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Publication Number: FHWA-RD-03-060

Concrete Mixture Optimization Using Statistical Methods

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APPENDIX A. Experiment Design and Data Analysis for Mixture Experiment

A.1 Experiment Design and Response Data

Table A-1. Mixture experiment design in terms of volume fractions of components

Standard Order Design ID Run Order A Water B Cement C Silica Fume D HRWRA E Coarse Agg F Fine Agg
3512710.17200.13950.01300.00600.40980.2598
71520.16560.15000.02700.00740.40000.2500
143830.17670.14170.02700.00460.40000.2500
112240.18500.14740.01300.00460.40000.2500
227150.16000.15000.01300.00740.40980.2598
41160.18500.13000.01300.00460.41740.2500
2570.16000.13000.02700.00460.42840.2500
81680.16000.13000.01300.00460.44240.2500
279190.18500.13000.01300.00740.40730.2573
29101100.17120.15000.01300.00460.41120.2500
2478110.18500.13000.02700.00740.40030.2503
37127120.17200.13950.01300.00600.40980.2598
920130.18500.13000.01300.00460.40000.2674
30103140.16000.15000.01300.00600.42100.2500
513150.16000.13000.01300.00460.40000.2924
1228160.16000.13000.02700.00460.40000.2784
2898170.16000.14000.01300.00460.43240.2500
36127180.17200.13950.01300.00600.40980.2598
2689190.16000.14000.01300.00460.40000.2824
39163200.16560.13430.01650.00670.42340.2536
31110210.17120.15000.01300.00460.40000.2612
15220.16000.13000.02700.00460.42840.2500
311230.18500.13000.01300.00460.41740.2500
2587240.16000.13000.01300.00460.42120.2712
2066250.16000.13000.02700.00740.41280.2628
1638260.17670.14170.02700.00460.40000.2500
1863270.17250.13000.02700.00740.41310.2500
1748280.17250.13000.01300.00740.40000.2771
2170290.16000.15000.01300.00600.40000.2710
1337300.16000.14000.02700.00740.40000.2656
1965310.16000.14000.02700.00740.41560.2500
38127320.17200.13950.01300.00600.40980.2598
32116330.17250.13000.02700.00740.40000.2631
1020340.18500.13000.01300.00460.40000.2674
2371350.16000.15000.01300.00740.40980.2598
33123360.16000.14000.02000.00600.41200.2620
34127370.17200.13950.01300.00600.40980.2598
615380.16560.15000.02700.00740.40000.2500
1538390.17670.14170.02700.00460.40000.2500

Table A-2. Mixture experiment: slump and 1-day strength data

Obs Design ID Run Order Slump (mm) 1-Day Strength (MPa)
32127150.854.024.324.424.0-
615225.431.824.622.925.9-
1338319.119.122.722.622.1-
1022469.963.521.121.421.8-
2071563.550.826.927.226.527.5
3116108.095.316.716.617.0-
15712.712.721.723.322.2-
716838.131.821.821.621.4-
25919203.2196.917.116.616.7-
271011025.419.126.426.526.8-
227811127.0127.019.219.319.0-
3312712101.695.320.821.921.9-
82013114.3120.718.318.018.3-
281031463.563.528.026.827.3-
5131563.550.822.721.721.0-
11281631.825.421.122.922.5-
26981731.831.825.226.424.2-
341271895.388.921.521.623.8-
24891938.138.118.123.124.0-
361632095.395.322.121.622.4-
291102150.850.824.525.124.4-
252225.425.422.124.024.2-
41123114.3114.317.116.515.9-
23872463.569.924.920.623.2-
18662576.276.225.024.125.1-
14382631.825.422.923.622.5-
166327101.695.321.622.021.4-
154828177.8165.122.722.723.7-
19702950.850.827.527.327.6-
12373038.131.826.827.827.4-
17653131.831.828.625.527.6-
3512732120.7120.722.922.222.1-
3011633114.3114.323.923.724.1-
92034127.0127.018.818.618.5-
217135114.3101.629.528.028.9-
311233676.269.926.226.427.3-
3212737101.6101.623.924.324.2-
6153850.850.829.227.629.5-
13383925.425.423.325.422.2-

Table A-3. Mixture experiment: 28-day strength and RCT charge passed data

Obs Design ID Run  Order 28-Day Strength (psi) RCT Charge Passed (coulombs)
32127151.052.554.9- - --
615259.458.759.5----
1338350.850.152.7----
1022447.948.048.8-120313101321
2071551.956.757.0-901790894
311647.351.746.6-114113081038
15746.149.449.9-422352-
716854.653.251.5-708-843
2591958.860.462.1-10921113877
271011057.052.351.6-730736767
22781151.055.448.6-474454549
331271251.152.347.4-853885789
8201351.950.250.7-995922793
281031460.456.646.255.1607576565
5131551.055.652.8-575719758
11281656.550.853.4-327268282
26981755.550.549.8-580579653
341271853.754.254.3-841852848
24891952.658.350.256.3677656826
361632061.161.060.2-544552566
291102152.352.954.4-716804857
252256.352.053.9-308441296
4112349.247.846.9-8941054956
23872453.050.449.4-751618732
18662557.162.659.7-326319303
14382657.451.850.3-450346375
16632756.654.554.5-324324257
15482858.958.057.3-622702723
19702954.752.556.4-496494524
12373049.257.859.757.4247254234
17653153.549.150.8-285350296
351273257.858.655.0-661567680
301163357.857.353.5-367358343
9203451.151.951.9-804901754
21713564.965.665.2-550566543
311233661.559.861.8-318312390
321273753.855.454.6-640614665
6153861.657.555.2-235232250
13383954.553.855.4-323379293

A.2  Data Analysis and Model Fitting

A.2.1 Slump

Table A-4. Mixture experiment: sequential model sum of squares for slump

Source Sum of Squares DF Mean Square F Value Prob > F
Mean212697.81212697.8--
Linear62543.72512508.7445.64< 0.0001
Quadratic4107.0715273.801.000.5016
Special cubic (aliased)2164.867309.271.270.3703
Cubic (aliased)0.000  --
Residual1950.918243.86--
Total283464.3367874.01--

Table A-5. Mixture experiment: lack-of-fit test for slump

Source Sum of Squares DF Mean Square F Value Prob > F
Linear6271.9322285.091.170.4335
Quadratic2164.867309.271.270.3703
Special cubic (aliased)0.000---
Cubic (aliased)0.000---
Pure error1950.918243.86--

Table A-6. Mixture experiment: model summary statistics for slump

Source Std. Dev. r2 Adj. r2 Pred. r2 PRESS
Linear16.560.88380.86440.840711272.43
Quadratic16.560.94180.86430.604927956.70
Special cubic (aliased)15.620.97240.8794-undefined
Cubic (aliased)----undefined

Table A-7. Mixture experiment: ANOVA for slump mixture model

Source Sum of Squares DF Mean Square F Value Prob > F
Model62543.72512508.7445.64< 0.0001
Linear mixture62543.72512508.7445.64< 0.0001
Residual8222.8430274.09--
Lack of fit6271.9322285.091.170.4335
Pure error1950.918243.86--
Corrected total70766.5635---

Table A-8. Mixture experiment: coefficient estimates for slump mixture model

Component Coeff. Estimate DF Std. Error 95% CI Low 95% CI High
Water155.68110.12135.03176.34
Cement-37.53113.34-64.77-10.30
Silica fume-80.39117.95-117.04-43.74
HRWRA1092.781100.02888.511297.04
Coarse aaggregate55.1418.8537.0773.22
Fine aaggregate71.0319.2952.0690.00

Table A-9. Mixture experiment:  adjusted effects for slump mixture model

Component Adjusted Effect DF Std. Error Approx. t for Ho Effect = 0 Prob > t
Water-138.04112.96-2.940.0063
Cement-139.80112.43-11.25< 0.0001
Silica fume-114.84110.42-11.02< 0.0001
HRWRA70.0016.8310.25< 0.0001
Coarse aaggregate-185.17120.68-8.95< 0.0001
Fine aaggregate-166.11122.02-7.54< 0.0001

Model equation in terms of pseudocomponents:

Slump  = 155.68*A - 37.53*B - 80.39*C   + 1092.78*D + 55.14*E + 71.03*F

Model equation in terms of real components:

Slump = 2166.5*water - 2390.5*cement - 3401.2*silica fume + 24267.7*HRWRA - 204.8*Coarse agg +169.9*Fine agg

Figure A-1. Mixture experiment: normal probability plot for slump. Chart. This is a normal probability plot of the residuals for the fitted slump model. The normal percent probability is plotted on the Y-axis against the studentized residuals on the X-axis. In this plot, most of the points fall on or close to a straight line, indicating that the normality assumption is reasonable.

Figure A-1. Mixture experiment: normal probability plot for slump

Figure A-2. Mixture experiment: residuals versus run for slump. Chart. In this figure, the residuals are plotted on the Y-axis against the corresponding runs on the X-axis. In this plot, the residuals show a slight upward trend with time, but probably not significant.

Figure A-2. Mixture experiment: residuals vs. run for slump

Figure A-3. Mixture experiment: Cook's distance for slump. Chart. In this figure, Cook's distance is plotted on the Y-axis against the corresponding runs on the X-axis. Cook's distance is a means of detecting points which have considerable influence on the least squares estimates. Usually points with Cook's distance greater than 1 are considered influential. None of the points in this plot meet that criterion.

Figure A-3. Mixture experiment: Cook's distance for slump

Figure A-4. Mixture experiment: trace plot for slump. Chart. This figure is a response trace plot for the slump model. The response, slump, is plotted on the Y-axis, and deviation from the reference blend is plotted on the X-axis. A line is plotted for each mixture component (labeled with the name of the component in the figure) showing the effect of deviating from the reference blend. A steeper slope indicates a greater effect. In this case, the order of effects (greatest to least) is HRWRA, silica fume, water, cement, coarse aaggregate, and fine aaggregate.

Figure A-4. Mixture experiment: trace plot for slump

Figure A-5. Mixture experiment: contour plot for slump in water, cement, and silica fume. Diagram. This figure shows a contour plot for slump in water, cement, and silica fume. Water is the top vertex of the triangular plot, with cement at the lower left and silica fume at the lower right. Each vertex represents the high setting of each component. The slump contours increase primarily from left to right with increasing silica fume and decreasing cement.

Figure A-5. Mixture experiment: contour plot for slump in water, cement, and silica fume

Figure A-6. Mixture experiment: contour plot of slump in water, cement, and HRWRA. Diagram. This figure shows a contour plot for slump in water, cement, and HRWRA. Water is the top vertex of the triangular plot, with cement at the lower left and HRWRA at the lower right. Each vertex represents the high setting of each component. The slump contours increase primarily from bottom to top with increasing water, with smaller effects for decreasing cement and decreasing HRWRA.

Figure A-6. Mixture experiment: contour plot of slump in water, cement, and HRWRA

A.2.2 1-Day Strength

Table A-10. Mixture experiment: sequential model sum of squares for 1-day strength

Source Sum of Squares DF Mean Square F Value Prob > F
Mean19278.40119278.40--
Linear351.33570.2757.34< 0.0001
Quadratic27.14151.812.820.0266
Special cubic (aliased)3.4370.490.630.7199
Cubic (aliased)0.000---
Residual6.1980.77--
Total19666.4836546.29--

Table A-11. Mixture experiment: lack-of-fit test for 1-day strength

Source Sum of Squares DF Mean Square F Value Prob > F
Linear30.57221.391.790.1992
Quadratic3.4370.490.630.7199
Special cubic (aliased)0.0000---
Cubic (aliased)0.0000---
Pure error6.1980.77--

Table A-12. Mixture experiment: summary statistics for 1-day strength

Source Std. Dev. r2 Adj. r2 Pred. r2 PRESS
Linear1.110.90530.88950.861153.89
Quadratic0.800.97520.94210.856655.65
Special cubic (aliased)0.880.98400.9302-undefined
Cubic (aliased)----undefined

Table A-13. Mixture experiment: ANOVA for 1-day strength mixture model

Source Sum of Squares DF Mean Square F Value Prob > F
Model 373.09 9 41.45 71.84 < 0.0001
Linear mixture 351.77 5 70.35 121.93 < 0.0001
AF 4.14 1 4.14 7.18 0.0126
BF 5.38 1 5.38 9.32 0.0052
CD 3.96 1 3.96 6.86 0.0145
DF 7.83 1 7.83 13.58 0.0011
Residual 15.00 26 0.58 - -
Lack of fit 8.81 18 0.49 0.63 0.8010
Pure error 6.19 8 0.77 - -
Corrected total 388.09 35 - - -

Table A-14. Mixture experiment: coefficient estimates for 1-day strength mixture model

Component Coeff. Estimate DF Std. Error 95% CI Low 95% CI High
water 12.93 1 0.62 11.64 14.21
Cement 35.42 1 0.83 33.72 37.12
Silica fume 24.89 1 1.15 22.52 27.25
HRWRA 10.83 1 8.88 -7.42 29.07
Coarse aaggregate 22.04 1 0.43 21.16 22.93
Fine aaggregate 21.50 1 0.57 20.32 22.67
Water-fine aaggregate 7.58 1 2.83 1.77 13.39
Cement-fine aaggregate -10.07 1 3.30 -16.86 -3.29
Silica fume-HRWRA 78.71 1 30.04 16.95 140.47
HRWRA-fine aaggregate 80.50 1 21.85 35.60125.41

Model equation in terms of pseudocomponents:

Slump = 12.93*A + 35.42*B + 24.89*C + 10.83*D + 22.04*E + 21.50*F + 7.58* A*F - 10.07*B*F + 78.71*C*D + 80.50*D*F

Model equation in terms of real components:

Slump =-1209.1*water + 1775.9*cement - 74.71*SF - 11969*HRWRA
+ 59.59*Coarse agg - 105.24*Fine agg + 4214.9*water*Fine agg
- 5603.1*cement*Fine agg + 43782 *SF*HRWRA + 44781*HRWRA*Fine agg

Figure A-7. Mixture experiment: normal probability plot for 1-day strength. Chart. This figure shows a normal probability plot of residuals for the 1-day strength model, with normal percent probability on the Y-axis and studentized residuals on the X-axis. Most of the points fall on a straight line, indicating that the normality assumption is reasonable.

Figure A-7. Mixture experiment: normal probability plot for 1-day strength

Figure A-8. Mixture experiment: residuals versus run for 1-day strength. Chart. This figure shows a plot of residuals versus run for 1-day strength. The residuals are plotted on the Y-axis against the corresponding runs on the X-axis. The residuals show a slight but insignificant curvelinear trend

Figure A-8. Mixture experiment: residuals vs. run for 1-day strength

Figure A-9. Mixture experiment: trace plot for 1-day strength. Chart. This figure shows a response trace plot for 1-day strength. The response, 1-day strength, is plotted on the Y-axis, and the deviation from reference blend is plotted on the X-axis. A line is plotted for each mixture component (labeled with the component name in the figure) showing the effect of deviating from the reference blend. A steeper slope indicates a greater effect. In this case, the order of effects (greatest to least) is HRWRA, water, cement, silica fume, coarse aaggregate, and fine aaggregate.

Figure A-9. Mixture experiment: trace plot for 1-day strength

Figure A-10. Mixture experiment: contour plot of 1-day strength in water, cement, and silica fume. Diagram. This figure shows a contour plot of 1-day strength in water, cement, and silica fume. Water is the top vertex of the triangular plot, with cement at the lower left, and silica fume at the lower right. Each vertex represents the high setting of each component. The strength contours increase primarily from top to bottom with decreasing water, with smaller effects for increasing cement and increasing silica fume.

Figure A-10. Mixture experiment: contour plot of 1-day strength in water, cement, and silica fume

Figure A-11. Mixture experiment: contour plot of 1-day strength in water, cement, and HRWRA. Diagram. This figure shows a contour plot of 1-day strength in water, cement, and HRWRA. Water is the top vertex of the triangular plot, with cement at the lower left, and HRWRA at the lower right. The vertices represent the high settings of each component. The strength contours increase primarily from top to bottom with decreasing water, with smaller effects for increasing HRWRA and increasing cement.

Figure A-11. Mixture experiment: contour plot of 1-day strength in water, cement, and HRWRA

Figure A-12. Mixture experiment: contour plot of 1-daystrength in water, cement, and fine aaggregate. Diagram. This figure shows a contour plot of 1-day strength in water, cement, and fine aaggregate. Water is the top vertex of the triangular plot, with cement at the lower left, and fine aaggregate at the lower right. The vertices represent the high settings of each component. The strength contours increase from top right to bottom left with increasing cement and decreasing water. Fine aaggregate has little effect.

Figure A-12. Mixture experiment: contour plot of 1-day strength in water, cement, and fine aaggregate

Figure A-13. Mixture experiment: contour plot of 1-day strength in silica fume, HRWRA, and fine aaggregate. Diagram. This figure shows a contour plot of 1-day strength in silica fume, HRWRA, and fine aaggregate. Silica fume is the top vertex of the triangular plot, with HRWRA at the lower left, and fine aaggregate at the lower right. The vertices represent the high settings of each component. The strength contours increase from right to left with increasing HRWRA, with smaller effects from increasing fine aaggregate and decreasing silica fume.

Figure A-13. Mixture experiment: contour plot of 1-day strength in silica fume, HRWRA, and fine aaggregate

Figure A-14. Mixture experiment: contour plot of 1-day strength in silica fume, coarse aaggregate, and fine aaggregate. Diagram. This figure shows a contour plot of 1-day strength in silica fume, coarse aaggregate, and fine aaggregate at the best settings of HRWRA, water, and cement. This plot indicates the best settings for 1-day strength. The top vertex is silica fume, with coarse aaggregate at the lower left, and fine aaggregate at the lower right. The vertices represent the high settings for each component. The strength contours increase from bottom to top of the graph, with increasing silica fume. There are no effects from coarse or fine aaggregate.

Figure A-14. Mixture experiment: contour plot of 1-day strength in silica fume, coarse aaggregate, and fine aaggregate

A.2.3  28-Day Strength

Table A-15. Mixture experiment: sequential model sum of squares for 28-day strength

Source Sum of Squares DF Mean Square F Value Prob > F
Mean106213.01106213.0--
Linear 257.52 5 51.505.460.0011
Quadratic 135.19159.010.92 0.5665
Special cubic (aliased)55.457 7.92 0.69 0.6826
Cubic (aliased) 0.00 0 - --
Residual92.17 8 11.52 --
Total 106753.3 36 2965.37 - -

Table A-16.  Mixture experiment: lack-of-fit test for 28-day strength

Source Sum of Squares DF Mean Square F Value Prob > F
Linear 190.64 22 8.67 0.75 0.7193
Quadratic 55.45 7 7.92 0.69 0.6826
Special cubic (aliased) 0.00 0 - - -
Cubic (aliased) 0.00 0 - - -
Pure error 92.17 8 11.52 - -

Table A-17. Mixture experiment: model summary statistics for 28-day strength

Source Std. Dev. r2 Adj. r2 Pred. r2 PRESS
Linear 3.07 0.4755 0.3894 0.2678 395.63
Quadratic 3.14 0.7268 0.3625 -1.5516 1378.72
Special cubic (aliased) 3.39 0.8294 0.2537 - undefined
Cubic (aliased) - - - - undefined

Table A-18. Mixture experiment: ANOVA for 28-day strength mixture model

Source Sum of Squares DF Mean Square F Value Prob > F
Model 257.52 5 51.50 5.46 0.0011
Linear mixture 257.52 5 51.50 5.46 0.0011
Residual 282.81 30 9.43 - -
Lack of fit 190.64 22 8.67 0.75 0.7193
Pure error 92.17 8 11.52 - -
Corrected total 540.33 35 - - -

Table A-19. Mixture experiment: estimated coefficients for 28-day strength mixture model

Component Coeff. Estimate DF Std. Error 95% CI Low 95% CI High
Water 48.60 1 1.88 44.77 52.43
Cement 54.30 1 2.47 49.25 59.35
Silica fume 50.36 1 3.33 43.56 57.15
HRWRA 134.13 1 18.55 96.24 172.01
Coarse aaggregate 52.14 1 1.64 48.79 55.50
Fine aaggregate 54.21 1 1.72 50.70 57.73

Table A-20. Mixture experiment: adjusted effects for 28-day strength mixture model

Component Adjusted Effect DF Std. Error Approx. t for Ho Effect = 0 Prob > t
Water -12.04 1 2.40 -5.01 < 0.0001
Cement -6.41 1 2.31 -2.78 0.0093
Silica fume -6.05 1 1.93 -3.13 0.0039
HRWRA 5.43 1 1.27 4.28 0.0002
Coarse aaggregate -16.17 1 3.84 -4.22 0.0002
Fine aaggregate -13.69 1 4.08 -3.35 0.0022

Model equation in terms of pseudocomponents:

28-Day Strength = 48.60*A + 54.30*B + 50.36*C + 134.13*D + 52.14*E + 54.21*F

Model equation in terms of real components:

28-Day Strength = -45.22*water + 89.15*cement - 3.809*silica fume + 1972*HRWRA + 38.36*Coarse agg + 87.19*Fine agg

Figure A-15. Mixture experiment: normal probability plot for 28-day strength. Chart. This figure shows a normal probability plot of residuals for the 28-day strength model. The normal percent probability is plotted on the Y-axis against the studentized residuals on the X-axis. Most of the points fall on a straight line, indicating that the normality assumption is reasonable.

Figure A-15. Mixture experiment: normal probability plot for 28-day strength

Figure A-16. Mixture experiment: residuals versus run for 28-day strength. Chart. This figure shows a plot of residuals versus run for 28-day strength. The residuals are plotted on the Y-axis against the corresponding runs on the X-axis. The residuals show no structure (they are randomly distributed about a horizontal line).

Figure A-16. Mixture experiment: residuals vs. run for 28-day strength

Figure A-17. Mixture experiment: Cook's distance versus run for 28-day strength. Chart. This figure shows Cook's distance (Y-axis) versus run (X-axis) for 28-day strength. Cook's distance assesses the influence that outlier data points have on the fitted regression function. According to this plot, the maximum value of Cook's distance for this model is 0.17, which indicates that the outlier points do not have significant effects on the regression model.

Figure A-17. Mixture experiment: Cook's distance vs. run for 28-day strength

Figure A-18. Mixture experiment: trace plot for 28-day strength. Chart. This figure shows a response trace plot for 28-day strength. The response is plotted on the Y-axis, and the deviation from reference blend is plotted on the X-axis. A line is plotted for each mixture component (labeled with the name of the component in the figure) showing the effect of deviating from the reference blend. A steeper slope indicates a greater effect. In this case, the order of effects (greatest to least) is HRWRA, cement, silica fume, coarse aaggregate, water, and fine aaggregate. The effect of HRWRA is significantly greater than the others.

Figure A-18. Mixture experiment: trace plot for 28-day strength

Figure A-19. Mixture experiment: contour plot of 28-day strength in water, silica fume, and HRWRA. Diagram. This figure shows a contour plot of 28-day strength in water, silica fume, and HRWRA. Water is the top vertex of the triangular plot, with silica fume at the lower left, andHRWRA at the lower right. Each vertex represents the high setting of each component. The strength contours increase primarily from left to right with increasing HRWRA.

Figure A-19. Mixture experiment: contour plot of 28-day strength in water, silica fume, and HRWRA

Figure A-20. Mixture experiment: contour plot of 28-day strength in water, silica fume, and coarse aaggregate. Diagram. This figure shows a contour plot of 28-day strength in water, silica fume, and coarse aaggregate. Water is the top vertex of the triangular plot, with silica fume at the lower left, and coarse aaggregate at the lower right. The vertices represent the high settings of each component. The strength contours increase primarily from top to bottom with decreasing water, with a smaller effect from increasing coarse aaggregate.

Figure A-20. Mixture experiment: contour plot of 28-day strength in water, silica fume, and coarse aaggregate

Figure A-21. Mixture experiment: contour plot of 28-day strength in cement, coarse aaggregate, and fine aaggregate. Diagram. This figure shows a contour plot of 28-day strength in cement, coarse aaggregate, and fine aaggregate. Cement is the top vertex of the triangular plot, with coarse aaggregate at the lower left, and fine aaggregate at the lower right. The vertices represent the high settings of each component. The strength contours increase from bottom left to upper right with increasing cement, decreasing coarse aaggregate, and increasing fine aaggregate.

Figure A-21. Mixture experiment: contour plot of 28-day strength in cement, coarse aaggregate, and fine aggregate

A.2.4  RCT Charge Passed

Table A-21. Mixture experiment: sequential model sum of squares for RCT charge passed

SourceSum of Squares DF Mean Square F Value Prob > F
Mean 1441.26 1 1441.26 - -
Linear 6.75 5 1.35 57.33 < 0.0001
Quadratic 0.38 15 0.03 1.17 0.3846
Special cubic (aliased) 0.11 7 0.02 0.57 0.7607
Cubic (aliased) 0.00 0 - - -
Residual 0.22 8 0.03 - -
Total 1448.72 36 40.24 - -

Table A-22. Mixture experiment: lack-of-fit test for RCT charge passed

Source Sum of Squares DF Mean Square F Value Prob > F
Linear 0.490 22 0.022 0.82 0.6666
Quadratic 0.109 7 0.016 0.57 0.7607
Special cubic (aliased) 0.000 0 - - -
Cubic (aliased) 0.000 0 - - -
Pure error 0.217 8 0.027 - -

Table A-23. Mixture experiment: model summary statistics for RCT charge passed

Source Std. Dev. r2 Adj. r2 Pred. r2 PRESS
Linear 0.15 0.9053 0.8895 0.8647 1.01
Quadratic 0.15 0.9563 0.8980 0.8044 1.46
Special cubic (aliased) 0.16 0.9709 0.8726 - undefined
Cubic (aliased) - - - - undefined

Table A-24. Mixture experiment: ANOVA for RCT charge passed mixture model

Source Sum of Squares DF Mean Square F Value Prob > F
Model 6.75 5 1.35 57.33 < 0.0001
Linear mixture 6.75 5 1.35 57.33 < 0.0001
Residual 0.71 30 0.024 - -
Lack of fit 0.49 22 0.022 0.82 0.6666
Pure error 0.22 8 0.027 - -
Corrected total 7.46 35 - - -

Table A-25. Mixture experiment: estimated coefficients for RCT charge passed mixture model

Component Coeff. Estimate DF Std. Error 95% CI Low95% CI High
Water 7.2 1 0.094 7.01 7.39
Cement 6.34 1 0.12 6.09 6.60
Silica fume 4.10 1 0.17 3.76 4.44
HRWRA 5.32 1 0.93 3.43 7.22
Coarse aaggregate 6.62 1 0.082 6.46 6.79
Fine aaggregate 6.45 1 0.086 6.28 6.63

Table A-26. Mixture experiment: adjusted effects for RCT charge passed mixture model

Component Adjusted Effect DF Std. Error Approx. t for Ho Effect = 0 Prob > t
Water 0.84 1 0.12 7.03 < 0.0001
Cement 0.19 1 0.12 1.65 0.1087
Silica fume -0.76 1 0.097 -7.83 < 0.0001
HRWRA -0.054 1 0.063 -0.85 0.3999
Coarse aaggregate 0.74 1 0.19 3.86 0.0006
Fine aaggregate 0.53 1 0.20 2.62 0.0138

Model equation in terms of pseudocomponents:

ln(RCT charge passed) = 7.20*A + 6.34*B + 4.10*C + 5.32*D + 6.62*E + 6.45*F

Model equation in terms of real components:

ln(RCT charge passed) = 20.82*water + 0.629*cement - 52.33*silica fume - 23.41*HRWRA + 7.235*Coarse agg + 3.190*Fine agg

Figure A-22. Mixture experiment: normal probability plot for RCT charge passed (no transform). Chart. This figure shows a normal probability plot of residuals for the RCT model. The normal percent probability is plotted on the Y-axis against the studentized residuals on the X-axis. The points appear to fall on a curve rather than a straight line, indicating that the normality assumption may not apply.

Figure A-22. Mixture experiment: normal probability plot for RCT charge passed (no transform)

Figure A-23. Mixture experiment: normal probability plot for RCT charge passed (natural log transform). Chart. This figure shows a normal probability plot of residuals for the RCT model using a natural logarithm (LN) transform. The normal percent probability is plotted on the Y-axis against the studentized residuals on the X-axis. In this case, most of the points fall on a straight line, indicating that the normality assumption is reasonable.

Figure A-23. Mixture experiment: normal probability plot for RCT charge passed (natural log transform)

Figure A-24. Mixture experiment: residuals versus predicted for RCT charge passed (no transform). Chart. This figure shows a plot of residuals versus predicted values for RCT (with no transform). The residuals are plotted on the Y-axis against the corresponding predicted values on the X-axis. The desired appearance of this plot is a uniform range in the residuals over the range of predicted values. With no transform, the range of residuals increases with increasing predicted value.

Figure A-24. Mixture experiment: residuals vs. predicted for RCT charge passed (no transform)

Figure A-25. Mixture experiment: residuals versus run for RCT charge passed (no transform). Chart. This figure shows residuals versus run for RCT (no transform). The residuals are plotted on the Y-axis against the corresponding runs on the X-axis. For these data, there appears to be a curvelinear structure to the residuals.

Figure A-25. Mixture experiment: residuals vs. run for RCT charge passed (no transform)

Figure A-26. Mixture experiment: residuals vs. predicted for RCT charge passed (natural log transform). Chart. This figure shows residuals versus predicted values for the RCT using the LN transform. The residuals are plotted on the Y-axis and the predicted values are plotted on the X-axis. In this case the range of residuals does not increase or decrease over the range of predicted values, indicating that the LN transform is useful in this case.

Figure A-26. Mixture experiment: residuals vs. predicted for RCT charge passed (natural log transform)

Figure A-27. Mixture experiment: residuals vs. run for RCT charge passed (natural log transform). Chart. This figure shows residuals versus run for the RCT using the LN transform. The residuals are plotted on the Y-axis against the corresponding runs on the X-axis. The residuals appear more randomly distributed (they show less structure) than without the transform.

Figure A-27. Mixture experiment: residuals vs. run for RCT charge passed (natural log transform)

Figure A-28. Mixture experiment: Cook's distance for RCT charge passed (natural log transform). Chart. This figure shows a plot of Cook's distance versus run for RCT using the LN transform. The calculated value of Cook's distance is on the Y-axis, and the run number is on the X-axis. There are a few points that appear to be outliers, but neither has a significant value of Cook's distance.

Figure A-28. Mixture experiment: Cook's distance for RCT charge passed (natural log transform)

Figure A-29. Mixture experiment: trace plot for RCT charge passed (natural log transform). Chart. This figure shows a response trace plot for RCT. The response is plotted on the Y-axis, and the deviation from reference blend is plotted on the X-axis. A line is plotted for each mixture component (labeled with the name of the component in the figure) showing the effect of deviating from the reference blend. A steeper slope indicates a greater effect. In this case, the order of effects (greatest to least) is silica fume, HRWRA, cement, coarse aaggregate, fine aaggregate, and water.

Figure A-29. Mixture experiment: trace plot for RCT charge passed (natural log transform)

Figure A-30. Mixture experiment: contour plot of LN (RCT charge passed) in water, silica fume, and coarse aaggregate. Diagram. This figure shows a contour plot of LN (RCT) in water, silica fume, and coarse aaggregate. Water is the top vertex of the triangular plot, with silica fume at the lower left, and coarse aaggregate at the lower right. The vertices represent the high settings of each component. The LN (RCT) contours increase primarily from bottom left to top right with increasing water and decreasing silica fume. Coarse aaggregate has little effect.

Figure A-30. Mixture experiment: contour plot of ln (RCT charge passed) in water, silica fume, and coarse aaggregate

Figure A-31. Mixture experiment: contour plot of LN (RCT charge passed) in water, silica fume, and HRWRA. Diagram. This figure shows a contour plot of LN (RCT) in water, silica fume, and HRWRA. Water is the top vertex of the triangular plot, with silica fume at the lower left, and HRWRA at the lower right. The vertices represent the high settings of each component. The LN (RCT) contours increase from bottom to top with increasing water and decreasing silica fume. HRWRA has little effect.

Figure A-31. Mixture experiment: contour plot of ln (RCT charge passed) in water, silica fume, and HRWRA

Figure A-32. Mixture experiment: contour plot of LN (RCT charge passed) in cement, HRWRA, and fine aaggregate. Diagram. This figure shows a contour plot of LN (RCT) in cement, HRWRA, and fine aaggregate. Cement is the top vertex of the triangular plot, with HRWRA at the lower left, and fine aaggregate at the lower right. The vertices represent the high settings of each component. The LN (RCT) contours increase from left to right with decreasing HRWRA, decreasing cement, and increasing fine aaggregate.

Figure A-32. Mixture experiment: contour plot of ln (RCT charge passed) in cement, HRWRA, and fine aaggregate

 

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The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT).
The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). Provide leadership and technology for the delivery of long life pavements that meet our customers needs and are safe, cost effective, and can be effectively maintained. Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
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