# Statistical analysis of design codes calculation methods for punching sheer resistance in column-to-slab connections.

Abstract. This paper analyses the compliance of the design codes calculation methods for punching shear resistance in reinforced concrete slabs STR 2.05.05:2005, E DIN 1045-1, ENV 1992-1-1 EC 2, prEN 1992-1 [Final draft] EC 2, Model Code CEB-FIP 1990, BS 8110, ACI 318-99 to the experimental data. It has been analysed whether the difference in the results of the mean punching shear resistance received according to these methods and through experiments is statistically significant, when the level of significance value is 0,05. To analyse the significance of the difference of the means Student t test was used. An analysis was carried out to find out which methods show the least different resistance results from the experimental data. According to this analysis, a classification of methods was made. Student t test was applied to analyse in which methods the ratio between the punching shear resistance results obtained and the punching shear resistance results received through experiment is statistically insignificant. The level of significance value considered was 0,05.

It has been determined that almost in all cases the difference between the punching shear resistance results received experimentally and theoretically is statistically significant. It has also been found out that generally the punching shear resistance can be calculated by applying the prEN 1992-1 [Final draft] EC 2 method. The best method to describe the punching shear resistance in minimally reinforced slabs is ACI 318. The worst results are obtained by applying ENV 1992-1-1 EC 2 and E DIN 1045-1 methods.

Keywords: punching shear resistance of concrete slabs, design codes of concrete slabs, statistical analysis of punching shear.

1. Introduction

Most works [1-4] analyse the impact of the main parameters on the punching shear resistance in reinforced concrete column-to-slab connections under axial forces. These parameters are: punching shear resistance [f.sup.n.sub.c], reinforcement ratios [rho], effective depth of the slab d, column geometry (transverse section c and form). Calculation methods for punching shear resistance of reinforced concrete constructions provided in design codes of different countries and international codes differ as well as the results obtained through these calculations.

Some works [1] provide a comparison of design codes punching shear resistance calculation methods. However, no statistical analysis of the compliance of the latest edition design codes calculation methods to the experimental data was found in the existing literature. Therefore, this work provides a statistical analysis of the Model Code CEB-FIP 1990 [5], E DIN 1045-1 [6], prEN 1992-1 [Final draft] EC 2 [7], ENV 1992-1-1 EC 2 [8], BS 8110 [9], ACI 318-99 [10] and STR 2.05.05 [11] calculation methods as well as experimental data. Further in this article the STR 2.05.05:2005 method is abbreviated to STR, E DIN 1045-1 to DIN, ENV 1992-1-1 EC 2 to EC2, prEN 1992-1 (Final draft) EC 2 to EC2Dr, Model Code CEB-FIP 1990 to MC, BS 8110 to BS and ACI 318-99 to ACI. Experimental data used were provided in [12]. It was analysed in which methods the difference between the mean punching shear resistance calculated theoretically and obtained experimentally is significant statistically. Also, it was examined which methods give the most precise calculation of the punching shear resistance.

2. Design codes

Punching shear of slabs under axial forces in column-to-slab connection occurs when a punching cone is formed. The area of the punching cone makes a 26,6[degrees] to 45[degrees] angle to the horizontal column face [2, 4]. Based on this failure mechanism, design codes of different countries and international design codes suggest to use a half empirical critical section method to calculate the punching shear resistance in a slab. This method is based on the assumption that the slab fails when there is a vertical section at a certain distance from the column face which extends to the whole perimeter of the column-to-slab connection. The perimeter of this section on the slab surface is called critical perimeter (u). The punching shear in slab occurs when punching shear strength in critical section exceed the punching shear resistance of the concrete.

The distance of the critical section from the column face as well as the geometry of the critical perimeter differ in design codes of different countries and international codes (Fig 1). Concrete punching shear strength dependence on approximation of cylindrical compressive strength also differs. These quantities are not precise in reinforced concrete theory. The values of these quantities provided in codes are empirical, based on experimental results [13].

[FIGURE 1 OMITTED]

The main code parameters of calculating punching shear in slabs are provided in Oable 1. In this Table [f.sub.c]--cylindrical compressive strength in concrete, [f.sub.cu] = 1, 25 [f.sub.c]--cubical compressive strength in concrete. As one can see from Table 1, all methods approximate punching shear resistance in concrete by function [f.sup.n.sub.c], only exponent quantities are different. Differently from other methods, ACI 318 does not evaluate the impact of the longitudinal reinforcement and the scale factor on the punching shear strength. ACI 318 admits that the maximal punching shear strengths in a slab 0,5d from the column surface are of constant size and direct distribution. Other calculation codes evaluate the non-linear distribution of tangent stresses in column-to-slab connection by increasing the distance of the critical section from the column surface.

Further we concisely present different punching shear calculation methods when axial forces are located centrally.

The punching strength by MC, DIN, EC2Dr and STR methods may be calculated by the following expressions:

V = [xi]k [(100[rho][f.sub.c]).sup.1/3] ud, (1)

where [xi] = 0,12 by MC 90, [xi] = 0,14 by DIN, [xi] = 0,18 by EC2Dr and STR, u = 2 ([c.sub.1] + [c.sub.2]) + 4[pi]d by MC and EC2Dr, u = ([c.sub.1] + [c.sub.2]) + 3[pi]d by DIN and STR, k--values are provided in Table 1. The punching strength according to EC2 method may be calculated by the following formulas:

V = [[tau].ub.R]k (1,2 + 40[rho]) ud, (2)

where [[tau].sub.R]--concrete shear strength (MPa) [8], u = 2([c.sub.1] + [c.sub.2]) + 3[pi]d. The k--value is given in Table 1. The punching strength by BS 8110 methods is as follows:

V = [xi][(100[rho]).sup.1/3] [k.sup.0,25] [([f.sub.c]/25).sup.1/3] ud, (3)

where [xi] = 0,12, u = 2([c.sub.1] + [c.sub.2]) + 12d. The resistance of punching shear force by ACI is as follows:

V = (2 + 4/[beta]) [square root of [f.sub.c]]ud (kips), (4)

V = ([[alpha].sub.s]d / u + 2) [square root of [f.sub.c]]ud (kips), (5)

V = ud [4 square root of [f.sub.c]] (kips), (6)

where u = 2([c.sub.1] + [c.sub.2]) + 4d, [beta] is the ratio the long side to the short side of the concentration load (or columns), [[alpha].sub.s] is 40 for interior column.

3. Statistical analysis of data

Due to the spread of the punching shear resistance data obtained through experiments, theoretically obtained values will always differ from the experimental results. However, this difference can be statistically insignificant. If the spread of the experimental data is only achieved because of an accidental error, as it is known, the bigger the number of experiments performed, the closer the mean of the accidental error to zero. Then the proximity of the theoretical method to the experimental data can be compared by applying the difference:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (7)

where [V.sub.exp] and [V.sub.calc] are experimentally obtained and theoretically calculated values of punching shear resistance, [[bar.V.sub.exp]] and [[bar.V.sub.calc]]--the means of the experimentally obtained and theoretically calculated values of punching shear resistance. The smaller [DELTA]V, the closer the theoretically obtained values to the experimental data. If n [right arrow] [infinity], then [DELTA]V [right arrow] 0. In reality, the number of experimental data is always limited, that is why to verify the equality of the means hypothesis [H.sub.0] is put against a competing hypothesis [H.sub.1]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

where

[[bar.V.sub.calc]] [member of] {[[bar.V.sub.STR]], [[bar.V.sub.DIN]], [[bar.V.sub.EC2]], [[bar.V.sub.EC2Dr]], [[bar.V.sub.MC]], [[bar.V.sub.BS]], [[bar.V.sub.ACI]}, [[bar.V.sub.STR]], [[bar.V.sub.DIN]], [[bar.V.sub.EC2]], [[bar.V.sub.EC2Dr]], [[bar.V.sub.MC]], [[bar.V.sub.BS]], [[bar.V.sub.ACI]]--means of punching forces calculated according to STR, DIN, EC2, EC2Dr, MC, BS and ACI [5-11] methods.

It is important to analyse which methods allow more precise calculations of punching shear resistance. The test of accuracy is the [V.sub.calc] / [V.sub.exp] ratio mean [[bar.V.sub.calc] / [V.sub.exp]]. The closer this ratio to 1, the more precisely allows the theoretical method to calculate the punching shear. According to the closeness of the obtained [[barV.sub.calc] / [V.sub.exp]] values to 1, theoretical calculation methods can be ranged into giving the best and the worst calculations of the punching shear resistance.

Sampling

Samples of 7 different slabs were chosen from [12] for statistical analysis. Parameters of slabs and characteristics of materials are given in Table 2. In this Table, c--measurements of transverse column section (m), [rho]--relative area of tensile reinforcement (%), d--the useful height of reinforced concrete slabs (m), [f.sub.y]--reinforcement yield point (MPa), [f.sub.c]--compressive cylinder strength of concrete (MPa). Literature source [12] offers the estimate of the average compressive strength [f.sub.c] in concrete of each slab; however, it does not provide the estimation of standard deviation and the number of tested samples. Without these data, it is not possible to evaluate the influence of distribution of compressive strength in concrete of each slab on the sheer strength [V.sub.calc] that is calculated according to the theoretical model. Therefore, it is agreed that, for the purpose of further analysis, the compressive strength of concrete of the [i.sup.th] slab is equal to the estimation of the average of the compressive strength of this slab provided in [12].

According to STR, DIN, EC2, EC2Dr, MC, BS and ACI methods, if we use parameters of each sample, it is possible to calculate the theoretical punching shear resistance of a slab [V.sub.calc]. Concrete strength [f.sub.c] in each slab is different. As for the formulas (1)-(6), the analyzed formulas evaluate the impact of [f.sub.c] on the punching shear strength. That is why the theoretical punching shear strength values [V.sub.calc] of a certain sample from a certain slab calculated with different [f.sub.c] values are compared to the experimental punching shear strength [V.sub.exp] of the same sample from the same slab. Since additionally to theoretical and experimental punching shear strength values we also analyze [[bar.V.sub.calc] / [V.sub.exp]], in each sample additionally to [V.sub.exp] and [V.sub.calc] variables, we shall have [V.sub.calc] / [V.sub.exp] variables.

The data normality is verified by applying the Shapiro-Wilk W test. As shown in [14], this test is the best to verify the normality of the data.

The main statistical variable estimates: the minimal and the maximal values, the mean, standard deviation as well as the values of estimation test of hypotheses on the normality of data are given in Table 2. In this Table W--the obtained Shapiro-Wilk test values, and P--Shapiro-Wilk test P values. The W was calculated according to the method described in [15] and P values were taken from [15]. As shown in Table 2, the P values of the W with all variables except for [f.sub.c] and [V.sub.calc] of sample 1 and for [V.sub.MC] of sample 4 is higher than the usually applied significance level [alpha] = 0,05. That is why the theoretically and experimentally obtained punching shear values [V.sub.exp] and [V.sub.calc], except in sample 1, do not contradict the hypothesis that the data are distributed in a normal distribution.

Verification of the hypothesis about the equality of the means obtained experimentally and theoretically

When data are in normal distribution (8), we can apply the Student t test for independent samples when general set variances are unequal [16] to verify the hypothesis. First of all, the t statistics is calculated applying the formula [16]:

t = [[bar.V.sub.exp]] - [[bar.V.sub.calc]] / [square root of [S.sup.2.sub.exp] / n + [S.sup.2.sub.calc] / n], (9)

where

[S.sub.calc] [member of] {[S.sub.STR], [S.sub.DIN], [S.sub.EC2], [S.sub.EC2Dr], [S.sub.MC], [S.sub.BS], [S.sub.ACI]} - [V.sub.STR] - [V.sub.BS] standard sample deviations (Table 2).

Hypothesis [H.sub.0] is rejected if |t| > [t.sub.[alpha]/2] (k). Here [t.sub.[alpha]/2](k) is the critical value of [alpha]/2 level in Student distribution with k degree of freedom. It is supposed that [alpha] = 0,05. Results of verifying (8) hypothesis is given in Table 3.

Calculation degree of freedom k, which is the smallest whole number satisfying the condition:

k [less than or equal to] [([S.sup.2.sub.exp] / n + [S.sup.2.sub.calc] / n).sup.2] / ([S.sup.4.sub.exp]/[n.sup.3] + [S.sup.4.sub.calc] / [n.sup.3]). (10)

As shown in Table 3, the difference of experimentally and theoretically obtained means of punching shear forces, except for sample 3 [[bar.V.sub.EC2Dr]] = [[bar.V.sub.exp]] is statistically significant. Therefore, generally we can make a conclusion that we cannot get an accurate calculation of punching shear force by applying the mentioned methods.

Analysis of the accuracy of calculation methods

Further this article analyses which methods allow to make the most accurate calculations of punching shear force. [[bar.V.sub.calc] / [V.sub.exp]] ratios as well as error bands of these rations are given in Table 4. The confidence intervals of means are calculated by using t test. The value of significance level is 0,05. The ratio difference calculated with different methods:

[[bar.V.sub.calc] / [V.sub.exp]] - [[bar.V.sub.calc1] / [V.sub.exp]], (11)

here [[bar.V.sub.calc1]] [member of] {[[bar.V.sub.STR], [[bar.V.sub.DIN]], [[bar.V.sub.EC2]], [[bar.V.sub.EC2Dr]], [[bar.V.sub.MC]], [[bar.V.sub.BS]], [[bar.V.sub.ACI]]} and [[bar.V.sub.calc]] [not equal to] [[bar.V.sub.calc1]] can be statistically insignificant. That is why to verify the significance of the difference [H.sub.0] hypothesis is put against a competing hypothesis [H.sub.1]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

Verification of (12) hypothesis is done similarly as for hypothesis (8) applying (9) and (10) formulas, only instead of [[bar.V.sub.exp]] and [[bar.V.sub.calc]] we use [[bar.V.sub.calc] / [V.sub.exp]] and [[bar.V.sub.calc1] / [V.sub.exp]], and instead of [S.sub.exp] and [S.sub.calc] we use [S.sub.calc/exp] and [S.sub.calc1/exp]. Here {[S.sub.calc/exp], [S.sub.calc1/exp]} [member of] {[S.sub.STR/exp], [S.sub.EC2Dr/exp], [S.sub.MC/exp] [S.sub.BS/exp], [S.sub.ACI/exp]}, [S.sub.STR/exp] - [S.sub.ACI/exp] are estimates of standard [V.sub.calc] / [V.sub.exp] deviations given in Table 2. Due to abundant data t values and [t.sub.[alpha]/2](k) critical value in verifying the hypothesis (12) are not provided. Table 5 provides the final results of the verifying hypothesis (12).

This Table also shows the theoretical methods used to calculate the punching shear force ranged by the proximity of the obtained punching shear values to the experimentally received punching shear values. First in a row are the methods where calculated punching shear force is the least different from the punching shear force obtained experimentally.

Column 3 in Table 5 shows the methods adequate to ranges 1, 2 etc. Column 4 shows the methods where the mean of the ratio between the theoretical punching shear values and the experimental punching shear values is insignificantly different from the mean of the ratio of the theoretical and experimental punching shear values of the method given in column 3, ie here hypothesis (12) [H.sub.0] is in force.

As the results given in Table 5 show almost in all cases the punching shear force calculated by the EC2Dr method is the closest to the results obtained experimentally.

The [[bar.V.sub.EC2Dr] / [V.sub.exp]] - [[bar.V.sub.calc] / [V.sub.exp]] difference in samples 1, 2, 5 is statistically significant in all methods except for ECDr. The [[bar.V.sub.EC2Dr] / [V.sub.exp]] - [[barV.sub.ACI] / [V.sub.exp]] difference in samples 3 and 4 is statistically insignificant. Therefore, in this case we can state that ACI and EC2Dr methods similarly accurately calculate the punching shear force in respect to experimental results.

Punching shear force calculated by the ACI method is the closest to the experimental punching shear results obtained in samples 6 and 7. Besides, the [[bar.V.sub.ACI]] / [V.sub.exp]] - [[bar.V.sub.calc] / [V.sub.exp]] difference is statistically significant when [V.sub.calc] is calculated applying all the methods except ACI. These samples are special because reinforcement of slabs is minimal. It is known that in a minimally reinforced slab punching shear cone is 45[degrees], which corresponds to the punching shear angle in ACI method. In limited reinforcement the shear force taken over by the longitudinal reinforcement is not big. Most part of the shear force is taken over by the concrete which is in the area of the punching shear cone. Therefore, absence of evaluation of reinforcement ratio [rho] in ACI method does not cause a significant calculation error.

In this case, therefore, experimental results confirm the theoretical presumptions. This allows to make a conclusion that ACI method is the best to calculate the punching shear force in slabs. It is possible to notice from the data provided in Table 5 that, when the reinforcement percentage is high calculation results of punching shear strength by applying the ACI method are less correct than applying other methods if compared to experimental results. This is clearly seen from comparison of samples 1and 5 [[bar.V.sub.ACI] / [V.sub.exp]] - [[bar.V.sub.calc] / [V.sub.exp]] (the Table 5). When the amount of reinforcement is approximately 1 %, the ACI and the ECDr methods are equally good to calculate the punching shear force.

Analysis of the results given in Table 5 also clearly shows that when reinforcement is strong, ie samples 1 and 5 in the second position, according to [[bar.V.sub.calc] / [V.sub.exp]] proximity to 1, is [V.sub.calc] values calculated by the BS method, and the third in a row is the STR method. Since [[bar.V.sub.BS] / [V.sub.exp]] - [[bar.V.sub.calc] / [V.sub.exp]] and [[bar.V.sub.STR] / [V.sub.exp]] - [[bar.V.sub.calc] / [V.sub.exp]] differences are statistically insignificant, it is possible to state that, when reinforcement is bigger than 1,6 %, BS and STR methods are second in a row to make an accurate calculation of punching shear. When reinforcement is small, better results than ACI and EC2Dr are obtained by applying the BS method, which is clearly seen from samples 6 and 7 (Table 5).

Punching shear values obtained using EC2 and DIN methods are the most different from the experimental results. This is clearly seen in Table 5.

4. Conclusions

Generally, the difference of the punching shear force in slabs calculated applying STR; DIN; EC2; EC2Dr; MC; BS; ACI methods from the punching shear force in slabs obtained experimentally is statistically significant. This shows that none of the analysed methods allows an accurate calculation of punching shear force.

Generally, the method allowing the most accurate calculation of punching shear force is EC2Dr method.

When reinforcement is minimal, less than 0,5 %, ACI is the best to make an accurate calculation of punching shear force.

When reinforcement is [rho] [greater than or equal to] 1,6 %, BS and STR methods are second in a row to calculate punching shear accurately.

References

[1.] Albrecht, U. Design of flat slabs for punching--European and North American practices. Cement & Concrete Composites, 24, 2002, p. 531-538.

[2.] Vainiunas, P.; Popovas, V.; Jarmolajev, A. Punching shear behavior analysis of RC flat floor slab-to-column connection. Journal of Civil Engineering and Management, 8(2), 2002, p. 77-82.

[3.] Vainiunas, P.; Popovas, V.; Jarmolajev, A. Non-linear 3d modelling of RC slab punching shear failure. Journal of Civil Engineering and Management, 10(4), 2004, p. 311-316.

[4.] Salna, R.; Mareiukaitis, G.; Vainiunas, P. Estimation of factors influencing the punching shear strength of RC floor slabs. Journal of Civil Engineering and Management, 10, Suppl 2, 2004, p. 137-142.

[5.] CEB-FIP Model Code 1990. London: Thomas Telford Ltd, 2001. 307 p.

[6.] E DIN 1045-1: Tragwerke aus Beton und Stahlbeton und Spannbeton, Teil 1: Bemessung und Konstruktion, 1999.

[7.] Eurocode 2: Design of concrete structures--Part 1: General rules and rules for buildings. European Committee for Standardization. Pr-EN 1992-1 (Final draft), October 2001. 230 p.

[8.] Eurocode 2: Design of concrete structures--Part 1: General rules and rules for buildings. European Committee for Standardization. ENV 1992-1-1, December 1991. 254 p.

[9.] BS 8110 (1985) Structural Use of Concrete, Part 1: Code of Practice for Design and Construction. British Standards Institution. London, 1985.

[10.] ACI Committee 318: Building Code Requirements for Reinforced Concrete. Detroit. American Concrete Institute, 1999.

[11.] STR 2.05.05:2005: Design of concrete and reinforced concrete structures (Betoniniu ir gelzbetoniniu konstrukciju projektavimas). 2005 (in Lithuanian).

[12.] Nolting, D. Das Durchstanzen von Platten aus Stahlbeton--Tragverhalten, Berechnung, Bemessung. Heft 62, Technische Universitat Braunschweig, 1984. 174 p.

[13.] Piel, W. Zur Erhohung der Durchstanztragfahigkeit von Flachdeken mit Stahlverbundsystemen. Dissertation, Bergischen Universitat Wuppertal, 2004. 277 p.

[14.] Lemeshko, B. Yu.; Lemeshko, S. B. Comparative analysis of goodness of fit tests for normal distributions. Metrology ([TEXT NOT REPRODUCIBLE IN ANSCII.]), No 2, 2005, p. 3-24 (in Russian).

[15.] Stepnov, M. N.; Shavrin, A. V. Statistical methods for analysis of observations data of mechanical tests. Handbook. ([TEXT NOT REPRODUCIBLE IN ANSCII.]). Moscow: Mashinostrojenije, 2005. 400 p.

[16.] Sheskin, D. J. Handbook of parametric and nonparametric statistical procedures. Boca Raton, Chapman & Hall/CRC, 2000. 982 p.

KOLONOS-PLOKSTES JUNGTIES BE SKERSINIO ARMAVIMO VEIKIANT SUTELKTAJAI APKROVAI PRASPAUDZIAMOJO STIPRIO NORMATYVINIU SKAIEIAVIMO METODIKU STATISTINE ANALIZE

D. Zabulionis, D. Sakinis, P. Vainiunas

Santrauka

Darbe nagrinejamas gelzbetoniniu plokseiu praspaudziamojo stiprio skaieiavimo normatyviniu metodiku STR 2.05.05:2005, E DIN 1045-1, ENV 1992-1-1 EC 2, prEN 1992-1 [Final draft] EC 2, Model Code CEB-FIP 1990, BS 8110, AC 318 atitikimas eksperimentiniams duomenims. Isanalizuota, ar pagal sias metodikas apskaieiuotu ir eksperimentiskai nustatytu praspaudziamojo vidutiniu stiprio reiksmiu skirtumas statistiskai reiksmingas, kai reiksmingumo lygmuo yra 0,05. Vidurkiu skirtumo reiksmingumo analizei naudotas Stjudento t kriterijus. Isnagrineta, pagal kurias metodikas apskaieiuotos stiprio reiksmes maziausiai skiriasi nuo eksperimentiniu duomenu. Tuo remiantis metodikoms priskirti rangai. Taikant Stjudento t kriteriju, isanalizuota, pagal kurias metodikas apskaieiuotu praspaudziamojo stiprio reiksmiu santykis su eksperimentiskai nustatytomis praspaudziamojo stiprio reiksmemis statistiskai nereiksmingas. Reiksmingumo lygmuo imtas 0,05.

Nustatyta, kad beveik visais atvejais skirtumas tarp eksperimentiskai ir teoriskai apskaieiuotu praspaudziamojo stiprio reiksmiu statistiskai reiksmingas. Taip pat nustatyta, kad tiksliausiai praspaudziamaji stipri galima apskaieiuoti pagal prEN 1992-1 [Final draft] EC 2 metodika.

Reiksminiai zodziai: gelzbetoniniu plokseiu praspaudziamasis stipris, gelzbetoniniu plokseiu projektavimo normos, praspaudziamojo stiprio statistine analize.

Darius Zabulionis (1), Dainius Sakinis (2), Povilas Vainiunas (2)

Vilnius Gediminas Technical University, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

(1) Dept of Bridges and Special Structures. E-mail: dariusz@st.vtu.lt

(2) Dept of Reinforced Concrete and Masonry Structures. E-mail: gelz@st.vtu.lt

Darius ZABULIONIS. Doctor, research worker at the Dept of Bridges and Special Structures of Vilnius Gediminas Technical University, Vilnius, Lithuania. Doctor (2003). Author of over 9 publications. Research interests: mechanics of composite materials and structures.

Dainius SAKINIS. MSc (CE), PhD student (from 2003) at the Dept of Reinforced Concrete and Masonry Structures, Vilnius Gediminas Technical University, Vilnius, Lithuania. Research interests: mechanics of reinforced concrete, design of buildings.

Povilas VAINIUNAS. Doctor, Professor. Dean of Civil Engineering Faculty at Vilnius Gediminas Technical University, Vilnius, Lithuania. PhD (1970) from Kaunas Politechnical Institute (presently Kaunas Technological University). Chairman of national group of International Association for Bridge and Structural Engineering (IABSE). Former vice-president (1992-95) and board member (since 1995) of Association of European Civil Engineering Faculties (AECEF). Chairman of scientific committee of biennial intern conference ,Modern building materials, structures and techniques" held at VGTU, Lithuania. Author and co-author of over 70 research papers. Research interests: mechanics of reinforced concrete, theory of durability and reliability, design of buildings, development of territory planning and building coder system of Lithuania and real estate assessment.

Received 30 Aug 2005; accepted 14 March 2006
```Table 1. Expressions of the main punching shear parameters in
calculation codes

Parameters MC DIN EC2Dr EC2

Shear resistance 3[square root ([f.sub.c])]

Reinforcement 3[square root ([rho])] 1, 2+40[rho]
ratios

Scale factor k 1 + [square root (200/d)] 1, 6-d

Critical section 2d 1, 5d 2d 1, 5d

Parameters BS ACI318

Shear resistance 3[square root [square root
([f.sub.cu])] ([f.sub.c])]

Reinforcement [square root --
ratios ([rho])]

Scale factor k 4[square root --
(400/d)]

Critical section 1, 5d 0, 5d

Parameters STR

Shear resistance 3[square root
([f.sub.c])]

Reinforcement 3[square root
ratios ([rho])]

Scale factor k 1 + [square
root (200/d)]

Critical section 1, 5d

Table 2. The main statistical sample estimates and the values of
verification test for hypotheses about the normality of data

Sample Slab pa- Sample Sample
number rameters size variables

1 2 3 4

1 c=0,015; 20 [f.sub.c]
[rho]=2,53; [V.sub.exp]
d= 0,046; [V.sub.STR]
[f.sub.y] [V.sub.DIN]
= 361-467 [V.sub.EC2]
[V.sub.EC2Dr]
[V.sub.MC]
[V.sub.BS]
[V.sub.ACI]
[V.sub.STR]/[V.sub.exp]
[V.sub.DIN]/[V.sub.exp]
[V.sub.EC2]/[V.sub.exp]
[V.sub.EC2Dr]/[V.sub.exp]
[V.sub.MC]/[V.sub.exp]
[V.sub.BS]/[V.sub.exp]
[V.sub.ACI]/[V.sub.exp]

2 c=0,287; 8 [f.sub.c]
[rho]=1,06; [V.sub.exp]
d=0,114; [V.sub.STR]
[f.sub.y] [V.sub.DIN]
=399-483 [V.sub.EC2]
[V.sub.EC2Dr]
[V.sub.MC]
[V.sub.BS]
[V.sub.ACI]
[V.sub.STR]/[V.sub.exp]
[V.sub.DIN]/[V.sub.exp]
[V.sub.EC2]/[V.sub.exp]
[V.sub.EC2Dr]/[V.sub.exp]
[V.sub.MC]/[V.sub.exp]
[V.sub.BS]/[V.sub.exp]
[V.sub.ACI]/[V.sub.exp]

3 c=0,287; 13 [f.sub.c]
[rho]=1,15; [V.sub.exp]
d=0,114; [V.sub.STR]
[f.sub.y] [V.sub.DIN]
=328 [V.sub.EC2]
[V.sub.EC2Dr]
[V.sub.MC]
[V.sub.BS]
[V.sub.ACI]
[V.sub.STR]/[V.sub.exp]
[V.sub.DIN]/[V.sub.exp]
[V.sub.EC2]/[V.sub.exp]
[V.sub.EC2Dr]/[V.sub.exp]
[V.sub.MC]/[V.sub.exp]
[V.sub.BS]/[V.sub.exp]
[V.sub.ACI]/[V.sub.exp]

4 c=0,115; 6 [f.sub.c]
[rho]=0,722; [V.sub.exp]
d=0,057; [V.sub.STR]
[f.sub.y] [V.sub.DIN]
=300 [V.sub.EC2]
[V.sub.EC2Dr]
[V.sub.MC]
[V.sub.BS]
[V.sub.ACI]
[V.sub.STR]/[V.sub.exp]
[V.sub.DIN]/[V.sub.exp]
[V.sub.EC2]/[V.sub.exp]
[V.sub.EC2Dr]/[V.sub.exp]
[V.sub.MC]/[V.sub.exp]
[V.sub.BS]/[V.sub.exp]
[V.sub.ACI]/[V.sub.exp]

5 c=0,115; 6 [f.sub.c]
[rho]=1,625; [V.sub.exp]
d=0,057; [V.sub.STR]
[f.sub.y] [V.sub.DIN]
=300 [V.sub.EC2]
[V.sub.EC2Dr]
[V.sub.MC]
[V.sub.BS]
[V.sub.ACI]
[V.sub.STR]/[V.sub.exp]
[V.sub.DIN]/[V.sub.exp]
[V.sub.EC2]/[V.sub.exp]
[V.sub.EC2Dr]/[V.sub.exp]
[V.sub.MC]/[V.sub.exp]
[V.sub.BS]/[V.sub.exp]
[V.sub.ACI]/[V.sub.exp]

6 c=0,402; 16 [f.sub.c]
[rho]=0,37; [V.sub.exp]
d=0,356; [V.sub.STR]
[f.sub.y] [V.sub.DIN]
=309 - 515 [V.sub.EC2]
[V.sub.EC2Dr]
[V.sub.MC]
[V.sub.BS]
[V.sub.ACI]
[V.sub.STR]/[V.sub.exp]
[V.sub.DIN]/[V.sub.exp]
[V.sub.EC2]/[V.sub.exp]
[V.sub.EC2Dr]/[V.sub.exp]
[V.sub.MC]/[V.sub.exp]
[V.sub.BS]/[V.sub.exp]
[V.sub.ACI]/[V.sub.exp]

7 31 [f.sub.c]
[V.sub.exp]
[V.sub.STR]
[V.sub.DIN]
[V.sub.EC2]
[V.sub.EC2Dr]
[V.sub.MC]
[V.sub.BS]

7 c=0,402; [V.sub.ACI]
[rho]=0,39; [V.sub.STR]/[V.sub.exp]
d=0,356; [V.sub.DIN]/[V.sub.exp]
[f.sub.y] [V.sub.EC2]/[V.sub.exp]
=309-515 [V.sub.EC2Dr]/[V.sub.exp]
[V.sub.MC]/[V.sub.exp]
[V.sub.BS]/[V.sub.exp]
[V.sub.ACI]/[V.sub.exp]

Minimal Maximal
value (MN) value (MN)
(for (for
Sample [f.sub.c]- [f.sub.c] -
variables (MPa)) (MPa))

4 5 6

[f.sub.c] 2,090 x [10.sup.1] 2,910 x [10.sup.1]
[V.sub.exp] 5,600 x [10.sup.-2] 8,940 x [10.sup.-2]
[V.sub.STR] 5,230 x [10.sup.-2] 5,840 x [10.sup.-2]
[V.sub.DIN] 4,070 x [10.sup.-2] 4,540 x [10.sup.-2]
[V.sub.EC2] 3,960 x [10.sup.-2] 4,420 x [10.sup.-2]
[V.sub.EC2Dr] 6,130 x [10.sup.-2] 6,840 x [10.sup.-2]
[V.sub.MC] 4,100 x [10.sup.-2] 4,600 x [10.sup.-2]
[V.sub.BS] 5,500 x [10.sup.-2] 6,200 x [10.sup.-2]
[V.sub.ACI] 4,100 x [10.sup.-2] 4,900 x [10.sup.-2]
[V.sub.STR]/[V.sub.exp] 6,359 x [10.sup.-1] 9,656 x [10.sup.-1]
[V.sub.DIN]/[V.sub.exp] 4,946 x [10.sup.-1] 7,510 x [10.sup.-1]
[V.sub.EC2]/[V.sub.exp] 4,813 x [10.sup.-1] 7,308 x [10.sup.-1]
[V.sub.EC2Dr]/[V.sub.exp] 7,451 x [10.sup.-1] 1,131
[V.sub.MC]/[V.sub.exp] 4,970 x [10.sup.-1] 7,540 x [10.sup.-1]
[V.sub.BS]/[V.sub.exp] 6,710 x [10.sup.-1] 1,019
[V.sub.ACI]/[V.sub.exp] 5,180 x [10.sup.-1] 8,030 x [10.sup.-1]

[f.sub.c] 2,050 x [10.sup.-1] 2,540 x [10.sup.-1]
[V.sub.exp] 3,114 x [10.sup.-1] 3,923 x [10.sup.-1]
[V.sub.STR] 2,394 x [10.sup.-1] 2,571 x [10.sup.-1]
[V.sub.DIN] 1,862 x [10.sup.-1] 2,000 x [10.sup.-1]
[V.sub.EC2] 1,700 x [10.sup.-1] 1,826 x [10.sup.-1]
[V.sub.EC2Dr] 2,804 x [10.sup.-1] 3,012 x [10.sup.-1]
[V.sub.MC] 1,870 x [10.sup.-1] 2,010 x [10.sup.-1]
[V.sub.BS] 2,390 x [10.sup.-1] 2,560 x [10.sup.-1]
[V.sub.ACI] 2,520 x [10.sup.-1] 2,810 x [10.sup.-1]
[V.sub.STR]/[V.sub.exp] 6,419 x [10.sup.-1] 7,688 x [10.sup.-1]
[V.sub.DIN]/[V.sub.exp] 4,993 x [10.sup.-1] 5,980 x [10.sup.-1]
[V.sub.EC2]/[V.sub.exp] 4,558 x [10.sup.-1] 5,459 x [10.sup.-1]
[V.sub.EC2Dr]/[V.sub.exp] 7,519 x [10.sup.-1] 9,005 x [10.sup.-1]
[V.sub.MC]/[V.sub.exp] 5,010 x [10.sup.-1] 6,000 x [10.sup.-1]
[V.sub.BS]/[V.sub.exp] 6,400 x [10.sup.-1] 7,670 x [10.sup.-1]
[V.sub.ACI]/[V.sub.exp] 6,910 x [10.sup.-1] 8,170 x [10.sup.-1]

[f.sub.c] 2,340 x [10.sup.1] 2,840 x [10.sup.1]
[V.sub.exp] 2,455 x [10.sup.-1] 3,714 x [10.sup.-1]
[V.sub.STR] 2,571 x [10.sup.-1] 2,742 x [10.sup.-1]
[V.sub.DIN] 2,000 x [10.sup.-1] 2,133 x [10.sup.-1]
[V.sub.EC2] 1,816 x [10.sup.-1] 1,937 x [10.sup.-1]
[V.sub.EC2Dr] 3,011 x [10.sup.-1] 3,212 x [10.sup.-1]
[V.sub.MC] 2,010 x [10.sup.-1] 2,140 x [10.sup.-1]
[V.sub.BS] 2,560 x [10.sup.-1] 2,730 x [10.sup.-1]
[V.sub.ACI] 2,700 x [10.sup.-1] 2,970 x [10.sup.-1]
[V.sub.STR]/[V.sub.exp] 7,179 x [10.sup.-1] 1,117
[V.sub.DIN]/[V.sub.exp] 5,583 x [10.sup.-1] 8,688 x [10.sup.-1]
[V.sub.EC2]/[V.sub.exp] 5,071 x [10.sup.-1] 7,890 x [10.sup.-1]
[V.sub.EC2Dr]/[V.sub.exp] 8,409 x [10.sup.-1] 1,308
[V.sub.MC]/[V.sub.exp] 5,610 x [10.sup.-1] 8,720 x [10.sup.-1]
[V.sub.BS]/[V.sub.exp] 7,160 x [10.sup.-1] 1,114
[V.sub.ACI]/[V.sub.exp] 7,670 x [10.sup.-1] 1,210

[f.sub.c] 2,640 x [10.sup.1] 3,100 x [10.sup.1]
[V.sub.exp] 8,180 x [10.sup.-2] 9,390 x [10.sup.-2]
[V.sub.STR] 5,180 x [10.sup.-2] 5,470 x [10.sup.-2]
[V.sub.DIN] 4,030 x [10.sup.-2] 4,250 x [10.sup.-2]
[V.sub.EC2] 3,980 x [10.sup.-2] 4,200 x [10.sup.-2]
[V.sub.EC2Dr] 6,160 x [10.sup.-2] 6,500 x [10.sup.-2]
[V.sub.MC] 4,100 x [10.sup.-2] 4,300 x [10.sup.-2]
[V.sub.BS] 5,500 x [10.sup.-2] 5,800 x [10.sup.-2]
[V.sub.ACI] 6,200 x [10.sup.-2] 6,700 x [10.sup.-2]
[V.sub.STR]/[V.sub.exp] 5,821 x [10.sup.-1] 6,595 x [10.sup.-1]
[V.sub.DIN]/[V.sub.exp] 4,528 x [10.sup.-1] 5,129 x [10.sup.-1]
[V.sub.EC2]/[V.sub.exp] 4,472 x [10.sup.-1] 5,066 x [10.sup.-1]
[V.sub.EC2Dr]/[V.sub.exp] 6,924 x [10.sup.-1] 7,844 x [10.sup.-1]
[V.sub.MC]/[V.sub.exp] 4,620 x [10.sup.-1] 5,230 x [10.sup.-1]
[V.sub.BS]/[V.sub.exp] 6,140 x [10.sup.-1] 6,960 x [10.sup.-1]
[V.sub.ACI]/[V.sub.exp] 7,140 x [10.sup.-1] 8,040 x [10.sup.-1]

[f.sub.c] 2,630 x [10.sup.1] 3,130 x [10.sup.1]
[V.sub.exp] 9,960 x [10.sup.-2] 1,254 x [10.sup.-1]
[V.sub.STR] 6,780 x [10.sup.-2] 7,190 x [10.sup.-2]
[V.sub.DIN] 5,270 x [10.sup.-2] 5,590 x [10.sup.-2]
[V.sub.EC2] 4,940 x [10.sup.-2] 5,230 x [10.sup.-2]
[V.sub.EC2Dr] 8,070 x [10.sup.-2] 8,550 x [10.sup.-2]
[V.sub.MC] 5,400 x [10.sup.-2] 5,700 x [10.sup.-2]
[V.sub.BS] 7,200 x [10.sup.-2] 7,600 x [10.sup.-2]
[V.sub.ACI] 6,200 x [10.sup.-2] 6,700 x [10.sup.-2]
[V.sub.STR]/[V.sub.exp] 5,731 x [10.sup.-1] 6,808 x [10.sup.-1]
[V.sub.DIN]/[V.sub.exp] 4,457 x [10.sup.-1] 5,295 x [10.sup.-1]
[V.sub.EC2]/[V.sub.exp] 4,174 x [10.sup.-1] 4,959 x [10.sup.-1]
[V.sub.EC2Dr]/[V.sub.exp] 6,816 x [10.sup.-1] 8,098 x [10.sup.-1]
[V.sub.MC]/[V.sub.exp] 4,540 x [10.sup.-1] 5,400 x [10.sup.-1]
[V.sub.BS]/[V.sub.exp] 6,050 x [10.sup.-1] 7,180 x [10.sup.-1]
[V.sub.ACI]/[V.sub.exp] 5,370 x [10.sup.-1] 6,200 x [10.sup.-1]

[f.sub.c] 2,120 x [10.sup.1] 2,870 x [10.sup.1]
[V.sub.exp] 1,837 2,309
[V.sub.STR] 1,065 1,178
[V.sub.DIN] 8,280 x [10.sup.-1] 9,160 x [10.sup.-1]
[V.sub.EC2] 8,528 x [10.sup.-1] 9,434 x [10.sup.-1]
[V.sub.EC2Dr] 1,314 1,45
[V.sub.MC] 8,760 x [10.sup.-1] 9,690 x [10.sup.-1]
[V.sub.BS] 1,181 1,307
[V.sub.ACI] 1,551 1,804
[V.sub.STR]/[V.sub.exp] 4,675 x [10.sup.-1] 6,297 x [10.sup.-1]
[V.sub.DIN]/[V.sub.exp] 3,636 x [10.sup.-1] 4,898 x [10.sup.-1]
[V.sub.EC2]/[V.sub.exp] 3,745 x [10.sup.-1] 5,045 x [10.sup.-1]
[V.sub.EC2Dr]/[V.sub.exp] 5,768 x [10.sup.-1] 7,771 x [10.sup.-1]
[V.sub.MC]/[V.sub.exp] 3,850 x [10.sup.-1] 5,180 x [10.sup.-1]
[V.sub.BS]/[V.sub.exp] 5,190 x [10.sup.-1] 6,990 x [10.sup.-1]
[V.sub.ACI]/[V.sub.exp] 6,810 x [10.sup.-1] 9,560 x [10.sup.-1]

[f.sub.c] 2,210 x [10.sup.-1] 2,980 x [10.sup.-1]
[V.sub.exp] 1,668 2,669
[V.sub.STR] 1,099 1,214
[V.sub.DIN] 8,544 x [10.sup.-1] 9,440 x [10.sup.-1]
[V.sub.EC2] 8,698 x [10.sup.-1] 9,610 x [10.sup.-1]
[V.sub.EC2Dr] 1,35 1,498
[V.sub.MC] 9,040 x [10.sup.-1] 9,980 x [10.sup.-1]
[V.sub.BS] 1,219 1,347

[V.sub.ACI] 1,583 1,838
[V.sub.STR]/[V.sub.exp] 4,470 x [10.sup.-1] 6,626 x [10.sup.-1]
[V.sub.DIN]/[V.sub.exp] 3,477 x [10.sup.-1] 5,153 x [10.sup.-1]
[V.sub.EC2]/[V.sub.exp] 3,539 x [10.sup.-1] 5,246 x [10.sup.-1]
[V.sub.EC2Dr]/[V.sub.exp] 5,516 x [10.sup.-1] 8,176 x [10.sup.-1]
[V.sub.MC]/[V.sub.exp] 3,680 x [10.sup.-1] 5,450 x [10.sup.-1]
[V.sub.BS]/[V.sub.exp] 4,960 x [10.sup.-1] 7,350 x [10.sup.-1]
[V.sub.ACI]/[V.sub.exp] 6,470 x [10.sup.-1] 9,580 x [10.sup.-1]

Standard
deviation
Mean (MN) (MN)
(for (for
Sample [f.sub.c] - [f.sub.c] -
variables (MPa)) (MPa))

4 7 8

[f.sub.c] 2,580 x [10.sup.1] 2,247
[V.sub.exp] 7,328 x [10.sup.-2] 9,844 x [10.sup.-3]
[V.sub.STR] 5,605 x [10.sup.-2] 1,660 x [10.sup.-3]
[V.sub.DIN] 4,361 x [10.sup.-2] 1,292 x [10.sup.-3]
[V.sub.EC2] 4,243 x [10.sup.-2] 1,253 x [10.sup.-3]
[V.sub.EC2Dr] 6,570 x [10.sup.-2] 1,948 x [10.sup.-3]
[V.sub.MC] 4,380 x [10.sup.-2] 1,361 x [10.sup.-3]
[V.sub.BS] 5,930 x [10.sup.-2] 1,895 x [10.sup.-3]
[V.sub.ACI] 4,580 x [10.sup.-2] 2,093 x [10.sup.-3]
[V.sub.STR]/[V.sub.exp] 7,772 x [10.sup.-1] 9,726 x [10.sup.-2]
[V.sub.DIN]/[V.sub.exp] 6,045 x [10.sup.-1] 7,565 x [10.sup.-2]
[V.sub.EC2]/[V.sub.exp] 5,882 x [10.sup.-1] 7,362 x [10.sup.-2]
[V.sub.EC2Dr]/[V.sub.exp] 9,106 x [10.sup.-1] 1,140 x [10.sup.-1]
[V.sub.MC]/[V.sub.exp] 6,071 x [10.sup.-1] 7,594 x [10.sup.-2]
[V.sub.BS]/[V.sub.exp] 8,205 x [10.sup.-1] 1,027 x [10.sup.-1]
[V.sub.ACI]/[V.sub.exp] 6,358 x [10.sup.-1] 7,689 x [10.sup.-2]

[f.sub.c] 2,344 x [10.sup.-1] 1,666 x [10.sup.-1]
[V.sub.exp] 3,622 x [10.sup.-1] 2,793 x [10.sup.-2]
[V.sub.STR] 2,502 x [10.sup.-1] 6,012 x [10.sup.-3]
[V.sub.DIN] 1,946 x [10.sup.-1] 4,680 x [10.sup.-3]
[V.sub.EC2] 1,777 x [10.sup.-1] 4,267 x [10.sup.-3]
[V.sub.EC2Dr] 2,931 x [10.sup.-1] 7,047 x [10.sup.-3]
[V.sub.MC] 1,954 x [10.sup.-1] 4,627 x [10.sup.-3]
[V.sub.BS] 2,495 x [10.sup.-1] 5,806 x [10.sup.-3]
[V.sub.ACI] 2,696 x [10.sup.-1] 9,812 x [10.sup.-3]
[V.sub.STR]/[V.sub.exp] 6,937 x [10.sup.-1] 4,581 x [10.sup.-2]
[V.sub.DIN]/[V.sub.exp] 5,396 x [10.sup.-1] 3,564 x [10.sup.-2]
[V.sub.EC2]/[V.sub.exp] 4,926 x [10.sup.-1] 3,253 x [10.sup.-2]
[V.sub.EC2Dr]/[V.sub.exp] 8,126 x [10.sup.-1] 5,364 x [10.sup.-2]
[V.sub.MC]/[V.sub.exp] 5,418 x [10.sup.-1] 3,572 x [10.sup.-2]
[V.sub.BS]/[V.sub.exp] 6,919 x [10.sup.-1] 4,571 x [10.sup.-2]
[V.sub.ACI]/[V.sub.exp] 7,471 x [10.sup.-1] 4,542 x [10.sup.-2]

[f.sub.c] 2,565 x [10.sup.-1] 1,626
[V.sub.exp] 3,092 x [10.sup.-1] 3,670 x [10.sup.-2]
[V.sub.STR] 2,650 x [10.sup.-1] 5,569 x [10.sup.-3]
[V.sub.DIN] 2,061 x [10.sup.-1] 4,328 x [10.sup.-3]
[V.sub.EC2] 1,872 x [10.sup.-1] 3,927 x [10.sup.-3]
[V.sub.EC2Dr] 3,104 x [10.sup.-1] 6,520 x [10.sup.-3]
[V.sub.MC] 2,068 x [10.sup.-1] 4,318 x [10.sup.-3]
[V.sub.BS] 2,643 x [10.sup.-1] 5,663 x [10.sup.-3]
[V.sub.ACI] 2,824 x [10.sup.-1] 8,732 x [10.sup.-3]
[V.sub.STR]/[V.sub.exp] 8,701 x [10.sup.-1] 1,212 x [10.sup.-1]
[V.sub.DIN]/[V.sub.exp] 6,767 x [10.sup.-1] 9,430 x [10.sup.-2]
[V.sub.EC2]/[V.sub.exp] 6,146 x [10.sup.-1] 8,562 x [10.sup.-2]
[V.sub.EC2Dr]/[V.sub.exp] 1,019 x 10 1,420 x [10.sup.-1]
[V.sub.MC]/[V.sub.exp] 6,793 x [10.sup.-1] 9,465 x [10.sup.-2]
[V.sub.BS]/[V.sub.exp] 8,677 x [10.sup.-1] 1,208 x [10.sup.-1]
[V.sub.ACI]/[V.sub.exp] 9,272 x [10.sup.-1] 1,355 x [10.sup.-1]

[f.sub.c] 2,908 x [10.sup.-1] 1,693 x [10.sup.-1]
[V.sub.exp] 8,575 x [10.sup.-2] 5,278 x [10.sup.-3]
[V.sub.STR] 5,347 x [10.sup.-2] 1,060 x [10.sup.-3]
[V.sub.DIN] 4,162 x [10.sup.-2] 8,329 x [10.sup.-4]
[V.sub.EC2] 4,107 x [10.sup.-2] 8,066 x [10.sup.-4]
[V.sub.EC2Dr] 6,363 x [10.sup.-2] 1,268 x [10.sup.-3]
[V.sub.MC] 4,250 x [10.sup.-2] 8,367 x [10.sup.-4]
[V.sub.BS] 5,667 x [10.sup.-2] 1,033 x [10.sup.-3]
[V.sub.ACI] 6,500 x [10.sup.-2] 2,000 x [10.sup.-3]
[V.sub.STR]/[V.sub.exp] 6,253 x [10.sup.-1] 3,013 x [10.sup.-2]
[V.sub.DIN]/[V.sub.exp] 4,863 x [10.sup.-1] 2,341 x [10.sup.-2]
[V.sub.EC2]/[V.sub.exp] 4,804 x [10.sup.-1] 2,313 x [10.sup.-2]
[V.sub.EC2Dr]/[V.sub.exp] 7,438 x [10.sup.-1] 3,582 x [10.sup.-2]
[V.sub.MC]/[V.sub.exp] 4,958 x [10.sup.-1] 2,363 x [10.sup.-2]
[V.sub.BS]/[V.sub.exp] 6,595 x [10.sup.-1] 3,192 x [10.sup.-2]
[V.sub.ACI]/[V.sub.exp] 7,587 x [10.sup.-1] 3,415 x [10.sup.-2]

[f.sub.c] 2,920 x [10.sup.-1] 1,815 x [10.sup.-1]
[V.sub.exp] 1,130 x [10.sup.-1] 8,486 x [10.sup.-3]
[V.sub.STR] 7,018 x [10.sup.-2] 1,491 x [10.sup.-3]
[V.sub.DIN] 5,460 x [10.sup.-2] 1,152 x [10.sup.-3]
[V.sub.EC2] 5,112 x [10.sup.-2] 1,065 x [10.sup.-3]
[V.sub.EC2Dr] 8,352 x [10.sup.-2] 1,747 x [10.sup.-3]
[V.sub.MC] 5,567 x [10.sup.-2] 1,033 x [10.sup.-3]
[V.sub.BS] 7,417 x [10.sup.-2] 1,472 x [10.sup.-3]
[V.sub.ACI] 6,500 x [10.sup.-2] 1,789 x [10.sup.-3]
[V.sub.STR]/[V.sub.exp] 6,234 x [10.sup.-1] 3,782 x [10.sup.-2]
[V.sub.DIN]/[V.sub.exp] 4,849 x [10.sup.-1] 2,943 x [10.sup.-2]
[V.sub.EC2]/[V.sub.exp] 4,541 x [10.sup.-1] 2,756 x [10.sup.-2]
[V.sub.EC2Dr]/[V.sub.exp] 7,415 x [10.sup.-1] 4,501 x [10.sup.-2]
[V.sub.MC]/[V.sub.exp] 4,943 x [10.sup.-1] 3,018 x [10.sup.-2]
[V.sub.BS]/[V.sub.exp] 6,578 x [10.sup.-1] 3,974 x [10.sup.-2]
[V.sub.ACI]/[V.sub.exp] 5,772 x [10.sup.-1] 3,138 x [10.sup.-2]

[f.sub.c] 2,489 x [10.sup.-1] 2,195 x [10.sup.-1]
[V.sub.exp] 2,114 1,382 x [10.sup.-1]
[V.sub.STR] 1,122 3,334 x [10.sup.-2]
[V.sub.DIN] 8,728 x [10.sup.-1] 2,593 x [10.sup.-2]
[V.sub.EC2] 8,989 x [10.sup.-1] 2,671 x [10.sup.-2]
[V.sub.EC2Dr] 1,385 4,115 x [10.sup.-2]
[V.sub.MC] 9,233 x [10.sup.-1] 2,738 x [10.sup.-2]
[V.sub.BS] 1,245 3,708 x [10.sup.-2]
[V.sub.ACI] 1,679 7,449 x [10.sup.-2]
[V.sub.STR]/[V.sub.exp] 5,329 x [10.sup.-1] 3,919 x [10.sup.-2]
[V.sub.DIN]/[V.sub.exp] 4,145 x [10.sup.-1] 3,048 x [10.sup.-2]
[V.sub.EC2]/[V.sub.exp] 4,269 x [10.sup.-1] 3,139 x [10.sup.-2]
[V.sub.EC2Dr]/[V.sub.exp] 6,576 x [10.sup.-1] 4,837 x [10.sup.-2]
[V.sub.MC]/[V.sub.exp] 4,386 x [10.sup.-1] 3,219 x [10.sup.-2]
[V.sub.BS]/[V.sub.exp] 5,914 x [10.sup.-1] 4,340 x [10.sup.-2]
[V.sub.ACI]/[V.sub.exp] 7,973 x [10.sup.-1] 6,466 x [10.sup.-2]

[f.sub.c] 2,526 x [10.sup.-1] 1,838 x [10.sup.-1]
[V.sub.exp] 2,202 2,136 x [10.sup.-1]
[V.sub.STR] 1,148 2,784 x [10.sup.-2]
[V.sub.DIN] 8,929 x [10.sup.-1] 2,164 x [10.sup.-2]
[V.sub.EC2] 9,090 x [10.sup.-1] 2,204 x [10.sup.-2]
[V.sub.EC2Dr] 1,417 3,436 x [10.sup.-2]
[V.sub.MC] 9,445 x [10.sup.-1] 2,287 x [10.sup.-2]
[V.sub.BS] 1,274 3,098 x [10.sup.-2]

[V.sub.ACI] 1,691 6,155 x [10.sup.-2]
[V.sub.STR]/[V.sub.exp] 5,258 x [10.sup.-1] 5,015 x [10.sup.-2]
[V.sub.DIN]/[V.sub.exp] 4,090 x [10.sup.-1] 3,900 x [10.sup.-2]
[V.sub.EC2]/[V.sub.exp] 4,163 x [10.sup.-1] 3,970 x [10.sup.-2]
[V.sub.EC2Dr]/[V.sub.exp] 6,488 x [10.sup.-1] 6,188 x [10.sup.-2]
[V.sub.MC]/[V.sub.exp] 4,326 x [10.sup.-1] 4,134 x [10.sup.-2]
[V.sub.BS]/[V.sub.exp] 5,833 x [10.sup.-1] 5,560 x [10.sup.-2]
[V.sub.ACI]/[V.sub.exp] 7,745 x [10.sup.-1] 7,367 x [10.sup.-2]

Sample
variables W P

4 9 10

[f.sub.c] 0,781 0,000
[V.sub.exp] 0,966 0,661
[V.sub.STR] 0,893 0,031
[V.sub.DIN] 0,889 0,026
[V.sub.EC2] 0,893 0,031
[V.sub.EC2Dr] 0,892 0,029
[V.sub.MC] 0,916 0,086
[V.sub.BS] 0,877 0,016
[V.sub.ACI] 0,874 0,014
[V.sub.STR]/[V.sub.exp] 0,963 0,601
[V.sub.DIN]/[V.sub.exp] 0,963 0,600
[V.sub.EC2]/[V.sub.exp] 0,963 0,600
[V.sub.EC2Dr]/[V.sub.exp] 0,963 0,601
[V.sub.MC]/[V.sub.exp] 0,962 0,594
[V.sub.BS]/[V.sub.exp] 0,962 0,603
[V.sub.ACI]/[V.sub.exp] 0,972 0,803

[f.sub.c] 0,956 0,776
[V.sub.exp] 0,923 0,456
[V.sub.STR] 0,950 0,710
[V.sub.DIN] 0,951 0,725
[V.sub.EC2] 0,952 0,734
[V.sub.EC2Dr] 0,952 0,730
[V.sub.MC] 0,959 0,802
[V.sub.BS] 0,944 0,655
[V.sub.ACI] 0,953 0,737
[V.sub.STR]/[V.sub.exp] 0,869 0,147
[V.sub.DIN]/[V.sub.exp] 0,869 0,146
[V.sub.EC2]/[V.sub.exp] 0,869 0,147
[V.sub.EC2Dr]/[V.sub.exp] 0,869 0,147
[V.sub.MC]/[V.sub.exp] 0,870 0,151
[V.sub.BS]/[V.sub.exp] 0,871 0,153
[V.sub.ACI]/[V.sub.exp] 0,902 0,303

[f.sub.c] 0,924 0,281
[V.sub.exp] 0,930 0,341
[V.sub.STR] 0,927 0,312
[V.sub.DIN] 0,929 0,327
[V.sub.EC2] 0,929 0,328
[V.sub.EC2Dr] 0,929 0,336
[V.sub.MC] 0,916 0,224
[V.sub.BS] 0,923 0,278
[V.sub.ACI] 0,930 0,339
[V.sub.STR]/[V.sub.exp] 0,883 0,078
[V.sub.DIN]/[V.sub.exp] 0,883 0,078
[V.sub.EC2]/[V.sub.exp] 0,883 0,078
[V.sub.EC2Dr]/[V.sub.exp] 0,883 0,078
[V.sub.MC]/[V.sub.exp] 0,882 0,076
[V.sub.BS]/[V.sub.exp] 0,883 0,077
[V.sub.ACI]/[V.sub.exp] 0,879 0,068

[f.sub.c] 0,878 0,261
[V.sub.exp] 0,794 0,052
[V.sub.STR] 0,886 0,296
[V.sub.DIN] 0,856 0,177
[V.sub.EC2] 0,883 0,285
[V.sub.EC2Dr] 0,864 0,205
[V.sub.MC] 0,701 0,006
[V.sub.BS] 0,915 0,473
[V.sub.ACI] 0,823 0,094
[V.sub.STR]/[V.sub.exp] 0,900 0,373
[V.sub.DIN]/[V.sub.exp] 0,900 0,373
[V.sub.EC2]/[V.sub.exp] 0,899 0,371
[V.sub.EC2Dr]/[V.sub.exp] 0,899 0,371
[V.sub.MC]/[V.sub.exp] 0,904 0,397
[V.sub.BS]/[V.sub.exp] 0,901 0,383
[V.sub.ACI]/[V.sub.exp] 0,950 0,739

[f.sub.c] 0,950 0,743
[V.sub.exp] 0,961 0,829
[V.sub.STR] 0,950 0,742
[V.sub.DIN] 0,944 0,691
[V.sub.EC2] 0,942 0,674
[V.sub.EC2Dr] 0,946 0,709
[V.sub.MC] 0,915 0,473
[V.sub.BS] 0,958 0,804
[V.sub.ACI] 0,933 0,607
[V.sub.STR]/[V.sub.exp] 0,988 0,984
[V.sub.DIN]/[V.sub.exp] 0,988 0,984
[V.sub.EC2]/[V.sub.exp] 0,988 0,984
[V.sub.EC2Dr]/[V.sub.exp] 0,988 0,984
[V.sub.MC]/[V.sub.exp] 0,987 0,982
[V.sub.BS]/[V.sub.exp] 0,988 0,984
[V.sub.ACI]/[V.sub.exp] 0,963 0,845

[f.sub.c] 0,963 0,721
[V.sub.exp] 0,948 0,461
[V.sub.STR] 0,957 0,613
[V.sub.DIN] 0,957 0,614
[V.sub.EC2] 0,957 0,610
[V.sub.EC2Dr] 0,957 0,613
[V.sub.MC] 0,957 0,607
[V.sub.BS] 0,958 0,626
[V.sub.ACI] 0,959 0,642
[V.sub.STR]/[V.sub.exp] 0,956 0,590
[V.sub.DIN]/[V.sub.exp] 0,956 0,589
[V.sub.EC2]/[V.sub.exp] 0,956 0,586
[V.sub.EC2Dr]/[V.sub.exp] 0,956 0,590
[V.sub.MC]/[V.sub.exp] 0,955 0,581
[V.sub.BS]/[V.sub.exp] 0,955 0,570
[V.sub.ACI]/[V.sub.exp] 0,930 0,244

[f.sub.c] 0,973 0,601
[V.sub.exp] 0,973 0,598
[V.sub.STR] 0,973 0,610
[V.sub.DIN] 0,973 0,618
[V.sub.EC2] 0,973 0,617
[V.sub.EC2Dr] 0,973 0,616
[V.sub.MC] 0,973 0,603
[V.sub.BS] 0,973 0,608

[V.sub.ACI] 0,974 0,624
[V.sub.STR]/[V.sub.exp] 0,954 0,195
[V.sub.DIN]/[V.sub.exp] 0,954 0,195
[V.sub.EC2]/[V.sub.exp] 0,954 0,197
[V.sub.EC2Dr]/[V.sub.exp] 0,954 0,196
[V.sub.MC]/[V.sub.exp] 0,953 0,190
[V.sub.BS]/[V.sub.exp] 0,954 0,197
[V.sub.ACI]/[V.sub.exp] 0,966 0,406

Table 3. Results of verifying hypothesis (8)

Sample Sample t [t.sub.[alpha]/2] [H.sub.0]
number variables

2 [V.sub.STR] -11,090 2,306 rejected
[V.sub.DIN] -16,742 2,365
[V.sub.EC2] -18,477 2,365
[V.sub.EC2Dr] -6,792 2,306
[V.sub.MC] -16,670 2,365
[V.sub.BS] -11,177 2,306
[V.sub.ACI] -8,848 2,262

3 [V.sub.STR] -4,293 2,160 rejected
[V.sub.DIN] -10,055 2,179
[V.sub.EC2] -11,918 2,179
[V.sub.EC2Dr] 0,116 2,160
[V.sub.MC] -9,983 2,179
[V.sub.BS] -4,355 2,160
[V.sub.ACI] -2,560 2,160

4 [V.sub.STR] -14,689 2,571 rejected
[V.sub.DIN] -20,231 2,571
[V.sub.EC2] -20,499 2,571
[V.sub.EC2Dr] -9,980 2,447
[V.sub.MC] -19,824 2,571
[V.sub.BS] -13,246 2,571
[V.sub.ACI] -9,005 2,447

5 [V.sub.STR] -12,178 2,571 rejected
[V.sub.DIN] -16,709 2,571
[V.sub.EC2] -17,729 2,571
[V.sub.EC2Dr] -8,340 2,571
[V.sub.MC] -16,433 2,571
[V.sub.BS] -11,049 2,571
[V.sub.ACI] -13,562 2,571

6 [V.sub.STR] -27,910 2,110 rejected
[V.sub.DIN] -35,311 2,120
[V.sub.EC2] -34,532 2,120
[V.sub.EC2Dr] -20,234 2,101
[V.sub.MC] -33,809 2,120
[V.sub.BS] -24,295 2,110
[V.sub.ACI] -11,100 2,069

7 [V.sub.STR] -27,260 2,040 rejected
[V.sub.DIN] -33,967 2,040
[V.sub.EC2] -33,543 2,040
[V.sub.EC2Dr] -20,226 2,037
[V.sub.MC] -32,609 2,040
[V.sub.BS] -23,961 2,040
[V.sub.ACI] -12,798 2,030

Table 4. Values of [bar.[V.sub.calc]/[V.sub.exp]] ratios

Sample Sample [bar.[V.sub.calc]/
number variables [V.sub.exp]]
values

1 [V.sub.STR]/[V.sub.exp] 0,777
[V.sub.DIN]/[V.sub.exp] 0,604
[V.sub.EC2]/[V.sub.exp] 0,588
[V.sub.EC2Dr]/[V.sub.exp] 0,911
[V.sub.MC]/[V.sub.exp] 0,607
[V.sub.BS]/[V.sub.exp] 0,821
[V.sub.ACI]/[V.sub.exp] 0,636

2 [V.sub.STR]/[V.sub.exp] 0,694
[V.sub.DIN]/[V.sub.exp] 0,540
[V.sub.EC2]/[V.sub.exp] 0,493
[V.sub.EC2Dr]/[V.sub.exp] 0,813
[V.sub.MC]/[V.sub.exp] 0,542
[V.sub.BS]/[V.sub.exp] 0,692
[V.sub.ACI]/[V.sub.exp] 0,747

3 [V.sub.STR]/[V.sub.exp] 0,870
[V.sub.DIN]/[V.sub.exp] 0,677
[V.sub.EC2]/[V.sub.exp] 0,615
[V.sub.EC2Dr]/[V.sub.exp] 1,019
[V.sub.MC]/[V.sub.exp] 0,679
[V.sub.BS]/[V.sub.exp] 0,868
[V.sub.ACI]/[V.sub.exp] 0,927

4 [V.sub.STR]/[V.sub.exp] 0,625
[V.sub.DIN]/[V.sub.exp] 0,486
[V.sub.EC2]/[V.sub.exp] 0,480
[V.sub.EC2Dr]/[V.sub.exp] 0,744
[V.sub.MC]/[V.sub.exp] 0,496
[V.sub.BS]/[V.sub.exp] 0,660
[V.sub.ACI]/[V.sub.exp] 0,759

5 [V.sub.STR]/[V.sub.exp] 0,623
[V.sub.DIN]/[V.sub.exp] 0,485
[V.sub.EC2]/[V.sub.exp] 0,454
[V.sub.EC2Dr]/[V.sub.exp] 0,742
[V.sub.MC]/[V.sub.exp] 0,494
[V.sub.BS]/[V.sub.exp] 0,658
[V.sub.ACI]/[V.sub.exp] 0,577

6 [V.sub.STR]/[V.sub.exp] 0,533
[V.sub.DIN]/[V.sub.exp] 0,414
[V.sub.EC2]/[V.sub.exp] 0,427
[V.sub.EC2Dr]/[V.sub.exp] 0,658
[V.sub.MC]/[V.sub.exp] 0,439
[V.sub.BS]/[V.sub.exp] 0,591
[V.sub.ACI]/[V.sub.exp] 0,797

7 [V.sub.STR]/[V.sub.exp] 0,526
[V.sub.DIN]/[V.sub.exp] 0,409
[V.sub.EC2]/[V.sub.exp] 0,416
[V.sub.EC2Dr]/[V.sub.exp] 0,649
[V.sub.MC]/[V.sub.exp] 0,433
[V.sub.BS]/[V.sub.exp] 0,583
[V.sub.ACI]/[V.sub.exp] 0,774

Sample [bar.[V.sub.calc]/ [bar.[V.sub.calc]/
variables [V.sub.exp]] [V.sub.exp]]
the lowest the highest

[V.sub.STR]/[V.sub.exp] 0,223 0,728
[V.sub.DIN]/[V.sub.exp] 0,396 0,567
[V.sub.EC2]/[V.sub.exp] 0,412 0,551
[V.sub.EC2Dr]/[V.sub.exp] 0,089 0,854
[V.sub.MC]/[V.sub.exp] 0,393 0,569
[V.sub.BS]/[V.sub.exp] 0,180 0,769
[V.sub.ACI]/[V.sub.exp] 0,364 0,597

[V.sub.STR]/[V.sub.exp] 0,657 0,730
[V.sub.DIN]/[V.sub.exp] 0,511 0,568
[V.sub.EC2]/[V.sub.exp] 0,467 0,518
[V.sub.EC2Dr]/[V.sub.exp] 0,770 0,855
[V.sub.MC]/[V.sub.exp] 0,513 0,570
[V.sub.BS]/[V.sub.exp] 0,656 0,728
[V.sub.ACI]/[V.sub.exp] 0,711 0,783

[V.sub.STR]/[V.sub.exp] 0,795 0,945
[V.sub.DIN]/[V.sub.exp] 0,618 0,735
[V.sub.EC2]/[V.sub.exp] 0,561 0,668
[V.sub.EC2Dr]/[V.sub.exp] 0,931 1,107
[V.sub.MC]/[V.sub.exp] 0,620 0,738
[V.sub.BS]/[V.sub.exp] 0,793 0,943
[V.sub.ACI]/[V.sub.exp] 0,843 1,011

[V.sub.STR]/[V.sub.exp] 0,598 0,653
[V.sub.DIN]/[V.sub.exp] 0,465 0,508
[V.sub.EC2]/[V.sub.exp] 0,459 0,502
[V.sub.EC2Dr]/[V.sub.exp] 0,711 0,777
[V.sub.MC]/[V.sub.exp] 0,474 0,517
[V.sub.BS]/[V.sub.exp] 0,630 0,689
[V.sub.ACI]/[V.sub.exp] 0,727 0,790

[V.sub.STR]/[V.sub.exp] 0,589 0,658
[V.sub.DIN]/[V.sub.exp] 0,458 0,512
[V.sub.EC2]/[V.sub.exp] 0,429 0,479
[V.sub.EC2Dr]/[V.sub.exp] 0,700 0,783
[V.sub.MC]/[V.sub.exp] 0,467 0,522
[V.sub.BS]/[V.sub.exp] 0,621 0,694
[V.sub.ACI]/[V.sub.exp] 0,548 0,606

[V.sub.STR]/[V.sub.exp] 0,511 0,555
[V.sub.DIN]/[V.sub.exp] 0,397 0,432
[V.sub.EC2]/[V.sub.exp] 0,409 0,444
[V.sub.EC2Dr]/[V.sub.exp] 0,631 0,685
[V.sub.MC]/[V.sub.exp] 0,421 0,457
[V.sub.BS]/[V.sub.exp] 0,567 0,616
[V.sub.ACI]/[V.sub.exp] 0,761 0,833

[V.sub.STR]/[V.sub.exp] 0,506 0,546
[V.sub.DIN]/[V.sub.exp] 0,393 0,425
[V.sub.EC2]/[V.sub.exp] 0,400 0,432
[V.sub.EC2Dr]/[V.sub.exp] 0,624 0,674
[V.sub.MC]/[V.sub.exp] 0,416 0,449
[V.sub.BS]/[V.sub.exp] 0,561 0,606
[V.sub.ACI]/[V.sub.exp] 0,745 0,804

Table 5. Results of verifying hypothesis (12)

Sample Range of Calculation Methods when (6)
number calculation method [H.sub.0] hypothesis is
method accepted

1 2 3 4

1 1 EC2Dr
2 BS STR
3 STR BS
4 ACI DIN; EC2; MC
5 MC DIN; EC2; ACI
6 DIN EC2; MC; ACI
7 EC2 DIN; MC; ACI

2 1 EC2Dr
2 ACI
3 STR BS
4 BS STR
5 MC DIN
6 DIN MC
7 EC2

3 1 EC2Dr ACI
2 ACI EC2Dr; STR; BS
3 STR BS; ACI
4 BS ACI; STR
5 MC DIN; EC2
6 DIN MC; EC2
7 EC2 DIN; MC

4 1 ACI EC2Dr
2 EC2Dr ACI
3 BS STR
4 STR BS
5 MC DIN; EC2
6 DIN MC; EC2
7 EC2 MC; DIN

5 1 EC2Dr
2 BS STR
3 STR BS
4 ACI
5 MC DIN
6 DIN EC2; MC
7 EC2 DIN

6 1 ACI
2 EC2Dr
3 BS
4 STR
5 MC EC2
6 EC2 MC; DIN
7 DIN EC2

7 1 ACI
2 EC2Dr
3 BS
4 STR
5 MC EC2
6 EC2 MC; DIN
7 DIN EC2
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