Guidelines for Concrete Mixtures Containing Supplementary Cementitious Materials to Enhance Durability of Bridge Decks (2007)

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76 Introduction The application of statistically based experimental design enables investigating a test space containing a large number of possible combinations of factors with a relatively small number of tests. The goal of this methodology is to conduct an experiment to determine the combination of the available and appropriate raw materials that will best produce the desired performance in a concrete mixture. The method re- quires mixing and testing several different concrete mixtures. In this step, the experimental matrix (i.e., the specific fac- tors to be investigated and the selected subset of the possible combinations for actual testing) will be chosen and tradeoffs will be made between desired and practical properties. The user should understand this step completely before trying to apply this portion of the methodology. Many approaches to the design and analysis of such experiments are possible; the “Orthogonal Design Method” is recommended. The method requires a minimum number of mixtures (usually 8 to 18) for a reasonably comprehensive investigation, while still allowing statistical modeling to predict the performance of mixtures that were not actually tested. Several methods for designing experiments to select con- crete mixtures together with their strengths and weaknesses (especially compared with the Orthogonal Design Method) follow: • Expert Opinion Method. In this method, experts look at the available materials and the potential levels of the fac- tors and identify a number of different concrete mixtures that are likely to perform well based on the experts’ expe- rience and knowledge. Samples of these mixtures are then cast and evaluated, and the concrete that performs the best is chosen. This method can produce many potentially very different types of mixtures, allowing for several chances of choosing a good mixture. Also, the number of mixtures is easily selected without constraints. However, if none of the chosen mixtures turn out to be acceptable, predicting how to adjust the mixtures to achieve better performance is dif- ficult. Also, investigating a limited number of mixes may not produce the optimum mixture. • One-Factor-at-a-Time Method. In this method, experts select a single mixture, called the “control mixture,” that is most likely to perform well. One factor is then varied from its level in the control mixture, and a new mixture (with only that factor changed) is cast and tested. The number of mixtures needed can be easily calculated because one addi- tional mixture is needed for each two-level factor, and two additional mixtures are needed for each three-level factor being considered. Some prediction of the performance of future mixtures can be made because each factor’s effect can be estimated using the mixtures with that varied factor. However, because only two mixtures (or three mixtures in the case of a three-level factor) are used to estimate the fac- tor effects, the estimates and prediction are likely to be quite poor. Also, because only one factor is varied at a time, the mixtures that are actually cast and tested are quite similar to each other and thus have a relatively high chance of all producing poor results if the control mixture is ill-chosen. In addition, for durability-based investigations that involve long-term testing, subsequent mixtures are not selected until testing is completed on the previous mixture, making this method probably the least-suited for development of mixtures where durability is a main objective. • Orthogonal Design Method. This method will be applied in this research and is recommended for most situations where there are a moderate number of factors to be stud- ied, the number of mixtures is limited (typically 8 to 16 mix- tures for three to seven factors), and reasonably good pre- diction of the performance of future mixtures is needed. In this method, a statistically designed experiment matrix that contains a list of mixtures to be tested is chosen. The rec- ommended design will be an orthogonal, main effects de- sign (for brevity, called orthogonal designs). Orthogonal S T E P 3 Generate the Experimental Design Matrix

77 designs are created so that each factor level is balanced with every other factor level such that an independent estimate of each factor effect is possible. The method allows for rel- atively simple statistical analysis and reasonably good esti- mates of the factor effects because all mixtures are used in the estimation of each factor effect. Also, because all factors are varied simultaneously, mixtures are likely to be quite different from each other, thus increasing the chance of finding a good mixture. The factor effects are estimated with much more precision than in the one-factor-at-a- time method, and thus it is much more likely to be able to predict a new (untested) mixture that has good or better performance than the tested mixtures. • A Response Surface Method. This method is an in-depth study that requires more mixtures (17 to 31 mixtures for three to five factors) than the other methods. This method is appropriate if the goal is to truly optimize the mixture (with less regard for high testing cost and time require- ments). This method is most likely to find the truly optimal mixture, but it requires a very large number of mixtures in most practical cases, making it less attractive to many prac- titioners. In addition, this method requires a high level of statistical sophistication to apply. In Step 1 (through the use of Figure S1.1 and Worksheet S1.1), the user will have identified the service environment, the appropriate concrete performance and testing require- ments, and reviewed the guidance on the influence of SCMs on these properties. In Step 2, the user will have summarized the durable sources of raw materials in Worksheet S2.1 and excluded some of these sources. (Excluded sources should have an “X” marked through them on Worksheet S2.1.) The information in Worksheets S1.1 and S2.1 will now be com- bined into Worksheet S3.1, which summarizes the potential factors and levels that could be tested in the orthogonal design experiment. Finally, the potential choices in Work- sheet S3.1 will be narrowed down to a final selection of fac- tors and levels to actually be tested, based on available time and resources. Generation of the Experimental Matrix The following steps should be followed to fill out Work- sheet S3.1 with the potential factors and levels (this list will be trimmed later based on the practical constraints of the investigation): 1. Type Factors (information from Worksheet S2.1) a. Any time that different types of materials are to be sub- stituted for each other in different mixtures, a type fac- tor should be created that describes the general category of the materials being substituted. For example, if differ- ent types of SCM (like slag or fly ash) are to replace each other in different mixtures, then a type factor called “Type of SCM” should be created. The levels of the type factor are the material names to be used as the SCM (if there is more than one SCM used in any mixture, the SCMs could be designated as the first SCM, the second SCM, etc.). The example presented in this step demon- strates the use of type factors for two SCMs. b. If there are two material sources to be chosen as a type factor, their names should be filled in as Levels 1 and 2 of that type factor on Worksheet S3.1, and Level 3 is left blank. If there are three material sources, they become the Levels 1 through 3 for that type factor. c. After filling out the row for the new type factor, the user places an asterisk beside the type that is most likely to perform very well in this application (if known). If there is high confidence that one of the types is best, then the user places a double asterisk next to that type. The asterisks will be useful later because some design matrices use certain levels more often than others. In these cases, levels that are expected to perform very well will be tested more often. 2. Source Factors (information from Worksheet S2.1) a. Any row in Worksheet S2.1 that has more than one source should be copied into the “Source Factors” sec- tion of Worksheet S3.1 (sources excluded by an “X” should be ignored). The name of the factor should con- sist of the material name and “Source.” For example, if Class C fly ash has two sources, then a new source fac- tor named “Class C fly ash Source” is to be created in Worksheet S3.1. b. If there are two sources to be investigated as a source factor, their names should be filled in as Levels 1 and 2 of that source factor, and Level 3 is left blank. If there are three sources, they become the Levels 1 through 3 for that source factor. c. After filling out the row for the new source factor, the user places an asterisk beside the source that is most likely to perform very well in this application (if known). If there is high confidence that one of the sources is best, then the user places a double asterisk next to that source. d. The previous steps are repeated for all rows in Work- sheet 2.1 that have more than one source listed. 3. Amount Factors (information from Worksheet S1.1) a. Any column in Worksheet S1.1 that has a range of pos- sible amounts in the summary should become a row in the “Amount Factors” section of Worksheet S3.1. b. The column heading and “Amount” (such as “Class F fly ash Amount”) become the new amount factor name.

78 c. Note: If two levels are examined in an amount factor, a trend can be established to describe the influence of the factor. However, testing three levels provides the abil- ity to determine an optimum (pessimum) level for that factor. For this reason, it is most advantageous if SCMs are tested at three levels. d. The user observes the range of amounts listed for the new amount factor in Worksheet S1.1. If the best per- formance is clearly expected at one or the other end of the range, then this factor should be assigned two levels. Typically, Level 1 will be the bottom of this range, and Level 2 will be the top of the range; however, any two val- ues in the range can be selected. e. If the best performance of a particular factor is likely to be at an amount somewhere inside the range listed in Worksheet S1.1, then that factor should be a three level factor. Level 1 will be the bottom of this range, Level 2 will be the middle of the range, and Level 3 will be the top of the range. However, for practical considerations, the number of three-level factors should be kept to a minimum, because three-level factors will increase the required testing. f. After filling out the row for the new amount factor, the user places an asterisk beside the level that is most likely to perform well in this application (if known). If there is high confidence that one of the levels is best, then the user places a double asterisk next to that amount. g. The previous steps are repeated for all columns in Worksheet S1.1 that have a range listed in the summary box. 4. Compound Factor (information from Worksheet S1.1) a. A compound factor is actually two factors: a type fac- tor and an amount factor that work together to define different amounts for different materials. This is pos- sible because, for the purposes of the experiment, the amount factor is defined generically. However, the generic definition is different for each level of the associated type factor. For example, suppose that Fac- tor 1 is a type factor for SCM, and its levels are fly ash or slag. Factor 2 then is selected as an amount factor whose levels are low and high. The amounts specified for low and high for each type of SCM can be different. Low and high for fly ash might be 15% and 40%, but low and high for slag might be 25% and 50%. Thus, the levels of Factor 2 change (from 15% and 40% to 25% and 50%) depending on the level of Factor 1 (either fly ash or slag). The reason for using a compound factor is that it allows more than one type of SCM to be used interchangeably while still testing at appropriate levels for that specific SCM. Without a compound factor, it would only be possible to set a type factor as multiple types of SCM and an amount factor at certain levels, but then those levels would apply to all of the SCM types. Relying on this non-compound factor is potentially problematic because, for example, slag is typically used at higher contents than fly ashes. While this greater flexibility may be of value in some situations, using compound factors adds to the complexity of the data analysis. In addition, the modeling is performed on each factor in- dividually, and so the responses from all mixtures with the same generic amount definition will be averaged to- gether even if the specific definitions are quite different. The modeling can still be performed on the amount factor component of the compound factor to allow interpolation; however, the levels must be scaled over an artificial range so the analysis can be performed over the same ranges. (More discussion on the analysis is given in Step 5.) b. For compound factors, the type factor and the amount factor will be named as before; however, the amounts to be used for each type must be separately specified. Worksheet S3.2 should be used to create a compound factor from a type factor and an amount factor. Exam- ples are provided in the “Example from Hypothetical Case Study” section of this chapter. 5. Type Constants (information from Worksheet S2.1) a. Type constants are those types of materials that are not to be investigated but are part of the mixture. These types of materials are not varied but are constant throughout the test. b. Type constants are recorded at one level only. For ex- ample, if only Type I cement is to be used, the type con- stant would be “type of cement,” and the only level to be recorded in Worksheet S3.1 would be Type I. The previous steps are repeated for all rows in Worksheet S2.1 that have a single type listed. 6. Source Constants (information from Worksheet S2.1) a. Source constants are materials that are not to be inves- tigated but are part of the mixture. These materials are not varied but are constant throughout the test. b. Any row in Worksheet S2.1 that has only one source should be copied into the “Source Constants” section of Worksheet S3.1 (sources excluded by an “X” should be ignored). The name of the constant should consist of the material name and “Source.” For example, if Class C fly ash has only one source, then a new source constant named “Class C fly ash Source” should be created in Worksheet S3.1. c. The single source should be copied into a “Source Constant” row. This single source will be used for all mixtures that include this material. These steps should be repeated for all rows in Worksheet S2.1 that have a single source listed.

7. Amount Constants (information from Worksheet S1.1) a. Amount constants are those types of materials that are not to be investigated but are part of the mixture. These types are not varied but are constant throughout the test. b. Any column in Worksheet S1.1 that has a single (nonzero) value in the summary box at the bottom of the column should become a row in the “Amount Constants” section of Worksheet S3.1. c. The column heading and “Amount” (such as “Class F fly ash Amount”) will become the new amount con- stant name. d. The value in the summary box in Worksheet S1.1 should be entered in the “Amount Constant” row. This value will be the amount of this material in all mixtures tested. If there is a zero in the summary box in Work- sheet S1.1, then this material will be left out of all mix- tures that are to be tested. Repeat for all columns in Worksheet S1.1 that have a single value listed in the summary box. Considerations in Selecting the Number of Mixtures Worksheet S3.1 should now summarize the potential fac- tors and levels that could be tested in the statistically designed experiment. Worksheet S3.1 is then used to determine how many factors and levels can actually be tested when the num- ber of test mixtures is limited because of time or budget constraints. Because the testing of any given concrete involves many tests and a great deal of time, the number of different concrete mixtures to be tested must be chosen carefully to reduce cost while still obtaining valid conclusions regarding the effect of each factor. Performing more testing will almost always pro- vide more information; however, use of a statistical design will help determine the minimum amount of testing neces- sary to produce the information needed for determining the best mixture possible. The tradeoffs between the size of the experiment and the number of factors and levels that can be studied is discussed in this section. In addition to cost consideration, the number of con- crete mixtures that can be examined relates to the shape of the experimental matrix because, for a specific number of factors and mixtures to be tested, only certain matrices qualify as orthogonal designs. Table S3.1 lists the minimum number of mixtures that are needed for an orthogonal design matrix with various combinations of two-level and three-level factors, including 4-, 8-, 9-, 12-, 16- and 18- mixture designs. For example, for a combination of three three-level factors and six two-level factors, 16 mixtures are needed for an orthogonal design. However, this number must be reduced if testing 16 mixtures is beyond the proj- ect budget. When reconciling the number of potential factors on Worksheet S3.1 with the amount of testing required, the user must consider the factors perceived to be the most important, the combinations that are likely to give the best performance, and the materials that are expected to provide a sufficiently wide scope of the experiment and maximize the chances of finding an optimum mixture. The two main options for reducing the number of mixtures are as follows: 1. Reduce a three-level factor to a two-level factor. An ex- periment with two three-level factors and three two-level factors will require a 16-mixture experiment. However, as shown in Table S3.1, an 8-mixture orthogonal design will accommodate one three-level factor and four two-level factors, which is close to the original experiment. The best way to convert a three-level factor to a two-level factor is to choose the two levels that are most likely to provide good performance. This approach restricts the range of the factor around the most promising area and often pro- vides a good tradeoff against testing a large number of mixtures. In Worksheet S3.1, any three-level factor that has a level marked with two asterisks is a good candidate for being reduced to two levels. When a three-level factor is to be reduced to a two-level factor, the user will elimi- nate one of the levels in Worksheet S3.1 by drawing “X” through that level or changing the levels as desired. 2. Hold some factors constant. To reduce testing further, the user may consider eliminating some factors. Eliminat- ing a factor is achieved by holding this factor constant in all tested mixtures at the level that the user feels is best (this level could be, but need not be, zero). The factors marked with double asterisks in Worksheet S3.1 would be good candidates for being held constant. In the example discussed previously, if one of the two-level factors is held constant, a 9-mixture orthogonal array is available for two three-level factors and two two-level factors. In this case, an “X” should be drawn through all the levels for that fac- tor except one on Worksheet S3.1. Another means for reducing the cost of the program while maintaining the same number of possible mixtures is to reduce the number of responses to be evaluated. Although Step 1 was designed to identify those tests that are relevant to the service environment, it is possible that some tests could be eliminated if the performance could be evaluated based on some other means. For example, freezing and thawing resistance could be assumed based on the presence of an adequate air void system in the hardened concrete. Because this is an assumption, the user may want to return to this test after the Best Concrete has been selected or perhaps during the confirmation testing to be 79

80 conducted in Step 6. Also, the electrical conductivity (AASHTO T 277/ASTM C 1202) test, which actually measures conductivity not permeability, is often used to evaluate chloride penetration resistance instead of the longer-term dif- fusion testing as a means for reducing cost and time. However, this approach is not recommended because the correlation between the two tests is mixture dependent, and the diffusion test provides a better representation of chloride ion penetra- tion conditions occurring in the field. Selection of the Design Matrix The next step is to set up a design matrix with the selected factors and levels. It is an iterative process that involves select- ing the factors, their levels, and the number of mixtures and usually requires some discussion and tradeoffs. Ultimately, the many possible alternatives must be reduced to a specific set of factors that will be included at a specific set of levels. When this decision is reached, Worksheet S3.1 should be fully updated by drawing an “X” through all levels of factors that are not to be used in the mixtures. Table S3.1 then can be used to deter- mine the number of mixtures that must be made. The section titled “Selected Orthogonal Design Matrices” presents generic orthogonal design matrices for every combination of two- level and three-level factors that can be accommodated in 4-, 8-, 9-, 12-, 16-, and 18-mixture orthogonal designs. These ta- bles show the combinations of levels for the factors that should be tested in each of the mixtures to be produced. The levels are referred to generically as 1, 2, or 3, so that the specific levels can be substituted in these tables. The following example explains how the shape of the design matrix is chosen and how the mixtures to be tested are defined. Suppose that two three-level factors and three two-level factors are considered for the experiment. Table S3.1 shows that 16 mixtures are needed for an orthogonal design with these fac- tors. If this number of mixtures was determined to be too many, one of the three-level factors could be reduced to a two- level factor and thus an 8-mixture matrix will accommodate the one three-level factor and four two-level factors. Table S3.2 (reproduced from “Selected Orthogonal Design Matrices”) shows the generic 8-mixture matrix defined to accommodate this experiment. The selected levels of each factor are referred to as Levels 1, 2, and 3 for the three-level factor and Levels 1 and 2 for the two-level factors. The specific levels are then sub- stituted into Table S3.2 for the level number under the factor to which they correspond; the table then defines the mixtures to be tested. This process is described in “Example from Hypo- thetical Case Study” using actual factor and level names. The tables of design matrices are applicable to a range of two-level factors; the two-level factor columns should be dis- regarded if they are not needed. For example, the matrix in Table S3.2 would also be used if there were one three-level factor and only three two-level factors, but the last column would simply be ignored. Note that the SEDOC tool dis- cussed in Step 5 supports only selected design matrices and does not permit dropping factors. After the generic design matrix is selected, a few more steps must be completed to customize the design matrix for use in a particular application: 1. Map the factors and their remaining levels in Worksheet S3.1 to specific columns of the generic design matrix. Sometimes a factor column in the design matrix has one level that appears more often than the others. That level should be shown in bold and underlined. For example, Level 2 is used four times in the first column of Table S3.2, and Levels 1 and 3 are only used twice each. The user should recognize this and use it to include more desirable levels in more of the mixtures. If there is a factor column that has Level 2 twice as often as Level 1, a factor that has a double asterisk on one of its levels should be chosen for that column if possible. The level with the double asterisk (or at least a level with a single asterisk) should be mapped to Level 2. 2. Sometimes orthogonal designs suggest concrete mixtures that are recognized by an experienced user as having some highly undesirable characteristics (such as very low work- ability). Therefore, the user should carefully check the overall reasonableness of the mixtures being proposed by the matrix after choosing a matrix and mapping the factor and levels onto it. If some mixtures appear to be undesir- able there are several possible corrections: a. Adjust the mapping of the levels or factors to avoid the undesirable mixture. b. Reevaluate the factors, levels, and matrix selection until all mixtures seem viable and likely to be suitable con- crete for the intended application. This step attempts to bring some of the strengths of the Expert Opinion Method into the Orthogonal Design Method. Experimental Details Mixture Proportioning and SCM Level Definition The amount of a particular SCM in a mixture is described as the percentage replacement of cement or as an addition, and the percentage is calculated in terms of mass or in terms of volume. It is probably most common to consider a per- centage replacement by weight and follow the guidelines of ACI Committee 211 (59) for developing the mixture pro- portions using the method of absolute volume mixture design. In this method, air content is chosen to meet a cer- tain type of exposure condition, the water content is

selected to produce the desired slump, and the w/cm is cho- sen based on a target strength or durability requirements. In combination, the water content and w/cm determine the cementitious materials content. The coarse aggregate con- tent is chosen based on the fineness modulus of the fine aggregate and the size of the coarse aggregates. Then, the balance of material required to produce a unit volume of concrete—after considering the volume of the predefined contents of cementitious materials, water, air and coarse ag- gregate—is made up with fine aggregate. Because many properties of the concrete ingredients, such as the aggregate shape and gradation, are not explicitly in- cluded in this process, trial mixtures are batched in the lab- oratory to verify that the mixture is workable and appropri- ate for field use and to adjust the mixture proportions if necessary. In the setting of this methodology, only the fac- tors included in the design matrix are varied during the Step 4 test program to eliminate complicating influences on performance. The dosage rates of admixtures are typically reported as volume per weight of cement, such as ounces per 100 lbs of cement (oz/cwt) or milliliters per 100 kg of cement. The ad- dition of other cementitious materials raises the question of whether they should be included in the dosage calculation (i.e., oz/cwt of total cementitious material). For example, superplasticizers are thought to act primarily on the surface of cement grains while AEAs, in general, act in the paste and water/air interfaces, suggesting not including them in the former case but including them in the latter. Nevertheless, admixture dosage should be reported in a consistent manner. The number of potential factors to be held constant for an experiment targeted at studying the effect of SCM contents is large; recommendations for factors such as w/cm and cement content can be found in the literature (56, 60). Control Mixtures A control mixture made with no SCMs but with similar other mixture characteristics may be included in the study. This mixture would provide a comparison to assess relative performance of mixtures with SCMs to conventional con- crete or to a mixture currently in use. Repeat Testing A replicate of the control mixture or another mixture may be added to provide an assessment of batch-to-batch vari- ability. Such a replicate will give the user a basis for evaluat- ing whether the differences in measured responses on the concrete mixtures result are due to differences in the concrete or to test variations. Example from Hypothetical Case Study This example shows how the design matrix is selected based on recommendations from Steps 1 and 2 summarized in Worksheets S1.1 and S2.1. The completed Worksheet S1.1 (Table S1.8) suggests that a large test program was necessary to characterize each mixture’s performance but it was con- strained (by the available budget) to a 9-mixture experiment. This number of mixtures controls the possible number of factors and levels to be evaluated as listed in Table S3.1. The next step is to select the factors and levels to include. To maximize the number of SCMs while limiting the size of the experimental test program to nine mixtures, a design matrix consisting of three three-level factors and one two- level factor was selected (Table S3.3). The factors and levels to be varied in the design matrix for the hypothetical case study were selected using the top half of Worksheet S3.1, completed as Table S3.4. While not apparent based on this table, several sets of factors and levels were proposed for testing, before the final selection shown was made. The specific factors for test- ing were chosen as “First SCM Type,” “First SCM Amount,” “Amount of Silica Fume,” and “w/cm.” Once the factors were selected, the range of investigation for each of the factors was chosen to span the upper and lower bounds where the optimum level is expected. For the exam- ple Worksheet S1.1 completed for Step 1 (Table S1.8), which considered a wide range of exposure conditions, when the recommended ranges of silica fume were compiled for all the desired properties in the “Summary” row of this worksheet, one level resulted: 5%. The same was true for GGBFS (30%) and Class F fly ash (25%). Because the objective of this re- search is to study SCMs, the test program was centered on the summary values shown in Table S1.8. For example, the levels for amount of silica fume were chosen to be 0%, 5%, and 8% (although the summary row of Table S1.8 recommends only 5%). Similarly, w/cm’s of 0.37 and 0.45 were included (although the summary of the level of w/cm from Table S1.8 recommended that the w/cm be less than 0.40). Ordinarily, an amount factor such as “First SCM Amount” would have simple numerical values given as levels. However, because the appropriate ranges for types of SCMs are depen- dent on that type, a compound factor was used. This factor links the definition of the amount factor to a type factor and allows flexibility in the definition of SCM contents. The levels of the “First SCM Type” factor were defined as slag, Class C fly ash, and Class F fly ash. Then, the “First SCM Amount” fac- tor was defined generically as low, medium and high—with different specific values of the SCM content associated with each slag or fly ash material. Despite the generic definition, the amount of SCM is an amount factor and the performance models are capable of interpolating between the levels tested. 81

82 The definitions of low, medium, and high were determined with Worksheet S3.2, which is completed in Table S3.5. (An example of a compound factor based on two type levels and two amount levels is given in Table S3.6.) After the factors and levels to be varied are chosen, the type, source and amount constants must be defined for the exper- iment. These characteristics of the mixture proportions will remain consistent throughout the experiment. For example, this step may include defining a constant cementitious con- tent (658 lb/yd3 [391 kg/m3]) and a constant coarse aggregate amount (1696 lb/yd3 [1007 kg/m3]). In summary, the top half of Table S3.4 lists the factors and levels for the hypothetical case study, while the bottom half of this table defines the constant values selected for this experi- ment. Table S3.5 defines the specific quantities for the general descriptions “low,” “medium,” and “high” used in the com- pound factor. The orthogonal design in this example requires nine specific mixtures be evaluated to provide sufficient informa- tion to optimize these factors and levels. These mixtures must be chosen according to the applicable table from “Selected Orthogonal Design Matrices.” The generic design matrix that applies for the 9-mixture (three three-level factor and one two-level factor) design is given in Table S3.7. The factors cho- sen in the top half of Table S3.4 were numbered Factor 1 to Factor 4 and the levels for each were also numbered Level 1 to Level 3. Table S3.8 lists the specific design matrix after the factor levels were substituted into the generic matrix accord- ing to this numbering. The batch weights for the mixtures to be tested corresponding to the selected levels and based on the source and amount constants are listed in Table S3.9. All SCM amounts were calculated as percentages by weight replace- ment of portland cement and changes in cementitious vol- umes were compensated by changes in fine aggregate content.

83 Worksheets for Step 3 Factor Level 1 Level 2 Level 3 Type Factors Source Factors Amount Factors Type Constants Source Constants Amount Constants Worksheet S3.1. Factors and levels to test.

84 Factor 1, Factor 2 Type Amount Type 1, Low Level Type 1, Medium Level Type 1, High Level Type 2, Low Level Type 2, Medium Level Type 2, High Level Type 3, Low Level Type 3, Medium Level Type 3, High Level Worksheet S3.2. Compound factor table. Number of 3-Level Factors Number of 2-Level Factors 0 1 2 3 4 5 6 7 0 3 9 9 9 16 18 18 1 2 8 9 9 16 18 18 18 2 4 8 9 16 16 18 18 >18 3 4 8 16 16 16 18 >18 >18 4 8 8 16 16 18 >18 >18 >18 5 8 16 16 16 >18 >18 >18 >18 6 8 16 16 16 >18 >18 >18 >18 7 8 16 16 >18 >18 >18 >18 >18 8 12 16 16 >18 >18 >18 >18 >18 9 12 16 16 >18 >18 >18 >18 >18 10 12 16 >18 >18 >18 >18 >18 >18 11 12 16 >18 >18 >18 >18 >18 >18 12 16 16 >18 >18 >18 >18 >18 >18 13 16 >18 >18 >18 >18 >18 >18 >18 14 16 >18 >18 >18 >18 >18 >18 >18 15 16 >18 >18 >18 >18 >18 >18 >18 Table S3.1. Number of mixtures required for an orthogonal design for various combinations of two- and three-level factors. Tables for Step 3

85 Mixture Factor 1 (3-Level) Factor 2 (2-Level) Factor 3 (2-Level) Factor 4 (2-Level) Factor 5 (2-Level) 1 1 1 1 2 2 2 2 1 2 1 2 3 2 1 2 2 1 4 3 1 1 1 1 5 1 2 2 1 1 6 2 2 1 2 1 7 2 2 1 1 2 8 3 2 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. Table S3.2. An 8-mixture design matrix for one three-level and four two-level factors. Number of 3-Level Factors Number of 2-Level Factors 0 1 2 3 4 5 6 7 0 3 9 9 9 16 18 18 1 2 8 9 9 16 18 18 18 2 4 8 9 16 18 18 >18 3 4 8 16 16 16 >18 >18 4 8 8 16 16 18 >18 >18 >18 5 8 16 16 16 >18 >18 >18 >18 6 8 16 16 16 >18 >18 >18 >18 7 8 16 16 >18 >18 >18 >18 >18 8 12 16 16 >18 >18 >18 >18 >18 9 12 16 16 >18 >18 >18 >18 >18 10 12 16 >18 >18 >18 >18 >18 >18 11 12 16 >18 >18 >18 >18 >18 >18 12 16 16 >18 >18 >18 >18 >18 >18 13 16 >18 >18 >18 >18 >18 >18 >18 14 16 >18 >18 >18 >18 >18 >18 >18 15 16 >18 >18 >18 >18 >18 >18 >18 16 18 Table S3.3. Selection of the 9-mixture design selected for the hypothetical case study.

86 Factor Level 1 Level 2 Level 3 Type Factors Source Factors Amount Factors Type Constants Source Constants Amount Constants Table S3.4. Completed Worksheet S3.1, factors, levels, and constants to test for hypothetical case study.

87 Factor 1, Factor 2 Type of SCM Amount of SCM Type 1, Low Level Type 1, Medium Level Type 1, High Level Type 2, Low Level Type 2, Medium Level Type 2, High Level Type 3, Low Level Type 3, Medium Level Type 3, High Level Factor 1, Factor 2 Type of SCM Amount of SCM Type 1, Low Level Type 1, Medium Level Type 1, High Level Type 2, Low Level Type 2, Medium Level Type 2, High Level Type 3, Low Level Type 3, Medium Level Type 3, High Level Note: Type 3 and medium are ignored because each factor only has two levels. Table S3.5. An example of Worksheet S3.2 filled out for a compound factor of three type levels and three amount levels. Table S3.6. An example of Worksheet S3.2 filled out for a compound factor of two type levels and two amount levels. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (3-Level) Factor 4 (2-Level) 1 1 1 1 1 2 1 2 2 2 3 1 3 3 2 4 2 1 2 2 5 2 2 3 1 6 2 3 1 2 7 3 1 3 2 8 3 2 1 2 9 3 3 2 1 If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. Table S3.7. The levels for the 9-mixture design matrix with three three-level factors and one two-level factor.

88 Mixture First SCM Type First SCM Amount Amount of Silica Fume w/cm 1 Fly Ash C Low (15%) 0% 0.45 2 Fly Ash C Medium (25%) 5% 0.37 3 Fly Ash C High (40%) 8% 0.37 4 Fly Ash F Low (15%) 5% 0.37 5 Fly Ash F Medium (25%) 8% 0.45 6 Fly Ash F High (40%) 0% 0.37 7 GGBFS Low (25%) 8% 0.37 8 GGBFS Medium (35%) 0% 0.37 9 GGBFS High (50%) 5% 0.45 Batch Weights (lb/yd3) Material Mixture 1 Mixture 2 Mixture 3 Mixture 4 Mixture 5 Mixture 6 Mixture 7 Mixture 8 Mixture 9 Water 296 243 243 243 296 243 243 243 296 Cement 559 461 342 526 441 395 441 428 296 Class C Fly Ash 99 165 263 0 0 0 0 0 0 Class F Fly Ash 0 0 0 99 165 263 0 0 0 GGBFS 0 0 0 0 0 0 165 230 329 Silica Fume 0 33 53 33 53 0 53 0 33 Fine Aggregate 1180 1300 1280 1294 1128 1261 1302 1316 1156 Coarse Aggregate 1696 1696 1696 1696 1696 1696 1696 1696 1696 Table S3.8. Experimental design matrix for hypothetical case study. Table S3.9. Trial mixture designs. A 4-mixture design matrix for two to three two-level factors. Mixture Factor 1 (2-Level) Factor 2 (2-Level) Factor 3 (2-Level) 1 1 1 1 2 2 1 2 3 1 2 2 4 2 2 1 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. An 8-mixture design matrix for four to seven two-level factors. Mixture Factor 1 (2-Level) Factor 2 (2-Level) Factor 3 (2-Level) Factor 4 (2-Level) Factor 5 (2-Level) Factor 6 (2-Level) Factor 7 (2-Level) 1 1 1 1 2 2 2 1 2 2 1 1 1 1 2 2 3 1 2 1 1 2 1 2 4 2 2 1 2 1 1 1 5 1 1 2 2 1 1 2 6 2 1 2 1 2 1 1 7 1 2 2 1 1 2 1 8 2 2 2 2 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. Selected Orthogonal Design Matrices

89 An 8-mixture design matrix for one three-level factor and one to four two-level factors. Mixture Factor 1 (3-Level) Factor 2 (2-Level) Factor 3 (2-Level) Factor 4 (2-Level) Factor 5 (2-Level) 1 1 1 1 2 2 2 2 1 2 1 2 3 2 1 2 2 1 4 3 1 1 1 1 5 1 2 2 1 1 6 2 2 1 2 1 7 2 2 1 1 2 8 3 2 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. A 9-mixture design matrix for two three-level and up to two two-level factors. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (2-Level) Factor 4 (2-Level) 1 1 1 1 1 2 1 2 2 2 3 1 3 2 2 4 2 1 2 2 5 2 2 2 1 6 2 3 1 2 7 3 1 2 2 8 3 2 1 2 9 3 3 2 1 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. A 9-mixture design matrix for three three-level factors and one two-level factor. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (3-Level) Factor 4 (2-Level) 1 1 1 1 1 2 1 2 2 2 3 1 3 3 2 4 2 1 2 2 5 2 2 3 1 6 2 3 1 2 7 3 1 3 2 8 3 2 1 2 9 3 3 2 1 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result.

90 A 12-mixture design matrix for 8 to 11 two-level factors. Mixture Factor 1 (2-Level) Factor 2 (2-Level) Factor 3 (2-Level) Factor 4 (2-Level) Factor 5 (2-Level) Factor 6 (2-Level) Factor 7 (2-Level) Factor 8 (2-Level) Factor 9 (2-Level) Factor 10 (2-Level) Factor 11 (2-Level) 1 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 2 2 1 1 1 2 1 3 1 1 2 1 2 2 2 1 1 1 2 4 2 1 1 2 1 2 2 2 1 1 1 5 1 2 1 1 2 1 2 2 2 1 1 6 1 1 2 1 1 2 1 2 2 2 1 7 1 1 1 2 1 1 2 1 2 2 2 8 2 1 1 1 2 1 1 2 1 2 2 9 2 2 1 1 1 2 1 1 2 1 2 10 2 2 2 1 1 1 2 1 1 2 1 11 1 2 2 2 1 1 1 2 1 1 2 12 2 1 2 2 2 1 1 1 2 1 1 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. A 16-mixture design matrix for 12 to15 two-level factors. Mixture Factor 1 (2-Level) Factor 2 (2-Level) Factor 3 (2-Level) Factor 4 (2-Level) Factor 5 (2-Level) Factor 6 (2-Level) Factor 7 (2-Level) Factor 8 (2-Level) Factor 9 (2-Level) Factor 10 (2-Level) Factor 11 (2-Level) Factor 12 (2-Level) Factor 13 (2-Level) Factor 14 (2-Level) Factor 15 (2-Level) 1 1 1 1 1 1 1 2 2 2 1 2 1 2 2 1 2 2 1 2 1 1 2 2 2 1 1 1 2 1 2 2 3 1 2 2 1 1 1 1 2 2 2 1 2 2 1 2 4 2 2 1 1 1 2 1 2 1 2 2 1 1 1 1 5 1 1 1 2 1 2 1 1 1 2 1 2 2 2 1 6 2 1 2 2 1 1 1 1 2 2 2 1 1 2 2 7 1 2 2 2 1 2 2 1 1 1 2 1 2 1 2 8 2 2 1 2 1 1 2 1 2 1 1 2 1 1 1 9 1 1 1 1 2 2 2 1 2 2 1 1 1 1 2 10 2 1 2 1 2 1 2 1 1 2 2 2 2 1 1 11 1 2 2 1 2 2 1 1 2 1 2 2 1 2 1 12 2 2 1 1 2 1 1 1 1 1 1 1 2 2 2 13 1 1 1 2 2 1 1 2 1 1 2 2 1 1 2 14 2 1 2 2 2 2 1 2 2 1 1 1 2 1 1 15 1 2 2 2 2 1 2 2 1 2 1 1 1 2 1 16 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. A 9-Mixture Design Matrix for four three-level factors. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (3-Level) Factor 4 (3-Level) 1 1 1 1 1 2 1 2 2 2 3 1 3 3 3 4 2 1 2 3 5 2 2 3 1 6 2 3 1 2 7 3 1 3 2 8 3 2 1 3 9 3 3 2 1 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result.

A 16-mixture design matrix for 1 three-level factor and 5 to 12 two-level factors. Mixture Factor 1 (3-Level) Factor 2 (2-Level) Factor 3 (2-Level) Factor 4 (2-Level) Factor 5 (2-Level) Factor 6 (2-Level) Factor 7 (2-Level) Factor 8 (2-Level) Factor 9 (2-Level) Factor 10 (2-Level) Factor 11 (2-Level) Factor 12 (2-Level) Factor 13 (2-Level) 1 1 1 1 1 2 2 2 1 2 1 2 2 1 2 2 1 1 2 2 2 1 1 1 2 1 2 2 3 2 1 1 1 1 2 2 2 1 2 2 1 2 4 3 1 1 2 1 2 1 2 2 1 1 1 1 5 1 2 1 2 1 1 1 2 1 2 2 2 1 6 2 2 1 1 1 1 2 2 2 1 1 2 2 7 2 2 1 2 2 1 1 1 2 1 2 1 2 8 3 2 1 1 2 1 2 1 1 2 1 1 1 9 1 1 2 2 2 1 2 2 1 1 1 1 2 10 2 1 2 1 2 1 1 2 2 2 2 1 1 11 2 1 2 2 1 1 2 1 2 2 1 2 1 12 3 1 2 1 1 1 1 1 1 1 2 2 2 13 1 2 2 1 1 2 1 1 2 2 1 1 2 14 2 2 2 2 1 2 2 1 1 1 2 1 1 15 2 2 2 1 2 2 1 2 1 1 1 2 1 16 3 2 2 2 2 2 2 2 2 2 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result.

92 A 16-mixture design matrix for two three-level factors and three to nine two-level factors. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (2-Level) Factor 4 (2-Level) Factor 5 (2-Level) Factor 6 (2-Level) Factor 7 (2-Level) Factor 8 (2-Level) Factor 9 (2-Level) Factor 10 (2-Level) Factor 11 (2-Level) 1 1 1 2 2 1 1 2 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 2 3 2 1 2 1 2 1 2 1 2 1 2 4 3 1 1 1 2 2 1 1 1 2 1 5 1 2 2 1 2 2 1 2 1 1 2 6 2 2 1 1 2 1 2 2 2 2 1 7 2 2 2 2 1 2 1 1 2 2 1 8 3 2 1 2 1 1 2 1 1 1 2 9 1 2 1 2 2 2 2 1 2 1 1 10 2 2 2 2 2 1 1 1 1 2 2 11 2 2 1 1 1 2 2 2 1 2 2 12 3 2 2 1 1 1 1 2 2 1 1 13 1 3 1 1 1 1 1 1 2 2 2 14 2 3 2 1 1 2 2 1 1 1 1 15 2 3 1 2 2 1 1 2 1 1 1 16 3 3 2 2 2 2 2 2 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. A 16-mixture design matrix for three three-level factors and two to six two-level factors. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (3-Level) Factor 4 (2-Level) Factor 5 (2-Level) Factor 6 (2-Level) Factor 7 (2-Level) Factor 8 (2-Level) Factor 9 (2-Level) 1 1 1 3 2 2 1 1 1 1 2 2 1 2 1 2 2 2 2 1 3 2 1 2 2 1 2 2 1 2 4 3 1 1 1 1 1 1 2 2 5 1 2 2 1 2 2 1 2 2 6 2 2 1 2 2 1 2 1 2 7 2 2 3 1 1 1 2 2 1 8 3 2 2 2 1 2 1 1 1 9 1 2 2 2 1 1 2 2 2 10 2 2 3 1 1 2 1 1 2 11 2 2 1 2 2 2 1 2 1 12 3 2 2 1 2 1 2 1 1 13 1 3 1 1 1 2 2 1 1 14 2 3 2 2 1 1 1 2 1 15 2 3 2 1 2 1 1 1 2 16 3 3 3 2 2 2 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result.

93 A 16-mixture design matrix for four three-level factors and one to three two-level factors. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (3-Level) Factor 4 (3-Level) Factor 5 (2-Level) Factor 6 (2-Level) Factor 7 (2-Level) 1 1 1 3 2 1 2 1 2 2 1 2 3 2 1 1 3 2 1 2 2 2 2 2 4 3 1 1 1 1 1 2 5 1 2 2 3 1 1 2 6 2 2 1 2 2 2 2 7 2 2 3 1 2 1 1 8 3 2 2 2 1 2 1 9 1 2 2 1 2 2 2 10 2 2 3 2 1 1 2 11 2 2 1 3 1 2 1 12 3 2 2 2 2 1 1 13 1 3 1 2 2 1 1 14 2 3 2 1 1 2 1 15 2 3 2 2 1 1 2 16 3 3 3 3 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. A 16-mixture design matrix for five three-level factors. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (3-Level) Factor 4 (3-Level) Factor 5 (3-Level) 1 1 1 3 2 2 2 2 1 2 3 2 3 2 1 2 2 3 4 3 1 1 1 1 5 1 2 2 3 1 6 2 2 1 2 3 7 2 2 3 1 2 8 3 2 2 2 2 9 1 2 2 1 3 10 2 2 3 2 1 11 2 2 1 3 2 12 3 2 2 2 2 13 1 3 1 2 2 14 2 3 2 1 2 15 2 3 2 2 1 16 3 3 3 3 3 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result.

94 An 18-mixture design matrix for five three-level factors and one to three two-level factors. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (3-Level) Factor 4 (3-Level) Factor 5 (3-Level) Factor 6 (2-Level) Factor 7 (2-Level) Factor 8 (2-Level) 1 1 1 1 1 1 1 1 1 2 1 2 3 3 1 2 2 1 3 1 3 2 3 2 1 2 1 4 1 2 2 1 3 2 1 2 5 1 3 1 2 3 2 2 2 6 1 1 3 2 2 2 2 2 7 2 2 2 2 2 2 1 1 8 2 3 1 1 2 2 2 1 9 2 1 3 1 3 2 2 1 10 2 3 3 2 1 1 1 2 11 2 1 2 3 1 2 2 2 12 2 2 1 3 3 1 2 2 13 3 3 3 3 3 2 1 1 14 3 1 2 2 3 1 2 1 15 3 2 1 2 1 2 2 1 16 3 1 1 3 2 2 1 2 17 3 2 3 1 2 1 2 2 18 3 3 2 1 1 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. An 18-mixture design matrix for four three-level factors and four two-level factors. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (3-Level) Factor 4 (3-Level) Factor 5 (2-Level) Factor 6 (2-Level) Factor 7 (2-Level) Factor 8 (2-Level) 1 1 1 1 1 1 1 1 1 2 1 2 3 3 1 2 2 1 3 1 3 2 3 2 1 2 1 4 1 2 2 1 2 2 1 2 5 1 3 1 2 2 2 2 2 6 1 1 3 2 2 2 2 2 7 2 2 2 2 2 2 1 1 8 2 3 1 1 2 2 2 1 9 2 1 3 1 2 2 2 1 10 2 3 3 2 1 1 1 2 11 2 1 2 3 1 2 2 2 12 2 2 1 3 2 1 2 2 13 3 3 3 3 2 2 1 1 14 3 1 2 2 2 1 2 1 15 3 2 1 2 1 2 2 1 16 3 1 1 3 2 2 1 2 17 3 2 3 1 2 1 2 2 18 3 3 2 1 1 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result.

95 An 18-mixture design matrix for six three-level factors and one to two two-level factors. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (3-Level) Factor 4 (3-Level) Factor 5 (3-Level) Factor 6 (3-Level) Factor 7 (2-Level) Factor 8 (2-Level) 1 1 1 1 1 1 1 1 1 2 1 2 3 3 1 2 2 1 3 1 3 2 3 2 1 2 1 4 1 2 2 1 3 3 1 2 5 1 3 1 2 3 2 2 2 6 1 1 3 2 2 3 2 2 7 2 2 2 2 2 2 1 1 8 2 3 1 1 2 3 2 1 9 2 1 3 1 3 2 2 1 10 2 3 3 2 1 1 1 2 11 2 1 2 3 1 3 2 2 12 2 2 1 3 3 1 2 2 13 3 3 3 3 3 3 1 1 14 3 1 2 2 3 1 2 1 15 3 2 1 2 1 3 2 1 16 3 1 1 3 2 2 1 2 17 3 2 3 1 2 1 2 2 18 3 3 2 1 1 2 2 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result. An 18-mixture design matrix for seven three-level factors and one two-level factor. Mixture Factor 1 (3-Level) Factor 2 (3-Level) Factor 3 (3-Level) Factor 4 (3-Level) Factor 5 (3-Level) Factor 6 (3-Level) Factor 7 (3-Level) Factor 8 (2-Level) 1 1 1 1 1 1 1 1 1 2 1 2 3 3 1 2 2 1 3 1 3 2 3 2 1 3 1 4 1 2 2 1 3 3 1 2 5 1 3 1 2 3 2 2 2 6 1 1 3 2 2 3 3 2 7 2 2 2 2 2 2 1 1 8 2 3 1 1 2 3 2 1 9 2 1 3 1 3 2 3 1 10 2 3 3 2 1 1 1 2 11 2 1 2 3 1 3 2 2 12 2 2 1 3 3 1 3 2 13 3 3 3 3 3 3 1 1 14 3 1 2 2 3 1 2 1 15 3 2 1 2 1 3 3 1 16 3 1 1 3 2 2 1 2 17 3 2 3 1 2 1 2 2 18 3 3 2 1 1 2 3 2 If not all factors are needed, unused columns can simply be ignored. If the font is underlined and bold, the level chosen for that factor should be the one expected to produce the best result.