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Page Predicting the Compressive JOINT
CSHRP/NEWFOUNDLAND BAYESIAN APPLICATION Model 1 Model Definition The Department of Works, Services and
Transportation of the Government of Newfoundland and
Labrador decided to develop a model to predict the
compressive strength of high performance silica fume
concrete. This model would be used by DWST personnel to
develop better scheduling of construction activities. Dependent Variable The dependent variable is the
compressive strength of silica fume concrete at 28 days,
f'c. Compressive strength tests are conducted on
concrete test cylinders 150 mm in diameter and 300 mm
high. Compressive strength is measured in megapascals
(MPa). Independent Variables A working paper entitled Factors
Influencing the Compressive Strength of Normal Portland
Cement and HighPerformance Silica Fume Concretes was
written to help select the independent variables. See
Appendix 1. Through literature searches, examination of
DWST files, and interviews with experts, the following 12
factors were identified as possible independent variables
for the model prediction: Several classical regressions and
correlation matrices were run on the "old" data
obtained from the laboratory mix designs to identify the
most statistically significant variables. After the first regression analysis,
the water cement ratio was found to be an insignificant
variable. Upon the insistence of the experts, the water
cement ratio remained while the cement and water contents
were disregarded as both are addressed in this ratio. Superplasticizer dosage was also found
to be statistically insignificant. However, the
literature suggested that the variable is important in
silica fume concrete whereas the database is comprised
largely of normal portland cement concrete. The slump was identified as a variable,
but was eliminated because it is used to measure the
consistency of a concrete mix, not its strength. After
consultation with Mr. Keith Foster, the fine to total
aggregate ratio variable was kept as well. The fine to total aggregate ratio is
obtained by dividing the weight of fine aggregate in a
concrete mix by the weight of the total aggregate. Fine
aggregate is defined as particles smaller than 5 mm in
diameter. Refer to the working paper in Appendix 1 for
information on the other variables. According to the experts, the following five variables would have the greatest impact on the compressive strength:
Model Type and Functional Form An empirical model type was selected by the experts as a mechanistic functional form could not be found. Their proposed equation for the model is the following simple additivelinear form: Model Inputs Appendix 2 and Excel File RUN1INPT.XLS
contain the model's worksheets NEWDATA, OLDDATA and
PRIOR. Quality assurance checks were carried out on the
data by the staff end Mr. Chris Walsh. The DWST staff,
involved in collecting the data, are certified as concrete technicians by the American
Concrete Institute. All compressive strength tests were
conducted on the same testing machine. NEWDATA consists of mix design data and
cylinder breaks from two bridge projects in Newfoundland:
Main Brook Bridge (MBB) and Holyrood Pond Bridge (HPB).
The worksheet contains 50 observations. Worksheet NEWDATA shows little
variation in the variables for fine to total aggregate
ratios and super plasticizer dosages. These variables
were set by the two mix designs and did not deviate much.
Silica fume content was 8% throughout the projects. The water cement ratio was determined
by the new AASHTO test, TP2393, Test Method for
Water Content of Freshly Mixed Concrete Using Microwave
Drying This test was used because the water cement
ratios given in the mix designs are not always complied
with. The moisture content of aggregates, workability,
and daily changes in the plant all affect the water
content of the mix and thus the water cement ratio. OLDDATA contains 98 observations from
concrete mix design data obtained at the Department's
Central Laboratory. Both normal portland cement and
silica fume concretes were tested. Silica fume contents
are 0% and 8% respectively. The water cement ratios are
the actual values used in the laboratory. The classical regression analysis
option in XLBAYES was run on OLDDATA to calculate
GPrior data. Estimates of the model coefficients,
degrees of freedom and residual variance were calculated.
The output is displayed in worksheet PRIOR. Calculation
of a constant was included in the regression. Bayesian regression was run using the
GPrior option. Worksheet NEWDATA contains the sample
data consisting of the five independent variables and one
dependent variable. Worksheet PRIOR contains the old data
consisting of the laboratory mix design data and
regression outputs. A GPrior value of 1 was assumed. The Posterior model for predicting Compressive Strengths was calculated as: Evaluation Table I and the following sections compare coefficient and model statistics for the Data, Prior, and Posterior regressions. Analysis is found in Appendix 3 and on Excel file XLRUN1A.XLS. Water Cement Ratio
The sign for the water cement ratio
coefficient is negative as expected. As the water cement
ratio increases, the compressive strength decreases. A
ttest examines whether the water cement ratio affects
compressive strength. The calculated t1 = 6.36 is greater
than t.025 = 1.96, signifying that the variable is
significant. The coefficients for the Prior and classical
regression of the Data (294.86 and 119.33) also have
correct signs and are statistically significant. Figure 1 is a normal probability density plot from XLBAYES showing the distribution of the water cement ratio coefficients for the Prior, Data and Posterior. The xaxis value represents the value of the coefficient while the spread of the plot suggests the variation in the data. As the spread increases, the calculated value of the coefficient varies more about the mean.
The plot indicates that the Posterior
is most affected by the Data: the Data is definitive. The
Posterior coefficient is almost identical to the
classical regression of the Data (NEWDATA). The only
effect of the Prior is to help decrease the uncertainty
and the standard deviation. Although the Data refutes the
Prior, this scenario is favourable. Air Content
The sign for the Posterior
coefficient is correct. An increase in air content
results in a decrease in compressive strength. The
calculated t = 5.99 < t .025, suggests that there
is a relationship between air content and compressive
strength. The air content coefficient is statistically
significant. There is little difference between the
coefficients for Prior and Posterior (1.68 vs. 1.73).
The Data, which is the classical regression of NEWDATA,
has least effect on the outcome and therefore little
value. The only effect of the Data is to decrease
confidence in the Prior. This increases the uncertainty
of the Posterior. Fine to Total Aggregate Ratio
Contrary to the findings in the working
paper, the sign for the coefficient, b3, positive. An
examination of Table 1 and the probability plot shows
that the sample Data has a negative coefficient (78.36)
whereas the Prior coefficient is positive (34.22).
According to Mr. Colin Crane, Senior Concrete Technician
with DWST, laboratory mix designs have implied that an
increase in the ratio will increase the compressive
strength. Review of worksheet NEWDATA shows that
the sample data provides only two values for fine to
total aggregate ratios, whereas the Prior provides
several. The Prior consists mainly of laboratory tests
conducted on normal portland cement concrete (0% Silica
Fume) where the role of fine aggregate is more crucial. The tstatistic suggests that the coefficient is not significant: b3 might equal 0.
The probability plot shows that the
Posterior is pulled toward the Prior. The Data is
therefore weak. Note that the Posterior crosses the axis
at 0 there is a probability that the coefficient b3 is
0. Silica Fume Content
According to Table 1, the classical
regression value for the Data is 117.59, while the values
for the Prior and Posterior are 3.29 and 2.32,
respectively. A review of the input file RUN1INPT.XLS shows
that the Prior consists of silica fume values of 0 % (90
readings) and 8 % (8 readings). The sample data input
NEWDATA contains 52 readings, all with a silica fume
concentration of 8 %. The problem is that there is very
little variation in values. It may have been better to
treat the Silica Fume Content variable as a categorical
variable. The NEWDATA appears to be "worthless" as it did not change the Posterior mean; it only lowered its uncertainty.
Super Plasticizer Dosage
A positive value for the Posterior
coefficient is correct. With the addition of a super
plasticizer admixture, the water content of a concrete
mix can be reduced, thereby increase the mix's
compressive strength. The change in signs between the Prior
and Posterior coefficients (0.66 vs. 0.57) indicates a
serious discrepancy between the Data and Prior.
Fortunately, the Data is definitive and ensures that the
Posterior has a positive sign ( i.e. correct sign).. The probability distribution plot suggests that all three coefficients may equal zero. Table 2 shows that the coefficients for the Data and Posterior are not statistically significant.
Constant (Intercept)
Review of Table 1 and the probability
distribution plot indicates another major discrepancy
between the Prior and the classical regression of the
Data. The Prior coefficient has a value of 151.43 while
the Data value is 788.73. It appears that the Data's
coefficient acts as a placeholder and is taking up slack
for other variables. The coefficient for the Data regression is wrong because the constant cannot be negative. If most of the other variables were minimized, the compressive strength would not be expected to be zero or negative. Model Statistics Classical regression was conducted on
the Prior and Data observations of RUN1INPT.XLS to
calculate their coefficient of determination, R2
. The coefficient of determination represents the
proportion of the total sample variability accounted for
by the model. For example, a value of R2 =
0.60 means that 60 % of the variation in the compressive
strength is explained by the model. The coefficient of
determination cannot be measured for the Posterior model
because determining whether observations from the Prior,
the Data, or both should be used is difficult. Table 1 shows that for the Prior model,
the model can explain 86 % of the variation in
compressive strength while the Data explains only 52 % of
the variation. Better quality control in the laboratory
and more observations may account for this difference. The standard error, Se, is the standard
deviation of values for compressive strength unexplained
by the models it is what the regression equations
cannot explain. The standard error equals zero when the
regression equation fits all the data points exactly. The
lower the Se, the more accurate the model is. By using
Bayesian statistics, the Posterior model has a lower
standard error than the Data model produced by classical
regression and appears to have greater predictive
abilities. Predictions for the three models were made based on the following values (base case):
High and low predictions were
calculated for a 95 % confidence limit (the base case
prediction +/ 1.96 * Se Predictions ranged from 60.85
MPA to 84.52 MPa for the Data function (classical
regression) while the Posterior (Bayesian regression) had
narrowed the prediction to values between 58.96 MPa and
79.03 MPa. Sensitivity Analysis A sensitivity analysis was carried out
to compare the Data, Prior and Posterior models. The
effect of changing the values for each independent
variable is shown in Figure 7. Appendix 4 and Excel File SENSTVTY.XLS
contain the input data. As the line plot shows, predictions from the Prior regression are usually higher than the Data and Posterior functions. According to the plot, a change in the water cement ratio has the greatest effect on compressive strength. Increasing the super plasticizer dosage has little effect on predicted strength. In fact, the compressive strength for the Prior regression decreases because its coefficient has the wrong sign.
The Posterior model predicts an
increase of only 19 MPa if silica fume concrete is used.
This is definitely wrong. Research has shown that it is
the silica fume content that increases strength by 30 to
100 MPa. The Data model for the silica fume variable
shows a minimum value for 0% silica fume content. Recall
that the Data input NEWDATA dealt only with silica fume
concrete at a cement content of 8 %. A DATA regression
equation for any other silica fume content is
meaningless. For the fine to total aggregate ratio
variable, the Prior and Posterior functions predict that
the compressive strength will increase with a larger
ratio, whereas conventional wisdom (and the Data
regression) predict otherwise. Recommendations for Model 1 After analyzing the model, the
following observations and recommendations were made by
DWST staff:
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