0

0.2

0.4

0.6

0.8

1

1.2

1.4

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

L=500 mmL=1020 mmL=1020 mmL=1540 mmL=1540 mm

cSd

z xz

NA f

η

χ ω η 0 2,

Totally 47 beams

cy Sd

LT xLT y Rd

MM

γ

χ ω.

. Flange in compressionWeb in compression

Buckling of T-Section Beam-Columns in Aluminium with or without Transverse Welds Stefan Edlund

TRITA-BKN. Bulletin 54, 2000 ISSN 1103-4270 ISRN KTH/BKN/B--54--SE

Doctoral Thesis

Buckling of T-Section Beam-Columns in Aluminium with or without Transverse Welds

Stefan Edlund

Department of Structural Engineering Royal Institute of Technology S-100 44 Stockholm, Sweden

TRITA-BKN. Bulletin 54, 2000 ISSN 1103-4270 ISRN KTH/BKN/B--54--SE Doctoral Thesis

©Stefan Edlund 2000 KTH, TS-Högskoletryckeriet, Stockholm 2000

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Abstract This thesis deals with buckling of T-section beam-columns in aluminium with or without transverse welds. Totally 26 unwelded and 39 transversely welded T-section beam-columns were tested. Five of the welded beams were clamped. All unwelded and the rest of the welded beams were simply supported for bending. The welding affects the load-carrying capacity of the beam-columns, because it introduces a heat-affected zone with reduced strength. All beam-columns had the same theoretical cross-section dimensions. The thickness of the flange and the web was 6 mm. The depth and the width were 60 mm. The theoretical beam lengths were 500, 1020 and 1540 mm, respectively. Tensile tests of both the parent and the heat-affected material were made in order to determine the material properties. Comparisons were made between the buckling tests and three codes, the European aluminium code Eurocode 9, the British aluminium code BS 8118 and the Swedish steel code BSK. Some interpretations of the codes had to be made, because the codes are not totally clear when applied on T-sections. Most problems are related to the fact that the section modulus is not the same for the two edges. In the interaction formulas, only the edge in compression was considered when the bending moment capacity was calculated. The chosen interpretation of the codes was often very conservative when compared with the buckling tests. The general-purpose finite element program Abaqus was used to develop numerical models of the tested beam-columns. Shell elements were used. The models were calibrated with the results from the buckling tests. The stress-strain curves used in the finite element calculations were obtained from the tensile tests. The results of the finite element calculations were satisfactory. The numerical models could predict the load carrying capacity accurate enough. A similar deformed shape of the tested and calculated beam-columns was also obtained. Different modifications of Eurocode 9 were analysed in order to improve the results. One modification was that the ultimate strength of the heat-affected zone was used instead of the yield strength of the parent material when the buckling reduction factors of a welded section were calculated. The calculation of the bending moment capacity in the interaction formulas was also modified. The plastic section modulus was used to calculate the bending moment capacity when the tip of the web was in tension. When the bending moment acted in the opposite direction, the calculation of the bending moment capacity was based on a modified classification of the web element. The investigation in this thesis indicates that Eurocode 9 is too severe in the classification of the cross-section. The way the bending moment capacity is calculated for unsymmetric cross-sections in the interaction formulas needs to be further analysed. Tensile failure at the tip of the web was also discussed. This thesis shows that the codes need to be improved when it concerns unsymmetric cross-sections. Some information how Eurocode 9 can be improved is given. Keywords: Buckling, buckling tests, Eurocode 9, aluminium beam-columns, beam-columns, T-sections, finite element analysis, transverse welds, codes, unsymmetric cross-sections.

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Preface The research presented in this doctoral thesis was carried out at the Department of Structural Engineering at the Royal Institute of Technology in Stockholm. Especially I would like to thank: My supervisor Prof. Torsten Höglund for his guidance, help and for being a discussion partner, assoc. Prof. Costin Pacoste for reading the manuscript and giving valuable comments, SkanAluminium and SAPA for the financial support, SAPA for delivering the aluminium profiles free of charge and the laboratory personnel, Mr. Stefan Trillkott, Mr. Claes Kullberg, Mr. Olle Läth and Mr. Daniel Hissing, for performing the tests. Stockholm, February 2000 Stefan Edlund

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Notations A Gross cross-section area

efA Effective cross-section area

E Modulus of elasticity G Shear modulus I xef . Effective second moment of area, major axis

xI Second moment of area, major axis

yI Second moment of area, minor axis

vK Saint-Venant torsion constant

wK Warping constant M Bending moment

crM Critical bending moment

RSxM Bending moment capacity according to BS 8118

RxdM Bending moment capacity according to BSK

y RdM . Bending moment capacity according to Eurocode 9

N Axial force, centric or eccentric crN Critical axial force

ef x cW . . Effective section modulus of the edge in compression, major axis

ef x tW . . Effective section modulus of the edge in tension, major axis

el x cW . . Elastic section modulus of the edge in compression, major axis

el x tW . . Elastic section modulus of the edge in tension, major axis

W yel . Elastic section modulus, minor axis

pl xW . Plastic section modulus, major axis

W ypl. Plastic section modulus, minor axis b Width of flange b f Width of flange element

bw Depth of web element e Load eccentricity

0 2,f Yield strength of parent material

uf Ultimate strength of parent material

hazf Ultimate strength of heat-affected material h Cross-section depth

1yk Unsymmetry factor used in Eurocode 9

Ll, Beam length

cl Buckling length r Radius between flange and web

ft Flange thickness

wt Web thickness

yt Cross-section constant

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midw Midspan deflection y gcef . Location of centre of gravity for an effective cross-section

gcy Location of centre of gravity

ply Location of plastic neutral axis

sy Distance between centre of gravity and shear centre

hazz Location of transverse weld

maxε Maximum strain

minε Minimum strain Abbreviations EC9 Eurocode 9 haz Heat-affected zone lap Load application point

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Contents Abstract Preface Notations Abbreviations 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Aim and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Buckling tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Tensile tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Comparisons between tests and different codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Buckling according to Eurocode 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.1 Cross -section classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.2 Effective cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2.3 Flexural buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2.4 Lateral-torsional buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Buckling according to BS8118 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.1 Cross -section classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.2 Effective cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.3 Flexural buckling (major axis buckling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.4 Lateral-torsional buckling (minor axis buckling) . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Buckling according to BSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.1 Cross -section classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.2 Effective cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4.3 Flexural buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.4 Lateral-torsional buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5 Critical loads according to the theory of elastic beam-columns . . . . . . . . . . . . . . . . . . . 54 3.6 Comparisons between buckling reduction factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4 Finite element calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.3 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4 Results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 Further analysis of Eurocode 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1 Buckling and tensile failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Section check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.3 Suggestions for improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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6 Conclusions, comments and further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Appendix A. Photos of the test equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Appendix B. Load-deflection curves of the tested beam-columns . . . . . . . . . . . . . . . . . . . . . 131 Appendix C. Example of Abaqus input files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 List of Bulletins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

1

1. Introduction 1.1 Aim and scope Load carrying constructions in aluminium are increasingly used in offshore and transportation industry. The main reasons are the extrusion technique, corrosion resistance and the lightness of the material. Examples of constructions in the offshore industry are helicopter decks, telescoping bridges, stair towers, cable ladders, accommodation modules and other parts exposed to the marine environment. In the shipping industry aluminium constructions are used in hulls for high-speed ferries and other vessels, topside parts of ships and large gas storage tanks. The car industry is using aluminium constructions in the space-frame concept and energy absorbing components. Aluminium constructions are used in train coaches. Furthermore, the use of aluminium constructions in the aircraft industry has a long history. Lot of research has been made on bi-symmetric I-sections, whereas unsymmetric cross-sections are not so widely studied. Most steel and aluminium codes are mainly written for bi-symmetric I-sections and rectangular hollow sections and are just partly adjusted to fit other cross-sections. The use of the extrusion technique makes unsymmetric cross-sections in aluminium more common than in steel. It is therefore especially important that aluminium codes can handle unsymmetric cross-sections. Extruded aluminium profiles can be adjusted to fit many different purposes and therefore they often end up unsymmetric. These are some reasons for studying buckling of aluminium beam-columns with unsymmetric cross-sections. The T-section is one of the most monosymmetric sections to be found. To choose a totally unsymmetric cross-section would likely have been a too large step. Most of the aluminium beam-columns in this thesis are transversely welded. When aluminium is welded there will be a heat-affected zone with reduced strength. This will affect the load carrying capacity of the beam-columns. When structural steel is welded there will normally not be a zone with reduced strength. For high strength steel, such a reduced strength zone may however occur, but the strength is not so drastically reduced as for aluminium. The width of the heat-affected zone is much smaller in steel than in aluminium due to the difference in heat transfer. The heat-affected zone makes buckling of aluminium beam-columns more complicated than steel beam-columns. The loss of strength in the heat-affected zone is a disadvantage for aluminium in comparison to steel. The effect of the loss of strength can to some extent be reduced by making a “smart” construction. The extrusion technique is a usefully tool for this. Suitable cross-sections can be made and welds can be located where they do less harm. Modern welding techniques such as friction stir welding cause less reduction of the strength in the heat-affected zone but is difficult to use for transverse welding. The content of this thesis is not only applicable on T-sections in aluminium. Parts of it are also applicable to other unsymmetric cross-sections, like monosymmetric I-sections. This type of cross-sections are perhaps more common than T-sections. A T-section can be seen as an extreme case of a monosymmetric I-section. All monosymmetric cross-sections have that in common that the section modulus for the two edges is not the same. How this should be treated is not clearly explained in the codes of today. At least not for the codes analysed in this thesis. Even though aluminium and steel are two different materials and have different qualities, the way of calculating aluminium and steel constructions have similarities. Parts of this thesis could therefore also be applicable on steel constructions. Hopefully this thesis can contribute to make the future codes better, especially aluminium codes but perhaps also steel codes.

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The use of beam-columns with monosymmetric I-sections is discussed in Hernelind et al. [25]. When the compressive axial load is eccentric, a monosymmetric cross-section can be more material effective than a symmetric one. One practical example where eccentric loads exist is a beam-column with loads from a roof and an overhead crane. Even a T-section can be more material effective than a bi-symmetric I-section under certain loading conditions. When considering that the usage of a building can be changed in the future, it can however be preferable to choose a not too unsymmetric cross-section. The objectives of this thesis are: 1. Perform a literature search to find out was has been done. A literature search increases the

knowledge of the topic being considered and avoids duplicate work. 2. Perform buckling tests as reference material when evaluating different calculation methods.

It is also important that they are documented so accurate that other future researchers can make use of them. Buckling tests are important when evaluating different design formulas. The tests should preferable be unique.

3. Evaluate different codes and compare the results from the buckling tests with the results

from the codes. Eurocode 9 is one of the most important codes. 4. Develop a numerical model and see if this model can predict the load carrying capacity of

the beam-columns accurate enough. The model should not only be able to predict the capacity, it is also important that it describes the physical behaviour of the tested beam-columns. Otherwise it can be more difficult to claim that it can predict the capacity for other not tested beam-columns. The goal of the development of the model is to have a tool that numerically can simulate a buckling test. It is time consuming to learn the chosen computer program, develop the numerical model and to calibrate the model with physical tests. To come up with realistic results the computer program has to be advanced and these programs tend to be non-user-friendly. The user must have knowledge about the theory used in the program. However, when the numerical model has been developed, it is fast and easy to use the model for extensive numerical simulations. Parametric studies and perhaps also shape optimisation can be performed.

5. Make suggestions for improved design of aluminium beam-columns with unsymmetric

cross-sections with or without transverse welds. 1.2 Previous work Below is a list of short descriptions of previous work related to the work presented in this thesis. The search of literature was focused on buckling of aluminium columns or beam-columns, with or without transverse welds. The references are chronologically ordered. The undated report by Gilson and Cescotto [23] was inserted where it was believed to be produced.

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Hill and Clark [26] made buckling tests on extruded I-section beam-columns. Four different lengths and load eccentricities were used. Totally 33 beam-columns were tested and they failed by flexural-torsional buckling. Comparisons with some theories were made. One conclusion was that an earlier proposed design formula gave unconservative results. The work by Hill and Clark was comment on in an appended discussion. In Clark [14] buckling tests on eccentrically compressed aluminium beam-columns were made. The cross-sections were rectangular, solid or hollow. Totally 36 buckling tests were made. The alloy was AA6061-T6. The failure was plane plastic buckling. Comparisons were made with various design formulas. One conclusion was that an existing design formula used for lateral buckling also could be used for the tested beam-columns. Hill et al. [27] discussed the design of aluminium beam-columns. Different interaction formulas were used. Comparisons were made with earlier buckling tests. The results of 12 eccentrically compressed aluminium beam-columns with circular tube cross-sections were presented. Some proposals were made. Only flexural buckling was considered. In an appended discussion the work was comment on. A theoretical study of how the shape of the stress-strain curve affects the load carrying capacity of compressed aluminium columns is given in Baehre [2]. Local and flexural-torsional buckling are not considered. Höglund [31] described a method to design beam-columns with respect to flexural buckling. The way Eurocode 9 treats the effect of transverse welds has its origin in this design method. In Bernard et al. [5] buckling of unwelded axially compressed aluminium columns with I- or O-sections were considered. Compression and tensile tests of the material were made. The residual stresses of the extruded profiles were measured. Normally the residual stresses were lower than 30 MPa, but a few peaks of 50 MPa were found. Totally 66 columns were tested. A computer program was used to calculate buckling curves, i.e. the buckling reduction factor as function of the slenderness parameter. The program could only consider in-plane buckling. Imperfections were considered. Ramberg-Osgood material model was used based on the material data from the compression tests. The buckling tests were used to check the buckling curves. This work was used by the ECCS committee 16, described by Frey and Mazzolani [21]. Klöppel and Bärsch [35] produced tables with buckling reduction factors for axially compressed aluminium columns. The tables were produced with help of a German code for steel structures and the results from an extensive test series with compressed aluminium columns. Only flexural buckling and extruded unwelded columns were considered. Among other things, the effect of imperfections, undesired load eccentricity, the shape of the cross-sections and different material properties were studied. The test series was likely the same as the flexural buckling tests presented in Klöppel and Bärsch [36].

4

Klöppel and Bärsch [36] presented results of a test series with 120 unwelded aluminium columns with I-, T- or tubular cross-sections. Only flexural buckling was considered for these columns. Three different alloys, three different slenderness values and some undesired load eccentricity was used. The results of a test series with 39 columns with channel sections were also presented. The axial load was applied at the centre of gravity and the column-ends were free to warp. The columns with low slenderness failed by local buckling whereas the other columns failed by flexural-torsional buckling. Also some tests with beams failing by lateral-torsional buckling was made as well as some buckling tests on shell cylinders. The results of the tests were compared with a new version of the German aluminium code DIN 4113 and good agreement was obtained. Frey and Mazzolani [21] presented the results of ECCS (European Convention for Constructional Steelwork) committee 16, “Aluminium Alloy Structures”. The main task for this committee was to establish buckling curves for axially loaded, unwelded and extruded aluminium columns. The effect of initial curvature, variation of wall thickness of hollow cross-sections and different material properties of the different alloys were considered. Valtinat and Müller [56] used a computer program for beam-columns to calculate the buckling load of longitudinally or transversely welded aluminium columns, centrically or eccentrically loaded. The alloy was AA6082-T6. Only bi-symmetrical I-sections were considered. By a parametric study the upper and lower bound of the influence of the welds on the strength of the columns was investigated. A design method for longitudinally welded columns was proposed. Gilson and Cescotto [23] made flexural buckling tests on extruded unwelded aluminium columns with T-sections. Tensile and stub column tests were made to determine the material properties. The dimensions of the cross-section and the imperfections were measured. Totally 14 buckling tests were made. The capacities from the tests were compared with the capacities from ECCS recommendations and some computer program. Good agreement was obtained between the capacities from the tests and the computer program. ECCS recommendations were conservative. Kitipornchai and Wang [34] made a theoretical study on lateral buckling of T-beams under moment gradient. An energy approach and Fourier series were used. Two assumptions were made: the material was linear elastic and the cross-section was rigid. Beam theory was used. One conclusion was that a design formula used in some codes was unsafe to use. One of these codes was Eurocode 3, the European steel code. Valtinat and Dangelmaier [57] dealt with buckling of unwelded and welded axially loaded aluminium columns mainly with I-, circular or square tube cross-sections. The results from interaction formulas of a German code and a German dissertation were compared with the results of an earlier theoretical study and with some previously made buckling tests. Some of the buckling tests were made on columns with hollow cross-sections with transverse welds. One of the results was that the position of the transverse welds only had a minor influence on the load-carrying capacity of the columns. The German dissertation gave in general better results than the German code.

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In Baehre and Riman [3] buckling tests on extruded aluminium columns with rectangular hollow cross-sections and transverse welds were made. Totally 123 columns were tested. A computer program was also used to calculate the load carrying capacity of aluminium columns. A reduction factor was proposed to be inserted in the interaction formulas of the draft aluminium code DIN 4113. The reduction factor takes into account the position of the transverse weld. A new interaction formula was also proposed. Bulson and Nethercot [11] gave some aspects on the draft version of BS 8118 regarding the design of aluminium columns, beams and beam-columns. Nethercot [44] presented aspects on the design of aluminium columns in the draft version of BS 8118. The interaction formulas of the draft code were presented and the results of them were compared with many buckling tests found in the literature. Hong [28] presented buckling design curves for aluminium columns failing by flexural buckling. Local and flexural-torsional buckling were not considered. The type of alloy, symmetry of the cross-section and welding condition affected the shape of buckling curves. The buckling curves were the results of a study at Cambridge during 1979-1983 and were recommended to the revisionary committee of the British aluminium code CP118. Likely these curves were at least partly adopted in BS 8118. Bradford [8] developed a finite element method that incorporated plate behaviour for modelling the lateral-distortional buckling of elastic T-beams. The theory does not assume that the cross-section is rigid. The method was verified with other theoretical methods. When the beam was subjected to equal end moments such that the tip of the web was in tension, lateral-distortional and lateral-torsional buckling gave similar result. Under the same loading condition but when the tip of the web was in compression, lateral-distortional buckling gave a lower moment capacity than lateral-torsional buckling. For a beam with length-to-height ratio of 15, the buckling moment was 36% lower. Lateral-torsional buckling assumes that the cross-section is rigid. Höglund [32] compared results from buckling tests found in the literature with different interaction formulas, among others a proposal by Mazzolani and a proposal based on the Swedish steel code BSK [7]. The results from 220 buckling tests were used. The tested beam-columns were all unwelded and centrically or eccentrically compressed. The shape of the cross-sections varied. The literature, where the tests were found, is included in the reference list of this thesis. Benson [4] tested 19 hollow square, thin-walled aluminium extruded beam-columns. The test specimens were eccentrically compressed. The alloy was AA6063-T6. The test results were used to evaluate the design methods in the Swedish Regulation for Light Gauge Structures, the ISO working draft ISO/TC 167/SC 3 N122E and a modified version of the Swedish steel code BSK. Höglund [33] presented interaction formulas for flexural and lateral-torsional buckling of I-beam-columns. The formulas are used in the Swedish steel code, BSK. Comparisons were made with tests on steel and aluminium beam-columns reported in the literature. The agreement was found to be excellent, especially for the aluminium tests.

6

Hong [29] used a computer program to simulate flexural buckling of transversely welded aluminium columns with solid rectangular cross-sections. Local and flexural-torsional buckling were not considered. Ramberg-Osgood material model was used. Transverse welding was simulated by depositing weld material on the four faces of the tubular section. The alloy was AA6061. To verify the results of the computer simulations buckling tests on aluminium columns with square tubular cross-sections were made. Especially the effect of the transverse weld’s position on the load carrying capacity was studied. Different buckling curves were drawn. One finding was that a sine curve was suitable to be used to describe the load carrying capacity of a column as function of the transverse weld’s position, similar to the calculations in Eurocode 9 and the modified Swedish steel code BSK. Only pin-ended columns were considered. The paper by Nethercot [45] deals with buckling of aluminium columns, beams and beam-columns with transverse welds. Numerical results from the two computer programs INSTAF and BIAXIAL were compared with a draft version of BS 8118. The two computer programs are described in Lai and Nethercot [37]. One result was that it is only somewhat conservative to design a transversely welded aluminium column as if the whole column consisted of heat-affected material, irrespective of the transverse weld’s position. Buckling curves for aluminium columns, beams, plates and shear webs were discussed in Marsh [41]. The effect of welds was also discussed. Lai and Nethercot [37] used two finite element computer programs to analyse welded and unwelded aluminium structural members. The first program, INSTAF, was used for in-plane analysis. It could consider geometric and material nonlinearity, the effect of residual stresses and strain hardening of the material. The results of the program were compared with some earlier made buckling tests and showed good agreement. The second program, BIAXIAL, was used to analyse the 3D-behaviour of beam-column elements. This program could consider the effect of twisting, warping, residual stresses and initial curvature. Both programs could consider the effect of longitudinal and transverse welds. A piecewise form of Ramberg-Osgood material model was used in both programs. Five unwelded and 22 welded beams were tested in 4-point bending tests to check parts of the programs. All tested beams had a rectangular cross-section, which expected the beams to fail in pure bending. Plates of different lengths were welded to the two flanges to create the heat-affected material. Tensile tests were made both on parent and heat-affected material. The beams were tested up to a reasonable high deflection but not up to failure. INSTAF was generally conservative with a maximum difference of 10% between the tested and calculated load. The two programs were used to make parametric studies on transversely welded beams and columns, with bi-symmetrical I-sections. The columns were pin-ended and subjected to axial load only. The beams were simply supported with a point load at midspan. Also some other structures were calculated. The parametric study showed that it is unsafe to neglect the softening effect of welds at the ends of columns. It also showed that the load carrying capacity of columns were lowered most when the welds were located at mid-height. The capacity was almost equal as if the whole column was made of heat-affected material.

7

The two computer programs INSTAF and BIAXIAL described above were used in Lai and Nethercot [38] to calculate various types of axially loaded aluminium columns. The purpose was to check the column design curves of a draft version of BS 8118. All cross-sections were “compact” for which no local buckling occurred. Imperfections were considered. The cross-sections were I- or T-shaped. The columns were unwelded, transversely welded or longitudinal welded. The strength of most types of columns was safely predicted by the method of the draft code. However, some improvements of the draft code were presented, especially for transversely welded columns when the welds were located at the column ends. Sanne et al. [49] describe buckling tests of unwelded aluminium columns with I-sections. A few beam-columns were tested with a centric load, but for most beam-columns the compressive load was applied eccentrically. The maximum eccentricity was 5 times the depth of the cross-section. Also tests with unequal load eccentricity at the two ends of the beam-column were made. Totally 40 beam-columns were tested. The alloy was AA6351. The material properties were determined by tensile tests. The load carrying capacity from the tests were compared with the capacities from the Swedish steel code BSK, the ECCS recommendations and the Norwegian aluminium code NS 3471. BSK gave the best results. In Sanne [50] 24 extruded aluminium beam-columns with I-sections were tested. The beam-columns were welded with one transverse weld at different sections along the beam-column. The load was applied with an eccentricity such that major axis bending and axial compression occurred. Some beam-columns were loaded centrically. The alloy was AA 6351-T6. The results were compared with a modified version of the Swedish steel code, BSK. The modifications made it possible to take into account the weakening effect of the weld. The comparisons showed that it was possible to use the modified BSK under the condition that the ultimate strength of the heat-affected zone was used as design strength. Using the yield strength will give very conservative results. The book by Sharp [52] deals with design of aluminium structures. Much of the research presented in the book was conducted at Alcoa Laboratories (Aluminium Company of America) during the last 30-40 years. Among other things, the design of columns, beams and beam-columns were discussed. Hellgren [24] presented the results of two test series with totally 28 extruded beam-columns with I-sections. The alloy was AA 6351-T6. The beam-columns were welded with one transverse weld at different sections along the beam-column. The compressive load was applied with an eccentricity in one or two directions. For some beam-columns the load was applied centrically. The results from the tests were compared with BS 8118 and a draft version of Eurocode 9. The draft Eurocode 9 was less conservative than BS 8118. Corona and Ellison [15] made an experimental and theoretical study of T-beams under pure bending. The investigation was focused on the case when the tip of the web was in compression. The tested beams were hot rolled and made of steel. They had length-to-height ratios ranging from 10 to 20. Tensile and compression tests were made to determine the material properties. Some theory was developed to calculate the moment-curvature response. The theory was only briefly described, but the stress-strain curve was trilinear. The theory showed good agreement with the tests. The work by Edlund [17,18,20] is included in this thesis.

8

Bradford [9] studied lateral-distortional buckling of elastic T-section cantilevers. The study was pure theoretical where the finite element method described in Bradford [8] was used. The case when the tip of the web was in compression was studied. Lateral-distortional buckling generally gave a lower buckling load than lateral-torsional buckling. Langhelle [39] studied aluminium structures exposed to fire. Eight buckling tests were made at room temperature. The aluminium columns in these tests had rectangular hollow cross-sections and were centrically compressed. The alloy was AA6082 and the temper was T4 or T6. Two columns were transversely welded at mid-height. The wall thickness was 5 or 7 mm. Global buckling occurred before local buckling for the columns with 7 mm wall thickness. For the columns with 5 mm wall thickness, global and local buckling occurred simultaneously. Besides the 8 tests at room temperature, 23 buckling tests were made at elevated temperature. Comparisons were made with Abaqus, another nonlinear finite element program and with three different codes, Eurocode 9, BS 8118 and the Norwegian aluminium code NS 3471. The paper by Rasmussen and Rondal [47] is dealing with column curves for extruded aluminium columns failing by flexural buckling. A column curve is the same as the previously used expression “buckling curve”.

9

2 Buckling tests 2.1 Introduction Buckling tests were performed. The tested beams were divided in two test series. The first series consisted of 26 unwelded beams, which were tested in 1996. The second series was larger, 39 welded beams tested in 1997 and 1998. All unwelded beams and 34 of the welded beams were simply supported for bending. The remaining five of the welded beams had clamped ends. Most welded beams were transversely welded either at the quarterspan or at the midspan, but for some beams two or three transverse welds were used. A transverse weld is perpendicular to the longitudinal axis of the beam. The lengths of the beams were 500, 1020 and 1540 mm, respectively. The theoretical cross-section dimensions were the same for all beams. The dimensions are shown in figure 2.1. The six load application points I-VI are also shown in the figure. The notation a in figure 2.1 is equal to the distance between the centre of gravity and the shear centre.

60

60 6

6 6

SC

GC

a

a

a

a

I

II

III

IV

V

VIa

Figure 2.1. Theoretical cross-section dimensions in mm and load application points. The beams were cut from profiles of five meters in length. All profiles were delivered free of charge from SAPA (Skandinaviska Aluminium Profiler AB) in Vetlanda. The alloy and temper for the welded beams were AA6082-T6. In order to get a sufficient large difference in strength between the parent and the heat-affected material it was necessary to use temper T6. The profiles from 1996 were taken from stocks so the alloy and temper were unknown, but probably the material was AA6082-T6 as for the profiles from 1997 and 1998. However, this was of no real importance since the material properties were determined by tensile tests. Figure 2.2 shows a sketch of the test equipment when the beam was simply supported for bending. In appendix A two photos of the test equipment are also presented. The compression force was applied by a hydraulic jack. With some time interval the load was increased by a suitable load step. Most beams were tested by using a time interval of two minutes during the whole test process. When quite many beams had been tested it was found out that it was necessary to speed up the test process. This was done by increasing the load instantaneously up to about 75% of the theoretical load carrying capacity and then use a two minutes interval. The theoretical load carrying capacity was calculated according to the Swedish steel code BSK [7], see section 3.4. The load step used in connection with the two minutes interval was chosen with help of the theoretical load carrying capacity. After each load step the distance between

10

the supports was fixed so the deflection in the direction of the applied load was constant during the whole load step. One reason for this was to avoid difficulties that otherwise would have occurred in conjunction with possible leaks in the hydraulic system. The measuring data was used to determine the failure load and to draw load-deflection diagrams. The reason for the time interval was that the transverse deflection of the beam should be stabilised. The value of the applied load changed somewhat during the two minutes interval due to the fixed deformation in the direction of the applied load. This can be seen in the load-deflection curves in appendix B.

2 HEA 800

mp3

Aluminium beam

jackload cellmp4

mp6

mp2 mp5

mp1

mp0mp9

mp8

mp7

steel platesteel plate

steel ballsteel ball

mp = measuring point

Figure 2.2. The test equipment seen from the side and from above, simply supported case. Three measurements of the deflection at the midspan were made. This made it possible to see if the beam twisted during the test. In 1996 and 1997, the beams were tested such that the force of gravity acted parallel with the web. During the testing in 1998 the force of gravity acted along the flange. The reason for this was that different persons from the laboratory personnel made the tests and also that there was a year between the different test series so details were likely forgotten by the laboratory personnel. The measurements are illustrated in figure 2.3. The deflection was also measured at the supports. By this way it was possible to compensate the midspan deflection from the possible movements of the steel ball at the supports when the applied load was increased. It was important to know if the deflection was parallel with the flange or the web. Any other information about the direction of the deflections was not considered as important. This means for instance that all measured deflection values could have been multiplied with –1 in a load-deflection diagram in appendix B, if this was considered to give a more “beautiful” diagram. All deflections were measured with rotational potentiometers.

11

1996, 1997

1998

Figure 2.3. Measurement of the deflection. Figure 2.4 shows the steel plate used to transfer the load into the beam. Only the most important dimensions are shown in the figure. To prevent the beam from sliding at the supports, seven eight mm screws with sharpened tips were used in each steel plate together with a 20 mm deep milled groove. Two of the screws are shown in the figure. The hinge at the supports was accomplished by a 14 mm steel ball. The friction was reduced by a sheet of teflon.

Figure 2.4. Steel plate used to transfer the load into the beam, dimensions in mm.

12

Five of the welded beams had clamped ends, which were obtained by removing the steel balls and the teflon sheets so the load was applied directly to the steel plates at the beam-ends. The load was applied at the centre of gravity of the T-section for the clamped beams. A steel ring was screwed to the back of the steel plates in order to fix the beam. The steel rings also prevented to some extent the beam from rotating at the supports. This adjustment of the supports was believed to give an accurate enough approximation of clamped ends. This fact was also confirmed to a certain extent by a visual inspection of the supports. When the load carrying capacities according to the different codes were calculated, the possible non-perfect clamped ends were considered by testing different buckling lengths. For all beams, the cross-section dimensions were measured with a slide-calliper and the beam length with a ruler. The results of the measurements are shown in table 2.7 and 2.8. In these tables, some data about the test series can also be found. The welds were obtained by introducing weld material on the surface around the beams. If the beams had been jointed with traditional butt welds, the initial curvature of the beams would have probably been larger. It was also more convenient to introduce the heat-affected material in this way. The initial curvature of aluminium profiles is very low, because immediately after the extrusion the profiles are stretched. The heating during the welding could have induced some initial curvature, but visually no curvature was seen. According to Mazzolani [42], the initial curvature of an extruded aluminium profile is about l 2000 (displacement at midspan), where l is the length of the profile. For the tested beams, the midspan deflection will then be 0,25-0,77 mm. In the codes the initial curvature is considered in the buckling formulas. For the Abaqus calculated beams, the initial curvature can be seen as included in the imperfections. It was decided that the initial curvature of both the unwelded and the welded beams were not necessary to measure. Residual stresses are very small in extruded profiles. The quenching, which is a part of the heat-treatment of temper T6, induce some residual stresses, but according to Mazzolani [42] they are very small (less than 20 MPa) and have negligible effect on the load carrying capacity. However, residual stresses in welded profiles are not small. All welded beams in this thesis were transversely welded. Residual stresses of transverse welds have less influence on the load carrying capacity than longitudinal welds because they are local and they are mostly directed perpendicular to the longitudinal axis of the beam. Residual stresses are mostly directed parallel with the welds. The effect of the residual stresses is at least partly included in the heat-affected material model based on the tensile tests. For the reasons given above, it was decided that the residual stresses were not necessary to measure. If the welds were longitudinal, residual stresses had to be considered. The weld itself was not studied but rather the affect of the heat-affected zone adjacent to the weld. MIG-welding was used for all welds, both the beams and the tensile tests.

13

2.2 Tensile tests The strength of the material was determined by tensile tests. All tensile tests were made for the material in the direction of the extrusion. According to Hopperstad [30] and Moen [43] the material is anisotropic, so the direction of measurement is of importance. From each profile a tensile test for the parent material was made. The reason for this was to determine the characteristics of the different materials as thoroughly as possible. Even if two profiles were made of the same alloy and temper, the material properties still could vary to a certain extent. The different alloying elements are allowed to vary in a quite wide range for a certain alloy. This means that two alloys with the same designation can have different material properties. When welded beams were cut from a profile, a tensile test for the heat-affected material was also made. The test specimens were tested in a Material Test System, MTS 311.21s, according to the Swedish standard. The heat-affected material for the welded tensile tests was obtained by introducing weld material on the surface of profile parts. The test specimens were then cut from the web of these parts. Two initial tensile tests were made, one where a traditional butt weld was used and one where the weld material was introduced on the surface. The failure for the jointed tensile test was brittle and the failure load was very low. The non-jointed tensile test behaved in a better way. For the welded tensile tests and beams, there were mainly two reasons for introducing weld material on the surface instead of using traditional butt welds: the result of the two initial tensile tests and that it was more convenient. The beams and corresponding tensile test must be welded as similar as possible. When the load carrying capacity according to different codes was calculated, only the yield strength f0.2 of the parent material and the ultimate strength fhaz of the heat-affected material were needed. The ultimate strength fhaz was evaluated as the maximum stress that occurred. When more complicated calculation methods are used, like the finite element calculations in chapter 4, the whole stress-strain curve is needed. During a buckling test it is the compression strength rather than the tensile strength that is most decisive for the load carrying capacity of the beam. However, it is difficult to perform good compression tests, and therefore the compression strength was approximated with the tensile strength. The most important reason for this fact is connected to difficulties in avoiding buckling of the specimen. For this reason the test specimens must be very short. This will in turn make the strain measurements more difficult. Moreover, the prevented lateral contraction at the supports will have a larger influence when the length of the specimens is short. For these reasons the compression strength was taken equal to the tensile strength. However, a brittle tensile test is a bad approximation of the compression strength. From the buckling tests of the welded beams it was possible to see that the load carrying capacity of the beams was not determined by the strength of the weld itself. For one beam where the bending induced compression in the flange and tension in the web, a crack appeared in the weld. However, this was most certainly a secondary failure caused by large displacements. The other welds were uncracked. For these reasons it was desirable that the failure of the welded tensile tests should be ductile and occur at the heat-affected zone beside the weld. In this section, nominal stresses and strains are shown. It means that the stresses were calculated as measured force divided by original cross-section area and the strains as measured

14

elongation divided by original length. The elongation was measured over a distance of 50 mm. This distance was the same as the original length. There exists other ways of calculating stresses and strains. In chapter 4, “true” or Cauchy stresses and log-strains are used and explained. The beams tested during 1996 were all unwelded. These beams were cut from six profiles. From each of these profiles a tensile test was made. The values of the yield strength and the modulus of elasticity from these tensile tests are shown in table 2.1. These values were evaluated graphically from the printouts of the testing machine. Table 2.1. f0,2 and E for the unwelded tensile tests from 1996. Tensile test 1 2 3 4 5 6 f0,2 [MPa] 323 268 326 324 320 313 E [GPa] 68,8 71,0 70,3 65,0 70,7 70,7 The yield strength for tensile test 2 was much lower than the rest so all profiles may not have been made of the same alloy. As already mentioned, it was however likely that the material was AA6082-T6. The modulus of elasticity should be close to 70 GPa. For tensile test number 4 the modulus of elasticity differed quite much from 70 GPa. It is difficult to give a reasonable explanation for this. The reason was likely not slip in the measuring device. The stress-strain curve would then have similar shape as the curve for the understiff behaviour in figure 2.5, but this was not the case. The slope of the stress-strain curve was constant in the elastic region, which can be seen in figure 4.2. All stress-strain curves from 1996 were left unmodified, because no detectable errors occurred during the testing. The tensile tests from 1997 were denoted A-G and AW-GW, where A-G were unwelded and AW-GW were welded. Corresponding notations from 1998 were 1-4 and 1W-4W. The profiles, from which the tensile tests and the beams were cut, were denoted in the same way as the parent material. Profile F was excluded. In table 2.2 the results from the first series of tensile tests from 1997 and the tensile tests from 1998 are shown. The failure for the welded tensile tests CW, DW and EW in table 2.2 occurred at the weld. For the other welded tensile tests in table 2.2, the failure occurred beside the weld in the heat-affected zone. In table 2.2, the failure strain at maximum load is shown for the welded tensile tests. The failures which occurred at the heat-affected zone were significant more ductile than the failures which occurred in the weld. The welded tensile test CW had significant lower ultimate strength than the other tensile tests. One explanation is that less weld material was used for tensile test CW than for the other welded tensile tests. There could also be welding defects like enclosed pores. Studying the welded zone thoroughly made it possible to observe that the penetration of the weld material was deeper for the specimens were failure occurred at the weld, as opposed to the cases when the failure occurred at the heat-affected zone. The reason why the failure did not occur more often at the weld was that the welds were overfilled in most cases, which in turn enlarged the cross-section area.

15

The materials which correspond to the tensile tests 1, 1W, 3W, 4 and 4W were slightly modified due to under- or overstiff behaviour at the start of the stress-strain curve, see figure 2.5.

0

50

100

150

200

250

300

350

Strain [-]

Understiff behaviour Overstiff behaviour

Figure 2.5. Under- and overstiff behaviour at the start of the stress-strain curve. The understiff behaviour was most likely due to slip in the test equipment. The overstiff behaviour was more difficult to explain. A modification was performed such that the start of the stress-strain curve was straightened and the whole curve was moved parallel with the strain-axis so the curve started at zero stress and strain. The values that are shown in table 2.2 are the values that were obtained after the modification due to under- or overstiff behaviour. Table 2.2. Tensile tests from 1997 (first series) and 1998. Tensile Unwelded tensile test Welded tensile test

Test f0,2 [MPa] E [GPa] fhaz [MPa] E [GPa] Failure strain, A50 [%] A, AW 296 71,1 227 76,6 4,96B, BW 293 71,1 213 71,7 4,50 C, CW 298 71,4 68 57,1 0,452 D, DW − − 167 71,0 2,04 E, EW − − 183 70,0 2,72 G, GW 276 70,5 205 76,3 4,79 1, 1W 312 70,7 188 90,9 7,42 2, 2W 307 71,4 189 65,6 7,40 3, 3W 283 69,8 183 99,4 7,36 4, 4W 321 69,8 191 76,2 7,42

From the longest beams tested during 1997, pieces were sawn out and used as tensile tests after the beams were tested. These tensile tests are shown in table 2.3. The failure for tensile test CW in table 2.3 occurred at the weld while the failure for tensile tests DW and EW occurred at the heat-affected zone.

16

Table 2.3. Second series of tensile tests from 1997. Tensile Unwelded tensile test Welded tensile test

Test f0,2 [MPa] E [GPa] fhaz [MPa] E [GPa] Failure strain, A50 [%] C, CW 293 74,0 137 73,3 1,74D, DW 284 71,2 179 79,0 6,73 E, EW 311 70,5 183 74,1 6,55

From table 2.2 and 2.3 it is clear that the values for the welded tensile tests were uncertain and highly dependent on how the welding was performed. Another question was if the beams and corresponding tensile test were welded in the same way. Even a small difference in, for instance, the depth of penetration seemed to affect the strength to a very high degree. The welds of the tensile tests from 1998 seemed to have more even quality, at least when considering the values of fhaz and the failure strain, but the wide scatter of the modulus of elasticity was undesired. For the welded tensile tests from 1997, the start of the stress-strain curve was non-linear, which made the evaluation of the modulus of elasticity rather arbitrary. This non-linearity was not present for the welded stress-strain curves from 1998. An example of a non-linear start of a stress-strain curve is given in figure 2.6. The value of f0,2 was possible to determine more accurate. The value of f0,2 for material C in table 2.2 and 2.3 does not differ much. The yield strength in table 2.1 is for all profiles except one, not so widely scattered, but the material was unknown so not too many conclusions can be drawn.

0

20

40

60

80

100

120

0.0000 0.0004 0.0008 0.0012 0.0016 0.0020 0.0024Nominal strain [-]

Figure 2.6. The start of the stress-strain curve for material BW. It is evident that there were more uncertainties with the welded tensile tests than the unwelded. There were several possible causes for that. One cause was that the section area was much enlarged in areas where the weld material was located. It was likely that the elongation only occurred for the parts where the section area was not enlarged. The result would be too low strain values, which means that the modulus of elasticity would be enlarged. Another cause could be residual stresses due to welding. The failures did not occur perpendicular to the longitudinal axis of the test specimens. This indicates that the strength was not uniformly

17

distributed over the width of the specimens. The residual stresses due to welding could also be one explanation why the start of some welded stress-strain curves was non-linear. A third cause could be eccentricity of the test specimens. The load was applied through two holes, see figure 2.7. If the holes were not located centric along the centre line of the test specimens there would be an additional moment to the axial force. How much the eccentricity affected the strain values depended on which side of the specimens the strains were measured. No information was given on the location of measurement points for the strains. The effect of the eccentricity was thus uncertain. Problems with this kind of eccentricity can also be present for unwelded tensile tests. It is likely that the different causes sometimes interacted and sometimes counteracted each other. This could be an explanation why the modulus of elasticity differed so much, but it is doubtful if this explanation is good enough, because it was not possible to see any significant difference in size and extension of the weld material. It would have been more realistic if the values were not so widely scattered.

20 mm

Figure 2.7. Test specimen for the tensile tests. The value of fhaz is not affected by any strain problems, since it is evaluated as the maximum stress value. The calculation according to different codes in chapter 3 only uses fhaz and is accordingly not affected by any strain problems. This is not the case for the finite element calculations in chapter 4. In this calculation the whole stress-strain curve is used. It appeared necessary in this context to correct at least some stress-strain curves due to unrealistic values of the modulus of elasticity. The uncertainties discussed above were present for all welded tensile tests and therefore it was decided that all stress-strain curves of the heat-affected material should be modified. The modification was performed such that all strain values for the heat-affected materials were multiplied with a factor so a modulus of elasticity close to 71 GPa was obtained. Before this modification could be made, it was necessary to decide which tensile tests that should represent the different heat-affected materials. For most materials there was only one choice. For material CW there were no satisfactory choices. Material A and C were almost identical, see figure 4.3. For this reason material CW was set equal to material AW. Both materials DW and EW were taken from table 2.3, probably because of the lower strain values in table 2.2. The value of fhaz for material DW in table 2.2 is also quite low. When the different heat-affected materials had been selected, the strains were multiplied with the factors that are given in table 2.4. Table 2.4 Strain multiplication factors. Material AW BW DW EW GW 1W 2W 3W 4W Factor 1,08 1,00 1,27 1,04 1,07 1,29 0,918 1,40 1,08 Some comments about material C must also be made. For this material there were two reasonable tensile tests. The yield strength f0,2 for material C was set equal to the mean value from table 2.2 and 2.3. In the finite element calculations in chapter 4, the stress-strain curve for

18

material C was taken from table 2.3. When considering the value of the modulus of elasticity probably it would have been better to choose the other tensile test, but this was not done. The strength values for the chosen and sometimes modified materials from 1997 and 1998 are summarised in table 2.5. These values were used when the load carrying capacities according to different codes were calculated. Table 2.5. Final nominal strength values for the materials from 1997 and 1998.

Profile A B C D E G 1 2 3 4 f0,2 [Mpa] 296 293 296 284 311 276 312 307 283 321 fhaz [MPa] 227 213 227 179 183 205 188 189 183 191 In section 4.2, the stress-strain curves for all tensile tests are shown. These curves are the material models used in the Abaqus calculations. They are shown after the modifications due to under- or overstiff behaviour at the start of the stress-strain curves and unrealistic values of the modulus of elasticity. The curves for the heat-affected materials are also modified at large strain values. Only the “true” stress and log-strain versions of the curves are shown. It was decided that these curves were more important to show than the nominal stress-strain curves, because they were used in the Abaqus calculations. To estimate the reasonableness of the strength values from the tensile tests, comparisons were made with values from the literature. The values are given in table 2.6 and are valid for the extruded alloy AA6082, temper T6, thickness 6 mm and MIG-welding. Table 2.6. Strength values from some literature. Literature f0,2 [MPa] fhaz [MPa] fu [MPa] Eurocode 9 [12] 260 202 * 310 BS 8118 [10] 255 128 ** 275 SAPA’s handbook [51] 260 310 BKR 99 [6] 245 180 290 “Goda råd…..” [54] 190 290 TALAT [40] 270 * haz haz uf f= ⋅ = ⋅ =ρ 0 65 310 202, MPa ** haz zf k f= ⋅ = ⋅ =0 2 0 5 255 128, , MPa Eurocode 9, BS 8118, SAPA’s handbook and BKR 99 give the same type of values and they can be considered as minimum guaranteed values. For the other literature no additional information about the values was found. BKR 99 does not distinguish between different welding techniques, so the value of fhaz does not require MIG-welding. It was not found in BS8118 that it is allowed to use fu when the strength of the heat-affected material is calculated, like it is in Eurocode 9 and BSK, and therefore the strength of the heat-affected material was set equal to zk f 0 2, . In section 5.5.2 of Eurocode 9 it is clearly stated that

f0,2 and fu should be multiplied with the same reduction factor ρhaz. Such a clear statement was

19

not found in BS 8118, but it was assumed that kz should be used in the same way. Both Eurocode 9 and BS 8118 say that f0,2 is lowered more than fu when the material is welded. It can therefore be questioned that both f0,2 and fu are lowered with the same reduction factor, but probably this should be seen as an acceptable approximation which has been adopted. Perhaps it would have been better to give values of the yield and ultimate strength of the heat-affected material instead of a reduction factor? A few conclusions can be drawn when the values in table 2.5 and 2.6 are compared. The lowest value of fhaz is given by BS 8118. Eurocode 9 gives a higher value of fhaz than quite many of the tensile tests. 2.3 Results In table 2.7 and 2.8 there are some data and measured results of the tested beams. From the two major series three minor test series were selected. These latter series were tested in connection with a course for the ungraduate students of the last year. There was a desire that the numeration of the three minor series should be continuous, which is the reason for the numeration in table 2.7 and 2.8. Among other things, the measurements showed that the thickness t for all beams was a bit larger ( 6 03 6 20, ,mm mm≤ ≤t ) than the theoretical (6 mm). The notations b, h, tw and tf in the tables are explained in figure 3.5. In table 2.8 the desired weld locations are shown. As can be seen, for some beams two or three transverse welds were used. When the beams were simply supported for bending, a weld location of 0 or lc was not possible. This location corresponded to the centre of the steel ball. It was not possible to locate the welds closer to the centre of the steel ball than about 65 mm. The buckling length and the beam length were denoted lc and l, respectively. zhaz is the distance between the point of contra flexure of the buckling curve and the most critical weld. No value of zhaz is given for the clamped beams since several buckling lengths were tested and for each buckling length a new value of zhaz was obtained. Both lc and zhaz are further explained in section 3.1. The beam length l is equal to the distance between the two groove bottoms of the steel plates at the beam-ends. One horizontal and one vertical load-deflection diagram is shown for each beam in appendix B. For all eccentrically loaded beams which were simply supported for bending, the load-deflection curves according to the first and second order beam theory were inserted in one of the two diagrams. The beam theory formulas are shown below. The buckling length lc is equal to the distance between the steel balls at the supports.

midcw

e lEI

N,1

2

8= , first order analysis

midE

w eN

P, cos22

12

1= −

π E

x

cP

E Il2

2

2=π

, second order analysis

20

From the diagrams in appendix B it can be seen that the load-deflection curves for the second order analysis coincide fairly well with the corresponding measured curves when the load level is reasonable low. Somewhat generalized, the course of events for the tests can be described as follows. The load-deflection diagrams in appendix B give additional information. Load application point I-II: During the test the beam bent in the direction of the web. The deflection parallel with the flange was generally small, except at failure when this deflection became very large. Load application point IV-VI: During the test the beam bent in the direction of the web. The deflection parallel with the flange was mostly small. At failure there was a local buckle in the web. Load application point III: The deflection parallel with the web was small. The deflection parallel with the flange was also small, except at failure when this deflection became very large. When the marks from the seven screws were studied at one of the two ends of PB-39 in table 2.8, it was possible to see that slip had occurred. The slip mark from all screws had the same shape. The most reasonable explanation for this was that the beam was not properly knocked into the bottom of the groove when the testing started. In the beginning when the load was applied, the beam sank uniformly into the bottom of the groove. The result was probably that PB-39 obtained too low load carrying capacity. This can also be seen when the failure load for PB-39 is compared with the failure load for PB-37. It is reasonable to believe that these two beams should have closer failure loads (67,4 kN and 51,3 kN). For the reasons given above PB-39 was not part of the evaluation of the test results. The slip for PB-22 in table 2.7 was more easily seen. In this case the screws and the groove could not withstand the moment caused by the applied load and therefore a rotation of the steel plates occurred. The load-deflection curves for the welded column PB-23 in table 2.8 are shown in appendix B. The deflection parallel with the web, i.e. the stiff direction, is large while the deflection parallel with the flange, i.e. the weak direction, is small. This situation seems not realistic and it does not occur for the other centrically compressed columns. When the test specimen for PB-23 was studied it was observed that local buckling had occurred at the tip of the web. Furthermore, it was observed that the penetration of the weld material in the flange was not so deep, which could mean that the heat-affected zone was small in the flange. In the web the penetration was larger and the softer web therefore buckled for the centric load. The deformation of the test specimen agreed with the information obtained from the load-deflection diagrams in appendix B. This buckling test indicates that it can be more unfavourable when just a part of the cross-section is welded than if the whole cross-section is welded. Due to these observations, the tested load carrying capacity for PB-23 could be too low. The welded column PB-23 is used in the further evaluation in this thesis, but it is given special comments.

21

Table 2.7. Unwelded beams from 1996. Beam Profile Lap l

[mm] b

[mm] h

[mm] tw

[mm]tf

[mm]N [kN]

test PB-1 2 II 500,0 60,00 60,00 6,06 6,10 104,8 PB-2 1 V 500,0 59,94 60,00 6,12 6,12 61,2 PB-3 1 IV 501,5 60,00 60,02 6,07 6,11 88,8 PB-4 3 V 502,0 60,00 60,00 6,08 6,09 61,6 PB-19 2 II 502,0 60,03 60,09 6,09 6,13 105,2 PB-20 1 V 501,0 60,04 60,03 6,08 6,13 61,1 PB-21 2 III 501,0 59,98 60,00 6,04 6,09 -H PB-22 3 I 500,5 60,02 60,02 6,12 6,13 -SL

PB-5 5 II 1020,0 60,04 60,00 6,04 6,08 73,7 PB-6 4 V 1019,5 59,98 59,97 6,04 6,03 42,4 PB-7 5 II 1020,5 59,96 59,97 6,11 6,12 67,2 PB-8 5 IV 1020,0 59,96 59,98 6,06 6,09 49,3 PB-9 4 III 1020,5 59,95 59,92 6,04 6,06 64,4 PB-10 3 IV 1019,0 60,01 59,99 6,11 6,12 58,5 PB-11 4 I 1019,5 60,02 60,01 6,05 6,07 52,7 PB-12 6 IV 1020,5 59,95 59,99 6,07 6,08 55,4 PB-23 6 III 1019,5 60,02 60,00 6,06 6,08 64,9 PB-24 6 V 1019,5 59,92 59,92 6,06 6,10 43,6

PB-13 2 II 1540,5 60,04 59,97 6,05 6,07 32,9 PB-14 1 V 1540,0 60,09 60,02 6,05 6,07 25,0 PB-15 1 II 1539,5 60,02 60,05 6,05 6,06 31,9 PB-16 3 III 1540,0 60,01 60,04 6,08 6,07 32,5 PB-17 2 I 1539,5 60,02 60,00 6,05 6,07 31,2 PB-18 5 V 1540,0 60,02 60,01 6,05 6,08 26,0 PB-25 4 I 1540,5 60,01 60,06 6,11 6,10 30,9 PB-26 6 III 1540,5 59,95 60,05 6,08 6,08 32,0

H = Failure load too high for the test equipment SL = Slip occurred Lap = Load application point

22

Table 2.8. Welded beams from 1997 and 1998. Beam Pro-

file Lap Weld-

locationzhaz

[mm]l

[mm] b

[mm] h

[mm] tw

[mm] tf

[mm] N [kN]

test PB-1 G I lc/2 273 499,5 60,05 60,13 6,09 6,20 59,9 PB-2 G IV lc/2 275 499,0 60,05 60,13 6,10 6,17 63,5 PB-3 G II lc/2 272 500,0 60,12 60,14 6,09 6,12 84,2 PB-4 G V lc/2 272 500,0 60,06 60,06 6,13 6,17 47,0 PB-5 G I lc/4 141 499,0 60,03 60,05 6,11 6,17 59,3 PB-6 G V lc/4 135 499,5 60,01 60,01 6,08 6,11 51,0 PB-19 4 II lc/2 275 500,0 60,05 60,13 6,10 6,14 90,7 PB-20 4 II lc/4 151 500,0 60,03 60,10 6,10 6,15 83,4 PB-21 4 V 0 , lc 66 499,0 60,05 60,10 6,10 6,16 55,2 PB-22 2 VI lc/4 150 500,0 60,00 60,05 6,05 6,15 36,9 PB-23 G III lc/2 275 499,0 60,00 60,03 6,09 6,13 98,9 PB-24 C I 0 , lc 65 500,0 59,95 59,98 6,03 6,10 54,7SL

PB-25 G VI lc/4 153 501,0 60,00 60,03 6,05 6,10 36,9

PB-7 A I lc/2 533 1020,0 60,10 60,18 6,09 6,14 38,4 PB-8 A V lc/2 532 1020,5 60,15 60,19 6,13 6,15 34,7 PB-9 A II lc/2 533 1019,0 60,07 60,12 6,09 6,12 51,2 PB-10 A IV lc/2 532 1019,0 60,07 60,12 6,07 6,11 47,4 PB-11 B I lc/4 270 1019,0 60,04 60,10 6,09 6,13 43,5 PB-12 B V lc/4 261 1019,0 60,04 60,08 6,16 6,18 34,8 PB-26 2 III lc/2 535 1018,0 60,00 60,10 6,10 6,13 62,9 PB-27 2 VI lc/4 281 1018,0 60,00 60,05 6,08 6,13 28,5 PB-28 3 I lc/2 537 1019,0 60,03 60,13 6,10 6,16 38,5 PB-29C 1 III 0 , l - 1019,0 60,05 60,08 6,11 6,15 105,2 PB-30C C III 0 , l/2 , l - 1019,0 60,00 60,00 6,08 6,10 89,2 PB-31 B VI lc/4 280 1019,0 60,00 60,00 6,06 6,10 28,0 PB-32C C III 0 , l/2 , l - 1019,0 60,00 60,00 6,10 6,11 117,9

PB-13 D I lc/2 790 1539,0 60,04 60,10 6,10 6,14 25,0 PB-14 E IV lc/2 795 1539,0 60,05 60,10 6,09 6,13 26,2 PB-15 C II lc/2 795 1539,0 60,04 60,06 6,09 6,13 29,7 PB-16 D V lc/2 793 1540,0 60,04 60,08 6,07 6,12 22,3 PB-17 D I lc/4 410 1539,5 60,07 60,11 6,10 6,12 27,4 PB-18 E V lc/4 400 1540,0 59,99 60,05 6,06 6,12 25,7 PB-33 4 I lc/2 795 1539,0 60,03 60,10 6,10 6,15 27,4 PB-34 3 VI lc/4 409 1540,0 60,10 60,15 6,10 6,16 20,0 PB-35 3 I lc/4 408 1539,0 60,05 60,13 6,11 6,18 26,2 PB-36 2 III lc/2 795 1539,0 60,00 60,05 6,08 6,13 26,9 PB-37C 1 III 0 , l - 1538,0 60,00 60,10 6,10 6,16 67,4 PB-38 1 VI lc/4 410 1539,0 60,00 60,08 6,10 6,15 21,3 PB-39C E III 0 , l - 1539,0 60,00 60,00 6,08 6,10 51,3SL

C = Clamped beam SL = Value not reliable due to slip Lap = Load application point

23

By observing the load application point, material, beam length and possible weld location(s) for the tested beams some conclusions could be drawn when the load carrying capacities were compared. The failure load for PB-2 and PB-20 in table 2.7 was almost the same (61,2 kN and 61,1 kN). The same applies for PB-1 and PB-19 (104,8 kN and 105,2 kN). These results are satisfactory, but it is undesirable that the failure load for PB-5 and PB-7 in the same table differ so much (73,7 kN and 67,2 kN). It is also undesirable that there is such a large difference of the failure load for PB-30 and PB-32 in table 2.8 (89,2 kN and 117,9 kN). In table 2.8, the beam PB-20 ought to be stronger than PB-19, but this is not the case. The two beams have the same material, load application point and length, but the weld was located at the quarterspan for PB-20 whereas it was located at the midspan for PB-19. In table 2.8, PB-1 is stronger than PB-5. For the same reason as PB-19 and PB-20, the result ought to be reverse. For PB-13 and PB-17 in table 2.8 the situation is the same. The two beams have the same length, material and load application point, but different weld location. In this case the result is more reasonable, the beam is stronger with a weld at the quarterspan than at the midspan. The comparisons made above show that the scatter of the test results was sometimes quite high and not always so reliable. The results in table 2.9 are an attempt of showing how much the load carrying capacity for a beam is reduced due to welding. The capacity will be reduced because the welding introduce a heat-affected zone with reduced strength. The table shows the percentage change in the load carrying capacity as a function of beam length, weld location and load application point. All unwelded and all welded beams with one transverse weld were sorted into the suitable categories in table 2.9. When more than one beam suited each category, an average load carrying capacity was calculated so one capacity before and one after the welding was obtained. From the capacities, the percentage difference was then calculated. The capacity differences could not always be calculated because all combinations of beam length, load application point and weld location were not tested. The differences in cross-section dimensions and especially the strength were not considered. It is difficult to draw any conclusions from table 2.9, but perhaps there is a tendency that the reduction is lower for the longer beams. This could be reasonable because for the longer beams, the failure is more of an instability type than a material strength type. At least two unrealistic situations can be found in table 2.9. It is not likely that the reduction should be higher when a short beam is welded at the quarterspan than at the midspan. It is also not likely that the capacity of a beam is increased when it is welded as the figure +0,78% indicates. The reason can be differences in strength and the scattering of the test results as discussed in accordance to table 2.7 and 2.8. Table 2.9. Change of the load carrying capacity due to welding. Beam length

[mm] Weld

location I II III IV V

500 lc/2 - -16,7% - -28,5% -23,3% lc/4 - -20,6% - - -16,8%

1020 lc/2 -27,0% -27,3% -2,71% -12,9% -1,93% lc/4 -17,5% - - - -1,91%

1540 lc/2 -15,6% -8,33% -16,6% - -12,5% lc/4 -13,7% - - - +0,78%

24

25

3 Comparisons between tests and different codes 3.1 Introduction The failure load from the tests was compared with the load carrying capacity from three different codes, the European aluminium code Eurocode 9 [12], the British aluminium code BS 8118 [10] and the Swedish steel code BSK [7]. It was necessary to use a modified version of BSK because the code does not describe how the local weakening at the heat-affected zone should be calculated. This is natural since for normal construction steel, welding will not induce a heat-affected zone that lowers the strength. In contrast, for high strength steel such a zone will occur. However, this zone is very narrow due to the lower thermal conductivity of steel in comparison to aluminium. Thus at least in principle there will not be any loss of strength due to the prevented lateral contraction. This has been verified by tensile tests on butt-welded joints in high strength steel. BSK 3:413 states that a local weakening may be calculated with respect to the ultimate strength instead of the yield strength. The codes are mainly written for double-symmetric I-sections and rectangular hollow sections and are just partly adjusted to fit other cross-sections. It is not obvious how the codes should be applied on T-sections. The largest problem is related to the fact that there are two section moduli and that the elastic section modulus can be larger than the plastic section modulus. How to handle these situations is not clearly described in the codes. Also some other situations occur in the codes that require some kind of interpretation. It would be desirable that the codes were so clearly written that a detailed description of the calculations was not needed. Unfortunately, this is not the case for a T-section when both axial force and bending moment are present. If the only applied load was an axial force, the codes would likely be clear enough. To give clear descriptions of the calculations was considered as necessary. These descriptions are given in sections 3.2-3.4. The calculations are described straightforward without too much discussion around the subjects. The reason is that additional information can be found in the codes, but also to save space. When something is considered to be unclear or when different interpretations have been made, detailed explanations and sometimes references to the codes are given. To get full understanding of this chapter, access to, or knowledge about, the codes is likely needed. The bending moment capacity is used in the interaction formulas for buckling. The codes are not totally clear in their description of how to handle the moment capacity for unsymmetric cross-sections. One interpretation of the codes could be to always use the moment capacity when both the compression and tension edges are considered, i.e. to use the lowest section modulus of the two edges. This interpretation has some drawbacks. The load carrying capacity for a T-section ought to be lower when the bending moment acts such that the web is in compression than in the opposite case. The results from the tests and the derivation of Mcr in section 3.5 also showed that this was the case. If both the compression and tension edges are considered when the moment capacity is calculated, it is not sure that there will be a difference in the load carrying capacity when the load eccentricity is equal. However, in the first case the bending moment causes compression in the web and in the second case compression in the flange. This is one drawback.

26

The chosen interpretation was to only consider the section modulus of the edge in compression when calculating the bending moment capacity in the interaction formulas. This is reasonable because the stresses caused by the axial force and the bending moment have the same sign at the edge in compression. At the edge where the bending moment causes tension, the axial force and the bending moment counteract each other. Such an effect is not found in the interaction formulas. This is a second drawback. It was found that the chosen interpretation gave better result for the tested beams loaded at position I and II, than the interpretation to use the lowest section modulus when calculating the bending moment capacity. This is a third drawback. For the tested beams it was found that the chosen interpretation gave conservative results. However, when the bending moment was large and the web was in tension it was found that the interaction formulas for flexural and lateral-torsional buckling gave unsafe values for the load carrying capacity. The reason can be large tensile stresses or insufficient buckling checks. This is further discussed in chapter 5. The chosen interpretation of how the bending moment capacity should be calculated, i.e. to only consider the section modulus of the edge in compression, will lead to the situation that

el x cW . . is greater than pl xW . when the bending moment causes compression in the flange and tension in the web. The formulas in the codes are not suited for such a situation. It would for instance mean that the bending moment capacity would not be highest in the most compact cross-section class. The situation for the shape factors ycα and ytα in Eurocode 9 is also unclear. These shape factors are explained in section 3.2.3. For these reasons some modifications of the codes were needed. The uncertainty about how the shape factors should be calculated has to do with unsymmetry of the cross-section and is not only present for the chosen interpretation of how the bending moment capacity should be calculated. Figure 3.1 and 3.2 show the variation of the bending moment capacity when only the edge in compression is considered as a function of the slenderness parameter β. The figures, which are out of scale, are valid for the Eurocode 9 calculations. The same figures are also valid for BS 8118 and BSK when class 1 and β1 are excluded. The names of the classes and the indices of β are also different. More details are found in section 3.2-3.4. The index x of the section moduli has been excluded. Figure 3.1 is used when the bending moment acts such that the tip of the web is in compression, i.e. when WW plcel <. . This figure is obtained from Eurocode 9 without any modifications. One disadvantage with this model is that the moment capacity is discontinuous when β is equal to β1. Such a discontinuity is physically impossible. Figure 3.2 is used when the top of the flange is in compression, i.e. when WW plcel >. . This figure is modified from what is found in the codes. The dashed lines show how the moment capacity should vary if the formulas from the codes were used without any modifications. A theoretical proof for the chosen way of treating the bending moment capacity is difficult to find, but some motivations for the chosen modification will be given here. It is not reasonable to use a lower moment capacity when the cross-section is stiffer, like the dashed line in figure 3.2 indicates. When WW plcel >. , it is difficult to find a suitable expression for the bending moment capacity, which is higher than fW cel 2,0. . The calculations based on the chosen

27

modification did not lead to any unrealistic results. There is a desire that the formulas in the codes should be simple to use. The chosen modification leads to simple formulas. For these reasons, the distribution in figure 3.2 was seen as a suitable choice. The shape of the moment-slenderness curve in figure 3.1 and 3.2 is uncertain for class 4 cross-sections and has not been investigated.

β3

β2

β1

M = W f0,2pl M = W f 0,2el.c

M

βclass 1 class 2 class 3 class 4

M = W f 0,2ef.c

M = 1,08 ( ) W fel.c 0,22,05

el.c

plWW

Figure 3.1. Moment capacity when only the edge in compression is considered, Wpl > Wel.c, Load application point IV, V or VI in figure 2.1.

β3

β2

β1

M = W f0,2el.c

M

βclass 1 class 2 class 3 class 4

M = W f 0,2ef.c

Figure 3.2. Moment capacity when only the edge in compression is considered, Wpl < Wel.c, Load application point I or II in figure 2.1. It would clearly be desirable if the codes were more precise in their description of how to handle the bending moment capacity when the cross-sections are unsymmetric. This is especially important in aluminium codes, because unsymmetric cross-sections are more common in aluminium than in steel. All three codes are using the buckling length lc in the calculations. For the simply supported beams, the steel balls at the supports were considered as hinges and the buckling length is then equal to the distance between the steel balls according to all codes. This distance was 50 mm longer than the beam length l. According to the theory of elastic beam-columns, the buckling length for the clamped beams is equal to 0,5 times the distance between the clamped ends. Both Eurocode 9 and BS 8118 give a value of 0,7. The value according to BSK is 0,6. These values can be seen as a compensation for partially clamped ends, which can be caused by the slenderness of surrounding construction elements. The ends of the tested beams were not perfectly clamped. However, it is

28

questionably if these conditions correspond to the values from the codes. According to Eurocode 9 and BS 8118, the clamping effect of the supports should be neglected when the welds are located at the ends of the clamped beams. All of the tested clamped beams were transversely welded at the supports. The calculation of the clamped beams according to Eurocode 9 and BS 8118 was performed for three values of the buckling length 0,5l, 0,7l and 1,0l. In the corresponding calculations according to BSK the buckling lengths 0,5l and 0,6l were used. Here, as before, the beam length is denoted by l. The reason for using different buckling lengths is just to see which one gives the best result. When the buckling length was 0,6l or 0,7l, the slope at the supports was theoretically not exactly zero. By making use of the beam length l in the formulas for the buckling length, it was assumed that the clamping effect was at the bottom of the grooves of the steel plates at the beam-ends. The same assumption was made when the clamped beams were calculated with Abaqus, as described in chapter 4. Figure 3.3 visualises the buckling length of a clamped beam. The weld location zhaz of the most critical weld was also needed in the calculations. This value represents the distance between the point of contra flexure of the buckling curve and the centre of the weld. For those beams which were simply supported, the point of contra flexure was located at the centre of the steel ball. For the clamped beams, zhaz for a weld location of 0 or l, see table 2.8, is calculated as ( )l lc− 2 , see figure 3.3. When the weld was located at the

midspan, zhaz was calculated as cl 2 . For the simply supported beams, zhaz are measured values. The values of zhaz are given in table 2.8. The notation zhaz is also used when the calculation according to Eurocode 9 is described, even though the co-ordinate z does not suit the co-ordinate system used in Eurocode 9. The weld location zhaz is not used by BS 8118. This way of calculating clamped beams is described in Eurocode 9 section 5.9.4.7.

l = 0,5 l, 0,6 l, 0,7 l or 1,0 lc

l

l c

l - lc2

l - lc2

3l - lc2

3l - lc2

Figure 3.3. Buckling model of the clamped beams. The critical loads from the theory of elastic beam-columns are used in the codes to calculate the slenderness parameters. This theory and the codes also have the names of the buckling phenomena in common. However, there is likely no standard regarding the names of the various buckling phenomena. Different names are used in different literature. The theory of elastic beam-columns is described in several books, Timoshenko and Gere [55], Chen and Atsuta [13], StBK-K2 [53] and Runesson et al. [48]. It should therefore be no doubt about the meaning of the phenomena even if the names of them may vary. Depending on the symmetry of the cross-section, the critical load for a centrically compressed column is determined by following buckling phenomena according to the theory of elastic beam-columns.

29

bi-symmetric: flexural buckling and torsional buckling monosymmetric: flexural buckling and flexural-torsional buckling unsymmetric: flexural-torsional buckling It is assumed that the torsion and bending of the beam-column are not restrained The meaning of the names of the buckling phenomena is quite obvious. Flexural buckling has only a bending mode with no torsion. Torsional buckling has only a torsion mode with no bending. The buckling mode for flexural-torsional buckling is a combination of bending and torsion. The information about how the symmetry of the cross-section affects the buckling is given because some confusing situations occur when the theory of elastic beam-columns is compared with the codes. These situations are related to the symmetry of the cross-section and are described below. In Eurocode 9 and BS 8118 there are two buckling phenomena mentioned for a centrically compressed column, flexural buckling and torsional buckling according to Eurocode 9 and column buckling and torsional buckling according to BS 8118, respectively. Column buckling and flexural buckling should most likely be seen as denoting the same phenomenon. The term flexural buckling is used here. If the torsional buckling section of Eurocode 9 and BS 8118 should be seen as a combination of torsional buckling and flexural-torsional buckling according to the theory of elastic beam-columns, it seems confusing that the codes specify that torsional buckling may be ignored for compact cross-sections with only radiating outstands, like a T-section. The calculation in section 3.5 showed that flexural-torsional buckling determined the capacity for all beam-columns. If on the other hand the torsional buckling section of the codes should be seen as torsional buckling according to the theory of elastic beam-columns, it seems confusing that cross-sections like angles and tees are mentioned. According to the theory of elastic beam-columns, torsional buckling can only occur for bi-symmetric cross-sections. When considering the theory of elastic beam-columns, it seems also confusing to apply a flexural buckling section according to some code on a member with an unsymmetric cross-section. It is not obvious if the flexural or the torsional buckling section of Eurocode 9 and BS 8118 consider the flexural-torsional buckling mode. It would be desirable if the codes treated the buckling phenomena more concordant with the theory of elastic beam-columns. When the calculations were performed according to Eurocode 9 and BSK, it was also necessary to check the load carrying capacity at the unwelded midspan for the welded beams, which were welded at the quarterspan. This calculation was obtained by setting

0 1ω ω ω ω= = = =xy xz xLT in Eurocode 9 and 0 1ρ ρ ρ ρ= = = =zx zy zLT in BSK. This has to

do with the sine expression, which takes into account the location of the transverse weld. When the weld is not located at the midspan, it is not sure that the welded section is the most critical section. Apart from the flexural and lateral-torsional buckling checks also a section check is generally needed. A section check verifies that the cross-section has enough capacity to resist the applied loads. In this check no buckling is considered. When xcω = 1 0, in BSK, the interaction formula for flexural buckling will be identical with the interaction formula for section check. In

30

Eurocode 9, it is somewhat uncertain how the interaction formula for lateral-torsional buckling looks like when the beam is stubby, i.e. all buckling reduction factors are equal to 1,0. When torsional buckling determines the capacity, Aef should not be used. When minor axis flexural buckling determines the capacity, Aef should be used. This assumes that the cross-section only consists of radiating outstands and belongs to class 4 for axial compression. If it is assumed that Aef should be used, the interaction formula for lateral-torsional buckling will be identical with the interaction formula for section check. In BS 8118 it is not possible to obtain the section check interaction formula by setting any buckling reduction factor equal to 1,0. For all three codes, the section check requirement will normally be automatically fulfilled in the checks for flexural and lateral-torsional buckling. One possible exception has to do with the above discussion about Aef. Another exception is in BS 8118 when all transverse welds are located close to a point of zero curvature in the buckled form of the strut. In this case 0 2,f is

used in the buckling checks whereas hazf is used in the section check. The section check in Eurocode 9 and BSK are very similar. Besides the different notations, the only difference is that the upper limit for the exponent of the axial force term is 2,0 in Eurocode 9 and 1,56 in BSK. The interaction formulas for flexural and lateral-torsional buckling are explained in section 3.2-3.4. The Eurocode 9 version of the interaction formula for section check is given in section 5.2. The interaction formula for section check of a welded section according to BS 8118 is given in section 3.3.3. The cross-section is built up by elements. Both Eurocode 9 and BS 8118 are somewhat vague in their description of when these elements should be considered as welded or unwelded. The interpretation of the codes was that longitudinal welds result in welded elements and transverse welds result in unwelded elements. Since only transverse welds were present for the tested beams, all elements were considered as unwelded. The calculations were performed as thoroughly as possible by using measured values of the cross-section dimensions, beam lengths, weld locations, yield strengths 0 2,f and the ultimate

strengths hazf . Also the 6 mm radius between the web and the flange was considered when the cross-section constants were calculated. In the calculations, the modulus of elasticity E was 70 GPa and the shear modulus G was 27 GPa. The used strength values are found in table 2.1 and 2.5. The cross-section dimensions, beam lengths and weld locations are found in table 2.7 and 2.8. The notations of the cross-section dimensions are shown in figure 3.5. The load eccentricity e is always positive in the formulas in this chapter. The Swedish steel code BSK was developed before Eurocode 9. Quite many of the buckling formulas in Eurocode 9 have been obtained from BSK. This is the reason why the calculations according to BSK and Eurocode 9 look so similar. The different codes do not use the same co-ordinate system, as can be seen in figure 3.4.

31

Figure 3.4. Co -ordinate systems used in Eurocode 9, BS 8118 and BSK. The notations according to the different codes were kept intact as much as possible. The different co-ordinate systems complicated the choice of some of the notations, because it was not desirable to have two notations for the same constant. The notations were chosen so they suited the co-ordinate system used in BS 8118 and BSK. This co-ordinate system was chosen as the main system throughout this chapter. The notations that are used in section 3.2 without being suited to the co-ordinate system used in Eurocode 9 are Ix, Iy, Ief.x, Wel.x.c, Wel.x.t, Wef.x.c, Wef.x.t, Wel.y, Wpl.x, Wpl.y, yef.gc, ygc, ys, ty, and zhaz. Below are the formulas for the cross-section constants A, ygc, Ix, Iy, Wpl.y, ypl, Wpl.x, ty, ys, Kv and Kw given. The plastic neutral axis for major axis bending was located in the flange for all beams. The formulas for ypl and Wpl.x are therefore sufficient. The co-ordinates x and y in the integral expression for ty are measured from the centre of gravity. ys is the distance between the centre of gravity and the shear centre, i.e. equal to the notation a in figure 2.1.

( )A bt h t t rf f w= + − + −

2 22π

( )( )gcf w

f f f fyA

bt t

h t h t r tr r

t rr

= + − + + +

− + −

12 2

22 2

43

22

2ππ

( ) ( ) ( )

( )

xf w f w

f f f

f f gc

Ib t t h t t

h t h t rr

tr

r rt r t r

rA y

= +−

+ − + + + +

−

− + + + −

−

3 32 2

2 2

2 2 2

3 12 42

12 2

2 483

ππ

( )

yf w w w w wI

t b b t r r t r r tr

tr

r= + +

+− + +

+ −

3 3 3 2 2

12 122

12 2 4 2 283

ππ

( ) ( )pl y

f w fw

wWh t t t b

r t rr t

rr

. =−

+ + + − + −

2 22

2

4 4 2 243

ππ

( )

pl

f f wy

b t h t t r

b=

+ − + −

2 22

2

π

when pl fy t≤

32

( )( ) ( )( )pl x f pl f pl

wf f

f fpl

Wb

t y t y th t h t

r tr r

t rr b y

. = − + + − + +

+ +

− + −

−

2 2

22 2

43 2

22 2

ππ

when pl fy t≤

( )y sx A

sx

t yI

y x y dA y II

= − +∫ = −21

23 2 s gcfy y

t= −

2

0 2x

trw= + 0y y t rgc f= − −

( ) ( )( ) ( ) ( ) ( )

( ) ( )[ ]( )

Ib y y t

b y y t

t y t h y t y t h y

ry t y r

y t r xt

rr r y x

gc gc f gc gc f

wgc f gc

wgc f gc

gc f gc fw

= − −

+ − −

+

+ − − −

+ − − −

+

+ − −

+ − − −

−

− + − +

4 24

4 24

2 32

8

25 2

4 4 3 2 2

4 4 3 2 2

4 40

30

3

23 2

0 0π ( )23

23 2

20

20 0 0 0 2

02

0r y x x y yx y+ −

+ +

π

( )[ ]v f w fK t b t h t= ⋅ ⋅ + ⋅ −11513

23 3,

( )

wKt b t h tf w f

=⋅

+⋅ −3 3 3 3

144

2

36

The warping, from which the warping constant Kw is calculated, consists of two parts, the contour warping and the thickness warping. The contour warping is normally so dominant that the thickness warping can be neglected. For a T-section, the contour warping is zero and therefore the expression for Kw was derived from the thickness warping. A brief derivation is shown below. Additional information is found in Edlund [19]. A point on the cross-section is described by the co-ordinates s and n , where s is directed along the centre line of the branches of the cross-section and n is directed perpendicular to the tangent of the centre line. The non-normalised sectorial co-ordinate with respect to the shear centre, scΩ , is equal to n hnsc , where nsch is the perpendicular distance from the shear centre to the normal of the tangent of the centre line of the cross-section. For the T-section in figure 3.5, the distance nsch is equal to –s, but the minus sign can be excluded.

m scA ns

b

t

t h t

t

t

AdA

An s dn ds

As ds n dn s ds n dn

f

f f

w

w

Ω Ω= ∫ = ∫∫ = ∫ ∫ + ∫ ∫

=−

−

−

1 1 12 0

0

2

2

2

0

2

2

2

33

( ) ( )

( )w sc m

A ns

t

tb

t

th tf w f

K dA n s dnds

s ds n dn s ds n dnt b t h t

f

f

w

wf

= −∫ = −∫∫ =

= ∫∫ + ∫∫ = +−

− −

−

2 2

2 2

2

2

0

22 2

2

2

0

2 3 3 3 3

0

2144

2

36

Ω Ω

s s

sshear centre

b

t

r

r y

b

t

f

w

w

f

gc

y

x

b

h

ypl

Figure 3.5. Cross-section notations and model for calculating Kw. 3.2 Buckling according to Eurocode 9 3.2.1 Cross-section classification The classification is performed separately for bending and axial compression. No classification is made for the combined state of stress, like it is in BS 8118. The cross-section can belong to different classes for axial compression and bending moment. There are four different classes 1, 2, 3 and 4, where class 4 corresponds to the highest slenderness. The most compact class in BS 8118 and BSK is divided in two classes in Eurocode 9, class 1 and 2, where both class 1 and 2 are used for plastic analysis but class 2 has limited rotation capacity. This is the reason why BS 8118 and BSK have three classes and Eurocode 9 has four. Bending moment: tip of the web in tension

ff

f

b

tβ = w

w

wg

bt

β =

g = +0 70 0 30, , ψ − < ≤1 1ψ

g =−

0 801

,ψ

ψ ≤ −1

The coefficient ψ is the ratio of the stresses at the edges of the web related to the maximum compressive stress. The stresses are calculated for the bending moment action. When checking for the limit between class 2 and 3 it is allowed to use the plastic neutral axis when the stresses are calculated. The interpretation of the code was that elastic stresses were used when ψ was calculated, but the elastic neutral axis was set equal to the plastic neutral axis. This appears like an odd mixture of the theories of elasticity and plasticity, but a better interpretation of the code was not found.

34

As an example, if plastic stresses were used and the plastic neutral axis was located in the flange but close to the web, ψ would have been +1. If on the other hand the plastic neutral axis was located in the web but close to the flange, ψ would have been −1. Such a discon-tinuity does not seem reasonable. When plastic stresses are used, ψ can only be +1 or –1. The plastic neutral axis was located in the flange for all beams. This means that the whole web was always in tension. Therefore, the web belonged to class 1. Bending moment: tip of the web in compression

flange = class 1 ww

w

bt

β =

Axial compression:

ff

f

b

tβ = w

w

w

bt

β =

Cross-section classification: The classification is made separately for the web and the flange. The whole cross-section is given the same class as the most slender part. When the flange is less highly stressed than the most severely stressed fibre in the section, it is allowed to use a modified higher value of ε for the classification of the flange for bending and also for evaluating the effective thickness of the flange, (see the formulas in this section and section 3.2.2). This feature is explained in section 5.4.4 of Eurocode 9. The stresses used in this check should be calculated for the effective cross-section, i.e. a cross-section reduced to local buckling. This seems complicated in a general case. The factor ε is used both during the classification and when the effective cross-section is calculated, but the value of ε is unknown. An iterative procedure is needed to determine ε . The calculation of the effective cross-section is often by itself an iterative procedure, because only the thickness of the compressed parts is reduced. This means that an iterative procedure is needed inside an iterative procedure. For the T-sections used in this thesis, the situation is simpler than in the general case. When the bending moment acted such that the flange was in tension and the web was in compression, the flange in tension belonged to class 1. When the bending moment acted in the reversed direction, the whole cross-section always belonged to class 1 and the effective cross-section was the same as the gross cross-section. It was shown that it was always possible to use the higher value of ε , for the flange under bending moment action. The absolute values of the stresses were compared. The maximum stress for bending always occurred at the tip of the web. In BS 8118 it is also allowed to use a higher value of ε for the flange, but the stresses are calculated for the gross section and for the combined state of stress. The stresses are calculated from either the plastic or the elastic neutral axis. The calculation is more precise described in section 3.3.1. The use of the gross section makes the calculation easier but the use of the combined state of stress and the mixture of the elastic and plastic neutral axis make the calculation more complicated than the one in Eurocode 9.

35

The slenderness β is referred to as either fβ or wβ .

Bending moment resistance Axial compression resistance β β≤ 1 : class 1 β β≤ 2 : class 1 or 2

1 2β β β< ≤ : class 2 2 3β β β< ≤ : class 3 2 3β β β< ≤ : class 3 3β β< : class 4

3β β< : class 4 1 3 0β ε= , 2 4 5β ε= , 3 6 0β ε= , heat-treated and unwelded outstand

ε =250

0 2,f flange for axial compression or web, (f0,2 in MPa)

( )

( )ε =−

−

250

20 2

h y

f y t

gc

gc f,

flange for bending, (f0,2 in MPa)

The notations bf, tf, bw, tw, ygc and h are defined in figure 3.5. 3.2.2 Effective cross-section If the cross-section belongs to class 4 according to bending, Wef.x.c and Wef.x.t need to be calculated. If the cross-section is slender for axial compression (class 4) also Aef must be calculated. The effective section modulus is calculated for the bending moment action and the effective area for the centric axial force action. The effective cross-sections are calculated by reducing the thickness of only the compressed parts. The calculations showed that the flange never belonged to class 4. The web belonged to class 4 for axial compression and for bending moment action when the web was in compression and the flange was in tension. The formulas are only shown for those cases when class 4 occurred. Web (unsymmetric, heat-treated and unwelded flat outstand):

xb

t fw w

w= =

β

ε 250 0 2, (f0,2 in MPa)

cρ = 1 0, when x ≤ 6

c x xρ = −

10 242 when x > 6 but c x

ρ ≤120

2

ef w c wt t. = ρ The elastic neutral axis is determined by bc, which is equal to the compressed part of the web. In a general case an iterative procedure is needed to find the elastic neutral axis. However, for the T-sections it is possible to find b c by solving a quadratic equation. The elastic neutral axis for the effective section always was located in the web, i.e. c wb b≤ .

36

Calculation of Wef.x.c and Wef.x.t:

( )eff f c f w c ef wA bt h b t t b t r= + − − + + −

.

2 22π

( )( )

ef gceff

f wf c f c ef w c

c

f f

yA

bt t

h t b h t b t b hb

r tr r

t rr

.

.=

+ − − + − + −

+

+ +

− + −

1 2 2 2

22 2

43

2

22π

π

c ef gcb h y= − .

( ) ( )( )

( )

ef xf w f c w

f c f cef w c

ef w cc

f

f f eff ef gc

Ibt t h t b t h t b h t b

t b

t b h br

rt

r

r rt r t r

rA y

..

.

.

= +− −

+ − − + − + +

+ −

+ + +

−

− + + + −

−

3 32

3

22

2 2

2 22

3 12 4 12

22

12 2

2 483

ππ

ef x cef x

ef gcW

Ih y. .

.

.=

− tip of the web

ef x tef x

ef gcW

Iy. .

.

.= edge of the flange

b

t

r

r y

b

t

f

w

w

f

ef.gc

b

h

tef.w

b c

Figure 3.6. Effective cross-section.

37

Calculation of Aef: The thickness for the whole web is reduced, i.e. bc is equal to bw in figure 3.6.

ef f w w ef wA bt r t b t r= + + + −

.

2 22π

3.2.3 Flexural buckling The load carrying capacity N is calculated from the formula below.

yc

NA f

N eMy xy y Rd

ξ

χ ω η ω0 2 01 00

, .,

+

⋅=

When the interaction formula above is compared with the equations in Eurocode 9 which deals with flexural buckling of axial compressed columns, i.e. no bending moment is present, there is a factor k1y missing in the interaction formula. This seems inconsistent. When the eccentricity e is zero, the interaction formula and the equations for flexural buckling of axial compressed columns ought to give the same result. The k1y factor has been left out in the interaction formula because it is left out in Eurocode 9. The factor k 1y takes into account the asymmetry of the cross-section. When the cross-section is symmetric about the buckling axis, k1y has no effect. The k 1y factor was calculated for all beams but it was not used when the load carrying capacity was calculated. The reason for calculating k1y was to give some information about what the effect of it should have been, if it had been used when the load carrying capacity was calculated. The obtained values of k 1y are shown in section 3.7.

( )( )1

2 2

2 21 2 4 1 21 1

ygc y

y y

ky

h= − −

+ +,

λ

λ λ

y Rd yc el x cM W f. . . ,= α 0 2

yc yξ ξ χ= 0 but ycξ ≥ 0 80,

0 1 00ξ = , (EC9 5.9.3.4)

( )xy

y yhaz

c

zl

ω ωχ χ π

= ⋅+ −

⋅

01

1 sin welded section

0 0 2ω = hazf f , but 0 1 00ω ≤ , welded section

0,10 == ωω xy unwelded section

yy

χφ φ λ

=+ −

12 2

but yχ ≤ 100,

38

yc

x

l A f

E Iλ π

η= 0 2,

( )[ ]φ λ λ= + − +0 5 1 0 2 0 1 2, , ,y y

η = 1 class 1, 2 and 3 for axial compression η = efA A class 4 for axial compression When the cross-section belongs to class 1, 2 or 3 for bending, y RdM . should not be less than

el x cW f. . ,0 2 . This has to do with the chosen interpretation of how y RdM . should be calculated.

The factor 0ξ in Eurocode 9 corresponds to 0γ in BSK, which is calculated differently as can be seen in section 3.4.3. The shape factors ycα and ytα are calculated as below. The classes according to bending were used when the shape factors were calculated. There are some differences from BSK. In BSK all shape factors have an upper limit of 1,25. In Eurocode 9 it is only the shape factor for minor axis bending which is limited to 1,25. The formula for the shape factors for class 1 cross-sections is described in Mazzolani [42]. It is derived for symmetric cross-sections and how well it suits T-sections is unknown. It is surely not suited for the situation when the elastic section modulus is greater than the plastic section modulus. This situation cannot occur for ycα when y RdM . is calculated because of the chosen

interpretation of how y RdM . should be calculated. This can be seen in figure 3.1 and 3.2. The shape factors ycα and ytα are used to calculate maxα . This factor is calculated as the

maximum value of the shape factors ycα and ytα , with the exception that the shape factors of class 1 are calculated according to class 2. This means that the formula for the shape factors for class 1 cross-sections will not cause any problems when maxα is calculated. maxα is only used to calculate η0 , which is used in connection with lateral-torsional buckling. The limits of η0 means indirect that the upper limit of maxα is 1,41.

( )( )

( )n e

e

=⋅ −

=log , , ,

log,

0 002 0 5 0 10 310 70000

260 31017 7732

0 ycpl x

el x c

W

Wα = .

. . 0 yt

pl x

el x t

W

Wα = .

. .

( ) ( )[ ]ycn

yc n yce eα α α= ⋅ =⋅ − ⋅0 21 10000 0 0796 0 0809 10 2 054943

010 1 079131, log , , log ,, class 1

( ) ( )[ ]ytn

yt n yte eα α α= ⋅ =⋅ − ⋅0 21 10000 0 0796 0 0809 10 2 054943

010 1 079131, log , , log ,, class 1

ycpl x

el x c

WW

α = .

. . yt

pl x

el x t

W

Wα =

.

. . class 2

ycpl x

el x c

WW

αβ ββ β= +

−−

−

1 13

3 2

.

. . yt

pl x

el x t

WW

αβ β

β β= +−−

−

1 13

3 2

.

. . class 3

39

ycef x c

el x c

W

Wα =

. .

. . yt

ef x t

el x t

W

Wα = . .

. . class 4

β = w

w

bt

tip of the web in compression

β = f

f

b

t tip of the web in tension

2 4 5β ε= , 3 6 0β ε= ,

ε =250

0 2,f tip of the web in compression, (f0,2 in MPa)

( )

( )ε =−

−

250

20 2

h y

f y t

gc

gc f,

tip of the web in tension, (f0,2 in MPa)

3.2.4 Lateral-torsional buckling The load carrying capacity N is calculated from the formulas below.

c cN

A fN e

Mz xz LT xLT y Rd

η γ

χ ω η χ ω0 2100

, .,

+

⋅

=

c zη η χ= 0 but cη ≥ 0 80,

02η α= max but 1 00 2 000, ,≤ ≤η

cγ = 1 00, (EC9 5.9.3.4)

( )xz

z zhaz

c

zl

ω ωχ χ

π= ⋅

+ −⋅

01

1 sin

welded section

( )

xLT

LT LThaz

c

zl

ω ωχ χ

π= ⋅

+ −⋅

01

1 sin welded section

0,1== ωω xLTxz unwelded section

LTLT LT LT

χφ φ λ

=+ −

12 2

but LTχ ≤1 00,

LTy Rd

cr

MMλ = .

Also here there is a difference between Eurocode 9 and BSK. The factor cγ in Eurocode 9

corresponds to cβ in BSK. However, these two factors are not calculated in the same way, as outlined in section 3.4.4.

40

There is an upper limit of y RdM . in the formula for LTλ . This upper limit is pl xW f. ,0 2 , but

when el x cW . . is greater than pl xW . , this limit can hardly be suitable. The interpretation of the code was that the higher value of ycα , which normally is obtained in class 1, is not admissible. The limit values are then given by y Rd pl xM W f. . ,≤ 0 2 tip of the web in compression, el x c pl xW W. . .<

y Rd el x cM W f. . . ,≤ 0 2 tip of the web in tension, el x c pl xW W. . .>

( )[ ]LT LT LT LT LTφ α λ λ λ= + − +0 5 1 02, ,

LT LTα λ= =01 0 60, ,, class 1 and 2 for bending

LT LTα λ= =0 2 0 40, ,, class 3 and 4 for bending η = 1 0, class 1, 2 and 3; torsional buckling of class 4 sections η = efA A minor axis flexural buckling of class 4 sections

When η is calculated, the classes are according to axial compression. Mcr is the elastic critical moment according to lateral-torsional buckling of beams and is calculated according to section 3.5. The notations ycα , ytα , maxα , 0ω and y RdM . are explained in section 3.2.3. The buckling reduction factor zχ should be chosen according to flexural buckling in the x-y plane, i.e. minor axis buckling, or lateral-torsional buckling of columns. In the section of Eurocode 9, which deals with resistance of axial compressed members, there are two buckling phenomena mentioned, flexural buckling and torsional buckling. There is no section dealing with lateral-torsional buckling of columns, which is rather confusing. According to Eurocode 9 it is not necessary to check torsional buckling for a member with a T-section when the section belongs to class 1 according to axial compression. The interpretation of the classification procedure in Eurocode 9 was that both class 1 and 2 exist for axial compressed columns, but there is no difference in the way the sections are calculated. It seems therefore confusing that only for class 1 cross-sections it is not necessary to check torsional buckling. The interpretation was that for both class 1 and 2 sections there is no need to check for torsional buckling. This is a difference from BS 8118. If a translation from BS 8118 is made, the result is that for class 1,2 and 3 there is no need to check for torsional buckling. The reduction factor zχ was calculated for both minor axis flexural buckling and torsional buckling and the lowest value was chosen except when the cross-section belonged to class 1 or 2 according to axial compression. For these cross-sections zχ according to minor axis flexural

buckling was chosen even if torsional buckling gave a lower value of zχ . However, the calculations showed that all cross-sections belonged to class 4 for axial compression.

41

If the rare situation should occur that the cross-section belongs to class 4 for axial compression and zχ according to minor axis flexural buckling and torsional buckling give the same value,

the least favourable situation is chosen, i.e. Aef is used. In the torsional buckling section of Eurocode 9 it is possible for a T-section to calculate the slenderness parameter both by using the theoretical load carrying capacity Ncr and by using the formulas shown below. By comparing these two ways of calculating the slenderness parameter it was possible to find an expression of Ncr, which is shown below under the headline torsional buckling. As a comparison, Ncr from this expression was compared with Ncr from the theory of elastic beam-columns. There are some uncertainties about what torsional buckling according to Eurocode 9 corresponds to according to the theory of elastic beam-columns, but most likely Ncr under the headline torsional buckling should be compared with the flexural-torsional buckling load Ncr which is calculated in section 3.5. The result of the comparison is shown in section 3.7. The origin of the expression for Ncr in Eurocode 9 is unknown. If zχ is chosen according to minor axis flexural buckling there is a factor k1z missing in the

interaction formula for lateral-torsional buckling in the same way as the factor k1y was missing in the interaction formula for flexural buckling. The T-section is symmetric around the minor axis and if k1z should have been considered it would have been reasonable to set the value equal to 1,0. In Eurocode 9 it is not clearly stated how the effective area Aef should be treated in connection with lateral-torsional buckling. In the lateral-torsional buckling section of Eurocode 9 it is written that Aef always should be used when the cross-section belongs to class 4 for axial compression. In the section of Eurocode 9 which deals with axially compressed members it is written that Aef should not be used in connection with torsional buckling when the cross-section belongs to class 4 for axial compression and the cross-section consists of entirely radiating outstands. The chosen interpretation was to follow the section in Eurocode 9 which deals with axially compressed members. This choice affected the result of the calculation, because all cross-sections belonged to class 4 for axial compression and zχ for torsional buckling was lower than zχ for minor axial flexural buckling for all tested beam-columns except one. Minor axis flexural buckling

zz

χφ φ λ

=+ −

12 2

but zχ ≤ 1 00,

( )[ ]φ λ λ= + − +0 5 1 0 2 01 2, , ,z z

zc

y

l A f

E Iλ π

η= 0 2,

η = 1 class 1, 2 and 3 for axial compression η = efA A class 4 for axial compression

42

Torsional buckling

zχφ φ λ

=+ −

12 2

but zχ ≤ 1 00,

( )[ ]φ λ λ= + − +0 5 1 0 2 0 6 2, , ,

λ =A f

N cr

0 2,

( )

cry

cN

E I

l

s s X s

X= ⋅

+ − + −

2

2

2 22 21 1 4

2

π

sl A

Ic

y=

0λ X

hb

hb

= − +

13 0 8 0 2

2

, , ,

0

1 5

1 4 15 1112 0

λ = + +

−+

h

tbh

hb t t

htw w f w

, , ,,

,

3.3 Buckling according to BS8118 3.3.1 Cross-section classification The cross-section should be given a single classification, fully compact, semi-compact or slender. During the classification, the slenderness parameters β for the elements in the cross-section are calculated for the combined action of the axial force and the bending moment. Fully compact and semi-compact cross-sections are considered as compact when the axial compression resistance is calculated. The distances y1 and y2 are used when ε for the flange is calculated. This procedure is described below. Bending moment: tip of the web in tension; Compression axial force:

ff

f

b

tβ = w

w

wg

bt

β =

g = +0 65 0 35, , ψ 0 1≤ ≤ψ g = +0 65 0 30, , ψ − ≤ ≤1 0ψ

g =−

0 701

,ψ

ψ ≤ −1

As in Eurocode 9, the coefficient ψ is the ratio of the stresses (with sign) at the edges of the web element, where the denominator contains the maximum compressive stress. The stresses should be calculated for the combined action of the axial force and the bending moment. In Eurocode 9 only the bending moment is considered. The ratio ψ can only be larger than 1,0 when the whole web is in tension and then the web is fully compact.

43

Check for the limit between fully compact and semi-compact The elastic stresses are calculated from the plastic neutral axis. The same type of calculation is also used in Eurocode 9. More information is found in section 3.2.1.

( )

( )ψ =− −

+ − −

eA h y I

eA t r y I

pl x

f pl x

r = 6 mm

1y h y pl= − when ( )− + − > − −1 1A

eI

h yA

eI

yx

plx

pl else 1y y pl=

2 2y y tpl f= −

Check for the limit between semi-compact and slender

( )

( )ψ =− −

− − − −

eA h y I

eA y t r I

gc x

gc f x

r = 6 mm

1y h ygc= − when ( )− + − > − −1 1A

eI

h yA

eI

yx

gcx

gc else 1y ygc=

2 2y y tgc f= −

Bending moment: tip of the web in compression; Compression axial force:

ww

w

bt

β =

Check for the limit between fully compact and semi-compact It was shown that ypl was always larger than ft 2 .

ff

f

b

tβ = when ( )e

Iy t

Axpl f− − <2

10 flange in compression

fβ = (fully) compact when ( )eI

y tAx

pl f− − ≥21

0 flange in tension

1y h y pl= − 2 2y y tpl f= −

Check for the limit between semi-compact and slender

ff

f

b

tβ = when ( )e

Iy t

Axgc f− − <2

10 flange in compression

fβ = (fully) compact when ( )eI

y tAx

gc f− − ≥21

0 flange in tension

1y h ygc= − 2 2y y tgc f= −

44

Compression axial force:

ff

f

b

tβ = w

w

w

bt

β =

Cross-section classification: The classification is made separately for the web and the flange. The whole cross-section is given the same class as the most slender part. The slenderness β is referred to as either fβ or

wβ . When the flange is not so highly stressed as the most severely stressed fibre in the section, it is allowed to use a higher value of ε for the classification of the flange. This is explained in section 4.3.3.5 of BS 8118. It was shown that the higher value of ε was always possible to use for the flange, except for the case of a centric axial force. y1 and y2 are the distances from the neutral axis of the gross section to the most severely stressed fibre and to the centre of the flange, respectively. When checking for the limit between fully compact and semi-compact, the plastic neutral axis is used. The elastic neutral axis is used when checking for the limit between semi-compact and slender. Bending moment resistance Axial compression resistance β β≤ 1 : fully compact β β≤ 0 : compact

1 0β β β< ≤ : semi-compact 0β β< : slender 0β β< : slender 0 7β ε= 1 6β ε= outstand and unwelded element

ε =250

0 2,f flange for axial compression and web, (f0,2 in MPa)

ε =250 1

0 2 2

yf y,

flange for bending and axial compression, (f0,2 in MPa)

The notations bf, tf, bw, tw, ygc, ypl and h are defined in figure 3.5. 3.3.2 Effective cross-section If the cross-section is considered as slender, two effective cross-sections should normally be calculated: one for the axia l force action and one for the bending moment action. When the cross-section has only radiating outstands (BS 8118 4.7.6.4), like a T-section, only one effective cross-section based on the bending moment action needs to be calculated. This calculation will result in the effective section modulus Wef.x.c. The effective area Aef is thus not needed.

45

When an element of a cross-section is slender and at least partly in compression, the thickness for the whole element is reduced. This is in contrast to Eurocode 9 and BSK for which only the compressed parts are reduced. The effective cross-section must be calculated only for the case when the bending moment causes compression in the web and tension in the flange. Only the thickness of the web is reduced.

xb

t fw w

w= =

β

ε 250 0 2, (f0,2 in MPa)

Lkx

=105

2 when 12,1x ≥

Lkx x

= −11 28

2 when 7 < 12,1x <

Lk = 10, when 7x ≤ ef w L wt k t. =

eff f w ef w wA bt b t t r r= + + + −

.

2 22π

( )

( )ef gc

eff

f ef ww f

f w f

yA

b t tb h t r

r tr

t rr

t rr.

.

=+ + + +

+ +

+ − + −

1 2 2

22

243

2

2ππ

( ) ( )

( )

ef xf ef w w ef w

w f w f

f f eff ef gc

Ibt t b t

b h t r r t rr

tr

r rt r t r

rA y

.. .

.

= + + + + + + + +

−

− + + + −

−

3 32 2 2

2 2 2

3 12 42

12 2

2 483

ππ

ef x cef x

ef gcW

Ih y. .

.

.=

− tip of the web

46

b

t

r

r y

bt

f

ww

f

ef.gc

b

h

tef.w

Figure 3.7. Effective cross-section. 3.3.3 Flexural buckling (major axis buckling) The load carrying capacity N is calculated from the formulas below.

N

PN e

MN e

P MRx RSx Rx RSx+

⋅+

⋅=

2

21 0,

Rx x zP Ak f= χ 0 2,

The cross-section was treated as unwelded and severely asymmetric when the expression for φ was obtained.

λπ

= c z

x

l A k f

E I0 2, φ

λ λ= + +

− ⋅

0 5 1

0 45 1 0 45 0 22,

, , ,

xχ φλ φ

= − −

1 112 2 but xχ ≤ 10,

The factor k z takes into account the softening of the heat-affected zone. For the beams with no transverse welds and for the beams where all transverse welds are located close to a point of zero curvature in the buckled form of the strut, k z is equal to 1,0. For all other beams k z is equal to hazf f 0 2, . This means that the welded simply supported beams PB-21 and PB-24, the welded clamped beams PB-29, PB-37 and PB-39 when the buckling length is equal to the beam length and all unwelded beams were calculated for zk = 10, . For the other beams,

z hazk f f= 0 2, . If the strength values would have been taken from the code, k z would have

been 0,5 instead of hazf f 0 2, . For the welded beams where zk = 10, , it was necessary to

perform a section check for a welded section. This section check is performed according to the formula below. The second factor of the last term is just to ensure that the moment capacity of a welded section is used. When the formula is applied on a clamped beam, the last term vanishes because the load eccentricity e is zero for clamped beams.

47

N

A fN e

M

k f

fhaz RSx

z

haz+

⋅⋅ =

0 21 0

,, section check at a welded section

The bending moment capacity RSxM is calculated as below. When the cross-section is fully compact or semi-compact, RSxM should not be less than el x c zW k f. . ,0 2 . This has to do with the chosen interpretation of how M RSx should be calculated. The β -values are calculated for the bending moment action only. RSx pl x zM W k f= . ,0 2 fully compact

( )RSx el x c z pl x el x c zM W k f W W k f= +−−

−. . , . . . ,0 20

0 10 2

β ββ β

semi-compact

RSx ef x c zM W k f= . . ,0 2 slender

β = w

w

bt

tip of the web in compression

β = f

f

b

t tip of the web in tension

0 7β ε= 1 6β ε=

ε =250

0 2,f tip of the web in compression, (f0,2 in MPa)

ε =250 1

0 2 2

yf y,

tip of the web in tension, (f0,2 in MPa)

The distances y1 and y2 are calculated according to section 3.3.1, where the elastic neutral axis is used when the stresses are calculated. 3.3.4 Lateral-torsional buckling (minor axis buckling) The load carrying capacity N is calculated from the formulas below.

N

PN eMRy Rx

+⋅

= 10,

Ry y zP A k f= χ 0 2,

The expression for φ assumes a symmetric unwelded cross-section. The cross-section is symmetric around the minor axis and the cross-section was treated as unwelded because there were no longitudinal welds. The lateral-torsional buckling moment M cr is calculated according to section 3.5 while k z is calculated according to section 3.3.3.

48

λπ

= c z

y

l A k f

E I0 2, φ

λ λ= + +

− ⋅

0 5 1

0 20 1 0 20 0 22,

, , ,

yχ φλ φ

= − −

1 112 2 but yχ ≤ 10,

Rx LT RSxM M= χ

LTRSx

cr

MMλ = LT

LT LTφ

λ λ= + +

− ⋅

0 5 10 1 1 0 1 0 6

2,, , ,

LT LTLT LT

χ φλ φ

= − −

1 112 2 but LTχ ≤ 1 0,

3.4 Buckling according to BSK 3.4.1 Cross-section classification The classification is performed separately for bending and axial compression. In contrast to BS 8118, no classification is made for the combined state of stress. The cross-section can belong to different classes for axial compression and bending moment. There are three different classes 1, 2 and 3, where 3 is the most slender class. There are no formulas given in BSK for the limit slenderness values of the cross-section classes when the stress gradient is non-uniform for the web. Therefore, the code had to be interpreted in some way. This is explained below. The classification is made separately for the web and the flange. The whole cross-section is given the same class as the most slender part. Bending moment: tip of the web in tension It is not likely that the tip of the web will buckle, because quite much of the web, including the tip, is in tension. The web was therefore considered as a web of an I-beam, i.e. a plate supported at four edges. The flange is calculated by considering local buckling of a plate supported at three edges. The plastic neutral axis was located in the flange for all beams. This means that the whole web was always in tension according to the theory of plasticity. Therefore, the web belonged to class 1. The classification of the flange is shown below.

ff

f

b

tβ = fpl

Ef

β = 0 30 2

,,

felE

fβ = 0 44

0 2,

,

class 1: f fplβ β≤

class 2: ffpl fel

β β β< ≤

49

class 3: fel fE

fβ β< < 2

0 2,

As can be seen, there is an upper limit for the slenderness β f of class 3. It is not allowed to

exceed this limit. Bending moment: tip of the web in compression The flange was in tension and therefore it belonged to class 1. The limit value wplβ was

calculated for the case when the whole web was uniformly compressed, whereas welβ was obtained from the expression of the slenderness parameter λ used when the effective thickness is calculated. By setting λ equal to 0,67, welβ is obtained. It seemed natural to use λ equal to 0,67 as the limit between class 2 and 3 because when λ is less than 0,67, the thickness is not reduced. The classification of the web is shown below.

ww

w

bt

β =

wplE

fβ = 0 3

0 2,

, ψ = −

− −

−gc f

gc

y t r

h y

welE

fβ

ψ=

+0 67

0 76 3 0 2

,, ,

when − < ≤3 1ψ

The limits of ψ are explained in section 3.4.2. class 1: w wplβ β≤

class 2: wwpl wel

β β β< ≤

class 3: wel wβ β< There ought to be an upper limit for class 3, but it was not known how this limit should be calculated. The calculations showed that wβ exceeded welβ with about 4% at most. When considering the upper limit of the slenderness of the flange β f , 4% should not be a problem.

Axial compression:

ff

f

bt

β = ww

w

bt

β =

The stress gradient is uniform and therefore the classification of the web and the flange can be performed in the same way as the flange was classified when the bending moment caused compression in the flange and a tension in the web. Essentially there are only two classes for

50

axial compression, slender or non-slender. The only difference is if the cross-section area should be calculated for the effective or the gross cross-section. The notations bf, tf, bw, tw, ygc and h are defined in figure 3.5. 3.4.2 Effective cross-section If the cross-section belongs to class 3, Wef.x.c and Wef.x.t (class 3 for bending) and Aef (class 3 for axial compression) must be calculated. The effective section modulus is calculated for the bending moment action and the effective area for the centric axial force action. The effective cross-sections are calculated by reducing the thickness of the compressed parts only. The calculations showed that the flange never belonged to class 3. The web belonged to class 3 for axial compression and for bending moment action when the tip of the web was in compression. The formulas are only shown for those cases when class 3 occurred. The formulas for Wef.x.c, Wef.x.t and Aef are found in section 3.2.2.

ef w wt t.,

= −

1 0 222λ λ

when λ > 0 67,

ef w wt t. = when λ ≤ 0 67, Bending moment: tip of the web in compression

ψ = −− −

−gc f

gc

y t r

h y

λψ

=+0 76 3 0 2, ,w

w

bt

f

E when − < ≤3 1ψ

When ψ ≤ −3 local buckling will not occur because the web is mostly in tension and the web belongs to class 1. The effective thickness will than be equal to the gross thickness. By observing the formula for ψ and figure 3.5 it can be seen that ψ can never be larger than 1,0. Axial compression:

λ =152 0 2, ,w

w

bt

f

E

51

3.4.3 Flexural buckling The load carrying capacity N is calculated from the formulas below.

xcN

NN eMRxcdzx Rxd

γ

ρ ρ

+

⋅=

01 00,

( )

zx

xc xchaz

c

zl

ρ ρ

ω ωπ

= ⋅+ −

⋅

01

1 sin welded section

0 0 2ρ = hazf f , but 0 1 00ρ ≤ , welded section

zx0 1 0ρ ρ= = , unwelded section

02γ η= max but 1 00 1 560, ,≤ ≤γ

maxη = maximum of xcη and xtη Rxd xc el x cM W f= η . . ,0 2

Rxcd xcN A f= ω 0 2, class 1 or 2 for axial compression

Rxcd xc efN A f= ω 0 2, class 3 for axial compression

xc xcγ γ ω= 0 but xcγ ≥ 0 80, When the cross-section belongs to class 3 for bending, the exponent 0 1 0γ = , , but this modification will not affect the value of 0γ . When the cross-section belongs to class 3 for

bending, maxη is less than 1,0 and the lower bound 1,0 of 0γ will make the calculation equivalent to a situation where 0γ is equal to 1,0. When the cross-section belongs to class 1 or

2 for bending, the bending moment capacity RxdM should not be less than el x cW f. . ,0 2 . This has to do with the chosen interpretation of how RxdM should be calculated. The buckling reduction factor xcω is calculated as:

xcc

cω

α α λ

λ=

− −2 2

24 4

2 2,

, but xcω ≤ 100,

cc

x

l A f

E Iλ

π= 0 2, class 1 or 2 for axial compression

cc ef

def x

l A f

E Iλ π

= 0 2,

. class 3 for axial compression

( )α λ λ= + − +1 0 34 0 2 11 2, , ,c c The second moment of area Idef.x is calculated for an effective cross-section. The effective thickness is not calculated in the same manner as it was when local buckling was considered. The index def comes from the word deformation, but Idef.x is only used in the ultimate limit state. It was only necessary to calculate the effective thickness for the web, tdef.w, and the

52

formula is shown below. It was shown that tdef.x always was equal to tw, which means that Idef.x was always equal to Ix.

def ww

wt

tb. =

18 2

but def w wt t. ≤

The shape factors xcη and xtη are calculated as below. There is an upper limit of 1,25 for all shape factors. The upper limit 1,0 of fwk has no effect. fwk is only used in class 2 for bending and then fwk can only vary between 0 and 1.

xcpl x

el x c

W

Wη = .

. . xt

pl x

el x t

W

Wη = .

. . class 1 for bending

xcpl x

el x cfw

WW

kη = + −

1 1

.

. . class 2 for bending

xtpl x

el x tfw

WW

kη = + −

1 1

.

. . class 2 for bending

xcef x c

el x c

WW

η = . .

. . xt

ef x t

el x t

W

Wη =

. .

. . class 3 for bending

Bending moment: tip of the web in tension

fwfel f

fel fplk =

−

−

β β

β β but fwk ≤ 1 00,

ff

f

b

tβ = fpl

Ef

β = 0 30 2

,,

felE

fβ = 0 44

0 2,

,

Bending moment: tip of the web in compression

fwwel w

wel wplk =

−−

β ββ β but fwk ≤ 1 00,

ww

w

bt

β = wplE

fβ = 0 3

0 2,

, ψ = −

− −

−gc f

gc

y t r

h y

welE

fβ

ψ=

+0 67

0 76 3 0 2

,, ,

when − < ≤3 1ψ

When the cross-section belongs to class 2 for bending, ψ cannot be less than or equal to –3. The limits of ψ are further explained in section 3.4.2. The notations b f, tf, bw, tw, ygc, r and h are defined in figure 3.5.

53

3.4.4 Lateral-torsional buckling The load carrying capacity N is calculated from the formulas below.

c cN

NN e

Mzy Rycd zLT Rxcd

α β

ρ ρ⋅

+

⋅⋅

= 100,

( )zy

yc ychaz

c

zl

ρ ρω ω

π= ⋅

+ −⋅

01

1 sin welded section

( )

zLT

xb xbhaz

c

zl

ρ ρω ω

π= ⋅

+ −⋅

01

1 sin welded section

zy zLTρ ρ= = 1 0, unwelded section Rxcd xb RxdM M= ⋅ω

Rycd ycN A f= ω 0 2, class 1 and 2 for axial compression

Rycd yc efN A f=ω 0 2, class 3 for axial compression

c ycα α ω= 0 but cα ≥ 0 80,

02 2

α η η= ⋅max y but 1 00 2 000, ,≤ ≤α

c yβ η= 2 but 1 00 1 56, ,≤ ≤cβ When the cross-section belongs to class 3 for bending, the exponents cα and cβ are calculated for max ,η η= =y 1 0 . Unlike the similar modification for section check and flexural buckling

( 0 1 0γ = , ), this modification will affect the load carrying capacity of the interaction formula. The result is not affected if all shape factors are less than 1,0. However, yη is not less than 1,0

for the T-sections, so the result will be affected. All cross-sections belonged to class 1 for minor axis bending. The exponents are thus calculated different in BSK and Eurocode 9. The notations RxdM , 0ρ and maxη are explained in section 3.4.3. The buckling reduction factors ycω and xbω are calculated as

ycc

ωλ

=+116

1 2,

but ycω ≤1 00,

xbb

ωλ

=+

1 02

1 4

, but xbω ≤ 100, (hot rolled beam)

ccr

A f

Nλ = 0 2, class 1 and 2 for axial compression

cef

cr

A f

Nλ = 0 2, class 3 for axial compression

54

bRxd

cr

MM

λ =

Ncr and Mcr are the critical loads according to the theory of elastic beam-columns and are calculated according to section 3.5. The shape factor yη is calculated as below. Class 2 and 3 did not occur for minor axis

bending.

ypl y

el y

W

Wη = .

. but yη ≤1 25, class 1 for bending

3.5 Critical loads according to the theory of elastic beam-columns Two critical loads were needed, Ncr and Mcr. The flexural-torsional buckling load Ncr is used in BSK and as a comparison to the critical load obtained in the torsional buckling section of Eurocode 9. In BS 8118, Ncr is not used at all. The critical moment Mcr is calculated from lateral-torsional buckling of beams and is used by all three codes. The co-ordinate system was chosen according to BS 8118 and BSK. The formulas are shown for the simply supported case. A clamped beam is calculated as a simply supported beam, with the exception that the warping is free at the beam-ends. The warping rigidity 4E K w will then be replaced with E K w in the formulas for Ncr and Mcr. Calculation of Ncr The only present load is an axial force applied at the centre of gravity. The governing differential equations and the corresponding boundary conditions are given below. The boundary conditions are believed to be reasonable when observing the steel plate in figure 2.4. The first equation gives the critical load according to major axis flexural buckling. The last two equations are coupled and give the flexural-torsional buckling load.

E I w N wxIV

y y+ ='' 0

( )E I w N w yyIV

x x s+ + ='' ' 'φ 0

E KIA

N G K N w ywIV p

v x sφ φ+ −

+ =' ' '' 0

( ) ( ) ( ) ( )x x c y y cw w l w w l0 0 0= = = = (deflection is zero)

( ) ( ) ( ) ( )' ' ' ' ' ' ''x x c y y cw w l w w l0 0 0= = = = (bending moment is zero)

( ) ( )φ φ0 0= =cl (restrained torsion)

( ) ( )' 'φ φ0 0= =cl (restrained warping)

55

The equations are solved by the assumptions shown below. These assumptions satisfy the boundary conditions.

( )yc

w z Az

l= 1sin

π ( )x

cw z A

zl

= 2 sinπ

( )φπ

z Az

lc= −

3 1

2cos

A non-trivial solution of the constants A1, A2and A3 requires that the determinant, of the two equation systems from which the constants are calculated, is zero. From these two conditions the two equations for Ncr are obtained. cr ExN N= major axis flexural buckling

( )( )Ey cr T cr p s crN N N N i y N− − − =2 2 2 0 flexural-torsional buckling

ppi

IA

= p x yI I I A ys= + + ⋅ 2

Tp

vw

cN

iG K

E Kl

= + 1 4

2

2

2π

torsional buckling load

Exx

cN

E Il

=2

2π

major axis flexural buckling load

Eyy

cN

E Il

=2

2π

minor axis flexural buckling load

The following equations hold for the flexural-torsional buckling load crN . ( )( ) ( )Ey cr T cr crN N N N a N− − − − =1 02

( )cr Ey T Ey T Ey TN a N N N N a N N= + ± + − 1

24

2

aI I

II I

I I A yx y

p

x y

x y s=

+=

+

+ + ⋅ 2 0 1< ≤a

( )cr Ey TN N N< min , or ( )cr Ey TN N N> max , when a ≠ 1 By observing the boundaries for a it can be seen that the expression inside the square root is always positive. This means that there are always two real roots of the quadratic equation for

crN . Due to the ±-sign crN is split into crN + and crN − , where the sign in the superscript shows if the plus or minus sign is used. When a = 1 following expressions for crN + and crN − are obtained.

( ) [ ] ( )cr Ey T Ey T Ey TN a N N N N N N+ = = + + − =112

max ,

( ) [ ] ( )cr Ey T Ey T Ey TN a N N N N N N− = = + − − =112

min ,

56

Since both crN + and crN − are continuous functions within the interval of a it means that

( )cr Ey TN N N+ ≥ max , and ( )cr Ey TN N N− ≤ min , . This shows that only crN − is of interest and that the flexural-torsional buckling load always is lower than, or equal to the lowest value of,

EyN and TN . This is also stated but not explained in Runesson et al. [48] and StBK-K2 [53]. Further observations: ( )crN a+

+→ = ∞0

( )crEy T

Ey TN a

N NN N

−+→ =

+0 ; ( ) ( ) ( )1

2 0min , min ,Ey T cr Ey TN N N a N N≤ → ≤−+

∂∂

crNa

+< 0 0 1< ≤a

This means that crN + = ∞ at a = +0 and then decreases steadily down to ( )max ,Ey TN N at

a = 1. The sign of the derivative ∂ ∂crN a− varies within the interval 0 1< ≤a . Calculation of Mcr The only present loads are two equal end moments, Mx. The differential equations are shown below. The assumptions and boundary conditions used when Ncr was derived are also used in the derivation of Mcr. The critical moment is calculated in the same way as Ncr, i.e. by setting the determinant equal to zero. Mx will then be equal to Mcr. E I w My

IVx x+ =' 'φ 0

E K G K M w t MwIV

v x x y xφ φ φ− + + =' ' ' ' ' ' 0

crc

y vc c

w

vM

lE I G K m l l m

E KG K

= ⋅⋅

+ + +

π π π1

1 42

2 2

mt

G KE Iy

v

y= −

2 tip of the web in compression

mt

G KE Iy

v

y=

2 tip of the web in tension

The formula for Mcr can also be found in annex H of Eurocode 9.

57

3.6 Comparison between the buckling reduction factors All three codes are using a slenderness parameter, cλ according to BSK and λ according to Eurocode 9 and BS 8118. In principle, this parameter is the same for all three codes. In this section no differences between the different slenderness parameters are made. More information about the slenderness parameters and the expressions for the buckling reduction factors are found in section 3.2-3.4. There is a parameter used in the different codes which is given by

( )[ ]φ α λ λ λ= + − +12

1 02 Eurocode 9

( )[ ]φλ

λ λ λ= + − +1

212 1

2c BS 8118

( )α β λ λ= + − +1 0 2 1112

c c, , BSK The expressions for the parameters contain the slenderness parameter and some constants. When the notation ( )α λ λ'= + − +1 1 2

2c c is introduced, where c1 and c2 are any constants, the parameters can be written as

φα

='

2 Eurocode 9

φαλ

='

2 2 BS 8118

α α≈ ' BSK The buckling reduction factors can then be written as

χα α

λ

α α λλ

=

+ −

=− −1

2 2

422

2

2 2

2' '

' ' Eurocode 9

χαλ

λαλ

α α λλ

= − −

=− −'

'

' '2

1 11

2

422

22

2

2 2

2 BS 8118

cc

cω

α α λλ

=− − ⋅

⋅' ' ,

,

2 2

24 11

2 11 BSK

The calculations show that basically the same formulas have been used in the different codes but different correction factors have been used. The factor 1,1 used in BSK is not used in the other codes. It is a safety factor and has its origin in the thickness variation of the flanges of rolled I-profiles in steel. The effect of the safety factor is that the buckling reduction factor will

58

be lowered most for beams with high slenderness. A reason for this is that a reduction of the flange thickness has largest influence for slender beams. It is somewhat amazing that the same formula has been rewritten in so different ways. 3.7 Results In table 3.1 and 3.2, the load carrying capacities, the bending moment capacities and some cross-section classes are shown for the three codes considered in this thesis. For Eurocode 9 and BSK, only the cross-section class for bending is shown. The classes fully compact, semi-compact and slender in BS 8118 have been replaced in the tables with 1, 2 and 3, respectively. In the tables the superscript F, L and S mark if flexural buckling, lateral-torsional buckling or section check determined the load carrying capacity, respectively. Lateral-torsional buckling determined the capacity for all beams except for a number of the short ones. For the quarterspan welded beams in Eurocode 9 and BSK, an additional buckling check at the unwelded midspan was needed. In BSK this check was not decisive for any beam. In Eurocode 9 the additional buckling check determined the capacity only for the clamped and welded beams PB-29, PB-37 and PB-39 when the buckling length was equal to the beam length. In this case, the calculation model corresponded to a simply supported beam with one transverse weld at each end. The buckling check at the weld will just turn into a section check without considering any buckling. Therefore the buckling check at the unwelded midspan determined the load carrying capacity. For a few welded beams in BS 8118, the strength of the parent material was used in the buckling check. For these beams it was necessary to perform a section check for a welded section. The section check was decisive only for the welded beams PB-21 and PB-24. The cross-section class for axial compression in Eurocode 9 and BSK was always equal to the most slender class, 4 in Eurocode 9 and 3 in BSK. In Eurocode 9, the ratio efA A varied between 0,923 and 0,945. Corresponding values in BSK were 0,955 and 0,979, i.e. somewhat higher than in Eurocode 9. The effective area efA was not needed in BS 8118. In Eurocode 9,

1yk varied between 0,944 and 0,957. If 1yk would have been used, its effect would have been small. The comments about efA A and 1yk apply to all beams, also the clamped ones. When the buckling reduction factors are calculated a slenderness parameter is used. For welded struts 0 2,f is used when this parameter is calculated according to Eurocode 9 and BSK whereas hazf normally is used in BS 8118, (see section 3.3.3). This is a major difference between the three codes. It is probably more correct to use hazf instead of 0 2,f when the buckling of a welded section is checked. The way Eurocode 9 and BSK are calculating the slenderness parameter can be seen as an acceptable simplification. For a welded beam where the transverse welds are not located at the midspan, two buckling checks are needed, one at the welded section and one at the unwelded midspan. This is further explained in section 3.1. The simplification lies in the fact that the same buckling reduction factors are used for both these buckling checks. This way of calculating the buckling will lead to the peculiar situation that the load carrying capacity will increase when 0 2,f is lowered and hazf is kept constant. The

59

reason for this is that hazf is used when the axial force and moment capacities are calculated but 0 2,f is used in the calculation of the buckling reduction factors. A lower value of 0 2,f will give a lower slenderness, which in turn will give a higher value of the buckling reduction factor, but this lower buckling reduction will not be compensated by a lower value of hazf . The different buckling reduction factors in the three codes could be compared in diagrams where the reduction factors are functions of the slenderness parameters. The slenderness parameters are in general not calculated in exactly the same way for the three codes and therefore it is unsuitable to draw these diagrams. Mostly it was difficult to draw any general conclusions about the value differences of the reduction factors between the codes, but some observations for the simply supported beams are given below. Reduction factor for major axis flexural buckling For the unwelded beams the highest values were obtained in Eurocode 9, the lowest in BS 8118. For the welded beams BSK mostly gave the lowest values. Reduction factor for minor axis flexural buckling The highest values were mostly obtained in BS 8118. Reduction factor for lateral-torsional buckling of beams The highest values were obtained in BS 8118, the lowest in BSK. When the clamped beams were calculated according to Eurocode 9 and BS 8118, three different buckling lengths were used, 0,5l, 0,7l and 1,0l, where l is the beam length. The value 0,5 comes from the theory of elastic beam-columns, whereas the values 0,7 and 1,0 come from the codes. For a beam with unwelded ends 0,7 is used and 1,0 is consequently used when the ends are welded. If these codes were followed strictly it means that 1,0 should be used, because all tested clamped beams were transversely welded at the ends. In the BSK calculations, two different buckling lengths were used, 0,5l and 0,6l. The origin of 0,5 is the same as for the other two codes. The value 0,6 comes from BSK. More information about the buckling lengths is found in section 3.1. When the criterion for approval is that the load carrying capacity according to the code should be lower than the tested load carrying capacity for all four clamped beams, the following conclusions can be drawn about the buckling length multipliers for clamped beams: Eurocode 9: 0,5 is not enough; 0,7 is enough; 1,0 is not needed or too conservative BS 8118: 0,5 and 0,7 are not enough; 1,0 is enough BSK: 0,5 is not enough; 0,6 is enough This means that the note in Eurocode 9 about neglecting the clamping effect for clamped beams with welded ends is not needed. However, too few clamped beams were tested to draw any secure conclusions. Cross-section classes are shown in table 3.1-3.2. At load application point I and II, the cross-sections belonged to the most compact cross-section class. At position IV, V and VI, the cross-sections belonged to the most slender class, except in BSK where some cross-sections belonged to the second most slender class. The limit between the most and the second most

60

slender class seems to be somewhat more generous in BSK than in the other two codes. The load application points are shown in figure 2.1. The interaction formulas for flexural and lateral-torsional buckling according to Eurocode 9 and BSK are very similar. However, one difference is that the exponents are not calculated in exactly the same way. The exponent cβ used in the interaction formula for lateral-torsional buckling according to BSK was equal to 1,00 when the cross-section belonged to the most slender class for bending and 1,56 otherwise. In Eurocode 9 corresponding exponent was always equal to 1,00. Both the exponents ycξ and xcγ were equal to 0,8 for all middle and long beams due to the lower bound of 0,8. For the short beams, the result varied. The comment about ycξ and xcγ also applies to cη and cα . In table 3.1-3.2, the bending moment capacities are shown. According to the chosen interpretation, these capacities are equal to the bending moment capacity when only the edge in compression is considered. The formulas for the bending moment capacities are shown in section 3.2-3.4. According to Eurocode 9 and BSK, the moment capacity is always calculated for an unwelded section. This is not the case in BS 8118. In order to make a more correct comparison between the three codes, RSxM in table 3.2 has been calculated for an unwelded section. This is a difference from what is found in section 3.3. At load application point I and II, the three codes give the same value for the moment capacity. This has to do with figure 3.2. At position IV, V and VI, Eurocode 9 gives the lowest value of the moment capacity and BSK the highest. At these positions, the cross-section mostly belongs to the most slender class and therefore the differences between the codes have a lot to do with the value of the effective cross-section modulus ef x cW . . . The shape factor for major axis bending has an upper limit of 1,25 according to BSK. Such an upper limit does not exist in Eurocode 9, as can be seen in section 3.2.3. This difference between the codes will not affect the result of the calculated beams. The reason for this can be seen in figure 3.1 and 3.2. At load application point I or II, the shape factor is 1,0. At position IV, V or VI, the cross-section belongs to class 4 for bending and the shape factor is less than 1,0.

61

Table 3.1. Some results for the unwelded beams from 1996. Eurocode 9 BS 8118 BSK Beam Lap Test

N [kN]

N [kN]

y RdM .

[kNm]CB N

[kN] RSxM

[kNm]CC N

[kN] RxdM

[kNm] CB

PB-1 II 104,8 95,6F 3,74 1 87,8F 3,74 1 93,3L 3,74 1 PB-2 V 61,2 35,7L 1,51 4 40,0F 1,61 3 37,4L 1,78 3 PB-3 IV 88,8 56,4L 1,50 4 62,4F 1,59 3 55,9L 1,76 3 PB-4 V 61,6 35,5L 1,51 4 39,9F 1,61 3 37,2L 1,78 3

PB-19 II 105,2 96,1F 3,76 1 88,3F 3,76 1 93,7L 3,76 1 PB-20 V 61,1 35,5L 1,50 4 39,8F 1,60 3 37,2L 1,77 3 PB-21 III -H 143,1L - - 144,3L - 3 111,4L - - PB-22 I -SL 82,3F 4,57 1 76,7F 4,57 1 82,4L 4,57 1

PB-5 II 73,7 46,3L 4,46 1 49,2L 4,46 1 51,3L 4,46 1 PB-6 V 42,4 24,0L 1,48 4 28,5L 1,58 3 25,9L 1,75 3 PB-7 II 67,2 46,5L 4,48 1 49,3L 4,48 1 51,5L 4,48 1 PB-8 IV 49,3 33,6L 1,48 4 38,5L 1,58 3 35,3L 1,74 3 PB-9 III 64,4 57,0L - - 58,8L - 3 56,7L - -

PB-10 IV 58,5 34,1L 1,52 4 39,1L 1,62 3 35,9L 1,79 3 PB-11 I 52,7 39,0L 4,51 1 42,2L 4,51 1 44,1L 4,51 1 PB-12 IV 55,4 33,3L 1,46 4 38,2L 1,56 3 35,0L 1,72 3 PB-23 III 64,9 57,4L - - 59,1L - 3 56,8L - - PB-24 V 43,6 23,9L 1,45 4 28,3L 1,55 3 25,7L 1,71 3

PB-13 II 32,9 23,7L 3,73 1 24,6L 3,73 1 27,7L 3,73 1 PB-14 V 25,0 16,1L 1,49 4 18,4L 1,59 3 17,8L 1,75 3 PB-15 II 31,9 24,0L 4,50 1 24,9L 4,50 1 28,4L 4,50 1 PB-16 III 32,5 27,7L - - 28,1L - 3 30,2L - - PB-17 I 31,2 20,9L 3,73 1 22,0L 3,73 1 24,9L 3,73 1 PB-18 V 26,0 16,0L 1,48 4 18,4L 1,57 3 17,7L 1,74 3 PB-25 I 30,9 21,4L 4,54 1 22,5L 4,54 1 25,9L 4,54 1 PB-26 III 32,0 27,6L - - 28,0L - 3 30,1L - - F = Flexural buckling L = Lateral-torsional buckling SL = Slip occurred H = Failure load too high for the test equipment CB = Cross-section class for bending CC = Cross-section class for combined action Lap = Load application point

62

Table 3.2. Some results from the welded beams from 1997 and 1998. Eurocode 9 BS 8118 BSK Beam Lap Test

N [kN]

N [kN]

y RdM .

[kNm]CB N

[kN] RSxM U

[kNm]CC N

[kN] RxdM

[kNm] CB

PB-1 I 59,9 53,3F 3,90 1 50,7F 3,90 1 56,4L 3,90 1 PB-2 IV 63,5 39,0L 1,35 4 42,6F 1,43 3 50,9L 1,68 2 PB-3 II 84,2 73,3F 3,88 1 69,8F 3,88 1 70,9L 3,88 1 PB-4 V 47,0 24,4L 1,35 4 26,9F 1,44 3 33,0F 1,70 2 PB-5 I 59,3 54,0F 3,89 1 50,6F 3,89 1 59,9L 3,89 1 PB-6 V 51,0 24,7L 1,34 4 26,6F 1,42 3 32,5F 1,66 2

PB-19 II 90,7 66,6F 4,52 1 65,5F 4,52 1 61,3L 4,52 1 PB-20 II 83,4 68,0F 4,52 1 65,5F 4,52 1 66,9L 4,52 1 PB-21 V 55,2 22,9L 1,50 4 27,2S 1,60 3 25,0L 1,77 3 PB-22 VI 36,9 15,8L 1,44 4 17,6F 1,53 3 17,2L 1,69 3 PB-23 III 98,9 108,9L - - 120,6L - 3 84,9L - - PB-24 I 54,7SL 59,8F 4,12 1 66,2S 4,12 1 69,0F 4,12 1 PB-25 VI 36,9 17,7L 1,33 4 19,4F 1,41 3 23,2F 1,64 2

PB-7 I 38,4 29,6L 4,17 1 38,0L 4,17 1 33,3L 4,17 1 PB-8 V 34,7 18,3L 1,43 4 24,2L 1,52 3 23,4L 1,72 2 PB-9 II 51,2 35,3L 4,16 1 45,7L 4,16 1 38,9L 4,16 1

PB-10 IV 47,4 25,4L 1,41 4 33,8L 1,50 3 31,6L 1,67 2 PB-11 I 43,5 32,7L 4,12 1 36,9L 4,12 1 37,1L 4,12 1 PB-12 V 34,8 19,5L 1,43 4 23,4L 1,52 3 25,2L 1,74 2 PB-26 III 62,9 35,7L - - 56,0L - 3 35,2L - - PB-27 VI 28,5 12,6L 1,45 4 16,1L 1,54 3 13,8L 1,70 3 PB-28 I 38,5 24,7L 3,99 1 34,6L 3,99 1 27,6L 3,99 1 PB-31 VI 28,0 14,4L 1,39 4 17,6L 1,48 3 18,3L 1,66 2

PB-13 I 25,0 13,4L 3,99 1 20,8L 3,99 1 16,1L 3,99 1 PB-14 IV 26,2 12,0L 1,46 4 19,4L 1,56 3 13,1L 1,72 3 PB-15 II 29,7 18,5L 4,15 1 24,5L 4,15 1 21,8L 4,15 1 PB-16 V 22,3 9,88L 1,36 4 14,9L 1,45 3 13,1L 1,66 2 PB-17 I 27,4 16,7L 3,99 1 20,7L 3,99 1 20,1L 3,99 1 PB-18 V 25,7 11,3L 1,45 4 14,9L 1,55 3 12,5L 1,71 3 PB-33 I 27,4 12,8L 4,52 1 21,1L 4,52 1 15,6L 4,52 1 PB-34 VI 20,0 9,78L 1,37 4 12,4L 1,46 3 13,0L 1,68 2 PB-35 I 26,2 17,3L 4,00 1 21,0L 4,00 1 20,9L 4,00 1 PB-36 III 26,9 17,2L - - 27,6L - 3 18,7L - - PB-38 VI 21,3 9,40L 1,47 4 12,4L 1,57 3 10,5L 1,73 3

F = Flexural buckling L = Lateral-torsional buckling S = Section check SL = Value not reliable due to slip CB = Cross-section class for bending CC = Cross-section class for combined action Lap = Load application point C = Clamped beam U = Calculated for an unwelded section

63

Table 3.2. (Continued). Eurocode 9 BS 8118 BSK Beam Lap Test

N [kN]

N [kN]

y RdM .

[kNm]CB N

[kN] RSxM U

[kNm]CC N

[kN] RxdM

[kNm] CB

PB-29C 0,5 III 105,2 98,7L - - 116,4L - 3 76,5L - - -″- 0,6 -″- -″- - - - - - - 71,4L - - -″- 0,7 -″- -″- 83,3L - - 96,0L - 3 - - - -″- 1,0 -″- -″- 63,3L - - 65,3L - 3 - - -

PB-30C 0,5 III 89,2 121,9L - - 134,5L - 3 94,1L - - -″- 0,6 -″- -″- - - - - - - 82,3L - - -″- 0,7 -″- -″- 84,7L - - 104,5L - 3 - - - -″- 1,0 -″- -″- 47,9L - - 62,1L - 3 - - -

PB-32C 0,5 III 117,9 122,3L - - 134,8L - 3 94,4L - - -″- 0,6 -″- -″- - - - - - - 82,6L - - -″- 0,7 -″- -″- 84,9L - - 104,7L - 3 - - - -″- 1,0 -″- -″- 48,0L - - 62,2L - 3 - - -

PB-37C 0,5 III 67,4 60,6L - - 89,1L - 3 53,1L - - -″- 0,6 -″- -″- - - - - - - 46,7L - - -″- 0,7 -″- -″- 48,0L - - 55,4L - 3 - - - -″- 1,0 -″- -″- 29,9L - - 30,3L - 3 - - -

PB-39C 0,5 III 51,3SL 58,4L - - 87,2L - 3 51,2L - - -″- 0,6 -″- -″- - - - - - - 45,0L - - -″- 0,7 -″- -″- 46,4L - - 54,6L - 3 - - - -″- 1,0 -″- -″- 29,5L - - 30,0L - 3 - - -

F = Flexural buckling L = Lateral-torsional buckling S = Section check SL = Value not reliable due to slip CB = Cross-section class for bending CC = Cross-section class for combined action Lap = Load application point C = Clamped beam U = Calculated for an unwelded section Conclusions about the load carrying capacities can be drawn directly from table 3.1 and 3.2. However, especially when many beams are calculated interaction diagrams are useful tools. These diagrams are just graphical representations of the interaction formulas. By making use of the load carrying capacities from the tests and the formulas in the codes, the interaction diagrams in figure 3.8-3.13 were constructed. For each code two interaction diagrams are shown, one for flexural buckling and one for lateral-torsional buckling. In each diagram, only those beams are shown, for which the buckling check was decisive for the capacity. No interaction diagram for section check according to BS 8118 is shown, because this check determined the capacity only for two beams and one of them could not be used in the diagrams due to slip at the support. Only the simply supported beams are included in the interaction diagrams. The clamped beams were mostly calculated to evaluate the effect of the different buckling lengths. The exponents cη in Eurocode 9 and cα in BSK are undetermined for centrically compressed columns. For these columns the exponents have been set equal to 1,0 in the interaction diagrams.

64

The notations SdN , SxdM and y SdM . are used in the numerators of the labels of the co-ordinate axes. These notations are just the dimensional loads and are not used elsewhere in this thesis. To make the diagrams more informative, the beams have been separated in the three different lengths. The diagrams also have two halves. The left one is for the beams where the bending moment caused compression in the web and tension in the flange. The right half is accordingly used when the bending moment acted in the opposite direction. Each marker in the diagrams corresponds to a tested beam. The straight lines are the graphical presentation of the interaction formulas. It is undesirable when a marker lies below the straight lines, but if the marker lies close below the straight lines the situation might not be so drastic. Normally, safety factors are used which have not been considered here. The critical load Ncr in section 3.2.4 was calculated both with the formulas in Eurocode 9 and with the formulas for the flexural-torsional buckling load in section 3.5. The average value of the quotient between Ncr from Eurocode 9 and the flexural-torsional buckling load was 0,9803. The standard deviation for the same quotient was 2,01%. Eurocode 9 thus gave a somewhat lower value of the critical load than the theory of elastic stability. In this analysis all tested beams were included, also the clamped. When zχ is calculated according to Eurocode 9 the lowest value of zχ according to minor axis flexural buckling and torsional buckling should be chosen. Minor axis flexural buckling determined the load carrying capacity only for the unwelded beam PB-19. For all other beams torsional buckling gave a lower value of zχ . When torsional buckling determines the capacity,

efA should not be used when the capacity according to lateral-torsional buckling is calculated, even if the cross-section has been considered as slender for axial compression. A condition for this is that the cross-section consists of entirely radiating outstands. This led to the situation that η , which is equal to efA A , was set equal to 1,0 for all beams except the unwelded beam PB-19 when the lateral-torsional buckling capacity was calculated. When the centrically compressed and welded column PB-23 was calculated according to Eurocode 9 and BS 8118, corresponding marker ended up quite much below the straight lines. This was not the case for the BSK calculations. The reason for this situation was different values of the buckling reduction factor and that efA was used in BSK and not in the other two codes. For the welded column PB-23 the load carrying capacity is calculated as:

N A fz haz= χ Eurocode 9 N A fy haz= χ BS 8118

N A fyc ef haz= ω BSK As described in section 2.3, the tested load carrying capacity for the welded column PB-23 might be too low. When the distribution of the markers in the interaction diagrams according to Eurocode 9 and BS 8118 is studied, it can be seen that the marker for PB-23 does not follow the pattern for the other markers. This increases the suspicion that the load carrying capacity

65

for PB-23 is too low. The marker for PB-23 is recognised as the square on the vertical axis. It could therefore be unsuitable to use the welded column PB-23 to draw any deeper conclusions. The best result is obtained when all markers lie close above the straight lines. From the interaction diagrams it is difficult to judge which code best could predict the load carrying capacity. The results in table 3.1-3.2 were used to calculate the quotients between the tested load carrying capacity and the capacity from the codes. The average value and the standard deviation were then calculated from these quotients. The results are shown in table 3.3, where it can be seen that the least bad results were obtained for BS 8118. Only the simply supported beams have been considered in table 3.3b. Table 3.3a. Unwelded beams from 1996

test ECN N 9 test BSN N 8118 test BSKN N Average 1,47 1,35 1,36 St. dev. 23,8% 14,8% 22,5%

Table 3.3b. Welded beams from 1997 and 1998

test ECN N 9 test BSN N 8118 test BSKN N Average 1,75 1,41 1,52 St. dev. 41,7% 31,6% 32,9%

From table 3.3 and figures 3.8-3.13 it can be seen that the result was not too good for any code. For most beams the result was too conservative. In section 5.3 different modifications of Eurocode 9 are discussed that improve the result of the calculated beams.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

L=500 mmTotally 7 beams

yc

Sd

y xy

NA f

ξ

χ ω η 0 2,

y Sd

y Rd

MM

.

.0ωWeb in compression Flange in compression

Figure 3.8. Flexural buckling according to Eurocode 9.

66

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

L=500 mmL=1020 mmL=1540 mm

cSd

z xz

NA f

η

χ ω η 0 2,

cy Sd

LT xLT y Rd

MM

γ

χ ω.

. Web in compression Flange in compression

Totally 50 beams

Figure 3.9. Lateral-torsional buckling according to Eurocode 9.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

L=500 mmTotally 16 beams

Sd

Rx

NP

Sxd

RSx

Sd

Rx

MM

NP

12

+ Web in compression Flange in compression

Figure 3.10. Flexural buckling according to BS 8118.

67

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

L=500 mmL=1020 mm=1540 mm

Totally 40 beams

Sd

Ry

NP

Sxd

Rx

MMWeb in compression Flange in compression

Figure 3.11. Lateral-torsional buckling according to BS 8118.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

L=500 mmTotally 3 beams

Web in compression Flange in compression

xcSd

zx Rxcd

NN

γ

ρ

Sxd

Rxd

MM0ρ

Figure 3.12. Flexural buckling according to BSK.

68

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

L=500 mmL=1020 mmL=1540 mm

Totally 54 beams

c

Sd

zy Rycd

NN

α

ρ

Web in compression Flange in compressionc

Sxd

zLT Rxcd

MM

β

ρ

Figure 3.13. Lateral-torsional buckling according to BSK.

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4 Finite element calculation

4.1 Introduction

For the numerical calculation the general-purpose finite element computer program Abaquswas chosen, [1]. The numerical calculation can also be based on other methods. Mazzolani [42]describes an iterative finite difference method. The method is applied to flexural buckling anddoes not consider any torsion, local buckling or out-of-plane buckling. However, the methodcan consider any uniaxial stress-strain material model. The stress-strain curve can for instancehave different shape on the compression and tension side. When the beams were loaded suchthat the tip of the web was in compression, load application point IV-VI in figure 4.6, thecompression stress at the tip of the web became quite high. This has to do with the unsymmetryof the T-section. For the welded beams another cause was that two different materials wereused. It was therefore likely that local buckling occurred. From the buckling tests it was alsopossible to see that local buckling occurred, but sometimes it was difficult to know if localbuckling was a primary or secondary failure. When the beams were tested the deflection wasmeasured such that it was possible to see that some torsion occurred, see the discussion inaccordance to figure 2.3. Lateral-torsional buckling determined the load carrying capacity formost of the beams, when the capacities according to the different codes were calculated. Forthe reasons given above, a drawback of the method described by Mazzolani was that it couldnot consider local buckling and out-of-plane buckling, and the method was therefore not analternative. Abaqus offers more refined calculation methods, which can describe the physicalbehaviour of the beams better. The method described by Mazzolani is from 1974 and shouldtoday most likely be seen as obsolete. The computers today are more powerful and morerefined calculation methods can thus be used.

4.2 Material model

The parent and the heat-affected material were modelled as an elastic-plastic material. The“true” or Cauchy stresses and logarithmic plastic strains used in this material model werecalculated from nominal stress-strain values obtained from uniaxial tensile tests from both theparent and the heat-affected material. In very simple terms, the Cauchy and nominal stressesare calculated as force divided by final area and as force divided by original area, respectively.Constant volume is assumed. The final area is the original area compensated for lateralcontraction. Log-strains are often used when the strains are large. Cauchy stresses areassociated with log-strains. In Crisfield [16], Cauchy stresses and log-strains are furtherdescribed. Equation 4.1 shows the relationship between the different stresses and strains.

( )true nom nomσ σ ε= +1 ( )log e nomε ε= +log 1 (4.1)

The chosen material model in Abaqus assumes that the stress-strain curve has the same shapeon the compression and tension side. This is probably not true for large strains. At the tensionside there will always be an ultimate strain level, where failure occurs. Such a strain level doesnot exist on the compression side. Sanne [50] made some compression and tensile tests onaluminium alloy AA6351-T6, both for the parent and the heat-affected material. The specimensused for the compression tests were just 20 mm long in order to prevent any buckling. The

70

stress-strain curves did not have the same shape on the compression and tension sides. Noultimate strength was reached for the compression tests but the maximum compression stresswas larger than the ultimate tensile strength. Sanne used nominal stresses and strains. Whentrue stresses and log-strains are used the difference between the compression and tension sidewill be smaller. This can be seen in equation 4.1 and figure 4.1. In equation 4.1, σnom and εnomare positive in tension and negative in compression. Even if “true” stresses and log-strains areused it is still uncertain if the stress-strain curve has the same shape on the compression andtension side at large strain levels.

Figure 4.1 shows the stress-strain curves from one compression test and one tensile test of theheat-affected material. The nominal compression stress-strain curve was graphically obtainedfrom Sanne [50]. It was also possible to graphically obtain a tensile test diagram, but thediagram was cut at 0,8% strain, so the diagram was not of great interest. Instead the stress-strain curve for the welded tensile test 4W was inserted in the diagram. The curve for thistensile test was somewhat representative for all tensile tests. There was not such a largedifference between the different stress-strain curves for the tensile tests when considering thescale of the diagram in figure 4.1. The alloys were not the same, but still, the stress-strain curveon the tension and compression side differed quite much in this example. The largest differencewas that the compression test did not have any ultimate strain level. In figure 4.1, also f 2,0 andf u for the two alloys are given. The strength values were obtained from SAPA’s handbook

[51]. Alloy 6351 seemed to be an obsolete alloy, because the strength values were only foundin an older version of SAPA’s handbook. For the tensile test, yielding started at a lower stresslevel than the compression test, even though that the strength values from the handbook werehigher. As discussed in section 2.2 good compression tests are difficult to perform. For thisreason, there is a risk that for instance the stress levels of the compression test in figure 4.1 arenot so accurate, but the important thing is that compression tests do not have an ultimate strainlevel as the tensile tests have. The slope of the stress-strain curve for tensile tests also goesdown when the maximum stress is reached. Such a downward slope does not seem natural for acompression test curve.

At high compression strain levels, the stress-strain curve from the tensile tests needs to becorrected. The compression strains were quite high when the beams were loaded at position IV-VI, see figure 4.6. For some beams the calculated maximum compression strain was muchhigher than the failure strain from the tensile tests. Some correction of the stress-strain curvesfrom the tensile tests at high strain levels was therefore needed before the curves could be usedas a material model in the calculations. This applies only for the heat-affected material, becausethe strains in the parent material will not be so high. This is natural because most of thedeformation will occur at the softer zones. In figure 4.1, there is a straight line, which is anapproximation of the true stress-strain relationship in the figure at high compression strains.The slope of this line was used as the stress-strain relationship at high strain levels for all heat-affected materials. This modification of the materials did not affect the tensile stresses, becausethe tensile strains were not so high.

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0

50

100

150

200

250

300

350

400

450

500

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36

Strain [-]

nominal stress

true stress

straight line

nominal stress

− −=−=−=− Compression test by Sanne, 6351-T6, f0,2=245 MPa, fu=290 MPa Material 4W, tensile test, 6082-T6, f0,2=260 MPa, fu=310 MPa

Figure 4.1 Compression and tensile test.

All heat-affected materials used in the Abaqus calculations are shown in figure 4.4. The upperlimit of 16% strain was chosen because it was believed that the maximum compression strainwould not exceed that value. However, no problems will occur if this limit is exceeded,Abaqus just assumes that the stress remains constant for plastic strains exceeding the last givenpoint on the stress-strain curve. When the Abaqus model later was used in chapter 5 the limit16% was exceeded. To give a more complete picture of the materials used in the Abaquscalculations, also the parent materials are shown in figures 4.2-4.3. To reduce the amount ofinput to the Abaqus calculations the stress-strain curves were represented by 12-19 straight linesegments.

From the discussion in this section and section 2.2 it is evident that there are uncertaintiesabout the material. It is possible that more testing of the materials should have been made, butfor instance good compression tests are difficult to perform. The uncertainties of the materialmodel can be said to be part of the overall uncertainties that all numerical models have.

As discussed in section 2.2, material AW was also used as material CW. The reason for thiswas that no satisfactory tensile test existed for material CW and material A and C were almostidentical, see figure 4.3.

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0

50

100

150

200

250

300

350

400

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Logarithmic strain [-]

2

6 5 1

43

Figure 4.2. Parent materials from 1996.

0

50

100

150

200

250

300

350

400

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Logarithmic strain [-]

G

4 1

D

E2

3B

A,C

Figure 4.3. Parent materials from 1997 and 1998.

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0

25

50

75

100

125

150

175

200

225

250

275

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16Logarithmic strain [-]

DW

AWBW

GW

1W

2W

3W

4W

EW

Figure 4.4. Heat-affected materials from 1997 and 1998.

Von Mises yield surface with isotropic hardening was used. The von Mises yield surface,which assumes that the yield is isotropic, was chosen even though Hopperstad [30] found outthat aluminium alloys exhibit anisotropic yield and hence, the von Mises yield surface shouldnot be valid. One reason for using von Mises yield surface was that the anisotropic yieldparameters were not found in the literature and there was no time for doing any extensivematerial tests. The isotropic yield surface was “dared” to be used because the tested beamswere used as a verification of the Abaqus model. It also became more evident that the materialwas anisotropic when Moen [43] was read. Some kind of check how the anisotropy affected thecalculations was therefore necessary. This check is shown in the next paragraph. The thesis byMoen was published in 1999 and it was not known when the calculations started. In other olderliterature it has not been found that the material should be anistropic. In Frey and Mazzolani[21] it is written that for extruded aluminium profiles, the maximum scatters of f 2,0 indifferent directions are low and are not significant for the load carrying capacity of compressedmembers. This statement was based on tests and it indicates that the material could beconsidered as isotropic. In Moen [43] there are some additional discussions about theanisotropic yield, where also the stress ratios rij defining the anisotropic yield surface are given.These stress ratios are used as input to the Abaqus calculations and are defined as the ratiobetween the yield stress in the direction ij divided by the reference yield stress, which waschosen in the direction of the extrusion. For each plate element composing the cross-section,Moen suggests that ryy=0,95, rxy=0,84 and rxx=rxz=ryz=rzz=1. The local x-, y- and z-co-ordinateof the plate are in the direction of the extrusion, perpendicular to the direction of the extrusionand in the direction of the thickness, respectively. The values of rxx and ryy are average valuesfrom tensile test whereas rxy is based on some calculations. The stress ratios given by Moenwere surely used for unwelded extruded material. It was not clearly found, but the stress ratioswere probably also used for the heat-affected material. According to Moen [43], the anisotropywas hardly affected by the shape of the section and the temper of the alloys. To give a correct

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material model of the heat-affected zone is very complex because the material is nothomogeneous due to the temperature difference during welding.

In order to investigate how the plastic anisotropy affected the load carrying capacity of thebeams, four unwelded beams of different lengths and load application points were calculatedwith all stress ratios equal to 1,0, i.e. von Mises yield surface or isotropic yielding, and with thestress ratios suggested by Moen by making use of Hill’s yield surface. The maximumdifference in the load carrying capacity was about 0,03%. Moen also made some sensitivitystudying on the effect of the plastic anisotropy. From two of he’s diagrams it could be seen thatthe anisotropy only had a minor influence on the maximum load carrying capacity, but in thepost-buckling area the capacity was affected to a larger extent. From these observations it wasdecided that the anisotropy did not affect the load carrying capacity to an extent that requiredrecalculation. Hence, the values from the isotropic models were kept. If any difference in theload carrying capacity could be seen, the isotropic material gave the higher value.

How changes in the material properties affected the load carrying capacity of the beams wasstudied by calculating a few beams with a modified stress-strain curve. From the second pointup to the last point on the stress-strain curve, the stress values were successively increased. Thestress at the last point was increased with 10%. The modulus of elasticity was not affected bythis modification. Figure 4.5 shows how the stress-strain curve was modified for one parentand one heat-affected material. The other materials were modified in the same way. For thewelded beams, only the heat-affected material was modified.

0

50

100

150

200

250

300

350

400

450

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16Logarithmic strain [-]

Origin parent materialModified parent materialOrigin heat-affected material

Modified heat-affected material

Figure 4.5. Stress-strain curves showing the effect of the material modifications.

Table 4.1 shows the increase of the load carrying capacity for the beams in this calculation. Thetheoretical beam lengths and load application points are also shown. From table 4.1, it can beseen that the load carrying capacity seems to be least affected for the longest beams. The reasonis most likely the lower strain levels. For the unwelded beams, the maximum imperfection was

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0,1 mm. The welded beams were calculated for a maximum imperfection of 1,0 mm and 30mm heat-affected zone. A few more unwelded beams were calculated, but there were somenumerical problems for these beams during the analysis. The reason for these problems wasunknown, but in some way they were related to the material modification because when thesebeams were calculated with the unmodified material no numerical problems occurred.

Table 4.1. Increase of load carrying capacity due to material modifications.Beam l [mm] lap Increase [%]PB-22 UW 500 I 3,1PB-9 UW 1020 III 0,0PB-15 UW 1540 II 0,0PB-18 UW 1540 V 0,0

PB-1 W 500 I 4,9PB-22 W 500 VI 6,7PB-7 W 1020 I 3,1PB-26 W 1020 III 0,91PB-31 W 1020 VI 2,7PB-15 W 1540 IV 0,29PB-34 W 1540 VI 1,3PB-36 W 1540 III 0,17UW = Unwelded beamW = Welded beam

Bauschinger effects were not considered. When the material is loaded into the plastic range inone direction, elastic unloaded in the reverse direction, yielding starts at a stress level less thanthe original yield stress. This is known as the Buschinger effect and it is further described inHopperstad [30] and Mazzolani [42]. Even if some parts of the beam can be unloaded when theapplied load is increased (due to local buckling for instance), it is not likely that the unloadingwill be so high that any Bauschinger effect will occur. The Bauschinger effect is present instructural metals as aluminium, steel and stainless steel. It is especially important to considerthe Bauschinger effect when the load is cyclic. It could be reasonable to believe that thecompressive strength should be lower than the tensile strength for the parent material due to theBauschinger effect. The reason for this is that the profiles are stretched after the extrusion. Thiscould be a motivation for not approximate the compression strength with the tensile strength,but as mentioned in section 2.2, good compression tests are difficult to perform.

4.3 Finite element model

Shell elements were used to model the beams. Abaqus divides the shell elements in thin, thickand general-purpose elements. From what was found in the Abaqus manuals, it was decidedthat a general-purpose shell element was best suited. It was assumed that the strains should belarge, because of the unsymmetry of the cross-section and that two different materials wereused. A finite strain element was therefore considered as necessary. The three node elementswere excluded because the strain is constant for these elements and a very dense mesh wouldbe required to obtain accuracy in the calculations. The geometry also does not requiretriangular elements. When the axisymmetric elements were excluded, only two possible

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elements remained, S4 and S4R. The element S4 has four integration points, while S4R hasone. Both elements have four nodes and six degrees of freedom at each node, threedisplacements and three rotations. Mesh instabilities like hourglass modes can occur for S4R.A fully integrated element, like S4, could have problems with locking. However, in the Abaqusmanuals it is written that locking cannot occur for S4. The term hourglass mode comes fromthe physical shape of the mesh. The element integrals are calculated exactly for a fullyintegrated element when the element is undeformed. Locking means that the element gets muchtoo stiff when the thickness of the element becomes small in comparison to the otherdimensions of the element. A condition for this is that the theory for “thick” elements is used.Six unwelded and six welded beams were calculated with both the elements S4 and S4R. Bothelements behaved well and no numerical problems occurred. It was found that the load carryingcapacity was 0,067% - 0,85% lower for S4R than for S4. The execution time was roughlyestimated to be three times longer for S4 than for S4R. The difference in the load carryingcapacity is very small when considering the overall uncertainty of the Abaqus model. If forinstance two tensile tests were made for each material and the average value was used, theresult would likely be more affected. It was decided to use S4R in the further calculations.

The steel plates used to transfer the load into the beam, see figure 2.4, were modelled with rigid3D beam elements, RB3D2. The model is shown in figure 4.6, where also the load applicationpoints I-VI are shown. These rigid elements have the same degrees of freedom at each node asthe shell elements S4R.

1

231

23

I

II

III

IV

V

VI

Figure 4.6. FE-model of the rigid connection at the beam-ends.

Two different widths of the heat-affected zone were used, represented by a bhaz value of 15 and25 mm in figure 4.7. The extension 25 mm was used first and the origin of it was much due tothe “one inch rule”. Eurocode 9 gives a value of 20 mm for a MIG butt-weld in a 6 mm thickaluminium plate. The weld material was located on the surface of the profiles and the cross-section was quite much enlarged where the weld material was located. The lump of the weldmaterial reduced the extension of the heat-affected zone. Even if the lump had reduced strengthdue to the heating it still was a reinforcement for the beam because of the larger cross-sectionarea. When the extension 25 mm was used the calculated load carrying capacities were too low.For the reasons given above it was also natural to use a lower value than 25 mm. It was decidedto also use a value of 15 mm. The width of the heat-affected zone is equal to twice the value ofbhaz, i.e. 30 and 50 mm, respectively.

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bhaz

Figure 4.7. Extension of the heat-affected zone.

The self-weight of the beam and the rigid connections at the beam-ends were not considered inthe calculations. The beams tested in 1996 and 1997 were tested such that the force of gravityacted along the web. During the testing in 1998 the force of gravity acted parallel with theflange. This was actually not a problem but it somewhat complicated the input to thecalculations. Figure 2.3 gives some additional information. As some kind of verification ofhow much the self-weight affected the deflection some hand calculations were made, see figure4.8 and the connected equations. For the longest beams with the weakest bending stiffness, forexample the welded beam PB-36, the midspan deflection was 0,23 mm. This showed that theself-weight did not affect the deflection so much. It is also possible to say that the effect of theself-weight was included in the imperfections used in the calculations.

( )q area density g N m= ⋅ ⋅ = ⋅ + ⋅ ⋅ ⋅ ⋅ =−60 6 54 6 10 2700 9 81 18 16 , ,M volume density g eccentricity Nm= ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ =−80 130 40 10 7800 9 81 0 025 0 7969 , , ,

midwq L

EIM L

EI= +

5384 8

4 2

yEI Nm= 7628 2xEI Nm= 16330 2 L m m m= 0 55 1 07 159, , , ,or

M M

L

q

EI

Figure 4.8. Model for the self-weight calculation.

In the calculations, imperfections were also considered. It was necessary to use imperfectionsin order to get a reasonable failure mode. A few eccentrically loaded beams were calculatedwith no imperfections. The failure mode was flexural buckling where the deflection wasparallel with the web. No torsion or deflection parallel with the flange occurred. The likelyreason was the symmetry of the numerical model. This shows that the imperfections areimportant. The shape of the imperfections was chosen equal to the shape of the firsteigenmode. It was difficult to define the magnitude of the imperfections. Abaqus’ manualssuggest a few percent of the shell thickness. The initial curvature according to Mazzolani [42],which is discussed in section 2.1, gives a midspan deflection of 0,25-0,77 mm for the testedbeams. These figures can also be a hint about the magnitude of the imperfections.

For the welded beams, the largest degree of freedom was set equal to 1,0. The unit wasassumed to be millimetres because a rotation of 1,0 radians is huge. For the unwelded beams,two different values were used, 1,0 and 0,1. The lower value 0,1 was an attempt to calibrate the

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magnitude of the imperfections so that the load carrying capacity according to Abaqus gave thesame capacity as the buckling tests.

The magnitude of the imperfections was the only parameter that was varied for the unweldedbeams. For the welded beams the magnitude of the imperfections was kept constant, instead thewidth of the heat-affected zone was varied. When Abaqus calculates the eigenmodes, it ignoresall nonlinear material data. The eigenmodes used as imperfections were thus based on linearelastic material. As an example of the importance of the imperfections, when the centric loadedcolumn PB-23 in table 4.3 was calculated with no imperfections the load carrying capacitybecame 126,5 kN. With a maximum imperfection of 0,1 mm the load carrying capacity became61,4 kN. The imperfections were necessary for the numerical calculations, but it is not claimedthat they were an approximation of the real initial curvature of the beams.

The imperfections are likely the most “mysterious” part of the finite element model. The choiceof the imperfections is mainly governed by two conditions. The first is that the numericalmodel should describe the physical behaviour of the beam from the test. If, for instance, a localbuckle occurred during the testing a local buckle should also occur during the Abaquscalculation. The second condition is that the calculated and tested load carrying capacity shouldbe equal within an accepted tolerance. It is necessary to have several buckling tests to comparewith in order to judge and calibrate the numerical model.

Residual stresses, which are further discussed in section 2.1, were not taken into account.

The mesh density will affect the results of the finite element calculations. A denser mesh willgive a more accurate result of the numerical solution. Eight of the welded and simply supportedbeams were calculated with two different mesh densities. The width of the heat-affected zonewas kept constant equal to 30 mm. The coarse mesh divided the web and flange in 16 elementseach and the reduced beam length in 90 elements, which totally gave 2880 shell elements. Thefine mesh used 100 elements along the reduced beam length and 20 elements along the flangeand the web, giving totally 4000 shell elements. The results of the calculations are shown intable 4.2. The reduced beam length is 40 mm shorter than the beam length. This is due to thegrooves of the steel plates at the supports. The beam lengths of the tested beams are found intables 2.7-2.8.

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Table 4.2. Some welded beams from 1997 and 1998, 30 mm haz.Coarse mesh Dense mesh

Beam Profile Lap* N [kN]test

N [kN] εmax atweld [%]

εmin atweld [%]

N [kN] εmax atweld [%]

εmin atweld [%]

PB-1 G I 59,9 56,7 4,4 -5,8 56,7 4,4 -5,8PB-22 2 VI 36,9 32,2 3,9 -11,9 32,1 4,2 -12,4PB-7 A I 38,4 40,3 2,3 -3,3 40,4 2,2 -2,9PB-26 2 III 62,9 51,8 0,40 -0,90 51,7 0,38 -0,77PB-31 B VI 28,0 30,1 2,3 -5,4 30,1 2,8 -6,9PB-15 C II 29,7 27,6 0,35 -0,81 27,6 0,35 -0,81PB-34 3 VI 20,0 20,3 2,0 -4,8 20,3 2,0 -4,7PB-36 2 III 26,9 27,7 0,16 -0,39 27,7 0,16 0,40*) Load application point.

The load carrying capacity was only affected little by using the dense mesh. The strains wereaffected more, but the strains were not used in any calculations. They were just used to seewhich part of the stress-strain curve that was used in the calculations, so the accuracy of thestrains was not so important. From the results in table 4.2, the conclusion was drawn that thecoarse mesh was accurate enough. One advantage of the coarse mesh was that the calculationtime was quite much faster. This was the only analyse about the effect of the mesh density thatwas made. The coarse mesh (2880 elements) was used for all unwelded and all welded beamswith one transverse weld. For the beams with two or three transverse welds, which alsoincluded all clamped beams, 100 elements were used along the longitudinal axis of the beam,which totally gave 3200 shell elements. It was decided that the more changes in the materialproperties for the beams with two or three transverse welds required a denser mesh to retain theaccuracy of the calculations. This is the explanation why 90 or 100 elements were used alongthe longitudinal axis of the beam. It is possible that the effect of the mesh density should havebeen studied more thoroughly, for instance by using an even more dense mesh, because themesh refinement is not high enough to guarantee that the decimals of the load carrying capacitycould not be changed. However, when considering the overall uncertainty of the model andhow the model is used, a change of for instance 0,1 kN of the load carrying capacity will not bevery important. For two beams the load carrying capacity was 0,1 kN higher for the coarsemesh than for the dense mesh. For one beam the situation was in the opposite case. This issomewhat confusing, because the finite element solution ought to converge from “the sameside” for all beams. One possible explanation for this situation could be not perfectly smoothload-displacement curves. It was found that the values of the parameters in Riks methodsomewhat affected the load carrying capacity. If a bad choice was made, the load-displacementcurves could be slightly jagged. If the curves were considered too jagged, the parameters werechanged and a new calculation was made. The shell elements were uniformly distributed alongthe beam length within each material region.

The clamped beams were also analysed with Abaqus. Only four clamped beams were testedwith useful result and there were also some uncertainties about the clamping effect of thesupports. It was therefore difficult to develop a reliable model for the clamped beams. Therewere so few tests to calibrate the model with. The numerical model for the clamped beams wasonly used in this chapter to make comparisons with the tested clamped beams. It was notconsidered to be so reliable that it could be used for any extensive numerical simulations. The

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clamped beams were modelled somewhat different than the simply supported beams. Thedifferences are related to the support conditions and are described below.

For the clamped beams, all degrees of freedom for the rigid connection at the two beam-endswere set equal to zero, except for the displacement along the longitudinal axis of the beam forthe rigid connection at one beam-end. In figure 2.4 a sketch of the rigid connection at thebeam-ends can be seen. The seven screws clamp the web but not the flange. When the beamwas deflected laterally, i.e. in the direction parallel with the flange, it was not reasonable tobelieve that there would be any tension stresses at the boundary between the flange and therigid connection at the beam-ends. For this reason, the boundary between the rigid connectionand the flange was modelled with truss elements that could resist compression but not tension.The compression modulus of elasticity of these truss elements was set equal to 1000 times thevalue of aluminium. By this way the flanges could not penetrate the rigid connections at thebeam-ends. The truss elements could not have zero length and therefore a small “deformation”of 0,1 mm was introduced for the rigid connections at the beam-ends. This “deformation” wasequal to the length of the truss elements. Some constraints were introduced between thedegrees of freedom at the ends of the truss elements. The twelve degrees of freedom for the twoend nodes were reduced to following eight: ux, uy, uz1, uz2, θx1, θx2, θy and θz. Only thedisplacement uz and the rotation θx were thus independent for the two nodes. The constraintsbetween the displacements and the rotations were introduced because it was believed that thegroove of the steel plates in figure 2.4 affected the movement of the flanges at the supports.The maximum “gap” that occurred was equal to the width of the two outermost shell elementson one side of the flange. Some additional information is found in figure 4.9.

y

z

0,1 mm x

Beam flange

Rigid connection

truss elementu z2

θ x2

u z1

θ x1

Figure 4.9. Part of the model between the beam flange and the rigid connection at thesupports of the clamped beams, (out of scale).

For the clamped beams, the whole beam length, i.e. also the part of the beam inside the 20 mmdeep grooves of the steel plates at the supports, was divided in shell elements. This means thatthe clamped beam length was 40 mm longer than the simply supported beam length, whichearlier was denoted as the reduced beam length. The modifications of the flange connectionand the extension of the beam length were introduced because the connection between the rigidconnection at the beam-ends and the beam was considered to be overstiff. It probably wouldhave been more correct to make these modifications also for the simply supported beams, butfor these beams the rigid connections at the beam-ends could move more freely and a overstiffconnection was therefore not considered so important.

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The imperfections were calculated first in a separate linear elastic buckling analysis.Thereafter, the load carrying capacities were calculated with the modified Riks method. It is anarc-length method and it can pass limit points like local maximum load or local maximumdeflection on the load-deflection curve. The load carrying capacity analysis was deformation-governed. More information about the modified Riks method is found in the Abaqus manuals[1]. Arc-length methods are further described in Crisfield [16]. There were no numericalproblems of importance.

Fairly many input files were needed for the Abaqus calculations. In order to simplify the work,a computer program was written which gave the input files as output. For each beam, two inputfiles were needed. The first file was used to calculate the imperfections and the second tocalculate the load carrying capacity. As an example, the content of the two input files for thewelded beam PB-35 with 50 mm heat-affected zone is shown in appendix C. The input files forthe unwelded beams and the beams with two or three transverse welds looked basically thesame as the given example of PB-35. The shell elements were just divided into differentmaterial regions. The input files for the clamped beams looked partly quite different. Theseinput files contained almost 1000 lines and therefore it was too much to show the wholecontent of these files. Instead, some parts of an input file for a clamped beam are shown inappendix C. The parts are taken from PB-37 with 50 mm heat-affected zone and are unique forthe clamped beams. In chapter 5 Abaqus is used to calculate the bending moment capacity ofbeams. Parts of the two input files for such a calculation are also shown in appendix C. Theparts are obtained from the two input files of a beam loaded such that the tip of the web was incompression. Only the parts which regard the loading conditions are shown, because thegeometry and the material parts are the same as in the other Abaqus calculations.

The rotation around the longitudinal axis of the beam was set equal to zero at both beam- ends.Some numerical experiments with rotational springs were made, but they did not turn outsatisfactory. At least visually, the first eigenmode was close to a rigid body rotation around thelongitudinal axis of the beam. This eigenmode was not suitable to be used as imperfectionwhen considering the deformation of the beam during the testing. A better result could perhapshave been obtained with a larger rotational stiffness. This was not tested. Reasonable resultswere obtained without any rotational springs, and then it was easier to just let the rotationsbeing locked. There were also uncertainties about which value of the rotational stiffness thatshould be chosen. Without any restrictions of the rotation, the numerical model would be amechanism.

When the beam is totally symmetric about the midspan, only half of the beam could bemodelled. However, this was not utilised. One reason was simplicity. The same type ofcomputer program was used for all simply supported beams, also for the beams with a weld atthe quarterspan. The location of the welds was also based on measured values, which meansthat very few of the welded beams were totally symmetric about the midspan. The first elasticbuckling mode was used as imperfection. Generally it is hazardous to utilise symmetry inconnection with buckling, because the unsymmetric buckling modes are left out. It was notknown if the first buckling mode always was symmetric. This is also one reason for not utilisethe symmetry in the calculations.

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4.4 Results and conclusions

As one example of the shape of the imperfections, the first eigenmode of the welded beam PB-4 is shown in figure 4.10. The shape of the eigenmodes was for instance affected by the beamlength and the boundary conditions.

Figure 4.10. First eigenmode of the welded beam PB-4, 50 mm haz.

Figure 4.11-4.13 shows the deformed shape at maximum load for three welded beams, PB-4,PB-10 and PB-35. The viewpoint of the pictures was chosen close to perpendicular to thelongitudinal axis of the undeformed beams, which means that the figures are quite informativeabout the mesh density. Only the outline of beam PB-35 is shown, otherwise the picture wouldjust be black. For this beam also the undeformed shape is shown. For PB-4, it can be seen thatthere was a local buckle at the weld. This buckle was probably the main cause for the failure.Local buckling seemed to have occurred also for PB-10, but the global buckling was moredominant. Both PB-4 and PB-10 were loaded such that the tip of the web was in compression.For the longer beams the global buckling seemed to be the main cause for the failure.

The applied load, i.e. the support reaction (the calculations were deformation-governed), wasplotted against the deflection of the point at the junction between the web and the flange at themidspan of the beam. The deflection was chosen both in the direction of the undeformed weband flange, i.e. two different diagrams were drawn. The selected point was not affected by localbuckling. The deflections of this point could therefore be used to describe the deflection of thewhole beam. This means that the two diagrams could be used to see if both twisting andbending of the beam occurred at the same time. This could be interesting to know becausewhen the load carrying capacities according to the different codes were calculated, lateral-torsional buckling determined the load carrying capacity for most beams. Lateral-torsionalbuckling is a buckling mode where both twisting and bending occurs at the same time. The twodiagrams showed that twisting and bending quite often occurred at the same time, at least closeto the maximum load.

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Figure 4.11. Displaced mesh for the welded beam PB-4, 50 mm haz.

Figure 4.12. Deformed mesh for the welded beam PB-10, 50 mm haz.

Figure 4.13. Outline of the undeformed and deformed mesh for the welded beam PB-35,50 mm haz.

In table 4.3 and 4.4 the load carrying capacities and the extreme values of the strains atmaximum load are shown. All shell elements and both E11 and E22 were considered in orderto find the extreme values of the strains. E11 and E22 are the two strain components in Abaqusthat are associated with normal stresses. From table 4.4 it can be seen that especially thecompression strains in the welds sometimes became very high. This had to do with the load

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application point, the unsymmetry of the cross-section and that two different materials wereused. When all beams and all shell elements were considered, the extreme values of the strainswere +5,6% and –16,5%. The strains in the parent material are not shown in table 4.4, but theywere considerably lower than the strains in the heat-affected zone. The extreme values of thesestrains were +1,93% for PB-21 and –3,51% for PB-4. Both these strains occurred at 30 mmheat-affected zone. From table 4.3 it can be seen that the strains for the unwelded beams wereeven smaller.

For the unwelded beams, the extreme values of the strains occurred at the midspan of the beamat the tip of the web or at the tip of the two flange halves, i.e. not where the web and the flangeare connected. In exceptional cases the extreme strains occurred at the supports.

Table 4.3. Some results of the unwelded beams from 1996.1,0 mm imperfection 0,1 mm imperfection

Beam Profile Lap N [kN]test

N [kN]Abaqus

εmax [%] εmin [%] N [kN]Abaqus

εmax [%] εmin [%]

PB-1 2 II 104,8 97,2 0,26 -0,54 105,0 0,29 -0,51PB-2 1 V 61,2 60,7 0,66 -1,50 60,8 0,64 -1,47PB-3 1 IV 88,8 88,0 0,49 -1,16 88,1 0,46 -1,11PB-4 3 V 61,6 60,8 0,62 -1,44 60,8 0,61 -1,41PB-19 2 II 105,2 97,7 0,27 -0,56 105,5 0,29 -0,52PB-20 1 V 61,1 60,4 0,63 -1,44 60,4 0,67 -1,53PB-21 2 III -H 144,7 0,15 -0,50 172,4 0,14 -0,58PB-22 3 I -SL 86,0 0,67 -0,72 89,9 0,73 -0,59

PB-5 5 II 73,7 57,9 0,30 -0,53 63,8 0,35 -0,47PB-6 4 V 42,4 37,6 0,61 -0,73 38,8 0,54 -0,71PB-7 5 II 67,2 58,1 0,29 -0,52 63,9 0,34 -0,44PB-8 5 IV 49,3 50,6 0,48 -0,55 52,3 0,46 -0,55PB-9 4 III 64,4 51,7 0,30 -0,54 53,9 0,22 -0,46PB-10 3 IV 58,5 53,6 0,48 -0,55 55,4 0,41 -0,53PB-11 4 I 52,7 48,2 0,63 -0,59 51,6 0,83 -0,50PB-12 6 IV 55,4 50,7 0,48 -0,55 52,5 0,43 -0,53PB-23 6 III 64,9 58,2 0,23 -0,47 61,4 0,16 -0,40PB-24 6 V 43,6 42,1 0,49 -0,95 42,1 0,37 -0,93

PB-13 2 II 32,9 28,1 0,17 -0,39 29,2 0,18 -0,38PB-14 1 V 25,0 24,7 0,46 -0,40 25,2 0,44 -0,41PB-15 1 II 31,9 29,6 0,19 -0,45 30,6 0,18 -0,40PB-16 3 III 32,5 29,1 0,34 -0,45 29,8 0,28 -0,40PB-17 2 I 31,2 27,2 0,34 -0,42 28,6 0,38 -0,40PB-18 5 V 26,0 23,6 0,50 -0,41 24,0 0,43 -0,41PB-25 4 I 30,9 24,8 0,37 -0,55 25,6 0,38 -0,49PB-26 6 III 32,0 27,9 0,34 -0,45 28,6 0,28 -0,40SL = Slip occurred H = Failure load too high for the test equipmentLap = Load application point

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Table 4.4. Some results of the welded beams from 1997 and 1998.50 mm haz 30 mm haz

Beam Profile Lap N [kN]test

N [kN]Abaqus

εmax atweld [%]

εmin atweld [%]

N [kN]Abaqus

εmax atweld [%]

εmin atweld [%]

PB-1 G I 59,9 51,0 2,9 -4,4 56,7 4,4 -5,8PB-2 G IV 63,5 59,1 2,7 -6,9 64,7 2,7 -7,0PB-3 G II 84,2 68,2 1,8 -4,2 76,0 2,1 -5,4PB-4 G V 47,0 41,8 3,4 -8,8 45,5 3,7 -9,5PB-5 G I 59,3 55,4 3,8 -5,1 61,0 5,2 -5,7PB-6 G V 51,0 41,5 2,6 -6,2 45,2 3,6 -9,7PB-19 4 II 90,7 60,8 2,2 -4,9 69,8 2,7 -6,7PB-20 4 II 83,4 66,5 2,8 -6,6 76,1 3,6 -9,1PB-21 4 V 55,2 44,6 5,1 -14,4 48,5 5,6 -16,5PB-22 2 VI 36,9 29,2 3,8 -8,4 32,2 3,9 -11,9PB-23 G III 98,9 92,7 0,99 -2,4 103,8 1,2 -3,1PB-24 C I 54,7SL 67,6 5,0 -6,6 71,9 4,8 -5,1PB-25 G VI 36,9 31,3 2,8 -6,3 34,2 3,8 -10,2

PB-7 A I 38,4 36,8 1,5 -2,1 40,3 2,3 -3,3PB-8 A V 34,7 34,4 2,3 -5,1 37,1 2,6 -5,4PB-9 A II 51,2 45,6 0,56 -1,3 49,0 0,96 -1,9PB-10 A IV 47,4 45,5 1,8 -4,0 49,1 1,8 -3,6PB-11 B I 43,5 40,8 2,0 -2,6 44,2 2,4 -3,1PB-12 B V 34,8 36,5 2,1 -4,8 38,8 2,3 -6,0PB-26 2 III 62,9 49,3 0,22 -0,54 51,8 0,40 -0,90PB-27 2 VI 28,5 25,2 3,0 -6,9 27,8 3,1 -7,3PB-28 3 I 38,5 29,4 1,6 -2,4 32,8 2,5 -3,5PB-29C 1 III 105,2 100,7 1,8 -5,0 111,2 2,2 -4,9PB-30C C III 89,2 97,8 0,98 -2,3 109,7 1,3 -3,1PB-31 B VI 28,0 28,2 2,0 -4,8 30,1 2,3 -5,4PB-32C C III 117,9 98,0 0,98 -2,3 110,0 1,3 -3,1

PB-13 D I 25,0 20,8 0,81 -1,1 22,4 1,2 -1,8PB-14 E IV 26,2 25,4 0,39 -0,79 25,6 0,32 -0,73PB-15 C II 29,7 27,0 0,18 -0,44 27,6 0,35 -0,81PB-16 D V 22,3 20,5 1,7 -3,1 21,7 1,2 -2,7PB-17 D I 27,4 22,9 0,98 -1,4 24,5 1,4 -2,1PB-18 E V 25,7 22,7 1,2 -2,6 23,3 0,92 -2,1PB-33 4 I 27,4 22,1 0,79 -1,2 23,6 1,1 -1,6PB-34 3 VI 20,0 19,0 2,6 -5,5 20,3 2,0 -4,8PB-35 3 I 26,2 23,5 0,78 -1,1 24,8 1,2 -1,9PB-36 2 III 26,9 27,4 0,13 -0,31 27,7 0,16 -0,39PB-37C 1 III 67,4 70,6 0,69 -1,4 76,7 0,83 -1,9PB-38 1 VI 21,3 19,4 2,3 -4,9 20,5 1,8 -4,2PB-39C E III 51,3SL 70,6 0,71 -1,4 76,9 0,81 -1,8C = Clamped beam SL = Value not reliable due to slip Lap = Load application point

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For all short and middle long beams, which were welded at the quarterspan, the extreme valuesof the strains in the parent material occurred close to the weld. For the longest of these beams,the position of the extreme strains was close to the weld, at the midspan or somewhere betweenthe weld and the midspan. When there was a weld at the midspan, the extreme values of thestrains in the parent material naturally occurred close to the weld at the midspan. No extremestrains occurred at the supports for the welded beams. The position in the section was mostly atthe tip of the web or at the tip of the two flange halves.

Some conclusions about the imperfections can be drawn from the results in table 4.3. When themagnitude of the imperfections was changed, the load carrying capacity was affecteddifferently depending on where the load was applied. When the load was applied at position Vthe load carrying capacity seemed to be affected least. Position I, II and III affected the capacitymost. The largest difference was obtained for a short centrically loaded column, PB-21. Thedifferent load application points are shown in figure 4.6.

In section 2.3 it was written that the tested load carrying capacity for the welded column PB-23might be too low. It is difficult to verify this suspicion with the evaluation made here.

Comparisons between the load carrying capacities according to Abaqus and the failure loadsfrom the tests are made in table 4.5. The best results are obtained when the quotient is as closeto one as possible. For the unwelded beams, a maximum imperfection of 0,1 mm gave betterresults than 1,0 mm imperfection. The load carrying capacity according to Abaqus was higherthan the tested one, for only four of the 24 unwelded beams when the imperfection was 0,1mm. This indicates that a better result probably would have been obtained with a smallerimperfection than 0,1 mm. For the welded beams, a width of the heat-affected zone of 30 mmgave better results than a width of 50 mm. When the width was 30 mm, the load carryingcapacity according to Abaqus was lower than the tested capacity for 23 beams. For the other 14beams the Abaqus load carrying capacity was accordingly higher. The results would probablybe better if the width of the heat-affected zone was chosen somewhat lower than 30 mm. Whenconsidering that the best results probably are obtained when Abaqus gives a higher loadcarrying capacity for about half of the beams, it is reasonable to believe that the result of theunwelded beams could be improved more than the result of the welded beams. When themagnitude of the imperfections and the width of the heat-affected zones are changed there is arisk that a few results become quite bad even if the overall result gets better. It is possible that adeeper study of the effect of the imperfections and the width of the heat-affected zone shouldhave been made, but the Abaqus calculations were quite time consuming and some limitationshad to be defined. The notations N 0,1 , N 1,0 , N 50 and N 30 denote the load carrying capacityat 1,0 mm maximum imperfection, 0,1 mm maximum imperfection, 50 mm heat-affected zoneand 30 mm heat-affected zone, respectively.

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Table 4.5. Comparisons between Abaqus and tests.Unwelded beams from 1996 Welded beams from 1997 and 1998

Beam Ntest/N1,0 Ntest/N0,1 Beam Ntest/N50 Ntest/N30PB-1 1,078 0,998 PB-1 1,174 1,057PB-2 1,008 1,007 PB-2 1,074 0,981PB-3 1,009 1,008 PB-3 1,234 1,108PB-4 1,014 1,012 PB-4 1,124 1,032PB-19 1,076 0,997 PB-5 1,071 0,972PB-20 1,012 1,011 PB-6 1,229 1,127PB-5 1,273 1,156 PB-19 1,491 1,300PB-6 1,127 1,094 PB-20 1,254 1,097PB-7 1,157 1,052 PB-21 1,239 1,139PB-8 0,974 0,942 PB-22 1,262 1,148PB-9 1,246 1,194 PB-23 1,067 0,953PB-10 1,092 1,055 PB-25 1,178 1,080PB-11 1,093 1,021 PB-7 1,043 0,952PB-12 1,092 1,055 PB-8 1,008 0,936PB-23 1,114 1,058 PB-9 1,122 1,045PB-24 1,036 1,036 PB-10 1,042 0,965PB-13 1,170 1,126 PB-11 1,067 0,984PB-14 1,013 0,992 PB-12 0,954 0,898PB-15 1,079 1,044 PB-26 1,276 1,215PB-16 1,120 1,093 PB-27 1,130 1,026PB-17 1,146 1,092 PB-28 1,310 1,174PB-18 1,103 1,082 PB-29C 1,045 0,946PB-25 1,245 1,206 PB-30C 0,912 0,813PB-26 1,147 1,120 PB-31 0,993 0,930

PB-32C 1,203 1,072PB-13 1,204 1,116PB-14 1,030 1,022PB-15 1,098 1,078PB-16 1,090 1,027PB-17 1,199 1,119PB-18 1,133 1,104PB-33 1,238 1,162PB-34 1,054 0,986PB-35 1,114 1,054PB-36 0,983 0,970PB-37C 0,954 0,878PB-38 1,098 1,037

C = Clamped beam

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The average values and the standard deviations for the quotients in table 4.5 are shown in table4.6 and 4.7. The unwelded beams were evaluated for two different cases: all beams and thecase where the three “worst” beams were excluded. The welded beams were evaluated for fourdifferent cases: all beams, all beams where the clamped beams were excluded, the clampedbeams and the case where also the two “worst” beams were excluded.

The “worst” beams were those beams where the calculated and tested load carrying capacitydiffered most. The values inside the parentheses are the number of beams that were used whenthe average values and standard deviations were calculated. The best results are obtained whenthe average value is close to one and the standard deviation is as small as possible. As can beseen in the tables, the unwelded beams gave somewhat better results. This was natural becausethere were quite many uncertainties about the heat-affected material. The standard deviationshows how wide the quotients in table 4.5 are scattered. If a constant was added to allquotients, the standard deviation would not be changed. Such a change of the quotients is notrealistic when the imperfections and the heat-affected zones are changed. This shows that thestandard deviation is affected by changes of the imperfections and the heat-affected zones. Thisis also shown by the results of table 4.6 and 4.7.

When the results of table 2.7 and 2.8 were discussed, it was shown that the scattering of the testresults sometimes was quite high and not always so reliable. A numerical model is always asimplification of the reality. These are two reasons why one cannot expect that the resultsshould be so exactly. The results were though considered so accurate that the Abaqus modelwas used in chapter 5 to calculate beams with larger load eccentricity.

When the clamped beams were tested, there were some uncertainties about the clamping effectof the supports. If some rotation of the supports occurred during the testing, the tested loadcarrying capacity ought to be lower than the capacity from the Abaqus calculation. In general, itwas difficult to draw any conclusions about the clamped beams, because so few of them weretested.

When tables 3.3 and 4.6-4.7 are compared, it can be seen that Abaqus could predict the loadcarrying capacity better than the codes.

Table 4.6. Evaluation of the unwelded beams in table 4.5.All (24) All – 3 (21)

Ntest/N1,0 Ntest/N0,1 Ntest/N1,0 Ntest/N0,1Average 1,101 1,106 1,079 1,043St. dev. 8,00% 6,56% 5,74% 4,71%

Table 4.7. Evaluation of the welded beams in table 4.5.All (37) All – clamped (33) Clamped (4) All – clamped – 2 (31)

Ntest/N50 Ntest/N30 Ntest/N50 Ntest/N30 Ntest/N50 Ntest/N30 Ntest/N50 Ntest/N30Average 1,127 1,041 1,146 1,058 1,029 0,9274 1,122 1,041St. dev. 11,8% 9,98% 11,5% 9,41% 12,9% 11,1% 9,07% 7,59%

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5 Further analysis of Eurocode 9 5.1 Buckling and tensile failure Introduction When the bending moment, caused by the applied compressive load, acts such that the tip of the web is in tension, it is likely that tensile failure could occur at the tip of the web. When the bending moment acts in the reverse direction, there is no risk for a tensile failure because the more slender web will buckle before the stiffer flange fails in tension. All tested beams where the bending moment caused tension in the web were sorted out. From these beams it was found that tensile failure was not necessary to check when the calculations were performed according to the three codes. The flexural and lateral-torsional buckling checks were sufficient. See further section 3.7. The load eccentricity of the beams that were sorted out was 14 or 28 mm. For a beam subjected to transverse loads the bending moment term of the interaction formulas will likely be more dominant than what it was for the tested beams. For this reason, it is desirable that the interaction formulas should be valid for wider combinations of the axial force and the bending moment than what was obtained from the tested beams. If for instance the axial force is zero, the allowable bending moment will be equal to the bending moment capacity when only the edge in compression is considered. However, the bending moment capacity when both the compression and tension edges are considered will be smaller when the tip of the web is in tension. This indicates that the flexural and lateral-torsional buckling checks may not be sufficient when the bending moment term of the interaction formulas is large. In this section different formulas are derived. The derivation is based on similar assumptions as the formulas in Eurocode 9 and BSK. The sine expression, which takes into account the location of the transverse weld, is for example used. Only Eurocode 9 is considered because it is believed to be the most important code of the three codes in chapter 3. Consequently, the notations follow Eurocode 9. Abaqus calculations The Abaqus numerical model described in chapter 4 was used to calculate beams with a larger load eccentricity than the tested ones in order to check the interaction formulas for flexural and lateral-torsional buckling. The larger load eccentricity increased the bending moment term of the interaction formulas. All of these beams were simply supported for bending and were either unwelded or welded with one transverse weld. For the unwelded beams the maximum imperfection was 0,1 mm. For the welded beams the maximum imperfection was 1,0 mm and the width of the heat-affected zone was 30 mm. These values were chosen because of the results of the Abaqus calculations in chapter 4. In chapter 4 more information about the Abaqus model is found. For most of the beams that were used in the Abaqus calculations, the load application point was equal to I, see figure 4.6. For these beams, the only difference from the earlier Abaqus calculations was that one co-ordinate of the load application point was changed and that the

90

parameters used by the Riks method was modified so they suited the new larger load eccentricity. It was of interest to know what the bending moment capacities of the beams were, i.e. when the only load was a pure bending moment applied at both beam-ends. For the unwelded beams and the midspan welded beams, this calculation was performed by gradually increase the rotation of the two beam-ends. The rotations were always equal but with opposite sign. The rotation was prescribed at load application point III, i.e. at the centre of gravity. The Riks method was not used here. The reason for this is that the Riks method only can prescribe the value of one degree of freedom and not two, as needed. This could probably have been overcomed in some way, but the selected method worked fine so no change of method was needed. The moment capacity was then obtained as the mean value of the maximum value of the two end moments. For the unwelded beams these two end moments were equal. When the weld was located at the midspan, the difference between the two end moments was small. When the weld was located at the quarterspan, the difference between the two end moments was so large that the method was not suitable to use. To gradually increase the applied end moments would most likely cause numerical problems. If a higher applied load is given than the maximum load, no numerical solution exists. There also exist more than one corresponding displacement to each load level. These are two possible numerical problems with load governed calculations. For the beams where the weld was located at the quarterspan, no pure bending moment calculation was performed. Instead, these beams were calculated with a load eccentricity of 1014 mm. In appendix C, parts of the two input files used in the Abaqus calculations of the bending moment capacities are shown. The parts are obtained from the two input files of a beam loaded such that the tip of the web was in compression. Only the parts which regard the loading conditions are shown, because the geometry and the material parts are the same as in the other Abaqus calculations. The prescribed displacement or rotation was applied through a system of rigid beam elements. This was probably one reason why no numerical problems occurred even when the load eccentricity was very large. The maximum positive strain became quite large for some of the Abaqus models, especially when the load eccentricity was large and when the applied load was given by two pure bending moments. For the welded beams the overall maximum positive strain became 19,27%. This value can for instance be compared with the minimum elongation according to Eurocode 9, which is 10%. It can also be compared with the failure strains from the tensile tests described in section 2.2, but it was difficult to know what the maximum strain was because for most tensile tests the measuring equipment stopped to measure the strain at 8,47% nominal strain. During a tensile test, the strain is measured over a distance of 50 mm. The strains from a tensile test therefore can be seen as mean values over this 50 mm distance. Locally, the strains are much higher. The high strain values from the Abaqus calculations were local values. It was therefore decided that the strain level was not a limiting factor for the load carrying capacity of the Abaqus calculations, i.e. the maximum load was chosen independently of the strain level. In order to see how the strain level affected the capacity of the beams, the load carrying capacity was also selected when the maximum positive strain was about 10%. The strains are obtained at each increment of the Abaqus calculation. The increment, which gave a maximum positive strain nearest above 10%, was chosen. The level 10% was obtained from Eurocode 9, as described above. This comparison showed that the capacity was lowered with maximum 3,72%, but for most beams the figure was much lower. The above discussion about the strains

91

considers the welded beams. The strains for the unwelded beams were likely smaller. This was concluded for the unwelded beams in chapter 4. The Abaqus calculations showed that the flexural and lateral-torsional buckling checks were insufficient. The chosen interpretation of Eurocode 9, which is described in section 3.2, therefore required a tensile failure check. The load carrying capacities from the Abaqus calculations are used in figure 5.2 and section 5.3. The Abaqus calculations give some information about when the flexural and lateral-torsional buckling checks in section 3.2 are insufficient. For the short beams, the buckling checks are insufficient when the load eccentricity is greater than or equal to 78 mm. Corresponding eccentricity for the middle long beams is 214 mm. For the long beams, the buckling checks were sufficient for all eccentricities. When the load eccentricity exceeds the values given above an additional failure check is thus needed. In table 5.1 the Abaqus calculated beams are shown. The models for the beams are the same as in chapter 4 with the exception that the load application points have been changed. Table 5.1. Abaqus calculated beams.

Beam e [mm] Beam e [mm] PB-2 UW pure moment PB-11 W 128 PB-22 UW 78 PB-11 W 214 PB-22 UW 128 PB-11 W 314 PB-22 UW 328 PB-11 W 1014 PB-9 UW pure moment PB-26 W pure moment PB-11 UW 78 PB-28 W 42,5 PB-11 UW 228 PB-28 W 164 PB-11 UW 628 PB-28 W 214 PB-16 UW pure moment PB-13 W 42,5 PB-17 UW 328 PB-13 W 128 PB-17 UW 628 PB-13 W 214 PB-17 W 42,5 PB-1 W 78 PB-17 W 128 PB-1 W 114 PB-17 W 214 PB-1 W 214 PB-17 W 414 PB-5 W 78 PB-17 W 1014 PB-5 W 114 PB-33 W 42,5 PB-5 W 214 PB-33 W 214 PB-5 W 1014 PB-35 W 214 PB-23 W pure moment PB-35 W 614 PB-7 W 128 PB-35 W 1014 PB-7 W 314 PB-36 W pure moment PB-11 W 42,5 UW = Unwelded beam W = Welded beam e = Load eccentricity

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Navier’s equation The simplest tensile failure check is probably Navier’s equation, which is shown below. The positive directions of N and 0M are found in connection to figure 5.1.

−

+ ≤N

A fM

f W el x t0 0 2

0

0 0 21 00

ω ω, , . ., (5.1)

= 0 0 2ω = hazf f , but 0 1 00ω ≤ , welded section 0 1 0ω = , unwelded section Navier’s equation is an elastic check of the cross-section where no buckling is considered. It was found that Navier’s equation was too conservative. It became less conservative when the elastic section modulus was replaced with the plastic section modulus, but the formula was still useless as a tensile failure criterion. This replacement of the section modulus meant that the curve for Navier’s equation in figure 5.2 and 5.5 was moved parallel to the right. The plastic section modulus was introduced with the motivation that Navier’s equation was seen as the sum of the degree of usage of the axial force and the bending moment. The denominator contains the force capacity and the numerator the corresponding acting force. Initial interaction formula for tensile check A suitable tensile failure interaction formula should most likely consider second order effects. This is not the case for Navier’s equation. An attempt to develop such an interaction formula will be presented in the remainder of this section. The formula is based on the similar assumptions as the flexural buckling interaction formula in Eurocode 9 and BSK. During the derivation of the formulas, it is assumed that the cross-section is not slender, i.e. the gross cross-section constants are used. If an interaction formula finally will be used as a tensile failure criterion, considerations regarding the slenderness of the cross-section should be made, i.e. the gross cross-section constants have to be replaced with the effective cross-section constants. The normal stress at the two edges of the strut in figure 5.1 can be written as

( ) ( ) ( )σ z

NA

M zW

NA

M N w zWel x c el x c

= − − = − −+

. . . .

0 edge in compression

( ) ( ) ( )σ z

NA

M zW

NA

M N w zWel x t el x t

= − + = − ++

. . . .

0 edge in tension

The axial force N is positive in compression. The bending moment 0M should always be taken positive. The direction of 0M is considered by selecting correct values of el x cW . . and

el x tW . . .

93

Figure 5.1. Strut. The expressions “edge in compression” and “edge in tension” refer to the stresses caused by the bending moment. At the edge in tension, the stress can be compressive if the axial force is dominant. The stresses are positive in tension and negative in compression. The stresses at the edge in compression are mainly used to make a comparison with the expression for flexural buckling in Eurocode 9. The deflection curve w(z) is chosen as the buckling curve of an unwelded Euler strut with pinned ends, i.e. a half sine wave. The amplitude wmax of the half sine wave is obtained from the condition that the yield stress f0,2 is reached at the midspan of the Euler strut, giving the conditions

( )σχ χ

cRd y Rd y

el x cl

NA

N wW

f2 0 2= − − = −max

. ., edge in compression

( )σχ χ

cRd y Rd y

el x cl

NA

N wW

f2 0 2= − + =max

. ., edge in tension

RdN A f= 0 2, where NRd is the axial force capacity of the cross-section. There is no need to check the compression stress at the edge in tension, because this will lead to a negative value of wmax, which is not possible. When wmax is negative the formulas for the stresses at the compression and tension edges are not valid. The following two expressions of wmax are obtained

( )max. . ,w

W f

Nel x c

Rd yy= −0 2 1χ

χ edge in compression

( )max. . ,w

W f

Nel x t

Rd yy= +0 2 1χ

χ edge in tension

The value of wmax should decrease when the beam gets stiffer, i.e. when yχ is increased. For

this reason the expression of wmax of the edge in tension is not realistic. yχ is the buckling reduction factor and it is positive with a maximum value of 1,0. The formulas for the stresses are based on the theory of elasticity and therefore the stresses must be within − ≤ ≤0 0 2 0 0 2ω σ ω, ,f f , where 0ω is equal to hazf f 0 2, when there is a weld at z and 1,0 otherwise.

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The normal stress at the two edges of the strut in figure 5.1 can then be written as

( )σπ

ωzNA W

M N wz

lf

el x c c= − − +

⋅

≥ −

10 0 0 2

. .max ,sin edge in compression

( )σπ

ωzNA W

M N wz

lf

el x t c= − + +

⋅

≤

10 0 0 2

. .max ,sin edge in tension

It is not likely that the compression stress at the edge in tension will determine the load carrying capacity. When the expressions for the stresses above are rewritten with help of the expression of wmax, the following interaction formulas are obtained.

( )NN

zl

Mf WRd y

y yc el x c0

0

0 0 21 1 00

ω χχ χ π

ω+ −

⋅

+ ≤sin ,

, . .

(5.2)edge in compression

( )NN

WW

zl

Mf WRd y

yel x c

el x ty

c el x t0

0

0 0 21 1 00

ω χχ χ π

ω− + −

⋅

+ ≤. .

. . , . .sin ,

(5.3)edge in tension

The interaction formula of the edge in tension can also be compared with the earlier presented Navier’s equation, eq. 5.1. By rewriting the interaction formula, it can easily be seen that the interaction formula is a more restrictive demand than Navier’s equation. This is also natural, because the interaction formula includes second order effects, which is not the case for Navier’s equation.

( )−+ + −

⋅ ≤

NA f

Mf W

NN

WW

zlel x t Rd y

el x c

el x ty

c0 0 2

0

0 0 2 01 1 00

ω ω ω χχ π

, , . .

. .

. .sin ,

edge in tension

The interaction formula for flexural buckling is given in section 3.2.3. When the exponent is excluded, the interaction formula is given by

( )NN

zl

MW fRd y

y yc yc el x c0

0

0 0 21 1 00

ω χχ χ π

ω α+ −

⋅

+ ≤sin ,

. . ,

As can be seen, this interaction formula is basically the same as the interaction formula of the edge in compression. The only difference is the denominator of the moment term. The difference can probably best be explained by seeing the interaction formulas in the codes as the sum of the degree of usage of the axial force and the bending moment as described in accordance with Navier’s equation. The derivation of the interaction formulas shown here is more or less just a way to find out what expression the axial force term, of the interaction formula, should be multiplied with. Navier’s equation was by it self too conservative and to use the more conservative interaction formula of the edge in tension as a failure criterion does not make sense. The interaction formula was therefore modified in some ways to make it less conservative. The quotient

el x c el x tW W. . . . was set equal to 1,0 to make the formula easier to use and to make it more

95

similar to the interaction formulas used in the codes. el x tW . . was replaced with the capacity pl xW . in the same manner as the interaction formulas in the codes. None of these

modifications gave any satisfactory results. Proposed interaction formula for tensile check After testing different types of interaction formulas it was found that the formula in equation 5.4 gave satisfactory results when compared with the Abaqus calculations. Equation 5.4 is not really a tensile failure check, because then the axial force and the bending moment terms should have opposite signs. At the edge where the bending moment causes tension, the axial force and the bending moment counteract each other. It is not really a section check either, because the buckling reduction factor yχ is included in the formula. Perhaps the expression second order section check could be used, if such an expression is possible. The equation could also be seen as an additional buckling check because of the similarities with the interaction formula for flexural buckling in Eurocode 9, see section 3.2.3. The differences are the exponent yc

'ξ and the way the bending moment capacity is calculated. Despite the improper look of the formula, equation 5.4 is denoted a tensile failure check. It was not possible to find a suitable “proper” interaction formula for tensile failure. The derivation of equation 5.4 was mostly based on testing different possible formulas and choosing the one, which gave the best result. A “proof” of the formula is hard to find, but some further comments are given here. When the axial force is zero the moment capacity according to equation 5.4 is equal to the plastic section modulus multiplied with the strength. This is reasonable when the bending moment acts such that the tip of the web is in tension. No lateral buckling will likely then occur. The derivation of equation 5.4 was also governed by the desire that it should look similar to the other interaction formulas in Eurocode 9. In equation 5.4 N and M should always be inserted with positive sign, i.e. in the same way as for the interaction formulas in the codes.

ycNA f

Mf Wxy y pl x

'

, , .,

ξ

ω χ η ω0 2 0 0 21 00

+ ≤ (5.4)

= yc y' 'ξ ξ χ= 0 but yc

' ,ξ ≥ 0 80

02'

maxξ α= but 1 00 1 560, ,'≤ ≤ξ The notations maxα , yχ , η , 0ω and xyω are explained in section 3.2.3. To make the

calculations more similar to the ones in section 3.2.3, an attempt was also made to set 0 10' ,ξ = , but this attempt did not turn out well. For the shortest beams, the curve for the “proposed tensile check” in figure 5.2 became concave instead of convex, which was not intended. For the longer beams there was no difference, yc

'ξ was not changed due to the lower bound 0,8.

96

By setting fWM xplRdy 2,0.. = , the interaction formula for flexural buckling in section 3.2.3 could be used as a tensile failure check, but it will be highly conservative for the shortest beams. The proposed interaction formula for the tensile check worked fine for the T-profiles used in this thesis. However, the interaction formula must be more extensively tested before using it as a general tensile failure check for any type of compressed member. It is hazardous and early to claim that it should be used in a code like Eurocode 9. For a welded beam, which is not welded at the midspan, it is necessary to check for flexural and lateral-torsional buckling both at the unwelded midspan and at the welded section closest to the midspan. Likely this procedure is also necessary for the tensile failure check, because the sine expression is also used in the proposed interaction formula for tensile failure. The sine expression takes into account the location of the transverse weld. When the weld is not located at the midspan, it is not sure that a welded section is the most critical section. Equation 5.4 is further discussed in section 5.3. Diagrams To get a better understanding of the different interaction formulas six diagrams were drawn. It was necessary to use six diagrams because all variables used by the interaction formulas must be constant for each diagram, if they are not included in the co-ordinates of the axes. If the theoretical cross-section dimensions, beam lengths and weld locations were used, the constant variables would be exactly the same for all beams in each diagram. However, measured values were used and therefore, the constant variables were calculated as the average value from the tested beams included in a diagram. In the diagrams, Navier’s equation (eq. 5.1), the initial interaction formula for the tensile check (eq. 5.3), the proposed interaction formula for the tensile check (eq. 5.4) and the interaction formulas for flexural and lateral-torsional buckling (see section 3.2) were inserted. The tested beams with a load application point equal to I or II (see figure 4.6) and the additional beams calculated with Abaqus are represented by markers (plus- and cross-signs) in the appropriate diagram. The co-ordinates of the markers are calculated by making use of the load carrying capacities from the tests and the numerical simulations. The radius between the web and the flange is not included in the Abaqus model and therefore,

el x cW . . and A are calculated without consideration of the radius for the Abaqus calculated beams in figure 5.2. For the tested beams, the radius is included in the calculation of el x cW . . and A. The load eccentricity in mm is shown in the diagrams for the beams with the largest load eccentricity. In the diagrams, the theoretical beam lengths, L, and the possible weld locations are shown. See also tables 2.7-2.8. As can be seen in the diagrams, Navier’s equation and the interaction formula for the initial tensile check (as well as their modified versions) are straight lines. From the distribution of the markers, it can be seen that a straight line would be a bad tensile check.

97

The straight lines representing Navier’s equation and the initial tensile check are cut at the position where the stress at the compression side reaches − 0 0 2ω ,f . For the initial tensile check, the stresses at the edge in compression have been calculated with equation 5.2. All cross-sections used here belonged to the most compact class for bending and the most slender class for axial compression. This means that the cross-section area A was replaced with the effective area Aef for all formulas in figure 5.2, except for Navier’s equation and lateral-torsional buckling. Navier’s equation was never intended to be used as a tensile failure criterion and therefore Aef was not used. It is just shown as a common formula that probably many engineers would use to check the tensile failure if no specific information was given. The effective area was not used in the formula for lateral-torsional buckling because torsional buckling determined the axial compression capacity for all columns except one. When torsional buckling determines the capacity, Aef should not be used. More information is found in section 3.2.4 and 3.7.

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1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Lateral-torsional buckling, EC9Flexural buckling, EC9Proposed tensile checkNavier's equationInitial tensile checkTestAbaqus

114214

78

28

L=500 mm, unwelded or a weld at the midspan

Mf W el x c0 0 2ω , . .

Nf A0 0 2ω ,

78

128328

Figure 5.2a. Graphic analysis of interaction formulas, symmetric short beams.

98

0

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0.3

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0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Lateral-torsional buckling, EC9Flexural buckling, EC9Proposed tensile checkNavier's equationInitial tensile checkTestAbaqus

114

2141014

78

28

L=500 mm, a weld at the quarterspan

Mf W el x c0 0 2ω , . .

Nf A0 0 2ω ,

Figure 5.2b. Graphic analysis of interaction formulas, unsymmetric short beams.

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0.1

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1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Lateral-torsional buckling, EC9Flexural buckling, EC9Proposed tensile checkNavier's equationInitial tensile checkTestAbaqus

164214

314

128

42,5

L=1020 mm, unwelded or a weld at the midspan

Mf W el x c0 0 2ω , . .

Nf A0 0 2ω ,

78228 628

Figure 5.2c. Graphic analysis of interaction formulas, symmetric middle long beams.

99

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Lateral-torsional buckling, EC9Flexural buckling, EC9Proposed tensile checkNavier's equationInitial tensile checkTestAbaqus

1014314214

128

42,5

L=1020 mm, a weld at the quarterspan

Mf W el x c0 0 2ω , . .

Nf A0 0 2ω ,

Figure 5.2d. Graphic analysis of interaction formulas, unsymmetric middle long beams.

0

0.1

0.2

0.3

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0.5

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0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Lateral-torsional buckling, EC9Flexural buckling, EC9Proposed tensile checkNavier's equationInitial tensile checkTestAbaqus

214128

L=1540 mm, unwelded or a weld at the midspan

Mf W el x c0 0 2ω , . .

Nf A0 0 2ω ,

42,5

328 628

Figure 5.2e. Graphic analysis of interaction formulas, symmetric long beams.

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Lateral-torsional buckling, EC9Flexural buckling, EC9Proposed tensile checkNavier's equationInitial tensile checkTestAbaqus

128

42,5

1014

614414214

L=1540 mm, a weld at the quarterspan

Mf W el x c0 0 2ω , . .

Nf A0 0 2ω ,

Figure 5.2f. Graphic analysis of interaction formulas, unsymmetric long beams. 5.2 Section check Introduction This section is mainly used to check the interaction formula for section check in Eurocode 9 when the axial force is compressive and the bending moment makes the tip of the web to be in tension. The section check is performed at the edge where the stresses from the axial force and the bending moment have the same direction, i.e. at the edge in compression. In the same way as in section 5.1 there is a suspicion about that the interaction formula may be insufficient when the bending moment is dominant and causes the tip of the web to be in tension. In this case the stresses at the tip of the web could be decisive for the capacity of the cross-section. The interaction formula for section check of a T-section is given below.

0

0 0 2

1 0

01 00

η

ω η ωNA f

MM y Rd,

,

., +

= (5.5)

η = 1 class 1, 2 and 3 for axial compression η = efA A class 4 for axial compression All cross-sections used here belonged to the most compact class for bending and the bending moment caused tension in the tip of the web. This means that y Rd el x cM W f. . . ,= 0 2 . The exponent 0η is explained in section 3.2.4.

101

Abaqus calculations The Abaqus model described in chapter 4 was also used to calculate stubby beams to check the interaction formula for section check. These beams were simply supported for bending and unwelded. Beams with three different lengths l and six different load eccentricities e were used, see table 5.2. Three different beam lengths were used in order to see how the load carrying capacity was affected. Even if all beam lengths were short, some kind of buckling might have occurred. Due to the steel plates at the supports, the distances between the pinned ends were 50 mm longer than the beam lengths in table 5.2, see section 3.1. The 100 mm beams were calculated first. These beams were calculated both with 0,1 mm as maximum imperfection and without imperfections. It was found that the imperfections had very little influence on the load carrying capacity. The 70 and 130 mm beams were calculated thereafter without any imperfections. The cross-section dimensions and material of the unwelded beam PB-1 in table 2.7 were used for all calculations. For the 100 mm beams in table 5.2, the load carrying capacities are from the calculations without imperfections, but the imperfections had so little influence that it would not be possible to see the difference without using more digits. Table 5.2. Some data and results for the Abaqus calculated beams. Beam l [mm] e [mm] εmax [%] εmin [%] N [kN]

1 70 14 23,5 -105 177,6 2 70 28 28,6 -20,7 112,0 3 70 39 21,3 -11,1 83,4 4 70 64 20,8 -4,49 50,5 5 70 114 19,1 -2,32 27,3 6 70 164 20,7 -2,30 18,5 7 100 14 5,83 -23,7 154,9 8 100 28 5,35 -12,3 101,8 9 100 39 5,67 -8,55 77,8

10 100 64 7,64 -4,32 48,1 11 100 114 12,3 -3,96 26,5 12 100 164 13,6 -4,21 18,1

13 130 14 4,30 -10,5 144,7 14 130 28 4,01 -4,94 96,8 15 130 39 3,36 -2,92 74,7 16 130 64 3,99 -1,91 46,7 17 130 114 5,73 -2,78 25,8 18 130 164 6,34 -2,97 17,6

Some numerical problems occurred for beam number 1 in table 5.2. It was not possible to pass the peak of the load-deflection curve, but several of the last increments gave a load equal to 177,6 kN. The likely reason was the large compressive strain, -105%. This strain occurred at a corner of the flange plate and when corresponding element was studied, it was possible to see that it was distorted. Perhaps the numerical problems could have been overcomed if the shell element S4 was used instead of S4R, but this was not tested. The element S4 is believed to be less sensitive for numerical problems than S4R. The elements S4 and S4R are described in section 4.3.

102

The cause of the failure was not always so easy to see. The failure mode for the beams with the largest load eccentricity was of a “bending-type”, while it was more of a “squashing-type” failure for the beams with the smallest load eccentricity. The difference between the capacities within the same load eccentricity can most likely be explained with some kind of flange curling. Especially for the longest beams, the load carrying capacity was reduced due to flange curling. When the beam is bent, the compressive force in the flange will have a component directed downwards. When this component bend the flange, the phenomenon is called flange curling and is visualised in figure 5.3. The flange curling effect ought to be more apparent for the beams with large load eccentricity because the bending moment is larger in these beams. Flange curling is dependent on the curvature, which is caused by the bending moment.

1

2 3

DISPLACEMENT MAGNIFICATION FACTOR = 5.00 ORIGINAL MESH DISPLACED MESH

RESTART FILE = pb139 STEP 1 INCREMENT 12

TIME COMPLETED IN THIS STEP 3.12 TOTAL ACCUMULATED TIME 3.12

ABAQUS VERSION: 5.7-1 DATE: 15-JUL-1999 TIME: 12:33:22

1

2 3

Figure 5.3. Flange curling for beam number 15 in table 5.2. Plastic analyses The cross-section itself was also analysed by making use of two plastic analyses. For both analyses, the axial force and the bending moment around the axis through the elastic centre of gravity were calculated from the stress distribution. The radius between the web and the flange was not considered in any of the two analyses, otherwise, the calculations would have been too complicated. The two plastic analyses assume that the cross-section is not slender. The axial force and the bending moment were thus calculated for the gross cross-section. To make plastic analyses of slender cross-sections would be too extensive in the scope of this thesis.

103

The first and simplest plastic analysis assumed that the stress was equal to f 2,00ω± over the whole cross-section and that the strain was unlimited, i.e. ± ∞ . The formulas for the axial force and the bending moment are shown below and are clarified by figure 5.4. This type of plastic analysis can for instance be found in Petersen [46]. When 0y varies between 0 and ply the axial force is in tension. The expressions for N and M should be multiplied with the strength

f 2,00ω , but in figure 5.5 it can be seen that the strength disappears in the division. The value of the strength does consequently not affect the result of this calculation.

pl fy y t≤ ≤0

( )N t h t bt b yw f f= − − − +2 0

( )[ ] ( )[ ]M th t y h t

b t y y t ywf gc f f gc f= − − − + − − −

22

22 2 22 2 2 2

0 0

ft y h≤ ≤0

( )N bt t h t t yf w f w= − + +2 0

( )[ ] ( )M bt yt t

h t y h t t y y yf gcf w

f gc f w gc= − + + − + + −2 2

2 22 20 0

ty

t

f

w

gc

y

x x

y

b

h

ypl y0

Figure 5.4. Cross-section with notations used for the two plastic analyses. In the second and more advanced plastic analysis, two material models were used, Ramberg-Osgood’s model obtained from Eurocode 9 where the “plastic range” was considered and the “true” stress-log strain model used in the Abaqus calculations in chapter 4. The strain was limited to ± 10% for both material models. The limit 10% was obtained as the minimum elongation for alloy 6082-T6 in Eurocode 9. Like in the Abaqus calculations, the stress is assumed to be constant if the strain exceeds the last given point on the stress-strain curve. The Ramberg-Osgood material model is given by

εσ σ

= + 70000

0 002260

17 77,

, σ in MPa

104

The stress-strain curve for both material models was assumed to be odd, i.e. ( ) ( )σ ε σ ε− = − . The calculations had to be performed numerically, so no formulas for the axial force and the bending moment can be shown. A linear strain distribution was assumed. The stresses were calculated from the strains. When Ramberg-Osgood’s model was used, this was done with Newton-Raphson iteration method. The flange and the web were divided in 10 and 55 elements, respectively. The stress was calculated at the centre of these elements. Plastic analyses, like the ones described here, could perhaps be used in codes, but likely they are too complicated. The usage of interaction formulas in codes is very common and to change this procedure is likely difficult. Plastic analyses could perhaps be used in codes as an alternative to interaction formulas. The likelihood that plastic analyses should be used in codes would increase if simplifying curve fitting or dimensioning diagrams were used. However, it has not been investigated if this is possible. For a general cross-section plastic analyses could be complicated and when computer programming is required, the likelihood for plastic analyses to be used in codes will decrease. In this thesis it is not claimed or intended that the plastic analyses should be used in any code, they are just shown to make comparisons with the Abaqus calculations and with the interaction formulas for section check in the codes. Diagrams In figure 5.5 different curves are inserted. The formulas for flexural and lateral-torsional buckling are presented in section 3.2. The formulas for Navier’s equation and the tensile checks are presented in section 5.1. It is somewhat uncertain how the interaction formula for lateral-torsional buckling looks like when the beam is stubby, i.e. all buckling reduction factors are equal to 1,0. When torsional buckling is decisive for the capacity Aef should not be used. When minor axis flexural buckling is decisive Aef should be used. This assumes that the cross-section only consists of radiating outstands and belongs to class 4 for axial compression. These conditions were also met. It was decided to include Aef, in the interaction formula for lateral-torsional buckling because this is the most unfavourable situation. The interaction formula for lateral-torsional buckling and section check will then be identical. The shortest beams (70 mm) from the Abaqus calculations were chosen. The reason for this was that the cross-section deformed least for these beams and thus, this calculation was most alike a plastic analysis. According to Eurocode 9, the buckling reduction factors yχ and LTχ became 0,995 and 0,951, respectively. Despite that, both of them were set equal to 1,0 in the interaction formulas for flexural buckling, lateral-torsional buckling and tensile checks. These factors are calculated for the buckling length, which is 50 mm longer than the beam length, see section 3.1. The reason why the curve for the second plastic analysis lies “outside” the curve for the first analysis is that the stresses in the second analysis quite often are higher than 0 2,f and lower than − 0 2,f . The only difference between Navier’s equation and the formula for the initial tensile check in figure 5.5 is that Aef is included in the initial tensile check but not in Navier’s equation. Like

105

the Abaqus calculated beams in figure 5.2, el x cW . . and A were calculated without consideration of the radius between the flange and the web for the beams in figure 5.5. For the two plastic analyses, the radius was not considered when N, M, el x cW . . and A were calculated. The load eccentricities in millimetres are shown in figure 5.5. In the Abaqus calculations the cross-section can deform whereas it is rigid in the two plastic analyses. This can be one explanation why the result differs between the two methods. The result shows that the interaction formula for section check in Eurocode 9 is insufficient when compared with the Abaqus calculated stubby beams. The proposed interaction formula for tensile check is sufficient as a section check, but perhaps it is too conservative. Section check is further discussed in section 5.3.

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1.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

L-T buckling & section check, EC9Flexural buckling, EC9Proposed tensile checkNavier's equationInitial tensile checkAbaqus

First plastic analysis

Stubby beams, plastic analysis

164

114

64

39

14

28

Nf A0 0 2ω ,

Mf W el x c0 0 2ω , . .

Second plastic analysis,Ramberg-Osgood

Second plastic analysis,"true" stress-log strain

Figure 5.5. Graphic analysis of different formulas. 5.3 Suggestions for improvements Results of the unmodified calculations Figures 5.6-5.9 show the interaction diagrams for flexural and lateral-torsional buckling when both the tested simply supported beams and the Abaqus calculated beams in section 5.1 are inserted. The unwelded and welded beams are treated separately in order to more clearly see the effect of the welding. The filled markers represent the Abaqus calculated beams. These beams were all simply supported for bending. The calculations strictly followed section 3.2. The tensile failure check in section 5.1 has not been considered. For the welded Abaqus calculated beams, which were unwelded at the midspan, the additional buckling check at the

106

unwelded midspan was not decisive for any beam. More information about this type of interaction diagrams is found in section 3.7.

0

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1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

L=500 mmL=500 mm

y Sd

y Rd

MM

.

.0ω

yc

Sd

y xy

NA f

ξ

χ ω η 0 2,

Web in compression Flange in compression

Totally 6 beams

Figure 5.6. Flexural buckling of unwelded beams, original calculation.

0

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2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

L=500 mmL=500 mm

yc

Sd

y xy

NA f

ξ

χ ω η 0 2,

y Sd

y Rd

MM

.

.0ωWeb in compression Flange in compression

Totally 13 beams

Figure 5.7. Flexural buckling of welded beams, original calculation.

107

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1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

L=500 mmL=1020 mmL=1020 mmL=1540 mmL=1540 mm

cy Sd

LT xLT y Rd

MM

γ

χ ω.

.

cSd

z xz

NA f

η

χ ω η 0 2,

Web in compression Flange in compression

Totally 29 beams

Figure 5.8. Lateral-torsional buckling of unwelded beams, original calculation.

0

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2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

L=500 mmL=1020 mmL=1020 mmL=1540 mmL=1540 mm

cSd

z xz

NA f

η

χ ω η 0 2,

cy Sd

LT xLT y Rd

MM

γ

χ ω.

.

Totally 53 beams

Web in compression Flange in compression

Figure 5.9. Lateral-torsional buckling of welded beams, original calculation. As can be seen from figures 5.6-5.9, the result is not too good. For many beams the result is very conservative. From the Abaqus calculated beams the conclusion can be drawn that the chosen interpretation of Eurocode 9 is unsafe for the short and middle long beams when the bending moment is dominant and causes the tip of the web to be in tension. This conclusion was also drawn from figure 5.2. For one welded short centrically compressed column (PB-23)

108

the result was also unsafe. However, this column should perhaps be excluded as described in section 2.3. It can also be concluded that the result for the unwelded beams was better than for the welded ones. This is perhaps most clearly shown for the centrically compressed columns. In some sense the welding seems to affect the accuracy of the calculations. In table 3.1 and 3.2 the calculated and tested load carrying capacity for each tested beam-column can be found. Possible modifications Some possible modifications to improve the result of the Eurocode 9 calculations will now be discussed. The moment capacity y RdM . is calculated as

y Rd ef x cM W f. . . ,= 0 2 tip of the web in compression (class 4 for bending)

y Rd el x cM W f. . . ,= 0 2 tip of the web in tension (class 1 for bending) In figure 3.2 it can be seen why y RdM . is calculated the way it does when the cross-section belongs to class 1 for bending. To increase the value of y RdM . for class 4 cross-sections would improve the result. The markers on the left-hand side of the interaction diagrams for lateral-torsional buckling would then move to the right. This indicates that the cross-section classification for bending is too severe when the bending moment causes the tip of the web to be in compression. When the tip of the web is in compression, typical values of the section moduli ef x cW . . ,

el x cW . . and pl xW . are 4719 mm3, 5536 mm3 and 9986 mm3, respectively. To replace ef x cW . . with el x cW . . for class 4 cross-sections would improve the result, but likely the increase of

y RdM . will not be enough when considering the distribution of the markers in the lateral-torsional buckling diagrams. To set y Rd pl xM W f. . ,= 0 2 for a class 4 cross-section seems confusing, but likely the result will be improved by this modification. This means that the fully plastic moment could be developed when the bending moment acts such that the tip of the web is in compression. It is difficult to motivate that the plastic section modulus should be used in the calculation of

y RdM . when the tip of the web is in compression. According to Eurocode 9, the cross-section then belongs to the most slender class. This modification was therefore not seen as a reasonable option. A higher value of y RdM . can also be obtained by modifying the classification of the web element. This classification is based on a model where the web element is seen as a long rectangular plate supported along three edges. The elastic critical stress for such a plate can be written as

( )btE

kw

wcr 22

22

112 νπσ σ

−=

109

where ν is Poisson’s ratio. The value of the buckling coefficient σk depends on the stress distribution and the boundary conditions at the edges. The expression of σ cr is used in BSK to calculate a slenderness parameter λ given by

( )

Ef

tb

k

f

w

w

cr

2,02

2,0 1121 ⋅⋅−

⋅==π

νσ

λσ

When the above expression of λ was compared with the expression of λ found in BSK, a value of σk was obtained. When this value was compared with values of σk found in the literature it was concluded that the three supporting edges of the web element were considered as simply supported in BSK. In Eurocode 9 this cannot so clearly be seen, but it is assumed that also Eurocode 9 consider the three supporting edges as simply supported. Values of σk are given in Galambos [22], Timoshenko and Gere [55] and Sharp [52]. A modification of the classification of the web element will now be described. The long supporting edge of the web element is considered as clamped instead of simply supported. This modification will lead to a less severe classification of the cross-section and thus a higher value of y RdM . . When the quotient tb ww , of a plate where the long supporting edge is clamped, is multiplied with the factor kk sc .. σσ , the slenderness parameter λ for this plate will be equal to λ for a plate where the long supporting edge is simply supported. The indices .c and .s of σk stand for clamped and simply supported, respectively. The quotient tb ww is known as the slenderness parameter β w in Eurocode 9. A reasonable modification of the classification of the web ele-

ment could therefore be to multiply the limit slenderness parameters 1β - 3β with kk sc .. σσ . Figure 5.10 shows the value of σk for different stress distributions and boundary conditions. The values are obtained from Galambos [22]. The values of σk in Timoshenko and Gere [55] and Sharp [52] sometimes differ somewhat from the values in figure 5.10. The reasons for this are unknown. The simply supported edges are shown as dashed lines.

σ

σ

σ

σk =0,42σ k =1,33σ

σ

σ

σ

σk =0,57σ k =1,61σ

Figure 5.10. The value of σk for different stress distributions and boundary conditions.

110

The parameters 1β and β 2 are used in the plastic state and therefore it is reasonable to use a constant stress distribution. The parameter β 3 is used in the elastic state. This means that the stress distribution is linear but not constant. When the peak compressive stress is at the tip of the web element, the linear stress distribution is not considered in Eurocode 9. In this case the same values of the slenderness parameters are used as if the compressive stress was constant. The triangular stress distribution in figure 5.10 is not identical with the stress distribution in the elastic state, but the difference is small. The factors to correct 1β - 3β are then given by

78,142,033,1

.

. ==kk

s

c

σ

σ used to correct 1β and β 2

96,142,061,1

.

. ==kk

s

c

σ

σ used to correct β 3

The modified classification of the web element was made as described in section 3.2.1, with the exception that slenderness parameters 1β - 3β were multiplied with the correction factors above. If the cross-section belonged to class 4 according to the modified classification, wβ was divided with 1,96 in the formula for x in section 3.2.2 when the effective cross-section was calculated. The value of ycα for a class 3 cross-section was for instance given by

ycw pl x

el x c

WW

αβ β

β β= +−

−− 1

1 961 96 1 78

13

3 2

,, ,

.

. .

If the modified classification of the web element should be adopted in Eurocode 9, some modifications of how the corrections are made might be necessary in order to agree with the calculation scheme of Eurocode 9. It would be desirable if it was possible to modify the calculations such that the tensile failure check in section 5.1 was not needed. The markers on the right-hand side of the interaction diagrams then would have to be moved to the right. This type of movement can be achieved by replacing el x cW . . with pl xW . when calculating y RdM . for a class 1 cross-section. In this

calculation el x cW . . and pl xW . will typically be 14023 mm3 and 9998 mm3, respectively. This means that y RdM . is lowered with 28,7%. This replacement will make the flexural buckling formula more similar to the tensile failure formula in section 5.1 To use the lowest section modulus of the two edges when y RdM . is calculated could also be a possible modification. In section 3.1 it was motivated that this interpretation of y RdM . was not suitable. For this reason this modification was not considered in this section. Table 5.3 shows the increase of the buckling reduction factors when hazf is used instead of

0 2,f . The calculation is made for η = =efA A 1. The evaluation is made for all beams in table 2.8 except for the clamped ones. When LTχ was calculated, the calculation of y RdM . strictly

111

followed section 3.2. The quotient between hazf and 0 2,f is not the same for all materials. This is one reason for the interval of the percentage increase. Table 5.3. Increase of the buckling reduction factors when 0 2,f is replaced with hazf . L [mm] Increase of yχ [%] Increase of zχ [%] Increase of LTχ [%]

500 3,2-6,1 9,1-21,3 0,0-3,6 1020 18,6-34,0 28,2-55,0 3,3-5,5 1540 26,3-58,2 29,2-66,4 7,4-31,9

When the buckling reduction factors are increased the load-carrying capacity will increase. This is positive for all tested beam-columns except for the welded and centrically compressed column PB-23. The reason is that the result was conservative for all tested beam-columns except for this one. In section 2.3 it was written that the tested load carrying capacity for this column might be too low. The Eurocode 9 calculations increase this suspicion. Table 5.3 shows that the increase is largest for the longest beams. This is positive because the result was most conservative for these beams. Some dramatic increases are shown in the table so this modification will have a major effect on the result. Another possible modification could be to change the shape of the χ λ− curves. When evaluating the curves for χ y and χ z , especially the centrically compressed columns are useful. However, few of these columns were tested. It is more difficult to evaluate the correctness of χ y and χ z from the eccentrically compressed beam-columns because more parameters are involved in the calculations. For the few centrically compressed columns that were tested, the result was better for the unwelded columns than for the welded ones. From this it is difficult to motivate a modification of the curves for χ y and χ z . The reduction factor for lateral-torsional buckling of beams LTχ could also be changed. In table 5.4, the value of LTχ as function of the load application point and the beam length is shown. All tested beams were considered, except the clamped ones. When LTχ was calculated, the calculation of y RdM . strictly followed section 3.2 and the yield strength 0 2,f was always used. When the bending moment causes the tip of the web to be in compression, a better result is obtained when LTχ is increased because the markers on the left-hand side of the interaction diagrams are moved to the right. When the bending moment causes compression in the flange, a better result is instead obtained when LTχ is decreased. It is difficult to fulfil the desire to both increase and decrease the value of LTχ . To change the values of χ LT will only have a minor effect on the distribution of the markers in the interaction diagrams, because it is not realistic to make any major changes to the shape of the curve for χ LT . The value of LTχ is determined by α LT , λ LT,0 and the quotient

MM crRdy. , see section 3.2.4.

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Table 5.4. LTχ in Eurocode 9 as function of the beam length and load application point. L=500 mm L=1020 mm L=1540 mm Lap I-II 1,00 0,91-0,94 0,69-0,79 Lap IV-VI 0,96-0,97 0,91-0,92 0,85-0,88 All cross-sections belonged to class 4 for axial compression. This means that there is an effective area efA . If efA is set equal to the gross area A, the load carrying capacity will generally increase. This is positive for most beam-columns. The load carrying capacity will not always increase because efA is not always used. This is explained in section 3.2.4. As a conclusion, the result will likely be improved by replacing A with efA , but the improvement will likely be small. This indicates that the cross-section classification for axial compression is too severe. The ratio efA A varied between 0,923 and 0,945 for the Eurocode 9 calculations. One motivation for not using efA is the discussion about the classification of the web element, which was described in connection with the calculation of y RdM . . The exponents of the interaction formulas only affect the markers, which do not lie on the co-ordinate axes. Since the distribution of the markers on the left-hand side of the interaction diagrams for lateral-torsional buckling almost is parallel with the straight line representing the interaction formula, it was considered that a change of the exponents was not a good idea. Also when considering the distribution of the markers on the right-hand side of the interaction diagrams it is doubtful that a change of the exponents could improve the result. For the short beams, the exponent ycξ varied between 0,86 and 0,88. The exponent cη varied between 1,33 and 1,51 for the short beams when the bending moment caused compression in the flange and tension in the web. For the non-mentioned beams the exponents ycξ and cη were equal to 0,8.

The exponent cγ was always equal to 1,0. It could not be observed that the result for the quarterspan welded beams was worse than for the midspan welded beams. No conclusions could therefore be drawn that the parameters

xLTω , xyω and xzω should be wrong in some way. One option is to change the material strengths 0 2,f and hazf . From the discussion in chapter 2 it can be concluded that the values of 0 2,f likely are quite accurate whereas the accuracy of

hazf can be discussed. The load carrying capacity of the welded beams is increased when

hazf is increased. This is positive for those welded beams where the calculated load carrying capacity was lower than the tested capacity, i.e. for most welded beams. To introduce some kind of unsymmetry factor to correct the interaction formulas is an undesirable option. The factor would complicate the calculations and it would be difficult to explain. There is always a risk that the factor gives bad results when applied on other cross-sections than the T-sections used here, especially when no theory verifies the factor. Even though the difference is not so dramatic, the result was still better for the unwelded beams than for the welded ones. This indicates that not only the unsymmetry is the reason for the not too good result, but also the material.

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Modified calculations The modifications were analysed, both separately and in combination. It has to be considered that the modifications interact on each other. The result of the analysis showed that the modifications below gave improved prediction of the load carrying capacity of the beam-columns. The modifications were present during the whole calculation.

• When the tip of the web was in tension, y RdM . was calculated as pl xW f. ,0 2 . maxα was calculated according to class 2. • When the tip of the web was in compression, the calculation of y RdM . was based on

the modified classification of the web element. • 1== AAefη • hazf was used to calculate the buckling reduction factors of a welded section • 0 2,f was used to calculate the buckling reduction factors of an unwelded section

(midspan). This is not a modification. The information is just shown to clarify how the calculations were made.

The modifications worked fine for the beams used here, but it is unknown how well they suit other beams and other cross-sections. It is hazardous and too early to suggest any modifications of Eurocode 9. The investigation in this thesis only gives indications of how the calculations could be modified. Together with other investigations there could be enough motivations to suggest modifications of Eurocode 9. To use f haz instead of 0 2,f when calculating the buckling reduction factors of a welded section was the only modification of the buckling reduction factors that could be motivated. Some additional calculations on unwelded and simply supported stubby beams were made with Abaqus to evaluate the value of the bending moment capacity when the bending moment caused compression in the web and tension in the flange. These bending moment capacities were compared with pl xW f. ,0 2 . The results of these calculations are shown in table 5.5. The material and cross-section dimensions were chosen from the three unwelded beams PB-1, PB-4 and PB-5 in table 2.7. The calculations were made for a maximum imperfection of 0,1 mm. Two different beam lengths were used, 100mm and 150 mm, respectively. Due to the steel plates at the supports, the buckling lengths were 50 mm longer than the beam lengths. At failure, the tip of the web buckled. The shell element S4 was used instead of S4R, because the element S4 is less sensitive for numerical problems. It seems that numerical problems more often occur for the stubby beams than for the longer beams. The beam length 70 mm was also used in some calculations, but then some numerical problems occurred. These problems were not overcomed even though the element S4 was used. The elements S4 and S4R are described in section 4.3. The bending moment capacity according to Abaqus was calculated as described in section 5.1.

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Table 5.5. Bending moment capacity when the tip of the web is in compression. l [mm] M [kNm] Abaqus 0 2,f [MPa] pl xW f. ,0 2 [kNm]

100 2,92 PB-1 150 2,73

268 2,66

100 3,45 PB-4 150 3,20

326 3,25

100 3,34 PB-5 150 3,09

320 3,17

It is not unrealistic that M according to Abaqus in table 5.5 mostly is higher than pl xW f. ,0 2 . One reason can be that the stresses in the Abaqus calculations are higher than 0 2,f . The bending moment capacity for class 1 cross-sections, as defined in Eurocode 9, is also higher than pl xW f. ,0 2 , see section 3.2.3. When the bending moment acted such that the tip of the web was in compression, the cross-section always belonged to class 3, but fairly close to class 2, according to the modified classification. This means that a lower value of y RdM . has been used than what was obtained from the Abaqus calculations. This indicates that the modified classification of the web element does not lead to unrealistic high values of y RdM . . When considering the results of the calculations, it can be concluded that the variation of the bending moment capacity in figure 3.1-3.2 was not satisfactory. For the T-sections used here, figure 3.1 will give a too low value of the bending moment capacity. This can be corrected by modifying the calculation of the slenderness parameters β, which has been described. Perhaps a constant value of fW pl 2,0 should be used in figure 3.2 instead of fW cel 2,0. when the cross-sections belong to class 1-3? The situation for class 4 cross-sections is then uncertain, because W cef . could be larger than W pl . The way the bending moment capacity is calculated for unsymmetric cross-sections in the interaction formulas needs to be further analysed. The markers in figure 5.14, which are also marked with a plus sign, represent those beams, which were welded at the quarterspan and for which the buckling check at the unwelded midspan determined the load carrying capacity. The capacity for these beams was calculated in the same way as for the unwelded beams, because the buckling check at the welded quarterspan was not decisive. The buckling check at the unwelded midspan was not decisive for all quarterspan welded beams. The result for the welded column PB-23 was deteriorated by the modifications. However, in section 2.3 it was written that the reliability of this test could be questioned. It was therefore considered that the worse result of PB-23 was not so important. The average values and the standard deviations for the quotient between the tested, or with Abaqus calculated, load carrying capacity and the capacity from the modified Eurocode 9 calculations are shown in table 5.6. All beams in figure 5.11-5.14 have been considered in this calculation. The notation Ntest is used for both the tested and Abaqus calculated beams. When the values in tables 3.3 and 5.6 are compared it can be seen that the result was improved, to a great extent, by the modifications.

115

Table 5.6. Result of the modified Eurocode 9 calculation. test ECN N 9 , unwelded beams test ECN N 9 , welded beams Average value 1,24 1,21 Standard deviation 19,8% 12,0%

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

L=500 mmL=500 mm

yc

Sd

y xy

NA f

ξ

χ ω η 0 2,

y Sd

y Rd

MM

.

.0ωWeb in compression Flange in compression

Totally 6 beams

Figure 5.11. Flexural buckling of unwelded beams, modified calculation.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

L=500 mmL=500 mm

yc

Sd

y xy

NA f

ξ

χ ω η 0 2,

y Sd

y Rd

MM

.

.0ωWeb in compression Flange in compression

Totally 19 beams

Figure 5.12. Flexural buckling of welded beams, modified calculation.

116

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

L=500 mmL=1020 mmL=1020 mmL=1540 mmL=1540 mm

cSd

z xz

NA f

η

χ ω η 0 2,

cy Sd

LT xLT y Rd

MM

γ

χ ω.

. Web in compression Flange in compression

Totally 29 beams

Figure 5.13. Lateral-torsional buckling of unwelded beams, modified calculation.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

L=500 mmL=1020 mmL=1020 mmL=1540 mmL=1540 mm

cSd

z xz

NA f

η

χ ω η 0 2,

cy Sd

LT xLT y Rd

MM

γ

χ ω.

. Web in compression Flange in compression

Totally 47 beams

Figure 5.14. Lateral-torsional buckling of welded beams, modified calculation.

117

Comparison between modified Eurocode 9 and the theory of elastic beam-columns The modified Eurocode 9 calculations were compared with the theory of elastic beam-columns. This comparison is shown in figure 5.15. Only unwelded and pin-ended beam-columns with restrained warping at the ends were considered. In the modified Eurocode 9 calculations, 0 2,f was set equal to 260 MPa. This value was obtained from Eurocode 9, see table 2.6. The cross-section constants were calculated for the theoretical cross-section dimensions where also the radius between the flange and the web was considered. The calculations were made for E = 70 GPa and G = 27 GPa. The six load application points are shown in figure 5.15 as vertical dashed lines. The three buckling lengths are also shown. The load eccentricity e is directed as the co-ordinate y in figure 3.5. Elsewhere in this thesis, crN is applied at the centre of gravity, but here crN is applied anywhere along the y-axis in figure 3.5. In the modified Eurocode 9 calculations, flexural buckling determined the load carrying capacity for the shortest beam-columns with large (both positive and negative) load eccentricity. For the other beam-columns, lateral torsional buckling was decisive. The value of crN from the theory of elasticity is obtained from the formula below. It is based on the same assumptions as the formulas in section 3.5. The different notations in the formula below are explained in section 3.5. The major axis flexural buckling load always gave a higher value of crN than what was obtained from the formula below.

( )2 2 0scp

y cr Eyp

yp

T crp

T Eyy eIA t e N N

IA t e

IA N N

IA N N− − + + −

+

− =

Some conclusions can be drawn from figure 5.15. According to the theory of elasticity, the highest value of crN is obtained when the load is applied at the shear centre. In Eurocode 9, the highest value is instead obtained at the centre of gravity. In the modified Eurocode 9 calculation, M Rdy. of the edge in compression was 2,52 kNm. Corresponding value for the edge in tension was 2,56 kNm. The difference between these two bending moment capacities is small and this is the reason why the curves in figure 5.15 for the modified Eurocode 9 calculations almost are symmetric about the centre of gravity. The theory of elasticity always gives a higher value of crN than the modified Eurocode 9 calculations. The largest difference is obtained for the shortest beam-columns. The reason for this is the material. The longest beam-columns behave more elastic.

118

0

50

100

150

200

250

300

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

Theory of elasticityModified Eurocode 9

lc=1590 mm

lc=550 mm

lc=1070 mm

[ ]crN kN

[ ]e mm

II IIIIIVVVI

Figure 5.15. The load carrying capacity as function of the load eccentricity. Modification of the section check in Eurocode 9 What has been written so far in section 5.3 has to do with buckling in Eurocode 9, but the calculation in section 5.2 showed that also the interaction formula for section check needs to be improved. When the bending moment is large and causes tension in the web, the chosen interpretation of the interaction formula for section check in Eurocode 9 is unsafe. The interaction formula for section check, equation 5.5, was modified by setting

fWM xplRdy 2,0.. = . The result of this modification is shown in figure 5.16. This figure is equivalent to figure 5.5, except that it is based on the modified calculation of section check and that some curves, which were not considered as important in this section, have been removed. The modification of M Rdy. was the only modification, which was considered in the section check calculations. To set 1== AAefη will improve the result but the improvement will be small. Since the buckling check almost always determine the load carrying capacity, it will not be so very important if the section check is highly conservative. The shape of the interaction formula curve has the limitation that the co-ordinate of the vertical axis never will increase when the co-ordinate of the horizontal axis increases. The co-ordinate axes are those found in figure 5.16. This means that the rounded shape found in figure 5.16 for the plastic analysis curves cannot be obtained with the interaction formulas. This is one reason why the interaction formula for section check cannot give so excellent result.

119

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Section check

Abaqus

First plastic analysis

Stubby beams, plastic analysis

164

114

64

39

14

28

Nf A0 0 2ω ,

Mf W el x c0 0 2ω , . .

Second plastic analysis,Ramberg-Osgood

Second plastic analysis,"true" stress-log strain

Figure 5.16. Section check, modified calculation. Some comments The results in figures 5.11-5.14 and 5.16 showed that a tensile failure check was not needed. Despite that, the tensile failure discussion in section 5.1 and 5.2 was kept. Perhaps future researchers have plans to deal with tensile failure checks and then, the content of section 5.1 and 5.2 can be useful. If another interpretation of the Eurocode 9 calculations was chosen than the modified versions shown here, a tensile failure check could be needed. If for instance the interpretation in section 3.2 is chosen, a tensile failure check is needed in some cases. These are some motivations for keeping the tensile failure discussion in section 5.1 and 5.2. When it concerns buckling, it is also difficult to know if there are two phenomena involved, tensile failure and buckling, or if all failures are due to buckling. Naturally, it is most convenient to only consider buckling, because this will reduce the number of interaction formulas. Tensile failure could exist in the physical reality even though it is not needed when the load carrying capacity is calculated. The buckling and section checks could have been “manipulated” in such a way that the tensile failure check is included.

120

121

6 Conclusions, comments and further research A thesis should contain unique parts. The buckling study in this thesis dealing with T-section beam-columns in aluminium with both an axial compressive force and a bending moment present is most likely unique. When also Eurocode 9 is involved, the study will be even more unique. Everything that has to do with Eurocode 9 is new, since it was published in 1998. Research has been made on buckling of centrically compressed T-section columns in aluminium, but no research has been found where also a bending moment is present. Generally the buckling tests worked fine. However, there were some difficulties related to the welding. For the welded tensile tests, it is important to obtain a ductile failure at the heat-affected zone and not a brittle failure at the weld. This was not always so easy to obtain. There were also some uncertainties about the clamping effect of the beam-ends of the clamped beams. When it concerns unsymmetric cross-sections, the codes need to be improved. This applies both for the usage of the codes and the results that the codes give. The codes are not always so easy to follow and sometimes information is lacking how to perform the calculations. A reason for this can be that it is unknown how certain structural components should be calculated. There is also a question how general a code should be. It is hardly economical to make a code for constructions that are never made. Beam-columns with unsymmetric cross-sections are considered to be reasonable structural components. Some motivations for this are given in section 1.1. It is more important that aluminium codes are adjusted to unsymmetric cross-sections, because unsymmetric cross-sections are more common in aluminium than in steel. This has to do with the extrusion technique used for aluminium profiles. It is most important that the codes do not give unsafe results, but it is uneconomical if the results are too conservative. When the codes are applied on unsymmetric cross-sections, all information is not given by the codes and therefore, some interpretations have to be made. At least this is true when both a compressive axial force and a bending moment are present at the same time. When only an axial force is present, the codes are clearer. The literature search showed that quite much research has been made on centrically compressed columns with unsymmetric cross-sections. Therefore, the codes ought to be able to handle these structural components fairly well. The chosen interpretations of the codes in section 3.2-3.4 were very conservative for many of the tested beams. The Abaqus calculations showed that the chosen interpretation of Eurocode 9 was unsafe when the bending moment was large and caused tension in the web. To handle this situation a tensile failure check or modified versions of the buckling checks could be used. This is discussed in chapter 5. The Abaqus calculations worked fine, but some comments for likely improvements are given here. It would have been better if the width of the heat-affected zone was determined by hardness tests. A more accurate value would then have been obtained than the two guesses 30 and 50 mm. Hopefully the tested width of the heat-affected zone would not have been so widely scattered. A constant value of the width could then have been used. The magnitude of the imperfections could then have been the only parameter that was varied for both the unwelded and welded beams. Likely, a better result could have been obtained if the imperfections were a function of the beam length. By adjusting the imperfections further, for

122

instance by having different shape of the imperfections for the different load application points, the result could likely further have been improved. By making the imperfections unique for each beam-column and calibrate the imperfections so the tested and calculated load carrying capacity coincided, a very excellent result could have been obtained. However, this would be “cheating”. When a non-tested beam-column was calculated such an excellent result could not be obtained, because there would not be a tested beam-column to calibrate the imperfections with. Generally the shell element S4R was a satisfactory choice. However, if numerical problems occurred for S4R or if there was a suspicion that numerical problems could occur, the element S4 was likely a better choice. These were just some observations that were made when the Abaqus calculations were finished. The reason for writing these observations is that future researchers hopefully can have some use of them. The main reason for not doing hardness tests was lack of time. In section 5.3 different modifications of Eurocode 9 were analysed. It was found that a better result was obtained when f haz was used instead of f 2,0 when the buckling reduction factors

of a welded section were calculated. The calculation of M Rdy . in the interaction formulas was also modified. When the tip of the web was in tension, W xpl . was used to calculate M Rdy. .

When the bending moment acted in the opposite direction, the calculation of M Rdy . was based on a modified classification of the web element. The investigation in this thesis indicates that Eurocode 9 is too severe in the classification of the cross-section. Also other modifications were analysed, but they only had minor effects on the result. Tensile failure at the tip of the web was also discussed. An interaction formula for tensile failure was presented. The tensile failure interaction formula was not needed when the modified versions of the buckling checks were used. The investigation in this thesis gives some information about how the bending moment capacity in the interaction formulas should be calculated for unsymmetric cross-sections. Likely more research is needed in this area. It could be interesting to study buckling of beam-columns with other unsymmetric cross-sections than T-sections. If the cross-section dimensions were changed, also T-sections could be further analysed. There is a risk that the findings in this thesis do not give so accurate results when applied on other unsymmetric cross-sections. It is the combination of unsymmetric cross-sections, compressive axial force and bending moment that is interesting, since very little research seems to have been made in this area. The investigation in this thesis about the clamped beams is small. To further analyse the behaviour of clamped beams could be an area of future research. A reason for this research could be to evaluate the correctness of the statement in Eurocode 9 that the clamping effect of clamped beams should be neglected when transverse welds are located at the clamped ends.

123

7 References [1] ABAQUS (1997). ABAQUS/Standard User’s manual Volume I-III and ABAQUS Post manual. Version 5.7. Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, USA. [2] Baehre, R., (1966). “Tryckta strävor av elastoplastiskt material – några frågeställningar”. Väg- och vattenbyggaren nr 3 1966, (in Swedish). [3] Baehre, R., Riman, R., (1986). “Traglastuntersuchung von Druckstäben aus Aluminium mit Quernähten”. Schweißen und Schneiden 38, Heft 8, (in German). [4] Benson, P., (1990). “Local and Flexural Buckling of Eccentrically Loaded Square, Thin-Walled Aluminium Alloy Columns”. Royal Institute of Technology, Department of Structural Engineering, Stockholm, Sweden. [5] Bernard, A., Frey, F., Janss, J., Massonnet, Ch., (1971). “Research on the behaviour of aluminium columns against buckling”. University of Liège, Laboratories for testing materials and stability of constructions, Liège, Belgium. [6] Boverket, (1998). “BKR 99, Boverkets konstruktionsregler 99”, BFS 1993:58, ISBN 91-7147-455-2, (in Swedish). [7] Boverket, (1999). “BSK 99, Boverkets handbok om stålkonstruktioner”, ISBN 91-7147-527-3, (in Swedish). [8] Bradford, M. A., (1990). “Lateral-Distortional Buckling of Tee-Section Beams”. Thin-Walled Structures, Vol. 10, No. 1, pp. 13-30. [9] Bradford, M. A., (1999). “Elastic distortional buckling of tee-section cantilevers”. Thin-Walled Structures, Vol. 33, No. 1, pp. 3-17. [10] BSI-British Standards Institution, (1991). “BS 8118, Structural use of aluminium”, Part 1. Code of practice for design, 2 Park Street, London, W1A 2BS, ISBN 0-580-19209-1. [11] Bulson, P. S., Nethercot, D. A., (1986). “New British Code for the Design of Aluminium Structures”. Colloquium on thin-walled metal structures in buildings, IABSE, Stockholm, Sweden, pp. 43-50. [12] CEN, European Committee for Standardization, (1998). “Eurocode 9, Design of Aluminium Structures”, ENV 1999-1-1:1998 E, May 1998. [13] Chen, W.-F., Atsuta, T., (1977). “Theory of Beam-Columns”, Volume 2, Space behavior and design, McGraw-Hill, Inc., USA. ISBN 0-07-010759-9. [14] Clark, J. W., (1955). “Eccentrically loaded aluminium columns”. ASCE Trans., Vol. 120, pp. 1116-1132.

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[15] Corona, E., Ellison, M. S., (1997). “Plastic Buckling of T-Beams under Pure Bending”. Journal of Engineering Mechanics, Vol. 123, No. 5, pp. 466-474. [16] Crisfield, M. A., (1991). “Non-linear Finite Element Analysis of Solids and Structures”, Volume 1, Reprinted 1994, John Wiley & Sons Ltd., Baffins Lane, Chichester, West Sussex PO19 1UD, England, ISBN 0-471-92956-5. [17] Edlund, S., (1997). “Buckling Tests of Unwelded Aluminium T-sections”, Royal Institute of Technology, Department of Structural Engineering, Stockholm, Sweden, Technical Report 1997:3, Steel Structures. [18] Edlund, S., (1997). “Buckling tests of welded aluminium T-sections”, Royal Institute of Technology, Department of Structural Engineering, Stockholm, Sweden, Technical Report 1997:18, Steel Structures. [19] Edlund, S., (1997). “Arbitrary Thin-Walled Cross Sections. Theory and Computer Implementation”, Licentiate Thesis, Bulle tin 33, Royal Institute of Technology, Department of Structural Engineering, Stockholm, Sweden. [20] Edlund, S., (1998). “Buckling Tests of Aluminium T-profiles with Transverse Welds”. Nordic Steel Construction Conference 98, Norwegian Steel Association, ISBN 82-91466-02-5. [21] Frey, F., Mazzolani, F. M., (1977). “Buckling behaviour of aluminium-alloy extruded members”. International Colloquium on Stability of Steel Structures, Liège, pp. 85-94. [22] Galambos, T. V., (1988). “Guide to Stability Design Criteria for Metal Structures, Fourth edition”. John Wiley & Sons Inc., USA. ISBN 0-471-09737-3. [23] Gilson, S., Cescotto, S., (undated). “Experimental research on the buckling of alu- alloys columns with unsymmetrical cross-section.” Laboratoire de Matériaux et Théorie des Structures, Université de Liège, Belgium. [24] Hellgren, M., (1995). “Lateral-Torsional Buckling of Aluminium Beam-Columns with Transverse Butt-Welds”. Proceedings of the third international conference on steel and aluminium structures, Istanbul, Turkey, Bogazici university, Department of civil engineering. [25] Hernelind, K., Höglund, T., Nylander, H., (1974). “Steel columns in buildings”, National Swedish Building Research, Report R50:1974, ISBN 91-540-2362-9, (in Swedish). [26] Hill, H. N., Clark, J. W., (1951). “Lateral buckling of eccentrically loaded I-section columns”. ASCE Trans., Vol. 116, pp. 1179-1196. [27] Hill, H. N., Hartmann, E. C., Clark, J. W., (1956). “Design of aluminium alloy beam-columns”. ASCE Trans., Vol. 121, pp. 1-21.

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[28] Hong, G. M., (1987). “Aluminium column curves”. Aluminium structures, design and construction. R. Narayanan Ed., Elsevier Applied Science Publishers, pp. 40-49. [29] Hong, G. M., (1991). “Effects of non-central transverse weld on aluminium columns”. International Conference on Steel and Aluminium Structures, ICSAS, Singapore. [30] Hopperstad, O. S., (1993). “Modelling of cyclic plasticity with application to steel and aluminium structures.” The Norwegian Institute of Technology, Department of Structural Engineering, Trondheim, Norway. [31] Höglund, T., (1968). “Interaction Design Method for Beam-Columns”. Royal Institute of Technology, Department of Structural Engineering, Stockholm, Sweden, Bulletin no. 77, (in Swedish). [32] Höglund, T., (1990). “Bending and Compression of Aluminium Members”. Royal Institute of Technology, Department of Structural Engineering, Stockholm, Sweden. [33] Höglund, T., (1991). “Flexural and lateral-torsional buckling of aluminium and steel beam-columns”. International Conference on Steel and Aluminium Structures, ICSAS, Singapore. [34] Kitipornchai, S., Wang, C. M., (1986). “Lateral buckling of tee beams under moment gradient”. Computers & Structures, Vol. 23, No. 1, pp. 69-76. [35] Klöppel, K., Bärsch, W., (1971).”Ein Beitrag zur Bemessung von Druckstäben aus Aluminium”. ALUMINIUM, No. 2, pp. 146-153, (in German). [36] Klöppel, K., Bärsch, W., (1973).”Versuche zum Kapitel ,,Stabilitätsfälle“ der Neufassung von DIN 4113”. ALUMINIUM, No. 10, pp. 690-699, (in German). [37] Lai, Y. F. W., Nethercot, D. A., (1992). “Strength of aluminium members containing local transverse welds”, Engineering Structures, Vol. 14, No. 4, pp. 241-254. [38] Lai, Y. F. W., Nethercot, D. A., (1992). “Design of aluminium columns”, Engineering

Structures, Vol. 14, No. 3, pp. 188-194. [39] Langhelle, N. K., (1999). “Experimental Validation and Calibration of Nonlinear Finite Element Models for Use in Design of Aluminium Structures Exposed to Fire”. Norwegian University of Science and Technology, Department of Marine Structures, Faculty of Marine Technology, N-7034 Trondheim, Norway. ISBN 82-471-0376-1. [40] Lundberg, S., (1995). “Design Philosophy”, TALAT Lecture 2204, Training in Aluminium Application Technologies, ATP−Aluminium Training Partnership, Brussels. [41] Marsh, C., (1991). “Unified treatment of buckling proposed for the ISO standard on aluminium structures”. International Conference on Steel and Aluminium Structures, ICSAS, Singapore.

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[42] Mazzolani, F. M., (1995). “Aluminium Alloy Structures”, Second edition, E & FN Spon, an imprint of Chapman & Hall, 2-6 Boundary Row, London SE1 8HN, UK, ISBN 0-419-17770-1.

[43] Moen, L. A., (1999). “Rotational capacity of aluminium alloy beams”, Norwegian University of Science and Technology, Department of Structural Engineering, N-7034 Trondheim, Norway. ISBN 82-471-0365-6. [44] Nethercot, D. A., (1987). “Aspects of column design in the new UK structural aluminium code”. Aluminium structures, design and construction. R. Narayanan Ed., Elsevier Applied Science Publishers, pp. 50-59. [45] Nethercot, D. A., (1991). “Behaviour and design of aluminium beams, columns and beam columns”. International Conference on Steel and Aluminium Structures, ICSAS, Singapore. [46] Petersen, C., (1994). “Stahlbau”, Grundlagen der Berechnung und baulichen Ausbildung von Stahlbauten, 3. überarbeitete und erweiterte Auflage, Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, Germany. ISBN 3-528-28837-X. (in German). [47] Rasmussen, K. J. R., Rondal, J., (1999). “Column Curve Formulation for Aluminium Alloys”. Fourth International Conference on Steel and Aluminium Structures, Espoo, Finland. Helsinki University of Technology, ISBN 0-08-043014-7. [48] Runesson, K., Samuelsson, A., Wiberg, N.-E., (1989). “Knäckning”, Chalmers University of Technology, Department of Structural Mechanics, Gothenburg, Sweden, (in Swedish). [49] Sanne, L., Benson, P., Höglund, T., (1992). “Lateral Torsional Buckling of Aluminium Beam Columns – Test and Design”. The Royal Institute of Technology, Department of Structural Engineering, Stockholm, Sweden, Report 1992:6. [50] Sanne, L., (1993). “Lateral Torsional Buckling of Aluminium Beam-Columns with Transverse Butt Welds.” The Royal Institute of Technology, Department of Structural Engineering, Stockholm, Sweden. [51] SAPA, (1995). “Handbok för konstruktörer, Hur man lyckas med aluminiumprofiler”, (in Swedish). [52] Sharp, M. L., (1993). “Behavior and Design of Aluminum Structures”, McGraw-Hill, Inc., ISBN 0-07-056478-7. [53] Statens Stålbyggnadskomitté, (1973). “StBK-K2, Kommentarer till stålbyggnadsnorm 70, knäckning, vippning, buckling”, AB Svensk Byggtjänst, (in Swedish). [54] Svetskommissionen, IVA, (1997). “Goda råd vid aluminiumsvetsning”, ISBN 91-630-5056-0, (in Swedish).

127

[55] Timoshenko, S. P., Gere, J. M., (1961). “Theory of elastic stability”, Second edition. McGraw-Hill Book Company, Inc., ISBN 0-07-085821-7. [56] Valtinat, G., Müller, R., (1977). “Ultimate load of beam columns in aluminium alloys with longitudinal and transversal welds”. Second Int. Coll. Stability, Prelim. Rep., Liège, pp. 393-402. [57] Valtinat, G., Dangelmaier, P., (1986). “Näherungsweise Berechnung der Traglasten von nichtgeschweißten und geschweißten Druckstäben aus Aluminium”, Bauingenieur, Vol. 61, pp. 507-513, (in German).

128

129

Appendix A. Photos of the test equipment.

Figure A.1. Test equipment.

Figure A.2. Detail of the test equipment.

130

Figure A.3. View of the test equipment.

Figure A.4. Detail of the test equipment.

Figure A.5. Detail of the test equipment.

131

Appendix B. Load-deflection curves of the tested beam-columns.

This appendix shows the load-deflection curves for all tested beams. For the beams tested in1996 and 1997, a horizontal deflection is parallel with the flange and a vertical deflection isaccordingly parallel with the web. The horizontal and vertical deflections for the beams from1998 are reversed. The deflection measurements are further described in accordance to figure2.3. At the end of the testing, the attachment of the measuring device could sometimes getloose from the beam. This is the reason why the load-deflection curves sometimes have an oddshape. Lap stands for “load application point”, see figure 2.1. The beam length is denoted L.The formulas for first and second order analysis are found in section 2.3.

Unwelded beams from 1996

PB-1, L = 500 mm, Lap = II

0

20

40

60

80

100

120

0 2 4 6 8 10 12Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-2, L = 500 mm, Lap = V

0

10

20

30

40

50

60

70

0 2 4 6 8 10 12 14 16Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-3, L = 500 mm, Lap = IV

0102030405060708090

100

0 2 4 6 8 10 12 14 16Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-1, L = 500 mm, Lap = II

0

20

40

60

80

100

120

0 4 8 12 16 20 24Horizontal midspan deflection [mm]

FlangeWeb

PB-2, L = 500 mm, Lap = V

0

10

20

30

40

50

60

70

-5 0 5 10 15 20 25 30 35Horizontal midspan deflection [mm]

FlangeWeb

PB-3, L = 500 mm, Lap = IV

0102030405060708090

100

-5 -4 -3 -2 -1 0 1Horizontal midspan deflection [mm]

FlangeWeb

132

PB-4, L = 500 mm, Lap = V

0

10

20

30

40

50

60

70

0 2 4 6 8 10 12 14 16 18Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-19, L = 500 mm, Lap = II

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-20, L = 500 mm, Lap = V

0

10

20

30

40

50

60

70

0 2 4 6 8 10 12 14 16 18 20

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-21, L = 500 mm, Lap = III

020

4060

80100

120140

160

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

Vertical midspan deflection [mm]

Test

The failure load was not reached

PB-4, L = 500 mm, Lap = V

0

10

20

30

40

50

60

70

-1 0 1 2 3 4 5Horizontal midspan deflection [mm]

FlangeWeb

PB-19, L = 500 mm, Lap = II

0

20

40

60

80

100

120

0 2 4 6 8 10 12Horizontal midspan deflection [mm]

FlangeWeb

PB-20, L = 500 mm, Lap = V

0

10

20

30

40

50

60

70

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

Horizontal midspan deflection [mm]

FlangeWeb

PB-21, L = 500 mm, Lap = III

020

40

60

80

100

120

140

160

-0.2 0 0.2 0.4 0.6 0.8 1Horizontal midspan deflection [mm]

FlangeWeb

The failure load was not reached

133

BP-22, L = 500 mm, Lap = I

0

1020

30

40

50

60

70

80

0 1 2 3 4 5 6 7 8 9 10Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

Slip occurred

PB-5, L = 1020 mm, Lap = II

0

10

20

30

40

50

6070

80

0 5 10 15 20 25Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-6, L = 1020 mm, Lap = V

05

1015202530354045

0 5 10 15 20 25 30 35 40 45 50

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-7, L = 1020 mm, Lap = II

0

1020

30

4050

60

7080

0 2 4 6 8 10 12 14 16 18

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-22, L = 500 mm, Lap = I

010

20

30

4050

60

70

80

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Horizontal midspan deflection [mm]

FlangeWeb

Slip occurred

PB-5, L = 1020 mm, Lap = II

0

1020

3040

5060

7080

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

Horizontal midspan deflection [mm]

FlangeWeb

PB-6, L = 1020 mm, Lap = V

05

1015202530354045

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Horizontal midspan deflection [mm]

FlangeWeb

PB-7, L = 1020 mm, Lap = II

0

1020

30

40

50

6070

80

-0.5 0 0.5 1 1.5 2 2.5 3

Horizontal midspan deflection [mm]

FlangeWeb

134

PB-8, L = 1020 mm, Lap = IV

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-9, L = 1020 mm, Lap = III

0

10

20

30

40

50

60

70

-1 0 1 2 3 4 5 6

Vertical midspan deflection [mm]

Test

PB-10, L = 1020 mm, Lap = IV

0

10

20

30

40

50

60

70

0 5 10 15

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-11, L = 1020 mm, Lap = I

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-8, L = 1020 mm, Lap = IV

0

10

20

30

40

50

60

-2 0 2 4 6 8 10 12 14 16

Horizontal midspan deflection [mm]

FlangeWeb

PB-9, L = 1020 mm, Lap = III

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40 45 50 55

Horizontal midspan deflection [mm]

FlangeWeb

PB-10, L = 1020 mm, Lap = IV

0

10

20

30

40

50

60

70

-5 0 5 10 15 20 25 30 35

Horizontal midspan deflection [mm]

FlangeWeb

PB-11, L = 1020 mm, Lap = I

0

10

20

30

40

50

60

-0.5 0 0.5 1 1.5 2Horizontal midspan deflection [mm]

FlangeWeb

135

PB-12, L = 1020 mm, Lap = IV

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14 16 18 20Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-23, L = 1020 mm, Lap = III

0

10

20

30

40

50

60

70

-0.5 0 0.5 1 1.5 2 2.5

Vertical midspan deflection [mm]

Test

PB-24, L = 1020 mm, Lap = V

0

10

20

30

40

50

0 5 10 15 20 25 30 35 40 45

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-13, L = 1540 mm, Lap = II

0

10

20

30

40

0 4 8 12 16 20 24Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-12, L = 1020 mm, Lap = IV

0

10

20

30

40

50

60

-2 0 2 4 6 8 10 12 14 16

Horizontal midspan deflection [mm]

FlangeWeb

PB-23, L = 1020 mm, Lap = III

0

10

20

30

40

50

60

70

-5 0 5 10 15 20 25 30 35 40

Horizontal midspan deflection [mm]

FlangeWeb

BP-24, L = 1020 mm, Lap = V

0

10

20

30

40

50

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Horizontal midspan deflection [mm]

FlangeWeb

PB-13, L = 1540 mm, Lap = II

0

5

10

15

20

25

30

35

-5 0 5 10 15 20 25 30

Horizontal midspan deflection [mm]

FlangeWeb

136

PB-14, L = 1540 mm, Lap = V

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-15, L = 1540 mm, Lap = II

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16 18 20 22

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-16, L = 1540 mm, Lap = III

0

5

10

15

20

25

30

35

-0.5 0 0.5 1 1.5 2 2.5 3 3.5

Vertical midspan deflection [mm]

Test

PB-17, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35 40Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-14, L = 1540 mm, Lap = V

0

5

10

15

20

25

30

-10 0 10 20 30 40 50 60 70 80

Horizontal midspan deflection [mm]

FlangeWeb

PB-15, L = 1540 mm, Lap = II

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35 40

Horizontal midspan deflection [mm]

FlangeWeb

PB-16, L = 1540 mm, Lap = III

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35 40 45

Horizontal midspan deflection [mm]

FlangeWeb

PB-17, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

35

-1 0 1 2 3 4 5 6 7 8

Horizontal midspan deflection [mm]

FlangeWeb

137

PB-18, L = 1540 mm, Lap = V

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-25, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35 40

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-26, L = 1540 mm, Lap = III

0

5

10

15

20

25

30

35

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Vertical midspan deflection [mm]

Test

Welded beams from 1997

PB-3, L = 500 mm, Lap = II

0102030405060708090

0 1 2 3 4 5 6 7 8 9Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-18, L = 1540 mm, Lap = V

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40 45 50

Horizontal midspan deflection [mm]

FlangeWeb

PB-25, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

35

-4 0 4 8 12 16 20 24 28 32Horizontal midspan deflection [mm]

FlangeWeb

PB-26, L = 1540 mm, Lap = III

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30Horizontal midspan deflection [mm]

FlangeWeb

PB-3, L = 500 mm, Lap = II

0102030405060708090

-2 0 2 4 6 8 10 12 14 16 18 20 22Horizontal midspan deflection [mm]

FlangeWeb

138

PB-4, L = 500 mm, Lap = V

05

101520253035404550

0 2 4 6 8 10 12 14 16 18 20 22Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-7, L = 1020 mm, Lap = I

05

1015202530354045

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-8, L = 1020 mm, Lap = V

05

10

15

20

25

30

3540

-5 0 5 10 15 20 25 30 35 40 45Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-9, L = 1020 mm, Lap = II

0

10

20

30

40

50

60

0 2 4 6 8 10 12Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-4, L = 500 mm, Lap = V

05

101520253035404550

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Horizontal midspan deflection [mm]

FlangeWeb

PB-7, L = 1020 mm, Lap = I

05

1015202530354045

-5 0 5 10 15 20 25 30 35Horizontal midspan deflection [mm]

FlangeWeb

PB-8, L = 1020 mm, Lap = V

0

5

10

15

2025

30

35

40

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Horizontal midspan deflection [mm]

FlangeWeb

PB-9, L = 1020 mm, Lap = II

0

10

20

30

40

50

60

-4 0 4 8 12 16 20 24 28 32Horizontal midspan deflection [mm]

FlangeWeb

139

PB-10, L = 1020 mm, Lap = IV

0

10

20

30

40

50

0 5 10 15 20 25 30 35 40Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-11, L = 1020 mm, Lap = I

0

10

20

30

40

50

0 5 10 15 20 25Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-12, L = 1020 mm, Lap = V

0

510

15

20

25

30

35

40

0 5 10 15 20 25 30 35

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-13, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-10, L = 1020 mm, Lap = IV

05

101520253035404550

0 1 2 3 4 5 6

Horizontal midspan deflection [mm]

FlangeWeb

PB-11, L = 1020 mm, Lap = I

0

10

20

30

40

50

-5 0 5 10 15 20 25 30 35Horizontal midspan deflection [mm]

FlangeWeb

PB-12, L = 1020 mm, Lap = V

0

5

10

15

20

25

30

35

40

-0.5 0 0.5 1 1.5 2 2.5 3 3.5

Horizontal midspan deflection [mm]

FlangeWeb

PB-13, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

-5 5 15 25 35 45 55

Horizontal midspan deflection [mm]

FlangeWeb

140

PB-14, L = 1540 mm, Lap = IV

0

5

10

15

20

25

30

-5 0 5 10 15 20 25 30

Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-15, L = 1540 mm, Lap = II

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12 14 16Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-16, L = 1540 mm, Lap = V

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-17, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

PB-14, L = 1540 mm, Lap = IV

0

5

10

15

20

25

30

-5 5 15 25 35 45 55Horizontal midspan deflection [mm]

FlangeWeb

PB-15, L = 1540 mm, Lap = II

0

5

10

15

20

25

30

35

-5 5 15 25 35 45Horizontal midspan deflection [mm]

FlangeWeb

PB-16, L = 1540 mm, Lap = V

0

5

10

15

20

25

-5 5 15 25 35 45 55Horizontal midspan deflection [mm]

FlangeWeb

PB-17, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

-5 0 5 10 15 20 25 30 35 40 45 50Horizontal midspan deflection [mm]

FlangeWeb

141

PB-18, L = 1540 mm, Lap = V

0

5

10

15

20

25

30

0 10 20 30 40 50Vertical midspan deflection [mm]

TestFirst order analysisSecond order analysis

Welded beams from 1998

PB-1, L = 500 mm, Lap = I

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5Vertical midspan deflection [mm]

Test

PB-2, L = 500 mm, Lap = IV

0

10

20

30

40

50

60

70

0 0.1 0.2 0.3 0.4 0.5 0.6Vertical midspan deflection [mm]

Test

PB-5, L = 500 mm, Lap = I

0

10

20

30

40

50

60

70

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Vertical midspan deflection [mm]

Test

PB-18, L = 1540 mm, Lap = V

0

5

10

15

20

25

30

-5 0 5 10 15 20 25 30 35 40 45 50Horizontal midspan deflection [mm]

FlangeWeb

PB-1, L = 500 mm, Lap = I

0

10

20

30

40

50

60

70

0 2 4 6 8 10 12 14Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-2, L = 500 mm, Lap = IV

0

10

20

30

40

50

60

70

0 2 4 6 8 10 12 14 16 18Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-5, L = 500 mm, Lap = I

0

10

20

30

40

50

60

70

0 2 4 6 8 10 12 14Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

142

PB-6, L = 500 mm, Lap = V

0

10

20

30

40

50

60

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Vertical midspan deflection [mm]

Test

PB-19, L = 500 mm, Lap = II

0

20

40

60

80

100

-5 0 5 10 15 20 25 30Vertical midspan deflection [mm]

Test

PB-20, L = 500 mm, Lap = V

0102030405060708090

0 2 4 6 8 10 12 14Vertical midspan deflection [mm]

Test

PB-21, L = 500 mm, Lap = V

0

10

20

30

40

50

60

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Vertical midspan deflection [mm]

Test

PB-6, L = 500 mm, Lap = V

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14 16Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-19, L = 500 mm, Lap = II

0

20

40

60

80

100

-2 0 2 4 6 8 10 12 14Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

BP-20, L = 500 mm, Lap = V

0102030405060708090

0 1 2 3 4 5 6 7 8 9Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-21, L = 500 mm, Lap = V

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

143

PB-22, L = 500 mm, Lap = VI

05

10152025303540

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Vertical midspan deflection [mm]

Test

PB-23, L = 500 mm, Lap = III

0

20

40

60

80

100

120

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Vertical midspan deflection [mm]

Test

PB-24, L = 500 mm, Lap = I

0

10

20

30

40

50

60

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3Vertical midspan deflection [mm]

Test

PB-25, L = 500 mm, Lap = VI

05

10152025303540

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3Vertical midspan deflection [mm]

Test

PB-22, L = 500 mm, Lap = VI

05

10152025303540

0 2 4 6 8 10 12 14 16Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-23, L = 500 mm, Lap = III

0

20

40

60

80

100

120

-2 0 2 4 6 8 10 12 14 16 18Horizontal midspan deflection [mm]

Upper edgeLower edge

PB-24, L = 500 mm, Lap = I

0

10

20

30

40

50

60

0 2 4 6 8 10Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-25, L = 500 mm, Lap = VI

05

10152025303540

0 2 4 6 8 10 12 14Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

144

PB-26, L = 1020 mm, Lap = III

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35Vertical midspan deflection [mm]

Test

PB-27, L = 1020 mm, Lap = VI

0

5

10

15

20

25

30

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Vertical midspan deflection [mm]

Test

PB-28, L = 1020 mm, Lap = I

05

1015202530354045

-2 0 2 4 6 8 10 12 14 16Vertical midspan deflection [mm]

Test

PB-29, L = 1020 mm, Lap = III

0

20

40

60

80

100

120

-5 0 5 10 15 20 25Vertical midspan deflection [mm]

Test

Clamped beam

PB-26, L = 1020 mm, Lap = III

0

10

20

30

40

50

60

70

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5Horizontal midspan deflection [mm]

Upper edgeLower edge

PB-27, L = 1020 mm, Lap = VI

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-28, L = 1020 mm, Lap = I

05

1015202530354045

0 5 10 15 20 25 30Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-29, L = 1020 mm, Lap = III

0

20

40

60

80

100

120

-6 -4 -2 0 2 4Horizontal midspan deflection [mm]

Upper edgeLower edgeClamped beam

145

PB-30, L = 1020 mm, Lap = III

0

20

40

60

80

100

-5 0 5 10 15 20 25 30Vertical midspan deflection[mm]

Test

Clamped beam

PB-31, L = 1020 mm, Lap = VI

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5 3Vertical midspan deflection [mm]

Test

PB-32, L = 1020 mm, Lap = III

0

20

40

60

80

100

120

140

-5 0 5 10 15 20 25 30 35 40

Vertical midspan deflection [mm]

Test

Clamped beam

PB-33, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40 45Vertical midspan deflection [mm]

Test

PB-30, L = 1020 mm, Lap = III

0

20

40

60

80

100

-1 0 1 2 3 4 5Horizontal midspan deflection [mm]

Upper edgeLower edge

Clamped beam

PB-31, L = 1020 mm, Lap = VI

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-32, L = 1020 mm, Lap = III

0

20

40

60

80

100

120

140

0 1 2 3 4 5 6 7 8 9Horizontal midspan deflection [mm]

Upper edgeLower edge

Clamped beam

PB-33, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

146

PB-34, L = 1540 mm, Lap = VI

0

5

10

15

20

25

0 1 2 3 4 5 6 7 8Vertical midspan deflection [mm]

Test

PB-35, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40 45Vertical midspan deflection [mm]

Test

PB-36, L = 1540 mm, Lap = III

0

5

10

15

20

25

30

0 10 20 30 40Vertical midspan deflection [mm]

Test

PB-37, L = 1540 mm, Lap = III

01020304050607080

-5 0 5 10 15 20 25 30 35Vertical midspan deflection [mm]

Test

Clamped beam

PB-34, L = 1540 mm, Lap = VI

0

5

10

15

20

25

0 10 20 30 40 50 60Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-35, L = 1540 mm, Lap = I

0

5

10

15

20

25

30

0 5 10 15 20 25 30Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-36, L = 1540 mm, Lap = III

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5 3 3.5Horizontal midspan deflection [mm]

Upper edgeLower edge

PB-37, L = 1540 mm, Lap = III

01020304050607080

0 1 2 3 4 5 6Horizontal midspan deflection [mm]

Upper edgeLower edge

Clamped beam

147

PB-38, L = 1540 mm, Lap = VI

0

5

10

15

20

25

0 10 20 30 40Vertical midspan deflection [mm]

Test

PB-39, L = 1540 mm, Lap = III

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70Vertical midspan deflection [mm]

Test

Clamped beam

Slip occurred

PB-38, L = 1540 mm, Lap = VI

0

5

10

15

20

25

0 10 20 30 40 50 60 70Horizontal midspan deflection [mm]

Upper edgeLower edgeFirst order analysisSecond order analysis

PB-39, L = 1540 mm, Lap = III

0

10

20

30

40

50

60

-2 0 2 4 6 8 10Horizontal midspan deflection [mm]

Upper edgeLower edge

Clamped beam

Slip occurred

148

149

Appendix C. Example of Abaqus input files. The first example shows the content of the two input files for the welded beam PB-35 with 50 mm heat-affected zone. The first and largest part is the same for the two input files, so this part is only shown once. The two input files were used to calculate the imperfections and the load carrying capacity, respectively. The second example shows some parts of the input files for the clamped beam PB-37 with 50 mm heat-affected zone. The third example shows some parts of the input files used when the bending moment capacity was calculated. *HEADING PB-35 50 mm haz mat 3, loadl=1, zhaz=408, len=1539, b=60.05, h=60.13, tw=6.11, tf=6.18, haz=50, (lmid=749.500000) *NODE 1, 57.04, 0., 0. 17, 0., 0., 0. 25, 0., -30.025, 0. 89, 0., 30.025, 0. 9001, 57.04, 0., 1499 9017, 0., 0., 1499 9025, 0., -30.025, 1499 9089, 0., 30.025, 1499 2001, 57.04, 0., 338 2017, 0., 0., 338 2025, 0., -30.025, 338 2089, 0., 30.025, 338 2301, 57.04, 0., 388 2317, 0., 0., 388 2325, 0., -30.025, 388 2389, 0., 30.025, 388 19001,-14.25,0,-45 19002,0,0,-45 19003,14.26,0,-45 19004,28.52,0,-45 19005,42.78,0,-45 19006,57.04,0,-45 19007,0,-15.0125,-45 19008,0,-30.025,-45 19009,0,15.0125,-45 19010,0,30.025,-45 19101,-14.25,0,1544 19102,0,0,1544 19103,14.26,0,1544 19104,28.52,0,1544 19105,42.78,0,1544 19106,57.04,0,1544 19107,0,-15.0125,1544 19108,0,-30.025,1544 19109,0,15.0125,1544 19110,0,30.025,1544 *NGEN,NSET=LWEB 1,17 *NGEN,NSET=LFLANGE1 17,25 *NGEN,NSET=LFLANGE2

17,89,9 *NGEN,NSET=RWEB 9001,9017 *NGEN,NSET=RFLANGE1 9017,9025 *NGEN,NSET=RFLANGE2 9017,9089,9 *NGEN,N SET=LWWEB 2001,2017 *NGEN,NSET=LWFLANS1 2017,2025 *NGEN,NSET=LWFLANS2 2017,2089,9 *NGEN,NSET=RWWEB 2301,2317 *NGEN,NSET=RWFLANS1 2317,2325 *NGEN,NSET=RWFLANS2 2317,2389,9 *ELEMENT,TYPE=RB3D2 5001,1,2 5017,17,18 5025,17,26 5033,19006,1 5034,19006,2 5035,19006,3 5036,19005,3 5037,19005,4 5038,19005,5 5039,19005,6 5040,19005,7 5041,19004,7 5042,19004,8 5043,19004,9 5044,19004,10 5045,19004,11 5046,19003,11 5047,19003,12 5048,19003,13 5049,19003,14 5050,19003,15 5051,19002,15 5052,19002,16 5053,19002,17 5054,19001,17 5055,19002,17 5056,19002,18

150

5057,19002,19 5058,19007,19 5059,19007,20 5060,19007,21 5061,19007,22 5062,19007,23 5063,19008,23 5064,19008,24 5065,19008,25 5066,19002,17 5067,19002,26 5068,19002,35 5069,19009,35 5070,19009,44 5071,19009,53 5072,19009,62 5073,19009,71 5074,19010,71 5075,19010,80 5076,19010,89 5077,19006,19005 5078,19005,19004 5079,19004,19003 5080,19003,19002 5081,19002,19001 5082,19002,19007 5083,19007,19008 5084,19002,19009 5085,19009,19010 5086,1,25 5087,1,89 5088,19006,19008 5089,19006,19010 5090,1,19008 5091,1,19010 5092,19001,25 5093,19001,89 5094,19001,19008 5095,19001,19010 *ELGEN 5001,16,1,1,1,1,1,1 5017, 8,1,1,1,1,1,1 5025, 8,9,1,1,1,1,1 *ELSET,ELSET=RIGIDLFT,GENERATE 5001,5095,1 *ELEMENT,TYPE=RB3D2 5101,9001,9002 5117,9017,9018 5125,9017,9026 5133,19106,9001 5134,19106,9002 5135,19106,9003 5136,19105,9003 5137,19105,9004 5138,19105,9005 5139,19105,9006 5140,19105,9007 5141,19104,9007 5142,19104,9008 5143,19104,9009

5144,19104,9010 5145,19104,9011 5146,19103,9011 5147,19103,9012 5148,19103,9013 5149,19103,9014 5150,19103,9015 5151,19102,9015 5152,19102,9016 5153,19102,9017 5154,19101,9017 5155,19102,9017 5156,19102,9018 5157,19102,9019 5158,19107,9019 5159,19107,9020 5160,19107,9021 5161,19107,9022 5162,19107,9023 5163,19108,9023 5164,19108,9024 5165,19108,9025 5166,19102,9017 5167,19102,9026 5168,19102,9035 5169,19109,9035 5170,19109,9044 5171,19109,9053 5172,19109,9062 5173,19109,9071 5174,19110,9071 5175,19110,9080 5176,19110,9089 5177,19106,19105 5178,19105,19104 5179,19104,19103 5180,19103,19102 5181,19102,19101 5182,19102,19107 5183,19107,19108 5184,19102,19109 5185,19109,19110 5186,9001,9025 5187,9001,9089 5188,19106,19108 5189,19106,19110 5190,9001,19108 5191,9001,19110 5192,19101,9025 5193,19101,9089 5194,19101,19108 5195,19101,19110 *ELGEN 5101,16,1,1,1,1,1,1 5117, 8,1,1,1,1,1,1 5125, 8,9,1,1,1,1,1 *ELSET,ELSET=RIGIDRGT,GENERATE 5101,5195,1 *RIGID BODY,ELSET=RIGIDLFT, REF NODE=19001

151

*RIGID BODY,ELSET=RIGIDRGT, REF NODE=19101 *NFILL,NSET=WEB1 LWEB,LWWEB,20,100 *NFILL,NSET=WEB2 LWWEB,RWWEB,3,100 *NFILL,NSET=WEB3 RWWEB,RWEB,67,100 *NFILL,NSET=FLANGE11 LFLANGE1,LWFLANS1,20,100 *NFILL,NSET=FLANGE12 LWFLANS1,RWFLANS1,3,100 *NFILL,NSET=FLANGE13 RWFLANS1,RFLANGE1,67,100 *NFILL,NSET=FLANGE21 LFLANGE2,LWFLANS2,20,100 *NFILL,NSET=FLANGE22 LWFLANS2,RWFLANS2,3,100 *NFILL,NSET=FLANGE23 RWFLANS2,RFLANGE2,67,100 *ELEMENT,TYPE=S4R 1,1,101,102,2 *ELGEN 1,90,100,1,16,1,90 *ELEMENT,TYPE=S4R 1441,17,117,118,18 *ELGEN 1441,90,100,1,8,1,90 *ELEMENT,TYPE=S4R 2161,17,117,126,26 *ELGEN 2161,90,100,1,8,9,90 *ELSET,ELSET=GRUNDW,GENERATE 1,20,1 24,90,1 91,110,1 114,180,1 181,200,1 204,270,1 271,290,1 294,360,1 361,380,1 384,450,1 451,470,1 474,540,1 541,560,1 564,630,1 631,650,1 654,720,1 721,740,1 744,810,1 811,830,1 834,900,1 901,920,1 924,990,1 991,1010,1 1014,1080,1 1081,1100,1 1104,1170,1 1171,1190,1

1194,1260,1 1261,1280,1 1284,1350,1 1351,1370,1 1374,1440,1 *ELSET,ELSET=GRUNDF,GENERATE 1441,1460,1 1464,1530,1 1531,1550,1 1554,1620,1 1621,1640,1 1644,1710,1 1711,1730,1 1734,1800,1 1801,1820,1 1824,1890,1 1891,1910,1 1914,1980,1 1981,2000,1 2004,2070,1 2071,2090,1 2094,2160,1 2161,2180,1 2184,2250,1 2251,2270,1 2274,2340,1 2341,2360,1 2364,2430,1 2431,2450,1 2454,2520,1 2521,2540,1 2544,2610,1 2611,2630,1 2634,2700,1 2701,2720,1 2724,2790,1 2791,2810,1 2814,2880,1 *ELSET,ELSET=WELDW,GENERATE 21,23,1 111,113,1 201,203,1 291,293,1 381,383,1 471,473,1 561,563,1 651,653,1 741,743,1 831,833,1 921,923,1 1011,1013,1 1101,1103,1 1191,1193,1 1281,1283,1 1371,1373,1 *ELSET,ELSET=WELDF,GENERATE 1461,1463,1 1551,1553,1 1641,1643,1 1731,1733,1

152

1821,1823,1 1911,1913,1 2001,2003,1 2091,2093,1 2181,2183,1 2271,2273,1 2361,2363,1 2451,2453,1 2541,2543,1 2631,2633,1 2721,2723,1 2811,2813,1 *SHELL SECTION,ELSET=GRUNDW, MATERIAL=ALUMIN 6.11 *SHELL SECTION,ELSET=GRUNDF, MAT ERIAL=ALUMIN 6.18 *SHELL SECTION,ELSET=WELDW, MATERIAL=SVETS 6.11 *SHELL SECTION,ELSET=WELDF, MATERIAL=SVETS 6.18 *MATERIAL,NAME=ALUMIN *ELASTIC 69343.5, 0.3 *PLASTIC 167.284, 0.0 210.598, 2.7873e-005 258.65, 0.000144117 267.677, 0.000263839 274.804, 0.000498352 280.236, 0.000919319 283.938, 0.00165304 286.243, 0.00272998 289.273, 0.00565603 300.376, 0.0188048 313.431, 0.0347786 323.691, 0.0512451 327.005, 0.0577758 *MATERIAL,NAME=SVETS *ELASTIC 69521, 0.3 *PLASTIC 67.8916, 0.0 84.9616, 0.000195629 90.7608, 0.000600164 96.2182, 0.0013416 101.343, 0.00232524 105.904, 0.00348518 110.264, 0.00476165 116.551, 0.00681665 121.429, 0.0088515 130.342, 0.0126703 138.263, 0.0166692 145.496, 0.0208451 152.021, 0.0250975 157.896, 0.0294807 168.315, 0.038412

184.316, 0.0573695 195.216, 0.0763269 202.069, 0.0951547 212.475, 0.156944 *BOUNDARY 19101,1,3 19101,6 19001,1,2 19001,6 *NSET,NSET=NPRINT 19001,19101,4517 *ELSET,ELSET=EPRINT 45,46 *RESTART,WRITE,FREQUENCY=1 *************************** *************************** ** **This part was used when the **imperfections were calculated ** *STEP *BUCKLE 1 *CLOAD 19001,3,100. *EL PRINT,ELSET=EPRINT S,E *NODE PRINT,NSET=NPRINT U RF *NODE FILE,GLOBAL=YES,MODE=1 U *END STEP *************************** *************************** ** **This part was used when the **load carrying capacity was calculated ** *IMPERFECTION,FILE=pb3550a,STEP=1 1,1.0 *STEP,NLGEOM,INC=25 *STATIC,RIKS 0.07,1.0,0.0,1.0,1000.,19001,3,20. *BOUNDARY,TYPE=DISPLACEMENT 19001,3,3,1.0 *EL PRINT,ELSET=EPRINT S,E *NODE PRINT,NSET =NPRINT U RF *END STEP ************************************* ************************************* *************************************

153

Second example, unique parts of the input files of a clamped beam. The parts are taken from PB-37 with 50 mm heat-affected zone. Extra nodes used by the rigid connections at the beam-ends. *NODE 17017,0,0,1538.1 17025,0,-30,1538.1 17089,0,30,1538.1 18017,0,0,-0.1 18025,0,-30,-0.1 18089,0,30,-0.1 *NGEN 17017,17025 17017,17089,9 18017,18025 18017,18089,9 Truss elements between the beam flanges and the rigid connections at the beam-ends. *ELEMENT,TYPE=T3D2,ELSET=LINK 6001,10018,17018 6002,10019,17019 6003,10020,17020 6004,10021,17021 6005,10022,17022 6006,10023,17023 6007,10024,17024 6008,10025,17025 6009,10026,17026 6010,10035,17035 6011,10044,17044 6012,10053,17053 6013,10062,17062 6014,10071,17071 6015,10080,17080 6016,10089,17089 6017,18,18018 6018,19,18019 6019,20,18020 6020,21,18021 6021,22,18022 6022,23,18023 6023,24,18024 6024,25,18025 6025,26,18026 6026,35,18035 6027,44,18044 6028,53,18053 6029,62,18062 6030,71,18071 6031,80,18080 6032,89,18089

Cross-section and material data for the truss elements. *SOLID SECTION,ELSET=LINK, MATERIAL=FEJK 100.0 *MATERIAL,NAME=FEJK *ELASTIC 7.E7, 0.3 *NO TENSION Constraints between the end nodes of the truss elements. Only the equations for one truss element are shown. Totally there were 32 truss elements. *EQUATION 2 10018,1,1.0,19103,1,-1.0 2 10018,2,1.0,19103,2,-1.0 2 10018,5,1.0,19103,5,-1.0 2 10018,6,1.0,19103,6,-1.0 Boundary conditions for the rigid connections at the beam-ends. *BOUNDARY 19103,1,6 19003,1,2 19003,4,6 ************************************* ************************************* *************************************

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Third example, parts used when the bending moment capacity was calculated. **T his part was used when the **imperfections were calculated ** *STEP *BUCKLE 1,1,5,50 *CLOAD 19003,5,-100. 19103,5,100. *EL PRINT,ELSET=EPRINT S,E *NODE PRINT,NSET=NPRINT U RF *NODE FILE,GLOBAL=YES,MODE=1 U *END STEP *************************** *************************** ** **This part was used when the **bending moment capacity was calculated ** *AMPLITUDE,NAME=KURVA 1,0.005, 2,0.01, 3,0.015, 4,0.02 5,0.025, 6,0.03, 7,0.035, 8,0.04 9,0.045, 10,0.05, 11,0.055, 12,0.06 13,0.065, 14,0.07, 15,0.075, 16,0.08 17,0.085, 18,0.09, 19,0.095, 20,0.1 21,0.11, 22,0.12, 23,0.13, 24,0.14 25,0.15, 26,0.16, 27,0.17, 28,0.18 29,0.19, 30,0.20, 31,0.21, 32,0.22 *IMPERFECTION,FILE=pb5a,STEP=1 1,0.1 *STEP,NLGEOM,INC=32 *STATIC 1.0,32.0,32.E-4,1.0 *BOUNDARY,AMPLITUDE=KURVA 19003,5,5,-1.0 19103,5,5,1.0 *EL PRINT,ELSET=EPRINT S,E *NODE PRINT,NSET=NPRINT U RF *END STEP

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List of Bulletins from the Department of Structural Engineering, The Royal Institute of Technology, Stockholm TRITA-BKN. Bulletin Pacoste, C., On the Application of Catastrophe Theory to Stability Analyses of Elastic Structures. Doctoral Thesis, 1993. Bulletin 1. Stenmark, A-K., Dämpning av 13 m lång stålbalk − "Ullevibalken". Utprovning av dämpmassor och fastsättning av motbalk samt experimentell bestämning av modformer och förlustfaktorer. Vibration tests of full-scale steel girder to determine optimu m passive control. Licentiatavhandling, 1993. Bulletin 2. Silfwerbrand, J., Renovering av asfaltgolv med cementbundna plastmodifierade avjämningsmassor. 1993. Bulletin 3. Norlin, B., Two-Layered Composite Beams with Nonlinear Connectors and Geometry − Tests and Theory. Doctoral Thesis, 1993. Bulletin 4. Habtezion, T., On the Behaviour of Equilibrium Near Critical States. Licentiate Thesis, 1993. Bulletin 5. Krus, J., Hållfasthet hos frostnedbruten betong. Licentiatavhandling, 1993. Bulletin 6. Wiberg, U., Material Characterization and Defect Detection by Quantitative Ultrasonics. Doctoral Thesis, 1993. Bulletin 7. Lidström, T., Finite Element Modelling Supported by Object Oriented Methods. Licentiate Thesis, 1993. Bulletin 8. Hallgren, M., Flexural and Shear Capacity of Reinforced High Strength Concrete Beams without Stirrups. Licentiate Thesis, 1994. Bulletin 9. Krus, J., Betongbalkars lastkapacitet efter miljöbelastning. 1994. Bulletin 10. Sandahl, P., Analysis Sensitivity for Wind-related Fatigue in Lattice Structures. Licentiate Thesis, 1994. Bulletin 11. Sanne, L., Information Transfer Analysis and Modelling of the Structural Steel Construction Process. Licentiate Thesis, 1994. Bulletin 12. Zhitao, H., Influence of Web Buckling on Fatigue Life of Thin -Walled Columns. Doctoral Thesis, 1994. Bulletin 13. Kjörling, M., Dynamic response of railway track components. Measurements during train passage and dynamic laboratory loading. Licentiate Thesis, 1995. Bulletin 14. Yang, L., On Analysis Methods for Reinforced Concrete Structures. Doctoral Thesis, 1995. Bulletin 15. Petersson, Ö., Svensk metod för dimensionering av betongvägar. Licentiatavhandling, 1996. Bulletin 16. Lidström, T., Computational Methods for Finite Element Instability Analyses. Do ctoral Thesis, 1996. Bulletin 17. Krus, J., Environment- and Function-induced Degradation of Concrete Structures. Doctoral Thesis, 1996. Bulletin 18. Editor, Silfwerbrand, J., Structural Loadings in the 21st Century. Sven Sahlin Workshop, June 1996. Proceedings. Bulletin 19.

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Ansell, A., Frequency Dependent Matrices for Dynamic Analysis of Frame Type Structures. Licentiate Thesis, 1996. Bulletin 20. Troive, S., Optimering av åtgärder för ökad livslängd hos infrastrukturkonstruktioner. Licentiatavhandlin g, 1996. Bulletin 21. Karoumi, R., Dynamic Response of Cable-Stayed Bridges Subjected to Moving Vehicles. Licentiate Thesis, 1996. Bulletin 22. Hallgren, M., Punching Shear Capacity of Reinforced High Strength Concrete Slabs. Doctoral Thesis, 1996. Bulletin 23. Hellgren, M., Strength of Bolt-Channel and Screw-Groove Joints in Aluminium Extrusions. Licentiate Thesis, 1996. Bulletin 24. Yagi, T., Wind-induced Instabilities of Structures. Doctoral Thesis, 1997. Bulletin 25. Eriksson, A., and Sandberg, G., (editors), Engineering Structures and Extreme Events − proceedings from a symposium, May 1997. Bulletin 26. Paulsson, J., Effects of Repairs on the Remaining Life of Concrete Bridge Decks. Licentiate Thesis, 1997. Bulletin 27. Olsson, A., Object-oriented finite element algorithms. Licentiate Thesis, 1997. Bulletin 28. Yunhua, L., On Shear Locking in Finite Elements. Licentiate Thesis, 1997. Bulletin 29. Ekman, M., Sprickor i betongkonstruktioner och dess inverkan på beständigheten. Licentiate Thesis, 1997. Bulletin 30. Karawajczyk, E., Finite Element Approach to the Mechanics of Track-Deck Systems. Licentiate Thesis, 1997. Bulletin 31. Fransson, H., Rotation Capacity of Reinforced High Strength Concrete Beams. Licentiate Thesis, 1997. Bulletin 32. Edlund, S., Arbitrary Thin -Walled Cross Sections. Theory and Computer Implementation. Licentiate Thesis, 1997. Bulletin 33. Forsell, K., Dynamic analyses of static instability phenomena. Licentiate Thesis, 1997. Bulletin 34. Ikäheimonen, J., Construction Loads on Shores and Stability of Horizontal Formworks. Doctoral Thesis, 1997. Bulletin 35. Racutanu, G., Konstbyggnaders reella livslängd. Licentiatavhandling, 1997. Bulletin 36. Appelqvist, I., Sammanbyggnad. Datastrukturer och utveckling av ett IT-stöd för byggprocessen. Licentiatavhandling, 1997. Bulletin 37. Alavizadeh-Farhang, A., Plain and Steel Fibre Reinforced Concrete Beams Subjected to Combined Mechanical and Thermal Loading. Licentiate Thesis, 1998. Bulletin 38. Eriksson, A. and Pacoste, C., (editors), Proceedings of the NSCM-11: Nordic Seminar on Computational Mechanics, October 1998. Bulletin 39. Luo, Y., On some Finite Element Formulations in Structural Mechanics. Doctoral Thesis, 1998. Bulletin 40. Troive, S., Structural LCC Design of Concrete Bridges. Doctoral Thesis, 1998. Bulletin 41.

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Tärno, I., Effects of Contour Ellipticity upon Structural Behaviour of Hyparform Suspended Roofs. Licentiate Thesis, 1998. Bulletin 42. Hassanzadeh, G., Betongplattor på pelare. Förstärkningsmetoder och dimensioneringsmetoder för plattor med icke vidhäftande spännarmering. Licentiatavhandling, 1998. Bulletin 43. Karoumi, R., Response of Cable -Stayed and Suspension Bridges to Moving Vehicles. Analysis methods and practical modeling techniques. Doctoral Thesis, 1998. Bulletin 44. Johnson, R., Progression of the Dynamic Properties of Large Suspension Bridges during Construction − A Case Study of the Höga Kusten Bridge. Licentiate Thesis, 1999. Bulletin 45. Tibert, G., Numerical Analyses of Cable Roof Structures. Licentiate Thesis, 1999. Bulletin 46. Ahlenius, E., Explosionslaster och infrastrukturkonstruktioner - Risker, värderingar och kostnader. Licentiatavhandling, 1999. Bulletin 47. Battini, J-M., Plastic instability of plane frames using a co-rotational approach. Licentiate Thesis, 1999. Bulletin 48. Ay, L., Using Steel Fiber Reinforced High Performance Concrete in the Industrialization of Bridge Structures. Licentiate Thesis, 1999. Bulletin 49. Paulsson-Tralla, J., Service Life of Repaired Concrete Bridge Decks. Doctoral Thesis, 1999. Bulletin 50. Billberg, P., Some rheology aspects on fine mortar part of concrete. Licentiate Thesis, 1999. Bulletin 51. Ansell, A., Dynamically Loaded Rock Reinforcement. Doctoral Thesis, 1999. Bulletin 52. Forsell, K., Instability analyses of structures under dynamic loads. Doctoral Thesis, 2000. Bulletin 53. Edlund, S., Buckling of T-Section Beam-Columns in Aluminium with or without Transverse Welds. Doctoral Thesis, 2000. Bulletin 54. The bulletins enumerated above, with the exception for those which are out of print, may be purchased from the Department of Structural Engineering, The Royal Institute of Technology, SE-100 44 Stockholm, Sweden. The department also publishes other series. For full information see our homepage http://www.struct.kth.se

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