.0 65.23 -11.1

0.312 65.450 0.59654384 0.184845725 63.49 -3.0 64.31 -1.7

0.328 64.450 0.613080883 0.191170931 61.12 -5.2 63.03 -2.2

0.329 60.980 0.614034759 0.191549456 60.97 0.0 62.95 3.2

0.337 60.040 0.621273259 0.194470262 59.83 -0.3 62.28 3.7

0.350 64.700 0.632022803 0.198964872 58.00 -10.4 61.11 -5.5

0.364 56.600 0.641731692 0.203184866 56.15 -0.8 59.81 5.7

0.400 53.700 0.66093175 0.211975837 51.49 -4.1 55.94 4.2

0.400 53.800 0.66093175 0.211975837 51.49 -4.3 55.94 4.0

0.400 55.400 0.66093175 0.211975837 51.49 -7.1 55.94 1.0

0.406 51.500 0.663263651 0.213083657 50.76 -1.4 55.25 7.3

0.468 47.200 0.676558111 0.219564837 43.80 -7.2 47.34 0.3

0.476 44.700 0.677135342 0.219852606 42.97 -3.9 46.24 3.5

0.500 41.200 0.6778092 0.220189215 40.58 -1.5 42.91 4.2

0.550 39.630 0.676231103 0.219402047 36.03 -9.1 35.93 -9.3

0.554 31.940 0.676048185 0.219311062 35.69 11.7 35.38 10.8

0.599 27.620 0.674931406 0.218756717 32.08 16.1 29.49 6.8

0.600 29.700 0.674944058 0.218762986 31.99 7.7 29.34 -1.2

0.600 29.200 0.674944058 0.218762986 31.99 9.6 29.34 0.5

0.700 21.500 0.694149925 0.228572135 25.21 17.3 19.44 -9.6

0.750 18.110 0.725096237 0.245604045 22.39 23.6 17.36 -4.2

0.750 18.070 0.725096237 0.245604045 22.39 23.9 17.36 -3.9

0.800 15.700 0.777240403 0.277718801 19.87 26.6 18.23 16.1

R2= 0.91 R

2= 0.9684

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Table 4 Empirical constants of proposed model obtained by nonlinear regression

A B C D

193.18 21.43 -41506.49 107.64

Table 5 Ranges of parameters in database

Parameters Range

w/cratio 0.238 - 0.8

Hydration degree (%) 48.5 -77.7

Capillary porosity (cc/g) 0.1544 - 0.277

28-day compressive strength (MPa) 15.7 - 74.3

Table 6 Results of the BPN model

Number of Hidden Layers :1 Number of Hidden Units :6 Train Cycles 10000

Learning Rate R2 RMS Train Set RMS Test Set RMS

1 0.9767 3.908 0.03322 0.04247

2 0.9767 3.894 0.03316 0.04231

3 0.9769 3.877 0.03306 0.04208

4 0.9769 3.866 0.03296 0.04181

5 0.9773 3.826 0.03288 0.04154

6 0.9773 3.821 0.03283 0.0413

7 0.9777 3.789 0.03279 0.04109

8 0.9778 3.772 0.03277 0.0409

9 0.9779 3.763 0.03276 0.04075

10 0.9782 3.732 0.03269 0.04052

11 0.9786 3.688 0.03249 0.04

12 0.9789 3.663 0.03244 0.03975

13 0.9788 3.667 0.03239 0.03968

14 0.9786 3.695 0.03238 0.04009

15 0.9791 3.642 0.03227 0.03943

16 0.9791 3.636 0.03224 0.03934

17 0.9719 4.338 0.03476 0.04707

18 0.975 4.091 0.03427 0.044519 0.9792 3.627 0.0322 0.03933

20 0.9796 3.598 0.03211 0.03901

21 0.9784 3.804 0.03248 0.04137

22 0.9775 3.869 0.03303 0.04197

23 0.9755 4.04 0.0342 0.04375

24 0.973 4.291 0.03563 0.04666

25 0.9701 4.54 0.03697 0.04927

26 0.0015 43.16 0.48358 0.46826

27 0.0015 43.16 0.4836 0.46828

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Table 7R2andRMS errors for different analytic models

Abrams' model Proposed model BPN model

Correlation Coefficient (R2) 0.91 0.9684 0.9796

Root of mean square (RMS) 5.2 3.94 3.598

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Fig.1 Relation between concrete strength and w/cratio using single-size aggregates[1]

Fig. 2 Relation between concrete strength and w/cratio using various sizes aggregates

0.2 0.4 0.6 0.8

Water - cement ratio

10

100

LogStrengthMPa

R2=0.78

28 Day Actual

28 Day Regression

0.3 0.4 0.5 0.6 0.7 0.8 0.9

Water - cement ratio

10

100

LogStrength

N/mm2

R2=0.9928 Day Actual

7 Day Actual

28 Day Regression

7 Day Regression

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Fig. 3 Influence of w/cratio and moist curing age on the degree of cement hydration [4]

Fig. 4 Relationship between total porosity and compressive strength [4]

0.2 0.3 0.4 0.5 0.6 0.7 0.8

W/C

30

40

50

60

70

80

90

DegreeofHydration%

Cure Temp. 21 oC90 Days

62 Days

28 Days

14 Days

7 Days

3 Days

1 Day

0 0.1 0.2 0.3 0.4 0.5

MIP Total Volume (cc/g)

0

10

20

30

40

50

Com.Strength(MPa)

S=78.9 x 10(-3.923xPt), R2=0.887

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Fig. 5 Relationship between capillary porosity and compressive strength [4]

Fig. 6 Relationship between hydration degree and capillary porosity for varying w/cratiosin paste [4]

0.3 0.4 0.5 0.6 0.7 0.8

Hydration Degree

0

0.2

0.4

0.6

0.8

Pc

(cc/g)

Cure Temp. 21

o

Cw/c = 0.71 y=0.58079-0.48393x R^2=0.94

w/c = 0.59 y=0.49830-0.54445x R^2=0.839

w/c = 0.47 y=0.44121-0.58396x R^2=0.944

w/c = 0.35 y=0.29885-0.43750x R^2=0.919

w/c = 0.23 y=0.19356-0.3844x R^2=0.902

0 0.1 0.2 0.3 0.4 0.5

MIP Capillary Pore Volume (cc/g)

0

10

20

30

40

50

Com.Strength(MPa)

S=510 x 10( -4.32xPc), R2 = 0.926

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Fig. 7 Comparison of actual strength and predicted strength by proposed model and Abrams' model

Fig. 8 A typical Back-Propagation neural network

Output LayerHidden Layernput Layer

Water cement

Hydration Degree

Capillary Porosity

Predicted Strength

0 20 40 60 80

Actural Strength (MPa)

0

20

40

60

80

PredictedStrength

(MPa)

Abram R^2 = 0.91

Predicted R^2 = 0.968

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0 2000 4000 6000 8000 10000

Learning Cycles

0

0.1

0.2

0.3

0.4

RMS

RMS error of Testing Examples

RMS error of Training Examples

Fig. 9RMS error convergence histories for training and testing set

Fig. 10 Comparison of actual strength and predicted strength by proposed model and BPN model

0 20 40 60 80

Actrual Strength (MPa)

0

20

40

60

80

PredictedStrength(MPa)

BPN Network R^2=0.9817

Predicted R^2=0.968