Recipe 4: Completely Randomized Block Designs from the literature
Recipes for the Design of Experiments
Zoe Konrad
Rensselaer Polytechnic Institute
Fall 2014 v1
1. Setting
System under test
The utilization of industrial waste and by-products in cementitious composites (concretes and mortars) has been attracting attention both due to the efforts for energy reduction and the rise of environmental consciousness.
The purpose of this work was to further investigate the replacement of natural aggregates in mortar mixtures by rubber waste (WRS) particles, its effect on the mortars properties and determine the optimum replacement amount.
x <- read.csv("~/Desktop/doe/Recipe 4.csv")
attach(x)
x## Run WRS W.C CI CS28
## 1 1 10 0.52 205 18.84
## 2 2 10 0.55 227 15.78
## 3 3 10 0.60 265 14.56
## 4 4 20 0.52 181 16.89
## 5 5 20 0.55 204 12.29
## 6 6 20 0.60 245 11.60
## 7 7 30 0.52 173 14.56
## 8 8 30 0.55 180 12.40
## 9 9 30 0.60 220 11.55Factors and Levels
Two factors were chosen. The factor of interest is the the WRS volume content and the blocking factor is the water/cement weight ratio (W/C). Each was run at three levels; 10, 20 and 30 vol.% and 0.52, 0.55 and 0.60, respectively.
WRS <- as.factor(WRS)
summary(WRS)## 10 20 30
## 3 3 3W.C <- as.factor(W.C)
summary(W.C)## 0.52 0.55 0.6
## 3 3 3Response variables
The response variables are the fresh mortar consistency index (CI) and the 28-day compressive strength of hardened mortars (CS28), both measured continuously in mm and MPa, respectively.
Acceptable values are CI less than 255 mm.
range(CI)## [1] 173 265Acceptable values are CS28 between 13–17 MPa.
range(CS28)## [1] 11.55 18.842. (Experimental) Design
This resulted in a 3^2 full factorial design, for a total of 9 experimental runs.
We use the factor W.C as a blocking factor to improve our model since it is a known and controllable nuisance variable.
The experimental runs were randomized and each reported result is the average of the test results for four specimens.
3. (Statistical) Analysis
Exploratory Data Analysis Graphics
plot(CI~WRS+W.C)
plot(CS28~WRS+W.C)
There appear to be significant differences in mean for both factors accross both response variables.
interaction.plot(WRS,W.C,CS28); interaction.plot(WRS,W.C,CI)
There are no significant interaction effects between the two factors for either response variable.
Testing
anova(aov(CS28~WRS))## Analysis of Variance Table
##
## Response: CS28
## Df Sum Sq Mean Sq F value Pr(>F)
## WRS 2 21.062 10.5312 2.0331 0.2118
## Residuals 6 31.080 5.1799anova(aov(CI~WRS))## Analysis of Variance Table
##
## Response: CI
## Df Sum Sq Mean Sq F value Pr(>F)
## WRS 2 2568.2 1284.11 1.473 0.3017
## Residuals 6 5230.7 871.78We fail to reject the null hypothesis using WRS alone to explain the variation in both CI and CS28.
model1<- aov(CS28~WRS+W.C)
anova(model1)## Analysis of Variance Table
##
## Response: CS28
## Df Sum Sq Mean Sq F value Pr(>F)
## WRS 2 21.0624 10.5312 21.776 0.007076 **
## W.C 2 29.1452 14.5726 30.133 0.003874 **
## Residuals 4 1.9344 0.4836
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1model2 <- aov(CI~WRS+W.C)
anova(model2)## Analysis of Variance Table
##
## Response: CI
## Df Sum Sq Mean Sq F value Pr(>F)
## WRS 2 2568.2 1284.11 47.658 0.0016221 **
## W.C 2 5122.9 2561.44 95.064 0.0004246 ***
## Residuals 4 107.8 26.94
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Introducing our blocking variable, we now see a significant P-value.
Thus, we reject the null hypothesis and conclude that randomization alone cannot account for the variability in the fresh mortar consistency index (CI) nor the 28-day compressive strength of hardened mortars (CS28.)
Estimation (of Parameters)
After verifying that the variation in both CI and CS28 can be explained by our model, we now can compare our experimental results to the specified acceptable values, accepting each experimental run as an appropriate estimation.
boxplot(CI~W.C*WRS, ylab='CI (mm)', xlab='treatment') #specified value less than 255 mm
abline(h=255)
boxplot(CS28~W.C*WRS, , ylab='CS28 (mPa)', xlab='treatment') #specified value 13-17 mPa
abline(h=13)
abline(h=17)
Clearly, at higher water/cement ratios (W.C 0.52 to 0.58) and replacement contents ( WRS < 20 vol.%), a specified 28-day compressive strength (13–17 MPa) can be achieved.
The paper conclueds that reusing waste vulcanized rubber scrap particles as aggregates in mortar production is not only a viable solution to the problem of inadequate concrete aggregates but also may help solving a vital environmental issue.
Diagnostics/Model Adequacy Checking
Neither of the models violate the assumption of normality.
qqnorm(residuals(model1)); qqline(residuals(model1)); qqnorm(residuals(model2)); qqline(residuals(model2))
4. References to the literature
Factorial design used to model the compressive strength of mortars containing recycled rubber from the Fifteenth International Conference on Composite Structures. The full paper can be found at: [http://www.sciencedirect.com.libproxy.rpi.edu/science/article/pii/S0263822309004760]