### 1. INTRODUCTION

⌅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 (11. EN 771-1:2011+A1:2015 Specification for masonry units - Part 1: Clay masonry units.

) and testing of resistance to freeze-thaw cycles according to EN 772-22 (22. EN 772-22:2018 Methods of test for masonry units - Part 22: Determination of freeze/thaw resistance of clay masonry units.

).
However, apart from the aforementioned direct method for testing brick
resistance to freeze-thaw cycles, international literature sources also
contain indirect procedures and cut-off/critical values per individual
procedure for assessing brick resistance to freeze-thaw cycles and they
prescribe the method to be followed in handling raw material in brick
production with the aim of achieving sufficient durability. American and
Canadian regulations prescribe cut-off values for a set of parameters
that the brick must meet to be regarded as resistant to freeze-thaw
cycles due to harsh conditions of exposure (33.
Straube, J.; Schumacher, C.; Mensinga, P. (2010) CP-1013: Assessing the
freeze-thaw resistance of clay brick for interior insulation retrofit
project. *Building Science Corporation*. Retrieved from https://www.buildingscience.com/documents/reports/rr-1013-freeze-thaw-resistance-clay-brick-interior-insulation-retrofits.

).
This set of parameters includes the minimal required compressive
strength, the maximum allowed water absorption, the maximum 5 h boiling
water absorption and the maximum allowed saturation coefficient, which
each individual brick from a group of five bricks must meet, as well as
the maximum/minimum median values of each listed parameter. Literature
sources provide the connection between each parameter contained in this
set and brick resistance to freeze-thaw cycles. The material’s
compressive strength is indirectly correlated to its resistance to
freeze-thaw cycles (44.
Mensinga, P. (2009) Determining the critical degree of saturation of
brick using frost dilatometry. A thesis at University of Waterloo.
Waterloo, Canada (2009). Retrieved from https://hdl.handle.net/10012/4638.

)
because stress occurs during the transition of water into ice and the
material must be able to resist it by its tensile strength, and a higher
tensile strength corresponds to a higher material compressive strength.
An increased water absorption signifies the presence of an increased
amount of water in the brick, and consequently, a higher tendency
towards damage when water freezes and turns into ice. During water
absorption testing under normal atmospheric pressure within 24 hours,
the easily accessible pores in the brick are filled. Therefore, 5 h
boiling water absorption was devised, during which the more difficult to
access pores in the brick are filled. The ratio of the amount of
absorbed water in the sample after submerging into water under normal
atmospheric pressure for 24 hours and the amount of absorbed water
during 5 h boiling water immersion is called the saturation coefficient
in the literature. The saturation coefficient determines the ratio of
pores that are easily filled with water and the total pore volume, and
it is an indicator of empty space in pore volume that remains after they
are filled with water, which can serve as an indicator of the water
volume generated by freezing. According to American and Canadian
regulations, a brick that does not meet the requirement set for the
aforementioned parameters can be subject to testing of direct resistance
to freeze/thaw cycles.

The authors in (55. Kung, J.H. (1985) Frost durability of canadian clay bricks. *Proceedings of the 7th International Brick Masonry Conference, Melbourne, Australia*.[1], 245-251. Retrieved from https://nrc-publications.canada.ca/eng/view/ft/?id=85a8f3ac-6e9e-43a4-aad1-51c6288e929b.

)
investigated the correlation between the saturation coefficient and the
initial water absorption and concluded that an increase in the
saturation coefficient proportionally increases the initial water
absorption, which means that the initial water absorption could be one
of the parameters that could serve for evaluating brick resistance to
freeze-thaw cycles.

The pore size in the brick’s material affects brick resistance to freeze-thaw cycles (55. Kung, J.H. (1985) Frost durability of canadian clay bricks. *Proceedings of the 7th International Brick Masonry Conference, Melbourne, Australia*.[1], 245-251. Retrieved from https://nrc-publications.canada.ca/eng/view/ft/?id=85a8f3ac-6e9e-43a4-aad1-51c6288e929b.

, 66. Hansen, W.; Kung, J.H. (1988) Pore structure and frost durability of clay bricks. *Mater Struct.* 21, 443-447. https://doi.org/10.1007/BF02472325.

).
According to literature sources, pores larger than 1 µm (large pores)
are easily filled with water and emptied of it, which leads to an
increase in the brick’s durability (55. Kung, J.H. (1985) Frost durability of canadian clay bricks. *Proceedings of the 7th International Brick Masonry Conference, Melbourne, Australia*.[1], 245-251. Retrieved from https://nrc-publications.canada.ca/eng/view/ft/?id=85a8f3ac-6e9e-43a4-aad1-51c6288e929b.

, 77.
Perrin, B.; Vu, N.A.; Multon, S.; Voland, T.; Ducroquetz, C. (2011)
Mechanical behaviour of fired clay materials subjected to freeze-thaw
cycles. *Constr Build Mater.* 25[2], 1056-1064. https://doi.org/10.1016/j.conbuildmat.2010.06.072.

). According to (55. Kung, J.H. (1985) Frost durability of canadian clay bricks. *Proceedings of the 7th International Brick Masonry Conference, Melbourne, Australia*.[1], 245-251. Retrieved from https://nrc-publications.canada.ca/eng/view/ft/?id=85a8f3ac-6e9e-43a4-aad1-51c6288e929b.

),
small pores (smaller than 0,1 µm) have a low impact on brick resistance
to freeze-thaw cycles because the water contained in them freezes only
at extremely low temperatures, and medium-sized pores are harmful.
Maage’s coefficient/factor (Fc) for predicting resistance to freeze-thaw
cycles, which is based on experimental results and a statistical model
with two main variables, total pore volume (P_{V}) and pore content of a specific diameter, i.e. pores larger than 3 µm (P3) (8-108.
Elert, K.; Culturone, G.; Rodríguez, C.; Pardo, E.S. (2003) Durability
of bricks used in the conservation of historic buildings-influence of
composition and microstructure. *J. Cult. Herit.* 4[2], 91-99. https://doi.org/10.1016/S1296-2074(03)00020-7.

9.
Bracka, A.; Rusin, Z. (2012) Comparison of pore characteristics and
water absorption in ceramics materials with frost resistance factor, Fc.
*Struc. Environ.* 3[4], 15-19. Retrieved from http://sae.tu.kielce.pl/12/S&E_NR_12_Art_3.pdf.

10.
Korenska, M.; Chobola, Z.; Sokolar R. (2006) Frequency inspection as an
assessment tool for the frost resistance of fired roof tiles. *Ceram. Silik.* 50[3], 185-192. Retrieved from https://www.irsm.cas.cz/materialy/cs_content/2006/Korenska_CS_2006_0000.pdf.

),
speaks in favour of the positive effect of large pores on brick
resistance to freeze-thaw cycles. In addition to Maage, an assessment of
brick resistance to freeze-thaw cycles by means of features of the pore
system in the material has been proposed by some other authors, such as
Koroth, Vincenzini, Franke and Bentrup, Litvan, and Nakamura (1111.
Koroth, S.R. (1997) Evaluation and improvement of frost durability of
clay bricks. A thesis in The Centre for Building Studies. Ottawa, Canada
(1997). Retrieved from https://spectrum.library.concordia.ca/282/.

, 1212.
Šveda, M.; Sokolar, R. (2004) The effect of firing temperature on the
irreversible expansion, water absorption and pore structure of a brick
body during freeze-thaw cycles. *Mater. Sci.* 19[4], 465-470. http://doi.org/10.5755/j01.ms.19.4.2741.

).
Most of these authors consider larger pores to have beneficial effects
on the resistance. Franke and Bentrup additionally introduced the median
pore radius as a parameter for assessing brick resistance to
freeze-thaw cycles (1111.
Koroth, S.R. (1997) Evaluation and improvement of frost durability of
clay bricks. A thesis in The Centre for Building Studies. Ottawa, Canada
(1997). Retrieved from https://spectrum.library.concordia.ca/282/.

, 1313.
Raimondo, M.; Dondi, M.; Ceroni, C.; Guarini, G. (2008) Durability of
clay roofing tiles: assessing the reliability of prediction models.
11DBMC International Conference on Durability of Building Materials and
Components. Istanbul, Turkey. Retrieved from https://www.irbnet.de/daten/iconda/CIB13184.pdf.

).

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.

### 2. EXPERIMENTAL PART

⌅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.

#### 2.1. Test methods

⌅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 (1414.
Netinger, I.; Vračević, M.; Ranogajec, J.; Vučetić, S. (2020) Influence
of pore-size distribution on the resistance of brick to freeze-thaw
cycles. *Materials*. 13 [10], 2364. https://doi.org/10.3390/ma13102364.

).
Bricks originated from uncontrolled production were bricks randomly
sampled from building material depots in Croatia, Bosnia and
Herzegovina, and Serbia as described in (1515.
Netinger, I.; Vračević, M.; Ducman, V.; Marković, B.; Szenti, I.;
Kukocz, A. (2020) Influence of the size and type of pores on brick
resistance to freeze-thaw cycles. *Materials*. 13 [17], 3717. https://doi.org/10.3390/ma13173717.

). The dimensions of all bricks were 250/120/65 mm.

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 (1616.
HRN B.D8.011:1987 Glinene opeke, blokovi i ploče - metode ispitivanja,
točka 9. Provjeravanje postojanosti prema mrazu. Croatian Standards
Institute. Zagreb, Croatia.

). This standard was chosen
for testing brick resistance to freeze-thaw cycles because it requires a
smaller number of testing samples in comparison to EN 772-22 (2), which
is usually used for testing brick resistance to freeze-thaw cycles.
According to this standard, the samples saturated with water were put
into a climate chamber and exposed to a temperature of -20±2°C for four
hours, after which they were submerged in water at between +15 and 20°C,
also for four hours. This cycle was repeated 25 times and samples were
checked after each cycle. A brick is considered resistant to freeze-thaw
cycles if none of the samples exhibit any signs of damage after 25
freeze-thaw cycles in water. The compressive strengths were determined
according to EN 772-1 (1717. EN 772-1:2011+A1:2015 Methods of test for masonry units - Part 1: Determination of compressive strength.

)
on a series of each type of brick before and after freezing, and the
mean values of compressive strength pre to post freezing were put into a
ratio to gain a coefficient as a quantitative indicator of brick
resistance to freeze-thaw cycles. The normalised compressive strength
was calculated according to Table A.1 of the EN 772-1 standard (1717. EN 772-1:2011+A1:2015 Methods of test for masonry units - Part 1: Determination of compressive strength.

). Water absorption was tested according to EN 772-21 (1818.
EN 772-21:2011 Methods of test for masonry units - Part 21:
Determination of water absorption of clay and calcium silicate masonry
units by cold water absorption.

), and the 5 h boiling water absorption was tested according to EN 772-7 (1919.
EN 772-7:1998 Methods of test for masonry units - Part 7: Determination
of water absorption of clay masonry damp proof course units by boiling
in water.

). The saturation coefficient is defined as
the ratio between water absorption and 5 h boiling water absorption. The
initial water absorption was determined according to EN 772-11 (2020.
EN 772-11:2011 Methods of test for masonry units -Part 11:
Determination of water absorption of aggregate concrete, autoclaved
aerated concrete, manufactured stone and natural stone masonry units due
to capillary action and the initial rate of water absorption of clay
masonry units.

). Pore distribution, median pore
radius, total pore content and total pore volume were determined by
Mercury Intrusion Porosimetry, and Maage’s factor was calculated
according to the description in (8-108.
Elert, K.; Culturone, G.; Rodríguez, C.; Pardo, E.S. (2003) Durability
of bricks used in the conservation of historic buildings-influence of
composition and microstructure. *J. Cult. Herit.* 4[2], 91-99. https://doi.org/10.1016/S1296-2074(03)00020-7.

9.
Bracka, A.; Rusin, Z. (2012) Comparison of pore characteristics and
water absorption in ceramics materials with frost resistance factor, Fc.
*Struc. Environ.* 3[4], 15-19. Retrieved from http://sae.tu.kielce.pl/12/S&E_NR_12_Art_3.pdf.

10.
Korenska, M.; Chobola, Z.; Sokolar R. (2006) Frequency inspection as an
assessment tool for the frost resistance of fired roof tiles. *Ceram. Silik.* 50[3], 185-192. Retrieved from https://www.irsm.cas.cz/materialy/cs_content/2006/Korenska_CS_2006_0000.pdf.

). Pore distribution for bricks from controlled production is shown in (1414.
Netinger, I.; Vračević, M.; Ranogajec, J.; Vučetić, S. (2020) Influence
of pore-size distribution on the resistance of brick to freeze-thaw
cycles. *Materials*. 13 [10], 2364. https://doi.org/10.3390/ma13102364.

) and pore distribution for bricks from uncontrolled production is shown in (1515.
Netinger, I.; Vračević, M.; Ducman, V.; Marković, B.; Szenti, I.;
Kukocz, A. (2020) Influence of the size and type of pores on brick
resistance to freeze-thaw cycles. *Materials*. 13 [17], 3717. https://doi.org/10.3390/ma13173717.

). Studies (1414.
Netinger, I.; Vračević, M.; Ranogajec, J.; Vučetić, S. (2020) Influence
of pore-size distribution on the resistance of brick to freeze-thaw
cycles. *Materials*. 13 [10], 2364. https://doi.org/10.3390/ma13102364.

, 1515.
Netinger, I.; Vračević, M.; Ducman, V.; Marković, B.; Szenti, I.;
Kukocz, A. (2020) Influence of the size and type of pores on brick
resistance to freeze-thaw cycles. *Materials*. 13 [17], 3717. https://doi.org/10.3390/ma13173717.

)
regarded pores larger than 3 µm as large pores, medium-sized pores
where those ranging from 0.1 to 3 µm and small pores are smaller than
0.1 µm.

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.

#### 2.2. Results of brick property testing

⌅Water absorption and desorption curves at a specific time are shown in Figure 1 for bricks from controlled production and Figure 2 for bricks from uncontrolled production. Every point of the curve is an average value of ten measuring results.

Figures 1 and 2 show that each brick has its water absorption and desorption trend at a time and that the biggest differences between brick series in the absorption curves exist for up to 30 minutes and in desorption curves for up to 540 minutes. Special attention was, therefore, given to water absorption rates at 10, 20 and 30 minutes and desorption rates at 180, 360 and 540 minutes and the following absorption [Equation 1] and desorption [Equation 2] coefficients were defined:

_{u,t0-t1}=(water absorption at t

_{1}- water absorption at t

_{0})/10 - water absorption coefficient [1]

_{o,t0-t1}=(water absorbed at t

_{1}- water absorbed at t

_{0})/180 - water desorption coefficient [2]

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 Table 1 for bricks from controlled production, and in Table 2
for bricks from uncontrolled production. Direct resistance to
freeze-thaw cycles was determined by observing the damages to the bricks
exposed to the freeze-thaw cycles on four sets of bricks (1 set is
comprised of 5 bricks), as prescribed by HRN B.D8.011. The compressive
strength pre to post freezing given here is the average value of ten
individual measured values. The ratio of compressive strengths pre- to
post-freezing was determined by putting the average values of
compressive strength pre- to post-freezing into a ratio. Water
absorption; 5 h boiling water absorption; saturation coefficient;
initial water absorption coefficient; water absorption coefficient at
10, 10-20 and 20-30 minutes, and water desorption at 180, 180-360 and
360-540 minutes shown in Tables 1 and 3
are mean values of the ten individual measured values. All the
parameters that were determined by Mercury Intrusion Porosimetry (large,
medium-sized, and small pore content, medium pore radius and total pore
volume) are the result of one measurement. The durability factor was
calculated according to Maage, as described in (8-108.
Elert, K.; Culturone, G.; Rodríguez, C.; Pardo, E.S. (2003) Durability
of bricks used in the conservation of historic buildings-influence of
composition and microstructure. *J. Cult. Herit.* 4[2], 91-99. https://doi.org/10.1016/S1296-2074(03)00020-7.

9.
Bracka, A.; Rusin, Z. (2012) Comparison of pore characteristics and
water absorption in ceramics materials with frost resistance factor, Fc.
*Struc. Environ.* 3[4], 15-19. Retrieved from http://sae.tu.kielce.pl/12/S&E_NR_12_Art_3.pdf.

10.
Korenska, M.; Chobola, Z.; Sokolar R. (2006) Frequency inspection as an
assessment tool for the frost resistance of fired roof tiles. *Ceram. Silik.* 50[3], 185-192. Retrieved from https://www.irsm.cas.cz/materialy/cs_content/2006/Korenska_CS_2006_0000.pdf.

).
Proportion of pores of a given size, total porosity, median pore radius
and total pore volume in the bricks as well as Maage’s factor are given
in Tables 2 and 4.

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 |

#### 2.3. Descriptive statistics

⌅Table 5 shows the descriptive statistics for all the tested parameters. The calculation for each parameter was made based on 7 mean values for non-resistant bricks and 9 mean values for resistant bricks.

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 |

#### 2.4 The potential of variables for discriminating freeze-thaw cycle resistant from non-resistant bricks

⌅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. (2121. Fawcett, T. (2006) An introduction to ROC analysis. *Pattern Recognit. Lett.* 27[8], 861-874. https://doi.org/10.1016/j.patrec.2005.10.010.

)).

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 (2222.
Sheskin, D.J. (2000) Handbook of parametric and nonparametric
statistical procedures. Second Edition, Chapman & Hall/CRC, United
States of America (2000). Retrieved from https://fmipa.umri.ac.id/wp-content/uploads/2016/03/David_J._Sheskin_David_Sheskin_Handbook_of_ParaBookFi.org_.pdf.

).
ROC analysis is a useful tool for classifier evaluation. It is based on
an ROC curve (receiver operating characteristic curve) which is a graph
showing the performance of a classification model at all classification
thresholds. This curve plots “True Positive Rate” versus “False
Positive Rate”. Area under the ROC curve (AUC) measures the entire area
underneath the entire ROC curve and is used as an aggregate measure of
performance across all possible classification thresholds. One way of
interpreting AUC is as the probability that the model ranks a random
positive example more highly than a random negative example. AUC ranges
in value from 0 to 1. A model whose predictions are 100% correct has an
AUC of 1. A more-detailed description of this method can be found in (2121. Fawcett, T. (2006) An introduction to ROC analysis. *Pattern Recognit. Lett.* 27[8], 861-874. https://doi.org/10.1016/j.patrec.2005.10.010.

).

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 Table 6.

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 Table 6 show the potential for classifying resistant and non-resistant bricks, the following variables can be included in the group of potentially good classifiers: median pore radius, water desorption coefficient at 180-360 minutes, compressive strength ratio pre- to post-freezing, large pore content, total pore volume, water absorption coefficient at 10-20 and 20-30 minutes, total pore content and Maage’s coefficient.

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 (2323. Vračević, M.; Ranogajec, J.; Vučetić S.; Netinger, I. (2014) Evaluation of brick resistance to freeze/thaw cycles according to indirect procedures.

*Građevinar*. 66[3], 197-209. https://doi.org/10.14256/JCE.956.2013.

), in which the authors presume that bricks with higher resistance to freeze-thaw cycles absorb and desorb water faster than bricks that have a lower resistance to freeze-thaw cycles due to the larger proportion of large pores in the bricks with higher resistance. It can be concluded that factors other than pore size, as presumed in (2323. Vračević, M.; Ranogajec, J.; Vučetić S.; Netinger, I. (2014) Evaluation of brick resistance to freeze/thaw cycles according to indirect procedures.*Građevinar*. 66[3], 197-209. https://doi.org/10.14256/JCE.956.2013.

), play a role in the brick’s water absorption and desorption rate.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 (33.
Straube, J.; Schumacher, C.; Mensinga, P. (2010) CP-1013: Assessing the
freeze-thaw resistance of clay brick for interior insulation retrofit
project. *Building Science Corporation*. Retrieved from https://www.buildingscience.com/documents/reports/rr-1013-freeze-thaw-resistance-clay-brick-interior-insulation-retrofits.

)
in the sense of the minimum required compressive strength, the maximum
allowed water absorption, maximum 5 h boiling water absorption, and the
maximum allowed saturation coefficient that the brick must meet. Three
out of four parameters were not proven statistically significant in this
research. Water absorption was proven as statistically significant, and
the upper threshold of this parameter at which a brick can be regarded
resistant to freeze-thaw cycles is 12.6%, whereas the upper threshold of
this parameter is 8% according to Canadian and American regulations.

#### 2.5. ROC analysis for potential classifiers

⌅A ROC analysis was carried out for each variable that proved to be a potential classifier in Table 6. AUC (area under the curve, cf. e.g. (2424.
Pouillot, R.; Delignette-Muller, M.L. (2010) Evaluating variability and
uncertainty in microbial quantitative risk assessment using two R
packages. *Int. J. Food Microbiol*. 142[3], 330-340. https://doi.org/10.1016/j.ijfoodmicro.2010.07.011.

)) values were calculated and are shown in Table 7. The calculations were carried out using the R software package ROCR.

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 Table 7, the same variables proved again to be the best classifiers (a very high AUC value), water absorption, median pore radius, desorption coefficient and the compressive strength ratio pre- to post-freezing.

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 (2525. Sing, T.; Sander, O.; Beerenwinkel, N.; Lengauer, T. (2005) ROCR: visualizing classifier performance in R. *Bioinformatics*. 21 [20], 3940-3941. https://doi.org/10.1093/bioinformatics/bti623.

).

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.

#### 2.6. Description of reliable classifiers for assessing brick resistance to freeze-thaw cycles by the pore system

⌅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 (2626.
Raimondo, M.; Ceroni, C.; Dondi, M.; Guarini, G.; Marsigli, M.;
Venturi, I.; Zanelli, C. (2009) Durability of clay roofing tiles: the
influence of microstructural and compositional variables. *J. Eur. Ceram. Soc.* 29 [15], 3121-3128. https://doi.org/10.1016/j.jeurceramsoc.2009.06.004.

).
Franke and Bentrup report 1.65 μm as the median pore radius cut-off
that separates resistant from non-resistant bricks, and bricks with a
median pore radius equal to or higher than 1.65 μm are resistant to
freeze-thaw cycles. However, no risk assessment for the wrong decision
was made for that cut-off. Our research showed that the cut-off can be
set to 1.2 μm with an extremely low risk of a wrong decision.

Since the pore system is regarded as responsible for brick resistance to freeze-thaw cycles in literature sources (5-75. Kung, J.H. (1985) Frost durability of canadian clay bricks. *Proceedings of the 7th International Brick Masonry Conference, Melbourne, Australia*.[1], 245-251. Retrieved from https://nrc-publications.canada.ca/eng/view/ft/?id=85a8f3ac-6e9e-43a4-aad1-51c6288e929b.

6. Hansen, W.; Kung, J.H. (1988) Pore structure and frost durability of clay bricks. *Mater Struct.* 21, 443-447. https://doi.org/10.1007/BF02472325.

7.
Perrin, B.; Vu, N.A.; Multon, S.; Voland, T.; Ducroquetz, C. (2011)
Mechanical behaviour of fired clay materials subjected to freeze-thaw
cycles. *Constr Build Mater.* 25[2], 1056-1064. https://doi.org/10.1016/j.conbuildmat.2010.06.072.

),
this section provides a description of reliable classifiers from
section 2.5 by means of the pore system (large, medium-sized, and small
pore content; total pore content; median pore radius; and total pore
volume). As the median pore radius itself is one of the parameters in
the pore system, the other two reliable classifiers from Section 2.5,
the brick’s compressive strength ratio pre- to post-freezing and the
water desorption coefficient in 180-360 minutes, were described using
pore system as quantitative indicators of brick resistance to freezing
cycles.

*2.6.1. Modelling pre- to post-freezing compressive strengths ratio of the brick*

⌅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 (2222.
Sheskin, D.J. (2000) Handbook of parametric and nonparametric
statistical procedures. Second Edition, Chapman & Hall/CRC, United
States of America (2000). Retrieved from https://fmipa.umri.ac.id/wp-content/uploads/2016/03/David_J._Sheskin_David_Sheskin_Handbook_of_ParaBookFi.org_.pdf.

).
As the initial indicator for the existence of a monotonic relationship
between compressive strength ratio pre- to post-freezing and the
variables from 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 Table 8.

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 (2727. Fox, J.; Weisberg, S. (2019) An R companion to applied regression. Third Edition. Sage, Thousand Oaks CA (2019).

) was used for model building, and the R software package boot (2828.
Davison, A.C.; Hinkley, D.V. (1997) Bootstrap methods and their
applications. Cambridge series in statistical and probabilistic
mathematics. Cambridge University Press, Cambridge (1997). https://doi.org/10.1017/CBO9780511802843.

, 2929. Canty, A.; Ripley, B.D. (2019) Boot: Bootstrap R (S-plus) functions. R package version 1.3-24, (2019).

) was used for bootstrapping.

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 Table 9.

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. Table 10 provides bootstrapped confidence intervals for the adjusted R^{2} model and the model’s coefficients.

^{2}and the coefficients of the presented model.

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 Figure 3.

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
(55. Kung, J.H. (1985) Frost durability of canadian clay bricks. *Proceedings of the 7th International Brick Masonry Conference, Melbourne, Australia*.[1], 245-251. Retrieved from https://nrc-publications.canada.ca/eng/view/ft/?id=85a8f3ac-6e9e-43a4-aad1-51c6288e929b.

),
according to which small pores have little influence on the brick’s
resistance to freeze-thaw cycles. The model also confirms that the
information on the large pore content, along with a known median pore
radius value, is important for the description of brick resistance to
cycles and that bricks with a higher large pore content have a better
resistance to freeze-thaw cycles, which is, again, in line with the
facts provided in literature sources (55. Kung, J.H. (1985) Frost durability of canadian clay bricks. *Proceedings of the 7th International Brick Masonry Conference, Melbourne, Australia*.[1], 245-251. Retrieved from https://nrc-publications.canada.ca/eng/view/ft/?id=85a8f3ac-6e9e-43a4-aad1-51c6288e929b.

, 77.
Perrin, B.; Vu, N.A.; Multon, S.; Voland, T.; Ducroquetz, C. (2011)
Mechanical behaviour of fired clay materials subjected to freeze-thaw
cycles. *Constr Build Mater.* 25[2], 1056-1064. https://doi.org/10.1016/j.conbuildmat.2010.06.072.

).

*2.6.2. Modelling of the water desorption coefficient in 180-360 minutes*

⌅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 Table 11.

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.

### 3. CONCLUSIONS

⌅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.