Research Article  Open Access
Siqi Li, Jinbo Yang, Peng Zhang, "WaterCementDensity Ratio Law for the 28Day Compressive Strength Prediction of CementBased Materials", Advances in Materials Science and Engineering, vol. 2020, Article ID 7302173, 8 pages, 2020. https://doi.org/10.1155/2020/7302173
WaterCementDensity Ratio Law for the 28Day Compressive Strength Prediction of CementBased Materials
Abstract
In the present contribution, the watercementdensity ratio law for the standard curing 28day compressive strength of cementbased materials including grout, normal concrete, ceramsite concrete, and foamed concrete is proposed. The standard curing 28day compressive strength of different grouts, normal concrete, ceramsite concrete, and foamed concrete was tested. Simulations for Abrams’ law, Bolomey’s formula, and watercementdensity ratio law were carried out and compared. The watercementdensity ratio law illustrates better simulations for the prediction of the 28day compressive strength of cementbased materials. The watercementdensity ratio law includes both the watercement ratio and relative apparent density of the cementbased material. Relative apparent density of the cementbased material is an important one of all the factors determining the compressive strength of the cementbased material. The watercementdensity ratio law will be beneficial for the precise and generalized prediction of the 28day standard curing compressive strength of cementbased materials.
1. Introduction
Portland cementbased materials such as grout, normal concrete, lightweight concrete, and foamed concrete have an extremely wide scale of application in civil engineering construction throughout the world and are expected to be used more extensively in the future as well. Properties of cementbased materials are determined by the whole component raw materials and their proportions. Clearly and correctly understanding the physical connection between property and composition of cementbased materials is very important. In other words, precise and generalized prediction of properties based on composition of cementbased materials will be beneficial for engineering design and application of cementbased materials. The 28day compressive strength is one of the first and foremost properties of cementbased materials for structural design of civil engineering. So far, how the 28day compressive strength is determined by the composition of cementbased materials is not understood clearly enough.
Since the invention of Portland cement in 1824, plenty of research works have proved that the 28day compressive strength of cementbased materials is determined by raw materials including cement, supplementary cementitious materials, aggregate etc., proportions including watercement ratio, sandaggregate ratio, cement content, etc., and curing conditions including ambient temperature, pressure, and humidity. In 1918, Abrams pronounced the watercement ratio law based on the observation that as the watercement ratio decreases, the strength of the concrete increases, accordingly. In 1935, Bolomey gave a formula to predict the compressive strength of cement mortar which expresses a linear relationship between the watercement ratio and compressive strength. Abrams’ law and Bolomey’s formula both indicate that compressive strength of cementbased materials is mainly dependent on the watercement ratio among all the other factors. Therefore, Abrams’ law and Bolomey’s formula are seen as different mathematical forms of watercement ratio law. For mortar, it has been found that cementsand ratio, supplementary cementitious material content, and cement type all will influence the parameters in Abrams’ law or Bolomey’s formula [1–4]. For normal concrete, cement type, curing age, and silica fume content all will influence the parameters in Abrams’ law or Bolomey’s formula [5–8]. For lightweight concrete, properties of the aggregates will influence the parameters in Abrams’ law or Bolomey’s formula [9]. For foamed concrete, curing conditions, cement type, foaming agent, and dry density will influence the parameters in Abrams’ law or Bolomey’s formula [10, 11]. However, so far, there were less efforts for generalization of Abrams’ law or Bolomey’s formula for cementbased materials including mortar, normal concrete, lightweight concrete, and foamed concrete all together.
In this contribution, the 28day compressive strength of cementbased materials including grout, normal concrete, lightweight concrete, and foamed concrete was investigated. Abrams’ law and Bolomey’s formula were applied for simulation of the relationship between watercement ratio and 28day compressive strength. A novel watercementdensity ratio law was proposed and verified. This work will be beneficial further for prediction of durability properties by composition of cementbased materials and durability study of cementbased material structure [12–18].
2. WaterCement Ratio Law and WaterCementDensity Ratio Law
The mathematical relationship between compressive strength and watercement ratio, according to Abrams’ law, is shown in the following equation:where σ_{c} is the compressive strength of cementbased material (MPa), is the watercement ratio, and a_{1} and a_{2} are parameters determined by cement type, cement content, curing condition, etc.
The mathematical relationship between compressive strength and watercement ratio, according to Bolomey’s formula, is shown in the following equation:where b_{1} and b_{2} are parameters determined by cement type, cement content, curing condition, etc.
The watercementdensity ratio law assumes that compressive strength of cementbased materials is mainly dependent on the watercementdensity ratio among all the other factors. The mathematical relationship between compressive strength and watercementdensity ratio, according to the watercementdensity ratio law, is shown in the following equation:where D_{r} is the apparent relative density of the cementbased material, is the watercementdensity ratio, and d_{1}, d_{2}, and d_{3} are parameters determined by cement type, cement content, curing condition, etc.
If cement and supplementary cementitious materials together are seen as binder, the waterbinderdensity ratio law assumes that compressive strength of cementbased materials is mainly dependent on the waterbinderdensity ratio among all the other factors. The mathematical relationship between compressive strength and waterbinderdensity ratio, according to the waterbinderdensity ratio law, is shown in the following equation:where is the waterbinder ratio, is the waterbinderdensity ratio, and f_{1}, f_{2}, and f_{3} are parameters determined by cement type, cement content, curing condition, etc.
3. Experimental
3.1. Materials
P.I 42.5 Portland cement (PC) was used in accordance with Chinese national standard GB1752007 (CEM I type Portland cement conforming to BS EN 1971: 2011). Silica fume (SF) used has specific surface area of 16000 m^{2}/kg. Ground granulated blast furnace slag (GGBS) has a particle size ranging from 0.5 μm to 125.8 μm. Fly ash with specific surface area of 380 m^{2}/kg was used. The chemical composition of PC, SF, GGBS, and FA is shown in Table 1. Water reducer used is polycarboxylic acid superplasticizer. Coarse aggregates were crushed limestone aggregates with a maximum diameter of 30 mm and density of 2615 kg/m^{3}. Local river sand with a maximum grain size of 5 mm and density of 2630 kg/m^{3} was adopted. Lightweight aggregates were expanded shale ceramsite aggregates with the maximum diameter of 20 mm and loose bulk density of 450 kg/m^{3}. Animal protein foaming agent was used for foamed concrete.

3.2. Mixture Proportions
Mixture proportions of grout, normal concrete, ceramsite concrete, and foamed concrete are shown in Tables 2–5, respectively. For grout, mixtures G0.20, G0.25, G0.30, and G0.40 are neat cement grouts. For normal concrete, silica fume is added in mixtures NC0.31, NC0.29, NC0.25, and NC0.20. For ceramsite concrete, the sand to aggregate ratio is changed from 32% to 42%. For foamed concrete, the water to cement ratio is changed from 0.60 to 1.43.




3.3. Test Methods and Sample Preparation
The compressive strength test was carried out conforming to BS EN 123903: 2009. The test specimens were prepared using standard metallic cube mould of size 10 cm × 10 cm × 10 cm and covered with plastic sheet after moulding for three days. After demoulding, all specimens were cured at >95% RH and 20 ± 2°C for 28 days. The compressive strength was calculated as the average of three test specimens. The specimens were experimented at room temperature of 20 ± 2°C with a relative humidity of 65%. Apparent relative density was determined after demoulding by weighting using electronic scale and volume calculation using checked dimensions.
4. Results and Discussion
4.1. Respective Results and Discussion for Different CementBased Materials
4.1.1. Grout
Simulations of Abrams’ law and Bolomey’s formula for grout were carried out, and the results are shown in Figure 1. Abrams’ law and Bolomey’s formula both have good simulations with Rsquare of 0.9733 and 0.9637, respectively. Parameters a_{1} and a_{2} of Abrams’ law are 245 and 66. Parameters b_{1} and b_{2} of Bolomey’s formula are 26 and 19. Simulations of watercementdensity ratio law for grout were carried out, and the results are shown in Figure 2. Watercementdensity ratio law has a good simulation with Rsquare of 0.9776. Parameters d_{1}, d_{2}, and d_{3} of watercementdensity ratio law are 2.4, 200, and 2174, respectively. Watercementdensity ratio law includes both the watercement ratio and apparent density of the cementbased material.
4.1.2. Normal Concrete
Simulations of Abrams’ law and Bolomey’s formula for normal concrete were carried out, and the results are shown in Figure 3. Abrams’ law and Bolomey’s formula both have good simulations with Rsquare of 0.9070 and 0.9293, respectively. Parameters a_{1} and a_{2} of Abrams’ law are 287 and 79. Parameters b_{1} and b_{2} of Bolomey’s formula are 33 and 33. Simulations of watercementdensity ratio law for normal concrete were carried out using the same simulation parameters of grout as shown in Figure 2, and the results are shown in Figure 4. Watercementdensity ratio law has a simulation with Rsquare of 0.6544 while parameters d_{1}, d_{2}, and d_{3} of watercementdensity ratio law are controlled to 2.4, 200, and 2174, respectively. It also seemed that while watercement ratio is higher than 0.3, the simulations are better because Abrams’ law is valid over the range of watercement ratios of 0.30 to 1.20 [19, 20]. For low watercement ratio, prediction of compressive strength is more complicated. The main reason is that for the cementbased material having high watercement ratio, a part of mixing water does not participate in cement hydration and finally forms pores in hardened hydration products which will determine the mechanical properties evidently. On the contrary, for the cementbased material having low watercement ratio, most of the mixing water participates in cement hydration and then does not form more pores in hardened hydration products. Generally speaking, for low porosity solid materials, more attention should be paid to the influence of chemical composition and load action form on material properties [21–23].
4.1.3. Ceramsite Concrete
Simulations of Abrams’ law and Bolomey’s formula for ceramsite concrete were carried out, and the results are shown in Figure 5. Abrams’ law and Bolomey’s formula both have bad simulations with Rsquare of 0.0696. Parameters a_{1} and a_{2} of Abrams’ law are 116 and 58. Parameters b_{1} and b_{2} of Bolomey’s formula are 15.4 and 15.5. Simulations of watercementdensity ratio law for ceramsite concrete were carried out using the same simulation parameters of grout as shown in Figure 2, and the results are shown in Figure 6. Watercementdensity ratio law has a simulation with Rsquare of 0.5112 while parameters d_{1}, d_{2}, and d_{3} of watercementdensity ratio law are controlled to 2.4, 200, and 2174, respectively.
4.1.4. Foamed Concrete
Simulations of Abrams’ law and Bolomey’s formula for foamed concrete were carried out, and the results are shown in Figure 7. Abrams’ law and Bolomey’s formula both have bad simulations with Rsquare of 0.1605 and 0.3985, respectively. Parameters a_{1} and a_{2} of Abrams’ law are 2.8 and 1.1. Parameters b_{1} and b_{2} of Bolomey’s formula are 0.35 and 2.12. Simulations of watercementdensity ratio law for foamed concrete were carried out using the same simulation parameters of grout as shown in Figure 2, and the results are shown in Figure 8. Watercementdensity ratio law has a simulation with Rsquare of 0.8154 while parameters d_{1}, d_{2}, and d_{3} of watercementdensity ratio law are controlled to 2.4, 200, and 2174, respectively.
4.2. Results and Discussion for General CementBased Materials
4.2.1. Abrams’ Law and Bolomey’s Formula
Simulations of Abrams’ law and Bolomey’s formula using watercement ratio for all cementbased materials were carried out, and the results are shown in Figure 9. Abrams’ law and Bolomey’s formula both have good simulations with Rsquare of 0.9391 and 0.8470, respectively. Parameters a_{1} and a_{2} of Abrams’ law are both 500. Parameters b_{1} and b_{2} of Bolomey’s formula are 28 and 29. Figure 9 shows that for ceramsite concrete and foamed concrete, simulations of Abrams’ law and Bolomey’s formula are not good. General simulation parameters are not uniform compared with the preceding individual simulations. Simulations of Abrams’ law and Bolomey’s formula using waterbinder ratio for all cementbased materials were carried out, and the results are shown in Figure 10. Abrams’ law and Bolomey’s formula have simulations with Rsquare of 0.6210 and 0.7952, respectively. Parameters a_{1} and a_{2} of Abrams’ law are 987 and 5930. Parameters b_{1} and b_{2} of Bolomey’s formula are 62 and 119. Comparison of Figures 9 and 10 shows that the watercement ratio law has a better simulation.
4.2.2. WaterCementDensity Ratio Law
Simulation of watercementdensity ratio law for all cementbased materials was carried out, and the results are shown in Figure 11. Watercementdensity ratio law has a good simulation with Rsquare of 0.9976. Parameters d_{1}, d_{2}, and d_{3} of watercementdensity ratio law are controlled to 2.4, 200, and 2174, respectively. General simulation parameters are uniform compared with the preceding individual simulations. Simulation of waterbinderdensity ratio law for all cementbased materials was carried out, and the results are shown in Figure 12. Waterbinderdensity ratio law has simulation with Rsquare of 0.9948. Parameters f_{1}, f_{2}, and f_{3} of waterbinderdensity ratio law are 1.6, 220, and 3333, respectively. Comparison of Figures 11 and 12 shows that the watercementdensity ratio law has a better simulation. It is reasonable to see that cement is still the main cementitious material in hydration and the hardening of cement hydration products determines mainly the mechanical properties of cementbased materials.
5. Conclusions
Based on the results identified in this study, the following conclusions are drawn. The novel watercementdensity ratio law is proposed based on watercement ratio law. Compared with Abrams’ law and Bolomey’s formula, the watercementdensity ratio law illustrates better simulations for the prediction of the 28day standard curing compressive strength of cementbased materials including grout, normal concrete, ceramsite concrete, and foamed concrete designed in this contribution over the range of watercement ratios of 0.20 to 1.40. The watercementdensity ratio law includes both the watercement ratio and relative apparent density of the cementbased material. Relative apparent density of the cementbased material is an important one of all the factors determining the compressive strength of the cementbased material. The watercementdensity ratio law will be beneficial for the precise and generalized prediction of the 28day standard curing compressive strength of cementbased materials. As a topic of future research, the effects of cement type and curing conditions on parameters of the watercementdensity ratio law should be studied.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The financial support for ongoing projects by the National Natural Science Foundation of China (51922052, U1706222, 51778309, and 51208013) and National Natural Science Foundation of Shandong Province (ZR2018JL018) is greatly acknowledged.
References
 G. A. Rao, “Generalization of Abrams’ law for cement mortars,” Cement and Concrete Research, vol. 31, no. 3, pp. 495–502, 2001. View at: Publisher Site  Google Scholar
 A. ElNemr, “Generating water/binder ratiotostrength curves for cement mortar used in Masnory walls,” Construction and Building Materials, vol. 233, Article ID 117249, 2020. View at: Publisher Site  Google Scholar
 G. A. Rao, “Role of waterbinder ratio on the strength development in mortars incorporated with silica fume,” Cement and Concrete Research, vol. 31, no. 3, pp. 443–447, 2001. View at: Publisher Site  Google Scholar
 A. Kargari, H. EskandariNaddaf, and R. Kazemi, “Effect of cement strength class on the generalization of Abrams’ law,” Structural Concrete, vol. 20, no. 1, pp. 493–505, 2019. View at: Publisher Site  Google Scholar
 T. S. Nagaraj and Z. Banu, “Generalization of Abrams’ law,” Cement and Concrete Research, vol. 26, no. 6, pp. 933–942, 1996. View at: Publisher Site  Google Scholar
 I.C. Yeh, “Generalization of strength versus watercementitious ratio relationship to age,” Cement and Concrete Research, vol. 36, no. 10, pp. 1865–1873, 2006. View at: Publisher Site  Google Scholar
 S. Bhanja and B. Sengupta, “Modified watercement ratio law for silica fume concretes,” Cement and Concrete Research, vol. 33, no. 3, pp. 447–450, 2003. View at: Publisher Site  Google Scholar
 E. M. Golafshani and A. Behnood, “Estimating the optimal mix design of silica fume concrete using biogeographybased programming,” Cement and Concrete Composites, vol. 96, pp. 95–105, 2019. View at: Publisher Site  Google Scholar
 N. P. Rajamane and P. S. Ambily, “Modified Bolomey equation for strengths of lightweight concretes containing fly ash aggregates,” Magazine of Concrete Research, vol. 64, no. 4, pp. 285–293, 2012. View at: Publisher Site  Google Scholar
 D. Falliano, D. De Domenico, G. Ricciardi, and E. Gugliandolo, “Experimental investigation on the compressive strength of foamed concrete: effect of curing conditions, cement type, foaming agent and dry density,” Construction and Building Materials, vol. 165, pp. 735–749, 2018. View at: Publisher Site  Google Scholar
 Z. M. Yaseen, R. C. Deo, A. Hilal et al., “Predicting compressive strength of lightweight foamed concrete using extreme learning machine model,” Advances in Engineering Software, vol. 115, pp. 112–125, 2018. View at: Publisher Site  Google Scholar
 F. Khademi, M. Akbari, S. M. Jamal, and M. Nikoo, “Multiple linear regression, artificial neural network, and fuzzy logic prediction of 28 days compressive strength of concrete,” Frontiers of Structural and Civil Engineering, vol. 11, no. 1, pp. 90–99, 2017. View at: Publisher Site  Google Scholar
 P. Zhang, F. H. Wittmann, M. Vogel, H. S. Müller, and T. Zhao, “Influence of freezethaw cycles on capillary absorption and chloride penetration into concrete,” Cement and Concrete Research, vol. 100, no. 10, pp. 60–67, 2017. View at: Publisher Site  Google Scholar
 P. Zhang, F. H. Wittmann, P. Lura, H. S. Müller, S. Han, and T. Zhao, “Application of neutron imaging to investigate fundamental aspects of durability of cementbased materials: a review,” Cement and Concrete Research, vol. 108, pp. 152–166, 2018. View at: Publisher Site  Google Scholar
 P. Zhang, D. Li, Y. Qiao, S. Zhang, C. T. Sun, and T. Zhao, “The effect of air entrainment on the mechanical properties, chloride migration and microstructure of ordinary concrete and fly ash concrete,” Journal of Materials in Civil Engineering, vol. 30, no. 10, Article ID 04018265, 2018. View at: Publisher Site  Google Scholar
 J. Bao, S. Li, P. Zhang et al., “Influence of the incorporation of recycled coarse aggregate on water absorption and chloride penetration into concrete,” Construction and Building Materials, vol. 239, Article ID 117845, 2020. View at: Publisher Site  Google Scholar
 S. Xue, P. Zhang, J. Bao, L. He, Y. Hu, and S. Yang, “Comparison of mercury intrusion porosimetry and multiscale Xray CT on characterizing the microstructure of heattreated cement mortar,” Materials Characterization, vol. 160, Article ID 110085, 2020. View at: Publisher Site  Google Scholar
 R. Buši´c, M. Benši´c, I. Miliˇcevi´, and K. Strukar, “Prediction models for the mechanical properties of selfcompacting concrete with recycled rubber and silica fume,” Materials, vol. 13, p. 1821, 2020. View at: Google Scholar
 L. K. A. Sear, J. Dews, B. Kite, F. C. Harris, and J. F. Troy, “Abrams law, air and high watertocement ratios,” Construction and Building Materials, vol. 10, no. 3, pp. 221–226, 1996. View at: Publisher Site  Google Scholar
 S. Popovics and J. Ujhelyi, “Contribution to the concrete strength versus watercement ratio relationship,” Journal of Materials in Civil Engineering, vol. 20, no. 7, pp. 459–463, 2008. View at: Publisher Site  Google Scholar
 K. Zhao, Y. Qiao, P. Zhang, J. Bao, and Y. Tian, “Experimental and numerical study on chloride transport in cement mortar during drying process,” Construction and Building Materials, vol. 258, Article ID 119655, 2020. View at: Publisher Site  Google Scholar
 J. Bao, S. Xue, P. Zhang, Z. Dai, and Y. Cui, “Coupled effects of sustained compressive loading and freeze–thaw cycles on water penetration into concrete,” Structural Concrete, pp. 1–11, 2020. View at: Publisher Site  Google Scholar
 Q. Song, H. Y. Zhao, J. W. Jia et al., “Pyrolysis of municipal solid waste with ironbased additives: a study on the kinetic, product distribution and catalytic mechanisms,” 2020 in Journal of Cleaner Production, vol. 258, Article ID 120682, 2020. View at: Publisher Site  Google Scholar
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Copyright © 2020 Siqi Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.