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Statistical tests for the goodness of fit of mortar compressive strength distributions L. Sorrentino, P. Infantino & D. Liberatore Sapienza University of Rome, Rome, Italy

ABSTRACT:  Increased computational capacities and augmented awareness about the uncertain nature of parameters and models involved in structural design are orienting current verifications in an explicit probabilistic direction. In order to perform such computations probabilistic distributions of mechanical parameters are necessary. Mortar compressive strength distributions are investigated herein with reference to previously published experimental results, related to three mixes, for which a reasonably large number of specimens were tested. Two distributions are considered: normal and log-normal, and three statistic tests are performed to assess which distribution is most appropriate: Kolmogorov-Smirnov, KolmogorovSmirnov-Lilliefors, and Shapiro-Wilk. Analysis results show that normal distribution is more effective for medium to high strength mortars, whereas log-normal distribution is recommended for low strength mixes, as can be encountered in existing buildings. Nonetheless, simplified distributions assumed in the literature cannot always match completely the laboratory values trends. 1  Introduction Increased computational capacities and augmented awareness about the uncertain nature of parameters and models involved in structural design are orienting current verifications in an explicit probabilistic direction. The “Instructions for the Reliability Assessment of the Earthquake Safety of Existing Buildings”, issued in Italy in 2014 for reinforced concrete and masonry structures are the last example, after previous contributions in this direction (ISO 2394:1998, JCSS 2001). However, their application is possible only if probability density functions for the parameters governing the problem, among which are material mechanical properties, are known. The need for such functions arises also from the calibration of semi-probabilistic limit states, which in the past have been tuned on previous experience (Heffler et al. 2008). Comparison of empirical and analytical distribution functions are not frequent in the literature for masonry components. Lawrence (1985), using probably the largest samples to date, considered three mechanical parameters (elastic modulus of masonry, mortar compressive strength, brick modulus of rupture) for which the best three analytical distributions are the normal, log-normal and Weibull, respectively. The only goodness of fit they discuss is the Kolmogorov-Smirnov.

The main advantage presented by the normal distribution is its simplicity, whereas the log-normal distribution avoids negative values which have no physical meaning. The Weibull distribution is often used in reliability engineering and failure analysis. Heffler et al. (2008) performed a large number of tests on flexural bond strength, determining the truncated normal distribution, fitted based on maximum likelihood, as the most effective, according to a Kolmogorov-Smirnov test. The sum of two normally (or the product of two log-normally) distributed random variables is again a normally (or log-normally) distributed random variable. This property of log-normal distribution is exploited by Schueremans and Van Gemert (2006), for estimating masonry compressive strength based on mortar and brick compressive strengths, as suggested in EC6 (2005). Finally, masonry compressive strength is essential both when designing new structures and when assessing existing constructions, particularly within an earthquake engineering setting (Sorrentino et al. 2014a-b, Addessi et al. 2015, Andreotti et al. 2015, Liberatore and Addessi 2015) and when tall buildings are investigated (Gizzi et al. 2014, Sorrentino et al. 2014c). Hence, a robust evaluation of masonry compressive strength is essential, and the selection of probability density functions needs to be evaluated carefully resorting to different normality tests.

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2  EXPERIMENTAL DATABASE

Table 3.  Compressive strength (MPa) of mortar specimens for the five samples.

Mortar compressive strength distributions are investigated herein with reference to previously published experimental results, related to three mixes: 1: 1: 6, 1: 1: 9, and 0: 1: 3 (cement: lime: sand), for which a reasonably large number of specimens were tested. The Agbabian, Barnes & Kariotis consortium (ABK 1981) tested two mortar mixes: 1: 1: 6 (Portland cement: hydraulic lime: sand, type N mortar), 1: 2: 9 (type O mortar). Specimens, varying in number between 30 and 45 (Table  1), had two shapes: cylindrical (51 mm diameter, 102 mm height), and cubic (51 mm side). Filardi et  al. (1996) considered a mortar mix having a 0:1:3 (cement: hydraulic lime: sand) proportion by weight, with an average water  / lime ratio of 0.78. The authors tested 30 couples of prismatic specimens resulting from the same number of bending tests on 40  ×  40  ×  200  mm3 elements, as suggested also by EN 1015–11 (2001). The compressive strengths are the average of the couple strengths. Average strength and coefficient of variation are presented in Table  2, whereas the complete database is reported in Table  3. As expected, higher strength correlates with higher cement content. Contrary to what is suggested for concrete specimens (EN 206-1:2011), the mortar cubic specimens considered here show a lower strength than cylindrical specimens (Tab.  2 and Fig. 1). Table 1.  Description of the experimental database.

Sample Reference

Mortar Mix (cement: lime: Specimen sand) shape

1 2 3 4 5

1: 1: 6 1: 1: 6 1: 2: 9 1: 2: 9 0: 1: 3

ABK (1981) ABK (1981) ABK (1981) ABK (1981) Filardi et al. (1996)

Cylindrical Cubic Cylindrical Cubic Prismatic

1

2

3

4

5

17.43 17.92 18.01 10.54 12.52 12.08 13.17 13.97 13.00   7.03   6.37   7.46 15.81 15.37 15.32 12.65 12.03 11.77 10.32 12.08 11.99 15.72 14.80 14.32 15.15 15.54 11.90 13.17 16.21 15.94 13.44 12.56 15.59   8.17   7.64   8.74   9.89   8.34   8.57 11.81 12.03 15.24

13.28 13.79 16.38   7.59   8.86 10.17 12.38 11.03 11.59   4.59   4.24   3.93   8.38 10.66 10.66   5.34   6.83   8.86   6.48   6.34   6.17   8.76   9.07   9.45 12.48 11.97 11.03 10.14 10.45   9.34   8.07   8.79   9.79   6.62   6.34   6.97   8.76   8.41   7.86 10.69 11.62 10.52

4.66 5.57 5.75 4.17 4.39 4.88 3.25 3.20 3.34 5.49 4.74 4.83 4.26 5.09 4.88 4.12 3.69 3.60 2.41 2.41 2.46 3.07 2.86 3.12 3.08 3.17 3.47 2.81 2.50 3.12 6.28 5.75 6.94 6.68 6.14 7.20 4.57 3.60 4.13 3.20 4.21 3.99 2.93 2.97 2.83

4.66 5.00 4.03 2.76 2.03 2.86 1.41 1.76 1.52 4.34 4.10 3.10 4.17 4.62 4.07 2.90 2.59 2.72 0.97 1.34 1.93 1.83 1.86 1.55 1.38 1.28 1.10 4.76 4.14 4.48

2.18 2.85 3.65 2.75 2.58 2.87 1.08 1.41 1.03 1.49 1.79 1.62 1.17 1.32 1.47 1.36 1.52 1.58 0.65 1.00 0.90 0.65 0.85 0.73 0.78 0.75 0.89 1.14 0.97 0.67

Table 2.  Mean compressive strength and coefficient of variation (CV) for different mixes and specimen shape.

Sample

Number of specimens

Average strength MPa

CV

1 2 3 4 5

42 42 45 30 30

12.66 9.16 4.13 2.84 1.45

0.24 0.29 0.31 0.47 0.54

3  Statistical analyses Previous results have been arranged according to strength classes, computing corresponding relative frequencies (Figs. 2–6). Two class amplitudes have been considered for each set of strengths. As a matter of fact, an excessive number of classes can

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Figure  2.  Relative frequency distribution of compressive strength of type N mortar, cylindrical specimen (ABK 1981). Class amplitude: a) 1 MPa, b) 2 MPa.

Figure  1.  Compressive strengths of mortar (ABK 1981), for different specimen shape. Mortar mix: a) type N, 1: 1: 6 (cement: lime: sand), b) type O, 1: 2: 9.

end up in zero frequency cases, whereas too few cannot actually describe a distribution. These discrete relative frequency distributions have been compared with two standard continuous distributions: normal and log-normal. Graphical comparisons show that in some instances normal distribution seems more appropriate (Fig. 2b, Fig. 3b), whereas in some others the lognormal distribution seems to be the one (Fig. 4a, Fig. 6a). In the remaining cases both distributions show a poor agreement with experimental values, somewhat suggesting that actual behaviours can deviate substantially from what is assumed even in the most advanced computation procedures. In order to establish which distribution is the most appropriate, or is appropriate at all, several quantitative goodness of fit tests have been performed. The χ2 test has been neglected because it is sensitive to the class discretisation. Hence, the following procedures have been preferred: - Kolmogorov-Smirnov; - Kolmogorov-Smirnov-Lilliefors; - Shapiro-Wilk.

Figure  3.  Relative frequency distribution of compressive strength of type N mortar, cubic specimen (ABK 1981). Class amplitude: a) 1 MPa, b) 2 MPa.

The Kolmogorov-Smirnov test can be used, as in the case at hand, to compare a sample with a reference probability distribution (Kolmogorov 1933, Smirnov 1948). The Kolmogorov-Smirnov

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Figure  4.  Relative frequency distribution of compressive strength of type O mortar, cylindrical specimen (ABK 1981). Class amplitude: a) 1 MPa, b) 2 MPa.

Figure  6.  Relative frequency distribution of compressive strength of mortar (Filardi et al. 1996). Class amplitude: a) 0.5 MPa, b) 1 MPa.

The Kolmogorov-Smirnov statistic, Dn, for a given cumulative distribution function F(x) is: Dn = max Fn ( x ) − F ( x )

(1)

Where Fn(x) = empirical distribution function for n observations, x = random variable. The cumulative distribution function F(x) should have predefined mean and standard deviation but in the following, as done by Lawrence (1985), it will be assumed as a normal distribution having same mean and standard deviation of the sample. The statistic Dn is then compared with a critical value depending on the number of observations and the level of significance α, i.e. probability of rejecting the null hypothesis given that it is true (“false positive”). The null hypothesis is not rejected at level α, i.e. the assumed distribution function is acceptable, if: Dn < Figure  5.  Relative frequency distribution of compressive strength of type O mortar, cubic specimen (ABK 1981). Class amplitude: a) 1 MPa, b) 2 MPa.

statistic quantifies the distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution.

Kα n

(2)

Where Kα is found from: Pr ( K ≤ Kα ) = 1 − α

(3)

and K is the Kolmogorov distribution. In the following α  =  5% will be assumed, as customary in

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the literature (Heffler et al. 2008), but more restrictive than the 20% assumed in Lawrence (1985). In addition to the already mentioned advantage of not requiring a preliminary lumping of data in discrete categories (Benjamin and Cornell, 1970), the Kolmogorov-Smirnov test has the benefit of being independent of the used variable: if ln(x) is used instead of x the critical values remain unchanged, provided that the assumed variable has a normal distribution. A disadvantage is related to the little sensitivity of the test to deviations between empirical and function distributions close to both tails. The results of the Kolmogorov-Smirnov test are presented in Table  4, with the critical values reported in the second last column. From the table it is possible to observe that the test is satisfied for all five samples and both distribution functions. A variation of the previous test is the so-called Kolmogorov-Smirnov-Lilliefors test, or just Lilliefors test (Lilliefors 1967). This test is a two-sided goodness-of-fit test suitable when a fully-specified distribution is unknown and its parameters must be estimated. In particular, the normal cumulative distribution function has mean and standard deviation equal to the mean and standard deviation of the empirical sample. The statistic is the same as defined in Eq. (1) and, given the assumption about the cumulative distribution function having parameters derived from the sample, F(x) has exactly the same meaning. However, the critical values change. Because the function distribution has been defined based on mean and variance of empirical data, the maximum deviation is taken smaller than it would have been if the procedure had singled out just one normal distribution, as the KolmogorovSmirnov test usually (but not here) assumes. Thus Lilliefors distribution of the test statistic, i.e. its probability distribution assuming the null hypothesis is true, is stochastically smaller than the Kol-

mogorov-Smirnov distribution, and is computed using Monte Carlo simulation. In particular, when n ≥ 30, as in the cases at hand, for α = 10%, 5%, 1% the critical values are approximately equal to 0.805 / √n, 0.886 / √n, 1.032 / √n. The results of the Kolmogorov-Smirnov-Lilliefors test are presented in Table 4, with the critical values reported in the last column. From the table it is possible to observe that this test is more severe compared to the previous one and is satisfied for just one distribution in the case of sample 1 and none in the case of sample 4. Comparisons between empirical, function and function ± Kolmogorov-Smirnov-Lilliefors critical value cumulative frequency distributions are presented from Figure  7 to Figure  11. The deviation between empirical and function distributions is evident, as well as the graphical meaning of the critical value. The last test used is that by Shapiro and Wilk (1965). The assumed distribution is considered acceptable if the following inequality holds: W > Wcr

(4)

where: 2

b   W =  σ n − 1 

(5)

Table 4.  Kolmogorov-Smirnov (KS) and KolmogorovSmirnov-Lilliefors (KSL) tests, for a level of significance equal to 0.05. Sample

Distribution

Dn

KS

KSL

1

Normal Log-normal Normal Log-normal Normal Log-normal Normal Log-normal Normal Log-normal

0.126 0.180 0.060 0.120 0.129 0.115 0.182 0.187 0.183 0.089

0.205 0.205 0.205 0.205 0.198 0.198 0.242 0.242 0.242 0.242

0.135 0.135 0.135 0.135 0.131 0.131 0.159 0.159 0.159 0.159

2 3 4 5

Figure  7.  Empirical, function and function ± Kolmogorov-Smirnov-Lilliefors critical value cumulative frequency distributions of compressive strength of type N mortar, cylindrical specimen (ABK 1981). Function distribution: a) Normal, b) Log-normal.

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Figure  8.  Empirical, function and function ± Kolmogorov-Smirnov-Lilliefors critical value cumulative frequency distributions of compressive strength of type N mortar, cubic specimen (ABK 1981). Function distribution: a) Normal, b) Log-normal.

Figure  9.  Empirical, function and function ± Kolmogorov-Smirnov-Lilliefors critical value cumulative frequency distributions of compressive strength of type O mortar, cylindrical specimen (ABK 1981). Function distribution: a) Normal, b) Log-normal.

Figure  10.  Empirical, function and function ± Kolmogorov-Smirnov-Lilliefors critical value cumulative frequency distributions of compressive strength of type O mortar, cubic specimen (ABK 1981). Function distribution: a) Normal, b) Log-normal.

Figure  11.  Empirical, function and function ± Kolmogorov-Smirnov-Lilliefors critical value cumulative frequency distributions of compressive strength of mortar (Filardi et  al. 1996). Function distribution: a) Normal, b) Log-normal.

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where σ = standard deviation of the sample, and: k

b = ∑ ai ( xn − i +1 − xi )

(6)

i =1

where the variables are sorted in ascending order, k  =  largest integer ≤ n/2, a is a function of the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution, and of the covariance matrix of those order statistics. Its values were tabulated by Shapiro and Wilk (1965) for samples having 2 ≤ n ≤ 50. The authors computed also the Wcr values, which depend on n and on the level of significance, α, for samples having 3 ≤ n ≤ 50. Current algorithms (Royston 1982) usually test the null hypothesis comparing the p-value with the level of significance. If the following inequality holds: p − value > α

(7)

the distribution in considered acceptable. The results of the Shapiro-Wilk test are presented in Table  5, both in terms of critical value Wcr and p-value. This test is the most severe among those considered here, with the normality being rejected in half of the cases. In particular, sample 4 has both normal and log-normal distributions being rejected. It is worth mentioning that this sample has neither the largest standard deviation nor the lowest mean strength, hence the outcome is a result of the scatter of the strength classes rather than of strength values alone. In the remaining samples normal distribution is acceptable in the first two cases, those having the highest mean strength (Tab.  2) and log-normal distributions in three, having comparatively lower strengths. It is worth mentioning that Lawrence found the lognormal distribution to be the most adequate for reproducing a mortar compressive strength probTable  5.  Shapiro-Wilk test, for a level of significance equal to 0.05. Sample

Distribution

W

Wcr (α = 5%)

p-value

1

Normal Log-normal Normal Log-normal Normal Log-normal Normal Log-normal Normal Log-normal

0.960 0.925 0.986 0.962 0.930 0.962 0.899 0.916 0.858 0.950

0.942 0.942 0.942 0.942 0.945 0.945 0.927 0.927 0.927 0.927

0.147 0.009 0.815 0.151 0.009 0.146 0.008 0.021 0.002 0.168

2 3 4 5

ability distribution, of a sample having mean equal to 2.68  MPa, reasonably close to average values of samples 3–5 herein. Lawrence (1985) provided no explanation for the positive skewness, but this can be explained with the negative values being bounded by the zero value, whereas such boundary does not exist for positive values, although large values becomes less and less likely to occur. 4  CONCLUSIONS Probabilistic design and assessment of structures, as well as probabilistic calibration of deterministic procedures require robust probability density functions of relevant parameters. Masonry compressive strength can be estimated based on components, mortar and unit, compressive strength. Two separate laboratory campaigns testing five reasonably large mortar samples (population varying between 30 and 45 elements), with average strength varying between 1.5 and 12.7 MPa, have been considered. As expected average compressive strength increases with cement content. For two mortar mixes both cylindrical and cubic specimens were available. Surprisingly cylindrical strength was larger in average by approximately 40%, whereas for concrete a reduction of about 20% is expected according to testing standards. Two probability density functions have been considered: normal and log-normal, and their acceptability against empirical distributions has been assessed trough statistical tests: KolmogorovSmirnov, Kolmogorov-Smirnov-Lilliefors, ShapiroWilk. The tests show themselves to be increasingly restrictive. According to the Kolmogorov-Smirnov both functions are suitable for all five samples. The Lilliefors test is passed only six times out of ten, and the Shapiro-Wilk only five. The tests are consistent between themselves, i.e. the distribution acceptable according to Shapiro-Wilk are acceptable also for the other two procedures. Normal distribution shows a better performance for higher strength mortar, whereas the opposite happens for low strength mortar. Hence, log-normal distribution is not always the best choice, as recent probabilistic codes suggest. Moreover, for one sample neither distribution passed both the Lilliefors and the Shapiro-Wilk tests, suggesting that additional distributions should be considered and that actual behavior can be even more scattered than what is assumed in current probabilistic procedures. Acknowledgements This work has been partially carried out under the program “Dipartimento di Protezione Civile—

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