Statistical Analysis of the Dynamic Performance of Reinforcement Steel at Elevated Temperatures: The Case of Johnson-Cook Model

Description of mechanical properties of reinforcement steel by means of mathematical models known as constitutive laws is considered. The attention is focussed on the Johnson-Cook (JC) model developed to express the stress-strain relation by considering the coupled effect of strain and strain rate hardening as well as thermal softening of steel. The JC model is analysed due to its prevailing role in the practice of constitutive relation of properties of reinforcement steels. The key element of this study is a new look at the JC model from the statistical viewpoint. The JC model is subjected to examination by confronting its deterministic nature with statistical variability of experimental data that can be acquired from stress-strain records. It is stated that to now this variability has been largely ignored. The current practice of fitting the JC model to individual and non-repetitive stress-strain records is analysed. It is suggested how to address the problem of the model fitting in the case where stress-strain data is obtained by repetitive measurements. A procedure for processing small-size statistical samples extracted from this data is proposed. The essential idea of this procedure is to fit components of the JC model to limits of one-sided confidence intervals calculated by means of the statistical technique known as bootstrap resampling.


Introduction
In situations of industrial accidents, military assaults and terrorist events, reinforced concrete components of building structures and elements of critical infrastructure can be exposed to a combined action of explosive loads and fire.Fires can not only act on structures but also trigger off explosions.A combined dynamic and thermal action on reinforced concrete structures is assessed by a coupled modelling of strain rate sensitive material properties and strength deterioration due to fire loading [1][2][3].As for reinforcing bars, constitutive laws of steel are used to consider both strain rate sensitivity and thermal softening.The practice of the coupled modelling of stress-strain relation of reinforcement bars is prevailed by the constitutive law known as the Johnson-Cook (JC) model [4,5].This model has been developed to consider the strain and strain rate hardening part of stress-strain relation and includes a component (singleparameter sub-model) accounting for thermal softening of steel.
The problem considered in the present study arises from the fact that the original version of JC model from the 1980s and its subsequent modifications are purely deterministic [4][5][6].In a series of studies, the deterministic JC model and several models of similar nature are fitted to experimental data as functions of strain rate and elevated temperature with fixed parameters called the material constants.An application of the deterministic models contradicts to an obvious statistical variability of experimental data expressed by strain-stress records.The present study seeks to examine the above contradiction by a closer look at the JC model from statistical point of view and to suggest recipes for alleviating this contradiction.It is proposed how to fit components of the JC model by applying repetitive data that consists of a relatively small number of stress-strain records.

A brief literature review
Concrete and steel of reinforced concrete structures may be exposed to dynamic loadings at a wide range of strain rates  .They range from the order of 10 -7 …10 -8 s -1 for quasi-static loading to the order 10 2 …10 3 s -1 for hard impacts and blast [7].The pair of the dynamic rate  and the static rate s  is the key information for quantifying an improvement in mechanical properties of steel (strength, modulus of elasticity, energy absorption).The improvement is expressed by dynamic increase factors (DIFs).In this study, these factors will be denoted by the symbol () DIF   [8].DIF is a ratio of material property  at a dynamic strain rate  to property at quasi-static strain rate s  , namely, the ratio of ()  to () s  .
with the static stress for instance, by Al Salahi and Othman [10] and Scholl et al. [11].
The function () yd   has been also developed to consider elevated temperatures of steel specimens.The yield stress () yd   is expressed as a function of absolute temperature of the specimen T, or dimensionless (homologous) temperature h T given by the ratio: where: Tmelt is the melting temperature of the specimen and Tr is the reference temperature that is usually taken equal to the room temperature Troom.H. Qian et al. [13], suggested to model the coupled influence of strain rate and temperature by introducing two separate DIFs: where: () are factors expressing the influence of strain rate and high temperature, respectively.How-ever, the prevailing approach is to use the constitutive equations, the arguments of which include both variables  and T .Four examples of such equations are given in   3)).The parameters c1, c2, c3 and c4 have to be determined from experimental results.The parameter c1 is the quasi-static yield stress.The parameters c2 and c3 describe the response to strain hardening.The parameter c4 represents the strain rate sensitivity.The parameter c5 models the thermal softening.


has been developed over the past 80 years.New models are still suggested or improved in the present time [11,14,15].Research on metals simultaneously subjected to large strains and high temperatures began in 1940s [14].However, the response of reinforcement bars to high strain-rate and elevated temperatures has been investigated mainly in the previous two decades [1][2][3]16].The general conclusion of this investigation is that the higher is the temperature, the larger is the strain rate effect.Despite relatively large quantity of experimental results obtained by investigating the combined effect of high temperature exposure and strain rate on reinforcement bars, the only constitutive equation ( , )     [6].
The Eyring model and its extension called the Ree-Eyring model were not developed specifically for reinforcement bars and later were not applied to this kind of steel [12].In addition, these models require to assess the relaxation activation energies Eα and Eβ, and therefore the implementation of these models is less practical when compared to the Johnson-Cook model.
The preceding assessment of the temperature-sensitive models ( , ) yd T  described in Table 1 leaves little choice but to look at the problem of strain rate sensitivity of reinforcement steel exposed to elevated temperatures in the light of the constitutive Johnson-Cook model.

Multiplicative composition and sensitivity analysis
The JC presented in Table 1 considers strain hardening, strain rate sensitivity and thermal softening.The lefthand side of this model is the true stress JC  that can be interpreted as the function of three arguments partially explained in Table 1: The argument p  of ( , , ) is the true plastic strain called also the equivalent or logarithmic plastic strain and   is the dimensionless plastic strain rate given by the ratio /   .The denominator   is assumed in a number of studies as the quasi-static strain rate s  with var- ying values 0.00025 s -1 , 0.001 s -1 , 0.002 s -1 or 0.0025 s -1 [1,14,17].However, normally   is taken as 1.0 s -1 and this value is called the reference quasi-static strain rate [1].The choice   = 1.0 s -1 means that the JC model will be appli- cable to relatively large strain rates  exceeding 1.0 s -1 and belonging to strain rate ranges related to hard impact and blast (Fig. 1).If the quasi-static stress is determined for the value the ratio   < 1.0, the JC model can be adjusted for   = 1.0 by increasing the quasi-static values of the con- stants 1 c and 2 c by the ratio of the stresses at   = 1.0 and   < 1.0 [5].represents the quasi-static flow.This factor can be regarded as the model for the quasi-static stress-strain curve [14].The parameter c1 can be the yield stress of hot rolled steels or the stress corresponding to 0.2% offset strain of cold drawn steels.For simplicity, the parameter c1 will express either of the stresses.The magnification factor + does not allow to describe a yield plateau and so JC models is suited only for strain-hardening stage (Fig. 1).
In JC, strain hardening, strain rate hardening and thermal softening are considered in a decoupled multiplication form.This allows to evaluate the parameters c1,…, c5 in tests of at least four kinds: quasi-static test at room the temperature .These tests are independent of each other, except that they must be carried out using the same sort of metal and, wherever possible, the same testing equipment and measuring procedures.
Table 2 Parameter values of the JC model related to the reinforcement steels B500A and B500B presented by Cadoni and Forni [1] as well as variation ranges of these parameters used for the global sensitivity analysis Examples of values of the parameters c1,…,c5 belonging to the JC model are given in Table 2. Eq. ( 5) indicates that this model is a function of three arguments

JC
  gener- ates a need to assess sensitivity of () JC   to these parame- ters.In our opinion, the global sensitivity analysis (GSA) is the best way to assess the dependence of () JC   on its pa- rameters [18].
In the deterministic context, input information for GSA can be expressed by uniform probability distributions over parameter variation ranges, say, ± 20% variation ranges [18].Values of the variation ranges are given in Table 2   .In line with the principles of GSA, the sensitivity analysis has been carried out for the function: (1 ((400 where: the symbols C1,…, C5 and r T are random parameter values and random room temperature, all uniformly distributed over the variation ranges given in Table 2, and () is the random stress expressed as function of the random variables C1 to C5 and r T .Results of GSA obtained for B500A and B500B steels and expressed by six first order GSA indices i S are given in columns 3 and 4 of Table 4.The same table contains also results of GSA indices computed for the case where three arguments  3. GSA has been carried out for the function that includes nine random variables, namely: ( , , , ... , , ) (1 (( )/(1500 )) ) Nine first order GSA indices Si computed for this function are presented in columns 5 and 6 of Table 3.The sum of the indices Si is close to 100%, and therefore higher order sensitivity indices are not presented.It turned out that the stress () JC   is influenced mainly by the yield stress c1, thermal softening parameter c5 and strain rate hardening parameter c4 in case of both B500A and B500B steels and both functions expressed by Eqs. ( 6) and (7).In case of the function given by Eq. ( 7), a relatively high sensitivity of () JC   with respect to the elevated temperature T was found.Table 3 Values of arguments of the JC model,
The value of the strain rate hardening parameter 4 c is determined by means of at least two schemes (see [15] and references cited therein).The first scheme is based on the difference between the quasi-static yield stress y  the dynamic yield stress 1 () Both y  and 1 () The value of c4 results from a linear fit of the function that in terms of regression analysis is given by: Examples of fitting the function given by Eq. ( 11) to the data ( , )   ck ck yx are presented by Lin et al. and Zeng et al. [15,17].
The thermal softening parameter c5 is strain rate independent in the JC model [4,5].In this case, a value of c5 is determined with static data, that is, on the basis of the equation: brought to the form: where: ( , )


(Fig. 3).The pairs of data used for a linear fit will have the form: ( , ) ( ( ), { }) where: () hl TDF T is the "thermal decrease factor" calculated for each hl T as an average of the stress ratios: The value of 5 c should be a result of a linear fit of the function that in notation of regression analysis can be expressed as: where: ( );  15).An example of fitting the liner function given by Eq. ( 17) to the data ( , ) Tl Tl yx is presented in the paper [17].


are the areas below true stress versus true strain records obtained at the temperatures Thl and Th, respectively (Fig. 5).An example of a calculation of the values 5 ( , )   hl cT  for five different temperatures and two values of strain rate is provided by Forni et al. [3].
The ratio ( , )/ ( , ) is interpreted as the thermal softening reduction factor [3]. Formally, this ratio expresses proportionality of two averages of the stress strain functions over a given interval of strain values, for instance, the interval [0, j pn  ] shown in Fig. 4.An application of this averaging procedure makes the estimate of the parameter c5 closely related to the averaging expressed by Eqs.(17).At the same time, the key difference between these two procedures is the use of static data in case of the ratios ( , )/ ( ,0) and dynamic data measured at the strain rates  in case of the ratios ( , )/ ( , ) . The parameter 5 ( , )   hl cT  is a bivariate function.In terms of a regression analysis, 5 ( , )   hl cT  can be expressed as: , where:  is a vector of re- gression parameters.Limited experimental data indicates that ( | )  f x  should be a nonlinear function of strain rate and temperature [1,3].Attempts to develop a nonlinear bivariate function ( | )  f  5) means that temperature and strain rate are considered in a decoupled form.This feature is considered to be one of the shortcomings of JC model, because flow stresses of metals are highly affected by the coupled effect of temperature and strain rate [19].However, any quantitative measures of this inconsistency are not provided by authors of this criticism.At the same time, the JC model is praised for its simplicity, a relatively simple parameter estimation and easiness of implementation in computer codes.The JC model was not developed to describe a yield plateau; however, it can be readily adapted to model the yield stress by assuming that the plastic strain εp is equal to zero, namely: .
Results of GSA presented in Sec.3.1 indicate that the JC model is most sensitive to the model component c1 (yield stress or 0.2% offset stress) and relatively insensitive to the parameter c4 and the strain rate   (Table 4).This naturally raises the question related to the influence of the variability Mechanical properties of steel are prone to a natural statistical variation.The variability expressed by the coefficient of variation (COV) for yield stress y  of general population of reinforcing bars is around 4-11% [20].The COV of the modulus of elasticity of reinforcement steels from the same population is equal to 3.3%.The strain-hardening modulus of construction steel sections has values of COV around 8% [21].COVs of yield strength of B500A and B500B reinforcing bars with diameters of 6, 8 and 10 mm are 2.9%, 6.6% and 20.3%, respectively [22].The above values of COVs are not excessively large; however, the random variation represented by them is not ignored in assessing the reliability of reinforced concrete structures [20].
Recorded stress-strain curves will inevitably differ randomly even in the case where test specimens are prepared from the bars having the same diameter, made of same steel and are subjected to the same a dynamic strain rate  .This random variation is illustrated for the 2. The stress related to any given strain   and given strain rate  is also "blurred" and represented by the statistical sample { , ( , ) With these two results, fitting the strain-hardening model 1 2

(
)  The first natural candidate for the conservative representation of random dynamic and static stress at given strain value p  are the low q-percentile , ( , ) true q p    and the high 1 q − -percentile ,1 ( , 1) true q p  − . They are illustrated in Fig. 7 for the true strain value   .The levels q and 1 q − can be equal, for instance, to 10 % and 90 %, respectively.

JC q p
    is the JC model fitted to the q-percen- tiles of dynamic stress and ,1 ( ,1)

JC q p
 − is the quasi-static JC model fitted to 1 q − -percentiles of this stress.A choice of the percentage q allows to control the conservativeness of the factor ( , ) p DIF .The greater is q and, the lesser and more conservative will be the factor ( , ) p DIF .


, given by Eq. ( 13).This allows to introduce a temperature dependent DIF that is an extension of the DIF defined by Eq. ( 22), namely: ,0.05 ( , , ) ( , , )/ ( , ), where: 0.05 and q are the percentages that should be interpreted in the same way as in case of the temperature-insen-sitive factor ( , ) p DIF  given by Eq. ( 22).The above con- siderations concerning the need to discretise the argument  of ( , ) p DIF  are also applicable to the second and third arguments of ( , , ) ph DIF T


. It is apparent that the complexity of discretisation problem in case of (., , ) h DIF T  doubles due to the presence of the additional argument Th.Arrangement of experiments on combined effect of high strain rate  and elevated temperature Th demonstrates the need for a discretisation of  and Th.For instance, Cadoni and Forni [1] have carried out such experiments on B500A steel specimens by imposing them to strain rates of 250, 500 and 900 s -1 and temperatures of 200, 400 and 600 °C.
The above considerations related to the statistical variability of the stress-strain records

Proposal for statistical implementation
In our opinion, reasons for the scarcity of statistical data related to strain hardening range are twofold: 1.The deterministic approach traditionally prevails in the field of dynamic increase modelling.The fact the records ,, true i true i

 −
are subject to statistical variation is not fully ignored.At the same time, attempts to express this variability explicitly have not been undertaken to the best of our knowledge.
2. The determination of uncertainty related to strain hardening properties for steel subjected to high strain rates and elevated temperatures is difficult.In addition, these properties have had relatively little attention, perhaps because they are seldom used for conventional design purposes.Reasons for that are mentioned by Melchers and Beck [20].
Despite the scarce data related to the repetitive records    and expressed by the statistical sample illus- trated in (Fig. 6), namely, , n}.
The sample size n will determine possibilities of modelling uncertainties related to dynamic stress-strain modelling in general and possibilities to estimate average values of ( , ) true pj k    and quantiles related to DIF ex- pressed by Eq. ( 22).Currently, the values of n that can be found in the literature are very small.Many repetitive tests were carried out for no more than three specimens, that is, n ≤ 3 (see, e.g., [14]).
In the ideal case of a large sample    with a pdf illustrated in Fig. 7.If the sample jk  seems to fit a particular probability distri- bution, a calculation of qth percentiles , ( , ) true q pj k    be- comes a trivial task.
In actual dynamic tests a question will be raised about the minimum size of the sample jk  .Procedures for the calculation of a required minimum sample size n avail- able in textbooks pertain to the estimation of such population parameters as mean and proportion as well as testing hypotheses about these parameters [24].
The minimum size of the sample jk  related to fit- ting a probability distribution for ( , ) pj k    is expressed by required samples sizes for goodness-of-fit (GoF) tests.However, practical tools are limited to a few highly specific procedures of distribution fitting, in particular to the parametric Pearson's test presuming large number of observations [25].Procedures for the required sample size calculation have been developed for individual probability distributions, for instance, the generalised extreme value (GEV) distribution [26].However, these procedures suppose naturally that the distribution type of the population under study is known in advance.Currently, this will not be the case in assessment of the probability distribution of ( , ) true q pj k    can be estimated directly from data rather than by q -quantiles of the probability distribution fitted to jk  .However, the calculation of empirical quantiles is problematic if data in the regions represented by them is absent or scarce.This will be precisely the case for a small size sample jk  consisting of, say, ten elements.An estimation of quantiles with low values of q will be impossible due lack or scarcity of data in the tail region of jk  .Thus, the esti- mation of the values , ( , ) true q pj k    by q-quantiles of the probability distribution fitted to jk  seems to be the most practicable, albeit not necessarily very accurate approach at the current practice and currently available possibilities of data acquisition.

Numerical example
Processing the data collected in dynamic tests on B500A steel is considered.The aim is to select a probability distribution of the random dynamic stress ( , )     The conservatively high 0.9-quantile of the static stress distribution N(622.1, 27.2) is equal to 657 MPa, whereas the conservatively low 0.1-quantile of the dynamic stress distribution GEV(647.1, 28.5, 0.1631) was calculated as 624 MPa.We see that the value of 657 MPa related to static stress exceeds the value of 624.0 MPa obtained for dynamic stress.This renders the specification of the factor value (0.03, 250) DIF as ratio of conservative distribution quantiles unreasonable, because the ratio 624/657 is less than one.
Goodness-of-fit results given in  .
An alternative and less conservative approach is to choose the value of (0.03, 250) DIF as a ratio of two limits of one-sided CIs calculated for means of the random stress values (0.03, 250)  and (0.03,1)  . The probability of 90 % was chosen as a not overly conservative confidence statistical procedure known as bootstrap resampling.

E. R. Vaidogas STATISTICAL ANALYSIS OF THE DYNAMIC PERFORMANCE OF REINFORCEMENT STEEL AT ELEVATED TEMPERATURES: THE CASE OF JOHNSON-COOK MODEL
S u m m a r y Description of mechanical properties of reinforcement steel by means of mathematical models known as constitutive laws is considered.The attention is focussed on the Johnson-Cook (JC) model developed to express the stressstrain relation by considering the coupled effect of strain and strain rate hardening as well as thermal softening of steel.The JC model is analysed due to its prevailing role in the practice of constitutive relation of properties of reinforcement steels.The key element of this study is a new look at the JC model from the statistical viewpoint.The JC model is subjected to examination by confronting its deterministic nature with statistical variability of experimental data that can be acquired from stress-strain records.It is stated that to now this variability has been largely ignored.The current practice of fitting the JC model to individual and non-repetitive stress-strain records is analysed.It is suggested how to address the problem of the model fitting in the case where stress-strain data is obtained by repetitive measurements.A procedure for processing small-size statistical samples extracted from this data is proposed.The essential idea of this procedure is to fit components of the JC model to limits of one-sided confidence intervals calculated by means of the statistical technique known as bootstrap resampling.
Most of the research devoted to the strain rate sensitivity of structural steel properties  is related to the yield stress () y .Due to the importance of () y  to the design of dynamically loaded reinforced concrete structures and in the interests of brevity, the present study will be limited by the dynamic yield stress of structural steels.Often cited examples of the factor y DIF  related to reinforcement steel and expressed by the static to dynamic yield stress ratio ( )/ ( ) yd ys s     are given by [9 the constitutive equations.The equations with the single argument s  do not consider the influence of temperature of specimens and in-situ reinforcement bars subjected to dynamic loading.It is simply assumed that this temperature is equal to the temperature of the room, Troom, in which dynamic tests are carried out (normal or room temperature).A review of various temperature-insensitive models ) the present time was the Johnson-Cook model presented in

Fig. 1 A
Fig. 1 A schematic depiction of the JC model and experimental records The right-hand side of the JC model is a product of three factors that consider three different effects.The factor difference between dynamic and static stress at the same strain increases with increasing strain rate.


,   and T if the dimensionless tempera- ture h T is developed according to Eq. (3).Thus, we have to | , ,..., ) JC p T c c c     that have five material-related parameters.If necessary, quasistatic strain rate s  and reference temperature Tr can be added to the set of parameters of () JC   .The presence of the relatively large number of parameters in () . GSA is based on a repetitive stochastic simulation of parameter values and computation of output values of () JC   .Results of GSA will depend on the choice of values of the arguments p  ,   and T. The number of combina- tions of these values is infinite.Consequently, GSA has been limited only by a set of three values of p  ,   and T given in the second columns of Table 3 and belonging to ranges of p  ,   and T used in dynamic tests of B500A and B500B steels and reported in the articles [1] Fig. 2 presents two graphs of the function ( ,400, 400) JC p  drawn for B500A and B500B steels as well as four graphs of ( , , ) JC p T     developed for two combinations (1, 20°C) and (400, 20°C) of the arguments ( , ) T

p
,   and T of () JC   were con- sidered random input variables p  ,   and T uniformly distributed over ± 20% variation ranges around the fixed values given in Table °C or Th, -400 (0.257)[320, 480] or [0.206, 0.308] * See the abscissa axis in Fig.2

Fig. 2
Fig. 2 Graphs of the function

Fig. 3
Fig. 3 The scheme of data used to determine parameters of the isothermal JC model by means of 1 k n + stress- used to estimate the slope parameter T  are defined by Eq. (

Fig. 4
Fig. 4 The scheme of data used to determine the parameter of thermal softening in the JC model An alternative approach to the evaluation of the thermal softening parameter c5 assumes that c5 depends on the strain rate  and temperature T [1].It is suggested to evaluate c5 by means of the equation: 5 ( , ) = (1 ( , ))/ ( ), hl hl hl c T log r T log T  − , (18a) x  or to transform recorded values of c5,  and hl T and apply new data to fitting a linear re- gression model T y     = x  with regressand T y   , regressors  x and regression coefficients   do not seem to be avail- model expressed by Eq. (

Fig. 5 Thl and Th 4 .
Fig. 5 An illustration of the areas below true stress versus true strain records, ( , ) hl T  and ( , ) h T  , obtained at the temperatures Thl and Th 4. Statistical examination of the Johnson-Cook model 4.1.Random variability of stress-strain records

1 .
true true − curves in Fig.2.The variation of the stress-strain records leads to variation of values related to them.If a set of n stress-strain records is considered and the index i used to refer to elements of this set, each record ,, true i true i −generates at least the following values related to the given strain rate  (Fig.2):pair of strains ( ( ), ( )) Values of onset and end of hardening stage represented by the statistical samples { () yi  , i = 1, 2, … , n} and { () ui  , i = 1, 2, … , n} are "blurred" and there is no single interval of plastic strain values for fitting the strainhardening model

Fig. 6
Fig. 6 Schematic illustration of the random variation of the records ,, true i true i  − related to a given strain rate  and resulting variation of specific values of stresses and strains Thirdly, the problem can be solved by fitting the models 12 () 3 c p cc  + and 1 2

Fig. 7
Fig. 7 Two possibilities of fitting the JC model to a set of n records ,, true i true i  − records: fitting the model to average stress values related to given strain values and fitting to conservative percentiles of stress valuesDespite the desirable separation ensured by the percentile levels q and 1 q − , the use of the percentiles , ( , ) true q p    and ,1 ( , 1)true q p  −

pDIFFig. 8 5 (
Fig. 8 The statistical situation related to uncertainties in static and dynamic stress expressed by random stress values ( ,1) p 


is 46.5 MPa and ranges of these samples overlap.Three hypothesized distributions have been used to fit probability distributions to , s jk  and jk  namely, two- parametric normal and lognormal distributions and threeparametric generalised extreme value (GEV) distribution.

Table
[10]s the activation energy of the α -relaxation; R is the universal gas constant; T is the absolute temperature of the specimen.In the Eyring model, the yield stress Comments: c1, c2, c3 and c4 are four material constants of the Ree-Eyring model; Eβ is the activation energy of the β-relaxation.The Ree-Eyring model assumes two relaxation processes and considers the increase of the strain rate sensitivity at high strain rate[10].
=+Comments: c1 and c2 are two material parameters of the Eyring model; Comments: εp is the true (equivalent) plastic strain; c1, c2, …, c5 are five material parameters of the Johnson-Cook model;   is the reference quasi-static strain rate commented in Sec.3.2 and Th is the homologous (dimensionless) temperature (Eq.(

Table 4 First
The parameters c2 and c3 are evaluated from the quasi-static true stress-strain record s  -s  shown in Fig.3.Values of c2 and c3 are results of a linear fit of the function that in typical symbols of regression analysis is expressed as: order global sensitivity indices Si computed for parameters and arguments of the JC model ( , , | , ,..., , )  and temperature h T .The dependence of c5 on p  can be handled similarly to the case of the strain rate hardening parameter c4.For a series of strain values, { pj  , j =1, 2, ..., nj} and a series of temperature values {Thl, l =1, 2, ..., nj}, the static data will be expressed by the stresses ( , ) ) [27]can run into tens or several hundreds.Sample sizes of this magnitude are not typical for dynamic tests of steel, although they are technically achievable.Literature on GoF procedures states that some nonparametric GoF tests can be applied even to the case where n is as low as 6 to 10 observations[27].Among them, the Anderson-Darling (AD) and Kolmogorov-Smirnov (KS) tests stand out.The AD test requires lesser number of data points in comparison to the KS test to properly reject the null hypothesis.Thus, the idea for quantifying uncertainty related to values of the stress ( , ) ,( , )

Table 5 .
Another pilot sample was composed of 10 values of the stress measured in static tests at was generated by means of a stochastic simulation and its average is relatively close to the stress value 610 MPa recorded in static test pj  = 0.03[23].
has been generated around the value 660 MPa by means of a stochastic simulation, that is, n = 10.The sample jk  is given in  jk  are given in Table 6.Difference between mean values of s , s jk Table 6 reveal that the best fit to the static sample