The aim of this research is to analyse the reliability of the existing methods, and find new ones, for assessing brick resistance to freeze-thaw cycles. A series of bricks were tested against a range of properties; compressive strength ratios pre- to post-freezing and Maage’s factor, were calculated. Using a database created in this way, an analysis of existing classifiers was carried out and new ones were established based on which bricks could be classified into resistant and non-resistant to freeze-thaw cycles. The median pore radius, the ratio of compressive strengths pre- to post-freezing and the water desorption coefficient at 180-360 minutes proved to be good classifiers with a clearly specified cut-off for the distinction between resistant and non-resistant bricks with an acceptable risk of a wrong decision. The ratio of compressive strengths pre to post freezing and the water desorption coefficient at 180-360 minutes were described using the pore system in the brick.
El objetivo de esta investigación es analizar la fiabilidad de los métodos existentes para evaluar la resistencia de los ladrillos a los ciclos de hielo-deshielo, así como encontrar nuevos métodos. Se analizaron las propiedades de distintos ladrillos; se calculó la variación de la resistencia a compresión antes y después de la congelación, así como el factor de Maage. A partir de una base de datos creada para talfin, se llevó a cabo un análisis de los criterios de clasificación existentes, estableciéndose nuevos criterios para clasificar a los ladrillos entre resistentes y no resistentes a los ciclos de hielo-deshielo. El radio medio de los poros, la relación de la resistencia a compresión antes y después de la congelación y el coeficiente de desorción de agua a 180-360 minutos, demostraron ser útiles para realizar dicha clasificación, mostrando un riesgo aceptable de error. La relación de la resistencia a compresión antes y después de la congelación y el coeficiente de desorción de agua a los 180-360 minutos se describieron utilizando el sistema de poros en el ladrillo.
Even though concrete is the most common material in construction, bricks/brick wall elements are still frequently used in constructing smaller buildings and restoring existing ones. As with any other material, bricks are susceptible to systematic breakdown when exposed to environmental conditions, which can pose a serious threat to the structure’s stability. Durability is, therefore, one of the main requirements set for bricks as a building material. According to European regulations, brick durability is considered through the initial salt determination according to EN 772-5:2003 (
The authors in (
The pore size in the brick’s material affects brick resistance to freeze-thaw cycles (
Brick resistance to freeze-thaw cycles is a neglected topic in newer literature sources. On the series of aforementioned brick properties that were also tested in the course of this study, the authors analysed the reliability of the existing methods and searched for new methods for assessing brick resistance to freeze-thaw cycles, the so-called reliable classifiers. Due to the aforementioned effect of the pore system on brick resistance to freeze-thaw cycles in literature sources, reliable brick classifiers according to freeze-thaw cycles identified in this study were described using the pore system.
The experimental part of this paper is divided into several units. Section 2.1 provides an overview of the methods used during the testing of brick properties given in this paper and Section 2.2 contains the results of brick features tested in such a way. Using the results of the tests of brick properties described in Section 2.2 as a database, Section 2.3 provides the descriptive statistics of these results. Section 2.4 deals with determining the potentials of variables/features for classification into resistant and non-resistant freeze-thaw cycles. Section 2.5 provides an overview of the procedure and results of the ROC analysis for each variable that proved to be a potential classifier in Section 2.4. Classifiers that proved to be reliable in Section 2.4 are the newly proposed methods for assessing brick resistance to freeze-thaw cycles in comparison with those proposed in literature sources and they were described in Section 2.6 using the pore system of the brick.
A series of properties were tested on a total of 16 different brick types (series); 8 brick series originated from controlled production (S1R1030-1.5h; S1R1030-0.5h; S2R1060-1.5h; S2R1060-0.5h; S1S10300-1.5h; S1S1030-0.5h; S2S1060-1.5h; S2S1060-0.5h) and 8 brick series originated from uncontrolled production (S1-S8). Bricks originated from controlled production were bricks produced in local factories under controlled conditions from raw materials whose chemical and mineral composition are presented in (
Direct brick resistance to freeze-thaw cycles, compressive strength (before and after exposure to freeze-thaw cycles), water absorption, 5 h boiling water absorption, saturation coefficient, initial water absorption, pore distribution, median pore radius, total pore content and total pore volume were tested in all the bricks. Their compressive strength ratios pre- to post-freezing and Maage’s factor were calculated.
Direct brick resistance to freeze-thaw cycles was determined according to the HRN B.D8.011 standard (
Furthermore, water absorption and desorption were measured in each brick series for a specific time. For the purpose of measuring water absorption, bricks were completely dried, after which they were submerged in water for 10, 20, 30, 40, 50, 100, 150 and 1440 minutes. After taking each brick out of the water, and before it was weighed, the superficial moisture, i.e. water film on the brick’s surface, was eliminated with a cloth. The amount of the absorbed water at the specific moment is given as a percentage for every brick. For the purpose of measuring water desorption, bricks were submerged in water for 24 hours. After 24 hours, they were taken out of the water, their surface was wiped with a dry cloth and they were weighed and put into a drying oven at 105 °C. At 0, 180, 360, 540, 720, 900, 1260, and 1440 minutes, the bricks were taken out of the drying oven, weighed, and their remaining water content percentage was calculated.
Water absorption and desorption curves at a specific time are shown in
a) water absorption curves; b) water desorption curves
a) water absorption curves; b) water desorption curves
These coefficients were also researched as parameters that describe brick resistance to freeze-thaw cycles.
The testing results given in Section 2.1 and the absorption and desorption coefficients at a given moment are provided in
Sample identification/Tested property | S1R1030-1.5h | S1R1030-0.5h | S2R1060-1.5h | S2R1060-0.5h | S1S10300-1.5h | S1S1030-0.5h | S2S1060-1.5h | S2S1060-0.5h |
---|---|---|---|---|---|---|---|---|
Direct resistance to freeze-thaw cycles | resistant | resistant | resistant | resistant | resistant | resistant | resistant | resistant |
Normalised compressive strength (N/mm^{2}) | 22.5 | 20.3 | 26.3 | 24.8 | 27 | 24.8 | 42 | 35.4 |
Compressive strength (N/mm^{2}) | 30.0 | 27.0 | 35.0 | 33.0 | 36.0 | 33.0 | 56.0 | 47.0 |
Compressive strength after freezing (N/mm^{2}) | 22.0 | 19.5 | 30.0 | 28.0 | 29.0 | 26.0 | 43.0 | 35.0 |
Ratio of compressive strengths pre to post freezing | 0.73 | 0.72 | 0.86 | 0.85 | 0.81 | 0.79 | 0.77 | 0.75 |
Water absorption (%) | 11.9 | 12.3 | 11.2 | 11.4 | 11.2 | 11.8 | 10.0 | 10.4 |
5 h boiling water absorption (%) | 16.3 | 16.9 | 13.0 | 13.4 | 13.7 | 14.7 | 12.9 | 13.6 |
Saturation coefficient | 0.69 | 0.76 | 0.73 | 0.75 | 0.74 | 0.75 | 0.83 | 0.84 |
Initial absorption coefficient [kg/(m^{2} x min)] | 1.1 | 2.0 | 1.0 | 1.8 | 2.5 | 2.8 | 2.5 | 2.5 |
Water absorption coefficient in 10 minutes (%/min) | 1.0570 | 1.1525 | 0.6350 | 0.7915 | 1.0336 | 1.0800 | 0.8680 | 0.8370 |
Water absorption coefficient in 10-20 minutes (%/min) | 0.0880 | 0.0544 | 0.2770 | 0.2230 | 0.0409 | 0.0288 | 0.0440 | 0.0975 |
Water absorption coefficient in 20-30 minutes (%/min) | 0.0100 | 0.0075 | 0.0820 | 0.0500 | 0.0036 | 0.0263 | 0.0080 | 0.0075 |
Water desorption coefficient in 180 minutes (%/min) | 0.0387 | 0.0349 | 0.0411 | 0.0391 | 0.0395 | 0.0409 | 0.0362 | 0.0376 |
Water desorption coefficient in 180-360 minutes (%/min) | 0.0126 | 0.0128 | 0.0109 | 0.0135 | 0.0153 | 0.0152 | 0.0117 | 0.0121 |
Water desorption coefficient in 360-540 minutes (%/min) | 0.0083 | 0.0108 | 0.0065 | 0.0081 | 0.0037 | 0.0062 | 0.0049 | 0.0033 |
Sample identification/Tested property | S1R1030-1.5h | S1R1030-0.5h | S2R1060-1.5h | S2R1060-0.5h | S1S10300-1.5h | S1S1030-0.5h | S2S1060-1.5h | S2S1060-0.5h | |
---|---|---|---|---|---|---|---|---|---|
Proportion of pores of a given size (%) | Large pores | 20.6 | 11.2 | 29.4 | 15.9 | 17.3 | 15.8 | 17.8 | 2.2 |
Medium pores | 76.2 | 84.8 | 65.9 | 78.7 | 66.0 | 72.8 | 74.3 | 86.4 | |
Small pores | 3.2 | 4.0 | 4.7 | 5.4 | 16.7 | 11.4 | 7.9 | 11.4 | |
Total pore content (%) | 19.4 | 20.4 | 17.2 | 17.8 | 16.2 | 19.1 | 22.2 | 28.3 | |
Median pore radius (µm) | 1.33 | 1.31 | 1.67 | 1.66 | 1.56 | 1.47 | 1.43 | 1.40 | |
Total pore volume (cm^{3}/g) | 0.0961 | 0.1131 | 0.1051 | 0.1093 | 0.0891 | 0.1075 | 0.0964 | 0.1471 | |
Maage’s durability factor | 83 | 55 | 100 | 69 | 77 | 68 | 76 | 27 |
Sample identification/Tested property | S1 | S2 | S3 | S4 | S5 | S6 | S7 | S8 |
---|---|---|---|---|---|---|---|---|
Direct resistance to freeze-thaw cycles | non-resistant | non-resistant | non-resistant | non-resistant | non-resistant | non-resistant | resistant | non-resistant |
Normalised compressive strength (N/mm^{2}) | 17.7 | 8.0 | 28.7 | 28.4 | 15.2 | 27.9 | 27.7 | 27.8 |
Compressive strength (N/mm^{2}) | 23.6 | 10.7 | 38.2 | 37.9 | 20.3 | 37.2 | 36.9 | 37.1 |
Compressive strength after freezing (N/mm^{2}) | 16.1 | 7.5 | 27.2 | 26.2 | 14.6 | 25.2 | 32.8 | 25.5 |
Ratio of compressive strengths pre to post freezing | 0.68 | 0.70 | 0.71 | 0.69 | 0.72 | 0.68 | 0.89 | 0.69 |
Water absorption (%) | 18.2 | 24.2 | 14.0 | 14.7 | 13.4 | 13.6 | 12.6 | 14.1 |
5 h boiling water absorption (%) | 23.6 | 31.5 | 18.6 | 19.6 | 17.8 | 17.9 | 13.6 | 20 |
Saturation coefficient | 0.77 | 0.77 | 0.75 | 0.75 | 0.75 | 0.76 | 0.93 | 0.74 |
Initial absorption coefficient [kg/(m^{2} x min)] | 2.6 | 6.3 | 1.4 | 1.5 | 1.3 | 1.5 | 2.7 | 1.6 |
Water absorption coefficient in 10 minutes (%/min) | 1.3410 | 2.3550 | 0.7560 | 0.7750 | 0.6980 | 0.8190 | 1.1270 | 0.6570 |
Water absorption coefficient in 10-20 minutes (%/min) | 0.2910 | 0.0233 | 0.3504 | 0.2850 | 0.2400 | 0.2860 | 0.0460 | 0.2790 |
Water absorption coefficient in 20-30 minutes (%/min) | 0.0580 | 0.0039 | 0.1504 | 0.1220 | 0.1050 | 0.1260 | 0.0140 | 0.1060 |
Water desorption coefficient in 180 minutes (%/min) | 0.0563 | 0.0686 | 0.0436 | 0.0439 | 0.0437 | 0.0325 | 0.0377 | 0.0359 |
Water desorption coefficient in 180-360 minutes (%/min) | 0.0226 | 0.0230 | 0.0254 | 0.0218 | 0.0206 | 0.0283 | 0.0169 | 0.0285 |
Water desorption coefficient in 360-540 minutes (%/min) | 0.0104 | 0.0240 | 0.0071 | 0.0108 | 0.0076 | 0.0033 | 0.0136 | 0.061 |
Sample identification/Tested property | S1 | S2 | S3 | S4 | S5 | S6 | S7 | S8 | |
---|---|---|---|---|---|---|---|---|---|
Proportion of pores of a given size (%) | Large pores | 4.2 | 15.4 | 3.8 | 2.8 | 13.1 | 3.9 | 56.1 | 2.4 |
Medium pores | 85.9 | 83.9 | 71.9 | 70.4 | 82.4 | 67.6 | 43.2 | 68.1 | |
Small pores | 9.9 | 0.7 | 24.3 | 26.8 | 4.5 | 28.5 | 0.7 | 29.5 | |
Total pore content (%) | 37.7 | 46.1 | 30.1 | 30.8 | 28.9 | 33.0 | 34.4 | 32.3 | |
Median pore radius (µm) | 0.25 | 1.01 | 0.26 | 0.08 | 0.51 | 0.09 | 1.45 | 0.05 | |
Total pore volume (cm^{3}/g) | 0.2325 | 0.3705 | 0.1685 | 0.1712 | 0.1534 | 0.1832 | 0.2075 | 0.8133 | |
Maage’s durability factor | 24 | 41 | 42 | 28 | 53 | 19 | 139 | 24 |
Property/Numerical features | Mean | Median | Std. dev. | Min | Max | |||||
---|---|---|---|---|---|---|---|---|---|---|
Resistance according to HRN B.D8.011 | NO | YES | NO | YES | NO | YES | NO | YES | NO | YES |
Normalized compressive strength (N/mm^{2}) | 22 | 27.8 | 27.8 | 26.3 | 8.3 | 6.8 | 8 | 20.3 | 28.7 | 42 |
Saturation coefficient | 0.754 | 0.777 | 0.75 | 0.75 | 0.0099 | 0.0743 | 0.74 | 0.69 | 0.77 | 0.93 |
Ratio of compressive strengths pre- to post-freezing | 0.696 | 0.797 | 0.69 | 0.79 | 0.0151 | 0.0602 | 0.68 | 0.72 | 0.72 | 0.89 |
Large pore content (%) | 6.51 | 20.7 | 3.9 | 17.3 | 5.36 | 15.1 | 2.4 | 2.2 | 15.4 | 56.1 |
Medium-sized pore content (%) | 75.7 | 72 | 71.9 | 74.3 | 7.98 | 13 | 67.6 | 43.2 | 85.9 | 86.4 |
Small pore content (%) | 17.7 | 7.27 | 24.3 | 5.4 | 12.3 | 5.05 | 0.7 | 0.7 | 29.5 | 16.7 |
Total pore content (%) | 34.1 | 21.7 | 32.3 | 19.4 | 6 | 5.97 | 28.9 | 16.2 | 46.1 | 34.4 |
Median pore radius (µm) | 0.321 | 1.48 | 0.25 | 1.45 | 0.342 | 0.13 | 0.05 | 1.31 | 1.01 | 1.67 |
Total pore volume (mm^{3}/g) | 299 | 119 | 183 | 108 | 239 | 37.1 | 153 | 89.1 | 813 | 208 |
Maage’s coefficient | 33 | 77.1 | 28 | 76 | 12.4 | 27.2 | 19 | 27 | 53 | 139 |
Water absorption (%) | 16 | 11.4 | 14.1 | 11.4 | 3.95 | 0.84 | 13.4 | 10 | 24.2 | 12.6 |
Initial water absorption coefficient (kg/m^{2} * min) | 2.28 | 2.1 | 1.45 | 2.47 | 1.81 | 0.651 | 1.25 | 1.03 | 6.25 | 2.77 |
Absorption coefficient in 0-10 (%/min) | 1.06 | 0.954 | 0.78 | 1.03 | 0.617 | 0.177 | 0.66 | 0.64 | 2.35 | 1.15 |
Absorption coefficient in 10-20 (%/min) | 0.251 | 0.1 | 0.29 | 0.05 | 0.105 | 0.0889 | 0.02 | 0.03 | 0.35 | 0.28 |
Absorption coefficient in 20-30 (%/min) | 0.0959 | 0.0233 | 0.11 | 0.01 | 0.0493 | 0.0262 | 0 | 0 | 0.15 | 0.08 |
Desorption coefficient in 0-180 (%/min) | 0.0466 | 0.0386 | 0.04 | 0.04 | 0.0123 | 0.0021 | 0.03 | 0.04 | 0.07 | 0.04 |
Desorption coefficient in 180-360 (%/min) | 0.0244 | 0.0136 | 0.02 | 0.01 | 0.0031 | 0.0019 | 0.02 | 0.01 | 0.03 | 0.02 |
Desorption coefficient in 360-540 (%/min) | 0.0099 | 0.0072 | 0.01 | 0.01 | 0.0068 | 0.0035 | 0 | 0 | 0.02 | 0.01 |
In the course of choosing good classifiers for determining bricks resistant to freeze-thaw cycles, the Mann-Whitney U test and ROC analysis were used (Receiver operating characteristics, cf. e.g. (
The Mann-Whitney U test is employed with rank-order data in a hypothesis testing situation involving a design with two independent samples. If the result of the Mann-Whitney U test is significant, it indicates there is a significant difference between the two sample medians, and it can be concluded that the samples represent populations with different median values. A more-detailed description of this method can be found in (
The results of the Mann-Whitney U test, which tested the existence of a difference between variable distributions in resistant and non-resistant bricks, are given in
Property/variable | statistic | p-value |
---|---|---|
Water absorption | 0.00 | < .001 |
Median pore radius | 0.00 | < .001 |
Water desorption coefficient in 180-360 | 0.00 | < .001 |
Ratio of compressive strengths pre to post freezing | 0.500 | 0.001 |
Large pore content | 9.00 | 0.016 |
Total pore volume | 4.00 | 0.002 |
Water absorption coefficient in 20-30 | 9.50 | 0.022 |
Water absorption coefficient in 10-20 | 10.00 | 0.023 |
Total pore content | 5.00 | 0.003 |
Maage’s coefficient | 4.00 | 0.004 |
Water desorption coefficient in 0-180 | 16.50 | 0.123 |
Small pore content | 17.50 | 0.152 |
Water desorption coefficient in 360-540 | 24.00 | 0.455 |
Water absorption coefficient in 0-10 | 24.00 | 0.470 |
Initial absorption coefficient | 26.00 | 0.596 |
Normalized compressive strength | 28.00 | 0.758 |
Medium-sized pore content | 25.00 | 0.779 |
Saturation coefficient | 28.50 | 0.791 |
Since variables that have a p-value lower than 0.05 in
At this level, the following were observed:
Water absorption of the brick is a good classifier since all the resistant bricks in the sample have water absorption values between 10 and 12.6% and all non-resistant bricks have a higher value, i.e. between 13.4 and 24.2%.
The median pore radius separates resistant from non-resistant bricks well, in such a way that all non-resistant bricks in the sample have a radius between 0.052 and 1.01 µm, while resistant bricks have a larger radius, between 1.31 and 1.67 µm.
The water desorption coefficient in 180-360 min proved to be a good classifier since all the values of this variable in non-resistant bricks are higher than those in resistant bricks-non-resistant bricks have values between 0.021 and 0.029 %/min and resistant bricks have values between 0.011 and 0.017 %/min. This result is contrary to the expectations presented in (
The compressive strength ratio pre- to post-freezing is a good classifier in the sense that all non-resistant bricks have a ratio between 0.68 and 0.72 and resistant bricks have a higher ratio, between 0.72 and 0.89.
Maage’s coefficient was confirmed as a good classifier and, on average, its value is higher in resistant than in non-resistant bricks. However, in this study, the cut-off value between resistant and non-resistant bricks for this coefficient was not clearly expressed.
Large pore content is higher on average in resistant than in non-resistant bricks.
The total pore volume and the total pore content are, on average, lower in resistant than in non-resistant bricks.
Desorption coefficients in 10-20 and 20-30 minutes are, on average, lower in resistant than in non-resistant bricks.
The other variables did not prove to be independently significant for brick classification into resistant or non-resistant classes in terms of freezing-thawing cycles.
As mentioned in the introductory part, American and Canadian regulations prescribe cut-off values for a set of parameters that have to be met by the brick to be regarded as resistant to freeze-thaw cycles due to harsh exposure conditions (
A ROC analysis was carried out for each variable that proved to be a potential classifier in
Property/variable | AUC |
---|---|
Water absorption | 1 |
Median pore radius | 1 |
Water desorption coefficient in 180-360 | 1 |
Ratio of compressive strengths pre- to post-freezing | 0.992 |
Total pore volume | 0.937 |
Maage’s coefficient | 0.937 |
Total pore content | 0.921 |
Large pore content | 0.857 |
Water absorption coefficient in 20-30 | 0.849 |
Water absorption coefficient in 10-20 | 0.841 |
According to the results shown in
To enable the use of these variables for brick classification into resistant and non-resistant, it is very important to define a cut-off for each chosen classifier and estimate the risk of a wrong decision. Due to the relatively small sample size, a Monte Carlo study was carried out for this purpose. The values of the selected variable from the estimated distributions were simulated separately for the resistant and non-resistant bricks, assuming normality for each.
Two-dimensional simulations were used in the sense that the expectation and the standard deviation of normal distributions were also simulated from the distributions of the parameter estimators. The simulations and analysis of the results were carried out using the R software package mc2d (
The study showed that the median pore radius is a reliable classifier. The cut-off that separates resistant from non-resistant bricks can be set to 1.2 µm. Bricks with a median pore radius lower than 1.2 µm can be classified as non-resistant and those with a median pore radius higher than 1.2 µm as resistant. The probability of a wrong decision is lower than 1% if the median pore radius is lower than 1.2 µm, and the probability of a wrong decision is lower than 10E-6 if the median pore radius is higher than 1.2 µm.
Water desorption coefficient in 180-360 minutes is also a reliable classifier. The separation cut-off can be set to 1.019 %/min. Bricks with a water desorption coefficient in 180-360 minutes lower than 1.019 %/min can be regarded as resistant to freeze-thaw cycles and bricks with higher values of this coefficient can be regarded as non-resistant. The probability of a wrong decision is lower than 3% if the coefficient is lower than 1.019 %/min, and the probability of a wrong decision is lower than 1% if the coefficient is higher than 1.019 %/min.
Monte Carlo simulations applied to the variable of compressive strength ratio pre- to post-freezing suggest a cut-off of 0.72 for classification into resistant and non-resistant bricks. For that cut-off, the probability of a wrong decision is lower than 5% if the brick, whose value of the ratio variable is greater than or equal to 0.72, is designated as resistant to freeze-thaw cycles and the probability of a wrong decision is lower than 9% if the brick, whose value of the ratio variable is lower than 0.72, is designated as non-resistant to freeze-thaw cycles.
Even though literature sources do not explicitly state it, in practice, it is usually believed that bricks with high water absorption are not resistant to freeze-thaw cycles. This study confirmed that, and it is clear that all resistant bricks in the sample have water absorption values between 10 and 12.6%, whereas all non-resistant bricks have higher values, between 13.4 and 24.2%. However, results that could clearly define a water absorption cut-off for separating resistant from non-resistant bricks with an acceptable risk of the wrong decision were not achieved. The standard deviation of water absorption in bricks that are non-resistant to freeze-thaw cycles is high, which, under the assumption of normality of water absorption distribution, allows for low values of water absorption in non-resistant bricks with a not so low probability. Therefore, based on this sample, a reliable cut-off cannot be set for the water absorption value with the aim of classifying bricks according to their resistance to freeze-thaw cycles.
The median pore radius proved to be an excellent classifier for assessing brick resistance to freeze-thaw cycles, which is in line with Franke and Bentrup’s results (
Since the pore system is regarded as responsible for brick resistance to freeze-thaw cycles in literature sources (
Spearman’s rank correlation coefficient is used here as a measure of association between two variables. It is based on an analysis of two sets of ranks and determines the degree to which a monotonic relationship exists between two variables. A more-detailed description of this method can be found in (
Property | Spearman’s rank correlation | p-value |
---|---|---|
Large pore content | 0.7354476 | 0.0011674 |
Medium-sized pore content | -0.3036116 | 0.2529648 |
Small pore content | -0.4369005 | 0.0906236 |
Total pore content | -0.6543863 | 0.0059524 |
Median pore radius | 0.9344164 | 1.18E-07 |
Total pore volume | -0.6234356 | 0.0098700 |
It can be seen here that there is a statistically significant increasing relationship between the compressive strength ratio and large pore content as well as between the compressive strength ratio and the median pore radius. A significant decreasing relationship was proven between the compressive strength ratio and total pore content and between the compressive strength ratio and total pore volume.
In the course of choosing a model that describes the compressive strength ratio using the pore system, classical methods of regression analysis were used. The aim was to establish a model that maximizes the adjusted R^{2}, minimizes the Akaike information criterion and exhibits a stable behaviour during the bootstrap method. The R software package car (
The best model achieved in the aforementioned sense is gained using the large pore content and median pore radius variables. The median pore radius does not enter the model linearly, but a piecewise linear function was used. The model coefficients are shown in
Regressor | Estimate | Std. error | p-value of t-test |
---|---|---|---|
Free member | 0.2970001 | 0.0604426 | 0.000461 |
Large pore content | 0.0024181 | 0.0003549 | 2.90e-05 |
Median pore radius | 0.3046612 | 0.0414089 | 1.43e-05 |
I (median radius ≤ 1.2) | 0.3878672 | 0.0610214 | 5.40e-05 |
median pore radius* I (median radius ≤ 1.2) | -0.3199488 | 0.0445659 | 1.80e-05 |
This model shows that the relationship between variables and compressive strength ratio is described in different ways for bricks with a median pore size of ≤1.2 μm than for bricks with a median pore size of >1.2 μm.
Namely,
for median pore radius ≤1.2 μm, the relationship can be described as:
compressive strength ratio =0.6849+0.0024*large pore content-0.0153*median pore radius
for median pore radius >1.2 µm, the relationship can be described as:
compressive strength ratio =0.297+0.0024*large pore content+0.3047*median pore radius
Even though the assumption of normality and homoscedasticity of the residual in the achieved model is supported (the Shapiro-Wilk test yields a p-value of 0.834, the Non-Constant Variance Score Test yields a p-value of 0.831), due to the relatively small data set, a bootstrap analysis of the suggested model was carried out as well, which shows that the key conclusions are stable.
Coefficient | Estimate | Percentile bootstrap confidence interval |
---|---|---|
Adjusted R^{2} | 0.9519899 | 0.9305, 0.9957 |
Intercept | 0.2970001 | 0.1374, 0.4485 |
Large pore content | 0.0024181 | 0.0001, 0.0030 |
Median pore radius | 0.3046612 | 0.2084, 0.4174 |
I (median pore radius <= 1.2) | 0.3878672 | 0.2279, 0.5472 |
Median pore radius* I (median pore radius <=1.2) | -0.3199488 | -0.4453, -0.1843 |
Based on the obtained model, it is clear that high values of the median pore radius (higher than 1.2μm) contribute much more to the description of the variable ratio than in the case of values lower than 1.2 μm. Spearman’s rank correlation coefficient for ratio and median pore radius, on the data subset for which median pore radius ≤ 1.2 μm, do not point towards the existence of a monotonic relationship (the p-value is 0.17) in that part. The aforementioned is shown in
If compressive strength ratio pre- to post-freezing is accepted as a quantitative indicator of brick resistance to freeze-thaw cycles, this model suggests that the median pore radius value, as long as it is lower than 1.2μm, does not significantly contribute to the resistance to freeze-thaw cycles but classifies bricks as non-resistant. As opposed to that, median pore radius values higher than 1.2 μm significantly contribute to resistance to freeze-thaw cycles. This fact is in line with the theory set forth in (
As the initial indicator of the existence of a monotonic relationship between the water desorption coefficient in 180-360 minutes to the variables in the pore system, the value of Spearman’s rank correlation and the pertaining p-value for testing the hypothesis on the non-existence of a monotonic relationship are shown in
Property | Spearman’s rank correlation | p-value |
---|---|---|
Large pore content | -0.510326 | 0.0434094 |
Medium-sized pore content | -0.153392 | 0.5705955 |
Small pore content | 0.443870 | 0.0850169 |
Total pore content | 0.707968 | 0.0021500 |
Median pore radius | -0.766965 | 0.0005266 |
Total pore volume | 0.777289 | 0.0003950 |
Here, it is noticeable that a statistically significant increasing relationship exists between the water desorption coefficient in 180-360 minutes and the total pore content as well as between the water desorption coefficient in 180-360 minutes and the total pore volume. A statistically significant decreasing relationship is proven between the water desorption coefficient in 180-360 minutes and large pore content and between water desorption coefficient in 180-360 minutes and median pore radius.
In the procedure of modelling the water desorption coefficient in 180-360 minutes using pore system variables, the only stable model relates the water desorption coefficient in 180-360 minutes with the classifier based on the median pore radius, i.e. with the indicator of the set (median pore radius <= 1.2 μm). Based on other variables of the pore system, it was not possible to extract a variable that would provide new information on the value of the water desorption coefficient in 180-360 minutes in addition to the one already included by the median pore radius. Since the indicator function of the set (median pore radius <= 1.2 μm) is also a classifier for resistant and non-resistant bricks, this model does not yield any significant new information that has not been presented in the previous chapters based on the analysis of the water desorption coefficient in 180-360 minutes as a classifier. Therefore, the only important conclusion about that coefficient based in the variables of the pore system is that it is on average 0.01356 (95% confidence interval (0.01211, 0.01500)) if the median pore radius is higher than 1.2 μm and 0.02443 (95% confidence interval (0.02161, 0.027246)) if the median pore radius is lower or equal to 1.2 μm.
During the search for new methods to classify bricks into resistant and non-resistant to freeze-thaw cycles, a variety of properties was tested on a series of bricks, including direct brick resistance to freeze-thaw cycles, compressive strength of bricks (before and after the brick’s exposure to freeze-thaw cycles), water absorption, 5 h boiling water absorption, saturation coefficient, initial water absorption, pore distribution, median pore radius, total pore content, total pore volume, compressive strength ratios pre- to post-freezing and Maage’s factor. Furthermore, water absorption and desorption were measured in each brick series for a specific time. The reliabilities of existing methods for classifying bricks into resistant and non-resistant to freeze-thaw cycles were analysed using a database generated in this way, and the compressive strength ratio pre- to post-freezing and a water desorption coefficient in 180-360 minutes were proposed as new qualitative indicators of brick resistance to freeze-thaw cycles.
The median pore radius and the proposed new measures, the water desorption coefficient in the period 180-360 min and the compressive strength ratios pre- to post-freezing, are highly reliable classifiers. For each of them, a cut-off was determined based on which the bricks could be classified as resistant and non-resistant. The risk of a wrong conclusion was also calculated using the proposed cut-off. For water absorption, which, expectedly, proved to be a good classifier into resistant and non-resistant to freeze-thaw cycles, no results were achieved based on this database that could clearly define the cut-off for separating resistant and non-resistant bricks with an acceptable risk of a wrong decision. Among all reliable classifiers, the water desorption coefficient in the period 180-360 min would stand out as the simplest and most profitable one.
A model for describing the compressive strength ratio pre- to post-freezing based on the pore system variables was also created. If compressive strengths pre- to post-freezing are accepted as a measure of brick resistance to freeze-thaw cycles, this model confirms that brick resistance in that sense can be very well characterized by the median pore radius and large pore content values.
In the effort to describe the water desorption coefficient in 180-360 minutes based on the pore system variables, it was confirmed that it is connected to these variables through its capacity to classify bricks into resistant and non-resistant to freeze-thaw cycles, i.e. the only statistically significant predictive variable in the model is the aforementioned classifier based on the median pore radius.
This research sheds light on the compressive strength ratio pre- and post-freezing as a reliable and until now completely unresearched classifier that classifies bricks into resistant and non-resistant to freeze-thaw cycles.
This work is supported by financial support from the Faculty of Civil Engineering and Architecture Osijek as well as by the Croatian Science Foundation through research grant IP-2016-06-6545.
Conceptualization: I. Netinger Grubeša, M. Benšić; Data curation: I. Netinger Grubeša, M. Benšić; Formal analysis: M. Benšić, M. Vračević; Funding acquisition: I. Netinger Grubeša; Investigation: I. Netinger Grubeša, M. Benšić, M. Vračević; Methodology: I. Netinger Grubeša, M. Benšić; Project administration: I. Netinger Grubeša; Resources: M. Benšić, M. Vračević; Software: M. Benšić; Supervision: I. Netinger Grubeša; Validation: I. Netinger Grubeša, M. Benšić; Visualization: I. Netinger Grubeša, M. Benšić; Roles/Writing, original draft: I. Netinger Grubeša, M. Benšić, M. Vračević; Writing, review & editing: I. Netinger Grubeša.