A measure of performance of self compacting
concrete mixtures
Sandra Nunes1 , Helena Figueiras1 , Paula Milheiro-Oliveira2 , Joana
Sousa-Coutinho1 , and Joaquim Figueiras1
1
2
Universidade do Porto, Faculdade de Engenharia, LABEST, P-4200-465 Porto,
Portugal snunes@fe.up.pt
Universidade do Porto, Faculdade de Engenharia, CEC, P-4200-465 Porto,
Portugal poliv@fe.up.pt
Summary. A measure of performance of self compacting concrete mixture in terms
of engineering robustness is proposed. This measure is based on the probability that
a mixture prepared in view of a specified target remains in a given neighbourhood of
that target. An application to a particular case of a self compacting concrete mixture
desired by a precast factory is presented. The study involves previously a factorial
design of experiments in a laboratory, the establishment of a statistical model for
the response variables and finally the computation of the performance measure. The
engineering robustness of the mixture is estimated in this particular case by means
of bootstrap resampling and a bootstrap interval for the estimate is given.
Key words: bootstrap resampling, factorial experiment design, regression models,
application in construction engineering.
1 Introduction
A robust self compacting concrete (SCC) mixture is an innovative technology in
the concrete industry and its success depends in part on the possibility of attaining
a target mixture in industrial production and on the possibility of comparing the
engineering robustness of different SCC mixes, as a criterion for decision making.
The performance of this new material is known to depend on several factors,
regarding mainly the properties of constituent materials that are used to prepare the
mixes and their relative proportions. Existing standards are not enough to ensure
a good performance of the material and, in practice, even if one is able to set a
target mixture, this target may not be attained due to random variability in the
constituent materials that occurs in current production. In engineering terms robust
mixes are desired, in the sense that they stay very close to the target response
properties eventhough in presence of random variations in the properties of the
constituent materials that are inherent to the production process. Thus a measure
of the engineering robustness of the mixture is needed. We will call it a measure of
performance of the SCC mixture to avoid confusion with the concept of statistical
1080
S. Nunes et al.
robustness. In this paper we propose a measure of performance of SCC mixtures
that translates the engineering concept discussed above and we explain how this
quantity can be estimated in practice. A case study regarding the production in a
Portuguese precast concrete factory is used to illustrate the method.
Since this measure of performance requires a model of the response of the SCC
mixes that did not exist to the date, a sequence of steps had to be carried out before
the measure could be computed in this case. In a first step, a central composite design
was carried out to analyse the influence of 5 mixture variables and their coupled
effects on deformability, passing and filling abilities and compressive strength of SCC
mixtures. The target SCC mix composition was selected to be the central point in
the factorial design. In a second step, regression models for the 5 response variables
were established. In a third step, data collected in the precast factory regarding the
constituent materials used in current production during the last year were analysed
in view of a statistical characterisation of the presence of these constituent materials
in production. Finally the derived models were used to estimate SCC properties while
random variations in the mixture variables were resampled from the data collected
in the factory. Bootstrap resampling was applied in order to produce samples of
the variables concerning the constituent materials, samples of the corresponding
response variables and samples of the frequency of rejection of the proximity of the
target. The SCC mix performance was estimated by the mean frequency of nonrejection of the proximity of the target. A bootstrap interval was computed for this
estimate.
2 Set up of the measure of performance and its
computation
Based on [EFNARC05] the definition of engineering robustness of SCC mixtures
can be set as the capacity of concrete to retain its performance requirements (fresh
and hardened properties, including durability) when small random variations in the
properties or quantities of the constituent materials occur.
In reality, a large number of independent variables influence concrete response
properties and a considerable amount of these properties must be assessed to verify
the fulfilment of engineering criteria. In the present study, following engineering
principles and standards on concrete production [OOO00], 5 independent variables
were considered, namely, X1 = V w/V p, X2 = wf /wc, X3 = Sp/p, X4 = s1/s and
X5 = V ap, where V w is the water volume, V p is the powder volume, wf is the
filler weight, wc is the cement weight, Sp is the superplasticizer weight, p is the
powder weight, s1 is the sand volume, s is the mortar volume and V ap is the solid
volume of the mix. They represent quantities and proportions of the constituent
materials that are used in the mixes. Response models relate Y1 = Df low, Y2 =
T 50, Y3 = T f unnel, Y4 = H and Y5 = fc,28 to the independent variables. The
response variables represent, respectively, final slump flow diameter, time necessary
for concrete to reach a 50cm diameter, flow time, filling height and compressive
strength at 28 days concrete age, resulting from Slump-Flow test, V-funnel test,
Box test and Compression test [EFNARC05].
i
i
Acceptance limits (Rinf
and Rsup
) for each response variable (Yi , i = 1, · · · , 5)
are established as well as an acceptance global criteria involving all 5 variables:
A measure of performance of self compacting concrete mixtures
T5 1081
i
i
Rinf
< Yi < Rsup
. Random fluctuations affect V w, V p, wf , wc, Sp, p, s1,
s and V ap in industrial production, leading us to consider these as independent
random variables. Data collected in the scope of the study case presented here (see
Section 3) supports this assertion. Engineering robustness of a SCC mixture as
defined on top of this section can then be interpreted in probabilistic terms as the
probability that the global acceptance criteria is fulfilled:
i=1
pr = P
5 n
\
i
Rinf
< Yi <
i
Rsup
o!
.
(1)
i=1
Contributions of each response variable, Yi , to engineering robustness of a SCC
mixture can as well be computed by the probability that the particular acceptance
limits regarding this response variable occur:
i
i
pri = P Rinf
< Yi < Rsup
, i = 1, · · · , 5.
(2)
Comparing values of these contributions is important, in practice, since they tell us
which of the responses are more sensitive or less sensitive in the production process.
Naturally, if the probability distributions of V w, V p, wf , wc, Sp, p, s1, s and
V ap, or of variables Xi , i = 1, · · · , 5 are known and the response model is known one
can theoretically compute the probabilities defined in (1) and (2). The computation
being quite hard, as the response model is generally of quadratic form and it is in fact
a regression model, a better approach would be to use Monte Carlo simulation. Our
view is that a even better approach is given by resampling methods. In practice, data
on the variables V w, V p, wf , wc, Sp, p, s1, s and V ap for a few target concrete
compositions is usually collected in the precast factory for quite long periods of
time. With these data one can produce samples and Bootstrap replicates of the
desired quantities. As a consequence estimation of the probabilities defined in (1)
and (2) and computation of Bootstrap percentile intervals demands a considerable
computational effort but presents no real difficulty.
3 Case study
An experimental study was carried out at LABEST laboratory, at the University
of Porto, in collaboration with a precast factory in the region. The goal of the
study is to quantify engineering robustness of a specified SCC mixture when applied
in industrial production. Our proposal is to use the probability defined in (1) to
measure the engineering robustness of the SCC mixture. As mentioned before, since
no response models can be found in the literature that describe key SCC properties
in this case, our first task will be to establish such models.
A 25−1 fractional factorial statistical design, corresponding to five parameters at
two levels, augmented with 10 axial runs (at levels ±2) plus 4 central runs (see [M01])
was used. This design was chosen because its central point corresponds to a SCC
mixture optimized in a previous study, the analysed region being relatively close to
the optimum. As a result a curvature should be expected in the response surface
(see [S01,S04] for similar studies regarding other types of concrete). Moulding, follow
up and tests on all SCC specimens in this experimental plan were carried out in the
laboratory.
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S. Nunes et al.
Quadratic regression models with interactions were derived for the dependence
between the response variables (Yi , i = 1, · · · , 5) and the variables Xi , i = 1, · · · , 5.
√
A transformation of the form 1/ y was introduced for the variable Y3 . Although
the fact that this transformation has no engineering meaning, it became necessary
in order to eliminate heteroscedasticity of the residuals. Model parameters and a
summary of the results of the analysis is presented in Table 1.
Response variables
Dflow (mm)
T50 (sec.)
Constant
650,53
58,67
2,936
-1,291
8,21
23,5
NS
-8,46
NS
10,44
-6,95
NS
[Tfunnel (sec.)]-0.5
H (mm)
fc,28 (MPa)
0,304
0,054
333,36
8,12
63,00
-3,29
-0,110
-0,355
-0,035
0,220
0,344
-0,451
0,316
NS
0,006
NS
0,004
NS
-0,009
NS
NS
NS
NS
2,79
NS
-3,54
NS
3,44
-3,95
NS
-1,11
1,25
1,26
-0,67
NS
-1,24
NS
-1,46
0
13,188
0
0,468
0
0,015
0
5,023
0
2,248
0,95
0,86
0,92
0,79
0,78
Parameter estimates
Vw/Vp
wf/wc
Sp/p
s1/s
Vap
(wf/wc)×(s1/s)
(Sp/p)×(Vap)
(Vw/Vp)2
(Vap)2
Residual error, ε
mean
standard deviation
R2
(NS) non-significant ( =5%)
α
Table 1. Estimated regression models for the response variables Df low, T 50,
[T f unnel]−0.5 , H and fc,28
Since this study was carried out prior to the industrial application of the particular mixes that constitute the factorial design at the precast factory there was
no available data of daily fluctuations concerning this SCC production. Therefore,
records of target and measured weights of each constituent material used in the
production of other concrete were used to derive records of absolute fluctuations of
these weights that characterise SCC production. The data collection was controlled
to ensure that data concerned the same constituent materials that were used in the
experimental factorial plan. A total of 132 observations were collected (see Fig. 1).
For the superplasticizer, no deviations were observed. Correlations found in the data
were non significant (Pearson’s correlation coefficients below 0,101, p > 20%).
These data was used to produce a sample of size 132 for the constituent materials
of a SCC mix (variables V w, V p, wf , wc, Sp, p, s1, s and V ap). This sample was
replicated producing 2000 Bootstrap samples of size 100 (see [ET93], for instance),
which is equivalent to 200000 replicates of size 1, and 2000 Bootstrap samples of
size 132. The regression models previously estimated and presented in Table 1 were
used to derive the corresponding Bootstrap samples of the responses Df low, T 50,
[T f unnel]−0.5 , H and fc,28 , including the error term in the regression model. Estimates of the probabilities defined in (1) and (2) can be given by the mean frequency
of occurrence of the correspondent events in the 2000 replications. The estimated
value of pr represents an estimate of the proposed measure of performance of the
A measure of performance of self compacting concrete mixtures
1083
relative deviation to target
3%
2%
1%
0%
-1%
cement
limestone filler
free water
total aggregates
-2%
-3%
0
6
12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132
case number
Fig. 1. Relative fluctuations of the weights of the constituent materials used in
current concrete production
mix (see Table 2). Estimates of pri , i = 1, · · · , 5 represent contributions of the different responses to the performance of the mix. The precision of these estimates
can be quantified by looking at the 95% Bootstrap intervals. Table 2 also shows the
estimates computed in the original sample as well as the correspondent asymptotic
95% confidance intervals. Estimates obtained from generating a large number (2000)
of Bootstrap samples of size 1 is also included in this Table. Fig. 2 illustrates the
sample of 2000 frequencies that were generated to estimate pr on samples of size
100. Other sample sizes close to 100 were also experimented with similar results.
According to the results the engineering robustness of the SCC mix under study
is estimated to be of 74% and the highest contribution to engineering robustness
of this SCC mix comes from the variable H, followed by variables Df low and T 50.
The variable exhibiting the most sensitivity to random fluctuations in the constituent
materials is fc,28 .
4 Comments
The measure of performance of SCC mixtures proposed here involves the computation of the probability of occurrence of an event that represents the conformity to
a criterion. We have shown how, in practice, using data on the constituent material
fluctuations inherent to the production process, variations of SCC target mix composition can be replicated, the estimated response models can be applied to estimate
SCC concrete properties and a measure of SCC performance can be estimated. The
accuracy of this estimate can also be assessed.
To increase the engineering robustness of a mix it would be necessary to reduce
constituent materials variations through more quality control, modernization of existing equipments, etc. In general, these improvements are difficult to implement and
very expensive so a more robust mixture is preferred. The methodology presented in
this work is particularly useful to evaluate and compare the engineering robustness
of different SCC mixtures and to assist the concrete producer in selecting mixes.
1084
S. Nunes et al.
Estimate
original
sample
95% confidance
interval
2000
Boostrap
Estimate
samples of
size 1
Estimate
2000
Boostrap
samples of 95%-percentile
size 100
interval
Estimate
2000
Boostrap
samples of 95%-percentile
size 132
interval
pr
pr1
pr2
pr3
pr4
pr5
77.3%
97.7%
96.2%
93.2%
100%
89.4%
[70%, 84%]
[95%, 100%]
[93%, 100%]
[89%,99%]
---
[84%,95%]
74.5%
96.2%
96.2%
92.1%
100%
87.7%
73.9%
96.5%
96.4%
91.8%
100%
86.4%
[65%, 82%]
[93%, 99%]
[93%, 99%]
[86%,97%]
---
[79%,93%]
74.1%
97.0%
96.7%
90.4%
100%
87.4%
[67%, 82%]
[94%, 99%]
[93%, 99%]
[85%,95%]
---
[82%,92%]
Table 2. Estimates of the engineering robustness of the SCC mix under study
250
Frequency
200
150
100
50
0
0,60
0,65
0,70
0,75
0,80
0,85
0,90
Fig. 2. Histogram of the 2000 replications of the frequency associated with pr
A measure of performance of self compacting concrete mixtures
1085
Acknowledgements
The authors are very grateful to the anonymous referees for their pertinent comments.
References
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[OOO00]
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