official exhibit - ner050-00-bd01 - reineck et al ...tests on structural concrete beams without and...

UNITED STATES OF AMERICA NUCLEAR REGULATORY COMMISSION

ATOMIC SAFETY AND LICENSING BOARD

In the Matter of NEXTERA ENERGY SEABROOK, LLC (Seabrook Station, Unit 1)

Docket No. 50-443-LA-2 ASLBP No. 17-953-02-LA-BD01

Hearing Exhibit

Exhibit Number:

Exhibit Title:

NER050

Reineck et al., “Research Report: Extended Databases with Shear Tests on Structural Concrete Beams without and with Stirrups for the Assessment of Shear Design Procedures” (Mar. 2010)

Research Report

PART 1: Survey on the extended databases with shear tests on structural concrete beams with and without stirrups

Karl - Heinz Reineck; Daniel A. Kuchma; Birol Fitik

Table of contents

1 Introduction p. 1-3

2 Overview of the databases and procedure p. 1-5

3 Conversion factors of strength values for concrete p. 1-9

3.1 Concrete compressive strength p. 1-9 3.1.1 Definition 3.1.2 Conversion of strength values derived from control specimens 3.1.3 Characteristic value of compressive strength

3.1.4 Conversion of the ACI cylinder compressive strength f´c 3.1.5 Control of conversion factors using the databases

3.2 Concrete tensile strength p. 1-19 3.2.1 Definition 3.2.2 Conversion of strength values of control specimens 3.2.3 Calculated values of the uniaxial concrete tensile strength

3.2.4 Characteristic value of the concrete tensile strength 3.2.5 Verification of conversion factors using the databases

4 Criteria for the evaluation and selection of tests p. 1-29

4.1 Generally valid criteria p. 1-29

4.2 Special criteria for beams with stirrups p. 1-30

4.3 Special criteria for prestressed concrete members p. 1-31

4.4 Assessment of calculated flexural failures (kon8) p. 1-31 4.4.1 General procedure

4.4.2 Calculation of ultimate bending moment of reinforced concrete beams 4.4.3 Calculation of ultimate bending moment of reinforced concrete beams with axial tension 4.4.4 Calculation of ultimate bending moment of prestressed concrete beams

4.5 Assessment of calculated anchorage failures at end support (kon11) p. 1-37

4.5.1 General Procedure

4.5.2 Reinforced concrete beams without stirrups

4.5.3 Reinforced concrete beams with stirrups

4.5.4 Prestressed concrete beams without stirrups 4.5.4.1 Determination of the tension chord force 4.5.4.2 Assessment without consideration of non-tensioned reinforcement 4.5.4.3 Assessment with consideration of the non-tensioned reinforcement

4.5.5 Prestressed concrete beams with stirrups

4.6 Determination of inner lever arm at shear failure p. 1-41

Research Report on extended shear databases - Part 1

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4.6.1 Introduction

4.6.2 Calculation of inner lever arm for reinforced concrete beams

4.6.3 Calculation of inner lever arm for p.c.c- beams without reinforcing steel

4.6.4 Calculation of the inner lever arm for p.c-beams with reinforcing steel

5 Procedure for evaluation of tests and of comparisons with design approaches p. 1-47

5.1 Consideration of self-weight of beams subjected to point loads p. 1-47

5.1.1 Problem

5.1.2 Location of the failure crack for reinforced concrete members without shear reinforcement

5.1.3 Determination of the shear force due to self-weight

5.2 Determination of the ultimate shear force of beams with a uniformly distributed load and consideration of the self-weight of the beams p. 1-51

5.2.1 Statement of problem

5.2.2 Slenderness of beams subjected to a uniformly distributed load

5.2.3 Location of the failure crack for reinforced concrete members without shear reinforcement 5.2.3.1 Determination of the distance xr of the crack 5.2.3.2 Determination of the distance xou of the crack

5.2.4 Calculation of the shear force due to self-weight

5.3 Determination of the distance xr of the crack for prestressed concrete members subjected to

point loads p. 1-59

5.4 Procedure of comparison with design equations p. 1-61

5.4.1 Definition of model safety factor

5.4.2 Statistical evaluations

References p. 1-63

Part I of the report comprises pages 1-1 to 1- 65

List of Tables: Table 1-1 .............................................................................................................................. p. 1- 6

Table 1-2 .............................................................................................................................. p. 1-11

Table 1-3 .............................................................................................................................. p. 1-19

Table 1-4 .............................................................................................................................. p. 1-50

Table 1-5 .............................................................................................................................. p. 1-58

Attachment 1-1: Notations .......................................................................................................... p. 1-8

Survey on the extended databases with shear tests on structural concrete beams with and without stirrups

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1 Introduction Part 1 covers overriding aspects that apply to all parts of the final report.

Chapter 2 provides a general overview of the various databases and the selected procedures.

Chapter 3 contains a detailed compilation of the many conversion factors for the concrete compressive strength and the concrete tensile strength derived from the various control specimens. It is very important that only a single value for each is handed over to the evaluation databases, that is, the uniaxial concrete compressive strength f1c and the uniaxial concrete tensile strength f1ct. With this approach, the inconsistencies of previous databases of various authors are eliminated. If then in the evaluation database for design equations the cylinder strength is required, it is recomputed from the uniaxial concrete compressive strength f1c. Finally, the conversion factors are verified by comparing them with the available data from all the databases using the strength values derived in parallel from different control specimens, e.g. cubes and cylinders for the concrete compressive strength or in parallel from the flexural tensile strength and tensile splitting strength.

Chapter 4 describes and justifies the many criteria for the control of the tests and their selection for use in evaluat-ing formula for calculating strength. The two most important criteria are, firstly as to whether a flexural failure is calculated to have occurred instead of a shear failure, and secondly as to whether the calculated anchorage of the longitudinal reinforcement was sufficient to ensure that an anchorage failure did not occur.

Chapter 5 explains the procedure for the evaluations of the tests and the comparisons with design equations. Thereby, a new proposal is made for how to consider member self-weight for the determination of the ultimate shear force. Furthermore in the case of test beams subjected to a uniformly distributed load, the ultimate shear force is defined as the shear force at the failure location in contrast to the shear force at the distance d from the face of the support.


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2 Overview of the databases and procedure All data has been recently extracted from either original test reports or from original author papers in respected professional journals such as the ACI Structural Journal. Recollecting data from original reports or papers was often essential since previously published summary tables have often been inconsistent and incorrect and mostly did not include all the information that is required to assess the value of the data. Furthermore, the majority of the previously published spreadsheets were limited to parameters that are used in design equations for shear capacity, such as in the case of members without shear reinforcement in which only the values for l, d, fc and a/d are often needed. Therefore, it would have been impossible to perform some control checks, such as for checking the flexural capacity if original data sources had not been used.

The data was collected and analyzed using the spreadsheet program Microsoft Excel. In the process several files have been generated, and an overview of the different files is provided in Table 1-1. Each file was given a distinct name so that it is possible to recognize what types of tests are contained and whether it is a data collection file, a data control file or a data evaluation file. Thus the file named vuct-PC-gl-DK_sl contains the data control of slender prestressed concrete members without stirrups subjected to a uniformly distributed load.


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Table 1-1: Overview of the databases and definitions of abbreviations

members without

stirrups

(vuct)

members

with stirrups

(vsw)

members with in-clined stirrups

(vswgB)

Reinforced Concrete (RC)

vuct-RC

vuct-RC-gl

vuct-RC-N

vsw-RC

vsw-RC-gl

vsw-RC-N

vswgB-RC

Prestressed Concrete (PC)

vuct-PC

vuct-PC-gl vsw-PC -

Light-weight Con-crete (LC)

vuct-LC vsw-LC -

Prestressed Light-weight Concrete

(LPC) vuct-LPC vsw-LPC -

The following abbreviations are used:

1. abbreviation: vuct = members without stirrups

vsw = member with stirrups

vsw-gB = members with inclined stirrups

2. abbreviation: RC = Reinforced concrete members

RC_Plain = RC-members with plain stirrups or bars

RC_Ribb = RC-members with ribbed stirrups or bars

PC = Prestressed concrete members

PC_Pre = Pretensioned prestressed concrete members

PC_Post = Post-tensioning prestressed concrete members

LC = Light-weight concrete

LPC = Prestressed light-weight concrete members

3. abbreviation (no abbreviation) = Beams subjected to point loads

gl = Beams subjected to uniformly distributed load

N = Beams with axial tension

4. abbreviation: DS = collection of data

DK_sl = data control of slender beams

A = evaluation of tests


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The data collection is subdivided into ten basic parts:

1) General information: In this part an internal number is assigned to the test beam, the author of the test paper and the assigned number of the beam is listed, and it is recorded whether the test paper is written in SI units or Im-perial units.

2) Section properties: The dimensions of the test specimen are listed. Further, it is taken into account that some beams are I-sections with flanges on top and bottom. The gross area and the distance of the center of gravity from the top of the section are determined.

3) Load position and geometry: The dimensions are entered into the file of the support plates and the loading plates, the overhang of the beam behind the support axis, the span and the distance of the loading point from the support. Additionally, the moment-shear force-ratio is determined. In case of beams subjected to a uniformly dis-tributed load, this ratio equals the distance:

[mm] (1-1)

4a) Longitudinal reinforcement (tension chord reinforcement): In this part the essential information that describes the reinforcement of the tension chord is entered: Spacing and area of reinforcement, ratio of reinforcement, type of anchorage, yield strength and tensile strength. In the case of reinforcement with different diameters and yield strengths, the average values for the yield strength are computed as area-weighted average values.

4b) Longitudinal reinforcement (compressive reinforcement): In this part the essential information of the rein-forcement of the compression chord is entered into the file (see 4a))

5a) Prestressing steel: Dimensions, areas and types of strands (or tendons), as well as yield stress and tensile strength are entered into the file. Subsequently, the center of gravity of the tendons (and if existent non-tensioned reinforcement) is computed and the effective depth of the total reinforcement of the tension chord is computed in proportion to the tensile forces.

5b) Prestress: In this part the prestress is summarized, including the prestressing force and the concrete stress at the centroid of the concrete section due to prestress.

5c) Axial force: In case of beams with axial tension or compression, the axial force is entered into the file and the concrete stress is computed.

6) Stirrup reinforcement: In case of beams with stirrups, the properties and spacing of stirrups are entered into the file. The ratio of reinforcement is computed and the type of the surface (ribbed or plain), the yield stress and the tensile strength are entered into the file.

7a) Concrete compressive strength: The uniaxial concrete compressive strength is derived from the values at the day of testing via conversion factors dependant on the dimensions of the test specimen. In the case of light-weight concrete beams additional data is included on the type of aggregate and the density of the concrete.

7b) Concrete tensile strength: The uniaxial concrete tensile strength (f1ct,test) is computed using the test values de-rived from test specimen if available. Furthermore, the calculated value of the uniaxial concrete tensile strength is computed according to PART 1, section 3.2.3.

8) Mechanical ratios of longitudinal reinforcement and stirrups are included if they were available.

9) Test: In this section all the results from the tests are summarized. Additionally, the shear forces due to the self-weight of the beam are determined and included. For beams with stirrups the ultimate measured stress in the stirrups is presented when available.

10) Control: At the end of the data collection the beams are checked in respect to completeness of all essential data (control konx, see PART 1, section 4.1) which are required to control and evaluate the data.

A detailed description of the data collection can be found in the appendix that is associated with each part of this report. The most important notations are given in attachment 1-1 below.


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The data collection file is followed by a separate file containing the control of slender beams and beams with a ratio of a/d > 2,4. The reason for this subdivision into two separate files is to keep the file sizes to be manageable and well organized. In the data control file, the beams are examined by a set of criteria, of which a more detailed de-scription is given in chapter 4 of PART 1. The remaining beams are passed onto another file for the evaluation of the tests and the comparisons with design equations according to standards or design proposals.

Attachment 1-1: Notations


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3 Conversion factors of strength values for concrete 3.1 Concrete compressive strength

3.1.1 Definition

The reference basis used in this report for the uniaxial concrete compressive strength f1c (units: [MN/m² ; N/mm² ; MPa]) is the strength of a slender prism. The design value is defined as the design value computed via the charac-teristic value of the cylinder strength:

where normally (1-2)

In the German standard DIN 1045-1, this value is consequently defined as the design value of fcd. This is not the case with the definition of the Eurocode EC2, PART 1 (1991), which does not include the coefficient . This has been adjusted in the edition of EC 2, PART 1 (2002), however, the recommended value of the coefficient is = 1.

The coefficient = 0,85 accounts for two influences, firstly, the conversion of the cylinder strength of the control specimen into the uniaxial compressive strength of a slender prism, and secondly, the influence of sustained load-

ing. The first influence is considered by taking so that the remaining reduction factor due to sus-

tained loading is approx. 0,90. These relationships are frequently used and due to the well-known research works by Rüsch (1960, 1968) and Grasser (1968) and by the definition of the characteristic values r as in DIN 1045 and DIN 4227. However, this seems to be forgotten by those many who consider the cylinder strength as the decisive design value for dimensioning. This is technically not correct and therefore in the following all the analyses are based on the uniaxial compressive strength which is defined as:

[MPa] (1-3)

Only on the basis of this value is it possible to reasonably derive design equations and create a coherent comparison between different standards.

The conversion factors defined in the following have been applied to both the research report DIBt- (1999), see CHAPTER 3 of Reineck (1999 a), and to fib Bull. 12 (2001) as well as to ACI-ASCE Committee 445, see Reineck, Kuchma et al. (2003). Therefore, these conversion factors can be regarded as internationally recognized.

3.1.2 Conversion of strength values derived from control specimens

The strength values derived from different control specimens of varying dimensions are converted to the uniaxial compressive strength f1c using the following conversion factors.

Cubes as control specimens:

[MPa] (1-4a)

where: fc,cu = fc,cu150 [MPa] (1-4b)

fc,cu150 = concrete compressive strength of cubes ( )

where: [MPa] (1-4c)


where: [MPa] (1-4d)



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Cylinders as control specimens:

[MPa] (1-5a)

where: fc,cyl = fc,cyl,150/300 [MPa] (1-5b)

fc,cyl,150/300 = cylinder concrete compressive strength ( )

where: [MPa] (1-5c)


where: [MPa] (1-5d)


where: [MPa] (1-5e)


where: [MPa] (1-5f)


Prisms as control specimen:

[MPa] (1-6)

fc,pr = prism concrete compressive strength ( )

There was no choice for the final determination of f1c for many reports, in which the compressive strength was de-rived from only one type of control specimen, such as in general from cubes in Germany and from cylinders in the United States. However, if strength values from different control specimen were reported, then in some cases quite significant discrepancies could occur between values of f1c derived from the different control specimens. Therefore the following rule of precedence was established:

prism prior to cylinder prior to cube.

However, this only applies if the same number of test specimens is tested for each type, i.e. in general three control specimens each; if more cubes are tested than cylinders, the value derived from the cube strength is considered to be more decisive.

By converting the values of the compressive strengths derived from different test specimen into the uniaxial com-pressive strength and by the rule of precedence in case of using varying test specimen for one test or a test series, it is warranted that only one single value of the compressive strength is handed over from the collection database _DS to the subsequent databases_DK for the control and databases_A for the evaluation. This single value is the uniaxial compressive strength f1c which is considered the essential basis for all dimensioning. Comparing the com-pressive strengths reported in previous databases that are derived from identical tests but by different authors, the discrepancies and inconsistencies are considerable in some cases. Of course, when comparing tests values with the shear capacity according to different standards or design proposals, form this value f1c the relevant strength value of the standard or approach is derived, e.g. from f1c the value fck is determined according to DIN 1045-1 or EC 2, Part 1 as in Eq.(3.8b) or correspondingly f´c according to ACI 318 as in Eq.(3.10).

Table 1-2 provides an overview of the conversions for the concrete compressive strength.


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Table 1-2: Overview of the conversion factors for the concrete compressive strength

strength Dimensions of control specimen [mm]

Conversion between different control specimens

Uniaxial compressive

strength

fc,pr pr ( ) ————————

fc,cyl cyl ( ) ————————

fc,cyl,100/300 cyl ( )

fc,cyl,170/150 cyl ( )

fc,cyl,120/360 cyl ( )

fc,cyl,100/200 cyl ( )

fc,cu cu ( ) ————————

fc,cu,200 cu ( )

fc,cu,100 cu ( )

From the uniaxial compressive strength the web compression strength fcwu, of the inclined struts in webs is deter-mined for struts parallel to inclined cracks:

[MPa] (1-7)

This strength value was established by the FIP Recommendations (1999), whereas the German standard DIN 1045-1 uses a more conservative factor of 0,75.

3.1.3 Characteristic value of the compressive strength

Since it is common that only an average value of the compressive strength is given in test reports, it is necessary to convert this value into the characteristic value used in standards.

In the case of site concrete, the concrete strength according to standards is defined as:

[MPa] (1-8)

For the evaluation of tests, the value of f is defined as f = 4 MPa for laboratory conditions, in accordance with the research report DIBt (1999). From this, it follows that:

[MPa] (1-9a)

and thus f1ck = 0,95 fck,cyl = 0,95 (fcm,cyl – 4) = (f1cm - 3,8) (1-9b)

There are no equivalent relationships available for the tensile strength of concrete so the characteristic values can only be determined for the calculated value related to the concrete compressive strength.

Correspondingly, the DIBt (1999) proposes for the reinforcing steel that:

[MPa] (1-10)


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3.1.4 Conversion of the ACI cylinder compressive strength fć

The value of fć as defined in American standards was set equal to the value of fcm,cyl.

The characteristic cylinder strength value fck as defined in European standards does not equal the cylinder compres-sive strength fć of the ACI 318, since fć only represents a 9%- fractile, whereas fck is a 5%-fractile value. According to Reineck (1999 a) and Reineck, Kuchma et al. (2003) the following relationship exists:

[MPa] (1-11)

Therefore, if tests are to be compared with the ACI 318 code, then the specified concrete compressive strength fć is determined via the average value f1c = f1cm reported in the databases in accordance with Eq.(1-3) as follows:

[MPa] (1-12)

3.1.5 Control of conversion factors using the databases

In this section the conversion factors are compared with the values from test reports, and altogether 3420 test beams were available. The tests were limited to those where in parallel the compressive strength values have been derived from different control specimens, e.g. from both prisms and cubes, respectively in the majority of cases from both cubes and cylinders. Beams which are part of a test series and thus are considered to have the same compressive strength were eliminated in order to avoid multiple nominations.

Fig. 1-1 shows all test values of the cube strengths plotted versus the tested cylinder strengths without subdividing between different dimensions of the control specimen. Thereby, normal-strength concrete is distinguished from high-strength concrete, and the parting line is defined at fc,cyl = 54 MPa and is shown as a dashed line. There is a set of 172 test beams for normal-strength concrete and a set of 48 for high-strength concrete. The lines shown as bold black lines are the trend lines of the data points. The dashed lines through origin illustrate the conversion factors of 100, 150 and 200 mm cubes into a cylinder with dimensions of 150/300 mm according to chapter 3.1.2.

It is noticeable that in the case of normal-strength concrete the trend line of all test results aligns very well with the line of origin representing the conversion of 150 mm cubes into cylinders. However, in the case of high-strength concrete the trend line is less sloped than the other lines, i.e. the differences between the strength values fc,cu and fc,cyl decrease. Thus, the conversion factors for normal-strength concrete should be different from conversion factors for high-strength concrete.

In the following, separate considerations are made for the different dimensions of cubes.


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Fig. 1-1: Cube strength versus cylinder strength distinguishing normal-strength concrete and

high-strength concrete


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Fig. 1-2 shows test values derived from both a cube with an edge length of 150 mm and a cylinder with dimensions of Ø150/h=300 mm. There are altogether 105 available values of which a total of 81 refer to normal-strength con-crete and the remaining 24 values refer to high-strength concrete. Additionally, the trend lines, a 45°-line, as well as the previously applied conversion with fc,cyl = 0,789·fc,cu are plotted. The conversion is defined as follows in accor-dance with equations Eq. 1-4a and Eq. 1-5a:

[MPa] (1-13)

The trend line of the test values coincide very well with this conversion for normal-strength concrete. However, in the case of high-strength concrete the values of fc,cu converge with the values of fc,cyl, so that the conversion factor according to Eq. (1-13) does not fit well.

Fig. 1-2: Cube strength (150 mm) versus cylinder strength (150/300 mm) distinguishing normal-strength concrete

and high-strength concrete

Consequently, there are two possibilities:

1.) An overall trend line as a line through the origin that has the following equation:

[MPa] (1-14)

Thus, the uniaxial compressive strength is:

. [MPa] (1-15)

This conversion factor is slightly lower than the factor of 0,864, given by König, Tue und Zink (2001), p. 50.


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2.) A bi-linear approach for the regions of normal-strength concrete and high-strength concrete with the same value at the partition line at fc,cyl = 54 MPa, as is presented in Fig. 1-3:

If fc,cyl 54 MPa: [MPa] (1-16)

and thus: [MPa] (1-17)

If fc,cyl > 54 MPa: [MPa] (1-18)

and thus: [MPa] (1-19)

Fig. 1-3: Newly proposed bi-linear approach for the conversion factor distinguishing normal-strength concrete

from high-strength concrete


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Fig. 1-4 shows data for the cubes with an edge length of 100 mm versus the data for the cylinders with Ø 150/ h=300 mm. There are altogether 87 values of which 64 refer to normal-strength concrete and the remaining 23 values refer to high-strength concrete. Additionally, Fig. 1-4 shows the trend lines, a 45°-line, as well as the previously applied conversion resulting in a magnitude of fc,cyl = 0,711·fc,cu defined as follows in accordance with equations Eq. 1-4a, Eq. 1-4d and Eq. 1-5a:

[MPa] (1-20)

Whereas the trend line of the normal-strength concrete coincides very well with this conversion, a different conver-sion factor is required for high-strength concrete.

Fig. 1-4: Cube strength (150 mm) versus cylinder strength (Ø 150/ h=300) mm distinguishing

normal-strength concrete and high-strength concrete

Consequently, there are two available possibilities:

1.) Considering the overall trend line as a line through origin yields the following equation:

fc,cyl = 0,815 · fc,cu100 (1-21)

This conversion factor of 0,815 according to Eq.(1-21) is a smaller value than the factor of 0,836, defined by König, Tue und Zink (2001), p. 50.

The conversion factor for the cube strength of cubes with an edge length of 150 mm corresponding to Eq. (1-14) is:

(1-22)


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2.) A bi-linear approach considering normal-strength concrete and high-strength concrete in sections keeping the parting line at fc,cyl = 54 MPa, as is presented in Fig. 1-5:

If fc,cyl 54 MPa:

[MPa] (1-23)

In accordance with Eq. (1-16), this results in the following conversion factor for the cube strength of cubes with an edge length of 150 mm:

[MPa] (1-24)

If fc,cyl > 54 MPa the equation of the trend line is as follows:

[MPa] (1-25)


[MPa] (1-26)

Fig. 1-5: Newly proposed approach for the conversion factor to distinguish normal-strength concrete



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Fig. 1-6 shows data derived from cubes with an edge length of 200 mm versus data derived from cylinders with dimensions of Ø150/h=300 mm. There are altogether 29 values of which 27 are normal-strength concrete and 2 are for high-strength concrete. Additionally, Fig. 1-6 shows the trend line, a 45°-line, as well as the previously applied conversion resulting in a magnitude of fc,cyl = 0,829·fc,cu defined as follows in accordance with equations Eq. 1-4a, Eq. 1-4c and Eq. 1-5a:

[MPa] (1-27)

The trend line of the normal-strength concrete coincides so far very well with the conversion.

The conversion factor according to Eq. (1-27) results in a smaller value (approx. 9%) than the factor of 0,909, de-fined by König, Tue und Zink (2001), p. 50.


[MPa] (1-28)

Table 1-3 provides an overview of the previously used and newly proposed approaches for the conversion between cube strength and cylinder strength

Fig. 1-6: Cube strength (200 mm) versus cylinder strength (150/300 mm) distinguishing normal-strength concrete



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Table 1-3: Summary of the previously used and the newly proposed conversion factors for the concrete compressive strength

New (bi-linear) strengths Previous New (linear)

fc,cu

fc,cu,100

fc,cu,200

3.2 Concrete tensile strength 3.2.1 Definition

Corresponding to the uniaxial compressive strength, the concrete tensile strength is defined as the uniaxial concrete tensile strength f1ct. Consequently again, conversion factors are defined in order to derive the uniaxial concrete ten-sile strength f1ct from the splitting tensile strength and the flexural tensile strength as measured from relevant con-trol specimens.

The coefficient of [-] is used to define the associated strength ratio.

3.2.2 Conversion of strength values of control specimens

The uniaxial concrete tensile strength was determined from the splitting tensile strength and the flexural tensile strength as follows:

- splitting tensile strength (in correspondence to EC 2, PART 1):

[MPa] (1-29)

fct,sp = splitting tensile strength derived from cylinders ( ) or prisms

- flexural tensile strength corresponding to CEB-FIP MC 90:

[MPa] (1-30)

where: h = height of control specimen [mm]

e.g.: h = 100 mm: f1ct = 0,600 fct,fl

h = 120 mm: f1ct = 0,630 fct,fl

h = 150 mm: f1ct = 0,666 fct,fl

h = 200 mm: f1ct = 0,709 fct,fl

In the following evaluations, the above given equation of the CEB-FIP MC 90 was applied since it is valid for dif-ferent heights of test specimens.

By this approach, determined test values of the uniaxial concrete tensile strength are referred to as f1ct,test in the databases. In cases where both test methods have been applied, the priority was set for the value f1ct derived from the splitting test corresponding to Eq. (1-29).

3.2.3 Calculated value of uniaxial concrete tensile strength


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In many cases, no control specimens were cast to determine the concrete tensile strength. The only remaining pos-sibility was then to calculate the concrete tensile strength from the compressive strength via empirically derived formulae. The different, known equations have been discussed in Reineck (1999 a) and the values defined there were adopted here. The calculated values of the uniaxial concrete tensile strength reported in the following are re-ferred to as f1ct,cal in the databases.

For low strength classes of concrete (up to fck = 50 MPa respectively f1c = 51,3 MPa) the average value as defined in CEB-FIP MC 90 is applicable:

[MPa] (1-31a)

In accordance with , this results in:

[MPa] (1-31b)

For high-strength concrete classes (up to fck > 50 MPa respectively f1c > 51,3 MPa) the German standard DIN 1045-1 (2001) proposes the following approach containing a logarithmical function for the uniaxial concrete tensile strength:

[MPa] (1-32)

Alternatively, where fcm = (fck - 8) the following relationship for fct,m depending on fck can be applied with very good accuracy in cases of high-strength concrete as shown in Fig. 1-7 (see Reineck (1999 a) and in chapter 6.4.3 in fib Bull. 12 (2001)):

[MPa] (1-33a)

in accordance with , this results in the mean value of:

[MPa] (1-33b)

ACI 318 proposes the following expression:

[MPa] (1-34a)

this results in:

[MPa] (1-34b)

Using the conversion of according to Eq.(1-12) , the following expression can be applied approxi-

mately:

[MPa] (1-34c)

To enable for comparisons, the expression defined in ACI 318 as shown in Fig. 1-7 and the other expressions were normalized so that the graphs of all the curves provide the same value at the border line of the high-strength con-crete fck = 50 MPa. This results in a modified expression of the ACI 318:

[MPa] (1-34d)

This expression was observed to provide too steep an increase for high-strength concrete.


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Fig. 1-7: Comparison of different calculated values of the concrete tensile strength

3.2.4 Characteristic value of the concrete tensile strength

The CEB-FIP MC 90 defines the lower and upper fractile values as follows:

5%- fractile: fct,k5% = 0,70 fctm [MPa] (1-35a)

95%- fractile: fct,k95% = 1,30 fctm [MPa] (1-35b)

These relationships are applied to both the test values of f1ct,test determined in accordance with section 3.2.2 and to the characteristic values of f1ct,cal according to section 3.2.3.

This results in a lower fractile value of the characteristic value of f1ct,cal:

- in case of normal-strength concrete: [MPa] (1-36a)

- in case of high-strength concrete: [MPa] (1-36b)

where f1c can be taken as either the mean value f1cm or the characteristic value f1ck = (f1cm - 3,8) [MPa] according to Eq.(1-9b).

3.2.5 Verification of conversion factors using the databases

The verification of the conversion factors for the uniaxial concrete tensile strength estimated from the splitting ten-sile strength and the flexural tensile strength is made using values given in the databases.

Out of the 3420 beams in the databases, only for 605 beams (17,7%) was data reported on the flexural tensile strength. For these tests, the ratio of the tested value for the flexural tensile strength f1ct,fl and the calculated concrete

tensile strength f1ct,cal (see section 3.2.3) was generated, i.e. , and in Fig. 1-8 this value is plot-

ted versus the uniaxial concrete compressive strength.


1 - 22

As was done for the compressive strength, the parting line for the high-strength concrete (fck = 50 MPa) was taken into account. Furthermore, a statistical evaluation of the generated ratio was carried out. The mean value is m = 0,892 and the variation coefficient is v = 0,248; the value of the lower fractile equals 0,528 and the upper equals 1,256. A trend line is shown and it is evident that the conversion method does not coincide very well for the high-strength concrete so that an adjustment is necessary.

Fig. 1-8: Diagram of the ratio for the flexural tensile strength versus the

uniaxial concrete compressive strength (with f1ct,cal acc. to section 3.2.3)

The verification of the conversion for the splitting tensile strength fct,spl was preformed similarly. A total of 518 beams (15,1%) included information about the splitting tensile strength. Again, the calculated value of the tensile

concrete strength is determined according to section 3.2.3. The ratio is again plotted versus

the uniaxial compressive strength (Fig. 1-9).

The ratios of some beams exceed the , and these outliers are identified separately and were examined

subsequently on the basis of the test reports. In the test series of Greco, Giliberti, Mele (1963) test specimens with dimensions of 100/100/300 mm have been used. These dimensions might be responsible for the resulting high magnitudes. The same reason is considered decisive with the test series of Chana (1981), since the dimensions of the test specimens were only 70/70/70 respectively 25/25/25 mm. The test series of Rajagopalan and Ferguson (1968) do not provide any information about the dimensions of the test specimens, so that it can only be assumed that the high magnitudes are again due to the small dimensions of the test specimens.


1 - 23

Fig. 1-9: Diagram of the ratio for the splitting tensile strength versus the

uniaxial concrete compressive strength (with f1ct,cal acc. to section 3.2.3)

The mean value of the ratio for the 518 beams is m = 1,012 and the coefficient of variation is

v = 0,206; the lower fractile value equals 0,669 whereas the upper fractile value equals 1,355. The trend line of all beams shows that the calculated value of the tensile strength coincides very well for the tensile splitting strength, so that no adjustment is necessary.

For reasons of comparison, Fig. 1-10 shows the results of the tests excluding the outliers to assert the trend respec-tively, to detect possible changes of the statistical evaluation. There are 498 remaining tests and an upward trend can be observed for the high-strength concrete classes. The mean value decreases marginally to m = 0,983 and the new variation coefficient equals v = 0,145; the lower fractile value equals 0,748 whereas the upper fractile value equals 1,217. It is obvious that the scatter has been distinctively reduced. The discrepancy of the trend line is sufficiently small so that an adjustment seems to be unnecessary for the determination of f1ct by means of the splitting tensile strength.


1 - 24

Fig. 1-10: Diagram of the ratio for the tensile splitting strength versus the uniaxial concrete

compressive strength without the outliers in Fig. 1-9 (with f1ct,cal acc. to section 3.2.3)

A further comparison is plotted in Fig. 1-11, where the uniaxial concrete tensile strength derived from the splitting tensile strength is plotted versus that derived from the flexural tensile strength. There are only 138 beams (4,0%) available, where both the splitting tensile strength and the flexural tensile strength was tested. The parting line for

the high-strength concrete is defined at and is shown as a dashed

line. A total of 29 beams are classified as being cast from high-strength concrete and the remaining 109 from nor-mal-strength concrete.

There are two separately determined trend lines that show distinctly different trends. In case of the normal-strength

concrete the trend line lies above the 45°-line, resulting in a value of . This means that the

splitting tensile strength results in a slightly smaller value for the uniaxial concrete tensile strength than the flexural tensile strength. Furthermore, in case of the normal-strength concrete the scatter is quite considerable. High-strength concrete beams, however, show a contrary trend: the flexural tensile strength results in essentially higher values for the uniaxial concrete tensile strength than the splitting tensile strength. The intersection point of the trend line and the 45° line lies approximately at 5,5 MPa.


1 - 25

Fig. 1-11: Diagram of the uniaxial concrete tensile strength derived from the splitting tensile strength

plotted versus that derived from the flexural tensile strength

In addition to this, the statistical evaluation of all beams is plotted for . The mean value equals m

= 1,151 and the variation coefficient equals v = 0,259; the lower fractile value is 0,665 whereas the upper value is 1,637. The considerable scatter is reflected by the statistical evaluation and the mean value proves that in general the tensile splitting strength results in a higher magnitude for the uniaxial concrete tensile strength than the flexural tensile strength.

As a final step, Fig. 1-12 and Fig. 1-13 show the uniaxial tensile strength versus the uniaxial concrete compressive strength. Fig. 1-12 shows the values for the 742 beams with data on the flexural tensile strength whereas Fig. 1-13 shows the values for the 655 beams with data on the splitting tensile strength. The beams are marked separately.

In Fig. 1-12 a considerable scatter can be observed which increases for high-strength concrete. Additionally, the figure shows the calculated tensile strength f1ct,cal according to Eqs. (1-31) to (1-33) versus the 5%- and the 95%- fractile (fct,k5% = 0,70 fctm and fct,k95% = 1,30 fctm ). It is obvious that for low strength beams, most of the beams are within theses boundaries. In the case of high-strength concrete classes, many beams exceed the 5%-fractile and the 95%-fractile. For reasons of comparison, the expression of the ACI f1ct,ACI according to Eq. (1-34c) is also plotted in Fig. 1-12 as a dashed line. Up to a concrete strength of fck = 25 MPa this expression results in higher magnitudes than the calculated values of f1ct,cal. Beyond fck = 25 MPa the magnitudes according to the ACI expression are es-sentially smaller than the calculated values of f1ct,cal until the graph of the ACI expression intersects the curve of f1ct,cal again at fck = 110 MPa.


1 - 26

Fig. 1-12: Dependency of the axial tensile strength f1ct = f1ct,fl derived from 742 flexural tests on the uniaxial com-pressive strength f1c and comparison with the values according to DIN 1045-1 and ACI 318

Fig. 1-13 shows the data points for the axial tensile strength derived from the splitting tensile strength plotted ver-sus the uniaxial concrete compressive strength. In contrast to Fig. 1-12 based on the flexural tensile strength, the scatter is considerably smaller. A few points are below the 5%- fractile whereas the data points of more beams ex-ceed the 95%- fractile. The data largely is within the boundaries and it can be observed that the splitting tensile strength provides more accurate results than the flexural tensile strength. The priority previously set in section 3.2.2 in case both tensile strength values are given, seems to be justified as illustrated by the results presented in Fig. 1-12 and Fig. 1-13.


1 - 27

Fig. 1-13: Dependency of the axial tensile strength f1ct = f1ct,sp derived from 655 splitting tests on the uniaxial compressive strength f1c and comparison with values according to DIN 1045-1 and ACI 318


1 - 28


1 - 29

4 Criteria for the Evaluation and selection of tests

4.1 Generally valid criteria In order to make use of experimental test data for the evaluation of code provisions and other formula for estimat-ing shear strength, it is necessary to be sure that the test results that are selected for this evaluation are complete, reliable, and suitable for use in these evaluations. To that end, criteria for accepting data into an “evaluation data-base” were developed. These criteria are implemented using control coefficients, or “koni” that have either a value of 0 or 1 for each test result as follows:

koni = 1 if “fulfilled”;

koni = 0 if “not fulfilled”

The beams of the data collection were checked for incomplete data using the first criteria:

konx = 1 fundamental data is available

This first criterion controls if fundamental data is available for the subsequent data control; e.g. for evaluating the flexural capacity of the beam or to check against a possible anchorage failure (f1c, fsy and Vu). In case of members with stirrups this also includes the important parameters fyw and w. In case of prestressed concrete members the initial prestressing force P and the yield strength of the prestressing steel fpy is required.

Only tests that fulfill the subsequently listed criteria are passed on to the control and evaluation databases. The con-trol criteria are all engineering-moderated or mechanically justified. In addition, no statistical selection - as sug-gested in Zilch, Staller and Brandes (1999) - of the tests took place to eliminate so-called outliers and not consider these for comparison with the design formulae.

The criteria in detail are:

kon1 compressive strength

kon2 compressive strength

kon3 width of web:

kon4 member height:

kon5 moment-shear force-ratio:



konx7 auxiliary condition:

kon7 height of compression zone at failure:


kon8 assessment of flexural failures:

The flexural capacity was calculated approximately using the stress block according to the CEB- FIP MC 90, which is also applicable for high-strength concrete members (see section 4.4).

kon81 extended assessment of flexural failures:

An examination is required for these beams using a more accurate design procedure and a review of the test report with respect to the behavior at failure.


1 - 30

kon10 ribbed reinforcing steel:

plain bars: fr = 0


kon11 no calculated anchorage failure:

The anchorage length was calculated according to the German standard DIN 1045-1 and using a bond stress of

. In the case of anchor plates used in some tests this coefficient was simplified to

(see section 4.5).

kon15 ”andbr” other types of failure (than flexural failure, named BB, or anchorage failure)

Failures that were other than calculated flexural failure or due to higher than permitted bond stress were identified in the column ”andbr” and were also excluded from the analysis. Examples of other types of failure are: - anchorage failure of the stirrups; - flange failure of T-beams.

4.2 Special criteria for beams with stirrups For members with stirrups, four additional criteria were added to the control criteria.

konx9 auxiliary condition

kon9 no test with excessive failure loads

Tests with a compressive stress of cwu > 1,00 f1c in the inclined struts in the web are inexplicable according to the truss analogy and were considered to have other load transfer actions.

kon101 ribbed reinforcing steel

kon102 ribbed stirrups

kon10 = kon101·kon102

kon12 measured stirrup stress at failure

The assessment of was carried out at approx. . If yielding of the stirrups was not measured to

have occurred, the stirrup stress at failure was inserted in the column ; this value was taken as the average

from three stirrups if possible by means of extrapolating the measured values of respectively in respect to

the ultimate load.

In cases where no data on stirrup stress was reported, the procedure was as follows:

a) If yielding of the stirrups was clearly reported in the test report, is inserted in column .

b) If there was no information in the report, no value is inserted into the file and the test data is excluded from fur-ther evaluation and a comparison with standards or design approaches.

kon133 nominal stirrup reinforcement ratio w,min

if : w,min

if (i.e. not given): w,min

This condition is an assortment criterion rather than an elimination criterion per se. It is only applied for comparing test results with a design approach according to DIN 1045-1 for members containing a higher web reinforcement ratio than the minimum required.


1 - 31

For comparing the test values with different design approaches for beams webs which did not exhibit a web com-pression failure, the following sortment criteria were defined for structural concrete members according to the de-sign approach considered:


kon141 if :

if :

if :

kon142 if :

if :


4.3 Special criteria for prestressed concrete members In case of prestressed concrete members the control criteria kon10 was extended to the type of ribs of the tensioning steel.

kon101 ribbed reinforcing steel

kon103 ribbed prestressing steel


Additionally, ribbed stirrups are considered in the case of prestressed concrete members with stirrups

kon101 ribbed reinforcing steel in tension chord

kon102 ribbed stirrups

kon103 ribbed prestressing steel

kon10 = kon101·kon102·kon103

4.4 Assessment of calculated flexural failures (kon8) 4.4.1 General procedure

The control of flexure failure is carried out via the following coefficient for the ratio of the attained ultimate flex-ural moment and the calculated value:

[ - ] (1-37)

A magnitude of means that the failure of a beam was calculated to be due to flexure.

Thereby is defined as follows:

[ - ] (1-38)

where: [kNm] ultimate flexural moment

[mm] width of compression zone [mm] effective depth

[MPa] uniaxial concrete compressive strength

4.4.2 Calculation of ultimate bending moment of reinforced concrete beams


1 - 32

In order to determine the dimension-free value of the ultimate bending moment, the equilibrium of the free-

body diagram was considered as shown in Fig. 1-14a where it is assumed that the reinforcement of the tension chord is yielding.

Fig. 1-14a: Free-body diagram and parameters for calculating the ultimate bending moment of r.c.-beams

The forces are:

[kN] tensile force of reinforcement of tension chord

[kN] compressive force of concrete (assumption of stress block)

where: [mm²] tension chord reinforcement

[MPa] yield strength of steel

[ - ] coefficient for stress block according to CEB-FIP MC 90 (1-39a)

[ - ] coefficient for location of neutral axis

[mm] width of compression chord

With it follows:

[ - ] (1-39b)

where: [ - ] mechanical reinforcement ratio of longitudinal reinforcement (1-40)


1 - 33

From , it follows that:

(1-41a)

where: [mm] inner lever arm

[ - ] coefficient for inner lever arm (1-42)

Thus, the calculated value for the dimension-free ultimate bending moment is:

(1-43)

These equations are only applicable for pure bending, whereby the compressive reinforcement is not considered

and the concrete strain is ‰.

In order to control the location of the neutral axis, a limiting value for the reinforcement ratio was calculated:

(1-44)

If the above given equations are applicable.

If the equations are as follows:

[ - ] (1-45a)

where: [ - ] coefficient for inner lever arm (1-45b)

[ - ] coefficient for location of neutral axis (1-45c)

s strain of longitudinal reinforcement

The strain s was calculated as follows:

‰] (1-46)

where: and


1 - 34

4.4.3 Calculation of ultimate bending moment of reinforced concrete beams with axial tension

The procedure for beams with axial tension was as previously described, but where Fig. 1-14b shows the free-body diagram.

Fig 1-14b: Free-body diagram and parameters for calculating the ultimate bending moment

of r.c.-beams with axial tension

The dimension-free axial force is defined as (N > 0 for tension):

[ - ] dimension-free axial force (1-47)

If :

:

[ - ] (1-48)

:

[ - ] (1-49a)

Thus, the calculated dimension-free ultimate flexural moment is determined as follows:

(1-49b)

If :

: (1-50)

From the strain distribution, it follows that: (1-51)


1 - 35

Thus the following equation can be solved for :

‰] (1-52)

where: ; ; ;

: [ - ]


(1-53)

4.4.4 Calculation of ultimate bending moment of prestressed concrete beams

Fig. 1-15 shows the free-body diagram for a prestressed concrete beam.

Fig. 1-15: Free-body diagram and parameters for calculating the ultimate bending moment

of prestressed concrete beams

The following terms can be defined:

[ - ] mechanical reinforcement ratio of prestressing steel (1-54a)

[ - ] mechanical reinforcement ratio of non-tensioned reinforcement (1-54b)


1 - 36

[ - ] mechanical reinforcement ratio of tension chord (1-55)

[ - ] limit of reinforcement ratio (1-56)

[ - ] effective depth (1-57)

If :

: [ - ] (1-58)

: [ - ]


(1-59)

If :

:

From the strain distribution, it follows that:

(1-51)

(1-60a)

(1-60b)

where: = additional strain at level of effective depth d (1-60c)

Thus the following equation can be solved for :

(1-61)

where: and respectively and

if: and


1 - 37

For members with non-prestressed reinforcement, it follows that: ; and :

(1-62)

: [ - ]

Thus, the calculated ultimate flexural moment is determined as follows:

(1-63a)

For members without non-prestressed reinforcement, it follows that: ; und :

(1-63b)

4.5 Assessment of calculated anchorage failures at end support (kon11)

4.5.1 General procedure

The assessment of adequate anchorage capacity at the end support is performed according to well known criteria.

The existent anchorage length is compared with the required anchorage length :

[ - ] (1-64)

A magnitude of means that the provided anchorage length is insufficient and that an anchorage failure may

have occurred.

Important input parameters in the process are the width of the support and the overhang of the beam behind the axis of support. For many beams this data is not specified in the reports, and therefore assumptions were made for the existent anchorage length lb,prov, in order to perform this check and to as not to unnecessary discard test data from use in evaluations:

[mm] if and

[mm] if and

[mm] if and

[mm] if and

where (see Attachment 1-1):

[mm] dimension of support plate

[mm] overhang of beam behind the support axis

[mm] effective depth


1 - 38

4.5.2 Reinforced concrete beams without stirrups

It is assumed that the concrete ties are inclined at an angle of 60° to the longitudinal axis and that the concrete struts are inclined by an angle of 30°, see Reineck (1990, 1991 c,d). Thus the tension chord force at the end support can be calculated from the truss analogy. The required anchorage ,length lb.req is determined as follows:

[mm] (1-65)

where:

[MPa] (1-66)

[kN] shear force at failure

[mm²] cross-sectional area of tension chord reinforcement

aA [mm] see attachment 1-1; if , the value = is assumed.

[ - ] type of anchorage (hook 0,7; straight 1; anchor plate 0,01)

[mm] diameter of tension chord reinforcement

[MPa] calculated value of the uniaxial concrete tensile strength

4.5.3 Reinforced concrete beams with stirrups

It is assumed that the concrete struts in the web are inclined by an angle of 30°, and thus the tension chord force at the end support can be calculated from the truss analogy. The required anchorage length is determined as follows:

[mm] (1-67)

where: [MPa]

[kN] shear force at failure

[mm²] cross-sectional area of tension chord reinforcement

aA [mm] see attachment 1-1; if , is replaced with .


[mm] diameter of the tension chord reinforcement



1 - 39

4.5.4 Prestressed concrete beams without stirrups

4.5.4.1 Determination of the tension chord force

Similar to r.c-beams, the force that has to be anchored at failure was determined for a truss with concrete struts inclined at an angle of about 24° (cot = 2,20) from the longitudinal axis and concrete ties inclined at an angle of 66°, and is as follows:

[kN] (1-68)

where: [kN] shear force at failure

and the associated stress:

[MPa] (1-69)

4.5.4.2 Assessment without consideration of non-tensioned reinforcement

In case of pretensioned structural members, the following equations are applicable:

- In the case of beams with 7-wire strands for prestressing steel:

[mm] (1-70a)

- In the case of beams with other types of prestressing steel:

[mm] (1-70b)

- In the case of post-tensioned beams, the equations are as follows for all types of prestressing steel:

[mm] (1-70c)

where: [MPa] stress in the prestressing steel due to prestress

[mm²] cross-sectional area of prestressing steel

aA [mm] see attachment 1-1; if , is replaced with .


[mm] diameter of prestressing steel


4.5.4.3 Assessment with consideration of non-tensioned reinforcement

At first, the non-tensioned reinforcement is considered and it is controlled if the non-tensioned reinforcement pro-vides a sufficient anchorage length. Therefore, a ratio was calculated of the actual force to be anchored and the yield force:

[ - ] (1-71)

If : [MPa] (1-72)


1 - 40

[mm] (1-73)

If : [mm] (1-74)

By means of equation Eq. (1-64) it is checked whether the existing anchorage length is sufficient:

[ - ] (1-64)

The ratio is used to used to compute the required anchorage force , that the prestressing steel has to resist. A

value of implies that the non-tensioned reinforcement was calculated to be insufficient to resist the

required anchorage force.

If : [kN] (1-75)

If : [kN] (1-76)

The required remaining anchorage force is determined as follows:

[kN] (1-77)

and the associated stress in the prestressing reinforcement:

[MPa] (1-78)

In case of pretensioned concrete members, the following equations are applicable:

- In the case of structural members with 7-wire strands as prestressing steel:

[mm] (1-79a)

- In the case of structural members with other types as prestressing steel:

[mm] (1-79b)

- In the case of post-tensioned beams, the equation for all types of prestressing steel is as follows:

[mm] (1-79c)

Thus, the new ratio can be generated:

[ - ] (1-80)


1 - 41

4.5.5 Prestressed concrete beams with stirrups

In case of prestressed concrete members with stirrups, it is only the actual force at failure that has to be

anchored. This differs from that given in section 4.5.3 due to consideration of vertical stirrups. The remaining procedure is equivalent to that given before.

The force to be anchored at the end support is as follows:

[kN] (1-81)

where: [kN] shear force at failure

4.6 Determination of inner lever arm at shear failure

4.6.1 Introduction

In the case of beams failing in shear, the maximum moment at midspan was typically much lower than the moment at the ultimate flexural capacity as researchers had chosen to use a large enough amount of longitudinal reinforce-ment so that any chance of a flexural failure was virtually eliminated. Therefore, at shear failure the state of stress and internal forces are typically quite different than at the ultimate flexural capacity, and the values for the depth x of the compression zone and for the inner lever arm z are different from that calculated in section 4.4. Many design approaches consider the inner lever arm, especially in case of members with stirrups, so that the inner lever arm ztest at shear failure is relevant and need be calculated.

4.6.2 Calculation of inner lever arm for reinforced concrete beams

For the determination of the inner lever arm, the equilibrium of the free-body diagram is considered (Fig. 1-16). At first, the stress of the reinforcing steel is determined.

Fig. 1-16: Free-body diagram and parameters for calculating the ultimate bending moment for r.c.-beams

[kN] tensile force of reinforcement of tension chord

[kN] compressive force of concrete (assumption of stress

block)

where: [mm²] tension chord reinforcement

[MPa] stress of reinforcing steel


1 - 42

[ - ] coefficient for stress block acc. to CEB-FIP MC90, Eq.(1-9a)

[ - ] coefficient for location of neutral axis


With it follows:

[ - ] (1-82)

where: [ - ] mech. reinforcement ratio of longitudinal reinforcement (1-40)

With it follows:

(1-83)

with: [mm] inner lever arm


This results in a dimension-free ultimate moment of:

(1-85)

Eq. (1-39b) is now inserted into Eq. (1-43b) and solved for the stress of the longitudinal reinforcing steel, resulting in:

[ MPa ] (1-86)

Given that the square root is not negative, respectively, μu/ c 0,5, it is possible to determine the stress. At the same time, the stress is limited by the yield limit. Now it is possible to determine the location of the neutral axis and the coefficient for the inner lever arm.

As an alternative solution, test can be determined directly from Eq. (1-43b):

[ - ] (1-87)

Subsequently, it is possible to determine the stress of the reinforcing steel and the coefficient for the inner lever arm.

[ MPa ] (1-88)

[ - ] coefficient for the inner lever arm (1-84)


1 - 43

4.6.3 Calculation of inner lever arm for p.c.c-beams without reinforcing steel The determination is carried out analogously as shown in the previous section. The following conditions are valid:

With it follows that:

[ - ] (1-89)

where: [ - ] mech. reinforcement ratio of prestressing steel (1-54a)

[ - ] effective depth (1-90)

With , the dimension-free ultimate moment results in:

(1-85)

The solution of the equation is again:

[ - ] (1-87)

Subsequently, it is possible to determine the stress of the prestressing steel and the coefficient for the inner lever arm.

[ MPa ] (1-88)

[ - ] coefficient for the inner lever arm (1-84)

4.6.4 Calculation of the inner lever arm for p.c.c-beams with reinforcing steel

The determination is carried out analogously as shown in the previous section. For the determination of the inner lever arm, the equilibrium of the free-body diagram is considered. Additionally, the strains are considered as shown in Fig. 1-17.

Fig. 1-17: Free-body diagram and parameters for calculating the ultimate bending moment for p.c.-beams


1 - 44

The following conditions are valid:

[kN] tensile force of reinforcing steel

[kN] tensile force of prestressing steel

[kN] compressive force of concrete (assumption of stress block)

where: [mm²] cross-section of reinforcing steel

[mm²] cross-section of prestressing steel

[‰] strain of reinforcing steel

[‰] additional strain in prestressing steel

[‰] strain in prestr. steel due to prestress

; [MPa] E- moduli in steel

[mm] effective depth

[ - ] coefficient for stress block according to CEB-FIP MC 90, Eq.(1-39a)

[ - ] coefficient for the location of the neutral axis



[ - ] (1-91)

where: [ - ] mech. reinforcement ratio of reinforcing steel (1-40)

[ - ] mech. reinforcement ratio of prestressing steel (1-54a)

[‰] yield strain of reinforcing steel

[‰] yield strain of prestressing steel

According to the theorem on intersecting lines, it follows:

[‰] strain of prestressing steel (1-92)


1 - 45

Inserting Eq. (1-92) in Eq. (1-91) results in:

[ - ] (1-93a)


[ - ] (1-94)

Inserting Eq. (1-92) in Eq. (1-94) results in:

(1-95a)

Since the difference of dp and ds is rather small, the following factors were established as a means of simplification after an internal evaluation:

[ - ] (1-96)

[ - ] 1-97)

Inserting Eq. (1-96) and Eq. (1-97) in equations Eq. (1-93a), respectively, Eq. (1-95a) results in the following sim-plified equations:

[ - ] (1-93b)

(1-95b)

Subsequently, Eq. (1-93b) is inserted in Eq. (1-95b) and solved for s,test:

[‰] (1-98)

where: [ - ]

[ - ]

[ - ]


1 - 46

After determining the strain s,test, the coefficient for the location of the neutral axis can be calculated from Eq. (1-93b). Concluding, it is possible to determine the coefficient for the inner lever arm.



1 - 47

5 Procedure for evaluation of tests and of comparisons with design approaches 5.1 Consideration of self-weight of beams subjected to point loads 5.1.1 Problem

Many test reports do not provide data with respect to the consideration of the self-weight of the beams. In the case of the predominantly tested small beams this is rather irrelevant, but in the case of large beams this can result in a difference up to 14 % in the ultimate shear force.

The shear force due to self-weight is determined for the failure section, as explained und justified in the later fol-lowing section 5.2.1 on beams with distributed load. The location of the failure crack is measured as the distance of the failure crack from the support axis xr in the axis of the beam (Fig. 1-18). The distance of the crack xr is thereby determined as an average of the distances of the cracks xr,o and xr,u measured on the upper and lower surface of the beam:

[mm] (1-100)

For some beams, xr was measured directly from the crack pattern in the beam axis (in h/2) (Fig. 1-16b).

a) crack pattern of beam BH50 from Podgorniak-Stanik (1998)

b) crack pattern of beam D4/1 from Leonhardt und Walther (1962)

Fig. 1-18: Determination of by measuring and respectively xr for the failure crack

for considering the self-weight of the beam for the ultimate shear force


1 - 48

If xr is known, the self-weight of the part of the beam between midspan and the failure crack may be calculated; see hatched area in Fig. 1-18b, and this results in:

[kN] (1-101)

where: [kN/m]

Ac [mm²]

c, a and xr [mm]


Many test reports do not provide crack patterns, so it is impossible to determine xr. Therefore in the following, an empirical formula is derived from the measurements of those test beams, for which crack patterns were provided and thus the distance xr of the crack could be determined.

Fig. 1-19 shows the distance xr of the failure crack versus the distance of the point load from the support axis for the 131 test beams for which xr could be measured. Furthermore, for reasons of comparison several lines of origin are given in Fig. 1-19 representing the ratio of xr and a. The magnitudes of the values scatter between xr/a = 0,25 and xr/a = 0,75; the mean value of the 131 beams equals xr/a = 0,533 and the trend line equals approx. xr/a = 0,60.

xr

[mm]

a[mm]

Fig. 1-19: Distance xr of the failure crack from the support axis plotted versus the distance a of the point load from

the support axis


1 - 49

In Fig. 1-20 the dimension-free values of xr/d are plotted versus the dimension-free values of a/d for the examined 131 test beams. Additionally, the lower and the upper boundaries are shown, since the inclined failure crack can only occur up to a distance up to xr,max= (a-d). Some beams exceeding the upper boundary are obviously due to flexural failure, since the failure cracks are close to midspan. The lower boundary is approx. xr,min = (0,25 a) ac-cording to the design model of Reineck (1990). For reasons of simplification, an alternative lower boundary of xr,min = d was established. The graph of the line xr = (0,50 a) coincides very well with the trend line for the test beams with a ratio of a/d up to a/d = 4,0. A more detailed examination of beams with a high slenderness of a/d > 6,0 ascertained that these beams were near flexural failures and that beams with a slenderness of approx. a/d = 8,0 were bending failures. Furthermore, beams of a small slenderness (a/d < 2,4) were eliminated, because the further evaluations are only valid for slender members.

Thus, Fig. 1-21 only shows the remaining 111 beams with a slenderness within 2,40 < a/d < 5,6. The mean value of these beams equals m = 0,512, confirming the trend of the test results within this range. Additionally, the upper and the lower boundary are shown, and the scatter band ranges from the lower boundary to the upper boundary.

According to these evaluations, the following expression for the distance of the failure crack from the support axis was taken for beams without stirrups and without data concerning the location of the failure cracks:

[mm] (1-102)

xr/d[-]

a/d[-]

Fig. 1-20: Diagram of measured values of 131 test beams plotted versus a/d for determining the

location of the failure crack


1 - 50

xr/d[-]

a/d[-]

Fig. 1-21: Diagram of measured values versus a/d and proposed relationship for the location of the failure

crack for beams with a slenderness of 2,4 < a/d < 5,6

5.1.3 Determination of the shear force due to self-weight

The shear force due to self-weight was determined according to equations Eq.(1-101) and Eq.(1-102) for beams subjected to point loads.

The following combinations of data concerning the consideration of self-weight occurred in the test reports:

1. Only the ultimate load F is given

2. The ultimate shear force is given including specifying the consideration of the self-weight.

3. The ultimate shear force is given without specifying if the self-weight was considered

Table 1-4 provides an overview to clarify the procedure of the database.

Table 1-4: Procedure of determining the ultimate shear force

case Vg F = Vu,F Vu,F+g,Rep Vu,Rep Vu,g+F

1 given

2 given

3

calc

ulat

ed

given

In case (1), in which only the ultimate load F is given explicitly, the self-weight was computed and added, resulting in the following ultimate shear force Vu,Rep:


1 - 51

[kN] (1-103a)

and [kN] (1-103b)

For test reports according to case (2) containing specifications of the self-weight, the given value for the shear force due to self-weight Vg,Rep was subtracted from the ultimate shear force and added to the shear force determined in accordance with Eq.(1-104):

[kN] (1-104)

This results in: [kN] (1-103a)

and [kN] (1-103b)

Most frequently, no specifications of the self-weight and the ultimate shear force were reported. In the following evaluation of this report, it is assumed that the reported ultimate shear force includes the self-weight as it ought to be. Thus, in this case (3) two values are reported for the ultimate shear force:

[kN] (1-105a)

and [kN] (1-105b)

In the following evaluation, however, the first equation was applied according to Eq. (1-64a), as it ought to be.

5.2 Determination of the ultimate shear force of beams with a uniformly distributed load and consideration of the self-weight of beams

5.2.1 Statement of problem

According to the majority of the standards, the check of the shear force capacity is performed in a distance d from the face of the support, such as in the German standard DIN 1045-1, 10.32 (1) or ACI 318-02, 11.1.3.1. However, this rule only applies to "direct" supports, i.e. if the support reaction is transferred to the beam via compression.

However, this rule only refers to the "loading side" of the following basic design equation:

VEd VRd,ct according to the German standard DIN 1045-1, 10.3.1 (1-106)

or Vn Vu according to the ACI 318-02 (1-107)

All standards determine the resistance side VRd,ct respectively Vu for structural concrete members without stirrups by an empirically derived relationship, which is meant to specify the shear force capacity at the failure crack. How-ever, this relationship was derived from single-span beams subjected to point loads since there are only a compara-tively small number of tests on beams subjected to a uniformly distributed load. In order to verify this relationship on basis of these few tests on beams with distributed load, it is essential to know the location of the failure crack xou (see Fig. 1-22), since only loads applied on the upper surface of the beam from the right side of the failure crack up to mid-span contribute to the failure of the different shear transfer actions in the failure crack.

For VRd,ct respectively Vu, it follows:

[kN] (1-108)

where: pu [kN/m] and xou [mm]

The measured distance of the failure crack from the support axis on the upper surface of the beam xou, is determined from the crack patterns. In the majority of cases, however, there are no available crack patterns or the test reports only provide values for the distance of the crack xr. Therefore, an approach was developed for these cases in the following section.


1 - 52

Fig. 1-22: Determination of the distance xr of the failure crack from the support axis in order to consider the con-

tribution of the self-weight of the beam to the ultimate shear force for beams subjected to a uniformly distributed load

5.2.2 Slenderness of beams subjected to a uniformly distributed load

For members subjected to a uniformly distributed load the following relationship exists for the so-called “shear slenderness” a/d as proposed by Kani (1958). In case of a single-span beam, the maximum moment Mmax and the maximum shear force Vmax equals:

[kNm] (1-109)

and [kN] (1-110)

Whereas for a single-span beam subjected to point load, it follows:

Mmax = Vmax · a (1-111)

If a is defined as a = Mmax/Vmax, it follows that:

[mm] (1-112)

and [-] (1-113)


5.2.3.1 Determination of the distance xr of the crack

The location of the failure crack is again determined from test reports if crack patterns were available.

The data collection includes a total of 128 beams. A total of 98 beams provide data concerning xr and 43 beams provide data concerning xou. The 55 beams examined by Krefeld und Thurston (1966) are considered conditionally since it is impossible to verify the specifications of xr. Krefeld und Thurston (1966) show the location of the failure crack and the type of determination in their system diagram (Fig. 1-23), but the crack patterns were not provided.

The distance of the crack in the axis of the beam for these beams is then estimated to:

[mm] (1-114)

where: [mm]


1 - 53

Fig. 1-23: Distance of the crack distance x1 according to Krefeld and Thurston (1966)

Fig. 1-24 provides a first overview plotting the distance of the crack xr versus a = l/4. One beam by Shioya (1989) has a very long span of a = 9000 mm and is identified with an arrow in Fig. 1-24. This beam and other long-span beams from this test series raise the trend considerably. Additionally, several lines through the origin are shown as in case of the beams subjected to point loads. The trend line is a line through the origin defined by the equation xr = 0,6187 a.

xr

[mm]

a[mm]

Fig. 1-24: Diagram of the measured distances xr of the failure crack plotted versus the shear span a for determining the location of the failure crack for considering self-weight of beams for the ultimate shear force

Fig. 1-25 exclusively shows slender beams with a/d > 2,40 and beams with a < 2000 mm in order to identify a reliable trend. The trend changes considerably and the equation of the new trend line is xr = 0,4637 a.

Additionally, the lower and upper boundaries for beams subjected to a uniformly distributed load are shown. The upper boundary for a uniformly distributed load equals:

[mm] (1-115)

[-] (1-116)

If a/d = 5,0, this results in an upper boundary of xr,max = 1,70 a. The lower boundary is determined to be xr,min = 0,25 a.


1 - 54

xr

[mm]

a[mm]

Fig. 1-25: Diagram of the measured distances of the crack plotted versus the shear span a for determining the

location of the crack for slender beams in order to consider the self-weigh for the ultimate shear force

Shioya (1989) presented the three different approaches plotted in Fig. 1-26. Whereas the equation of Higai resem-bles a parabola, the approach of Niwa appears to be of higher order. The approach according to Kani, however, is a simple line through the origin defined by the equation y = 1/8 x. These different approaches and all test beams are plotted in Fig. 1-26 in the dimension-free format of xr/d versus l/d. Even though a considerable scatter is identifi-able, the trend line of the data points (parabola of second order) coincides well with the approach of Higai for a low slenderness. The total trend line and the trend line derived from beams from Leonhardt und Walther (1962) are very close to each other. The large scatter of beams from Krefeld und Thurston (1966) is inexplicable since the crack patterns are unavailable. The trend line of these beams, however, coincides very well with the overall trend line (Fig. 1-27).

In Fig. 1-26 all 98 beams are plotted and the results reraise the question of excluding long-span and respectively less slender beams.

The following bi-linear approach is proposed:

if : [mm] (1-117a)

if : [mm] (1-117b)


1 - 55

xr/d[-]

l/d[-]

Fig. 1-26: Diagram of measured values and proposed relationships for plotted versus l/d in order to deter-

mine the location of the failure crack

Fig. 1-27 shows this proposal (dashed) together with the 98 test results and the respective trend lines. This approach is applied to beams without data concerning the failure crack xr since the chosen approach seems to be quite rea-sonable and coincides very well with the respective trend lines.

xr/d[-]

l/d[-]

Fig. 1-27: Diagram of measured values plotted versus l/d and comparison with proposed and other relation-

ships in order to determine the location of the failure crack


1 - 56

5.2.3.2 Determination of the distance xou of the crack

In the following a relationship is determined for the distance xou of the failure crack measured on the upper surface of the beam. Leonhardt und Walther (1962) propose the following value for the upper distance xou of the crack:

[mm] (1-118)

[-] (1-119)

The proposed values is based on their test series whereby xou,Leo is determined as shown in Fig. 1-28. It is clearly identifiable that Leonhardt und Walther (1962) define the distance of the crack xou,Leo differently. Therefore, it would be wrong to determine the directly transferred part of load near the end support by means of this distance xou,Leo. Additionally, Fig. 1-28 shows the crack distance xou, which was used in the databases for determining the directly transferred part of load near the end support and thus the ultimate shear force.

Fig. 1-28: Demonstration of the directly transferred parts of the load with the xou,Leo and xou at the crack pattern of

beam 13/1 from Leonhardt und Walther (1962)

Fig. 1-29 shows that the relationship (dashed line) proposed by Leonhardt und Walther (1962) coincides very well with the trend line derived from their reported values of xou,Leo. However, the self-measured distances of the cracks xou (Fig. 1-28) are far below the values given by Leonhardt und Walther; the trend lines show a discrepancy of up to xou/d = 1,5 in the medium range. Therefore, for further evaluations the self-measured distances of the cracks xou are used.

In Fig. 1-30 all measured distances of the cracks xou of the test by Leonhardt, Walther (1962) are plotted in dimen-sion-free format xou/d versus l/d. The individual trend lines of the tests are shown together with the overall trend line (equation of second order) for all 43 test beams. In order to determine a lower boundary, the following ap-proach was adapted assuming an angle of 60° degree from the axis of the beam:

[mm] (1-120)

For d/h = 0,9 this results in a lower boundary for xou of:

- if :

[-] (1-121a)

- if :

[-] (1-121b)


1 - 57

xou/d[-]

l/d[-]

Fig. 1-29: Diagram of measured values and proposed relationships for the distances xou/d respectively xou,Leo/d of the failure crack

xou/d[-]

l/d[-]

Fig. 1-30: Diagram of measured values xou/d versus l/d and comparison with proposed and other relationships for the location of the failure crack

In Fig. 1-30 this lower boundary is plotted as a dashed line. After considering the slenderness, it is possible to con-servatively use the shear force on the load side at a distance of 1,92d respectively 2,0d in contrast to the previously common value of 1,0d for beams with a/d = 3. The above given formula may also be applied to less slender beams. For beams with a the slenderness of a/d = 1,0 (l/d = 4,0) a distance of the failure crack of still (1,0 d) is obtained.


1 - 58

The approach of the database and thus of the resistance side according to Eqs.(1-106) and (1.107), however, follows corresponding to the mean value. In Fig. 1-30 the chosen approach is plotted dash-dotted and is defined for d/ h = 0,9 as follows:

- for :

[-] (1-122a)

- for :

[-] (1-122b)

The results is an angle of about 35° between the axis of the beam to the upper crack distance xou.

5.2.4 Calculation of the shear force due to self-weight

The shear force due to self-weight is calculated as follows for beams subjected to a uniformly distributed load:

[kN] (1-123)

where: [kN/m] ; Ac [mm²] ; l and xr [mm]

The following combinations occurred in the test reports and Table 1-5 provides an overview to clarify the procedure:

- case 1: The ultimate load (2 F) is given;

- case 2: The uniformly distributed load pu is given at failure;

- case 3: The ultimate shear force at the support is given without data on the consideration of self-weight.

Table 1-5: Procedure for determining the ultimate shear force

Fall g Vg 2F pu Vu,F pu,Rep Vu,xou Vu,g+F

1 give

n

2 given

3

kalk

ulie

rt

kalk

ulie

rt

give

n

In case (1), for which only the total force 2 F is given, it is assumed according to the definition that the self-weight was not taken into account. Thus, at first the ultimate uniformly distributed load pu is computed followed by the shear force at a distance of xou:

[kN/m] (1-124)


1 - 59

and [kN] (1-125)

and [kN] (1-126)

where: pu [kN/m] ; l and xou [m]

If test reports according to case (2) only specifying the ultimate uniformly distributed load pu, it was assumed ac-cording to the definition that the self-weight is not considered and identical formulae were applied.

In case (3), only the shear force at the support Vu,F was reported and the test reports did not mention data concern-ing the consideration of the self-weight. Now two assumptions are possible. Firstly, it was assumed that the self-weight is considered in the determination of this ultimate shear force. In this case (3a) the ultimate uniformly dis-tributed load pu is determined from the given shear force Vu,F:

[kN/m] (1-127)

From this, the ultimate shear force Vu,xou may now be determined at the distance xou of the failure crack whereby the self-weight is considered:

[kN] (1-128)

Another assumption may be made as case (3b) when the self-weight was not considered in the test report and thus the shear force Vu,F was directly calculated from the distributed load pu,Rep at failure:

[kN/m] (1-129)

From this, the ultimate shear force Vu,g+F may now be determined at the distance xou of the failure crack whereby the self-weight is considered

[kN] (1-130)

Finally it is pointed out that the evaluations carried out later were performed with Vu,xou according to case (3a). Thus it was assumed that the self-weight was considered for the determination of the shear force at the support, and this is on the safe side. However, the database contains both values for Vu,xou and Vu,g+F, so that also an evaluation may be done for the second case (3b).

5.3 Determination of the distance xr of the crack for prestressed concrete members subjected to point loads

For prestressed concrete members without stirrups, the identical relationships are applicable as used for reinforced concrete members, but a different magnitude of the distance xr of the crack from the support is to be expected. This is verified by a small evaluation of 15 beams for which the distance of the crack xr could be determined from the reported crack patterns in accordance with section 5.1 and respectively section 5.2.

In Fig. 1-31, the distance of the failure crack xr is plotted versus the distance a of the point load from the support for these 15 beams. For reasons of comparison, several lines through the origin are plotted representing different ratios of xr /a. The scatter is quite considerable and the number of test results is small, but the trend lines favor an ap-proach.


1 - 60

In order to determine a simple relationship from Fig. 1-32, the distance of the failure crack is plotted versus the distance of the point load from the support in the dimension-free format of xr/d and a/d. Two beams with a/d < 2,4 are eliminated to create the same basis as presented in section 5.1. The lower boundary remains xr,min = 0,25 a; the upper boundary is adjusted since the evaluation of the crack pattern shows that the distance of the failure crack reaches a value up to xr,max = a - d/2. The mean value equals m = 0,671, corresponding to the trend of the test data in this report. According to these evaluations, the following relationship is proposed for beams without stirrups for which no data was provided on the location of the failure crack:

[mm] (1-131)

xr

[mm]

a[mm]

Fig. 1-31: Measured values xr/d of the failure crack versus the distance a of the point load from the support axis for prestressed concrete beams

xr/d[-]

a/d[-]

Fig. 1-32: Diagram of measured values of versus a/d and proposed relationship for the location of the

failure crack for prestressed concrete beams with a slenderness of a/d > 2,4


1 - 61

5.4 Procedure of comparison with design equations 5.4.1 Definition of model safety factor

For reasons of comparing with design approaches the model safety factor is defined as:

[-] (1-132)

Thereby, the measured shear forces from the test reports are referred to as Vu,test and the calculated shear forces are referred to as Vu,cal . The shear forces Vu,cal are calculated according to several design approaches such as e.g. for structural members without shear reinforcement according to the German standard Eq.(70) of DIN 1045-1 (2001) or according for the proposals of Reineck (2002) or Loov (2003).

This factor qualifies the safety with which a relationship for the ultimate shear meets the test results. If a sufficient amount of data is provided, a statistical analysis can be carried out as shown in section 5.4.2.

A different approach has been proposed by Bazant and Yu (2004). However, the above definition of the model safety factor mod and the further procedure has traditionally been used for a long time and can be regarded as established and approved. A reference is made to the textbook of MacGregor (1988), sect. 2-4, in which the

variability of the ultimate flexural moments observed in tests Mtest is compared with the calculated values of Mn. The theoretical basis can be followed up in e.g. Schneider (1997), sect. 3.3.2 and the publication of the appropriate committees of CEB and fib, such as the CEB Bulletins 219 (1993) and 224 (1995). A special reference is made to König und Fischer (1995), who with this method derived the design values for structural concrete members without shear reinforcement (see section 5.2.2 of PART 2.1).

In the case of tests on beams subjected to point loads it is Vu,test = Vu,Rep, according to Eq. (1-103b) respectively Eq. (1-105a), see section 5.1.2. For members subjected to a uniformly distributed load, the shear force of the test beam is Vu,test = Vu,xou according to Eq. (1-108) respectively Eq. (1-113), see section 5.1.4, whereby Vu,xou is de-fined by the location of the failure crack as shown in Fig. 1-33a and defined in section 5.2.

Contrarily, the shear force VEd in the condition VEd Vu,cal in general refers to the distance d from the face of the support as it is shown in Fig. 1.33b and section 5.2.1. However, this is no contradiction since the load carried by the beam is directly compared with the allowable load according to the respective design approach:

- load carried in test: qu (l/2 - xou) = Vu,xou = Vu,test (1-133)

- load according to the design approach: qd [l/2 - (aA/2 +d)] = Vu,cal (1-134)

This results in: qu (l/2 - xou) Vu,test mod = = (1-135) qd [l/2 - (aA/2 +d)] Vu,cal

a) test b) code for directly supported beams

Fig. 1-33: Relevant shear force for members with uniformly distributed load

5.4.2 Statistical evaluations


1 - 62

A statistical evaluation of the distribution of the model safety factor according to Eq.(1-132) was performed using the general known notations and formulae (see PART 2.1, section 5.2.2):

total number of beams

values of of the individual beams i

arithmetic mean

variance

standard deviation

vx = sx / mx variation coefficient

5%- fractile for normal distribution

95%- fractile for normal distribution

The different design expressions yielded different results. Some approaches held a as 5%- fractile

value and thus are on the safe side, but generally the approaches resulted in a 5%- fractile value of

mod < 1,0. Therefore, the coefficients according to the expressions were adjusted using adjustment coefficients so that in all cases the lower 5 %- fractile values was equal to 1,00 assuming a normal distribution. Thus the ap-proaches are identically safe and comparable with each other.

If the number of test values was sufficient, such as in case of reinforced concrete members without stirrups, the existing distribution of mod was compared with the normal distribution, the logarithmical normal distribution and a normal distribution of the values below the median value reflected on the median (see PART 2.1, section 5.3). All cases showed that the normal distribution did not apply and that there were considerably more values of a very high

magnitude than ones of a very small, so that not 5% of the values were below the criterion . But in

general this was a considerably smaller percentage if the coefficient in a relationship was determined by assuming a normal distribution. Therefore, the coefficient was consequently increased by a second adjustment coefficient, to

guarantee that exactly 5 % of the model safety factors definitely fulfills the criterion ; i.e. the beams

were counted and the second adjustment coefficient is taken as the value of mod for the first of the 5% of all values. For example, if there are 792 test beams available, 5% of 792 beams are 39,6 beams. The second adjustment coeffi-

cient is determined, so that applies exactly to 39,6 tests using a linear interpolation between the two

first values. The further analysis is carried out using these newly adjusted coefficients to compare the model safety factors of different relationships or the role of influencing parameters.


1 - 63

References


1 - 64


1 - 65

PART 2.1: Database with shear tests on reinforced concrete beams without stirrups subjected to point loads


Contents

1 Introduction p. 2.1-3

2 The database vuct-RC-DS of tests on beams subjected to point loads p. 2.1-5

3 The database vuct-RC-DK_sl for tests on slender beams with point loads p. 2.1-9

3.1 Results of the evaluation by means of criteria p. 2.1-9

3.2 Selection of tests for evaluation p. 2.1-9

4 The evaluation database vuct-RC-A p. 2.1-13

4.1 Introduction p. 2.1-13

4.2 Presentation of the database vuct-RC-A p. 2.1-13

5 Comparison of the test results with design approaches for slender beams subjected to

point loads p. 2.1-17


5.2 Comparison of test results with the design approach of the German standard

DIN 1045-1 p. 2.1-17

5.2.1 Approach of the German standard DIN 1045-1 and determination of

coefficients

5.2.2 Direct determination of the design value

5.2.2.1 Input values 5.2.2.2 Determination of the design value according to the approximation in appendix D, EN 1990 (2002)

5.2.2.3 Determination of the design value according to the exact formula in appendix D, EN 1990 (2002)

5.2.2.4 Summary of the determinations of the design value 5.2.3 Statistical evaluation of the values of mod of the dataset

5.2.4 Dependency of the model safety factor on different parameters

5.3 Comparison of the test results with the design proposal by Reineck (2002) p. 2.1-34

5.4 Comparison of the test results with the design proposal by Loov (2003) p. 2.1-39

References p. 2.1-45

The report comprises pages 2.1-1 to 2.1- 45

List of Tables:

Research Report on extended shear databases - Part 2.1

2.1 - 2

Table 2.1-1.......................................................................................................................... p. 2.1-9

Table 2.1-2.......................................................................................................................... p. 2.1-10

Table 2.1-3.......................................................................................................................... p. 2.1-11

Table 2.1-4.......................................................................................................................... p. 2.1-18

Table 2.1-5.......................................................................................................................... p. 2.1-34

Table 2.1-6.......................................................................................................................... p. 2.1-39

Attachments

Attachment 2.1-1: Notation and Formulary for the shear data bases Vuct-RC-DS for the data collection and Vuct-RC-DS for the data control of reinforced concrete beams without stirrups under point loads

Attachment 2.1-2: References for the collection database vuct-RC-DS

Database with shear tests on reinforced concrete beams without stirrups subjected to point loads

2.1-3

1 Introduction The new extended database for tests on reinforced concrete beams without stirrups subjected to point loads com-prises presently 1220 test beams. The majority of the beams had rectangular cross-section, i.e 1.145 beams, and there are only 75 tests on T-beams.

In the process of the data collection and data evaluation the following files were generated:

- vuct-RC-DS = data collection file

- vuct-RC-DK_sl = data control file for slender structural members with a/d 2,40

- vuct-RC-A = evaluation and comparison with design expressions

The flow chart shown in Fig. 2.1-1 provides an overview of these files and of the distribution of the remaining beams and the eliminated beams.

Fig. 2.1-1: Selection of the beams according to the primary selection criteria

In order to assure that all data required for the data collection is provided, as many original test reports were ob-tained as possible for the extended database for tests on reinforced concrete beams without stirrups subjected to point loads.

By means of the criterion konx it was ascertained that the test data of 20 out of the 1220 beams did not include important data such as the uniaxial concrete compressive strength f1c, the yield strength of the reinforcing steel fsy or the ultimate shear force Vu. This data is essential for carrying out important assessments in the data control such as to check the flexural capacity and that there was adequate anchorage at the end support.

In a second step, the beams were checked for slenderness as shown in Fig. 2.1-1 using the selection criterion kon 61. According to this criterion, 284 beams had a slenderness of a/d < 2,4 and are thus transferred to the file vuct-RC-DK_24.

The remaining 916 tests on slender beams were transferred to the file vuct-RC-DK_sl.

kon61

1200 tests

20 tests

1220 tests

konx

916 tests 284 tests


2.1 - 4


2.1-5

2 The database vuct-RC-DS for tests on beams subjected to point loads In the following section the characteristics of the database vuct-RC-DS are examined by plotting the number of beams versus the most important test parameters as classified into intervals.

In Fig. 2.1-2 the number of beams n is plotted versus the concrete compressive strength f1c subdivided in class intervals of f = 5 MPa. Since no data was provided concerning the concrete compressive strength in the case of three beams, these beams were eliminated by means of the criteria konx. Most of the 1217 beams feature a uniaxial concrete compressive strength of 25 to 30 MPa. A total of 371 beams (30%) feature a concrete compressive strength of f1c < 25 MPa, and only 173 beams (approx. 15%) feature a concrete compressive strength of f1c > 60 MPa, and these are classified as high-strength concrete beams.

Fig. 2.1-2: Number of beams plotted versus concrete compressive strength for the database vuct-RC-DS

The yield strength of the longitudinal reinforcement used in each test beam was considered as an additional parame-ter. In Fig. 2.1-3 the number of beams is plotted versus the yield strength fsy subdivided in class intervals of

f = 50 MPa. For all beams, the grade or yield strength of the reinforcement was provided.

In recent years, low steel grades were applied in many test beams and thus 505 tests, i.e. 41%, resulted in yield strengths of fsy < 400 MPa. The peak (295 beams, 24%) of the distribution appears at yield strengths between 400 and 450 MPa. In the case of 136 test beams (11 %) the yield strengths were between 500 and 600 MPa and thus corresponds to the steel grades reported in the German standard DIN 1045-1 (2001). Higher yield strengths of 600 < fsy < 1.000 MPa were applied in 106 test beams, and in 28 test beams high-strength steel with a yield strength of fsy > 1.350 MPa was applied, likely as a means of avoiding flexural failures.


2.1 - 6

Fig. 2.1-3: Number of beams plotted versus the yield strength of the longitudinal reinforcement

for the database vuct-RC-DS

The next Fig. 2.1-4 shows the number of beams versus the longitudinal (or geometrical) reinforcement ratio subdi-vided in class intervals of l = 0,25 %. In the case of 17 beams, no data was reported concerning the provided reinforcement ratio. The scatter in Fig. 2.1-4 illustrates that the tests were carried out on beams that had a large range of reinforcement ratios.

Only a few test beams with low reinforcement ratios of l < 0,5 % were carried out (68, i.e. just under 6%), and only a total of 277 test beams (23%) fulfill the criterion l < 1,0 %. Thus, this range is under-represented which is particularly decisive for slabs in structural engineering. The peak of the distribution (197 test beams, approx. 16%) appears at l = 1,75 up to 2,00 %, and 474 tests were carried out on beams with a reinforcement ratio of l > 2,0 %, which is almost 40 % of the available test data.

l

Fig. 2.1-4: Number of beams plotted versus the geometrical reinforcement ratio l for database vuct-RC-DS


2.1-7

The geometrical reinforcement ratio does not characterize the flexural capacity as well as the mechanical rein-forcement ratio l = l fsy/f1c. Therefore, in Fig. 2.1-5 the number of beams is plotted versus the mechanical reinforcement ratio of the longitudinal reinforcement subdivided into class intervals of l = 0,05. The scatter in Fig. 2.1-5 shows, that the tests on the beams were carried out for a large range of mechanical reinforcement ratios. Only 152 beams (approx. 13%) had values of l < 0,10, whereas 331 beams (approx. 28%) featured values between 0,10 < l < 0,20. Thus more than 50% of the test beams were highly reinforced or very highly reinforced with me-chanical reinforcement ratios of l > 0,20.

Fig. 2.1-5: Number of beams plotted versus the mechanical reinforcement ratio l for database vuct-RC-DS

Another important parameter is the slenderness = a/d. In Fig. 2.1-6 the number of beams is plotted versus the slenderness subdivided into class intervals of = 0,4. The slenderness of all beams was reported so that no beam was eliminated.

The scatter in Fig. 2.1-6 illustrates that tests on beams were carried out for a large range of the slenderness levels. However, the range of beams with a slenderness between 2,4 < < 3,6 is emphasized (583 test beams, i.e. approx. 48%). Apparently, this is due to the orientation of the researchers on the "shear valley" by Kani (1964, 1966). Most beams, i.e. 269 beams (22%), feature a slenderness between 2,8 - 3,2. Tests on beams with a slenderness of < 2,4 are selected by the selection criterion kon61, and passed on to the database vuct-RC-DK-24 (see Fig. 2.1-1).


2.1 - 8

Fig. 2.1-6: Number of beams plotted versus the slenderness for the database vuct-RC-DS

A very important parameter is the effective depth d. In Fig. 2.1-7 the number of beams is plotted versus the effec-tive depth subdivided into class intervals of d = 100 mm. In the case of 17 beams no data concerning the effective depth was provided, so that it was impossible to consider these beams in the further evaluation.

The distribution illustrates that predominantly beams with an effective depth of d < 400 mm were tested, i.e. 1013 of 1203 beams, that is 84%. Most beams (557 beams, 46 %) featured an effective depth between 200 - 300 mm. Only 64 beams (5,3%) featured an effective depth higher than d > 600.

Fig. 2.1-7: Number of beams plotted versus the effective depth d for the database vuct-RC-DS


2.1-9

3 The database vuct-RC-DK_sl for tests on slender beams with point loads 3.1 Results of the evaluation by means of criteria The database vuct-RC-DK_sl for slender beams subjected to point loads comprises 916 beams. The beams are ex-amined by means of the control criteria and the remaining beams are transferred to the data evaluation file. Individ-ual criteria were defined and checked for:

koni = 0 not fulfilled; not transferred to evaluation file;

koni = 1 fulfilled; transferred to evaluation file

The individual criteria and the results of the data control are listed in Table 2.1-1.

Table 2.1-1: Results of the evaluation of the tests with respect of the individual criteria koni

Individual criteria Fulfilled number % of 916 Not fulfilled

kon1 898 98,0 18

kon2 908 99,1 8

kon3 880 96,1 36

kon4 901 98,4 15

kon5 741 80,9 175

kon6 175 19,1 741

kon7 888 96,9 28

kon8 744 81,2 172

kon81 85 9,3 831

kon10 : gerippt 839 91,6 77

kon11 893 97,5 23

kon15 ”andbr” 916 100 0

3.2 Selection of tests for evaluation In order to be transferred to the evaluation file, several or all constraints must be fulfilled simultaneously for a beam, i.e.:

KONAi = kon1 · kon2 · kon3 · ... · koni

According to the individual criteria, this means:

KONAi = 0 no transfer to evaluation file;

KONAi = 1 transfer

All of the evaluated tests have to fulfill the following constraint:

KONA0 = kon1 · kon3 · kon4 · kon7 · kon10 · kon15 · kon11


2.1 - 10

758 tests fulfilled this combined criterion KONA0 and were transferred to the dataset A0. The flowchart in Table 2.1-2 as well as Fig. 2.1-8 illustrates the subsequently following application of the criteria. It is obvious that the criteria KONA0a and kon10 (ribbed reinforcement) considerably influence KONA0.

Fig. 2.1-8: Diagram of the individual criteria according to their subsequent application

Table 2.1-2: Subsequent application of the individual selection criteria KONA0

Selection criterion

Combination of the individual criteria

Added criterion Remaining of

916 difference

KONA0a kon1 · kon3 · kon4 · kon7 844 72

KONA0b KONA0a · kon10 : ribbed 777 67

KONA0c KONA0b · kon15 ”andbr” 777 0

KONA0 KONA0c · kon11 758 19

Different alternatives were selected in the evaluation file for sorting the evaluated tests in which the following crite-ria KONA0i were defined.

For the evaluation dataset A21, the following product of criteria was defined:

KONA21 = KONA0 · kon5 · kon8

where: kon5 = criterion for slenderness ( )

kon8 = criterion for flexural failures ( ).

For the evaluation dataset A22 in the dataset KONA21 the criterion kon8 was replaced with kon81:

0 tests kon15

kon10

kon1·kon3·kon4·kon7 72 tests

916 tests

844 tests

67 tests

777 tests

777 tests

kon11 19 tests

758 tests


2.1-11


where: kon81 = criterion for tests with

For the third evaluation dataset A31, the criterion kon6 was taken instead of kon5:


where: kon6 = criterion for test beams with slenderness of

For the fourth dataset A32, the criterion kon8 in the dataset KONA31 was replaced with kon81:


The result reported in Table 2.1-3 shows the configuration of the 688 out of 758 beams fulfilling the criteria KONA0. The flow chart in Fig. 2.1-9 is used to describe the generated datasets. The dataset A2 comprises of 540 remaining test beams, whereas the dataset A3 comprises 148 test beams, and thus the merged dataset (A2&A3) comprises of 688 test beams. All these datasets may separately be used for a statistical evaluation.

Table 2.1-3: Result of the evaluation of test beams on reinforced concrete beams by means of the selection criteria KONAi for the evaluation files KONA

Selection criterion Combination of the individual criteria

Fulfilled number % of 758

KONA21 KONA0 · kon5 · kon8 484 63,9




file (A2&A3) 688

Fig. 2.1-9: Selection criteria for the datasets A2, A3 and (A2&A3)

kon6·kon81

758 tests

kon6·kon8 kon5·kon81 kon5·kon8

11 tests 484 tests 56 tests 137 tests

540 tests 148 tests

688 tests


2.1 - 12


2.1-13

4 The evaluation database vuct-RC-A 4.1 Introduction The tests selected according to the criterion KONA2 were passed onto the evaluation database. This file comprises 540 tests, i.e. 44,3 % out of the 1220 tests in the collection database.

In this database, values for the dimension-free ultimate shear force are computed, which differ depending on the parameters they are related to:

[ - ] (2.1-1)

[ - ] (2.1-2)

[ - ] (2.1-3)

[ - ] (2.1-4)

[ - ] (2.1-5)

In this evaluation database, different design relationships for reinforced concrete members without shear reinforce-ment are calculated and then compared with test results. However, the values for the concrete compressive strength used in these relationships have first to be derived from the values for the uniaxial concrete compressive strength f1c, because only values for f1c were handed over to the evaluation database:

[MPa] = characteristic uniaxial compressive strength (2.1-6)

[MPa] = mean cylinder strength (2.1-7)

[MPa] = characteristic concrete compressive strength (2.1-8)

.4.2 Presentation of the database vuct-RC-A The number of the beams is plotted versus important test parameters that are subdivided into class intervals for the evaluation database vuct-RC-A with the dataset (A2&A3) containing 688 tests in order to obtain an overview of the distribution of the test parameters.

In Fig. 2.1-10 the number of beams n is plotted versus the uniaxial concrete compressive strength f1c subdivided into class intervals of f = 5 MPa. The peak of the distribution appears at tests with a uniaxial concrete compressive strength of f1c = 25 up to 30 MP, i.e. 167 test beams, respectively, 24 %. The predominant number of test beams features a uniaxial concrete compressive strength of f1c < 40 MPa, i.e. 502, respectively 73 % of the test beams. Only 132 beams (19 %) feature a uniaxial concrete compressive strength of f1c > 55 MPa and these are classified as high-strength concrete beams.


2.1 - 14

Fig. 2.1-10: Number of beams plotted versus the uniaxial compressive strength f1c for database vuct-RC-A

In Fig. 2.1-11 the number of beams is plotted versus the yield strength of the longitudinal reinforcement fsy of the beams subdivided in class intervals of fsy = 50 MPa.

In the evaluation database, 286 test beams (approx. 42 % of 688 beams) contained low steel grades with yield strengths of fsy < 400 MPa. A total of 234 beams (34 %) had longitudinal reinforcements with yields strengths in the range of 400 < fsy < 500 MPa. Only 103 beams (15 %) contained steel with yield strengths in the range of 500 < fsy < 600 MPa, corresponding to the German standard DIN 1045-1 (2001). A total of 62 test beams had high steel grades of fsy > 600 MPa, and from these 5 test beams had a yield strength of fsy > 1.500 MPa.

Fig. 2.1-11: Number of beams plotted versus the yield strength for the database vuct-RC-A


2.1-15

In Fig. 2.1-12 the number of beams is plotted versus the geometrical reinforcement ratio subdivided in class inter-vals of l = 0,25 %. The peak of the distribution (altogether 313 test beams, approx. 45% of 688 beams) appears at reinforcement ratios of 1,5 % < l < 3,0 %. A total of 120 test beams (approx. 17 %) were more highly or very highly reinforced resulting with reinforcement ratios of l > 3,0 %. Only 19 test beams (approx. 3 %) featured a very low reinforcement ratio of l < 0,5 %, whereas 105 test beam (approx. 15 %) were classified as having low reinforcement ratios in the range between 0,5 % < l < 1,0 %.

l

l

Fig. 2.1-12: Number of beams plotted versus the geometrical reinforcement ratio of the longitudinal reinforcement

of the beams for the database vuct-RC-A

In Fig. 2.1-13 the number of beams is plotted versus the mechanical reinforcement ratio of the longitudinal rein-forcement subdivided into class intervals of l = 0,05. Most test beams were highly or very highly reinforced: 292 beam (approx. 42 %) were carried out for the range of high mechanical reinforcement ratios of 0,20 < l < 0,40, and 118 beams (approx. 17 %) were very highly reinforced with a mechanical reinforcement ratio of l > 0,40. Only 82 beams (approx. 12 %) featured a mechanical reinforcement ratio of l < 0,10, thus representing the range of lightly reinforced beams that are common in practice.

Fig. 2.1-13: Number of beams plotted versus the mechanical reinforcement ratio l for database vuct-RC-A


2.1 - 16

In Fig. 2.1-14 the number of beams is plotted versus the slenderness = a/d subdivided in class intervals of

= 0,4. Most tests were carried out on beams with a slenderness of = 2,4 up to 3,2 (i.e. 315 of 688 beams, respectively approx. 46 %). A second peak (246 test beams, respectively approx. 36%) appeared at a slenderness of 3,2 < < 4,4. A total of 40 test beams (approx. 6 %) featured a high slenderness of > 6.

Fig. 2.1-14: Number of beams plotted versus the slenderness for the database vuct-RC-A

In Fig. 2.1-15 the number of the beams is plotted versus the effective depth d subdivided into class intervals of d = 100 mm. More than 50 % of the 688 test beams in the evaluation file featured an effective depth between

d = 200 and 300 mm, i.e. 350 beams, respectively 51 %. A total of 59 test beams (just under 9 %) featured an effective depth of d > 600 mm.

Fig. 2.1-15: Number of beams plotted versus the effective depth for the database vuct-RC-A


2.1-17

5 Comparison of the test results with design approaches for slender beams subjected to point loads 5.1 Introduction For the two datasets A2 and A3 as well as for their combination (A2&A3), the model safety factor was determined defined as follows (see PART 1, section 5.4, Eq.(1-132)):

[-] (2.1-9)

Thereby, the shear forces determined from the test reports are referred to as Vu,test and the theoretical shear forces are referred to as Vu,cal. The theoretical shear forces are determined according to different design approaches, such as e.g. the German standard DIN 1045-1, Reineck (2002) or Loov (2003).

The model safety factor was evaluated statistically and the different design approaches resulted in different results.

Some approaches featured a as 5%- fractile value and thus were on the safe side, but in general the

5%- fractile value of mod did not fulfil the criterion . The coefficients of the different relationships

were then adjusted by means of a first adjustment coefficient, so that assuming a normal distribution, the lower 5%-fractile value equals 1,00. All approaches are then equally safe.

However, all cases proved that a normal distribution does not accurately characterize the test data (see Chapter 5.3)

and considerably less values than 5% had a magnitude of . Therefore, the coefficient was increased by

a second adjustment coefficient to assure that 5% of the test data had a model safety factor . This

means that the values were counted and the second adjustment coefficient is then taken as the value of mod for the last value of the 5% (in cases where the number results in a decimal value, a linear interpolation was performed between adjacent whole-number values). The newly determined coefficients were then taken as the basis for the further evaluations.

5.2 Comparison of test results with the design approach of the German standard DIN 1045-1

5.2.1 Approach of the German standard DIN 1045-1 and determination of the coefficients

The design value for structural members without shear reinforcement and N = 0 is defined as follows according to Eq.(70) of the German standard DIN 1045-1:

[kN] (2.1-10)

where: for normal concrete

= coefficient for the influence of the height d of the member (size effect)

lw = As/(bw d) 0,02 = longitudinal reinforcement ratio [-]

fck = characteristic cylinder strength [MPa]

= minimum width of section within tension zone in [mm]

d = effective depth [mm]

The slenderness a/d of the structural member is not considered in this expression.

The coefficient of 0,10 in the expression for the design value is generally referred to as d. The characteristic value of VRk,ct including the coefficient k is determined according to Eq.(1-9a), PART 1 with fck = (fcm,cyl - 4) [MPa]. In the following evaluations the strength limitation of lw 0,02 is not respected.

Values of mod >> 1 were calculated and this resulted in a high mean value. By means of the first adjustment coeffi-cient, it is ascertained that according to a normal distribution the 5%-fractile value of the model safety factors is


2.1 - 18

mod,5% = 1,00 (see section 5.1). In the case of the dataset A2 with 540 tests (see Fig. 2.1-9) this first adjustment coefficient was determined to be 1,3318 (5 %). The second adjustment coefficient was determined to be 1,0713 fulfilling the criterion that exactly 5% of all tests, i.e. 27 tests, had a value of mod 1,00. This resulted in a total adjustment coefficient of (1,3318 1,0713) = 1,427, and thus the characteristic coefficient equals k = 0,1427. The magnitude of this value is considerably higher than the magnitude of the value of approx. 0,13 according to the evaluation of the database in Reineck (1999a).

The evaluation with mean values of the compressive strength using the previously determined characteristic coeffi-cient of 0,1427 in Eq.(2.1-10), respectively, Eq.(70) of the German standard DIN 1045-1 (2001), resulted in the following statistical values for the model safety factor mod = Vu,test/Vu,cal:

- number of tests: n = 540

- mean value: = 1,2895

- standard deviation: = 0,2164

- coefficient of variation: v = = 0,1678

- 5 % - fractile: x95% = 0,9335

- 95 % - fractile: x5% = 1,6455

The characteristic coefficient equals k = 0,1427.

In the case of dataset A3 which contained 148 test beams with a slenderness of 2,40 = a/d 2,89, the statisti-cal values were considerably different. The values are listed in Table 2.1-4 in comparison with the values for the dataset A2 as well as for the merged dataset (A2&A3).

Table 2.1-4: Statistical values for the model safety factors mod = Vu,test /Vu,cal for the datasets A2, A3 and (A2&A3)

statistical value Dataset

A2 A3 A2 & A3

coefficient k 0,1427 0,1402

n 540 148 688

1,2895 1,5543 1,3644

0,2164 0,4104 0,2898

v 0,1678 0,2640 0,2124

5% 0,9335 0,8792 0,8876

95% 1,6455 2,2294 1,8411

The evaluation of the dataset A3 results in essentially higher values for the mean value, yet also for the scatter, and the magnitude of the coefficient of variation (v = 0,264) is considerably higher than the coefficient of variation for the dataset A2 where v = 0,167. In the case of the merged dataset (A2&A3), this results in a lower fractile value and also in a lower coefficient.


2.1-19

For the merged dataset (A2&A3), the adjustment coefficients are as follows: first adjustment coefficient: 1,2443 (5 %); second adjustment coefficient: 1,1267 (34,4 beams); and thus the total adjustment coefficient is 1,402. This leads to a characteristic coefficient of k = 0,1402 for (A2&A3) and its magnitude is thus approx. 2 % smaller than

k = 0,1427 in case of A2 .

Implying a safety factor of c = 1,50, the following design values are determined applying the above given relation:

- d = 0,0951 for A2;

- d = 0,0935 for A2 & A3.

In the subsequent section 5.2.2, the design value is directly determined according to the European standard EN 1990.

5.2.2 Direct determination of the design value according to appendix D of EN 1990 (2002) 5.2.2.1 Input values The two datasets A2 & A3 comprise altogether 688 beams and feature adjustment coefficients of 1,2443 (5 %) and 1,1267 (34,4 beams) according to the approach of the German standard DIN 1045-1 as shown above. Initially, some notations are defined:

total number of beams

value of the individual beams i

arithmetic mean

variance

standard deviation

According to Heinhold and Gaede (1972) the log-normal distribution is a one-side unsymmetrical distribution. In

practice, however, the distribution is not applied directly, but a transformation of is always performed

and the thus generated normal distribution of y is used (see Fig. 2.1-16).

Fig. 2.1-16: Relation of the log-normal distribution according to König and Tue (1998)


2.1 - 20

The following definitions are applicable:

natural logarithm of of the individual beams i

arithmetic mean of yi

variance of yi

standard deviation of yi

In the case of a logarithmical normally distributed random variable, the following equations are valid:

arithmetic mean of the log-normal distribution

variance of the log-normal distribution

standard deviation of the log-normal distribution

Furthermore, it follows:

coefficient of variation

The value of , being undercut or exceeded with a probability of p%, is referred to as one-side p-quantile.

Instead of p- quantile, the term fractile is also common.

5%- fractile value of a distribution

95%- fractile value of a distribution

In respect of the formulae, a reference is made to König; Tue (1998) and Heinold; Gaede (1972).

Subsequent to the determination of this input value, there are two possibilities for the determination of the design value:

1. According to the approximation in appendix D of EN 1990 (2002)

2. According to the exact formula in appendix D of EN 1990 (2002)

5.2.2.2 Determination of the design value according to the approximation in appendix D, EN 1990 (2002)

Step 1: Develop a design model

[kN] (2.1-11)

The coefficient of d (in König et al. (1999) and Hegger et al. (1999) referred to as cd) is set to

d = 1,2443·1,1267·0,10 = 0,1402 considering the two adjustment coefficients (see 5.2.1), so that precisely 5% of

all tests fulfil the criterion .

Step 2: Compare experimental and theoretical values

[-] (2.1-12)

In Fig. 2.1-17 the test values Vu,test taken from the reports are plotted versus the calculated shear forces Vu,cal. The

deviation from the 45°-line is clearly noticeable. The continuously drawn trend line (y = 1,1068 x)

defines at the same time the deviation of the mean value (see step 3). The dashed-dotted line

(y = 1,3636 x) shows the mean value of the log-normal distribution.


2.1-21

Vu,test - Vu,cal - Diagramm

Vu,cal

[kN]

Vu,test [kN]

Fig. 2.1-17: Comparison of experimental with calculated values

Step 3: Estimate the mean value correction b

The estimator for the mean value correction b is determined by comparing the theoretical values rt with the ex-perimental values re. Firstly, or each specimen i ( i = 1 to n) the correction term bi is determined (see step 2). From

these tests a realization of the estimator for the mean value is calculated by:

[-] (2.1-13)

The deviation of the mean is as follow:

[-] (2.1-14)

Step 4: Estimate the coefficient of variation of the random error term

[-] (2.1-15)

[-] (2.1-16)

[-] (2.1-17)

[-] (2.1-18)

[-] (2.1-19)


2.1 - 22

Alternative for the determination for the coefficient of variation of the random error term

[-] (2.1-20)

[-] (2.1-21)

[-] (2.1-22)

[-] (2.1-23)





Step 5: Analyze the compatibility

Analyze the compatibility of the test population with regard to the assumptions made in the resistance function (see Fig. 2.1-17).

Step 6: Determine the coefficient of variation of the basic variables in the resistance function

Concrete compressive strength:

Contrary to the following approach and according to the standards and Hegger et al. (1999)

a magnitude of f = 4 MPa is applied to the evaluation of the tests (see PART 1, section 3.1.3), resulting in:

By means of the database, the mean value of the cylinder strength is determined as follows:

mean value

The standard deviation and the coefficient of variation can now be determined to be:

standard deviation


According to Kraemer et al. (1975) und Hegger et al. (1999) the standard deviation is estimated to be:

.


2.1-23

Effective depth:

According to Kraemer et al. (1975) the standard deviation is defined assuming a slab depth h and a concrete cover of 6 mm. The standard deviation for the effective depth is also assumed to be 6 mm according to Hegger et al. (1999), and these values are used also here.

mean value

standard deviation


Width of web:

Whereas Kremer et al. (1975) assume a constant width of slab and provide no information concerning the width of web, Hegger et al. (1999) assume a standard deviation for the width of web bw of 5 mm, which was taken here as well.

mean value

standard deviation


Step 7: Determine characteristic value rk of the resistance function

Coefficient of variation for all uncertainties of the structural member:

(2.1-24)

Coefficient of variation:

(2.1-25)

Standard deviation:

(2.1-26)

characteristic value rk of the resistance function:

(2.1-27)

where: magnitude of the fractile factor kn

If according to König; Tue (1998) the following approximation is valid: standard deviation equals

coefficient of variation:


2.1 - 24

König et al. (1999) assume that the coefficient of variation of the model uncertainty is and the deviation

of geometry is . In this case, this results according to König et al. (1999) in the following standard

deviation Q, respectively, coefficient of variation Vr:

(2.1-28)

Step 8: Determine the design value rd of the resistance function

Design value rd of the resistance function:

(2.1-29)

where: magnitude of fractile factor kd,n

reference period 50 years, probability of collapse


Hegger et al. (1999), König; Tue (1998) and König et al. (1999) apply a safety index of , as reported in the

European standard Eurocode 1. However, König et al. (1999) criticize, that the safety index of does not

consider the influence of the failure mode for the type of failure (ductile failure with reserve strength capacity re-sulting from strain hardening; ductile failure with no reserve capacity; brittle failure).

Brittle types of failure, such as shear failure in case of structural members without stirrups, are considered by increasing the safety index to , according to a recommendation of the Probabilistic Model Code by

JCSS (2001), p. 18.

In the following, the results for both safety indexes are shown.

For the determination of the design value follows with :

(2.1-30)

Step 9: Final choice of the characteristic values and the partial safety factor R

A first estimate of the partial safety factor results in:

(2.1-31)

It is additionally considered that in the design approach to use a characteristic value of and

instead of the so far reported mean values of the concrete compressive strength fcm.

(2.1-32)

(2.1-33)

(2.1-34)


2.1-25

(2.1-35)

The design value of the coefficient in Eq. (70) of DIN 1045-1 is thus determined to be for a safety

index of considering brittle failure.

A safety index of results in a design value of , i.e. in an approx. 10% higher magnitude.


The approximate formula is applicable in the case of a large number of tests ( ) starting from step 7. For a

small number of tests, the exact formula should be applied since it considers additional parameters. This dataset contains 688 beams and is thus very well covered by the approximate formula. However, for reasons of control and completeness, the calculation according to the exact formula is presented here too.



(2.1-24)


(2.1-25)

Standard deviation:

(2.1-36)

(2.1-37)

(2.1-26)

Characteristic value rk of the resistance function:

(2.1-38)

where: value of the fractile factor kn

fractile factor

coefficient

coefficient



(2.1-39)


2.1 - 26

The fractile factor in Hegger et al. (1999) is multiplied by the ratio (4,4/3,8) of the safety indexes:


Step 9: Determination of the partial safety factor


(2.1-31)

It is additionally considered that in the design approach to use a characteristic value of instead of the so far

reported mean values of the concrete compressive strength fcm.

(2.1-32)

(2.1-33)

(2.1-34)

(2.1-35)

The design value of the coefficient according to the exact formula is thus determined to be d = 0,0937 for a safety index of considering brittle failure. (i.e. the magnitude of this design value is approx. 2% lower than the

design value according to the approximate formula where ).

A safety index of results according to the exact formula in a design value of , i.e. an approx.

11% higher value.

5.2.2.4 Summary of the determinations of the design value

The exact determination according to section 5.2.2.3 resulted for the dataset (A2&A3) in a design value of

d = 0,0937 when applying a safety index of to consider brittle failure The magnitude of this design value

is 2% smaller than the value of determined from the approximate formula according to the section

5.2.2.2. Applying a safety index of only = 3,8 as suggested in the European Eurocode EC 2 resulted in values of

approx. 10 % higher magnitudes; according to the approximate formula, respectively,

according to the exact formula.


2.1-27

The statistical evaluation of the databases resulted in a characteristic coefficient of k = 0,1402 for the merged data-set (A2&A3) and an approx. 2 % higher magnitude of k = 0,1427 for the dataset A2. Implying a safety factor of c = 1,50, the coefficients of the expression for the design value according to Eq.(2.1-10), respectively, Eq.(70) of the German standard DIN 1045-1 (2001) are determined to d = 0,0951 for the dataset A2 and d = 0,0935 for the merged dataset (A2&A3). The latter coincides almost exactly with the design value of 0,0937 determined applying a safety index of = 4,4, whereby the partial safety factor according to Eq. (2.1-34) was 1,4956, i.e. almost exactly 1,5.

The design value of d = 0,10 in Eq.(70) of the German standard DIN 1045-1 appears to be in between the results for the safety indexes of = 4,4 and 3,8, thus corresponding to a partial safety factor of approximately c = 1,43.

5.2.3 Statistical evaluation of the values of mod of the dataset

Fig. 2.1-18 shows the number of all mod- values for the dataset (A2&A3) according to the approach of the German standard DIN 1045-1. The mod-values are plotted versus the frequencies of occurrence of the beams. The total

number of the dataset comprises 688 beams. The maximum (77 beams) appears at . After a small

decrease, the second maximum (66 beams) appears at . It is noticeable that the distribution of

the dataset is sinistral and does not entirely match a normal distribution. The normal distribution ( ) opens too far on the right and the mean value is located too far on the right side.

For an additional comparison, the log-normal distribution ( ) is plotted. The mean value of

the logarithmical distribution appears at the maximum values and coincides well with the sinistral distribution of the tests.

Finally, the median-normal distribution was plotted ( ). According to Weber (1992) the

median (central value) of a sorted statistical series is defined as:

a) in case of an uneven number n, the value in the middle;

b) in case of an even number n, the arithmetic mean of the two values in the middle.

Thus, at first the median is determined and subsequently all values smaller than the median are reflected, so that the test number does not change:

(2.1-40)

The median is not identical with the mean value of the test series since more test values are smaller than the mean value so that if the mean is taken a new dataset of 806 beams would be generated.

After having determined the mean value ( ) for the dataset (A2&A3), the values were reflected and the

standard deviation was determined ( ). Subsequently, it is possible to draw the curve (see Fig. 2.1-18).

This procedure of using the median-normal distribution was also proposed by Collins (2003) in the ACI Committee 445-F.


2.1 - 28

Fig. 2.1-18: Statistical evaluation of all mod- values for the dataset (A2&A3) according to their frequencies of oc-

currence subdivided in class intervals of 0,05 and in comparison with the normal distribution, the log-normal distribution and the median-normal distribution.

In Fig. 2.1-19, the sum frequency of the mod- values is plotted for the dataset (A2&A3). For reasons of comparison, a normal distribution, a log-normal distribution and a median-normal distribution is plotted. The best match with the dataset is achieved by the median-normal distribution. Additionally, the line of the 5%- fractile value (34,4 beams) is shown.

Fig. 2.1-19: Statistical evaluation of all mod- values of the dataset A2 & A3 for the cumulative frequency subdi-

vided into class intervals of 0,05 and in comparison with the normal distribution, the log-normal distribution and the median-normal distribution.


2.1-29

In order to clarify the region with in Fig. 2.19, the following Fig. 2.1-20 shows an enlarged cut-out of

the left region of Fig. 2.1-19. It is noticeable, as mentioned earlier, that the median-normal distribution with approx.

38 beams coincides best with the 5%-fractile value of 34,4 tests for the range below . In the case of the

logarithmical distribution, 50 beams fall short of the lower boundary, and in case of the normal distribution, 72 test fall short of the lower boundary, i.e. more twice the boundary of 34,4.

Fig. 2.1-20: Enlarged cut-out of the left region of Fig. 2.1-19 for a better assessment of the different

distributions in comparison to the real distribution

5.2.4 Dependency of the model safety factor on different parameters

In the following figures the model safety factor is plotted versus the essential parameters in

order to obtain further information of the influence of the different parameters and the quality of the relationship according to Eq. (70) with respect to considering these parameters.

Considering the adjustment coefficients of 1,07126 and 1,3318, the characteristic value of Eq.(2.1-10) is for normal concrete as follows:

(2.1-41)

In each diagram the relevant parameter is subdivided into individual ranges. Thereby, the magnitude of the individ-ual ranges is freely determined with respect to the parameter. For example, in case of the concrete compressive strength, three class intervals were generated each for normal-strength concrete and high-strength concrete. The boundaries are at the values of fck = 20, 35 and 50 MPa in the case of normal-strength concrete and fck = 65 and 80 MPa in the case of high-strength concrete. The values of f1c = 25,3, respectively 41,0, respectively, 56,8 as well as 72,6 and 88,4 result from the equation f1c = (fck + 4)/0,95, see PART 1, sections 3.1.2 and 3.1.3.

In each of these ranges the mean and the lower and upper fractile value is determined for all the data in this range. Accompanying each of these figures is a table in which the ranges with their statistical values determined in sec-tions are listed and can be compared with the also listed statistical values for the whole dataset.


2.1 - 30

In Fig. 2.1-21 for the dataset A2 the model safety factor for Eq.(70) of the German standard

DIN 1045-1 is plotted versus the uniaxial concrete compressive strength f1c , which is taken as the prism strength

. The trend of the mean values as shown in Fig. 2.1-21 is almost horizontal; this means that the

influence of the concrete compressive strength (respectively, of the concrete tensile strength indirectly via ) is

well captured. The values for the coefficients of variation in the Table below in Fig. 2.1-21 show that the coeffi-cients of variation increase with increased concrete compressive strength and that in case of high-strength concrete the scatter is considerably larger than in the case of normal-strength concrete. For the types of concrete as classified in the ranges D and E, Eq. (70) is clearly unsafe and the coefficient ought to be considerably reduced, since the magnitude of the 5%-fractile value is considerably smaller than the magnitude of the 5%-fractile value for all tests: 0,785 for range D and 0,854 for range E in contrast to 0,934 for all tests.

Fig. 2.1-21: Model safety factor mod for Eq.(70) of the German standard DIN 1045-1 plotted versus the uniaxial

concrete compressive strength f1c for the dataset A2 containing 27 values of mod 1,0


2.1-31

In Fig. 2.1-22 the model safety factor for Eq.(70) of the German standard DIN 1045-1 is plot-

ted versus the longitudinal reinforcement ratio for the dataset A2. The associated Table shows that the ranges C,

F and G feature a small uncertainty. The test beams with very high reinforcement ratios are likewise ones with high-strength concrete, allowing a high reinforcement ratio and nonetheless the flexural failure is characterized by yielding. This is the reason that in the data selection that the height of the compression zone instead of the rein-forcement ratio is restricted.

Fig. 2.1-22: Model safety factor mod for Eq.(70) of the German standard 1045-1 plotted versus the longitudinal

reinforcement ratio for the dataset A2 containing 27 values mod 1,0


2.1 - 32

In Fig. 2.1-23 the model safety factor for Eq.(70) of the German standard DIN 1045-1 is plot-

ted versus the effective depth d for the dataset A2. The predominant number of tests was carried out on beams with an effective depth up to 300 mm, and only a few tests results are available from beams with larger effective depths. The diagram shows a considerable decrease of mod with an increasing effective depth. A clear answer to possible uncertainties is provided by examining the values in the different ranges reported in the Table below, and it can be seen that the uncertainties appears in the range of 600 < d < 1000 mm (range D and E).

Fig. 2.1-23: Model safety factor mod for Eq.(70) of the German standard DIN 1045-1 plotted versus the effective depth d for the dataset A2 containing 27 values of mod 1,0


2.1-33

In Fig. 2.1-24 the model safety factor for Eq.(70) of the German DIN 1045-1 is plotted versus

the slenderness a/d for the merged dataset (A2&A3), which contains the dataset A2 with the ranges B to D. Fig. 2.1-24 clearly demonstrates the higher shear capacities of the tests in comparison to those predicted by Eq.(70) for small magnitudes of a/d. The associated table illustrates this increase in the range of low a/d is even better. The clearly lower scatter and the coefficients of variation for beams with a/d > 2,89 are remarkable, being significantly smaller with magnitudes of s = 0,1052 to s = 0,2515, respectively v = 0,0915 to 0,1835, than the magnitudes of s = 0,4104, respectively v = 0,2640 for the range of a/d < 2,89.

Accordingly striking is that from a magnitude of a/d = 2,89 the lower fractile values of the individual ranges are in between 0,9569 and 1,0424. Only a few tests (considerably less than 5%) appear below the 5%-fractile values.

Fig. 2.1-24: Model safety factor mod for Eq.(70) of the German standard DIN 1045-1 plotted versus the slender-ness a/d for the merged dataset (A2&A3) containing 34,4 values of mod 1,0


2.1 - 34

5.3 Comparison of the test results with the design proposal by Reineck (2002) Within the discussion of the ACI subcommittee 445-F about the consideration of the size effect in the empirical relation Vc = 2 f´c bw d [Imperial units] of the ACI 318 for structural concrete members without shear reinforcement, several relationships were proposed.

The "Empirical Proposal 1" by Reineck (2002) is for Imperial units:

[kips] (2.1-42a)

respectively in SI units with fck [MPa], w [-] , bw und ds [mm]:

[kN] (2.1-42b)

The first adjustment coefficient is determined to 1,0352 (5 %) and the second adjustment coefficient is determined to 1,0655 (for 27 beams) for the dataset A2. Thus the total adjustment coefficient equals 1,1030, which results in an increased coefficient of 0,280 in Eq. (2.1-42b). For the merged dataset (A2&A3), the following adjustment coefficients are determined: first adjustment coefficient: 0,9890 (5 %); second adjustment coefficient: 1,1200 (34,4 beams). This results in a total adjustment coefficient of 1,1077 and thus in a coefficient of 0,281 in the Eq. (2.1-42b), which leads to the following relationship:

[kN] (2.1-42c)

The relevant statistical values are given in Table 2.1-5. In case of the dataset A2, the statistical values for the rela-tionship by Reineck (2002) are almost identical with those for Eq.(70) of the German standard DIN 1045-1 (2001). In case of the merged dataset (A2&A3) the statistical values are slightly better, e.g. the magnitude of the coefficient of variation v = 0,2063 is approx. 3 % smaller than the coefficient of variation of 0,2124 for Eq.(70) of the German standard DIN 1045-1.

The evaluation of the dataset A3 yielded considerably higher magnitudes of the mean, yet also the scatter, and the coefficient of variation (v = 0,264) are considerably higher than the coefficient of variation for the dataset A2 where v = 0,167. For the merged dataset (A2&A3) this results in a lower fractile value; however, it does not influ-ence the coefficient in Eq. (2.1-42c).

Table 2.1-5: Statistical values for the model safety factor mod = Vu,test / Vu,cal for datasets A2, A3 and (A2&A3)

1,2976

0,2183

0,1682

0,9385

1,6567


2.1-35

In Fig. 2.1-25 the model safety factor for Eq.(2.1-42c) is plotted versus the uniaxial concrete

compressive strength for the dataset A2, which is taken as the prism strength .

The trend of the mean values is almost horizontal for this approach as well. The high-strength concrete classes fea-ture a considerably higher scatter than the normal-strength concrete classes as shown by the significantly higher magnitudes of the variations coefficients of the ranges D and E. For these ranges of the concrete compressive strengths, the relationship of Eq.(2.1-42c) is slightly unsafe, as shown by the slightly lower fractile values of 0,837, respectively 0,845 in contrast to 0,938 of all tests.

Fig. 2.1-25: Model safety factor mod for Eq.(2.1-42c) by Reineck (2002) plotted versus the uniaxial concrete compressive strength f1c for the dataset A2 containing 27 values of mod 1,0


2.1 - 36

In Fig. 2.1-26 the model safety factor for Eq.(2.1-42c) by Reineck (2002) is plotted versus the

longitudinal reinforcement ratio for the dataset A2. It can be seen that the uncertainties occur especially in the ranges C and D. The results from the test beams featuring very high reinforcement ratios are at the same time predominantly ones in high-strength concrete, allowing a high reinforcement ratio and nonetheless the flexural failure is characterized by yielding; Thus, in the data selection, the height of the compression zone instead of the reinforcement ratio is restricted.

Fig. 2.1-26: Model safety factor mod for Eq.(2.1-42c) by Reineck (2002) plotted versus the longitudinal

reinforcement ratio for the dataset A2 containing 27 values of mod 1,0


2.1-37

In Fig. 2.1-27 the model safety factor for Eq.(2.1-42c) by Reineck (2002) is plotted versus the

effective depth d for the dataset A2. Altogether, the influence of the effective depth is captured pretty well as illus-trated by the comparatively low variation of the mean values of the different ranges. However, with increasing ef-fective depth a low increase is detectable and the unsafe values almost exclusively appear in range A and B with d < 300 mm. Large scatter bands, respectively high magnitudes of the coefficients of variation appear in range C, i.e. in the range of medium values for the effective depth.

Fig. 2.1-27: Model safety factor mod for Eq.(2.1-42c) by Reineck (2002) plotted versus the effective depth d

for the dataset A2 containing 27 values of mod 1,0


2.1 - 38

In Fig. 2.1-28 the model safety factor for Eq.(2.1-42b) by Reineck (2002) is plotted versus the

slenderness a/d for the merged dataset (A2&A3), which contains the dataset A2 with the ranges B to D. Range A with a slenderness of 2,4 < a/d < 2,89 features a considerably higher scatter, respectively coefficients of variation as well as mean values, and the mean values steadily decrease with increasing slenderness. This is obviously due to not considering the slenderness in the approach. The unsafe values appear in all ranges.

Fig. 2.1-28: Model safety factor mod for Eq.(2.1-42c) by Reineck (2002) plotted versus the slenderness a/d for the

merged dataset (A2&A3) containing 34,4 values of mod 1,0


2.1-39

5.4 Comparison of the test results with the design proposal by Loov (2003) Within the discussion of the ACI subcommittee 445-F about the consideration of the size effect in the empirical relation Vc = 2 f´c bw d [Imperial units] for structural members without shear reinforcement of the ACI 318, several relations were proposed. The proposal of Loov (2003) corresponds to the proposal of Reineck (2002), however, additionally, the slenderness a/d is considered resulting in the following equation:

[kips] (2.1-43a)

respectively in SI units with fck [MPa], w [-] , bw und ds [mm]:

[kN] (2.1-43b)

where: = coefficient for the size effect (2.1-44)

In case of the dataset A2, the first adjustment coefficient is determined to be 1,0870 (5%) (see section 5.1) and the second adjustment coefficient is determined to be 1,0458 (27 beams). Thus the total adjustment coefficient equals 1,1368, resulting in an increased coefficient of 1,574 in Eq.(2.1-43b) and thus to the following relationship for the characteristic shear capacity:

[kN] (2.1-43c)

For the merged dataset (A2&A3), the following adjustment coefficients are determined: first adjustment coeffi-cient: 1,0389 (5 %); second adjustment coefficient: 1,0485 (34,4 beams). This results in a total adjustment coeffi-cient of 1,0893 and thus in a coefficient of 1,509 in Eq. (2.1-43b).

The statistical values are given in Table 2.1-6. The statistical values for the relationship by Loov (2003) are only marginally lower than the values by Eq.(2.1-42c) for Reineck (2002) in case of the dataset A2. However, in case of the dataset A3, the approach by Loov (2003) provides considerably improved values. For example, the magnitude of the mean value of 1,27 is considerably lower than the mean value of 1,57 for Eq.(2.1-42b) by Reineck (2002) as well as the coefficient of variation of 0,216 in contrast of 0,230 according to Reineck. This is obviously due to the consideration of the slenderness a/d of the approach. This beneficially affects the evaluation of the merged dataset (A2&A3), resulting in a coefficient of variation of only v = 0,1707 in contrast to 0,2063 according to Reineck, i.e. in an approx. 17% lower value.

Table 2.1-6: Statistical values for the model safety factor mod = Vu,test / Vu,cal for the datasets A2, A3 and (A2&A3)

1,2856

0,2002

0,1557

0,9562

1,6149


2.1 - 40

In Fig. 2.1-29 for the dataset A2 the model safety factor for Eq.(2.1-43c) by

Loov (2003) is plotted versus the uniaxial concrete compressive strength f1c, which is taken as the prism

strength . The trend of the mean values is almost horizontal for this approach, too. The high-

strength concrete classes again feature a considerably higher scatter than the normal-strength concrete classes, as shown by the considerably higher magnitudes of the coefficients of variation of the ranges D and E. For these ranges of the concrete compressive strength, the approach is unsafe which is illustrated by the considerably lower fractile values of 0,788, respectively 0,880 in contrast to 0,956 of all tests.

Fig. 2.1-29: Model safety factor mod for Eq.(2.1-43c) by Loov (2003) plotted versus the uniaxial concrete compressive strength f1c for the dataset A2 containing 27 values of mod 1,0


2.1-41

In Fig. 2.1-30 the model safety factor for Eq.(2.1-43c) by Loov (2003) is plotted versus the

longitudinal reinforcement ratio for the dataset A2. It can be seen that the unsafe values appear in several

ranges. Many of the test beams feature very high reinforcement ratios and at the same time are predominantly ones in high-strength concrete that allow for a high reinforcement ratio and thereby flexural failures are characterized by yielding. Thus, in the data selection the height of the compression zone instead of the reinforcement ratio is restricted.

Fig. 2.1-30: Model safety factor mod for Eq.(2.1-43c) by Loov (2003) plotted versus the longitudinal

reinforcement ratio for the dataset A2 containing 27 values of mod 1,0


2.1 - 42

In Fig. 2.1-31 the model safety factor for Eq.(2.1-43c) by Loov (2003) is plotted versus the

effective depth d for the dataset A2. Altogether, the influence of the effective depth is captured pretty well as illus-trated by the comparatively low variation of the mean values for the different ranges. Large scatters, respectively high magnitudes of the coefficients of variation appear in the ranges A, C, and E; i.e. in the whole range of the tested effective depths.

Fig. 2.1-31: Model safety factor mod for Eq.(2.1-43c) by Loov (2003) plotted versus the effective depth d for the

dataset A2 containing 27 values of mod 1,0


2.1-43

In Fig. 2.1-32 the model safety factor for the proposal by Loov (2003) is plotted versus the

slenderness a/d for the merged dataset (A2&A3), which gives a coefficient of 1,509 instead 1,574 in Eq.(2.1-43c). The mean values of the individual ranges alter only slightly, and no considerable increase is detectable in Fig. 2.1-32 in range A. Thus, the selected approach seems to very well consider the influence of the shear-slenderness a/d.

Fig. 2.1-32: Model safety factor mod for Eq.(2.1-43c) by Loov (2003) plotted versus the slenderness a/d for the

merged dataset (A2&A3) containing 34,4 values of mod 1,0


2.1 - 44


2.1-45

References of PART 2.1

Att

achm

ent

2.1-

1:

Not

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n an

d F

orm

ular

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ear

data

bas

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data

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s w

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unde

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load

s

D

ecem

ber

2008

N

r. (

no.)

runn

ing

num

ber,

Lit.

(A

utho

r)

re

fere

nce:

aut

hor,

yea

r

Bez

. (T

est s

peci

men

)

sp

ecim

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s na

med

by

auth

or

Ein

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nits

): d

ual i

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in I

mpe

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uni

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r SI

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; Im

p. u

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are

con

vert

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to S

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ll ca

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SI-

units

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Que

rsch

nitts

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prop

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s)

b

b

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idth

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[i

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of

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ight

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ight

of

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Las

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Geo

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ular

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ism

s

f1cp

rism

f 1c,

pris

[ksi

→ M

Pa]

unia

xial

com

pr. s

tren

gth

deri

ved

from

fcp

r f1

c

f 1

c

[ksi

→ M

Pa]

un

iaxi

al c

ompr

. str

engt

h of

con

cret

e

Bet

onzu

gfes

tigk

eit (

conc

rete

tens

ile

stre

ngth

)

fctf

l

f c

t,fl

[k

si →

MPa

] m

odul

es o

f ru

ptur

e

Pkf

l

[i

n →

mm

] di

men

sion

of

cont

rol s

peci

men

f1ct

fl

f 1ct

,fl

[k

si →

MPa

] ax

ial t

ensi

le s

tren

gth

deri

ved

from

fct

fl

fcts

p

f c

t,sp

[k

si →

MPa

] sp

litt

ing

tens

ile

stre

ngth

Pks

p

[i

n →

mm

] di

men

sion

of

cont

rol s

peci

men

f1ct

sp

f 1

ct,s

p

[ksi

→ M

Pa]

axia

l ten

sile

str

engt

h de

rive

d fr

om f

ctsp

f1ct

test

f 1

ct,te

st

[k

si →

MPa

] te

st v

alue

for

axi

al te

nsil

e st

reng

th

beta

ctte

st

β

ct,te

st =

f1c

t,tes

t / f

1c

[

- ]

ra

tio

f1ct

cal

f 1

ct,c

al

[M

Pa]

ca

lcul

ated

val

ue o

f te

nsil

e st

reng

th

beta

ctca

l

βct

,cal =

f1c

t,cal /

f1c

[ -

]

rati

o

Atta

chm

ent 2

.1-1

: N

otat

ion

and

form

ular

y fo

r th

e sh

ear

data

bas

es V

uct_

RC

-DS

and

Vuc

t_R

C-D

K f

or r

.c.-

beam

s w

ithou

t stir

rups

4

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

- P

art 2

.1

Rei

neck

, K.-

H.;

Kuc

hma,

D.A

.; Fi

tik, B

. -

ILE

K, U

nive

rsity

of

Stu

ttgar

t and

Uni

vers

ity

of I

llin

ois

Dec

200

8

mec

hani

sche

Bew

ehru

ngsg

rade

(m

ecah

nica

l rei

nfor

cem

ent r

atio

s)

oms

100

f

f

1c

sys

s⋅⋅

=

[ -

]

mec

h. r

einf

. rat

io o

f re

info

rcin

g st

eel

oml

sl

=

[ -

]

mec

h. r

einf

. rat

io o

f te

nsio

n ch

ord

Ver

such

(te

st)

vu

Vu

[k

lbf

→ k

N]

ulti

mat

e sh

ear

forc

e

tute

st

τ

u,te

st =

d

b

1000

V

w

u

⋅

⋅

[ksi

→ M

Pa]

ulti

mat

e „s

hear

str

ess“

vute

st

c

w

u

ctest

ute

stu

fd

b

V

f1

1,,

⋅⋅

==

τυ

[

- ]

di

men

sion

fre

e te

st v

alue

of

ulti

mat

e sh

ear

forc

e

vute

stct

test

=tu

test

/f1c

test

test

,ct1

test

,ute

st,u

fτ=

υ

[

- ]

ra

tio

vute

stca

l = tu

test

/ f1

ctca

l

cal

,ct1

test

,ute

st,u

fτ=

υ

[

- ]

ra

tio

beta

r

β

r

[ °

]

angl

e of

incl

ined

cra

cks

xr

x r

[-

]

dist

ance

of

crac

k fr

om s

uppo

rt a

xis

sslm

ax

σ

ssl,m

ax

[-]

m

axim

um m

easu

red

stee

l str

ess

ssla

σ

sl,a

[-

]

mea

sure

d st

eel s

tres

s ne

ar e

nd s

uppo

rt

xsla

x s

l,a

[-

]

dist

ance

of

mea

sure

men

t fro

m s

uppo

rt a

xis

vxsl

a

v x

sl

[-

]

load

at m

easu

red

stee

l str

ess

Ver

sage

nsar

t

fail

ure

type

Bem

erku

ngen

re

mar

ks

andb

r

[

- ]

an

dbr

= o

ther

fai

lure

type

s

Atta

chm

ent 2

.1-1

: N

otat

ion

and

form

ular

y fo

r th

e sh

ear

data

bas

es V

uct_

RC

-DS

and

Vuc

t_R

C-D

K f

or r

.c.-

beam

s w

ithou

t stir

rups

5

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

- P

art 2

.1

Rei

neck

, K.-

H.;

Kuc

hma,

D.A

.; Fi

tik, B

. -

ILE

K, U

nive

rsity

of

Stu

ttgar

t and

Uni

vers

ity

of I

llin

ois

Dec

200

8

Kon

trol

le B

iege

bruc

h (c

heck

of

flex

ural

cap

acit

y)

oml

10

0f

f

1c

syl

l⋅⋅

ρ=

ω

[

- ]

m

echa

nica

l rei

nfor

cem

ent r

atio

of

tens

ion

chor

d

kapc

250

f1

c1c

−=

κ

[

- ]

co

effi

cien

t for

max

imum

str

ess

of s

tres

s bl

ock

zeta

2/1

c

1

κ

ω−

=ζ

[ -

]

coef

fici

ent f

or in

ner

leve

r ar

m

muf

lex

10

0f

f

1c

syl

l⋅⋅

ρ=

ω

[

- ]

di

men

sion

s fr

ee m

omen

t at f

lexu

ral f

ailu

re

vufl

ex

)

b/b(

wfl

ex,u

flex

,u⋅

κ

μ=

υ

[

- ]

di

men

sion

fre

e sh

ear

forc

e at

fle

xura

l fai

lure

beta

flex

= v

utes

t / v

ufle

x

fl

ex,u

test

,ufl

exυυ

=β

[ -

]

rati

o

BB

re

mar

k: B

B =

fle

xura

l fai

lure

Ver

anke

rung

(an

chor

age

at e

nd s

uppo

rt)

lbvo

rh f

or a

a ≠

0; b

a ≠

0:

)(

2/,

dh

ba

prov

lbA

A−

−+

=

[in

→ m

m]

prov

ided

anc

hora

ge le

ngth

for

aa =

0; b

a ≠

0:

Ab

prov

lb=

,

[i

n →

mm

]

fo

r aa

≠ 0

; ba

= 0

:

d

apr

ovlb

A⋅

+=

1,0,

[i

n →

mm

]

fo

r aa

= b

a =

0:

dpr

ovlb

⋅=

25,0,

[i

n →

mm

]

ssla

u

+

+⋅

=58,0

73,15,0

,zd

za

AVu

sla

lA

sluσ

[-

]

stee

l str

ess

= v

test

*100

0* (

0,5*

aa/(

zeta

*d)

+ 1

,73

* (h

-d)/

(ze

ta*d

) +

0,5

8)/ A

sl

for

aa =

0:

ssla

u =

vu*

1000

*[0,

5*(0

,2*d

)/(z

eta*

d)+

1,73

*(d/

0,9-

d)/(

zeta

*d)+

0,57

7)/A

sl)

if

alf

aa n

ot a

vail

able

, the

n al

faa

= 1

.

lber

f =

alf

aa*d

st*s

slau

/(9*

f1ct

cal)

()

cal

ctu

sla

sta

erf

fd

lb,

1,

9/

⋅⋅

⋅=

σα

[in

→ m

m]

requ

ired

anc

hora

ge le

ngth

beta

lb =

lber

f/lb

vorh

= lb

req/

lbpr

ov

[

- ]

ra

tio

Atta

chm

ent 2

.1-1

: N

otat

ion

and

form

ular

y fo

r th

e sh

ear

data

bas

es V

uct_

RC

-DS

and

Vuc

t_R

C-D

K f

or r

.c.-

beam

s w

ithou

t stir

rups

6

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

- P

art 2

.1

Rei

neck

, K.-

H.;

Kuc

hma,

D.A

.; Fi

tik, B

. -

ILE

K, U

nive

rsity

of

Stu

ttgar

t and

Uni

vers

ity

of I

llin

ois

Dec

200

8

VB

re

mar

k: V

B =

anc

hora

ge f

ailu

re

Kon

trol

len

(cri

teri

a fo

r da

ta s

elec

tion)

exp

lana

tion

see

Rep

ort,

Part

1

kon1

ko

n2

kon3

ko

n4

kon5

ko

n61

kon6

2 ko

n7

kon8

ko

n9

kon9

1 ko

n10

konA

0 ko

nA11

K

onA

12

Kon

A2

konA

3 U

mre

chnu

ngsf

akto

ren

(Con

vers

ion

fact

ors)

1 in

ch

=

25,

4 m

mm

1 po

und

=

4,4

5 N

1 ki

p

=

4,4

5 kN

1 kl

bf f

t

= 1

,36

kNm

1 ps

i

=

1/1

45 *

MPa

1 ks

i

=

100

0/ 1

45 *

MPa

1 kp

/cm

2

= 1

00/9

,81

MPa

1 M

p =

1.0

00 k

p

= 1

/9,8

1 kN

Research Report on extended shear databases: PART 2.1

Attachment 2.1-2: References for the collection database vuct-RC-DS

Acharya, D.N.; Kemp, K.O. (1965): Significance of dowel forces on the shear failure of rectangular reinforced concrete beams without web reinforcement. ACI Journal, V. 62 (1965), No.10, 1265-1279

Adebar, P.E. (1989): Shear Design of Concrete Offshore Structures. A Thesis submitted in conformity with the requirements for the Degree of Doctor of Philosophy, University of Toronto 1989

Adebar, P.; Collins, M.P. (1994): Shear design of concrete offshore structures. ACI Structural Journal, V. 91, No. 3, May/June 1994, 324-335

Adebar, P.; Collins, M.P. (1996): Shear Strength of members without transverse reinforcement. Canadian Journal of Civil Engineering 23 (1996), No. 1, 30-41

Ahmad, S.H.; Khaloo, A.R.; Poveda, A. (1986): Shear Capacity of Reinforced High-Strength Concrete Beams. ACI Journal V.83 (1986), No. 2, March/April, 297-305

Ahmad, S.H.; Lue, D.M. (1987): Flexure-shear interaction of reinforced high-strength concrete beams. ACI Structural Journal, V.84 (1987), No. 4, July/Aug., 330-341

Ahmad, S.H.; Park, F.; El-Dash, K. (1995): Web reinforcement effects on shear capacity of reinforced high-strength concrete beams. Magazine of Concrete Research 47 (Sep. 1995), No. 172, 227-233

Al-Alusi, A.F. (1957): Diagonal tension strength of reinforced concrete T-beams with varying shear span, ACI Journal, May 1957, S. 1067-1077

Angelakos, D.; Bentz, E.C.; Collins, M.P. (2001): Effect of Concrete Strength and Minimum Stirrups on Shear Strength of Large Members. ACI Structural Journal, V.98 (2001), No.3, May/June, 290-300

Aster, H.; Koch, R. (1974): Schubtragfähigkeit dicker Stahlbetonplatten. BuStb 69 (1974), H.11, 266-270

Baldwin, J.W.; Viest, I.M. (1958): Effect of Axial Compression on Shear Strength of Reinforced Concrete Frame Members. ACI Journal, V.30 (1958), Nov., 635-654

Bazant, Z.P., Kazemi, M.T. (1991): Size effect on diagonal shear failure of beams without stirrups. ACI Structural Journal, V.88 (1991), May/June, 268-276

Bernander, K. (1957): An investigation of the shear strength of concrete beams without stirrups or diagonal bars. RILEM-Symp., Stockholm, Vol.1, 1957

Bernhardt, C.J.; Fynboe, C.C. (1986): High strength concrete beams. Nordic Concrete Research, Norske Betongforening, Oslo 1986

Bhal, N.S. (1968): Über den Einfluß der Balkenhöhe auf die Schubtragfähigkeit von einfeldrigen Stahlbetonbalken mit und ohne Schubbewehrung. Otto-Graf-Institut, H.35, Stuttgart, 1968

Bresler, B.; Scordelis, A.C. (1963): Shear strength of reinforced concrete beams. ACI Journal, V.60 (1963), No.1, 51-74

Cladera, A.; Mari A. R. (2002): Shear Strength of Reinforced High-Strength and Normal-Strength Concrete Beams. A New Simplified Shear Design Method., Spain

Cao, Shen (2000): Size Effect and the Influence of Longitudinal Reinforcement on the Shear Response of Large Reinforcement Concrete Members. A Thesis in conformity with the requirements for the degree of Masters of Applied Science Graduate Department of Civil Engineering University of Toronto, 2000

Cederwall, K.; Hedman, O.; Losberg, A. (1970): Shear strength of partially prestressed beams with pretensioned reinforcement of high grade deformed bars. Division of concrete structures, Chalmers University of Technology, Gothenburg, Sweden, Publication 70/6

Cederwall, K.; Hedman, O.; Losberg, A. (1974): Shear strength of partially prestressed beams with pretensioned reinforcement of high grade deformed bars. SP 42 - 9

Chana, P.S. (1981): Some aspects of modelling the behaviour of reinforced concrete under shear loading. Techn. Report 543, Cement and Concrete Association, Wexham Springs, 1981

Attachment 2.1-2: References for the collection database vuct-RC-DS 2

Research Report on extended shear databases - Part 2.1 Reineck, K.-H.; Kuchma, D.A.; Fitik, B. Dec 2008

Chang, T.S.; Kesler, C.E. (1958): Static and fatigue strength in shear of beams with tensile reinforcement, ACI Journal, June 1958

Clark, A.P. (1951): Diagonal Tension in Reinforced Concrete Beams. ACI Journal, V.48 (1951), No.2, 145-156 and Discussion

Collins, M.P.; Kuchma, D. (1999): How Safe Are Our Large, Lightly Reinforced Concrete Beams, Slabs, and Footings? ACI Structural Journal, V.96 (July-Aug 1999), No.4, 482-490

Diaz de Cossio, R.; Siess, C.P. (1960): Behavior and strength in shear of beams and frames without web reinforcement. ACI Journal, V.31 (1960), No.8, Feb., 695-735

Drangshold, G.; Thorenfeldt, E. (1992): High Strength Concrete. SP2 – Plates and Shells. Report 2.1, Shear Capacity of High Strength Concrete Beams. SINTEF Structural Engineering – FCB, August 1992, STF70 A92125

Elzanaty, A.H.; Nilson, A.H; Slate, F.O. (1986): Shear Capacity of Reinforced Concrete Beams Using High-Strength Concrete. ACI Journal, V.83 (1986), No.2, March-April, 290-296

Feldman, A.; Siess, C.P. (1955): Effect of Moment Shear Ratio on Diagonal Tension Cracking and Strength in Shear of Reinforced Concrete Beams. Univ. of Illinois Civil Eng. Studies, Structural. Research Series No. 107, 1955

Ferguson, P.M.; Thompson, J.N. (1953): Diagonal Tension in T-Beams without stirrups. Journal of the American Concrete Institute (Mar. 1953), S. 655-676

Ferguson, P. M. (1956): Some Implications of Recent Diagonal Tension Tests. ACI Journal V. 28, No. 2, Aug., 1956 page

Foster, S.J.; Gilbert, R.I. (1996): Tests on High Strength Concrete Deep Beams. University of New South Wales, Sydney, Australia June 1996

Gabrielsson, H. (1993): High Performance Concrete Beams Tested in Shear. Comparison Between the Traditional Approach and the Modified Compression Field Theory. 169-176 in: Holand, I.; Sellevold, E. (Editors): Utilization of High Strength Concrete, 3rd Symp. in Lillehammer, Norway, 20.-23. June 1993, Proc. V.1. Norwegian Concrete Assoc., Oslo 1993. ISBN 82-91341-00-1

Gaston, J.R.; Siess, C.P.; Newmark, N.M. (1952): An investigation of the load deformation characteristics for reinforced concrete beams. Univ. of Illinois, Structural Research Series, No.4c, 1952

Ghannoum, W.M. (1998): Size Effect on Shear Strength of Reinforced Concrete Beams. Department of Civil Engineering and Applied Mechanics, McGill University Montréal, Canada, November 1998

Grimm, R. (1996/97): Einfluß bruchmechanischer Kenngrößen auf das Biege- und Schubtragverhalten hochfester Betone. Diss., Fachb. Konstr. Ingenieurbau der TH Darmstadt, 1996 und DafStb H.477, Beuth Verlag GmbH, Berlin 1997

Haddadin, M.J., Hong, S., Mattock, A.H. (1971): Stirrup Effectiveness in Reinforced Concrete Beams with Axial Force. Proceedings of the ASCE; V.97 ST9 (Sept. 1971), 2277-2297

Hallgren, M. (1994): Flexural and Shear Capacity of Reinforced High Strength Concrete Beams without Stirrups. KTH, Stockholm, TRITA-BKN. Bull.9, 1994, 1-49

Hallgren, M. (1994): Shear Tests on Reinforced High and Normal Strength Concrete Beams without Stirrups. Royal Institute of Technology, Stockholm, February 1994

Hallgren, M. (1996): Punching shear capacity of reinforced high strength concrete slabs. Doctoral thesis, KTH Stockholm und TRITA-BKN: Bulletin 23, Stockholm, 1996

Hamadi, Y.D. (1976): Force transfer across cracks in concrete structures. PhD-thesis, Polytechnic of Central London, 1976

Hanson, J.A. (1958): Shear Strength of Lightweight Reinforced Concrete Beams. ACI, Title No. 55-24, (1958), 387- 403

Hanson, J.A. (1961): Tensile Strength and Diagonal tension resistance of Structural lightweight Concrete. ACI Journal, V.58 (1961), No.1, July, 1-39

Hedmann, O.; Losberg, A. (1978): Design of Concrete structures with regard to shear forces. p.184-209 in: CEB-Bull. 126, Paris, June 1978



Islam, M.S.; Pam, H.J.; Kwan, A.K.H. (1998): Shear Capacity of high-strength concrete beams with their point of inflection within the shear span. Proc. Instn Civ. Engrs Structs & Bldgs. 128 (Feb. 1998), 91-99

Johnson, M.K.; Ramirez, J.A. (1989): Minimum Shear Reinforcement in Beams with Higher Strength Concrete. ACI Structural Journal, V.86 (1989), No.4, 376-382

Kani, G.N.J. (1967): How safe are our large reinforced concrete beams? ACI Journal, V.64 (1967), No.3, 128-141 Disc. in ACI-Journal, Sept. 1967, 602-613

Kani, M.W.; Huggins, M.W.; Wittkopp, R.R. (1979): Kani on shear in reinforced concrete. Dep. of Civil Engineering, Univ. of Toronto, 1979

Kawano, H.; Watanabe, H. (1998): Shear strength of reinforced concrete columns under cyclic load reversal. 1499-1508 in Concrete under Severe Conditions 2. Proceedings of the 2nd International Conference on CONSEC '98. Vol. III. Editors: Gjørv, O.E.; Sakai, K., Banthia, N.(1998), E & FN Spon, London and New York

Kim, J.-K.; Park, Y.-D. (1994): Shear strength of reinforcement high strength concrete beams without web reinforcement. Magazine of Concrete research 46 (1994), No. 166, 7-16

Kim, D.; Kim, W.; White, R. (1999): Arch Action in reinforced Concrete Beams- A Rational Prediction of Shear Strength. ACI Structural Journal, V.96 (July-August 1999), No.96, 586-593

Krefeld, W.J.; Thurston, Ch.W. (1966): Studies of the shear and diagonal tension strength of simply supported r.c.-beams. ACI Journal, V.63 (1966), April, 451-475

Küng, R. (1985): Ein Beitrag zur Schubsicherung im Stahlbetonbau. Betonstahl in Entwicklung, H.33, TOR-ISTEG Steel Corp., Luxemburg

Kützing, L. (2000): Tragfähigkeitsermittlung stahlfaserverstärkter Betone., Institut für Massivbau und Baustofftechnologie, Univ. Leipzig , 2000

Kuhlmann, U. Ehmann, J. (2001): Versuche zur Ermittlung der Querkrafttragfähigkeit von Verbundplatten unter Längszug ohne Schubbewehrung- Versuchsbericht. Institut für Konstruktion und Entwurf Stahl-, Holz-, und Verbundbau, Universität Stuttgart, Nr. 2001- 6X, Februar 2001

Kuhlmann, U. et al (2002): Querkraftabtragung in Verbundträgern mit schlaff bewehter und aus Zugbeanspruchung gerissener Stahlbetonplatte ohne Schubbewehrung- Mitteilungen. Institut für Konstruktion und Entwurf Stahl-, Holz-, und Verbundbau, Universität Stuttgart, Nr. 2002- 2

Kulkarni, S.M.; Shah, S.P. (1998): Response of reinforced Concrete Beams at High Strain Rates. ACI Structural Journal V.95 (Nov.-Dec. 1998), No. 6, 705-714

Lambotte, H.; Taerwe, L.R. (1990): Deflection and cracking of High-Strength Concrete Beams and Slabs. SP 121-7, Page 109–128 in Utilization of High Strength Concrete, Symposium Berkeley 5/1990

Laupa, A.; Siess, C.P.; Newmark, N.M. (1953): The shear strength of simple-span reinforced concrete beams without web reinforcement. Univ. of Illinois, Structural Research Series, No.52, 1953

Leonhardt, F.; Walther, R. (1962a): Schubversuche an einfeldrigen Stahlbetonbalken mit und ohne Schubbewehrung. DAfStb H.151, Berlin, 1962

Lubell, A.; Sherwood, T.; Bentz, E.; Collins, M. (2004): Safe Shear Design of Large Wide Beams. Concrete International, V. 26, January (2004), No.1, 62-78

Manuel, R.F.; Slight, B.W.; Suter, G.T. (1971): Deep Beam Behavio Affected by Length and Shear Span Variations. ACI Journal No. 68-81 (Dec. 1971), 954-958

Marti, P.; Pralong, J.; Thürlimann, B. (1977): Schubversuche an Stahlbeton-Platten. IBK-Bericht Nr. 7305-2, ETH Zürich, Sept. 1977

Maruyama, K.; Rizkalla, S.H. (1988): Shear design consideration for pretensioned prestressed beams. ACI-Journal, V.85 (1988), No.5, Sep.-Oct., 492-498

Mathey, R.G.; Watstein, D. (1963): Shear strength of beams without web reinforcement containing deformed bars of different yield strengths, ACI-Journal, V.60 (1963), No.2, February. 1963, 183-207

Moayer, M.; Regan, P.E. (1974): Shear strength of prestressed and reinforced concrete T-beams. 183-214 in: Shear in reinforced concrete. Vol. 1, Publ. SP 42, ACI Detroit. 1974



Moody, K.G.; Viest, I.M.; Elstner, R.C.; Hognestad,E. (1954/1955): Shear strength of r.c.-beams. Part 1 – Tests of Simple Beams. ACI-Journal V.26, (1954), No.4, Dec. 1954, 317-332 Part 2 – Tests of Restrained Beams without Web Reinforcement. ACI-Journal V.26, (1955), No.5, Jan. 1955, 417-434 Part 3 – Tests of Restraine Beams with Web Reinforcement. ACI-Journal V.26, (1955), No.6, Febr. 1955, 525-539

Moretto, O. (1945): An investigation of the strength of welded stirrups in reinforced concrete beams. ACI-Journal, Nov. 1945

Morrow, J.D.; Viest, F.M. (1957): Shear strength of r.c.-frame members without web reinforcement. ACI-Journal, V.28 (1957), No.9, March, 833-869

Mphonde, A.G.; Frantz, G.C. (1984): Shear tests of high- and low- strength concrete beams without stirrups. ACI-Journal, V.81 (1984), July-Aug., 350-357

Niwa, J.; Yamada, K.; Yokozawa, K.; Okamura, M. (1987): Revaluation of the equation for shear strength of r.c.-beams without web reinforcement. Proc. JSCE No.372/V-5 1986-8 Translation in: Concrete Library of JSCE, No. 9, June 1987

Olesen, S.O.; Sozen, M.A.; Siess, C.P. (1967): Investigation of prestressed reinforced concrete for highway bridges, part IV: Strength in shear of beams with web reinforcement. University of Illinois, Bulletin No. 493, V.64, No.134, July 5, 1967

Palaskas, M.N.; Attiogbe, E.K.; Darwin, D. (1981): Shear Strength of Lightly Reinforced T-Beams. ACI-Journal, V.78 (1981), Nov. - Dez., 447-455

Podgorniak-Stanik, B.A. (1998): The Influence of Concrete Strength, Distribution of Longitudinal Reinforcement, Amount of Transverse Reinforcement and Member Size on Shear Strength of Reinforced Concrete Members. University of Toronto (1998)

Rajagopalan, K.S.; Ferguson, P.M. (1968): Exploratory shear tests emphasizing percentage of longitudinal reinforcement. ACI-Journal, V.65 (1968), No.8, 634-638

Regan, P.E. (1971 a): Shear in Reinforced Concrete – an analytical study. CIRIA-Report, April 1971

Regan, P.E. (1971 b): Shear in Reinforced Concrete – an experimental study. CIRIA-Report, April 1971

Regan, P.E. (1971 b): Behaviour of reinforced and prestressed concrete subjected to shear forces. Institution of civil engineers, Proceedings, Paper 7441S

Rehm, G.; Eligehausen, R.; Neubert, B. (1978): Rationalisierung der Bewehrungstechnik im Stahlbetonbau - Vereinfachte Schubbewehrung im Balken. Betonwerk + Fertigteil-Technik (1978), H.3, 147-155 u. H. 4, 222-227

Reineck, K.-H.; Koch, R.; Schlaich, J. (1978): Shear Tests on Reinforced Concrete Beams with axial compression for Offshore Structures – Final Test Report. Stuttgart, July 1978, Institut für Massivbau, Univ. Stuttgart (unveröffentlicht)

Remmel, G. (1991): Zum Zugtragverhalten hochfester Betone und seinem Einfluß auf die Querkrafttragfähigkeit von schlanken Bauteilen ohne Schubbewehrung. Diss., TH Darmstadt, 1992

Rogowsky, D.M.; MacGregor, J.G.; Ong, S.Y. (1983): Test of reinforced concrete deep beams. Structural Engineering Report No. 109 (Nov. 1983); 1-178

Rogowsky, D.M.; MacGregor, J.G. (1983): Shear strength of deep reinforced concrete continuous beams. Structural Engineering Report No. 110, Department of Civil Engineering, University of Alberta, Edmonton, Alberta; Canada (Nov. 1983); 1-178

Rosenbusch, J.; Teutsch, M. (2002): Trial Beams in Shear. Brite/Euram project 97-4163, Final Report Sub task 4.2, Institut für Baustoffe, Massivbau und Brandschutz, TU Braunschweig, January, 2002

Rüsch, H.; Haugli, F.R.; Mayer, H. (1962): Schubversuche an Stahlbeton-Rechteckbalken mit gleichmäßig verteilter Belastung. DafStb H.145, W. Ernst & Sohn, Berlin, 1-30

Salandra, M.A.; Ahmad, S.H. (1989): Shear Capacity of reinforced Lightweight High-Strength Concrete Beams. ACI Structural Journal, V. 86, (Nov-Dec. 1989), No. 6, 697-704



Scholz, H. (1994): Ein Querkrafttragmodell für Bauteile ohne Schubbewehrung im Bruchzustand aus normalfestem und hochfestem Beton. Berichte aus dem Konstruktiven Ingenieurbau Heft 21, Technische Universität Berlin 1994

Shin, S-W.; Lee, K-S; Moon, J-I; Ghosh, S. K. (1999): Shear Strength of Reinforced High-Strength Concrete Beams with Shear Span-to-Depth Ratios between 1.5 and 2.5. ACI Structural Journal, V.96-S61 (1999), July-August, 549-556

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Suter, G.T.; Manuel, R.F. (1971): Diagonal Crack Width Control in Short Beams. ACI Journal, V.68 (June 1971), No.6, 451-455

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Taylor, H.P.J. (1968): Shear stress in reinforced concrete beams without shear reinforcement. Cement and Concrete Ass., Techn. Rep. No.407, February, 1968

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Walraven, J.C. (1978): The influence of depth on the shear strength of lightweight concrete beams without shear reinforcement. Report S-78-4, Stevin Lab., Delft Univ., 1978

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Part 2.2: Databases with shear tests on reinforced concrete beams without stirrups subjected to a uniformly distributed load


Contents

1 Introduction p. 2.2-5

2 The database vuct-RC-gl-DS for the data collection of test beams subjected to a uniformly distributed load p. 2.2-7

3 The database vuct-RC-gl-DK_sl for the data control of tests on slender beams subjected

to a uniformly distributed load p. 2.2-11

3.1 Results of the evaluation by means of criteria p. 2.2-11

3.2 Selection of the tests for evaluation p. 2.2-12

4 The evaluation database vuct-RC-gl-A p. 2.2-15


4.2 Presentation of the database vuct-RC-gl-A p. 2.2-15

5 Comparison of test results with design approaches for slender beams subjected to a uniformly distributed load p. 2.2-19


5.2 Comparison of test results with the design approach of the German standard DIN 1045-1 p. 2.2-19

5.2.1 Approach of the German standard DIN 1045-1 and determination of coefficients

5.2.2 Model safety factors plotted versus different parameters

5.2.3 Summary of the results for the approach according to the German standard

DIN 1045-1

5.3 Comparison of the test results with the design approach by Reineck (2002) p. 2.2-26

5.4 Comparison of the test results with the design approach by Loov (2003) p. 2.2-31


2.2- 2

6 Comparison of the design approaches with the test results of all slender beams subjected to point

loads or to a uniformly distributed load p. 2.2-37


6.2 Comparison of the test results with the design approach of the German standard DIN 1045-1 p. 2.2-37

6.2.1 The design approach of the German standard DIN 1045-1 and determination of coefficients

6.2.2 Model safety factors with respect to different parameters

6.2.3 Direct determination of the design value according to appendix D, EN 1990 (2002)

6.2.3.1 Input values

6.2.3.2 Determination of design value according to the approximation in Appendix D, EN 1990 (2002)

6.2.3.3 Determination of design value according to the exact formula in appendix D, EN 1990 (2002)

6.2.3.4 Summary of the determination of design values

6.3 Comparison of the test results with the design approach of Reineck (2002) p. 2.2-50

6.3.1 The design approach of Reineck (2002) and determination of the coefficients


6.4 Comparison of the test results with the design approach of Loov (2003) p. 2.2-55

6.4.1 The design approach of Loov (2003) and determination of the coefficients


References of part 2.2 p. 2.2-61

The report comprises the pages 2.2-1 to 2.2-61

Tables:

Table 2.2-1.......................................................................................................................... p. 2.2-11

Table 2.2-2.......................................................................................................................... p. 2.2-13

Table 2.2-3.......................................................................................................................... p. 2.2-14

Table 2.2-4.......................................................................................................................... p. 2.2-20

Table 2.2-5.......................................................................................................................... p. 2.2-26

Table 2.2-6.......................................................................................................................... p. 2.2-31

Table 2.2-7.......................................................................................................................... p. 2.2-38

Table 2.2-8.......................................................................................................................... p. 2.2-50

Table 2.2-9.......................................................................................................................... p. 2.2-55

Databases with shear tests on reinforced concrete beams without stirrups subjected to uniformly distributed load

2.2-3

Attachments

Attachment 2.2-1: Formulary for the collection database (vuct-RC-gl-DS) of tests on reinforced concrete beams without stirrups subjected to a uniformly distributed load

Attachment 2.2-2: References of the collection database vuct-RC-gl-DS

Attachment 2.2-7: Evaluation Database vuct-RC-gl-A2 + A3&(A2 + A3)_gl for slender reinforced concrete beams without stirrups subjected to point loads and uniformly distributed loads


2.2- 4


2.2-5

1 Introduction The new extended database for tests on beams without stirrups subjected to a uniformly distributed load comprises 128 tests on beams with a rectangular cross-section.

In the process of the data collection and data evaluation the following databases were generated:

- vuct-RC-gl-DS = database for the collection of data

- vuct-RC-gl-DK_sl = database for the control of data of slender structural members with a/d 2,40

- vuct-RC-gl -A = evaluation database for comparison with design relationships

The flow chart as shown in Fig. 2.2-1 provides an overview of these different databases and of numbers of the re-maining and the eliminated beams.

Fig. 2.2-1: Selection of the beams according to the primary selection criteria

In order to assure that all data required for the data collection is provided, as many original test reports were ob-tained as possible for the extended database for tests on beams without stirrups subjected to a uniformly distributed load.

By means of the criterion konx (see Fig. 2.2-1) it was ascertained in a first step that all essential data was available such as the uniaxial concrete compressive strength f1c, the yield limit of the reinforcing steel fsy or the ultimate shear force Vu. This data was provided for all of the available tests.

In a second step, the test beams were selected according to their slenderness, applying the selection criterion kon61. According to this criterion, 69 tests were on slender beams, remained, and these test results were then transferred to the database vuct-RC-gl-DK_sl. In five of these tests the longitudinal reinforcement was staggered according to Shioya et al. (1989). For these five tests, the assessment of potential flexural failure was carried out in a separate file. The remaining 59 beams with a (shear) slenderness of a/d < 2,4 were transferred to the database vuct-RC-gl_DK_24.

staggered reinforcement 5 tests

59 tests

through reinforcement 64 tests

69 tests

kon61 128 tests

0 tests

128 tests

konx


2.2- 6


2.2-7

2 The database vuct-RC-gl-DS for tests on beams subjected to a uniformly distributed load An overview of the database vuct-RC-gl-DS for the data collection is presented by plotting the number of beams versus the most important test parameters subdivided in class intervals.

In Fig. 2.2-2 the number of beams n is plotted versus the concrete compressive strength f1c subdivided in class intervals of f = 5 MPa. The majority of the 128 beams features a uniaxial concrete compressive strength of f1c = 15 up to 30 MPa, i.e. (30 + 36) = 66 beams. The concrete compressive strength of all tests lies below f1c = 40 MPa, and thus there are no tests on high-strength concrete beams subjected to a uniformly distributed load.

Fig. 2.2-2: Number of beams plotted versus the concrete compressive strength for database vuct-RC-gl-DS

The steel grades of the reinforcement used in the tests are considered as the next parameter, and in Fig. 2.2-3 the number of beams is plotted versus the yield strength fsy subdivided in class intervals of f = 50 MPa. In almost 50% of the tests, i.e. 56 tests (approx. 44 %), the beams were reinforced with low yield strengths of fsy < 400 MPa. The emphasis of the distribution (51 tests) appears at yield strengths of 400 - 500 MPa. Only 21 tests (approx. 16 %) were carried out on beam with reinforcing steel complying with the German standard DIN 1045-1 (2001), i.e. for yield strengths of fsy > 500 MPa.

Fig. 2.2-3: Number of beams plotted versus the yield strength for the database vuct-RC-gl-DS


2.2- 8

In Fig. 2.2-4 the number of beams is plotted versus the geometrical (or longitudinal) reinforcement ratio subdivided in class intervals of l = 0,25 %. The distributions shows that the tests were predominately carried out for very high magnitudes of the reinforcement ratio: 54 tests (42 %) featured a reinforcement ratio of 2,0 % < l < 3,0 %, and 45 tests (35 %) even had reinforcement ratios of l > 3,0 %. Thus, only very few tests were carried out on members with a low reinforcement ratio which is in general very common in practice, that is only 15 tests were carried out for a reinforcement ratio of < 1,0 %.

Fig. 2.2-4: Number of beams versus the geometrical reinforcement ratio l for database vuct-RC-gl-DS

The geometrical reinforcement ratio does not characterize the flexural capacity as well as the mechanical rein-forcement ratio l = l fsy/f1c. Therefore, in Fig. 2.2-5 the number of beams is plotted versus the mechanical rein-forcement ratio of the longitudinal reinforcement subdivided in class intervals of l = 0,05. The distribution in Fig. 2.2-5 shows that the tests were carried out on beams with a large range of the mechanical reinforcement ratio. Only 15 beams featured a reinforcement ratio of l < 0,20, and 51 beams featured a reinforcement ratio of 0,20 < l < 0,40. Thus more that 50% of the beam should be considered very highly reinforced or over-reinforced with an reinforcement ratio of l > 0,40.

l

Fig. 2.2-5: Number of beams versus the mechanical reinforcement ratio l for database vuct-RC-gl-DS


2.2-9

Another important parameter is the slenderness = a/d, and in Fig. 2.2-6 the number of beams is plotted versus the slenderness subdivided in class intervals of = 0,4. Tests were carried out on beams with a large range of the slenderness and the distribution features two peaks:

- 43 beams feature a slenderness in between 2,4 and 3,2;

- 53 beams feature a slenderness in between 1,2 and 2,4; these beams were transferred to the database vuct-RC-gl-DK_24 by means of the criterion kon61.

Fig. 2.2-6: Number of beams plotted versus the slenderness a/d for the database vuct-RC-gl-DS

A very important parameter for structural concrete members without shear reinforcement is the effective depth d (= ds). In Fig. 2.2-7 the number of beams is plotted versus the effective depth subdivided in class intervals of

d = 100 mm. The distribution shows that the tests were predominately carried out for beams with an effective depth of 200 < d < 300 mm, that is 100 out of the 128 beams. Only 8 beams had effective depths of d 600 mm.

1d=3000

Fig. 2.2-7: Number of beams plotted versus the effective depth d for the database vuct-RC-gl-DS


2.2- 10


2.2-11

3 The database vuct-RC-gl-DK_sl for tests on slender beams subjected to a uniformly distributed load 3.1 Results of the evaluation by means of criteria The database for tests on slender beams subjected to a uniformly distributed load comprises of 69 beams. The beams are examined by means of control criteria and the remaining beams are transferred to the evaluation data-base.

The results of the data control for the individual criteria are listed in Table 2.2-1. Almost all beams fulfilled the first four criteria kon1 through kon4 as well as kon10, kon11 and kon15; only 4 beams from the test series of Krefeld, Thurston (1966) featured a uniaxial concrete compressive strength smaller than 12 MPa (kon1) and thus were eliminated from further evaluation.

However, it is noteworthy that only 22 beams, i.e. approx. 32%, clearly fulfil the criterion kon8 for the assessment

of the calculated flexural failure ( ). However, 29 additional tests (42 %) just missed to

fulfill this criterion (kon81,i.e. 1,0 flex < 1,10), so that possibly 51 of the 69 tests can be considered for further evaluation.

Table 2.2-1: Result of the evaluation of the tests with respect to the individual criteria koni

Individual criteria Fulfilled number % of 69 Not fulfilled

kon1 65 94,2 4

kon2 69 100,0 0

kon3 69 100,0 0

kon4 69 100,0 0

kon5 53 76,8 16

kon6 16 23,2 53

kon7 53 76,8 16

kon8 22 31,9 47

kon81 29 42,0 40

kon10 : ribbed 69 100,0 0

kon11 69 100,0 0

kon15 ”andbr” 69 100,0 0


2.2- 12

3.2 Selection of the tests for evaluation

In order to be transferred to the evaluation database, several or all criteria must be fulfilled simultaneously for a beam, i.e.:


As with individual criteria, this means that if:

KONAi = 0 no transfer to evaluation database

KONAi = 1 transfer to evaluation database

All of the evaluated tests have to fulfill the following criterion:

KONA0 = kon1 · kon3 · kon4 · kon7 · kon10 · kon15 · kon11

The result of this evaluation is shown in the flowchart in Fig. 2.2-8 and in Table 2.2-2, which demonstrate the in-fluence of the subsequent application of the criteria. In the first step, the primary criterion that all test beams con-sidered for the evaluation have to fulfill, were ascertained by means of the criterion KONA0a. Primary criteria are

such as: kon1 ( ), kon3 ( ), kon4 (h > 70 mm) and kon7 ( test 0,5). In this process,

17 tests results were eliminated. Applying the additional criteria kon10, kon15 and kon11, kon15 results in two additional eliminations, so that 50 tests remained in the database A0 for the further controls.

Fig. 2.2-8: Diagram of the subsequent application of the primary criteria that all tests of the database A0 have to fulfill

2 tests kon15

kon10

kon1·kon3·kon4·kon7 17 tests

69 tests

52 tests

0 tests

52 tests

50 tests

kon11 0 tests

50 tests


2.2-13

Table 2.2-2: Subsequent application of the individual selection criteria for KONA0

selection criterion combination of the individual criteria

added criterion remaining of

69

KONA0a kon1 · kon3 · kon4 · kon7 52

KONA0b KONA0a · kon10 : ribbed 52

KONA0c KONA0b · kon15 ”andbr” 50

KONA0 KONA0c · kon11 50

Different alternatives for the configuration of the evaluated tests were selected in the evaluation database and the following criteria KONA0i were defined, as can be seen in Fig. 2.2-9 and in Table 2.2-3.

For the evaluation A21 the two following products were defined:


where: kon5 = criterion for slenderness ( )

kon8 = criterion for flexural failures ( ).

For the evaluation A22 the approach of KONA21 was adjusted, replacing factor kon8 with factor kon81, i.e. only

tests with values of were selected:


For the third evaluation dataset A31, the factor kon6 was taken instead of kon5, i.e. tests with a slenderness of

were considered:


For the fourth evaluation dataset A32, the criterion kon81 was selected instead of kon8:


The results of these evaluations are shown in Table 2.2-3 and in Fig. 2.2-9. Thus, altogether 40 beams remain in the dataset (A2&A3) to be considered for the evaluation of a design formula; i.e. 31,2 % of the altogether 128 available tests.


2.2- 14

Table 2.2-3: Result of the evaluation of tests on reinforced concrete beams without stirrups and with uniformly distributed load by means of the selection criteria KONAi for the datasets KONA

Selection criterionCombination of the individual crite-

ria Fulfilled number % of 50





Sum 40

Fig. 2.2-9: Selection criteria for the evaluation datasets A2 and A3

kon6·kon81

52 tests

kon6·kon8 kon5·kon81 kon5·kon8

6 tests 13 tests 15 tests 6 tests

28 tests 12 tests

40 tests


2.2-15

4 The database vuct-RC-gl-A for the evaluation of the tests

4.1 Introduction The tests selected according to the criteria KONA2 and KONA3 were transferred to the evaluation database vuct-RC-gl-A. This database contains altogether 40 tests, i.e. 32% of the 128 collected tests. The number of tests are again presented in diagrams versus the different test parameters.

4.2 Presentation of the database vuct-RC-gl-A In Fig. 2.2-10 the number of beams n is plotted versus the uniaxial concrete compressive strength f1c subdivided in class intervals of f = 5 MPa. All tests were carried out for a uniaxial strength of concrete between 10 and 40 MPa, and thus no tests are carried out for high-strength concrete beams subjected to a uniformly distributed load.

Fig. 2.2-10: Number of beams plotted versus the uniaxial concrete compressive strength f1c for the

database vuct-RC-gl-A

In Fig. 2.2-11 the number of beams is plotted versus the yield strength of the longitudinal reinforcement fsy that was used in the beams subdivided in class intervals of fsy = 50 MPa. More than 50% of the 40 tests featured a yield strength of fsy < 400 MPa, that is 21 beams (52,5 %). In the case of 11 tests (28 %) the yield strength was within the range of 400 and 500 MPa. Only 7 tests featured values of fsy > 500 MPa, thereby complying with the German DIN 1045-1.

Fig. 2.2-11: Number of beams plotted versus the yield strength for the database vuct-RC-gl-A


2.2- 16

In Fig. 2.2-12 the number of beams is plotted versus the geometrical reinforcement ratio subdivided in class inter-vals of l = 0,25 %. A total of 17 tests featured a reinforcement ratio 2,0 % < l < 3,0 %, while 13 test beams (33 %) were highly or very highly reinforced featuring a reinforcement ratio of l > 3,0 %. Only 7 tests had low reinforcement ratios of l < 1,0 %, representing the range that is more common in practice.

Fig. 2.2-12: Number of beams versus the geometrical reinforcement ratio l for database vuct-RC-gl-A

In Fig. 2.2-13 the number of beams is plotted versus the mechanical reinforcement ratio l of the longitudinal reinforcement subdivided in class intervals of l = 0,05. Almost all test beams lie in the range of

l = 0,20 to 0,45, and only 7 tests featured a mechanical reinforcement ratio of l < 0,10, thus representing the range of reinforcement levels that are more common in practice.

l

Fig. 2.2-13: Number of beams versus the mechanical reinforcement ratio l for database vuct-RC-gl-A


2.2-17

In Fig. 2.2-14 the number of beams n is plotted versus the slenderness = a/d = l/4 d (see Part 1, section 5.2.2) subdivided in class intervals of = 0,4.

Most tests (12 + 17) of the overall 40 tests appear in the range between = 2,4 to 3,2, i.e. between l/d = 9,6 through 12,8. Out of the 26 tests on slender beams with a slenderness of = 3,2 through 5,6 (see Fig. 2.2-6) only 11 tests remained, and the highest magnitude of the slenderness was reached for 3 beams: approx. = 4,6; i.e. l/d = 18,4.

Fig. 2.2-14: Number of beams plotted versus the slenderness a/d for the database vuct-RC-gl-A

In Fig. 2.2-15 the number of beams n is plotted versus the effective depth d subdivided in class intervals of d = 100 mm. Almost all tests (that is 33) lie in between d = 200 and 300 mm, and 7 remained for the evaluation

out of the 8 tests with high effective depths of d > 600 mm (see Fig. 2.2-7).

1d=3000

Fig. 2.2-15: Number of beams plotted versus the effective depth d for the database vuct-RC-gl-A


2.2- 18


2.2-19

5 Comparison of the test results with design approaches for slender beams subjected to a uniformly distributed load 5.1 Introduction For the two datasets A2 and A3, for each test the model safety factor was determined for different design relation-ships. Thereby, the model safety factor is defined as follows (see Part 1, section 5.4, Eq.(1-132)):

[-] (2.2-1)

In Eq.(2.2-1) Vu,test is the ultimate shear force of the test, which is determined according to Part 1, section 5.2.3.2, respectively, Eq.(1-81) via the distance xou from the support axis to the failure crack, measured on the upper surface of the beams.

The calculated shear forces are referred to as Vu,cal, which are determined as characteristic values of the different design approaches, such as the German standard DIN 1045-1, Reineck (2002) or Loov (2003), as explained in Part 1, section 5.4.

The model safety factor was evaluated statistically and the different design approaches yielded different results.


5%- fractile value of mod did not fulfil the criterion . The coefficient of each design relationship for

Vu,cal were then adjusted by means of a first adjustment coefficient, so that assuming a normal distribution the lower 5%-fractile value equals 1,00.

However, all cases proved that a normal distribution is not valid (see Part 1, section 5.2.3) and in general consid-

erably fewer values than 5% featured a magnitude of . Therefore, the coefficient was consequently

increased by a second adjustment coefficient to assure that definitely 5% of the model safety factors fulfill the crite-

rion . To do this, the values were counted and the second adjustment coefficient is then taken as the

value of mod for the last value of the 5%. The newly determined coefficients were then applied to the further evaluations, and thus, all approaches are equally safe and comparable to each other.

The statistical characteristic values are in the following determined for reasons of completeness, although their sig-nificance must be taken conditionally due to the small number of tests.

5.2 Comparison of test results with the design approach of the German standard DIN 1045-1 5.2.1 Approach of the German standard DIN 1045-1 and determination of the coefficients

The design value for reinforced concrete members (i.e. N = 0) without shear reinforcement is defined as follows according to Eq.(70) of the German standard DIN 1045-1:

[kN] (2.2-2)

where: = coefficient for the influence of the height d of the member

l = As/(bw d) 0,02 = longitudinal reinforcement ratio [-] fck = characteristic cylinder strength [MPa]

= minimum width of section within tension zone [mm]

d = effective depth [mm]

The coefficient of 0,10 in the expression for the design value is generally referred to as d. The characteristic value of VRk,ct including the coefficient k is determined according to Eq.(1-9), Part 1 with fck = (fcm,cyl - 4) [MPa]. In the following evaluations the limit of lw 0,02 is ignored.


2.2- 20

The relevant statistical values were calculated and are listed in Table 2.2-4. Hereby, the dataset A3 forms range A of the merged dataset (A2&A3), see Fig. 2.2-19.

The first adjustment coefficient (see section 5.1) was determined to be 1,0014 (5%-fractile value) for the dataset A2 containing 28 tests, and the second adjustment coefficient was determined to be 0,8347 (1,4 beams counted). Thus, the total adjustment coefficient equals 0,8359 resulting in a characteristic coefficient of only k = 0,0836. The mag-nitude of this value is only 59 % of the magnitude of the value of k = 0,1427 derived for the dataset A2 of the tests on beams subjected to point loads (see Part 2.1, section 5.2.1).

For the total dataset (A2&A3) containing 40 tests, the adjustment coefficients are as follows: first adjustment coef-ficient 1,0660 (5 %); second adjustment coefficient: 0,8960 (2 beams counted). This results in a total adjustment coefficient of 0,9551 and respectively a characteristic coefficient of k = 0,0955. Whereas, in case of the tests on beams subjected to point loads according to part 2.1, section 5.2.1, the coefficient was determined to k = 0,1402. Thus, the tests on beams subjected to a uniformly distributed load result in only 68 % of the magnitude of the coef-ficient derived from tests on beams subjected to point loads. The reasons for the considerably lower coefficients are now explained.

Table 2.2-4: Relevant statistical values for the model safety factor mod = Vu,test / Vu,cal for the datasets A2, A3 and (A2&A3)

Relevant statistical value Datasets

A2 A3 A2&A3

Adjustment coefficient for 0,8359 0,9551 0,9551

n 28 12 40

1,8752 2,1083 1,7813

0,4117 0,3072 0,4044

v 0,2195 0,1457 0,2270

5% 1,1980 1,6030 1,1161

95% 2,5524 2,6137 2,4465

5.2.2 Model safety factors plotted versus different parameters

In the following figures the model safety factor is plotted versus the essential parameters in

order to obtain further information of the influence of the different parameters and the quality of the approach according to Eq. (70) of the German standard DIN 1045-1, respectively, Eq.(2.2-2). In each diagram the parameter is subdivided into individual ranges, and for each range the mean value is calculated as well as the upper and the lower fractile value. Below each of the diagrams, a table summarized the relevant statistical values for the whole dataset as well as for the different ranges.

However, it should be recognized that the number of tests is comparably small especially in some ranges. Thus, some of the conclusions might only be valid conditionally.


2.2-21

In Fig. 2.2-16 the model safety factors according to Eq.(2.2-2) or Eq.(70) of the German stan-

dard DIN 1045-1 are plotted for the dataset A2 versus the uniaxial concrete compressive strength f1c , which is

taken as the prism strength .

In range A the scatter is remarkably large due to tests on beams with a large effective depth and low reinforcement ratios, as shown in the following figures Fig. 2.2-17 and Fig. 2.2-18.

Fig. 2.2-16: Model safety factors mod according to the German standard DIN 1045-1 plotted versus the uniaxial concrete compressive strength f1c for dataset A2 containing 1,4 values of mod 1,0


2.2- 22

In Fig. 2.2-17 the model safety factors according to Eq.(70) of the German standard DIN

1045-1 are plotted versus the longitudinal reinforcement ratio for dataset A2. All unsafe test values appear in

range A, and these beams are from the test series of Shioya et al. (1989) and Shioya (1989) on beams with very low reinforcement ratios and large effective depths. In contrast to this, the other ranges show an almost horizontal trend for the mean values, and they all lie approximately around m = 2,0.

Fig. 2.2-17: Model safety factors mod according to the German standard DIN 1045-1 plotted versus the

longitudinal reinforcement ratio for dataset A2 containing 1,4 values of mod 1,0


2.2-23

In Fig. 2.2-18 the model safety factors according to Eq.(2.2-2) or Eq.(70) of the German stan-

dard DIN 1045-1 are plotted versus the effective depth d for dataset A2.

The diagram illustrates that all tests with a low effective depth (range B) are very well on the safe side with a 5%-fractile value of mod = 1,198. This means, that the low coefficient in Eq.(2.2-7) for all tests is only due to the 7 tests with an effective depth of d 600 mm from Shioya et al. (1989) and Shioya (1989), which with their very low val-ues for the model safety factor cause the low coefficient in Eq.(2.2-2) for all tests. Simultaneously, these tests fea-ture a very low reinforcement ratio lw as demonstrated earlier in Fig. 2.2-17.

d = 3000


effective depth d for dataset A2 containing 1,4 values of mod 1,0


2.2- 24

In Fig. 2.2-19 the model safety factors according to the approach in Eq.(70) of the German

standard DIN 1045-1 is plotted versus the slenderness a/d for the merged dataset (A2&A3).

Fig. 2.2-19 and the chart show in range A for low a/d a higher model safety factors than in the other ranges. The reason obviously is that Eq.(70) of DIN 1045-1 does not consider the influence of a/d. The tests featuring low model safety factors in the range B are again beams from the test series of Shioya et al (1989) with large effective depths and low reinforcement ratios.

Fig. 2.2-19: Models safety factors mod according to the German standard DIN 1045-1 plotted versus the slender-

ness a/d for the merged dataset (A2&A3) containing 2 values of mod 1,0

5.2.3 Summary of the results for the approach according to the German standard DIN 1045-1

The characteristic coefficient according to Eq.(70) of the German standard DIN 1045-1, respectively, Eq.(2.2-7) was determined to be k = 0,0836 for the dataset A2 containing 28 tests, and thus it is only 59% of the correspond-ing coefficient determined to be k = 0,1426 for the 540 tests on beams subjected to point loads (see Part 2.1, sec-tion 5.2.1).

The evaluation of the total dataset (A2&A3) containing 40 tests and resulting in a characteristic coefficient of

k = 0,09551, which is only approx. 68% of the magnitude of k = 0,1402 which is the coefficient determined for the 688 tests on beams subjected to point loads.


2.2-25

These low coefficients for the tests on beams subjected to a uniformly distributed load in Eq.(2.2-7), respectively, Eq.(70) of the German standard DIN 1045-1 are caused by the 7 tests of Shioya (1989) and Shioya et al. (1989), featuring effective depths of d 600 mm and very low reinforcement ratios, see Fig. 2.2-16 and Fig. 2.2-17 for the dataset A2. This is once again demonstrated by Fig. 2.2-20 in which the model safety factors are plotted versus the effective depth d for the total dataset (A2&A3) containing all 40 tests. The Fig. 2.2-20 shows that all tests with a low effective depth (in range B) are on the safe side with a 5%-fractile value of approx. mod = 1,52. If only these tests were considered, this would result in a coefficient of k = (0,0836 1,52) = 0,1270 in Eq.(2.2-7), which is almost in the same range of the coefficient of k = 0,1402 for the tests on beams subjected to point loads (approx. 90 %). This means that the 7 tests featuring effective depths of d 600 mm and very low reinforcement ratios lw result in very low magnitudes of the model safety factor mod 0,76 (range F) and thus cause the low coefficient in Eq.(2.2-7) for all 40 tests.

In section 6.2 these tests are evaluated together with all tests on beams subjected to point loads for the dataset (A2&A3) in order to determine the exact value for the coefficient in Eq.(2.2-2).

d = 3000


effective depth d for the merged dataset (A2&A3) containing 2 (i.e. 5% of 40) values of

mod 1,0 and the coefficient of 0,0955 in Eq.(70) of the German standard DIN 1045-1


2.2- 26

5.3 Comparison of the test results with the design approach according to Reineck (2002) The approach according to Reineck (2002) is as follows (see Part 2.1, section 5.3):

[kN] (2.2-3)

The first adjustment coefficient (see section 5.1) is determined to be 0,94853 (5%) for dataset A2 and the second is determined to be 0,82275 (1,4 beams). This results in a total adjustment coefficient of 0,7804, and thus this value is only 71 % of the magnitude of the coefficient of 1,103 for tests on beams subjected to point loads. The characteris-tic coefficient in Eq.(2.2-3) is thus k = 0,1982.

For the total dataset (A2&A3) the following adjustment coefficients were determined: first adjustment coefficient: 0,95503 (5 %), and second adjustment coefficient: 0,91583 (2 beams). The total adjustment coefficient is 0,87465, and this value is only 79 % of the magnitude of 1,1076 which was the coefficient for tests on beams subjected to point loads. The characteristic coefficient in Eq.(2.2-3) is thus k = 0,2222 for the total dataset (A2&A3).

The statistical values are given in Table 2.2-5. Hereby, the dataset A3 forms range A of the merged dataset (A2&A3), see Fig. 2.2-24.

Table 2.2-5: Relevant statistical values for the model safety factors mod = Vu,test / Vu,cal according to the approach of Reineck (2002) for the datasets A2, A3 and (A2&A3)

statistical value Dataset

A2 A3 A2 & A3

Adjustment coefficient 0,7804 0,8746 0,8746

n 28 12 40

1,6317 1,7993 1,5590

0,2531 0,2657 0,2839

v 0,1551 0,1477 0,1821

5% 1,2154 1,3623 1,0919

95% 2,0481 2,2364 2,0260

In comparison with the values according to Eq.(70) of the German standard DIN 1045-1 reported in Table 2.2-4, the approach according to Reineck (2002) results in significantly lower coefficients of variation, e.g. the coefficient of variation is only v = 0,182 for the dataset (A2&A3) instead of v = 0,227 according to the German standard DIN 1045-1. This is possibly due to the different function for the size effect.

In section 6.3 these tests are evaluated together with all tests on beams subjected to point loads for the dataset (A2&A3) in order to determine the exact adjustment coefficient and the exact coefficient.


2.2-27

In Fig. 2.2-21 the model safety factors according to the approach of Reineck (2002) are plotted

for the dataset A2 versus the uniaxial concrete compressive strength f1c, which is taken as the prism strength of

. The test values in range A with a model safety factor close to mod = 1,0 are once again beams

from the test series of Shioya et al. (1989) featuring effective depths of d 600 mm and very low reinforcement ratios.

Fig. 2.2-21: Model safety factors mod according to Reineck (2002) plotted versus the uniaxial concrete

compressive strength f1c for dataset A2 containing 1,4 values of mod 1,0


2.2- 28

In Fig. 2.2-22 the model safety factors for the approach by Reineck (2002) are plotted versus

the longitudinal reinforcement ratio for dataset A2. The 7 tests of Shioya et al. (1989) on beams with large

effective depths and low reinforcement ratios appear in range A. All other tests feature high reinforcement ratios, a mean value of approx. 1,75 and a 5%-fractile value of approx. 1,55.

Fig. 2.2-22: Model safety factors mod according to Reineck (2002) plotted versus the longitudinal

reinforcement ratio for the dataset A2 containing 1,4 values of mod 1,0


2.2-29

In Fig. 2.2-23 the model safety factors for the approach by Reineck (2002) are plotted versus

the effective depth d for the dataset A2.

Fig. 2.2-23 clearly demonstrates that low model safety factors and thus a low characteristic coefficient are caused by tests with high effective depths of d > 600 mm. In comparison to the approach of the German standard DIN 1045-1 (see Fig. 2.2-20); however, the difference of the tests in range B featuring low effective depth is considera-bly smaller: the mean values of range B and F are determined to be 1,747 (range B) versus 1,274 (range F) accord-ing to the approach of (2002) in contrast to 1,93 (range B) versus 1,02 (range F) according to the approach of the German standard DIN 1045-1, see Fig. 2.2-20. This means that for these tests the size effect is slightly better captured by Eq.(2.2-8) than in the expression of the German standard DIN 1045-1.

d = 3000

Fig. 2.2-23: Model safety factors mod for the approach by Reineck (2002) plotted versus the effective depth d for

the dataset A2 containing 1,4 values of mod 1,0


2.2- 30

In Fig. 2.2-24 the model safety factors according to the approach of Reineck (2002) are plotted

versus the slenderness a/d for the merged dataset (A2&A3). Hereby, range A comprises dataset A3.

30 of the 40 tests appear within the ranges A and B, and therefore the low scatter within the other ranges C through E are only marginally meaningful due to the respectively smaller number of tests. The mean value and the 5%-fractile value are of significantly higher magnitude in range A containing the dataset A3 than in the other ranges, since the approach does not consider the influence of slenderness a/d.

Fig. 2.2-24: Models safety factors mod according to Reineck (2002) plotted versus the slenderness a/d for the

merged dataset (A2&A3) containing 2 values of mod 1,0


2.2-31

5.4 Comparison of the test results with the design approach according to Loov (2003)

The approach according to Loov (2003) is as follows (see Part 2.1, section 5.4):

[kN] (2.2-4)

where: = coefficient for the influence of the slenderness (2.2-5)

The evaluated values for the relevant statistical values are listed in Table 2.2-6. Hereby, the dataset A3 forms range A of the merged dataset (A2&A3), see Fig. 2.2-28.

The first adjustment coefficient is determined to be 0,8699 (5 %) for the dataset A2 and the second adjustment co-efficient is determined to be 0,82899 (1,4 beams). Thus, the total adjustment coefficient is 0,7212, and thus the characteristic coefficient in Eq.(2.2-4) is k = 0,9989.

The evaluation of the merged dataset (A2&A3) results in the following magnitudes: first adjustment coefficient: 0,9000 (5 %) and second adjustment coefficient: 0,8982 (1,4 tests). This then results in a total adjustment coeffi-cient of 0,8083 and a characteristic coefficient of k = (0,8083 1,385) = 1,1195 in Eq.(2.2-4). This is only 74% of the coefficient of k = 1,509 for tests on beams subjected to point loads (see Part 2.1, section 5.4).

Table 2.2-6: Relevant statistical values for the model safety factors mod = Vu,test / Vu,cal according to the approach of Loov (2003) for the datasets A2, A3 and (A2&A3)

Relevant statistical value Datasets

A2 A3 A2 & A3

Adjustment coefficient 0,7271 0,8083 0,8083

n 28 12 40

1,7467 1,5305 1,5501

0,3285 0,1959 0,2655

v 0,1881 0,1280 0,1713

5% 1,2063 1,2082 1,1134

95% 2,2872 1,8528 1,9869


2.2- 32

In Fig. 2.2-25 for the dataset A2 the model safety factors according to the approach of Loov

(2003) are plotted versus the uniaxial concrete compressive strength f1c, which is taken as the prism strength of

. The results are very similar to the results according to the approach of Reineck (2002), see

Fig. 2.2-19.

Fig. 2.2-25: Model safety factors mod for Loov (2003) plotted versus the uniaxial concrete

compressive strength f1c for the dataset A2 containing 1,4 values of mod 1,0


2.2-33

In Fig. 2.2-26 the model safety factors for the approach of Loov (2003) are plotted versus the

longitudinal reinforcement ratio for dataset A2. The low model safety factors in range A are the 7 test values

from the test series of Shioya et al. (1989) on beams with large effective depths, i.e. d > 600 mm.

Fig. 2.2-26: Model safety factors mod for the approach by Loov (2003) plotted versus the longitudinal reinforce-

ment ratio for dataset A2 containing 1,4 values of mod 1,0


2.2- 34


effective depth d for the dataset A2. Fig. 2.2-27 is almost identical with Fig. 2.2-23 for the approach of Reineck (2002), and thus the same conclusions can be drawn.

d = 3000

Fig. 2.2-27: Model safety factors mod for the approach by Loov (2003) plotted versus the effective depth d for the

dataset A2 containing 1,4 values of mod 1,0


2.2-35


slenderness a/d for the merged dataset (A2&A3). This approach results in an almost horizontal trend line for beams with a slenderness of a/d > 2,9 and only range A and range B slightly decrease. Hence, the influence of the slenderness is a little overemphasized in this approach.

Fig. 2.2-28: Models safety factors mod for the approach by Loov (2003) plotted versus the slenderness a/d for the

merged dataset (A2&A3) containing 2 values of mod 1,0


2.2- 36


2.2-37

6 Comparison of the design approaches with the test results of all slender

beams subjected to point loads or to a uniformly distributed load

6.1 Introduction Comparing the test results with the design approaches resulted in significantly smaller characteristic coefficients for tests on beams subjected to a uniformly distributed load than for tests on beams subjected to point loads. E.g. the coefficient in Eq.(70) of the German standard DIN 1045-1 (2001) for test on beams subjected to a uniformly dis-tributed load for the dataset (A2&A3) containing 40 tests added up to only approx. 68 % of the magnitude of the coefficient for tests on beams subjected to point loads. For the approach of Reineck (2002) it was only approx. 79 % whereas for the approach of Loov (2003) it was only approx. 74 % for the dataset (A2&A3).

It is now examined how these few tests on beams subjected to a uniformly distributed load featuring low coeffi-cients, i.e. 40 tests for the total dataset (A2&A3), affect the approaches if they are evaluated together with the 688 tests on beams subjected to point loads for the total dataset (A2&A3). Subsequently, the two adjustment coeffi-cients are then determined for all 728 tests, as show in section 5.1, in order to determine the exact value for the characteristic coefficient k, for which 5% of the model safety factors of all 728 tests, i.e. 36,4 tests, fulfill the crite-rion mod 1,0.

6.2 Comparison of the test results with the design approach of the German standard DIN 1045-1 6.2.1 The approach of the German standard DIN 1045-1 and determination of coefficients

The approach of the German standard DIN 1045-1 was presented in section 5.2.1 of part 2.1 and is defined as fol-lows in case of normal-strength concrete and N=0:

[kN] (2.2-6)

where: = coefficient ( d = 0,10 as design value)

The relevant statistical values for the evaluation of the different datasets are given in Table 2.2-4.

For the total dataset (A2&A3) containing 40 tests on beams subjected to a uniformly distributed load, the character-istic coefficient is k = 0,09551 according to the approach of the German standard DIN 1045-1 corresponding to Eq.(2.2-2). Whereas the evaluation of the 688 tests on beams subjected to point loads according to part 2.1, section 5.2.1, resulted in a characteristic coefficient of k = 0,1402. Thus, the tests on beams subjected to a uniformly dis-tributed load result in a coefficient which has a magnitude that adds up to only 68% of the coefficient for tests on beams subjected to point loads.

The evaluation of all tests (688 + 40) = 728 tests resulted in the following adjustment coefficients: first adjustment coefficient: 1,23018; second adjustment coefficient: 1,12599. The total adjustment coefficient is thus 1,3852, and this results a characteristic coefficient of k = 0,1385, which is only approx. 1 %, i.e. only marginally smaller than the magnitude of k = 0,1402 for the evaluation of only the tests on beams subjected to point loads.


2.2- 38

Table 2.2-7: Relevant statistical values for the model safety factors mod = Vu,test / Vu,cal in Eq.(70) of the German standard DIN 1045-1 for the datasets (A2&A3)

Relevant statistical value

datasets (A2&A3)

uniformly distributed load point load all

coefficient k 0,0955 0,1402 0,1385

n 40 688 728

1,7813 1,3644 1,3725

0,4044 0,2898 0,2944

v 0,2270 0,2124 0,2145

5% 1,1161 0,8876 0,8881

95% 2,4465 1,8411 1,8568

Altogether, the influence of the tests on beams subjected to a uniformly distributed load featuring low model safety factors is rather small, as was to be expected due to the small number of tests. The coefficients of variation, too, differ only slightly comparing test on beams subjected to point loads with tests on beams subjected to a uniformly distributed load.

However, it still is remarkable, that the approach according to Eq.(70) of the German standard DIN 1045-1 (2001) is unsafe for tests on beams subjected to a uniformly distributed load, since this is the most common type of loading in practice. Correspondingly, a decrease of the coefficient would be required to approx. 68%, resulting in a design value of only approx. d = 0,0955 / 1,5 = 0,0637, implying a safety index of

= 4,4 (see section 5.2.2.4 of Part 2.1).

The model safety factors are again plotted versus the essential parameters, showing the 40 tests on beams subjected to a uniformly distributed load as filled-in circles, so that they can be compared with the 688 tests on beams sub-jected to point loads.


2.2-39

6.2.2 Model safety factors in respect to different parameters

In Fig. 2.2-29 the 728 model safety factors mod for Eq.(70) of the German standard DIN 1045-1 are plotted versus the uniaxial concrete compressive strength f1c for all tests on beams subjected to a uniformly distributed load (40 beams) and subjected to point loads (688 beams) for the dataset (A2&A3).

The tests on beams subjected to a uniformly distributed load are shown by the filled-in circles that coincide very well with the total dataset except for the 7 known Japanese tests featuring large effective depths. However, on average they appear below the mean value of all tests, by approx. 11% as calculated in section 6.2.1.

Fig. 2.2-29: Model safety factors mod for Eq.(70) of the German standard DIN 1045-1 plotted versus the uniaxial

concrete compressive strength f1c for all beams subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) for the dataset (A2&A3) containing 36,4 values of mod 1,0


2.2- 40

In Fig. 2.2-30 the model safety factors mod for Eq.(70) of the German standard DIN 1045-1 are plotted versus the reinforcement ratio lw for all tests on beams subjected to a uniformly distributed load (40) and subjected to point loads (688) for the dataset (A2&A3).

The tests on beams subjected to a uniformly distributed load (filled-in circles) lie within the values of the tests on beams subjected to point loads in case of very high reinforcement ratios of lw > 2 %. However, this is not the case for the range of low reinforcement ratios. The number of unsafe tests is especially significantly increased for the range of low reinforcement ratios of lw < 1 % by the 7 very unsafe tests from the test series of Shioya et al. (1989) on beams with high effective depths subjected to a uniformly distributed load. All of these tests values range in low, i.e. they feature low model safety factors mod which decrease down to 0,54.

Fig. 2.2-30: Model safety factors mod in Eq.(70) of the German standard 1045-1 plotted versus the reinforcement

ratio lw for all test beams subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) for the datasets (A2&A3) containing 36,4 values of mod 1,0


2.2-41

In Fig. 2.2-31 the model safety factors mod for Eq.(70) of the German standard DIN 1045-1 are plotted versus the effective depth d = ds for all 728 tests on beams subjected to a uniformly distributed load (40) and subjected to point loads (688).

Fig. 2.2-31 clearly demonstrates that the unsafe tests appear predominately for high effective depth. Out of the 36,4 tests, 20 tests feature an effective depth of d 600 mm, and out of these 20 tests, 7 tests are on beams subjected to a uniformly distributed load. Further, out of these 20 tests, 9 tests feature a model safety factor of mod < 0,80, which suggests that the approach of the effective depth according to Eq.(70) of the German standard DIN 1045-1 should be revised.

d = 3000

Fig. 2.2-31: Model safety factors mod in Eq.(70) for the German standard DIN 1045-1 plotted versus the effective

depth d = ds for all test beams subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) for the datasets (A2&A3) containing 36,4 values of mod 1,0


2.2- 42

In Fig. 2.2-32 the model safety factors mod for Eq.(70) of the German standard DIN 1045-1 are plotted versus the slenderness a/d for all tests on beams subjected to a uniformly distributed load and subjected to point loads.

The unsafe test values appear predominately in the range of a/d = 2,9. The mean value increases considerably for a decreasing slenderness a/d, since the influence of the slenderness is obviously not considered in the approach.

Fig. 2.2-32: Model safety factors mod for Eq.(70) of the German standard DIN 1045-1 plotted versus the slender-

ness a/d for all test beams subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) for the datasets (A2&A3) containing 36,4 values of mod 1,0


2.2-43

6.2.3 Direct determination of the design value according to appendix D, EN 1990 (2002) 6.2.3.1 Input values The two datasets (A2&A3) comprise altogether 728 tests. The adjustment coefficients were 1,2302 (5 %) and 1,1260 (34,4 tests) as shown above, so that the total adjustment coefficient is 1,3852, leading thus to a characteristic coefficient of k = 0,1385.

The definitions of the different terms and parameters were given in Part 2.1, section 5.2.2.1 and are not repeated here.

With given input values, there are two possibilities for the determination of the design value:

1. According to the approximation in appendix D, EN 1990 (2002);

2. According to the exact formula in appendix D, EN 1990 (2002).

6.2.3.2 Determination of the design value according to the approximation in Appendix D, EN 1990:2002

Step 1: Develop a design model

[kN] (2.2-7)

The coefficient of d (in König et al. (1999) and Hegger et al. (1999) referred to as cd) is set to 1,2302·1,1260·0,10 = 0,1385 considering the two adjustment coefficients so that precisely 5% of all tests fulfil the criterion

.

Step 2: Compare experimental and theoretical values

[-] (2.2-8)

In Fig. 2.2-32 the test values Vu,test taken from the reports are plotted versus the calculated shear forces Vu,cal.

The deviation from the 45°-line is clearly noticeable. The continuously drawn trend line (y = 1,0095 x)

defines at the same time the deviation of the mean value (see step 3). The dashed-dotted line

(y = 1,3719 x) shows the mean value of the log-normal distribution.

Step 3: Estimate the mean value correction b

The estimator for the mean value correction b is determined by comparing the theoretical values rt with the ex-perimental values re. Firstly, or each specimen i ( i = 1 to n) the correction term bi is determined (see step 2). From

tests a realization of the estimator for the mean value is calculated by:

[-] (2.2-9)

The deviation of the mean is as follows:

[-] (2.2-10)


2.2- 44

Vu,test - Vu,cal - Diagramm

Vu,cal

[kN]

Vu,test

[kN]

Fig.2.2-33: Comparison of experimental with calculated values

Step 4: Estimate the coefficient of variation of the random error term

[-] (2.2-11)

[-] (2.2-12)

[-] (2.2-13)

[-] (2.2-14)

[-] (2.2-15)

Alternative for the determination for the coefficient of variation of the random error term

[-] (2.2-16)


2.2-45

[-] (2.2-17)

[-] (2.2-18)

[-] (2.2-19)





Step 5: Analyze the compatibility

Analyze the compatibility of the test population with regard to the assumptions made in the resistance function (see Fig. 2.2-32).

Step 6: Determine the coefficient of variation of he basic variables in the resistance function

Concrete compressive strength:

Contrary to the following approach according to the standards and Hegger et al. (1999)

a magnitude of f = 4 MPa is applied to the evaluation of the tests (see PART 1, section 3.1.3), resulting in:

By means of the database, the mean value of the cylinder strength is determined as follows:

mean value

The standard deviation and the coefficient of variation can now be determined to be:

standard deviation


According to Kraemer et al. (1975) und Hegger et al. (1999) the standard deviation is estimated to be:

Effective depth:

According to Kraemer et al. (1975) the standard deviation is defined assuming a slab depth h and a concrete cover of 6 mm. The standard deviation for the effective depth is also assumed to be 6 mm according to Hegger et al. (1999) and this is used also here.

mean value; standard deviation


2.2- 46


width of web:

Whereas Kremer et al. (1975) assume a constant width of slab and provide no information concerning the width of web, Hegger et al. (1999) assume a standard deviation for the width of web bw of 5 mm, which was carried over as well.

mean value

standard deviation



Coefficient of variation for all uncertainties of the structural concrete member:

(2.2-20)


(2.2-21)

Standard deviation:

(2.2-22)

characteristic value rk of the resistance function:

(2.2-23)

where: magnitude of the fractile factor kn

If according to König and Tue (1998) the following approximation is valid: standard deviation equals

coefficient of variation:

König et al. (1999) assume that the coefficient of variation of the model uncertainty is and the deviation

of geometry is . In this case, this results according to König et al. (1999) in the following standard devia-

tion Q, respectively, coefficient of variation Vr:

(2.2-24)




2.2-47

(2.2-25)

where: magnitude of fractile factor kd,n



Hegger et al. (1999), König and Tue (1998) and König et al. (1999) apply a safety index of according to

Eurocode 1. However, König et al. (1999) criticize that the safety index of does not consider the influence

of the failure mode for the type of failure (ductile failure with reserve strength capacity resulting from strain hard-ening; ductile failure with no reserve capacity; brittle failure).

Brittle types of failure, such as shear failure in case of structural members without stirrups, are considered by in-creasing the safety index to ; according to a recommendation of the Probabilistic Model Code of the JCSS

(2001), p. 18.

In the following, the determination of the design value is carried out for both safety indexes.

For the determination of the design value with :

(2.2-26)

Step 9: Final choice of the characteristic values and the partial safety factor R


(2.2-27)

It is additionally considered that in the design approach a characteristic value of and

is applied, instead of the so far reported mean values of the concrete compressive strength fcm.

(2.2-28)

(2.2-29)

(2.2-30)

(2.2-31)

The design value of the coefficient for Eq.(70) of DIN 1045-1 is thus determined to be for a safety

index of considering brittle failure.

A safety index of results in a design value of which is about an 11% higher value.



2.2- 48

The approximate formula is applicable for a large number of tests ( ) starting from step 7. For a small num-

ber of tests, the exact formula should be applied since it considers additional parameters. This dataset contains 688 beams and is thus very well covered by the approximate formula. However, for reasons of control and complete-ness, the calculation according to the exact formula is presented here too.



(2.2-20)


(2.2-21)

Standard deviations:

(2.2-32)

(2.2-33)

(2.2-22)

Characteristic value rk of the resistance function:

(2.2-34)

where: value of the fractile factor kn

fractile factor

coefficient

coefficient



(2.2-35)

The fractile factor in Hegger et al. (1999) is multiplied by the ratio (4,4/3,8) of the safety indexes:


Step 9: Determination of the partial safety factor



2.2-49

(2.2-36)

It is additionally considered that in the design approach a characteristic value of is applied instead of the so far

reported mean values of the concrete compressive strength fcm.

(2.2-37)

(2.2-38)

(2.2-39)

(2.2-40)

The design value of the coefficient according to the exact formula is thus determined to be for a safety

index of considering brittle failure. I.e., the magnitude of this design value is approx. 2% lower than the

design value according to the approximate formula where .

A safety index of results according to the exact formula in a design value of , i.e. an approx.

11% higher value.

6.2.3.4 Summary of the determination of the design values

The exact determination according to section 6.2.3.3 resulted for the dataset (A2&A3) in a design value of

d = 0,0910 when applying a safety index of considering brittle failure. The magnitude of this design value

is 2% smaller than the value of d = 0,0927 determined from the approximate formula according to the section 6.2.3.2. Applying a safety index of only = 3,8 as suggested in the European Eurocode EC 2, resulted in values of

approx. 11 % higher magnitudes; according to the approximate formula, respectively d = 0,1008

according to the exact formula.

The statistical evaluation of the databases resulted in a characteristic coefficient of k = 0,1385 for the merged data-set (A2&A3). Applying a safety factor of c = 1,50, the coefficients of the expression for the design value according to Eq.(70) of the German standard DIN 1045-1 (2001) are determined to d = 0,0923 for the merged dataset (A2&A3). The latter coincides almost exactly with the design value applying a safety index of = 4,4 whereby the partial safety factor according to Eq. (2.1-30) was about 1,52.

The design value of d = 0,10 for Eq.(70) of the German standard DIN 1045-1 appears to be near the result for the safety index of = 3,8, thus corresponding to a partial safety factor of approximately c = 1,39.


2.2- 50

6.3 Comparison of the test results with the design approach of Reineck (2002) 6.3.1 The approach of Reineck (2002) and determination of the coefficients

The approach of Reineck (2002) is as follows (see section 5.3):

[kN] (2.2-41)

The relevant statistical values were evaluated for the different datasets and are listed in Table 2.2-8.

For the overall evaluation of all tests on beams subjected to point loads and subjected to a uniformly distributed load, the first adjustment coefficient was determined to 0,9848 (see section 5.1) and the second adjustment coeffi-cient was determined to 1,118. This results in a total adjustment coefficient of 1,1007 and thus a coefficient in Eq.(2.2-41) of = 0,2796, i.e. approx. 0,280 instead of 0,254. This adjustment coefficient of 1,017 thus almost equals the total adjustment coefficient of 1,108 determined for tests on beams subjected to point loads; i.e. the 39 tests on beams subjected to a uniformly distributed load affect the results only marginally.

Table 2.2-8: Relevant statistical values for the model safety factors mod = Vu,test / Vu,cal for datasets (A2&A3)

statistical value Datasets (A2&A3)

Uniformly distributed load Point load All

Coefficient k 0,222 0,281 0,2796

n 40 688 728

1,5590 1,3515 1,3532

0,2839 0,2788 0,2790

v 0,1821 0,2063 0,2062

5% 1,0919 0,8929 0,8942

95% 2,0260 1,8100 1,8121

It should be noted that in Table 2.2-8 the listed mean and fractile values are valid for the reported different coeffi-cients. Applying the coefficient of approx. k = 0,280 of all tests also to the 40 tests on beams subjected to a uni-formly distributed load would result in values of approx. 22% lower magnitude:

- mean value: 1,238

- 5% fractile value: 0,867


Thus, on average the 40 tests on beams subjected to a uniformly distributed load lie approx. 8,4% below the mean value 1,352 of all tests.


2.2-51

6.3.2 Model safety factors in respect to different parameters

In Fig. 2.2-34 the 728 model safety factors mod for the approach by Reineck (2002) are plotted versus the uniaxial concrete compressive strength f1c for all tests subjected to a uniformly distributed load (40 marked �) and subjected to point loads (688 marked �)) for the datasets (A2&A3).

The tests on beams subjected to a uniformly distributed load shown by the filled-in circles coincide very well with the total dataset except for the 7 known tests featuring large effective depth. However, on average they appear be-low the mean value of all tests, by approx. 8% as calculated in section 6.3.1.

Fig. 2.2-34: Model safety factors mod for the approach by Reineck (2002) plotted versus the uniaxial concrete

compressive strength f1c for all test beams of the datasets (A2&A3) subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) containing 36,4 values of mod 1,0


2.2- 52

In Fig. 2.2-35 the model safety factors mod according for approach by Reineck (2002) are plotted versus the rein-forcement ratio lw for all tests on beams subjected to a uniformly distributed load (40) and subjected to point loads (688) for the datasets (A2&A3). The results are similar with the results according to Eq.(70) of the German stan-dard DIN 1045-1.

The tests on beams subjected to a uniformly distributes load (filled-in circles) lie within the values of the tests on beams subjected to point loads in case of very high reinforcement ratios of lw > 2 %; however, the number of un-safe tests is significantly increased in the range of low reinforcement ratios of lw < 0,55 % by 4 tests (not 7 as in case of the German standard DIN 1045-1) from the test series of Shioya et al. (1989) on beams with high effective depths subjected to a uniformly distributed load. All of these tests values range in low, i.e. they feature low model safety factors mod which decrease to 0,65.

Fig. 2.2-35: Model safety factors mod for the approach by Reineck (2002) plotted versus the reinforcement ratio lw

for all test beams of the datasets (A2&A3) subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) containing 36,4 values of mod 1,0


2.2-53

In Fig. 2.2-36 the model safety factors mod for the approach of Reineck (2002) are plotted versus the effective depth d = ds for all 728 tests on beams subjected to a uniformly distributed load (40) and subjected to point loads (688).

Fig. 2.2-36 demonstrates that in contrast to the comparison with the German standard DIN 1045-1 the unsafe test value are approximately evenly distributed to high and low effective depths, and only 11 tests of the overall 66 tests featuring an effective depth of d > 600 mm are unsafe. Out of these 11 tests only 2 feature a model safety factor of

mod < 0,80 in contrast to 9 in case of the German standard DIN 1045-1.

d = 3000

Fig. 2.2-36: Model safety factors mod according for the approach by Reineck (2002) plotted versus the effective

depth d = ds for all test beams of the datasets (A2&A3) subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) containing 36,4 values of mod 1,0


2.2- 54

In Fig. 2.2-37 the model safety factors mod for approach by Reineck (2002) are plotted versus the slenderness a/d for all tests on beams subjected to a uniformly distributed load and subjected to point loads.

The unsafe values appear as well around the range of a/d = 2,9 as in the range of a high slenderness a/d including 13 tests with a slenderness of a/d > 4,4. The mean value increases for a decreasing slenderness a/d, since the influ-ence of the slenderness is not considered in the approach.

Fig. 2.2-37: Model safety factors mod for the approach by Reineck (2002) plotted versus the slenderness a/d for all

test beams of the datasets (A2&A3) subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) containing 36,4 values of mod 1,0


2.2-55

6.4 Comparison of the test results with the design approach of Loov (2003) 6.4.1 The approach of Loov (2003) and determination of the coefficients

The approach according to Loov (2003) is as follows (see Part 2.1, section 5.4):

[kN] (2.2-42)

where: coefficient for the influence of the slenderness (2.2-43)

The relevant statistical values were evaluated for the different datasets of the tests and are listed in Table 2.2-9.

For the overall evaluation of all tests on beams subjected to point loads and subjected to a uniformly distributed load the first adjustment coefficient was determined to be 1,02494 (see section 5.1) and the second adjustment coef-ficient was determined to be 1,04607. This results in a total adjustment coefficient of 1,07216, i.e. it adds up to 98% of the total adjustment coefficient for tests on beams subjected to point loads. This results in a characteristic coefficient of = 1,485 in Eq.(2.2-42) instead of 1,385.

Table 2.2-9: Relevant statistical values for the model safety factors mod = Vu,test / Vu,cal for the approach by Loov for the datasets (A2&A3)

statistical value Datasets (A2&A3)

Uniformly distributed load Point load All

coefficient k 1,1195 1,509 1,485

n 40 688 728

1,5501 1,3261 1,3374

0,2655 0,2263 0,2319

v 0,1713 0,1707 0,1734

5% 1,1134 0,9537 0,9560

95% 1,9869 1,6984 1,7189

It is notified that in Table 2.2-9, the listed mean and fractile values are valid for the reported different coefficients. Applying the coefficient of approx. k = 1,485 of all tests to the 40 tests on beams subjected to a uniformly distrib-uted load would result in values of approx. 25 % lower magnitude:

- mean value: 1,168



Thus, on average the 40 tests on beams subjected to a uniformly distributed load lie approx. 12,7 % below the mean value of all tests, similar to the approaches of the German standard DIN 1045-1 (11%) and Reineck (2002) (8,4%).


2.2- 56


In Fig. 2.2-38 the model safety factors mod for the approach by Loov (2003) are plotted versus the uniaxial concrete compressive strength f1c for all tests on beams subjected to a uniformly distributed load (40 marked �) and subjected to point loads (688 marked �)) for datasets (A2&A3).

The tests on beams subjected to a uniformly distributed load shown by the filled-in circles coincide very well with the total dataset except for the 7 known tests featuring large effective depth.

Fig. 2.2-38: Model safety factors mod for the approach by Loov (2003) plotted versus the uniaxial concrete com-

pressive strength f1c for all test beams of datasets (A2&A3) subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) containing 36,4 values of mod 1,0


2.2-57

In Fig. 2.2-39 the model safety factors mod according for the approach by Loov (2003) are plotted versus the rein-forcement ratio lw for all tests on beams subjected to a uniformly distributed load (40) and subjected to point loads (688) for datasets (A2&A3). The results are similar to the results for Eq.(70) of the German standard DIN 1045-1.

The tests on beams subjected to a uniformly distributed load (filled-in circles) lie within the values of the tests on beams subjected to point loads in case of very high reinforcement ratios of lw > 2 %; however, the number of un-safe tests is significantly increased in the range of low reinforcement ratios of lw < 0,55 % by 5 tests from the test series of Shioya et al. (1989) on beams with high effective depths subjected to a uniformly distributed load. All of these tests values range in low, i.e. they feature low model safety factors mod which decrease to 0,618.

Fig. 2.2-39: Model safety factors mod for the approach by Loov (2003) plotted versus the reinforcement ratio lw

for all test beams of datasets (A2&A3) subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) containing 36,4 values of mod 1,0


2.2- 58

In Fig. 2.2-40 the model safety factors mod for the approach by Loov (2003) are plotted versus the effective depth d = ds for all 728 tests on beams subjected to a uniformly distributed load (40) and subjected to point loads (688).

Fig. 2.2-40 demonstrates that in contrast to the comparison with the German standard DIN 1045-1 the unsafe test value are approximately evenly distributed for high and low effective depths, and only 15 tests of the overall 65 tests for an effective depth of d > 600 mm are unsafe. Out of these 15 tests only 3 feature a model safety factor of

mod < 0,80 in contrast to 9 in case of the comparison with the German standard DIN 1045-1. The lowest magnitude is 0,618.

d = 3000

Fig. 2.2-40: Model safety factors mod for the by Loov (2003) plotted versus the effective depth d = ds for all test

beams of datasets (A2&A3) subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) containing 36,4 values of mod 1,0


2.2-59

In Fig. 2.2-41 the model safety factors mod for the approach by Loov (2003) are plotted versus the slenderness a/d for all tests on beams subjected to a uniformly distributed load and subjected to point loads.

Almost all of the unsafe values appear in the range around and below a slenderness of a/d = 2,9. The mean value is hardly influenced by the slenderness a/d except for range C, since this parameter a/d is considered in the approach.

Fig. 2.2-41: Model safety factors mod for the approach by Loov (2003) plotted versus the slenderness a/d for all test beams of datasets (A2&A3) subjected to a uniformly distributed load (40 tests marked �) and point loads (688 tests marked �) containing 36,4 values of mod 1,0


2.2- 60


2.2-61

References of Part 2.2

DIN 1045-1 (2001): Deutsche Norm: Tragwerke aus Beton, Stahlbeton und Spannbeton - Teil 1: Bemessung und Konstruktion. S. 1 - 148. (Concrete, reinforced and prestressed concrete strutures - Part 1: Design). Normenausschuss Bauwesen (NABau) im DIN Deutsches Institut für Normung e.V. Beuth Verl. Berlin, Juli 2001

Krefeld, W.J.; Thurston, Ch.W. (1966): Studies of the shear and diagonal tension strength of simply supported r.c.-beams. ACI-Journal, V.63 (1966), April, 451-475

Loov, R.E. (2003): ACI Quickfix. Proposal for a ACI shear equation to ACI Subcommittee 445-F. 14.01.2003. Univ. of Cal-gary.

Reineck, K.-H. (2002): Proposal for "quick fix". Proposal to ACI Subcommittee 445-F. 15 Dec. 2002

Shioya, T. (1989): Shear Properties of Large Reinforced Concrete Member. Special Report of Institute of Technology., Shimizu Corp., No.25, February 1989

Shioya, T.; Iguro, M.; Nojiri, Y.; Akiayma, H.; Okada, T. (1989): Shear Strength of Large Reinforced Concrete Beams. p.259-279 in: Fracture Mechanics: Application to Concrete, SP-118, ACI, Detroit,


2.2- 62

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Attachment 2.2-2: References for database vuct-RC-gl-DS

Feldman, A.; Siess, C.P. (1955): Effect of Moment Shear Ratio on Diagonal Tension Cracking and Strength in Shear of Reinforced Concrete Beams. University of Illinois Civil Engineering, Structural Research Series, No. 107, 1955

Iguro, M.; Shioya, T.; Nojiri, Y.; Akiayma, H. (1985): Experimental studies on shear strength of large reinforced concrete beams under uniformly distributed load. Concrete Library International, JSCE No.5, 137-154 (translation of article in JSCE in 1984)

Krefeld, W.J.; Thurston, Ch.W. (1966): Studies of the shear and diagonal tension strength of simply supported r.c.-beams. ACI Journal, V.63 (1966), April, 451-475

Leonhardt, F.; Walther, R. (1962a): Schubversuche an einfeldrigen Stahlbetonbalken mit und ohne Schubbewehrung. DAfStb H.151, Berlin, 1962

Leonhardt, F.; Walther, R. (1962b): Versuche an Plattenbalken mit hoher Schubbeanspruchung. DAfStb H.152, Beuth Verlag Berlin, 1962

Leonhardt, F.; Walther, R. (1962c): Versuche an schlaff bewehrten Rechteck- und Plattenbalken mit Schubbewehrung; 1. Teil; Forschungsreihe über die Tragfähigkeit auf Schub von Stahl- und Spannbetonbalken, FMPA (Otto-Graf-Institut), Stuttgart, März 1962

Leonhardt, F.; Walther, R. (1962d): Versuche an schlaff bewehrten Rechteckbalken und Platten ohne Schubbewehrung; 2. Teil; Forschungsreihe über die Tragfähigkeit auf Schub von Stahl- und Spannbetonbalken, FMPA (Otto-Graf-Institut), Stuttgart, März 1962

Leonhardt, F., Walther, R. (1962e): Versuche an schlaff bewehrten Rechteckbalken und Platten ohne Schubbewehrung; Zusammenfassung und vorläufige Schlußfolgerungen; Forschungsreihe über die Tragfähigkeit auf Schub von Stahl- und Spannbetonbalken, FMPA (Otto-Graf-Institut), Stuttgart, April 1962




Reineck, K.-H.; Koch, R.; Schlaich, J. (1978): Shear Tests on Reinforced Concrete Beams with axial compression for Offshore Structures – Final Test Report. Stuttgart, July 1978, Institut für Massivbau, Univ. Stuttgart (internal report)

Rüsch, H.; Haugli, F.R.; Mayer, H. (1962): Schubversuche an Stahlbeton-Rechteckbalken mit gleichmäßig verteilter Belastung. DafStb H.145, W. Ernst & Sohn, Berlin, 1-30

Shioya, T. (1989): Shear Properties of Large Reinforced Concrete Member. Special Report of Institute of Technology, Shimizu Corp., No.25, February, 1989

Shioya, T.; Iguro, M.; Nojiri, Y.; Akiayma, H.; Okada, T. (1989): Shear Strength of Large Reinforced Concrete Beams. p.259-279 in: Fracture Mechanics: Application to Concrete, SP-118, ACI, Detroit, 1989

Part 3: Database with shear tests on prestressed concrete beams without stirrups subjected to point loads


Table of contents

1 Introduction p. 3-3

2 The database vuct-PC-DS for tests on p.c.-beams subjected to point loads p. 3-5

3 The database vuct-PC-DK_sl for the control of the tests on slender p.c.- beams subjected to point loads p. 3-11

3.1 Results of the evaluation with respect to criteria p. 3-11

3.2 Selection of the tests for evaluation p. 3-12

3.3 Pre-tensioned concrete beams with plain reinforcing steel (vuct-PC-Pre-Plain-A) p. 3-13

3.4 Pre-tensioned concrete beams with ribbed reinforcing steel (vuct-PC-Pre-Ribb-A) p. 3-14

3.5 Post-tensioned concrete beams with plain reinforcement (vuct-PC-Post-Plain-A) p. 3-15

3.6 Post-tensioned concrete beams with ribbed reinforcement (vuct-PC-Post-Ribb-A) p. 3-16

3.7 Total dataset p. 3-17

3.8 Summary p. 3-18

4 The evaluation database vuct-PC-A p. 3-19

4.1 Introduction p. 3-19

4.2 Presentation of the evaluation database vuct-PC-A p. 3-19

5 Comparison of the test results with design approaches for slender beams subjected to point loads p. 3-25

5.1 Introduction p. 3-25

5.2 Comparison of the test results with the design approach for the German standard DIN 1045-1 p. 3-25

5.2.1 The approach of the German standard DIN 1045-1 and determination of

coefficients


5.2.3 Summary of the results for the German standard DIN 1045-1

References of Part 3 p. 3-34

The report comprises the pages 3-1 through 3-34


3 - 2

Tables:

Table 3-1............................................................................................................................. p. 3-11

Table 3-2............................................................................................................................. p. 3-14

Table 3-3............................................................................................................................. p. 3-14

Table 3-4............................................................................................................................. p. 3-14

Table 3-5............................................................................................................................. p. 3-15

Table 3-6............................................................................................................................. p. 3-15

Table 3-7............................................................................................................................. p. 3-15

Table 3-8............................................................................................................................. p. 3-16

Table 3-9............................................................................................................................. p. 3-16

Table 3-10........................................................................................................................... p. 3-16

Table 3-11........................................................................................................................... p. 3-17

Table 3-12........................................................................................................................... p. 3-17

Table 3-13........................................................................................................................... p. 3-17

Table 3-14........................................................................................................................... p. 3-18

Table 3-15........................................................................................................................... p. 3-18

Table 3-16........................................................................................................................... p. 3-26

Table 3-17........................................................................................................................... p. 3-27

Attachments of Part 3

Attachment 3-1: Notation and formulary for the shear databases Vuct-PC-DS for the data collection and Vuct-PC-DK_sl for the data control of p.c.- beams without stirrups subjected to point loads

Attachment 3-5: Evaluation database vuct-PC-A for slender p.c.- beams without stir-rups subjected to point loads

Database with shear tests on prestressed concrete beams without stirrups subjected to point loads

3-3

1 Introduction The extended database with tests on prestressed concrete beams without stirrups subjected to point loads comprises 403 tests.

For the data collection and the data evaluation the following files were generated:

- vuct-PC-DS = file for data collection

- vuct-PC-DK-sl = file for data control for slender members with a/d 2,40

- vuct-PC-A = file for evaluation and comparison with standard approaches

The flowchart in Fig. 3-1 provides an overview of the tests subdivided in these files.

Fig. 3-1: Selection of the beams according to the primary selection criteria

In case of 14 out of the 403 tests collected in the database vuct-PC-DS, essential data was missing such as the uniaxial concrete compressive strength f1c, the yield strength of the prestressing steel fpy , the prestressing force P or the ultimate shear force Vu. This data is required for assessing flexural or end anchorage failures within the two files for data control vuct-PC-DK. As a first step, these beams were identified via the criterion konx and eliminated.

In the second step, the beams were checked for their slenderness via the criterion kon61. A total of 69 beams featuring a slenderness of a/d < 2,4 were transferred to the database vuct-PC-DK-24. This left 320 beams that were transferred to the database vuct-PC-DK-sl.

In the data collection, tendons (or strands) in the top layer and bottom layer of the beam were distinguished. The prestressing force was determined from data provided in the reports and when possible at the time of the test. Unfortunately, the type of ribs of reinforcement was not provided in many reports; in case this information was missing, plain prestessing steel was assumed. With respect to the type of prestress, pre-tensioning (Pre) and post-tensioning (Post) was distinguished.

kon61 389 tests

14 tests

403 tests

konx

320 tests 69 tests


3 - 4


3-5

2 The database vuct-PC-DS for tests on p.c.-beams subjected to point loads 2.2 Presentation of the database vuct-PC-DS In the following section, the database vuct-PC-DS is presented by plotting the number of beams versus the most important parameters subdivided in class intervals.

In Fig. 3-2 the number of tests n is plotted versus the uniaxial concrete compressive strength f1c, subdivided into class intervals of f1c = 5 MPa. In case of 3 beams no data was provided for the compressive strength and so that these beams were eliminated by means of the criterion konx.

Most beams (85 tests, i.e. 21%) had a uniaxial concrete compressive strength of 30-35 MPa, and 154 beams (ap-prox. 39%) had a uniaxial concrete compressive strength < 30 MPa. Only 14 tests (approx. 4%) feature a uniaxial concrete compressive strength of f1c > 60 MPa and these were classified as high-strength concrete beams.

Fig. 3-2: Number of beams plotted versus concrete compressive strength for the database vuct-PC-DS

The steel grade of the prestressing steel applied in the tests is now considered. In Fig. 3-3 the number of beams n is plotted versus the yield strength fpy subdivided into class intervals of f = 50 MPa. The required data concerning the steel grade is provided for all beams.

The predominant number of tests (279 tests, corresponding to 69%) featured yield strengths in the range of 1350 < fpy < 1600 MPa, and the peak (123 tests) appears at yield strengths 1400 < fpy < 1500 MPa. In case of 65 tests prestressing steel with a yield strength of 800 < fpy < 900 MPa was utilized.


3 - 6

Fig. 3-3: Number of beams plotted versus the yield strength fpy of the prestressing steel for

the database vuct-PC-DS

In Fig. 3-4 the number of beams n is plotted versus the reinforcement ratio l = lw = ( s + p) (related to the width of web) subdivided into class intervals of l = 0,25 %. As the distribution in Fig. 3-4 shows, most tests are carried out for a comparably small range of reinforcement ratios up to 1,5%. Only a few tests were carried out for low rein-forcement ratios of l < 0,5 %, i.e. 47 tests, i.e. approx. 12 %. Many tests appear in the range of 0,5 < l < 1,0 %, so that below a reinforcement ratio of l < 1,0 % there are altogether 191 tests (47%). A total of 136 tests (approx. 34 %) appear in the range of 1,0 < l < 1,5 %, so that 76 tests (approx. 19 %) had very high reinforcement ratios.

lw

l

Fig. 3-4: Number of beams plotted versus the reinforcement ratio l = lw = ( s + p) (related to the width of web)

for the database vuct-PC-DS


3-7

In Fig. 3-5 the number of beams n is plotted versus the mechanical reinforcement ratio p = lp fpy/f1c of the prestressing reinforcement subdivided into class intervals of l = 0,05. In the case of 4 beams, the required data was not provided.

The distribution shows a peak at reinforcement ratios of 0,10< l < 0,15 and then the number approximately line-arly decreases up to approx. l = 0,60. Only 47 beams (approx. 12%) feature magnitudes of l < 0,10, and 134 beams (approx. 34%) feature a reinforcement ratio of 0,10 < l < 0,20. Thus, more than 50% of the tests are carried out for highly, respectively, very highly reinforced beams with l > 0,20.

p

Fig. 3-5: Number of beams plotted versus the mechanical reinforcement ratio p = lp fpy/f1c of the prestressing

reinforcement for the database vuct-PC-DS

A further important parameter is the slenderness = a/d, and in the Fig. 3-6 the number of beams n is plotted versus the slenderness subdivided into class intervals of = 0,4. Data concerning the slenderness is provided for all beams.

The distribution in Fig. 3-6 shows that tests are carried out beams for a large range of slenderness. However, the emphasis of the distribution appears in the range of 3,2 < < 4,4 with 186 tests, i.e. 46%. The magnitudes of the slenderness of the prestressed members were somewhat higher than the magnitudes of the slenderness of the rein-forced (non-tensioned) members (see Part 2.1) with an emphasis of the distribution in the range of 2,4 < < 3,6.

Tests with a slenderness of < 2,4 were selected using the criterion kon61 and transferred to the database vuct-RC-DK-24 (see Fig. 3-1).

In Fig. 3-7 the number of beams n is plotted versus the effective depth d subdivided into class intervals of d = 100 mm. The required data concerning the effective depth was provided for all beams.

The distribution shows that the tests on beams were predominately carried out for the small range of effective depths of 200 < d < 300 mm, that is 340 tests of the 403 tests, i.e. 84%. Only 6 beams, i.e. 1,5%, featured an effec-tive depth of d > 600 mm; however, these tests featured a low slenderness of a/d < 2,4. Thus, they were transferred to the file vuct-DK-24 (see Fig. 3-1) and are hence not considered in the following evaluations.


3 - 8

Fig. 3-6: Number of beams plotted versus the slenderness a/d for the database vuct-PC-DS

Fig. 3-7: Number of beams plotted versus the effective depth d for the database vuct-PC-DS

The magnitude of the prestress of the test beams is now discussed. In the case of 10 beams this data was missing. In Fig. 3-8a the number of beams n is plotted versus the longitudinal concrete stress = P / Ac subdivided into class intervals of = 0,5 MPa. This is a parameter that is often used in design equations. The magnitude of the prestress was rather evenly distributed in the range of 2,0 < < 6,0 MPa, and 241 tests, respectively, 61% appeared in this range. An additional peak (63 tests, i.e. 16%) of the distribution appeared in the range of high prestress of 9,0 < < 12,0 MPa.

In the two following figures, Fig. 3-8b and Fig. 3-8c, the distribution of the magnitude of the prestress is plotted versus the dimension-free axial forces related to the uniaxial concrete compressive strength f1c as well as the uniaxial concrete tensile strength f1ct,cal. In Fig. 3-8b, the number of beams n is plotted versus the dimensions-free axial force P = /f1c subdivided into class intervals of P = 0,05. Correspondingly, in Fig. 3-8c the number of beams is plotted versus the dimension-free axial force P,ct = /f1ct related to the uniaxial tensile strength f1ct sub-divided into class intervals of P,ct = 0,25.


3-9

In contrast to Fig. 3-8a, both distributions are somewhat more normal. The peak values appear at approx.

P = - 0,30, respectively, at P,ct = /f1ct = - 3,4. The emphases of the distributions are at

P = - 0,125 (124 tests, 31%), respectively, in the range of -1,0 > /f1ct > -1,75 (159 tests, 39%).

cp (-)

a) number of beams plotted versus the longitudinal concrete stress = P / Ac

b) number of beams plotted versus the dimension-free axial force P = /f1c

c) number of beams plotted versus the dimension-free axial force P,ct = /f1ct in relation to f1ct

Fig. 3-8: Number of beams plotted versus the longitudinal concrete stress subdivided in class intervals


3 - 10


3-11

3 The database vuct-PC-DK-sl for the control of the tests on slender p.c.-beams subjected to point loads 3.1 Results of the evaluation with respect to the criteria The database for slender tests on beams subjected to point loads comprises of 320 beams, see Fig. 3-1. The tests are examined by means of control criteria, and the remaining beams are transferred to the evaluation database. Table 3-1 provides an overview of the results of the data control which are now discussed.

At first, the basic criteria were checked such as e.g. , and . The first

two criteria were fulfilled for nearly all beams, but only 78,4% of the beams featured a web width of .

Only 83 test beams, i.e. 74%, fulfilled the criterion kon8 for assessing that no flexural failure occurred. However, out of these 83 beams, 44 beams just missed to fulfill this criterion, see kon81.

The predominate number of tests contained plain reinforcing steel according to the criterion kon103, that is 208 tests or 66% of all tests.

Table 3-1: Result of the evaluation of the tests with respect to the individual criteria koni

individual criterion fulfilled number % of 320 not fulfilled

kon1 319 99,7 1

kon2 320 100,0 0

kon3 251 78,4 69

kon4 320 100,0 0

kon5 263 82,2 57

kon6 57 17,8 263

kon7 309 96,6 11

kon8 237 74,1 83

kon81 44 13,8 ---

kon101 : ribbed 44 13,8 276

kon103 : ribbed 112 35,0 208

kon10 oder 136 42,5 184

kon11 80 25,0 240

kon15 ”andbr” 320 100,0 0

It is also noteworthy that only 80 tests (25%) fulfilled the criterion kon11 for the assessment for the anchorage of the tension chord reinforcement at the end support. This is doubtless as too short of overhangs of the beams behind the axis of the support were provided for the predominately used plain prestressing steel.


3 - 12

For further evaluation, the 320 beams were firstly separated according to their type of prestressing as well as according to their type of reinforcement (plain or ribbed), and subdivided into four groups:

- vuct-PC-Pre-Plain: Prestressed pre-tensioned concrete with plain prestressing and reinforcing steel.

- vuct-PC-Pre-Ribb: Prestressed pre-tensioned concrete with ribbed prestressing or reinforcing steel.

- vuct-PC-Post-Plain: Prestressed post-tensioned concrete with plain prestressing and reinforcing steel.

- vuct-PC-Post-Ribb: Prestressed post-tensioned concrete with ribbed prestressing or reinforcing steel.

The flow chart in Fig. 3-9 provides an overview of the results and in the following the individual criteria koni are applied to the four groups. Out of the 320 tests, 156 beams (48,8%) were pre-tensioned and 164 beams (51,3%)

were post-tensioned. Subsequently, the beams were further split down by means of the criteria kon101 ( ) and

kon103 ( ), thus capturing the beams with ribbed reinforcing steel separately. Out of the 156 pre-tensioned

beams, 73 had plain reinforcement while the remaining 83 beams had ribbed reinforcement. In case of the post-tensioned beams, 111 beams had plain reinforcement while the remaining 53 beams had ribbed reinforcement.

Fig. 3-9: Overview of the configuration of the files

3.2 Selection of the tests for evaluation

In order to be transferred to the evaluation database, several or all criterions must be fulfilled simultaneously for a beam, i.e.:


As with the case of the individual criteria, the conditions are:

KONAi = 0 no transfer to evaluation file

KONAi = 1 transfer to evaluation file

All of the evaluated tests have to fulfill the following criterion:

KONA0 = kon1 · kon3 · kon4 · kon7 · kon15 · kon11

All of the four files shown in Fig. 3-9 were separately checked for this criterion and the results are presented in the following sections.

Furthermore, different alternatives for the configuration of the evaluated tests were chosen in the evaluation file and the following criteria KONA0i were defined.

For the evaluation A21, the following combined criterion was defined:


Where: kon5 = criterion for the slenderness ( )


3-13

kon8 = criterion for flexural failure ( ).

For the evaluation file A22, the criterion KONA21 was adjusted and the factor kon8 was replaced by kon81, i.e.

only tests with magnitudes of were selected:


For the third evaluation file A31, the factor kon6 was taken instead of kon5, i.e. tests with a slenderness levels of

were considered:


For the fourth and last evaluation file A32, the criterion KONA31 was adjusted by replacing factor kon8 with factor kon81:


This process is referred to as sorting procedure "sort-flex" and is shown in Fig. 3-10.

Fig. 3-10: The sorting procedure “sort-flex”

3.3 Pre-tensioned concrete beams with plain reinforcing steel (vuct-PC-Pre-Plain-A) The evaluation of the tests is carried out in step by step manner for the 73 pre-tensioned beams and with plain rein-forcement as shown in Fig. 3-9. In order to determine the process for the evaluation of which beams are subse-quently eliminated, the following tables show the effect of applying individual criteria to each evaluation file. Table 3-2 provides an overview of the results for the subsequent application of the selection criteria KONA0, see section 3.2, which have to be fulfilled by all tests in order to be considered in the evaluation. According to the criteria KONA0a, only 47 tests remain and none of the tests fulfills the criterion kon11 (anchorage failure).

The criteria which need be additionally considered were applied to the two datasets KONA0a and KonA0b in order to avoid the immediate elimination of tests with possible anchorage failures. At first, the individual criteria for the slenderness and for flexural failures are applied to the 47 beams of the evaluation file KONA0a corresponding to the sorting procedure "sort-flex" according to Fig. 3-10 and the result is provided in Table 3-3. By this, a total of 41 (35+3+2+1) tests remain for the evaluation.

Table 3-2: Subsequently following application of the individual selection criteria for KONA0

Selection criterion


added criterion

Remaining of 73

Difference


kon6·kon8 kon5·kon81 kon5·kon8 kon6·kon81


3 - 14

KONA0b KONA0a kon15 andbr 47 26

KONA0 KONA0b · kon11 0 73

Table 3-3: Result of the evaluation of the tests on prestressed concrete beams according to the sorting procedure "sort-flex" for the evaluation files KONAa

Selection crite-rion


Fulfilled number % of 47 Not fulfilled

KONA21a KONA0a · kon5 · kon8 35 74,5 12




After adding the following criterion kon15 (different type of failure) no beams are eliminated and the configuration is as shown in Table 3-3. After considering the anchorage failures (kon11), all of the so far remaining 41 beams were eliminated.

3.4 Pre-tensioned concrete beams with ribbed reinforcing steel (vuct-PC-Pre-Ripp-A) The evaluation of the tests is carried out step by step for the 83 pre-tensioned beams that had ribbed reinforcement as shown in Fig. 3-9. Table 3-2 provides an overview of the results for the subsequent application of the selection criteria KONA0 (see section 3.2), which have to be fulfilled by all tests in order to be considered in the evaluation. According to the criteria KONA0a, only 70 tests remain and only 16 tests fulfill the criterion kon11 (anchorage failure).


Selection criterion


Added criterion Remaining

of 83 Difference


KONA0b KONA0a kon15 ”andbr” 70 13


In order to determine the process for the evaluation of how many beams are subsequently eliminated, the following tables show the effect of applying the individual criteria to each evaluation file of Table 3-4. At first, the individual criteria for the slenderness and for flexural failures are applied to the 70 beams of the evaluation file KONA0a and the result is provided in Table 3-5. Out of the 70 tests, 59 (31+7+21+0) remain in the evaluation. By applying the criterion kon15 (other types of failure) no beams are eliminated, see Table 3-4.

Table 3-5: Result of the evaluation of the tests on prestressed concrete beams according to the sorting proce-dure "sort-flex" for the evaluation files KONAa






3-15




After considering calculated anchorage failures (kon11), the evaluation file KONA0 containing 16 test beams is used to apply the sorting procedure “sort-flex” and the result is provided in Table 3-6. No additional beams were eliminated and the remaining 16 tests correspond to 5% of all 320 slender beams.





KONA21 KONA0a · kon5 · kon8 16 100 0




3.5 Post-tensioned concrete beams with plain reinforcement (vuct-PC-Post-Plain-A) The evaluation of the tests is carried out in a step by step manner for the 111 post-tensioned beams that had with plain reinforcement as shown in Fig. 3-9. Table 3-7 provides an overview of the results for the subsequent applica-tion of the selection criteria KONA0 (see section 3.2), which have to be fulfilled by all tests in order to be consid-ered in the evaluation. According to the criteria KONA0a only 91 tests remain and only 55 tests fulfill the criterion kon11 (anchorage failure). Table 3-7: Subsequently following application of the individual selection criteria for KONA0

Selection criterion



of 111 Difference




In order to determine the process of the evaluation for determining which beams are subsequently eliminated, in the following tables the sorting procedure “sort-flex” are applied to each evaluation file of Table 3-7. At first, these criteria are applied to the 91 beams of the evaluation file KONA0a, and (63+17+5+4) = 89 beams remain for the evaluation. After adding the following criterion kon15 (different types of failure) no additional beams are elimi-nated, see Table 3-7.


3 - 16









After considering calculated anchorage failures (kon11), the sorting procedure “sort-flex” is applied to the evalua-tion file containing 55 test beams and the result is reported in Table 3-9. Only 53 (42+8+3+0) beams remain in the evaluation, corresponding to approx. 13% of all 413 beams.

Table 3-9: Result of the evaluation of the tests on prestressed concrete beams according to the selection criterion "sort-flex" for the evaluation files KONAa




KONA21 KONA0a · kon5 · kon8 42 76,4 13




3.6 Post-tensioned structural members and ribbed reinforcement (vuct-PC_Post_Ripp-A) The evaluation of the tests is carried out step by step for the 53 post-tensioned beams that had ribbed reinforcement as shown in Fig. 3-9. Table 3-10 provides an overview of the results for the subsequent application of the selection criteria KONA0, see sect. 3.2, which have to be fulfilled by all tests in order to be considered in the evaluation. According to the criterion KONA0a only 31 tests remain and only 23 tests fulfill the criterion kon11. Table 3-10: Subsequently following application of the individual selection criteria for KONA0

Selection criterion



of 53 Difference




In order to determine the process for evaluating which beams are to be subsequently eliminated, the following ta-bles show the individual criteria applied to each evaluation file. At first, the individual criteria for the slenderness and for flexural failures are applied to the 53 beams of the evaluation file KONA0a. A total of 31 beams remain for the evaluation.


3-17

In the following tables the sorting procedure “sort-flex” were applied to each evaluation file of Table 3-10. At first, these criteria are applied to the 31 beams of the evaluation file KONA0a, and 28 (19+5+3+1) beams remain for the evaluation. After adding the following criterion kon15 (different types of failure) no further beams are eliminated, see Table 3-10.

Table 3-11: Result of the evaluation of the tests on prestressed concrete beams according to the selection criterion "sort-flex" for the evaluation files KONAa








After adding the following criterion kon15 (different types of failure) no beams are eliminated and the configura-tion is as shown in Table 3-11.

Considering the calculated anchorage failures (kon11), the sorting procedure “sort-flex” is applied to the evaluation file KONA0 containing 23 test beams and the result is reported in Table 3-12. Only 5 (5+0+0+0) beams remain for the evaluation, corresponding to 1,24% of all 403 beams.

Table 3-12: Result of the evaluation of the tests on prestressed concrete beams according to the sorting procedure "sort-flex" for the evaluation files KONA








3.7 Total dataset Examining the total dataset with respect to these criterions results in the values reported in Table 3-13 for the evaluation criteria KONA0. It is obvious that the criterion kon11 (i.e. assessment of potential anchorage failure at end support) influences the evaluation of the still available data significantly. Therefore, the evaluation is also per-formed for tests that did not satisfy the anchorage criterion in order to ascertain whether these tests feature a lower shear capacity.

The result is provided in Table 3-14, and it shows the configuration of the 76 beams fulfilling the criterion KONA0.

All of these beams feature a slenderness of . Hence, only 74 (63+8+3+0) beams without calculated

anchorage failures remain overall for KONA2 and KONA3, which can be evaluated statistically.


3 - 18



Combination of the individ-ual criteria


of 320 Difference


KONA0b KONA0a · kon15 ”andbr” 239 81


Table 3-14: Result of the evaluation of the tests on prestressed concrete beams according to the sorting procedure "sort-flex" for the evaluation files KONA




KONA21 KONA0 · kon5 · kon8 63 82,9 13

KONA22 KONA0 · kon5 · kon81 8 10,5 68

KONA31 KONA0 · kon6 · kon8 3 3,9 73

KONA32 KONA0 · kon6 · kon81 0 0,0 76

3.8 Summary Table 3-15 provides a concluding overview of the result of the evaluation of the four files shown in Fig. 3-8 accord-ing to the sorting procedure "sort-flex". Out of the altogether 217 tests (Table 3-3a) only 74 tests (i.e. 34%) fulfill the criterion for having adequate anchorage at the end support. The following evaluation is carried out for the data-set (A2b+A3b); however, inadequate anchorage as well as the type of prestressing is identified.

Table 3-15: Result of the sorting procedure "sort-flex" of the four files in Fig. 3-8

a) number of tests for the datasets KON21b through KON32b with calculated anchorage failure

Pre Post All All Criterion Plain 73 Ripp 83 Plain 111 Ripp 53 320 320

KON21b 35 31 63 19 148

KON22b 3 7 17 5 32 KON2b 180

KON31b 2 21 5 3 31

KON32b 1 0 4 1 6 KON3b 37

sum 41 59 89 28 sum 217 b) number of tests for the datasets KON21b through KON32b without calculated anchorage failure

Pre Post All All Criterion

Plain 73 Ripp 83 Plain 111 Ripp 53 320 320

KON21 0 16 42 5 63

KON22 0 0 8 0 8 KON2 71

KON31 0 0 3 0 3 KON32 0 0 0 0 0

KON3 3

sum 0 16 53 5 74 sum 74


3-19

4 The evaluation database vuct-PC-A 4.1 Introduction

Out of the overall 403 collected tests, 320 beams with a slenderness of remained for the data evaluation.

Furthermore, these 320 tests were subdivided in four groups (see Fig. 3-8), and after applying the previously pre-sented control criteria the evaluation databases could be generated.

At first, the evaluation files were generated in detail, i.e. for each of the four groups distinguishing the type of prestressing, an evaluation dataset A2, respectively, A3 was generated. In some cases only a few test beams re-mained in these evaluation datasets. To compensate for this, the evaluation datasets A2b and A3b were generated, which comprise altogether 217 tests and include all tests that did not satisfy the anchorage criterion. The chosen classifications allow for the evaluation of design formula for beams in two categories, one including and the other excluding beams that did not satisfy the anchorage criteria. This is done separately in the figures that follow.

As a second step, an evaluation dataset containing all four groups was generated. By means of this evaluation data-set, the individual types of prestressing and the types of ribs of steel (plain or ribbed) can directly be compared with each other. Once again, a merged evaluation dataset (A2b+A3b) including an identification of the anchorage fail-ures is generated in addition to the two evaluation datasets A2 and A3 containing 74 tests excluding anchorage failures.

4.2 Presentation of the evaluation database vuct-PC-A In the following section, the number of tests n is plotted versus the most important parameters for the database vuct-RC-A, in order to provide an overview of the distribution of the tests transferred to the evaluation file. In the diagrams, the 74 tests without calculated anchorage failures (dark boxes on bottom) are separately shown from the 143 tests with anchorage failures (light boxes on top). In Fig. 3-11 the number of tests n is plotted versus the uniax-ial concrete compressive strength f1c subdivided into class intervals of f1c = 5 MPa. Most beams (85, i.e. 39%) featured a uniaxial concrete compressive strength of 25 to 35 MPa, and 59 tests (approx. 27%) had a strength of f1c < 25 MPa. Thus, the tests were predominately carried out for beams with comparably low concrete strengths. In case of 48 tests (approx. 22%) the concrete compressive strength lay in the range of 35 < f1c < 45 MPa. Only 14 tests (i.e. 6,4 %) were performed with a concrete compressive strength of f1c >60 MPa and these were classified as high-strength concrete beams. After all, 12 beams out of the 14 beams did have a calculated anchorage failure.


3 - 20

In Fig. 3-12 the number of beams is plotted versus the yield strength of the prestressing steel subdivided into class intervals of f = 50 MPa. The required data with respect of the steel grade is provided for all beams.

In case of 40 test beams (approx. 18%), the prestressing steel had a low yield strength of approx. 850 MPa. Most tests (119 tests, approx. 55%) were on beams that had prestressing steel with yield strengths in the range of 1.400 < fpy < 1.550 MPa.

In Figs. 3-13a through c, the distribution of the magnitude of the prestressing force of the beams is shown. In Fig. 3-13a the number of tests is plotted versus the longitudinal concrete stress = P / Ac subdivided into class intervals of = 0,5 MPa. The magnitude of the prestressing force is rather evenly distributed for a longitudinal concrete stress in the range of 2,0 < < 5,0 MPa, and 124 tests, respectively 57% of all tests, appeared in this range. A total of 26 beams, i.e. 12% of all beams, featured high magnitudes of the prestressing force for longitudi-nal concrete stresses in the range of 9,0 < < 12,5 MPa.

In the two following figures, Fig. 3-13b and Fig. 3-13c, these distributions are plotted versus the dimension-free axial force P, respectively, P,ct, which is the magnitude of related to the uniaxial concrete compressive strength f1c, respectively, the uniaxial concrete tensile strength f1ct,cal.

In Fig. 3-13b the number of beams n is plotted versus the dimension-free axial force P = /f1c subdivided into class intervals of P = 0,05. Correspondingly, in Fig. 3-13c the number of beams is plotted versus the dimension-free axial force P,ct = /f1ct, related to the uniaxial concrete tensile strength f1ct, subdivided into class intervals of

P,ct = 0,25. In contrast to Fig. 3-13a, both distributions are more “normal”. The peak values appear at approx. P = - 0,30, respectively, at P,ct = /f1ct = - 3,4. The emphases of the distributions are at P = - 0,125 (85 tests, approx. 39% of all 217 tests), respectively, in the range of -1,0 > /f1ct > -1,75 (99 tests, 46%).

In case of the 74 tests without calculated anchorage failures, the emphasis of the distribution of P,ct = /f1ct is shifted somewhat to the left side, i.e. in direction of lower prestress.


3-21

cp

a) Number of beams plotted versus the longitudinal concrete stress = P / Ac

b) Number of beams plotted versus the dimension-free axial force P = /f1c

c) Number of beams plotted versus the dimension-free axial force P,ct = /f1ct in relation to f1ct

Number of beams plotted versus the longitudinal concrete stress


3 - 22

In Fig. 3-14, the number of beams is plotted versus the longitudinal reinforcement ratio l = lw = ( s + p), subdivided into class intervals of l = 0,25 %. The distribution of the ranges shown in Fig. 3-4 is as follows: - 38 tests (approx. 18%) with reinforcement ratios of l < 0,5 %;

- 81 tests (37%) with reinforcement ratios of 0,5 < l < 1,0 %;

- 52 tests (approx. 24 %) with reinforcement ratios of 1,0 < l < 1,5 %.

Hence, only 16 tested beam, i.e. approx. 7% (in contrast to the 19% as shown in Fig. 3-4) had very high reinforce-ment ratios.

l

The geometrical reinforcement ratio does not characterize the flexural capacity as well as the mechanical rein-forcement ratio l = l fsy/f1c. Therefore, in Fig. 3-15 the number of beams n is plotted versus the mechanical rein-forcement ratio of the longitudinal reinforcement l = s + p subdivided into class intervals of l = 0,05. The distribution in Fig. 3-15 shows that the tests on the beams were carried out for a large range of l. Only 11 beams (approx. 5%) featured a reinforcement ratio of l < 0,10, and 55 beams (approx. 25%) featured a reinforcement ratio of 0,10 < l < 0,20. Thus considerably more that 50% of the tests (70%) are highly, or very highly reinforced, with an reinforcement ratio of l > 0,20. The highest magnitudes are around l = 0,60.

p


3-23

In Fig. 3-16 the number of beams n is plotted versus the (shear) slenderness = a/d subdivided into class intervals of = 0,4. The distribution in Fig. 3-16 shows that the tests on the beams are carried out for a large range of the slenderness levels. The emphasis of the distribution (126 tests, i.e. 58%) appears between a slenderness of 3,2 < = a/d < 4,4. These are somewhat higher magnitudes than for the reinforced concrete beams with an emphasis of the distribution between 2,4 < < 3,2 see Fig. 2.1-14, part 2.1. A total of 47 tests (approx. 22 %) were carried out for a large slenderness of = a/d > 4,4 up to a/d = 6,8.


3 - 24


3-25

5 Comparison of the test results with design approaches for slender beams subjected to point loads 5.1 Introduction For the different datasets, the model safety factors were determined, defined as follows (see Part 1, section 5.4, Eq.(1-132)):

[-] (3-1)

In Eq. (3-1) Vu,test is the reported ultimate shear force. The calculated shear forces are referred to as Vu,cal, which are determined as characteristic values according to the different design approaches, such as, e.g. the German standard DIN 1045-1, see Part 1, section 5.4.

The model safety factor was evaluated statistically and the different design approaches resulted in different results.


5%- fractile value of mod did not fulfil the criterion . The approaches were standardized by means of

adjustment coefficients, so that the 5%-fractile value equals 1,00 for all cases.

For the evaluation of the prestressed concrete beams, the two adjustment coefficients are adopted from the data evaluation for reinforced concrete beams. In order to achieve the same level of the 5%-fractile value for the total approach, the term for the influence of the prestressing was multiplied by another adjustment coefficient, so that 5%

of the model safety factors fulfill .

5.2 Comparison of the test results with the design approach according to the German standard DIN 1045-1

5.2.1 The approach of the German standard DIN 1045-1 and determination of the coefficients

The characteristic value of the shear capacity of structural concrete members without shear reinforcement is defined for normal concrete as follows according to Eq.(70) of the German standard DIN 1045-1:

[kN] (3-2)

where: = minimum width of section within tension zone in [mm]

fck = characteristic cylinder strength [MPa]

= coefficient for the influence of the depth d of the member d [mm]

l = Asl /(bw d) = longitudinal reinforcement ratio [-] where Asl = As + Ap

= design value of the longitudinal concrete stress (compression < 0)

The factor of 1,402 results from the first two adjustment coefficients 1,2443 and 1,1267, which are adopted from the database vuct-RC-A2&A3 for reinforced concrete beams (see section 5.2.1 and Table 2.1-4 in Part 2.1). The subsequently determined third adjustment coefficient is only related to the term cd.

Table 3-16 provides the relevant statistical values of the model safety factors mod = Vu,test / Vu,cal which are deter-mined for the datasets A2b as well as for different datasets (A2&A3). The evaluation of the dataset A2b containing 180 tests with a slenderness of a/d > 2,89 results in lower magnitudes of the third adjustment coefficient, the mean value, and the coefficient of variation than the total dataset containing 217 tests with a slenderness of a/d > 2,40. Thus, the 37 tests with a low slenderness in the range of 2,40 < a/d < 2,89 feature higher magnitudes of the mean value and the coefficient of variation than the tests with a slenderness of a/d > 2,89 in comparison to the reinforced concrete beams in Part 2.1.

The third adjustment coefficient of 0,916, respectively, 0,951 shows that the above given approach for the consid-eration of the influence of the prestress is marginally unsafe.


3 - 26

Table 3-16: Statistical characteristic values for the model safety factors mod = Vu,test/Vu,cal for the different datasets A2, A3 and (A2b&A3b)

datasets Characteristic value A2b A2b&A3b A2&A3 only VB A2&A3

3. adjustment coefficient

0,916 0,9513

N 180 217 143 74

M 1,6623 1,7024 1,7084 1,6909

S 0,448 0,494 0,462 0,554

V 0,2695 0,2902 0,2704 0,3274

5% 0,925 0,890 0,948 0,780

95% 2,399 2,515 2,468 2,602

Furthermore, Table 3-16 shows the results of the separate evaluations for the datasets containing only calculated anchorage failures and the dataset without calculated anchorage failures of the datasets (A2b&A3b), while keeping the coefficient of all 217 tests constant. The mean values of the different datasets differ only marginally so that it does not seem to be justified to not consider the tests with calculated anchorage failures in the evaluation. This is also justified, since the coefficient of variation of the 74 tests excluding calculated anchorage failures is signifi-cantly higher than the one of the 143 tests of the dataset containing only calculated anchorage failures.

Determining the third adjustment coefficient is separately for each of the datasets (A2&A3), results in the following values: - (A2b&A3b): n = 143 tests with only calculated anchorage failures (VB): 0,9579; - (A2&A3): n = 73 tests without calculated anchorage failures (no VB): 0,9419. This evaluation, too, results in only very marginal differences of the datasets containing beams with and without calculated anchorage failures VB, and the first dataset (containing only VB) even features a somewhat lower third adjustment coefficient or model safety factor than the latter dataset without VB.

In the following, these conclusions are to be ascertained for the different datasets subdivided according to the type of prestress. In the following tables, Table 3-17a through c, the relevant statistical values of the different datasets (A2b&A3b) with and without calculated anchorage failures (VB) are compared with the statistical values of the datasets subdivided according to the types of prestress. At the bottom of the table, the ratio of the mean value of the partial dataset and the respective total dataset, is also provided in an additional row.

Comparing Table 3-17 b and Table 3-17 c shows that in case of the datasets "PreRibb", the tests of the dataset con-taining only calculated anchorage failures feature about the same mean value as for the tests of the dataset without calculated anchorage failures. Whereas, the 43 tests of the dataset containing only calculated anchorage failure fea-ture a considerably higher coefficient of variation than the 16 tests of the dataset without calculated anchorage fail-ures. This comparison was not performed for the datasets "PostRibb" due to the small number of tests (5 tests). In case of the datasets "PostPlain", however, the mean value is of higher magnitude for the tests without calculated anchorage failures, yet the coefficient of variation is considerably higher resulting in a lower 5%-fractile value.

The Tables 3-17b and c also allow comparisons with the datasets "Pre" with pretensioned beams and "Post" with post-tensioned beams. All tests "Post" feature significantly higher coefficients of variation than the tests "Pre". For all tests "Pre", the 5%-fractile value is higher than 1,0, and there are no major differences between the tests "Ribb" with ribbed reinforcement and the tests "Plain" with plain reinforcement. Whereas, in case of the tests "Post" all tests "Plain" featured lower mean values than the tests "Ribb"; however, the coefficients of variation are signifi-cantly higher, so that the 5%-fractile values are lower.

Overall, these comparisons do not provide unambiguous conclusions and no contradiction with respect to the above given conclusion not to eliminate the tests with calculated anchorage failures from the further evaluation. Thus, the


3-27

following evaluation is carried out for all 217 tests, and in the diagrams, the tests "Post" and Plain" as well as "Pre" und "Ribb" are labeled separately.

Table 3-17: Statistical values for the model safety factors mod = Vu,test/Vu,cal for the different datasets subdivided according to the type of prestress

a) Total dataset (A2b&A3b) with and without calculated anchorage failures (VB)

total Characteristic value

1,7024 1,6360 1,8264 1,9929 1,5979 s 0,4940 0,3141 0,3778 0,6012 0,5569

v = s/m 0,2901 0,1920 0,2068 0,3017 0,3485 0,05 0,890 1,119 1,205 1,004 0,682 0,95 2,515 2,153 2,448 2,982 2,514

b) Datasets only with calculated anchorage failures

total Characteristic value

1,7084 1,6107 1,8264 2,1423 1,4121 s 0,4620 0,3341 0,3778 0,5469 0,3688

v = s/m 0,2704 0,2074 0,2068 0,2553 0,2612 0,05 0,948 1,061 1,205 1,243 0,805 0,95 2,468 2,160 2,448 3,042 2,019

c) Datasets only without calculated anchorage failures

total Without calculated anchorage failures VB Characteris-tic value A2&A3 PreRibb PrePlain PostRibb PostPlain

n 74 16 0 5 53 m 1,6909 1,7042 - 1,3058 1,7301 s 0,5536 0,2495 - 0,2759 0,6294

v = s/m 0,3274 0,1464 - 0,2113 0,3638 0,05 0,780 1,294 - 0,852 0,695 0,95 2,602 2,115 - 1,760 2,766

m/mges 0,9932 1,0010 - 0,7670 1,0163


3 - 28


In the following figures the model safety factor for Eq.(70) of the German standard

DIN 1045-1 is plotted versus the essential parameters in order to obtain further information on the influence of the different parameters and the quality of their coverage by Eq. (70). In each diagram the parameter is subdivided into individual ranges. For each of these groups, the mean value as well as the upper and the lower fractile value is listed. Below the diagrams, tables are given in which the relevant statistical values are listed for each range in com-parison to the values for the whole dataset.

In the figures, the calculated anchorage failures are labeled with special symbols in red. Furthermore, the magni-tudes of the model safety factors are separately marked according to the type of prestress as well as to the type of reinforcement (plain or ribbed).

There is no diagram showing the distribution of the model safety factors versus the effective depth since the tests were only carried out for a small range of effective depth d, as demonstrated by Fig. 3-6 as well as Fig. 3-16.

In Fig. 3-17 the model safety factors for Eq.(70) of the German standard DIN 1045-1 are plot-

ted versus the uniaxial concrete compressive strength f1c for the dataset (A2b&A3b). It can be seen that the scatter of the model safety factors of this dataset is predominately due to the high values of the model safety factor since significantly more values exceed the 95%-fractile value than fall below the 5%-fractile value. The trend of the mean values is nearly horizontal over the ranges, and thus the influence of the concrete compressive strength

(respectively, of the concrete tensile strength indirectly by means of ) is accurately captured. The highest scatter

appears for the range of normal-strength concrete (f1c < 45 MPa), but there are only a few tests available for high-strength concrete beams (f1c > 60 MPa).

The model safety factors for the tests of the dataset containing only calculated anchorage failures (VB) are distrib-uted similarly to the dataset of tests without calculated anchorage failures. Thus, Fig. 3-17 also verifies the conclu-sions drawn in section 5.2.1.


3-29


3 - 30


ted versus the longitudinal reinforcement ratio l = lw = ( s + p) for the dataset (A2b&A3b). All the unsafe tests appear in the ranges A and B with low reinforcement ratios. The mean values of the ranges increase with increasing reinforcement ratio.

The associated table illustrates that range D exhibits a large scatter; overall, however, only 8 beams lie in this range. Out of these 8 beams, 2 beams are post-tensioned with ribbed reinforcement. The other beams are pre-tensioned, and 5 beams belong to the test series of Kar (1968). These beams are I-shapes and thus the reinforcement ratio is calculated using the thin width of web.


3-31


ted versus the (shear) slenderness a/d for the dataset (A2b&A3b).

Fig. 3-19 shows a considerable increase of the shear capacity for decreasing a/d, which is also shown by the mean values in the associated table. Equally striking are the considerably low scatters (s = 0,3098 to s = 0,3545) and coefficients of variation (v = 0,1892 to 0,2527) for a shear slenderness of a/d > 3,40, which are significantly below the magnitudes of s = 0,6254, respectively, v = 0,3093 for a shear slenderness of a/d < 2,89.

All of the tests below the 5%-fractile value appear in the ranges E and F with a high slenderness a/d.


3 - 32

In Fig. 3-20 the model safety factors for Eq.(70) of the German DIN 1045-1 are plotted versus

the magnitude of the prestressing force which is described by the following three parameters:

- the longitudinal concrete stress for Eq.(70) of the German standard DIN 1045-1 = P / Ac (Fig. 3-20a).

- the dimension-free axial force P = cP/f1c related to the uniaxial concrete compressive strength f1c (Fig. 3-20b).

- the dimension-free axial force P,ct related to the uniaxial concrete tensile strength f1ct,cal (Fig. 3-20c).

Firstly, the figures demonstrate again the distribution of the tests versus the parameters (see Fig. 3-13) as well as the considerable scatter of the model safety factors mod. The unsafe, respectively, low magnitudes of the model safety factor mod are distributed over the whole range of the respective parameters , P und P,ct for the prestressing force in all three diagrams. Accordingly, the trend lines are nearly horizontal and are almost identical with the lines representing the mean value of the model safety factors mod of all tests. Obviously, the influence of the prestressing is overall sufficiently captured.

The quadratic trend curves are slightly curved; for low magnitudes of , P, respectively, P,ct they differ only marginally (approx. 15 %) and for high magnitudes of these parameters they differ approx. 30% from the mean value. This indicates, that the approach for the influence of the prestressing in Eq.(70) of the German standard 1045-1, could be enhanced.

a) Model safety factors mod plotted versus the longitudinal concrete stress = P / Ac


3-33

b) Model safety factors mod plotted versus the dimension-free axial force P = /f1c

c) Model safety factors mod plotted versus the dimension-free axial force P,ct = /f1ct in relation to f1ct


3 - 34

5.2.3 Summary of the results for the German standard DIN 1045-1

For the evaluation the characteristic value for the coefficient k = 0,1402 in Eq.(3-2) was adopted from the evalua-tions of the tests on reinforced concrete beams of the dataset (A2&A3). The additional adjustment coefficient for adapting to the 5%-fractile value (3,7 beams) only relates to the term of (0,12 cd) in Eq.(3-2) or Eq.(70) of the German standard, and it was determined to be 0,916 for the dataset A2b (180 tests), and respectively to be 0,951 for the dataset (A2b&A3b) (217 tests). Hence, this term is slightly unsafe.

The 217 beams transferred to the data evaluation also include tests with calculated anchorage failures (VB), since the results of the tests with calculated anchorage failures do not differ essentially and unambiguous from those of test without calculated anchorage failures.

The high scatter of v = 0,291 is noteworthy. The highest magnitudes of the scatter appear for normal-strength concrete (f1c < 45 MPa); however, only a few tests (17) are carried out for high-strength concrete beams (f1c > 60 MPa).

The evaluation database does not contain test beams with effective depths of d < 400 mm. All unsafe test values appear in the range of 200 < d < 300 mm and feature low reinforcement ratios ( l < 1,1 %). The mean values of the class intervals of the reinforcement ratios increase with an increasing reinforcement ratio.

The model safety factors show a significant increase for decreasing slenderness a/d. Equally striking are the considerably low scatter and the coefficients of variation for a slenderness of a/d > 3,40. All tests below the 5%-fractile value appear in the range E and F with a high slenderness of a/d > 5,0.

Overall, the influence of the prestress respectively of axial compression is satisfactorily captured in Eq.(70) of the German standard DIN 1045-1, although it is slightly unsafe.

References of Part 3

DIN 1045-1 (2001): Deutsche Norm: Tragwerke aus Beton, Stahlbeton und Spannbeton - Teil 1: Bemessung und Konstruktion. 1 - 148. Normenausschuss Bauwesen (NABau) im DIN Deutsches Institut für Normung e.V. Beuth Verl. Berlin, Juli 2001

Kar, J.N. (1968): Diagonal cracking in prestressed concrete beams. Journal of the Structural Division, Proc. of ASCE 94 (1968), St 1, January, 83 - 109

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ge p

late

0.0

1)

rhos

10

0d

bAs

s⋅

⋅=

[%]

ge

omet

rica

l per

cent

age

of lo

ng. r

einf

orce

men

t

rhos

w

10

0d

bA w

ssw

⋅⋅

=

[%]

ge

om..

perc

. of

long

. rei

nfor

cem

ent r

elat

ed to

b w

fsy

f sy

[k

si →

MPa

] yi

eld

stre

ngth

of

stee

l

esy

ε sy

= f s

y /

Es

[‰]

st

eel s

trai

n at

yie

ld (

Es =

200

.000

MPa

)

ftk

f tk

[k

si →

MPa

] ch

arac

teri

stic

tens

ile

stre

ngth

ftk/

fsy

f tk/f

sy

[

- ]

ra

tio

euk

ε uk

[‰

]

stee

l str

ain

at m

axim

um s

teel

str

ess

Län

gsdr

uckb

eweh

rung

(lo

ngitu

dina

l com

pres

sion

rei

nfor

cem

ent)

Stab

_D

ba

rs

[

- ]

nu

mbe

r of

bar

s

ds2

d s2

[i

n →

mm

] di

stan

ce o

f co

mpr

ess.

rei

nfor

c. f

rom

com

pres

s.

edge

dst2

∅

st2

[i

n →

mm

] av

erag

e di

amet

er o

f co

mpr

. bar

s

As2

A

s2

[i

n² →

mm

2 ]

area

of

com

pr. b

ars

Atta

chm

ent 3

-1:

Not

atio

n an

d fo

rmul

ary

for

the

data

base

s vu

ct-P

C-D

S an

d vu

ct-P

C-D

K_s

l for

p.c

.- b

eam

s w

ithou

t stir

rups

sub

ject

ed to

poi

nt lo

ads

-3-

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

– P

art 3

Rei

neck

, K.-

H.;

Kuc

hma,

D. A

.; F

itik

, B.

– U

nive

rsit

y of

Stu

ttga

rt a

nd U

nive

rsit

y of

Ill

inoi

s

D

ec 2

008

fsy2

f s

y2

[k

si →

MPa

] yi

eld

stre

ngth

of

com

pres

sion

bar

s

Span

nsta

hl (

pres

tres

sing

ste

el)

dpbo

t

pb

otd

[in

→ m

m]

effe

ctiv

e de

pth

of p

rest

ress

ing

stee

l at b

otto

m

dpto

p

pt

opd

[in

→ m

m]

effe

ctiv

e de

pth

of p

rest

ress

ing

stee

l at t

op

type

ty

pe

[

- ]

nu

mbe

r an

d ty

pe o

f pr

estr

essi

ng

pres

tr. m

etho

d

pre

/ pos

t

pre-

or

post

tens

ioni

ng

dsp

d sp

[m

m]

no

min

al d

iam

eter

fRp

rpf

[

- ]

r

= r

ibbe

d, 0

= p

lain

Apb

ot

pb

otA

[in²

→ m

m2 ]

ar

ea o

f bo

tton

pre

str.

ste

el

Apt

op

pt

opA

[in²

→ m

m2 ]

ar

ea o

f to

p pr

estr

. ste

el

Ap

ptop

pbot

pA

AA

+=

[mm

2 ]

area

of

pres

tres

sing

ste

el

rhop

10

0d

bAp

p⋅

⋅=

ρ

[%]

ge

om. r

einf

. rat

io o

f pr

estr

. ste

el

rhop

w

10

0d

bA w

pbot

pw⋅

⋅=

[%

]

geom

. rei

nf. r

atio

of

pres

tr. s

teel

rel

ated

to b

w

rhol

p

sl

+=

[%]

ge

om. r

einf

. rat

io

rhol

w

pwsw

lw+

=

[%

]

geom

. rei

nf. r

atio

Ep

Ep

[M

Pa]

yo

ung’

s m

odul

us o

f pr

estr

essi

ng s

teel

(if

not g

iven

Ep=

200

.000

MPa

)

fpy

f py

[k

si →

MPa

] yi

eld

stre

ngth

= f

p0,1

k

epy

p

pyE

f=

pyε

[

- ]

st

eel s

trai

n at

yie

ld

fpk

f pk

[k

si →

MPa

] ch

arac

teri

stic

tens

ile

stre

ngth

fpk/

fpy

f p

k/f p

y

[ -

]

rati

o

epuk

ε p

uk

[

- ]

st

eel s

trai

n at

max

imum

ste

el s

tres

s

Atta

chm

ent 3

-1:

Not

atio

n an

d fo

rmul

ary

for

the

data

base

s vu

ct-P

C-D

S an

d vu

ct-P

C-D

K_s

l for

p.c

.- b

eam

s w

ithou

t stir

rups

sub

ject

ed to

poi

nt lo

ads

-4-

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

– P

art 3

Rei

neck

, K.-

H.;

Kuc

hma,

D. A

.; F

itik

, B.

– U

nive

rsit

y of

Stu

ttga

rt a

nd U

nive

rsit

y of

Ill

inoi

s

D

ec 2

008

lam

bda

(

)(

)sy

spy

p

pyp

fA

fA

fA

⋅+

⋅

⋅=

λ

[

- ]

pr

estr

essi

ng r

atio

d

()

()

()

()

sys

pyp

ssy

sp

pyp

fA

fA

df

Ad

fA

d⋅

+⋅

⋅⋅

+⋅

⋅=

[m

m]

av

erag

e ef

fect

ive

dept

h of

tens

ion

chor

d

Vor

span

nung

(pr

estr

ess)

Pbo

t

bo

tP

[kip

→ k

N]

pres

tres

sing

for

ce o

f bo

ttom

tend

ons

Pto

p

to

pP

[kip

→ k

N]

pres

tres

sing

for

ce o

f to

p te

ndon

s

zpbo

t

pb

otz

[mm

]

dist

ance

of

botto

m te

ndon

s fr

om C

GS

zpto

p

pt

opz

[mm

]

dist

ance

of

top

tend

ons

from

CG

S

P

P

[

kN]

pr

estr

essi

ng f

orce

sigc

p

σ

cp

[M

Pa]

ax

ial c

oncr

ete

stre

ss a

t CG

S

sigc

p/f1

c

σ

cp /

f 1c

[

- ]

ra

tio

mp

Mp

[k

Nm

]

mom

ent d

ue to

pre

stre

ss

sigp

p

pA

P=

ppσ

[MPa

]

stee

l str

ess

due

to p

rest

ress

Vp

Vp

[k

ip →

kN

] ve

rtic

al c

ompo

nent

of

effe

ctiv

e pr

estr

essi

ng

forc

e

Nor

mal

kraf

t (ax

ial f

orce

)

n

N

[kN

]

axia

l for

ce

sigc

n

σ

c,N =

N /

Ac

[MP

a]

axia

l con

cret

e st

ress

at C

GS

sigc

n/f1

c

σ

c,N

/ f 1

c [

- ]

di

men

sion

s-fr

ee a

xial

for

ce

sigc

/f1c

= (

sigc

p +

sig

cn)/

f1c

σ

c / f

1c

[ -

]

rati

o fo

r to

tal a

xial

for

ce s

tres

s w

ith

σc =

σcp

+

σc,

N

Bet

ondr

uckf

estig

keit

(con

cret

e co

mpr

essi

ve s

tren

gth)

diaa

∅

a

[in

→ m

m]

max

. dia

met

er o

f ag

greg

ates

fccy

l

f c

,cyl

[psi

→ M

Pa]

cylin

der

stre

ngth

of

conc

rete

PKcy

l

Pr

üfkö

rper

[i

n →

mm

] di

men

sion

of

cylin

ders

Atta

chm

ent 3

-1:

Not

atio

n an

d fo

rmul

ary

for

the

data

base

s vu

ct-P

C-D

S an

d vu

ct-P

C-D

K_s

l for

p.c

.- b

eam

s w

ithou

t stir

rups

sub

ject

ed to

poi

nt lo

ads

-5-

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

– P

art 3

Rei

neck

, K.-

H.;

Kuc

hma,

D. A

.; F

itik

, B.

– U

nive

rsit

y of

Stu

ttga

rt a

nd U

nive

rsit

y of

Ill

inoi

s

D

ec 2

008

f1cc

yl

f 1

c,cy

l

[MPa

]

unia

xial

com

pr. s

tren

gth

deri

ved

from

fcc

yl

fccu

be

f c

,cub

e

[psi

→ M

Pa]

cube

str

engt

h of

con

cret

e

PKcu

be

Prüf

körp

er

[in

→ m

m]

dim

ensi

on o

f cu

bes

f1cc

ube

f 1

c,cu

be

[M

Pa]

un

iaxi

al c

ompr

. str

engt

h de

rive

d fr

om f

ccub

e fc

pr

f c

,pri

sm

[p

si →

MPa

] pr

ism

str

engt

h of

con

cret

e

PKpr

is

Prüf

körp

er

[in

→ m

m]

dim

ensi

on o

f pr

ism

s

f1cp

r

f 1

c,pr

is

[M

Pa]

un

iaxi

al c

ompr

. str

engt

h de

rive

d fr

om f

cpr

f1c

f 1c

[M

Pa]

un

iaxi

al c

ompr

. str

engt

h of

con

cret

e

met

hod

[ -

]

test

ing

met

hod

(cyl

; cu;

pr)

Bet

onzu

gfes

tigke

it (c

oncr

ete

tens

ile

stre

ngth

)

fctf

l

f c

t,fl

[p

si →

MPa

] m

odul

us o

f ru

ptur

e

PKfl

Pr

üfkö

rper

[i

n →

mm

] di

men

sion

of

cont

rol s

peci

men

f1ct

fl

f 1ct

,fl

[M

Pa]

ax

ial t

ensi

le s

tren

gth

deri

ved

from

fct

fl

fcts

p

f c

t,sp

[p

si →

MPa

] sp

litt

ing

tens

ile

stre

ngth

PKsp

Pr

üfkö

rper

[i

n →

mm

] di

men

sion

of

cont

rol s

peci

men

f1ct

sp

f 1

ct,s

p

[MP

a]

axia

l ten

sile

str

engt

h de

rive

d fr

om f

ctsp

f1ct

test

f 1

ct,te

st

[M

Pa]

te

st v

alue

for

axi

al te

nsil

e st

reng

th

met

hod

[ -

]

cont

rol t

est f

lexu

re o

r sp

lit

beta

ctte

st

β

ct,te

st =

f1c

t,tes

t / f

1c

[

- ]

ra

tio

f1ct

mca

l

f 1ct

m,c

al

[MP

a]

calc

ulat

ed v

alue

of

axia

l ten

sile

str

engt

h

beta

ctca

l

βct

,cal =

f1c

tm,c

al /

f1c

[ -

]

rati

o

mec

hani

cal r

einf

orce

men

t rat

ios

oms

1csys

sf

db

fA

⋅⋅

⋅=

[

- ]

m

ech.

rei

nf. r

atio

of

rein

f. s

teel

omp

1cpybo

tp,

pf

db

fA

⋅⋅

⋅=

[

- ]

m

ech.

rei

nf. r

atio

of

pres

tr. s

teel

oml

p

sl

+=

[ -

]

mec

h. r

einf

. rat

io o

f te

nsio

n ch

ord

Atta

chm

ent 3

-1:

Not

atio

n an

d fo

rmul

ary

for

the

data

base

s vu

ct-P

C-D

S an

d vu

ct-P

C-D

K_s

l for

p.c

.- b

eam

s w

ithou

t stir

rups

sub

ject

ed to

poi

nt lo

ads

-6-

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

– P

art 3

Rei

neck

, K.-

H.;

Kuc

hma,

D. A

.; F

itik

, B.

– U

nive

rsit

y of

Stu

ttga

rt a

nd U

nive

rsit

y of

Ill

inoi

s

D

ec 2

008

Ver

such

(te

st v

alue

s)

g

24

Ag

s⋅

=

[kip

/in→

kip/

ft→

kN/m

] se

lf w

eigh

t

Vg

))x

(ac

(0,5

gV

rg

−+

⋅⋅

=

[kip

→ k

N]

shea

r fo

rce

due

to s

elf

wei

ght

F

F

[kip

→ k

N]

failu

re lo

ad

Vu,

f

f

u,V

[kip

→ k

N]

shea

r fo

rce

at f

ailu

re w

ithou

t sel

f w

eigh

t

Vu,

Rep

Rep

u,V

[kip

→ k

N]

shea

r fo

rce

at f

ailu

re w

ith s

elf

wei

ght (

from

repo

rt)

Vu,

g+f

f

gu,

V+

[kip

→ k

N]

shea

r fo

rce

at f

ailu

re w

ith s

elf

wei

ght

beta

r

r

[ °

]

crac

k an

gle

xr,m

ess

x r

,mea

s

[ -

]

mea

sure

d di

stan

ce o

f cr

ack

from

sup

port

axi

s

xr

rx

[ -

]

calc

ulat

ed d

ista

nce

of c

rack

fro

m s

uppo

rt a

xis

ssla

a

sl,

[ -

]

stee

l str

ess

near

end

sup

port

xsla

a

sl,

x

[

- ]

di

stan

ce f

rom

sup

port

axi

s

vxsl

a

v x

sl

[-

]

load

at m

easu

red

stee

l str

ess

br

type

of

failu

re

bem

re

mar

ks

konx

kon

61

konx

1

----

----

----

----

----

----

en

d of

dat

abas

e vu

ct-D

S -

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

----

---

Bru

chsc

hnitt

größ

en (

test

val

ues)

tu,d

db

1000

V

w

Rep

u,d

u,⋅⋅

=

[MP

a]

ulti

mat

e „s

hear

str

ess“

rel

ated

to d

vu,d

1c

w

Rep

u,

1c

du,

du,

fd

bV

f⋅

⋅=

=

[ -

]

dim

ensi

on-f

ree

valu

e of

ulti

mat

e sh

ear

forc

e

Atta

chm

ent 3

-1:

Not

atio

n an

d fo

rmul

ary

for

the

data

base

s vu

ct-P

C-D

S an

d vu

ct-P

C-D

K_s

l for

p.c

.- b

eam

s w

ithou

t stir

rups

sub

ject

ed to

poi

nt lo

ads

-7-

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

– P

art 3

Rei

neck

, K.-

H.;

Kuc

hma,

D. A

.; F

itik

, B.

– U

nive

rsit

y of

Stu

ttga

rt a

nd U

nive

rsit

y of

Ill

inoi

s

D

ec 2

008

tu

z

b

1000

V

w

Rep

u,u

⋅⋅=

[M

Pa]

ul

tim

ate

„she

ar s

tres

s“ r

elat

ed to

z

vuu

1c

w

Rep

u,

1cu

uf

zb

Vf

⋅⋅

==

[ -

]

dim

ensi

on-f

ree

valu

e of

ult

imat

e sh

ear

forc

e

Mu

1000

aV

MR

epu,

u

⋅=

[k

Nm

]

max

.mom

ent a

t fai

lure

muu

1c2

uu

fd

bM

⋅⋅

=

[ -

]

dim

ensi

on-f

ree

valu

e of

ult

imat

e m

omen

t

tute

st

te

stwRep

u,te

stu,

zb

1000

V

⋅⋅=

[M

Pa]

ul

tim

ate

„she

ar s

tres

s“ r

elat

ed to

zte

st

vute

st

1c

test

w

Rep

u,

1ctest

u,te

stu,

fz

bV

f⋅

⋅=

=

[

- ]

di

men

sion

-fre

e va

lue

of u

ltim

ate

shea

r fo

rce

Kon

trol

le B

iegu

ng (

chec

k of

fle

xura

l cap

acity

)

kapc

= 1

- f1

c/25

0

25

0

f1

c1c

−=

κ

[ -

]

coef

fici

ent f

or m

axim

um s

tres

s of

str

ess

bloc

k

epp

ppp

ppE

=

[‰]

st

ress

in p

rest

r. s

teel

due

to p

rest

ress

esy

=sy

fsy

/ E

s [‰

]

yiel

d st

rain

of

rein

forc

ing

stee

l

omgr

d

d0,

4s

cgr

⋅⋅

=

[ -

]

lim

itin

g re

info

rcem

ent r

atio

zeta

1 =

1-o

ml/k

apc/

2

(om

l<om

gr)

cl

κωζ

*21

11

−=

[

- ]

fa

ctor

for

inne

r le

ver

arm

z1

acal

1 no

rei

nfor

cing

ste

el:

1,

0a ca

l1=

[

- ]

au

xili

ary

fact

or

w

ith

rein

forc

ing

stee

l:

sypb

ot

ss

pypca

l1d

da

⋅⋅+

=

[ -

]

bcal

1 no

rei

nfor

cing

ste

el:

3,

5b

ppca

l1+

=

[

- ]

au

xili

ary

fact

or

Atta

chm

ent 3

-1:

Not

atio

n an

d fo

rmul

ary

for

the

data

base

s vu

ct-P

C-D

S an

d vu

ct-P

C-D

K_s

l for

p.c

.- b

eam

s w

ithou

t stir

rups

sub

ject

ed to

poi

nt lo

ads

-8-

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

– P

art 3

Rei

neck

, K.-

H.;

Kuc

hma,

D. A

.; F

itik

, B.

– U

nive

rsit

y of

Stu

ttga

rt a

nd U

nive

rsit

y of

Ill

inoi

s

D

ec 2

008

w

ith

rein

forc

ing

stee

l:

−⋅

⋅⋅

++

⋅=

1dd

23,

53,

5b

pbots

sys

py

ppp

cal1

[

- ]

ccal

1 no

rei

nfor

cing

ste

el:

p

pyc

ppca

l1

3,5

3,5

c⋅

⋅−

⋅=

[

- ]

au

xili

ary

fact

or

w

ith

rein

forc

ing

stee

l:

d

d3,

51

dd3,

53,

5c

pbot

cpb

ots

sy

2

spy

ppp

cal1

⋅⋅

−−

⋅⋅

+⋅

⋅=

[

- ]

ed

[‰]

st

rain

in p

rest

r. s

teel

es

[‰]

st

rain

in r

einf

. Ste

el

delta

ep

ca

l1

cal1

cal1

2ca

l1ca

l1p

a2

ca

4b

b

⋅

⋅⋅

−+

−=

[‰

]

stra

in o

f pr

estr

. ste

el

xsi

no

rei

nfor

cing

ste

el:

p

3,53,

5

+=

[

- ]

fa

ctor

for

dep

th o

f co

mpr

essi

on z

one

wit

h re

info

rcin

g st

eel:

p

pbot

3,53,

5d

d

+=

[

- ]

zeta

2

21

12

⋅−

=

[

- ]

fa

ctor

for

z2

zeta

[

- ]

fa

ctor

for

z

z_

dz

⋅=

[m

m]

in

ner

leve

r ar

m

muf

lex1

1

lfl

ex1

u,⋅

=

[ -

]

dim

ensi

onsf

ree

mom

ent

muf

lex2

no

rei

nf. S

teel

:

2py

ppp

lfl

ex2

u,⋅

+⋅

=

[ -

]

dim

ensi

onsf

ree

mom

ent

wit

h re

inf.

ste

el:

(

)⋅

−⋅

−+

⋅⋅

+⋅

−⋅

+⋅

=21

dd3,

53,

5dd

21d

ds

ppb

ots

syspb

ot

py

ppp

pfl

ex2

u,

muf

lex

flex

u,

[

- ]

di

men

sion

-fre

e m

omen

t at f

lexu

ral f

ailu

re

Atta

chm

ent 3

-1:

Not

atio

n an

d fo

rmul

ary

for

the

data

base

s vu

ct-P

C-D

S an

d vu

ct-P

C-D

K_s

l for

p.c

.- b

eam

s w

ithou

t stir

rups

sub

ject

ed to

poi

nt lo

ads

-9-

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

– P

art 3

Rei

neck

, K.-

H.;

Kuc

hma,

D. A

.; F

itik

, B.

– U

nive

rsit

y of

Stu

ttga

rt a

nd U

nive

rsit

y of

Ill

inoi

s

D

ec 2

008

Mu_

flex

[ ]

mom

ent a

t fle

xura

l fai

lure

Vu_

flex

[ ]

shea

r fo

rce

at f

lexu

ral f

ailu

re

beta

flex

flex

[ -

]

ratio

of

atta

ined

to c

alc.

mom

ent

Bbe

m

rem

ark

(BB

= f

lexu

ral f

ailu

re)

ccal

no r

einf

, ste

el:

[ -

]

auxi

liary

val

ue

wit

h re

inf.

ste

el:

uc

pppyp

sc

2 pp2 py2 p

cal

2dd

0,96

802

0,96

80c

⋅⋅

+⋅

⋅⋅

⋅⋅

−⋅

⋅=

este

st

cal

cal

cal

2ca

lca

lst

est

a2

ca

4b

b

⋅

⋅⋅

−−

−=

[‰]

st

rain

in r

einf

. ste

el

delt

aept

est

d

d

dd

test

s

test

pbot

stes

tpt

est

⋅−

⋅−

⋅=

[‰

]

stra

in in

pre

str.

ste

el

sigp

no r

einf

, ste

el:

⋅−

−⋅

⋅=

cu

p

pyc

p2

11

f

[MPa

]

stre

ss in

pre

str.

ste

el

wit

h re

inf,

ste

el:

()

ptes

tpp

pp

E+

⋅=

[M

Pa]

st

ress

in r

einf

. ste

el

xsit

est

no

rei

nf. s

teel

: py

c

pp

test

f ⋅⋅=

[

- ]

fa

ctor

for

dep

th o

f co

mpr

essi

on z

one

wit

h re

inf.

ste

el:

pyc

ppp

pyp

sys

c

stes

tte

st0,

9578

⋅⋅+

⋅+

⋅=

[

- ]

zeta

test

te

stte

st21

1⋅

−=

[

- ]

fa

ctor

for

inne

r le

ver

z tes

t

z_te

st

d

zte

stte

st⋅

=

[m

m]

in

ner

leve

r ar

m z

test

Ver

anke

rung

(an

chor

age)

lbvo

rh

for

aa

≠ 0

; ba

≠ 0

:

d)(h

b/2

avo

rhlb

,A

A−

−+

=

[mm

]

prov

ided

anc

hora

ge le

ngth

for

aa

= 0

; ba

≠ 0

:

Ab

vorh

lb,

=

[mm

]

Atta

chm

ent 3

-1:

Not

atio

n an

d fo

rmul

ary

for

the

data

base

s vu

ct-P

C-D

S an

d vu

ct-P

C-D

K_s

l for

p.c

.- b

eam

s w

ithou

t stir

rups

sub

ject

ed to

poi

nt lo

ads

-10

-

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

– P

art 3

Rei

neck

, K.-

H.;

Kuc

hma,

D. A

.; F

itik

, B.

– U

nive

rsit

y of

Stu

ttga

rt a

nd U

nive

rsit

y of

Ill

inoi

s

D

ec 2

008

fo

r a

a ≠

0; b

a =

0:

lb

,vor

h =

aA +

0,1

d

[m

m]

fo

r a

a =

ba

= 0

:

d0,

25vo

rhlb

,⋅

=

[mm

]

Fsa

+−

+⋅

=0,

873

zd

h2,

20za

0,5

VF

A

Rep

u,sa

[k

N]

st

eel f

orce

to b

e an

chor

ed

alph

a

sys

sa

fA

F ⋅=

[-]

ra

tio

ssla

u

ssasl

auAF

=

[M

Pa]

st

eel s

tres

s ne

ar e

nd s

uppp

ort

lber

f1

()

cal

1ctm

,u

sla,

sta

berf

1f

9/

dl

⋅⋅

⋅=

[mm

]

requ

ired

anc

hora

ge le

ngth

lber

f2

(

)ca

l1c

tm,

syst

abe

rf2

f9

/f

dl

⋅⋅

⋅=

[m

m]

re

quir

ed a

ncho

rage

leng

th

beta

lb1

bv

orh

berf

2

bvor

h

berf

1lb

1ll

bzw

.ll

=

[

- ]

ra

tio o

f re

quir

ed to

pro

vide

d an

chor

age

leng

th

Fsav

orh

lb

1

sys

lb1

savo

rhsa

,

fA

bzw

.F

F⋅

=

[kN

]

prov

ided

tens

ion

forc

e at

end

sup

port

delt

aFsa

,p

vo

rhsa

,sa

vorh

sa,

FF

F−

=

[kN

]

diff

eren

ce f

orce

spau

for

Pre:

psa

pau

AF=

[MPa

]

stre

ss in

pre

str.

ste

el d

ue to

ΔFs

t

for

Pos

t:

p

sapa

uA

PF

+=

[M

Pa]

lber

f3

fo

r S

WS

: (

)pa

upp

cal

1ctm

,

spbe

rf0,

80,

5f

0,55

4

dl

⋅+

⋅⋅

⋅⋅

=

[m

m]

re

quir

ed a

ncho

rage

leng

th

Atta

chm

ent 3

-1:

Not

atio

n an

d fo

rmul

ary

for

the

data

base

s vu

ct-P

C-D

S an

d vu

ct-P

C-D

K_s

l for

p.c

.- b

eam

s w

ithou

t stir

rups

sub

ject

ed to

poi

nt lo

ads

-11

-

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

– P

art 3

Rei

neck

, K.-

H.;

Kuc

hma,

D. A

.; F

itik

, B.

– U

nive

rsit

y of

Stu

ttga

rt a

nd U

nive

rsit

y of

Ill

inoi

s

D

ec 2

008

fo

r ot

her

case

s:

()

pau

ppca

l1c

tm,

spbe

rf1,

00,

7f

0,64

14

dl

⋅+

⋅⋅

⋅⋅

=

[mm

]

requ

ired

anc

hora

ge le

ngth

lber

f4

()

cal

1ctm

,pa

usp

abe

rf1

f9

/d

l⋅

⋅⋅

=

[mm

]

requ

ired

anc

hora

ge le

ngth

beta

lb =

lber

f3/lb

vorh

bzw

. lbe

rf4/

lbvo

rh

bv

orh

berf

4

bvor

h

berf

3lb

1ll

bzw

.ll

=

[ -

]

ratio

of

requ

. to

prov

. anc

hora

ge le

ngth

rem

ark

VB

re

mar

k: V

B =

anc

hora

ge f

ailu

re

Sel

ekti

onsk

rite

rium

(cr

iter

ia f

or d

ata

sele

ctio

n)

konx

=W

EN

N(O

DE

R(f

1c=

0;fp

y=0;

Vu=

0;P=

0);0

;1)

kon1

=W

EN

N(f

1c>

12;1

;0)

kon2

=W

EN

N(f

1c<

100;

1;0)

kon3

=W

EN

N(b

w>

50;1

;0)

kon4

=W

EN

N(h

>70

;1;0

)

kon5

=W

EN

N(k

ap>

2,89

;1;0

)

kon6

=W

EN

N(k

ap<

2,89

;1;0

)

konx

7

=W

EN

N(r

hol=

0;0;

1)

kon7

=W

EN

N(k

onx7

=0;

0;W

EN

N(r

hol<

3;1;

0))

konx

8

=W

EN

N(o

ml=

0;0;

1)

konx

9

=W

EN

N(b

etaf

lex=

0;0;

1)

kon8

=W

EN

N(O

DE

R(k

onx8

=0;

konx

9=0)

;0;W

EN

N(o

ml*

beta

flex

<0,

4;1;

0))

kon9

=W

EN

N(k

onx9

=0;

0;W

EN

N(b

etaf

lex<

1;1;

0))

kon9

1

=W

EN

N(b

etaf

lex>

1;W

EN

N(b

etaf

lex<

=1,

1;1;

0);0

)

kon1

0

=W

EN

N(f

R=

"0";

0;1)

konx

11

=W

EN

N(b

etal

b=0;

0;1)

kon1

1

=W

EN

N(k

onx1

1=0;

0;W

EN

N(b

etal

b<1;

1;0)

)

KO

NA

0

= k

on1

⋅ kon

3 ⋅ k

on4

⋅ kon

8 ⋅ k

on10

⋅ ko

n11

KO

NA

11

= K

ON

A0

⋅ kon

5 ⋅ k

on7

⋅ kon

9

Atta

chm

ent 3

-1:

Not

atio

n an

d fo

rmul

ary

for

the

data

base

s vu

ct-P

C-D

S an

d vu

ct-P

C-D

K_s

l for

p.c

.- b

eam

s w

ithou

t stir

rups

sub

ject

ed to

poi

nt lo

ads

-12

-

Res

earc

h R

epor

t on

exte

nded

she

ar d

atab

ases

– P

art 3

Rei

neck

, K.-

H.;

Kuc

hma,

D. A

.; F

itik

, B.

– U

nive

rsit

y of

Stu

ttga

rt a

nd U

nive

rsit

y of

Ill

inoi

s

D

ec 2

008

KO

NA

12

= K

ON

A0

⋅ kon

5 ⋅ k

on7

⋅ kon

91

KO

NA

2

= K

ON

A0

⋅ kon

5 ⋅ k

on9

KO

NA

3

= K

ON

A0

⋅ kon

6 ⋅ k

on9

KO

NA

2a

= K

ON

A0

⋅ kon

9

KO

NA

2b

= K

ON

A2a

⋅ ko

n5

KO

NA

2c

= K

ON

A0

⋅ kon

5

KO

NA

2d

= K

ON

A2c

⋅ ko

n9

Um

rech

nung

sfak

tore

n (c

onve

rsio

n fa

ctor

s)

1 in

ch

=

25,

4 m

m

1 po

und

=

4,4

48 N

1 ki

p

=

4,4

48 k

N

1 kl

bf f

t

= 1

,36

kNm

1 ps

i

=

1/1

45 *

MPa

1 ks

i

=

100

0/ 1

45 *

MPa

1 kp

=

9,8

1 N

1 kp

/cm

2

= 9

,81/

100

MPa

= 9

,81/

100

N/m

m2

Vor

span

nart

(ty

pe o

f pr

estr

essi

ng)

SWS/

270

=

Se

ven-

Wir

e St

rand

(fp

k =

270

ksi

)

SWS/

250

=

Se

ven-

Wir

e St

rand

(fp

k =

250

ksi

)

TFW

S

=

T

hree

- an

d Fo

ur-

Wir

e St

rand

(fp

k =

250

ksi

)

PW

=

P

rest

ress

ing

Wir

e

SPB

/145

=

Smoo

th P

rest

ress

ing

Bar

s (f

pk =

145

ksi

)

SPB

/160

=

Smoo

th P

rest

ress

ing

Bar

s (f

pk =

160

ksi

)

DP

B

=

Def

orm

ed P

rest

ress

ing

Bar

s

Research Report on extended shear databases – Part 3

Attachment 3-2: References for collection database vuct-PC-DS

Arthur, P.D. (1965): the shear strength of pre-tensioned I beams with unreinforced webs. Univ. Of Glasgow, Magazine of

concrete research: Vol. 17, No. 53, Dec. 1965



Cederwall, K.; Hedman, O.; Losberg, A. (1974): Shear strength of partially prestressed beams with pretensioned reinforcement of high grade deformed bars. SP 42 - 9

Elzanaty, A.H.; Nilson, A.H.; Slate, F.O. (1985): Shear Critical High-Strength Concrete Beams. Research Rep. No. 85-1, Dept. of Struct. Eng., Cornell University, Ithaca, Febr. 1985, 216pp

Evans, R.H.; Schuhmacher, E.G. (1963): Shear Strength of Prestressed Beams without Web Reinforcement. Journal of the American Concrete Institute, Vol. 60, (Nov. 1963), No. 11, 1621-1642

Giliberti, A.; Radogna, E. F.; Ermolli, E. R. (1966): Esperienze sul comportamento a taglio di travi precompresse a cavi post-tesi. Estratto da “L’ Industria Italiana del Cemento” Gruppo III, November, 1966

Hicks, A. B. (1958): The influence of shear span and concrete strength upon the shear resistance of a pre-tensioned prestressed concrete beam. Uni. of London, Imperial College of science and Technology, 1958

Kar, J.N. (1968): Diagonal cracking in prestressed concrete beams. Journal of the Structural Division; January 1968, page 83 - 109.

Kar, J.N. (1969): Shear strength of prestressed concrete beams without web reinforcement. Magazine of Concrete Research, Vol. 21, No. 68, September 1969

MacGregor, J.G. (1958): Effect of draped reinforcement on behavior of prestressed concrete beams. Civil engineering studies, Structural research series No. 154, Uni. of Illinois, May 1958

Mahgoub, M.O. (1975): Shear strength of prestressed concrete beams without web reinforcement. Magazine of Concrete Research Vol. 27 (1975), No. 93, December, 219-228

Radogna, E. F. (1962): Esperienze di rottura al taglio su travi a doppio t con armatura pre-tesa. Nota I: Rapporto sulla prima serie di diciotto travi. Universita`degli studi di Roma, Instituto di scienza delle costruzioni, No. II-48, Roma-Napoli, 1962

Radogna, E. F. (1962): Esperienze di rottura al taglio su travi a doppio t con armatura pre-tesa. Nota II: Rapporto sulla seconda serie di diciannove travi. Universita`degli studi di Roma, Instituto di scienza delle costruzioni, No. II-49, Roma-Napoli, 1962




Nielsen, M.P.; Bræstrup, M.W. (1978): Shear strength of prestressed concrete beams without web reinforcement. University of Denmark, 1978

Olesen, S.O.; Sozen, M.A.; Siess, C.P. (1967): Investigation of prestressed reinforced concrete for highway bridges, part IV: Strength in shear of beams with web reinforcement. University of Illinois, Bulletin No. 493, V.64, No.134, July 5, 1967

Shahawy, M.A.; Batchelor, B. (1996): Shear Behavior of Full- Scale Prestressed Concrete Girders: Comparison between AASHTO Specifications and LRFD Code. PCI Journal, Vol. 41 (1996), No. 3, May/June, 72-93

Attachment 3-2: References for collection database vuct-PC-DS 2

Research Report on extended shear databases - Part 3 Reineck; Kuchma; Fitik Dec 2008

Sozen, M.A.; Zwoyer, E.M.; Siess, C.P. (1959): Strength in shear of beams without web reinforcement. University of Illinois Bulletin No. 452, V.56, No.62, April 1959

Teng, S.; Kong, F.K.; Poh, S.P. (1998): Shear strength of reinforced and prestressed concrete deep beams. Part 1: current design methods and a proposed equation. Proc. Instn. Civil Engineers. Structures & Buildings (1998), No. 128, May, Paper No. 11196, 112-123

Teng, S.; Kong, F.K.; Poh, S.P. (1998): Shear strength of reinforced and prestressed concrete deep beams. Part 2: the supporting evidence. Proc. Instn. Civil Engineers. Structures & Buildings (1998), No. 128, May, Paper No. 11197, 124-143

Zwoyer, E.M. (1953): Shear Strength of Simply Supported Prestressed Concrete Beams. University of Illinois, Doctor Thesis (1953)

official exhibit - ner050-00-bd01 - reineck et al ...tests on structural concrete beams without and...

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