PSI - Issue 78

Annalisa Franco et al. / Procedia Structural Integrity 78 (2026) 1245–1252

1250

Table 3. Values of the fitting parameters of the probability distribution models (Normal, Lognormal, 3-p Lognormal) for each specified dataset.

N ORMAL

L OGNORMAL

3- P L OGNORMAL

D ATASET

P ROPERTY

μ

σ

μ

σ

μ

σ

ϑ

GLASS + LM

σ u σ f σ u σ f σ u σ u σ f σ u σ u σ f

998

94 79

6,90 6,97 7,27 7,26 7,98 7,98 8,02 8,03 8,01 8,15

0,09 0,07 0,12 0,07 0,07 0,04 0,04 0,11 0,10 0,12

7,32 5,91 6,89 7,37 4,41 5,17 5,09

0,06 0,21 -0,17 0,07 -0,57 -0,52 -0,31 0,003

-509

GLASS

1066 1444 1432 2944 2918 3052 3093 3024 3500

690

BASALT + LM

165 106 196 106 129 347 309 415

2441 -157 3107 3119 3274

BASALT

STEEL + LM STEEL + CM

STEEL

PBO + LM PBO + CM

11,71

-118202

6,46 9,78

0,44 0,02

2319

PBO

-14199

3.2. Best fitting probabilistic model: the Kolmogorov-Smirnov test The Kolmogorov-Smirnov (K-S) test, which allows to have a quantitative estimate of the best fitting probabilistic model, is based on the comparison between the maximum absolute difference between the experimental CDF (ECDF), F(x) and the CDF associated with each theoretical probability distribution, φ(x). Such a difference is calculated as:

(5)

max ( ) ( ) F x x  

MAX D



The hypothesis regarding the distributional form is rejected if D MAX is greater than a certain critical value, which depends on the specimen size, N, and the level of significance, α, of the test. There are several values available in the literature for the level of significance; some of them (Massey, 1951) are reported in Table 4 for a level of significance of 0,05, which means that in 5% of cases the null hypothesis (the estimate of the difference between the empirical and the theoretical CDF is based solely on chance) is rejected. The 3-p Lognormal distribution appears to be the best fitting distribution of the experimental data, since it generally provides the lowest value of D MAX in most cases (see Table 4, values in bold). For glass and PBO with lime-based mortar, the values are almost comparable among lognormal and 3- p lognormal for the composite strength σ u , and the same occurs for the basalt and PBO dry fabrics in terms of σ f .

Table 4. K-S test for σ u and σ f : values of D MAX

GLASS + LM BASALT + LM STEEL + LM STEEL + CM PBO + LM PBO + CM  = 0,05  = 0,05  = 0,05  = 0,05  = 0,05  = 0,05 0,136 0,176 0,215 0,215 0,215 0,176

PROBABILITY DISTRIBUTION

PROP.

0,052 0,048 0,049

0,132 0,145 0,103

0,240 0,258 0,114

0,108 0,110 0,070

0,095 0,084 0,088

0,192 0,175 0,109

Normal

σ u

Lognormal

3-p Lognormal

GLASS  = 0,05

BASALT  = 0,05

STEEL  = 0,05

PBO

PROBABILITY DISTRIBUTION

PROP.

 = 0,05

 = 0,05

0,196 0,164 0,153 0,145

0,179 0,067 0,052 0,055

0,161 0,134 0,137 0,120

0,201 0,127 0,124 0,121

Normal

σ f

Lognormal

3-p Lognormal