Eindhoven University of Technology MASTER Bending moment, shear and normal force interaction of I-shaped steel cross-sections Hamidi, S.

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Eindhoven University of Technology

MASTER

Bending moment, shear and normal force interaction of I-shaped steel cross-sections

Hamidi, S.

Award date:

2018

Link to publication

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Where innovation starts

Bending Moment, Shear and Normal Force Interaction of I-shaped Steel

Cross-sections

____________________________________________________

Assessment of the cross-sectional design rules by means of a numerical and statistical evaluation

Author

S. (Shams) Hamidi Silenenstraat 24 5212 XH Den Bosch 0785449

Supervisors:

prof. ir. H.H. (Bert) Snijder ir. R.W.A. (Rianne) Dekker dr. ir. P.A. (Paul) Teeuwen

Date July 2018 Our Reference A-2018-228

______________________________________________________________________________________

Department of the Built Environment Structural Design

Den Dolech 2, 5612 AZ Eindhoven P.O. Box 513, 5600 MB Eindhoven The Netherlands

www.tue.nl

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1

Where innovation starts

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2 Preface

This research report is written in the context of my study Structural Design at Eindhoven University of Technology as final thesis for the master program Architecture Building and Planning. This research project focuses on the resistance of a steel cross-section subjected to the combination of bending, shear and normal force with the aim to assess the cross-sectional design rules for I-shaped cross-sections regarding M-N-V interaction by means of a numerical and statistical evaluation.

This master research project is supervised by prof.ir. H.H. (Bert) Snijder, Professor of Steel Structures at Eindhoven University of Technology; ir. R.W.A. (Rianne) Dekker, TU/e doctoral candidate (PhD) for the assessment of cross-sectional design rules regarding ductile failure modes and dr.ir. P.A. (Paul) Teeuwen, structural engineer at Witteveen+Bos.

I would like to thank my supervisors: prof.ir. H.H. Snijder, ir. R.W.A. Dekker and dr.ir. P.A. Teeuwen for offering me this research opportunity and for their supervision during this project.

S. Hamidi

Eindhoven, July 2018

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3 Abstract

To ensure the safety of a structure it is important to determine its structural resistance. A single force or multiple forces can work on structural elements. The Eurocode provides rules to determine the resistance of cross-sections for bending moment, shear force, axial forces and all possible combinations of these. The design rules for steel cross-sections are described in NEN-EN 1993-1-1 [1].

Because the cross-sections subjected to multiple internal forces could react differently than predicted by theory, which is mostly based on mechanics, a reassessment of the design rules is necessary to ensure safety. The aim of this research project is as follows:

This research project focuses on the resistance of a cross-section subjected to the combination of bending, shear and normal force with the aim to assess the cross-sectional design rules for I-shaped cross-sections regarding M-N-V interaction by means of a numerical and statistical evaluation. As a result, the magnitude of the partial safety factor 𝛾𝑀0 will be reassessed.

An analytical solution is proposed for an I-shaped cross-section subjected to combined bending, shear force and normal force. The calculation is done by moving the neutral axis over the cross- section. The position of the neutral axis has been taken into account which can be situated in the web, in the roots at the intersection of web and flang or in the flange.

An M-N-V model is made for the numerical simulations in Abaqus which represents a four-point bending test. In the middle of the beam a symmetry axis is applied to reduce the calculation time.

The used solution method is “Static Riks” with element C3D8R. For the important part of the beam, the mesh size L = 4 mm is chosen. Because this research project focuses on class 1 and 2 cross- sections, non-linearity (NL GEOM) is ON. The buckling length is checked for large lengths of the beam where the relative slenderness should be smaller than 0.2 according the Eurocode [1]. In the case of a short beam, the distance between the vertical force and the support point, should be larger than 1.5 times the height of the beam. In order to get a big moment on the beam, a constant moment is added at the end of the beam. Extra step is used to enable calculation for higher ratios nV.

To investigate the influence of varying the geometrical properties and material properties of I-shaped steel cross-sections, a parametric study is performed. The validated numerical model is used to generate a database of numerical simulations. The analytical solution is conservative compared with the numerical results obtained from Abaqus. Only for small shear force and normal force, the analytical solution is less conservative.

For the statistical evaluation, the procedure of Annex D of EN 1990-1-1 [15] is used to determine the partial safety factor 𝛾_𝑀0. All statistical results do not exceed the acceptance limit proposed by the RFCS-project Safebrictile. From the results of statistical evaluation it can be concluded that the partial safety factor 𝛾𝑀,𝑡𝑎𝑟𝑔𝑒𝑡 = 1.00 for the investigated I-shaped cross-sections with steel grade S235, S355 and S460 is acceptable.

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4 Samenvatting

Om de veiligheid van een constructie te waarborgen, is het belangrijk om de constructieve weerstand ervan te bepalen. Een enkele kracht of meerdere krachten kunnen gelijktijdig werken op de constructie-elementen. De Eurocode biedt regels om de weerstand te bepalen van de doorsneden voor buigend moment, dwarskracht, axiale krachten en alle mogelijke combinaties hiervan. De ontwerpregels voor stalen dwarsdoorsneden worden beschreven in NEN-EN 1993-1-1 [1].

Omdat de doorsneden onderworpen aan meerdere interne krachten anders zouden kunnen reageren dan voorspeld door de theorie, die grotendeels gebaseerd is op mechanica, is een herbeoordeling van de ontwerpregels noodzakelijk om de veiligheid te garanderen. Het doel van dit onderzoek is als volgt:

Dit onderzoeksproject concentreert zich op de weerstand van een dwarsdoorsnede onderworpen aan de combinatie van buiging, dwarskracht en normaalkracht met als doel de weerstandsregels voor I- vormige doorsneden betreffende M-N-V interactie te bepalen door middel van een numerieke en statistische evaluatie. Als resultaat zal de grootte van de partiële veiligheidsfactor 𝛾_𝑀0 opnieuw worden beoordeeld.

Een analytische oplossing is voorgesteld voor een I-vormige doorsnede onderworpen aan gecombineerde buiging, dwarskracht en normaalkracht. De berekening wordt uitgevoerd door de neutrale as over de doorsnede te verschuiven. Er is rekening gehouden met de positie van de neutrale as die zich in het lijf, in de afrondingstraal of in de flens kan bevinden.

Voor de numerieke simulaties is een M-N-V-model in Abaqus gemaakt, dat een vierpuntsbuigproef representeert. In het midden van de balk wordt een symmetrie as gebruikt om de rekentijd te verkorten. De gebruikte oplossingsmethode is "Static Riks" met element C3D8R. Voor het belangrijkste deel van de balk wordt de mesh met L = 4 mm gekozen. Omdat dit onderzoek zich richt op klasse 1 en 2 doorsneden, is non-lineariteit (NL GEOM) AAN. De kniklengte wordt gecontroleerd bij de grote lengtes van de balk waar de relatieve slankheid kleiner zou moeten zijn dan 0.2 volgens de Eurocode [1]. In het geval van een korte balk moet de afstand tussen de verticale kracht en de steunpunt groter zijn dan 1.5 keer de hoogte van de balk. Om een groot moment op de balk te krijgen, wordt aan het uiteinde van de balk een constant moment toegevoegd. Extra stap wordt gebruikt om berekeningen met hogere verhoudingen van nV mogelijk te maken.

Om de invloed van variërende geometrische eigenschappen en materiaaleigenschappen van I- vormige stalen doorsneden te onderzoeken, wordt een parametrische studie uitgevoerd. Het gevalideerde numerieke model wordt gebruikt om een database met numerieke simulaties te genereren. De analytische oplossing is conservatief vergeleken met de numerieke resultaten verkregen met Abaqus. Alleen voor kleine dwarskracht en normaalkracht is de analytische oplossing minder conservatief.

Voor de statistische evaluatie wordt de procedure van bijlage D van EN 1990-1-1 [15] gebruikt om de partiële veiligheidsfactor 𝛾_𝑀0 te bepalen. Alle statistische resultaten overschrijden de door het RFCS- project Safebrictile voorgestelde acceptatielimiet niet. Uit de resultaten van statistische evaluatie kan worden geconcludeerd dat de partiële veiligheidsfactor 𝛾𝑀,𝑑𝑜𝑒𝑙= 1.00 voor de onderzochte I-vormige doorsneden met staalsoort S235, S355 en S460 aanvaardbaar is.

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4. Numerical investigation ... 40

4.1 Validation of the FE model (M-V) ... 40

4.1.1 Numerical test set-up ... 40

4.1.2 Load and boundary conditions ... 41

4.1.3 Validation numerical model by experimental results ... 42

4.1.4 Mesh study ... 43

4.1.5 Element study ... 46

4.2 M-N-V models ... 48

4.2.1 M-N-V model ... 48

4.2.2 Two intermediate models ... 49

4.3 Modified M-N-V model ... 50

4.3.1 Simulation set-up M-N-V ... 50

4.3.2 Loads and Boundary conditions ... 50

4.3.3 Beam length ... 52

4.3.4 Order effects... 53

4.4 Conclusions regarding FE analyses ... 54

5. Parametric study ... 55

5.1 Resistance functions and parameters ... 55

5.2 Numerical test sets ... 58

5.2.1 Test set regarding the utilization ratio n ... 58

5.2.2 Test set regarding the type of cross-section ... 59

5.2.3 Test set regarding the steel grade ... 61

5.3 Numerical results from Abaqus ... 63

5.3.1 Numerical results of HEA240 cross-section... 63

5.3.2 Numerical results of HEB200 cross-section ... 66

5.3.3 Numerical results of HEM160 cross-section ... 68

5.3.4 Numerical results of IPE330 cross-section ... 70

5.3.5 Conclusion regarding numerical results ... 72

6. Statistical evaluation ... 73

6.1 Methodology of Annex D ... 73

6.2 Statistical results ... 76

6.3 Conclusions regarding statistical results ... 79

7. Conclusions & Recommendations ... 80

7.1 Conclusions ... 80

7.2 Recommendations ... 81

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7

References ... 82

APPENDIX ... 83

Appendix A: Python script for Abaqus ... 84

Appendix B: Spreadsheet analytical solution ... 96

Appendix C: Length of the beam (four-point bending test) ... 104

Appendix D: Equilibrium check ... 105

Appendix E: Script Matlab ... 109

Appendix F: Statistical results ... 112

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8 Nomenclature

aw ratio between area of the web and total area of a cross-section = (A - 2btf)/A < 0.5 -

A total area of the cross section mm²

Af area of the flanges (2 ∙ 𝑏 ∙ 𝑡𝑓) mm²

Ar area of the roots (4𝑟²(1 − 0.25𝜋)) mm²

AV shear area mm²

Aw area of the web (𝑡𝑤∙ ℎ𝑤) mm²

𝐴_𝑟(𝛽) surface area of the roots which is determined by the distance β mm²

b width of the cross-section mm

b correction factor -

c height of the cross-section minus the flanges and the radii (in case of internal parts) mm

d depth measured between the roots mm

fy yield strength N/mm²

fy,r reduced yield strength N/mm²

h overall depth of a cross-section mm

hf depth measured between the center of the flanges mm

hw clear depth measured between flanges mm

hv unknown distance in y-direction where shear force is applied mm

I second moment of area mm⁴

My,Ed design value of the bending moment about the y-axis kNm

My,V,Rd reduced bending resistance due to shear about the y-axis kNm

Mpl,y,Rd plastic design resistance for bending about strong axis of the cross-section kNm

MN,V,y,Rd the design resistance of the cross-section to combination of moment and axial force about

the y-axis kNm

n utilization ratio -

NEd design value of the axial force kN

Npl,y,Rd design plastic resistance to normal force of the gross cross-section kN

NVz,Rd reduced normal force resistance due to shear about the z-axis kN

r root radius mm

𝑟_𝑑,𝑖 the design values of the resistance kNm

rt,i theoretical resistance values kNm

re,i experimental resistance values kNm

S first moment of area at that location mm³

𝑠∆2 standard deviation -

t plate thickness at location of shear stress determination mm

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9

tf flange thickness mm

tw web thickness mm

X basic variables m or kN/m²

V_Ed design value of the shear force kN

Vpl,y,Rd plastic resistance of shear force kN

𝑉_𝑟 coefficient of variation -

𝑉𝑟,𝑡2 the coefficient of variation with respect to the theoretical resistance function -

𝑉_𝛿 coefficient of variation of the error term -

Wel elastic section modulus mm³

WM plastic section modulus regarding the bending moment mm³ WN plastic section modulus regarding the normal force mm³ WNrec plastic section modulus of the rectangular regarding normal force mm³

Wpl,y plastic section modulus mm³

Wpl,y,Av plastic section modulus at the area where shear forces are present in the cross-section mm³

Wpl,y,r addition of the four roots to the plastic section modulus mm³

Wrec plastic section modulus of the rectangular mm³ 𝑊𝑟(𝛽) plastic section modulus of the roots in the distance Β , see Figure 3.13 mm³

𝑊_{𝑟,𝑡𝑜𝑡} plastic section modulus of the total roots mm³

Wv plastic section modulus regarding the shear force mm³

α shape factor -

Β distance in the roots where shear force is applied mm

𝛾_{𝑡𝑎𝑟𝑔𝑒𝑡} desired partial safety factor -

𝛾_𝑀∗ required partial safety factor for resistance of cross-sections - 𝛾_𝑀0 partial safety factor for resistance of cross-sections - δ vertical distance in the flange where the shear force is applied mm

δi the error terms of simulation i -

∆𝑖 the natural logarithm of the error terms -

∆𝑚𝑒𝑎𝑛 the estimated mean value of the error terms -

ᵨ

reduction factor due to shear -

σM,Ed design value of the normal stresses due to the bending moment [N/mm²]

τEd design value of the elastic shear stress [N/mm²]

σN,Ed design value of the normal stresses due to the axial force [N/mm²]

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10 1. Introduction

1.1 Problem definition

To ensure the safety of a structure it is important to determine its structural resistance. By means of structural design rules the structural safety can be ensured in the built environment. A single force or multiple forces can work simultaneously on the structural elements, see Figure 1.1.

Figure 1.1 Single force and multiple forces

The Eurocode provides rules to determine the resistance of cross-sections for bending moment, shear force, axial forces and all possible combinations of these. The design rules for steel sections are described in NEN-EN 1993-1-1 [1].

Because the cross-sections subjected to multiple internal forces could react differently than predicted by theory, which is mostly based on mechanics, a reassessment of the design rules is necessary to ensure safety.

1.2 Goal

This research project focuses on the resistance of a cross-section subjected to the combination of bending, shear and normal force with the aim to assess the cross-sectional design rules for I-shaped cross-sections regarding M-N-V interaction by means of a numerical and statistical evaluation. As a result, the magnitude of the partial safety factor 𝛾𝑀0 will be reassessed.

1.3 Scope

In this research project the cross sectional design rules regarding M-N-V interaction are limited to class 1 and 2 cross-sections and it is aimed at plastic modes of the cross-section without interference of stability issues like local buckling. This project starts with only I-shaped and H-shaped beams of steel quality S235. Later also higher steel grades S355 and S460 are a part of this research.

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11 1.4 Approach

This research report is divided in seven chapters which are explained per chapter.

Chapter 2: Literature study

In this chapter the available literature is discussed, which is used for this research project. In this chapter also the design rules of steel sections are described from de Eurocode [1].

Chapter 3: Analytic derivation of the a solution

Because de Eurocode does not provide explicit design rules for de combination of bending moment, normal force and shear force, this chapter describes an analytical derivation of a solution.

Chapter 4: Numerical investigation

In this chapter the Finite Element (FE) Model is described which is used for this research. This chapter also includes the validation of the FE-model, the mesh study and the element study.

Chapter 5: Parametric study

To investigate the influences of varying the geometrical properties and material properties of I- shaped steel cross-sections, a parametric study is performed. The validated numerical model is used to generate a database of test results. The test results are shown in this chapter.

Chapter 6: Statistical evaluation

For the statistical evaluation the procedure of Annex D of EN 1990-1-1 [15] is used to determine the partial safety factor 𝛾𝑀0.

Chapter 7: Conclusions & Recommendations

In the last chapter, the conclusions and recommendations regarding M-N-V interaction by means of a numerical and statistical evaluation are described.

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12 2. Literature survey

In this chapter the bending moment, shear and normal force interaction (M-N-V) is introduced. After that, the elastic and plastic theory is explained and the cross-sectional classification according Eurocode [1] is described. Finally the proposals for design rules are summarized.

2.1 Bending, shear and axial force interaction

A structural system which is loaded by forces induces strains and stresses in its members. Stress distributions and the value of the maximum stress in the critical cross-section are crucial to the verification of the ultimate limit state. The design situation when only one type of internal force is present is seldom. An interaction between different internal forces exists in most cases.

The rules for the determination of the resistance of cross-sections are described in EN 1993-1-1, clause 6.2.10 [1]. These rules are for the combination of bending moment, shear force and normal force.

2.2 Stress distribution

Stress distributions in steel structures can be based on elastic, elasto-plastic and plastic behavior.

Once the bending moment has a value greater than zero, the elastic deformation starts. Increasing the bending moment will lead to yielding of the steel in the outer fibers though the stress distribution remains elastic. Increasing the bending moment even further will lead to yielding of inner fibers and eventually to yielding of the entire cross-section.

The allowed stress distribution of a cross-section is dependent on the profile classification: Class 1 up to Class 4. This investigation has been limited to Class 1 and 2 of I-shaped cross-sections.

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13 2.3 Elastic theory

For Class 1 up to Class 4 cross-sections the elastic theory can be used. If the determination of the cross-sectional resistance is based on an elastic stress distribution, the following 3 conditions need to be met according Krachtwerking [3]:

 the stress distribution in a cross-section should be in equilibrium with internal forces acting on the cross section;

 the yield criterion may not be exceeded at any location in the cross-section;

 the stress-strain relation should be linear.

2.3.1 Equilibrium

This condition requires equilibrium of the stresses, which are integrated over the cross-sectional area and the internal forces.

Cross-section subjected to a normal force:

σ_N,Ed=^N^Ed

A

Cross-section subjected to bending moment:

σ_M,Ed=^M^Ed

W_el 

M_Ed= σ_M,Ed∙ W_el (2.2)

Cross-section subjected to a shear force:

τ_ED=^V^{Ed ∙ S}

I ∙ t  V_Ed=^{I ∙ t}

S ∙ τ_ED (2.3)

Where:

A area of the cross-section mm²

σN,Ed design value of the normal stresses due to the axial force N/mm²

MEd design value of the bending moment about the y-axis kNm

Wel elastic section modulus mm³

σM,Ed design value of the normal stresses due to the bending moment N/mm²

VEd design shear force kN

I second moment of area mm⁴

t plate thickness at location of shear stress determination mm

S first moment of area at that location mm³

τEd design value of the elastic shear stress N/mm²  N_Ed = σ_N,Ed∙ A (2.1)

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14

2.3.2 Yield criterion

The second condition is that the yield criterion may not be exceeded at any location in the cross- section. When regarding an elastic stress distribution for axial stress in one direction, the yield strength should not be exceeded in any fiber. But, when dealing with stress in multiple directions, it gets more complicated. In that case shear stresses may also occur next to the normal stresses, see Figure 2.1.

Figure 2.1 Stresses acting on an infinitesimal fiber particle in (2D) and (3D) [2]

The yield criteria is:

(^σ^x,Ed

f_y/γ_M0)

2

+ (^σ^z,Ed

f_y/γ_M0)

2

− (^σ^x,Ed

f_y/γ_M0) (^σ^z,Ed

f_y/γ_M0) + 3 ( ^τ^Ed

f_y/γ_M0)

2

≤ 1 (2.4)

When γM0= 1.0 the equation can be rewritten as:

√σx,Ed2+ σz,Ed2− σx,Ed∙ σz,Ed+ 3τEd2≤ fy (2.5)

When the combination of only longitudinal and shear stress is present, equation (2.5) can be simplified to equation (2.6).

√σx,Ed2+ 3τ𝐸𝑑≤ fy (2.6)

Absence of the shear stress in equation (2.6) results in equation (2.7) and absence of the normal stresses in equation (2.8)

σ_{x,Ed ≤} 𝑓_𝑦 (2.7)

√3 ∙ τ𝐸𝑑 ≤ fy  τ𝐸𝑑 ≤ ^𝑓^𝑦

√3 (2.8)

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15

2.3.3 Elastic stress distribution

For an elastic stress distribution, the stress-strain relation should be linear, i.e. only the linear elastic part of the steel stress-strain curve may be used, see Figure 2.2 left graph. In this case the relation between the stress and strain is determined by the modulus of elasticity E according to Hooke’s law (σ = E*ε), see left part of Figure 2.2.

Figure 2.2 Linear part stress-strain curve (left), elastic stress distribution by three individual load cases (right) [3]

In case that a structural member is only loaded elastically, the stress distributions are as illustrated by the right picture of Figure 2.2.

2.4 Plastic theory

For Class 1 and 2 cross-sections also the plastic theory can be used. If the determination of the cross- sectional resistance is based on a plastic stress distribution, the following 3 conditions need to be met according to Krachtswerking [3]:

 the stress distribution in a cross-section should be in equilibrium with section forces acting on the cross section;

 the yield criterion may not be exceeded at any location in the cross-section;

 the stresses may be distributed about the cross-section in the most favorable manner, as long as the emerging deformations relate to the plastic stress distribution.

2.4.1 Equilibrium

For any cross-section loaded by an axial force, where equilibrium is met, both the elastic and plastic stress distribution are uniform. Therefore the equation of equilibrium is similar, see equation (2.1).

For a cross-section which is loaded by a bending moment the stresses are uniformly distributed over the section, this results in a higher section modulus (Wpl):

𝜎_{𝑀,𝐸𝑑}=^𝑀^𝐸𝑑

𝑊_𝑝𝑙  𝑀_𝐸𝑑 = 𝜎_{𝑀,𝐸𝑑}∙ 𝑊_𝑝𝑙 (2.9)

Wpl plastic section modulus mm³

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16

Equation (2.10) and (2.11) show the plastic section modulus for I-shaped cross-sections regarding the strong and weak axis.

𝑊_{𝑝𝑙,𝑦} = 𝑏 ∙ 𝑡_𝑓(ℎ − 𝑡_𝑓) +¹₄∙ 𝑡_𝑤(ℎ − 2 ∙ 𝑡_𝑓)² (2.10)

𝑊_{𝑝𝑙,𝑧}=¹₂(𝑏²∙ 𝑡_𝑓) +¹₄∙ 𝑡²_𝑤(𝑑 − 2 ∙ 𝑡_𝑓) (2.11)

b width of a cross-section mm

tf flange thickness mm

h overall depth of a cross-section mm

tw web thickness mm

The plastic moment resistance, Mpl is reached when the yield stress is reached in all fibers. The shape factor α which is the ratio between the plastic and elastic moment resistance, is displayed in equation (2.12).

𝛼 =^𝑀^𝑝𝑙

𝑀_𝑒𝑙 = ^𝑊^𝑝𝑙

𝑊_𝑒𝑙 (2.12)

α shape factor -

Squared section: α = 1.5.

I-shaped cross-section (y-axis): α ≈ 1.15.

In cross-sections loaded by a shear force, the shear stresses should be uniformly distributed over the shear area, see equation (2.13).

𝜏𝐸𝑑 =^𝑉_𝐴^𝐸𝑑

𝑉  𝑉𝐸𝑑 = 𝐴𝑉 ∙ 𝜏𝐸𝑑 (2.13)

𝐴_𝑉 shear area mm²

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17

2.4.2 Yield criterion

The yield criterion may not be exceeded at any location in the cross-section according Krachtswerking [3]. In the case of a plastic stress distribution the normal stresses are maximum equal to the yield strength. The formulas for the yield criterion as previously described are still valid. The relative reduced yield strength by Von Mises can be written as Equation (2.14).

𝑓𝑦,𝑟 = √𝑓𝑦2− 3𝜏_𝐸𝑑² (2.14)

𝑓𝑦,𝑟 reduced yield strength N/mm²

Rewriting this formula results in equation (2.15).

𝑓_𝑦,𝑟

𝑓_𝑦 = √1 −^3𝜏^𝐸𝑑²

𝑓_𝑦² (2.15)

The relative reduced yield strength by Von Mises can be written as equation (2.16). The plastic design shear resistance is equal distributed over the cross-section, see equation (2.17).

𝑓_𝑦,𝑟

𝑓_𝑦 = √1 − ( ^𝑉^𝐸𝑑

𝑉_{𝑝𝑙,𝑅𝑑})

2

(2.16)

𝑉_{𝑝𝑙,𝑅𝑑}= 𝐴_𝑉∙ ^𝑓^𝑦

√3 (2.17)

𝑉𝑝𝑙,𝑅𝑑 plastic design shear resistance kN

By using substitution, the formula can be simplified to equation (2.18). In this simplification the amount of the present shear force regarding the maximum plastic capacity of the cross-section is expressed in the utilization ratio, see equation (2.19).

𝑓_𝑦,𝑟

𝑓_𝑦 = √1 − 𝑛² (2.18)

𝑛 = ^𝑉^𝐸𝑑

𝑉_{𝑝𝑙,𝑅𝑑} (2.19)

𝑛 utilization ratio -

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18

2.4.3 Plastic stress distribution

The last condition states that stresses may be distributed in the most favorable manner within the cross-section, as long as the deformations and plastic stress distribution relate to each other. Figure 2.3 illustrates plasticity in the stress-strain curve by means of a horizontal line, through the yield plateau.

Figure 2.3 Plastic part stress-strain curve [3]

Three cases where single forces are present in the cross-section are displayed in figure 2.4. Every fiber of the cross-section was used, where every fiber reaches the yield strength fy or shear strength τy.

Figure 2.4 Plastic stress distributions by three individual cases [3]

When multiple forces are present instead of a single force, the possible distributions can be shown as figure 2.5. In the case of multiple internal forces the stresses must still remain within the limits of the yield criterion [3].

Figure 2.5 Possible plastic stress distribution in the case of multiple internal forces [3]

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19 2.5 Classification of cross-sections

NEN-EN 1993-1-1, article 5.5 [1] defines the cross-sectional classes based on the capacity to deform plastically and the ability to redistribute the moments. Figure 2.6 shows the classification in a moment-rotation diagram. Cross-sections will be classified by the c/t ratio of the internal (web) and external (flanges) parts.

t thickness of the flange or web mm

c height of the cross-section minus the flanges and the radii (in case of internal parts) mm

Figure 2.6 Moment rotation behavior for I-shaped beams subjected to bending moment [3]

The four classes of cross-sections are:

- Class 1: plastic sections can form plastic hinges with sufficient rotation capacity to redistribute the moments (strain hardening) before local buckling occurs;

- Class 2: compact sections are able to develop the plastic moment resistance, but the rotation capacity is limited because of local buckling;

- Class 3: semi-compact sections are able to reach the elastic moment resistance, while local buckling prevents the development of the plastic moment resistance;

- Class 4: slender sections are not able to attain the yield stress, because local buckling occurs in one or several parts of the cross-section.

Figure 2.7 shows the different calculation models and corresponding classification of cross-sections.

Figure 2.7 Classification of cross-sections [3]

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20 2.6 Code requirements

2.6.1 Eurocode 3: EN 1993-1-1

The Eurocode provides rules to determine the resistance of cross-sections for bending moment, shear force, axial forces and all possible combinations of these. The design rules for steel sections are described in NEN-EN 1993-1-1, article 6.2.10 [1].

“Where shear and axial force are present, allowance should be made for the effect of both shear force and axial force on the moment resistance. If the design value of the shear force VEd does not exceed 50% of the design plastic shear resistance Vpl,Rd, no reduction of the resistances defined for bending and axial force need to be made, except where shear buckling reduces the section resistance.

In the case that VEd exceeds 50% of Vpl.Rd, the design resistance of the cross-section to combinations of moment and axial force should be calculated using a reduced yield strength for the shear surface.”

𝑓_𝑦,𝑟 = (1 − 𝜌)𝑓_𝑦 (2.20)

𝜌 = (^2𝑉^𝐸𝑑

𝑉_{𝑝𝑙,𝑅𝑑}− 1)

2

(2.21)

𝜌 Reduction factor due to shear -

Instead of reducing the yield strength also the plate thickness of the relevant part of the cross- section may be reduced [1].

2.6.2 Dutch National Annex

The Dutch National Annex (NA) of NEN-EN 1993-1-1 article 6.2.10 provides in rules to determine the interaction between bending, shear and normal force. For I-shaped sections of class 1 and 2 the interaction about the strong axis may be checked as follows:

𝑀_{𝑦,𝐸𝑑} 𝑀_{𝑦,𝑉,𝑅𝑑}+

𝑁𝐸𝑑 𝑁𝑉𝑧,𝑅𝑑−^𝑎2

2

1−^𝑎2

2

≤ 1,0 (2.24)

with:

𝑁𝑉𝑧,𝑅𝑑= 𝑁𝑝𝑙,𝑅𝑑 − (𝜌×𝐴𝑣×𝑓𝑦 ) 𝛾𝑀𝑂

(2.25)

𝑎₂= 𝑎₁× (1 − 𝜌) (2.26)

𝑎₁= 𝑚𝑖𝑛 (^{𝐴−(2×𝑏}^𝑓^×𝑡^𝑓⁾

𝐴 ; 0,5) (2.27)

Where:

My,Ed design value of the bending moment about the y-axis kNm

My,V,Rd reduced bending resistance due to shear about the y-axis kNm

NVz,Rd reduced normal force resistance due to shear about the z-axis kN

Npl,Rd design plastic resistance to normal forces of the gross cross-section kN

ᵨ

reduction factor due to shear -

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21

Av shear area mm²

fy yield strength N/mm²

𝛾_𝑀0 partial factor for resistance of cross-sections -

2.7 Results from earlier research

2.7.1 Research by Sherbourne & Oostrom, 1972

The investigation of Sherbourne & Oostrom [4] attempted a plastic analysis of castellated beams by developing a computer model which duplicated behavior under incremental, proportional loading.

Moment, shear and axial force interaction surfaces were developed for critical sections of castellated beams and any other I-section beam with a hole in the web. The sections considered the rectangular and T-section. The interaction curves for these shapes were compared with that for the continuous I- section in order to obtain a clearer understanding of the change deriving from a change in shape.

Furthermore, interaction curves were plotted for a complete range of castellated beams in order to assess the effect of changes in size and profile of the web elements and their attendant discontinuities.

Figure 2.8 Castellated beam [4]

To make a 3-dimensional diagram, the shape of the m-v, the m-n and n-v curves as end points were used. The 3-dimensional interaction diagram is shown in Figure 2.9.

Figure 2.9 Geometrical interaction surface [4]

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22

This research about the castellated beams [4] is not relevant for the current investigation, because old codes where used in the research and the castellated beam differed largely from the I-shaped cross-section. The way of making the 3D-diagram can be used for this research.

2.7.2 Research by Goczek & Supel, 2014

The analytical research of Goczek and Supel [5] includes the resistance of steel cross-sections subjected to bending, shear and axial forces. The aim of their investigation was to analyze the Eurocode 3 rules for the combined bending, shear and axial force interaction. The investigation has been limited to Class 1, 2 and 3 cross-sections. The results showed that the most uncertain design situation arose when bending and axial force were combined with a significant shear force that cannot be neglected. The comparative analysis of interaction diagrams is presented, see Figure 2.10 and 2.11.

Figure 2.10 M-N-V interaction graph – Class 1 and 2 cross-sections[5] Figure 2.11 M-N-V interaction graph – Class 3 cross-sections [5].

The analytical research conducted by Goczek & Supel [5] on the resistance of steel cross-sections subjected to bending, shear and axial forces showed good agreement for Class 1 and 2 cross-sections and poor for Class 3 cross-sections. The pure elastic approach assumed in the Huber-Mises yield criterion, and applied to Class 3 cross-sections excluded partial plastic stress distribution, which is permitted in elastic design by Eurocode 3.

2.7.3 Research by Neal, 1961

Neal, B.G. [6] investigated analytically the effect of shear and normal forces on the fully plastic moment of an I-beam. Therefore a cantilever of I-section subjected to both normal and shear forces at its free end is considered. The results of his investigation are presented in the form of interaction relations between the shear and normal forces and bending moment at the clamped end of the cantilever at failure. The applied forces are shown in Figure 2.12.

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23

Figure 2.12 Cantilever I-beam with applied forces [6]

In this research only two cases were investigated, namely neutral axis in the web and neutral axis in the flange. The roots were not taken into acount. The relation between normal force, shear force and bending moment is shown in Figure 2.13.

Figure 2.13 Interaction relations between M-N-V [6]

The two curves which are indicated with (i) and (ii) are from earlier research where bending moment is limited to be zero. This research about normal force, shear force and bending is not relevant for current investigation, because the roots are not taken into account.

2.8 Conclusion regarding literature survey

The Eurocode provides rules to determine the resistance of cross-sections for bending moment, shear force, axial forces and all possible combinations of these. The design rules of M-N-V for steel sections are described in NEN-EN 1993-1-1, clause 6.2.10 [1]. There are two cases:

 V_Ed≤ 0,5 ∙ V_pl,Rd the shear force (V) can be neglected

 V_Ed> 0,5 ∙ V_pl,Rd the design resistance of the cross-section to combination of moment and axial force should be calculated using a reduced yield strength for the shear area

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24 3. Analytical solution

In this chapter an analytical solution is presented for a rolled I-shaped cross-section subjected to combined bending, shear force and normal force. The cross-section was divided in several parts to cope with the shear force, normal force and bending moment, see Figure 3.1. The center of the cross-section is reserved for the shear force. The remaining parts of the cross-section are reserved for the normal force and bending moment. Dependent on the amount of applied shear force the size of these areas changes. Figure 3.1 (right picture) shows the geometrical parameters of an I-shaped cross-section.

Figure 3.1 Areas reserved for M-N-V (left) and geometrical parameters of a rolled I-shaped cross-section (right)

Where:

Af area of the flanges (2 ∙ 𝑏 ∙ 𝑡𝑓) mm²

Ar area of the roots (4𝑟²(1 − 0.25𝜋)) mm²

Aw area of the web (𝑡𝑤∙ ℎ𝑤) mm²

As an example an HEA240 cross-section was used to demonstrate the analytical solution. To enable the calculation of M-N-V interaction, three regions and several cases were introduced. Table 3.1 shows an overview of those regions & cases.

Table 3.1 All possible cases for the interaction of M-N-V

When distributing shear force, the regions presented themselves namely:

I) The shear force is only in the web II) The shear force is in the web & roots

III) The shear force is in the web, roots and in the flange

Figure 3.2 shows the three regions where the shear force can be situated.

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25

Figure 3.2 Possible positions of the shear force region I, II and III

When distributing the shear force, normal force and bending moment, three general cases presented themselves, namely:

1) The neutral axis of the cross-section is situated in the web 2) The neutral axis of the cross-section is situated in the roots 3) The neutral axis of the cross-section is situated in the flange Three possible cases are shown in Figure 3.3.

Figure 3.3 Possible positions of the neutral axis [7]

Besides these three general cases two subcases emerged, namely 3A and 3B, see Figure 3.4. When the shear force has reached the flange, which is indicated in black in Figure 3.4, then case 3A and 3B would be applied. Flange1 and flange2 in case 3A and 3B are the distances where the normal force can be varied. The difference between the flange and flange1/flange2 is that in case of the flange the shear force is located in web or in the roots and in case of flange1/flange2 the shear force is located in the flange itself. Table 3.1 shows the possible cases.

Figure 3.4 Possible positions of the neutral axis in the flange, Case 3A (left) and Case 3B (right)

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26 3.1 M-N-V interaction in region I (case 1, 2 and 3)

Case 1: neutral axis in web

In this case, the neutral line is located in the web. As mentioned earlier the shear force is taken by the center part of the web over the height (ℎ_𝑦). The normal force is taken by another part of the web which height is indicated with α. The remaining part of the cross-section beared the bending moment, see Figure 3.5.

Figure 3.5 Yield stress distributions in case 1

To determine which part of the cross-section (hy) is reserved for the shear force, equation (3.1) and (3.2) were used.

𝑛_𝑉∙𝑉_𝑝𝑙

𝑓𝑦⁄√3 = 𝐴 with 𝐴 = 𝑡_𝑤∙ ℎ_𝑦 (3.1)

ℎ_𝑦 =

𝑛𝑉∙𝑉𝑝𝑙 𝑓𝑦 √3⁄

𝑡𝑤 (3.2)

For region I, ℎ_𝑦 ≤ 𝑑

Figure 3.6 shows the distances form the center of the cross-section to the center of the gravity of each surface in Case 1.

Figure 3.6 Distance from the center of the cross-section to the center of the gravity of each surface, in Case 1

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27

In Equations (3.3) to (3.7) the design bending moment with the corresponding plastic section modulus regarding the shear force, normal force and bending moment is described.

𝑊_𝑉=¹

4∙ 𝑏 ∙ ℎ²=¹

4∙ 𝑡_𝑤∙ ℎ_𝑦² (3.3)

𝑊_𝑁= (2 ∙ 𝑡_𝑤∙ 𝛼) ∙ (^𝛼

2+^ℎ^𝑦

2) (3.4)

The design value of the axial force is given by equation (3.10).

𝑁_𝐸𝑑 = 𝑓_𝑦∙ 𝐴_𝑁 = 𝑓_𝑦∙ (2 ∙ 𝑡_𝑤∙ 𝛼) (3.5)

With:

hv unknown distance in y-direction where shear force is applied mm Wv plastic section modulus regarding the shear force mm³ WN plastic section modulus regarding the normal force mm³ WM plastic section modulus regarding the bending moment mm³

My,Ed design bending moment kNm

𝑊_𝑀= 𝑊_𝑝𝑙− 𝑊_𝑉− 𝑊_𝑁 (3.6)

𝑀_{𝑦,𝐸𝑑}= 𝑓_𝑦∙ 𝑊_𝑀= 𝑓_𝑦∙ (3.7)

(𝑏 ∙ 𝑡_𝑓(ℎ − 𝑡𝑓) +¹₄∙ 𝑡_𝑤(ℎ − 2 ∙ 𝑡_𝑓)²+ (4𝑟²(1 − 0.25𝜋) ∙ (^ℎ₂− 𝑡_𝑓− 0.223 ∙ 𝑟)) − (¹₄∙ 𝑡_𝑤∙ ℎ_𝑦²) − (2 ∙ 𝑡_𝑤∙ 𝛼) ∙ (^𝛼

2+^ℎ^𝑦

2) Case 2: neutral axis in roots

In this case the neutral axis is located in the roots. A function is defined to describe the roots, see equation (3.8). This equation can be used to define the area of the roots or to calculate the neutral point of the root, illustrated by Figure 3.7.

(3.8)

(3.9) 𝑦 = 𝑟 − √2 ∙ 𝑥 ∙ 𝑟 − 𝑥²

𝑧₁=∫ 𝑥 ∙ (𝑟 − √2 ∙ 𝑥 ∙ 𝑟 − 𝑥₀^𝑟 ²)𝑑𝑥

∫ (𝑟 − √2 ∙ 𝑥 ∙ 𝑟 − 𝑥₀^𝑟 ²)𝑑𝑥 = (1 − 2 3⁄

4 − 𝜋) 𝑟 ≈ 0.223 ∙ 𝑟

Figure 3.7 Area and neutral point of the root [7]

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28

Equation (3.8), (3.9) and (3.10) originate from the literature survey of Rombouts, I.M.J. [7].

For the calculation of the root area which is given in equation (3.10), the integral of equation (3.8) is taken.

(3.10)

This results in a slightly different yield stress distribution, see Figure 3.8.

Figure 3.8 Yield tress distributions in case 2

To calculate the plastic section modulus, the distance from the center of the cross-section to the center of gravity of each surface is needed. These distances are shown in Figure 3.9.

 

   



 

      

   

  

       

        

     

     

 

 

       

        

     

     

 

 



²

0

2 32

0

3

2 2

4 2

2 1 1

2 arctan 2 2 2 2

4 2

2 1 1

2 arctan 2 2 2 2

4 2

Ar r xr x dx

x r x

r x r x x r x r x x r x

x r x x rx

r r rr r r

r r

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29

Equation (3.11) and (3.12) describes the section modulus regarding normal force. In equation (3.13) the normal force is described.

𝑊_𝑁= (2 ∙ 𝑡_𝑤(𝛽 +^𝑑₂−^ℎ₂^𝑦)) ∙ ((^(𝛽+

𝑑 2−^ℎ𝑦

2)

2 ) +^ℎ₂^𝑦) + (𝐴_𝑟(𝛽)∙ ((𝛽 − 0.223 ∙ 𝛽) + 𝑑 2⁄ )) (3.11)

𝑊𝑁,𝑟(𝛽)= 𝐴𝑟(𝛽)∙ 𝑎 = (𝐴𝑟(𝛽)∙ ((𝛽 − 0.223 ∙ 𝛽) + 𝑑 2⁄ )) (3.12)

𝑁_𝐸𝑑 = 𝑓_𝑦∙ 𝐴_𝑁 = 𝑓_𝑦∙ ((2 ∙ 𝑡_𝑤∙ (𝛽 +^𝑑₂−^ℎ₂^𝑦)) + 𝐴_𝑟(𝛽)) (3.13)

𝐴_𝑟(𝛽) surface area of the roots which is determined by the distance β mm²

In equation (3.14), the section modulus regarding bending moment is described which is used in equation (3.15) to calculate the bending moment.

𝑊_𝑀= 𝑊_𝑝𝑙− 𝑊_𝑉− 𝑊_𝑁 (3.14)

𝑀_{𝑦,𝐸𝑑}= 𝑓_𝑦∙ 𝑊_𝑀 (3.15)

𝑀_{𝑦,𝐸𝑑}= 𝑓_𝑦∙ (𝑊_𝑝𝑙− 𝑊_𝑉− 𝑊_𝑁)

= 𝑓_𝑦 (

(𝑏 ∙ 𝑡_𝑓(ℎ − 𝑡_𝑓) +1

4∙ 𝑡_𝑤(ℎ − 2 ∙ 𝑡_𝑓)²+ (4𝑟²(1 − 0.25𝜋)) ∙ (ℎ

2− 𝑡_𝑓− 0.223 ∙ 𝑟))

−(1

4∙ 𝑡_𝑤∙ ℎ_𝑦²) − (2 ∙ 𝑡_𝑤(𝛽 +𝑑 2−ℎ_𝑦

2)) ∙ (

( (𝛽 +𝑑

2 − ℎ_𝑦

2 )

2 ) +ℎ_𝑦

2 )

+ (𝐴_𝑟(𝛽)∙ ((𝛽 − 0.223 ∙ 𝛽) + 𝑑 2⁄ )) )

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30

Case 3: neutral axis in flange

In case three the neutral axis is located in the flange. The yield stress distribution in case three is shown in Figure 3.10 (𝛿 ≤ 𝑡𝑓).

Figure 3.10 Yield stress distributions in case 3

The area where normal force and bending moment can be varied is shown in Figure 3.11.

For the calculation of bending moment and axial force equation (3.16) and (3.17) can be used.

𝑀_{𝑦,𝐸𝑑}= 𝑓_𝑦∙ ((𝑡_𝑓− 𝛿)𝑏 ∙ 2) ∙ (^(𝑡^𝑓^−𝛿)

2 + 𝛿 +^ℎ^𝑤

2) (3.16)

𝑁_𝐸𝑑 = 𝑓_𝑦∙ 𝐴_𝑁 = 𝑓_𝑦∙ ((2 ∙ 𝑡_𝑤∙ (^ℎ^𝑤

2 −^ℎ^𝑦

2)) + (2 ∙ 𝑏 ∙ 𝛿) + 𝐴_𝑟) (3.17)

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31

Figure 3.12 shows the M-N-V interaction diagrams of HEA240 in region I

Figure 3.12 M-N-V interaction diagrams of HEA240 in Region I

3.2 M-N-V interaction in region II (case 2 and 3)

To determine the shear area in Region II, equation 3.18 is used.

𝑛_𝑉∙𝑉_𝑝𝑙

𝑓_𝑦⁄√3 = (𝑡_𝑤∙ d) + 𝐴_{𝑉,𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙} (3.18)

AV,residual= ^𝑛_𝑓^𝑉^∙𝑉^𝑝𝑙

𝑦⁄√3− (𝑡_𝑤∙ d) = (2 ∙ (𝑡_𝑤∙ 𝛽)) + 𝐴_𝑟(𝛽) (3.19)

Since equation (3.24) cannot be solved for β, the dimensions of an HEA240 cross-section were substituted in equation (3.15) and (3.24). For example, when 𝑛_𝑉 = 0.5, the value of β can be found numerically:

(0.5 ∙ 341.6 ∙ 10³

235 √3⁄ ) − (7.5 ∙ 164) = 𝟐𝟖. 𝟕𝟖 𝐦𝐦^𝟐 = (2 ∙ (7.5 ∙ 𝛽)) + 𝐴_𝑟(𝛽) => 𝜷 = 𝟎. 𝟑𝟐𝟐𝟔 𝒎𝒎

Eindhoven University of Technology MASTER Bending moment, shear and normal force interaction of I-shaped steel cross-sections Hamidi, S.

Bending Moment, Shear and Normal Force Interaction of I-shaped Steel

Cross-sections

____________________________________________________

______________________________________________________________________________________

1

2

Preface

3

Abstract

4

Samenvatting

5

Table of contents

6

7

8

Nomenclature

9

ᵨ

10

1. Introduction

1.1 Problem definition

1.2 Goal

1.3 Scope

11 1.4 Approach

12

2. Literature survey

2.1 Bending, shear and axial force interaction

2.2 Stress distribution

13 2.3 Elastic theory

14

15

2.4 Plastic theory

16

17

18

19 2.5 Classification of cross-sections

20 2.6 Code requirements

ᵨ

21

2.7 Results from earlier research

22

23

2.8 Conclusion regarding literature survey

24

3. Analytical solution

25

26 3.1 M-N-V interaction in region I (case 1, 2 and 3)

27

28

 

   

   



29

30

31

3.2 M-N-V interaction in region II (case 2 and 3)