MATERIAL AND GEOMETRICAL CHARACTERISTICS OF STRUCTURAL STEELS BASED ON STATISTICAL ANALYSIS OF METALLURGICAL PRODUCTS

The experimental research results of material and geometrical characteristics of Czech steel are given in the presented paper. Measurements are carried out for the yield strength of plates made of steel grade S355. The values measured are used to compute axial resistances which are compared with the resistance resulting from standard (nominal) values. In this paper, there was used a statistical information for evaluating the results of strength measurements on a series of plates and hot-rolled profiles under tension to determine the adequate partial γM0 safety factor.


Introduction
Steel structures have a long tradition and great history in Czech Republic. Žďákovský bridge over the Vltava from the 1960's belongs to the largest arched bridges in the world. Former Czechoslovakia was from the Second World War till the end of the 20 th century one of the world's largest producers of iron and steel in terms of production per inhabitant. This was due to political reasons. Change in political situation, subsequent fall of Czechoslovakia and the creation of Czech Republic was accompanied by decrease in steel production. At present, production of steel lays predominantly in the hands of foreign owners and management. 700 kg of steel per inhabitant is produced yearly in the Czech Republic. This is approximately double the number in other developed European countries. There is a large competition between producers. The decisive criterion for foreign management is price, leading to the shift of production volume to countries with lower production costs and more tolerant ecological regulations. Due to this and other reasons we can objectively expect that a decrease of production in Czech smelting plants will continue.
The development of the economic situation initiated and initiates a number of non-commercially aimed research projects aimed at the long time monitoring of the change in quality of metallurgical products, not only in central Europe. These researches provide detailed statistical information on the material and geometric characteristics of steel products and are important in terms of the verification of the indicators of reliability of the EUROCODE standards. Non-commercially aimed researches are not identical with activities performed by producers, which are aimed more at the adherence of standard production procedures and optimization of economical aspects.
Traditionally, the material yield strength, tensile strength and ductility have been studied among the mechanical characteristics of structure steels. For dimensioning the structures, yield strength is the most important above all. 95% of valid observations of yield strength should be higher than the nominal value of yield strength listed in EUROCODE 3. If this is not met, it is necessary to recommend product quality improvements. Reliable conclusions require a high number of valid observations. Extensive research of material characteristics of Czechoslovakian steel in the 1960's and 1970's (Mrázik, Sadovský 1992) produced valuable information on the quality and reliability of steel structures. One of the first comprehensive papers (Rozlívka et al.) publishing the most important material characteristics of steel products of a dominant Czech manufacturer subsequently became the source of input data for a number of reliability studies using the Monte Carlo method, whose popularity interrelated with the quick development of potential of computers. The most important results of experimental research are yield strength, tensile strength, ductility of steel plates and flanges of cross-sections IPE and geometrical characteristics of cross-sections IPE from 1989 till 2004 are published (Melcher et al. 2004). With regard to international cooperation we can in this period accentuate mainly the comparison of the above-listed material characteristics of the dominant Czech and Austrian producers Strauss et al. 2006). In an independent manner, material characteristics of steel products of both Austrian and Czech makers were statistically evaluated at the Vienna and the Brno Universities with the aim of guaranteeing the maximum objectivity of the results obtained and of the conclusions drawn by them Strauss et al. 2006). It is also necessary to mention of the extensive experimental research performed in Hungary (Szalai, Papp 2009).
Attention is paid lately in the Czech Republic to the research of material characteristics of highly utilized structures of steel S355 (Melcher et al. 2008). The aim of the paper is to provide new information on the listed characteristics from the year 2007. The studies published here link up with the long-term non-commercial experimental research, the results of which were published in (Mrázik, Sadovský 1992;Rozlívka et al.;Melcher et al. 2004;Kala et al. 2005;Strauss et al. 2006;Szalai, Papp 2009;Melcher et al. 2008).

Material characteristics of steel S355 plates
The yield strength is the fundamental parameter representing the basis for derivation of design characteristics and partial safety factors of steels. The mutual difference of yield strength values of plates in dependence on their thickness was studied in the course of experimental research.  We obtained 2138 samples of P4-16, 1368 samples of P16-40, and 689 samples of P40-100. The approximations of the measured data of Gauss density probability functions are depicted in Fig. 1 and Fig. 2. The comparisons of the Hermite and the Gauss density probability functions are to be seen in Fig. 3 to Fig. 5.
The Hermite density probability function is a fourparametric function (accessible, e.g., in the computer programme Statrel, version 3.10), which takes into con-sideration both skewness and kurtosis of quantities measured. The largest difference between the Gauss and the Hermite density probability functions has been found for the plates with thicknesses of 4 to 40 mm (Fig. 3.) This difference is caused by a relatively high value of kurtosis, 5.3305. The survey of the yield strength statistical characteristics is given in Table 1.  The real probability model of yield strength was obtained from measured data (Fig. 6). As the manufacturing process is concerned, the frequency of the occurrence of yield strength being lower than the nominal value is controlled as a parameter characterizing the production quality. The yield strength nominal value for plates with thickness lower than 40 mm is 355 MPa, for plates with thickness higher than 40 mm, it is 335 MPa, see Standard EUROCODE 3. If the manufacturing quality is convenient, the number of values lower than the nominal value must be less than 5%.
In effect, it is compared, whether the 5% quantile (characteristic value) is higher than the nominal value. To assess this quantile with a reliability, it would be necessary to have a large number of measured values at the disposal. If the number of measured samples is rather low, one of the ways how to obtain an assessment of the characteristic value is the approximation of the set containing the data measured by an appropriate type of the probability density distribution function.
The characteristic values obtained from some most frequently used probability density distribution functions are presented in Table 2.

Design values of yield strength and partial safety factors
The design value of material resistance is determined as the product of random variables (Melcher et al. 2008): where f is yield strength; ϕ A -non-dimensional random area (cross-section area divided by nominal value: A/A n ); ϕ 0 deviation of theoretical model from the results of tests. When calculating the material strength on the basis of yield strength random values, ϕ 0 = 1.0 is usually considered (Melcher et al. 2008). For plates, the mean value 1.0 and the standard deviation 0.03 are usually assumed for ϕ A . For the hot-rolled profiles non-presented here, the standard deviation for ϕ A can be considered to be 0.02 (Melcher et al. 2008;JCSS … 2001).
For calculating design values R (1), 400 000 simulation runs of the Monte Carlo method were used. It has been supposed that the yield strength has the Gauss, lognormal, shift lognormal probability density functions (Tables 3 and 4) and that the coefficient ϕ A has the Gauss probability density distribution function, and that ϕ 0 = 1.0. In compliance with the articles of (JCSS … 2001;EN 1990EN … 2003, the design value R d was evaluated from the basic probability definition as the 0.1% quantile (Melcher et al. 2004). Let us introduce a clear procedure of partial safety factor γ M verification, e.g. for plates with thickness 4 -16 mm. The computer algorithm of calculation γ M can be described in the following way: − 400 000 runs of the yield strength f are generated by application of the Monte Carlo method. Statistical characteristics are given in Table 1. As the number of measurements is relatively low and does not enable us to determine the density distribution function of yield strength accurately, this function must be selected (e.g. Gauss).
− By application of the Monte Carlo method, 400 000 runs of ϕ A are generated; it is the coefficient having the Gauss density distribution function with mean value 1.0 and the standard deviation 0.03. The ϕ 0 is deterministic value of 1.0. − According to the relation (1), 400 000 runs of material resistance R = f⋅ ϕ A are calculated. 400 000 runs of R are rank-ordered from the lowest value to the highest one. The lowest realization of R is the first value, the highest, the last one. The design value R d , calculated as 0.1 percent quantile, is the 400th value in the ordered set.
− Partial safety factor γ M is calculated from the rela- where 355· 10 6 Pa is nominal value of yield strength for thickness 4-16 mm (for thickness 40-100 mm, this value is 335· 10 6 Pa).
The results of partial safety factor γ M are in Table 4. Partial safety factors γ M in Table 4 are given in dependence on the density distribution function of yield strength. The number of yield strength observations is relatively high; nevertheless, more observations are needed to get a satisfactorily accurate statistic analysis of R d and/or γ M . A statistically acceptable mistake would be obtained only after more than 100 000 observations; however, it is very expensive from the economic and organisation points of view. The values γ M are therefore sensitive to the epistemic uncertainty (Kala 2007b(Kala , 2008a(Kala , 2008b, nevertheless, they provide basic information on intervals within which γ M lies with high probability. The sensitivity of the type of density function manifested itself, to maximum degree, for thickness 4 -16 mm (Table 4), where a relatively high kurtosis 5.33 (Table 1) was found for yield strength. For the other thicknesses, relatively low skewness and kurtosis values were found (Table 1), and therefore it can be inclined to the results obtained on behalf of the Gauss density functions (Table 4).
The results from Table 1 to Table 4 cannot be generalized for yield strength of hot-rolled profiles. In the case of yield strength of hot-rolled profiles, it is therefore required to know the yield stress distribution along the cross-section in more detail (see e.g. Kala, Z., Kala, J. 2002). The only right way to obtain information on yield strength variability in a hot-rolled profile is to carry out material tests by taking the samples from different profile points. A suitable auto-correlation function and correlation length of a random field could be recommended as the output, see e.g. (Kala 2003).

Geometric characteristics of hot-rolled profiles
Observance of geometric dimensions and weight of bars are pre-eminently checked in production. Tolerances on geometrical shape and dimensions are listed in the Tolerance Standard EN 10034:1993, see Figs 7-9. The permitted limit deviation of the actual weight from the theoretical weight is given as ±4% for individual bars. The variables h, b 1 , b 2 , t 1 , t 21 , t 22 , see Figs 7-8, were statistically evaluated by experimentally obtained data.
We obtained 369 valid observations. The measured valid observations h, b 1 , b 2 , t 1 , t 21 , t 22 are depicted in Figs 10-15. Relative geometrical characteristics were evaluated. The relative geometrical characteristic is given as the ratio of the real characteristic obtained from the measurement of cross-sectional geometry (e.g. crosssection depth h) to the nominal dimension of the crosssectional one (Melcher et al. 2004).  The relative (non-dimensional) statistical characteristics are listed in Table 5. The results of sensitivity analyses (Kala 2004(Kala , 2005(Kala , 2007a(Kala , 2008c(Kala , 2009) clearly illustrate that the flange thickness is, for ultimate limit state, the dominant variables of cross-section. The results (Kala 2009;Szalai 2003) show the typical shapes of the sensitivity diagrams as the function of the slenderness. The flanges are rolled approximately evenly within the positive and negative tolerances (mean value and median are close to one).
For the two-axially symmetrical cross-section, statistical characteristics of quantities t 21 and t 22 should be approximately identical, when supposing the high number of samples measured. Small differences between statistical characteristics of the quantities mentioned observed in Table 6 are probably due to a relatively low number of measurements. As it will be shown further on, there exists a strong correlation between t 21 and t 22. The use of couples of random quantities t 21 , t 22 and b 1 , b 2 in reliability calculations is singular. In reliability studies, statistical characteristics of the symmetric cross-section are frequently applied, see, e.g., (Kala 2004(Kala , 2005(Kala , 2007a(Kala , b, 2008a(Kala , b, c, 2009Szalai 2003). In practice, it means that (i) section widths b 1 and b 2 are evaluated as one random quantity b and (ii) flange thicknesses t 21 and t 22 are evaluated as one random quantity t 2 . The statistical characteristics of the quantity b given in Table 6 were evaluated, based on 738 valid observations; the same procedure was applied to flange thicknesses t 2 .
The histograms of non-dimensional section width b and non-dimensional flange thickness t 2 are depicted in Figs 16, 17.  The correlation matrix (2) was evaluated, based on a relatively low number of measurements (369), which will have to be completed, and the results evaluated to be specified in future.  Geometrical characteristics k 1 , k 2 , f, m were not measured. The variability of geometric characteristics k 1 , k 2 , f, m has a relatively small influence on the ultimate limit states and can be considered as deterministic.

Reliability of steel structures
The results of statistical analysis represent important input data for reliability studies, techniques and optimisation problems in contemporary computational mechanics; see e.g. Jatulis et al. 2007;Juozapaitis et al. 2006Juozapaitis et al. , 2008aJuozapaitis, Norkus 2007;Karkauskas, Norkus 2001Kaklauskas et al. 2008;Mang et al. 2009a, b;Mang 2009;Strauss et al. 2008). With the aim to improve the reliability of load-bearing structures, it is necessary to work towards both to the improvement of processes to prove the reliability, and for development of quality in manufacturing the steel structures.
For the purpose of aiming at these objectives, the probabilistic assessment can be also enlarged by the sensitivity analysis, (see e.g. Kala 2004, 2005, 2007a, 2008c, Szalai 2003 and (Mang et al. 2009). New results of sensitivity analysis of systems illustrate that the interaction effects between input imperfections can have a significant influence (Kala 2008c). The sensitivity analysis identifies these quantities with a dominant effect on reliability, which should be controlled with intensified accuracy during the manufacturing process (Kala 2008c). Moreover, the sensitivity analysis also supplies important background for formulation, verification, and calibration of the reliability indices of Standards for designing, and for parameters of Tolerance Standards.

Conclusion
It can be observed from the results of experimental yield strength measurements of plates divided into three categories of thicknesses that, for plates of higher thickness, the yield strength mean value is lower. The difference between the yield strength mean value of plates with thickness of 4-16 mm and of those with thickness of 40-100 mm is approximately 10%. The differences occur, as well, between standard deviation values, but it is necessary to take into consideration the statistical error due to small number of measurements. The reliability of steel structures is also influenced by the yield strength skewness and kurtosis. The profiles U65 to U140 of steel grade S355 show, in comparison with the plates, a lower standard deviation (see Melcher 2004Melcher , 2008. These conclusions have been confirmed by a long-term study of material characteristics (Mrázik, Sadovský 1992;Rozlívka et al.;Melcher et al. 2004;Kala et al. 2005;Strauss et al. 2006;Szalai, Papp 2009;Melcher et al. 2008).
The mean value of non-dimensional geometrical characteristics h, b 1 , b 2 , t 1 was higher than one, the mean value of geometrical characteristics t 21 , t 22 approaches to one very closely. It means that average values of these characteristics approach the nominal values given by the manufacturer. Higher statistical characteristics, the standard deviation in particular, are also important for the reliability of structures. From the perspective of dimension tolerance, namely thickness, there exists the tendency to roll within the negative part of the tolerance limit. Improving technical possibilities enables that the deviation limits provided by standards are not used. In the future, it is necessary to continue monitoring of these characteristics with the aim of the analysis of their influence on the reliability of newly designed structures. It is possible to find only by means of the probabilistic reliability analysis, whether the structure design is reliable or is not, see e.g. (Kala 2003).
When formulating the conclusions, also the specialist's point of view and the long-term experience are of great importance. It is necessary both to continue studying the material and geometrical characteristics of industrially manufactured structures and to aim at international cooperation of specialists concerned.
The authors gratefully acknowledge the partial support provided by (i) the Czech Science Foundation within the projects of GACR No. 103/07/1067 and (ii) the Czech Ministry of Education, Youth and Sports within the Research Project MSM0021630519. We are grateful to C. Eng. Miroslav Fajkus and Dr. Lubomír Rozlívka for furnishing with the data on material characteristics.