Eindhoven University of Technology

MASTER

Lateral torsional buckling of a spreader beam in a hoisting construction

Ploegmakers, D.G.

Award date:2017

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Download date: 06. Aug. 2018

Lateral torsional buckling of a spreader beam in a hoisting construction

A/O-0017.180

D.G. Ploegmakers

Hoenderbosplein 78

5406 AJ Uden

Student number: 0728608

Phone: 06-53512164

e-mail:d.g.ploegmakers@student.tue.nl

Supervisors:

Prof. ir. H.H. Snijder

Prof. dr. ir. J. Maljaars

Ir. R.W.A. Dekker

27-01-2017

Structural Design

Department of the Built Environment

Eindhoven University of Technology

2

PREFACE

This report concludes the Master’s research project about the lateral torsional buckling resistance of an I-shaped spreader beam, performed at the Eindhoven University of Technology for the master track Architecture, Building and Planning with the specialization Structural Design at the Department of the Built Environment.

I would like to thank my graduation committee: Prof. ir. H.H. Snijder, ir. R.W.A. Dekker and Prof. dr. ir. J. Maljaars for their professional supervision. And ing. H.L.M. Wijen and A.W. van Alen for their support during the experiment.

I also want to thank my wife Laura and my parents Iedje and Geert for the support throughout my whole academic career.

Finally, I want to mention that this graduation project could not have been done without the help of other students who were always there to answer questions, help me and show interest during this project

Dirk Ploegmakers

Uden, January 2017

3

ABSTRACT

I-shaped beams that are subjected to a bending moment might be sensitive to lateral torsional buckling. Currently the Eurocode and the Dutch national annex provide calculation methods to determine the lateral torsional buckling resistance of an I-shaped beam. However, it is only allowed to use the method to determine the critical moment provided in the Dutch National Annex, when fork ended conditions are present. This is not the case for spreader beams. Therefore, the currently available calculation methods cannot be used.

Extensive research on lateral torsional buckling has been conducted. However, almost all research focused on beams with fork ended conditions. Therefore, little literature on the subject is included.

A calculation method to determine the lateral torsional buckling resistance of an I-shaped spreader beam numerical tests has been developped. Different types of beams, different lengths and different configurations in the way the beam is suspended are considered. The used numerical model is validated using an experiment.

First a method to determine the critical moment of a spreader beam is developed. This method uses the factors C1 and C2, which are also used in the method provided in the Dutch National Annex, therefore making implementation very straightforward. A statistical evaluation has been performed on this calculation method to determine partial safety factors. This safety factor is within the acceptance limit for γM1=1.0.

The calculation method to determine the lateral torsional buckling resistance in the Eurocode uses buckling curves to determine a reduction factor. Therefore, the applicable buckling curve is determined by choosing the least conservative curve of which the determined safety factor is within the acceptance limit. In this case buckling curve c for steel grades S355 and S460, and buckling curve d for beams with steel grades S235.

4

TABLE OF CONTENTS

Preface .................................................................................................................................... 2

Abstract .................................................................................................................................. 3

Notations ................................................................................................................................ 6

1 Introduction ......................................................................................................................... 8

1.1 Problem statement ..................................................................................................................... 8

1.2 Aim ............................................................................................................................................ 10

2 Literature survey ................................................................................................................ 11

2.1 Analytical research ..................................................................................................................... 11

2.2 Experimental research ............................................................................................................... 14

2.3 Numerical research .................................................................................................................... 14

2.4 Numerical models ...................................................................................................................... 16

3 Experiment ......................................................................................................................... 18

3.1 Introduction ............................................................................................................................... 18

3.2 Tensile test ................................................................................................................................. 19

3.3 Experiment ................................................................................................................................ 28

4 Finite Element Model .......................................................................................................... 36

4.1 Material properties .................................................................................................................... 37

4.2 Boundary conditions .................................................................................................................. 38

4.3 Loads ......................................................................................................................................... 39

4.4 Element type .............................................................................................................................. 39

4.5 Solving methods ......................................................................................................................... 40

4.6 validation with experiment ........................................................................................................ 41

4.7 Adaptions for Steel profiles ........................................................................................................ 42

4.9 Mesh .......................................................................................................................................... 48

4.10 Validation plastic moment resistance ....................................................................................... 49

5 Development design rule .................................................................................................... 50

5.1 Parameters ................................................................................................................................ 50

5

5.2 Change from simply supported beam to the final model ............................................................ 57

5.3 design a method to determine the critical moment .................................................................... 65

5.4 Develop a method to determine the ultimate LTB resistance ..................................................... 78

5.4 Summary .................................................................................................................................... 96

5.5 Calculation example ................................................................................................................... 98

6 Conclusions and recommendations ................................................................................... 100

6.1 Conclusions .............................................................................................................................. 100

6.2 Recommendations ................................................................................................................... 100

References .......................................................................................................................... 102

Appendix ............................................................................................................................. 104

Appendix A: script .......................................................................................................................... 104

Appendix B: Various lengths and ratios for considered beams ....................................................... 142

Appendix C .................................................................................................................................... 144

Appendix D: overview of different models used. ............................................................................ 150

Appendix E: PARTIAL DERIVATIVES OF Mcr WITH RESPECT TO THE BASIC VARIABLES ...................... 170

Appendix F: 153 models used for evaluation of the buckling curves ............................................... 180

Appendix G: PARTIAL DERIVATIVES OF GRT,I WITH RESPECT TO THE BASIC VARIABLES .................. 184

Appendix H: plots of subsets .......................................................................................................... 190

Appendix I: Acceptance limit plots ................................................................................................. 200

6

NOTATIONS

All symbols are in order of occurrence.

Mb,Rd : Lateral torsional buckling resistance of a prismatic beam in bending (Nmm)

LTχ : Reduction factor (-)

Wy : Applicable section modulus (mm3) fy : Yield strength (N/mm2)

1Mγ : Partial safety factor for resistance of elements with relation to stability (-)

LTΦ : Value used to determine the reduction factor LTχ (-)

LTλ : Relative slenderness for lateral torsional buckling stability (-)

LTα : Imperfection factor (-)

Mcr : Critical moment (Nmm) h : Height of the beam (mm) tw : Thickness of the web (mm) Lg : Length of the beam between fork ended supports (mm) E : Modulus of elasticity (N/mm2) Iz : Moment of inertia around the z-axis (mm4) G : Shear modulus (N/mm2) It : torsional moment of inertia (mm4) Iwa : Camber moment of inertia (mm4) Fb : Allowable bending stress (N/mm2) Cb : Bending coefficient dependent upon moment gradient (-) Nd : Design factor (-) Lb : Distance between cross sections braced against twist or lateral

displacement of the compression flange (mm)

d : Depth of the section (mm) Af : Area of the compression flange (mm2) Fy : Specified minimum yield strength (N/mm2) CLTB : Factor describing the ratio between the two models in figure 2.7 (-) Iy : moment of inertia around the y-axis (mm4) J : St. Venant torsion constant (mm4) b : Width of the section (mm)

ν c : crosshead separation rate (mm/min)

Lc : parallel length, in this case 75mm (mm)

cLe : estimated strain rate over the parallel length (1/s)

tf : thickness of the flanges (mm) Wpl,y : plastic section modulus (mm3) Wel,y : elastic section modulus (mm3)

rσ : residual stress where a positive sign indicates a tensile stress and negative sign a compressive stress

(N/mm2))

C : Coefficient dependent on beam length, cross sectional properties, nature and application point of load

(-)

7

C1 : Coefficient dependent on the nature of the load (-) C2 : coefficient dependent on the application point of the load relative to

the neutral axis (-)

,cr LTBσ : Critical stress for lateral torsional buckling (N/mm2)

rt,i : Theoretical resistance values (Nmm) re,i : Experimental resistance values (Nmm) ri : Resistance values (Nmm) b : Least-square approximation factor (-) δi : Error terms (-) Δi : Natural logarithm of error terms (-) µ∆ : Estimated mean value of the error terms (-)

sΔ : Estimated standard deviation of the error terms (-)

iVδ : Coefficient of variation with respect to the error terms (-)

,V

t ir : Coefficient of variation with respect to the resistance values (-)

Vir : Coefficient of variation with respect to the resistance values (-)

iQδ : Lognormal coefficient with respect to the error terms (-)

,t irQ : Lognormal coefficient with respect to the theoretical resistance values (-)

irQ : Lognormal coefficient with respect to the resistance values (-)

,d ir : Design resistance values (-)

,dk ∞ : Design fractile factors for n>100 single test results (-)

,ndk : Design fractile factors for n single test results (-)

*1Mγ : Corrected partial safety factor (-)

fa : Acceptance limit (-) γM,target : Target value of the partial safety factor, already existing safety factor (-)

Rdγ : Partial factor only accounting for the model accuracy (-)

8

Figure 1.1: mechanical schematization of a suspended spreader beam

1 INTRODUCTION

During construction of a building usually hoisting of numerous materials is needed. A spreader beam may be used for this purpose. In general practice an I-shaped steel beam is used as spreader beam, the beam is suspended by cables and loads act on both beam ends. This results in a bending moment during hoisting. Subsequently, lateral torsional buckling (LTB) might be decisive when designing an I-shaped spreader beam for hoisting purposes.

1.1 PROBLEM STATEMENT

In the case that an I-shaped steel beam is subjected to a bending moment, lateral torsional buckling should be taken into account as a failure mechanism. A typical configuration of an I-shaped hoisting spreader beam, see figure 1.1, is connected to a crane by cables and is subjected to two loads at both beam ends representing the force of the weight lifted.

EN 1993-1-1 [1] clause 6.3.2.2 [1] describes the design rule used for determination of the lateral torsional buckling resistance of prismatic beams loaded with a bending moment, see equation (1.1) to (1.4).

,1

LT y yb Rd

M

W fM

χγ

= (1.1)

Mb,Rd : Lateral torsional buckling resistance of a prismatic beam in bending

LTχ : Reduction factor

Wy : Applicable section modulus fy : Yield strength

1Mγ : Partial safety factor for resistance of elements with relation to stability

22

1 ; 1,0LT LT

LTLT LT

χ χλ

= ≤Φ + Φ −

(1.2)

LTΦ : Value used to determine the reduction factor LTχ

LTλ : Relative slenderness for lateral torsional buckling stability

9

LTχ can also be determined using the buckling curves given in figure 6.4, clause 6.3.1.2 of EN 1993-1-1.

( ) 20,5 1 0,2LT LTLT LTα λ λ Φ = + − + (1.3)

y yLT

cr

W fM

λ = (1.4)

LTα : Imperfection factor

Mcr : Critical moment

For a beam with h/tw≤75 and a constant bending moment:

2

21 wacr z t

g g t

EIM EI GIL L GI

ππ = +

(1.5)

h : Height of the beam tw : Thickness of the web Lg : Length of the beam between fork ended supports E : Modulus of elasticity Iz : Moment of inertia around the z-axis G : Shear modulus It : torsional moment of inertia Iwa : Camber moment of inertia

In order to make use of this calculation procedure, the critical moment Mcr is required. The Dutch National Annex describes a procedure to obtain this critical moment, see equation 1.5. However, in paragraph 6.3.2.5 “supports and lateral restraints” (“opleggingen en zijdelingse steunen”) of the Dutch National Annex of EN 1993-1-1 [1] it is stated that the beam has to have specific boundary conditions to allow for this calculation. The spreader beam in a hoisting structure does not fulfill this requirement. Therefore, the national annex in its current form cannot be used to calculate the critical moment. Subsequently, the lateral torsional buckling resistance cannot be determined. Because, there is no procedure given for calculating the critical moment of a spreader beam in a hoisting structure.

Taras, et al. [2] proposed for a new design rule similar to the currently used method following the Dutch National Annex of EN 1993-1-1. The main difference is presented by introducing a new procedure for calculating the reduction factor. However, still no method to calculate Mcr is added. Therefore, it still is impossible to use this design procedure for a suspended spreader beam in bending.

10

1.2 AIM

The goal of this project is to develop a method to calculate the lateral torsional buckling resistance of spreader beams in a hoisting structure. In order to achieve this, the currently used buckling curves applicable for this structure need to be assessed.

11

2 LITERATURE SURVEY

Although the lateral torsional buckling behavior of symmetric I-beams has been researched extensively, these studies were primarily aimed at beams with forked end supports. For example recent research by Dahmani et al [3] compared numerical models with the EN 1993-1-1 [1] calculation method to define the ultimate lateral torsional buckling load of steel beams. The current EN 1993-1-1 design rule corresponded well with the numerical results, leading to the conclusion that the EN 1993-1-1 method is an accurate and conservative method.

Despite of this good accordance between the method provided in the Eurocode and the numerical simulations. Steel beams that are suspended by cables or chains have not received much attention.

2.1 ANALYTICAL RESEARCH

Dux et al [4] presented an analytical solution of the lateral torsional buckling resistance of a suspended I-shaped beam. This research first states that since the problem is symmetrical there are two possible buckling mode shapes, see figure 2.1. In this figure the lateral displacement of the middle of the beam is visualized.

The research continued with a setup of the differential equations for a suspended I-shaped beam in bending based on the buckling shapes. Four different regions of the beam were regarded (following figure 2.2):

region 1: (from the right point load to the end of the beam)

region 2: (from the cable attachment point to the right point load)

region 3: (from the left point load to the cable attachment point)

region 4: (from the middle of the beam to the left point load)

Figure 2.1: possible buckling modes, Dux et al. [4]

12

Only half the beam is shown because symmetry is used. Note that both point loads at the bottom of the beam are not situated at the end of the beam. Since the problem statement of this research is aimed at only one point load at the end of the beam, region 1 is not considered. And because of the fact that λ1=0, region 3 and region 4 will merge. This means only region two and three will be considered. The force is defined as W and λ1 and λ2 are defined as the ratio between the two point loads, λ2 will be considered as 1.0. In the differential equations they considered a uniformly distributed load w (which is not displayed in figure 2.2). This uniformly distributed load may represent the self-weight of the beam. In figure 2.3 a representation is shown of the displaced cross section with associated symbols used.

Figure 2.2: geometry of suspended beam, Dux et al. [4]

Figure 2.3: displaced cross section, Dux et al. [4]

The differential equations for region two are equations 2.1 and 2.2, for region three equations 2.3 and 2.4 are the differential equations.

( )2 2

2 8 2 2yL LZ ZM W Z wϕ α ϕ

= − − − +

(2.1)

( )2 2 2

2 2 2 8 2 2

L

z A AZ

du L du L LZ ZM W Z u u a w udZ wu Z w

dZ dZα ϕ= − + − − − + − + − +

∫ (2.2)

( ) ( ) ( )2 2

2 1 18 2 2 2ya L LZ Z wL aM W a W a u w a ub b

ϕ α ϕ ϕ ϕ

= − + − − − + + −

(2.3

13

12

tan

ωπ

µθ

=

=

EIK

L GJhL

( )2

2 2 1

2 2

1

2

8 2 2 2

L

z A AZ

du a du LM W a u a Wa w udZ wu ZdZ b dZ

du L LZ Z wL aw u a a ZdZ b

α ϕ = − − − − − + −

+ − + − + + −

∫ (2.4)

These differential equations were solved using the Finite Integral Method. Two relevant numerical examples are discussed.

First the influence of the load position along the beam length and the angle in which the cables are connected to the beam is discussed. In Figure 2.4 a graph with associated equations made by Dux et al. [4]. The vertical axis shows a non-dimensional buckling load and the horizontal axis refers to the load position on the beam. Where on the left of the horizontal axis at Z/L=0 the load is in the middle of the beam and on the right of the horizontal axis the load is at the end of the beam.

Figure 2.4: Beams with variable cable Angle and load position, Dux et al. [4]

The load position varies from the exact middle of the beam to the end of the beam which is the case in this research. There are four different graphs in figure 2.4 that indicate different cable angles (from top to bottom: 90°, 60°, 45° and 30°). It may be obvious that the angle of the cables has a large impact on the buckling strength of the beam when the load is applied between Z/L=0.1 and Z/L=0.3. If the load is placed towards the end of the beam this influence becomes negligible. Which may indicate that the cable angle and thus the normal force in the beam has no influence on the buckling strength if the load is located at the end of the beam.

Another important factor in the ultimate buckling strength of the suspended I-beam is the distance between the actual beam and the cable- and load attachment on the beam. These distances have two effects. Increasing the distance of the load attachment increases the torsional restraint and therefore increases the maximum buckling strength of the beam. But at the same time increasing the height of the cable attachment increases the maximum major axis moments and thus lowering the maximum buckling capacity of the beam. In figure 2.5 four graphs are drawn. The dotted lines have an increased distance of the load attachment and the two graphs most right have an increased distance of the cable attachment.

14

Figure 2.5: influence of cable and load attachment height, Dux et al. [4]

2.2 EXPERIMENTAL RESEARCH

In the research of Dux et al. a number of experiments were conducted. These were small scale experiments using an extruded aluminum profile with a height of 75,2mm and a width of 31,4mm. Figure 2.6 shows the set-up and the clamping method of the cables and the loads to the beam. In this experiment the beam was laterally fixed at the cable attachment points. This is allowed because these cables will not move laterally. The beam was loaded using lead weights.

Figure 2.6: experiment set up and attachment details, Dux an Kitipornchai (1990) [4]

They compared there experimental values with the theoretical ones obtained from the differential equations. They were all within 5% of each other which was probably due to additional restrained caused by friction in the attachment devices.

2.3 NUMERICAL RESEARCH

Duerr [5] developed numerical models to find a correction factor for suspended beams in order to calculate the buckling strength of these beams using existing commonly used equations like the ones given by ASME [6], see equation 2.5. The numerical models represented torsional restrained beams and suspended beams.

15

( )0.66F = ≤ yb

bdd b f

FECNN L d A (2.5)

Fb : Allowable bending stress Cb : Bending coefficient dependent upon moment gradient Nd : Design factor Lb : Distance between cross sections braced against twist or lateral

displacement of the compression flange d : Depth of the section Af : Area of the compression flange Fy : Specified minimum yield strength

In equation 2.5 the distance between cross sections braced against twist or lateral displacement of the compression flange is needed. Subsequently this equation cannot be used to design a suspended beam in bending for lateral torsional buckling since there are no cross sections that are braced against twist or lateral displacement.

Figure 2.7 gives a representation of the numerical models. Note that the beam section after the load attachment point is excluded, for this part has no influence on the buckling strength. Moreover, there is a big difference between this research and the one performed by Dux et al. [4]. Where they had two cables under an angle, Duerr modelled a so called spreader beam which has only one attachment at the top. This model shows that you can laterally restrain the beam at certain places which reduces the complexity and inaccuracy of the model, because no springs have to be added.

Figure 2.7: model for beam-buckling analyses, Duerr (2015)

Fifty different beams were modelled this way and through the ratio between these models Duerr found a so called CLTB-factor, see equation 2.6.

2

2,000,275 1,00= + ≤

y

LTBb

EIGJC

Lb

(2.6)

16

CLTB : Factor describing the ratio between the two models in figure 2.7 Iy : moment of inertia around the y-axis J : St. Venant torsion constant b : Width of the section

Subsequently, this factor was used to compare one experimental result from the research of Dux et al. with a calculation using the method of ASME [6] (equation 2.5) from which the final result was altered with the CLTB. The difference in the maximum allowable bending stress according to ASME and the maximum bending stress in the experiment of Dux et al. was 3.6%. So it is safe to say that the model created by Duerr is a good working model and the idea of restraining the bottom flange at certain places is a useful tool in creating a numerical model of buckling analyses of a suspended beam.

2.4 NUMERICAL MODELS

As stated before, a Finite Element (FE)-program cannot work with too many degrees of freedom. The research of Duerr [5] is very helpful in showing that some points of the lower flange of the beam can be laterally restrained.

Several studies Kucukler et al. [7], Jovic [8] and van der Aa [9] have documented about the build-up of a model in which an I-beam shows lateral torsional buckling.

All the models in these studies were build up out of shell elements where, Jovic and van der Aa used 8 node elements and Kuculer used 4-node elements. Though, none of them explains why the specific shell element was chosen.

All of these studies indicate that a small error occurs in their model. The fillet radius is not modelled, because of the use of shell elements. Jovic stated that this is a small error and that it is possible to continue with this error, but when comparing results of his model with analytical solutions this should be considered likewise. Kuculer solved this error by adding beam elements at the flange-web intersection so that the section properties in steel section tables are achieved. Van der Aa showed that it is not necessary to compensate for these fillet radii using the study of Taras [10]. This showed that there is a minor difference between models with or without the fillet radius using an IPE2400 section. This is important to know before making the model, because determining the right beam elements is an iterative and therefore time consuming process.

Jovic and Kuculer also take note of a geometrically error that exists in model of I-beams which consists out of shell elements. This occurs because of the fact that when thickness is added to a shell element this is done in two directions the flange of the beam than overlaps the web of the beam, so a small are is accounted for twice as can be seen in figure 2.8. Jovic states that this error is small and not of a big influence on the results. While Kuculer states that this error is easily solved by offsetting the flange, so overlapping will be avoided.

17

In actual beams residual stresses with a triangular shape are present. All three studies have modelled residual stresses by dividing the cross section into small parts where a straight residual stress pattern is modelled with the average value of the triangular stress pattern from the original cross section. The final residual stress pattern corresponds well with the triangular stress pattern.

Big differences between the three studies were regarding the width-to-length ratio of the shell elements. Where Kuculer used a maximum ratio of 20, Jovic used a ratio of approximately 5 and van der Aa used a ratio of 4. Only in the study of Jovic the ideal ratio was determined using a mesh density study. The abaqus documentation [11] however defined the length to width ratio to never exceed 10. There is no right answer for the question what ratio to use. The maximum ratio of the abaqus manual should not be exceeded, but a mesh density study will be necessary to determine the optimal ratio. Furthermore a mesh density study will be needed to determine the maximum element size needed for reliable results.

Figure 2.8: real beam cross compared with FE-model, Jovic [8]

18

3 EXPERIMENT

3.1 INTRODUCTION

To define the buckling shape of a suspended beam in bending an experiment was executed. Based on the buckling shape the boundary conditions, which can be used in the FE-model, are determined. The test results (for instance the lateral displacement with associated vertical loads) are used in validation of the numerical model. The experiment consists of a small aluminum (alloy AW-6060 T6), prismatic, I-section beam which is suspended by two steel cables and subjected to two loads at the ends of the beam, see figure 3.1. Where point A and E are the sections where the loads are attached, point B and D are the sections where the cables are attached, point C is the middle section, S is the suspension point and α is the angle between the cables. The different locations within a section are identified with subscripts, where Aupper is the top surface of the upper flange, Abottom is the bottom surface of the bottom flange and Aatt is the position where the load, or cable at points B and D, is attached.

In order to define the material properties tensile tests on coupons originating from the flanges and web are executed. This results in stress strain diagrams presenting the material properties.

Figure 3.1: Scheme of the experiment

19

3.2 TENSILE TEST

The tensile test coupons are loaded in tension to determine the material properties, for instance the Young’s modulus (E), 0.2% proof stress (f0.2,p) and ultimate tensile strength (fu). Three coupons originate from the beam, one from each flange and one from the web.

3.2.1 TENSILE TEST SET-UP

In the international standard ISO 6892-1 [12] it is prescribed what the different conditions of a coupon tensile test should be.

The tensile tests are executed in the Pieter van Musschenbroek Laboratory at the Eindhoven University of Technology following ISO 6892-1. This standard describes the test conditions, including geometry of the test specimens. A 250 kN Instron testing machine was used, it is equipped with a video extensometer to define the longitudinal strains. The three coupons were marked with two dots to indicate the original gauge length (L0). The video extensometer registers the deformations between the two dots and therefore the elongation. Figure 3.2 shows a picture of the test setup.

Figure 3.2: Test set up of tensile coupon

20

3.2.1.1 DIMENSIONS OF THE TEST COUPON

The dimensions of the coupons are based on the international standard ISO 6892-1 [1]. Both flanges are 1.5 mm in thickness and the web is 2.0 mm thick, see figure 3.3. Annex B of ISO 6892-1 was used to determine the dimensions of the coupons, see figure 3.4 and 3.5. The chosen configuration required only 250mm of the beam leaving adequate length to perform the experiment. The three coupons are taken from the same beam with which the experiment is conducted.

Figure 3.3: Cross section of the aluminum beam

Figure 3.4: Tensile coupons originating from the web

Figure 3.5: Tensile coupons originating from the flanges

21

To be able to determine the stress-strain diagrams the actual dimensions of the coupons are needed. The widths and thicknesses of the coupons were measured at three places on the coupons, see figure 3.5. Table 1 present these measurements.

Table 3.1: Cross-sectional dimensions of the different coupons

Coupon 1 Upper flange

Coupon 2 Bottom flange

Coupon 3 Web

Thickness 1 1.5 mm 1.5 mm 2.0 mm Width 1 12.6 mm 12.5 mm 12.5 mm Thickness 2 1.5 mm 1.5 mm 2.0 mm Width 2 12.6 mm 12.5 mm 12.5 mm Thickness 3 1.5 mm 1.5 mm 2.0 mm Width 3 12.5 mm 12.5 mm 12.5 mm Average thickness 1.5 mm 1.5 mm 2.0 mm Average width 12.6 mm 12.5 mm 12.5 mm Average area 18.9 mm2 18.8 mm2 25.0 mm2

3.2.1.3 THE STRAIN RATE

ISO 6892-1 [12] gives guidelines to determine the strain rate. For different purposes different rates are given with sometimes changes in rate at the various stages of the tensile test. For instance high accuracy is required when the Young’s modulus is determined therefore a low strain rate is needed. When regarding the ultimate tensile strength lower accuracy is required consequently higher strain rates are allowed. In this case a constant strain rate was maintained during the tests.

Section 10.3 of ISO 6892-1 gives two possibilities to control the testing rate based on strain rate control. Where ‘method A’ controls the rate of the test using the specified strain rate (ėLe). Which means that the crosshead separation rate (νc) is adapted during the test depending on the strain rate. This method is not possible with the testing machine used, because no extensometer is clamped to the test piece. Therefore ‘method B’ is used. This method uses the estimated strain rate (ėLc) over the parallel length (Lc) to calculate the crosshead separation rate which can be inserted in the machine. The cross head separation rate used in testing is calculated using formula 3.1.

Figure 3.5: Positions used for cross-sectional measurements

22

ν = cc c LL e (3.1)

ν c : crosshead separation rate

Lc : parallel length, in this case 75mm

cLe : estimated strain rate over the parallel length

ISO 6892-1 [12] gives an estimated strain rate of 0.00007s-1 with a relative tolerance of 20%. Using equation 3.1 the maximum and minimum crosshead separation rate can be calculated:

1

,

,lower

,chosen

75 0.00007 0.00525 0.315 minupper limit:

120% 0.378 minlower limit:

80% 0.252 min0.300 min

ν

ν ν

ν ν

ν

−= ⋅ = =

= ⋅ =

= ⋅ =

=

c

c upper c

cc

c

mm mmmm s s

mm

mm

mm

The test was stopped for about two minutes at a strain of approximately 2%, 4% and 6%, according to the technical memorandum [13]. This can be used to determine the static stress values which exclude the influence of strain rate.

23

Figure 3.7: Stress-strain diagram coupon tensile test: enlargement elastic branch

3.2.2 TENSILE TEST RESULTS

The tensile tests result in stress-strain diagrams, which can be used to determine the material properties like the E, f0.2,p, and fu.

The three stress-strain diagrams of the tensile test of the coupons are shown in figure 3.6. In figure 3.7 an enlargement of the elastic branch of the stress-strain diagram is shown.

Figure 3.6: stress-strain diagram coupon tensile tests: whole test

24

3.2.2.1 DETERMINING THE YOUNG’S MODULUS

The beam is made of aluminum alloy AW-6060 T6. The nominal material properties for this alloy given in the Eurocode EN 1993‐1‐1 [1] are:

• fu = 170 N/mm2

• f0.2,p = 140 N/mm2

• E = 70 kN/mm2.

The Young’s modulus of the material can be determined by calculating the slope of the elastic branch of the stress-strain diagram. ISO 6892-1 [12] states that the elastic branch of the diagram has its lower limit at 10% and its upper limit at 50% of the nominal 0.2% proof stress. However Huang et al. [14] recommends an interval between 25% and 35% of the nominal 0.2% proof stress for an aluminum T6 alloy.

When comparing the stress-strain diagrams, figure 3.6, from the tensile tests and the given nominal value of the 0.2% proof stress a big difference is expected. The nominal value of the 0.2% proof stress is given as 140 N/mm2, but in figure 3.7 it is clear that the 0.2% proof stress is a lot higher for all coupons tested. This is the reason for initial visually determination of the 0.2% proof stress from figure 3.7, instead of using the nominal value. The 25-35% interval of these values are used to determine the Young’s modulus. With this Young’s modulus the 0.2% proof stress is determined. This results in a new interval for determination of the Young’s modulus, see table 2. A loop arises in which the Young’s modulus is needed to determine the 0.2% proof stress and conversely the 0.2% proof stress is used to determine the interval at which the Young’s modulus is determined. This loop is repeated if the determined Young’s modulus shows a significant difference with the previous iteration.

To calculate the slope of the elastic branch a linear trend line is produced which fits the stress and strain data on the used interval the best. The R2-value is the correlation coefficient and when this value is above 0.95 the correlation between the values from the test and the linear trend line is very high and thus gives a reliable value of the Young’s modulus.

25

Table 3.2: Determination of the Young’s modulus using 25-35% interval of visual determined f0.2,p

Coupon 1 Upper flange

Coupon 2 Bottom flange

Coupon 3 Web

Visually determined 0,2% proof stress 200 N/mm2 200 N/mm2 215 N/mm2

25%-35% interval 50-70 N/mm2 50-70 N/mm2 53,75-75,25 N/mm2

Young’s modulus 65976 N/mm2 66218 N/mm2 70074 N/mm2 R2-value 0,993 0,978 0,990 0,2% proof stress belonging to this Young’s modulus 200,7 N/mm2 201,0 N/mm2 212,8 N/mm2

25%-35% interval 50,18-70,25 N/mm2

50,25-70,35 N/mm2

53,2-74,5 N/mm2

Young’s modulus 66065 N/mm2 66218 N/mm2 70004 N/mm2 R2-value 0,993 0,978 0,990 Difference between first and second E- modulus +0,13% - -0,10%

More iterations needed no no no

A big difference is present in the determined Young’s modulus of the flanges and the web. Where for the flanges the Young’s modulus is around 66000 N/mm2 the web showed a much larger value namely 70000 N/mm2. Therefore the Young’s modulus will be determined using other intervals, namely the 10-50% and the 25-35% interval of the nominal value of the 0,2% proof stress of 140 N/mm2 and the 10-50% interval of the visual determined 0,2% proof stress, see table 3.3, 3.4 and 3.5.

Table 3.3: Determination of the Young’s modulus using 10-50% interval of the nominal f0.2,p

Coupon 1 Upper flange

Coupon 2 Bottom flange

Coupon 3 Web

nominal 0,2% proof stress 140 N/mm2 140 N/mm2 140 N/mm2 10%-50% interval 12-60 N/mm2 12-60 N/mm2 12-60 N/mm2 Young’s modulus 67839 N/mm2 66504 N/mm2 68735 N/mm2 R2-value 0,999 0,997 0,999

Table 3.4: Determination of the Young’s modulus using 25-35% interval of the nominal f0.2,p

Coupon 1 Upper flange

Coupon 2 Bottom flange

Coupon 3 Web

nominal 0,2% proof stress 140 N/mm2 140 N/mm2 140 N/mm2 25%-35% interval 35-49 N/mm2 35-49 N/mm2 35-49 N/mm2 Young’s modulus 68529 N/mm2 64470 N/mm2 70125 N/mm2 R2-value 0,989 0,967 0,985

26

Table 3.5: Determination of the Young’s modulus using 10-50% interval of visual determined f0.2,p

Coupon 1 Upper flange

Coupon 2 Bottom flange

Coupon 3 Web

Visually determined 0,2% proof stress 200 N/mm2 200 N/mm2 215 N/mm2

25%-35% interval 20-100 N/mm2 20-100 N/mm2 21,5-107,5 N/mm2

Young’s modulus 67404 N/mm2 65928 N/mm2 69258 N/mm2 R2-value 0,999 0,999 0,999 0,2% proof stress belonging to this Young’s modulus 200,6 N/mm2 201,1 N/mm2 212,9 N/mm2

25%-35% interval 20,1-100,3 N/mm2

20,1-100,5 N/mm2

21,29-106,5 N/mm2

Young’s modulus 67401 N/mm2 65912 N/mm2 69284 N/mm2 R2-value 0,999 0,999 0,999 Difference between first and second E- modulus -0,00% -0,02% +0,04%

More iterations needed no no no

It is clear that the determination of the Young’s modulus on different intervals of the stress-strain diagram gives a wide range of values. Where the lowest Young’s modulus, with a value of 62817 N/mm2, is found in the bottom flange on the 25-35% interval of the nominal f0.2,p and the highest value, which is 70004 N/mm2, is found in the web on the 25-35% interval of the visual determined f0.2,p. With a margin greater than 10% between these values. This does not give one value to use in the FE-model. To choose a value which can be used the method described in the international standard ISO 6892-1 [12] will be used. According to this standard the Young’s modulus needs to be determined using a 10-50% interval of the nominal f0.2,p, see table 3.3. This results in an average value of 67693 N/mm2.

27

Figure 3.8: dynamic and static stress-strain diagrams of coupon 1

3.2.2.2 DETERMINING THE ULTIMATE TENSILE STRENGTH

The ultimate tensile strength (fu,max) is determined for the three coupons. The displacement of the testing machine was kept constant three times during the test at a strain of 2, 4 and 6%. The difference between the dynamic and static values is determined at these interruptions. The average difference at these strains is used to lower the dynamic values of the stress-strain diagram to obtain the static stress-strain diagram, see figure 3.8. The maximum stress value of the dynamic stress-strain diagram is the dynamic ultimate tensile strength (fu,d,max). The maximum stress value of the lowered stress-strain diagram is the static ultimate tensile strength (fu,s,max). These values are indicated by the red and blue dot on the diagram in figure 3.8. The values of the static and dynamic ultimate tensile stresses of the three coupons are given in table 3.6.

Table 3.6: dynamic and static ultimate tensile strengths of all three coupons

Coupon fu,d,max fu,s,max

Coupon 1 216.75 N/mm2 205.29 N/mm2

Coupon 2 219.95 N/mm2 208.62 N/mm2

Coupon 3 232.29 N/mm2 220.47 N/mm2

Average 223.00 N/mm2 211.46 N/mm2

28

3.3 EXPERIMENT

An experiment is conducted to explore the behavior of a suspended, prismatic, I-section beam in bending. The buckling mode can give information about possible boundary conditions to use in the FE-model. Furthermore the experiment will be used in verification of the FE-model by comparison of the loads and associated lateral displacements.

Four points are expected to show behavior which can be used as boundary conditions in the numerical model. Namely the two points were the cable is attached to the beam (Batt and Datt) and the two points where the load is attached to the beam (Aatt and Eatt), see figure 3.1. To determine the boundary conditions to be used in the numerical model the experiment is executed four times with different boundary conditions on these points. During the experiments the points are either constrained from moving in the lateral direction using wire, see figure 3.9, or they are free to move.

Figure 3.9: Lateral restrained point using wire

When the test results show that the different boundary conditions on these points have no influence on the behavior of the beam these points can be laterally constrained in the FE-model. The four different situations are:

• Test 1: all four points laterally constrained, ux,Aatt= ux,Batt = ux,Datt= ux,Eatt =0.

• Test 2: two cable attachment points laterally constrained, ux,Batt= ux,Datt =0.

• Test 3: two load attachment points laterally constrained, ux,Aatt= ux,Eatt =0.

• Test 4: no lateral constraints on the beam.

29

3.3.1 EXPERIMENT SET-UP

An aluminum beam is suspended by two steel cables connected to the top flange. The axial stiffness of the cables is very large in comparison to the forces acting on them. Two buckets are attached to the bottom flange of the beam to apply the loads. Weights are added to the buckets to increase the load gradually and no connection between the beam and the fixed world is necessary. The cables are hung 825mm above the beam. The attachments of the cables on the beam are 1650mm apart. This results in a cable angle of 45 degrees. The two buckets are placed 50mm from the ends of the beam. A scheme of the set-up of the experiment can be seen in figure 3.1 and an overview pictures is seen in figure 3.10.

3.3.1.1 DIMENSIONS OF THE ALUMINUM BEAM

The beam is a 2750mm long, prismatic, I-section, aluminum beam with an AW-6060 T6 alloy. The height of the beam is 53mm, Both flanges are 1.5 mm in thickness and the web is 2.0 mm thick, see figure 3.3.

Figure 3.10: Overview picture of the experiment

30

3.3.1.2 CONECTIONS ON THE BEAM

The suspension point of the cable consists of a steel rod that is bent in a way that causes the cables to be on the same height and in line with the beam, see figure 3.11. The connection between the cable and the beam is constructed so the beam is able to rotate with minimum friction, see figure 3.12. The buckets are connected to the beam in similar manner also being able to rotate freely and minimizing the friction, see figure 3.13.

Figure 3.11: Suspension point of cables

Figure 3.12: Connection between cable and beam

Figure 3.13: Connection between bucket and beam

S

31

3.3.1.3 LOAD STEPS DURING EXPERIMENT

The load on the beam is gradually increased by adding equal weights simultaneously in the two buckets. After each increase in weight a picture is taken to determine the associated lateral displacements. At the start of the experiment the weight of the buckets itself are already present (step 0). During the experiment the weight added each step is decreased to evaluate the buckling and post buckling behavior with high accuracy, see table 3.7.

Table 3.7: Load steps during the experiment

step Ftotal (N) ∆F (N) step Ftotal (N) ∆F (N)

0 3.9 3.9 17 121.2 3.7 1 13.9 10.0 18 125.0 3.7 2 23.9 10.0 19 128.7 3.7 3 33.9 10.0 20 130.6 1.9 4 43.9 10.0 21 132.5 1.9 5 53.9 10.0 22 134.3 1.9 6 61.4 7.5 23 136.2 1.9 7 68.9 7.5 24 138.1 1.9 8 76.3 7.5 25 140.0 1.9 9 83.8 7.5 26 140.9 0.9

10 91.3 7.5 27 141.9 0.9 11 98.8 7.5 28 142.8 0.9 12 102.5 3.7 29 143.7 0.9 13 106.3 3.7 30 144.7 0.9 14 110.0 3.7 31 145.6 0.9 15 113.7 3.7 32 146.6 0.9 16 117.5 3.7 33 147.5 0.9

32

3.3.1.4 MEASURING LATERAL DISPLACEMENTS

In order to verify the FE-model comparison is needed between the load and associated displacements. Subsequently the displacements of the beam need to be determined. This is done taking a picture from above the beam at each load step, see figure 3.14.

Figure 3.14: Picture of the beam used to determine the locations of the red marks (test 1, load step 21)

Five red dots are marked on the top flange of the beam (Atop, Btop, Ctop, Dtop and Etop)

During the experiment the complete beam is able to rotate, therefore no absolute displacements can be determined. To determine displacements that correspond with the loads a picture is taken every load step, see figure 3.14. This picture is a top view of the beam. Software is used to determine the center coordinates of the red markers on the beam, using the bottom left corner of the picture as origin. Subsequently the displacements of a red marker relative to the other markers can be determined.

For verification of the numerical model the relative displacement between Btop, Ctop and Dtop are considered. This relative displacement is determined by evaluating the height (Δ) of the triangle constructed by these three points for every load step, see figure 3.15.

Figure 3.15: Scheme of determination of relative displacement h

33

3.3.2 RESULTS

During the experiment lateral torsional buckling occurred, see figure 3.16. This picture shows lateral displacement and twist in torsion of the beam which confirms lateral torsional buckling.

During the experiment without any lateral constraints it was possible to visually verify the fact that the four points where the cables and buckets were connected (Aatt, Batt, Datt and Eatt) can be laterally constrained. Although this is not definite no lateral displacement was observed at these positions, see figure 3.18. Although the top view scheme in figure 3.15 suggests that section A deflects laterally the scheme in figure 3.17 shows that Aatt does not.

Figure 3.16: Rotation about length axis, test 4 load step 21

34

Figure 3.17: Scheme showing no lateral displacement at point A, B and C

Figure 3.18: Visual confirmation of boundary conditions, test 4 load step 21

Section C

Section B

Section A

35

3.3.2.1 FORCE-DISPLACEMENT DIAGRAMS

The results of the four experiments are force-displacement diagrams, see figure 3.19. On the horizontal axis the relative displacement between the middle of the beam and the cable attachment point is set out. On vertical axis the force that acts on one end of the beam is set out. The four graphs represent the four experiments with different boundary conditions namely:

• Test 1: all four points laterally constrained.• Test 2: two cable attachment point laterally constrained.• Test 3: two load attachment points laterally constrained.• Test 4: no lateral constraints on the beam.

The four force-displacement diagrams show little differences. Subsequently it can be concluded that the boundary conditions applied to the beam during the different experiments have no influence on the behavior of the beam. Therefore the point where the cables and buckets are attached to the beam can be modelled laterally restrained in the numerical model.

Figure 3.19: force displacement diagrams of four experiments conducted.

36

4 FINITE ELEMENT MODEL

To achieve the goal of this research and develop a calculation method a FE-model is constructed.

For verification a numerical model representing the aluminum beam is constructed, see figure 4.1. The research on the suspended beams contains steel, rolled, I-section beams, therefore during the parameter study the material properties of the model are adapted. In rolled steel beams residual stresses may need to be taken into account. These properties may be added during the development of the new calculation method.

The numerical model is built in the FE-software ABAQUS 6.14. Appendix A holds the script of the model.

Figure 4.1: Numerical model that is used

37

4.1 MATERIAL PROPERTIES

4.1.1 ALUMINUM

During the analysis of the model representing the experiment, discussed in §3.3, stresses remain in the elastic range of the stress-strain diagram. Subsequently, no plasticity is modelled. The Young’s modulus is the average of the earlier, §3.2.2.2 table 3.2, determined Young’s moduli resulting in a Young’s modulus of 67692 N/mm2, see figure 4.2. The Poisson’s ration (ν) used for aluminum is 0.33.

4.1.2 STEEL

The numerical models with steel beams contain a Young’s modulus of 210000 N/mm2 and a The Poisson’s ration of 0.3. In the Linear Buckling Analysis (LBA) no plasticity is considered. However, stresses may exceed the yield strength of 235 N/mm2 during Geometrical and Material Nonlinear Imperfection Analysis (GMNIA). Subsequently, this yield strength is added resulting in a bilinear true stress-strain relationship, see figure 4.2.

Figure 4.2: True stress and strain diagrams of steel and aluminum

38

4.2 BOUNDARY CONDITIONS

4.2.1 SYMMETRY

The model is symmetric over the middle of the beam. Subsequently, half the beam can be modelled to decrease the computation time. Therefore, a symmetry boundary condition is placed at the end of the beam, see figure 4.3. All the points at the end of the beam have restricted movement in the z-direction and rotation about the x- and y-axis (U3=UR1=UR2=0).

To ensure that half the beam may be modelled the whole beam was also modelled, see figure 4.4. Both models show the exact same results when performing a LBA and a GMNIA, see figure 4.5. Subsequently the model using only half the beam in combination with a symmetry boundary condition may be used.

Figure 4.4: Numerical model of the whole beam Figure 4.3: symmetry boundary condition

Figure 5.5: Figure comparing the results of the whole and half beam

39

4.2.2 REFERENCE POINTS

Three reference points in the model are indicated as RP-1, RP-2 and RP-3, see figure 4.2. These correspond with A, B and S, see figure 3.1. The suspension point is constraint from movement in the x-, y- and z-direction (Ux=Uy=Uz=0). Points A and B are restrained from lateral movement (Ux=0) as discussed in paragraph 3.3.2.1

4.2.3 RIGID BODIES

The connection between the load and the beam is modelled using rigid bodies between the points Aatt and Abottom. The connection between the cable and beam is modelled the same using a rigid body between the points Batt and Bupper. A rigid body connection between two points causes the relative position of the nodes to stay constant throughout the analysis. Subsequently, the rigid body between the nodes cannot deform, but can undergo rotations.

4.2.4 LINK

The suspension cables between point S and Batt are modelled using a link constrained between the two corresponding reference points. This constrained connects associated points with hinges and keeps the distance between the points constants during the analysis, see figure 4.6.

4.3 LOADS

In an LBA the load, which is applied at Aatt, is used to indicate the direction of the load. Subsequently the program calculates at which load the bifurcation point of the load displacement relation is reached.

In the GMNIA the load is applied as a vertical displacement on Aatt. Subsequently the associated load is gained by measuring the vertical reaction force in point S. It is chosen to use a displacement instead of a force, because the model handles a maximum in the force displacement graph better.

4.4 ELEMENT TYPE

Shell elements are used for the model. The length and width of the beams is large compared to the thickness of the flanges and web besides this the stresses in the thickness direction of the elements are negligible, subsequently shell elements can be used. The elements used are called S4R-elements (shell, 4 nodes and reduced integration) in abaqus. These are so called general-purpose shells with four nodes at the corners and five integration points over the thickness in the middle of the element. All nodes have six degrees of freedom, three displacement and three rotational degrees of freedom. The element uses reduced integration which means there is only one integration point in a plane instead of four, Therefore calculation time of a model is reduced significantly. The thickness of the different elements is defined in section properties.

Figure 4.6: Scheme of link constrained [11]

40

4.5 SOLVING METHODS

During this project two different types of calculations are used in abaqus. Several different solving methods are possible for an LBA and GMNIA.

4.5.1 LINEAR BUCKLING ANALYSIS

To determine Mcr of a beam an LBA is executed. This analysis finds Mcr by calculating the bifurcation point. This point is calculated by solving an eigen value problem and determining when the stiffness matrix becomes singular and gives nontrivial solutions. This analysis is executed in abaqus using the linear perturbation procedure type with buckling. Abaqus gives Lanczos and subspace iteration as two possible solving methods for this analysis. However the Lanczos method cannot be used in combination with rigid bodies. Subsequently the subspace iteration method is used.

4.5.2 GEOMATRICAL AND MATERIAL NON-LINEAR ANALYSIS

A GMNIA is used to determine the maximum resistance of a model. During a GMNIA the calculations performed by the program are done using an iteration method. During this research the modified RIKS method in abaqus is used. This method is recommended to use when analyzing post buckling behavior, because this method is capable of calculating for example snapback in force or displacement. The Riks method includes nonlinear effects of large displacements in the model. The general calculation method in abaqus solves for either the displacement or the force using the other as a known quantity. Where the RIKS method solves both displacement and force simultaneously. Subsequently the arc-length is introduced as an additional quantity to be able to solve for both displacement and force simultaneously. This arc-length is used by the program to create a spherical path, using the last solution point as center, on which the next solution point is calculated throughiterations in a force-displacement space, see figure 4.7. Figure 4.7: visualization of the arc-length

41

4.6 VALIDATION WITH EXPERIMENT

The behavior of the FE-model including the links, rigid bodies and lateral restrained points is validated using the experiments discussed in paragraph 3.3. The beam in the experiment is an extruded aluminum profile, subsequently no residual stresses are considered. The distance between Abottom and Aatt or the eccentricity of the load is 15 mm as is the distance between Bupper and Batt or the eccentricity of the cable. The imperfection used in the GMNIA is obtained from an LBA. The imperfection shape and magnitude is unknown for the actual beam used for the experiment. Therefore a range of different maximum magnitudes of the imperfection shape is used. The modulus of Elasticity is set to 68014 N/mm2 as discussed in paragraph 3.2.2.1. To compare the FE-model with the experiment the result is plotted including the results of the experiment from figure 3.19, see figure 4.8. The behavior of the beam in the FE-model compares very well to the behavior of the beam during the experiment, subsequently this model can be used to evaluate other configurations.

Figure 4.8: Result of the FE-model compared to the experimental results

42

4.7 ADAPTIONS FOR STEEL PROFILES

4.7.1 FILLET RADIUS

The currently used calculation methods to determine the lateral torsional buckling resistance use Iy, Iz, It, Iwa and Wy as cross sectional properties of a beam. Subsequently the same cross sectional properties will be used in the calculation method for the lateral torsional buckling resistance of spreader beams in a hoisting structure. Therefore it is not needed to incorporate the fillet radius in the FE-model. It is a time consuming process to add the fillet radius of a profile in the FE-model and a number of different profiles are used to develop the calculation method. Common profiles such as HEA400 without the fillet radius will be indicated as HEA400*.

4.7.2 CROSS SECTIONAL PROPERTIES

When an I-shaped beam is constructed in abaqus using shell elements it is possible to offset the flanges to avoid a geometrical error where the flanges and web overlap. This is done during the section assignment by offsetting the top flange to the top surface and the bottom flange to the bottom surface.

The four parameters to construct an I-shaped profile, which is symmetrical over the height and width, are the height of the web (hw), the thickness of the web (tw), the width of the flanges (b) and the thickness of the flanges (tf). These parameters are shown for an HEA400*, see figure 4.9.

Using these parameters the cross sectional properties Iy, Iz, It, Iwa and Wy of a beam can be calculated using formula’s 4.1 To 4.6.

Figure 4.9: Four parameters in a HEA400* cross section

43

Figure 4.10: Eigen mode

23 31 1 1 1( 2 ) 2 2 ( 2 )

12 12 2 2y w f f f f fI t h t bt t b h t t = − + ⋅ + ⋅ − +

(4.1)

3 31 12 ( 2 )12 12z f f wI t b h t t= ⋅ + − (4.2)

3 31 12 ( 2 )3 3t f f wI bt h t t= ⋅ + − (4.3)

2 31 ( 2 )24wa f fI t h t b= − (4.4)

( ) 2,

1( 2 ) ( 2 )4pl y f f f w fW t b t h t t h t= + − + − (4.5)

, 12

yel y

IW

h= (4.6)

tf : thickness of the flanges Wpl,y : plastic section modulus Wel,y : elastic section modulus

4.7.3 IMPERFECTIONS

The Eigen mode obtained from the LBA, see figure 4.10 where the blue crosses are the reference points, as discussed in §4.2.2, is inserted into GMNIA as imperfection. The magnitude of the maximum displacement, located at Cbottom, is defined by determining the difference in displacement in the z-direction of Bbottom and Cbottom. This distance is then scaled to a factor of the length between B and D (LBD). Two different scaling factors are used. One when residual stresses are incorporated in the model and another when residual stresses are not in the model. For the model including residual stresses an imperfection of L/1000 is used. This a generally accepted approach. When residual stresses are neglected the Eurocode [1] gives the values of the imperfections that should be taken into account, see table 4.1. Although the Eurocode gives more information about imperfections along with this table it is chosen to use the eigen mode as an imperfection with a magnitude of L/250.

44

Table 4.1: values of the initial imperfection for beams [1]

Buckling curve

According to table 6.2 [1]

Elastic calculation Plastic calculation

eo/L eo/L

a0 1/350 1/300

a 1/300 1/250

b 1/250 1/200

c 1/200 1/150

d 1/150 1/100

The two different methods to introduce imperfection with or without residual stresses is compared using a simply supported beam with fork end supports, see figure 4.11. Different lengths of the beam are modelled to ensure different slendernesses are considered. The results are plotted in figure 4.10 Together with the buckling curves that are currently used in EN 1993-1-1 [1]. The slenderness and reduction factor are calculated using equations 1.4 and 1.1.

Figure 4.11: Scheme of a simply supported beam with fork ended support

45

The results of these calculations are plotted together with the according buckling curves, see figure 4.12. The red markers show the results of the models where residual stresses are taken into account and an imperfection of L/1000 is used. The blue markers show the results of the model with the bigger imperfection of L/250. The difference between the red markers and the buckling curve is negligible compared to the difference between the blue markers and the buckling curve in the region of low slenderness. When slenderness is higher both results are comparable accurate. Therefore during this project an imperfection of L/1000 including residual stresses will be modelled.

Figure 4.12: Results of comparison between two different imperfection possibilities

46

4.7.4 RESIDUAL STRESSES

As a result of the production process of hot rolled steel I-shaped beams residual stresses remain in the beam. These compressive and tensile stresses in the longitudinal direction can be simplified to linear triangular shapes, see figure 4.13. Equations 4.6 and 4.7 are used to calculate the values of σr.

Figure 4.13: residual stress pattern of a hot rolled, Figure 4.14: visualization of a discrete steel, I-shaped beam residual stress pattern.

<1.2 =0.5fr yhb

σ→ (4.6)

>1.2 =0.3fr yhb

σ→ (4.7)

rσ : Residual stress where a positive sign indicates a tensile stress and negative sign a compressive stress

The program abaqus does not allow for an linear distributed residual stress pattern. Therefore, the residual stresses are approximated using a discrete pattern, see figure 4.14. This example shows a distribution where the flanges and web are divided in eight smaller areas

To ensure results are accurate a small study is done to determine how many partitions are needed for the discrete initial stress pattern. Four different patterns are used, see figure 4.15. The results show that the pattern with 32 partitions is very accurate when compared to the analytical calculation using the Eurocode [1], see figure 4.16. Subsequently, this pattern is used.

47

Figure 4.15: visualization of discrete residual stress patterns using 4, 8, 16 and 32 partitions

Figure 4.16: results residual stress pattern study

48

4.9 MESH

To ensure the results of the FE-model are accurate a mesh convergence study is executed. This is done using a simply supported HEA400* beam with fork ended supports, a length of 9000 mm including residual stresses, see figure 4.10. The Eigen mode of the LBA is inserted in the GMNIA as imperfection with a maximum displacement of Lg/1000. The analytical answer for Mcr and Mb,Rd is known for this system, subsequently it is possible to ensure accurate results from the model.

Using equations 1.1 to 1.5 the analytical values of Mcr and Mb,Rd of this beam can be determined.

6

6,

643 10

405 10cr

b Rd

M Nmm

M Nmm

= ⋅

= ⋅

During the mesh convergence study three different meshes are used, see figure 4.17. The meshes have 16, 32 and 64 elements over the length of the flanges and web. Residual stresses must be constant on one element. Subsequently the model using 16 elements has a residual stress pattern including 16 partitions. The length of the elements is approximately 5 times the width of the elements. The different meshes are used to evaluate Mcr and Mb,Rd, these are compared to the analytical values, see table 4.2. The results show a mesh with 32 elements along the flanges and web is accurate and is chosen to be used.

Figure 4.17: Visualization of the three meshes used during the mesh convergence study

49

Table 4.2: mesh convergence study

Number of elements along the flanges and web

Mcr using LBA (kNm)

Mcr analytical (kNm)

Difference Mb,Rd using GMNIA (kNm)

Mb,Rd analytical (kNm)

Difference

16 659.34 643.07 2.53% 392.85 405.17 -3.04%

32 657.45 643.07 2.24% 405.42 405.17 0.06%

64 655.98 643.07 2.01% 403.85 405.17 -0.33%

4.10 VALIDATION PLASTIC MOMENT RESISTANCE

Initially the cross sectional dimensions are validated by determining the plastic moment resistance around the y-axis (Mpl,y) analytically, using equation 4.5, and using the FE-model. The difference between the analytical value of 577.0∙106 Nmm and the value obtained with the FE-model of 575.2∙106 Nmm is 0.3%. Although this difference is negligible the value obtained using the FE-model is too low. This can be explained using figure 4.18, which shows a part of the beam at the maximum moment. In this figure the elements that are colored red are already plasticized and the blue elements are still in the elastic phase. Few elements in the middle of the web did not reach plasticity which explains the lower plastic moment resistance.

Figure 4.18: Visualization of a part the beam in bending. Elements coloured red are in the plastic branch of the stress strain diagram and the blue coloured elements are in the elastic branch.

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5 DEVELOPMENT DESIGN RULE

In this chapter a design rule for the lateral torsional buckling resistance of an I-shaped spreader beam is developed.

At first the boundaries of the parameters are determined for an I-shaped spreader beam. To be able to develop a design rule, the behavior of a spreader beam is investigated. This is done by first modelling a beam with fork ended supports and a constant bending moment and changing this in small steps to the final model of an I-shaped spreader beam.

Currently the Eurocode NEN-EN 1993-1-1 [1] provides a method to determine the lateral torsional buckling resistance of a beam, see equations 1.1 to 1.4. As stated, this method requires the critical moment to be known. A method to determine the critical moment is provided in the Dutch National Annex of the code. Since these two calculation methods are currently provided in two different codes, two separate design rules are developed. One to determine the critical moment and one to determine the lateral torsional buckling resistance. Both of the design rules are developed in a way that implementation in the current calculation methods is straightforward.

5.1 PARAMETERS

5.1.1 SECTION PROPERTIES OF THE BEAM In this project the section dimensions of common (IPE, HEA and HEB) I-shaped steel beams are considered. When designing a spreader beam, deflections of the beam are not taken into account. Therefore, only beams with the capability of reaching the plastic moment capacity are considered, subsequently only beams of class 1 and 2 are considered.

5.1.2 CABLE ANGLE

According to Dutch safety regulations [15], the angle a between the two cables may not exceed 120°. This is because of the fact that above this angle the tension force in the cable becomes bigger than the weight of the load lifted. Therefore it is too difficult to estimate what forces are in the cables. This leads to an angle a with a maximum of 120°.

Fourteen different models using all ranges of all parameters have been created and modelled using an angle a of 120°. The same models have been created using an angle between the cable and beam of 90°. These are two extreme values of this parameter and causes a difference in the normal force between the cable attachment points. The difference between the critical moment, obtained via an LBA, of both models and the maximum moment resistance, obtained via a GMNIA, of both models is negligible, see table 5.2. When the cable angle a is smaller the normal force in the beam gets smaller, subsequently the beams with the cable perpendicular to the beam have a higher moment resistance. Therefore, only a

51

cable angle of 120° will be considered during the development of the calculation methods for both Mcr and Mb,Rd.

This negligible difference might be caused by the fact that the occurring axial compressive force is not present in critical parts of the cross section of the beam. Therefore, the axial stresses in different parts of the cross sections will be compared between two models.

Lateral torsional buckling is caused by lateral buckling of the flange and a small part of the web in the compression zone, in case of a spreader beam, the bottom flange. The normal force in the beam is caused by the cables. These cables are attached at the top of the beam in the tension zone. This axial compressive force, only occurring when the angle between cable and beam is below 90o, might therefore only influence the tension zone.

The compressive axial forces in the bottom flanges are approximately the same for a model with an angle a of 120o and a model where the cables and the beam are perpendicular. This fact is checked by comparing the axial stresses of the top and bottom flanges in the middle of the beam for a model with an angle a of 120o and a model where the cables and the beam are perpendicular. Although the force and moment are comparable, deflections are different. Therefore, stresses are different between the two models. Subsequently, this comparison is done with two models which have lateral restraints in the top flange, so no actual lateral torsional buckling occurs. The model used for comparison is the model in the top row of table 5.2. The axial stresses are obtained at the nodes in the middle of the beam in the top and bottom flange. This value is the average of the stresses in the integration points of the elements attached to the node. The results show a 1.1% higher average axial compressive stress in the bottom flange in the model with an a of 120o and a 7.9% lower average axial tensile stress in the top flange, see table 5.1. This shows that the normal force induced by the cables under an angle is almost completely taken by the top flange in the tension zone. Subsequently the compression stresses in the bottom flange are not much higher when the cables are under an steeper angle. Therefore the lateral torsional buckling resistance is nearly equal.

Table 5.1: Axial stresses of a model with α=120o and a model with perpendicular beams and cables

Perpendicular Beam and cables average stress (N/mm2)

α=120o

average stress (N/mm2)

Difference

Top flange (tension)

238.00 219.11 -7.9%

Bottom flange (compression

235.63 238.20 1.1%

52

Table 5.2: comparing models with different cable angles

profile fy (N/ mm2)

LAE(m)

LBD(m)

Ecc (mm)

Mcr,α=120(N)

Mcr,perp

(N) Mb,Rd,α=120(N)

Mb,Rd,α=perp(N)

HEA450* 235 14.5 7.26 133 4.91∙108 4.92∙108 0.2% 3.83∙108 3.89E+08 1.6% HEB1000* 355 15.0 7.5 162 1.50∙109 1.52∙109 1.1% 1.32∙109 1.34E+09 2.1% HEB1000* 355 15.0 7.5 43 1.31∙109 1.35∙109 2.5% 1.18∙109 1.21E+09 2.6% HEB1000* 460 15.0 3.45 162 1.63∙109 1.63∙109 0.1% 1.44∙109 1.45E+09 0.5% HEB1000* 460 15.0 3.45 77 1.47∙109 1.47∙109 0.3% 1.33∙109 1.33E+09 0.5% HEB200* 355 3.0 0.69 61 9.38∙108 9.23∙108 -1.5% 1.96∙108 1.95E+08 -0.5% HEB200* 460 3.0 0.69 10 7.46∙108 7.47∙108 0.2% 2.46∙108 2.52E+08 2.6% HEB600* 460 9.0 4.5 162 2.08∙109 2.03∙109 -2.3% 1.50∙109 1.53E+09 2.4% HEB600* 460 9.0 4.5 26 1.68∙109 1.71∙109 1.9% 1.35∙109 1.36E+09 1.3% IPE100* 235 1.5 0.75 30 1.57∙107 1.54∙107 -1.7% 6.89∙106 6.92E+06 0.5% IPE360* 235 5.4 1.24 28 2.45∙108 2.45∙108 0.1% 1.33∙108 1.33E+08 0.2% IPE360* 355 5.4 1.24 16 2.34∙108 2.34∙108 0.2% 1.59∙108 1.59E+08 -0.4% IPE600* 235 9.0 2.07 26 4.69∙108 4.71∙108 0.3% 3.17∙108 3.15E+08 -0.4% IPE600* 355 9.0 2.07 162 5.97∙108 5.93∙108 -0.7% 4.28∙108 4.41E+08 3.0%

5.1.3 TOTAL BEAM LENGTH

The maximum total beam length (LAE) is 28 meters which is the longest beam that is generally available. Rules of thumb are used in preliminary structural designs to determine beam heights. Commonly these rules have L/h ratios of 20 to 40 [16]. To cover all commonly used beam lengths a higher ratio of 50 is used, subsequently the maximum length of a beam can be determined using equation 5.1.

,max min(50 ;28000 )AEL h mm= ⋅ (5.1)

As stated before the beams are able to reach the plastic moment capacity. The maximum moment is between point B and D, see figure 5.11. Besides the moment, a shear force is present between points A and B. When shear force and a bending moment are combined on a steel I-shaped beam, the Eurocode [1] prescribes that the moment resistance of the beam does not have to be reduced if the shear force acting on the beam is lower than half the plastic shear force resistance of the beam. When LAB has a sufficient length reduction of the moment resistance is not needed, see equations 5.2 to 5.5. Equation 5.3 seems very conservative, because the Eurocode allows for a part of the flanges to be added to the total shear area. However, in the models during this project no fillet radii are modelled. This minimum value for LAB is used to determine the minimum value for the total length of the beam and the maximum value of the ratio between LBD and LAE. Extreme values are used to define the boundaries of the parameters. Subsequently, not all beams will have shear forces lower than half the plastic resistance, this will also result in beams failing in shear.

53

EdF V= (5.2) 1V =t h f3pl w w y (5.3)

max ,max1 1 1V =V = t h f 2 2 3Ed pl Ed w w yV F≤ → (5.4)

, , ,,min

max=1 1 1 1t h f t h

2 23 3

pl y pl y pl yAB

w w y w w

M M WL

F= =

(5.5)

F : Force at the end of the beam VEd : Design shear force Vpl : Plastic shear force capacity

The minimum length of a beam, and the maximum ratio between the values of LBD and LAE is determined using equations 5.6 to 5.8. All the results of these equations for all different sections are shown in the table in appendix B. An absolute minimum length of approximately 1500mm is found. Besides this, the ratio between LAE,min/hbeam is considered and the lowest ratio is approximately 15. Subsequently, the minimum value of LAE can be determined using equation 5.9.

,max ,min

,maxmax

2AE ABBD

AE AE

L LLL L

− ⋅ =

(5.6)

,min ,min

max

12AE ABBD

AE

L LLL

= ⋅

(5.7)

2BD AE ABL L L= − ⋅ (5.8)

,min min(15 ;1500 )AEL h mm= ⋅ (5.9)

5.1.4 DISTANCE BETWEEN CABLE ATTACHMENT POINTS

The distance between the two cable attachment points (LBD) is smaller than the distance between the loads, because if LBD=LAE only normal force will act on the beam. Subsequently, lateral torsional buckling is not a relevant failure mechanism. The maximum ratio of LBD/LAE is calculated using equations 5.6 and 5.7.

Appendix B shows the table with this ratio for all cross sections considered, calculated using equation 5.6. The maximum ratio is 0.77, subsequently LBD,max=0.77∙LAE

When the distance between the cable attachments is very small the use of one cable in the middle would become more suitable. Besides this, when the angle between the cables is at 120° the distance between the beam and the suspension point decreases when LBD decreases, see figure 5.1. In this space the connections have to be made. Therefore, a minimum distance of 100mm is chosen to ensure making the connections is still possible.

54

Figure 5.1: Visualization of the biggest and smallest ratio LBD/LAE with a cable angle a of 120o

For an angle between cable and beam of 30°, the minimum value for LAE of 1500mm and a minimum distance between beam and suspension point of 100mm the minimum value for LBD is 346mm. Subsequently, LBD,min =0.23∙LAE. The biggest and smallest ratio of LBD/LAE are shown in figure 5.1.

5.1.5 EXCENTRICITIES OF THE CABLE AND LOAD

A spreader beam usually has a strip on the top and bottom flange with holes to attach the cables or chains. Subsequently, an eccentricity is introduced where the loads and cables are attached. When beams for small loads are used, the loads and the cables are accordingly thin. Therefore the eccentricity may be small for both the cables and loads. When loads increase, so will the attachment equipment.

To determine the minimum and maximum eccentricity for the considered beams, first the maximum and minimum tension force in the cables are calculated using equation 5.9 to 5.12.

,max

,min

pl y

AB

MF

L= (5.9)

,max,max max

,min2 2

cos60pl y

t oAB

MFF FL

= = = (5.10)

,max ,maxmax

0.5 1 BDAB AE

AE

LL LL

= − ⋅

(5.11)

55

, ,,min min

,max,max

max

0.5 1

pl y pl yt

AB BDAE

AE

M MF F

L L LL

= = =

− ⋅

(5.12)

Ft : Tension force in the cables

The calculation of a pin connection described in the Eurocode [21] is used to estimate the maximum and minimum dimensions of the connections between beams, loads and cables. The resistance of a pin connection is calculated using equation 5.13.

2

,0.6

F upV Rd

m

Afγ

= (5.13)

, 235 2360up SNf

mm=

, 355 2490up SNf

mm=

, 460 2570up SNf

mm=

FV,Rd : Shear force resistance in connection in this case equal to Ft,max or Ft,min

fup : tensile strength

Using equation 5.13 and inserting the maximum and minimum tension force for the resistance of the pin the minimum and maximum required diameter of the circular pin can be calculated, see equations 5.14 to 5.16. Using the diameter of the pin, the maximum and minimum eccentricity can be calculated using equations 5.17 and 5.18. In these equations the diameter is enlarged because of hole clearance. A common way of producing a spreader beam includes a strip welded on top with holes in it, so different configurations can be used for various situations, see figure 5.2. For this reason the maximum eccentricity is increased with 50% in equation 5.18. An absolute minimum for the eccentricity is found when a small ring is welded on a small beam where cables can be hooked on, see figure 5.3. This figure shows an eccentricity of 10.0mm which seems fairly small, subsequently emphasizing that the whole range is considered. In appendix C.1, C.2 and C.3 these calculations are done for all beams considered for steel grades S235, S355 and S460. From these tables it can be derived that the minimum eccentricity e=h/23 and the maximum eccentricity e=h/3.3. Including the absolute minimum of 10.0mm and the absolute maximum found in the table of approximately 162.0mm the minimum and maximum eccentricities are calculated using equations 5.19 and 5.20. Where the maximum value of the eccentricity seems very large this only emphasizes that the whole range of possible eccentricities is considered.

2 ,min,min

F0.6M t

pinup

Af

γ= (5.14)

56

2 ,max,max

F0.6M t

pinup

Af

γ= (5.15)

4pin

ADπ

= (5.16)

min 3e D= + (5.17)

( )max 3 1.5e D= + ⋅ (5.18)

min min ;8.023he =

(5.19)

max min ;162.03.3he =

(5.20)

Apin : Cross sectional area of the pin in the connection

2Mγ : Partial safety factor for the resistance on cross sections considering fracture because of tensile stress

Dpin : Diameter of the circular pin e : eccentricity of the attachment of cables and loads

Figure 5.2: Visualization of an HEA550 beam with the maximum eccentricity using a strip with holes

Figure 5.3: Visualization of an IPE80 beam with the minimum eccentricity using rings welded on the beam

5.1.5 YIELD STRESS

In this project three different yield stresses are considered. The most commonly used are 235 N/mm2, 355 N/mm2 and 460 N/mm2.

57

Table 5.3: Overview results changing the model from simply supported with fork ended supports to the final model

5.2 CHANGE FROM SIMPLY SUPPORTED BEAM TO THE FINAL MODEL

The simply supported model with fork ended supports, see figure 4.11, is changed step by step to the final model to evaluate the influence of the different changes on the model. This is done in seven steps. The models created in these steps are calculated using an LBA to determine the critical buckling force and associated Eigen mode. Subsequently, this Eigen mode is inserted as imperfection into a GMNIA calculation. The maximum magnitude of this imperfection is determined by scaling the difference in lateral deflection of Bbottom and Cbottom to LBD/250. The mechanical schemes show half the beam, which can be done because of symmetry. However, the mechanical scheme of the original is the full scheme. The results of the different models are shown in table 5.3. For comparison purposes the vertical forces are converted to moments by multiplying them with the distance LAB, in this case 3500mm.

Mcr LBA (kNm)

Mb,R GMNIA (kNm)

Imperfection (mm)

Mechanical scheme

Original 641 422 36 mm

Step 1 190 258 97 mm

Step 2 251 239 98 mm

Step 3 294 249 83 mm

Step 4 353 270 95 mm

58

5.2.1 STEP ZERO: ORIGINAL BEAM WITH FORK ENDED SUPPORTS

The original beam with fork ended supports is used in §4.7.3 to determine the way of implementing the imperfections, see figure 4.11. The HEA400* beam with a length of 9000mm is used for evaluation of the influence of the changes in the next six sections. The LBA of this model gives a critical moment of 641 kNm and the GMNIA gives a maximum lateral torsional buckling resistance of 422 kNm.

5.2.2 STEP ONE: NO FORK ENDED SUPPORTS In the first step the beam is longer and the fork ended conditions are changed. In the original scheme the top and bottom flange were laterally restrained at one cross section. However in this step the bottom flange is supported at the end of the beam where later the vertical load is applied. The same constant bending moment is applied on the same length of the beam, see figure 5.4. Both Mcr and Mb,Rd decrease significantly. The critical moment decreases 70.4% and the maximum resistance decreases 38.9%. This decrease is caused because of the increasing beam length and the fact that the beam has more freedom to rotate. When fork ended conditions are present the beam can not rotate around its own axis, Subsequently when only the top or bottom flange is supported laterally the beam can.

Figure 5.4: Mechanical scheme of step 1

Step 5 377 279 85 mm

Step 6 382 278 86 mm

59

Δ (mm)

Because this decrease in resistance is very drastic the change to this scheme is done in steps. The length between the two supports is increased gradually. This is done to make sure no unexpected behavior occurred in between the two schemes. The results show no abnormalities, see figure 5.5. This figure shows the relation of the moment and the relative displacement Δ of the beam modelled with different LAB. It is clear that the maximum lateral torsional buckling resistance decreases when LAB increases. Moreover, the stiffness decreases as well. This can be seen by looking at the slope of the lines in the elastic branche of the graphs.

5.2.3 STEP TWO: VERTICAL FORCE INSTEAD OF CONSTANT MOMENT

In this step the cable support at the top flange of the beam is introduced and the constant bending moment is replaced by a vertical force on the end of the beam. Subsequently, a linearly distributed bending moment and a constant shear force act on the beam between the cable attachment and the vertical force, see figure 5.6. Although extra bending moment and shear force act on the beam compared to the previous step the critical moment increases with 32.1%. This is caused by the vertical force which “pulls” the beam back into a straight position. However the maximum resistance in this step decreases slightly with 7.4% compared to the previous step.

The effect of the force pulling the beam straight at the end is minimized when the force is applied at the middle of the web. To show that this effect is the cause of the increase in critical moment the force is first applied at the middle of the beam and lowered until it is applied at the bottom. The results show a linear increase in both Mcr and Mb,Rd, see figures 5.7 And 5.8. This ensures the straightening effect of the vertical force causes the increase.

Figure 5.5: figure showing the relations between models with increasing LAB

60

Δ (mm)

Figure 5.6: mechanical scheme of step 2

Figure 5.7: Figure showing the relations when the force is applied at different heights

61

5.2.4 STEP THREE: INCLUDING EXCENTRICITY TO THE FORCE

In this step the eccentricity is added to the point where the force is applied, see figure 5.9. This results in a 17.1% higher critical moment and 4.2% increase of the maximum resistance. This increase is caused by the same effect as the increase in the previous step. The effect of the force pulling the beam back into a straight position is increased. This eccentricity is gradually increased up to 200mm. The results of this gradual increase shows the effect increases when the eccentricity is enlarged, see figure 5.11, where the relation of the increase in eccentricity appears linear when compared to Mcr and Mmax.

Figure 5.9: Mechanical scheme step three

Figure 5.8: Figure showing Mb,Rd and Mcr for different heights of applying the load

62

5.2.5 STEP FOUR: INCLUDING EXCENTRICITY TO THE CABLE ATTACHMENT

In this step the eccentricity is added at the point where the cable is attached to the beam. However, the eccentricity at the load point is removed, see figure 5.12. Subsequently, this step will be compared to step 2. This change shows the same effect as step three. However, the influence is bigger. The increase of the critical moment is 40.6% and the increase of the maximum resistance is 13.0%. The maximum lateral displacement of the beam when lateral torsional buckling occurs is at the middle. The cable attachment point are situated more to the middle of the beam. Therefore, the “pulling” effect is presumably stronger when the eccentricity is added at the cable attachments when compared to the added eccentricity at the load attachment points. This eccentricity is applied gradually increasing in size. This shows a linear distribution of the increase of both the critical moment and the maximum resistance, see figure 5.13.

Figure 5.12: Mechanical scheme of step 4

Figure 5.11: Relation of Mb,Rd and Mcr with increasing load eccentricity

63

Figure 5.13: Relation between Mb,Rd and Mcr with increasing cable eccentricity

5.2.6 STEP FIVE: COMBINING BOTH EXCENTRICITIES

The second to last step is combining the eccentricities at the loads and the cables, see figure 5.14. When compared to step 2 the increase of the critical moment is 50.2%. This is close to the sum of increase in step 3 and 4 which is 57.8%. The increase of the maximum resistance compared to step 2 is 16.7%. This difference between this increase and the sum of the increase in step 3 and 4, which is 17.2%, is also very small. This result shows that the effect of the forces pulling the beam back into a straight position when both eccentricities are present is almost the sum of the influence of the two individual eccentricities.

Figure 5.14: Mechanical scheme of step 5

64

5.2.7 STEP SIX: CHANGING CABLE ANGLE

In the final step the angle between the cable and beam is changed from 90o to 30o, see figure 5.15. As stated before in §5.1.2 this change causes a normal force in the beam between the cable attachment points. Although the critical moment shows an increase of 1.3% the maximum resistance shows a decrease of 0.4% it may be concluded that the influence of the cable angle is negligible for this model.

5.2.7 CONCLUSIONS

Step 1 shows that removing the fork ended support has a big negative influence on the lateral torsional buckling resistance of a beam. This is caused by the increase of the total length of the beam, but also because of the different placement of the lateral restraints. At no cross section along the beam length rotation of the beam is prevented.

Steps two to five clearly show that the location, with respect to the neutral axis, where the force is applied has a very large influence on both the critical moment and the ultimate lateral torsional buckling resistance of a spreader beam. The critical moment increases each step. The ultimate LTB resistance decreases slightly between steps two and three slightly, which might be because of the added moment on part AB. However, when the eccentricities of the cables and loads are introduced the ultimate resistance increases because the force “pulls” the beam back into a straight position.

The final step shows that the influence of the cable angle is negligible. This was earlier discussed in §5.1.2.

Figure 5.15: Mechanical scheme of step 6

65

5.3 DEVELOP A METHOD TO DETERMINE THE CRITICAL MOMENT

In the currently used Dutch national annex of NEN-EN 1993-1-1 [1] a procedure to determine the critical moment is provided, see equations 5.21 to 5.23. However, this calculation method may only be used if the beam has fork ended conditions and the load is not applied above or below 0.1h of the beam. Subsequently, this method cannot be used for a spreader beam. The calculation method in the Dutch national annex uses two coefficients, C1 and C2, to determine the critical moment. Where C1 is dependent on the mechanical scheme of the structure and C2 is dependent on the height where the force is applied compared to the neutral axis of the beam. In this research that same calculation method is used. Subsequently, C1 and C2 need to be determined to create a new method which can be used to obtain the critical moment of a spreader beam. Table NB.6 of the calculation method in the Dutch national annex [1] shows that C1 and C2 change when the type of loading changes. Subsequently, C1 and C2 for a spreader beam will be developed as equations with different variables. During this project multiple ways were attempted to create this calculation method. The sets of models used to determine C1 and C2 are collections of all LBA done during this project and include a very wide range of the different parameters defined in §5.1.

,M=cr z tg

CM EI GIL

(5.21)

( )2 2

1 22 22

CC= 1 1g

kip kipkip

L S SC CL LL

π π π + + +

(5.22)

S= w

t

EIGI

(5.23)

Lkip g AEL L= = (5.24) C : Coefficient dependent on beam length, cross sectional properties, nature

and application point of load

First the coefficient C1 is determined. To do this a set of 421 LBA have been used in which the load is applied in the heart of the flange and eccentricities are zero, see figure 5.17, and the critical moment is calculated, see appendix D.1 for a table with all models. Combining equations 5.21 to 5.24 and solving this for C1 gives equation 5.25. Using equation 5.25 the coefficient C1 can be determined for all different models. However in this equation the coefficient C2 is still a variable. Subsequently, a provisionally coefficient for C2 is determined first to make the determination of C1 possible. This first determination of C2 will not be used later to determine the final coefficient for C2, Instead it is just used as a tool to make determination of C1 possible.

66

When the coefficient C1 is known the coefficient C2 is determined using a set of 295 models in which various eccentricities of the cables and loads are present, see appendix D.2 for a table with all models. Combining equations 5.21 to 5.24 and solving this for C2 gives equation 5.26. Using equations 5.26 and 5.35 which is determined in §5.3.2 the coefficient C2 can be determined for all different models.

Finally when the coefficients of C1 and C2 are known a full calculation method to determine the critical moment for a spreader beam within the parameters discussed in §5.1 is can be set out. This calculation method will be reviewed using a statistical analysis comparing it with the set of 295 models including eccentricities.

( )1

2 2 22

C

11

cr kip

w

w tz t

t kip kip

M L

EICEI C GIEI GI

GI L L

πππ

= + + +

(5.25)

2 2 4 2 2 2 4 21 1

22 2

1

1C2

w z t z kip kip cr

wkip cr z t

t

C E I I C EGI I L L MEIC L M EI GIGI

π π

π

+ −=

(5.26)

C1 : Coefficient dependent on the nature of the load C2 : coefficient dependent on the application point of the load relative to the

neutral axis

5.3.1 DETERMINE PROVISIONAL COEFFICIENT C2

In the calculation method to determine the critical moment provided by the Dutch national annex of NEN-EN 1993-1-1 [1] the coefficient C2 is zero when loads are applied at the neutral axis of a beam. Using this, the critical moment obtained via an LBA and equation 5.25 the coefficient C1 can be determined for a beam. Two simple models have been created, see figure 5.16 and 5.17. These models have been created using an HEA400 beam with a length of 9 meters. The load in the first model (figure 5.16) is applied on the neutral axis of the beam and as explained before can be used to determine C1. Since C1 changes when the mechanical scheme changes the distance LBD is varied from 3LAE/4 to LAE/4 or 6.75m to 2.25m. For all different lengths C1 is determined using equation 5.25, see table 5.4. When C1 is known for different lengths of LBD C2 can be obtained using the second model (figure 5.17). In this model the load is applied on the bottom flange. C2 also changes when the mechanical scheme changes. Subsequently the same varying length of LBD is used as in the first model. For all lengths of LBD C2 is determined using equation 5.26, see table 5.4. Finally the least square method is used to determine a formula that describes C2 with LBD/LAE as variable, see equation 5.27. This is the provisional value for C2 used to determine coefficient C1.

2

2, =-0.6 0.056 0.558BD BDprovisional

AE AE

L LCL L

− +

(5.27)

67

Figure 5.16: Scheme of FE-model to determine C1

Figure 5.17: Scheme of FE-model to determine C2,provisional and C1

Table 5.4: C1 and C2 determined using simple models with varying LBD/LAE

LBD/LAE Mcr (kNm) first model

(load on neutral axis)

C1 (-) Mcr (kNm) second model

(load on bottom flange)

C2 (-)

12/16 593.96 0.924 662.34 0.182 11/16 579.26 0.901 666.25 0.234 10/16 563.23 0.876 668.16 0.286 9/16 547.95 0.852 669.33 0.337 8/16 534.81 0.832 670.75 0.381 7/16 524.65 0.816 673.33 0.421 6/16 518.02 0.806 677.86 0.453 5/16 515.36 0.801 685.28 0.483 4/16 517.10 0.804 696.45 0.505

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5.3.2 DETERMINE COEFFICIENT C1

Using equation 5.25 C1 can be determined for all 421 models used, see figure 5.14. A new factor is introduced to ensure the calculation method works well for all beam types and all lengths within the parameters defined in §5.1. The new factor fsb=LAEtf/bh is similar to the equation of the lateral torsion buckling stresses, see equation 5.28. Both factors were tested and the new factor works better when creating the calculation model for a spreader beam.

,0.66E=cr LTB

f

Lhbt

σ(5.28)

,cr LTBσ : Critical stress for lateral torsional buckling

When using only the variable LAEtf/bh a large deviation in the calculated C1 is found, see figure 5.18. Subsequently another variable is needed to create an accurate calculation method. C1 is supposed to be dependent of the nature of load application on the beam, the length and type of beam is already incorporated in the first variable. Subsequently the other variable is the ratio between the lengths LBD and LAE. When all results as shown in figure 5.18 are separated by this ratio, different lines of points can be distinguished. Different ratios have been given different colors in the graph. These different lines can be approximated using a second degree polynomial with the factor LAEtf/bh as a variable, see equation 5.29 and 5.30. In equation 5.29 the parameters aC1, bC1 and cC1 can be determined using the least square method. However, they will be different for each line of points considered. Subsequently a variable needs to be incorporated in these parameters. This is the factor LBD/LAE which is inserted as a linear function, see equations 5.31 to 5.33. The least square method has been used to determine the regression coefficients. This equation gives an average absolute difference of 2.2%.

1 1 1

21 C sb C sb CC a f b f c= + + (5.29)

AE fsb

L tfbh

= (5.30)

1a = 0.019 0.0185BD

CAE

LL

− + (5.31)

1=0.184 0.248BD

CAE

LbL

− (5.32)

1c = 0.117 1.11BD

CAE

LL

− + (5.33)

69

Figure 5.15: Visualization of all models comparing the factor LAEtf/bh to the calculated C1

5.3.3 DETERMINE COEFFICIENT C2

The coefficient C2 is developed in a similar way as C1, using a set of 295 models including eccentricities of loads and cable attachments, see figure 5.19 for model type and appendix D.2 for the full table of models. The same factor fsb=LAEtf/bh is used to ensure the calculation method works for all beams within the parameters. When the calculated values for C2, using equations 5.26 and 5.29, of all models are set out using the factor LAEtf/bh very large deviations occur, see figure 5.20. The calculation model of C2 is supposed to be dependent on the location where the loads are applied relative to the neutral axis of the beam. Therefore in this case the models have been separated in 6 sets using h/e as factor. In figure 5.20 the different groups of points are given different colors and several relations between C2 and the factor LAEtf/bh become visible. Therefore, the same procedure is used as during the development of the formula for C1. Subsequently the initial equation 5.34 is very similar to equation 5.29. The parameters aC2, bC2, cC2 and dC2 can be determined for a set of points. However, these parameters will be different for every set. Therefore these parameters are substituted for quadratic functions with the factor h/e as variable, see equations 5.36 to 5.39. The least square method has been used to determine the regression coefficients in these equations.

LBD/LAE

70

Figure 5.19: Scheme of FE-model to determine C2

Using the equations 5.21, 5.22, 5.23, 5.29 and 5.34 The critical moment of any spreader beam within the parameters discussed in §5.1 can be determined. When the calculation method is compared to the results from the set of 295 FE-models the average absolute difference is 8.7%.

2 2 2 2

3 22 C sb C sb C sb CC a f b f c f d= + + + (5.34)

AE fsb

L tfbh

= (5.35)

2

2

a =0.000184 0.004 0.032Ch he e

− −

(5.36)

2

2

=-0.0013 0.027 0.29Ch hbe e

+ +

(5.37)

2

2

c =0.0032 0.076 0.43Ch he e

− −

(5.38)

2

2

=-0.0003 0.005 0.9Ch hde e

− +

(5.39)

71

5.3.4 ASSESMENT OF CALCULATION METHOD USING STATISTAL ANALYSIS

To evaluate the accuracy and safety of the calculation method to determine the critical moment of a spreader beam for lateral torsional buckling a statistical evaluation is conducted.

The Eurocode EN-1990-1-1 Annex D [17] provides a safety assessment procedure which is used to perform this statistical evaluation. In this procedure experimental results are compared with a calculation model.

The different steps of the procedure are set out in §5.3.4.1 to §5.3.4.7.

Figure 5.20: Visualization of all models comparing the factor LAEtf/bh to the calculated C2

72

5.3.4.1 STEP 1: DEFINE A CALCULATION MODEL

The theoretical calculation model has nine basic variables (Xj) b, h, tf, tw, E, G, LAE, LBD and e. inserting equations 4.2, 4.3, 4.4, 5.22, 5.23, 5.29 and 5.35 into equation 5.21 gives thetheoretical resistance function rt,i, see figure 5.21. re,i is the critical moment obtained from the set of 295 finite element models, see appendix D.2.

Figure 5.21: Equation 5.40 obtained using the program maple

rt,i : Theoretical resistance values (Nmm) re,i : Experimental resistance values (Nmm)

5.3.4.2 STEP 2: COMPARE THEORETICAL AND EXPERIMENTAL VALUES

The theoretical and experimental critical moment is compared for all 295 models, see figure 5.22 with re,i on the horizontal axis and rt,i on the vertical axis. The blue line shows when both the experimental and theoretical critical moment would be the same.

73

Figure 5.22: Comparison of the theoretical and experimental values of the critical moment

5.3.4.3 STEP 3: ESTIMATE CORRECTION FACTOR B

Using linear regression the relation between the theoretical and the experimental resistance values can be modelled. The regression line is assumed to be linear and given in equation 5.41. In this equation the slope is defined by the least-square approximation of factor b which can be obtained using equation 5.42.

,i t i ir br δ= (5.41)

, ,1

2,

1

n

t i e ii

n

t ii

r rb

r

=

=

=∑

∑(5.42)

ri : Resistance values (Nmm) b : Least-square approximation factor (-) δi : Error terms (-)

74

5.3.4.4 STEP 4: ESTIMATE THE COEFFICIENT OF VARIATION WITH RESPECT TO THE ERROR TERMS

The coefficient of variation with respect to the error terms, which is defined as the scatter of the pairs in figure 5.18, is estimated. Initially, the error term is defined in equation 5.43 and the natural logarithm is taken in equation 5.44. Then the estimated mean value is calculated using equation 5.45. In equation 5.46 the estimated standard deviation is defined. Finally the coefficient of variation with respect to the error terms can be determined using equation 5.47.

,

,

e ii

t i

rbr

δ = (5.43)

( )lni iδ∆ = (5.44)

1

1 n

iin

µ∆=

= ∆∑ (5.45)

( )2

2

1

11

n

ii

sn∆

=

= ∆ − ∆− ∑ (5.46)

21

i

sV eδ∆= − (5.47)

Δi : Natural logarithm of error terms (-)

µ∆ : Estimated mean value of the error terms (-)

sΔ : Estimated standard deviation of the error terms (-)

iVδ : Coefficient of variation with respect to the error terms (-)

5.3.4.5 STEP 5: ESTIMATE THE COEFFICIENT OF VARIATION WITH RESPECT TO THE THEORETICAL RESISTANCE MODEL

The coefficient of variation with respect to the theoretical resistance model Vrt,i is defined as the scatter of the values rt,i compared to the mean values of the theoretical resistance model grt,i(μX), see equations 5.48 and 5.49. The partial derivatives with respect to the basic variables are given in appendix E.1 through E.6 and are obtained using the program maple. The mean value and standard deviation for the variables E is obtained from the research by Taras et al [19]. The variable G is dependent of E, subsequently the same mean value and standard deviation is considered. The statistical distributions for the yield strength are obtained from the safebrictile project [20]. The nominal values of the yield strength are split in two by the thickness of the flange. Where for flanges thinner than 16mm the nominal values for S235 is 235 N/mm2, for S355 it is 355 N/mm2 and for S460 it is 460 N/mm2. However for flanges thicker than 16mm the nominal values are 225 N/mm2, 345 N/mm2, and 440 N/mm2. Two different sets of mean values and standard deviations for the basic variables b,h, tf and tw are used. The first set, referred to as SB (Safebrictile), is obtained from the research executed by da Silva et al [20]. This set of statistical distributions is

75

developed using the database of measurements provided by the Safebritile project supplemented with other literature. The second set of statistical distributions, referred to as SP (steel producers), is provided by steel producers. Among them are Arcellor Mittal, Dillinger and Salzgitter AG. The steel producers provided measurements and these results are processed by RWTH Aachen. All mean values and standard deviations for the basic variables are shown in table 5.5.

Table 5.5: Different mean values and standard deviations for six basic variables

mean c.o.v. source

Emean/Enom 1.0 5.0% [19]

Gmean/Gnom 1.0 5.0%

fy,mean/fy,nom (S235) 1.2651 5.76%

[20] fy,mean/fy,nom (S355) 1.1814 4.83%

fy,mean/fy,nom (S460) 1.1323 5.15%

SB SP SB SP

bmean/bnom 1.0 1.0015 0.9% 0.6%

[20] hmean/hnom 1.0 0.9965 0.9% 0.05%

tf,mean/ tf,nom 0.975 0.962 2.5% 1.7%

tw,mean/ tw,nom 1.0 0.993 2.5% 2.5%

Using these values and the partial derivatives the coefficient of variation with respect to the theoretical values can be calculated using equation 5.55.

, ,t ir t ig r= (5.48)

( )( )

, ,

,

,

2

222,

1=V t i t i

t i j

t i

r rr X

jt ir X

VARg X g

Xrgσ

µ

∂ ≅ ∂ ∑ (5.49)

,V

t ir : Coefficient of variation with respect to the resistance values (-)

76

5.3.4.6 STEP 6: DETERMINE THE DESIGN VALUE OF THE RESISTANCE

The coefficient of variation with respect to the resistance value is defined using equation 5.50. The lognormal coefficient with respect to the error terms can be determined using equation 5.51 to 5.53. Using this the design resistance can be determined using equation 5.54. The design failure factor kd,n is obtained EN 1990-1-1 table D2 [17], see table 5.6, and has a value of 3.04 when the number of test results is higher than 100. When the number of test results is in between two values the lower n was taken since there is no description which states that the value for kd,n may be interpolated in any way and this is the conservative approach.

,

2 2 2i t i ir rV V Vδ= + (5.50)

( )2ln 1i i

Q Vδ δ= + (5.51)

( ), ,

2ln 1t i t ir rQ V= + (5.52)

( )2ln 1i ir rQ V= + (5.53)

22, 2

, ,n

,

0.5

, ( )

rt i id d ri

r ri i

t i

QQk k Q

Q Q

d i r Xr bg e

δ

µ∞

− − − =

(5.54)

Vir : Coefficient of variation with respect to the resistance values (-)

iQδ : Lognormal coefficient with respect to the error terms (-)

,t irQ : Lognormal coefficient with respect to the theoretical resistance values (-)

irQ : Lognormal coefficient with respect to the resistance values (-)

,d ir : Design resistance values (-)

,dk ∞ : Design fractile factors for n>100 single test results (-)

,ndk : Design fractile factors for n single test results (-)

Table 5.6: Values for kd,n according to EN 1990-1-1 table D2 [17]

N 1 2 3 4 5 6 8 10 20 30 ∞ (n≥100)

Vx known 4.36 3.77 3.56 3.44 3.37 3.33 3.27 3.23 3.16 3.13 3.04

77

5.3.4.7 STEP 7: DETERMINE THE CORRECTED PARTIAL SAFETY FACTOR

In this step the corrected partial safety factor γM1* is determined. In EN 1990-1-1 [17] the equation to determine the safety factor is defined as equation 5.55.

nom,*1

,1

1 ni

Md ii

rn r

γ=

= ∑ (5.55)

*

1Mγ : Corrected partial safety factor (-)

5.3.5 RESULTS OF THE STATISTICAL ANALYSIS

The results of the statistical analysis are presented in table 5.7. It shows that the number of models considered, the factor b and the coefficient of variation with respect to the error terms are the same for the two different mean values in combination with standard deviations considered. The partial safety factors are evaluated in §5.3.5.1. This shows that both options result in a partial safety factor that is below the acceptance limit.

Table 5.7: Results of the statistical evaluation considering the calculation method for Mcr

n b Vδi Vrt,i Vr,i γ*Mo

SB 295 1.0317 0.0414

0.0555 0.0692 1.000

SP 0.0477 0.0631 1.015

5.3.5.1 ACCEPTANCE LEVELS FOR γM0*

One way to evaluate the corrected partial safety factors obtained through the statistical analysis is given in the research by Taras et al [19]. This research states that the corrected partial safety factor can be compared with a target safety factor γM,target, which in this case is γM1=1.0, using equation 5.56. In this equation fa is the acceptance limit provided by the research of Taras et al [19], see figure 5.23. The black continuous line is the acceptance limit fa, the blue dot represents SB and the red dot SP. On the vertical axis the ratio between the corrected partial safety factor and the target partial safety factor is set out and on the horizontal axis the Coefficient of variation with respect to the resistance values is set out. Both results are within the acceptance limit. Therefore, γM1=1.0 is accepted

*1

,target

Ma

Mf γ

γ≤ (5.56)

fa : Acceptance limit (-)

γM,target : Target value of the partial safety factor, already existing safety factor (-)

78

Figure 5.23: Acceptance limit plot of the partial safety factors determined for Mcr

5.4 DEVELOP A METHOD TO DETERMINE THE ULTIMATE LTB RESISTANCE

To develop a calculation method for the lateral torsional buckling resistance of a suspended spreader beam a variety of different compositions using the parameters will be modelled. The results of these models will be compared to the currently used buckling curves. Finally a statistical evaluation is performed to determine a corresponding safety factor. The GMNIA models are set up using the initial imperfections discussed in §4.7.3, the residual stresses discussed in §4.7.4 and the mesh determined in §4.9.

5.4.1 DIFFERENT COMPOSITIONS USING THE PARAMETERS

The table below, table 5.8, shows the different compositions of beams modelled.

Table 5.8: Different configurations considered during the assessment of the buckling curves

Profile type

Profile size LAE (max length is 28m.)

LAB/LAEec,el (min e=10.0mm,

max e=162.0mm) Steel grade

IPE

100*

360*

600* 15h

33h

50h

0.23

0.50

0.77

h/3.3

h/13.0

h/23.0

S235

S355

S460 HEA

100*

450*

900*

79

HEB

200*

600*

1000*

15h

33h

50h

0.23

0.50

0.77

h/3.3

h/13.0

h/23.0

S235

S355

S460

5.4.2 CURRENTLY USED BUCKLING CURVES

Mcr can be determined using equations 5.21 to 5.23 and the equations for C1 and C2 developed in §5.3.2 and §5.3.3. However, the critical moment can also be obtained numerically using an LBA. The maximum moment capacity of a model is found by determining the maximum force F during a GMNIA calculation and multiplying this with length LAB. The buckling curves as provided in the Eurocode NEN-EN 1993-1-1 [1] §6.3.1.2 figure 6.4, see figure 5.23, are developed using a relative slenderness and a corresponding reduction factor. The relative slenderness of the different models is determined using equation 5.58. The corresponding reduction factor is determined using values obtained from a GMNIA and equation 5.59. The buckling curves in the Eurocode are presented as a figure with which the reduction factor can be visually determined using a known slenderness. However, the Eurocode also provides equations for buckling curves a through d to determine the reduction factor of a beam, see equation 5.60 And 5.61. In equation 5.61 the difference between the curves a, b, c and d is made by using different imperfection factors αLT. The different imperfection factors are given in the Eurocode NEN-EN 1993-1-1 [1] §6.3.1.4 table 6.3, see table 5.9.

80

Figure 5.24: buckling curves as provided by the Eurocode [1]

Table 5.9: Recommended values for imperfection factors for the buckling curves

y yLT

cr

f WM

λ = (5.58)

max,,FEM

FEM ABLT

y y

F LW f

χ = (5.59)

22

1 1.0LT

LTLT LT

χλ

= ≤Φ + Φ −

(5.60)

( ) 20.5 1 0.2LT LTLT LTα λ λ Φ = + − +

(5.61)

Buckling curve a b c d

imperfection factor αLT 0.21 0.34 0.49 0.76

81

5.4.3 OBTAINING RESULTS

Not all configurations have given results which can be compared to the buckling curves. During the LBA a number of configurations give errors, see figure 5.25, although this model was built up using the same python script. It is not clear why Abaqus gives this error or how to resolve this. One possible way to resolve this would be to use a different solving method. However as discussed in §4.5.1 only the subspace method can be applied in this model. Subsequently, Mcr cannot be determined. Moreover, no initial imperfection can be inserted in the corresponding GMNIA calculations.

Figure 5.25: Print screen of message area in abaqus during an LBA resulting in a common error

Another reason why some results are not used for comparison is the failure mode in the GMNIA. This project focusses on lateral torsional buckling. Therefore only beams that fail due to lateral torsional buckling are taken into account. However, some slender beams fail because of weak axis bending and some stocky beams fail because of shear failure. To determine which models are not taken into account in the development of the design rule two methods are used. Initially, visual determination of the failure mode is used. This is done by looking at the image showing the von Mises stresses of the deformed beam at the moment of the highest reaction force. Shear failure can be determined with high certainty, see figure 5.26. Some cases of weak axis bending failure can be visually determined as well, see figure 5.27. In this figure it is clear that the beam is rotated approximately 90 degrees and the stresses between the cable attachment points at both flange tips are high compared to the point where the web and flanges meet. A typical lateral torsional buckling failure shows high stresses on one side of the top flange and high stresses on the opposing side in the bottom flange, it also shows a smaller rotation of the beam, see figure 5.28. However the visual difference between lateral torsional buckling and weak axis bending can become fairly small. Subsequently, hard to determine visually. In this case the reduction factor is compared to the factor Mz,pl/My,pl. When the difference is small, below 10%, the failure mode is considered as weak axis bending.

82

Figure 5.26: An image of a spreader beam showing shear failure

Figure 5.27: An image of a spreader beam showing weak axis bending failure

Figure 5.28: An image of a spreader beam showing lateral torsional buckling failure

83

Figure 5.29: All results plotted with the buckling curves a-d

5.4.4 RESULTS

Results are obtained from 153 models, see appendix F for the full table, These results are presented together with the four buckling curves a through d, see figure 5.29. In this figure the buckling curves a through d are represented as purple, yellow, red and blue lines, the grey line indicates the Euler curve (ΧLT=1/λ2LT) and the dots are the numerical values of the different models.

5.4.5 PROCEDURE OF STATISTICAL EVALUATION

Two types of statistical evaluations regarding the total set of 153 models and subsets is performed to determine the partial safety factor, similar to the evaluation executed in §5.3.4.

In the research executed by da Silva et al [18] a statistical evaluation is performed. A comparison is made between numerical results and the buckling curves. Two assumptions are considered in the research of da Silva [18]:

• Vrt=0. There is no calculation method used to determine Mcr, but each value for λLT and ΧLT are determined numerically considering one case. Subsequently no partial derivatives can be estimated.

• Basic variable Xm is considered as Xnom because of the first assumption. Which means that the nominal properties of for example the yield stress are considered in this evaluation.

84

In the second statistical evaluation equations 5.58 to 5.62 are used as a calculation model to determine the ultimate lateral torsional buckling resistance of a beam. In this procedure the slenderness of a beam is determined using the critical moment obtained via equations 5.21 to 5.23 and 5.29 to 5.39. The Mcr obtained using these equations is inserted as a value. Subsequently, the average value and the Coefficient of variation with respect to the resistance values determined in §5.3.4.6 are taken in to account during the evaluation.

, ,=b Rd LT pl y yM W fχ (5.62)

The different steps of the procedure are set out in §5.4.5.1 to §5.4.5.7. However, because of the two different methodologies some steps have two different approaches. When this is the case the step will be split up into two explanations one considering the procedure applied by da Silva et al [18] and the other considering the method provided by the Eurocode EN 1990-1-1 annex D [17]. Moreover, since the procedures are very similar to the followed procedure in §5.3.4 only differences will be discussed.

5.4.5.1 STEP 1: DEFINE A CALCULATION MODEL

5.4.5.1.1 DA SILVA ET AL [18]

There is no calculation model to determine Mcr. Subsequently, the theoretical resistance rt,I is considered as the reduction factor of the different buckling curves, which are calculated using equations 5.60 and 5.61. Using the numerically determined critical moment to determine the slenderness. The experimental resistance values are obtained via numerical models and using equation 5.59. This results in equation 5.63 for the theoretical resistance and equation 5.64 for the experimental resistance. Where equation 5.63 contains four equations (5.63a to 5.63d) for the different buckling curves.

,i LT,buckling curvetr χ= (5.63)

,i, LT,buckling curve at ar χ= (5.63a)

,i,b LT,buckling curve btr χ= (5.63b)

,i,c LT,buckling curve ctr χ= (5.63c)

,i,d LT,buckling curve dtr χ= (5.63d)

, ,e i LT FEMr χ= (5.64)

85

5.4.5.1.2 EUROCODE EN 1990-1-1 ANNEX D [17]

Combining equations 4.5 and 5.58 to 5.62 gives the theoretical calculation model rt,i, see equation 5.65 and figure 5.19, obtained using the program maple. This calculation model has seven basic variables (Xj) b, h, tf, tw, fy, Mcr and αLT. The imperfection factor is given in table 5.5. In §5.4.5.1.1. The theoretical resistance model is split into four different equations. The same can be done with equation 5.66 changing αLT for the different buckling curves. The experimental resistance is the maximum moment obtained using GMNIA, see equation 5.67.

,i ,t b Rr M= (5.65)

Figure 5.19: equation 5.66, resistance of a beam considering lateral torsional buckling

, max,e i FEMr M= (5.67)

5.4.5.2 STEP 2: COMPARE THEORETICAL AND EXPERIMENTAL VALUES

For all 153 models the theoretical resistance rt,i and the experimental resistance re,i are known. All corresponding pairs are plotted in a scatter plot with rt,i on the horizontal axis and re,i on the vertical axis. This is done for all four buckling curves, see figure 5.20 and 5.21. In this figure the blue line represents the values where the theoretical and experimental resistance is equal.

86

5.4.5.2.1 DA SILVA ET AL [18]

Figure 5.30 below shows the comparison between rt,I and re,I using the method of da Silva et al [18] for all buckling curves.

Figure 5.30: Comparison of re,I and rt,i of buckling curves a through d (da Silva et al [18])

87

5.4.5.2.2 EUROCODE EN 1990-1-1 ANNEX D [17]

Figure 5.31 below shows the comparison between rt,I and re,I using the method provided by the Eurocode EN 1990-1-1 annex D [17] for all buckling curves.

5.4.5.3 STEP 3: ESTIMATE CORRECTION FACTOR B

See §5.3.4.3

5.4.5.4 STEP 4: ESTIMATE THE COEFFICIENT OF VARIATION WITH RESPECT TO THE ERROR TERMS

See §5.3.4.4

Figure 5.31: Comparison of re,I and rt,i of buckling curves a through d (Eurocode [17])

88

5.4.5.5 STEP 5: ESTIMATE THE COEFFICIENT OF VARIATION WITH RESPECT TO THE THEORETICAL RESISTANCE MODEL

5.4.5.5.1 DA SILVA ET AL [18]

As stated in §5.4.5, because of the fact that there is no theoretical resistance model no partial derivatives can be derived. Subsequently this coefficient of variation is zero, see equation 5.68.

,0

t ir rtV V= = (5.68)

5.4.5.5.2 EUROCODE EN 1990-1-1 ANNEX D [17]

Since this method does have a theoretical resistance model the same procedure as described in §5.3.4.5 can be used.

The six partial derivatives with respect to b, h, tf, tw, fy and Mcr are provided in appendix G.1 to G.6. For the basic variable αLT no statistical distribution is possible. Therefore, no partial derivative is calculated.

The statistical distributions used are given in table 5.5. However, the mean value and coefficient of variation of Mcr are not in this table. These are determined during the statistical analysis set out in §5.3.4, see table 5.10. The mean value (μMcr)is calculated by dividing the critical moment obtained using an LBA by the critical moment calculated with the formula created in §5.3.1 to §5.3.3, see equation 5.69. Only one coefficient of variation is used, since the difference between the two options is negligible. Moreover, the biggest coefficient is chosen which is the most conservative option.

Table 5.10: Mean value and coefficient of variation for Mcr

μMcr c.o.v.

Mcr,FEM/Mcr,formula 1.0232 6.92% (see table 5.7)

,

,1

1cr

ne i

Mt ii

rn r

µ=

= ∑ (5.69)

5.4.5.6 STEP 6: DETERMINE THE DESIGN VALUE OF THE RESISTANCE

See §5.3.4.6.

89

5.4.5.7 STEP 7: DETERMINE THE CORRECTED PARTIAL SAFETY FACTOR

5.4.5.7.1 DA SILVA ET AL [18] Using this method a partial factor γRd is determined which only accounts for the model accuracy, see equation 5.70. This factor is different from γM1*, which also accounts for the statistical distributions of the material and geometrical properties of the members.

( )2, 0.5

1d

Rd k Q Qbeγ

∞− −= (5.70)

Rdγ : Partial factor only accounting for the model accuracy (-)

5.4.5.7.2 EUROCODE EN 1990-1-1 ANNEX D [17]

See §5.3.4.7.

5.3.6 RESULTS OF THE STATISTICAL EVALUATION

5.3.6.1 PARTIAL SAFETY FACTORS FOR THE DIFFERENT (SUB)SETS

The partial safety factors are determined for both methods. For the method provided by the Eurocode two partial safety factors are determined one for each option of the statistical distributions, see table 5.11. Several subsets were made during this statistical analysis. At first S235, S355 and S460 have been calculated as subsets, see table 5.12. Also IPE, HEA and HEB beams have been calculated as subsets, see table 5.13. Several subsets have been made using the slenderness. Three different variations in range are used, because in §V.1 in the research of Taras et al [19] small ranges in the slenderness are recommended, see table 5.14. This leads to very small numbers of tests in some subsets. Subsequently bigger ranges have been used were one range is chosen so the number of tests is approximately evenly divided, see table 5.15 and one range divides the whole range of the slenderness exactly in half, see table 5.16. The last subsets made are two sets divided by the h/b ratio, see table 5.17. This break-up is also used in the current calculation method for the lateral torsional buckling resistance in the Eurocode.

Table 5.11: Results of the statistical evaluations using the whole set

(sub)set n Bucklingcurve

b γRd γ*M1 Da Silva [18]

Eurocode [17]

Da Silva [18]

Eurocode [17] SB

Eurocode [17] SP

All 153

a 0.97 0.99 1.30 1.23 1.23 b 1.06 1.07 1.19 1.13 1.13 c 1.14 1.15 1.11 1.05 1.05 d 1.28 1.30 1.02 0.95 0.95

90

Table 5.12: Results of the statistical evaluations using subsets divided by strength class

(sub)set n Bucklingcurve

b γRd γ*M1 Da Silva [18]

Eurocode [17]

Da Silva [18]

Eurocode [17] SB

Eurocode [17] SP

S235 75

a 0.98 0.96 1.35 1.31 1.31 b 1.07 1.04 1.23 1.19 1.19 c 1.16 1.13 1.14 1.10 1.10 d 1.31 1.29 1.03 0.98 0.98

S355 38

a 0.97 0.98 1.27 1.21 1.21 b 1.05 1.06 1.16 1.10 1.11 c 1.14 1.15 1.07 1.03 1.03 d 1.29 1.29 0.97 0.93 0.93

S460 40

a 0.97 1.02 1.24 1.14 1.14 b 1.04 1.09 1.16 1.06 1.06 c 1.11 1.17 1.12 1.00 1.00 d 1.23 1.30 1.06 0.92 0.92

Table 5.13: Results of the statistical evaluations using subsets divided by beam type

(sub)set N Bucklingcurve

b γRd γ*M1 Da Silva [18]

Eurocode [17]

Da Silva [18]

Eurocode [17] SB

Eurocode [17] SP

IPE 55

a 0.94 0.94 1.38 1.32 1.33 b 1.03 1.02 1.23 1.18 1.19 c 1.13 1.11 1.11 1.07 1.08 d 1.30 1.26 0.96 0.94 0.94

HEA 34

a 1.04 1.00 1.13 1.13 1.14 b 1.15 1.08 1.05 1.07 1.07 c 1.25 1.17 0.98 1.01 1.01 d 1.43 1.31 0.89 0.92 0.93

HEB 64

a 0.98 0.99 1.27 1.20 1.21 b 1.04 1.06 1.21 1.12 1.12 c 1.11 1.15 1.16 1.06 1.06 d 1.22 1.29 1.11 0.97 0.98

91

Table 5.14: Results of the statistical evaluations using subsets divided by 5 ranges of slenderness

(sub)set n Buckling curve

b γRd γ*M1 Da Silva [18]

Eurocode [17]

Da Silva [18]

Eurocode [17] SB

Eurocode [17] SP

λLT<0.5 6

a 0.95 0.98 1.15 1.03 1.04 b 0.98 1.04 1.10 1.00 1.01 c 1.02 1.11 1.06 0.98 0.99 d 1.09 1.21 1.00 0.96 0.97

0.5<λLT<1.0 21

a 0.95 0.95 1.30 1.18 1.19 b 1.03 1.05 1.18 1.07 1.07 c 1.13 1.15 1.09 0.98 0.99 d 1.28 1.32 0.99 0.88 0.88

1.0<λLT<1.5 51

a 1.00 0.96 1.35 1.34 1.34 b 1.11 1.06 1.23 1.21 1.21 c 1.22 1.17 1.12 1.05 1.05 d 1.40 1.34 0.98 0.96 0.96

1.5<λLT<2.0 50

a 0.99 0.98 1.24 1.22 1.22 b 1.07 1.06 1.14 1.12 1.12 c 1.15 1.14 1.06 1.03 1.03 d 1.30 1.28 0.93 0.90 0.90

2.0<λLT<3.0 25

a 1.04 1.03 1.03 1.02 1.03 b 1.10 1.09 0.97 0.96 0.96 c 1.16 1.16 0.91 0.90 0.90 d 1.28 1.28 0.83 0.82 0.82

Table 5.15: Results of the statistical evaluations using subsets divided by 3 ranges of slenderness

(sub)set N Buckling curve

b γRd γ*M1 Da Silva [18]

Eurocode [17]

Da Silva [18]

Eurocode [17] SB

Eurocode [17] SP

λLT<1.2 47

a 0.96 0.96 1.31 1.23 1.24 b 1.04 1.06 1.21 1.12 1.13 c 1.12 1.17 1.15 1.04 1.05 d 1.26 1.35 1.06 0.95 0.95

1.2<λLT<1.7 51

a 0.99 0.96 1.34 1.32 1.32 b 1.09 1.05 1.22 1.21 1.21 c 1.19 1.14 1.12 1.11 1.11 d 1.36 1.30 0.98 0.98 0.98

1.7<λLT<3.0 55

a 1.03 1.01 1.11 1.11 1.12 b 1.10 1.08 1.04 1.04 1.04 c 1.18 1.15 0.97 0.97 0.97 d 1.31 1.28 0.88 0.88 0.88

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Table 5.16: Results of the statistical evaluations using subsets divided by 2 ranges of slenderness

(sub)set N Bucklingcurve

b γRd γ*M1 Da Silva [18]

Eurocode [17]

Da Silva [18]

Eurocode [17] SB

Eurocode [17] SP

λLT<1.5 78

a 0.97 0.96 1.37 1.23 1.23 b 1.05 1.06 1.26 1.13 1.13 c 1.14 1.17 1.18 1.05 1.05 d 1.28 1.34 1.07 0.94 0.94

1.5<λLT<3.0 75

a 0.99 1.00 1.19 1.17 1.17 b 1.07 1.07 1.10 1.07 1.07 c 1.16 1.15 1.02 0.99 0.99 d 1.30 1.28 0.90 0.88 0.89

Table 5.17: Results of the statistical evaluations using subsets divided by the factor h/b

(sub)set N Bucklingcurve

b γRd γ*M1 Da Silva [18]

Eurocode [17]

Da Silva [18]

Eurocode [17] SB

Eurocode [17] SP

h/b≤2.0 73

a 0.98 0.99 1.29 1.22 1.22 b 1.07 1.09 1.20 1.10 1.11 c 1.15 1.20 1.14 1.02 1.02 d 1.29 1.38 1.07 0.92 0.92

h/b>2.0 80

a 0.94 0.99 1.29 1.23 1.23 b 1.03 1.06 1.11 1.10 1.11 c 1.12 1.14 1.00 1.00 1.00 d 1.27 1.27 0.87 0.88 0.88

Table 5.18: Results of the statistical evaluations using subsets IPE and excluding strength class S235

(sub)set N Bucklingcurve

b γRd γ*M1 Da Silva [18]

Eurocode [17]

Da Silva [18]

Eurocode [17] SB

Eurocode [17] SP

IPE excluding

S235 47

a 0.95 0.96 1.34 1.30 1.30 b 1.05 1.03 1.20 1.18 1.18 c 1.16 1.12 1.09 1.04 1.04 d 1.33 1.27 0.95 0.94 0.94

1.2<λLT<1.7 excluding

S235 13

a 0.99 1.00 1.34 1.13 1.13 b 1.09 1.10 1.22 1.03 1.03 c 1.19 1.20 1.12 0.94 0.94 d 1.36 1.38 0.98 0.82 0.82

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5.3.6.2 ACCEPTANCE LEVELS FOR γM0*

The acceptance check described in §5.3.5.1 is done for all test results and the subsets using γm1,target=1.0. The safety factors that are below the acceptance line are shown green and the others are red in tables 5.11 to 5.17. All acceptance limit plots from the method used in the research of da Silva et al [18] are provided in Appendix I.1. All acceptance limit plots obtained using the method provided by Eurocode EN 1990-1-1 annex D [17] are given in appendix I.2. The differences between the two options within the method provided by the Eurocode are negligible. Subsequently, only the plots using the statistical distributions provided by the safebrictile project (SB) are provided, see figure 5.32 for the acceptance limit plot of the complete set. The black continuous line is the acceptance limit fa and the four dots show the position of the corrected partial safety factor of the four different buckling curves. On the vertical axis the ratio between the corrected partial safety factor and the target partial safety factor is set out and on the horizontal axis the Coefficient of variation with respect to the resistance values is set out.

Figure 5.32: Acceptance limit plot of the whole set using the method including geometrical and material statistical distributions, see table 5.11

94

5.3.7 CONCLUSIONS

The difference between the two methods provided by da Silva [18] and the Eurocode [17] is whether the statistical distributions of the basic variables are taken into account. Subsequently, it is likely that the assessment including the statistical distributions is more accurate. The mean value of the yield strengths, see table 5.5, is high. Subsequently, the results of the statistical analysis including the statistical distributions are less conservative. Only the results of the analysis including the statistical distributions is considered for drawing conclusions.

The differences between the two different sets of statistical distributions within the method provided by the Eurocode are negligible. Therefore, these results will be considered as one. The only difference between the two options is for the subset λLT<0.5 where SB gives an accepted partial safety factor for all buckling curves and SP gives an accepted value for buckling curves b, c and d, see figure 5.33.

a b

Figure 5.33: Acceptance limit plots of the subset λLT<0.5 for both different statistical distributions, see table 5.14 a) SB b) SP

When the whole set is considered the partial safety factor belonging to buckling curve c is within the acceptance limit. However, for the subsets S235, IPE and 1.2<λLT<1.7 only buckling curve d is applicable, see figure 5.34. When the strength class S235 is removed from the subsets IPE and 1.2<λLT<1.7, see table 5.18, buckling curve c is applicable for subset IPE and buckling curve b for the subset 1.2<λLT<1.7, see figure 5.35. Subsequently, it can be concluded that buckling curve d is recommended when a beam is designed using strength class S235 and buc. Kling curve c is recommended when using strength class S355 or S460

95

a b c

Figure 5.34: Acceptance limit plots of subsets with statistical distribution SB, see tables 5.12, 5.13 and 5.15 a) S235 b) IPE c) 1.2<λLT<1.7

a b

Figure 5.35: Acceptance limit plots of subsets with statistical distribution SB, see table 5.18 a) IPE excluding S235 b) 1.2<λLT<1.7 excluding S235

The partial safety factors obtained using the different subsets only result in a less conservative buckling curve for really stocky beams, λLT<0.5, and really slender beams, λLT>2.0. Subsequently, when the slenderness range is split in three parts buckling curve b may be used for λLT<0.5, buckling curve c for 0.5λLT<2.0 and buckling curve a for 2.0<λLT. However, beams designed with a slenderness lower than 0.5 will be very rare in practice, because a beam within this range is likely to fail in shear. This is also the reason why there are only six results in this range. In the range 2.0<λLT<3.0 the differences between the buckling curves are very small. Therefore the use of a less conservative buckling curve gives a low increase in resistance.

Using one buckling curve for the whole range gives the most simple design rule. This may seem less conservative when compared to the previously discussed splitting of the slenderness range, but in practice this difference is negligible. Therefore, buckling curve c is recommended for strength class S355 and S460 and buckling curve d is recommended for strength class S235.

96

5.4 SUMMARY

The critical moment can be obtained using equations 5.21 to 5.23 and 5.29 to 5.39 as shown below. Equations 5.21 to 5.23 are obtained from the Dutch National Annex from the Eurocode [1]. Therefore, implementation in the currently used calculation method is straightforward. Equations 5.29 to 5.39 are developed during this project and are means to determine the factor C1 and C2 which are needed to be able to use the method currently provided.

,M=cr z tg

CM EI GIL (5.21)

( )2 2

1 22 22

CC= 1 1g

kip kipkip

L S SC CL LL

π π π + + +

(5.22)

S= w

t

EIGI

(5.23)

1 1 1

21 C sb C sb CC a f b f c= + + (5.29)

AE fsb

L tfbh

= (5.30)

1a = 0.019 0.0185BD

CAE

LL

− + (5.31)

1=0.184 0.248BD

CAE

LbL

− (5.32)

1c = 0.117 1.11BD

CAE

LL

− + (5.33)

2 2 2 2

3 22 C sb C sb C sb CC a f b f c f d= + + + (5.34)

AE fsb

L tfbh

= (5.35)

2

2

a =0.000184 0.004 0.032Ch he e

− −

(5.36)

2

2

=-0.0013 0.027 0.29Ch hbe e

+ +

(5.37)

2

2

c =0.0032 0.076 0.43Ch he e

− −

(5.38)

2

2

=-0.0003 0.005 0.9Ch hde e

− +

(5.39)

97

The lateral torsional buckling resistance of a spreader beam can be calculated using equations 5.58 to 5.62. All these equations are provided by the Eurocode [1], which causes this method to be almost exactly the same. However, in the currently used calculation method other buckling curves are prescribed. For spreader beams with steel grade S355 and S460 buckling curve c is applicable and for beams with steel grade S235 buckling curve d is applicable.

y yLT

cr

f WM

λ = (5.58)

max,,FEM

FEM ABLT

y y

F LW f

χ = (5.59)

22

1 1.0LT

LTLT LT

χλ

= ≤Φ + Φ −

(5.60)

( ) 20.5 1 0.2LT LTLT LTα λ λ Φ = + − +

(5.61)

, ,=b Rd LT pl y yM W fχ (5.62)

Statistical evaluations were done comparing these methods with the finite element calculations. The results are within the acceptance limits provided by Taras et al [19]. Subsequently the theoretical methods can be used using a partial safety factor of γM1=1.0.

98

5.5 CALCULATION EXAMPLE

This project was induced by a question from an engineering company, Vissers&Vissers B.V., who were calculating a spreader beam. However, they could not determine the lateral torsional buckling resistance with high certainty. In this paragraph that beam will be calculated as an example taken from practice.

An HEA400 beam with steel grade S235 was used, This is a class 1 profile. The total length, LAE, is 12m and the distance between the cable attachments, LBD, is 8m. The eccentricity of the cables and loads, e, are 85mm. Since this beam is designed using steel grade S235 buckling curve d will be used to determine the lateral torsional buckling resistance.

12000 19 1.95300 390sb

mm mmfmm mm

⋅= =

⋅ (5.35)

2

2390 390a =0.000184 0.004 0.032 0.046585 85C

mm mmmm mm

− − = −

(5.36)

2

2390 390=-0.0013 0.027 0.29 0.38785 85C

mm mmbmm mm

+ + =

(5.37)

2

2390 390c =0.0032 0.076 0.43 0.71185 85C

mm mmmm mm

− − = −

(5.38)

2

2390 390=-0.0003 0.005 0.9 0.87185 85C

mm mmdmm mm

− + =

(5.39)

3 22 0.0465 1.95 0.387 1.95 0.711 1.95 0.871 0.611C = − ⋅ + ⋅ − ⋅ + = (5.34)

1

8000a = 0.019 0.0185 0.0058312000C

mmmm

− + = (5.31)

1

8000=0.184 0.248 0.12512000C

mmbmm

− = − (5.32)

1

8000c = 0.117 1.11 1.0312000C

mmmm

− + = (5.33)

21 0.00583 1.95 0.125 1.95 1.03 0.808C = ⋅ − ⋅ + = (5.29)

5 9 62

4 4 42

2.1 10 2894 10S= 2001.0

7.93 10 191.4 10

N mmmm mmN mmmm

⋅ ⋅ ⋅=

⋅ ⋅ ⋅(5.23)

( )( )

( )22

22

2001.0 2001.0C=0.808 1 0.611 1 0.611 3.791200012000

mm mmmmmm

π ππ ⋅ ⋅ ⋅ + + + =

(5.22)

5 4 4 4 4 4 62 2,M

3.79= 2.1 10 8564 10 7.93 10 191.4 10 522 1012000cr

N NM mm mm Nmmmm mmmm

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ (5.21)

99

3 32

6

235 2562 101.074

522 10LT

N mmmm

Nmmλ

⋅ ⋅= =

⋅ (5.58)

( )( )20.5 1 0.76 1.074 0.2 1.074 1.41LTΦ = + − + = (5.61)

2 2

1 0.496 1.01.41 1.41 1.074

LTχ = = ≤+ −

(5.60)

3 3 62, =0.496 2562 10 235 298.6 10b Rd

NM mm Nmmmm

⋅ ⋅ ⋅ = ⋅ (5.62) 6

6,

300 10. . 1.0047 1.0298.6 10

Ed

b Rd

M NmmU CM Nmm

⋅= = = >

⋅ (5.62)

The lateral torsional buckling resistance of the beam in this example is 298.6 kNm. Subesequently, the maximal value of a load on one side of the beam is Mb,Rd/LAB=149.3kN. This beam has been designed for a load of 100kN. However all these calculations are static, where the behavior of the beam during hoisting is dynamic. Therefore the calculation was done using a safety factor of 1.5 on the load. This would lead to a load of 150kN and a moment of 300kN. Subsequently, the design of the beam has a unity check of 1.0047, see equation 5.63. Therefore just a little unsafe, but almost perfect.

100

6 CONCLUSIONS AND RECOMMENDATIONS

6.1 CONCLUSIONS

The goal of this project was to create a calculation method to determine the lateral torsional buckling resistance of a steel I-shaped spreader beam.

During the literature research only two researches were found that treated lateral torsional buckling of a suspended beam without fork ended supports.

Using the finite element method, models were made to determine the critical moment of a spreader beam using linear buckling analysis. Moreover, the ultimate lateral torsional buckling resistance has been determined determined using geometrical and material, nonlinear analysis including imperfections. These finite element models have been validated using an experiment that has been done during this project. This experiment was done using a small aluminum I-shaped beam.

Using the results of the finite element models two calculation methods have been developed. One to determine the critical moment of a spreader beam and another to determine the ultimate lateral torsional buckling resistance.

Both methods that have been developed have been evaluated using a statistical analysis. This has confirmed that the methods are safe when using a partial safety factor of γM1=1.0.

One example calculation has been made using a beam that is used in practice.

6.2 RECOMMENDATIONS

An enrichment to this project would be to evaluate the parameters used during this project. For example one could investigate at what maximum LBD/LAE ratio it becomes more economic to use different types of beams like square hollow sections.

During this project only one basic load case is considered. Therefore, a study can be performed where different load cases are considered. For example three loads, two at the ends of the beam and one in the middle.

Furthermore different types of beams can be tested. For example welded beams or beams including stiffeners.

Besides the extension of the research, experimental research can be conducted to validate the numerical results of this research since only one experiment is done using a small aluminum beam.

101

102

REFERENCES

[1] EN 1993‐1‐1 including Dutch national annex, Eurocode 3: Design of steel structures. Part 1‐1: General rules and rules for buildings, Brussels, Belgium, 2011.

[2] Taras, A., Greiner, R., Unterweger, H., Proposal for amended rules for member buckling and semi-compact cross-section design, Doc. CEN-TC250-SC3_N1898, 2013.

[3] Dahmani, L., Boudjemia, A., LATERAL TORSIONAL BUCKLING RESPONSE OF STEEL BEAM WITH DIFFERENT BOUNDARY CONDITIONS AND LOADING, UDC 539.4. (2014), 46(3), 429–432.

[4] Dux, P. and Kitipornchai, S. (1990) ”Buckling of Suspended I‐Beams” J. Struct. Eng., 116(7), 1877–1891.

[5] Duerr, D., and Asce, M. (2014). Lateral – Torsional Buckling of Suspended I-Shape Lifting Beams, (1), 2–5. http://doi.org/10.1061/(ASCE)SC.1943-5576.0000263.

[6] The American Society of Mechanical Engineers. (2006). Design of Below-the-Hook Lifting Devices.

[7] Kucukler, M., Gardner, L., & Macorini, L. (2015). Lateral–torsional buckling assessment of steel beams through a stiffness reduction method. Journal of Constructional Steel Research, 109, 87–100. http://doi.org/10.1016/j.jcsr.2015.02.008

[8] Jovic, M. (2015) “Lateral torsional buckling analysis of multiple laterally restrained I-beams in bending” Master thesis A-2015.102. Eindhoven University of Technology - Department of the Built Environment - Structural Design.

[9] Van der Aa, R.P. (2015) “Numerical assessment of the design imperfections for steel beam lateral torsional buckling” Master thesis A-2015.102. Eindhoven University of Technology - Department of the Built Environment - Structural Design.

[10] Taras, A., Contribution to the Development of Consistent Stability Design Rules for Steel Members, PhD thesis, Graz University of Technology, 2010.

[11] ABAQUS (2011) `ABAQUS Documentation', Dassault Systèmes, Providence, RI, USA.

[12] ISO. (2009). ISO INTERNATIONAL STANDARD ISO 6892-1: First edition 2009-08-15, Metallic materials – Tensile testing – part 1: Method of test at room temperature. Geneva, Switzerland: International Organization for Standardization (ISO).

[13] Ziemian, R. (2010). Guide to stability design criteria for metal structures. Sixth edition. 22 February 2010.

[14] Huang, Y. and Young, B. (2014). The art of coupon tests. Journal of Constructional Steel Research 2014(96), 159-175. doi:10.1016/j.jcsr.2014.01.010

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[15] Blommers, R. (2015). Lesboek veilig hijsen en aanslaan van lasten: Theorieboek. Uitgavenummer 3- Rev: 3. Hengelo: BLOM Opleindingen.

[16] Barendse, P (12-03-2001), Vuistregels bij het ontwerpen van een draagconstructies, Gedownload op 18-10-2016, van http://wiki.bk.tudelft.nl/mw_bk-wiki/images/5/59/Vuistregels_dc.pdf

[17] EN 1993‐1‐1, Eurocode 3: Basis of Structural Design, Brussels, Belgium, 2011.

[18] da Silva, L.S., Marques, L., Tankova, T., Andrade, A., Canha, J. (2016). SAFEBRICTILE: Standardization of Safety Assessment Procedures across Brittle to Ductile Failure Modes Deliverable D4.1, Report of the safety assessment of the general method in EC3-1-1, University of Coimbra, Portugal.

[19] Taras, A., Dehan, V., da Silva, L.S., Marques, L., Tankova, T. (2016). SAFEBRICTILE: Standardization of Safety Assessment Procedures across Brittle to Ductile Failure Modes Deliverable D1.1, Guideline for the safety assessment of Design Rules for Steel Structures in Line with EN 1990.

[20] Taras, A., Dehan, V., da Silva, L.S., Marques, L., Tankova, T. (2016). SAFEBRICTILE: Report on the statistical distribution of the variables analyzed from the test results Deliverable D2.1.

[21] EN 1993‐1‐8+C2 including Dutch national annex, Eurocode 3: Design of steel structures. Part 1‐8: Design of joints, Brussels, Belgium, 2011.

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APPENDIX

APPENDIX A: SCRIPT

# -*- coding: mbcs -*- from part import * from material import * from section import * from assembly import * from step import * from interaction import * from load import * from mesh import * from optimization import * from job import * from sketch import * from visualization import * from connectorBehavior import *

### input ### LAE=16000.0 LAB=3500.0

hw=352.0 b=300.0 tf=19.0 tw=11.0

e=100.0

elementspersurface=2

bigseed=(5*b)/(32*elementspersurface)

Emod=210000.0 poisson=0.3 fy=235.0

alpha=30

nameLBA='1_LBA' nameRIKS='1_RIKS'

### calculations ### load=50.0 length=LAE+100.0 LAB=LABe+50.0 halfb=b/2 excentricitycable=200.0 excentricityload=200.0 RPcable=hw+excentricitycable

105

RPload=-excentricityload cable2=length-LABe load2=length-load halflength=length/2 LBC=0.5*(length-2*LABe) pi=3.14159 alpha1=(alpha*pi)/180 h=(tan(alpha1))*LBC heightattachment=h+RPcable h=hw+2*tf if h/b<1.2: factor=0.5 if h/b>1.2: factor=0.3 resstress1=fy*0.9375*factor resstress2=fy*0.8125*factor resstress3=fy*0.6875*factor resstress4=fy*0.5625*factor resstress5=fy*0.4375*factor resstress6=fy*0.3125*factor resstress7=fy*0.1875*factor resstress8=fy*0.0625*factor ### LBA ### ### create beam ### mdb.models['Model-1'].ConstrainedSketch(name='__profile__', sheetSize=200.0) mdb.models['Model-1'].sketches['__profile__'].Line(point1=(0.0, 0.0), point2=( 0.0, hw)) mdb.models['Model-1'].sketches['__profile__'].VerticalConstraint(addUndoState= False, entity=mdb.models['Model-1'].sketches['__profile__'].geometry[2]) mdb.models['Model-1'].sketches['__profile__'].Line(point1=(-halfb, 0.0), point2= (halfb, 0.0)) mdb.models['Model-1'].sketches['__profile__'].HorizontalConstraint( addUndoState=False, entity= mdb.models['Model-1'].sketches['__profile__'].geometry[3]) mdb.models['Model-1'].sketches['__profile__'].Line(point1=(-halfb, hw), point2=(halfb, hw)) mdb.models['Model-1'].sketches['__profile__'].HorizontalConstraint( addUndoState=False, entity= mdb.models['Model-1'].sketches['__profile__'].geometry[4]) mdb.models['Model-1'].Part(dimensionality=THREE_D, name='beam', type= DEFORMABLE_BODY) mdb.models['Model-1'].parts['beam'].BaseShellExtrude(depth=length, sketch= mdb.models['Model-1'].sketches['__profile__']) del mdb.models['Model-1'].sketches['__profile__'] mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#6 ]', ), ), name='bottom flange')

106

mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#18 ]',

), ), name='top flange') mdb.models['Model-1'].parts['beam'].Set(faces=

mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#1 ]', ), ), name='web')

### materials ### mdb.models['Model-1'].Material(name='steel') mdb.models['Model-1'].materials['steel'].Elastic(table=((Emod, poisson), )) mdb.models['Model-1'].materials['steel'].Plastic(table=((fy, 0.0), ))

### create sets ### mdb.models['Model-1'].parts['beam'].Set(faces=

mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#1 ]', ), ), name='web') mdb.models['Model-1'].parts['beam'].Set(faces=

mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#6 ]', ), ), name='bottom flange')

mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#18 ]', ), ), name='top flange')

### sections ### mdb.models['Model-1'].HomogeneousShellSection(idealization=NO_IDEALIZATION,

integrationRule=SIMPSON, material='steel', name='flange', numIntPts=5, poissonDefinition=DEFAULT, preIntegrate=OFF, temperature=GRADIENT, thickness=tf, thicknessField='', thicknessModulus=None, thicknessType= UNIFORM, useDensity=OFF)

mdb.models['Model-1'].HomogeneousShellSection(idealization=NO_IDEALIZATION, integrationRule=SIMPSON, material='steel', name='web', numIntPts=5, poissonDefinition=DEFAULT, preIntegrate=OFF, temperature=GRADIENT, thickness=tw, thicknessField='', thicknessModulus=None, thicknessType= UNIFORM, useDensity=OFF)

### section assignment### mdb.models['Model-1'].parts['beam'].SectionAssignment(offset=0.0, offsetField=

'', offsetType=BOTTOM_SURFACE, region= mdb.models['Model-1'].parts['beam'].sets['bottom flange'], sectionName= 'flange', thicknessAssignment=FROM_SECTION)

mdb.models['Model-1'].parts['beam'].SectionAssignment(offset=0.0, offsetField= '', offsetType=TOP_SURFACE, region= mdb.models['Model-1'].parts['beam'].sets['top flange'], sectionName= 'flange', thicknessAssignment=FROM_SECTION)

mdb.models['Model-1'].parts['beam'].SectionAssignment(offset=0.0, offsetField= '', offsetType=MIDDLE_SURFACE, region= mdb.models['Model-1'].parts['beam'].sets['web'], sectionName='web', thicknessAssignment=FROM_SECTION)

### create partitions for residual stresses ### mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces=

mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#8 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[10], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint(

107

mdb.models['Model-1'].parts['beam'].edges[12], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#20 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[18], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[16], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#8 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[11], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[13], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#20 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[19], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[17], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#10 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[16], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[17], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#1 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[3], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[1], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#40 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[22], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[23], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#20 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[21], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[19], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#1000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[39], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[38], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#100 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[30], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[29], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#2000 ]',

108

), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[43], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[41], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#20 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[21], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[20], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#80 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[26], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[25], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#40 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[24], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[23], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#4000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[48], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[47], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#800 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[40], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[38], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#40000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[59], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[58], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#4000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[49], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[48], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#80000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[63], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[61], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#100 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[30], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint(

109

mdb.models['Model-1'].parts['beam'].edges[32], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#800000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[72], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[73], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#100000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[66], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[68], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#1000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[43], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[44], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#4000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[49], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[50], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#10000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[86], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[87], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#800000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[75], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[77], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#40000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[61], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[63], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#8000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[56], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[54], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#400000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint(

110

mdb.models['Model-1'].parts['beam'].edges[75], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[73], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#1000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[80], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[79], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#20000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[63], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[62], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#20000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[63], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[62], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#2000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[85], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[83], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #2 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[106], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[105], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#80000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[70], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[69], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#80000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[70], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[68], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #1 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[104], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[102], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #80 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[122], MIDDLE), point2=

111

mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[121], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#200000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[77], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[76], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#200000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[77], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[76], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #20 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[119], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[117], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #400 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[132], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[131], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#800000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[84], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[82], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#80 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[28], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[26], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#10000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[99], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[97], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #200 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[132], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[131], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces=

112

mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#200 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[35], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[34], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#200 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[35], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[34], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #40000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[155], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[154], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #1 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[112], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[111], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#800 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[42], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[41], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#800 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[42], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[41], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #4 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[119], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[117], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #40000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[158], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[157], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#2000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[49], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[48], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#2000 ]',

113

), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[49], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[48], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #2000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[177], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[176], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #40 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[132], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[131], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#8000 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[56], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[54], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#80000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[108], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[106], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #10000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[159], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[157], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #800000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[176], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[175], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #2 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[115], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[114], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #2 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[115], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[114], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces=

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mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #8000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[187], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[186], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #100000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[172], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[171], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #8 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[122], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[121], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #8 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[122], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[121], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #400000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[179], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[177], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #40000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[198], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[197], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #20 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[129], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[128], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #20 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[129], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[128], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #100 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[223], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint(

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mdb.models['Model-1'].parts['beam'].edges[222], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #4000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[192], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[191], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(('[#0 #80 ]', ), ), point1=mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[136], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[135], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #1 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[205], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[207], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #1000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[190], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[191], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #1000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[190], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[192], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #20000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[246], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[247], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #4000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[240], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[242], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #4000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[197], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[198], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces=

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mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #4000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[197], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[199], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #100 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[229], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[230], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #400 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[234], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[235], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #10000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[204], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[205], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #10000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[204], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[206], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #4000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[272], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[273], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #200000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[261], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[263], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0 #40000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[211], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[212], MIDDLE))

mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask((

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'[#0 #40000000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[211], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[213], MIDDLE)) mdb.models['Model-1'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #40000 ]', ), ), point1= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[257], MIDDLE), point2= mdb.models['Model-1'].parts['beam'].InterestingPoint( mdb.models['Model-1'].parts['beam'].edges[259], MIDDLE)) ### create datum planes for loads and cables ### mdb.models['Model-1'].parts['beam'].DatumPlaneByPrincipalPlane(offset=load, principalPlane=XYPLANE) mdb.models['Model-1'].parts['beam'].DatumPlaneByPrincipalPlane(offset=LABe, principalPlane=XYPLANE) mdb.models['Model-1'].parts['beam'].DatumPlaneByPrincipalPlane(offset=cable2, principalPlane=XYPLANE) mdb.models['Model-1'].parts['beam'].DatumPlaneByPrincipalPlane(offset=load2, principalPlane=XYPLANE) mdb.models['Model-1'].parts['beam'].DatumPlaneByPrincipalPlane(offset=length/2.0, principalPlane=XYPLANE) ### create partitions for loads and cables ### mdb.models['Model-1'].parts['beam'].PartitionFaceByDatumPlane(datumPlane= mdb.models['Model-1'].parts['beam'].datums[102], faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#ffffffff:3 ]', ), )) mdb.models['Model-1'].parts['beam'].PartitionFaceByDatumPlane(datumPlane= mdb.models['Model-1'].parts['beam'].datums[101], faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#aaaaaaaa #aaaaffff #8000aaaa #ff00 #ff7fff #75830ff0 ]', ), )) mdb.models['Model-1'].parts['beam'].PartitionFaceByDatumPlane(datumPlane= mdb.models['Model-1'].parts['beam'].datums[103], faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#aaaaaaaa #aaaaffff #aa80aaaa #0:3 #ff00 #7f1fff #758307f0 ]', ), )) mdb.models['Model-1'].parts['beam'].PartitionFaceByDatumPlane(datumPlane= mdb.models['Model-1'].parts['beam'].datums[100], faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#aaaaaaaa #aaaaffff #aaab6aaa #0:6 #ff00 #3f07ff #750307f0 ]', ), )) mdb.models['Model-1'].parts['beam'].PartitionFaceByDatumPlane(datumPlane= mdb.models['Model-1'].parts['beam'].datums[99], faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#aaaaaaaa #6aaaffff #ab6db6db #0:9 #ff00 #1f01ff #550307f0 ]', ), )) ### assembly ### mdb.models['Model-1'].rootAssembly.DatumCsysByDefault(CARTESIAN) mdb.models['Model-1'].rootAssembly.Instance(dependent=ON, name='beam-1', part= mdb.models['Model-1'].parts['beam']) mdb.models['Model-1'].rootAssembly.rotate(angle=90.0, axisDirection=(0.0, 1.0, 0.0), axisPoint=(0.0, 0.0, 0.0), instanceList=('beam-1', )) ### create rp ### mdb.models['Model-1'].rootAssembly.ReferencePoint(point=(halflength, heightattachment, 0)) mdb.models['Model-1'].rootAssembly.ReferencePoint(point=(load, RPload, 0))

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mdb.models['Model-1'].rootAssembly.ReferencePoint(point=(LABe, RPcable, 0)) mdb.models['Model-1'].rootAssembly.ReferencePoint(point=(cable2, RPcable, 0)) mdb.models['Model-1'].rootAssembly.ReferencePoint(point=(load2, RPload, 0))

### create sets ###

#points on beam mdb.models['Model-1'].parts['beam'].Set(name='Abottom', vertices=

mdb.models['Model-1'].parts['beam'].vertices.getSequenceFromMask(( '[#0:15 #4 ]', ), )) mdb.models['Model-1'].parts['beam'].Set(name='Ebottom', vertices=

mdb.models['Model-1'].parts['beam'].vertices.getSequenceFromMask(( '[#0:5 #8000 ]', ), ))

mdb.models['Model-1'].parts['beam'].Set(name='Bbottom', vertices= mdb.models['Model-1'].parts['beam'].vertices.getSequenceFromMask((

'[#0:11 #80000000 ]', ), )) mdb.models['Model-1'].parts['beam'].Set(name='Btop', vertices=

mdb.models['Model-1'].parts['beam'].vertices.getSequenceFromMask(( '[#0:10 #2000 ]', ), ))

mdb.models['Model-1'].parts['beam'].Set(name='Dbottom', vertices= mdb.models['Model-1'].parts['beam'].vertices.getSequenceFromMask((

'[#0:9 #4 ]', ), )) mdb.models['Model-1'].parts['beam'].Set(name='Dtop', vertices=

mdb.models['Model-1'].parts['beam'].vertices.getSequenceFromMask(( '[#0:2 #2 ]', ), ))

mdb.models['Model-1'].parts['beam'].Set(name='Ctop', vertices= mdb.models['Model-1'].parts['beam'].vertices.getSequenceFromMask((

'[#0:7 #10 ]', ), )) mdb.models['Model-1'].parts['beam'].Set(name='Cbottom', vertices=

mdb.models['Model-1'].parts['beam'].vertices.getSequenceFromMask(( '[#0:9 #8 ]', ), ))

#residual stresses mdb.models['Model-1'].parts['beam'].Set(faces=

mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#3 #38000 #10 #3 #40038000 #0 #3', ' #38000 #4000 #3 #38000 #800000 #3 #38000', ' #80000000 #1800000 #8001 #90000 ]'), ), name='f1')

mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#c #c4000 #80 #c #c4000 #2 #c', ' #c4000 #20000 #c #c4000 #2000000 #c #c4000', ' #40000000 #8001 #18810000 ]'), ), name='f2')

mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#30 #302000 #400 #30 #302000 #10 #30', ' #302000 #80000 #30 #302000 #8000000 #30 #302000', ' #20000000 #4002 #0 #6900 ]'), ), name='f3')

mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#c0 #c01000 #2000 #c0 #c01000 #80 #c0', ' #c01000 #200000 #c0 #c01000 #20000000 #c0 #c01000', ' #10000000 #2400000 #4002 #81 ]'), ), name='f4')

mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#300 #3000800 #10000 #300 #3000800 #400 #300', ' #3000800 #800000 #300 #3000800 #80000000 #300 #3000800',

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' #8000000 #4200000 #2004 #42 ]'), ), name='f5') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#c00 #c000400 #80000 #c00 #c000400 #2000 #c00', ' #c000400 #2000000 #c00 #c000400 #40000000 #c00 #c000400', ' #4000000 #2004 #0 #10d00000 ]'), ), name='f6') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#3000 #30000200 #400000 #3000 #30000200 #10000 #3000', ' #30000200 #8000000 #3000 #30000200 #10000000 #3000 #30000200', ' #2000000 #1008 #24420000 ]'), ), name='f7') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#c000 #c0000100 #1000000 #c000 #80000100 #80001 #c000', ' #c0000100 #20000000 #c000 #c0000100 #4000000 #c000 #c0000100', ' #1000000 #8100000 #1008 #42000000 ]'), ), name='f8') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#30000 #80 #4000003 #30000 #80 #40000c #30000', ' #80 #80000003 #30000 #80 #1000003 #30000 #80', ' #800003 #10080000 #810 #84000000 ]'), ), name='f9') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#c0000 #40 #1000000c #c0000 #40 #2000060 #c0000', ' #40 #4000000c #c0000 #40 #40000c #c0000 #40', ' #40000c #810 #42240000 ]'), ), name='f10') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#300000 #20 #40000060 #300000 #20 #8000300 #300000', ' #20 #10000030 #300000 #20 #200030 #300000 #20', ' #200030 #420 #0 #29200000 ]'), ), name='f11') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#c00000 #10 #80000300 #c00000 #10 #20001800 #c00000', ' #10 #40000c0 #c00000 #10 #1000c0 #c00000 #10', ' #1000c0 #20040000 #420 #24 ]'), ), name='f12') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#3000000 #8 #20001800 #3000000 #8 #8000c000 #3000000', ' #8 #1000300 #3000000 #8 #80300 #3000000 #8', ' #80300 #40020000 #240 #18 ]'), ), name='f13') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#c000000 #4 #800c000 #c000000 #4 #40060000 #c000000', ' #4 #400c00 #c000000 #4 #40c00 #c000000 #4', ' #40c00 #240 #0 #9600 ]'), ) , name='f14') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#30000000 #2 #2060000 #30000000 #2 #10300000 #30000000', ' #2 #103000 #30000000 #2 #23000 #30000000 #2', ' #23000 #180 #81180000 ]'), ), name='f15') mdb.models['Model-1'].parts['beam'].Set(faces= mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#c0000000 #1 #b00000 #c0000000 #1 #5800000 #c0000000', ' #1 #58000 #c0000000 #1 #1c000 #c0000000 #1', ' #1c000 #80010000 #180 #60000 ]'), ), name='f16')

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# sets reference points mdb.models['Model-1'].rootAssembly.Set(name='RP S', referencePoints=( mdb.models['Model-1'].rootAssembly.referencePoints[4], )) mdb.models['Model-1'].rootAssembly.Set(name='RP Aatt', referencePoints=( mdb.models['Model-1'].rootAssembly.referencePoints[8], )) mdb.models['Model-1'].rootAssembly.Set(name='RP Batt', referencePoints=( mdb.models['Model-1'].rootAssembly.referencePoints[7], )) mdb.models['Model-1'].rootAssembly.Set(name='RP Datt', referencePoints=( mdb.models['Model-1'].rootAssembly.referencePoints[6], )) mdb.models['Model-1'].rootAssembly.Set(name='RP Eatt', referencePoints=( mdb.models['Model-1'].rootAssembly.referencePoints[5], )) mdb.models['Model-1'].rootAssembly.Set(name='RP load', referencePoints=(

mdb.models['Model-1'].rootAssembly.referencePoints[5], mdb.models['Model-1'].rootAssembly.referencePoints[8]))

### BC ###

mdb.models['Model-1'].DisplacementBC(amplitude=UNSET, createStepName='Initial', distributionType=UNIFORM, fieldName='', localCsys=None, name='RP S', region=mdb.models['Model-1'].rootAssembly.sets['RP S'], u1=SET, u2=SET, u3=SET, ur1=UNSET, ur2=UNSET, ur3=UNSET)

mdb.models['Model-1'].DisplacementBC(amplitude=UNSET, createStepName='Initial', distributionType=UNIFORM, fieldName='', localCsys=None, name='RP Aatt', region=mdb.models['Model-1'].rootAssembly.sets['RP Aatt'], u1=UNSET, u2= UNSET, u3=SET, ur1=UNSET, ur2=UNSET, ur3=UNSET)

mdb.models['Model-1'].DisplacementBC(amplitude=UNSET, createStepName='Initial', distributionType=UNIFORM, fieldName='', localCsys=None, name='RP Batt', region=mdb.models['Model-1'].rootAssembly.sets['RP Batt'], u1=UNSET, u2= UNSET, u3=SET, ur1=UNSET, ur2=UNSET, ur3=UNSET)

mdb.models['Model-1'].DisplacementBC(amplitude=UNSET, createStepName='Initial', distributionType=UNIFORM, fieldName='', localCsys=None, name='RP Datt', region=mdb.models['Model-1'].rootAssembly.sets['RP Datt'], u1=UNSET, u2= UNSET, u3=SET, ur1=UNSET, ur2=UNSET, ur3=UNSET)

mdb.models['Model-1'].DisplacementBC(amplitude=UNSET, createStepName='Initial', distributionType=UNIFORM, fieldName='', localCsys=None, name='RP Eatt', region=mdb.models['Model-1'].rootAssembly.sets['RP Eatt'], u1=UNSET, u2= UNSET, u3=SET, ur1=UNSET, ur2=UNSET, ur3=UNSET)

### create constraint ### ### rigid bodies ### mdb.models['Model-1'].RigidBody(name='Constraint-1', refPointRegion=

mdb.models['Model-1'].rootAssembly.sets['RP Aatt'], tieRegion= mdb.models['Model-1'].rootAssembly.instances['beam-1'].sets['Abottom']) mdb.models['Model-1'].RigidBody(name='Constraint-2', refPointRegion=

mdb.models['Model-1'].rootAssembly.sets['RP Batt'], tieRegion= mdb.models['Model-1'].rootAssembly.instances['beam-1'].sets['Btop'])

mdb.models['Model-1'].RigidBody(name='Constraint-3', refPointRegion= mdb.models['Model-1'].rootAssembly.sets['RP Datt'], tieRegion=

mdb.models['Model-1'].rootAssembly.instances['beam-1'].sets['Dtop']) mdb.models['Model-1'].RigidBody(name='Constraint-4', refPointRegion=

mdb.models['Model-1'].rootAssembly.sets['RP Eatt'], tieRegion= mdb.models['Model-1'].rootAssembly.instances['beam-1'].sets['Ebottom'])

### link ### mdb.models['Model-1'].ConnectorSection(name='ConnSect-1', translationalType=

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LINK) mdb.models['Model-1'].rootAssembly.WirePolyLine(mergeType=IMPRINT, meshable=OFF , points=((mdb.models['Model-1'].rootAssembly.referencePoints[4], mdb.models['Model-1'].rootAssembly.referencePoints[6]), ( mdb.models['Model-1'].rootAssembly.referencePoints[4], mdb.models['Model-1'].rootAssembly.referencePoints[7]))) mdb.models['Model-1'].rootAssembly.Set(edges= mdb.models['Model-1'].rootAssembly.edges.getSequenceFromMask(('[#3 ]', ), ) , name='Wire-1-Set-1') mdb.models['Model-1'].rootAssembly.SectionAssignment(region= mdb.models['Model-1'].rootAssembly.sets['Wire-1-Set-1'], sectionName= 'ConnSect-1') ### create job ### mdb.Job(atTime=None, contactPrint=OFF, description='', echoPrint=OFF, explicitPrecision=SINGLE, getMemoryFromAnalysis=True, historyPrint=OFF, memory=90, memoryUnits=PERCENTAGE, model='Model-1', modelPrint=OFF, multiprocessingMode=DEFAULT, name=nameLBA, nodalOutputPrecision=SINGLE, numCpus=1, numGPUs=0, queue=None, resultsFormat=ODB, scratch='', type= ANALYSIS, userSubroutine='', waitHours=0, waitMinutes=0) ### create step ### mdb.models['Model-1'].BuckleStep(maxIterations=300, name='lba', numEigen=2, previous='Initial', vectors=4) ### create load ### mdb.models['Model-1'].ConcentratedForce(cf2=-1.0, createStepName='lba', distributionType=UNIFORM, field='', localCsys=None, name='Load-1', region= mdb.models['Model-1'].rootAssembly.sets['RP load']) ### MESH ### mdb.models['Model-1'].parts['beam'].seedEdgeByNumber(constraint=FINER, edges= mdb.models['Model-1'].parts['beam'].edges.getSequenceFromMask(( '[#b6db6d55 #6db6db6d #db6db6db #b6db6db6 #db56db6d #b6db6db6 #db6adb6d', ' #b6db6db6 #b6db6dbd #cccccc95 #cccccccc #5555555c #cccccca5 #d5965964', ' #ad5ab56a #66649ad6 #66666666 #aaaae666 #66652aaa #66666666 #66672666', ' #9926b5ad #99999999 #aab99999 #994aaaaa #99999999 #55599999 #49ad6b55', ' #66666666 #ae666666 #52aaaaaa #66666666 #56666666 #56db5555 #a525294a', ' #a5294a94 #5293294 #33301980 #7eefed6b #3e ]'), ), number=elementspersurface) mdb.models['Model-1'].parts['beam'].seedEdgeBySize(constraint=FINER, deviationFactor=0.1, edges= mdb.models['Model-1'].parts['beam'].edges.getSequenceFromMask(( '[#492492aa #92492492 #24924924 #49249249 #24a92492 #49249249 #24952492', ' #49249249 #49249242 #3333336a #33333333 #aaaaaaa3 #3333335a #2a69a69b', ' #52a54a95 #999b6529 #99999999 #55551999 #999ad555 #99999999 #9998d999', ' #66d94a52 #66666666 #55466666 #66b55555 #66666666 #aaa66666 #b65294aa', ' #99999999 #51999999 #ad555555 #99999999 #a9999999 #a924aaaa #5adad6b5', ' #5ad6b56b #fad6cd6b #cccfe67f #81101294 #1 ]'), ), size=bigseed) mdb.models['Model-1'].parts['beam'].setElementType(elemTypes=(ElemType( elemCode=S4R, elemLibrary=STANDARD, secondOrderAccuracy=OFF, hourglassControl=DEFAULT), ElemType(elemCode=S3, elemLibrary=STANDARD)), regions=(mdb.models['Model-1'].parts['beam'].faces.getSequenceFromMask(( '[#ffffffff:18 ]', ), ), )) mdb.models['Model-1'].parts['beam'].generateMesh()

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### NODE FILE OUTPUT ### mdb.models['Model-1'].keywordBlock.synchVersions(storeNodesAndElements=False) mdb.models['Model-1'].keywordBlock.replace(124,

'\n*Output, field, variable=PRESELECT\n*NODE FILE\nU')

mdb.models['Model-1'].rootAssembly.regenerate()

mdb.jobs[nameLBA].submit(consistencyChecking=OFF)

### RIKS ###

mdb.Model(modelType=STANDARD_EXPLICIT, name='Model-2')

### create beam ### mdb.models['Model-2'].ConstrainedSketch(name='__profile__', sheetSize=200.0) mdb.models['Model-2'].sketches['__profile__'].Line(point1=(0.0, 0.0), point2=( 0.0, hw)) mdb.models['Model-2'].sketches['__profile__'].VerticalConstraint(addUndoState= False, entity=mdb.models['Model-2'].sketches['__profile__'].geometry[2]) mdb.models['Model-2'].sketches['__profile__'].Line(point1=(-halfb, 0.0), point2= (halfb, 0.0)) mdb.models['Model-2'].sketches['__profile__'].HorizontalConstraint(

addUndoState=False, entity= mdb.models['Model-2'].sketches['__profile__'].geometry[3]) mdb.models['Model-2'].sketches['__profile__'].Line(point1=(-halfb, hw),

point2=(halfb, hw)) mdb.models['Model-2'].sketches['__profile__'].HorizontalConstraint(

addUndoState=False, entity= mdb.models['Model-2'].sketches['__profile__'].geometry[4]) mdb.models['Model-2'].Part(dimensionality=THREE_D, name='beam', type=

DEFORMABLE_BODY) mdb.models['Model-2'].parts['beam'].BaseShellExtrude(depth=length, sketch= mdb.models['Model-2'].sketches['__profile__']) del mdb.models['Model-2'].sketches['__profile__'] mdb.models['Model-2'].parts['beam'].Set(faces=

mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#6 ]', ), ), name='bottom flange') mdb.models['Model-2'].parts['beam'].Set(faces=

mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#18 ]', ), ), name='top flange')

mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#1 ]', ), ), name='web')

### materials ### mdb.models['Model-2'].Material(name='steel') mdb.models['Model-2'].materials['steel'].Elastic(table=((Emod, poisson), )) mdb.models['Model-2'].materials['steel'].Plastic(table=((fy, 0.0), ))

### create sets ### mdb.models['Model-2'].parts['beam'].Set(faces=

mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#1 ]', ), ), name='web')

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mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#6 ]', ), ), name='bottom flange') mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#18 ]', ), ), name='top flange') ### sections ### mdb.models['Model-2'].HomogeneousShellSection(idealization=NO_IDEALIZATION, integrationRule=SIMPSON, material='steel', name='flange', numIntPts=5, poissonDefinition=DEFAULT, preIntegrate=OFF, temperature=GRADIENT, thickness=tf, thicknessField='', thicknessModulus=None, thicknessType= UNIFORM, useDensity=OFF) mdb.models['Model-2'].HomogeneousShellSection(idealization=NO_IDEALIZATION, integrationRule=SIMPSON, material='steel', name='web', numIntPts=5, poissonDefinition=DEFAULT, preIntegrate=OFF, temperature=GRADIENT, thickness=tw, thicknessField='', thicknessModulus=None, thicknessType= UNIFORM, useDensity=OFF) ### section assignment### mdb.models['Model-2'].parts['beam'].SectionAssignment(offset=0.0, offsetField= '', offsetType=BOTTOM_SURFACE, region= mdb.models['Model-2'].parts['beam'].sets['bottom flange'], sectionName= 'flange', thicknessAssignment=FROM_SECTION) mdb.models['Model-2'].parts['beam'].SectionAssignment(offset=0.0, offsetField= '', offsetType=TOP_SURFACE, region= mdb.models['Model-2'].parts['beam'].sets['top flange'], sectionName= 'flange', thicknessAssignment=FROM_SECTION) mdb.models['Model-2'].parts['beam'].SectionAssignment(offset=0.0, offsetField= '', offsetType=MIDDLE_SURFACE, region= mdb.models['Model-2'].parts['beam'].sets['web'], sectionName='web', thicknessAssignment=FROM_SECTION) ### create partitions for residual stresses ### mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#8 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[10], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[12], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#20 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[18], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[16], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#8 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[11], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[13], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#20 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[19], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint(

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mdb.models['Model-2'].parts['beam'].edges[17], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces=

mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#10 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[16], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[17], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#1 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[3], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[1], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#40 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[22], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[23], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#20 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[21], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[19], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#1000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[39], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[38], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#100 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[30], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[29], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#2000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[43], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[41], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#20 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[21], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[20], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#80 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[26], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[25], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#40 ]', ),

125

), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[24], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[23], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#4000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[48], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[47], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#800 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[40], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[38], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#40000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[59], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[58], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#4000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[49], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[48], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#80000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[63], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[61], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#100 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[30], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[32], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#800000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[72], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[73], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#100000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[66], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[68], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#1000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint(

126

mdb.models['Model-2'].parts['beam'].edges[43], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[44], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#4000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[49], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[50], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#10000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[86], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[87], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#800000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[75], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[77], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#40000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[61], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[63], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#8000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[56], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[54], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#400000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[75], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[73], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#1000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[80], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[79], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#20000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[63], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[62], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#20000 ]',

127

), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[63], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[62], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#2000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[85], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[83], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #2 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[106], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[105], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#80000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[70], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[69], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#80000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[70], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[68], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #1 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[104], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[102], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #80 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[122], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[121], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#200000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[77], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[76], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#200000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[77], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[76], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #20 ]',

128

), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[119], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[117], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #400 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[132], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[131], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#800000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[84], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[82], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#80 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[28], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[26], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#10000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[99], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[97], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #200 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[132], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[131], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#200 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[35], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[34], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#200 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[35], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[34], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #40000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[155], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[154], MIDDLE))

129

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #1 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[112], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[111], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#800 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[42], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[41], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#800 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[42], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[41], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #4 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[119], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[117], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #40000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[158], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[157], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#2000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[49], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[48], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#2000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[49], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[48], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #2000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[177], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[176], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #40 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[132], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[131], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces=

130

mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#8000 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[56], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[54], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#80000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[108], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[106], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #10000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[159], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[157], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #800000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[176], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[175], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #2 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[115], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[114], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #2 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[115], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[114], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #8000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[187], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[186], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #100000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[172], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[171], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #8 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[122], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint(

131

mdb.models['Model-2'].parts['beam'].edges[121], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #8 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[122], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[121], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #400000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[179], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[177], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #40000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[198], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[197], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #20 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[129], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[128], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #20 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[129], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[128], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #100 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[223], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[222], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #4000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[192], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[191], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(('[#0 #80 ]', ), ), point1=mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[136], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[135], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #1 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint(

132

mdb.models['Model-2'].parts['beam'].edges[205], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[207], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #1000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[190], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[191], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #1000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[190], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[192], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #20000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[246], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[247], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #4000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[240], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[242], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #4000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[197], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[198], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #4000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[197], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[199], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #100 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[229], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[230], MIDDLE))

mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #400 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[234], MIDDLE), point2=

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mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[235], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #10000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[204], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[205], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #10000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[204], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[206], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #4000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[272], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[273], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #200000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[261], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[263], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #40000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[211], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[212], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0 #40000000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[211], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[213], MIDDLE)) mdb.models['Model-2'].parts['beam'].PartitionFaceByShortestPath(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#0:2 #40000 ]', ), ), point1= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[257], MIDDLE), point2= mdb.models['Model-2'].parts['beam'].InterestingPoint( mdb.models['Model-2'].parts['beam'].edges[259], MIDDLE)) ### create datum planes for loads and cables ### mdb.models['Model-2'].parts['beam'].DatumPlaneByPrincipalPlane(offset=load, principalPlane=XYPLANE) mdb.models['Model-2'].parts['beam'].DatumPlaneByPrincipalPlane(offset=LABe, principalPlane=XYPLANE)

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mdb.models['Model-2'].parts['beam'].DatumPlaneByPrincipalPlane(offset=cable2, principalPlane=XYPLANE) mdb.models['Model-2'].parts['beam'].DatumPlaneByPrincipalPlane(offset=load2, principalPlane=XYPLANE) mdb.models['Model-2'].parts['beam'].DatumPlaneByPrincipalPlane(offset=length/2.0,

principalPlane=XYPLANE)

### create partitions for loads and cables ### mdb.models['Model-2'].parts['beam'].PartitionFaceByDatumPlane(datumPlane=

mdb.models['Model-2'].parts['beam'].datums[102], faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#ffffffff:3 ]', ), ))

mdb.models['Model-2'].parts['beam'].PartitionFaceByDatumPlane(datumPlane= mdb.models['Model-2'].parts['beam'].datums[101], faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#aaaaaaaa #aaaaffff #8000aaaa #ff00 #ff7fff #75830ff0 ]', ), ))

mdb.models['Model-2'].parts['beam'].PartitionFaceByDatumPlane(datumPlane= mdb.models['Model-2'].parts['beam'].datums[103], faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#aaaaaaaa #aaaaffff #aa80aaaa #0:3 #ff00 #7f1fff #758307f0 ]', ), ))

mdb.models['Model-2'].parts['beam'].PartitionFaceByDatumPlane(datumPlane= mdb.models['Model-2'].parts['beam'].datums[100], faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#aaaaaaaa #aaaaffff #aaab6aaa #0:6 #ff00 #3f07ff #750307f0 ]', ), ))

mdb.models['Model-2'].parts['beam'].PartitionFaceByDatumPlane(datumPlane= mdb.models['Model-2'].parts['beam'].datums[99], faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#aaaaaaaa #6aaaffff #ab6db6db #0:9 #ff00 #1f01ff #550307f0 ]', ), ))

### assembly ### mdb.models['Model-2'].rootAssembly.DatumCsysByDefault(CARTESIAN) mdb.models['Model-2'].rootAssembly.Instance(dependent=ON, name='beam-1', part= mdb.models['Model-2'].parts['beam']) mdb.models['Model-2'].rootAssembly.rotate(angle=90.0, axisDirection=(0.0, 1.0,

0.0), axisPoint=(0.0, 0.0, 0.0), instanceList=('beam-1', ))

### create rp ### mdb.models['Model-2'].rootAssembly.ReferencePoint(point=(halflength, heightattachment, 0)) mdb.models['Model-2'].rootAssembly.ReferencePoint(point=(load, RPload, 0)) mdb.models['Model-2'].rootAssembly.ReferencePoint(point=(LABe, RPcable, 0)) mdb.models['Model-2'].rootAssembly.ReferencePoint(point=(cable2, RPcable, 0)) mdb.models['Model-2'].rootAssembly.ReferencePoint(point=(load2, RPload, 0))

### create sets ###

### points on beam ### mdb.models['Model-2'].parts['beam'].Set(name='Abottom', vertices=

mdb.models['Model-2'].parts['beam'].vertices.getSequenceFromMask(( '[#0:15 #4 ]', ), )) mdb.models['Model-2'].parts['beam'].Set(name='Ebottom', vertices=

mdb.models['Model-2'].parts['beam'].vertices.getSequenceFromMask(( '[#0:5 #8000 ]', ), ))

mdb.models['Model-2'].parts['beam'].Set(name='Bbottom', vertices= mdb.models['Model-2'].parts['beam'].vertices.getSequenceFromMask((

'[#0:11 #80000000 ]', ), )) mdb.models['Model-2'].parts['beam'].Set(name='Btop', vertices=

mdb.models['Model-2'].parts['beam'].vertices.getSequenceFromMask((

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'[#0:10 #2000 ]', ), )) mdb.models['Model-2'].parts['beam'].Set(name='Dbottom', vertices= mdb.models['Model-2'].parts['beam'].vertices.getSequenceFromMask(( '[#0:9 #4 ]', ), )) mdb.models['Model-2'].parts['beam'].Set(name='Dtop', vertices= mdb.models['Model-2'].parts['beam'].vertices.getSequenceFromMask(( '[#0:2 #2 ]', ), )) mdb.models['Model-2'].parts['beam'].Set(name='Ctop', vertices= mdb.models['Model-2'].parts['beam'].vertices.getSequenceFromMask(( '[#0:7 #10 ]', ), )) ### residual stresses ### mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#3 #38000 #10 #3 #40038000 #0 #3', ' #38000 #4000 #3 #38000 #800000 #3 #38000', ' #80000000 #1800000 #8001 #90000 ]'), ), name='f1') mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#c #c4000 #80 #c #c4000 #2 #c', ' #c4000 #20000 #c #c4000 #2000000 #c #c4000', ' #40000000 #8001 #18810000 ]'), ), name='f2') mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#30 #302000 #400 #30 #302000 #10 #30', ' #302000 #80000 #30 #302000 #8000000 #30 #302000', ' #20000000 #4002 #0 #6900 ]'), ), name='f3') mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#c0 #c01000 #2000 #c0 #c01000 #80 #c0', ' #c01000 #200000 #c0 #c01000 #20000000 #c0 #c01000', ' #10000000 #2400000 #4002 #81 ]'), ), name='f4') mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#300 #3000800 #10000 #300 #3000800 #400 #300', ' #3000800 #800000 #300 #3000800 #80000000 #300 #3000800', ' #8000000 #4200000 #2004 #42 ]'), ), name='f5') mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#c00 #c000400 #80000 #c00 #c000400 #2000 #c00', ' #c000400 #2000000 #c00 #c000400 #40000000 #c00 #c000400', ' #4000000 #2004 #0 #10d00000 ]'), ), name='f6') mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#3000 #30000200 #400000 #3000 #30000200 #10000 #3000', ' #30000200 #8000000 #3000 #30000200 #10000000 #3000 #30000200', ' #2000000 #1008 #24420000 ]'), ), name='f7') mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#c000 #c0000100 #1000000 #c000 #80000100 #80001 #c000', ' #c0000100 #20000000 #c000 #c0000100 #4000000 #c000 #c0000100', ' #1000000 #8100000 #1008 #42000000 ]'), ), name='f8') mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#30000 #80 #4000003 #30000 #80 #40000c #30000', ' #80 #80000003 #30000 #80 #1000003 #30000 #80', ' #800003 #10080000 #810 #84000000 ]'), ), name='f9')

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mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#c0000 #40 #1000000c #c0000 #40 #2000060 #c0000', ' #40 #4000000c #c0000 #40 #40000c #c0000 #40', ' #40000c #810 #42240000 ]'), ), name='f10')

mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#300000 #20 #40000060 #300000 #20 #8000300 #300000', ' #20 #10000030 #300000 #20 #200030 #300000 #20', ' #200030 #420 #0 #29200000 ]'), ), name='f11')

mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#c00000 #10 #80000300 #c00000 #10 #20001800 #c00000', ' #10 #40000c0 #c00000 #10 #1000c0 #c00000 #10', ' #1000c0 #20040000 #420 #24 ]'), ), name='f12')

mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#3000000 #8 #20001800 #3000000 #8 #8000c000 #3000000', ' #8 #1000300 #3000000 #8 #80300 #3000000 #8', ' #80300 #40020000 #240 #18 ]'), ), name='f13')

mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#c000000 #4 #800c000 #c000000 #4 #40060000 #c000000', ' #4 #400c00 #c000000 #4 #40c00 #c000000 #4', ' #40c00 #240 #0 #9600 ]'), ) , name='f14')

mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#30000000 #2 #2060000 #30000000 #2 #10300000 #30000000', ' #2 #103000 #30000000 #2 #23000 #30000000 #2', ' #23000 #180 #81180000 ]'), ), name='f15')

mdb.models['Model-2'].parts['beam'].Set(faces= mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#c0000000 #1 #b00000 #c0000000 #1 #5800000 #c0000000', ' #1 #58000 #c0000000 #1 #1c000 #c0000000 #1', ' #1c000 #80010000 #180 #60000 ]'), ), name='f16')

### sets reference points ### mdb.models['Model-2'].rootAssembly.Set(name='RP S', referencePoints=( mdb.models['Model-2'].rootAssembly.referencePoints[4], )) mdb.models['Model-2'].rootAssembly.Set(name='RP Aatt', referencePoints=( mdb.models['Model-2'].rootAssembly.referencePoints[8], )) mdb.models['Model-2'].rootAssembly.Set(name='RP Batt', referencePoints=( mdb.models['Model-2'].rootAssembly.referencePoints[7], )) mdb.models['Model-2'].rootAssembly.Set(name='RP Datt', referencePoints=( mdb.models['Model-2'].rootAssembly.referencePoints[6], )) mdb.models['Model-2'].rootAssembly.Set(name='RP Eatt', referencePoints=( mdb.models['Model-2'].rootAssembly.referencePoints[5], )) mdb.models['Model-2'].rootAssembly.Set(name='RP load', referencePoints=(

mdb.models['Model-2'].rootAssembly.referencePoints[5], mdb.models['Model-2'].rootAssembly.referencePoints[8]))

### BC ###

mdb.models['Model-2'].DisplacementBC(amplitude=UNSET, createStepName='Initial', distributionType=UNIFORM, fieldName='', localCsys=None, name='RP S',

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region=mdb.models['Model-2'].rootAssembly.sets['RP S'], u1=SET, u2=SET, u3=SET, ur1=UNSET, ur2=UNSET, ur3=UNSET) mdb.models['Model-2'].DisplacementBC(amplitude=UNSET, createStepName='Initial', distributionType=UNIFORM, fieldName='', localCsys=None, name='RP Aatt', region=mdb.models['Model-2'].rootAssembly.sets['RP Aatt'], u1=UNSET, u2= UNSET, u3=SET, ur1=UNSET, ur2=UNSET, ur3=UNSET) mdb.models['Model-2'].DisplacementBC(amplitude=UNSET, createStepName='Initial', distributionType=UNIFORM, fieldName='', localCsys=None, name='RP Batt', region=mdb.models['Model-2'].rootAssembly.sets['RP Batt'], u1=UNSET, u2= UNSET, u3=SET, ur1=UNSET, ur2=UNSET, ur3=UNSET) mdb.models['Model-2'].DisplacementBC(amplitude=UNSET, createStepName='Initial', distributionType=UNIFORM, fieldName='', localCsys=None, name='RP Datt', region=mdb.models['Model-2'].rootAssembly.sets['RP Datt'], u1=UNSET, u2= UNSET, u3=SET, ur1=UNSET, ur2=UNSET, ur3=UNSET) mdb.models['Model-2'].DisplacementBC(amplitude=UNSET, createStepName='Initial', distributionType=UNIFORM, fieldName='', localCsys=None, name='RP Eatt', region=mdb.models['Model-2'].rootAssembly.sets['RP Eatt'], u1=UNSET, u2= UNSET, u3=SET, ur1=UNSET, ur2=UNSET, ur3=UNSET) ### create constraint ### ### rigid bodies ### mdb.models['Model-2'].RigidBody(name='Constraint-1', refPointRegion= mdb.models['Model-2'].rootAssembly.sets['RP Aatt'], tieRegion= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['Abottom']) mdb.models['Model-2'].RigidBody(name='Constraint-2', refPointRegion= mdb.models['Model-2'].rootAssembly.sets['RP Batt'], tieRegion= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['Btop']) mdb.models['Model-2'].RigidBody(name='Constraint-3', refPointRegion= mdb.models['Model-2'].rootAssembly.sets['RP Datt'], tieRegion= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['Dtop']) mdb.models['Model-2'].RigidBody(name='Constraint-4', refPointRegion= mdb.models['Model-2'].rootAssembly.sets['RP Eatt'], tieRegion= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['Ebottom']) ### link ### mdb.models['Model-2'].ConnectorSection(name='ConnSect-1', translationalType= LINK) mdb.models['Model-2'].rootAssembly.WirePolyLine(mergeType=IMPRINT, meshable=OFF , points=((mdb.models['Model-2'].rootAssembly.referencePoints[4], mdb.models['Model-2'].rootAssembly.referencePoints[6]), ( mdb.models['Model-2'].rootAssembly.referencePoints[4], mdb.models['Model-2'].rootAssembly.referencePoints[7]))) mdb.models['Model-2'].rootAssembly.Set(edges= mdb.models['Model-2'].rootAssembly.edges.getSequenceFromMask(('[#3 ]', ), ) , name='Wire-1-Set-1') mdb.models['Model-2'].rootAssembly.SectionAssignment(region= mdb.models['Model-2'].rootAssembly.sets['Wire-1-Set-1'], sectionName= 'ConnSect-1') ### create job ### mdb.Job(activateLoadBalancing=False, atTime=None, contactPrint=OFF, description='', echoPrint=OFF, explicitPrecision=SINGLE, getMemoryFromAnalysis=True, historyPrint=OFF, memory=80, memoryUnits= PERCENTAGE, model='Model-2', modelPrint=OFF, multiprocessingMode=DEFAULT, name=nameRIKS, nodalOutputPrecision=SINGLE, numCpus=2, numDomains=2, parallelizationMethodExplicit=DOMAIN, queue=None, resultsFormat=ODB,

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scratch='', type=ANALYSIS, userSubroutine='', waitHours=0, waitMinutes=0)

### create step ### mdb.models['Model-2'].StaticStep(name='equilibrium stresses', nlgeom=ON, previous='Initial') mdb.models['Model-2'].StaticRiksStep(initialArcInc=10000.0, maxArcInc=1e+36,

maxNumInc=100000, minArcInc=1.0, name='RIKS', previous= 'equilibrium stresses', totalArcLength=100000.0) mdb.models['Model-2'].steps['RIKS'].setValues(dof=2, maximumDisplacement=

-1000.0, nodeOn=ON, region= mdb.models['Model-2'].rootAssembly.sets['RP Aatt'])

### create load ### mdb.models['Model-2'].DisplacementBC(amplitude=UNSET, createStepName='RIKS',

distributionType=UNIFORM, fieldName='', fixed=OFF, localCsys=None, name= 'load', region=mdb.models['Model-2'].rootAssembly.sets['RP load'], u1=UNSET , u2=-1.0, u3=UNSET, ur1=UNSET, ur2=UNSET, ur3=UNSET)

### MESH ### mdb.models['Model-2'].parts['beam'].seedEdgeByNumber(constraint=FINER, edges=

mdb.models['Model-2'].parts['beam'].edges.getSequenceFromMask(( '[#b6db6d55 #6db6db6d #db6db6db #b6db6db6 #db56db6d #b6db6db6 #db6adb6d', ' #b6db6db6 #b6db6dbd #cccccc95 #cccccccc #5555555c #cccccca5 #d5965964', ' #ad5ab56a #66649ad6 #66666666 #aaaae666 #66652aaa #66666666 #66672666', ' #9926b5ad #99999999 #aab99999 #994aaaaa #99999999 #55599999 #49ad6b55', ' #66666666 #ae666666 #52aaaaaa #66666666 #56666666 #56db5555 #a525294a', ' #a5294a94 #5293294 #33301980 #7eefed6b #3e ]'), ), number=elementspersurface)

mdb.models['Model-2'].parts['beam'].seedEdgeBySize(constraint=FINER, deviationFactor=0.1, edges= mdb.models['Model-2'].parts['beam'].edges.getSequenceFromMask(( '[#492492aa #92492492 #24924924 #49249249 #24a92492 #49249249 #24952492', ' #49249249 #49249242 #3333336a #33333333 #aaaaaaa3 #3333335a #2a69a69b', ' #52a54a95 #999b6529 #99999999 #55551999 #999ad555 #99999999 #9998d999', ' #66d94a52 #66666666 #55466666 #66b55555 #66666666 #aaa66666 #b65294aa', ' #99999999 #51999999 #ad555555 #99999999 #a9999999 #a924aaaa #5adad6b5', ' #5ad6b56b #fad6cd6b #cccfe67f #81101294 #1 ]'), ), size=bigseed)

mdb.models['Model-2'].parts['beam'].setElementType(elemTypes=(ElemType( elemCode=S4R, elemLibrary=STANDARD, secondOrderAccuracy=OFF, hourglassControl=DEFAULT), ElemType(elemCode=S3, elemLibrary=STANDARD)), regions=(mdb.models['Model-2'].parts['beam'].faces.getSequenceFromMask(( '[#ffffffff:18 ]', ), ), ))

mdb.models['Model-2'].parts['beam'].generateMesh()

### insert residual stress ###

mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-1', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f1'], sigma11= -resstress1, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None)

mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-2', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f2'], sigma11= -resstress2, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None)

mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-3', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f3'], sigma11=

139

-resstress3, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-4', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f4'], sigma11= -resstress4, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-5', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f5'], sigma11= -resstress5, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-6', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f6'], sigma11= -resstress6, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-7', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f7'], sigma11= -resstress7, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-8', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f8'], sigma11= -resstress8, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-9', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f16'], sigma11= resstress1, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-10', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f15'], sigma11= resstress2, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-11', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f14'], sigma11= resstress3, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-12', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f13'], sigma11= resstress4, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-13', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f12'], sigma11= resstress5, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-14', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f11'], sigma11= resstress6, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-15', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f10'], sigma11= resstress7, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) mdb.models['Model-2'].Stress(distributionType=UNIFORM, name= 'Predefined Field-16', region= mdb.models['Model-2'].rootAssembly.instances['beam-1'].sets['f9'], sigma11= resstress8, sigma12=0.0, sigma13=None, sigma22=0.0, sigma23=None, sigma33=None) ### output ### mdb.models['Model-2'].HistoryOutputRequest(createStepName='RIKS', name= 'reaction force', rebar=EXCLUDE, region=

140

mdb.models['Model-2'].rootAssembly.sets['RP S'], sectionPoints=DEFAULT, variables=('RF2', )) mdb.models['Model-2'].HistoryOutputRequest(createStepName='RIKS', name=

'Uz Ctop', rebar=EXCLUDE, region= mdb.models['Model-2'].rootAssembly.allInstances['beam-1'].sets['Ctop'], sectionPoints=DEFAULT, variables=('U3', ))

mdb.models['Model-2'].HistoryOutputRequest(createStepName='RIKS', name= 'Uz Btop', rebar=EXCLUDE, region= mdb.models['Model-2'].rootAssembly.allInstances['beam-1'].sets['Btop'], sectionPoints=DEFAULT, variables=('U3', ))

### input imperfection ### mdb.models['Model-2'].keywordBlock.synchVersions(storeNodesAndElements=False) mdb.models['Model-2'].keywordBlock.replace(132, '\n** ----------------------------------------------------------------\n** \n** STEP: equilibrium stresses\n** \n*IMPERFECTION, FILE=change, step=1\n2,change')

mdb.models['Model-2'].rootAssembly.regenerate()

141

142

APPENDIX B: VARIOUS LENGTHS AND RATIOS FOR CONSIDERED BEAMS

profile name

tw (mm)

Hw(mm)

Wy,pl (mm3)

LAB,min (mm)

LAE,min (mm) ,minAE

beam

Lh

LAE,max(mm)

max

BD

AE

LL

IPE 80* 3.8 69.6 2.25E+04 589.2 1571.3 19.6 4000 0.71

IPE 100* 4.1 88.6 3.76E+04 717.3 1912.8 19.1 5000 0.71

IPE 120* 4.4 107.4 5.85E+04 858.1 2288.4 19.1 6000 0.71

IPE 140* 4.7 126.2 8.58E+04 1001.7 2671.1 19.1 7000 0.71

IPE 160* 5 145.2 1.19E+05 1135.2 3027.1 18.9 8000 0.72

IPE 180* 5.3 164 1.61E+05 1282.1 3419.0 19.0 9000 0.72

IPE 200* 5.6 183 2.10E+05 1417.4 3779.8 18.9 10000 0.72

IPE 220* 5.9 201.6 2.73E+05 1591.8 4244.7 19.3 11000 0.71

IPE 240* 6.2 220.4 3.46E+05 1754.3 4678.1 19.5 12000 0.71

IPE 270* 6.6 249.6 4.61E+05 1936.9 5165.0 19.1 13500 0.71

IPE 300* 7.1 278.6 6.02E+05 2108.9 5623.6 18.7 15000 0.72

IPE 330* 7.5 307 7.63E+05 2295.1 6120.3 18.5 16500 0.72

IPE 360* 8 334.6 9.74E+05 2520.3 6720.7 18.7 18000 0.72

IPE 400* 8.6 373 1.24E+06 2674.5 7132.1 17.8 20000 0.73

IPE 450* 9.4 420.8 1.62E+06 2844.3 7584.9 16.9 22500 0.75

IPE 500* 10.2 468 2.11E+06 3058.5 8155.9 16.3 25000 0.76

IPE 550* 11.1 515.6 2.66E+06 3222.7 8593.9 15.6 27500 0.77

IPE 600* 12 562 3.38E+06 3468.3 9248.9 15.4 28000 0.75

HEA 100* 5 80 7.84E+04 1357.9 3621.1 37.7 4800 0.43

HEB 100* 6 80 9.96E+04 1437.6 3833.6 38.3 5000 0.42

HEB 120* 6.5 98 1.59E+05 1734.6 4625.7 38.5 6000 0.42

HEB 140* 7 116 2.39E+05 2035.7 5428.5 38.8 7000 0.42

HEB 160* 8 134 3.42E+05 2208.2 5888.5 36.8 8000 0.45

HEB 180* 8.5 152 4.67E+05 2506.5 6683.9 37.1 9000 0.44

HEB 200* 9 170 6.20E+05 2807.6 7487.0 37.4 10000 0.44

HEB 220* 9.5 188 8.02E+05 3111.2 8296.5 37.7 11000 0.43

HEB 240* 10 206 1.02E+06 3416.8 9111.4 38.0 12000 0.43

HEB 260* 10 225 1.23E+06 3787.2 10099.3 38.8 13000 0.42

HEB 320* 11.5 279 2.07E+06 4460.6 11894.8 37.2 16000 0.44

143

profile name

tw (mm)

Hw(mm)

Wy,pl (mm3) LAB,min (mm)

LAE,min (mm) ,minAE

beam

Lh

LAE,max

max

BD

AE

LL

HEB 340* 12 297 2.32E+06 4507.9 12021.1 35.4 17000 0.47

HEB 360* 12.5 315 2.59E+06 4554.1 12144.2 33.7 18000 0.49

HEB 400* 13.5 352 3.13E+06 4556.7 12151.1 30.4 20000 0.54

HEA 450* 11.5 398 3.10E+06 4685.1 12493.5 28.4 22000 0.57

HEB 450* 14 398 3.86E+06 4801.5 12804.0 28.5 22500 0.57

HEA 500* 12 444 3.81E+06 4959.1 13224.3 27.0 24500 0.60

HEB 500* 14.5 444 4.68E+06 5035.7 13428.6 26.9 25000 0.60

HEA 550* 12.5 492 4.47E+06 5037.5 13433.3 24.9 27000 0.63

HEB 550* 15 492 5.44E+06 5107.4 13619.7 24.8 27500 0.63

HEA 600* 13 540 5.19E+06 5117.4 13646.4 23.1 28000 0.63

HEB 600* 15.5 540 6.26E+06 5181.6 13817.7 23.0 28000 0.63

HEA 650* 13.5 588 5.96E+06 5198.4 13862.4 21.7 28000 0.63

HEB 650* 16 588 7.14E+06 5257.8 14020.7 21.6 28000 0.62

HEA 700* 14.5 636 6.84E+06 5136.1 13696.3 19.8 28000 0.63

HEB 700* 17 636 8.13E+06 5210.8 13895.6 19.9 28000 0.63

HEA 800* 15 734 8.42E+06 5299.1 14131.0 17.9 28000 0.62

HEB 800* 17.5 734 9.95E+06 5366.9 14311.8 17.9 28000 0.62

HEA 900* 16 830 1.05E+07 5475.6 14601.5 16.4 28000 0.61

HEB 900* 18.5 830 1.23E+07 5535.6 14761.7 16.4 28000 0.60

HEA 1000* 16.5 928 1.25E+07 5642.8 15047.4 15.2 28000 0.60

HEB 1000* 19 928 1.45E+07 5698.3 15195.3 15.2 28000 0.59

min 589.2 1571.3 15.2 0.42

max 5698.3 15195.3 38.8 0.77

144

APPENDIX C

APPENDIX C.1: LIMITS OF E FOR CONSIDERED BEAMS IN S235

profile name

Ft,max Ft,min min Apen

min Dpen

min e

h/ min e

max Apen

max Dpen

max e h/ max e

IPE 80* 17942.0 3432.5 19.9 5.0 8.0 10.0 103.8 11.5 21.7 3.7

IPE 100* 24643.1 4591.3 26.6 5.8 8.8 11.3 142.6 13.5 24.7 4.0

IPE 120* 32057.8 5954.6 34.5 6.6 9.6 12.5 185.5 15.4 27.6 4.4

IPE 140* 40237.8 7477.8 43.3 7.4 10.4 13.4 232.9 17.2 30.3 4.6

IPE 160* 49250.9 9075.8 52.5 8.2 11.2 14.3 285.0 19.0 33.1 4.8

IPE 180* 58965.4 10909.2 63.1 9.0 12.0 15.0 341.2 20.8 35.8 5.0

IPE 200* 69521.1 12797.4 74.1 9.7 12.7 15.7 402.3 22.6 38.4 5.2

IPE 220* 80690.0 15164.2 87.8 10.6 13.6 16.2 467.0 24.4 41.1 5.4

IPE 240* 92700.2 17600.0 101.9 11.4 14.4 16.7 536.5 26.1 43.7 5.5

IPE 270* 111754.7 20822.9 120.5 12.4 15.4 17.5 646.7 28.7 47.5 5.7

IPE 300* 134188.9 24501.0 141.8 13.4 16.4 18.3 776.6 31.4 51.7 5.8

IPE 330* 156198.5 28216.9 163.3 14.4 17.4 18.9 903.9 33.9 55.4 6.0

IPE 360* 181590.5 33019.9 191.1 15.6 18.6 19.4 1050.9 36.6 59.4 6.1

IPE 400* 217612.8 37793.0 218.7 16.7 19.7 20.3 1259.3 40.0 64.6 6.2

IPE 450* 268337.2 44054.4 254.9 18.0 21.0 21.4 1552.9 44.5 71.2 6.3

IPE 500* 323834.6 51451.2 297.8 19.5 22.5 22.3 1874.0 48.8 77.8 6.4

IPE 550* 388251.5 59089.9 342.0 20.9 23.9 23.0 2246.8 53.5 84.7 6.5

IPE 600* 457503.9 73598.0 425.9 23.3 26.3 22.8 2647.6 58.1 91.6 6.6

HEA 100* 27135.5 9969.7 57.7 8.6 11.6 8.3 157.0 14.1 25.7 3.7

HEB 100* 32562.6 12159.0 70.4 9.5 12.5 8.0 188.4 15.5 27.7 3.6

HEB 120* 43213.2 16224.8 93.9 10.9 13.9 8.6 250.1 17.8 31.3 3.8

HEB 140* 55085.0 20804.5 120.4 12.4 15.4 9.1 318.8 20.1 34.7 4.0

HEB 160* 72723.0 26069.1 150.9 13.9 16.9 9.5 420.9 23.1 39.2 4.1

HEB 180* 87647.5 31700.7 183.5 15.3 18.3 9.8 507.2 25.4 42.6 4.2

HEB 200* 103793.1 37845.7 219.0 16.7 19.7 10.2 600.7 27.7 46.0 4.3

HEB 220* 121159.8 44504.2 257.5 18.1 21.1 10.4 701.2 29.9 49.3 4.5

HEB 240* 139747.6 51676.1 299.1 19.5 22.5 10.7 808.7 32.1 52.6 4.6

HEB 260* 152637.0 57749.3 334.2 20.6 23.6 11.0 883.3 33.5 54.8 4.7

HEB 320* 217660.3 78805.8 456.1 24.1 27.1 11.8 1259.6 40.0 64.6 5.0

HEB 340* 241777.0 83262.6 481.8 24.8 27.8 12.2 1399.2 42.2 67.8 5.0

HEB 360* 267114.7 87767.3 507.9 25.4 28.4 12.7 1545.8 44.4 71.0 5.1

HEB 400* 322369.3 95384.9 552.0 26.5 29.5 13.6 1865.6 48.7 77.6 5.2

HEA 450* 310497.5 85873.8 497.0 25.2 28.2 15.6 1796.9 47.8 76.2 5.8

HEB 450* 377997.0 104759.5 606.2 27.8 30.8 14.6 2187.5 52.8 83.7 5.4

145

HEA 500* 361444.4 95014.2 549.9 26.5 29.5 16.6 2091.7 51.6 81.9 6.0

HEB 500* 436745.3 114250.7 661.2 29.0 32.0 15.6 2527.5 56.7 89.6 5.6

HEA 550* 417207.7 101090.7 585.0 27.3 30.3 17.8 2414.4 55.4 87.7 6.2

HEB 550* 500649.3 120755.9 698.8 29.8 32.8 16.8 2897.3 60.7 95.6 5.8

HEA 600* 476227.4 113035.4 654.1 28.9 31.9 18.5 2755.9 59.2 93.4 6.3

HEB 600* 567809.6 136464.6 789.7 31.7 34.7 17.3 3285.9 64.7 101.5 5.9

HEA 650* 538503.3 129840.5 751.4 30.9 33.9 18.9 3116.3 63.0 99.0 6.5

HEB 650* 638226.1 155642.3 900.7 33.9 36.9 17.6 3693.4 68.6 107.4 6.1

HEA 700* 625608.1 149035.3 862.5 33.1 36.1 19.1 3620.4 67.9 106.3 6.5

HEB 700* 733471.6 177272.6 1025.9 36.1 39.1 17.9 4244.6 73.5 114.8 6.1

HEA 800* 746903.6 183577.6 1062.4 36.8 39.8 19.9 4322.4 74.2 115.8 6.8

HEB 800* 871387.5 216914.1 1255.3 40.0 43.0 18.6 5042.8 80.1 124.7 6.4

HEA 900* 900897.4 228800.2 1324.1 41.1 44.1 20.2 5213.5 81.5 126.7 7.0

HEB 900* 1041662.6 267452.3 1547.8 44.4 47.4 19.0 6028.1 87.6 135.9 6.6

HEA 1000* 1038745.5 271865.0 1573.3 44.8 47.8 20.7 6011.3 87.5 135.7 7.3

HEB 1000* 1196131.2 316134.4 1829.5 48.3 51.3 19.5 6922.1 93.9 145.3 6.9

146

APPENDIX C.2: LIMITS OF E FOR CONSIDERED BEAMS IN S355

profile name

Ft,max Ft,min min Apen

min Dpen

min e

h/ min e

max Apen

max Dpen

max e h/ max e

IPE 80* 27103.8 5185.3 22.0 5.3 8.3 9.6 115.2 12.1 22.7 3.5 IPE 100* 37226.8 6935.7 29.5 6.1 9.1 11.0 158.3 14.2 25.8 3.9 IPE 120* 48427.8 8995.2 38.2 7.0 10.0 12.0 205.9 16.2 28.8 4.2 IPE 140* 60784.8 11296.2 48.0 7.8 10.8 12.9 258.4 18.1 31.7 4.4 IPE 160* 74400.2 13710.3 58.3 8.6 11.6 13.8 316.3 20.1 34.6 4.6 IPE 180* 89075.3 16479.9 70.1 9.4 12.4 14.5 378.7 22.0 37.4 4.8 IPE 200* 105021.2 19332.2 82.2 10.2 13.2 15.1 446.5 23.8 40.3 5.0 IPE 220* 121893.4 22907.5 97.4 11.1 14.1 15.6 518.3 25.7 43.0 5.1 IPE 240* 140036.4 26587.2 113.0 12.0 15.0 16.0 595.4 27.5 45.8 5.2 IPE 270* 168820.9 31455.8 133.7 13.0 16.0 16.8 717.8 30.2 49.8 5.4 IPE 300* 202710.9 37012.1 157.4 14.2 17.2 17.5 861.9 33.1 54.2 5.5 IPE 330* 235959.4 42625.5 181.2 15.2 18.2 18.1 1003.2 35.7 58.1 5.7 IPE 360* 274317.6 49881.1 212.1 16.4 19.4 18.5 1166.3 38.5 62.3 5.8 IPE 400* 328734.3 57091.5 242.7 17.6 20.6 19.4 1397.7 42.2 67.8 5.9 IPE 450* 405360.4 66550.3 283.0 19.0 22.0 20.5 1723.5 46.8 74.8 6.0 IPE 500* 489197.0 77724.2 330.5 20.5 23.5 21.3 2079.9 51.5 81.7 6.1 IPE 550* 586507.6 89263.5 379.5 22.0 25.0 22.0 2493.7 56.3 89.0 6.2 IPE 600* 691122.9 111179.9 472.7 24.5 27.5 21.8 2938.4 61.2 96.2 6.2 HEA 100* 40991.9 15060.6 64.0 9.0 12.0 8.0 174.3 14.9 26.8 3.6 HEB 100* 49190.2 18367.8 78.1 10.0 13.0 7.7 209.1 16.3 29.0 3.5 HEB 120* 65279.6 24509.8 104.2 11.5 14.5 8.3 277.5 18.8 32.7 3.7 HEB 140* 83213.5 31428.1 133.6 13.0 16.0 8.7 353.8 21.2 36.3 3.9 HEB 160* 109858.2 39381.0 167.4 14.6 17.6 9.1 467.1 24.4 41.1 3.9 HEB 180* 132403.7 47888.2 203.6 16.1 19.1 9.4 562.9 26.8 44.7 4.0 HEB 200* 156793.9 57171.1 243.1 17.6 20.6 9.7 666.6 29.1 48.2 4.1 HEB 220* 183028.7 67229.7 285.8 19.1 22.1 10.0 778.2 31.5 51.7 4.3 HEB 240* 211108.1 78063.9 331.9 20.6 23.6 10.2 897.6 33.8 55.2 4.3 HEB 260* 230579.3 87238.3 370.9 21.7 24.7 10.5 980.4 35.3 57.5 4.5 HEB 320* 328806.0 119047.1 506.2 25.4 28.4 11.3 1398.0 42.2 67.8 4.7 HEB 340* 365237.6 125779.7 534.8 26.1 29.1 11.7 1552.9 44.5 71.2 4.8 HEB 360* 403513.7 132584.7 563.7 26.8 29.8 12.1 1715.6 46.7 74.6 4.8 HEB 400* 486983.4 144092.0 612.6 27.9 30.9 12.9 2070.5 51.3 81.5 4.9 HEA 450* 469049.5 129724.3 551.5 26.5 29.5 14.9 1994.3 50.4 80.1 5.5 HEB 450* 571016.7 158253.7 672.8 29.3 32.3 13.9 2427.8 55.6 87.9 5.1 HEA 500* 546011.7 143532.1 610.3 27.9 30.9 15.9 2321.5 54.4 86.1 5.7 HEB 500* 659764.1 172591.5 733.8 30.6 33.6 14.9 2805.1 59.8 94.1 5.3

147

HEA 550* 630250.0 152711.5 649.3 28.8 31.8 17.0 2679.6 58.4 92.1 5.9 HEB 550* 756300.0 182418.5 775.6 31.4 34.4 16.0 3215.6 64.0 100.5 5.5 HEA 600* 719407.3 170755.7 726.0 30.4 33.4 17.7 3058.7 62.4 98.1 6.0 HEB 600* 857754.9 206148.6 876.5 33.4 36.4 16.5 3646.9 68.1 106.7 5.6 HEA 650* 813483.6 196142.0 833.9 32.6 35.6 18.0 3458.7 66.4 104.0 6.2 HEB 650* 964128.8 235119.2 999.7 35.7 38.7 16.8 4099.2 72.2 112.9 5.8 HEA 700* 945067.5 225138.4 957.2 34.9 37.9 18.2 4018.1 71.5 111.8 6.2 HEB 700* 1108010.2 267794.7 1138.6 38.1 41.1 17.0 4710.9 77.4 120.7 5.8 HEA 800* 1128301.2 277319.4 1179.1 38.7 41.7 18.9 4797.2 78.2 121.7 6.5 HEB 800* 1316351.4 327678.7 1393.2 42.1 45.1 17.7 5596.7 84.4 131.1 6.1 HEA 900* 1360930.1 345634.3 1469.5 43.3 46.3 19.2 5786.3 85.8 133.2 6.7 HEB 900* 1573575.4 404023.7 1717.8 46.8 49.8 18.1 6690.4 92.3 142.9 6.3 HEA 1000* 1569168.7 410689.7 1746.1 47.2 50.2 19.7 6671.6 92.2 142.7 6.9 HEB 1000* 1806921.6 477564.7 2030.5 50.8 53.8 18.6 7682.5 98.9 152.9 6.5

148

APPENDIX C.3: LIMITS OF E FOR CONSIDERED BEAMS IN S460

profile name

Ft,max Ft,min min Apen

min Dpen

min e

h/ min e

max Apen

max Dpen

max e h/ max e

IPE 80* 35120.4 6719.0 24.6 5.6 8.6 9.3 128.4 12.8 23.7 3.4 IPE 100* 48237.5 8987.1 32.8 6.5 9.5 10.6 176.3 15.0 27.0 3.7 IPE 120* 62751.5 11655.7 42.6 7.4 10.4 11.6 229.4 17.1 30.1 4.0 IPE 140* 78763.4 14637.4 53.5 8.3 11.3 12.4 287.9 19.1 33.2 4.2 IPE 160* 96405.9 17765.5 64.9 9.1 12.1 13.2 352.4 21.2 36.3 4.4 IPE 180* 115421.6 21354.2 78.0 10.0 13.0 13.9 421.9 23.2 39.3 4.6 IPE 200* 136083.8 25050.2 91.6 10.8 13.8 14.5 497.4 25.2 42.2 4.7 IPE 220* 157946.4 29683.0 108.5 11.8 14.8 14.9 577.3 27.1 45.2 4.9 IPE 240* 181455.6 34451.0 125.9 12.7 15.7 15.3 663.2 29.1 48.1 5.0 IPE 270* 218753.9 40759.7 149.0 13.8 16.8 16.1 799.5 31.9 52.4 5.2 IPE 300* 262667.7 47959.4 175.3 14.9 17.9 16.7 960.0 35.0 56.9 5.3 IPE 330* 305750.3 55233.1 201.9 16.0 19.0 17.3 1117.5 37.7 61.1 5.4 IPE 360* 355453.8 64634.6 236.2 17.3 20.3 17.7 1299.2 40.7 65.5 5.5 IPE 400* 425965.6 73977.7 270.4 18.6 21.6 18.6 1556.9 44.5 71.3 5.6 IPE 450* 525255.7 86234.2 315.2 20.0 23.0 19.5 1919.8 49.4 78.7 5.7 IPE 500* 633889.0 100713.1 368.1 21.6 24.6 20.3 2316.8 54.3 86.0 5.8 IPE 550* 759981.6 115665.4 422.8 23.2 26.2 21.0 2777.7 59.5 93.7 5.9 IPE 600* 895539.5 144064.1 526.6 25.9 28.9 20.8 3273.2 64.6 101.3 5.9 HEA 100* 53116.2 19515.2 71.3 9.5 12.5 7.7 194.1 15.7 28.1 3.4 HEB 100* 63739.5 23800.5 87.0 10.5 13.5 7.4 233.0 17.2 30.3 3.3 HEB 120* 84587.6 31759.2 116.1 12.2 15.2 7.9 309.2 19.8 34.3 3.5 HEB 140* 107825.9 40723.7 148.8 13.8 16.8 8.4 394.1 22.4 38.1 3.7 HEB 160* 142351.5 51028.9 186.5 15.4 18.4 8.7 520.3 25.7 43.1 3.7 HEB 180* 171565.4 62052.3 226.8 17.0 20.0 9.0 627.1 28.3 46.9 3.8 HEB 200* 203169.6 74080.9 270.8 18.6 21.6 9.3 742.6 30.7 50.6 4.0 HEB 220* 237163.9 87114.6 318.4 20.1 23.1 9.5 866.8 33.2 54.3 4.0 HEB 240* 273548.6 101153.2 369.7 21.7 24.7 9.7 999.8 35.7 58.0 4.1 HEB 260* 298778.8 113041.2 413.2 22.9 25.9 10.0 1092.0 37.3 60.4 4.3 HEB 320* 426058.5 154258.2 563.8 26.8 29.8 10.7 1557.2 44.5 71.3 4.5 HEB 340* 473265.6 162982.1 595.7 27.5 30.5 11.1 1729.8 46.9 74.9 4.5 HEB 360* 522862.8 171799.9 627.9 28.3 31.3 11.5 1911.0 49.3 78.5 4.6 HEB 400* 631020.8 186710.8 682.4 29.5 32.5 12.3 2306.4 54.2 85.8 4.7 HEA 450* 607782.4 168093.4 614.4 28.0 31.0 14.2 2221.4 53.2 84.3 5.2 HEB 450* 739909.0 205061.2 749.5 30.9 33.9 13.3 2704.3 58.7 92.5 4.9 HEA 500* 707508.1 185985.2 679.8 29.4 32.4 15.1 2585.9 57.4 90.6 5.4 HEB 500* 854905.6 223639.7 817.4 32.3 35.3 14.2 3124.7 63.1 99.1 5.0

149

HEA 550* 816662.0 197879.7 723.2 30.3 33.3 16.2 2984.9 61.6 97.0 5.6 HEB 550* 979994.3 236373.3 863.9 33.2 36.2 15.2 3581.9 67.5 105.8 5.2 HEA 600* 932189.7 221260.9 808.7 32.1 35.1 16.8 3407.1 65.9 103.3 5.7 HEB 600* 1111457.0 267122.2 976.3 35.3 38.3 15.7 4062.3 71.9 112.4 5.3 HEA 650* 1054091.5 254155.8 928.9 34.4 37.4 17.1 3852.7 70.0 109.6 5.8 HEB 650* 1249293.6 304661.5 1113.5 37.7 40.7 16.0 4566.1 76.2 118.9 5.5 HEA 700* 1224594.6 291728.7 1066.3 36.8 39.8 17.3 4475.9 75.5 117.7 5.9 HEB 700* 1435731.6 347001.6 1268.3 40.2 43.2 16.2 5247.6 81.7 127.1 5.5 HEA 800* 1462024.1 359343.4 1313.4 40.9 43.9 18.0 5343.7 82.5 128.2 6.2 HEB 800* 1705694.8 424597.8 1551.9 44.5 47.5 16.9 6234.3 89.1 138.1 5.8 HEA 900* 1763458.7 447864.2 1636.9 45.7 48.7 18.3 6445.4 90.6 140.4 6.3 HEB 900* 2038999.1 523523.6 1913.5 49.4 52.4 17.2 7452.5 97.4 150.6 6.0 HEA 1000* 2033289.1 532161.3 1945.0 49.8 52.8 18.8 7431.6 97.3 150.4 6.6 HEB 1000* 2341363.2 618816.2 2261.8 53.7 56.7 17.6 8557.6 104.4 161.1 6.2

150

APPENDIX D: OVERVIEW OF DIFFERENT MODELS USED.

APPENDIX D.1: 421 MOELS USED TO CREATE C1

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-) HEA 400* 17.00 0.23 214.5 0.514 0.606 HEA 400* 16.00 0.24 239.6 0.509 0.627 HEA 360* 5.40 0.25 1477.6 0.507 0.884 IPE 400* 6.00 0.25 247.1 0.507 0.854 IPE 400* 6.00 0.25 247.1 0.507 0.854 HEA 400* 7.00 0.25 1106.6 0.507 0.854 IPE 500* 7.50 0.25 325.9 0.507 0.839 HEB 200* 3.00 0.25 678.1 0.507 0.839 HEB 400* 6.00 0.25 1907.1 0.507 0.833 IPE 200* 3.00 0.25 53.5 0.507 0.826 HEA 400* 8.00 0.25 864.6 0.507 0.829 HEA 800* 12.00 0.25 1192.5 0.507 0.805 IPE 140* 2.10 0.25 24.2 0.507 0.796 HEA 400* 9.00 0.25 695.5 0.507 0.802 HEA 360* 9.00 0.25 575.4 0.507 0.797 IPE 100* 1.50 0.25 12.0 0.507 0.765 HEA 400* 10.00 0.25 572.5 0.507 0.775 HEA 450* 11.25 0.25 572.2 0.507 0.748 HEA 400* 11.00 0.25 480.1 0.507 0.748 IPE 400* 10.00 0.25 95.9 0.507 0.726 HEA 400* 12.00 0.25 408.7 0.507 0.722 HEB 200* 5.00 0.25 265.4 0.507 0.716 HEB 400* 10.00 0.25 745.3 0.507 0.704 IPE 200* 5.00 0.25 20.7 0.507 0.680 HEA 400* 13.00 0.25 352.3 0.507 0.697 IPE 400* 12.00 0.25 68.3 0.507 0.666 HEA 400* 14.00 0.25 307.0 0.507 0.672 HEA 800* 20.00 0.25 464.1 0.507 0.657 IPE 500* 15.00 0.25 90.1 0.507 0.640 HEA 360* 14.40 0.25 240.9 0.507 0.656 HEA 400* 15.00 0.25 270.0 0.507 0.649 IPE 600* 18.00 0.25 120.0 0.507 0.611 HEA 400* 16.00 0.25 239.3 0.507 0.626 HEA 400* 17.00 0.25 213.6 0.507 0.605 IPE 140* 4.20 0.25 6.7 0.507 0.572 HEA 450* 18.00 0.25 238.8 0.507 0.593 IPE 400* 16.00 0.25 39.8 0.507 0.565

151

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-) HEA 400* 18.00 0.25 191.8 0.507 0.585 HEA 400* 19.00 0.25 173.2 0.507 0.566 IPE 100* 3.00 0.25 3.3 0.507 0.530 HEB 200* 8.00 0.25 110.4 0.507 0.557 HEA 800* 28.00 0.25 246.8 0.507 0.539 HEB 400* 16.00 0.25 309.8 0.507 0.544 IPE 200* 8.00 0.25 8.6 0.507 0.514 HEB 260* 3.90 0.25 1333.4 0.506 0.859 HEA 500* 7.35 0.25 1521.4 0.506 0.847 HEB 400* 6.00 0.25 1906.4 0.506 0.833 HEA 600* 8.85 0.25 1406.0 0.506 0.832 HEB 180* 2.70 0.25 512.6 0.506 0.831 HEA 700* 10.35 0.25 1327.2 0.506 0.816 HEB 140* 2.10 0.25 264.6 0.506 0.806 HEB 500* 7.50 0.25 1830.1 0.506 0.800 HEA 1000* 14.85 0.25 1106.9 0.506 0.780 HEB 600* 9.00 0.25 1675.4 0.506 0.785 HEB 100* 1.50 0.25 111.9 0.506 0.759 HEB 1000* 15.00 0.25 1288.0 0.506 0.735 HEB 260* 7.80 0.25 372.9 0.506 0.700 HEA 900* 20.03 0.25 557.2 0.506 0.671 HEA 1000* 22.28 0.25 523.3 0.506 0.658 HEA 500* 14.70 0.25 424.1 0.506 0.669 HEB 220* 6.60 0.25 244.3 0.506 0.669 HEB 400* 12.00 0.25 531.4 0.506 0.644 HEA 600* 17.70 0.25 391.2 0.506 0.641 HEB 180* 5.40 0.25 143.0 0.506 0.642 HEB 1000* 22.50 0.25 607.4 0.506 0.603 HEA 700* 20.70 0.25 368.2 0.506 0.611 HEB 140* 4.20 0.25 73.7 0.506 0.603 HEB 500* 15.00 0.25 509.1 0.506 0.593 HEB 600* 18.00 0.25 464.6 0.506 0.571 HEA 900* 28.00 0.25 297.5 0.506 0.555 HEB 1000* 28.00 0.25 402.3 0.506 0.527 HEB 100* 3.00 0.25 30.9 0.506 0.535 HEB 260* 13.00 0.25 143.3 0.506 0.524 HEA 700* 28.00 0.25 208.1 0.506 0.505 HEA 500* 24.50 0.25 162.2 0.506 0.491 HEB 220* 11.00 0.25 93.4 0.506 0.491 HEB 400* 20.00 0.25 202.5 0.506 0.466 HEB 180* 9.00 0.25 54.5 0.506 0.463

152

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-)

HEA 400* 15.00 0.26 269.4 0.503 0.648 HEA 100* 4.80 0.26 9.9 0.503 0.449 HEA 400* 7.00 0.27 1097.7 0.499 0.851 HEA 400* 14.00 0.28 305.2 0.496 0.671 HEB 340* 5.10 0.30 1923.0 0.487 0.846 HEA 400* 7.00 0.30 1086.8 0.487 0.849 IPE 220* 3.30 0.30 68.6 0.487 0.826 IPE 180* 2.70 0.30 41.5 0.487 0.813 HEA 400* 9.00 0.30 685.2 0.487 0.799 HEB 600* 9.00 0.30 1649.5 0.487 0.781 HEB 700* 10.50 0.30 1543.7 0.487 0.765 HEA 400* 11.00 0.30 474.2 0.487 0.746 HEA 400* 12.00 0.30 404.1 0.487 0.721 IPE 330* 9.90 0.30 48.7 0.487 0.680 IPE 270* 8.10 0.30 31.8 0.487 0.663 HEB 340* 10.20 0.30 542.2 0.487 0.681 HEA 400* 14.00 0.30 304.2 0.487 0.672 HEA 400* 15.00 0.30 267.8 0.487 0.649 IPE 220* 6.60 0.30 19.1 0.487 0.626 HEA 400* 16.00 0.30 237.5 0.487 0.626 HEA 400* 17.00 0.30 212.2 0.487 0.605 HEA 400* 18.00 0.30 190.7 0.487 0.585 HEA 400* 19.00 0.30 172.3 0.487 0.566 HEB 700* 21.00 0.30 430.2 0.487 0.546 HEA 450* 22.00 0.30 162.7 0.487 0.523 HEB 900* 28.00 0.30 350.8 0.487 0.501 HEB 340* 17.00 0.30 209.1 0.487 0.506 HEB 600* 24.00 0.30 268.3 0.487 0.471 HEB 700* 28.00 0.30 249.4 0.487 0.448 HEA 400* 6.00 0.32 1432.4 0.480 0.869 HEA 400* 12.00 0.33 402.5 0.476 0.721 HEA 400* 8.00 0.35 839.0 0.465 0.825 HEA 400* 10.00 0.35 559.2 0.465 0.774 HEA 400* 12.00 0.35 401.3 0.465 0.723 HEA 400* 13.00 0.35 346.8 0.465 0.699 HEA 400* 15.00 0.35 266.9 0.465 0.652 HEA 400* 16.00 0.35 237.0 0.465 0.630 HEA 400* 17.00 0.35 211.9 0.465 0.610 HEA 400* 19.00 0.35 172.4 0.465 0.571 HEA 400* 11.00 0.35 469.9 0.463 0.749 HEA 400* 10.00 0.39 555.8 0.445 0.778

153

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-)

HEA 400* 6.00 0.40 1391.6 0.440 0.868 HEA 400* 8.00 0.40 829.3 0.440 0.828 HEA 400* 9.00 0.40 670.9 0.440 0.804 HEA 400* 10.00 0.40 555.1 0.440 0.779 HEA 400* 12.00 0.40 399.9 0.440 0.729 HEA 400* 13.00 0.40 346.2 0.440 0.705 HEA 400* 14.00 0.40 302.8 0.440 0.682 HEA 400* 15.00 0.40 267.2 0.440 0.660 HEA 400* 16.00 0.40 237.6 0.440 0.638 HEA 400* 17.00 0.40 212.7 0.440 0.617 HEA 400* 18.00 0.40 191.5 0.440 0.598 HEA 400* 19.00 0.40 173.4 0.440 0.579 HEA 400* 19.00 0.42 173.9 0.431 0.582 HEA 400* 9.00 0.43 667.4 0.421 0.809 HEA 400* 18.00 0.44 192.9 0.418 0.606 HEA 450* 6.75 0.45 1370.9 0.411 0.857 HEA 400* 7.00 0.45 1041.0 0.411 0.855 HEA 400* 8.00 0.45 820.8 0.411 0.834 HEA 400* 9.00 0.45 665.8 0.411 0.811 HEA 400* 10.00 0.45 552.3 0.411 0.787 HEA 400* 11.00 0.45 466.4 0.411 0.763 HEA 400* 12.00 0.45 399.6 0.411 0.739 HEA 400* 14.00 0.45 303.8 0.411 0.693 HEA 400* 15.00 0.45 268.5 0.411 0.671 HEA 400* 16.00 0.45 239.2 0.411 0.649 HEA 450* 18.00 0.45 240.0 0.411 0.617 HEA 400* 18.00 0.45 193.4 0.411 0.609 HEA 400* 19.00 0.45 175.3 0.411 0.591 HEA 400* 8.00 0.49 814.9 0.388 0.840 HEA 400* 16.00 0.49 241.3 0.384 0.662 HEA 900* 15.00 0.50 908.3 0.380 0.775 HEA 650* 15.00 0.50 584.6 0.380 0.734 HEA 600* 15.00 0.50 522.2 0.380 0.723 IPE 450* 15.00 0.50 64.5 0.380 0.654 HEB 320* 15.00 0.50 246.4 0.380 0.585 HEA 450* 15.00 0.50 335.2 0.380 0.690 HEB 600* 15.00 0.50 651.2 0.380 0.664 HEB 450* 15.00 0.50 433.1 0.380 0.617 IPE 360* 15.00 0.50 33.2 0.380 0.593 IPE 330* 15.00 0.50 23.3 0.380 0.574 HEA 360* 5.40 0.50 1344.4 0.380 0.882

154

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-)

IPE 400* 6.00 0.50 229.4 0.380 0.862 IPE 400* 6.00 0.50 229.4 0.380 0.862 HEA 400* 7.00 0.50 1027.5 0.380 0.862 HEB 220* 3.30 0.50 810.3 0.380 0.851 IPE 500* 7.50 0.50 304.1 0.380 0.848 HEB 200* 3.00 0.50 630.1 0.380 0.846 HEB 400* 6.00 0.50 1772.3 0.380 0.839 IPE 200* 3.00 0.50 50.2 0.380 0.837 HEA 400* 8.00 0.50 812.9 0.380 0.843 IPE 180* 2.70 0.50 39.7 0.380 0.830 HEA 800* 12.00 0.50 1120.4 0.380 0.815 IPE 140* 2.10 0.50 23.0 0.380 0.811 HEA 400* 9.00 0.50 661.4 0.380 0.821 HEA 360* 9.00 0.50 549.1 0.380 0.818 HEB 600* 9.00 0.50 1579.8 0.380 0.795 HEA 1000* 15.00 0.50 1029.9 0.380 0.790 HEA 400* 10.00 0.50 550.2 0.380 0.798 HEA 800* 15.00 0.50 760.2 0.380 0.766 HEB 1000* 15.00 0.50 1233.8 0.380 0.750 HEA 400* 11.00 0.50 465.9 0.380 0.774 HEA 700* 15.00 0.50 651.7 0.380 0.742 HEA 400* 12.00 0.50 400.3 0.380 0.751 HEB 800* 15.00 0.50 927.6 0.380 0.718 HEB 400* 10.00 0.50 730.3 0.380 0.731 IPE 600* 15.00 0.50 166.8 0.380 0.704 IPE 330* 9.90 0.50 48.8 0.380 0.713 IPE 200* 5.00 0.50 20.5 0.380 0.710 HEA 400* 13.00 0.50 348.0 0.380 0.729 IPE550* 15.00 0.50 121.8 0.380 0.694 IPE 400* 12.00 0.50 68.0 0.380 0.699 HEA 550* 15.00 0.50 462.8 0.380 0.710 HEB 700* 15.00 0.50 800.5 0.380 0.689 HEA 400* 14.00 0.50 305.7 0.380 0.706 HEA 800* 20.00 0.50 460.3 0.380 0.686 HEB 220* 6.60 0.50 242.9 0.380 0.702 IPE 500* 15.00 0.50 90.2 0.380 0.674 IPE 500* 15.00 0.50 90.2 0.380 0.674 HEA 500* 15.00 0.50 406.2 0.380 0.694 HEB 650* 15.00 0.50 723.5 0.380 0.678 HEA 400* 15.00 0.50 270.8 0.380 0.685 HEA 400* 15.00 0.50 270.8 0.380 0.685

155

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-)

HEA 320* 15.00 0.50 176.6 0.380 0.682 HEA 340* 15.00 0.50 199.9 0.380 0.680 HEA 360* 15.00 0.50 224.7 0.380 0.679 HEA 400* 16.00 0.50 241.6 0.380 0.664 IPE 400* 15.00 0.50 45.8 0.380 0.626 HEA 400* 17.00 0.50 217.0 0.380 0.644 IPE 140* 4.20 0.50 6.8 0.380 0.608 HEB 500* 15.00 0.50 514.7 0.380 0.627 IPE 400* 16.00 0.50 40.8 0.380 0.604 HEA 400* 18.00 0.50 196.1 0.380 0.625 HEB 400* 15.00 0.50 357.9 0.380 0.605 HEB 200* 8.00 0.50 113.5 0.380 0.596 HEA 800* 28.00 0.50 253.6 0.380 0.576 HEB 360* 15.00 0.50 303.2 0.380 0.592 HEB 340* 15.00 0.50 274.1 0.380 0.588 HEB 400* 16.00 0.50 318.9 0.380 0.583 IPE 200* 8.00 0.50 8.9 0.380 0.553 HEB 220* 11.00 0.50 97.7 0.380 0.532 HEB 600* 24.00 0.50 281.3 0.380 0.508 IPE 180* 9.00 0.50 4.6 0.380 0.463 HEB 550* 15.00 0.50 581.7 0.380 0.647 HEB 900* 15.00 0.50 1094.6 0.380 0.731 HEB 340* 5.10 0.50 1805.7 0.379 0.854 HEA 600* 8.85 0.50 1305.9 0.379 0.838 HEB 180* 2.70 0.50 478.2 0.379 0.840 HEB 140* 2.10 0.50 248.9 0.379 0.818 HEB 500* 7.50 0.50 1716.5 0.379 0.808 HEA 1000* 14.85 0.50 1047.8 0.379 0.793 HEB 600* 9.00 0.50 1579.5 0.379 0.796 HEB 900* 13.50 0.50 1313.7 0.379 0.758 HEB 1000* 15.00 0.50 1233.7 0.379 0.750 HEA 450* 13.20 0.50 419.3 0.379 0.729 HEB 260* 7.80 0.50 367.4 0.379 0.731 HEA 900* 20.03 0.50 549.3 0.379 0.698 HEB 340* 10.20 0.50 541.8 0.379 0.713 HEA 1000* 22.28 0.50 517.4 0.379 0.686 HEA 500* 14.70 0.50 420.9 0.379 0.701 HEA 600* 17.70 0.50 391.1 0.379 0.674 HEB 180* 5.40 0.50 143.3 0.379 0.676 HEB 900* 20.25 0.50 648.4 0.379 0.643 HEB 1000* 22.50 0.50 608.2 0.379 0.633

156

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-)

HEB 140* 4.20 0.50 74.6 0.379 0.638 HEA 1000* 28.00 0.50 345.7 0.379 0.613 HEB 500* 15.00 0.50 514.9 0.379 0.628 HEB 600* 18.00 0.50 472.3 0.379 0.605 HEA 900* 28.00 0.50 304.0 0.379 0.590 HEB 1000* 28.00 0.50 412.2 0.379 0.560 HEA 450* 22.00 0.50 169.7 0.379 0.563 HEB 260* 13.00 0.50 148.8 0.379 0.566 HEB 900* 28.00 0.50 363.9 0.379 0.536 HEB 340* 17.00 0.50 218.8 0.379 0.546 HEA 500* 24.50 0.50 169.5 0.379 0.531 HEB 180* 9.00 0.50 57.3 0.379 0.504 HEB 140* 7.00 0.50 29.6 0.379 0.466 HEB 600* 28.00 0.50 211.9 0.379 0.457 HEB 500* 25.00 0.50 204.4 0.379 0.456 HEA 400* 6.00 0.55 1322.2 0.346 0.882 IPE 270* 4.05 0.55 106.3 0.346 0.869 HEA 400* 7.00 0.55 1013.6 0.346 0.870 HEB 400* 6.00 0.55 1748.4 0.346 0.846 IPE 220* 3.30 0.55 64.6 0.346 0.850 IPE 600* 9.00 0.55 403.6 0.346 0.840 HEA 400* 8.00 0.55 805.0 0.346 0.853 HEA 700* 10.35 0.55 1227.9 0.346 0.832 HEA 400* 9.00 0.55 657.3 0.346 0.832 IPE 100* 1.50 0.55 11.5 0.346 0.795 HEA 400* 10.00 0.55 548.6 0.346 0.810 HEA 400* 11.00 0.55 465.9 0.346 0.788 HEA 400* 12.00 0.55 401.5 0.346 0.766 HEA 400* 13.00 0.55 350.0 0.346 0.744 IPE 270* 8.10 0.55 32.2 0.346 0.712 HEA 400* 14.00 0.55 308.2 0.346 0.723 HEA 400* 15.00 0.55 273.7 0.346 0.702 IPE 220* 6.60 0.55 19.6 0.346 0.677 IPE 600* 18.00 0.55 122.4 0.346 0.662 HEA 400* 16.00 0.55 244.8 0.346 0.682 HEA 700* 20.70 0.55 375.2 0.346 0.661 HEA 400* 17.00 0.55 220.4 0.346 0.662 HEA 400* 18.00 0.55 199.5 0.346 0.643 HEA 400* 19.00 0.55 181.5 0.346 0.625 IPE 270* 13.50 0.55 13.1 0.346 0.542 HEB 400* 20.00 0.55 218.7 0.346 0.525

157

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-)

HEA 400* 7.00 0.56 1011.6 0.341 0.872 HEA 400* 14.00 0.56 309.1 0.335 0.728 HEA 400* 6.00 0.60 1296.7 0.308 0.889 HEA 400* 8.00 0.60 796.4 0.308 0.864 HEA 400* 9.00 0.60 652.8 0.308 0.845 HEA 400* 10.00 0.60 546.8 0.308 0.825 HEA 400* 11.00 0.60 466.1 0.308 0.804 HEA 400* 12.00 0.60 402.9 0.308 0.783 HEA 400* 13.00 0.60 352.4 0.308 0.762 HEA 400* 14.00 0.60 311.2 0.308 0.742 HEA 400* 15.00 0.60 277.2 0.308 0.722 HEA 400* 16.00 0.60 248.6 0.308 0.702 HEA 400* 18.00 0.60 203.6 0.308 0.665 HEA 400* 19.00 0.60 185.6 0.308 0.647 HEA 400* 13.00 0.61 352.8 0.302 0.765 HEA 400* 19.00 0.63 188.0 0.288 0.660 HEA 400* 6.00 0.65 1268.4 0.268 0.896 HEA 400* 6.00 0.65 1268.4 0.268 0.896 HEA 400* 7.00 0.65 981.5 0.268 0.888 HEA 400* 8.00 0.65 786.3 0.268 0.875 HEA 400* 10.00 0.65 544.3 0.268 0.839 HEA 400* 11.00 0.65 465.7 0.268 0.820 HEA 400* 14.00 0.65 314.2 0.268 0.762 HEA 400* 15.00 0.65 280.8 0.268 0.743 HEA 400* 16.00 0.65 252.6 0.268 0.725 HEA 400* 17.00 0.65 228.6 0.268 0.707 HEA 400* 18.00 0.65 208.0 0.268 0.689 HEA 400* 19.00 0.65 190.2 0.268 0.672 HEA 400* 12.00 0.66 404.1 0.261 0.803 HEA 400* 18.00 0.66 209.0 0.259 0.695 IPE 400* 6.00 0.70 214.8 0.225 0.896 IPE 400* 6.00 0.70 214.8 0.225 0.896 HEA 400* 7.00 0.70 961.5 0.225 0.896 HEB 400* 6.00 0.70 1654.7 0.225 0.867 IPE 180* 2.70 0.70 37.7 0.225 0.866 HEA 900* 13.35 0.70 1057.8 0.225 0.833 HEA 400* 9.00 0.70 639.6 0.225 0.870 HEB 600* 9.00 0.70 1499.5 0.225 0.826 HEB 700* 10.50 0.70 1419.5 0.225 0.812 HEA 400* 10.00 0.70 540.2 0.225 0.853 HEB 900* 13.50 0.70 1263.3 0.225 0.790

158

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-)

HEA 400* 11.00 0.70 464.0 0.225 0.836 IPE 400* 10.00 0.70 93.8 0.225 0.814 HEA 400* 12.00 0.70 404.2 0.225 0.818 IPE 330* 9.90 0.70 50.1 0.225 0.783 HEA 450* 13.20 0.70 426.4 0.225 0.796 HEA 900* 20.03 0.70 554.8 0.225 0.753 IPE 400* 12.00 0.70 70.2 0.225 0.770 HEA 400* 14.00 0.70 316.7 0.225 0.782 HEA 400* 15.00 0.70 283.9 0.225 0.765 HEB 400* 12.00 0.70 551.1 0.225 0.748 HEA 400* 16.00 0.70 256.3 0.225 0.748 HEA 400* 17.00 0.70 232.8 0.225 0.731 HEA 400* 17.00 0.70 232.8 0.225 0.731 HEA 450* 18.00 0.70 260.9 0.225 0.716 IPE 400* 16.00 0.70 44.3 0.225 0.690 HEA 400* 18.00 0.70 212.5 0.225 0.714 HEB 600* 18.00 0.70 501.7 0.225 0.678 HEA 400* 19.00 0.70 194.9 0.225 0.699 HEB 700* 21.00 0.70 473.5 0.225 0.654 IPE 330* 16.50 0.70 22.0 0.225 0.635 HEA 450* 22.00 0.70 189.0 0.225 0.657 HEB 400* 20.00 0.70 242.4 0.225 0.600 HEB 700* 28.00 0.70 296.4 0.225 0.567 IPE 180* 9.00 0.70 5.4 0.225 0.555 HEB 600* 28.00 0.70 244.0 0.225 0.545 HEA 400* 11.00 0.72 462.9 0.208 0.841 HEA 400* 16.00 0.74 258.8 0.184 0.767 HEA 360* 5.40 0.75 1185.8 0.179 0.906 HEA 400* 6.00 0.75 1199.4 0.179 0.905 IPE 270* 4.05 0.75 98.3 0.179 0.899 HEA 400* 7.00 0.75 944.7 0.179 0.909 IPE 500* 7.50 0.75 279.0 0.179 0.886 HEB 200* 3.00 0.75 569.4 0.179 0.875 HEB 400* 6.00 0.75 1608.1 0.179 0.868 IPE 220* 3.30 0.75 60.1 0.179 0.879 IPE 200* 3.00 0.75 46.3 0.179 0.876 HEA 400* 8.00 0.75 757.3 0.179 0.892 HEA 800* 12.00 0.75 1034.1 0.179 0.849 HEA 400* 9.00 0.75 628.8 0.179 0.879 HEA 360* 9.00 0.75 525.9 0.179 0.881 HEB 600* 9.00 0.75 1460.0 0.179 0.826

159

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-)

IPE 100* 1.50 0.75 10.9 0.179 0.822 HEA 400* 10.00 0.75 533.3 0.179 0.864 HEA 400* 11.00 0.75 460.0 0.179 0.848 HEA 400* 12.00 0.75 402.3 0.179 0.833 HEB 200* 5.00 0.75 258.2 0.179 0.815 IPE 200* 5.00 0.75 20.8 0.179 0.785 HEA 400* 13.00 0.75 355.8 0.179 0.816 IPE 270* 8.10 0.75 33.0 0.179 0.783 HEA 400* 14.00 0.75 317.6 0.179 0.800 HEA 800* 20.00 0.75 468.1 0.179 0.759 IPE 500* 15.00 0.75 94.0 0.179 0.761 HEA 400* 15.00 0.75 285.9 0.179 0.785 IPE 220* 6.60 0.75 20.4 0.179 0.749 IPE 600* 18.00 0.75 127.5 0.179 0.733 HEB 600* 15.00 0.75 666.7 0.179 0.736 HEA 400* 16.00 0.75 259.0 0.179 0.769 HEA 400* 17.00 0.75 236.1 0.179 0.754 IPE 140* 4.20 0.75 7.3 0.179 0.700 HEA 400* 18.00 0.75 216.2 0.179 0.739 HEA 400* 19.00 0.75 199.0 0.179 0.724 IPE 100* 3.00 0.75 3.7 0.179 0.659 HEB 200* 8.00 0.75 126.0 0.179 0.707 HEA 800* 28.00 0.75 279.1 0.179 0.675 HEB 400* 16.00 0.75 355.2 0.179 0.692 IPE 200* 8.00 0.75 10.0 0.179 0.663 IPE 270* 13.50 0.75 14.9 0.179 0.645 HEB 600* 24.00 0.75 323.1 0.179 0.615 IPE 220* 11.00 0.75 9.2 0.179 0.606 IPE 140* 7.00 0.75 3.2 0.179 0.552 IPE 100* 5.00 0.75 1.6 0.179 0.509 HEB 260* 3.90 0.75 1089.5 0.178 0.884 HEB 340* 5.10 0.75 1616.3 0.178 0.879 HEA 700* 10.35 0.75 1139.4 0.178 0.857 HEB 140* 2.10 0.75 228.5 0.178 0.849 HEB 500* 7.50 0.75 1575.8 0.178 0.838 HEA 1000* 14.85 0.75 972.4 0.178 0.824 HEB 100* 1.50 0.75 100.6 0.178 0.813 HEB 1000* 15.00 0.75 1155.5 0.178 0.778 HEB 260* 7.80 0.75 372.0 0.178 0.812 HEB 340* 10.20 0.75 553.1 0.178 0.796 HEB 220* 6.60 0.75 249.9 0.178 0.788

160

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) C2,provisional (-) C1,FEM(-)

HEA 600* 17.70 0.75 406.0 0.178 0.758 HEA 100* 2.88 0.75 27.7 0.178 0.757 HEB 1000* 22.50 0.75 620.3 0.178 0.695 HEB 500* 15.00 0.75 545.5 0.178 0.715 HEB 1000* 28.00 0.75 443.3 0.178 0.641 HEB 100* 3.00 0.75 34.9 0.178 0.669 HEB 260* 13.00 0.75 170.4 0.178 0.688 HEA 700* 28.00 0.75 244.7 0.178 0.652 HEB 340* 17.00 0.75 252.9 0.178 0.668 HEB 220* 11.00 0.75 114.0 0.178 0.657 HEA 600* 28.00 0.75 200.1 0.178 0.638 HEB 180* 9.00 0.75 67.9 0.178 0.628 HEA 100* 4.80 0.75 12.5 0.178 0.615 HEB 100* 5.00 0.75 15.5 0.178 0.522 HEA 1000* 22.28 0.75 519.0 0.178 0.749

161

162

APPENDIX D.2: 295 MODELS USED TO DETERMINE C2

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) h/e (-) C1,formula (-) C2,FEM (-) HEB180* 8.10 0.47 103.0 3.30 0.607 1.394 IPE180* 8.10 0.48 8.0 3.30 0.569 1.253 IPE180* 2.70 0.65 50.5 3.30 0.875 0.700 IPE180* 3.60 0.65 31.2 3.30 0.827 0.755 HEB220* 7.70 0.47 264.7 3.30 0.706 1.148 HEB220* 8.80 0.47 211.6 3.30 0.666 1.241 HEB220* 5.50 0.65 436.7 3.30 0.821 0.763 HEB220* 6.60 0.65 327.4 3.30 0.783 0.833 HEB260* 11.70 0.47 260.0 3.30 0.654 1.278 HEB260* 13.00 0.48 217.5 3.30 0.623 1.355 HEB260* 5.20 0.65 930.3 3.30 0.872 0.667 IPE270* 9.45 0.47 34.2 3.30 0.695 1.139 IPE270* 10.80 0.47 27.3 3.30 0.654 1.225 IPE270* 6.75 0.65 57.3 3.30 0.813 0.776 IPE270* 8.10 0.65 42.8 3.30 0.775 0.837 HEB340* 13.60 0.47 468.4 3.30 0.671 1.228 HEB340* 8.50 0.64 967.8 3.30 0.822 0.763 HEB340* 10.20 0.65 722.7 3.30 0.786 0.822 IPE400* 20.00 0.47 39.5 3.30 0.584 1.398 HEB400* 20.00 0.48 315.4 3.30 0.565 1.511 HEB400* 6.00 0.65 2219.5 3.30 0.889 0.646 IPE400* 6.00 0.65 286.5 3.30 0.897 0.662 IPE400* 8.00 0.65 175.7 3.30 0.855 0.715 HEA450* 17.60 0.47 358.3 3.30 0.678 1.206 HEA450* 11.00 0.65 739.1 3.30 0.828 0.746 HEA450* 13.20 0.65 552.3 3.30 0.791 0.812 HEA100* 4.80 0.47 15.8 3.30 0.562 1.533 HEA100* 1.92 0.64 69.7 3.30 0.843 0.747 HEA100* 1.50 0.65 103.9 3.30 0.883 0.673 HEA500* 22.05 0.47 296.1 3.30 0.612 1.376 HEA500* 24.50 0.48 247.3 3.30 0.580 1.470 HEA500* 7.35 0.65 1741.8 3.30 0.894 0.631 HEA500* 9.80 0.65 1078.9 3.30 0.852 0.702 IPE100* 3.50 0.47 3.8 3.30 0.595 1.317 HEB140* 5.60 0.47 66.4 3.30 0.614 1.369 IPE100* 4.00 0.47 3.0 3.30 0.550 1.436 HEB140* 4.20 0.65 104.9 3.30 0.744 0.903 IPE100* 2.50 0.65 6.5 3.30 0.743 0.857 IPE100* 3.00 0.65 4.9 3.30 0.694 0.937

163

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) h/e (-) C1,formula (-) C2,FEM (-)

HEB400* 15.00 0.45 514.7 3.33 0.651 1.274 HEB400* 10.00 0.60 974.9 3.33 0.793 0.840 HEB140* 4.00 0.60 112.8 3.50 0.745 0.910 HEB180* 8.00 0.45 102.7 3.60 0.605 1.327 IPE600* 21.00 0.47 127.1 3.70 0.654 1.096 IPE600* 24.00 0.47 100.9 3.70 0.611 1.177 HEB600* 21.00 0.47 504.4 3.70 0.607 1.248 HEB600* 24.00 0.47 400.4 3.70 0.562 1.370 IPE600* 18.00 0.64 162.4 3.70 0.740 0.813 IPE600* 15.00 0.65 217.0 3.70 0.785 0.743 HEB600* 15.00 0.65 865.3 3.70 0.751 0.793 HEA700* 24.15 0.47 378.4 4.26 0.641 1.062 HEA700* 27.60 0.48 289.0 4.26 0.601 0.990 HEA700* 20.70 0.65 485.3 4.26 0.732 0.768 HEA700* 17.00 0.60 661.9 4.60 0.770 0.725 HEA900* 28.00 0.47 385.8 5.49 0.642 0.865 HEA900* 28.00 0.47 385.8 5.49 0.642 0.865 HEA900* 17.80 0.65 808.0 5.49 0.802 0.572 HEB1000* 28.00 0.47 514.2 6.17 0.620 0.779 HEB1000* 28.00 0.47 514.2 6.17 0.620 0.779 HEA100* 4.00 0.35 18.4 6.40 0.589 0.953 IPE270* 4.50 0.70 96.6 7.71 0.886 0.426 IPE400* 14.00 0.29 61.9 8.00 0.655 0.848 HEB180* 6.30 0.29 132.4 8.00 0.642 0.864 IPE270* 10.80 0.29 22.8 8.00 0.606 0.886 HEA450* 17.60 0.29 303.3 8.00 0.633 0.919 IPE180* 6.30 0.29 10.6 8.00 0.600 0.867 HEB340* 13.60 0.29 393.2 8.00 0.625 0.910 HEB220* 8.80 0.29 176.4 8.00 0.620 0.895 IPE600* 24.00 0.29 85.2 8.00 0.559 0.892 HEB1000* 28.00 0.29 491.7 8.00 0.569 0.923 HEB1000* 28.00 0.29 491.7 8.00 0.569 0.923 HEA450* 19.80 0.29 244.4 8.00 0.592 0.964 IPE270* 12.15 0.29 18.3 8.00 0.564 0.925 HEB340* 15.30 0.29 316.7 8.00 0.583 0.954 HEB220* 9.90 0.29 142.0 8.00 0.578 0.935 HEB260* 13.00 0.29 177.8 8.00 0.568 0.968 HEA700* 27.60 0.29 262.7 8.00 0.544 1.004 HEA700* 28.00 0.29 255.7 8.00 0.538 1.011 IPE600* 27.00 0.29 68.4 8.00 0.516 0.939 HEB140* 5.60 0.29 53.7 8.00 0.561 0.917

164

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) h/e (-) C1,formula (-) C2,FEM (-)

HEB600* 24.00 0.29 333.5 8.00 0.505 1.063 IPE100* 4.00 0.29 2.4 8.00 0.492 0.889 HEB400* 20.00 0.29 252.3 8.00 0.505 1.057 HEB180* 9.00 0.29 68.4 8.00 0.516 0.991 HEB140* 6.30 0.29 43.1 8.00 0.519 0.967 HEA100* 4.80 0.29 12.5 8.00 0.505 0.983 IPE180* 9.00 0.29 5.5 8.00 0.471 1.012 HEB600* 27.00 0.29 267.3 8.00 0.463 1.147 IPE100* 4.50 0.29 1.9 8.00 0.451 0.935 HEA900* 28.00 0.30 352.8 8.00 0.595 0.821 HEB400* 14.00 0.30 487.3 8.00 0.635 0.878 HEA900* 28.00 0.30 361.8 8.00 0.595 0.901 IPE400* 20.00 0.30 31.2 8.00 0.532 0.840 HEA500* 24.50 0.30 200.2 8.00 0.524 1.046 HEA100* 3.00 0.60 29.6 8.00 0.740 0.547 HEB140* 2.10 0.76 261.9 8.00 0.889 0.326 HEA450* 8.80 0.76 895.8 8.00 0.878 0.366 HEB400* 10.00 0.76 819.0 8.00 0.821 0.378 HEB140* 2.80 0.77 165.7 8.00 0.849 0.332 HEB220* 4.40 0.77 522.3 8.00 0.873 0.345 IPE100* 1.50 0.77 12.2 8.00 0.864 0.297 HEB340* 6.80 0.77 1157.0 8.00 0.875 0.346 HEB260* 6.50 0.77 548.9 8.00 0.852 0.369 HEB180* 4.50 0.77 221.5 8.00 0.828 0.362 IPE600* 9.00 0.77 423.1 8.00 0.889 0.331 HEA700* 13.80 0.77 813.0 8.00 0.841 0.339 IPE600* 12.00 0.77 266.0 8.00 0.848 0.333 IPE270* 5.40 0.77 69.2 8.00 0.868 0.364 HEB260* 7.80 0.77 415.5 8.00 0.821 0.394 IPE400* 10.00 0.77 103.3 8.00 0.834 0.373 HEA500* 13.00 0.70 592.7 8.17 0.807 0.456 HEA100* 4.32 0.23 14.9 9.60 0.526 0.970 HEA100* 4.80 0.24 12.2 9.60 0.490 0.996 HEA100* 3.84 0.36 18.3 9.60 0.606 0.663 HEA100* 3.84 0.41 18.4 9.60 0.620 0.598 HEA100* 4.32 0.41 14.8 9.60 0.580 0.590 HEA100* 2.40 0.53 42.0 9.60 0.781 0.534 HEA100* 2.88 0.53 30.8 9.60 0.735 0.528 HEA100* 2.88 0.59 31.0 9.60 0.748 0.492 IPE100* 3.50 0.23 2.9 10.00 0.523 0.861 IPE100* 4.00 0.23 2.3 10.00 0.473 0.920

165

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) h/e (-) C1,formula (-) C2,FEM (-)

IPE100* 4.50 0.35 1.8 10.00 0.471 0.546 IPE100* 5.00 0.35 1.5 10.00 0.436 0.525 IPE100* 3.50 0.41 2.9 10.00 0.577 0.469 IPE100* 5.00 0.42 1.5 10.00 0.459 0.344 IPE100* 1.50 0.52 13.1 10.00 0.833 0.506 IPE100* 2.00 0.53 8.0 10.00 0.772 0.481 IPE100* 2.50 0.58 5.5 10.00 0.726 0.414 IPE100* 2.00 0.59 8.0 10.00 0.783 0.439 IPE100* 1.50 0.71 12.4 10.00 0.857 0.345 IPE400* 15.00 0.45 51.6 11.43 0.671 0.522 HEB180* 6.00 0.40 134.7 12.00 0.684 0.547 IPE180* 7.20 0.41 7.8 13.00 0.588 0.438 IPE180* 8.10 0.41 6.2 13.00 0.547 0.411 HEB180* 8.10 0.41 77.5 13.00 0.589 0.458 IPE180* 2.70 0.58 42.9 13.00 0.869 0.430 IPE180* 5.40 0.58 13.3 13.00 0.722 0.390 HEB180* 2.70 0.59 514.3 13.00 0.888 0.406 HEB180* 5.40 0.59 162.4 13.00 0.755 0.417 HEA450* 15.40 0.41 357.0 13.00 0.704 0.574 HEA450* 8.80 0.59 915.4 13.00 0.860 0.436 HEA450* 11.00 0.59 630.2 13.00 0.819 0.452 HEB140* 4.90 0.41 63.6 13.00 0.641 0.465 HEB140* 7.00 0.42 33.1 13.00 0.522 0.365 HEB140* 3.50 0.58 115.8 13.00 0.774 0.411 HEB140* 2.80 0.59 167.5 13.00 0.823 0.415 HEA700* 24.15 0.41 312.6 13.00 0.625 0.541 HEA700* 28.00 0.41 238.8 13.00 0.575 0.534 HEA700* 13.80 0.59 823.1 13.00 0.814 0.429 HEA700* 17.25 0.59 568.0 13.00 0.764 0.430 IPE270* 9.45 0.41 27.0 13.00 0.681 0.522 IPE270* 13.50 0.41 14.1 13.00 0.563 0.468 IPE270* 5.40 0.59 70.4 13.00 0.847 0.438 IPE270* 6.75 0.59 48.4 13.00 0.804 0.439 IPE400* 18.00 0.42 35.9 13.00 0.605 0.425 HEB400* 18.00 0.42 286.8 13.00 0.583 0.503 IPE400* 6.00 0.59 244.1 13.00 0.893 0.420 HEB400* 12.00 0.59 598.8 13.00 0.748 0.431 IPE400* 12.00 0.59 75.9 13.00 0.764 0.430 HEB260* 10.40 0.41 247.7 13.00 0.675 0.545 HEB260* 11.70 0.42 200.6 13.00 0.640 0.524 HEB260* 7.80 0.58 409.6 13.00 0.787 0.454

166

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) h/e (-) C1,formula (-) C2,FEM (-)

HEB260* 3.90 0.59 1300.5 13.00 0.908 0.390 HEB1000* 28.00 0.41 457.4 13.00 0.603 0.481 HEB1000* 28.00 0.41 457.4 13.00 0.603 0.481 HEB1000* 20.00 0.58 833.0 13.00 0.746 0.378 HEA500* 19.60 0.41 281.2 13.00 0.634 0.568 HEA500* 22.05 0.41 227.2 13.00 0.594 0.566 HEA500* 14.70 0.59 458.8 13.00 0.759 0.394 HEB600* 21.00 0.41 398.0 13.00 0.589 0.523 HEB600* 28.00 0.41 234.1 13.00 0.490 0.509 IPE600* 21.00 0.42 101.8 13.00 0.641 0.457 HEB600* 12.00 0.58 1060.7 13.00 0.790 0.420 HEB600* 15.00 0.58 731.7 13.00 0.736 0.414 IPE600* 12.00 0.59 269.1 13.00 0.822 0.417 IPE600* 15.00 0.59 185.4 13.00 0.774 0.404 HEB340* 11.90 0.41 462.6 13.00 0.697 0.552 HEB340* 17.00 0.41 242.9 13.00 0.581 0.525 HEB340* 8.50 0.59 819.5 13.00 0.815 0.442 HEB220* 11.00 0.41 108.8 13.00 0.576 0.481 HEB220* 7.70 0.42 207.6 13.00 0.694 0.522 HEB220* 4.40 0.59 535.3 13.00 0.854 0.429 HEB220* 5.50 0.59 369.4 13.00 0.812 0.437 HEB400* 15.00 0.40 397.4 13.33 0.638 0.532 HEB140* 4.90 0.23 63.7 14.00 0.593 0.715 HEB140* 5.60 0.24 49.5 14.00 0.547 0.714 HEB140* 6.30 0.35 39.4 14.00 0.537 0.467 HEB140* 7.00 0.36 32.4 14.00 0.503 0.430 HEB140* 2.10 0.71 255.2 14.00 0.884 0.291 HEB140* 4.20 0.71 86.1 14.00 0.758 0.300 HEA900* 24.00 0.55 443.5 14.83 0.708 0.392 HEB1000* 27.00 0.40 475.9 16.67 0.613 0.415 HEB1000* 28.00 0.24 445.7 18.00 0.554 0.677 HEB1000* 28.00 0.24 445.7 18.00 0.554 0.677 HEB1000* 15.00 0.71 1262.1 18.00 0.832 0.219 HEB340* 5.10 0.70 1783.8 18.00 0.906 0.276 HEB340* 10.20 0.70 590.4 18.00 0.796 0.326 IPE270* 9.45 0.23 26.6 18.00 0.638 0.662 IPE180* 8.10 0.23 6.0 18.00 0.491 0.684 IPE180* 9.00 0.23 4.9 18.00 0.452 0.717 IPE270* 10.80 0.24 20.7 18.00 0.593 0.651 HEB180* 8.10 0.24 74.6 18.00 0.539 0.638 HEB180* 9.00 0.24 61.1 18.00 0.500 0.658

167

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) h/e (-) C1,formula (-) C2,FEM (-)

HEB180* 7.20 0.35 92.3 18.00 0.613 0.435 IPE180* 6.30 0.36 9.5 18.00 0.619 0.418 IPE180* 7.20 0.36 7.4 18.00 0.573 0.383 IPE180* 5.40 0.53 12.9 18.00 0.710 0.337 HEB180* 5.40 0.53 157.1 18.00 0.743 0.374 IPE180* 4.50 0.53 17.6 18.00 0.758 0.380 IPE270* 8.10 0.70 35.0 18.00 0.785 0.317 HEB180* 3.60 0.71 300.9 18.00 0.856 0.291 HEB180* 4.50 0.71 211.9 18.00 0.818 0.294 IPE270* 4.05 0.71 106.9 18.00 0.900 0.301 IPE180* 3.60 0.71 25.1 18.00 0.836 0.289 IPE180* 4.50 0.71 17.6 18.00 0.793 0.274 HEA500* 22.05 0.23 220.6 18.00 0.542 0.778 HEA500* 24.50 0.24 180.3 18.00 0.505 0.791 HEA500* 9.80 0.71 870.3 18.00 0.859 0.303 HEA500* 12.25 0.71 611.1 18.00 0.820 0.315 HEA700* 24.15 0.23 305.9 18.00 0.575 0.729 HEA700* 27.60 0.24 237.3 18.00 0.529 0.742 HEA700* 20.70 0.70 414.2 18.00 0.744 0.293 HEA900* 28.00 0.23 329.9 18.00 0.576 0.712 HEA900* 28.00 0.23 329.9 18.00 0.576 0.712 HEA900* 22.25 0.70 499.7 18.00 0.763 0.263 HEA900* 17.80 0.71 709.0 18.00 0.812 0.264 IPE600* 21.00 0.23 99.4 18.00 0.590 0.640 IPE600* 24.00 0.23 77.2 18.00 0.541 0.655 IPE400* 18.00 0.23 35.3 18.00 0.550 0.689 HEB600* 21.00 0.23 386.8 18.00 0.536 0.748 HEB400* 18.00 0.23 276.7 18.00 0.526 0.740 HEB600* 24.00 0.23 300.3 18.00 0.486 0.801 IPE400* 20.00 0.24 28.9 18.00 0.514 0.686 HEB400* 20.00 0.24 223.5 18.00 0.490 0.702 HEB600* 9.00 0.70 1603.4 18.00 0.861 0.268 HEB400* 8.00 0.70 1113.8 18.00 0.851 0.302 HEB400* 10.00 0.70 782.4 18.00 0.811 0.308 IPE400* 10.00 0.70 99.4 18.00 0.823 0.321 IPE600* 9.00 0.71 409.0 18.00 0.883 0.286 IPE400* 8.00 0.71 141.9 18.00 0.862 0.312 HEA450* 13.20 0.70 452.8 18.00 0.801 0.342 HEB220* 3.30 0.71 799.7 18.00 0.905 0.270 HEA450* 6.60 0.71 1373.9 18.00 0.909 0.277 HEB220* 6.60 0.71 266.6 18.00 0.795 0.308

168

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) h/e (-) C1,formula (-) C2,FEM (-)

HEB260* 13.00 0.23 160.2 18.01 0.551 0.717 HEB260* 5.20 0.71 753.0 18.01 0.878 0.299 HEB260* 6.50 0.71 528.3 18.01 0.844 0.315 HEA500* 18.00 0.40 315.2 19.60 0.658 0.470 HEB220* 9.90 0.35 124.3 22.00 0.595 0.393 HEB220* 11.00 0.35 102.1 22.00 0.558 0.367 HEB220* 4.40 0.52 524.3 22.00 0.846 0.408 HEB220* 3.30 0.53 856.5 22.00 0.894 0.390 HEB340* 15.00 0.40 288.9 22.67 0.621 0.377 HEB340* 15.00 0.40 288.9 22.67 0.621 0.377 IPE600* 27.00 0.35 60.2 23.00 0.534 0.297 IPE600* 28.00 0.35 56.2 23.00 0.522 0.286 HEB600* 27.00 0.36 234.2 23.00 0.486 0.365 HEB600* 28.00 0.36 218.6 23.00 0.474 0.357 HEB600* 9.00 0.53 1670.0 23.00 0.841 0.379 HEB600* 12.00 0.53 1022.8 23.00 0.781 0.367 IPE600* 9.00 0.53 428.8 23.00 0.867 0.393 IPE600* 12.00 0.53 261.0 23.00 0.813 0.377 HEA900* 28.00 0.36 309.3 23.00 0.611 0.324 HEA900* 28.00 0.36 309.3 23.00 0.611 0.324 HEA900* 26.70 0.52 355.4 23.00 0.670 0.296 HEA900* 22.25 0.52 488.0 23.00 0.722 0.340 IPE270* 13.50 0.35 13.2 23.00 0.545 0.354 IPE270* 12.15 0.36 16.1 23.00 0.585 0.366 IPE270* 4.05 0.52 113.3 23.00 0.888 0.412 IPE270* 5.40 0.53 68.6 23.00 0.840 0.407 HEB1000* 20.00 0.53 798.7 23.00 0.735 0.310 HEB1000* 15.00 0.53 1306.6 23.00 0.804 0.354 HEA700* 28.00 0.36 224.9 23.00 0.560 0.416 HEA700* 28.00 0.36 224.9 23.00 0.560 0.416 HEA700* 10.35 0.53 1307.6 23.00 0.860 0.394 HEA700* 13.80 0.53 798.5 23.00 0.804 0.391 HEA450* 22.00 0.35 176.5 23.00 0.572 0.461 HEA450* 6.60 0.53 1471.7 23.00 0.899 0.391 HEA450* 8.80 0.53 896.4 23.00 0.854 0.413 IPE400* 14.00 0.35 54.6 23.00 0.669 0.443 IPE400* 16.00 0.35 42.7 23.00 0.625 0.413 HEB400* 14.00 0.35 427.7 23.00 0.648 0.447 HEB400* 16.00 0.36 334.6 23.00 0.606 0.409 IPE400* 12.00 0.52 72.5 23.00 0.750 0.366 HEB400* 10.00 0.53 779.5 23.00 0.781 0.380

169

Profile LAE (m) LBD/LAE (-) Mcr,FEM (kNm) h/e (-) C1,formula (-) C2,FEM (-)

HEB400* 12.00 0.53 570.4 23.00 0.735 0.353 IPE400* 10.00 0.53 99.4 23.00 0.795 0.390 HEB340* 5.10 0.53 1900.1 23.00 0.896 0.384 HEA500* 17.15 0.35 339.9 23.00 0.662 0.493 HEA500* 12.25 0.53 614.7 23.00 0.791 0.403 HEA500* 14.70 0.53 450.0 23.00 0.746 0.387 HEB260* 9.10 0.35 299.3 23.01 0.702 0.476 HEB260* 10.40 0.35 234.5 23.01 0.661 0.453 HEB260* 6.50 0.53 536.7 23.01 0.819 0.403 HEB260* 7.80 0.53 392.9 23.01 0.779 0.388

170

APPENDIX E: PARTIAL DERIVATIVES OF MCR WITH RESPECT TO THE BASIC VARIABLES

APPENDIX E.1: PARTIAL DERIVATIVE OF MCR WITH RESPECT TO B

171

172

APPENDIX E.2: PARTIAL DERIVATIVE OF MCR WITH RESPECT TO H

173

174

APPENDIX E.3: PARTIAL DERIVATIVE OF MCR WITH RESPECT TO TF

175

176

APPENDIX E.4: PARTIAL DERIVATIVE OF MCR WITH RESPECT TO TW

177

APPENDIX E.5: PARTIAL DERIVATIVE OF MCR WITH RESPECT TO E

178

APPENDIX E.6: PARTIAL DERIVATIVE OF MCR WITH RESPECT TO G

179

180

APPENDIX F: 153 MODELS USED FOR EVALUATION OF THE BUCKLING CURVES

profile Fy (N/mm2)

LAE (m) LBD (m) Ecc (mm) Mcr,FEM (kNm)

λLT Mmax,GMNIA (kNm)

ΧLT

IPE100* 235 1.5 0.35 30 17.5 0.71 6.4 0.73 IPE100* 235 1.5 0.35 10 14.2 0.79 6.3 0.71 IPE100* 235 1.5 0.75 30 15.7 0.75 6.9 0.78 IPE100* 235 1.5 0.75 10 13.1 0.82 6.6 0.75 IPE100* 235 1.5 1.16 30 14.1 0.79 6.9 0.78 IPE100* 235 1.5 1.16 10 11.9 0.86 6.6 0.75 IPE100* 235 3.3 0.76 30 4.3 1.43 3.6 0.40 IPE100* 235 3.3 0.76 10 3.3 1.64 3.0 0.34 IPE100* 235 3.3 2.54 30 4.1 1.47 3.5 0.40 IPE100* 235 3.3 2.54 10 3.5 1.58 2.9 0.32 IPE360* 235 5.4 1.24 109 299.8 0.87 165.3 0.72 IPE360* 235 5.4 1.24 28 244.8 0.97 132.9 0.58 IPE360* 235 5.4 1.24 16 233.9 0.99 130.3 0.57 IPE360* 235 11.9 2.73 109 75.2 1.74 67.4 0.29 IPE360* 235 11.9 9.15 109 66.6 1.85 59.9 0.26 IPE360* 235 11.9 9.15 28 56.8 2.01 53.3 0.23 IPE360* 235 11.9 9.15 16 55.2 2.04 52.0 0.23 IPE600* 235 9.0 2.07 162 597.2 1.15 355.2 0.45 IPE600* 235 9.0 2.07 46 492.0 1.27 328.1 0.41 IPE600* 235 9.0 2.07 26 469.5 1.30 316.8 0.40 IPE600* 235 9.0 6.93 162 483.1 1.28 336.2 0.42 IPE600* 235 9.0 6.93 46 403.4 1.40 319.8 0.40 IPE600* 235 9.0 6.93 26 389.5 1.43 313.2 0.39 IPE600* 235 19.8 15.25 162 136.4 2.41 130.8 0.16 HEA100* 235 3.2 2.44 29 30.7 0.78 15.7 0.85 HEA100* 235 3.2 2.44 10 26.6 0.83 15.0 0.81 HEA100* 235 4.8 2.40 29 15.9 1.08 12.8 0.69 HEA100* 235 4.8 2.40 10 12.5 1.21 11.7 0.63 HEA450* 235 14.5 7.26 133 491.1 1.22 382.7 0.53 HEA450* 235 14.5 7.26 34 396.0 1.36 338.4 0.47 HEA450* 235 14.5 7.26 19 378.8 1.39 329.4 0.45 HEA450* 235 14.5 11.18 133 461.3 1.26 361.4 0.50 HEA450* 235 14.5 11.18 34 394.2 1.36 331.6 0.46 HEA450* 235 14.5 11.18 19 382.7 1.38 323.9 0.45 HEA900* 235 13.4 6.68 68 1229.0 1.42 952.4 0.39 HEA900* 235 13.4 6.68 39 1180.7 1.45 922.2 0.37

181

profile Fy (N/mm2)

LAE (m) LBD (m) Ecc (mm) Mcr (kNm)

λLT Mmax,GMNIA (kNm)

ΧLT

HEA900* 235 13.4 10.28 162 1223.2 1.42 937.4 0.38 HEA900* 235 13.4 10.28 68 1105.4 1.49 901.8 0.37 HEA900* 235 13.4 10.28 39 1066.8 1.52 881.1 0.36 HEA900* 235 28.0 21.56 162 388.4 2.52 375.8 0.15 HEA900* 235 28.0 21.56 162 388.4 2.52 375.8 0.15 HEB200* 235 3.0 0.69 61 937.7 0.39 134.1 0.92 HEB200* 235 3.0 0.69 15 775.1 0.43 134.0 0.92 HEB200* 235 10.0 2.30 61 115.6 1.12 92.8 0.64 HEB200* 235 10.0 5.00 61 113.2 1.13 95.0 0.65 HEB200* 235 10.0 5.00 15 86.6 1.30 85.1 0.58 HEB200* 235 10.0 7.70 61 116.3 1.12 90.9 0.62 HEB200* 235 10.0 7.70 15 98.1 1.22 82.9 0.57 HEB200* 235 10.0 7.70 10 95.4 1.24 81.5 0.56 HEB600* 235 9.0 4.50 162 2081.4 0.84 1030.0 0.70 HEB600* 235 9.0 4.50 46 1753.1 0.92 991.0 0.67 HEB600* 235 9.0 4.50 26 1682.2 0.94 979.8 0.67 HEB600* 235 19.8 4.55 162 579.2 1.59 507.8 0.35 HEB600* 235 19.8 4.55 46 448.0 1.81 435.3 0.30 HEB600* 235 19.8 4.55 26 423.0 1.86 422.3 0.29 HEB600* 235 19.8 9.90 162 557.6 1.62 505.8 0.34 HEB600* 235 19.8 9.90 46 449.5 1.81 438.6 0.30 HEB600* 235 19.8 9.90 26 427.9 1.85 424.9 0.29 HEB600* 235 19.8 15.25 162 547.2 1.64 485.2 0.33 HEB600* 235 19.8 15.25 46 472.4 1.76 433.2 0.29 HEB600* 235 19.8 15.25 26 457.4 1.79 421.2 0.29 HEB1000* 235 15.0 3.45 162 1632.0 1.45 1165.8 0.34 HEB1000* 235 15.0 3.45 77 1466.8 1.52 1084.4 0.32 HEB1000* 235 15.0 3.45 43 1395.5 1.56 1039.3 0.30 HEB1000* 235 15.0 7.50 162 1498.3 1.51 1186.1 0.35 HEB1000* 235 15.0 7.50 77 1368.9 1.58 1113.3 0.33 HEB1000* 235 15.0 7.50 43 1312.7 1.61 1072.1 0.31 HEB1000* 235 15.0 11.55 162 1337.8 1.60 1093.8 0.32 HEB1000* 235 15.0 11.55 77 1235.8 1.66 1050.5 0.31 HEB1000* 235 15.0 11.55 43 1193.9 1.69 1023.1 0.30 HEB1000* 235 28.0 21.56 162 517.7 2.57 495.5 0.15 HEB1000* 235 28.0 21.56 77 482.1 2.66 465.1 0.14 HEB1000* 235 28.0 21.56 43 466.9 2.70 452.1 0.13 HEB1000* 235 28.0 21.56 162 517.7 2.57 495.5 0.15 HEB1000* 235 28.0 21.56 43 466.9 2.70 452.1 0.13 IPE100* 355 1.5 0.35 30 17.5 0.87 8.7 0.65

182

profile Fy (N/mm2)

LAE (m) LBD (m) Ecc (mm) Mcr (kNm)

λLT Mmax,GMNIA (kNm)

ΧLT

IPE100* 355 1.5 0.35 10 14.2 0.97 8.3 0.62 IPE100* 355 1.5 0.75 30 15.7 0.92 9.0 0.67 IPE100* 355 1.5 0.75 10 13.1 1.01 8.6 0.64 IPE100* 355 1.5 1.16 10 11.9 1.06 8.2 0.62 IPE100* 355 3.3 0.76 30 4.3 1.76 4.3 0.32 IPE100* 355 3.3 2.54 30 4.1 1.80 3.8 0.29 IPE360* 355 5.4 1.24 109 299.8 1.07 227.1 0.66 IPE360* 355 5.4 1.24 28 244.8 1.19 164.4 0.48 IPE360* 355 5.4 1.24 16 233.9 1.22 159.3 0.46 IPE600* 355 9.0 2.07 162 597.2 1.42 428.1 0.36 IPE600* 355 9.0 2.07 46 492.0 1.56 380.9 0.32 IPE600* 355 9.0 2.07 26 469.5 1.60 366.9 0.31 IPE600* 355 9.0 6.93 162 483.1 1.58 374.9 0.31 IPE600* 355 9.0 6.93 46 403.4 1.72 353.7 0.30 IPE600* 355 9.0 6.93 26 389.5 1.75 345.2 0.29 HEA100* 355 3.2 2.44 29 30.7 0.95 20.0 0.72 HEA100* 355 3.2 2.44 10 26.6 1.02 18.9 0.68 HEA450* 355 14.5 7.26 133 491.1 1.50 453.5 0.41 HEA450* 355 14.5 11.18 133 461.3 1.54 408.5 0.37 HEA900* 355 13.4 6.68 68 1229.0 1.74 1066.1 0.29 HEA900* 355 13.4 6.68 39 1180.7 1.78 1029.0 0.28 HEA900* 355 13.4 10.28 162 1223.2 1.75 1031.6 0.28 HEA900* 355 13.4 10.28 68 1105.4 1.84 985.4 0.26 HEA900* 355 13.4 10.28 39 1066.8 1.87 958.9 0.26 HEB200* 355 3.0 0.69 61 937.7 0.48 196.1 0.89 HEB200* 355 3.0 0.69 15 775.1 0.53 196.0 0.89 HEB600* 355 9.0 4.50 162 2081.4 1.03 1316.9 0.59 HEB600* 355 9.0 4.50 46 1753.1 1.13 1243.3 0.56 HEB600* 355 9.0 4.50 26 1682.2 1.15 1209.7 0.54 HEB1000* 355 15.0 3.45 162 1632.0 1.78 1337.1 0.26 HEB1000* 355 15.0 3.45 77 1466.8 1.87 1228.4 0.24 HEB1000* 355 15.0 7.50 162 1498.3 1.85 1316.1 0.26 HEB1000* 355 15.0 7.50 77 1368.9 1.94 1223.8 0.24 HEB1000* 355 15.0 7.50 43 1312.7 1.98 1177.4 0.23 HEB1000* 355 15.0 11.55 162 1337.8 1.96 1185.0 0.23 HEB1000* 355 15.0 11.55 77 1235.8 2.04 1127.9 0.22 HEB1000* 355 15.0 11.55 43 1193.9 2.08 1096.4 0.21 IPE100* 460 1.5 0.35 30 17.5 1.00 10.3 0.59 IPE100* 460 1.5 0.35 10 14.2 1.10 9.6 0.56 IPE100* 460 1.5 0.75 30 15.7 1.05 10.4 0.60

183

profile Fy (N/mm2)

LAE (m) LBD (m) Ecc (mm) Mcr (kNm)

λLT Mmax,GMNIA (kNm)

ΧLT

IPE100* 460 1.5 0.75 10 13.1 1.15 9.8 0.56 IPE100* 460 1.5 1.16 10 11.9 1.21 9.2 0.53 IPE100* 460 3.3 2.54 10 3.5 2.21 3.5 0.20 IPE360* 460 5.4 1.24 109 299.8 1.22 266.5 0.59 IPE360* 460 5.4 1.24 28 244.8 1.35 182.5 0.41 IPE360* 460 5.4 1.24 16 233.9 1.38 176.6 0.39 IPE600* 460 9.0 2.07 162 597.2 1.61 469.8 0.30 IPE600* 460 9.0 2.07 46 492.0 1.78 414.2 0.27 IPE600* 460 9.0 2.07 26 469.5 1.82 399.3 0.26 IPE600* 460 9.0 6.93 162 483.1 1.79 395.0 0.25 IPE600* 460 9.0 6.93 46 403.4 1.96 370.3 0.24 IPE600* 460 9.0 6.93 26 389.5 2.00 360.7 0.23 HEA100* 460 3.2 2.44 29 30.7 1.08 22.7 0.63 HEA100* 460 3.2 2.44 10 26.6 1.16 21.2 0.59 HEA450* 460 14.5 11.18 133 461.3 1.76 437.8 0.31 HEA900* 460 13.4 6.68 68 1229.0 1.98 1126.7 0.23 HEA900* 460 13.4 6.68 39 1180.7 2.02 1087.0 0.23 HEA900* 460 13.4 10.28 162 1223.2 1.99 1081.2 0.22 HEA900* 460 13.4 10.28 68 1105.4 2.09 1027.9 0.21 HEA900* 460 13.4 10.28 39 1066.8 2.13 999.7 0.21 HEB200* 460 3.0 0.69 61 937.7 0.55 248.4 0.87 HEB200* 460 3.0 0.69 15 775.1 0.61 246.8 0.87 HEB200* 460 3.0 0.69 10 746.1 0.44 245.8 0.86 HEB200* 460 3.0 1.50 15 699.1 0.46 249.2 0.87 HEB200* 460 3.0 1.50 10 677.5 0.46 248.2 0.87 HEB600* 460 9.0 4.50 162 2081.4 1.18 1498.1 0.52 HEB600* 460 9.0 4.50 46 1753.1 1.28 1389.0 0.48 HEB600* 460 9.0 4.50 26 1682.2 1.31 1346.1 0.47 HEB1000* 460 15.0 3.45 162 1632.0 2.02 1442.3 0.22 HEB1000* 460 15.0 3.45 77 1466.8 2.13 1327.0 0.20 HEB1000* 460 15.0 3.45 43 1395.5 2.19 1276.7 0.19 HEB1000* 460 15.0 7.50 162 1498.3 2.11 1384.6 0.21 HEB1000* 460 15.0 7.50 77 1368.9 2.21 1283.3 0.19 HEB1000* 460 15.0 7.50 43 1312.7 2.25 1236.6 0.19 HEB1000* 460 15.0 11.55 162 1337.8 2.23 1231.0 0.18 HEB1000* 460 15.0 11.55 77 1235.8 2.32 1167.2 0.17 HEB1000* 460 15.0 11.55 43 1193.9 2.36 1131.9 0.17

184

APPENDIX G: PARTIAL DERIVATIVES OF GRT,I WITH RESPECT TO THE BASIC VARIABLES

APPENDIX G.1: PARTIAL DERIVATIVE OF GRT,I WITH RESPECT TO B

185

APPENDIX G.2: PARTIAL DERIVATIVE OF GRT,I WITH RESPECT TO H

186

APPENDIX G.3: PARTIAL DERIVATIVE OF GRT,I WITH RESPECT TO TF

187

APPENDIX G.4: PARTIAL DERIVATIVE OF GRT,I WITH RESPECT TO TW

188

APPENDIX G.5: PARTIAL DERIVATIVE OF GRT,I WITH RESPECT TO FY

APPENDIX G.6: PARTIAL DERIVATIVE OF GRT,I WITH RESPECT TO MCR

189

190

APPENDIX H: PLOTS OF SUBSETS

191

192

193

194

195

196

197

198

199

200

APPENDIX I: ACCEPTANCE LIMIT PLOTS

APPENDIX I.1: METHOD OF DA SILVA ET AL [18]

201

202

203

204

205

206

207

208

209

210

APPENDIX I.2: METHOD PROVIDED BY EUROCODE [17] SB

211

212

213

214

215

216

217

218

219