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Publication Number: FHWA-RD-03-060

Concrete Mixture Optimization Using Statistical Methods

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APPENDIX B. Experiment Design and Data Analysis for Factorial Experiment

B.1 Experiment Design and Response Data

Table B-1. Factorial experiment: design by volume fraction of factors

Std Order Run Order Point Factor Aw/c Factor B Fine Agg Factor C Coarse Agg Factor DHRWRA Factor E Silica Fume
17 1 Center 0.39525 0.2712 0.4212 0.0060 0.0200
11 2 Fact 0.3576 0.2853 0.4071 0.0069 0.0247
7 3 Fact 0.3576 0.2853 0.4353 0.0051 0.0247
10 4 Fact 0.4329 0.2571 0.4071 0.0069 0.0247
4 5 Fact 0.4329 0.2853 0.4071 0.0051 0.0247
12 6 Fact 0.4329 0.2853 0.4071 0.0069 0.0153
2 7 Fact 0.4329 0.2571 0.4071 0.0051 0.0153
1 8 Fact 0.3576 0.2571 0.4071 0.0051 0.0247
18 9 Center 0.39525 0.2712 0.4212 0.0060 0.0200
8 10 Fact 0.4329 0.2853 0.4353 0.0051 0.0153
9 11 Fact 0.3576 0.2571 0.4071 0.0069 0.0153
6 12 Fact 0.4329 0.2571 0.4353 0.0051 0.0247
3 13 Fact 0.3576 0.2853 0.4071 0.0051 0.0153
14 14 Fact 0.4329 0.2571 0.4353 0.0069 0.0153
15 15 Fact 0.3576 0.2853 0.4353 0.0069 0.0153
13 16 Fact 0.3576 0.2571 0.4353 0.0069 0.0247
19 17 Center 0.39525 0.2712 0.4212 0.0060 0.0200
5 18 Fact 0.3576 0.2571 0.4353 0.0051 0.0153
16 19 Fact 0.4329 0.2853 0.4353 0.0069 0.0247
25 20 Axial 0.39525 0.2712 0.4494 0.0060 0.0200
21 21 Axial 0.47055 0.2712 0.4212 0.0060 0.0200
23 22 Axial 0.39525 0.2994 0.4212 0.0060 0.0200
27 23 Axial 0.39525 0.2712 0.4212 0.0078 0.0200
20 24 Axial 0.31995 0.2712 0.4212 0.0060 0.0200
31 25 Center 0.39525 0.2712 0.4212 0.0060 0.0200
26 26 Axial 0.39525 0.2712 0.4212 0.0042 0.0200
28 27 Axial 0.39525 0.2712 0.4212 0.0060 0.0106
24 28 Axial 0.39525 0.2712 0.3930 0.0060 0.0200
29 29 Axial 0.39525 0.2712 0.4212 0.0060 0.0294
22 30 Axial 0.39525 0.243 0.4212 0.0060 0.0200
30 31 Center 0.39525 0.2712 0.4212 0.0060 0.0200

Table B-2. Factorial experiment: slump and 1-day strength data

Run Point Slump (mm) 1-Day Strength (MPa)
1Center767016.616.016.2
2Fact444422.222.523.0
3Fact131322.517.922.0
4Fact10210215.816.716.8
5Fact575716.416.716.1
6Fact14014613.414.213.2
7Fact706413.911.113.7
8Fact131317.722.820.5
9Center898317.918.618.7
10Fact10210215.215.215.2
11Fact14014020.920.720.4
12Fact323213.818.818.9
13Fact131324.523.324.7
14Fact767617.317.217.2
15Fact131322.719.521.8
16Fact131320.820.621.4
17Center516419.418.918.1
18Fact322521.422.222.1
19Fact383216.016.116.3
20Axial383818.719.318.9
21Axial12111414.514.914.0
22Axial706418.017.817.5
23Axial646419.320.020.4
24Axial191326.025.427.7
25Center837620.217.319.6
26Axial646418.919.518.9
27Axial15215216.417.016.9
28Axial959520.120.819.7
29Axial383217.220.118.0
30Axial10210217.717.017.7
31Center767616.819.618.8

Table B-3. Factorial experiment: 28-day strength and RCT charge passed data

Run Point 28-Day Strength (MPa) RCT Charge Passed (coulombs)
1 Center 54.0 63.0 - - 263 296 300
2 Fact 59.4 60.2 - - 186 472 268
3 Fact 53.5 52.8 51.7 - 151 151 178
4 Fact 62.7 60.8 55.1 62.9 280 329 279
5 Fact 52.6 56.6 55.8 - 273 236 262
6 Fact 60.4 52.1 60.6 61.3 553 585 485
7 Fact 50.1 49.4 51.5 - 468 550 490
8 Fact 51.0 56.3 47.2 55.2 253 240 208
9 Center 63.5 62.5 - - 247 315 305
10 Fact 53.8 54.2 56.3 - 460 437 439
11 Fact 61.6 64.5 60.8 - 415 427 393
12 Fact 56.5 55.3 56.8 - 258 258 240
13 Fact 58.1 50.2 54.4 - 343 362 317
14 Fact 52.3 46.6 52.1 - 596 527 481
15 Fact 60.3 58.7 58.6 - 218 288 330
16 Fact 60.2 58.5 62.9 - 208 194 216
17 Center 59.9 54.5 55.5 - 299 327 318
18 Fact 58.4 60.1 56.3 - 360 364 340
19 Fact 65.0 59.9 63.9 - 242 243 206
20 Axial 56.7 57.0 62.1 - 168 242 224
21 Axial 53.9 51.1 56.7 - 463 461 449
22 Axial 60.7 62.9 63.6 - 319 305 257
23 Axial 68.4 66.8 67.1 - 272 280 251
24 Axial 62.1 59.4 54.0 - 190 184 192
25 Center 58.2 55.8 56.8 - 239 246 287
26 Axial 52.9 48.2 51.6 - 258 281 280
27 Axial 51.3 55.2 56.8 - 704 766 644
28 Axial 50.0 55.2 54.8 - 268 302 351
29 Axial 55.5 53.8 56.2 - 163 170 153
30 Axial 50.4 49.0 51.6 - 340 304 351
31 Center 55.8 53.3 56.5 - 262 294 274

B.2 Data Analysis and Model Fitting

B.2.1 Slump

Table B-4. Factorial experiment: sequential model sum of squares for slump

Source Sum of Squares DF Mean Square F Value Prob > F
Mean130296.61130296.6--
Linear 31972.3856394.489.10< 0.0001
2FI 10539.29101053.932.250.0755
Quadratic 2598.705519.741.180.3856
Cubic (aliased)2316.865463.371.100.4593
Residual2104.405420.88--
Total 179828.3315800.91--

Table B-5. Factorial experiment: lack-of-fit test for slump

Source Sum of Squares DF Mean Square F Value Prob > F
Linear 17103.6121814.467.150.0345
2FI 6564.3211596.765.240.0618
Quadratic 3965.626660.945.800.0553
Cubic (aliased) 1648.7511648.7514.470.0190
Pure error 455.644113.91--

Table B-6. Factorial experiment: ANOVA for slump model

Source Sum of Squares DF Mean Square F Value Prob > F
Model 40109.97 7 5730.00 13.99 < 0.0001
A 12138.75 1 12138.75 29.63 < 0.0001
B 606.52 1 606.52 1.48 0.2360
C 6048.38 1 6048.38 14.77 0.0008
D 2426.07 1 2426.07 5.92 0.0231
E 10752.67 1 10752.67 26.25 < 0.0001
AB 1837.19 1 1837.19 4.48 0.0452
CD 6300.39 1 6300.39 15.38 0.0007
Residual 9421.67 23 409.64 - -
Lack of fit 8966.03 19 471.90 4.14 0.0887
Pure error 455.64 4 113.91 - -
Cor total 49531.64 30 - - -

Table B-7. Factorial experiment: summary statistics for slump model

Std. Dev. 20.24R-Squared0.8098
Mean 64.83Adj R-Squared0.7519
C.V. 31.22Pred R-Squared0.6275
PRESS 18452.19Adeq Precision15.6200

Table B-8. Factorial experiment: coefficient estimates for slump model

Factor Coefficient Estimate DF Standard Error 95% CI Low 95% CI High
Intercept64.8313.6457.3172.35
A (w/c)22.4914.1313.9431.04
B (fine agg)-5.0314.13-13.573.52
C (coarse agg)-15.8714.13-24.42-7.33
D (HRWRA)10.0514.131.5118.60
E (silica fume)-21.1714.13-29.71-12.62
AB10.7215.060.2521.18
CD-19.8415.06-30.31-9.38

Model equation for slump in terms of coded factors:

Slump = 64.83 + 22.49*A - 5.03*B - 15.87*C + 10.05*D - 21.17*E + 10.72*A*B - 19.84*C*D

Model equation for slump in terms of actual factors:

Slump = -1365.5 - 4876.9*w/c - 8334.7*fine agg + 8256.5*coarse agg
+ 6.6982 x 105*HRWRA - 4503.6*silica fume + 20185*w/c*fine agg
- 1.564 x 106 *coarse agg*HRWRA

This figure shows a normal probability plot of residuals for slump from the factorial experiment. 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 B-1. Factorial experiment: normal probability plot for slump

This figure shows a raw data plot for slump. The slump (in millimeters) is plotted on the Y-axis against corresponding runs on the X-axis. Two data points are shown for each run. The data points for control runs are shown as hollow squares, while the other data points are shown as filled triangles. The plot indicates that the control mixes showed little variation, and there were no obvious trends in the data with time.

Figure B-2. Factorial experiment: raw data plot for slump (hollow squares indicate control runs)

This figure shows a scatterplot of slump (Y-axis) versus water-cement ratio (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of water-cement ratio, as defined by the experiment design. There is wide scatter in the data with a slight upward trend. A best-fit line indicates the upward trend.

Figure B-3. Factorial experiment: scatterplot of slump vs. w/c

This figure shows a scatterplot of slump (Y-axis) versus coarse aggregate volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of coarse aggregate volume fraction, as defined by the experiment design. There is wide scatter in the data with a slight downward trend. A best-fit line indicates the downward trend.

Figure B-4. Factorial experiment: scatterplot of slump vs. coarse aggregate

This figure shows a scatterplot of slump (Y-axis) versus fine aggregate volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of fine aggregate volume fraction, as defined by the experiment design. There is wide scatter in the data with a very slight downward trend. A best-fit line indicates the trend.

Figure B-5. Factorial experiment: scatterplot of slump vs. fine aggregate

This figure shows a scatterplot of slump (Y-axis) versus HRWRA volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of HRWRA, as defined by the experiment design. There is wide scatter in the data with a slight upward trend. A best-fit line indicates the upward trend.

Figure B-6. Factorial experiment: scatterplot of slump vs. HRWRA

This figure shows a scatterplot of slump (Y-axis) versus water-cement ratio (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of silica fume volume fraction, as defined by the experiment design. There is wide scatter in the data with a downward trend. A best-fit line indicates the upward trend.

Figure B-7. Factorial experiment: scatterplot of slump vs. silica fume

This figure shows a set of means plots for slump. In the graph, slump is on the Y-axis, and the five variables (factors) in the experiment are shown on the X-axis. From left to right, the factors are water-cement ratio, coarse aggregate, fine aggregate, HRWRA, and silica fume. For each factor, there are five data points shown, indicating the mean values for slump at each of the five settings of the factor. The data points are plotted from lowest setting (coded value negative 2) on the left to highest setting (coded value positive 2) on the right. The data points are connected by a line. The lines show the following general trends: slump increases with increasing water-cement ratio, slump is relatively unaffected by coarse aggregate, fine aggregate, and HRWRA, and slump decreases with increasing silica fume content.

Figure B-8. Factorial experiment: means plots for slump

This figure shows a plot of slump (Y-axis) versus run sequence (X-axis). This plot is similar to figure B-2, the raw data plot for slump. In this plot, the mean values of slump for each run are shown. The plot is used to assess any trends in slump with time. This plot shows a random distribution of slump with run, indicating no significant trends with time.

Figure B-9. Factorial experiment: slump vs. run sequence

Chart. This figure shows a lag plot for slump. In the lag plot, slump for a given run is plotted on the Y-axis, and the slump for the previous run is plotted on the X-axis. The X-axis value is referred to as the "lag slump." The purpose of plot is to identify whether any two consecutive runs appear be more alike than those taken farther apart. For random data, plots should show no structure or pattern. If significant exists in plot, assumption randomness may violated. There apparent this indicating that assumptions are valid.

Figure B-10. Factorial experiment: lag plot for slump

Table B-9. Factorial experiment: sequential model sum of squares for 1-day strength

Source Sum of Squares DF Mean Square F Value Prob > F
Mean  10780.54 1 10780.54 - -
Linear  215.97 5 43.19 22.15 < 0.0001
2FI 27.35 10 2.73 1.92 0.1236
Quadratic 13.60 5 2.72 3.48 0.0441
Cubic (aliased) 1.86 5 0.37 0.31 0.8865
Residual 5.95 5 1.19 - -
Total  11045.26 31 356.30 - -

Table B-10. Factorial experiment: lack-of-fit test for 1-day strength

Source Sum of Squares DF Mean Square F Value Prob > F
Linear  43.98 21 2.09 1.75 0.3129
2FI  16.63 11 1.51 1.27 0.4444
Quadratic 3.03 6 0.51 0.42 0.8339
Cubic (aliased) 1.18 1 1.18 0.98 0.3772
Pure error 4.78 4 1.19 - -

Table B-11. Factorial experiment: ANOVA for 1-day strength model

Source Sum of Squares DF Mean Square F Value Prob > F
Model 240.87 8 30.11 27.76 < 0.0001
A 213.26 1 213.26 196.66 < 0.0001
B 0.48 1 0.48 0.45 0.5113
C 0.043 1 0.043 0.040 0.8433
E 2.06 1 2.06 1.90 0.1819
A2 6.20 1 6.20 5.72 0.0257
AC 5.15 1 5.15 4.75 0.0404
AE 7.16 1 7.16 6.60 0.0175
BC 6.51 1 6.51 6.00 0.0227
Residual 23.86 22 1.08 - -
Lack of fit 19.08 18 1.06 0.89 0.6248
Pure error 4.78 4 1.19 - -
Cor total 264.72 30 - - -

Table B-12. Factorial experiment: summary statistics for 1-day strength model

Std. Dev. 1.04R-Squared0.9099
Mean 18.65Adj R-Squared0.8771
C.V. 5.58Pred R-Squared0.8294
PRESS 45.16Adeq Precision22.583

Table B-13. Factorial experiment: coefficient estimates for 1-day strength model

Factor Coefficient Estimate DF Standard Error 95% CI Low 95% CI High
Intercept 18.29 1 0.24 17.80 18.79
A (w/c) -2.98 1 0.21 -3.42 -2.54
B (fine agg) 0.14 1 0.21 -0.30 0.58
C (coarse agg) 0.043 1 0.21 -0.40 0.48
E (silica fume) 0.29 1 0.21 -0.15 0.73
A2 0.46 1 0.19 0.061 0.86
AC 0.57 1 0.26 0.027 1.11
AE 0.67 1 0.26 0.13 1.21
BC -0.64 1 0.26 -1.18 -0.098

Model equation for 1-day strength in terms of coded factors:

1-day strength = 18.29 - 2.98*A + 0.14*B + 0.043*C + 0.29*E + 0.46*A2 + 0.57*A*C + 0.67*A*E - 0.64*B*C

Model equation for 1-day strength in terms of actual factors:

1-day strength =-63.8 - 860.8*w/c + 1361.3*fine agg + 450.8*coarse agg - 1431.5*silica fume
+ 323.9*(w/c)2 + 1068*w/c*coarse agg + 3780*w/c*silica fume
- 3208*fine agg*coarse agg

This figure shows a normal probability plot of residuals for 1-day strength from the factorial experiment. 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 B-11. Factorial experiment: normal probability plot for 1-day strength

This figure shows a raw data plot for 1-day strength. The strength (in megapascals) is plotted on the Y-axis against corresponding run on the X-axis. Three data points are shown for each run (each data point representing an individual cylinder break). The data points for control runs are shown as hollow squares, while the other data points are shown as filled triangles. The plot indicates that the control mixes showed little variation, and there were no obvious trends in the data with time.

Figure B-12. Factorial experiment: raw data plot for 1-day strength (hollow squares indicate control runs)

This figure shows a scatterplot of 1-day strength (Y-axis) versus water-cement ratio (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of water-cement ratio, as defined by the experiment design. The data show a definite downward trend. A best-fit line indicates the downward trend.

Figure B-13. Factorial experiment: scatterplot of 1-day strength vs. w/c

This figure shows a scatterplot of 1-day strength (Y-axis) versus fine aggregate volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of fine aggregate volume fraction, as defined by the experiment design. There is wide scatter in the data with virtually no trend. A nearly horizontal best-fit line indicates the lack of any trend.

Figure B-14. Factorial experiment: scatterplot of 1-day strength vs. fine aggregate

This figure shows a scatterplot of 1-day strength (Y-axis) versus coarse aggregate volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of coarse aggregate volume fraction, as defined by the experiment design. There is wide scatter in the data with no trend, as indicated by the horizontal best-fit line.

Figure B-15. Factorial experiment: scatterplot of 1-day strength vs. coarse aggregate

This figure shows a scatterplot of 1-day strength (Y-axis) versus HRWRA volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of HRWRA, as defined by the experiment design. There is wide scatter in the data with no significant trend, as indicated by a horizontal best-fit line.

Figure B-16. Factorial experiment: scatterplot of 1-day strength vs. HRWRA

This figure shows a scatterplot of 1-day strength (Y-axis) versus silica fume volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of silica fume volume fraction, as defined by the experiment design. There is wide scatter in the data with virtually no trend, as indicated by the nearly horizontal best-fit line.

Figure B-17. Factorial experiment: scatterplot of 1-day strength vs. silica fume

This figure shows a set of means plots for 1-day strength. In the graph, 1-day strength is on the Y-axis and the five variables (factors) in the experiment are shown on the X-axis. From left to right, the factors are water-cement ratio, coarse aggregate, fine aggregate, HRWRA, and silica fume. For each factor, there are five data points shown, indicating the mean values for 1-day strength at each of the five settings of the factor. The data points are plotted from lowest setting (coded value negative 2) on the left to highest setting (coded value positive 2) on the right. For each factor, the data points are connected by a line. The lines show the following general trends: 1-day strength decreases with increasing water-cement ratio, 1-day strength is relatively unaffected by coarse aggregate, fine aggregate, HRWRA, and silica fume.

Figure B-18. Factorial experiment: means plot for 1-day strength

This figure shows a plot of 1-day strength (Y-axis) versus run sequence (X-axis). In this plot, the mean values of 1-day strength for each run are plotted. The plot is used to assess any trends in 1-day strength with time. This plot shows a random distribution of 1-day strength with run, indicating no significant trends with time.

Figure B-19. Factorial experiment: 1-day strength vs. run sequence

This figure shows a lag plot for 1-day strength. In the lag plot, the 1-day strength for a given run is plotted on the Y-axis, and the 1-day strength for the previous run is plotted on the X-axis. The X-axis value is referred to as the "lag 1-day strength." The purpose of plot is to identify whether any two consecutive runs appear be more alike than those taken farther apart. For random data, plots should show no structure or pattern. If significant exists in plot, assumption randomness may violated. There apparent this indicating that assumptions are valid.

Figure B-20. Factorial experiment: lag plot for 1-day strength

B.2.3. 28-Day Strength

Table B-14. Factorial experiment: sequential model sum of squares for 28-day strength

Source Sum of Squares DF Mean Square F Value Prob > F
Mean 100300.01100300.0--
Linear 292.28558.465.710.0012
2FI 129.241012.921.530.2208
Quadratic 27.5355.510.560.7318
Cubic (aliased) 61.25512.251.620.3050
Residual 37.8457.57--
Total  100848.1313253.17--

Table B-15. Factorial experiment: lack-of-fit test for 28-day strength

Source Sum of Squares DF Mean Square F Value Prob > F
Linear219.79 21 10.47 1.16 0.4972
2FI 90.55 11 8.23 0.91 0.5942
Quadratic63.02 6 10.50 1.16 0.4620
Cubic (aliased)1.76 1 1.76 0.20 0.6813
Pure error36.08 4 9.02 - -

Table B-16. Factorial experiment: ANOVA for 28-day strength model

Source Sum of Squares DF Mean Square F Value Prob > F
Model300.21 475.057.870.0003
A16.57116.571.740.1990
D223.411223.4123.43< 0.0001
E7.4717.470.780.3842
AE52.76152.765.530.0265
Residual247.94269.54--
Lack of fit211.86229.631.070.5389
Pure error36.0849.02--
Cor total548.1530---

Table B-17. Factorial experiment: summary statistics for 28-day strength model

Std. Dev. 3.09R-Squared0.5477
Mean 56.88Adj R-Squared0.4781
C.V. 5.43Pred R-Squared0.3997
PRESS 329.05Adeq Precision9.964

Table B-18. Factorial experiment: coefficient estimates for 28-day strength model

Factor Coefficient Estimate DF Standard Error 95% CI Low 95% CI High
Intercept 56.88 10.5555.7458.02
A (w/c) -0.83 10.63-2.130.46
D (HRWRA) 3.05 10.631.764.35
E (silica fume) 0.56 10.63-0.741.85
AE 1.82 10.770.233.40

Model equation for 28-day strength in terms of coded factors:

28-day strength = 56.88 - 0.83*A + 3.05*D + 0.56*E + 1.82*A*E

Model equation for 28-day strength in terms of actual factors:

28-day strength = 124.0 - 227.3*w/c + 3390*HRWRA - 3937.5*silica fume + 10262*w/c*silica fume

This figure shows a normal probability plot of residuals for 28-day strength from the factorial experiment. 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 B-21. Factorial experiment: normal probability plot for 28-day strength

This figure shows a raw data plot for 28-day strength. The strength (in megapascals) is plotted on the Y-axis against corresponding runs on the X-axis. Three data points are shown for each run (each data point representing an individual cylinder break). The data points for control runs are shown as hollow squares, while the other data points are shown as filled triangles. The plot indicates that the control mixes showed little variation, and there were no obvious trends in the data with time.

Figure B-22. Factorial experiment: raw data plot for 28-day strength (hollow squares indicate control runs)

This figure shows a scatterplot of 28-day strength (Y-axis) versus water-cement ratio (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of water-cement ratio, as defined by the experiment design. The data show a very slight downward trend. A best-fit line indicates the trend.

Figure B-23. Factorial experiment: scatterplot of 28-day strength vs. w/c

This figure shows a scatterplot of 28-day strength (Y-axis) versus fine aggregate volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of fine aggregate volume fraction, as defined by the experiment design. There is wide scatter in the data with a very slight upward trend. A best-fit line indicates the trend.

Figure B-24. Factorial experiment: scatterplot of 28-day strength vs. fine aggregate

This figure shows a scatterplot of 28-day strength (Y-axis) versus coarse aggregate volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of coarse aggregate volume fraction, as defined by the experiment design. There is wide scatter in the data with a very slight upward trend, indicated by the nearly horizontal best-fit line.

Figure B-25. Factorial experiment: scatterplot of 28-day strength vs. coarse aggregate

This figure shows a scatterplot of 28-day strength (Y-axis) versus HRWRA volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of HRWRA, as defined by the experiment design. There is wide scatter in the data with a definite upward trend, indicated by a best-fit line.

Figure B-26. Factorial experiment: scatterplot of 28-day strength vs. HRWRA

 This figure shows a scatterplot of 28-day strength (Y-axis) versus silica fume volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of silica fume volume fraction, as defined by the experiment design. There is wide scatter in the data with a very slight upward trend, as indicated by the nearly horizontal best-fit line.

Figure B-27. Factorial experiment: scatterplot of 28-day strength vs. silica fume

This figure shows a set of means plots for 28-day strength. In the graph, 28-day strength is on the Y-axis and the five variables (factors) in the experiment are shown on the X-axis. From left to right, the factors are water-cement ratio, coarse aggregate, fine aggregate, HRWRA, and silica fume. For each factor, there are five data points shown, indicating the mean values for 28-day strength at each of the five settings of the factor. The data points are plotted from lowest setting (coded value negative 2) on the left to highest setting (coded value positive 2) on the right. For each factor, the data points are connected by a line. The lines show the following general trends: 28-day strength is slightly affected by water-cement ratio; relatively unaffected by coarse aggregate, fine aggregate, and silica fume; and significantly affected by HRWRA.

FIgure B-28. Factorial experiment: means plot for 28-day strength

This figure shows a plot of 28-day strength (Y-axis) versus run sequence (X-axis). In this plot, the mean values of 28-day strength for each run are plotted. The plot is used to assess any trends in 28-day strength with time. This plot shows a random distribution of 28-day strength with run, indicating no significant trends with time.

Figure B-29. Factorial experiment: 28-day strength vs. run sequence

This figure shows a lag plot for 28-day strength. In the lag plot, the 28-day strength for a given run is plotted on the Y-axis, and the 28-day strength for the previous run is plotted on the X-axis. The X-axis value is referred to as the "lag 28-day strength." The purpose of plot is to identify whether any two consecutive runs appear be more alike than those taken farther apart. For random data, plots should show no structure or pattern. If significant exists in plot, assumption randomness may violated. There apparent this indicating that assumptions are valid.

Figure B-30. Factorial experiment: lag plot for 28-day strength

Table B-19. Factorial experiment: sequential model sum of squares for RCT charge passed

Source Sum of Squares DF Mean Square F Value Prob > F
Mean315586713155867--
Linear393517.2578703.4327.43< 0.0001
2FI15156.50101515.650.400.9252
Quadratic44410.7658882.157.310.0040
Cubic (aliased)10046.8352009.374.770.0557
Residual2104.615420.92--
Total362110331116809.8--

Table B-20. Factorial experiment: lack-of-fit test for RCT charge passed

Source Sum of Squares DF Mean Square F Value Prob > F
Linear69614.70213314.996.300.0432
2FI54458.20114950.759.410.0221
Quadratic10047.4461674.573.180.1410
Cubic (aliased)0.6110.61.0010.9745
Pure error2104.004526.00--

Table B-21. Factorial experiment: ANOVA for RCT charge passed model

Source Sum of Squares DF Mean Square F Value Prob > F
Model441455.3 6 73575.89 74.25 < 0.0001
A81666.67 1 81666.67 82.42 < 0.0001
B6868.17 1 6868.17 6.93 0.0146
C11440.67 1 11440.67 11.55 0.0024
E292604.2 1 292604.2 295.30 < 0.0001
E238369.41 1 38369.41 38.72 < 0.0001
AE10506.25 1 10506.25 10.60 0.0034
Residual23780.54 24 990.86 - -
Lack of fit21676.54 20 1083.83 2.06 0.2537
Pure error2104.00 4 526.00 - -
Cor total465235.9 30 - - -

Table B-22. Factorial experiment: summary statistics for RCT charge passed model

Std. Dev. 31.48R-Squared0.9489
Mean 319.06Adj R-Squared0.9361
C.V. 9.87Pred R-Squared0.8784
PRESS 56577.00Adeq Precision34.166

Table B-23. Factorial experiment: coefficient estimates for RCT charge passed model

Factor Coefficient Estimate DF Standard Error 95% CI Low 95% CI High
Intercept291.11 1 7.22 276.20 306.01
A (w/c)58.33 1 6.43 45.07 71.59
B (fine agg)-16.92 1 6.43 -30.18 -3.66
C (coarse agg)-21.83 1 6.43 -35.09 -8.57
E (silica fume)-110.42 1 6.43 -123.68 -97.16
E236.11 1 5.80 24.14 48.09
AE-25.63 1 7.87 -41.87 -9.38

Model equation for RCT charge passed in terms of coded factors:

RCT charge passed = 291.11 + 58.33*A - 16.92*B - 21.83*C - 110.42*E + 36.11*E2 - 25.63*A*E

Model equation for RCT charge passed in terms of actual factors:

RCT charge passed = 635.4 + 4445.6*w/c - 1199.8*fine agg - 1548.5*coarse agg - 31651*silica fume + 1.635 x 106*(silica fume)2 - 1.448 x 105*w/c*silica fume

This figure shows a normal probability plot of residuals for RCT from the factorial experiment. 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 B-31. Factorial experiment: normal probability plot for RCT charge passed

This figure shows a raw data plot for RCT. The charge passed (in coulombs) is plotted on the Y-axis against corresponding run on the X-axis. Three data points are shown for each run (each data point representing an individual test run). The data points for control runs are shown as hollow squares, while the other data points are shown as filled triangles. The plot indicates that the control mixes showed little variation, and there were no obvious trends in the data with time. However, run 27 has a significantly higher charge passed than the others and could be an outlier.

Figure B-32. Factorial experiment: raw data plot for RCT charge passed (hollow squares indicate control runs)

This figure shows a scatterplot of RCT (Y-axis) versus water-cement ratio (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of water-cement ratio, as defined by the experiment design. The data show an upward trend, as indicated by a best-fit line.

Figure B-33. Factorial experiment: scatterplot of RCT charge passed vs. w/c

This figure shows a scatterplot of RCT (Y-axis) versus fine aggregate volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of fine aggregate volume fraction, as defined by the experiment design. There is wide scatter in the data with a slight downward trend, shown by a nearly horizontal best-fit line.

Figure B-34. Factorial experiment: scatterplot of RCT charge passed vs. fine aggregate

This figure shows a scatterplot of RCT (Y-axis) versus coarse aggregate volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of coarse aggregate volume fraction, as defined by the experiment design. There is wide scatter in the data with a slight downward trend, as indicated by the nearly horizontal best-fit line.

Figure B-35. Factorial experiment: scatterplot of RCT charge passed vs. coarse aggregate

This figure shows a scatterplot of RCT (Y-axis) versus HRWRA volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of HRWRA, as defined by the experiment design. There is wide scatter in the data with no significant trend, as indicated by a horizontal best-fit line.

Figure B-36. Factorial experiment: scatterplot of RCT charge passed vs. HRWRA

This figure shows a scatterplot of RCT (Y-axis) versus silica fume volume fraction (X-axis). Each data point represents an experimental run. The data are plotted at five distinct settings of silica fume volume fraction, as defined by the experiment design. The data show a definite downward trend, which is clearly indicated by a best-fit line.

Figure B-37. Factorial experiment: scatterplot of RCT charge passed vs. silica fume

This figure shows a set of means plots for RCT. In the graph, RCT is on the Y-axis and the five variables (factors) in the experiment are shown on the X-axis. From left to right, the factors are water-cement ratio, coarse aggregate, fine aggregate, HRWRA, and silica fume. For each factor, there are five data points shown, indicating the mean values for slump at each of the five settings of the factor. The data points are plotted from lowest setting (coded value negative 2) on the left to highest setting (coded value positive 2) on the right. For each factor, the data points are connected by a line. The lines show the following general trends: RCT increases with increasing water-cement ratio, RCT is relatively unaffected by coarse aggregate, fine aggregate, and HRWRA; and RCT decreases with increasing silica fume.

Figure B-38. Factorial experiment: means plot for RCT charge passed

This figure shows a plot of RCT (Y-axis) versus run sequence (X-axis). In this plot, the mean values of RCT for each run are plotted. The plot is used to assess any trends in RCT with time. This plot shows a random distribution of RCT with run, indicating no significant trends with time.

Figure B-39. Factorial experiment: RCT charge passed vs. run sequence

This figure shows a lag plot for RCT. In the lag plot, the RCT for a given run is plotted on the Y-axis, and the RCT for the previous run is plotted on the X-axis. The X-axis value is referred to as the "lag RCT." The purpose of the lag plot is to identify whether any two consecutive runs appear to be more alike than those taken farther apart. For random data, lag plots should show no structure or pattern. If significant structure exists in the lag plot, the assumption of randomness may be violated. There is no apparent structure in this lag plot, indicating that randomness assumptions are valid.

Figure B-40. Factorial experiment: lag plot for RCT charge passed

 

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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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