lateral torsional buckling

Lateral ... torsional buckling of

laterally restrained steel beams

Supervising committee:

Technische Universiteit Eindhoven Prof. ir. H.H. (Chairman) Dr. ir. M.c'M. Bakker Dr. ir. J.C.D. ~V"'lUvl

TNOB&O II'. H+M.G.M. ,.I>P" .... ''''r<pn

Report nr: A-2007.7 0-2007.8

mg. Bruins

Lateral Torsional restrained steel beams

Preface This report has written to conclude the graduation project at Eindhoven University of Technology (TU/e). This graduation project was carried out at the Structural Design and Construction Technology unit of the faculty of Architecture, and Planning.

Gratitude is to the supervising committee: H.H. drjr. M.C.M Bakker, droir J.C.D Hoenderkamp and ir. H.M.G.M Steenbergen (TNO B&O). Their comments and reviews during the past year have been proven to be valuable in the completion of the presented research.

Gratitude is also expressed to everyone who has motivated me in the past II years of my educational career to achieve what is presented in this Sincere gratitude is expressed to my parents who have always kept supporting me and have made me remember: "quand tu veux, tu peux! ". Also I would to thank Wouter ten Napel and Roel for the necessary discussions and laughs the graduation Last but not least I would like to thank Mariska for here patience and the last years.

Bruins June 2007

III

Lateral TU/e

Table of contents Pre/ace ___________________________________________________ III

____________________________________________________ VIII

Definitions ______________________________ XlII

1. Introduction ------------------------------1.1. Problem statement ____________________________ ~

1.2. Objective ________________________________ -

1.3.

1.4. Assumptions ______________________________ -

1.5. Outline of

2. Study o/literature ___________________________ __

2.1. Load bearing capacity according to Eurocode 3 __________________ -

2.2. Methods of determining Mer _________________________ -

2.3. Load and support cases ___________________________ -

2.4. Discussion _______________________________ ~

3. Comparison o/numerical model to experiment: Unrestrained

3.1.

3.2. Finite element model ___________________________ -1

Comparison with experiment ________________________ ~-

4. Comparison 0/ GMNIA to the method: Unrestrained beams _________ _

4.1.FE-model ______________________________ -

4.2. Illustration of Method __________________________ -

4.3. Results of nv',,,tiu<>f-;fln ____________________________ --

5. Comparison o/GMNIA to EC3: Single concentrated load with a single elastic restraint _25

5.1. Load and support case: __________________________ -~

5.2. FE-model ----------------------------------5.3. Presentation of results ---------------------------

Elastic critical m(.m~em: "----------------------------5.5. GMNIA _______________________________________ -

5.6. Comparison of GMNIA to

6. Comparison o/GMNIA to EC3: Two concentrated loads with two elastic res:lrtl~mjrs ___ j

6.1. Load and support case: __________________________ -

6.2. FE-model _______________________________ -

6.3. Elastic critical II.Hun:". ---------------------------" 6.4. GMNIA _________________________________________ -

Comparison of GMNIA to

7. Comparison o/munerical model to second experiment: beam with two elastic restraints_39

v

Table Contents

7.1. 39

7.2. FE-model ____________________________ 39

7.3. Comparison of FE-model to 40

8. Parameter study: Beams with spring-stiffness restraints _________ _

8.1. cases ______________________________ 43

8.2. FE-model 44

8.3. Determination of 44

8.4. Parameter study 45

9. Development of a design rule to determine the strength requirements of restraints ___ 57

9.1. Current design 57

Proposed design 57

10. Discussion 61 --------------------------------10.1. Modelling of structural shapes using shell elements 61

10.2. Validation of numerical model 61

10.3. Effect of restraint location 61

10.4. Load capacity the clauses of the 61

10.5. Strength requirements of restraints 63

10.6. Procedures 64

11. Conclusions and recommendations 65 ________________________________________________________ 67

Appendix Tables CO~1JI4~lelus ______________________________________ 69

Appendix B. Comparison of formulae for M erc __________________ 71

B.1. Greiner 71

B.2. Clark and 71

B.3. NEN 6771 71

B.4. 73

Appendix C. Determination of kefactor 75

Appendix D. Nominal dimensions of [PE sections 77

Appendix Finite element 79

E.l. Elements 79

E.2. Cross-sectional properties 81

E.3. Compensation of the discretisation error of the cross-section 85

E.4. FE models 95

E.5. 103

E.6. Input file 104

analysis using BEAM """' ...... ' .. _________________________ 111

Appendix Plastic St-Vernant Torsional CU1JU(;'lt _________________________ 113

7 Appendix G. Development ofa tool to calculate Mer using the method. ------

VI

Lateral Torsional TUle

G.t. Energy equations ____________ .. ~ _______________ ~

G.2. Trial functions 17 ------------------------------G.3. Illustration 18

GA. Testing the

Appendix H. Results ofsimulations without restraints _____________ _

Appendix 1. Results of simulations for 1 load and 1

1.1. Tabular overview ____________________________ -

1.2. Load-displacement graphs _________________________ --

Appendix J. Results of simulations for 2 loads and 2 rp.~,trill!nt_~ ___________ -

J.t. Tabular overview ____________________________ --

J.2. Load-displacement

Appendix K. Results of simulation from parameter

K.t. Tabular overview results from study ________________ .~-/

K.2. Graphs of parameter

----~---------------

VII

Summary

Summary Structural wide-flange beams are commonly used structural cross-section in building structures. To be able to obtain the load-bearing capacity of these beams design rules have been derived. The design rules have been developed for unrestrained beams. To increase the load bearing capacity of these beams lateral restraints are applied (Figure A). However no reference has been found on the performance of these design rules when restraints are applied. The objective ofthe research is to determine how the clauses of the Eurocode 3 can be used to determine the influence of lateral restraints on the load bearing capacity of steel beams in bending that fail through lateral-torsional buckling. Emphasis has been put on varying the location and stiffness of the restraint.

Modelling o'bearn cross-section using FEM To reduce the computational time when using the Finite Element Method (FEM) shell elements (SHELL181) are used to model the cross-section instead of applying solid elements. By using shell elements a discretisation error of the actual cross-section is made. This leads to large differences between the torsional properties of the FE-model and those of the actual cross-section (Figure B left). Using different compensation methods that have been found in the literature, the missing properties have been compensated. The elementary structural behaviour of each compensation method has been investigated. From these investigations the RHS method (elastic plastic rectangular hollow sections using BEAM 1 88 elements see Figure B right) has shown to be the only compensation method that performs well in each of the elementary tests.

Elastic buckling The elastic critical moment (Figure C) is required when using the buckling curves to determine the slenderness to obtain a reduction factor. In the literature no accurate methods and applicable calculation methods have been found. To be able to obtain controllable results an analytical determination of the elastic critical moment has been performed using the Ritz-method. Using this method a powerful, accurate and easy to use tool is obtained to compute the elastic critical moment for a large number of load cases and restraint stiffnesses and locations.

Lateral torsional buckling using FEM The failure load has been determined using geometrical and material non-linear imperfect analysis (GMNIA) with the FEM. The model behaviour has been compared to two investigatory experiments performed in earlier research at the TU/e. Using GMNIA the influence of the restraint location, restraint stiffness, impelfection pattern, load case and span have been determined and have been compared to the results obtained with the buckling curves (Figure D.)

VIII

k

Figure A Elastic partial restraint

1 r- 1 l

Real beam section FE-Cross-section

Figure B Difference between actual and Finite element cross-section

M' Unstable I'" 1

Neutral e uilibrium Mu ~--~~~~~~~~

Bifurcation oint

Stable equilibrium

Out -of-plane deflection / u/f{J rotation

Figure C Elastic Lateral-torsional buckling

1.1 ..---- ---_- _ __ ~

10.9

~

J 0.8

o.s

0.4

0.3 0.4 0.5 0 .6 0.7 0.8 0.9 1.1

XcCJ M,,;95 %

.115.1

,§ x 57.55

~ .. 0

! ~ x ·57.55 0:

• -115.1

Figure D Influence of restraint location for typical load case using the general method of EC3

Lateral Torsional Buckling of laterally restrained steel beams TUfe

Buckling curves The EN-1993-I-l (Eurocode 3, see Figure E) allows the designer to use a number of buckling curves. The slenderness, depending on the elastic critical moment and plastic moment, leads to a reduction factor using the appropriate buckling curve. The general method is the straight-forward approach which is based on the colul1m buckling curves. The specific methods have been developed based on numerical simulations performed on rolled beams. From the GMNIA it can be concluded that the general method may be used as a design rule. Furthermore serious doubts have been raised on the performance of the specific methods when applied to unrestrained beams. Also it has been concluded that they should not be used with laterally restrained beams.

Stiffness requirements of restraints The stiffness of the elastic restraints has influence on the elastic critical moment. As true rigid restraints can not be applied, stiffness requirements need to be set. These stiffness requirements have been based on the spring-stiffness needed to allow a 5% reduction of the elastic critical moment when applying rigid restraints (see Figure F). Using the 95% spring-stiffness has proven to give acceptable results using the general method when considering this as the lower bound spring-stiffness of a rigid restraint.

Strength requirements of restraints During loading, forces occur in the restraint, this leads to strength requirements that have to be applied to the restraints. From the numerical simulations, the force in the restraint is obtained and is compared to the clauses ofNEN6770. It has been found that the clauses of NEN6770 give a large underestimation of the force in the restraint at failure and they have to be altered. A new design rule (1) has been derived to account for the findings presented in this report, concerning the strength requirement of restraints. This design rule depends on four coefficients (Table A), and depending on the nondimensional restraint location the force in the restraint can be determined.

Table A Coefficients of the resented desi n rule.

(qrJ ho Al A2 A3 A4 1 -0.2 0.076 0.026 -0.124

0.5 -0.228 0.091 0.034 -0.137 0 -0.197 0.087 0.038 -0.110

-0.5 -0.128 0.060 0.028 -0.069 -1 -0.014 0.021 0.017 0.007

1.1

0.9

O.B

0.7

~:1 0.3

0.2

0.1

Buckling curves according to Eurocode 3

0.' I'

O~~~~_~~_~~~~~

0.0 0.2 0.4 0.6 0.8 10 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

-A

-e

- c

-D

-'''pI

.. .. Eul8f

Figure E Buckling curves according to Eurocode 3

Figure F Restraint spring-stiffness requirements

X, x, For -<0.25 -A -+ A I 'J I '''2

IX

Lateral Torsional ,..",,,,trg,,,,,.EI steel beams

Nomenclature b d e

f /y /y,d g h

k kc kd,'n

kred

lew I lbeamparl

19 h mx q r seg t

tp tw U

Un

V

W

E F

Width of rectangular hollow cOlmpem,atllon element of

Distance from the shear center to centroid (positive if shear center lies between centroid and compression flange, negative otherwise) Modification factor for XLT Yield strength

yield strength Distance from the load to the shear center if load is below shear center) Height Height between centroids of the Lateral bending coefficient

'''rr'p"t",,, factor for moment distribution Design fractile factor Reduction factor (NEN677 1) Lateral warping coefficient

I'\Pt'WPf'fl supports between and restraint between the

Length between inflection points Mean of value X Uniformly distributed load Root-radius

of torsional spring (Malj aars' COJmp,enlsatllon method) Thickness of SHSIRHS con:lPelt1Sat:ion Flange thickness Web thickness Plane displacements of web Displacements in axis denoted by subscript Out-of-plane displacements of web Width of flange Restraint location in x-direction Restraint location in y-direction (positive above centroid, below "",n,tr""i

Distance from the load to the shear center if load is below shear 1

-e - + y2)dA Wagner factor

Equivalent moment coefficient (Nethercot) Coefficients used in simplified rule Coefficient to take account for the load height (Nethercot) Equivalent bending moment coefficient Coefficient to take account for the load Coefficient to take account for the load height to NEN6771 Coefficient to take account for the asymmetry of cross-sections Young's modulus of Elasticity Force Elastic critical load Plastic load Ultimate load Shear modulus Minor bending stiffness Major bending stiffness Torsional constant

constant

fy(x2 + y2)dA Wagner factor

Ul",-'Ml.HH''''''' of restraint to reach 95% Spring-stiffness of restraint to reach 99% of Mer

rigid restraints rigid restraints

mm mm mm

Nlmm 2

Nlmm2

mm mm mm

mm mm mm mm

mm mm mm mm mm mm mm mm mm mm mm mm mm

Nlmm 2

kN kN kN kN Nlmm2

4 mm

Nmm Nmm

TUle

XI

Kro/

Ksp

M Mer

Mcr;95%

Mer; 100%

Mp/

M.,

T

Ur

V

Wy

Wel;y

Wpl;y

W.w;'y

Xd a aLI'

f3 X XGMNIA

XLI'

XL1;mod

XEC3

<5

U

U

Uj;y;d

<PLI'

Rotational spring-stiffness (Maljaars' compensation method) of restraint

Moment Elastic critical moment 95% of Elastic critical moment when using restraints 99% of Elastic critical moment when using rigid restraints Elastic critical moment using restraints Plastic moment Ultimate moment Force in restraint at failure Torque (torsional moment) Plastic torque Extemal energy (Ritz method) .",,<,LAU.'U.H.,O energy (Ritz method) Strain energy (Ritz rnPTnr,f1

Coefficient of variation Section modulus (y-axis) Elastic Section modulus (y-axis) Plastic Section modulus

Coefficient to take account for the load height and moment diagram (Nethercot) Imperfection factor according to EC3 Correction factor for the lateral torsional buckling curve of rolled section Reduction factor Reduction factor obtained from OMNIA Reduction factor obtained method Modified Reduction factor obtained method Reduction factor obtained Eurocode 3 Virtual operator (Ritz method) Strain Logarithmic strain Rotation Factor to cover uncertainties not covered tests Dimensionless (Nethercot) Section parameter of torsion Plateau length of the lateral torsional UU •• l\.llll/"i curve of rolled sections

Slenderness ratio obtained using analytical methods

Slenderness ratio from OMNIA

Stress Standard deviation True stress Initial stress to NEN6771 Occuring stress (NEN6770) Value to obtain reduction factor

Calculation of Differences:

Diff V

Where:

V,. Reference value

V Value which the difference to be

XII

Nlrad Nmm kNm kNm kNm kNm kNm kNm kNm kN kNm kNm

rad

Lateral Torsional Buckling of laterally restrained steel beams TU/e

Definitions

Coordinate system: The applied coordinate system is shown in Figure 1. The x-axis is the axis along the length of the member. The yaxis is the axis that is perpendicular to the web of the member and the z-axis is parallel to the web, where the positive direction is taken above the centroid.

z

x

Elevation Sideview

1 X. I---y ------'

Figure 1 Coordinate system

Major/Minor axis: The major axis is the y-axis, the minor axis is the z-axis taken according to Figure 1.

Deformations of beam: In-plane displacements (u) are perpendicular to the major axis (Figure 2 (a)). The out-of-plane displacements (v) are perPendicular to the minor axis (Figure 2 (b)). Torsional rotations are the rotations around the x-axis (Figure 2 (c)). And distortion is the deformation of the cross-sectional shape from its original shape (Figure 2 (d))

Figure 2 Deformations of a beam

Stiff beam element: Stiff beam element elements have high bending stiffnesses (Iy = Iz = J(lmm\ while having no torsional properties (I/ = Iw;::: 0), a cross-sectional area is only given when local failure is expected (stocky beams). The lack of the torsional properties prevents the restraining of warping and torsion at the supports (see Figure 3).

x

y

Figure 3 Allowance of warping of the cross-section at the supports

Fork support: Fork supports are supports, where displacements (both in-plane and out-of-plane) as well as torsional rotations and distortion of the cross-section are prevented, but where warping is not restrained. Displacements are prevented by applying three constraints , for out-of-plane displacement and the rotation of the cross-section by

XIII

Definitions

at the intersection to O. The are displatceiment at the centroid to 0 in z-direction (Figure 4a). The distortion of the cross-

section is prevented stiff beam elements 4b). This cause plane sections to remain plane. And only allow linear deflections due to warping of the cross-section.

System imperfections

o

centroid

lIy 0 (a) Constraints applied to prevent

displacements and rotation

Stiff beam elements

(b) Stiff beam elements to prevent distortion of the cross-section

applied at supports

Figure 4 Modelling of fork ~lIflnnrr~.

System imperfections are imperfections in y-direction (out-of-plane displacements and rotation along x-axis) See Figure 5.

Imperfection X __ .... ----'11<

Figure 5 System imperfection

Cross-sectional imperfections Cross-sectional are imperfection over the cross-section. These can be: lack thickness

initial etc.

Residual/initial stresses the manufacturing process of steel wide flanged beams the of the beam occurs unequally;

of the flange cools faster than the intersection of web and flanges. Therefore compression occurs in the tips and tension at the intersection. These stresses are present at the unloaded

LBA Linear analysis predicts the theoretical bifurcation of an ideal linear elastic structure.

GMNIA Geometrical and material non-linear imperfect analysis predicts the limit point of a structure of a geometrical and material non-linear imperfect structure.

XIV


1. Introduction

1.1. Problem statement When a perfectly straight elastic member is subjected to bending, and the material can not yield, a point is reached where the equilibrium alters from stable equilibrium to a neutral equilibrium; this point is the elastic critical moment Mer (Figure 6). This phenomenon is called lateral-torsional buckling.

M' Unstable I'"

Neutral e uilibrium

Bifurcation oint

Stable -'W--1_

-,-U

-1_

equilibrium

Out -of-plane deflection I u/q> rotation

Figure 6 Lateral-torsional buckling

In bending two physical limits are given, this is the full plastic moment and the Euler buckling moment (elastic critical moment). These limits are shown in Figure 7 using the dashed lines. These are theoretical upper bound limits of the load bearing capacity. Due to imperfections these limits can not be reached. This has lead to the buckling curves (Figure 7) used in Eurocode to detennine the load bearing capacity of a beam. The use of the buckling curves consists of determining a slenderness ratio (Il), this slenderness ratio is the root of the section capacity (Wy '/y) divided by the elastic critical moment Mer. With this ratio a reduction factor (x) is obtained, which is then multiplied by the section capacity to obtain the ultimate load M.,. The elastic critical moment depends on the support conditions, the load case, the cross-sectional properties and the number and position of the restraints.

1.1

0.9

0.8

0.7

0.6

X 0.5

0.4

0.3

0.2

0.1

Buckling curves according to Eurocode 3

+--:--;- :....- ~ t ..... __ +- _ J

. "

O~+-~-r-+~--~~~~-+~--.-~+-~

0.0 0 .2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

A

Figure 7 Buckling curves according to EC3 [CI]

-A

-6

-c

-0

- 'Mpl

- - Euler

The slenderness ratio has influence on the buckling resistance and therefore also on Mu. Thus when restraining an unrestrained beam, the slenderness ratio of this beam can be reduced and therefore increasing Mu. The restraining of beams can be accomplished in a number of ways; all of these can be categorized in two groups:

Continuous restraints. Discrete restraints.

In these two groups different types ofrestraints can be distinguished (Figure 8), full restraints (type a) prevent outof-plane displacement and rotations at their points of application. The use oflateral restraints (type b) restrains the out-of-plane displacements and the torsional restraints (type c) restrain the rotation of the beam at its point of application. Further a classification is introduced restraints can either fully restrain the displacement and/or

Introduction

rotations, or allow some displacement and/or rotations. The first class of restraints are denoted as rigid restraints k and c = 00, the second being flexible or elastic restraints (Figure 9).

<a) Full restraint (b) Lateral restra int (c) Torsionat restrainl

Figure 8 Types of restraints

Figure 9 Classification of restraints

The method described in the Eurocode has been developed for unrestrained beams [Rl] (Figure lO). The Eurocode does not mention restraints or how Mer should be calculated and therefore it is unknown how these curves perform when using this method for restrained beams (Figure 11).

1.2. Objective

~F\oSF I

~ : Load c:ase "" 1: At I R ;. I

I : Level ()f resuaint I I

T200 0: Position of reslrain1 "If

o

" ...

Figure 10 First buckling shape of an unrestrained steel beam

~F+~l ~3000 * 110lid C&9C

"... 7200 "v I I

: Levelofm.tnlinl

I I

: POliition ofrertraint ** __ ...... Z2.9""O ...... -,j<"v I

Figure 11 First buckling shape of a restrained steel beam

The objective of this research is to determine how the clauses in the Eurocode 3 can be used to determine the influence of restraints on the load bearing capacity of steel beams in bending that fail by lateral-torsional buckling. The emphasis will be put on the variation of the position and of the number of discrete lateral restraints (see Figure 8 type b).

2

Lateral Torsional Buckling oflaterally restrained steel beams TU/e

1.3. Approach The research is divided in two parts: analytical and numerical (Figure 12). The analytical study consists of the calculation of the elastic critical moment which is needed for the use of the buckling curves. As will be mentioned in the literature study, different methods can be used to detelmine Me/) and each of these methods will result in a reduction factor and therefore also in a load bearing capacity (see Figure 13).

Analytical (Eurocode )

NlJ!IIerical Finite Element Methotl

(ANSYS)

Analytical

Linear-buckling analysis (LBA)

Geometrical and Material Nonlinear Imperfection

Analysis (GMNIA)

Buckling curves

Load bearing capacity

Figure 12 Outline of research

The numerical study will be performed using ANSYS ViO.O. This is a general purpose finite element method (FEM) program. With this program, two types of analyses will be performed. The first is a linear bucking analysis (LBA). With this analysis the elastic critical moment can be determined which can be used to determine a load bearing capacity in accordance with the EC3. The second analysis will be a geometrical and material nonlinear imperfection analysis (GMNlA), which allows the load bearing capacity to be determined by simulating the real beam behaviour (see Figure 13). Using the buckled shape (from the LBA), the system imperfections will be modelled according to the recommendations in the EC3. The same will be done for the residual stresses and the material law. The numerical model will be compared to experiments that have been performed by Swart & Sterrenburg [Pi].

M

M .. FromEC3

fHrp Figure 13 Interpretation of the results

Furthermore the stiffness of the restraint is a point of interest. First the stiffness of the elastic restraint will be varied. After which in the parameter study the required spring-stiffness (K95%) needed to reach 95% of the elastic critical moment is used as a finite spring-stiffness of the elastic restraint. The use of this spring-stiffness is twofold; firstly the use of rigid-restraint in practice is impossible; secondly no stiffness requirements are given in the current codes and therefore a validation will be performed on the used spring-stiffness of K95% as a rigid restraint.

1.4. Assumptions The models used to validate the clauses of the Eurocode are complex. A number of phenomena have been deliberately avoided. Only failure due to lateral torsional buckling is investigated. Distortion of the cross-section due to loads and restraints will not be taken into account. Global behaviour is of interest and therefore local buckling and failure have been prevented. Furthermore only class I and 2 cross-sections are taken into consideration as these fail with respect to lateral torsional buckling.

1.5. Outline of report Chapter 2: Study of literature In this chapter the findings of a literature study are presented. The current design rules in the Eurocode 3 are discussed and the different methods of applying these design rules have investigated. None of the presented application methods that have been found can incorporate the placement of restraints at less favourable locations.

Chapter 3: Comparison of numerical model to experiment Experimental test of a single unrestrained beam has been performed at the TU/e. The behaviour of the FE-model has been compared to the behaviour of the experiment. The steps in the comparison that have been taken are discussed in this chapter.

3

Introduction

Chapter 4: of GMNIA to the EC3 method: Before restraints can be the of the analytical methods is to the GMNIA simulations. This comparison has been performed in this chapter for three load cases and three spans.

concentrated load with a elastic restraint For an case the influence of a res traint is In this H""'."~'" the stiffness of the restraint is varied. Also the influence of the shapes has been looked into.

Chapter 6: Comparison ofGMNIA to EC3: Two concentrated loads with two elastic restraints To be able to compare the FE-model to a second another is performed. Here two loads and two restraints have been In this both the influence of shape and the stiffness of the restraint have been investigated.

Chapter 7: Comparison of numerical model to second experiment of a beam load with a concrete slab has been performed at the TU/e. The behaviour of the

FE-model has been compared to the The in the comparison that have been taken are discussed in this chapter

Chapter 8: Parameter study In this the parameter study is performed. For three load-cases and different spans the behaviour of beams with a elastic restraint with K95% has been In this only the location of the restraints are varied.

Chapter 9: of a rule to determine the requirements of restraints The current design rule in the NEN6770 governing the detennination of the strength requirements of restraints was found to lead to unsafe results. In this chapter a new design rule has been derived to account for the findings in the parameter study.

Chapter 10: Discussion In this discussions are given on of the VV",ll,'"'''' results.

Chapter 11: Conclusions and recommendations The report is finalised with the conclusions and recommendations.

4

restrained steel TU/e

Study of literature In the chapter the findings of the literature study are First the load bearing capacity according to Eurocode 3 is shown, then the methods of Mer are reviewed and the found load and support cases are surnmarised.

2.1. Load bearing capacity according to Eurocode 3 In the Eurocode two methods are available to determine the load bearing capacity. These methods are the method and the method. Both are discussed here.

For both me1thol[Js the u .... rulll'/5 resistance moment of an unrestrained beam should be taken as:

M Rd = XIIWy/y (2.1)

Where is the appropriate section modulus:

for Class 1 or 2 cross-sections

The slenderness ratio is given by (2.2) and is used in the to the reduction It should be noted that Mer is not specified in the Eurocode. The determination hereof is left to the designer.

Au = JW,f, (2.2) Mer

2.1.1. General Method This method has been derived for the buckling of columns and was made suitable to be used on beams. According to the the reduction factor XII is as follows:

1 XII =----;:====:<;;1 and when :<;; 0.4 ~ XLT == 1

Where

<1>/1 0.5[I+aLT (lLT-O.2)+ LTJ

L'<>fJ<OllUlll>; on the height to width ratio the choice of the IJU'-''''Ull~ curve is done <If'£'rwl1.n to Table 1 where the imperfection factor is taken from (Table 6)

2.1.2. method This method was developed for the lateral torsional buckling of rolled or equivalent sections. According to the "fJ~;'-ll'''' method, the reduction factor is detennined to:

Where

<D/t 0.5[1 + a/I

And

)+

;hT,O 0.4 (maximum value)

J3 = 0.75 value)

--~r======:<;; I and :<;; 1

When lateral restraints into account the reduction factor may be modified:

5

of the discretisation error of the

f defined:

f = 1 - kJ[l 2(A-LT 0.8)2J S 1

kc should be taken L./VV''''~UA5 on the to width ratio the choice of the buckling curve is done to Table 3 where the mn,prt·Pl'tICln factor is taken from Table 4.

Table 2 Correction factors kc taken from ICI]

Moment distribution k.:

1111111111111111111111111111111 1,0 '1'=1

nnmlllilillTTm"'''~ 1

-I .. 1jJ:O; ! 1,33 - 0.33'1'

0,94

....... ....... 0,90

.... 0,91

-u· 0,86

...... ........ ........ 0,77

....... ....... 0,82

Table 3 Recommended values for lateral-torsional buckling curves [C11 Cross-section Limits Buckling curve Rolled I-sections hIb:::: 2 b

hIb>2 c

lateral-torsional buckling curves [CII

2.2. Methods of determining Mer As mentioned in the introduction the elastic critical moment is needed for the computation of the slenderness ratio (2.2). Different methods have been found in the literature. These can be divided in two groups: and application methods. With the analytical methods the exact value or an accurate approximation for is obtained. With the application methods these values have been translated to coefficients to use in the application formulae such that these can be reported and therefore made available to others. The application methods and their coefficients can be found in literature or in codes. These fast and accurate results of the elastic critical moment.

2.2.1. Analytical methods

Under analytical methods two different approaches can be followed: the exact method and method.

2.2.1.1. Exact method

approximation

Timoshenko gives the general governing differential equation of lateral-torsional buckling of a beam under a constant bending moment (2.5), this equation will result in the elastic critical moment for this bending moment This is the only available closed form solution (2.6).

M2 GIl +-Y ¢=O (2.5)

Elz

6

beams TU/e

2.2.1.2. Approximation methods

To solve the differential equation for other cases researchers have used different approximation methods as no closed form solutions are available. In Table 5 an overview is given of the methods that have been used. When used correctly it is possible to obtain accurate results for Mer.

Greiner [Rl], Trahair [2],

Each of the different approximation methods has a different approach. The Ritz method used Trahair and the energy method used by Timoshenko are based on trial functions to calculate Mer. The Ritz method used in

uses displacement fields that can describe any displacement. The finite element method is a nlP"·P •• U!I

implementation the Ritz-method. The finite difference method is a method of differential equation.

2.2.2. Application methods

The application methods are quite similar. When with double symmetrical sections the U,fJl0'H,_U,LJ,Vll

methods three characteristic coefficients: a moment a coefficient which includes the load and a coefficient that includes that support conditions at the supports. These can then be used in the equations. The coefficients can be found in the literature and can also be derived using numerical or analytical methods.

2.2.2.1. Clark & Hill [P8] Clark & Hill have developed a general method to compute the elastic critical moment for a number of cases. The general equation (2.7) can be used to compute the elastic critical moment. The coefficients

, C2 and for distance between shear centre and centroid) can be taken from tables (Appendix A,

where no values are given).

+ + (2.7)

Where: g distance between load introduction and shear center (negative ifload in below shear center)

K 1

+ l)dA 0 double symmetrical cross-sections

k = Out-of-plane rotation coefficient, 1 = free 0.5 restrained out-of rotation

e = Distance from the shear center to centroid (positive if shear center lies between centroid and compression flange, otherwise)

From the equation given by Clark & Hill and adapted versions have been derived. These are in Table 6. These equations use conventions, and the equation given by et aL has an additional coefficient kw; which accounts for the out-of-plane warping. A value of 1 will indicate that is not

while a value of 0.5 indicates that warping is restrained.

7

Compensation of the discretisation error of the cross-section

Table 6 Equations for the elastic critical moment based on Clark & Hill equation

Bijlaard & Steenbergen [P2]

Greiner et al. [RI]

M =c ff2

Elz [ cr 1 12

M = C ff

2

Elz [ er 1 (kl/

In [P2] Bijlaard & Steenbergen have made an important side note in the use of this method. When dealing with linear moment gradients and a unifonnly distributed load (see Figure 14). When -1 < P < -O.S, the Cl factors are highly influenced by the moment distribution (see Figure IS).

q

M(M£ 1 I 1 I I ~ M'J-PM

) I .. ..

Figure 14 End moment and uniformly distributed load. (Taken from [P2])

Figure 15 C] factor as function of Band P (Taken from [P2]) Where:

B= 8M 81MI+q12

2.2.2.2. Nethercot

Nethercot [P4, PS, P6] has developed his own approach. Herein the critical elastic moment is defined as:

ffa ff2 EI

Mer =-1 ~ElzGII 1+ {2G/ (2.8) I

The factor a is the lateral buckling coefficient whose value depends on the loading conditions. Simple expressions for a are available in the literature [P4, PS, P6]. u is the expression for two factors (A and B), A and B need to be multiplied when the load is applied on the lower flange, divided if the load is applied on the top flange and when applied at the shear centre only A should be taken.

Nethercot also made this method suitable to compute Mer for restrained beams. To compute Mer for a restrained beam a dimensionless parameter has to be determined first:

8

beams TU/e

48Ely

is the of the lateral restraint position at mid-span, Next the factor R2 I E1w has to be

calculated, With A and a magnification factor (c) is obtained using different figures and tables. This magnification factor accounts for the increase of Mer due to the restraint.

2.2.2.3. NEN6771 [C3}

According to the Dutch code the elastic critical moment is calculated to 10)

Mer = ~ ~ElzGI, (2.10)

Where:

C

S JElw

GI,

The coefficients for C1 and

interpolated for the load

between the inflection points,

g

+

are for specific cases, these can be fOlmd in the literature and needs to be

19 is the between the and fLT is given by (2, II) or the distance

(1.4 0.8P)I"eamporl (2.11 )

hT is restricted to 1.0:::; :::; 1.4, where lbeamparl is the distance between the fork and the lateral restraint

and P = MY;I;s;d I M y;2;s;d' Moreover this is only valid if the restraint is positioned where the I'-'vu"",, ..

Ul"f.H"'\A';'CllVlllA> of an unrestrained is Pv,,"' .... tPi1 to occur.

2.3. Load and support cases The lateral-torsional buckling problem is influenced by the load case, the support case and the cross-sectional properties. Each of the load and support cases (referred to as cases) have specific condition that have to be applied, These conditions are: the load case, load application height and the type and number and location of restraints, In Table 7 an overview of the studied cases in the literature is presented, In this overview is not shown how the results are reported, Nethercot [P4,PS] his finding for the use of his specific method while Tirnoshenko [1] his in tables to be used with his methods, Bijlaard & [P2] and Aswandy [P7] have their results as performed for the unifonn moment

"w."v"uv,,<, while the results for non-uniform as numerical values for an IPESOO,

References ~~----~--~~------~~-

Lateral Level of Torsional restraints

9

of the discretisation error of the cross-section

2.4. Discussion In this section the determination in general is discussed. Also the Mer of restrained beams is looked into.

of Mer As seen in the previous Mer can be determined in different ways. Only one exact solution is available to be used on a load and support case. This leads to approximation methods to compute Mer for every other case. Each of the methods are known to good results [P19, PI]]. However the analytical energy methods strongly on the choice of trial functions or fields for their accuracy [1,2, PI, The application methods are as accurate as the used approximation methods used to

the coefficients. However the application methods can only be used for cases for which the coefficients have been detennined.

2.4.2. Mer of restrained beams The determination of Mer for unrestrained beams has been thoroughly investigated. For almost every moment diagram coefficients have been provided. For restrained beams few coefficient are Aswandy in [P7] has provided coefficients for the uniform bending for non-uniform bending calculated results of

for an IPE500 have been provided. Nethercot also provides a set of coefficients but this is only for use with his own approach. Trahair also provides values for the computation of the elastic critical moment; however the restraints applied here are full restraints.

In Appendix B, the different for Mer have been The Dutch code has its method such that the effect of restraints can be accounted for. This has been done by introduction two different terms III and which are introduced to account for not fully restraining the beam. However it is shown in Appendix B.3; that substituting II/ = kl this will lead to the identical equation as the ones used by & Hill. This makes the Clark & Hill method also suitable to compute the elastic critical moment for restrained beams. However none of the methods found incorporates the location of the restraint other than positioning it at the most-favourable location. application methods can not be employed and analytical method will be used to compute the elastic critical moment.

10


3. Comparison of numerical model to experiment: Unrestrained beam A numerical model will be used to simulate the behaviour of steel beams (Appendix E). To make sure that this model is able to simulate the behaviour of steel beams properly, comparison has to be performed on the model behaviour to actual beam behaviour. The goal is not to accurately predict the failure load, but to investigate whether the behaviour of the Finite Element Model corresponds to the behaviour of a real beam. As a master project Swart & Sterrenburg [PI] have performed experimental research on the influence of an uncoupled concrete slab on the load bearing capacity of a steel beam. In this research two experiments have been performed; the first without the concrete plate (unrestrained) and the second with a concrete slab (restrained). In this chapter the first experiment that was performed will be discussed, then the numerical model is discussed briefly and the comparison of the behaviour of the FE-model to the experiment is performed.

3.1. Experiment The first experiment will be used to validate the model for the unrestrained case. The experiments where performed on a simply supported IPE240 with a length of nOOmrn. Two loads were applied at 3000mm from both the supports (Figure 16), chains (I = 6000mrn) where used to apply these loads. The supports consisted of two rollers being applied at both sides of the beam.

YlFl tF 15. 3000 ) 209 3000 l::.

7200

Figure 16 Mechanical model for the experiment

The strength of the material was tested; these have been used in the numerical model. Only cross-sectional imperfections have been measured, the system imperfections have not been measured. The system imperfections have great influence on the load bearing capacity, the amplitude of the imperfections have been fitted to obtain matching load-displacement curves. Also no residual stress measurements have been performed.

3.2. Finite element model In this section the Finite Element Model is briefly discussed, this has been done in more detail in Appendix E (ModelS in §EA).

Figure 17 Finite element model

3.2.1. Elements and mesh density

The FE-model consists of two flanges and a web modelled using 4 node SHELLI81 elements (Based on MindlinReissner shell theory). In doing so, the cross-sectional properties of the FE-model will not match that of an actual cross-section; therefore RHS (rectangular hollow section) compensation elements using BEAMl88 (based on the

II

Comparison of numerical model to experiment

Timoshenko beam theory) have been applied (see Appendix E §E.3). The web and flanges each consist of 8 elements, and including the 2 compensation element will lead to 26 elements per segment. The span of the beam is modelled using 72 segments.

3.2.2. Loading and support conditions

The applied load case is shown in Figure 16. The supports have been modelled according to the rollers that have been applied in the experiment. On one side of the each flange the out-of-plane displacement have been restrained (Figure 18).

Figure 18 Support conditions

To introduce the load, stiffeners (Figure 19) have been applied to prevent distortion of the section due to the load, however these were not present in the experiment but were added to prevent peak-stresses in the FE-model, the load was applied using following rods to model the chains that have been applied in the experiments (see Figure 138 Appendix E). These rods are modelled as being rigid as no information is available on the stiffness and area of these chains.

Figure 19 Stiffeners located at the load introduction

3.2.3. Material law

The material law has been taken according to one of the material tests performed by Swart & Sterrenburg [PI]. The stress-strain relations have been converted from engineering stresses to true stresses. After this step a 12point approximation of the stress-strain relationship has been made; which is used as the material law in the FE-model. The Poisson ratio is taken u = 0.3 according to EuroCode 3 [CI]. Young's modulus of elasticity is taken from the material law shown in Figure 20.

12

~ 300 :;

.,/ ~

-.... ~ .. ~

~ - Enginering stress

-True stress

_12 point approximation

Figure 20 Stress-strain relationship of the tinite element model


3.2.4. Imperfections and residual stresses

A single type of imperfection has been implemented in the analyses. The system imperfections have been chosen according to the first buckling shape, Three imperfection amplitudes have been investigated: 1I2000L, lll500L and III OOOL; effectively leading to imperfections of 3.6,5.4 and 7.2mm for a span of 7200mm. The residual stresses have been applied according to the idealised pattem found in the NEN677 1 (see Figure 21) where

(J"ni = 1/ 3 fy .

v ~ Gin;

Figure 21 Residual stresses pattern according to NEN6771

3.3. Comparison with experiment To compare the model to the experiments, an imperfection analysis is performed. The imperfection analysis was performed to detennine which load-displacement behaviour matches the behaviour obtained from the experiment. Three amplitudes where investigated for the model, the shape of the imperfection was taken as the first buckling mode, as the actual longitudinal imperfections were not measured during the experiments.

The influence of the imperfection on the load-displacement path (in-plane deflections) can be seen in Figure 22. Here it is seen that the imperfection of Ll2000 and Lll500 causes the load-displacement path to have a distinct point at which the displacement bifurcate. The load-displacement path with the imperfection of Lll 000 has a more gradual behaviour; this is affine with the experiment. However as can be seen in Figure 23, the out-of-plane displacement behaviour greatly differs . This is caused by the initial out-of-plane deflections. The most unfavourable imperfection shape is chosen (first buckling mode) as elastic buckling is seen in the loaddisplacement paths. Physically meaning that the imperfection shape has the same shape as the failure mode, therefore less energy is needed to deform the beam up to failure and leading to a lower failure load.

As a next step the cross-sectional imperfections have been modelled. By taking the mean of the different imperfection measurements (see Appendix E §EA.7) the newly obtained dimensions of the beam modelled. The measured dimensions are larger than the nominal dimensions therefore a slight increase of strength and stiffness occurs. It can be seen in Figure 24 and Figure 25 similar behaviour is found for both in- and out-of-plane deformations. However the shape of the imperfection has influence on the failure load as will be shown in the continuation of the research. This shows that the FE-model performs well, although most of the imperfections have been left unknown in this comparison.

30

25

Z 20

=. ., 15 I: 0 u. 10 "t ·"

5 ~"

./' .r 0

p ., o 10 20 30 40 50 60 70 80

Displacement u [mm)

1/2 F l l1/2 F

is 3000 1200 3000 ~ ;1' 71"X X

7200 v }\

FIrst ord er bending

--Experiment ___ M1U2000

---t-M 1 U1500

M lUtlOO

Figure 22 In-plane deformations of the simulations using nominal dimensions compared to the experiment

13

30

25

Z 20

=. Q> 15 ~ 0

u.. 10

5

0 0 10 20

Comparison of numerical model to experiment

-

30 40 50 60 70

Displacement u [mm]

80

1I2Fl l1/2F

C; 3000 1200 3000b. A' ;01< A< X

7200

--Experlrrent

_M1U2000

--+--M 1 U1500

--M1UlJOO

Figure 23 Out-of-plane deformations of the simulations using nominal dimensions compared to the experiment

30 1/2Fl11/2F

25 15. 3000 1200 3000 21. ;V ,r ;I' O!

" Y 7200 "v 20 z

I~t =.

I -CIl 15 ~ 0

u.. 10

5 -_First order bending

0 --Experiment

0 10 20 30 40 50 60 70 80 _ M 1 UlJOO With measured

Displacement u [mm] cross-sectional dimensions

Figure 24 In-plane deformations of the simulation using measured dimensions compared to the experiment

30

25

Z 20

=. Q> 15 u ... 0

u.. 10

5

0 0 10 20

e

30 40 50 60 70 Displacement u [mm]

• •

1/2 F 1 f/2 F

c; 3000 1200 3000 21. At ,r)' "O!

7200 v

"

--Exp ... lrnent

80 - M1UlJOOWlthmeasured cros&-sectlonal dimensions

Figure 25 Out-of-plane deformations of the simulation using measured dimensions compared to the experiment

14


4. Comparison of GMNIA to the EC3 method: Unrestrained beams Before restraints are applied, a number of unrestrained beams will be investigated. In this investigation the loadcases and the length of the beam will be varied. This is performed to determine how the analytical methods perform for these cases before restraints are applied. In this investigation three load cases will be investigated (see Figure 26). The first load case consist of a beam loaded with a concentrated load at mid-span, the load is applied at the most unfavourable location (intersection of web and top flange). The second load case is a uniformly distributed load applied at the most unfavourable location. The third load case is a statically indeterminate system; this load case is identical to the first load case only here one of the supports is clamped such that warping is not restrained. For each of the load cases three span to height ratio will be investigated (l/h = 15, 22.5 and 30), the height ofIPE240 section is 240mm thus resulting in spans of 3600, 5400 and nOOmm. In this chapter the load bearing capacity will be determined using GMNIA and the analytical methods. First the FE-model is discussed, after which an illustration of the method is performed. Then the results from the GMNIA are shown. And this is concluded with a comparison of the GMNIA to the analytical methods.

Fi /fi- h}

"I! 1f21 ,t 1f21 ;t

,," ,,'l.

(a) load case 1

4.1. FE-model

q

1 I I I I I I *1 "I{. ?It£.

(b) load case 2

Figure 26 Load cases used

~ Fi

7#-,,'l.

1f21 "Ii.

1f21 "It.

"V "I! (clload case 3

In this section the Finite-Element model is discussed. The applied support, and loading conditions are discussed. Hereafter the material law, residual stress pattern, imperfection shape and amplitude are discussed. A more detailed description of the used FE-models is given in Appendix E (Models 1,2,3 in §EA).

4.1.1. Support, and loading condition

In the investigation two types of support conditions have been modelled, one being a simple support, the other being a clamp without the restraining of warping. The first type of support is a fork support (see Figure 27a). The clamped support (see Figure 27b) will be used for load case 3. Special attention has been taken to allow warping of the cross-section, allowing warping ofthe section has been chosen as this is the condition on which the current codes and the LTB-Tool base the calculation of the elastic critical moment and hereby effectively only clamping the web of the section. This will also comply with the conditions of the system as fork supports are denoted by the use of the Eurocode.

Stiff elements Stiff elements

(a) Simple support (b) Clamped support

Figure 27 Support conditions applied in the FE-model

15

of GMNIA to the EC3 method: Beams

The stiff elements as shown in the previous are applied to distortion of the cross-section. For stocky an area is to these elements. During simulations it has been found that local failure is

And as is stated in the introduction local failure is not to be accounted for. For slender beams no local failure has been seen. The loads have been as follows: the concentrated load has been applied to a node, a stiffening plate is applied here as to prevent deformation of the cross-section due to the load and to peak the uniformly distributed load has been discretisized using 73 individual nodal loads on the intersection of the web and the individual loads at the nodes (Figure 28).

FFFFFFFFFFFF FFFFFFF

-'/2 FI2

1 J '1

28 applying II. uniformly distributed load on the FE-model (4 node shell elements)

4.1.2. Material law

The steel has been chosen as 8235 which was used in the GMNIA that have been to derive the buckling curves. A bilinear model has been taken: modulus of has been taken as 210000 N/rrun2 The yield strength has been set to

4.1.3. Residual stresses Residual stresses were added <>rr'".-,1,,\ the in the NEN6771 see Figure 21, where

4.1.4. System imperfections

4.1.4.1. Shape

As denoted in the Eurocode 3, the first LfU"",.IUJ; shape should be used as an imperfection shape. This will be used in the simulations.

4.1.4.2. Amplitude

An amplitude of L/300 as described in the Eurocode should be used as an This imperfection all

etc.). In the literature R2 1, P12, Pl3, an amplitude ofL/lOOO has been used in numerical to predict the behaviour of beams, in these simulations both residual stresses and imperfection have been to the models. As this is also the case in the present research, an amplitude of L/IOOO will be used. This is applied at the location where the largest displacements are found in the LBA.

4.2. Illustration of Method The methods used to obtain and compare the results will be illustrated in this This will be np,'1n''tYlF'r!

analytical and application methods. The application methods will only be used in this paragraph to illustrate their workings. In the further course of the research the LTB-Tool program (Appendix G) will be used to determine the elastic critical moment. The illustration is on load case 1 with a span of nOOmm.

4.2.1. Elastic critical moment

First the determination of Mer is performed; this is performed a different number of methods. The methods that will be used are: Linear Analysis (shell element and beam elements) ANSY8 VIO.O, NEN677 I , Nethercot's method, the Clark &Hill method and with the use of the program and the LTB-Tool program.

I Available from http://www.cticm.fr/docs/iogicieis/L TBeam -,lost.zip - .... ~~ ... ~-~~-~~----~---

16


4.2.1.1. LBA

4.2.1.1.1 LBA-beam

The linear buckling analysis (or eigenvalue analysis) has been performed using a ANSYS beam model, the model consists of 72 BEAMI88 elements over the length of the span, and the cross-section has been arbitrary inputted using the ASEC2 command. To account for the level of the load, stiff elements are used to incorporate the load height. The boundary conditions have been chosen such that the beam is free to warp at the supports, only out-ofplane rotations are restrained, thus obtaining fork support. In Appendix E §E.7 the input-file used is given. The program outputs a critical force. This can then be translated to a critical moment.

The total critical force obtained is: Fer = 21.746 kN

The reaction force is: R=I/2Fer

The arm is 3600mm (half span) Mer=R. a

This leads to the following: Mer= 1800 . Fer Mer =39.143 kNm

Figure 29 First buckling shape of system using a linear buckling analysis with beam elements

4.2.1.1.2 LBA-Shell

The FE-model also provides an elastic critical load. This model uses shell-elements and is discussed in more detail in Appendix E,

Figure 30 First buckling shape of system using LBA with shell elements

2 This command allows for arbitrary input of the cross-sectional properties and therefore the actual properties can be inputted.

17

The cri tical load "ht, .. ",>,;

=22.368 kN Mer= 1800. Mer :=40.264 kNm

4.2.1.2. NEN6771

of GMNIA to the method: Unrestrained

is then converted to the elastic critical moment.

When determining the elastic critical moment NEN6771, this should be calculated

C ~EdJzGdJ( 19

Where:

+

The coefficients and need to be determined from table 9 of NEN677 1. This table is where for this case the following are obtained:

1.35 0.55

to 1)

in r\ '''U<OllU1A A

The coefficient which accounts for the load height be taken if the load is above the centroid. -0.55

Next C and S need to be determined

C Jrl.35· 7200 7200

865

+ Jr·-0.55·865 =3.737 7200

Then the elastic critical moment can be determined:

Mer 3.737 40.636kNm 7200

4.2.1.3. Clark & Hill

is the one used equation is used. This

1)

(4.2)

Using and the coefficients

=40.543kNm

18

Lateral Torsional Buckling of laterally restrained steel beams

4.2.1.4. Nethercot

When using this method the elastic critical moment should be calculated according to eg. (4.3)

Mer = 1m ~EJ.GJI 1+ 1(:EJ(J} L LGJI

The factor a has to be computed according to the load height, since the load is located on the top flange. a is computed by dividing the two coefficients A and B. For this load case the following holds:

A = 1.35 and B = 1- 1.779 + 2.039 R2 R

Where:

R2 = L2GJ / EJ I (J)

D .. R2 eterrmmng :

R2 = 7200280769 ·127400 = 69.25

210000.36.68.109

Detennining ex:

a - A _ 1.3 5 = 1.1 07 - B - 1- 1.779 + 2.039

69.25 ..) 69.25

M = 1(1.107..)210000.2836000.80769.127400 1+ 1(2

210000.36.68.109

=40.423kNm er 7200 7200280769.127400

4.2.1.5. LTB-Tool

TU/e

(4.3)

The LTB-Tool has been written to compute the elastic critical moment using the Ritz-method. A detailed description of this method is given in Appendix G. This tool calculates the elastic critical moment and displays the buckling shape (Figure 31). The elastic critical moment has been calculated as: Mer = 40.340 kNm

Figure 31 Buckling shape obtained using LTB-Tool

4.2.1.6. LT-beam

The LT -beam program has been written by the "Centre Technique Industriel de la Construction M6tallique", a French steel research institution. The program is able to compute Mer for any number of different load cases using the finite element method. With the program the following Mcrhas been obtained:

Mer = 40.399kNm

4.2.1.7. Discussion of results obtained for the elastic critical moment

Each of the different methods results in different elastic critical moment. The results are obtained and summarized in Table 8. As a reference value the average is taken. It can be seen that the LT -beam and LTB-tool give the same results . The LBA-beam gives the lowest results, the LBA-Shell model gives a slightly lower elastic critical moment compared to the LTB-Tool and LT-beam; this is caused as shear deformations and distortion of the crosssection is possible. All of the application methods give similar results; the NEN6771 and the Greiner method have slight differences caused by the interpretation of the load-height (seen in Appendix B). In the detennination of the analytical failure load the LTB-tool is used as the application methods will results incorrect values of Mer

19

Comparison of GMNIA to the EC3 method: Unrestrained Beams

T bl 8 R It f th d t a e esu so e e ermma IOn 0 f e e as IC cn Ica momen usmO! t e I eren me f th I f 'f I h d'n th d o s

LT beam Diff LBA -beam Diff LBA-shell Diff Nethercot Diff iNEN6771 Diff Greiner Diff LTB-too1 Diff

Fcr[kNl 21.921 22.369

Me,. [kNm 40.340 0.1% 39.458 -2.1% 40.264 -0.1% 40.423 0.3% 40.636 0.9% 40.543 0.6% 40.339 0.1%

4.2.2. Numerical results

During each simulation two loads are calculated: using linear buckling analysis the elastic critical load is obtained, using GMNIA the failure load is obtained. Using the elastic critical moment the slenderness ratio is obtained using eq.(4.4) . Using the failure moment, the reduction factor can be calculated (4.5). These values can be plotted in 'A-X diagrams along with the buckling curves. The load-displacement diagram is shown in Figure 32.

- ffip, /1,= -

Mer

The elastic critical moment from the numerical simulation is already given as (LBA-shell): 40.264kNm

The plastic section capacity can be determined using:

Mp/ = WpJy = 366.6 . 103 .235 = 86.l51kNm

The relative slenderness is calculated as:

~ = ~Mpi = 86.151 = 1.463 Mer 40.264

The failure moment is calculated from the failure load: FII = 19003 kN Mu = 19003 x 1800 = 34.206 kNm

The reduction factor can now be determined:

= Mil = 34.206 = 0 397 XGMNlA Mp' 86.151 .

Z 6 (I) o (;

LL

20

f 7200 1/ I /]

v --

-In-plane (u)

- out·of -plane (v O ~----~--------~~--~-----.----~----~----~---------~·

o 15 30 45 60 75 90 105 120

DisplacelT'ent (rrm]

Figure 32 Load-displacement curve Ml L 7200 UR3

(4.4)

(4.5)

3 Simulation coding is performed as follows: M#, denotes the model number including load case (see Appendix E), L# denoted the span, UR denotes that no restraints are applied.

20

Lateral restrained steel TU!e

4.2.3. Comparison of results from GIVINIA and EC3 methods

The elastic critical moment detennined using the LTB-Tool will be used to ",tM-rn .. ,,,, the slendernessratio for the use of the buckling curves. Three methods will be used to determine the reduction method and the methods with and without moment distribution cornpems!lhon.

4.2.3.1. General method

The reduction factor the method is determined using(4.6): 1

----;=====:<,; 1

Where

<1>1t=0.5[1+0.21ptll 0.2)+ IIJ

The slenderness ratio is detennined the elastic critical moment from the LTB-Tool:

--=1.461

Then CP/I can be detennined:

<1>1/ 0.5[ 1 + 0.21(1.461- 0.2) + 1.46f ] == 1.699

And by substituting CP/I in (4.6) results in:

__ ---r==l===:::::_ == 0.388

method with-out moment distribution

"11<::;\...111\... method the reduction factor is detennined a ...... vl~llll14 to:

XIi == ----;======:<,; 1 and :<,;

<1>'1 + Where

<1>'1 0.5[1+0.34(~'1 0.4)+0.75~21IJ Then <l>it can be determined:

<1>1J = + 0.34(1.461 0.4) + ·1

And by <l>lt in results in:

__ ---;===1 ==== = 0.443

] =1

4.2.3.3. Specific method with moment distribution

1

When taking lateral restraints into account the reduction factor may be modified:

XLT,mod == f

defined as:

f 1- 0.5(1- kc )[1 2(A-LT 0.8)2]:<,; 1

For this load case kc = 0.86 thus leading to: f = 1 0.5(1 0.86)[1 2(1.461 0.8)2] 0.99:<'; 1

And resulting to the reduction factor:

== 0.443 == 0.447 0.99

these being the k~U~'""

(4.6)

(4.7)

21

Comparison of GMNIA to the EC3 method: Unrestrained Beams

4.2.3.4. Comparison

Each of the methods give different results, these are summarized in Table 9. It can be seen that the general method gives the safest prediction of the reduction factor (and ultimate load). The other two methods give a large underestimation of the reduction factor.

Table 9 Reduction factors obtained usin GMNIA and anal tical methods GMNIA General Diff Specific Diff Specific Diff

method without with 0.397 0.388 2% 0.443 -11.8% 0.447 -12.83%

4.3. Results of investigation In this investigation three height to span ratios and three load cases have been investigated, thus leading to 9 results to be compared. First the results from the numerical results are shown, and the comparison is made between each of the analytical methods.

4.3.1. Results from GMNIA

From the results of the GMNIA the slenderness and the reduction factor can be obtained. In Figure 33 this shown along with the two buckling curves (general and specific method). It can be seen that the results obtained show expected behaviour. For tabular results see Appendix H.

1.2

1

0.8

X 0.6

0.4

0.2

- EC3 curve General method curve A

---EC3 curve Specific method curve B

- Euler curve

Load case 1, GMNIA L11000

Load case 2, GM\l1A L11000

Load case 3, GM"JIA L11000

O +----,----,----,----~----.---_.----._--_T----._--~----,_--_,

o 0.25 0.5 0.75 1.251 1.5 1.75 2 2.25 2.5 2.75 3

Figure 33 Results from numerical simulation of unrestrained beams

4.3.2. Comparison of numerical simulation to the general method

To compare the results from the numerical simulation to the clauses of the Eurocode 3 the following is performed: First the reduction factor from the GMNIA is determined. After which the LTB -tool is used to compute the elastic critical moment, this as it a method for which controllable results have been obtained. With this elastic critical moment the reduction factor is determined using the clauses of the Eurocode. The comparison is visualized in Figure 34. On the horizontal axis the analytical determined reduction factor is shown. On the vertical axis the numerically determined reduction factor is shown. The thick black line is the 1: I line, physically meaning that the results from the simulation and the clauses of the Eurocode match pelfectly. Two other lines are seen, these are the 5% over- and lUlderestimation lines. In this figure it is seen that the general method performs well. For load case 1 an overestimation is made once, the same is seen for load case 2. A 5%+ underestimation is made for load case 3 on all cases (Table 10).

22


Figure 34 Comparison of GMNlA to the general method

4.3.3. Comparison of numerical simulation to the specific method

The same comparison is performed using the specific methods. The specific method with-out moment distribution compensation (§4.2.3.2) is shown in Figure 35a, here it can be seen that for only load case 3 an accurate estimation is made. For the other two load case a >5% overestimation is made. When compensating for the moment distribution (§4.2.3.3) a single estimation is accurate, in all other cases a 5%+ overestimation is made, this is shown in Figure 35b (See also Table 10).

0.9

0.8

« ~ 0.7 o ~

0.6

0.5

0.4

Safe

-5%

• Load case I

:.: Load Case 2

Unsafe • Load Case 3

Safe

0.9

0.8

« ~ 0.7

-5%

o ~ 0.6

• Load case I

0.5 :.: Load Case 2

0.4

Unsafe • Load Case 3

0.3 .jt2'~r---i----+--.,---+===:::::t====r1 0.3 ~=-,---l---,..---,.----,'===+===n 0.3 0.4 0.5 0.6 O~ 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9

X EC3 X EC3

(a) Specific method without moment distribution (b) Specific method with moment distribution compensation compensation

Figure 35 Comparison of GMNIA to the specific method

T bl 10 D'ff a e 1 erences b t e ween d'ff eren 1 tGMNIA d an I f 1 analYllca me th d 0 S

Load case 1 Load case 2 Load case 3

1 = 7200 1 = 5400 1 = 3600 1 = 7200 1 = 5400 1 = 3600 1=7200 1 = 5400 1=3600

General method 2.02% -0.60% -3.810/. 2.44% -0.84% -6.13% 14.00% 9.37% 6.22%

Specific method -11.83% -11.37% -9.490/. -12.83% -13.31% -13.320/. 3.17% 0.79% 1.50%

Specific method -J 2.82% -15.90% -16.860/. -12.83% -14.61% -16.220/. -1.41% -8.40% -11.62%

23


5. Comparison of GMNIA to EC3: Single concentrated load with a single elastic restraint

In this chapter the load-bearing capacity of a system with a single concentrated load with a single elastic restraint is investigated . First the load and support case are discussed. Then the Finite Element model is discussed. Hereafter the influence of the spring-stiffness of the restraint on the elastic critical moment is shown. After which the load-bearing capacity is compared with the buckling curves from the Eurocode 3.

5.1. Load and support case: A simply supported beam (IPE240) with a length of 7200mm is loaded at mid-span with a concentrated load which is applied on the centroid of the top-flange. An elastic lateral restraint is applied at mid-span which is attached to the centroid of the top-flange (see Figure 36). An elastic restraint is chosen as a rigid restraint (K~) cannot be made in practice. In this comparison the influence of the spring-stiffness is investigated.

F F

z Ksp

x y L ~ .-i, ~

I

Elevation I -z I I

P Ksp Sideview

x I I

I I

Y 'hI I I

IL 71

IL 71

X X Planview

Figure 36 Plan-view, elevation and side-view of load and support case 2.

5.2. FE-model In this section the Finite-Element model is discussed. The applied support loading and restraining condition are discussed briefly; hereafter the material law, residual stress pattern and the imperfections are discussed. This is done in more detail in Appendix E (Modell in §E.4)

5.2.1. Support, loading and restraining condition

The support are modelled as being fork supports, the load is introduced using stiffening-plates as is common in practice. The elastic restraint has been modelled using a single COMBINl4 element. This element is attached to the cross-section and has a free end. At the free end the displacement in the y-direction is set to zero. The displacement in the z and x-direction correspond to the displacements of the end which is attached to the crosssection (Figure 37).

Restraint

Figure 37 Modelling of restraint.

25

Comparison of GMNIA to EC3: Single concentrated load with a single elastic restraint

5.2.2. Material law

The steel grade has been chosen as 8235 which was used in the numerical simulations that have been perfonned to derive the buckling curves. A bilinear model has been taken: the Young's modulus of elasticity has been taken as 210000 N/mm2 The yield strength has been set to 235N/mm2 (See Figure 122).

5.2.3. Residual stresses

Residual stresses were added according to Figure 126, where (aini = 1 13/y).


5.2.4.1. Shape

In the simulations three imperfection shapes are looked into. The first shape is the buckling shape of the unrestrained beam ( ...-= ). The second shape is the first buckling shape of a beam with a rigid restraint ( .....-==-. -=---- ). The third imperfection shape is the intennediate buckling shape belonging to the partially restrained system as denoted by the clauses of the Eurocode (5.3.3) (these shapes are shown in Figure 41 thru Figure 46). These imperfection shapes have been chosen to see how the imperfection shape influences the load bearing capacity.

5.2.4.2. Amplitude

In the simulations three imperfection shapes are analyzed. For the first and third shape (intennediate buckling shape), the length between the supports is taken as L (Figure 38b). For the second shape the length between the support and restraint is taken. Therefore here the imperfection is taken as Ll2000 (Figure 38c). The amplitude of the imperfections is taken at the point where the greatest displacements are found in the linear buckling analysis . The rotation of the cross-section in the LBA is also scaled (Figure 38a). Applying the imperfection amplitude at the centroid will cause the rotational vector to also be scaled and will lead to inconsequent results as will be shown in §5.5 .1.

(a) Displacement vector (b) First Imperfection shape

(c) Second imperfection shape

Figure 38 Displacement vector and amplitude of the imperfections used

5.3. Presentation of results During each simulation two loads are calculated: using linear buckling analysis the elastic critical load is obtained, using GMNIA the failure load is obtained. Using the elastic critical moment the slenderness ratio is obtained using eq. (5.1) . Using the failure moment, the reduction factor can be calculated (5.2). These values can be plotted in 'A.-X diagrams along with the buckling curves; which provides insight into the influence of the variations of the imperfections .

(5.1)

M Mil = MplX ---7 X = Mil (5.2)

pI

As an example simulation "M1 L 7200 X/'l 3600 Yrl 115.1 Kl 500 IS-l ,,4 (simulation number 31 see Table 25 in Appendix 1) is used. The in-plane and out-of-plane load-displacement curve of the simulation are shown in Figure 39. For this case the failure load is 45 .885 kN, along with a reaction force in the restraint of7.332 kN. The elastic

4 Coding of simulation is perfolmed as follows: M# denotes the model number and load case (see Appendix E), L denotes the span, Xrn is the position along x axis of restraint number n, Ym is the position along x axis of restraint number n, KII is the spring-stiffness of restraint number n, IS denotes the imperfection shape.

26


critical load has been calculated as Fer = 87.426 kN. With the failure load and the elastic critical load the slenderness ratio and the reduction factor is computed.

First the plastic moment capacity of the section needs to be determined.

M pi = WpJy = 366.6 .103 ·235 = 86.151kNm

Then Fer and F" are transformed to moments. Using the elementary formulae of a single load:

M=FI 4

M = F;) =F 1800=87426·1800=157.367 kNm cr 4 cr

Mu = Fu1800 =45.885 ·1800= 82.59 kNm

Then leading to

A ~ r " ~ '--8-6-.1-51- = 0.740 Mer 157.367

X = Mu = 82.59 = 0.959 Mpi 86.151

w,-----------------------------------,--------,--------, 45 jo--=-<~..::----____:_:__:___= ..... -o-_

40

35

Z 30 C Q) 25 u '-0 u. 20

15

10

5

0 0 20 40 60 80 100 120

Displacement [mm]

1 F

~ = 500 Z>.

/IV 3600 ,r 3600 ,r v 7200 ,r

u

I ~v

--In-plane displacementv

__ Out-or-plane displacem ent u

-Reaction In restraint

Figure 39 Load-displacement curve of simulation "MI L 7200 X,l 3600 Y,l 115.1 K1 SOO IS-I"

5.4. Elastic critical moment The spring-stiffness of the restraint highly influences the elastic critical moment. Therefore the spring-stiffness of the restraint will be varied to determine its influence. To set spring-stiffness requirements two values of the spring-stiffness will be determined; these values are the spring-stiffness needed to reach 95% and 99% of Mer when using rigid restTaints; these can be used as stiffness requirements for use in the Codes.

The computation of the elastic critical moment has been performed using three different methods. The LTB-Tool program and the LT-beam program both have given identical results, whereas the LBA-Shell (without weld-in plates at the load inh·oduction, as these restrain warping) model has given slightly lower results (1 .0%). This is due to the deformations of the cross-section which is not accounted for in the LTB-Tool and LT -beam programs (see Table II.) It can also be seen in Figure 40, that at a spring-stiffness of Ksp;:::: 480 N/mm the second buckling mode occurs, where the Ksp-Mer relationship bifurcates. This can also be seen in Table II (denoted with the double division line). In Figure 41 thm Figure 46 the buckling shapes are shown, here the displacements at the cenh·oid are shown. This combined with the rotational vector the displacements of the upper and lower flanged are visualized. On the vertical axis the displacements of the centroid and flanges are shown. These have been

27

Comparison of GMNIA to EC3 : Single concentrated load with a single elastic restraint

detennined using the LTB-Tool program. Here it can be seen that before the second buckling shape occurs the displacements occurring at the upper and bottom flanges increases, this indicates that the rotational vector of the buckling shape increases. For example in Figure 41 the displacement vector of the upper flange equals 1.5 at midspan, whilst in Figure 45 the displacement vector ~ 6.

M" (kNmJ

14 0

120

100

80

60

40

0( First buckling shape Second buckling shape

20 '--____ ~ ____ ~ _ _L_ _______ ~ ____ K" [N/mmJ

200 400 600 800

Figure 40 Influence of the spring-stiffness of a restraint of the elastic critical moment

Table 11 Results of M.:r using the different methods

Mer [kNm]

Ksp LTB-Tool LT-beam Diff* LBA-Shell Diff*

0 40.340 40.339 <l)

0.0% 39.950 1.0% -0 0

100 75.669 75 .663 0.0% 75 .298 0.5% S 200 102.565 102.560 0.0% 102.049 0.5%

bO .S

300 124.579 124.570 0.0% 123.836 0.6% ~ u

400 142.927 142.920 0.0% 141.881 0.7% ;:l .I:J

438; 95% of Mer 149.070 149.060 0.0% 147.892 0.8% .... .~

480; 99% of Mer 155.378 155.370 0.0% 154.013 0.9% ~

500 156.826 156.820 0.0% 155.984 0.5% bO

600 156.826 156.820 0.0% 156.130 0.4% -g .S 1000 156.826 156.820 0.0%

o ~ <l)

156.137 0.4% u U-o <l) ;:l 0

1 0000000 ~ 00 156.826 156.820 0.0% 156.139 0.4% CZl.I:JS

* The results from LTB-Tool are taken as a reference

2.5

1.5

0.5

28

1.4

1.2

0.8

0.6

0.4

0 .2 ... '" __ -- --- -- ---- - ~ ......... -¥~~~~~----~~--~--~--~--~--~--~~~~~- 1

1000 2()()() 3000 4000 5000 6000 7()()()

Figure 41 Buckling shape when Ksp = 0

1.5

0.5

1000 2000 3000 4000 5000 6000 7000

Figure 42 Buckling shape when Ksp = 100 N/mm

--:-:..... - - "lOGO -- _ _ _ . 2000_ _ _ _ --3000 _ 0-· __ 5000 _ ._ . ...6000-.--_ . 7()()()


Centroid

Centroid

w..oer Flange

Centroid

UJ:fer FlaNJe


u

~~ ____ ~~~~--~~~~----~--~~--___ ~~~~~~_ 1 6000 --rooo - ~ 1000 2000 3000 4000 sooo

-1

Figure 44 Buckling shape when Ksp = 300 N/mm u

3000 4000 sooo - - 6000 - - .-.... ::.·1600 -2 -~---.-- ...--

Figure 45 Buckling shape when Ksp = 400 N/mm u

1. 5

_.- - -- ..... -0.5

- 0.5 1000 2000 3000 - 5000

-1

-1. 5


5.5. GMNIA

5.5.1. Load bearing capacity

l.cA-.er Flange

Centroid

UJ:per Flange

l.cA-.er Flange

Centroid

UJ:per Flange

l.cA-.er Flange

Centroid

UJ:per Flange

TU/e

In total 30 GMNIA simulations have been performed on the presented load and support case. Ten different springstiffnesses (Ksp= 0, 100,200,300,400,438 (95%), 480 (99%), 500, 600 and 1000 N/mm) of the elastic restraints have been used on each of the three different imperfection shapes (as mentioned in §5.2.4). The results are listed in Table 25(Appendix I), and are shown in Figure 47 along with the two governing EC3 buckling curves.

In Figure 47 the influence of the imperfection shape can be seen graphically. The intermediate buckling mode (belonging to the pal1ially restrained system) gives the lowest failure loads. However unexpected behaviour occurs when the imperfection amplitude is applied at the centroid; between the restraint stiffness of 200N/mm and 500N/mm. Here the failure load decreases, while an increase should be expected. This unexpected behaviour is caused by the increase of the rotational vector of the buckling shape (this is seen in Figure 41 thru Figure 46, and mentioned in §5.4). Therefore the imperfection has been applied at where the greatest displacements are found in the LBA. Doing this causes the behaviour to be consistent. The imperfection shape of the unrestrained system shows consistent behaviour. When the second buckling mode occurs the failure load still increases; this however does not occur with the other imperfection shapes. This is caused by occurrence of the second buckling mode; where the elastic critical moment remains constant.

When no restraint is applied and the imperfection is taken as the buckling shape of the fully restrained system, the failure load equals the elastic critical moment. This is expected as the no initial out-of-plane and rotational imperfections are present at where the load is applied . And during loading no large out-of-plane displacements occur and therefore when the elastic critical load is reached the beam buckles elastically, as would be expected. When the restraint stiffness exceeds 200N/mm, increasing the restraint stiffness has no significant influence on the failure load .

29

0.9

0.6

0.7

X 0.6

0.5

0.4

0.3

0.2 0.6

Comparison of OMNIA to EC3: Single concentrated load with a single elastic restraint

K = 1000 R;' 600 K= 500

0.7

K= 300

K= 400 K = 200

0.6 0.9

Amplitude applied at centroid

Arrl>lltude applied at Range centroid

1.1 1.2 1.3 1.4

K=O

1.5

is. ~ K ::::tL % 3600 A' 3600 ;t:

7200 -- EC3 CUM General method

cun.eA

--EC3 CUM Specific method curwB

- -. GMNIA intermediate buckling shape l/1000L

• GMNIA buckling shape of unrestrained beam 1I1000L

~- GMNIA buckling shape or res(ralned beam 1/2000L

- - - - Euler CUM

...•.. . GMNIA Internedlate buckling shape 1/1OCOl applied al centroid

Figure 47 Graphical representation of results obtained from simulations

5.5.2. Force in restraint

The reaction forces in the restraints are plotted along with the spring-stiffness in Figure 48. The highest reaction forces are obtained by the use of the intermediate imperfection shape up to Ksp = 400N/rnm (intermediate buckling mode).

The analytical determination of the force in the restraint at failure will be determined using the clauses of the NEN6770 (art. 12.2.4.2) [C2], this as no clauses are given in the Eurocode 3. These rules are denoted as the 1 % and 0.5% rule. These rules represent a percentage of the force in the compression flange including the root-radii and 1/3 of the web taken between the root-radii. The I % rule (5.3) is calculated using the largest stress obtained in the section, and for the 0.5% rule (5.4) the yield strength is taken.

N,p = 0.0 1 AjlO"j;s;d

Nsp = 0.005Ajlfy;d

(5.3)

(5.4)

To determine the force in the compression flange including the root-radii and 1/3 of the web taken between the root-radii the cross-section is divided in 4 sections. For each of these sections the stress at the centroid is determined and multiplied by the area of that section. Doing this leads to the following equation and giving the force

h-tj' (( Jr) h-2tjl-(r-2/(12-3Jr)r)] btjl--+2 1-- r + ....

h 4 h (5.5)

h-2tjl-(1I3(h-2tjl-2r)+r)12 tw(1/3(h-2tjl-2r)+r) h

The largest stress can be calculated as:

(Yj;s;d = ::" (5.6) y;el

By applying the nominal dimensions of the IPE240 cross-section (Appendix D) and substituting (5.5) and (5.6) in (5.3) this will lead to the equation

Nsp = Mil 15.3082mm2 = Mllh 15.3082mm2 (5 .7) WCI;y Iy 2

By applying the nominal dimensions of the IPE240 cross-section and substituting (5.5) 'l.I1d/y;d = 235 N/mm2 in (5.4) this will lead to the following:

Nsp = 1. 856kN (5.8)

The greater of(5.7) and (5.8)will be the force in the restraint at failure.

30


10

9 -+- Unrestrained Imperfection shape

Z 8 e

-II- Restrained imperfection shape

-..-Intermediate imperfection shape

~ 7 2

J2

-*""1% Rule NEN6770

"""*-"0.5% Rule NEN6770

IV 6

" ~ 5 UI ~

.= 4 .. ~ J! c: 3 0

1l IV 2 Q)

Il:

° 100 200 300 400 500 600 700 800 900 1000

Sprlngstlffness [N/mm]

Figure 48 Graphical representation of Table 25 of appendix I

5.6. Comparison of GMNIA to EC3 Comparing the results obtained from GMNIA to the Eurocode 3 is some what shown in Figure 47. In this figure it can be seen how the results obtained from the LBA and GMNIA are compared to the buckling curves. In the following figures on the vertical axis the results from the simulation are plotted, on the horizontal axis the reduction factor calculated using the values of Mer found with the LTB-Tool program (see Table 11) are plotted. This has been performed for both the general and specific methods (see §2.I).

5.6.1. General method

In Figure 49 the results of the simulations are compared with the analytical results obtained using the L TB-Tool and the clauses of the Eurocode. It can be seen that the general method always underestimates the reduction factor obtained from GMNIA.

Saf.

0.9

0.8

~ 07

'" >< 0.6

0.5

0.4

0 .3

0.3 0.4 0.5 0 .6 0.7

XEC3

I • "'~

• G\NA buckling shape of unrestraned Beam 1/1000L

X G~ buckling shape of reslraloed beam 1/1000L

• GM'JIA Interrredlate buckling shape 1I1000l

0.8 0 .9

Figure 49 Comparison of GMNlA using the general method

5.6.2. Specific method without moment distribution compensation

The specific method uses a correction factor for the moment diagram between lateral restraints. The most conservative correction is using a linear moment diagram. Here no COlTection is applied on the reduction factor obtained with this method. In Figure 50 the results of the numerical simulation are compared to the analytical results obtained with the LTB-Tool and the clauses of the Eurocode 3. It can be observed that the failure load is overestimated once and an accurate approximation is made once.

31

Comparison of GMNIA to EC3: Single concentrated load with a single elastic restraint

0.9

0.8

< § 0.7

" >< 0.6

0 .5

0.4

Safe

Unsafe

• G'-NA bucldi1g rn:xIe of unrestrained Beam 1/1000L

:t GMNA.. bucking rn:>de of restrained beam 1/1000L

• GM\IA tlterrredlate buckling rrod 1I1000L

0,3 ~----+---~-~--~-~--~----1 0.3 0 .4 0.5 0.6 0.7 0 .8 0.9

XEeJ

Figure 50 Comparison of GMNIA using the specific metbod without moment distribution compensation

5.6.3. Specific method with moment distribution compensation

The clauses of the Eurocode allow for compensation of the moment distribution between lateral restraints. In the Eurocode no reference is made to define a lateral restraint. Therefore two different interpretations can be followed. The first interpretation is to define a partial restraint as a restraint. The second to defme full-restraints as lateral restraints or in this case the fork-supports.

Using the first interpretation of lateral restraints the moment distribution between the restraints is linear (nil to M). This moment distribution leads to a correction factor of kc = 0.75. In Figure 51 the results of the comparison are shown. It can be seen that for the simulation using the intermediate buckling shape as an imperfection shape a large (>5%) overestimation of the failure load is computed. The estimation made using the buckling shape of the umestrained beam is performed well for small reduction factors (small = 1, large = 0). For the second interpretation (where the lateral restraints are defined as full-restraints), the correction factor of the overall moment diagram may be taken as kc = 0.86. In Figure 51 the results are shown using this interpretation of the clauses of the Eurocode 3. It can be seen that the estimation of the failure load is more accurate. However for XEC3

<0.8 an overestimation is made, where as for XEC3 > 0.8 an accurate prediction is obtained.

O,g

0 ,8

~ 07

" >< 0,6

0 .5

0.4

Safe

Unsare

% GMNA.. bucldlng rmde of restrained beam 1I1000L

• GM'-UA Interrredlate buckfrng Il'Od lfl000L

0.3l'-L-~--";'--~--~-~--~----1

0.3 0.4 0.5 0.6 0 .7 0.8 0 .9

XEeJ

0.9

0,8

~ 07

" >< 0.6

0.5

0.4

Safe

• G~IA buckling rn:xIe of unrestrained Beam 1I1000L

x GMNA buckJing rrode of reslralned beam 1/1000L

• Q.,ofiIA nlenredlate bucklilg rrod 1I100OL

0.3 ~=---~-_--~--4=======-'1 0,3 0 .4 0.5 0,6 0.7 0.8 O,g

XEe)

(a) Linear moment diagram between support and (b) True moment diagram between the supports restraint

Figure 51 Comparison of GMNIA using the specific method with moment distribution compensation

32


6. Comparison of GMNIA to EC3: Two concentrated loads with two elastic restraints

6.1. Load and support case: A simply supported beam (IPE240) with a length ofnOOmm is loaded At 3000mm from the supports with two concentrated load which are applied at the centroid of the top-flange. Elastic lateral restraints are applied at 3000mm from the supports, which are attached to the centroid of the top-flange (see Figure 52).

z

x L ~

Elevation

x

J /l

v 71

Planview

3000

Y2F Y2F

i i i i

Ksp

1200 /

7200

i !

Ksp

!

l 3000

.-l, ~

I I

v /

x

y

-z

Sideview

Figure 52 Plan-view, elevation and side-view of load and support "case 4".

6.2. FE-model The Finite-Element model is identical to the one presented in §5.2 and discussed in more detail in appendix E.

6.2.1. Imperfection shape and amplitude

Two imperfection shapes will be investigated. The first shape is the buckling shape of the unrestrained beam with an amplitude of //1000 (see §5.2.4). The second imperfection shape, is the first buckling shape of the partially restrained system (denoted as intermediate buckling shape in §5.2.4) with an amplitude of //1000.

6.2.2. Load control vs. Displacement control

The system is asymmetric, thus applying two equal loads using displacements is not possible. However the asyrnrnetry is considered to be small. Therefore a small number of simulations have been performed using both load and displacement control. Here it was found that no difference has been found in the load displacement paths. In the simulation small differences between the each of the loads where found using displacement control, these differences are very small and therefore it can be concluded that using displacements as load for this load case will not cause the results to be inaccurate.

6.3. Elastic critical moment The elastic critical moment has been determined using the LTB-Tool program. The other methods (LT-beam and the ANSYS shell model) have not been looked upon, as these give similar results. The Ksp-Mcr relationship is presented graphically in Figure 53 . In this figure the different buckling shapes are shown graphically herein it can be seen that the when the slope-angle changes a different buckling shape is normative. These buckling shapes are presented in Figure 54 thru Figure 59.

Mer

200

150

100

Third buck lino sila e ;0 ~~~----~~--~~~=====---------------------~

~--~~--~-~-.~-~. ---' S pdngstiITncssN/mm 500 1000 1500 2000 2500 3000 3500 -1000

Figure 53 Influence of the spring-stiffness Ksp on the elastic critical moment Mer

33

34

Comparison of GMNIA to EC3: Two concentrated loads with two elastic restraints

I.q

1.2

0.8

0.6

/~------ -----. ,..,... ...... - .- - .............. ........

l.o.<er Flange

centroid

o A tJwer Flange

0 .2 ~-

1000

2.5

1.5

0.5

-0.5

-2

1.5

0.5

-0.5 1000

-1

-1. 5

0.5

1000

-0 .5

-1

0.2

-0.2

-OA

-0.6

-0. 8

-1

-1.2

2000 3000 qOOO SOOO 6000 7000

Figure 54 Buckling shape of case 4 when Ksp = 0 N/mm

__ .4000 __ _ _


3000 4000


... ---- -------_ .....

2000 3000


2000 3000 qOOO 00 6000

Figure 58 Buckling shape of case 4 when KSD = 1000 N/mm

3000 qOOO 5000 6000


l.o.<er Flange

Centroid

Centroid

\.Wer Flange

l.o.<er Flange

Centroid

tJwer Flange

l.o.<er F1 arqe

Centroid

l.o.<er flange

Centroid

UfFer Flange


6.4. GMNIA

6.4.1. Load bearing capacity

In total 12 OMNIA simulations have been perfonned on this case. For each of the two imperfection shape, six spring-stiffnesses have applied (Ksp= 0, 125,250,500, 1000 and 8000 N/mm). The results are listed in Table 26(Appendix J), and are shown in Figure 60 along with the two governing EC3 buckling curves.

In Figure 60 the influence of the imperfection pattern is shown graphically. It can be seen that the use of the buckling shape of the partially restrained system shows consistent behaviour, however a "peak" is found when the spring-stiffness of the springs equals 500N/mm. Also it shows that the use of the first buckling shape does not always give the lowest ultimate load. This is only valid when the imperfection shape applied is affine with the failure mode. The use of the first buckling shape of the unrestrained system shows consistent behaviour.

0.9

.--:--- ..---==>. . 0.8

0.7

0.6

X K = 250

0.5

0.4

0.3 ···_ ·1

112Fl 11/2F

--EC3 CUI'\e General ' method cune A

--EC3 CUM specific method CUI'\e B

-+-GMNIA buckling shape of unrestrained beam 1/1000L

--lI:- GMNIA first buckling shape 01 partially restrained

........ Wuf!~~~OOOl 0.2 +---~-----+---~--~---~--~--~--~--_--~_-~

0.5 0.6 0.7 0.8 0.9 1.1 -X 1.2 1.3 1.4 1.5 1.6

Figure 60 Graphical representation of results obtained from simulations performed

6.4.2. Force in restraint

The forces in the restraint at failure are shown in Figure 61. It can be seen that the clauses of the NEN6770 give a better prediction of the force in the restraint compared to the previous comparison. This is likely mere coincidence. The highest reaction forces are obtained with the use of the imperfection shape of the partially restrained system. The reaction forces at Ksp = 500N/mm being opposite for the imperfection shape of the partially restrained system is caused by the imperfection shape (see Figure 57).

I: 'iii J:; IJl e .':

'" e .g

10.---~--~---.---------~------.-----~

8

0

1

-2

-8

-10

0 200 500 600

I

~ I r.=::::=: ! I

I

7 0 800 1000 11 0

--+- Unrestrained imperfection shape Restraint 1

- _ Unrestrained imperfection shape Restraint 2

-.- Partially restrained system imperfection shape Restraint 1

___ Partially restrained system imperfection shape Restraint 2

~1% Rule NEN6770

-*""0.5% Rule NEN6770 .L~lm~p=e~rfe=ct~io~n~sh~a~pe~~~e~n~K~=5=Ol9~N/~m~m~ ___ -===========~===================r---l

Springstiffness [N/mm]

------------------------- --------------~ Figure 61 Reaction forces in restraint at failure

35

Comparison of GMNIA to EC3: Two concentrated loads with two elastic restraints

6.5. Comparison of GMNIA to EC3

6.5.1. General method

In Figure 62 the results of the comparison of the numerical results are compared with the analytical results obtained with the LTB-Tool and the general method. It can be seen that all the results obtained give a good estimation of the failure load of the system.

0.9

0.8

<:

§ 0.7 0

><

0.6

0.5

0.4

0.3

0.3 0.4 0.5

• GM'IA Buckling fTOde of the unrestrained system 1I1000L

X GM'IA Bucking mode of the partiat restrained system 1I100OL

0.7 0.8 0.9

Figure 62 Comparison of GMNIA to the analytical methods using the general method

6.5.2. Specific method without moment distribution compensation

In Figure 63 the results of the comparison of the numerical results are compared with the analytical results obtained with the LTB-Tool and the specific method with-out the moment distribution correction. It can be seen that all the results obtained give a good estimation of the failure load of the system for XEC3 > 0.7. For XEc3 < 0.7 an underestimation (-5%) is made.

0.9

0.8

<:

~ 0.7 0

><

0.6

0.5

0.4

0.3

0.3 0.4 0.5

• GM\lLI\. Buckling of the unrestrained system 1I1000L

X Gt.NA Buckling fTOde of the parOalty restrained system 1I1000L

0.7 0.8 0.9

Figure 63 Comparison of GMNIA to the analytical methods using the specific method without moment diagram correction

6.5.3. Specific method with moment distribution compensation

The correction of the moment distribution will be performed twice. This will follow the same interpretation as was mentioned in §5.6.3. The first interpretation is to define a partial restraint as a lateral restraint. The second to defme full-restraints as lateral restraints or in this case the fork-supports.

36


However the correction factor (ke see Table 2 §2.1.2) of the actual load case is not given in the Eurocode. It has been found that the correction factor can be determined using (6.1) (see also Appendix C). This leads to a correction factor of ke = 0.917. The first interpretation is given in Figure 64a, the second interpretation is given in Figure 64 b. In these figures it can be seen that using the first interpretation of the clauses of the Eurocode will lead to unsatisfactory results. Using the second interpretation will however lead to more satisfying results. Therefore the second interpretation of the clause in the Eurocode will be followed in the comparison of results when applying the correction of the moment distribution in the specific method. Thus using the moment distribution between full restraints or the supports.

0.9

0.8

~ 07

'" ><

0.6

0.5

0.4

0.3 0.3 0.4 0.5

k =H; c C I

• GM\I~ Buckling of the unrestrained system 1/100OL

X GM\PA Bucking rrode of the partial)' restrained system 1/1000L

0.7 0.8 0.9

0.9

0.8

~ 07

'" ><

0.6

0.5

0.4

0.3 0.3 0.4 0.5

(6 .1)

• G~ Sucking of the unrestrained system 111000L

:0: GI.NA Buckling trode of tile partially reslralned system 1/1 OOOL

0.7 0.8 0.9

(a) Moment diagram between support and one of the (b) True moment diagram between supports partial lateral restraints

Figure 64 Comparison of GMNIA with the specific method with moment diagram correction

37


7. Comparison of numerical model to second experiment: beam with two elastic restraints

Swart & Sterrenburg [P 1] have performed two experiments to determine the influence of a concrete slab on the load bearing capacity; the reference experiment has been described in chapter 3. In this chapter results from the experiment with concrete slab will be compared to the FE-model. The objective of this comparison is not to simulate the experiment, but to validate whether the behaviour of the FE-model is similar to that of the experiment. First the experiment is discussed, and then the FE-model will be discussed. After which a comparison will be made between the results obtained using the FE-model and the results from the experiments.

7.1. Experiment The second experiment consisted of a beam (IPE240) with a length between the supports of nOOmm. The supports were left identical to the first experiment (rollers on both side of the section). The load was applied using a concrete slab with a width of l200mm located at mid-span. The load is transferred from the slab to the beam using a rubber strip (100x20mm) with unknown properties (see Figure 65). In [P 1] the load of the slab was discretisized to two concentrated loads; this leading to the mechanics scheme that is used in chapter 3. This was chosen to predict the failure load, and to quantify the influence of the slab compared to the first experiment. As was done with the first experiment the material properties where determined; the cross-sectional imperfections where measured and other imperfections (residual stresses and system imperfections) were left unknown.

Figure 65 Loading of concrete slab on beam

7.2. FE-model In tills section the FE-model is described. The loading, support and restraining conditions are described first; then the material law and the residual stresses are described. Then the applied imperfection shapes and amplitudes are described. This is done in more detail in Appendix E (Model 6 in §EA)

7.2.1. Loading, support and restraining conditions

The load case applied is as is shown in Figure 16. The supports have been modelled according to the rollers that have been applied in the experiment. On one side of the each flange the out-of-plane displacement have been restrained (Figure 66).

Figure 66 Support conditions

39

Comparison of numerical model to second experiment

To introduce the load, stiffeners (Figure 19) have been applied to prevent distortion of the section due to the load, however these were not present in the experiment but were added to prevent peak-stresses in the FE-model. The concrete slab is discretisized using two rigid restraints . This as no information has been made available on the stiffnesses of the slab and surrounding load frame as shown in Figure 65.

7.2.2. Material law

The material law was taken according to one of the material tests performed by Swart & Sterrenburg (Figure 67). The stress-strain relations have been converted from engineering stresses to true stresses. After this step a 12 point approximation of the stress-strain relationship has been made; which is used as the material law in the FE-model.

600

_Engineering alress

""'-True Slress

200

100

0.05 0.1 0.15 0.2 0.25 0.3

StraIn

Figure 67 Stress-strain relationship of the finite element model


The system imperfections have been chosen according to three different buckling shapes. The first imperfection shape is taken that of an unrestrained beam( -= =-- ). The second imperfection shape is taken that of the second buckling shape of tbe unrestrained beam ( -==> -===-- ). The third imperfection shape is taken the first buckling shape of the actual system. The amplitude are taken respectively as 111 OOOL, 1/2000L and 111 OOOL, this to illustrate the possible imperfection shapes of the beam.

7.2.4. Residual stresses

The residual stresses have been applied according to the idealised pattern found in the NEN677 1 where

(Tini = 1I3fy ·

7.3. Comparison of FE-model to experiment The load-displacement shapes obtained from the experiment and the FE-simulation are shown in Figure 68 and Figure 69. Here it can be seen that the in-plane behaviour of both the experiment and the FE-model behave alike. The out-of-plane deflections behave significantly different. This as the rubber used in the experiment which constrains the out-of-plane displacements at mid-span and is therefore expected as the FE-model does not model the slab in combination with the rubber strip. The load-displacement behaviour of the FE-model behaving similar to the experiment shows that the numerical model behaves as would be expected.

Z ~ <I> U (; LL

40

80

70

60

50

20

0

0 20 40 60 80

Displacement [mm]

- Experiment

-+-- Buckling shape of unrestrained beam

- _ Second buckling shape of unrestrained beam

--.- actual buckling shape

100 120 140

Figure 68 In-plane displacement of simulations compared to experiment


80.-------------------------------~--------------~

~ 50 .. ~ 40 u.

30 I r-------~--------L-------------~

- Experiment

20 -+- Buckling shape of unrestrained beam

10 --Second buckling shape of unrestrained beam

--*- Actual buckling shape O~====~======r======r====~~----~----_Lt

-12 -10 -8 -6

Displacement [mm]

-4 -2 o

Figure 69 Out-or-plane displacement or simulations compared to experiment

TUie

41


8. Parameter study: Beams with K95% spring-stiffness restraints The parameter study will consist of varying the load-case and also varying the position of restraint along the length of the beam (xr see Figure 70) and its position up the height of the cross-section (yr see Figure 70). Rigid restraints can not be made in practice; therefore stiffness requirements will need to be set. An investigation will therefore be perfonued on the influence of spring-stiffness. First the results will be obtained using rigid restraints(Kc,,), 99% (K95 %) and 95% (K95%) spring-stiffnesses of the spring to achieve 99% (Mer;99%) and 95% (Mer.95%) of MC/. when using rigid restraints (see Table 12, analysis types a thru c). This investigation will only be performed on a single load case and span. For other spans and IQad cases the K95% spring-stiffness is used (Table 12, analysis type c). Using the K 95% spring-stiffness will not lead to 5% reduction of the failure load. In this parameter study it will be shown if using the K95% spring-stiffness can be considered rigid or not (Table 12, analysis type d). This is performed to validate possible stiffness-requirements for use in codes.

Table 12 Analyses performed in parameter study and objective

Type Mer Spring-stiffness Objective Load case Span a. M er-Joo% K ", Reference I 7200mm b. M er; 99% K99% Determine influence of using K99% I 7200mm C. Mcr;95% K95% Detennine influence of using K95% All All d. McrjlOO% K95% Validation of using K95% as rigid restraint All All

First the load cases are given, then the FE-model and how the results are obtained is discussed. Then the elastic critical moment is detennined together with the K95% and K99% spring-stiffenesses. The results from OMNIA will be presented and compared to the analytical results .

~Xr ~

Figure 70 Coordinate system used in the determination of the location of the restraint

8.1. Load cases In the parameter study three load cases will be investigated (see Figure 71). The first load case on which a parameter study will be penonned is shown in Figure 71 a. For this load case three spans (7200nun, 5400mm, 3600mm) will be investigated. The concentrated load is applied at mid-span at the upper-flange web intersection. The location of the restraint is varied along the length and height of the system. As achlal rigid restraints can not be made in practice an elastic spring will be used. To set stiffness requirement for this spring the K95% and K99%

spring-stiffnesses will be detennined. This stiffness wiII be varied for the first span (7200mm); here K 95%, K99%

and Koo (rigid restraint) spring-stiffness will be used in OMNIA. For the two remaining spans only K95% will be used in the OMNIA. This as no significant difference was found between the behaviour of the results from OMNIA using K 95%. K99% and K ", spring-stiffnesses. Furthetmore the spring-stiffnesses needed to reach M er;99%

are significantly larger (l0% to 600%) than the ones needed to reach M er;95%.

The second load case on which a parameter study will be performed is shown in Figure 71 b. This consists of an IPE240 with a length of 7200ml11 loaded with a uniformly distributed load applied on the upper-flange web intersection. The location of the restraint is varied along the length and height of the system. The third load case on which a paranleter study will be performed is shown in Figure 71c. This consists of an IPE240 with a length of 7200m111 clamped at one end and simply supported at the other loaded with a concentrated load applied on the upper-flange web intersection. The location of the restraint is varied along the length and height of the system. In the OMNIA K 95% spring-stiffnesses are used.

43

Parameter study

~

Fi A

JrV 'hI "v 'hI

;t 'hI 'hI

"'~ '111£

(a) load case 1 (b) load case 2 (e) load case 3

Figure 71 Load cases used in the parameter study

B.2. FE-model Three load cases will be studied, each of these load cases have different loading and support conditions. The material law and the residual stresses are identical to the previous numerical investigations. Therefore only the support and loading conditions and the system imperfection are discussed here. This is done in more detail in Appendix E (Models 1,2 and 3 in §E.4)

8.2.1. Loading and support conditions

In the parameter study two types of support conditions have been modeled, one being a simple support, the other being a clamp without the restraining of warping. The concentrated load and uniformly distributed load have been applied as was done in §4.1.1.


In the previous investigations the system imperfections and their amplitude have been varied. It has been observed that using ~he first buckling shape of the unrestrained system gives the most consistent results. To compare results consistent results are favourable. Each of the restraint locations will result in different and distinctive buckling shapes (see Figure 74), the amplitude of the impelfection must be varied to obtain comparable results. This will always cause the imperfection to vary from one simulation to another. Therefore it has been decided to constantly always apply the first buckling shape of the unrestrained system, and therefore always use the same imperfection shape (small differences apply due to the location of the stiffening-plates used to introduce the load and the attachment of the restraint). For the load cases I and 2 the imperfection shape is shown in Figure 72a, for load case 3 the imperfection shape is shown in Figure 72b. In this figure it is observed that an asymmetrical pattern is applied.

(a) Load case I and 2 (b) load case 3 Figure 72 Imperfection shape applied to simulations for load cases 1 and 2

B.3. Determination of results

8.3.1. Numerical simulations

From each simulation two loads are calculated; using LBA the elastic critical load is obtained, using OMNIA the failure load is obtained. Using the elastic critical moment the slendemess ratio is obtained using eq (8.1). Using the failure moment, the reduction factor can be calculated (8.2) . These values can be plotted in A-X diagrams along with the buckling curves. See also (§5 .3)

44

M M =MIX~X=-I( I( P . M

pi

(8.1 )

(8.2)


8.3.2. Comparison of analytical values and simulations

The comparison of analytical values and simulations is performed as follows. Thc 95% spring-stiffuess of the restraint is inputted in the finite element model; from this simulation a failure load is obtained. This failure load is rewritten as a failure moment. From this failure moment a reduction factor is obtained, thus obtaining the numerical reduction factor XOMNIA. Using the LTB-Tool the elastic critical moment using rigid restraints (Mer. JOO%)

and the 9S% value are obtained which are used in the determination of the reduction factor (XEC3) using the clauses of the Eurocode. Thus meaning that the failure load in the numerical simulation will be obtained using K95%

spring-stiffnesses and are compared to the results using M er.95% and M er; Joo%, Using M er;95% the clauses of the Eurocode will be validated. Using M er;Joo% the validation of whether K95% can be used as a stiffness requirement for the building codes to allow the designer to determinc whether the restraint may be considered rigid (K = 00).

8.3.3. Analytical force in restraint at failure

The analytical determination of the force in the restraint at failure will be determined using the clauses of the NEN6770, this as no clauses are given in the Eurocode 3. These rules have been described earlier in the report as the 1 % and O.S% rule (See §S.S.2) .

8.4. Parameter study As mentioned the parameter study consists of 3 load cases and a number of spans. As an example the different findings of this study is reported only for load case 1 with a span of 7200mm. For the other load cases and spans similar results have been found. The appropriate figures for these spans and load cases are found in Appendix K.

~

Fi

"v Ifzl v Ifzl Ji

v Ji

Figure 73 Load case 1

8.4.1. Elastic critical moment using rigid restraints (Mcr;100%)

The LTB-tool is used to compute the elastic critical moment. This has been evaluated for 5 locations along the height of the section (web-flange intersections, Centroid and halfway centroid to web-flange intersections).

In Figure 74 the influence of the restraint position on the elastic critical moment is shown. On the horizontal axis the location of the restraint is shown, whereas on the vertical axis the elastic critical moment is shown. Each of the plotted lines represents the application height of the restraint. To show the complexity of the buckling shapes four are randomly illustrated. The location of the restraint over the height of the cross-section is dependent on the value of ho (height between centroid of flanges) for an IPE240, ho = 230.l (see Appendix D). Using the coordinate system, the upper flange web intersection is + 115. 1, the centroid equals 0, and the bottom flange-web intersection is -IIS.I. These values will be used in the subsequent chapter to denote the location of the restraint over the height of the cross-section.

It can be seen that applying the restraint at the bottom flange (-IIS.I) has no significant influence on the elastic critical moment. The samc applies for when the restraint is located halfway from the bottom flange to the centroid (-S7 .55). More influence is seen when the restraint is applied at the centroid and up. The greatest influence is seen when the restraint is located at mid-span and when it is applied above the centroid .

45

180

160

E z =. 140

'E Q)

120

E 0 100 E Cii 80 u ,., '5 60 u ~ 40

'" iii 20

0

Parameter study

=

0 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 7200

PositJon of restraint over length of system (x)

M cr;/ OO%

Height of restraint on crosssection

-115.1 IT - 57.55---+

-0-

- -57.55-+"

--115.1

Figure 74 Effects of rigid restnint location on the elastic critical moment for load case 1 1= nOOmm

For other spans the similar behaviour is found. For load case 3 no second buckling shape is found. See Figure 205 thru Figure 208 (Appendix K, pages 133 and 134). For shorter spans the influence of placing the restraint at and below the centroid will not have large effects, for longer spans placing the restraint above the centroid can have large influence

8.4.2. K95% and K99% spring-stiffnesses of the restraint In Fi.gure 75 the required values of the spring-stiffnesses are shown; here it can be seen that the spring-stiffnesses needed greatly increase when restraints are positioned in proximity of the supports. Another phenomenon occurs at mid-span, where the spring-stiffness either rises or drops depending on the location of the restraint. To quantify the behaviour of the restraint location and its influence on Mer and the spring-stiffness needed to achieve a percentage of Mer is done by applying a rigid restraint at mid-span and varying the restraint location over the height. This will show the influence of the restraint location on the elastic critical moment. After which the springstiffnesses needed to reach a percentage of the elastic critical moment will be investigated.

30000

25000

E i 20000

~ 15000 tE :; g> 10000 .~

f/)

5000

0

/ - 7200mm

~"~-------------------------'--------~------~. ' . , 't , t •

" '.: :.' : • I , . ' • • ' . , t , · '. " , · ' . . ' ,-

I , 11. I ' , . \ , " . \. .", I 1 , ' I

',' " -~ : .. . - , ... .. ., ',,' , " .. ,. , to """ ..... .... ~ . .. _. ,: •• ", , ... ..... . "' ....... • _ .. ·."'N'

... ... . .. .. ,. .......... .. ...... ... , .. , . .... ... - .. .... .. ~ ... -. ..... --.. -, ,


• - - -115.1 • • • • 57.55 • - - - 0

~ :::::~~;~ --- 115.1

~ --57.55 --0 ---57 .55

0 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 7200

Position of restraint along length of system (x)

Figure 75 K95% and K99% spring-stiffnesses for I = nOOmm

Varying the location at mid-span of the rigid restraint along the height of the cross-section will cause a point where the second buckling shape is the lowest goveming elastic critical moment and applying a restraint at a location above this point has no influence on M"".. This is visualized in Figure 76. The same can be seen in Figure 74 where on two locations at mid-span the same value of M er is obtained.

The spring-stiffhess needed to achieve M er; /oo% is highly dependent on the governing buckling mode. When for the position of the restraint the second buckling shape is the lowest governing shape the spring-stiffness needed drops, for the first buckling shape the high stiffnesses needed of 109 N/mm is the spring-stiffness applied in the LTBTool as rigid . To achieve Mcr;95%different behaviour is found , here the spxing-stiffness needed rise up to where the buckling shapes change from the first to the second. The spring-stiffnesses needed increases for the percentage that needed to be calculated. This can be seen in Figure 77, where for a number of percentages of Mer when using rigid restraints the spring-stiffness has been determined . Therefore leading to conclude that the increase of

46


restraining needed to reach a certain percentage is under influence of the lowest govcrning buckling shape. As can be seen in Figure 75, two application heights of the restraint show a decrease of the elastic critical load. These are the two application levels for which at mid-span the second buckling shape is effective.

Upper flange centroid

115.1.---.--------..,.---~--_;_--,..._----.,....,...--_,

E oS 86.325 :E '" ~ 57.55

'" c 0 a; 28.775 c: 'E 1ii 0

frecond buckling . [:~e!' __ :~I~: -I ------------------------. :~:~~uckling

+--~~--.--_r--~-~~--r_-~--_r--~ ~ f 4.0 60 100 140 160 1 0 o 5 -28.775 ~ u .3 -57.55

-86.325

_115.1-'--------'--''--_________ ''--_---' __ --'-__ -' Lower flange centroid

Elastic critical moment [kNm]

Figure 76 Effect of rigid restraint position over height of section on the elastic critical moment

:E '" ~ 57.55

-86.325

-115.1

115.1 Upper flange centroid

E 11 ~ l i . II : II I I I ,III II, oS 86.325-1-+-"-i+'l.~--'--++m+lt__t+n+ilt_-""_,_'+Hltl__+_"'++tr.It-+__+_:+"11t1--_+_H++I+!I

I I ~ I II I II II! r:Second bucklin L I I ~ II i I I I I I 1 II I shape

~ T i i- - - i it I I III i I I. I II I I Fi~t-;'~c~ing a; 28.775 -I-+-t+h-!tlt--++++Hlt_+-tft+l'rlt--+-H-r'-fJ"t--HflH-l+Ht_t-i-l-++Ht--+f-l++HII

t 0 ceUi~ II : I I I ;1 I I III II I I \ V s~ f .f 02 \1' ,' 03 1.Ii J II ~ II' '011 . 11111' ;11 Ii ' 07 1 11" 'of.. 1 09 '5 I I I I / I 05 i i' ~ r1 r If I Springstiffness

~ -2B.775~i+iint-/,Y-1+'tlfI ,H+-t-H1 -'-rf1J ;t---,---t-tl 1+mi111l,IItf+-ri-jl +Htl11+-+-+++mlltHli---+H+lIlll #% of Mer

.3 -57.55 . I I I I l i I lM I 11 I~ -95%

!,I I I I: I' ~ II I I ~ I I: I =::~9% l1 II ! I1I11 1111' I 11111 ! I I 11' I =~::'99%

Lower nange centroid Springstiffness [Nmm]

Figure 77 Spring-stiffness of the restraint needed to achieve a percentage of Mer

For the other two spans and the other two load cases similar behaviour is found . The figures for these are found in Appendix K (Figure 209 thm Figure 212, pages 134 thm 135).

8.4.3. 'Load bearing capacity using K95% spring-stiffness

In the GMNIA the restraint location is varied along the length and the height of the system. A limited number of simulations are perfomled due to computational effort. The location of the restraint is varied over 5 locations over the height of the section. These locations are identical to the ones used in the computation of the elastic critical moment. The locations over the length of the system are varied over 6 locations (from the support up to midspan). The computation of the slenderness and the reduction factor is performed according to the procedurcs illustrated in §8.3 .

For each of the restraint locations the ultimate load has been detemlined. These are shown for load case 1 with a length ofnOOmm in Figure 78 (Tabular results are given in Appendix K). lt can be secn that thc influence of the restraint can have a large influence on the failure load. Placing the restraint below the centroid will not lead to large increase of the failure load . Placing in at and above the centroid can have large increase of the failure load. In Figure 79 the results obtained using GMNlA are plotted along with the governing Eurocode 3 huckling curves. It can be observed in this figure that the simulation performed show consistent behaviour; this is expected as consistent system imperfections have been applied to the FE-model.

47

50

45

40

35

30 1

! 25 ::I u.

20

15

10

5

0

600

Parameter study

1=7200 [mm)

1200 1800 2400 3000

Restraint position along length of system (x) [mm)

Figure 78 Ultimate load obtained from GMNIA for load case II = nOOmm

I = 7200mm

3600

__ 115.1

_ 57.55 _ 0

_ .07.55 _-115.1

-Unrestrained I

1.2~----------~~--------------------------~==~l-~~~~ ..... --General Method Cur-..e A

.......... --Specific method cur-..e B - - - - - - - - - - - - - - - - - - - - - ......- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Euler CUM

: ............ - - - -Mpl ..... ..... • +115.1

..... , . ~~ ..... • 0

0.8 r=~· ::::!::::::=::-r--.. -< ' ..... ...... • -57.55

• -115.1 ~ 0.6 o

X

0.4

0.2

o +------,----.-------.------,--------r.-------r------r------~

0.7 0.8 0.9 1.1

AOMNIA

1.2 1.3 1.4 1.5

Figure 79 Graphical representation of results from GMNIA for load case II = nOOmm

For the other two spans and the other two load cases similar behaviour is found. The figures for these are found in Appendix K (Figure 213 thru Figure 220, pages 136 thm 138)

8.4.4. Force in restraints

From the OMNIA the force in the restraint at failure is obtained. Analytical values have been obtained using the 1% and 0.5 % rule of the NEN6770 (see §5.5.2). For load case 1 with a length of 7200mm the results are shown in Figure 80 and Figure 82 and in tabular form in Appendix K. When the restraint is applied near proximity of the support or at mid-span the force of the restraint rises. No distinction can be seen between the forces in the restraints at failure due to the second buckling shape. This is caused by the imperfection shape which has been applied according to the first buckling shape.

48

12

10 Z i'E. ~ .2 8 j2 1;;

" E 6 ~ u ;;; .. Gi 4

-= ~ 0 u. 2

0

600


1 - 7200ml1l

1200 1800 2400 3000

location 01 restraint along length of system x [mm)

3600

1ti.1

&;1, .. -+-NEN 67711% rule 115.1

__ NEN67711% rule 57.55

-.-NEN 67711% rule 0

__ NEN 6771 1% rule -57.55

___w__NEN67711% rule-115.1

__ NEN 67710.5% rule

__ K95_115.1

-+-K95_57.55

-+-K95_0

-+-K95_-57.55 __ K95_-115.1

Figure 80 Force in restraint at failure for 95% for 1= nOOmm

TU/e

It can also be seen that the force in the restraint is not the greatest where would be expected (upper flange-web attachment) but at the centroid (0) and at midpoint between centroid and upper flange-web attachment (57.55). The force in the elastic restraint is dependent on the scheme shown in Figure 81 . The load F has an arm a to the rotational centre, the elastic restraint has an ann b to the rotational centre, the force in the restraint is dependent on the displacement u (F=k u). Due load F a moment Fa occurs, this moment causes an additional or negative force in the restraint of Fa/b, which in his tum causes an additional or negative displacement. The force Fa/b is the smallest at the flange-web intersections; here it causes a positive inward force and lowering the force or causing a negative outward force and increasing the force. However the most influence is found at the centroid of the web (b ~ 0) the force then becomes infinite. This therefore leads to higher forces in the restraints at the centroid and at mid-point between centroid and upper-flange. The arm a in its turn is influenced by the displacement lI, which is dependent on the spring-stiffness of the restraint. This is also varied in this case.

F

u Elastic restraint

Rotational b center

Figure 81 Scheme to illustrate influence of restraint location

The force in the restraint for the 99% spring-stiffness shows the same behaviour as that of the 95% springstiffness. It can also be seen that the force in the restraint at failure is also dependent on the spring-stiffness. It cannot be said that the force at failure for the 99% stiffness is always greater than the one obtained with the 95% spring-stiffness and vice versa. However in both cases the clauses of the NEN6770 do not give a good prediction of the force in the restraint at failure.

49

12

10 Z ~

f .::! 8 ]! ... E £ 6 '" f 0 .., '" .. "ii 4 .: CD l::! 0

LL 2

0 600

Parameter study

/ = 7200mm

1200 1800 2400 3000

Location of restraint along length of system x [mm]

3600

115.1

-+-NEN 67711% rule 115.1

___.__ NEN 6771 1% rule 57.55 __ NEN 6771 1% rule 0

__ NEN 6771 1% rule -57.55

___ NEN 6771 1% rule -115.1

___ NEN 6771 0.5% rule

-+-K99_115.1 __ K99_57 .55

-t-K99_0

-+- K99_·57.55 __ K99_-115.1

Figure 82 Force in restraint at failure for 99% for I = nOOmm

For shorter span (load case 1) the rise and drops of the failure force will tend to only drop. For other load cases similar behaviour is found. This is shown in Appendix K (Figure 221 thru Figure 224, pages 138 thru 139).

8.4.5. Comparison of numerical results to EC3

The I-esuits from GMNIA will be compared to the EC3 using three different clauses. The first is the general method, the second is the specific method without correcting for the moment distribution and the third is the specific method with con·ection for the moment distribution. Only the results obtained with K95% are used. Each of the figures contains two graphs, graph (a) is used to validate the clauses of EC3, graph (b) is used to determine if K95%, may be used as a lower bound spring-stiffness of the restraint. In contrary to the previous sub-section each of the different load cases is reviewed here.

8.4.5.1. General method

In Figure 83a thru c the results from GMNIA and the results obtained using the general method of the EC3 are compared. On the horizontal axis the reduction factor obtained using the appropriate percentage of M er (Mer;lOO%

Mer:99% and M er,95%) and the clauses of the Eurocode are shown (Table 12, analysis type a thru c). On the vertical axis the results from the numerical simulation are shown using the appropriate spring-stiffness (K"" K99')., K95%)'

Here it can be seen that the majority of the results lead to an underestimation of the reduction factor. For 'X.EC3 < 0.7 this underestimation is small; for 'X.EC3 > 0.7 this underestimation becomes larger. The numerical results show a reduction factor greater than 1, this is caused by the use of the stiffening plates app'lied at the load inh-oduction.

In Figure 83d the results from GMNIA and the results obtained using the general method of the EC3 are compared. On the horizontal axis the reduction factor obtained using M er:JOO% and the clauses of the Eurocode are shown. On the vertical axis the results from the numerical simulation are shown using K95% (Table 12, analysis type d). This is performed to validate if K95% may be used as the lower bound spring-stiffness to declare a restraint as rigid . Here it can be seen that a slight shift of the results is obtained. Here the general method still does not yield to any large overestimations (+5%) of the reduction factor.

Similar figures have been made for load cases 2 and 3. These are given in Figure 84 and Figure 85. Here the same observations can be made as for load case 1.

50

I 0.8

J 0.7

< ~ 06

" 0.5

0.4


0.9

!1 o.8

·1 0.7 1

~ 06

" 0.5

0.4

Safe •

0.3 ~-~~_=:::::::::;===--~--I 0.3 ~---..---!::====::=::::;_~-.-I 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3

(a) Koo and Mcr;Joo% for / = 7200mm (b) K99% and M cr;99% for I = 7200mm

1.1 .----------------~ 1.1 r:o-:------- ----------,i

,4 09

"'I 0 .8

:5 0 .7

~ " 0.6

1"<

0 .5

0.4

.115.1

B X 57.55

E .. 0 .~ ~ :0: -57.55 (l tr

• -115.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.1

d 09

"I 0.8

:5 0 .7 z ::;:

;: 0 .6

0.5

0.4

0.3 0.4 0.5 0.6 0.7 0.8 0 .9

y Mcr;JOO% "'ECl __ _

(c) K95% and M cr;95% (d) K95% and Mcr;Joo%

1.1

Figure 83 Comparison or GMNIA to analytical results using the general method load case 1

q

r I i r Safe

TU/e

.115.1

.B ~

X 57.55

II AO .G ~ :0:-57.55 Ki tr

.-115.1

.115.1

-I x 57.55 12 .j;; A 0 f" ji :0: -57.55 tr

.-115.1

q

I i 0.9

l ,e: .t

l ,e: ,I' 0.9

,4 0.8

"I 0.7

~, 0.8

0.7

< ~ 06

"

.115.1

B 1l X 57.55

< ~ 0,6 ·

"

.115.1 c

~ X 57.55 12 :l! AO g Ki :0: -57.55 tr

1"< 0.5

<5%

12 1,: .. 0

~ " :0: -57.55 tr

0.5

0.4 • -115.1

0.4 • -115.1

0.3

0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 O.S 0,7 0.8 0.9

XEC3 XECJ A1c:r: IOO%

---.... ---....

(a) K95% and Mer, 95% (b) K95% and Mcr;Joo%

Figure 84 Comparison or GMNIA to analytical results using the general method load case 2

51

0.9

~ 0.8

0.7

-<

~ 0.6

'" >< 0.5

0.4

0.3 0.4

§ 1ii II .~ ~ ~

'"

0.5 0.6 0.7 0.8 0.9

Parameter study

.115.1

X 57.55

. 0

x -57.55

.-115.1

0.9

""~ 0.8

'I 0.7

-< ~ 0.6

'" >< 0.5

0.4

0.3 0.3

Stocky

Slender

0.4 0.5 0.6 0.7 0.8

~O M cr: 100 %

(a) K95% and Mcr;95% (b) K95% and Mcr;100%

0.9

Figure 85 Comparison of GMNIA to analytical results using the general method load case 3

8.4.5.2. Specific method without moment distribution correction

.115.1

.~ X 57.55 II .~ . 0

" '" ~ x -57 .55

'" .-115.1

In Figure 86a the results from GMNIA and the results obtained using the general method of the EC3 are compared. On the horizontal axis the reduction factor obtained using M er.95% and the clauses of the Eurocode are shown. On the vertical axis the results from the numerical simulation are shown using K 95%. Here it can be seen that the majority of the results lead to a large (+5%) overestimation of the reduction factor; for XE C3 < 0.75 this overestimation is large; for XEC3 > 0.75 this overestimation becomes an underestimation. The numerical results show a reduction factor greater than 1, this is caused by the use of the stiffening plates applied at the load introduction.

In Figure 86b the results from GMNIA and the results obtained using the general method of the EC3 are compared. On the horizontal axis the reduction factor obtained using M er; 100% and the clauses of the Eurocode are shown. On the vertical axis the results from the numerical simulation are shown using K95%. This is pelformed to validate if K95% may be used as the lower bound spring-stiffness to declare a restraint as rigid. Here it can be seen that a slight shift of the results is obtained.

Similar figures have been made for load cases 2 and 3. These are given in Figure 87 and Figure 87. Here the same observations can be made as is done for load case 1.

1.1

safe

~0.9 , 0.8

-< 0.7

~ '" 0.6

><

0.5 +5%

0.4

0.3 · 0.3 0.4 0.5

, .. 360P ll\111 I:

Stocky

$lender

0.6 0.7 0.8 0.9

XEC3 f.,1 cr;95 %

1 F 1.1 r---- ---,------- --7!

~~~-X-71~i~l-a~t =-~~~

.115.1

~ X 57.55

1l 10 . 0

~ I: x-57.55

'" • -115.1

1.1

f ·g "'] 0.8

-< 0.7 ;Z ::< '" 0.6

><

0.5

0.4

0.3

.115.1 c ~ X 57.55

1l

x-57 .55

. -11 5.1

0.3 0.4 O.S 0.6 0.7 0.8 0.9 1.1

XEO M cr:/ OO %

(a) K95% and Mcr•95% (b) K 95% and Mcr;100%

Figure 86 Comparison of GM.NIA to analytical results using the specific method without moment distribution correction. Load case 1

52


q q

0.9

1 J l 1 ,r: I Ie-- ,1'

1 J l 1 ,r: I Ie-- ,1'

0.9

~ 0.8

0.7 ~1

0.8

0.7

< § 0.6

'" ,.., 0.5

.115.1 c

~ x 57.55

.l! C .. 0 ~ ~ X -57.55

<

~ 0.6

'" ,.., 0.5

.115.1 c

~ x 57.55

.l! C ' 0 ';; '.;: X-57.55 " a:

0.4 .-115.1 0.4 .-115.1

0.3 fZ-~---....!::;:::=::;::::==:.--~--l 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9

'XED M cr;95 % XED M cr; l 00 0;.

(a) K95% and Mcr;95% (b) K95% and Mcr, IOO%

Figure 87 Comparison of GMNIA to analytical results using the specific method without moment distribution correction. Load case 2

0.9

d 08

~ I 0.7

0.5

0.4

.115.1

~ x 57.55 " .l! .;;; . 0 ~ !1 x-57.55 a:

.-115.1

0.4 0.5 0.6 0.7 0.8 0.9

0.9

d 0.8

"I 0.7

<

§ 0.6

'" 0.5

0.4

0.3 W--~--!--....!::;:::==:;:::=::;.......-~~ 0.3 0.4 0.5 0.6 0.7 0.8 0.9

~CJ Mcr;/oO %

(a) K95% and M cr;95% (b) K95% and Mcr,IOO%

.115.1

~ x 57.55 '" .l!

"I . 0

8! X -57.55

• -115.1

Figure 88 Comparison of GMNIA to analytical results using the specific method without moment distribution correction. Load case 3

8.4.5.3. Specific method with moment distribution correction In Figure 89a the results from GMNIA and the results obtained using the general method of the EC3 are compared. On the horizontal axis the reduction factor obtained using M cr;95% and the clauses of the Eurocode are shown. On the vertical axis the results from the numerical simulation are shown using K 95%. Here it can be seen that the majlority of the results lead to a large (+5%) overestimation of the reduction factor; for XEC3 < 0.9 this overestimation is large; for XEC3 > 0.9 this overestimation becomes an underestimation. The numerical results show a reduction factor greater than 1, this is caused by the use of the stiffening plates applied at the load introduction.

In Figure 89b the results from GMNIA and the resu Its obtained using the general method of the EC3 are compared. On the horizontal axis the reduction factor obtained using M cr. JOO% and the clauses of the Eurocode are shown. On the vertical axis the results from the numerical simulation are shown using K95 %' This is pelformed to validate if K95% may be used as the lower bound spring-stiffness to declare a restraint as rigid. Here it can be seen that a slight shift of the results is obtained.

I

Similar figures have been made for load cases 2 and 3. These are given in Figure 90 and Figure 91. Here the same observations can be made as is doue for load case 1.

53

1.1 r::--:---~--------::------:>I

~ 09

""I 0.8

< 0.7 :z ;;:

.: 0.6

0.5

0.4 SIOCky I

Slender

0.4 0.5 0.6 0.7 0.8 0.9

XECl

M c/';9J%

c .Q 1i .k! :~ ~ ~ 0:

1.1

Parameter study

1.1 r Sa=--fe---------------:>I

.115.1

X 57.55

A D

X-57.55

.-115.1

~ 0.9

~

" 0.8

< 0.7

~ <:! 0.6

N

0.5

0.4 Siocky

Slender

0.4 0.5 0.6 0.7 0.8 0.9

XED M cr;JOO %


.115.1 c B B X 57.55

.E . 0

~ & X -57.55

.-115.1

1.1

Figure 89 Comparison of GMNIA to analytical results using the specific method with moment distribution correction. Load case 1

q

0.9

i I i ! ,~ ,1'

0.9

"1 0.8

0.7 "1 0.8

0.7

:s ~ 0.6 <:! ,..,

0.5

.115.1

.§ X 57 .55

B A D

i ~ X -57.55 0:

<

~ 0.6 <:!

N 0.5

.115.1

B X 57.55 .. ~ A O

1 ~ X -57.55 0:

0.4 .-115.1 .-115.1

0.3 f£::"-~--~:::==::;::==::.--~-~ 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9

XnC3 M"r;95 % XECJ

M cr: JOO%


Figure 90 Comparison of GMNIA to analytical results using the specific method with moment distribution correction. Load case 2

"1 < § <:! ,..,

54

0.9

0.8

0.7

0.6

0.5

0.3 0.4

.115.1 c ,g X 57.55 .. .!l .E A D .. '" ~ X -57.55 0:

.-115.1

0.5 0.6 0.7 0.8 0.9

0.9

~ 08

"'I 0.7

-<

~ 0.6 · <:!

N

0.5 ·

0.4

Safe

Siocky

Slender 0.3 .. ~---~...::::;::::=:;:::=:._-~~

0.3 0.4 0.5 0.6 0.7 0.8 0.9

XECJ ,wcr'; JOOO/O

(a) K95% and M Cr;95% (b) K9.5% and Mcr,IOO%

.115.1

g X 57.55 '" ~

4 0

~ ~ X -57.55 ~ 0:

.-115.1

Figure 91 Comparison of GMNIA to analytical results Llsing the specific method with moment distribution correction. Load case 3


8.4.5.4. System effect with use of the general method for statically indeterminate structures

Load case 3 is a statically indeterminate struCll.lre. Procedures can be followed to incorporate system effects.

The plastic failure load can be obtained using the first order plasticity theory. 6Mp' Fp, =-1-

The elastic critical load is calculated as:

F = 16Mcr

CI' 31

(8.3)

(8.4)

Where as for statically detenninate slmctures a linear relationship between the plastic moment and force is valid where:

~PI ftPI A= -=-Mer F::r

(8.5)

For statically indeterminate structures however: r----

6Mp'

,<- JF", - -1- - ] JM", * JM" - Fel' - I 6 Mer - 2J2 Mer Mer

(8.6)

31 For statically determinate structures the numerical reduction factor is obtained as:

Mu P" Mu =Mp'X-" x= M =P

pi pi

(8.7)

For statically indeterminate structures however:

16M"

x=~= 31 = 8M" =t M" Fp' 6Mp' 9Mp' Mp,

(8.8)

1 Using the plastic force and elastic critical force leads to the results shown in Figure 92b; where for convenience the results from the standard approach is given in Figure 92a. It can be seen from eq (8.6) that the slendemess increases and therefore the reduction factor will decrease. The same applies to the reduction factor determined using GMNIA and (8.8). Here it can be seen that the general method gives an accurate estimation of the failure load. The clauses of the Eurocode do not describe this procedure; therefore the standard approach will be followed in the remainder of the comparisons. More research should be carried out using this approach in the failure of statically indeterminate structures.

0.9

"rl 0.8

'I 0.7

:!i ~ 0.6

'" >< 0.5

0.4

0.9

~ 0.8

" 0.7

< Z

0.6 ::;:

'" >< 0.5

0.4

0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(a) without syslem effecl identical to Figure 85a (b) with system effects Figure 92 Comparison of GMNIA to analytical results IIsing the general method

• 115.1

x 57.55

.. 0

:0: -57.55

• -115.1

55


9. Development of a design rule to determine the strength requirements of restraints

In the parameter study performed in chapter 9, the restraint force at failure has been obtained. It was shown that in the majority of the studied cases, the current rule in the NEN6770 leads to unsafe results . A new design rule is therefore needed. In this chapter the current design is regarded first, a proposed design rule is given.

9.1. Current design rule The current design rule for strength requirements of restraints is found in NEN6770 (art. 12.2.4.2) [e2]. This design rules is named the percentage rule. The rule represents a percentage of the force in the compression flange including the root-radii and 113 of the web taken between the root-radii. The I % rule (9.1) is calculated using the largest stress obtained in the section, and for the 0.5% rule (9.2) the yield strength is taken. The largest value according to equations (9.1) and (9.2) should be used.

Nsp =O.OIAflO"J;s;d

Nsp = 0.005Afl il";d

(9.1 )

(9.2)

9.2. Proposed design rule The current design rules are based on the percentage rule. These are easy to use and therefore will be altered according to the obtained results. The current design rule does not regard the location of the restraint.

For each of the spans and load cases the location of the restraint has been altered in steps of 1112, and has been taken over 5 locations over the height of the cross-section. Load cases I and 2 are symmetrical therefore the locations of the restraint have been taken up to mid-span. For load case 3 this was done for the complete span.

Each of the restraint locations are grouped such that 6 horizontal restraint locations will be found, effects of asymmetry are neglected for load case 3 as too few results are available using asymmetric load-cases. For each of the 6 groups the maxima have been obtained for the each application heights of the restraint. This has been done as the sample size is considered to be too small (upper probability values can also be used when more results have been obtained).

Using these values two approaches can be followed. A 4th order polynomial curve fit can be applied and an equation for the percentage of the force in the restraint is obtained. The second approach is a simplified fit, where two regions are applied. The inner half span from 114 to 3114, and the region between the supports and 114. For the inner half of the beam a constant will be calculated. For the outer quarter-span regions a linear fit is performed. This is shown in Figure 93. Only the bilinear design rule will be proposed as it is felt that a 4th order polynomial equation does not fit the teon "ease-of-use".

1/21 1/41

Bilinear design rule

1112/ 1/4/ 1/2/ 3/4/ 11112/

Fligure 93 Design rule forms

57

Development of a design rule to determine the strength requirements of restraints

The bilinear design rule consists of three regions: support to ~ span, ~ to 14 span and 14 span to support. The two outer regions will consist of a linear equation; the ilmer region is represented by a conslant(9.3).

x x For -'-- < 0.25 -A -'-- + A

/ 'I / "'2

x Nsp = For 0.25 ~ -;- ~ 0.75,A3

Each of the coefficients can easily be determined. First coefficient A.I should be detemlined; thjs is the plateau between //4 and /3/4. The maximum value of the percentage of the compression force at failure at this p,lateau should be taken as A3.

(9.3)

A3 = maximum %; For / /4> x < /3/4 (9.4)

The coefficient A I denotes the slope of the linear equation between x = 0 and x = //4. The maximum value of the percentage at x = 1112 is taken; the slope A I can be determined as:

(Max(% at /112) - A)) AI = (1112 -114) (9.5)

The coefficient A2 is determined according to: (9.6)

The coefficient A4 is determined according to: (9.7)

These coefficients can be determined for each of the non-dimensional vertical locations of the restraint. Based on the current data the following constants may be applied .

endent on the vertical position of the restraint

(2~, ) AI A2 A3 A4

-0.2 0.076 0.026 -0.124 0.5 -0.228 0.091 0.034 -0.137 0 -0.197 0.087 0.038 -0.110

-0.5 -0.128 0.060 0.028 -0.069 -1 -0.014 0.021 0.017 0.007

Using these coefficients the results are shown in Figure 94. In case inefficient locations of restraint near the supports are neglected together with restraints at the tension side, the percentage rules will result in a percentage of2.6% for attachment at the upper-flange, 3.4% at half the distance upper-flange to centroid and a percentage of 3.8% when applied at the centroid.

58


10% .. • 115.1 :S Z • 57.55 9%

~ 2:

" 8% -

E ... 0 0

~, <l:: II x -57.55

7% :l

~." \j ~ -115.1

6% :.:

~,,~ 5%

--NEN6770

~~~ 4% " --1

~ '" "2 3% ... c --0.5

'00 '" ~~ • • OJ

--0 2% -

"0 "0

X '" III i <.::I ---0.5 1% x 0.

i I - E ~ :.: Ui ---1

0% 0.000 0.083 0.167 0.250 0.333 0.417 0.500

Figure 94 Performance of proposed design rule and the current design rule

59

----- --


10. Discussion

10.1. Modelling of structural shapes using shell elements In chapter 3 and Appendix E the Finite Element Model and the steps in calibrating have been described. Shellelements were used to model the cross-section in FEM. Shell-clements require far less computational time compared to solid elements and therefore shell-elements are preferred in this research. In doing this the geometrical propeliies of the FE-model will not match that of the actual cross-section. The root-radii are ignored and they have large influence on the torsional properties. In [R2, 3] and the present document compensation elements have been added to compensate for the error made in the model. To be able to make choices in the compensation method, elementary behaviour of the model has been investigated. These investigations can not be performed on actual sections. So the behaviour of the compensated models is compared to that of uncompensated models. However due to computational limitation and time no elementary non-linear behaviour of sections using solid models has been perfonned and these would be preferred as these offer the most accurate description of the section. It is therefore recommended that these elementary non-linear tests should be performed on models with solid-elements. Chapter 11, recommendation 1.

10.2. Validation of numerical model To validate the numerical model two experiments that have been perfonned in earlier research were provided. However no verification of the model could be performed as a large number of essential data (imperfection size and shape) were omitted in this research. Therefore mere comparisons of behaviour could be perfonned. Also a number (stiffening plates, etc) of alterations have been perfonned to the FE-model to eliminate local "numerical" effects to produce reliable results. Therefore additional experimental research should be performed to validate the numerical results and the boundary conditions hereof should be taken as those used in the numerical model (fork supports, stiffeners, conservative loading, restraints, etc) and vice versa (material law, imperfection shape and size, etc). Chapter II, recommendation 2.

10.3. Effect of restraint location In the previous chapter an investigation has been perfonned on the influence of the position of elastic lateral restraints on the elastic critical moment and load bearing capacity. For the influence of the restraint location on the elastic critical moment few comparative results have been found. An internal TC85 document has been made available. In this document similar graphs have been presented, however here a different span and cross-section have been investigated. Therefore only a qualitative comparison can be pelformed. In this comparison it has been observed that the expected behaviour of the elastic critical moment and the 95% and 99% spring-stiffnesses show similar behaviour as for the earlier perfonned cases here. For the influence of restraint location on the loadbearing capacity no literature has been found.

In the three load cases which have been investigated in the parameter study, the restraint had the most influence when applied at mid-span and at the upper flange-web intersection; the most favourable location as it increases the load-bearing capacity the most. Depending on the span, the influence of the restraint can increase the capacity by a factor of2.5. Other less favourable restraint locations do have significant influence on the load bearing capacity of the considered system. When applying the restraint above the centroid and near mid-span (±//6) large increase of the load bearing capacity of the system can be obtained. Other locations also have influence but do not have any significance. Chapter 11, conclusion 1.

10.4. Load bearing capacity using the clauses of the Eurocode

10.4.1. Unrestrained beams

For unrestrained beams thc determination of the load-bearing capacity has been thoroughly investigated by others. However for the small number of simulations that has been perfOlmed in chapter 5 worrisome results havc been obtained on the validity of the specific methods. In [R2, P 15] the load bearing capacity has been also determined for a number of unrestrained beams (see Figure 95), here identical results can be found . The specific mcthod will always lead to smaller reduction factors and it can be concluded that tbe specific method can give a large overestimation of the load-bcaring capacity of umestrained beams . Chapter II, conclusion 2.

5 Technical committee 8 Stahility of the "Bouwen Illet staal"

61

Mb•Rd

Wy ' fy

i

Discussion

1.0

0.8

0.6

0.4

0.2

o IPE 160. upper flange 6. 1 PE 160, cen troid II IPE 500, upper flange ... IPE 500, cenlroid

- cross-s~tion capacity - elastic critical buckling - buckling curve 'u' - buckling curve 'b'

0.0 'i ii

0.0 0.5 1.0 1.5 2.0 2.5

-----. JWy ' fy

Mer

Figure 9S Results for unrestrained beams taken from [PIS)

10.4.2. Restrained beams

Varying the position of the restraint has been perfomled over length and height of the beam. K95% spring-stiffness has been used to obtain a finite spring-stiffness of the restraint for use in GMNIA. The Km. spring-stiffness was chosen to be validated whether it can be used as a lower bound spring-stiffness for restraints.

From the performed GMNIA analyses results have been compared using the clauses of the Eurocode 3. First a validation was performed of the results using the K95% spring-stiffness and using Mcr;95% The frequency of error made in the detelmination of the reduction factor is shown in Figure 96. Here it can be seen that the general method performs exceptionally well, a very small number of simulations will lead to unsafe results. The specific methods however will lead to possibly large (11 % and 15%) errors in the determination of the reduction factor. And therefore the general method may be used to detelmine the load bearing capacity when applying lateral restraints, Chapter 11, conclusions 3 and 4.

35

30

10

Unsafe Safe

• General method

• Specific method wilhout moment diagram compensation

o Specific method W'ith moment diagram compensation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Error In %

Figure 96 Frequency of error made in the estimation of the reduction factor using K95% and Mcr;95%

The same type of comparison will be performed to detelmine whether the K95% spring-stiffness may be used as a lower bound spring-stiffness to consider the restraint rigid. In Figure 97 the estimation elTor of the clauses is shown with respect to the frequency of the error made using K95% and Mcr;IOO%. It can be seen that the general method leads to acceptable results and will lead to a maximum error of 5%. Therefore it can be concluded that the K95% spring-stiffness may be used as a lower bound spring-stiffness of the restraint to consider it rigid in analytical determination of the elastic critical moment. Chapter J 1, conclusion 5.

In the Eurocode 3 a correction may be applied on the obtained reduction factor using the specific method. This is dependent on the moment distribution between "lateral restraints"; the definition of a lateral restraint is not given. In chapter 6 and 7 two interpretations of " lateral restraints" have been followed. From this investigation it is felt that the definition of "lateral restraints" should be defmed as full restraints. A lateral restraint as used in this research only restrains the displacement of the beam at a single point over the height of the section. Full-restraint fully prevents rotations and displacements at a point along the length of the beam, Chapter 11, recommendation 3.

62

l O

~ 20 ;

Lateral Torsional Buckling oflaterally restrained steel beams

Unsafe Safe

[] General metnod

• Specific method without moment diagram compensation

o Specific method with moment diagmm compensation

o

1::-+--+--4-.'-'_1.+ .... ~~j~~~~~. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Enorln %

Figure 97 Frequency of error made in the estimation of the reduction factor using K95% and Mcr;IOO%

10.5. Strength requirements of restraints

TUle

The scope of the research is to determine whether the clauses of the Eurocode can be used to determine the failure load. The force in restraints has been obtained from the GMNIA and has been compared to the clauses of the current Dutch code of practice (NEN6770). This has been done as a side step to determine the validity of these clauses. It has been found that on nearly all cases the clauses give a large underestimation of the force in the restraint at failure . Chapter 11, conclusion 6. The force in the restraint at failure is dependent on a majority of factors:

Imperfection shape (§5.5.2 and §6.4.2) Spring-stiffness (§5 .5.2, §6.4.2 and §8.4.4) Span (§8.4.4) Location (§8.4.4) Load case (§5.5.2, §6.4.2 and §8.4.4) Number of restraints (§6.4.2)

The percentage rules used in NEN6770 are easy to use. However these give a large underestimation of the load in the restraint at failure. To account for these fIndings a new design rule has been made. This design rule is dependent on four coefficients; however too few results have been obtained to create a valid design rule for all load-cases and spans. Therefore additional research should be performed using different cross-sections, spans, load cases and resh·aint locations. Chapter 11, recommendation 4.

It should be taken into consideration that applying restraints at unfavourable locations leads to higher stiffness and strength requirements and does not necessarily lead to large increase in load bearing capacity. And therefore the need for a design rule that incorporates all of the major factors in the restraint force at failure should be taken into consideration. Furthennore in the cun·ent research, elastic restraints have been applied which have an infInite capacity. Research should be perfonned on restraints with a finite strength; this as a 50% reduction of the strength of the restraint will not lead to a 50% reduction of the failure load see Figure 98. Chapter 11, recommendation 5.

50000

45000

40000

35000

30000

~ ~ 25000 0 u. 20000

15000 - / 10000

20 40 60

Disptacement [mm]

80 100

K = 500

3600 J600

7100

--In plane displacement u

--out of plane displacement v

- - - - Reaction force in restraints

120

Figure 98 Influence of 50% strength 011 the faillll·e load

63

Discussion

10.6. Procedures In thc course of the pelfonned work procedurcs havc been followed to determine thc cffects of the restraint location, restraint stiffness, and imperfection amplitude and its shape. It is to determine results for each possible imperfection shape and amplitude together with a specific restraint location and restraint stiffness. However this would takc large computational effort and lead to results that would be difficult to compare. In order to obtain comparative results rational choices have been made. By only varying the location of the restraint, and applying the 95% spring-stiffness comparative results have been obtained.

It has been found that the influence of the stiffeners which have been applied at the load introduction and at the restraint locations have an influence larger that has been expected (seen in stocky beams). These have been positioned to prevent distortion of the cross-section, local failure (see chapter 1) and peak stresses. A more elegant solution could have been used (beam elements, constraint equations) to prevent these local effects.

In the presented research a limited number of spans, load cases, restraint locations, material laws and crosssections have been investigated. To complete the work perfonned in this research, firstly more cross-sections should be investigated. Then load cases, spans and restraint locations should be increased. Chapter II, recommendation 6.

In the presented research only beams with lateral restraints have been taken into consideration. However a broader scope of research should be perfonned on the validity of the Eurocode 3 with respect to partial rotational restraints on beams, and the use of partial restraints on beam-columns. Chapter II, recommendation 7.

Furthermore the procedures followed in §8.4.5.4 to incorporate the system effects in statically indeterminate struchlres should be investigated for possible use with the clauses of the Eurocode 3. Chapter II, recommendation 8.

64


11. Conclusions and recommendations Based on the presented research the following can be concluded:

1. Restraints positioned at other than favourable locations can have significant influence on the load bearing capacity.

2 . Using the FE-model presented in chapter 5, the specific methods in the EN-1993-1-1 have not given satisfactory results for unrestrained beams.

3. The general method given in the EN-1993-1-1 may be used to determine the influence of partial elastic restraints.

4. The clauses of the EN-1993-1-1 using the specific methods give an unsatisfactory prediction of the numerical failure load in the investigated cases. Therefore they should not be used when dealing with partial restrained systems in bending.

5. The spring-stiffness of an elastic restraint to achieve 95% of Mer when using rigid restraints may be used as a lower bound spring-stiffness when using the general method given in the EN-1993-1-1.

6. The strength requirements for restraints in the clauses ofNEN6770 give unsafe results when dealing with beams in bending.

Based on the findings in the presented research the following recommendation can be made:

1. Numerical research should be performed on the elementary first-order plastic and geometrically and physically non-linear behaviour of cross-sections modelled with solids and should be compared to the methods used in this research.

2. Experimental research should be performed to further validate the numerical simulations performed in this research.

3. A distinction should be made in the EN1993-1-1 between full and partial lateral restraints.

4. Additional research should be performed to determine the coefficients of the design rule to detennine the strength requirement of restraints in bending.

5. Additional research should be performed to determine the influence of restraints with finite strength.

6. Additional numerical research should be performed on different height to span ratios, cross-sections and steel grades.

7. Experimental or numerical research should be performed to detennine the validity of the EN-1993-1-1 with respect to:

• Partial rotational restraints on beams and beam columns; • Partial lateral restraints on beam-columns;

8. Effects of using the 1st order plasticity theory should be detennined when using the clauses ofthe Eurocode when dealing with statically indeterminate structures.

65

________________ ~L~a~t~er~a~I:T~o~rs~i~on~a~I_~,~~~~~~~r~e,~s~tr~a:i,nedste_e_l_be_a_m_s ________________ __

References

Books

[I) Timoshenko Gere J.M., Theory of elastic stability, edition, 1961 , McGraw-Hili, New York

(2) N.S., Flexural-torsional buckling 1993, E&FN Spon, London

[3] La Poutre D.B, Inelastic spatial stability wide flanged steel arches, 2005, Eindhoven

[4] York

Budynas "Roark's formulas for stress and strain 1h edition ",2002, McGraw-Hili, New

[5] Bruneau M, Whittaker "Ductile design sln/clures", 1998, McGraw-Hill, New York

Codes

I] EN 1993-1-1 : 2004 Eurocode 3: of steel structures part 1-1 General rules and rules for 49 2004 CEN Brussels.

[C2} NEN Staalconstructies TGB 1990,

NEN 6771, Staalconstructies TGB 1990, Stabiliteit.

Papers I] Swart Sterrenburg R, Unpublished "Article about the experiments perfonned on the stability of a concrete

slab loaded on a steel beam"

Bijlaard, between the

"Lateral torsional stability of members with lateral restraints at various locations ,Proceedings Eurosteel 2002 pp. 191-200

Serna A., Puente 1., Yong D.l," Equivalent unifonn moment factors for lateral-torsional buckling of steel Members", Journal of Constructional Steel Research, VOL 2006, pp. 566-580

[P4] Nethercot D.A, "Buckling of laterally or torsionally restrained beams", Journal of the Engineering Mechanics 1973, vol. 99, No. 04, American Society of Civil pp.773-791

[P5]Nethercot Rockey "A unifed Approach to the Elastic Lateral buckling of Columns and Beams ", Structural Hn,ry'n.(>or Vol. No.7, July 1971, pp. 321-329

Nethercot Rockey K.C., "The lateral buckling of beams having discrete intennediate restraints", The Structural Engineer, Vol. No. 10, October 1972, pp. 391-403

Aswandy , Greiner R, of members under l)v"'UllJ'",- and axial f'''t'1'' .. mp<c~, with intermediate lateral , Proceedings Eurosteel 2005 Maastricht

[P8] Clark lW., Hill "Lateral buckling of beams", Journal of the Structural Division, Vol 86, No: July 1960, American Society of Civil pp 175- 196.

[P9] Nethercot Rockey K.C., "Finite element solutions for the buckling of columns and beams", International Journal of Mechanical SCiences, 1971, Vol. pp 945-949.

[PIO] Schmidt, L.E,"Restraints against elastic lateral bucking", Journal of the Engineering Mechanics Division,VoI91, No. EM6, December pp.1-10

[PIll R. "Buckling check me.lnuer:-; and frames based on numerical simulations ", 2003

67

[P 12] ""'''''J'Ul''l A. "Flexural behavior of beam-columns with difl'prpnf boundary conditions -"substitutive members ", 2003

information on the beam-column interaction formulae at level 1 " ECCS TC-8 1J'"j'''I.I,,,1 2001.

[P14] Lechner, A. Greiner,R "Application of the equivalent column methodfor flexural buckling according to new EC3 ndes" Eurosteel 2005 Maastricht.

Stppnt,pr'Jf"n H.M.G.M. Abspoel, R. numerical model for UU"l\J.llll', of coped beams" Journal of Constructional Steel pp 1576-1593.

J. J. 1evelooment and validation Vol 61. No: 10,2005,

[P 16] La Poutre Snijder Wluie-f,ranvea beams with FEM", int Symposium in Civil '-'U"'HI." .... '

Reports

[RI] Greiner R , Ofner R, """"'J',"''''''' G, "Lateral torsional buckling of beam-columns" Theoretical background 5. ECCS-Validation 1999

Maljaars "Graduation Thesis: Lateral torsional stability ", TU 2002.

Website

[WI] http://www.bouwenmetstaal.n1I06_3_hoofd_tabelz.lasso, Steel cross-section library.

Manuals

[Ml] ANSYS VIO.O Manual

68


Appendix A. Tables of coefficients Tabel9 from NEN6771 [C3]

gcval belasMg C1 C2

eM P~) 1,75 - I,OS fJ + 0,3 JP indien C 1 > 2,3

1 '" Deem dan C1 = 2,3 0

l I st L '1 '1

H = eindmoment met grootste absolute waarde -1~P~1

llq 2 11 } I 1 1,13 0,45 -l I ~t ~ 11

F

3 i 1,35 O.sS :::A. .A l -1 0.5/101 O,51s t ~

7J '1 " F F

4 ~ l 1,04 0,42

"* A.

l l -O,25i dL 02 1st ~O.Z5's' " " ... '1

I H) S 1 0

l , l >a " 1st =21

I rJ I I I lq 1,68 0,78 6 l L 1

" 1\

1st :0,821

F

7 I ! 1,45 0,56

J L I

" I It = 1,131

69

restrained steel beams TUie

Appendix B. Comparison of formulae for Mer

In this appendix a is made of the different simplified methods that are available. First the reference equation is described. Hereafter the different other equations are look into. To compare these equations, the conditions are normalized to a double cross-sections = K and supports(k = 1).

B.1. Greiner The formula that Greiner uses will be used as a reference(B.l). When applying the previously set normalisation (B. 1 ) becomes (B.2). And results to the equation used by Bijlaard & ~leent)el"J~en

1)

(B.2)

B.2. Clark and Hill The basic formula used by Clark and Hill is given in [P8] (B.3). After substitution of K=O (double symmetry) and k = 1 (simple supports), (B.4) is obtained.

+ +

+

Clark and Hill use a different coordinate system, and as g

Which is identical to the reference equation

B.3. NEN 6771 According to the NEN 6771 [C3] is written as:

Where:

c

s

C = k ~J

r"" I g

When substituting C and S in 1.7 eq. (B.8) is obtained and

(B.3)

(B.4)

1 (B.S)

(B.6)

(B.7)

I ,II/ kl and kred 1 .

(B.8)

71

is simplified to:

12 GI

kl (B.9)

(II h),JElz

__________ ~~----~----~~------+--------~--~I

The Dutch code uses C2 in a manner than the other on the load and should be interpolated. When applying the coordinate and substitution is valid:

the coefficient c:lepiem1S the following interpolation

2

Also by multiplying the PYr,rp""U1.n

kl 10)

from the term under the root leads to:

This is rewritten to:

1 The Dutch code also that the warping constant may be determined as follows:

lw = 4

This leads to:

The following substitution is possible:

This is the same as the equation k=l

1 by Clark and Hill (using the other sign convention)

1)

(B.!

(B.

c,z, 1 (B.14)

However it should be noted that this only applies when the approximation of 1,. When the correct values for 1w in factor S (see equation the height oftlle load application will be taken differently compared to the reference equation. And therefore give different results.

72

Lateral

BA. Nethercot The formulae used by Nethercot is derived using (B.2) and

Multiplying the inner tenn with 1CE1, the following I

Simplifying gives:

Simplifying once more

This can be rewritten to:

M =1Ca cr I

M =1Ca cr I

TU/e

o which leads to (RI

(B. 15)

6)

(B.

(B.18)

19)

By letting a = 1 (pure bending), (B.19) is equal to (2.6). The method used by Nethercot does not use the C2 factor and can therefore not be to match the reference method.

73

Lateral Torsional Buckling of laterally restrained steel beams TUle

Appendix C. Determination of kc factor The Eurocode 3 allows for moment distribution compensation when using the specific method. This compensation is performed using the kc factor. This factor can be computed using (C.l). The results of using this equation is seen in Table 14 where the kc factors prescribed in the Eurocode are found along with the Cl coefficient and the kc factor determined with (C. 1). No difference is found between (C.l) and the values given in the EC3

k =Jt c C. (C.1 )

Table 14

Moment distribution kc [EC3] C j Jt 1111111111111111111111111111111

0.94 1.13 0.941

b cd 0.9 1.240 0.898

0.91 1.204 0.911

0.86 1.35 0.858

0.77 1.71 0.764

0.82 1.498 0.817

75


Appendix D. Nominal dimensions of IPE sections

--- --- ------ ------ --,---.-"---- - -----,

h ho

----- - --- -- -------- --f---------------j

b : ~

Figure 99 Dimensions of a wide-flanged beam

.. !.~~!~ .. ~ ~.~~~!~~.~ .~!.~~!l:.~!~~.~. ~!}.~.J?!:~J?~!:~~~~ .. ~r.~~ I PE240 [WI] Dimensions h 240 mm b 120 mm tjl 9.8 mm tw 6.2 mm r 15 mm .............................................................................

··?~gp.~~~:'!·························"4'···············4·· .............. . Iy 3892 x 10 mm Iz 283.6 xl04 mm4

1/ 12.74 x 104 mm4

Iw 36.68 xl09 mm6

W el,y 324.3 X 103 mm3

Wpl,y 366.6 x 103 mm3

Web 47.27 X 103 mm3

, 3 3

.. !fr!;: ................ _.1? ;??.~ . .l. 2 ....... _.!?!!'!. ................. .

77


Appendix E. Finite element models The accuracy of a simulation is dependent of the care that is taken in the modelling of this simulation. In this appendix the choices that have lead to the finite element models will be discussed. The FE-program used is ANSYS VIO.O.

In this appendix first the elements used are described. Then the determination of the cross-sectional properties is discussed. Hereafter the FE-model used in the simulations will be explained. Then the solution methods are summarized and finally with the Input-file listing is give.

E.1. Elements In this section the elements that have been used in the FE-model are described.

E.1.1. Shell181

The SHELL18I element is a 4node 3dimensional shell element based on the Mindlin-Reissner shell theory. It has 6 degrees-of-freedom (DOFS) at each node (translation and rotations). This element incorporates bending, membrane and shear deformations. The element also allows a choice in the number of integration points thru thickness (9 is chosen) and the integration scheme (reduced integration is used). It is used to model the web, flange, and stiffeners.

E.1.2. BEAM188

.20 I •

Z

lto = Bement x-axis ifESYS is not provided.

x = FJement x-axis ifESYS is provided.

~ /iK.L .LlJ

Triangular Option (not recanmended)

Figure 100 SHELL181 geometry [Ml]

The BEAM188 element is a 2 node 3dimensional beam element based on the Timoshenko beam theory and has 7 DOFs per node (translation, rotation and warping). This element incorporates bending, axial and shear deformation. It allows for arbitrary cross-sectional input as well as a large number of common parametric crosssections. Arbitrary input is used to model the stiff beam elements at the supports and the compensation elements.

z

Figure 101 BEAM188 element geometry [Ml]

79

E Finite element models

E.1.3. SHELL93

102 SHELL93 element geometry [MIl

.4. LlNK8

The LINK8 element is a 3D spar element. It is a uniaxial v1vJ"lv'," with only 3 DOF (displacement only) at each node. It is used as the rod.

}-

l I

'f I L_

x -_ ---Figure 103 SHELL93 element geometry [MIl

E.1.S. COMBIN14

The COMBIN14 element is a 3 dimensional spring-damper. As used it has no and tension. It is used as the elastic restraint.

ro~ I ~f\\;~, I .t4't;~ I : r't.e., :

I: II I><!t; I

)-

' I I I y l---____ l l_____ :

:If

104 COMBIN14 element geometry [Ml]

E.1.S. COMBIN39

The COMBIN39 element is unidirectional non-linear spring. It has longitudinal and torsional ''';:''~'''V'Lll and 3 dimensional For more information see [MI], this element is used in the method to model the rotational spring.

1.

'(

x

Figure 105 COMBIN39 element geometry IMl]

80

compression

Lateral TU/e

E.1.7. MESH200

The MESH200 element is a "mesh-only" element. It does not contribute to the solution. It is used to model the restraint when a LBA is to obtain the unrestrained shape .

.J

,/ KEYOPT [Ii = (I 2·0 11M willi '2 ftO","

Figure 106 2-D line MESH200 element geometry [MIl

Cross~sectional properties The detelmination of the properties will be done according to the used La Poutre et al.[P16]. This will be done two different element first this is performed a She1l93 as this was the element used by La Poutre, the second will be a Shell lSI element as this is the element that will be used in the fmal FE-model. This element is used since it has the ability to describe initial stress states. A reference is created using the SHELL93 element for which values are known. Using the same the nr{\",o,rt"",

determined SHELLlSl elements. This is it is then sectional nrl'1,np,",!", for any other cross-section.

In structural I -shapes four of stiffenesses (major axis stiffness Iy, minor axis bending stiffness torsional constant It and warping constant Iw) are needed for the lateral-torsional buckling problem. Each of these will be determined separately in order to accurate results.

E.2.1. Bending

In bending of beams with flanges two properties are relevant, major and minor axis bending stiffnesses. The calculations of both of these properties are moments which creates pure l}vlj'Ul1j,l'. (Figure 107). The conditions are applied to lOS. At both of the model, the out-of-plane displacements are constrained at the shear centre and at the centroid of the flanges, dis:pl~lceme:nts are constrained at the shear centre at both ends ofthe model. The moments are applied

dis:pl~lcemejnts and a rotation at the shear centre6; this is shown in Figure 109.

M

Uz= 0

108 Constraints applied for the computation of the and minor axis bending stiffness

6 This has been ", .. ·t .... ,·"',.r! due to the fact that the use of constraint V'l"""'VU" did not apply the rotation correctly, This is due to artificial stiffnesses of the element when applying in-plane rotations ANSYS user manual)

SHELLl81 element description in the

81

E Finite element

~I I "4f , I I , , I

-u,

Figure 109 Applying a rotation on shell elements.

Each of the applied results in a reaction force. Using these forces the moment can be determined. As rotations are known, the stiffuess can be determined 1).

1 Ml

2Erp

For a HEAl 00 modelled with 4 She1l181 elements in each flange and 4 in the web. With a length of 600mm, an Young's modulus of elasticity of 210000N/mm2

, and <p == lmm I 1I2~ 1144. M = 2«70000 * 2 + 140000 * 2 + 169000) * 44 + 38498 * + 135410

M

2«reaction in u1)* y;: ho + reaction in U2 * 32Nmm

53661322 * 600

2 * 210000 *1144

11.0)+ Moment at (,Pt1ltrAIf1

The same is performed for minor axis <p = Imm 1112~ 1/50. M = 1 *2 *50+2*70141 *25)+ 16574

2(reaction in u1 *2* y;: 11.0 + in u2 *2* ~ ho)+ Moment at centroid

M 17596874Nmm

1 = 17596874*600 1256920~1.257*106mm4 z 2 * 210000 * 1150

E.2.2. Torsion

St-Vern ant torsion

In beams with wide-flanges St-Vernant torsion occurs when warping is not restrained. This is achieved using the scheme in 110. This is translated to the FE-model using the conditions and loads according to

111. The displacements of the centroid at are constrained in three axes, while at the upper and lower flange centroids the in the z-direction are The load is introduced stiff beam elements and loaded by a couple at each of the ends (nodes 3 and 11).

T T

Figure 110 Mechanical scheme for St-Vernant torsion

82

1)


F

Ann ~ ho

V2

Figure 111 Constraints and loads applied on the FE-model for the computation of the torsional stiffness

The average rotation occurring at the load introduction is used to compute the torsional stiffness with eq.(E.2).

M(~l) I = (E.2)

I Grp

0.00099845

0.0009984

0.00099835 +--+-".... .. ->-f--<---+--T+---.,...~-..

1 2 3 4 5 6 7 8 9 10 11 1213

Figure 112 Rotation of the nodes in uniform torsion

The average rotation is 0.000998355 the couple applied is 10000N, the length is 600mm and with the relation ship of G=21 0000/(2*(l+v) with v = 0.3 G = 80769 which leads to:

I = 10000*300 = 37204mm4

I 80769 * 0.000998355

E.2.2.2. Mixed torsion In beams with wide-flanges mixed torsion occurs when warping is restrained. This is achieved using the scheme in Figure 113. This is translated to the FE-model using the restraining conditions and loads according to Figure 114. This is achieved applying the boundary condition shown in, here all the nodes at x = 0 are fully restrained (both rotations and displacement). The load is introduced using stiff beam elements and loaded by a couple.

~~ _____________________ T~~~

L .. Figure 113 Mechanical scheme for mixed torsion

Full restrained

Ann = ho

Figure 114 Constraints applied for the computation of the warping constant

83

Appendix E Finite element models

In mixed torsion the rotation is calculated using (E.3) [3]. First the torsional stiffness needs to be calculated using the previous test. Then the factor It is chosen such that the rotation matches the value from the average rotation value of the Finite element analysis (PEA). Then the warping constant is calculated:

rp= M AL cosh( AL) - sinh( AL)

GIlA cosh(AL) Where:

$EI GI A = __ I and thus EI = -21

EI W A W

0.0007 8387 _"-:':='--t_'-~=--"""I--::'t<t=---. ..

0.00078385

0.00078383

0.0007 8381 +----.--.-.---.--.--.--.--r-,---,---;~

1 2 3 4 5 6 7 8 9 1011 1213

-+-- Rotation of the nodes

__ Average node rolation

Figure 115 Rotation of the nodes in mixed torsion

(E.3)

As it is not possible to simply determine the warping constant, a value of A. has to be chosen such that the then obtained rotation of the cross-section equals the average rotation of the FE-model. This has been done using a simple iteration method. This method is shown in Figure 116. Herein Si are the steps taking in the iteration.

Obtain value' of lambda (1,) for the smallest positive

difference

-Obtain value of lambda (1,) for the smallest poSitive

difference

Obtain value of lambda (7.) for the smallest positive

difference

Figure 116 Iteration procedure for J...

The model used is the same as in the previous illustration. The couple applied is 10000N the length is 600mm, the average value of the rotation obtained is 0.000783836. For the stiffness the Poisson ratio has been set to 0.3 with E = 210000, G = 80769 . The torsional constant has already been determined and is It == 37204. In total 8 iteration steps are performed. The results hereof are presented in Table 16.

84


Table 16 Iteration of Iw

Iteration ste A Iw mm Diff %)

0.001993208 0.0000000000000000 1.4309E+38 99.7963954 2 .001926651 0.0000000000000000 1.4309E+38 99.7963954 3 .000748749 0.0023750000000000 2536819500 4.4763108 4 0.000789719 0.0024750000000000 2335965309 0.0613490 5 0.000785375 0.0024762500000000 2333607534 0.0070676 6 0.000783882 0.0024764062500000 2333313063 0.0002840 7 0.000783839 0.0024764125000000 2333301286 0.0000127 8 0.000783836 0.0024764127734375 2333300770 0.0000000

E.2.3. Discussion

In this section the results of the detennination of the cross-sectional properties are discussed. In earlier research [PI6] the tests that have been described previously have been perfonned using SHELL93 elements. In this research however SHELL181 elements will be used. To be certain that the properties are detennined correctly, the results from [P 16] will be taken as a reference and the determination of the cross-sectional properties will be perfonned using both the SHELL93 and SHELL181 elements.

In Table 17 the results are shown for the perfonned tests and of the results from [PI6]. It can be seen that with the use of the same elements the results are alike. Except for the torsional constant where the results are not matched correctly. In the detennination of the constant, the stiffness of the load introduction has a great influence on the results; due to the fact that it is unknown which stiffnesses have been used, the results could not be matched unless by applying "trial and error", the use of stiff beam elements has given acceptable results. It is also seen that the SHELL93 element gives better results then the SHELL181 element. This is caused by the fact that the SHELL93 is an 8-node element while the SHELL 181 is a 4-node element. With the SHELL 181 element the more elements used in the cross-section the better the solution is. Each of the flanges and web all will be modelled using 8 elements each giving 24 elements to model the cross-section.

Table 17 com(!arison of the results of the research and 1P16Hwithout root-radii~ Section ProEerty Model

SHELL93 SHELL93 diff SHELL181 diff diff [P5] 12 elements 12 elements % 12 elements % 24 elements %

Iy[mm4] 3.39.106 3.39.106 (0.0) 3.373.106 (0.5) 3.386.106 (0.1) Iz [mm4] 1.334.106 1.334.106 (0.0) 1.257.106 (5.8) 1.316.106 (1.3)

HEAI00 II [mm4J 35939 35977 (-0.1 ) 37204 (-3.5) 36756 (-2.3)

lw[mm ] 2.485.109 2.485.109 ~0 .02 2.333.109

~6.12 2.444.109 ~1.62

E.3. Compensation of the discretisation error of the cross-section Using shell-elements to model the cross-section causes a discretisation error in the modelling of the cross-section (see Figure 117). This discretisation error particularly affects the torsional properties due to not modelling the root-radii at the web-flange intersections; also a small area of the web is taken into account twice. This discretisation error can be compensated by addition of specific elements. This can be achieved using different methods. Each of the different compensation methods will be discussed, and then their first-order elastic and plastic behaviour is looked into. Then their geometrical and material non-linear behaviour is looked into. Finally a choice is made which method will be used in the FE-model.

(a) Real beam

~ Area that is accounted twice

(b) Discrelisalion error lhal is occuring due to the use of shell element

Figure 117 Discretisation that is occurring due to the use of shell elements

85

· Compensation methods

In the literature two "VJ,U!-"\.,ll"atlVU

the La Poutre [3] and the compensation methods consist of elements at the Location of Wen-l]arlQe intersection 118). These elements are compensation elements

the appropriate to compensate for the lacking properties (properties that are discretisized in the model). La Poutre has used a single element hollow section 118 location of compensation elements (SHS»). The method developed by Maljaars uses two one element is the bending properties and area, and the torsional H.<vA'>'U"i'> are compensated (bi-)Iinear rotational In this an alternative model has been made. In the so called alternative method three different elements are used to compensate the cross-sectional Axial elements with bi-linear material properties are used to compensate the major-axis bending properties and the plastic moment capacity. An elastic beam compensates the [ninor-axis bending stiffness; an elastic torsional beam is used to the torsional constant. The so-called RHS method is an version of the La Poutre a a rectangular hollow is used. In Table 18 an overview is given of the four different compensation models, It should be noted that the of the warping constant is not this is to the nature of this constant, this constant includes individual bending of the And compensating the constant will cause the minor-axis bending stiffness to be overcompensated.

Table 18 Overview of compensation methods Property ~M_e_th_o_d~ __________ ~ ____________ ~ __ ~ ________________________ _

La Poutre Mal'aars Alternative Iy Elastic-plastic SHS Elastic Beam Axial element with RHS

bi-linear material model Ix " Elastic Beam Elastic Beam " 1/ (Bi-)Linear Elastic torsional Beam "

rotational s rin

E.3.!.!. Determination of the properties of the compensation elements using the different methods Each method has specific methods of determining the properties of the different elements that are used, In paragraph an overview is for the determination of the properties of the compensation "l"Ul\.oIUi>,

In the nrp(:Pt1,tpri equations the used subscripts denote the following: comp given to the compensation elements, FEM obtained from the Finite Element Model. (-) The analytical values of the cross-sectional nrf'",p,-t1

La Poutre Method In simple equations are given to calculate the properties for the square hollow "'''''''<lV,U, the height, width and thickness (Figure 1 are given according to:

b

Where:

Iy;comp (E.6)

2

86

Lateral Torsional steel beams

It was found when using the equations by La Poutre that the properties did not match accurately. The element used in and the hereby belonging cross-section uses numerical to compute the different properties of the here it was found that the use of the computes a different torsional constant. Also the compensation for major axis bending is taken inaccurately. Therefore these equations have been left aside and a new procedure has been followed to detennine the

for the square hollow section.

In [4] a more accurate equation for the torsional constant of a SHS has been found: (b -t)3 t

b

119 Dimensions

ora SHS

TU/e

(E.B)

The bending stiffuess of a SHS is calculated including the Steiner part at a distance of llzho from the centroid as:

b4 -(b-2t)2 1102

12 12 4 (E.9) -'----'-- + -'-------'---

Both these equation have two unknowns (b and t) and three knowns , ho). Thus solving band t from the following equation will the dimensions for the SHS. cause major axis-bending compensation to be exact; the torsional compensation however will not match exactly. This can be matched accurately iteration methods.

{

(t-brb

b4 -

~-~+~--------~-12 12

10)

Maljaars Method This method has been developed as unexpected behaviour was found when using elastic elements using the FE-software DIANA. Here both major and minor bending-stiffness are axial and bending element. The torsional constant is compensated a (bi-) linear rotational

Bending element

The bending stiffness of the element is taken as half the different between the reference value and the one of the FE-model The minor axis bending stiffuess of the bending element is by:

Torsional element

1 -z :::::---'---

2

The torsional used in this method only accounts for the lacking torsional properties. First the for each element is calculated to Then the characteristics of the rotational

(E.ll)

either linear or bi-linear. The linear characteristics are calculated to (E.7) and (E.12). The bi-linear characteristics are calculated (E.7) and (E.l2) after which the plastic torsional moment of a circular bar is taken as the point where the spring becomes plastic (El3). The rotation at which the becomes can be taken as CfJp/ Tp/I K"ol . This is valid as the rotations are considered smalL

T

f/Jp/ f/J

Figure 120 NOll-linear spring properties

The rotational stiffness of each has been determined by using the following equations:

7 No references have been found on how the torsional "rf\npl't1p~ are calculated in the ANSYS theoretical manual.

seg The plastic moment of the rotational is determined with [R2J:

Where: seg == of the rotational

Alternative method

models

(E.12)

The alternative method is an adaptation of the Maljaars here three elements are used to COlmp,em:ate the lacking properties. Each element is used to compensate each of the three to be compensated "rru,,,,rh,,,,

The area of the compensation element for the major Vvl,lUU1); stiffness is calculated using:

IfA-AFEM > 0 (to overcompensation of the plastic section modulus) the need to be and an additional element bending needs to be introduced. This element will be given the remainder of the lacking major-bending

The stiffuess of the element that compensated the minor bending properties is determined according to 11) and for the torsional element equation is used.

RHSMethod This method is an adaptation of the La Poutre method, instead of using a square hollow section a rectangular hollow is used. The dimensions of the section are defined

three parameters: the width and thickness of the section 121). these three the properties can be matched It is then

to all the properties minor-axis V,","'Ulll,);

taken into account in the La Poutre method. These are to be obtained using the following """,£,"""

The torsional constant of a rectangular hollow section 2t2

The major LJ\.;"lUHjl); stiffness is

12 4

The minor bending stiffness is given by:

Figure 121 Dimensions

ofa RHS

(E.16)

The three previous lead to the ge()mt:trv of the cross-section, these equations have 7 four are known and three are unknown. Thus the (the properties of the element are substituted with the

can be solved according to:

88

Lateral

(E.18)

The same here as for the method used La the torsional constant will not match, and therefore iteration will have to be np,7r"·IT\~·rl

E.3.1.2. Elements used in the FE-model Each different FE-model uses different elements depending on the compensation method. The cross-section is first modelled using SHELL 181 elements with 9 integration point tbm the thickness. The La Poutre and the RHS method both use a BEAM188 element. The Maljaars method uses a COMBIN39 element as a rotational and a BEAM 1 88 element to the properties. The alternative method uses a LINK8 element to cornp4~nsate for the a BEAM188 to the minor-axis bending property and a BEAM4 element used to the torsional constant. An overview is in Table 19.

Table 19 Elements used in the different compensation methods Compensation for I----'E.:;;,.le:..:,m::.e.:;;..n.:;;..t;;;...s..:.;,u.:;,.se.:;;..d'--__________________ _

La Poutre Mal' aars Alternative RHS BEAM188 BEAM188 LINK8 BEAM188 Elastic-Plastic Elastic Elastic-Plastic Elastic-Plastic

Iz " BEAM188 BEAM 1 88 Elastic Elastic

1/ " COMBIN39 BEAM4 " Bi- linear Elastic

E.3.2. First order elastic behaviour

In Appendix E the determination of the elastic cross-sectional is discussed in more detail. In Table 20 it is shown how the are using the different compensation methods. The FE-model consists of a beam with a of 7200mrn, where 8 elements are used for each of the and web. The beam is in 72 givin~ each element a of lOOmrn. The material is given a Young's modulus of of 21 OOOON/mrn . The modelled cross-section is identical to the one used in the experiment used to calibrate the FE-model (IPE240). See Appendix D

The La Poutre method uses only the two properties (Major-axis bending and pure torsion) to compensate for the "' ..... ''UJ.''IS properties. The for minor-axis are not included in the compensation. As the SHS elements have the same for both and minor the for minor vvllUHIt:.

C01npcmsatect. The method developed by Maljaars, the RHS and the Altemative method each However their first order plastic behaviour differ from each other. This is

shown in the next section. The method developed by La Poutre will not be investigated in the rest of the comparison of the behaviour of the as the RHS method is accurate elastically.

Table 20 Cross-sectional compensation using the different methods Property Analytical Model

[WI] No comp- Dill' La Poutre Dill' MaJjaars Diff Alternative Diff RHS

*The analytical values are taken as the reference values for the calculation of the difference.

Diff


E.3.3. First order plastic behaviour

The use of restraints will cause the slenderness to decrease hereby making plastic failure eminent. Therefore the first order plastic behaviour is analyzed. This is done using pure bending for the two bending axes and pure torsion for St-Vernant torsion. The same model is used as presented earlier, only here the material model has been altered according to Figure 122 (E = 21 OOOON/rrun2 ,/y = 235N/rrun2

).

o

I::

Figure 122 Material law used in the FE-model for pure bending and pure-torsion

E.3.3.1. Major Axis

In Figure 123 it is shown how the different compensation methods perfonn. Here it can be seen that the Maljaars method exceeds the plastic moment. This is caused by the use of the elastic beam elements.

E z ~

" .. E 0 :Ii

100

90

80

70

60

50

40

30

20

10

oa---r-~--~--~--~~---+--~--~--~~---r--~

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13

Rotation [rad)

-Exact

-No compensation __ RHS Method

__ Mal)sars method

__ AltemaU\{l method

Figure 123 First order plastic behaviour of the different compensation methods

E.3.3.2. Minor axis In Figure 124 it is shown how the different compensation methods perfonn using first order plasticity. Here it can be seen that their influence is small, however it can be seen that the RHS method shows similar behaviour as the uncompensated model. The paths of the alternative method and the Maljaars do not have a horizontal assyrnptote; this is caused by the elastic beam elements. This influence here is rather small as the compensation elements are placed near the neutral axis.

90


20

18

16r---~~~~ 14

E 12 z .!!:. E 10 " E ~ 8

6

4

2

0.05 0.1 0.15

Rotation [rad)

0.2 0.25 0.3

-EXACT

- No compensation ___ RHS method

~Maljaars method

AltemaU"" method

Figure 124 First order plastic behaviour of the different compensation methods

E.3.3.3. St-Vernant Torsion

TUle

In Figure 125 it is shown how the different compensation methods perform under first order plastic St-Vemant torsion. The computation of the Plastic St-Vemant torsional moment is performed in Appendix F. Here it can be seen that this influence is rather large. Two rather different moment-rotation paths can be seen. The uncompensated model and the method using RHS have similar paths. The Altemative and the Maljaars method behave significantly different. It can also be seen when the Bi-linear rotational spring in the Maljaars method becomes plastic. The angular paths of the Maljaars method and the Altemative method show a bilinear path. These are the two elastic properties, the initial angle is the elastic stiffuess of the system, and the other angle is the elastic stiffness of the torsional compensation elements.

4 .---------------~--------------._~~~~----~

3.5

3

I 2.5 r------~;f!'.,;;...-====::==::====::::;:::t ';; 2 ::I

EO ~ 1.5

0.5

- Tpl crosssection

__ No com pens ation

__ RHS Method

__ Maljaars method linear spring

__ Maljaars method bl-linear spring

O~ ______ .-______ +-______ .-______ .-______ .-~~·~~~re~rn~a~ti~~m~e~th=od~ ____ --J

o 0.5 1.5 2 2.5 3

Rotation [rad)

Figure 125 First order plastic behaviour of compensation models in St-Vernant torsion.

E.3_4_ Geometrical and material non-linear imperfect behaviour

The geometrical and material non-linear behaviour of the different compensation models is done using GeometTical and Material Non-linear Imperfect Analysis (GMNIA). The investigation is performed for three situations (height-to-span ratios) each subjected to pure bending (Figure 107). First a height-to-span ratio of lOis analyzed, from this span it is expected that the beam will fail due to plasticity. Then a height-to-span ratio of30 is analyzed, here the beam will fail elastically; as will be expected for the third ratio where the height-to-span ratio is taken 50. The height of the section is 240mm, therefore respectively leading to the following spans: 2400, 7200, 12000mm. Eight elements are used to model each of the flanges and the web, and have a length of 100nun. The model is given the same material law as was used in the investigation of the plastic behaviour. Residual stresses were added according to Figure 126, where (ailli = 1/3 1;,). The out-of plane system imperfection has been taken as 1/1000 of the span where the greatest displacements were found in the linear buckling analysis.

91


Figure 126 Residual stress pattern according to NEN6771 le3)

E.3.4.1. GMNIA Pure bending span = IOh (2400mm)

In Figure 127 and Figure 128 the behaviour of the different models is shown, along with the uncompensated model. Here it can be seen that the Maljaars method (either with linear or bi-linear springs) behave alike. The same is found for the a'ltemative method; however it gives a slightly lower fai}ure load. The RHS method shows slightly different behaviour. The RHS method also shows affinity with the model without compensation where the moment-displacement path has a horizontal assymptote.

92

80

70

M

C?~V ;1/ 2400mm /IV

-M

60

E 50 z .!!. i: 40 " E 0 :E 30

- No compensation 20

___ RHS Method

10 ........ Maliaars v.ith linear spring

0 __ Maliaars v.ith bi· linear spring

0 2 3 4 5 6 7 8 ___ Alternative method Displacement u [mm]

Figure 127 Geometrical and material non-linear behaviour of an IPE240 under a uniform moment with an imperfection of 1110001 with a length of 2400mm (In-plane displacement)

80

70

60 -

M ·M

(ol--------'V 2400mm

:tV X

'E SO z .!!. i: 40 .. E 0 :E 30

- No compensation 20

___ RHS Method

10 -.it- Maliaars v.ith linear spring

0 __ Maliaars v.ith bi-linear spring 0 2 3 5 6 7 8 9 10

Dlsptacom cnt u [mm] --Alternative method

Figure 128 Geometrical and material non-linear behaviour of an IPE240 under a uniform moment with an imperfection of 1110001 with a length of 2400mm (Out-of-plane displacement)


E.3.4.2. GMNIA Pure bending span = 30h (7200mm)

In Figure 129 and Figure 130 it can be seen that all the compensation methods behave alike, also the momentdisplacement slopes of the compensated models and the uncompensated model are similar. Also no significant differences have been found between the uses of linear or bi-linear rotational springs.

40 ,-----------------------------------------------, M oM G----v 35

30

E 25 z ~ - 20

~ ~ 15

10

7200mm 711/ v /1

- No compensation

-+-- RHS Method

~ Matjasrs wth linear spring

o~~--------~----------~----------~--------~ ~ Maljaars with bi-linear spring o 10 20 30 40

---+- Alternative method Displacement u (mm]

~---.---------------~----~~--------------~============~ Figure 129 Geometrical and material non-linear behaviour of an IPE240 under a uniform moment with an

imperfection of 1110001 with a length of 7200mm (In-plane displacement)

40

35

30

E 25 z ~ C 20

~ ~ 15

10

0 0 10 20 30

Displacement u [mm]

40 50

M -M

G-v 7200mm /IV v

/1

- No corJl)ensation

-+--RHS Method

~ Maljaars with linear spring

60 ~ Maljaars wth bi-linear spring

--- Alternative method


E.3.4.3. GMNIA Pure bending span = SOh (12000mm)

In Figure 131 and Figure 132 it can be seen that all the compensation methods behave alike, also the momentdisplacement slopes of the compensated models and the uncompensated model are similar. Also no significant differences have been found between the used of linear or bi-linear rotational springs. Only here the RHS method has a slightly lower moment-displacement path.

93


25r-----------~--------------------------------__.

20

-M

(o~v ~/ 12000mm 7V

M

- f\b corTl>ensation

-+- RHS Meihod

___ Maljaars with linear spring

.....0<- Maljaars wth b ... ,inear spring o 10 20 30 40 50 60 70 80 90 100 110 120

Displacement u [mmJ -+-- Alternative method

Figure 131 Geometrical and material non-linear behaviour of an IPE240 under a uniform moment with an imperfection of 1110001 with a length of 12000mm (In-plane displacement)

25.----------------~----------~----------------__. -M

20

M

(o-------<v 12000mm

X v 71

- f\b compensation

-+-RHS Meihod

~ Mafjaars "",th linear spring O~~--__ --____ --~ __ --__ ----~--~ __ --__ --__ ~~

o 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 ~ Matjaars W'ith bi-linear spring

Displacement u [mm] --- Alternative method


E.3.S. Discussion and final choice

All the different compensation models each compensate the lacking cross-sectional properties accurately. The plastic capacity differs mostly for major axis bending. The method developed by Maljaars has no finite capacity describing the plastic moment caused by the use of elastic elements. The major-axis plastic moment of inertia is accurately modelled by the alternative and RHS method . The RHS will start to yield before the flanges start to yield, this is compensated as the bottom part of the cross-section will yield later as is it positioned at an equal

. distance below the centroid of the top flange. Yielding in the alternative method will occur suddenly as the axial element used does not account for bending.

The elastic beams in both the alternative method and the Maljaars method do not have a large influence on the plastic capacity in the minor-axis. Since these elements are placed in the neutral axis. The plastic St-Vemant torsional behaviour of the RHS method is the only method where its behaviour is similar to the one of the uncompensated model. The incorporation of plasticity in torsion of the Maljaars method does not give a good prediction of the plastic St-Vernant torsional capacity. The point at which the rotational spring becomes plastic can be recognized . However in the testing of the different models where torsion dominates (30h and 50h) no difference has been found between the different models, and the moment-displacement paths show similar behaviour. Therefore it is concluded that in the range of height-to-span ratios (:S 30h) that will be studied; torsional failure will not occur.

94


In Geometrical and material non-linear behaviour stresses are combined, the RHS method will combine the stresses for the three major moments (My, Mz, T see Figure 133) in the RHS element and therefore when yielding occurs in a part of the section, the influence hereof is taken into account according to the Von Mises stress criterion. This does not occur in the Maljaars method, as the elements will not yield. In the alternative

M~y ---a-I) T

method, only the elements that contribute to the major bending will yield, Figure 133 Major moment in the and the effects of plasticity in this element will not affect the other RHS element elements. The influence hereof is best seen in Figure 127, where the in-plane displacements are shown for a stocky beam. The moment-displacement path of the RHS method changes to horizontal before the other compensation methods. While the alternative method follows the Maljaars method, up to were the compensation element becomes plastic. Failure occurs here suddenly, while with the RHS method this occurs gradually.

The RHS method will be used in the rest of the research, as it is the only compensation method which gives good elastic, plastic and geometrical and material non-linear behaviour in each of the performed tests. It can also be noted that the Maljaars method can be altered to correctly determine the plastic torsional capacity and the first order plastic torsional behaviour. However large effort is needed to achieve this.

E.4. FE models

I Boundary conditions

~ Load type I span I restraint I

~ Cross-sectional prope,r:tles I

5~ ~ ~ Iii Closs sectloh

h (!)c:

/ System imperfections I '--__ -..-__ ....,..-!.N.."'" -~

' lo :n ~,§ : I Matenal properties

a.5 c

\ ~ Cross-sectional I \.~=im=perf=ecti=ons~

., Residual stresses J

Material bEihavour I

Figure 134 Dependencies of the finite element model.

Models 1,2 and 3 have been used for the investigation of unrestrained beams (chapter 5) and the parameter study (chapter 9). Modell has also been used to determine the influence of a single restraint at mid-span (chapter 6). Model 4 has been used in the determination of the influence of two restraints on the behaviour of the system (chapter 7). Models 5 and 6 have been used to validate the behaviour of the FE-model to the experiments that have been performed in earlier performed research. An overview hereof is given in Table 21 this together with each of the variables shown in Figure 134. Each of the different variables are given in the appropriate chapter and sections in this appendix.

95

element models

Table 21 Different FE-models used Modell Model 2 Model 3 Model 4 Model 5'" Model 6'" Fork Fork Clamped-fork, Fork Semi-fork Semi-fork

fork Stiffener Stiffener Stiffeners Stiffener Stiffeners

(§EA.3) load UDL load Two two Two loads loads follower

loads nOOmm, nOOmm nOOmm noOmm nOOmm 5400mm , 5400mm 5400mm 3600mm 3600mm 3600mm

Cross-section IPE240 IPE240 IPE240 IPE240 IPE240 IPE240 Cross-sectional None None None None Nonel None

EA.7 (5,9) 151 lSI (4) 151 (8)

Int(7) 2nd (8) (8)

111000 111000 l/l000 111000 111000 111000 112000 (2nd

) 111500 112000 112000

1I3/y l/3/y 1I3/y 1/3/y 1I3/y lI3/y

Restraints (§EA.5) None or 1 None 2 elastic none 2 restraints restraints

Material law S235 S235 S309 S305 5,6,9 7 4 8

'" Models used to compare the behaviour of the FE-model to the experiment.

E.4.1. Support conditions Three different support conditions have been modelled: A fork 135a), a support (Figure 1 and a semi-fork U5c).

Sliff elements Stiff elements· "- No Stiff elements

.. ..

(a) Fork support (b) Clamped-fork support (c) Semi-fork

135 Support types used in simulation

E.4.1.1. Fork supports Fork are supports, where (both and rotations and distortion of the cross-section are but where warping is not restrained. The displacements are using three constraints, for out-of-plane displacement and the rotation of the cross-section by setting the displacement in the

,rp.I'rWIn at the intersection to O. The in-plane displacements are by the at the centroid to 0 in z-direction. The distortion of the cross-section is ..... ,·c'u"''''''', ..

elements. This will cause plane sections of remain And only allow linear cross-section due to

96


E.4.1.2. Clamped fork support

Clamped-Fork supports are supports, where displacements (both in-plane and out-of-plane), rotations and distortion of the cross-section are prevented but where warping is not restrained. Also rotations and displacement of the web have been restrained. The displacements are prevented using three constraints, for out-of-plane displacement and the rotation of the cross-section by setting the displacement in the y-direction at the web-flange intersection to O. The in-plane displacements and rotations are prevented by setting the displacement of the web to o in z-direction. The distortion of the cross-section is prevented using stiff beam elements. This will cause plane sections of remain plane. And only allow linear deflections due to warping of the cross-section due to loading.

E.4.1.3. Semi-fork

In the experiments no forks could be created at the supports. But a semi-fork was created using two roller placed at either side of the section, these were used in order to prevent out-of-plane displacements of the flanges. A simple support was placed at intersection of the lower flange and the web. It can be seen that a single restraints has been positioned at the flange tips, positioned at either side will create an out-of-plane rotational restraint.

E.4.2. Load introduction

When applying loads in a FE model, care has to be taken in order to apply these correctly. As in practice it is common to use stiffeners at the load introduction. In this model the same will be done, these plates also prevent the cross-section to distort under the load. This can be seen in Figure 136. The stiffeners have an arbitrary thickness of 10mm.

Figure 136 Stiffeners located at the load introduction

E.4.3. Load type

E.4.3.1. Force load

The loads have been applied as follows: the concentrated load has been applied to a single node, a stiffening plate is applied here as to prevent deformation of the cross-section due to the load and to prevent peak stresses.

E.4.3.2. Uniformly distributed load

The uniformly distributed load is discretisized using 73 individual nodal loads applied on the intersection of the web and the upper-flange using individual loads at the appropriate nodes. This has been chosen as applying a surface load on SHELLl81 or BEAM188 element, will be treated as follower loads and giving load-stiffening effects.

FFFFFFFFFFFF FFFFF F F

12

r I l FI2

It

Figure 137 Applying a uniformly distributed load on FE-model using 4 node shell elements

97


E.4.3.3. Follower load

In the experiment described in chapter 4 the load has been applied using chains (Figure 138a). These chains have a length of 6000mm. When load is increased and out-of-plane displacement increase these chains will excite and inward force (Figure 138b). This inward force has a positive effect on the load bearing capacity as the beam is "pushed" back due to this force. In model 5 these chains have been modelled using LINK8 truss element with large properties.

Inward force due to

follower rod

Follower rod

Vertical force due to

follower rod

(a) Chains used in the experiment as loading rods. (b) Influence of follower rod Figure 138 Follower load

E.4.4. Residual stresses

During the manufacturing process of steel wide flanged beams the cooling of parts of the beam occurs unequally: the tip of the flange cools faster than the intersection of web and flanges. Therefore compression occurs in the flange tips and tension at the intersection. Due to the lack of experiments residual stress measurements can not be performed. However some existing stress distributions are available. The pattern that will be used is the pattern given in the Dutch code NEN6771, which is presented in Figure 139.

Figure 139 Residual stresses pattern according to NEN6771

When using FEM the stress distribution has to be modelled. The ideal pattern given in Figure 139 needs to be inputted where (5 . = ±113 f . However in the FEM model the stresses are applied at the centroid of the elements.

III/ J),

98

Lateral Torsional Buckling of laterally restrained steel beams TUle

The load has been applied on a 26 (8 for each flange and web and 2 for the box-sections) element cross-section. First the load is applied using step wise loading by taking the average value of the stress distribution over the width of the elements. Using the average values has caused the stresses at the flange tips and the intersection of web and flanges to be smaller than the ideal distribution (see Figure 140, in this figures 0 is the midpoint of the flange). This is caused due to the fact that the stresses cannot exceed the inputted values. It can also be seen that the stresses applied are average at the nodes. Using this knowledge a new load distribution has been created. The loading at the flange tips and the two elements at the web-flange intersection are set at ±1 I 3 f y ' this will cause the

stresses at -1, 0 and 1 to be according the ideal pattern, the other elements are set to 0 this will cause the stresses to average out according to the ideal pattern (Figure 141 and Figure 142). This loading pattern will be used as it gives perfect representation of the ideal residual stress pattern.

Position

,-.-Ideafised stress distribution

___ App6ed stresses

_ Results

Figure 140 Residual stress distribution in the bottom flange at mid-span for the first loading type

Position

-a-ldea6sed stress distribution

___ Apptied stresses

_ Results

Figure 141 Residual stress distribution in the bottom flange at mid-span for the second loading type

99

., ., !!

<ii -1 0


Position

--a- k:tealised stress distribution

__ Applied stresses

10 _ Resufts

Figure 142 Residual stress distribution in the web at mid-span for the second loading type

E.4.S. Restraint

The elastic restraint has been modelled using a single COMBIN14 element. This element is attached to the crosssection and has a free end. At the free end the displacement in the y-direction is set to zero. The displacement in the z and x-direction correspond the displacements of the end which is attached to the cross-section

Restraint

E.4.6. Material law

In the FE-models three different material models have been used. The material model S235 is the material model used in all models except those used to compare the FE-model to the experiments.

E.4.6.1. S235

The S235 steel grade is used as the material law of choice_ It has been used in the development of the buckling curves for the EC3. This elastic-full plastic material is shown in Figure 143 is used where E = 210000N/mm2,/y= 235N/rom2

.

E.4.6.2. S309

E

Figure 143 Material law used in the FE-model

Model 5 uses the S309 material law obtained from the stress-strain relations obtained from tensile tests . The SHELL 181 element used in the simulation is based on true stress and logarithmic strain measures. The stressstrain relations obtained from the tensile test have been determined using the nominal dimensions of the coupon. The stress-strain relationships obtained from these tensile-tests are called engineering stress and strain. True stress and logarithmic strain are calculated according to (E.19).

100


(J,rue = (J eng (I + £ eng)

£Iog = !n(l + £ eng)

TU/e

(E.19)

As is not possible to input each of the points obtained in the stress-strain measurement, a 12 point-approximation has been made of the true stress logarithmic strain curve up to the necking region (Figure 144). A magnification of the yield plateau is given in Figure 145.

600

500

400

VI

~ 300

;;;

200

100

350

340

330

320

310

::! 300 I!! ;;;

290

280

270

260

250

.. --J~

~

!-- ~

1/- "-

~

4

0 .05 0.1 0.15

strain

0 .2 0.25 0.3

Figure 144 Stress-strain relationship for S309

",~

/~ -'110. L V .. ..: ..... If~ ~- --~~ ~ '- .. ,

,...~ ~

~ V'-/

0.005 0 .01 0.015 0 .02 0.025

strain

Figure 145 Magnification of yield-plateau

--Enginering stress

--True stress


-- Enginering stress

- True stress


101


E.4.6.3. S305

Model 6 uses the S305 material law, these have also been determined using the procedures followed for the S309 material law. The 12 point approximation is shown up to the necking region in Figure 146 and the magnification of the yield plateau in Figure 147.

600

500

400

., e 300

u;

200

"IlO

o

350

340

330

320

3"1l

., !II 300 ~ u;

290

280

270

260

250

- ... ---V-~ e---- ~

L-V

0.05

~ ~-1I!IIIe -.--

~

~

~

~ ~

s. 0.005

~

0 .1 0 .15

strain

0 .2 0 .25 0.3

Figure 146 Stress-strain relationship for S305

.,-/~~

1'>. ~ ~ -- ~ ~ i-"

~ ~ ~ ~ <...,.;>

0.01 0 .015 0 .02 0.025

strain

Figure 147 Magnification of yield-plateau

E.4.7. Cross-sectional imperfections

--Engineering stress

--True Stress

_12P Approximation

--Engineering stress

- True Stress

_12P Approximation

In the experiments used to compare the behaviour of the FE-model cross-sectional imperfections have been measured. These measurements are given in Table 22, Average values have been determined. These dimensions have been used in model 5. All other model have nominal dimensions.

102


Location x 300 1300 2100 3300 3600 3900 4200 4500 5700 6500 7500

Height 240.8 240.8 240.1 240.4 240.6 240.6 240.3 240.2 240.1 240.7 240.5

Width upper 120.95 121.03 120.95 121.07 120.97 120.9 120.94 120.9 121.06 120.68 121.02

Width bottom 120.96 120.95 120.97 120.97 121.2 120.92 120.98 120.91 121.16 120.94 120.88

lt1 upper left 9.73 9.73 9.78 9.74 9.9 9.71 9.78 9.94 9.76 9.9 9.82

lt1 upper right 9.83 9.65 9.67 9.69 9.59 9.72 9.83 9.71 9.6 9.6 9.69

to bottom left 9.8 10.05 10.03 9.8 10.14 10.25 9.83 9.88 9.76 10.2 9.93

to bottom right 9.8 9.96 10.03 9.91 9.87 9.88 9.98 10.01 9.88 9.96 9.97

tw 6.48 6.5 6.48 6.49 6.49 6.48 6.49 6.49 6.48 6.48 6.47

The average values of the height, the width, the upper- and lower flanges and the web are: h = 240.46 mm w= 120.97 mm tft = 9.85 mm tw= 6.48 mm

E.5. Solving Two types of simulation have been used using the FE-model. The first is a linear buckling analysis (LBA). The second is a geometrical and material non-linear imperfect analysis. For each simulation both are performed. Using the LBA the elastic critical force/moment is determined. Also the buckling shape is used as an imperfection shape forGMNIA

E.S.1. Linear buckling analysis

With a linear buckling analysis the buckling point of a perfect linear structure is determined. In bending this is the Euler buckling load c.q. Euler buckling moment. The FE-software allows for two solving methods. The chosen method the subspace iteration, as is recommended in the manual, this has been chosen as few mode-shapes have to be determined and well-formed shell-elements have been used in the FE-model.

E.S.2. Geometrical and material non-linear imperfect analysis

The geometrical and material non-linear imperfect analysis is build-up of three components. First the system imperfections are applied, the buckling shape is obtained from the LBA the amplitude is determined appropriately. Then the appropriate material model is defined. After this the residual stresses are applied and solved. After this step the loads are applied and solved. Depending on the load type two different solving methods have been used. First the two solving methods using either displacement control or force control. Then the convergence criteria are discussed.

E.S.2.1. Iteration methods In the GMNIA, solutions are obtained using iteration methods. The Newton-Raphson method determines the tangent stiffuess matrix at each iteration step. The modified Newton-Raphson method determines the tangent stiffness matrix once every step and needs more iteration steps compared to that of the Newton-Raphson method. However less computational effort is needed with the reduced Newton-Raphson method. In the simulation the Newton-Raphson method is used primarily as it is recommended in the manual [MI] with the use of SHELL 181 and BEAMI88 elements. The arc-length method uses spherical iterations to determine the solution. Only suitable results have been found in applying the total load and letting the automatic time-step algoritlun determine the number and step of the load step.

F

Step size

u (a) Newton-Raphson iteration

Step size

F

Iterations

u (b) Modified Newton-

Raphson iteration

Figure 148 Iteration procedures

Step size

F

u

(c) Arc-length iteration

103

element models

E.5.2.2. Loading A concentrated load can be applied using two methods either a force or a displacement. When a force the

pla,cenaenlts are calculated. If performed with a a reaction force is obtained for the applied displacement. However GMNIA, the load-displacement curve will have a horizontal at the limit point and stop to converge. However non-convergence is not a sign that the limit point is reached. This is not favourable due to the number of simulations that needs to be performed. Using displacement controlled load the limit can be reached without non-convergence and show the after-critical behaviour. Therefore

control is used when possible.

Applying a distributed load is uncommon and large effort to achieve. To be able to obtain the limit point a different approach is followed. Making use of the arc-length method makes it

to obtain the limit and the after-critical behaviour. This method however is strongly on the numerical stability of the asymmetry) and is used with care.

E.S.2.3. Convergence criteria When results are the iteration process is The criterion that whether the results are satisfactory is known as the convergence criterion. The solution is when the difference between the inputted force (displacement) and the calculated force (displacement) is smaller than the criterion that has been set. In the simulations the force convergence criterion is set to 0.5% of the applied force and a U'''IYl<',\''vll1vlU

criterion of 1 %.

E. 6. Input file The ANSYS allows use of so-called input-files. are of following which create the FE-model. These files are the equivalent automated version of the procedures used to graphically implementing the model. The model is created the ANSYS Parametric Design (APDL) which allows the use of loops (*do) and (*if,). this, a model is created depending on a number of parameters. As an one of the input files used is listed below. Comments in the files are by an exclamation

by the software.

104

Lateral Torsional steel beams

INPUT FILE Listin : Variables and PARAMETERS MODEL 1 IMP =1/1000

L = 7200

LN = 1 LXI 3600 LYI 115.1

RN = 1 RXI 900 RYl 115.1

2 Xl RXI X2 LXI

SPRINGSTIFFNESS CRn:' 95

10000

!Model number !Amplitude of imperfection

!span

'Number of loads (add LX~ and LY~ accordingly) !Location of load ~1 !Vertical location of load #1

!Number of restraints (add RX# and RY# accordingly) !Location of restraint over length !Vertical position

!Number of stiffeners(add X# accordingly) !Location of stiffener# over length (for restraint) Location of stiffener#2 over length (for load)

!Springstiffness of restraint !99% 95% of Mcr, used in naming convention of files

TU/e

MH 8 !Number of elements in each of the flanges, must be an even value, and greater of equal to 4

MW 8 !Number of elements in the web, must be an even value, and greater of equal to 4

M = 72 !number of elements over the length of the model

!Cross-sectional properties for an IPE240 TF 9. !Flange thickness TW = 6.2 !web thickness HEIGHT 240 !height the real section WIDTH 120 ! width of the section

!Properties for compensation db 25.8119 bb 58.20583 tbuis 0.343917

element from l't order elastic !Height of RHS element !Width of RHS element !Thickness of RHS element

analysis

!Variables fy 235

for the material properties

E 210000 POlS O.

!Parameters for nodes H2 = (HEIGHT-TF) I 2

. W2 = WIDTH /2 'PART = LIM

fprep7

!Yield strength !Young's modulus of elasticity !Poisson ratio

and elements !distance from centroid to 1/2 thickness of flange !1/2 width of section !Distance between nodes

!Element type for the flanges and web !Reduced integration scemme

!Defining elements ET,1,SHELL181 KEYOPT,l,3,0 ET,2,BEAM188 KEYOPT,2,l,1 KEYOPT,2,2,1 KEYOPT,2,3,0

!Element used for compensation element and stiff elements at support !Warping included in deformation

ET,3,COMBIN14 KEYOPT,3,3,0

ET,4,MESH200

KEYOPT,4,1,2

SECTYPE,l,SHELL SECDATA,TF,1,,9

SECTYPE,2,SHELL iSECDATA,TW,1,,9

ISECTYPE,3,BEAM,ASEC 0.00001 10**8

!Section is assumed rigid (no axial stretching) !Linear polynomial shape function (no internal node)

!Element used for restraint !3 dimensional longitudinal spring-damper

!Dummyelement, has no influence on results (used to apply zero springstiffness for LBA for imperfection shape)

!2Node 3 dimensional element

!properties for flanges 'Thickness of shell, material 1 and 9 integration points

!properties for web !Thickness of shell, material I and 9 integration points

the elements at the supports

105

106

models

!A ,ry ,1pol, 1z, Iw, It , shear distances

4,BEAM,HREC !properties of compensation elements BB,DB,TBUIS,TBUIS,TBUIS,TBU1S

!properties for stiffeners !Thickness of shell, material and 9 integration points

!Springstiffness of restraint, no damping

!Material properties !Material properties

of cross-section

NN n,k ,i ,-H2 ,-W2 n,k+MW,i ,-H2 ,+W2 n,k+MW+MH,i,H2,-W2 n,k+2*MW+MH,i,H2,W2 FILL,k,K+MW,MW-l,

!Startup !Looping 'Using a !Node in !Node in

value i from to in steps of PART second trigger for the numbering of nodes lower left corner lower right corner

!Node in upper left corner !Node in !Filling

upper right corner the nodes between corners

FILL, k+MW+MH, k+2*MW+MH,MW-l, FILL,K+MW/2,k+l.5*MW+MH,MH-l,K+MW+l,l,l,l

elements for cross-section

,1,NN*M,NN MAT = 1 TYPE,l SECNUM,l

Looping steps of NN (number of nodes in each "section" !Material number 1, used for section only (will incorporate yielding) !Element type 1 !Section number 1

*do,k,l,MW,l !Loop for the bottom flange e,k+i-l,k+i-l+NN,k+i-l+l+NN,k+l+i-l

*enddo

SECNUM,2 ! Section number 2, (web) e,i+MW/2,i+NN+MW/2,i+NN+MW+l,i+MW+l *do,k,2,MH-l,1

e,k+i-l+MW, 1+MW+NN,k+i-l+l+MW+NN,k+i-l+1+MW *enddo e,i+MW+MH-l,i+MW+MH-l+NN,i+l.5*MW+MH+NN,i+l.5*MW+MH

SECNUM,l !Section number 2 *do,k,l,MW,l !Loop for the upper flange

e,k+i-l+MW+MH,k+i-l+NN+MW+MH,k+i-l+l+NN+MW+MH,k+l+i-l+MW+MH *enddo

ELEMENTS

AT SUPPORTS

!Material type 1 Element type 2

!Real constant 2 (preventing error messages) !Section number 4

!Element type 2 !Real constant 3 !Section number 3 !Material 2

elements of flange

!Element at intersection bottom flange and web !Looping elements of web

Lateral steel beams TU/e

e,MW+MH,+MW+MW*3/2+1,MW+MH+NN !Element at intersection upper flange and web *do,i,MW+MH+l,MW*2+MH,l !Looping element of upper flange e,i,i+l,i+NN *enddo

*do,i,MW+MH+l,MW*2+MH,l e,i+NN*(M),i+l+NN*(MI,i+NN *enddo e, (M) *NN+MW/2+l,MW+2+NN* (M) ,MW/2+l+NN *do,i,MW+2+NN*(MI,MW+MH-l+NN*(M),l e,i,i+l,i-NN *enddo e,MW+MH+NN*(M),NN*(M)+MW*3/2+MH+l,MW+MH+NN *do,i,l+NN*(M),MW+NN*(M), e,i,i+l,i-NN *enddo

!STIFFENERS

*do,a,l,SN, MAT, 2 TYPE,l REAl,l SECNUM,

*GET,N,NODE,O,NUM,MAXD, Xl X%a% Nl N+l MI XI/PART*NN

!Repeating procedures depending on number of stiffeners !Material model 2 (no yielding) !Element type 1 !Real constant !Section number 5

!Obtain largest node number !Location of restraint depending on step !Add I to largest node number !Node number of cross-section at location of stiffener

the nodes for the stiffeners left hand side of section g *do, k, 0, (MH-l) *MH/2-1,MH/2 g g + 1 f ~ MH/2+1

*enddo

g

*do,i,Nl+l+k,Nl+{MW)/2+k,l

*enddo

f f 1 n,i,Xl,H2*(g)/(MH/2),-W2*(f)/(MW/2)

the nodes for the stiffeners right hand side of section

*do,k,O, (MH-l)*MH/2-1,MH/2

*enddo

g g + 1

f ° *do,i,Nl+l+«MH+l)*(MW+l)-NN)/2+k,Nl+«MH+l)*(MW+l)-NN)/2+(MWI/2+k,

*enddo

f = f + 1 n,i,Xl,H2*(g)/(MH/2),W2*(f)/(MW/2)

!Defining the elements in the intersection of web and flanges e,Ml+MW/2+1,Ml+MW+2,Nl+MW/2,Ml+MW/2 e,Ml+MW/2+2,Nl+l+«MH+l)* (MW+l)-NN)/2,Ml+MW+2,Ml+MW/2+1 e,Nl+«MH+l)*(MW+l)-NN) (MH-2)/2,Ml+MW*3/2+MH+2,Ml+MW*3/2+MH+l,Ml+MW+MH e,MI+MW+MH,Ml+MH+MW*3/2+1,Ml+MH+MW*3/2,Nl+«MH+l *(MW+l)-NN)/2

!Defining the elements at the bottom flange *do,i,l,MW/2-1,l

e,Ml+i,Ml+i+I,Nl+i+l,NI+i *enddo *do,i,O,MW/2-2,l

e,Ml+i+MW/2+2,Ml+i+3+MW/2,Nl+i+2+«MH+l)*(MW+l)-NN)/2,Nl+i+«MH+l) (MW+l)-NN)/2+1 *enddo

'Defining the elements at the top flange *do,i,O,MH/2+l+(MH/2-4),l f MH/2

e,Nl+f*(i+l),Ml+i+MH+2,Ml+i+MH+3,NI+(i+2)*f *enddo

*do,i,O,MH/2+1+(MH/2-4),1 f = MH/2

e,Nl+«MH+l)*(MW+l)-NN)/2+l+f*i,Nl+(MH+l)*(MW+l)-NN)/2+1+f*(i+l),Ml+i+MH+3,Ml+i+MH+2 *enddo

!Defining the elements at the web *do, ,O,MW/2-2, e,Nl+«MH+l)*{MW+11-NN)/2-(MH/2)+1+i,Nl+({MH+l)*(MW+l)--NN)/2-(MH/2)+2+i,Ml+MH+MW+2+i,Ml+MH+MW+l+i

107


*enddo *do,i,0,MW/2-2,l e,Nl+((MH+l)*(MW+l)-NN)-MW/2+l+i,Nl+((MH+l)*(MW+l)-NN)-MW/2+2+i,Ml+MW*3/2+MH+3+i,Ml+MW*3/2+MH+2+i *enddo

!Filling the rest o f the elements *do ,k,0,MH-3,l

*do,i,0,MW/2-2,1 e ,Nl+l+i+(k*MW/ 2),N l+i+2+ (k*MW/2) ,Nl+i+MW/2+2+(k*MW/2),Nl+l+i+MW/ 2 +(k *MW/2 )

*enddo *enddo

*do, k, 0,MH-3, 1 *do,i,O+( (MH+l)*(MW+l)-NN)/2,MW/2-2+((MH+l)*(MW+l)-NN)/2,l

e,Nl+l+i+(k*MW/2) ,Nl+i+2+(k*MW/2 ) ,Nl+i+MW/2+2+(k*MW/ 2 ) ,Nl+l+i+MW/2+ (k *MW/2 )

*enddo *ENDDO

*enddo

!Elastic restraint *do,a, 1,RN, 1 N,29999+ a ,RX%a% ,RY%a%,lOOO D,29999+a,UZ

Ml M2

RX%a%/PART*NN (RY%a%+H2)/(H2*2)*MH+MW+l

*if,M2,EQ,MW+l,THEN M2 = MW!2+1 *ENDIF

*if,M2,EQ,MW+MH+l,THEN M2 = MH+MW*3/2+l *ENDIF

TYPE,4 REAL,7 E,Ml+M2,299 99 +a

CP,NEXT,UX,Ml+M2,29999+a CP,NEXT,UY,Ml+M2,29999+a *enddo

!Bo undary conditions at supports !Support 1 D,MN,UX,O D,MN,UY,O D,MN,UZ,O D,l+MW/2,UZ,0 D,1+MW*1 .5 +MH,UZ,0

!Support 2 D,MN+NN*M,UY,O, D,MN+NN*M,UZ,O, D,MW/2+1+NN*M,UZ,0 D,l+MW*1.5+MH+NN*M,UZ,O

!Intermediate loads *do,a,l,NL,l,

Ml M2

LX%a %/ PART*NN (LY%a %+H2)/(H2*2)*MH+MW+l

*if,M2,EQ,MW+l,THEN M2 = MW/2+1 *ENDIF

*if,M2,EQ,MW+MH+1,THEN M2 = MH+MW*3/2+l *ENDIF F,M1+M2 ,FY,-l/NL *enddo

FINISH

108

!Node at location RX#,RY# and 1000 !Boundary condition of node

!Node number at location of restraint part !Node number at locat ion of restraint part 2

!Exclusion if l ocat i on is at web-flange intersection

!Element type 3 !Real constant 6

!Coupling of displacement of either end of restraint !Coupling of displacement of either end of restraint

!Centroid !Centroid !Centroid !Lower flange intersection !Upper flange intersect ion

!Centroid !Centroid !Lower flange intersection !Upper flange intersection

!Node number at location o f restra int part 1 !Node number at location o f restraint part 2

'Exclusion if location is at web-flange intersect i on

!Load total will equal 1

Lateral

INPUT FILE Listing: Linear buckling analysis (Performed twice with and without restraint) /FILNAME,%MODEL% %R¥l% %RXl%_K%CRIT% %SPRINGSTIFFNESS% LBA !Static analysis

!Changing filename

/SOLU NSUB,l ANTYPE,STATIC PSTRES,ON SOLVE FINISH

!Buckling analysis /SOLU ANTYPE,BUCKLE BUCOPT,SUBSP,l SOLVE FINISH

/POSTl SE'I', LIST SET, LAS'!' PLDISP

!Updating geometry NSORT,U,Z, *GET,UMAXLOC,SORT,O,IMAX, *GET,IMPFAC,NODE,UMAXLOC,UZ

*GET,UMAXLOCl,SORT,O,IMIN, *GET,IMPFACl,NODE,UMAXLOCl,UZ

IMPFACI -l*IMPFACl

*if,IMPFAC1,GT,IMPFAC,THEN IMPFAC = IMPFACI UMAXLOC UMAXLOCI *endif

FINISH

!Static analysis !Pre-stress effects

!Analysis, buckling !Use subspace iteration

!Post processing !List result !Take last result !Plot deformed shape

!Sorting the out-of-plane displacements !Getting greatest displacement location !Getting greatest displacement

'Getting smallest displacement location !Getting smallest displacement

!Smallest displacement * -1

!Finding greatest displacement

/FILNAME,%MODEL%_%RYl% %RXl% K%CRIT% %SPRINGSTIFFNESS% LBA REAL /PREP7

!Changing filename

*get,RESTRAINT,ELEM,O,NUM,MAXD EMODIF,RESTRAINT,TYPE,3 EMODIF,RESTRAINT,REAL,6

!Static analysis /SOLU NSUB,l ANT¥PE,STATIC PSTRES,ON SOLVE FINISH

Buckling analysis /SOLU ANTYPE,BUCKLE BUCOPT,SUBSP,l SOLVE FINISH

/POSTl SET,LIST SET,LAST PLDISP

SET,last *set,eigenvals,O *GET,EIGENVALS,ACTIVE,O,SET,FREQQ

FINISH

!Get highest elem number (that of the restraint) !Changing this elements type (MESH200 to COMBIN14) !Changing real constant (7 to 6 (springstiffness)

!Static analysis !Pre-stress effects

!Analysis, buckling !Use subspace iteration

!Post processing !List result Take last result

!Plot deformed shape

!Setting variable to zero ! Setting variable to value obtained

/FILNAME,,,,MOflRT,'" _%RYl% %RXl% K%CRIT% %SPRINGS'l'IFFNESS% !Changing filename

TUie

109

","","pn,C!1V E Finite element models

/PREP7 UPGEOM, (l/IMPFAC)*(IMP*L)", '%MODEL%_%RYl%

geometry according to buckling shape !Updating

FDEL,ALL

TB,BKIN TBTEMP,20 TBDATA, 1, fy, 0

/SOLU ANTYPE,STATIC NLGEOM,ON NROPT,FULL ARCLEN,OFF CNVTOL,F"O.005,,1000 NEQIT,lOO NSUBST,10

:OUTRES,ALL,ALL

ESEL,S",l,1728 ISFILE, READ, 'isfileI728', 1st" 0, esel,all SOLVE CNVTOL,U"O.OI,

!Load steps : *do, ,10,120,10 :d,MI+M2, -I SOLVE

:*enddo

110

!Removing of loads

!Material non-linearity

!Static analysis !Geometric non-linearity on !Full Newton-raptson !No arc-length method !Converengence criteria 0.5% of force, with a minimum of 1000 !MAXIMUM ITERATIONS !NUMBER OF SUBSTEPS !OUTPUT all results

!Select first 1728 element (shells only) !Read initial stress file !Select all elements SOLVE

!Converengence criteria 1% of displacement

!Looping of application of displacements !Applying load !Solve

Lateral Torsional restrained steel TU!e

E. 7. Linear buckling analysis using BEAM elements Using ANSYS a linear buckling analysis has been np"j-",'",,·/1 in this appendix the file is on how this

been pelfwrmeu.

IPREP7 !Pre-processor !Material properties E=210000, G = E/(2(1+1/3) !Element type BEAMI88, a two node beam element

UIMP,1,EX"PRXY,210000,,0.3 ET,1,BEAM188 SECTYPE,l,BEAM,ASEC SECDATA,3912,38920000

!cross-section definition for the application of ,0,2836000,3.668*10**10 ,127400 , 0, 0, 0, 0 !Properties

!Area, Iy,Iyz,Iz,Iw,It, !Using 7 degrees of freedom including warping

the load height of the beam,

KEYOPT,l,l,1 KEYOPT,1,3,2 KEYOPT,1,2,1 SECTYPE,2,BEAM,RECT SECDATA,SO,SO

!cross-section definition for the application of the load height !Stiffness of the previously mentioned element

L 1200 M 12 PART LIM

n,l,O,O,O n,M+l,L,O,O FILL n,M+2,0,-1,0

*do,i,l,M,l e,i,i+l,M+2 *enddo

D,l,UX,O D,l,UY,O D,l,UZ,O D,l,ROTX,O D,M+l,UY,O D,M+l,UZ,O D,M+l,ROTX,O

SECNUM,2

n,M+3,3600,11S.1 e,M+3,3600/PART+l, F,M+3,FY,-1

/SOLU ANTYPE,STATIC PSTRES,ON SOLVE FINISH

/SOLU ANTYPE,BUCKLE BUCOPT,SUBSP,l SOLVE FINISH

! /POSTl

I'SET,LIST SET, LAST

: PLDISP

lLength of the model !number of elements !Length of the elements

!first node !last node !nodes between first and last node !orientation node

!defining elements

!Boundary condition on first node

!Boundary condition on last node

!Using the second cross-section

!Node for the load height !element for the load height !force on the node

!Solver !Static analystic !Prestresses on !Solving

!Solver !Buckling analysis !Mode extraction method, with 1 node !Solving

[Post-processor !List results !Open last result !Plot displacement shape

111


Appendix F. Plastic St-Vernant Torsional capacity. The calculation of the plastic St-Vernant torsional capacity can be performed using Nadai's Sand-heap analogy (Figure 149). The plastic torque is given by (F. 1). The stress function can be detennined using the Sand-Heap analogy.

Tp/ = 2 (Vohune under stress function) (F.I)

Fully Plastic CondItion

Figure 149 Graphical representation of the Sand-heap analogy figure obtained from [5)

The sand heap figure of the cross-section with equivalent roots is shown in Figure 151, where equivalent roots have been calculated (see Figure ISO). To calculate the volume the section is divided twice, once thru the major axis and once thru the minor axis. This leads to 1I4th of the total volume. The section that is left have the division is once again divided in 4 parts. For three of those parts the volume is easily calculated. For a single part (Web-flange intersection and the root-radius) this is a bit more complicated. The different parts are shown in Figure 152.

Figure 151 Sand-heap analogy applied to model with equivalent roots

req

~~, Equivalent area Figure 150 Equivalent roots

113

Appendix F Plastic St-Vemant Torsional capacity.

Part 4

Figure 152 Parts in which tbe volume can be divided.

For part 1 the volume is calculated according to:

1 lit fl3 V. =-t ·-t ·t -=

I 2 fI 2 fI fI 3 12

For part 2 the volume is calculated according to:

Part 3

V. =!(t .!t .(!b-r _!t _!t )J=!t 2(!b-r _!t _!t) 22 fl2f1 2 eq 2f1 2 w 4f1 2 e<J 2f1 2 w

. . ~ -4r2 - ffr2 reg = eqwvalent root-radIUs length = J2

Part 4

Figure 153 Part 4

114

Volume 8 Volume 6

(F.2)

(F.3)

(F.4)

For part 4 more work is needed to compute the volume. First volume 4 is calculated.

1 1 1)2 1 ( 1 V4 +tjl +2 t w hpyromid "6 r~ +tjl + 2 ·0.414 (F.5)

The height of the pyramid is unknown. Two This angle of the diagonal is then:

are known, the angle perpendicular to the "root" equals 45.

9'= tan-l (sin(45°)tan(45")) 35.26°

Using the sine rule the height of the

c

1 + tjl +

2 sin(35.26 +

can be calculated.

this two volumes (Volume 5 and 6 are subtracted, VI"'"''_''' are already ac(:oull1t~~d for. pyramids have a of:

1 1 2 Vs:::: 0.4141j1

32 0.414 3

12

0.414 3 --I

6 jI

Hereafter two small volumes (volumes 7 and 8) are added, these have a volume of:

0,1701/ 0.01

By adding and subtracting each nr"'"n,.".lrp volume. The total volume can be calculated. Which result in:

V::::4(V;+V2+V3+ V4- VS V6 +V7 +VS )

The Saint-Vemant torsional moment is the calculated as:

+ ..... 1384(t 3 +t 2(~_~J\)+ jI w 2 2.fi 2 2

1 fl }. - 0.11761.'

substituting the cross-sectional ,."rn,,,,p"11 of an IPE240 (F.9), leads to the

~)I 184681"

(F.7)

(F.8)

(F.9)

115

Appendix G. method.

Lateral Torsional

Development

steel beams TU!e

a tool to calculate Mer using the Ritz

Using Mathematica 5.2 a tool Lateral Torsional Buckling-Tool) was developed to calculate Mer and its buckling mode for any moment distribution including restraints. In the case oflateral-torsional buckling two trial functions have to be chosen: one for the rotational field and one for the displacement field. These need to be substituted in the energy First the energy equations are and then the trial an illustration of the method is shown; this is for an umestrained and exact load case. After this the tool will be tested against the LT-beam program to check whether it functions

G.1. Energy equations The energy equations have been obtained from Trahair is by:

The work equation is given by

I L

2" f{ElzuII2 +

o

~82V =~ Lf2M k."dz+ 1 2 2 xV'''' 2

o

here 3 equations are The strain energy

1)

(G.2)

In which Mx the function for the bending moment, Qy the concentrated load, Y Q ! Yq is distance of the load to the

shear centre, qy is the distributed load.

And the work performed by the restraints is:

1 1

Where:

{d} = {U,¢}T

[ab ] [:g 2 ]

In which Ksp elastic stiffness of restraint

These equations are then added and neutral equilibrium occurs when:

~82(U + + V) =0 2

G.2. Trial functions In the equation (G.l) thru(G.3) two trial functions have to substituted, the accuracy ofthe method

(G.3)

(GA)

depends on the trial function. These equations have to comply with the kinematic boundary these are in the case of simply supported the displacements and rotations at the supports are equal to zero.

UO,l = CPo'! =: 0 (G.5) In [P2] a similar tool was made, here series of sine where used as trial With these series the kinematic boundary conditions are satisfied. The use of these series also gives good approximation of Mer as is shown 111 The LTB-Tool program allows for any number of sines to be set. For accuracy purposes the number is set to 25.

This leads to the following trial functions:

(G.6)

en

117

Ritz method.

G.3. Illustration To illustrate the mel[llOIJ, a uniform moment diagram is taken (c::::J) as for this case an exact solution is known for.

For this the moment diagram is:

The trial function will be taken as a single sine term:

M= x

::::: o-sin(.rrz) I

lP(z) =

(G.7)

(G.8)

Substituting (G.7) and in (G.1) and where the Q and q equal zero, lead to:

2

1 M.rr 2o-e 2 2 41

2 (U+V)

2

+V) ~{~r

Solving this to zero is done the non-trivial solution. By is the determinant of the Matrix and solving it for zero will lead to the solution.

2/ o

is obtained:

This is the same equation as the one that Timoshenko has derived for this case.

118

Lateral Torsional beams TU/e

GA. Testing the tool In order to check if the tool (denoted as LTB-Tool) works accurately, it will be tested. This is done using the LTbeam program. In total 4 moment distributions will be tested, the uniform bending diagram case 1), a non-linear diagram a uniform load at the shear center case 3) and a single concentrated load on the top at ,case Also 4 restraint will be at Y4 span at the top flange and centroid, and at Yz span at the bottom flange and Y2 the distance between the centroid and bottom with a

"t,ttnp"" of 500kN/m. The span of the beam is taken as 7200 and the beam is an IPE240

T hi 23 R a e f h fi esu ts 0 t e lrst two restrammg cases

Restraint location quarter-span top quarter-span centroid

LTB-Tool LT-BEAM diff LTB-Tool LT-BEAM diff Case 1 69.945 69.946 0.0% 69.350 69.348 0.0% Case 2 109.306 109.3 0.0% 101.282 101.28 0.0% Case 3 84.655 84.654 0.0% 79.893 79.893 0.0% Case 4 81.787 81.782 0.0% 63.663 63.655 0.0%

Table 24 Results of the last two restraming cases Restraint location

mid-span bottom !mid-span Yz bottom to centroid LTB-Tool LT-BEAM diff LTB-ToolILT-BEAM diff

Case 1 56.302 56.299 0.0% 65.978 65.955 0.0% Case 2 100.98 100.98 0.0% 98.061 98.059 0.0% Case 3 63.066 63.064 0.0% 74.423 74.4 0.0% Case 4 47.938 47.936 0.0% 55.68 55.662 0.0%

Here it can be seen that the tool the computation of the elastic critical moment very the rest of the use of this tool, extra comparisons will be made to assure accuracy of the tooL

During

119

Lateral Torsional Buckling oflaterally restrained steel beams

Appendix H. Results of simulations without restraints Coding of simulation is performed as follows: M Denotes the model number L Denotes the span UR Unrestrained Xrn position along x axis of restraint number n

Ym position along y axis of restraint number n Kn Springstiffness of restraint number n IS Denotes the impelfection shape (when used)

num Simulation code

Ml L7200 UR 2 Ml L5400 UR

3 Ml L3600 UR

4 M2 L7200 UR 5 M2 L5400 UR

6 M2 L3600 UR

7 M3 L7200 UR 8 M3 L5400 UR 9 M3 L3600 UR

Force [kN) 20 .

15

o

Fu[N]

19012 30978 60160

qu 4.706 10.320 30.460

Fu 32200 49254 90988

Mcr;NUM Mcr;ANA

[kNm] [kNm] 40.282 40.340 53.347 53.539 82.200 82.999

34.987 35.063 46.652 46.881 72.511 73.493

44.970 46.042 57 .925 58.980 85.098 86.875

1 I

--In-pi,.,.,

--out-ol·pl,.,.,

o 15 30 45 60 75 90 1>5 120 Displacement [mm)

Figure 154 Ml L 7200 UR

60

50

40

30

20 1> --hpl,.,.,

--out-ol-pl,.,.,

o t--~--__ --~--_~==~==~==~ o 15 30 45 60 75 90 1>5 12

Displacement [mm) -------------------------------~

Figure 156 Ml L3600 UR

XGMNIA XGM XSM XSM;MOD

0.397 0.389 0.448 0.444 0.485 0.488 0.563 0.541 0.628 0.652 0.734 0.688

0.354 0.345 0.399 0.399 0.437 0.440 0.500 0.495 0.573 0.608 0.666 0.649

0.505 0.434 0.512 0.489 0.579 0.525 0.628 0.574 0.713 0.669 0.796 0.702

Fo~~e [kN)

30

20

10 --In-plane

--out-ol-plane

o 15 30 45 60 75 90 1>5 12 Displacement [mm)

Figure 155 Ml L5400 UR F'!;rce per unll length [kNfm)

o 15 30 45 60 75 90 105 12 Displacement [mm)

Figure 157 M2 L7200 UR

TU/e

121

Appendix H Results of simulations without restraints

F~{ce per unit length (kN/m]

~ ~~~~~~~---------9 8 7 6

4

3 2 --I!)-plane

--out-ol-plane

0!---~--~--------~====~===1 o 10 20 30 40 50 60

Displacement (mm]


F".{8" (k~]

32

24

:!-__ 4-__ -r __ --1_++ _·-_-=~-=-=-=-=ru=I~~=I-=PI=aM==~ o 10 20 30

Displacement (mm]


F'lf(e (k~L

56

40

32

24

16

40 50 60

o ~---+----+----+---=~===+====~ o 10 20 30 40 50 60

Dlsplacemenl [mm]


122

F~~ce per unit length [kN/m]

24

16

8

O+---,----_--~-

o 20 30

I --In-plane l --oul-ol-plane I

40 50 I

60 Dlsplacemenl [mm]


r F'!I~e IkN!

L 48

40

32

24

8 t --In-plane

--Qut-of-plane 0

0 10 20 30 40 50 60 Displacement (mm]



Appendix I. Results of simulations for 1 load and 1 restraint

1.1. Tabular overview

Table 25 Results from GMNIA for second load and su~~ort case Reaction

In Spring Restraint

Simulation Simulation Stiffness Imperfection Fer Fu at failure num code [NmmJ shaee [NJ [NJ A /'l; rNJ

10 Ml L7200 UR IS-I 0 UNREST 22369 19003 1.463 0.397 11 Ml L7200 UR KO IS-2 " REST 22369 22665 1.463 0.474 12 Ml L7200 UR KO IS-3 BEAM 22369 19003 1.463 0.397 13 MI L7200xrJ 3600YrJ 115.1 K,100 IS-l 100 UNREST 41951 32026 1.068 0.669 5302 14 MIL 7200 XrJ 3600 YrJ 115.1 K,100 IS-2 REST 41951 37089 1.068 0.775 816 15 Ml LnOO XrJ 3600 l:rJ 115.1 K,100 IS-3 BEAM 41951 31540 1.068 0.659 5849 16 MIL 7200 Xd 3600 YrJ 115.1 K, 200 IS-l 200 UNREST 56834 39142 0.918 0.818 7828 17 Ml L7200xrJ 3600YrJ 115.1 K,200 IS-2 REST 56834 44304 0.918 0.926 1404 18 Ml L7200xd3600l:rlI15.1 KI 200 IS-3 BEAM 56834 38137 0.918 0.797 8081 19 MIL 7200 XrJ 3600 YrJ 115.1 K, 300 IS-l 300 UNREST 68997 42913 0.833 0.897 7858 20 Ml LnOO Xd 3600 YrJ 115.1 K,300 IS-2 REST 68997 45199 0.833 0.944 57 21 Ml L7200 XrJ 3600 l:rJ 115.1 KI 300 IS-3 BEAM 68997 41483 0.833 0.867 8595 22 Ml L7200xrJ 3600YrJ 115.1 K,400IS-1 400 UNREST 79110 44836 0.778 0.937 7685 23 Ml LnOO XrJ 3600 YrJ 115.1 KI400 IS-2 REST 79110 45199 0.778 0.944 20 24 Ml L7200 XrJ 3600 l:rJ 115.1 KI400 IS-3 BEAM 79110 43050 0.778 0.899 9296 25 M 1 LnOO Xrl 3600 Yrl 115.1 K, 438 IS-l 438 UNREST 82512 45317 0.762 0.947 7797 26 MIL 7200 XrJ 3600 YrJ 115.1 K, 438 IS-2 REST 82512 45199 0.762 0.944 16 27 Ml L7200xrJ3600l:rlI15.1 KI438 IS-3 BEAM 82512 43397 0.762 0.907 9184 28 Ml L7200 XrJ 3600 YrJ 115.1 KI 480 IS-l 480 UNREST 85972 45729 0.746 0.955 7478 29 MIL 1200 XrJ 3600 YrJ 115.1 K, 480 IS-2 REST 85972 45199 0.746 0.944 14 30 Ml LnOO XrJ 3600 l:rJ 115.1 KI 480 IS-3 BEAM 85972 43675 0.746 0.913 9037 31 Ml LnOO Xd 3600 YrJ 115.1 KI 500 IS-l 500 UNREST 87426 45885 0.740 0.959 7332 32 Ml L7200 xrJ 3600 YrJ 115.1 K,500 IS-2 " REST 87426 45199 0.740 0.944 13 33 MI L7200 XrJ 3600 l:rJ 115.1 K,500 IS-3 BEAM 87426 43656 0.740 0.912 7218 34 MIL 1200 Xd 3600 YrJ 115.1 K, 600 IS-l 600 UNREST 87811 46430 0.738 0.970 6716 35 MIL 1200 XrJ 3600 YrJ 115.1 KI 600 IS-2 REST 87811 45199 0.738 0.944 11 36 Ml L7200 Xd 3600 l:rJ 115.1 KI600 IS-3 BEAM 87811 43127 0.738 0.901 209 37 Ml L 1200 XrJ 3600 YrJ 115.1 K, 1000 IS-l 1000 UNREST 87816 47208 0.738 0.986 5458 38 Ml L 1200 XrJ 3600 Yd 115.1 KI 1000 IS-2 " REST 87816 45199 0.738 0.944 8 39 Ml LnOO XrJ 3600 YrJ 115.1 K, 1000 IS-3 BEAM 87816 43097 0.738 0.900 44

1.2. Load-displacement graphs

Fo;ge [kN}

I Force [kN}

50

40

I I

30 I

.~ I .. 20 -- -- L - - !

"/ I I

-in-plane

·-out·of-plare

0 Force in restraint

40

:~~ I

-=--

-in-plare

-+ - out·of-plane

I Force in restraint O~"

0 15 30 45 60 7 ~u 1Uo UU Displacement [mm}

0 15 30 45 60 75 90 "'1)5 120 Displacement [mm}

-Figure 163 M1 L7200 UR IS-1 Figure 164 M1 L 7200 XrJ 3600 Yri 115.1 K.100 IS-1

123

Appendix I Results of simulations for 1 load and 1 restraint

Force [kN] 50

40

30

20

15 30 45 60 75 Displacement [mm]

--In-plane

--out-aI-plane

--Force in rest raint

90 1J5 12

Figure 165 M1 L 7200 X r] 3600 Yr] 115.1 KJ 200 IS-1 Force [kN]

50

40

30

20

10 --In-plane

--out-aI-plane

--Force In restraint o ~ ___ --~----_--~--~L-__________ ~

o 15 30 45 60 75 90 105 12 Dtsplacement [mm]

Figure 167 Ml L7200xr] 3600Yr] 115.1 KJ 400 IS-1 Foroe[kNJ

50

40

30

20

10 --In-plane

--out-aI-plane

o ~ ____ -,~_~ ____ ~~~~~~Fo:r:ce~ln:r~es:tr~a:'nt~

o 15 30 45 60 75 90 105 12 Dlsptacement [mm]

Figure 169 M1 L 7200 X r] 3600 Yr] 115.1 KJ 480 IS-1 F°50ce[kN]

40

30

20

10 --In-plane

--out-aI-plane

o -l"'---:----::;;---:;---;;;---::J:-:-=-=~~~Fo:..:r~ce~ln~r.:es~tr~ai:Jnt o 15 30 45 60 7

Displacement [mm]

Figure 171 Ml L 7200 X r] 3600 Yr] 115.1 KJ 600 IS-1

Force [kN] 50

,

40 , ,

30 I

"/ I ;

:

'KJ -

I --irrplane

--out-aI-plane o , ~ ~ -- ~

0 15 30 45 60 7~ Displacement Imm

90 105 120

Figure 173 M1 L7200 UR IS-2

124

Force [kNJ 50

40

30

20 --in-plane

--out-aI-plane

O~'------~----_--_--__ '---_-_--rl o 15 30 45 60 75 90 1J5

Displacement [mm]

Figure 166 Ml L 7200 X r] 3600 Yr] 115.1 KJ 300 IS-l Force [kN]

50

40

30

20

--In-plane

--out-aI-plane

--Force in restraint O~--~--~~--~--~L-__ ~~~~

o 15 30 45 60 75 90 'KJ5 120 Displacement [mm]

Figure 168 Ml L 7200 X r] 3600 Yr] 115.1 KJ 438 IS-l FOri: IkN]

40

30

20

--In-plane

--alA-aI-plane

O~-_-~ --Force In restraint

o 15 30 45 60 7 Displacement [mm]

Figure 170 Ml L7200xr] 3600Yr] 115.1 KJ 500 IS-l

40

30

20

--in-plane

--out-ol-plane

o -I<---:-~;;---:;---;;;---::J:-:-=-=~~~Fo:..:r~ce~in~r::es~t r~a:lnt~ o 15 30 45 60 7

Displacement Imm]

Figure 172 Ml L7200xr] 3600Yr] 115.1 KllOOO IS-1

Force [kN] 50

l I 40 ~

30

L I -- ,

I

"---4 20

--In-plane

'KJ J.. ---out-aI-plane

- ---Force in restraint I 0

0 15 30 45 60 75 90 105 120

1

o Isp lacement [mm]

Figure 174 M1 L7200 X r ] 3600 Yr]l1S.1 K I 100 IS-2


Force [kN] 50

40

30

20 --In-plane

--ol1-ol-plane

--Force in restraint

o 15 30 45 60 75 90 105 120 Displacement [mm]

Figure 175 M1 L noo XrJ 3600 YrJ 115.1 K t 200 IS-2 Force [kN]

50

Displacement [mm]

Figure 177 M1 L noo XrJ 3600 YrJ 115.1 K) 400 IS-2 Force [kN]

50

40

30

o 15 30

--in-plane

--out-ai-plane


45 60 75 90 105 12 Displacement [mm]

Figure 179 M1 L noo XrJ 3600 YrJ 115.1 K) 480 IS-2 Forc~kN]

40

30

20

--In-plane 10

--out-ai-plane

O*--~--~-__ --__ --~ _ ______ ~ o 15 90 105

Figure 181 M1 L noo XrJ 3600 YrJ 115.1 K) 600 IS-2

Force [kN] 50

40

30

20

"~ I 0 - , , o 15 30

I!,I -- in-plane

II --out-ai-plane i

45 60 75 o isp lacement [mm) 90 105 120

Figure 183 M1 L noo UR IS-3

Force [kN] 50

40

30

20

o 15

--In-plane

--out-ol-plane

--Force In restraint

30 45 60 75 90 105 12 Displacement [mm]

Figure 176 M1 L noo XrJ 3600 YrJ 115.1 K)300 IS-2 Force [kN[

50

40

30

--In-plane

--oul-ol-plane


45 60 75 90 105 120 Displacement [mm]

Figure 178 M1 L noo Xrl 3600 YrJ 115.1 K\ 438 IS-2 Force [kN]

50

40

30

20

o 15 30

--In-plane

--out-ai-plane


45 60 75 90 105 120 Displacement [mm]

Figure 180 M1 L noo XrJ 3600 YrJ 115.1 K1SOO IS-2 Forge [kN]

40

30

20

--In-plane

10 --out-ai-plane


Figure 182 M1 LnOO XrJ 3600 YrJ 115.1 K)1000 IS-2

Force [kN] 50

40

30

20

o 15 30

--in-plane

--out-ai-plane


45 60 75 90 105 Displacement [mm]

Figure 184 M1 LnOOxrl 3600YrJ 115.1 K)IOO IS-3

125

Appendix I Results of simulations for 1 load and 1 restraint

Force [kN] 50

40

30

20

o 15 30

--In-pia,..,

--out-ot-pla,..,


45 60 75 90 "XIS 12 o Isp I acement [mm]

Figure 185 M1 L 7200 X,J 3600 Y,J 115.1 Kl 200 IS-3 Force (kN]

50

40

30

20

o 15 30

--In-plane

--out-ot-plare

--Force in restrairt

45 60 75 90 105 Displacement (mm]

Figure 187 M1 L 7200 Xrl 3600 Y,J 115.1 Kl 400 IS-3 Force [kNI

50

--Ir>-plME>

--out-ot-pla....


60 75 90 "XIS 12 Displacement [mm]

Figure 189 M1 L7200 X,J 3600 Y,J 115.1 Kl 480 IS-3 Force [kN]

50

40

30

20 ! --in-plane

I --out-ot-pla....

I --Forcelntestraint

O¥-~--__ --__ ~--~~~~~ o 15 30 45 60 75 90 "XIS 12

Displacement [mml

Figure 191 M1 L7200 X,J 3600 Yrl11S.1 K l 600 IS-3

126

Force [kN] 50

40

30

20 --In-plane

--out-ot-pIMa


o 15 30 45 60 75 90 "XI5 120 Dispiacement [mm]

Figure 186 M1 L 7200 X,J 3600 Y,J 115.1 K l 300 IS-3 Force [kN]

50

40

30

20 --ir>-piare

--out-ot-plane


o 15 30 45 60 75 90 "XI5 12 Displacement [mm]

Figure 188 M1 L 7200 X,J 3600 Y,J 115.1 Kl 438 IS-3 Force [kN]

50

--in-piane

--out-ot-plare

--Forcelnresttalnl

60 75 90 "XIS 12 Dlspiacement [mm]

Figure 190 M1 L 7200 X,J 3600 Yrl 115.1 K1SOO IS-3 Force [kN]

50

40

30

20

o 15 30

--ir>-plane

--out-ot-piME>


45 60 75 90 105 120 Displacement [mm]

Figure 192 M1 L7200 X,J 3600 Y,J 115.1 K 1 1000 IS-3


Appendix J. Results of simulations for 2 loads and 2 restraints

J.1. Tabular overview

Table 26 Results from GMNIA for third load and su~~ort case

Spring Simulation Simulation Stiffness

num code ~mmJ 40M4 L7200 UR IS-l 0

41M4 L7200 UR IS-2 M4 L7200x" 3000 Xd 4200 Y,';' 115.1 K' ;2125 IS-

421 125 M4 L7200 x" 3000 x,, 4200 Y,';' 115.1 K' ;2 125 IS-

432 M4 L7200 x,, 3000 x" 4200 Y";' I 15 .1 K';2 250 IS-

441 250 M4 L7200 x" 3000 x" 4200 y,,;,115.1 K' ;2 250 IS-

452 M4 L7200 x" 3000 x" 4200 y,,;, 115.1 K';2 500 IS-

461 500 M4 L7200x" 3000 x"4200y,,., I 15.1 K';2 500 IS-

472 M4 L7200 x" 3000 x,,4200 Y".1 115.1 K';2 1000

48IS-1 1000 M4 L7200x,dOOO x,, 4200 y"., 115.1 K';2 1000

49IS-2 M4 L7200 x" 3000 x,, 4200 J,j ;' 115.1 K';2 8000

501S-1 8000 M4 L7200x" 3000 x" 4200 y'J;' I 15.1 K' ;2 8000

511S-2

J.2. Load-displacement graphs

,-_ . Fo<ce [1<N]

60

50 I

I 40

30 I

20 ; I

I

10 --In-plare v- --out-ai-plane

0 I

0 15 30 45 60 75 90 105 12C Displacement [mm}

Figure 193 M4 UR 18-1

Force [kNI 60

50

40

30 --in-p''''''

20 --out-ai-pi"'"

10

O~--~--~--~--~--__ ----~--~

o 15 30 45 60 75 90 105 Dlsplacomenl [mm]

Figure 195 M4 L7200 xr/ 3000 Xr2 4200 Yrl;2115.1Kl ;2 250 18-1

Reaction In

Restraint I Imperfection F" Fill Fill at failure

sha~e (NL (Nl (Nl ). ~ ~J

UNREST 23832 10262 10263 1.552 0.357

INT 23832 10262 10263 1.552 0.357

UNREST 64993 22108 22109 0.940 0.770 4410

!NT 64993 21537 21551 0.940 0.750 4547

UNREST 90455 25193 25195 0.7970.877 4268

!NT 90455 24158 24237 0.797 0.843 4997

UNREST 1115192623026231 0.7180.913 3037

INT 11151926646264210.7180.924 8118

UNREST 133878 26623 26623 0.655 0.927 2890

INT 133878 25299 25298 0.655 0.881 3140

UNREST 145169 26846 26845 0.629 0.935 2454

!NT 145169 25522 25522 0.629 0.889 2814

Force [kN1 60

50

40

30 --in-plane

20 --out-ai-plane

Reaction In

Restraint 2 at failure

L!:!J

4410

4547

4270

4959

3033

-9014

2887

3137

2452

2810

10 --Total force in restraints

o 15 30 45 60 75 90 105 12 Displacemont [mm]

Figure 194 M4 L7200 xr/ 3000 xr14200 Yrl;1115.1 Kl;2

12518-1 Force [kN]

60

50

40

30

20

10

o .jL----+---~--~

I '0,'- ' --out-ai-plane

--Total force in restraints

L-__ 0 ____ 15 ____ 3_0 ____ 4_5 ___ 6 __ 0 ___ 7_5 ____ 90 ____ 1O_5 1201 _ Displacement [mm] ~

Figure 196 M4 L7200 xr/3000 xr14200 Yrl;1115.1Kl;2 50018-1

127

Appendix J Results of simu lations for 2 loads and 2 restraints

Force [kN) 60

50

40

30

20

o 15 30

--In-plane

--out-of-plane

-- Total force in restraints

45 60 75 90 105 120 Dlsplacemen! [mm)

Figure 197 M4 L7200 x,J3000 X,2 4200 Y,J;2115.1K1;2 1000 IS-1

Force [l<NJ 60

50

40

30

2~0~1 __ ~ __ r-__ ~ __ ~ ____ ~ ___ l __ ~~~~_i~_p_lane ____ -4 V --out-of-plane

o

Force [kN) 60

50

40

30

20

10

o

15 30 45 60 75 90 105 120 Displacement (mm)

Figure 199 M4 L 7200 UR [S-2

----In-phl'e

--out-of-plane

----To!aI force In restraints

15 30 45 60 75 90 105 120 Dlsplacemen! [mm)

Figure 201 M4 L 7200 X,J 3000 X,2 4200 Y,J;2115.1 K 1;2

2501S-2 Force [kNJ

60

50

40

30

20

10

0 0 15 30

t o --i~plane

--out-of-plane

--Total force in restraint s

45 60 75 90 105 120 Dlsplacemen! [mm]

Figure 203 M4 L7200 x,J3000 X,2 4200 Y,J;2115.1 [(1;2 10001S-2

128

Force [kN) 60

50

--In-plane

---out-of-plane

--Total force In restraint s

45 60 75 90 105 120 Displacement [mm]

Figure 198 M4 L7200 x,J3000 X,2 4200 Y,J;2115.1 K1;2

8000 IS-1

Force (kN) 60

50

40

30

20

10

0~~~--~----+----+----~--4---~--~

45 60 75 90 105 Dlsplacemen! [mm)

Figure 200 M4 L7200 x,J3000 X,2 4200 Y,J;2115.1 K1;2

125 [S-2 Force (kN)

60 -I - , I ! 50 r----- o o o ~

,

1 40 V 30

/ --I~plane

20 --out-of-plane

10 --To!aI force In restraints

0 0"-1

0 15 30 45 60 75 90 105 120 Dlsplacemen! [mm)

Figure 202 M4 L7200 x,J3000 X,2 4200 Y,J;2115.1 K 1;2

500 [S-2 Force [kN)

60

50

40

30 ----in-plane

20 --out-of-plane

---- Total force in restraints 10

30 45 60 75 90 105 120 Displacement [mm)

Figure 204 M4 L7200 x,J3000 X,2 4200 Y,J;2115.1 K 1;2

80001S-2

beams TU/e

Appendix K. Results of ulation from parameter study

K.1. Tabular overview results from parameter study num

44 Mil nOOXr=1200Yr -115.1 K 23095%

45 MII=7200Xr=1800Yr=-115.1 K 23495%

46 MI 1= noo Xr =2400 Yr -115.1 K 25695%

47 Mil = noo Xr=3000 Yr = -115.1 K = 281 95%

48 Mil 7200Xr=3600Yr=-115.1 K=29295%

49 MI I noo Xr =600 YI' = -57.55 K = 2263 95%

50 Mil noo Xr=1200 Yr -57.55 K 107595%

51 Ml 1= noo Xr =1800 Yr -57.55 K 90695%

52 MIl = 7200 Xr =2400 Yr -57.55 K 96395%

53 Mll=7200Xr=3000Yr -S7.SSK 112795%

54 MII=nooXr=3600Yr=-57.55K= 123495%

55 Mil 7200Xr=600Yr=OK 484295%

56 Mil nOOXr=1200Yr OK=211995%

57 Mil nOOXI'=1800Yr=OK= 175695%

58 Ml 1= 7200 Xr =2400 YI' 0 K = 1998 95%

59 MI 1= 7200 Xr =3000 Yr 0 K 289795%

60 Ml!=nOOXr=3600Yr=OK 408995%

61 Mil 7200Xr=600Yr=57.55K=591795%

62 Mil nooXr=!200Yr=57.55K=251495%

63 Mil 7200 Xl' =1800 Yr 57.55 K = 2006 95%

64 Mil 7200XI'=2400YI'=57.55K 216295%

65 MI 1= 7200 Xr=3000 YI'= 57.55 K 268895%

66 M I I = 7200 XI' =3600 Yr = 57.55 K = 1792 95%

67 Ml I =7200 XI' =600 YI'= 115.1 K 528895%

68 Mil 7200Xr=!200Yr= 115.1 K=223895%

69 MIl 7200 Xr=1800 Yr liS.! K 169495%

70 MII=7200Xr=2400Yr 115.1 K 158295%

71 MII=7200Xr=3000YI' 115.1 K 123295%

72 MI 1= 7200 Xr =3600 Yr = liS.! K 43895%

73 MI 1 = 5400 Xl' =450 Yr = -115.1 K 095%

73 Mil 5400Xr=900YI'=-115.1 K=095%

73 MI I 5400 Xr =1350 YI' = -115.1 K = 0 95%

73 Mil 5400 Xr=1800YI'=-IIS.l K 5095%

74 Mil = 5400 Xr =2250 Yr= -115.1 K 8895%

75MII=5400Xr=2700Yr -115.IK 10195%

76 MII=5400Xr=450Yr -57.55K 299195%

17MII 5400XI'=900Yr=-57.55K 149795%

78 MI I 5400 Xr =1350 Yr -57.55 K = 128095%

79 MI I 5400 Xr =1800 Yr -57.55 K 133495%

80 MI 1= 5400 XI' =2250 Yr -57.55 K 148695%

81 MIl = 5400 Xr =2700 Yr -57.55 K 157995%

82 MI 1= 5400 XI =450 Yr 0 K = ]055495%

83 MIl 5400Xr=900Yr=OK 463795%

84 Ml I 5400 Xr =1350 YI' = 0 K 383795%

85 MIl 5400XI'=1800Yr OK=432195%

86 Ml I 5400 XI' =2250 Yr 0 K = 5934 95%

87 MI 1= 5400 Xl' =2700 Yr 0 K = 761995%

88MII=5400XI'=450Yr 57.55K 1476895%

89 Mil 5400 XI' =900 Yr = 57.55 K 631395%

230

234

256

281

292

2263

1075

906

963

1127

1234

4842

2119

1756

1998

2897

4089

5917

2514

2006

2162

2688

1792

5288

2238

1694

1582

1232

438

o o o 50

88

101

2991

1497

1280

1334

1486

1579

10554

4637

3837

4321

5934

7619

XGMJlL, XGM XSM XSM:MOD

19869 41.948 44.156 1380 0.415 0.419 0.474 0.484

20251 43.060 45.326 1626 0.423 0.428 0.483 0.495

20624 44.223 46.551 1771 0.431 0.438 0.492 0.505

20885 45.178 47.556 2049 0.436 0.445 0.500 0.514

20912 45.560 47.958 1918 0.437 0.448 0.503 0.518

21002 44.994 47.362

21715 47.146 49.627

22630 49.994 52.625

23704 53.484 56.299

24755 57.069 60.073

25213 58.820 61.916

22867 50.601 53.264

24151 54.584 57A57

25925 60.530 63.716

28363 69.445 73.1 00

31454 82.594 86.941

33562 92.941 97.833

24464 55.606 58.533

26338 61.609 64.852

28930 71.021 74.759

5280 0.439 0.444 0.498

3928 0.454 0.461 0.514

3917 0.473 0.482 0.535

3934 0.495 0.507 0.558

4662 0.517 0.532 0.581

4798 0.527 0.543 0.591

8540 0.478 0.486 0.539

6099 0.505 0.515 0.565

5259 0.542 0.554 Q.601

6370 0.593 0.606 0.647

7248 0.657 0.669 0.702

8193 0.701 0.708 0.737

9387 0.5 \I 0.522 0.572

5752 0.550 0.561 0.607

5156 0.604 0.614 0.655

32754 86.638 91.198 6139 0.684 0.685 0.717

38265 114.722 120.760 6845 0.800 0.769 0.792

42713 148.990 156.832 8343 0.892 0.828 0.849

25324 58.346 61.417 8834 0.529 0.540 0.588

27654 65.988 69.461 5218 0.578 0.587 0.630

30785 77.780 81.874 5067 0.643 0.648 0.684

35350 96.990 102.095 4868 0.739 0.722 0.749

41236 127.451 134.159 5038 0.862 0.795 0.816

45317 148.990 156.832 7797 0.947 0.828 0.849

o 0.000 0.000 0 0.000 0.000 0.000

o 0.000 0.000 0 0.000 0.000 0.000

o 0.000 0.000 0 0.000 0.000 0.000

31509 54.406 56.855 298 0.494 0.511 0.562

31631 54.622 57.367 509 0.496 0.514 0.565

31563 54.564 57.566 575 0.495 0.515 0.566

0.512

0.532

0.556

0.583

0.610

0.622

0.561

0.592

0.633

0.687

0.751

0.790

0.599

0.641

0.696

0.767

0.851

0.912

0.619

0.668

0.730

0.803

0.878

0.912

0.000

0.000

0.000

0.587

0.591

0.592

33137 58.140 60073

33974 59.979 62.104

34967 62.258 64.695

36111 64.815 67.690

37195 67.089 70.480

5214 0.519 0.532 0.581 0.610

3809 0.532 0.544 0.592 0.623

3520 0.548 0.560 0.606 0.640

470 I 0.566 0.577 0.622 0.658

4588 0.583 0.592 0.635 0.673

37583 67.914 71.760 5023 0.589 0.599 0.641

36703 66.811 69.453 11580 0.575 0.587 0.630

38582 71.687 74.535 9200 0.605 0.613 0.654

41108 78.683 81.939 8253 0.644 0.648 0.684

44419 88.700 92.669 9063 0.696 0.691 0.721

48513 101.724 106.986 9934 0.760 0.736 0.762

51094 109.963 116.772 10021 0.801 0.760 0.784

0.680

0.668

0.695

0.730

0.772

0.817

0.842

14768 39935 75.660 79337 14990 0.626 0.636 0.674 0.718

6313 42790 84.570 88.448 8087 0.671 0.675 0.708 0.757

129

num

91 MI I 5400 Xr =1800 Yr 57.55 K = 5663 95%

92 MI I 5400 Xr =2250 Yr = 57.55 K = 7651 95%

93 Mil = 5400 Xr =2700 Yr 57.55 K = 6661 95%

94 MI 1= 5400 Xr =450 Yr = 115.1 K 1322095%

95 MI 1=5400Xr=900Yr= I 15.1 K 559195%

96 MIl 5400Xr=1350Yr 115.1 K 424595%

97 Mil 5400 Xr =1800 Yr= 115,1 K = 3986 95%

98 MIl 5400 Xr =2250 Yr 115.1 K = 3097 95%

99 MI I 5400 Xr =2700 Yr 115.1 K 1024 95%

100 MIl = 3600 Xr =300 Yr = ·57,55 K = 1138 95%

101 MII=3600Xr=600Yr=·57.55K 117295%

102 Ml 1 3600 Xr =900 Yr = ·57.55 K 126795%

103 MI I 3600 Xr =1200 Yr = -57.55 K = 143695%

I04Mll 3600Xr=1500Yr -57.55K=161995%

105 MI I 3600 Xr =1800 Yr -57,55 K = 1711 95%

106 MI 1= 3600 Xr =300 Yr 0 K = 30448 95%

107 MI 1= 3600 Xr =600 Yr = 0 K = 1334695%

108 MI 1= 3600 Xr =900 Yr= 0 K 1088995%

109 Mil 3600Xr=1200Yr=OK 1179495%

110 Mil 3600 Xr =1500 Yr 0 K 1479495%

III Mil 3600 Xr=1800 Yr OK=I726795%

112 MI I 3600 Xr =300 Yr 57.55 K = 54915 95%

113 MI 1= 3600 Xr =600 Yr = 57.55 K 2353095%

114 MII=3600Xr=900Yr=57,55K 1921095%

115MII 3600Xr=1200Yr=57.55K 2195795%

116 MI I 3600 Xr =1500 Yr 57.55 K = 32570 95%

117 Mil 3600Xr=1800Yr=57.55K=7382595%

118 Mil 3600 Xr =300 Yr 115.1 K = 49927 95%

119 MI 1= 3600 Xr =600 Yr 115.1 K 20890 95%

120 MII=3600Xr=900Yr= 115,1 K 1580595%

121 Mil 3600 Xr =1200Yr= 115.1 K 1481695%

122 Ml I 3600 Xr =1500 Yr = 115.1 K 1136695%

123 Mil 3600Xr=1800Yr 115,1 K=345195%

124 Mil 7200Xr=600Yr -115.1 K=516099%

125 MII=7200Xr=1200Yr=·115.1 K=244499%

126 MII=7200Xr=1800Yr=-1l5.1 K 197099%

127 Mll=7200Xr=2400Yr=-115.l K 191599%

128 Mil 7200 Xr=3000 Yr -115.1 K 198299%

129 Mil 7200 Xr=3600 Yr -115.1 K=2025 99%

130 Mil 7200 Xr=600 Yr ·57,55K= 1601999%

131 MI 1= 7200 Xr =1200 Yr -57,55 K = 6996 99%

132 MI 1 = 7200 Xr =1800 Yr -57.55 K 557099%

133 Mll=7200Xr=2400Yr=-57.55K 570499%

134MI 7200Xr=3000Yr=·57.55K 653699%

135 MI! 7200XI=3600Yr=·57.55K=711399%

136 Mil 7200 Xr =600 Yr 0 K = 2928199%

137MI! 7200Xr=1200Yr OK=1231699%

138 MII=7200Xr=J800Yr OK=989699%

139 Mll=7200Xr=2400Yr=OK 1100899%

140 Mil 7200 Xr =3000 Yr = 0 K 1575599%

141 MI 1 7200 Xl =3600 Yr 0 K 2227499%

142 Mil 7200 XI =600 Y, 57.55 K = 33958 99%

143 MI 1= 7200 Xr =1200 Yr 57,55 K = 1399299%

144 Ml 1= 7200 Xr =1800 Yr= 57.55 K = 1086899%

130

XOMNIA XOM XSM XSM;MOD

52108 121.812 127.132 9001

58726 164,852 172.220 8407

63343 221.595 237.904 8935

0.817 0,782 0.804

0.920 0.844 0,866

0.993 0,889 0.916

0,864

0,930

0.979

5663

7651

6661

13220

5591

4245

3986

41576 80,842 85.127 12486 0.652 0.661 0,696 0,743

45152 92.895 97.446 8077 0,708 0.707 0.736 0.789

49840 111.600 116.691 6715 0.781 0.760 0.784 0.842

55573 142.876 148.821 6645 0.871 0.817 0.838 0.901

3097 61516 193.310 200,500 5202 0.964 0,868 0.891 0.955

1024 65295 223,593 237,904 6173 1.023 0.889 0,916 0,979

1138

1172

1267

1436

61671 85.234 88,054

62678 86.527 89.575

63792 87,920 91.476

64904 89.309 93.566

1282 0.644 0.673 0.706

2426 0.655 0.679 0.711

2838 0.666 0,686 0,718

3146 0,678 0.694 0.724

1619 65929 90.374 95.383 4137 0,689 0.700 0,730

1711 66056 90.529 96.172 4310 0.690 0,703 0.732

30448 69885 100,583 104.719 20977 0,730 0.729 0.756

13346 72930 106,898 111.372 13307 0,762 0.747 0,772

10889 76556 115.343 120.711 10962 0.800 0.769 0,792

11794 80900 126.439 133.342 12975 0,845 0.794 0,815

14794 85040138.950148,174 11869 0.8880.816 0.837

17267 86719 145,442 156.757 11415 0.906 0.827 0.849

54915 71331 120.970 128.051 23647 0.808 0,784 0.806

23530 81762 136.554 144.147 17598 0,854 0.811 0.832

19210 87162 160.412 169.678 12612 0.911 0,842 0.863

21957 92138 200.664 213.695 10575 0.963 0,876 0,901

32570 96483 275.567 298,621 8801 1.008 0,912 0,945

73825 99420 394.237 457.417 7930 1.039 0.944 0.987

49927 80606 134.703 143.497 20183 0,842 0,810 0,831

20890 85676 157.715 167.439 14445 0,895 0.839 0,861

0755

0.761

0.768

0,716

0,782

0,785

0.811

0.829

0.851

0.876

0.900

0.912

0.866

0.894

0.927

0.965

1.000

1.000

0.893

0.924

15805 90975 193.669 205.567 9373 0.950 0.871 0.895 0.959

14816

11366

3451

5160

2444

1970

1915

1982

2025

16019

6996

5570

5704

6536

7113

29281

12316

9896

!l008

15755

22274

33958

13992

10868

95196 254.439 270.612

98070 354,148 378.051

99902 413.686 457.417

20212 42,737 43.169

20516 43,714 44.156

20875 44.873 45.326

21237 46,085 46.551

21457 47.080 47.556

21509 47.479 47.959

21651 46,888

22357 49,132

23268 52.099

24336 55,736

25377 59.472

25828 61.297

23536 52.731

24830 56.882

26596 63,078

29028 72.369

32062 86,071

34111 96,854

25168 57,947

27034 64,203

29608 74,01 I

47.362

49.628

52,625

56.299

60.073

61.916

53,264

57.457

63,715

73,100

86,940

97.832

58.532

64,852

74,759

7471

5812

3564

4055

2897

2554

2697

3018

2708

0.994 0.903 0,933

1,025 0.931 0.969

1.044 0.944 0,987

0.422 0.412 0.467

0.429 0.419 0.474

0.436 0.428 0.483

0.444 0.438 0.492

0.448 0.445 0.500

0.449 0.448 0.503

7063 0.452 0,444 0.498

5296 0.467 0.461 0.514

4560 0.486 0.482 0.535

4992 0.508 0.507 0.558

5095 0.530 0.532 0.581

5193 0,540 0.543 0,591

9831 0.492 0.486 0.539

6752 0.519 0.515 0,565

5605 0.556 0.554 0.601

6643 0.606 0,606 0.647

7417 0,670 0.669 0,702

8236 0,713 0,708 0,737

10256 0.526 0.522 0,572

6024 0.565 0.561 0,607

5234 0,619 0,614 0.655

0,995

1.000

1.000

0.475

0.484

0.495

0,505

0.514

0.518

0,512

0.532

0.556

0.583

0,610

0,622

0.561

0,592

0,633

0,687

0.75\

0,790

0,599

0,641

0,696

Lateral Torsional

num

146 Mil = 7200 Xr =3000 Yr= 57,55 K 1336999%

147 Mil 7200Xr~3600Yr=57.55K=232299%

148 Mil = 7200 Xr=600 Yr = 115,1 K =29766 99%

149MII=7200Xr=1200Yr 115.lK 1221999%

150MII 7200Xr=1800Yr 115.1K=898499%

151 Mll=7200Xr=2400Yr=1I5.lK 807399%

152 MIL = 7200 Xr =3000 Yr= 115.1 K = 5503 99%

153 MI I 7200 Xr =3600 Yr= 115.1 K = 480 99%

154 MII=7200Xr=600Yr=-115.1 K 100%

155 MI1=7200Xr=1200Yr -115.1 K= 100%

156MI! 7200Xr=1800Yr=-115.1K= 100%

157 Mll=7200Xr=2400Yr=-115.l K 100%

158 MII=7200Xr=3000Yr -115.1 K= 100%

159 Mil 7200Xr=3600Yr=-115.l K= 100%

160 MIl 7200 Xr=600 Yr = -57,55 K = 100%

161 MI 1= 7200 Xr =1200 Yr -57.55 K 100%

162MII 7200Xr=1800Yr=-57.55K= 100%

163 Mll=7200Xr=2400Yr=-57.55K 100%

164 MII=7200Xr=3000Yr -57.55K= 100%

165 MI I 7200Xr=3600Yr=-5755K= 100%

166 MJ 1= 7200 Xr =600 Yr = 0 K 100%

167 MII=7200Xr=J200Yr OK= 100%

168 MI I 7200Xr=1800Yr=OK 100%

169 Mll=7200Xr=2400Yr=OK= 100%

170 Mil = 7200 Xr=3000 Yr 0 K = 100%

171 Mil 7200Xr=3600Yr=OK 100%

172 MII=7200Xr=600Yr=5755K= 100%

173MII 7200 Xr=1200 Yr 57.55K= 100"t.

174MII 7200 Xr=1800 Yr=57.55 K= 100%

175 MII=7200Xr=2400Yr=57.55K 100%

176 MII=7200Xr=3000Yr 57.55K 100%

177 MIl 7200 Xr =3600 Yr = 57.55 K = 100%

178 MI 1= 7200 Xr =600 Yr = 115.1 K = 100%

179MII=7200Xr=1200Yr 115.IK 100%

180 Mil 7200 Xr=1800Yr= 115.1 K= 100%

181 Mil = 7200 Xr=2400 Yr= 115.1 K= 100%

182 Mil 7200 Xr=3000 Yr 115.1 K 100%

183 MI I 7200 Xr=3600 Yr= 115.1 K = 100%

185M21 7200Xr=1200Yr=-15.1K=46395%

186M21=7200Xr=1800Yr -IIS.lK 43095%

187 M2 I 7200 Xr =2400 Yr -115.1 K = 448 95%

188 M21 7200 Xr =3000 Yr = -115.1 K 477 95%

189 M2 I = 7200 Xr =3600 Yr -115.1 K 49095%

190M21 7200Xr=600Yr -57.55K=355395%

191 M21 7200 Xr =1200 Yr= -57.55 K 170595%

192 M21=7200Xr=1800Yr ·57.55K 145195%

193 M21 7200 Xr =2400 Yr = -57.55 K = 156295%

194 M2 I 7200 Xr =3000 Yr = -57.55 K 185795%

195 M2 1= 7200 Xr =3600 Yr -57.55 K 205695%

196 M2 I 7200 Xr =600 YI' 0 K 666995%

197 M2 I = 7200 Xl' =1200 Yr = 0 K = 2881 95%

13369

2322

29766

12219

8984

8073

5503

480

463

430

448

477

490

3553

1705

1451

1562

1857

2056

6669

2881

lau XSM XSM:MOD

38953 119.552 120.760 6869 0.814 0.769 0,792

0.900 0.828 0.849

0.544 0.540 0.588

0.593 0.587 0.630

0.658 0,648 0.684

0,754 0.722 0.749

0.877 0.795 0.816

0.955 0.828 0.849

0.851

0.912

0,619

0.668

0.730

0.803

0.878

0.912

43089 155.263 156.831 8131

26037 60.803 61.417 8603

28368 68.766 69.461 5329

31508 81.055 81.874 5060

36077 101.073 102.094 4699

41986 132.818 134.160 4576

45729 155.263 156.831 7478

20376 50.601 43.169 0.426 0.412 0.467 0.475

20668 54.584 44.156

21023 60.530

21390 69.445

21635 82.594

21648 92.941

21809 50.601

22518 54.584

23421 60.530

24488 69.445

25522 82.594

25967 92.941

45.326

46.551

47.556

47.959

47.362

49.628

52.625

56.299

60.073

61.916

23700 50,601 53.264

24990 54.584 57.457

26764 60.530 63.715

29183 69.445 73.100

32213 82.594 86.940

34206 92.941 97 .832

25334 50,601 58.532

27185 54584 64.852

29775 60.530 74.759

33561 69.445 91.198

39084 82.594 120.760

44340 92.941

26210 50,601

28517 54.584

31680 60.530

156.831

61.417

69.461

81.874

36240 69.445 102.094

42129 82.594 134.160

47941 92.941 156.831

q"

5.066

5.198

5.319

5.404

5.433

5.433

5.689

6.002

6.358

6.675

6.828

6,021

6.444

37.683

39.076

40.473

41.559

41.975

41.073

43.800

47.349

51.602

55.803

57.786

47.236

51.989

39.666

41.133

42.603

43.746

44.184

43.235

46.105

49.841

54.318

58,740

60.827

49.722

54.725

0.432 0.419 0.474 0.484

0.439 0.428 0.483

0.447 0.438 0.492

0.452 0.445 0.500

0.452 0.448 0.503

0.456 0.444 0.498

0.470 0.461 0.514

0.489 0.482 0.535

0.5 12 0.507 0.558

0.533 0.532 0.581

0.543 0.543 0.591

0.495

0.505

0.514

0,518

0.512

0.532

0.556

0.583

0,610

0.622

0.495 0.486 0.539 0,561

0.522 0.515 0.565 0.592

0.559 0.554 0.60 I 0.633

0.610 0.606 0,647 0.687

0.673 0.669 0.702 0,751

0.715 0.708 0.737 0.790

0.529 0.522 0.572 0.599

0.568 0.561 0.607 0,641

0.622 0.614 0.655 0.696

0.701 0.685 0.717 0.767

0.817 0.769 0.792 0.851

0.926 0.828 0.849 0.912

0.548 0.540 0.588 0.619

0.596 0.587 0,630 0.668

0.662 0.648 0.684 0.730

0.757 0,722 0.749 0.803

0.880 0.795 0.816 0.878

1.000 0.828 0.849 0.912

2506 0.381 0.384 0.439

2588 0.391 0.396 0.451

2693 0.400 0.407 0.462

2847 0.406 0.4 1 6 0.471

2904 0.409 0.420 0.475

8126 0.409 0.412 0.467

5760 0.428 0.434 0.489

5056 0.451 0.462 0.516

5246 0.478 0.494 0.546

5570 0.502 0.523 0.573

5662 0.514 0.536 0585

11874 0.453 0.461 0.515

7670 0.485 0.496 0.548

0.440

0.453

0.465

0.475

0.479

0.471

0.494

0.523

0.555

0.584

0597

0.522

0.558

131

Appendix K Results of simulation from parameter study

num Simulation Code

198 M2! = 7200 Xr =1800 Yr = 0 K = 228895%

199 M2 I = 7200 Xr =2400 Yr = 0 K = 2379 95%

200 M2 1= 7200 Xr =3000 Yr = 0 K = 2883 95%

201 M2 I = 7200 Xr =3600 Yr = 0 K = 2942 95%

202 M21 = 7200 Xr =600 Yr = 57,55 K = 6898 95%

203 M2 I = 7200 Xr =1200 Yr = 57.55 K = 2678 95%

204 M21 = 7200 Xr =1800 Yr = 57,55 K = 179695%

205 M2 ! = 7200 Xr =2400 Yr = 57.55 K = 138295%

206 M2 I = 7200 Xr =3000 Yr = 57.55 K = 88395%

207 M2! = 7200 Xr =3600 Yr = 57.55 K = 42095%

208 M2! = 7200 Xr =600 Yr = 115.1 K = 5439 95%

209 M2 I = noo Xr =1200 Yr = 115,1 K = 2004 95%

210 M2 i= 7200 Xr=1800 Yr= 115,1 K= 121995%

211 M2 I = 7200 Xr =2400 Yr = 115.1 K = 819 95%

212 M2!=nooXr=3000Yr= 115,1 K=455 95%

213 M21 = 7200 Xr=3600 Yr = 115,1 K = 22195%

214 M3 1 = 7200 Xr =2400 Yr = -57.55 K = 66 95%

215 M3! = 7200 Xr =3000 Yr = -57.55 K = 13795%

216 M3 1= 7200 Xr =3600 Yr = -57,55 K = 21095%

217 M31 = noo Xr =4200 Yr = -57.55 K = 258 95%

218 M31 = 7200 Xr =4800 Yr = -57.55 K = 280 95%

219 M31 = 7200 Xr =5400 Yr = -57.55 K = 304 95%

220 M3 1 = 7200 Xr =6000 Yr = -57.55 K = 376 95%

221 M3 1= 7200 Xr =6600 Yr = -57.55 K = 74595%

222 M3 I = noo Xr =600 Yr = 0 K = 203 95%

223 M3 1 = 7200 Xr =1200 Yr = 0 K = 208 95%

224 M3 1 = 7200 Xr =1800 Yr = 0 K = 258 95%

225 M3 1= 7200 Xr =2400 Yr = 0 K = 353 95%

226 M3 ! = 7200 Xr =3000 Y r = 0 K = 510 95%

227 M3 1= 7200 Xr =3600 Yr = 0 K = 700 95%

228 M3 1= 7200 Xr =4200 Yr = 0 K = 793 95%

229 M3 1= 7200 Xr =4800 Yr = 0 K = 79395%

230 M31 = 7200 Xr =5400 Yr = 0 K = 84195%

231 M3 1 = 7200 Xr =6000 Yr = 0 K = 110995%

232 M3 1 = 7200 Xl' =6600 Yr = 0 K = 2631 95%

233 M3 1 = 7200 Xr =600 Yr = 57,55 K = 138595%

234 M3 1 = 7200 Xl' =1200 Yr = 57.55 K = 692 95%

235 M3 1 = 7200 Xr =1800 Yr = 57,55 K = 63595%

236 M3 1= 7200 Xr =2400 Yr = 57,55 K = 758 95%

237 M3 1 = 7200 Xl' =3000 Yr = 57,55 K = 1065 95%

238 M3 1 = 7200 Xr =3600 Yr = 57,55 K = 146395%

239 M3 1= 7200 Xr =4200 Yr = 57.55 K = 148495%

240 M3 1 = 7200 Xr =4800 Yr = 57.55 K = 130595%

241 M3 1= 7200 Xr =5400 Yr = 57.55 K = 129795%

242 M3 I = 7200 Xr =6000 Yr = 57,55 K = 168J 95%

243 M3 1= 7200 Xr =6600 Yr = 57,55 K = 401895%

244 M3 1 = 7200 Xr =600 Yr = 115.1 K = 2417 95%

245 M3 1= 7200 Xr =1200 Yr = 115,1 K = 1151 95%

246 M3 1= 7200 Xr =1800 Yr = 115,1 K = 997 95%

247 M3 1= 7200 Xr =2400 Yr = 115.1 K = 115295%

248 M3 1 = 7200 Xr =3000 Yr = 115,1 K = 164095%

249 M3 1 = 7200 Xr =3600 Yr = 115,J K = 218095%

250 M3 1 = 7200 Xl' =4200 Yr = 115, I K = 180795%

132

Spring q" M,,:NUM Mcr;ANA Fin XGMNIA XGM XSM XSM,MOO

Stiffness [kN/m] [kNm] [N/mm]

[kNm] restraint at failure

2288

2379

2883

2942

6898

2678

1796

1382

883

420

5439

2004

1219

819

455

221

66

137

210

258

280

304

376

745

203

208

258

353

510

700

793

793

841

1109

2631

1385

692

635

758

1065

1463

1484

1305

1297

1681

4018

2417

1151

997

1152

1640

2180

1807

7,005

7,727

8,628

58,829 61,925

68,708 72.324

83,129 87.504

9.439 99.999 105.262

6.418

6.947

7.627

8.465

51.810

57,898

66.390

77.950

54.537

60.945

69.884

82,053

9.421 91.842 96,676

10,281 99.999 105,262

6,566 53.576 56.396

7.158

7.880

7,880

60.317

69.195

69.195

63.492

72.837

72.837

9.607 93.199 98.104

10.730 99.999 105.262

F,,[N]

[N] 6556

6695

7359

8901

11302

6595

4993

4319

4897

9411

8908

5596

4010

4010

3769

8427

0.527 0,543 0,591

0,581 0.602 0,644

0,649 0,671 0,704

0,710 0,731 0,757

0.483 0.495 0.547

0.523 0.537 0.586

0.574 0.589 0.632

0,637 0.648 0.685

0.709 0.704 0,734

0.773 0.731 0,757

0.494 0.508 0.559

0.538 0.553 0,600

0.593 0,605 0,646

0.593 0.605 0,646

0.723 0.709 0,738

0,807 0.731 0,757

0,604

0.660

0,724

0,780

0,557

0.598

0,648

0,703

0,755

0.780

0.569

0.613

0.663

0.663

0,760

0.780

31641 47.442 49.939 757 0.496 0.463 0.516 0.550

31763 49.305 51.900 1639 0.498 0.477 0,530 0.568

32033 51.036 53.722

32424 51.743 54.466

32416 51.296 53.996

1915 0.502 0.490 0.542 0.584

2166 0.508 0.495 0.547 0,590

2205 0,508 0.492 0.544 0.586

32187 50.241 52.885 2317 0.504 0.484 0,536 0.577

31914 49,022 51.602

31717 47.922 50.444

31435 46,616 49.069

2517 0.500 0.475 0.528 0.565

3183 0.497 0.467 0.520 0.554

949 0.493 0.457 0.510 0,541

31954 47,896 50.417 1531 0.501 0.466 0.520 0.554

32182 50.062 52.697

33211 53.360 56.168

35005 57.884 60,931

36835 62.233 65.508

37164 63.314 66,646

36232 61.133

35168 58.003

34371 55.042

33448 52,686

32207 49.928

32959 52.358

34773 56.372

37462 62.711

42037 72.314

47107 82.230

45751

41988

38898

36741

35571

33961

35546

38158

82.481

75.203

67.886

62,092

57,886

53,680

57,736

64,256

64.351

61.056

57.939

55.459

52.556

55.114

59.339

66.012

76,120

86,558

86,822

79,161

71.459

65.360

60,933

56.505

60.775

67.638

2521 0.504 0.483 0.535

2924 0.520 0.506 0.557

3481 0.549 0.537 0.586

4250 0.577 0.564 0.610

4772 0.582 0.571 0,616

4411

4286

4999

6462

5050

4077

4214

4709

6256

8067

7019

6082

5903

6668

9761

6626

5618

5520

42892 74,959 78.904 6101

0,568 0.558 0,604

0.551 0.538 0.586

0.539 0.518 0,568

0.524 0.501 0.553

0,505 0.482 0.534

0.516 0.499 0,551

0.545 0.527 0.576

0.587 0.567 0,613

0.659 0.621 0.661

0.738 0.667 0,70]

0,717 0,668 0,702

0,658 0.635 0,673

0,610 0.597 0.640

0,576 0.564 0.610

0,557 0.537 0.586

0,532 0.508 0,559

0.557 0.536 0,585

0,598 0.577 0,621

0,672 0.634 0,672

0,796 0,708 0.737

0.943 0,764 0,787

0.848 0,750 0,774

50790 93.005 97,900 7584

60152 112.521 118.443 8836

54142 106,754 112.373 6697

0.575

0.605

0.643

0.676

0.684

0.668

0,644

0.619

0,599

0.574

0.596

0,630

0,679

0,742

0,795

0,796

0,758

0.715

0.675

0,643

0,608

0,642

0,690

0.757

0,840

0,902

0,886


num Simulation Code Spring FII Nlcr;NUM M cr;ANA Fin X GMNIA

Stiffness [N) [kNm) [kNm) restraint [N/mm) at failure

~~

[NJ 251 M31=7200Xr=4800Yr = 115.1 K-145495% 1454 47037 90.074 94.815 5585 0.737

252 M3 1= 7200 Xr =5400 Yr = 115.1 K = 140695% 1406 42177 77.167 81.228 5749 0.661

253 M31 = 7200 Xl' =6000 Yr = 115.1 K = 1793 95% 1793 38833 68.081 71.664 6813 0.609

254 M3 1= 7200 Xr =6600 Yr = 115.1 K = 416795% 4167 36893 61.778 65.029 10052 0.578

K.2. Graphs of parameter study

K.2.1. Elastic critical moment

K.2.1.1. Load case 1

1=5400111111 250 .-----~------------~--------------------------------~----_,

XCM X SM

0.698 0.728

0.645 0.681

0.598 0.641

0.562 0.608

Mer; J O()%


-0 --- -57.55-+

==:~55:1 0+-____ -r ____ _.------.-----~------~----_r----_.------;_----~~~------1~1~5.~1--~

o 600 1200 1800 2400 3000 3600 4200 4800 5400

Position of restraint over tength of system (xl

TU/e

XSM; MOO

0.829

0.769

0.716

0.673

Figure 205 Effects of rigid restraint location on the elastic critical moment for load case 1 1= 5400mm

1 = 3600nrm

500,------------------------------------------------------------,

450

E 400 ! C 350 CI>

E 300 0 E 250 iO u 200 :e t;

150 u "" --- ----II> 100 .. iii

50

0

0 600 1200 1800 2400 3000 3600

Position of restraint over length of system (xl

Mer; J 00%


=:~5~~[ -0 __

- -57.55-+

--115.1

Figure 206 Effects of rigid restraint location on the elastic critical moment for load case 1 1= 3600mm

133



120 .-----~------------------------------------~----~--------------__.

E 100 z ~

c: 80 ., E 0 E 60

~ :e t; 40

" ~ IV

iii 20

0

0

r, M cr: /OO"/'


-115.1 IT --57.55 _ --0 _

---57.55-

---115.1

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 7200

Position of restraint over length of system (x)

Figure 207 Effects of rigid restraint location on the elastic critical moment for load case 2


140

120 E z ~ 100 c: ., E 80 0 E

~ 60 :e t; .2 40 U; IV

iii 20

0

0

~= ! I

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 7200


M cr:/OO%

Height of res trai nt on crosssection

-" IIS.IIT - - 57.55-

--0 -+

---57 .5~

---115.1

Figure 208 Effects of restraint location on the elastic critical moment for load case 3

K.2.2. K95% and K99% spring-stiffnesses


1=5400111111 100000

90000

E 80000

E 70000

~ 60000 en

en ., 50000 :E

~ 40000

'" " ." 30000 Q.

U) 20000

10000 -

0

0

I ' .,

I ' .. , " • 'I, ... " 11

• 'I , I ' t "1-· " , • 1 It I .. \,

.. ... .. . , ' " ... .. -. .......

, ' to III. ,. "

~\. ': :~;~ . ~ :; ~~:: -- .... / • .. .. , • r .- .. _-_ . _---- ........... . . .

600 1200 1800 2400 3000 3600


4200

"I • .,' . -.1 .

I ~ ,

,.' . I ,' • .. ' , .. ' I .. ' t

" ' . , ,' •

4800

Figure 209 [(95% and K99% spring-st.iffnesses for 1= 5400mm

134

Height or reslraint on crossseclion

5400

•• • -115 .1 • ••• 57 .55 ••• ·0

1ll - - • • -57.55 ~ •••• -115 .1

00 --115.1

gl __ 57 .55

--0 -57.55 -11 ~ 1


1=3600",,,,

200000 .-g;--~--------------------~~---------------------------~,--,.-.

E

180000

160000

E 140000

~ 120000 en en ~ 100000

';; 80000 Cl

.s 60000 a. V) 40000

. , , ' . ,

"'" ........ . . -

. , ,

.. ...

.............

,. . I .

, . , . . , Height of restraint on crosssection

• • . • 115.1 • ••• 57.55 • ••• 0 • •• · -57 .55 • ••• ·115.1 --115.1 -_57.55 --0

2000: 1-____ ~~::~~~:::::;::::::::~~~~~::::~::::::~::;:~=-----_J~--~-:-:-::-~.157~.~55~ o 600 1200 1800 2400 3000 3600


Figure 210 K95% and K99% spring-stiffnesses for I =3600mm


30000 nrnr~~--~--------------------------------------~--------~1 7, -,~nn

II.. ".

1 pi E ~ 25000

t;

~ 20000

~

m 15000 .E ::: ~ 10000

~ Cl <: 5000 '§. V)

, \ . \ .. --, \ ~ 4 ',

, I • '\ ' , ... • \ . t

. , ... - ... , ... ,. .. ... ..

". 111 ..... .-\0. ': , ... -" _ . ... . ""

.1,- _ .. ..

'1 ' ,

" " I, '. I, I.

" , I

, ' . f, .. ~ I • ..

" ..' 11-..... ... .. . ,. ... " . . ,:' . . . '

r= Height of restraint on crosssection

· .. ·115.1 • •• • 57.55 • ••• 0 • - . --57.55

~ - - - .·115.1 on ---115.1 ~ --- 57.55 'r ---0

l~~~==~~-~-~'~':-~'~'~'~'~-'g'~'~'~~~::j:~~~~-~-~'~'~'~'~·~-~~~~~~~~Jl~;~;-~~-5~7~.525--J o ---115.1

o 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 7200


Figure 211 K95% and K99% spring-stiffnesses


20000 ~r.-~----------------~--------------~----------~---------.--~,,, 1 F ': : . I. I i;:: ~~x ,~

18000 - :, , ' ' . '. JI2II/I2/ • • t • I , • •

16000 • • • • , I I ,

E E 14000

~ 12000 ::: ~ 10000

~ 8000 <: .~ 6000 V)

4000

, , , . l I · , , Height of restraint

: : ~ , I I I: : . on crosssection · , . ' 1 /.,: _ .. ·115.1 : : 1 Il, ,,' ~ ...... " '" , . I '. • .... 57.55 • , ~ - , ;1 ,. :: _ ...... 0

• , '" ... .... __ .... .. r- ," .,

: '. , • • •• - - • • • • • • • • •• - •• - - •• - - - • -' • , ' , : ~ ::::: ~; ~~~ I

: " '. " ,.' ,' -. ' :: "'_ ':~~' j)' " : Sl - - - - -57.55 '~. ' • - , • • - - • • , • , , ,,- - • --115.1

'.. .. .. ... ... ". r t# •

200:tl~~"~':_~_ ~_~'~- ~-~'~-~-~'~'~-~-~'~'~'~~--'~~~~~~~' ~-~·~·~·~-~-~' -~~~-~.~.~.~'~·~'~~I~::::=~~~~.~.5~~~ o 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600 7200


Figure 212 K95% and K99% spring-stiffnesses

TU/e

135


K.2.3. Loadbearing capacity

K.2.3.1. Loadcase 1

Z ~

" LL.

136

~

1=5400 [mmJ

70 r------------------------------------------------------------------,

20

10

-t-115.1

-'57.55

......... -57.55

_-1! 5.1

- Unrestrained

O ~. ------------~------------~------------__ ----------__ ------------~ 450 900 1350 1800 2250 2700

Restraint posilion along length of systom (x) [nvn)

Figure 213 Ultimate load obtained from GMNIA for load case 11 = 5400mm 1=54000101

1.2

" ----Gene"'l Method Cu"" A

" --Specific method CUI\e 8 .... 1 • ......... _-_ .. __ ....

- - Eulercu"" .. .......... .... . - ................................................... - • ..... • - - - - Mpl

• ..... +115.1 ...... • 0.8 • • ..... +57.55 ..... • ..... 0 ....... •

~ 0.6 ~ • ·57.55

• -115.1 0

X ---0.4 1 F

0.2 ~ 0

0.5 0.6 0 .7 0.8 - 0.9 1 1.1 1.2 1.3

"oMNIA

Figure 214 Graphical representation of results from GMNIA for load case 11 = 5400mm

1=3600 [mm] 1 F 120

! b 100 ! I

80 I

60

40 __ 115. 1

_ 67.55 ..... 0

20 __ ·57.55

__ · 11 5.1

-Unrestrained

a 300 600 900 1200 1500 1800

Restraint position along length of system (x) [mmJ

Figure 215 U timate load obtained from GMNlA for load case 1/= 3600mm


I = 3600mm

1.2

I " --General Melhod Curve A .. " --Specific melhod curve B

1 ;-.... _____ t ____ • _ • __ •.. ____ . __ _ ________ .. ______ . ___ ...... _

- - Euler cur.e -. - - • - Mpl t. • • , • • +115.1

0.8 • +57.55

<: • 0 J ~ ~ 0.6

• -57 .55

• -115.1 0

X

0.4 ! F

0.2 ~ 0

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

AGMNIA Figure 216 Graphical representation of results from GMNIA for load case 11 = 3600mm


12

10

8

2

o 600

1=7200 [mm]

1: I q J

~ , ~ 1

..., --,

I ---t-115.1

I _ 57.55 _ 0

~-57.55

_ -115.1

-Unrestrained

I 1200 1800 2400 3000 3600

Restraint position along length of system (x) [mm]

Figure 217 Ultimate load obtained from GMNIA for load case 2 1= nOOmm I =7200nun

1.2 .-- - -----=-, - - - -:----- --- - - - - - - - - - - ... :..:....----:::G-en-er-a:-:I MC"e-::th-od"CC::O-u-rve----:-A'1

, " --Specific method curve B _______ • ___ • __ ":' ~ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - Euler curve

" ---.M~ , • +115.1

" • +57.55 .............. "' • 0 0.8 1=::::::::::::::::::::~:~:!r __

:5 ~0.6 -- • -57.55

• -115.1 o X ---

0.4

0.2

O +-------.-------,,-------r-------r------~-------,------~

0.8 0.9 1..1 1.2 1.3 1.4 1.5

AOMNIA

Figure 218 Graphicall-epresenl"ation of the results from GMNIA for load case 2 1= nOOmm

TU/e

"

137



1=7200 [mm) 70 ··r--------------------------------------------------------------------------------~

20

10

O~----~------~------~------~----~------__ ------~----~------__ ----~ 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600

Restraint posilion along length 01 system (x) (mmJ

Figure 219 Ultimate load obtained from GMNIA for load case 3 1= nOOmm

1=72000101

_115.1

_ 57.55 _ 0

_ ·57.55 __ ·115.1

- Unrestrained

': L ------- -'>~~~- -------- -- -- T -- -- ------- -- -----General Method CuM A

...... .... • ....

0.8 r==~S-~ :s ~~_&"_~

~ 0.6 CJ

?-<

0.4

0.2

--Specific method CUM B

- - Euler CUM

•••• Mpl

• +115.1 .... - • +57.55 -.... • 0 • -57.55 • -115.1

o +--------,--------.-------__ --------.-----------------,-------~ 0.8 0.9 1.1 1.2 1.3 1.4 1.5

AGMNIA

Figure 220 Graphical representation of the results from the GMNIA for load case 3 1= nOOmm

K.2.4. Force in restraint at failure

K.2.4.I. Load case 1

138

I - 5400mm

16 ,---------------------------------~----__,

14

z ~ 12

~ ~ 10

c

S) 8 e u

! 6 0; c

450 900 1350 1800 2250 2700

Location of reslraint OIlong length of system )( Imm]

I F

I t (" i: 0

-W.I

~NEN 67711% rule 115.1

____ NEN 6711 1% rule 57.55

-&- NEN 6771 1% role 0

~ NEN 6771 1% rule -57.55

_____ NEN 67711"'~ rule -115.1

~NEN 67710.5% rule

_K95_"5.1

--.-K95_ 57.55

-+-K95.0 -+- K95_,57.55

____ K95_ .115.1

Figure 221 Force ill rest-raint at failure for 95% for I =5400mm


I - 3600nml

25 r-------------------~------------------------·--_. l5.'

300 600 900 1200 1500

location or restraint along length of system x [mmJ

1600

-+-NEN 67711% rule 11 6.1

_ NEN 67711% rule 57.55

-.- NEN 6771 1% rule 0

-oo- NEN 67711% rule -67.55

.-... NEN 6771 0.5% rule

__ K95_115.1

-.-K95_57.55

-I-K95_0

-+-K95_-57.55

Figure 222 Force in restraint at failure for 95% for / =3600mm


14r--------------------------------------------------,

12

z ~ 10 ~ 2 'iii LL B

.S e 4 o

LL

o+-------~------~--------+_------~------~ 800 1200 1800 2400 3000 3600

Location of restraints along length of system

7 r: " ...

10

· 1151

- -:1:- - NEN 67711% rule 115.1

- ~. - NEN 6771 1% rule 57.55

- -A- - NEN 6771 1 % rule 0

• ... - NEN 6771 1% rule -57.55

. .... . . NEN 67711% rule -115.1

..... .. NEN 6771 0.5% rule

_K95_115.1

-lIE-K95 57.55

.....-K95_0

-+-K95 -57.55

-4-K95_-115.1

Figure 223 Force in restraint at failure for 95% critical spring-stiffness


l5.'

I· • -x- - NEN 6771 1% rule 115.1

- -x- - NEN 6771 1% rule 57.55

- ,.. _ NEN 6771 1% rule 0

• .. . NEN 6771 1% rule -57.55

- ... . NEN 6771 0.5% rule

--+--K95_·S7.55

600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600

location of restraint over length of sys tem x (mOl]

Figure 224 Force in rest.-aint at failure for 95% critical spring-stiffness

TU/e

139

lateral torsional buckling

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