문제 보고This book has a different problem? Report it to us
"네" 선택하시는 조건: "네" 선택하시는 조건: "네" 선택하시는 조건: "네" 선택하시는 조건:
파일 열기 성공했습니다
파을 내용은 책 (또는 만화)입니다
책 내용이 적당합니다
파일의 제목, 작성자와 언어가 책 설명과 일치합니다. 다른 필드는 보조이므로 무시하셔도 좋습니다.
"아니요" 선택하시는 조건: "아니요" 선택하시는 조건: "아니요" 선택하시는 조건: "아니요" 선택하시는 조건:
- 잘못된 파일입니다
- 이 파일이 DRM으로 보호돼 있습니다
- 파일은 책이 아닙니다 (예: xls, html, xml)
- 파일은 기사입니다
- 파일은 책에 일부입니다
- 파일은 잡지입니다
- 파일은 시험지 또는 테스트입니다
- 파일은 스팸입니다
책의 내용이 적당하지 않으며 차단되어야 한다고 생각합니다
파일의 제목, 작성자와 언어가 책 설명과 일치하지 않습니다. 다른 필드는 무시하셔도 좋습니다.
Change your answer
Construction and Building Materials, Vol. 10, No. 3, pp. 221-226, 1996 Copyright 8 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 095&0618/96 $15.00+0.00 09504618(95)00079-8 ELSEVIER Abrams law, air and high water-to-cement ratios L. K. A. Sear*t, J. Dews*, B. Kite*, F. C. Harris* and J. F. Troy* *Tarmac Topmix limited, Milifieids Road, Ettingshali, Wolverhampton, West Midlands, WV4 6JP, UK #University of Woiverhampton, Wulfruna Street, Woiverhampton, West Midlands WVI, UK Received 24 February 1995; revised 30 September 1995; accepted 10 June 1995 Abrams law linearly relates the logarithm of strength to the water to cement ratio. This article shows that between water to cement ratios of 0.30 and 1.20 significant improvements to the linear regression correlation coefficient can be made by the inclusion of the volume of entrained air, but not entrapped air, in the mix to the water content. However, the inclusion of higher water and air to cement ratios above 1.20 in the analysis produced unreliable regression coefficients. Further investigation showed an independent, second linear relationship existed at high water plus air cement ratios with a different slope and intercept. This change of slope coincides with a turning point predicted by the Dewar method of mix design. Comparison is also made of the logarithm of strength and porosity, as a fundamental relationship. The augmented Abrams law was found in all cases to improve the accuracy of regression analysis. Keywords: concrete; air-entrained; water-to-cement Abrams’ law Admixtures, particularly air entraining agents, degrade the correlation coefficient found from the linear regression. Production concrete is normally considerably more variable. Plotting the water to cement ratio against the logarithm of strength for production data is very different to laboratory trial mix data, as the normal variation found decreases the correlation coefficient markedly. Additionally the effect of taking the logarithm of the stren; gth on a normal variable3 like concrete strength, makes the relationship V-shaped, as seen in Figure 2. This type of relationship is called multiplicative3. Abrams law has limitations with concrete containing admixtures which entrain air, such as air entrainers and some plasticizers. These materials create spherical air bubbles normally assumed to range between 0.25 mm and 0.1 mm in diameter. Air entraining agents are specifically designed to create stable air bubbles within this size range to give frost resistance to the concrete. Some plasticizers entrain air as a side effect, particularly at higher workabilities. This effect is tolerated in readymixed concrete as the increased air content corrects the yield of the plasticized concrete back to that of an equivalent non-plasticized concrete. Additionally the entrained air reduces the water content, increasing the apparent water reduction achieved due to the addition of the plasticizer. All the concretes researched contained commercially available admixtures. The measurement of bubble spacing and bubble Duff Abrams in 1919 developed the following law’ for the strength of concrete. Strength = e K (1) K,F where K, and K2 are constants, W is the mass of free water and C the mass of cement per unit volume. This resolves to: Log(strength) = K, + K, x 5 (2) The formula is valid over the range of water to cement ratios of 0.30 to 1.20, which is satisfactory for structural concretes. Taking a typical set of trial mix results, the relationship is clearly seen as being valid as seen in Figure I. The constants K, and K2 are calculated by linear regression’ on the log strength ver3U.rwater to cement ratio. With trial mix data the correlation coefficient, which is a measure of the goodness of fit, can be 0.99 if the trials are carefully carried out on a single aggregate size and workability and exclude the use of admixtures. t Correspondence Wolverhampton, to L. K. A. Sear at: 10 Bromley West Midlands WV8 IBE, UK Gardens, ratios Codsall, 221 Abrams law, air and high water-to-cement 222 Water to Cement Ratio vs LO&IStrength Typical Laboratory Trial Mix ata ii I .F 0.40 0.50 0.60 0.70 Free Water to Cement z 7 Day Actual - Data from laboratory 0.90 Rabo _ 7 Day Regression 26 Day Actual Figure 1 0.80 26 Day Regresston trial mixes Production 1OOmm Test Cube Data l/6/69 Concretes made with Portland Cement from to 14/2/92 with common aggregate and cement ratios: L. K. A. Sear et al. Additionally at water to cement ratios greater than 1.2 the relationship changes and the linear regression coefficients K, and K2 derived at lower water to cement ratios cannot be used. Such problems have been found by others”’ and various augmentations have been suggested to improve the situation. Popovics’ suggests using multiple terms including the water to cement ratio, the cement content, the water content, the cement content plus the water content, the logarithm of the cement content and others. Shilstone’ questions whether it is the volume of water to the bulk volume of cement that Abrams law refers to rather than the masses of these constituents. Studying these proposals gives no clear indication of a satisfactory approach which could be used to predict the strength of both production and trial mix concretes. The following augmentations were studied, developed and found to be universally beneficial. Augmentingfbr source air content Material scientists relate the strength to the porosity as a primary law Neville’ gives examples of a number which follow a single power function Strength = Strength,, (1 - p) I 1 .oo Water to Cement Regression : Data Slope = 3.7066 Correlatton Coeffuent Ratlo Intercept = -0.7093 = 2.0563 Figure 2 Data extracted from a database of test cube data used for routine quality control from a number of readymixed concrete plants. All concretes are based on common coarse and fine aggregate and cement sources size was not relevant as the research carried out was designed to be wide ranging, covering a number of admixtures, aggregates types and cement sources. The air entraining agents used were based on either Vinsol resin or synthetic based products and the plasticizers were ligno-sulphonate based materials. All admixtures were commercially available and in common usage. Entrapped air is air that is left in the concrete due to lack of compaction as opposed to being created by surfactants like air entraining agents. The entrapped air bubble has a random shape and size. Some entrapped air is always present in concrete even with normal compaction techniques, whether in the structure or in test cube specimens. This residual air is normally assumed to be in the range 0.5 to 1.5% at normal workabilities3. Strengths are invariably reduced by the presence of both entrained and entrapped air. Entrapped air exists in all normally compacted concrete and therefore is included in the Abrams law linear regression coefficients, K, and K2. However, concretes with air entraining admixtures are consistently found to fall below the regression line for mixes of the same aggregate and cement type, but without admixtures. of many materials for strength, and of brittle materials as in Equation (3). (3) The influence of the volume of pores, p, on strength can be expressed as a power function, where Strength,, is the strength for zero porosity. n is a coefficient, which may NOT be a constant. Abrams law could be considered to be a modified version of this law. The porosity of the material is analogous to the free water content which has not chemically combined with the cement plus any air. The strengthgiving material is the cement. The aggregates act as filler making the cement paste extend as far as practicable. This presumes the aggregate strength is greater than that of the cement paste. The missing term in the Abrams law is the air content of the concrete. This adds to the the porosity, reducing strength by typically 5% for each 1% air3. All normal concretes contain some entrapped air even when ‘fully’ compacted in test cubes. This residual air is a function of the workability, the degree of compactive effort applied, and the presence of admixtures. Analysis of laboratory trial mix data carried out at the principal author’s laboratory showed that a significant improvement in correlation coefficients for Abrams relationship could be achieved by the addition of the volume of entrained air in the mix. The air content of the concrete, in litres, is added to the free water content. This residual air can be calculated by the absolute volume of constituents in the concrete using their relative densities and computing the theoretical maximum density with trial mix data. The difference between this theoretical density and the measured plastic density must be the residual air content. For production data accurate calculation of residual air contents is considerably more difficult. Obviously the direct measurement Abrams law, air and high water-to-cement of the air content, using a pressure air meter, would give a reliable indication of the air content air. Where direct measurement is not available the entrapped and entrained air content can be approximated by making some general assumptions for various workabilities and admixtures. A simple empirical relationship was tested which could estimate entrapped air contents for various measured slumps as in Equation (4). ratios: 1. K. A. Sear et al. Typical 1.2 laboratory 1 ” \ / Air content = Ai%( l- dslumg Water mm] : (4) o Augmgntinq for Air Content tn I m!x resu s. 1OOmm cubes 8 Portland FW:*UOIIAW ” 0.4 0.3 ” 1” laboratory Augmentinkfor Air Content tn I mx resu 1OOmm cubes 8 Portland Re:~uOI(R12 l 20mm IOmm 0.9 for air content) agg. 8 air entrainer aggregates Regression Coefficient I ” 0.6 Line = 0.9657 reduce the correlation coefficient. Various forms of Equation (4) with differing values for the zero slump air content and a cubic root form were tested and all found to reduce the correlation coefficient. The reason adjusting for entrapped air in production concrete did not improve log strength to water to cement ratio relationship is probably due to the following. The relationship between slump and strength with readymixed concrete is invariably found to be very weakg. Figure 5 shows typical results taken from the production database. This would imply slump has little bearing on the water to cement ratio or strength of production concrete and the variability observed is due to other factors. Such factors are aggregate and cement properties, batching accuracy, compensation for moisture content and the repeatability of the slump test. In the production data analysed to date, by the authors, it can be shown that augmenting for entrapped air or slump is never beneficial. More complex analysis of the other variables involved would appear to be required to improve Abrams’ relationship further. Augmenting for high water to cement ratios When carrying out linear regression on trial mix results, if results are included with water to cement ratios which Comparison of Slump versus Streng Portland Cement, 2 Cement o 1 ” 0.7 Ratio (i.e. adjusted agg. mixes agg. Pump mixes agg. B Plasticise~ Cement. Figure 4 Adding the volume of air to the water content improves the accuracy of regression coefficient 70.0 Typical 1 ” 0.6 plus air to Cement 20mm 20mm 20mm 1” ” 0.5 COrrelatiOn where 300 represents the height of the slump cone in mm and Air, represents the maximum entrapped air content at zero measured slump. A suitable Air,, value could be 3.0%. The effect of admixtures is somewhat easier to estimate as manufacturers’ data usually states the degree of air entrainment in the literature for the admixture. For example, ligno-sulphonate based plasticizers typically entrain 1.0 to 2.0% air at standard dosages at 50 mm slump; however, at higher workabilities the manufacturers’ published values can be misleading, with the true air contents being considerably higher. The trial mix data shown in Figures 3 and 4 show the effect of augmenting for entrapped and entrained air. The trials include normaI 20 mm maximum aggregate size mixes, pump mixes, plasticized mixes, air entrained mixes and 10 mm aggregate mixes. Augmenting for air improves the coefficient of determination in the above trial mixes by 7.6%, that is the correlation coefficient increases from 0.93 to 0.97, and allows the inclusion of air entrained concretes into the regression analysis. Of all the trial data analysed to date, improvements in coefficients of determination of between 5% and 8% have been seen. Similar improvements were found when analysing a large production test cube database of some 2100 results. The improvements in the coefficient of determination were typically 5.2% to 6.6% for Portland cement based concretes when augmenting for entrained air, i.e. from air entraining admixtures and platicizers. However, augmenting for entrapped air based on the concrete workability using Equation (4) was found to 223 E i 1OOmm t&eswhk,data @m3 comected to I ., 60.0 2 g 50.0 d .%40.0 z X P 30.0 = $ v) 20.0 I .L 0.3 0.4 Water 0.5 to Cement 0.7 0.6 Ratio (Not adjusted 0.6 0.9 I 10.0 30 0 for air content) 60 Measured E o 20mm 20mm 20mm agg. mixes egg. Pump mixes egg. & Plasticii~ Correlatiin o 20mm egg. 8 air entrainer l l0mm aggregates Coeffiiient Regression Line = 0.9265 Figure 3 Trial mix test cube data including air entrained and plasticized mixes not adjusted for air content x Measured Slump’- 90 120 150 Slump in mm Linear Regression 5 Comparison of measured slump of concrete against strength at 28 days extracted from a single readymixed concrete plant production using common aggregate and cement source throughout the period Figure 224 Abrams law, air and high water-to-cement are greater than 1.2 one finds the correlation coefficient reduces dramatically and the slope of the line invariably is lowered. Inclusion of such results gives a shallow linear relationship and therefore relatively low theoretical strengths at lower water to cement ratios. This makes Abrams law unusable when such data is included. To investigate this phenomenon, a series of trial mixes was carried out to include a full range of water to cement ratios from 0.3 to 4.5. Figure 6 shows the relationship(s) found from this series of trial mixes at seven and 28 days. Clearly a second linear relationship exists at higher water to cement ratios with a different slope and intercept. This data was confirmed by further laboratory trials mixes on different material sources. By detailed analysis of 145 laboratory trial mixes the same pattern was observed. The errors associated with measuring low strengths are higher when expressed logarithmically. The degree of scatter in the data at higher water to cement ratios is to be expected. As suggested above, the porosity of the concrete mix should be related to the strength. As water plus air representi the volume of material NOT contributing to the solid phases, this must be effectively the porosity of the mix. When the voidage is calculated from the relative densities of the aggregates and cement, this relationship is clearly seen as in Figure 7. Some tendency to being linear is seen, but a breakdown of the relationship occurs at cement contents greater than 400 kg/m3 (water and air/cement ratio = 0.465). However, some of the water in the mix must be taken out of the system due to hydration of the cement. By adjusting the voidage for hydration the low water to cement ratio mixes become generally more linear. Figure 8 shows the effect of allowing for 45% of the cement being hydrated. Effectively this is excluding the combined water from the voidage. This may or may not be chemically reacted with the cement as some water will be interstitial which contributes to the strength by allowing a chemical reaction to continue. The figure 45% was found by inspection of the data to achieve the best linear relationship. Figure 9 shows the 95% confidence limits based around a standard deviation of trial mix and testing of 1.5 N/mm’. This clearly shows that linear regression over two ranges of water to cement ratios is a solution to predicting strength from cement contents over the widest range of practicable strengths. Other factors Abrams law explains only part of the true relationship for the strength of concrete. The true relationship is likely to be simply that the log of strength is inversely proportional to the porosity, as with other materials. However, the porosity is just as complex to calculate accurately as the water to cement ratio and the terminology of water to cement ratios are well understood by concrete technologists. Therefore, there would appear little or no benefit in using the porosity. ratios: L. K. A. Sear et al. Water & Air to Cement Ratio versus Strength High and Low Water/Cement Cement. Ratio. 1OOmm test cubes, Portfan R*:HIOHWCI* i 1.0 2.0 4.0 3.0 Water 8 Air lo Cement ’ Figure 6 26 day strengths Laboratory trial o 5.0 Ratio 7 days strengths mix data including high water/cement ratio data Percenta e voids based on Relative Densi Laboratory #aI mixes, 100mm test cubes, Portfand Cemat!Y FkHKIHWC,B P 16.0 19.0 20.0 Percentage q Figure 7 Voidage of the Perct;+ag 22.0 21 .o Voids baaed on abclute l WC>l.O concrete 23.0 volume. WC<=l.O compared to logarithm of strength Voids based on Relative Density for hydratlcn of cement (45% hydrated) co/ - 1.0 I, 12.0 ” I”” ” I 16.0 14.0 I 16.0 Percentage x WC>l.O O ” ” I ” 20.0 ” / ” 22.0 ” 4 214.O Voids WC-zl.0 Adjusting the void content for the hydration of cement improves the relationship Figure 8 Of interest is the cause of the change of slope observed at water to cement ratios greater than 1.2 (or thereabouts). Those familiar with the Dewar” method of mix design will observe the water to cement ratio of 1.2 coincides closely with one of the turning points in the water demand. The Dewar method relates the process of mix design of concrete to the voidage between various particles, i.e. cement, fine and coarse aggregates. Using complex calculations on the bulk density, relative density and logarithmic mean particle size of the materials, a series of batchweights can be Abrams law, air and high water-to-cement Water & Air to Cement ratio versus 28 day strength 95%Conti&mcelimitsbasedR:HHYIWCIG on SO 1.5Nhm2 trialhestrngvartat n I ~o”“‘l”““““““““” L 5.0 4.0 3.0 2.0 1.0 0.0 Water8 Airto CementRatio - x Resutt - Regress - 95%Min ratios: L. K. A. Sear et al. 225 to completely fill the voids between the coarser fine aggregate particles. This free space is therefore filled with water. As cement contents are reduced below point B, i.e. at higher water and air cement ratios, increasingly large amounts of water are required to fill these voids. The water demand of the concrete consequently increases. The strength of the cement paste is then diluted by the voidage and should follow the porosity/strength relationship. Close examination of the logarithm of strength versus water plus air plot may even suggest a correlation between other turning points and the augmented Abram relationship. 95% Max Figure 9 Relationship with 95% confidence limits for strength values Conclusions Table 1 Dewar mix design analysis from trial mix data in Figure 9 Augmentations to Abrams law are consistently beneficial to the accuracy of the relationship as follows. Firstly, adding the volume of entrained air in the mix to the volume of water in the water to cement ratio term gives consistent improvements. A further addition of entrapped air to trial mix data is beneficial, but not to readymixed concrete production data. Secondly, there are two linear relationships. A second relationship exists at high water to cement ratios above 1.2 which concurs with one of the turning points predicted by the Dewar method of mix design. Further turning points may also exist which are coincident with other Dewar points. To date the augmented Abrams relationship would appear to be a major improvement and correlate with the Dewar mix design method. Using the augmented Abrams relationship with production data also gives significant improvements in linear regression correlation coefficients adding weight to the theory. Further work to determine the relationship between regression slopes and intercepts at the cross-over point is required. These relationships would appear to be complex from the data analysed to date. Additionally whether the other turning points do exist, as suggested by the Dewar mix design method, could be researched. Though high water to cement ratios are not used for structural concrete, accurate relationships are required for commercial reasons to generate cement content versus strength charts at low characteristic strengths, that is below 20 N/mm’, for readymixed concrete plants. Cement content (kg/m’) 0.0 154.8 290.0 403.3 624.2 830.5 Coarse aggregate (kg/m’) 940.3 1063.8 1112.9 1143.3 1046.6 1154.8 Fine Aggregate (kg/m? Water Demand (kg/m’) 912.6 914.7 802.3 663.5 458.4 0.0 210.1 197.0 111.9 182.8 225.8 292.9 Water air/ cement ratio N/A 1.370 0.665 0.490 0.386 0.371 Turning point letter A B C D E F computed. These are not a continuous series of material weights but five linear relationships which are connected at six turning points. Table I shows the Dewar mix design analysis from the trial mix data as shown in Figure 9. Turning point B occurs at 1.370 and indicates the point where a significant change in the material contents is required for higher water and air cement ratios. The value at which the two linear regression lines cross from the augmented Abram analysis is 1.374. This would imply there is close correlation between Dewar’s theories and the augmented Abrams law. Figure IO compares graphically the water demand verms water and air cement ratio overlaid with the logarithm of strength. Point B is that point at which there are insufficient fines in the mixture, from the fine aggregate and the cement, B Water and Air to Cement ratio versus 28 day strength f Comparisonwtth Dewar mix de&i turningpain s FL:HIO(IWCV E References 0 5 Water 8 Air to Cement Ratio ’ - Result Regress - 95% Min 95% Max - Water Demand Figure 10 Turning point in Abrams relationship Dewar mix design method turning point B 2 coincides with Neville, A. M. Properties of’ Concrete, 3rd edn, Longman Scientific & Technical, Harlow, 1987, pp. 268-275 Mendenhal, W. and Sincich T. Stutistics fbr the Engineering und Computer Sciences New York, 1992 Dewar, J. D. and Anderson, R. Manual oJ Ready-Mi\-erl Concrete, Chap 5, Blackie, Glasgow, 1988 Wagner-Grey, U. Water/cement ratio and compressive strength of concrete - findings and results of external quality control centers. Betomverk & Fertigeil-Technik 1983, 49 (11) 691-695 Shilstone, Snr. J. M. The water-cement ratio - which one and where do we go. Concr. Intern. September 1991, 64 226 6 7 Abrams law, air and high water-to-cement Kosmakta, S. H. In defense of the water-cement ratio. Concr. Intern September 1991, 6549 Popovicsi S. Analysis of the concrete strength versus watercement ratio relationshio. ACI Muter. J. Seutember/October 1990, 517-529 8 Barton, AL47 .” 9 ratios: L. K. A. Sear et al. R. B. Water-cement ratio is passe. Concr. Prod. 92 (11). ._ Shilstone November Snr, J. M. Interpreting 1988, 68-70 the slump test. Concr. Intern.