Cidect hollow sections in structural applications

J. Wardenier, J.A. Packer, X.-L. Zhao and G.J. van der Vegte

HOLLOW SECTIONS IN STRUCTURAL APPLICATIONS

ISBN 978-90-72830-86-9 Š CIDECT, Geneva, Switzerland, 2010 The publisher and authors have made careful efforts to ensure the reliability of the data contained in this publication, but they assume no liability with respect to the use for any application of the material and information contained in this publication.

Printed by Bouwen met Staal Boerhaavelaan 40 2713 HX Zoetermeer, The Netherlands P.O. Box 190 2700 AD Zoetermeer, The Netherlands Tel. Fax E-mail

+31(0)79 353 1277 +31(0)79 353 1278 info@bouwenmetstaal.nl

ii

PREFACE The global construction market requires a world-wide coordination of product-, testing-, design- and executionstandards, so that contracts for delivery of products and for engineering- and construction services can be agreed on a common basis without barriers.

The mission of CIDECT is to combine the research resources of major hollow section manufacturers in order to create a major force in the research and application of hollow steel sections world wide. This forms the basis of establishing coordinated and consistent international standards.

For the ease of use of such standards, it is however necessary to reduce their content to generic rules and to leave more object-oriented detailed rules to accompanying non-conflicting complementary information, that have the advantage to be more flexible for the adaptation to recent research results and to be useable together with any international code.

The book by J. Wardenier, J.A. Packer, X.-L. Zhao and G.J. van der Vegte "Hollow sections in structural applications" is such a source, developed in an international consensus of knowledge on the topic. It incorporates the recently revised design recommendations for hollow sections joints of the International Institute of Welding, IIW (2009) and CIDECT (2008 and 2009). Both are consistent with each other and are the basis for the Draft ISO standard for Hollow Section Joints (ISO 14346) and may form the basis for future maintenance, further harmonisation and further development of Eurocode 3 (EN 1993-1-8), AISC (ANSI/AISC 360) and the CISC recommendations.

For the use together with EN 1993-1-8 and ANSI/AISC 360, both being based on the previous IIW (1989) recommendations, the main differences to these rules are highlighted.

The authors are all internationally recognized experts in the field of tubular steel structures, three of them having been chairmen of the IIW-Subcommission XV-E on "Tubular Structures" since 1981. This committee is the pre-eminent international authority producing design recommendations and standards for onshore tubular structures.

This book should therefore be an invaluable resource for lecturers, graduate students in structural, architectural and civil engineering, explaining the important principles in the behaviour of tubular steel structures. It is also addressed to designers of steel structures who can find in it the special items related to the use of hollow sections, in particular joints, their failure modes and analytical models as supplements to more general design codes.

Aachen, Germany, August 2010 Prof. Dr.-Ing. Dr.h.c. Gerhard Sedlacek

iii

ACKNOWLEDGEMENTS This book gives the background to design with structural hollow sections in general and in particular for joints to hollow sections. For the latter, the recently updated recommendations of the International Institute of Welding (IIW, 2009) and CIDECT (2008 and 2009) are adopted.

The background to design recommendations with the relevant analytical models is especially important for students in Structural and Civil Engineering, whereas the design recommendations themselves serve more as an example. Since the available hours for teaching Steel Structures, and particularly Tubular Structures, vary from country to country, this book has been written in a modular form. The presentation generally follows European codes, but the material is readily adapted to other (national) codes.

Since the first edition of this book was used not only by students but also by many designers, this second edition was needed due to the recent update of the recommendations by IIW and the subsequent revision of the CIDECT Design Guides Nos. 1 and 3 in 2008 and 2009.

The new IIW (2009) recommendations and the revised CIDECT Design Guides Nos. 1 and 3 (2008 and 2009) are consistent with each other and are the basis for the Draft ISO standard for Hollow Section Joints (ISO 14346). Although the current Eurocode 3 (EN 1993-1-8, 2005) and AISC (2010) recommendations are still based on the previous IIW (1989) and CIDECT (1991 and 1992) recommendations, it is expected that in the next revision these will follow the new IIW and CIDECT recommendations presented in this book.

Besides the static design recommendations and background for hollow section joints, information is given for member design in Chapter 2, composite structures in Chapter 4, and fire resistance in Chapter 5. These chapters fully comply with the latest versions of the Eurocodes (EN 1993 and EN 1994). Further, fatigue design of hollow section joints is covered in Chapter 14.

We wish to thank our colleagues from the IIW Sub-commission XV-E "Tubular Structures" and from the CIDECT Project Working Group and the CIDECT Technical Commission for their constructive comments during the preparation of this book.

We are very grateful that Prof. J. Stark and Mr. L. Twilt were willing to check Chapters 4 and 5 respectively on composite members and fire resistance.

Appreciation is further extended to the authors of CIDECT Design Guides Nos. 1 to 9 and to CIDECT for making parts of these Design Guides or background information available for this book.

Finally, we wish to thank CIDECT for the initiative to update this book.

Delft, The Netherlands, September 2010 Jaap Wardenier Jeffrey A. Packer Xiao-Ling Zhao Addie van der Vegte

iv

CONTENTS 1.

1.1 1.2 1.3

Introduction History and developments Designation Manufacturing of hollow sections

1 1 2 2

2.

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Properties of hollow sections Mechanical properties Structural hollow section dimensions and dimensional tolerances Geometric properties Drag coefficients Corrosion protection Use of internal void Aesthetics

9 9 10 11 14 14 15 15

3.

3.1 3.2 3.3 3.4 3.5 3.6

Applications Buildings and halls Bridges Barriers Offshore structures Towers and masts Special applications

29 29 29 29 30 30 30

4.

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Composite structures Introduction Design methods Axially loaded columns Resistance of a section to bending Resistance of a section to bending and compression Influence of shear forces Resistance of a member to bending and compression Load introduction Special composite members with hollow sections

37 37 37 37 39 39 39 39 41 41

5.

5.1 5.2 5.3 5.4 5.5 5.6

Fire resistance of hollow section columns Introduction Fire resistance Unfilled hollow section columns Concrete filled hollow section columns Water filled hollow section columns Joints

49 49 50 52 53 55 56

6.

6.1 6.2 6.3 6.4 6.5

Design of hollow section trusses Truss configurations Joint configurations Limit states and limitations on materials General design considerations Truss analysis

65 65 65 66 67 68

7.

7.1 7.2 7.3

Behaviour of joints General introduction General failure criteria General failure modes

75 75 77 77 v

7.4

Joint parameters

77

8.

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

Welded joints between circular hollow sections Introduction Modes of failure Analytical models Experimental and numerical verification Basic joint strength formulae Evaluation to design rules Other types of joints Design charts Relation to the previous recommendations of IIW (1989) and CIDECT (1991) Concluding remarks

81 81 81 81 83 83 84 85 86 87 87

9.

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

Welded joints between rectangular hollow sections Introduction Modes of failure Analytical models Experimental and numerical verification Basic joint strength formulae Evaluation to design rules Other types of joints or other load conditions Design charts Concluding remarks

103 103 103 104 106 106 107 107 109 109

10.

10.1 10.2 10.3 10.4 10.5 10.6

Welded joints between hollow sections and open sections Introduction Modes of failure Analytical models Experimental verification Evaluation to design rules Joints predominantly loaded by bending moments

129 129 129 129 131 131 131

11.

11.1 11.2 11.3 11.4 11.5 11.6 11.7

Welded overlap joints Introduction Modes of failure Analytical models for RHS overlap joints Analytical models for CHS overlap joints Analytical models for overlap joints with an open section chord Experimental and numerical verification Joint strength formulae

141 141 141 141 143 143 143 144

12.

12.1 12.2 12.3 12.4 12.5 12.6

Welded I beam-to-CHS or RHS column moment joints Introduction Modes of failure Analytical models Experimental and numerical verification Basic joint strength formulae Concluding remarks

151 151 151 151 153 153 154

13.

13.1 13.2 13.3 13.4

Bolted joints Flange plate joints End joints Gusset plate joints Splice joints

161 161 161 162 162 vi

13.5 13.6 13.7 13.8 13.9 13.10

Beam-to-column joints Bracket joints Bolted subassemblies Purlin joints Blind bolting systems Nailed joints

162 163 163 163 163 163

14.

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8

Fatigue behaviour of hollow section joints Definitions Influencing factors Loading effects Fatigue strength Partial factors Fatigue capacity of welded joints Fatigue capacity of bolted joints Fatigue design

175 175 175 176 177 177 177 179 180

15.

15.1 15.2 15.3 15.4 15.5 15.6

Design examples Uniplanar truss of circular hollow sections Uniplanar truss of square hollow sections Multiplanar truss (triangular girder) Multiplanar truss of square hollow sections Joint check using the joint resistance formulae Concrete filled column with reinforcement

193 193 197 197 199 199 200

16.

References

209

Symbols

221

CIDECT

229

vii

viii

together too voluminous for educational purposes and do not give the theoretical background, it was decided to write this book especially to provide background information for students and practitioners in Structural and Civil Engineering.

1. INTRODUCTION Design is an interactive process between the functional and architectural requirements and the strength and fabrication aspects. In a good design, all these aspects have to be considered in a balanced way. Due to the special features of hollow sections and their joints, it is here even of more importance than for steel structures of open sections. The designer should therefore be aware of the various aspects of hollow sections.

This book is written in a limit states design format (also known as LRFD or Load and Resistance Factor Design in the USA). This means that the effect of the factored loads (the specified or unfactored loads multiplied by the appropriate load factors) should not exceed the factored resistance of the joint or member.

The factored resistance expressions, in general, already include appropriate material and joint partial safety factors (γM) or joint resistance (or capacity) factors (). This has been done to avoid interpretation errors, since some international structural steelwork specifications use γM values  1,0 as dividers (e.g.

Eurocodes), whereas others use  values  1,0 as multipliers (e.g. in North America and Australia). In general, the value of 1/γM is almost equal to .

Many examples in nature show the excellent properties of the tubular shape with regard to loading in compression, torsion and bending in all directions, see Figs. 1.1 and 1.2. These excellent properties are combined with an attractive shape for architectural applications (Figs. 1.3 and 1.4). Furthermore, the closed shape without sharp corners reduces the area to be protected and extends the corrosion protection life (Fig. 1.5).

Another aspect which is especially favourable for circular hollow sections is the lower drag coefficients if exposed to wind or water forces. The internal void can be used in various ways, e.g. to increase the bearing resistance by filling with concrete or to provide fire protection. In addition, heating or ventilation systems sometimes make use of the hollow section columns.

1.1 HISTORY AND DEVELOPMENTS The excellent properties of the tubular shape have been recognised for a long time; i.e. from ancient time, nice examples are known. An outstanding example of bridge design is the Firth of Forth Bridge in Scotland (1890) with a free span of 521 m, shown in Fig. 1.6.

This bridge has been built up from tubular members made of rolled plates which have been riveted together, because at that time, other fabrication methods were not available for these sizes.

Although the manufacturing costs of hollow sections are higher than those for other sections, leading to higher unit material cost, economical applications are achieved in many fields. The application field covers all areas, e.g. architectural, civil, offshore, mechanical, chemical, aeronautical, transport, agriculture and other special fields. Although this book will be mainly focused on the background to design and application, in a good design not only does the strength have to be considered, but also many other aspects, such as material selection, fabrication including welding and inspection, protection, erection, in service inspection and maintenance.

In the same century, the first production methods for seamless and welded circular hollow sections were developed. In 1886, the Mannesmann brothers developed the skew roll piercing process (Schrägwalzverfahren), shown in Fig. 1.7, which made it possible to roll short thick walled tubulars. This process, in combination with the pilger process (Pilgerschrittverfahren, Fig. 1.8), developed some years later, made it possible to manufacture longer thinner walled seamless hollow sections.

One of the constraints initially hampering the application of hollow sections was the design of the joints. However, nowadays design recommendations exist for all basic types of joints, and further research evidence is available for many special types of joints.

In the first part of the previous century, an Englishman, Whitehouse, developed the fire welding of circular hollow sections. However, the production of welded circular hollow sections became more important after the development of the continuous welding process in 1930 by the American, Fretz Moon (Fig. 1.9).

Especially after the Second World War, welding processes have been perfected, which made it possible for hollow sections to be easily welded

Based on the research programmes carried out, CIDECT (Comité International pour le Développement et l'Etude de la Construction Tubulaire) has published Design Guides Nos. 1 to 9 for use by designers in practice. Since these nine Design Guides are all 1

together.

1.2 DESIGNATION

The end cutting required for fitting two circular hollow sections together was considerably simplified by the development of a special end preparation machine by MĂźller (Fig. 1.10).

The preferred designations for structural applications are: - Circular hollow sections (CHS) - Rectangular hollow sections (RHS) - Square hollow sections (SHS)

For manufacturers who did not have such end cutting machines, the end preparation of circular hollow sections remained a handicap.

In Canada and the USA, it is common to speak about Hollow Structural Sections (HSS), whereas in Europe also the term Structural Hollow Sections (SHS) is used.

A way of avoiding the connection problems was the use of prefabricated connectors, e.g. in 1937 Mengeringhausen developed the Mero system. This system enabled the fabrication of large space structures in an industrialized way (Fig. 1.11).

1.3 MANUFACTURING OF HOLLOW SECTIONS

In 1952, the rectangular hollow section was developed

by Stewarts and Lloyds (now Corus Tubes). This section, with nearly the same properties as the circular hollow section, enables the connections to be made by straight end cuttings.

As mentioned, hollow sections can be produced seamless or welded. Seamless hollow sections are made in two phases, i.e. the first phase consists of piercing an ingot and the second step considers the elongation of this hollow bloom into a finished circular hollow section. After this process, the tube can go through a sizing mill to give it the required diameter.

More information about other processes, most of them based on the same principle, is given by Dutta (2002).

In the fifties, the problems of manufacturing, end preparation and welding were all solved and from that point of view the way to a successful story was open.

The remaining problem was the determination of the strength of unstiffened joints.

Nowadays, welded hollow sections with a longitudinal weld are mainly made employing either electrical resistance welding processes or induction welding processes, shown in Fig. 1.12. A strip or plate is formed by rollers into a cylindrical shape and welded longitudinally. The edges are heated, e.g. by electrical resistance, then the rollers push the edges together, resulting in a pressure weld. The weld protrusion on the outside of the tube is trimmed immediately after welding.

The first preliminary design recommendations for truss connections between circular hollow sections were given by Jamm in 1951. This study was followed by several investigations in the USA (Bouwkamp, 1964; Natarajan & Toprac, 1969; Marshall & Toprac, 1974), Japan (Togo, 1967; Natarajan & Toprac, 1968), and Europe (Wanke, 1966; Brodka, 1968; Wardenier, 1982; Mang & Bucak, 1983; Puthli, 1998; Dutta, 2002).

Rectangular hollow sections are made by deforming circular hollow sections through forming rollers, as shown in Fig. 1.13. This forming process can be done hot or cold, using either seamless or longitudinally welded circular hollow sections. Although it is common practice to use longitudinally welded hollow sections, for the very thick sections, seamless sections may be used.

Research on joints between rectangular hollow sections started in Europe in the sixties, followed by many other experimental and theoretical investigations. Many of these were sponsored by CIDECT. Besides these investigations on the static behaviour, in the last 25 years much research was carried out on the fatigue behaviour and other aspects, such as concrete filling of hollow sections, fire resistance, corrosion resistance and behaviour under wind loading.

Square or rectangular hollow sections are sometimes made by forming a single strip to the required shape and closing it by a single weld, preferably in the middle of a face.

Large circular hollow sections are also made by rolling plates through a so-called U-O press process shown in Fig. 1.14. After forming the plates to the required 2

shape, the longitudinal weld is made by a submerged arc welding process.

Another process for large tubulars is to use a continuous wide strip, which is fed into a forming machine at an angle to form a spirally formed circular cylinder, see Fig. 1.15. The edges of the strip are welded together by a submerged arc welding process resulting in a so-called spirally welded tube.

More detailed information about the manufacturing processes and the limitations in sizes can be obtained from Dutta (2002).

3

Fig. 1.1 Reeds in the wind

Fig. 1.2 Bamboo

Fig. 1.3 Airport Bangkok, Thailand

Fig. 1.4 Ripshorster Bridge, Germany

4

Fig. 1.6 Firth of Forth Bridge, Scotland

Fig. 1.5 Paint surface for hollow sections vs open sections

Fig. 1.8 Pilger process (Pilgerschrittverfahren)

Fig. 1.7 Skew roll piercing process (Schrägwalzverfahren)

5

welding rollers

welded CHS

heating forming rollers

coil heating

Fig. 1.10 End cutting machine Fig. 1.9 Fretz Moon process Welded CHS Pressure rollers

inductor

Fig. 1.11 Mero connector

Fig. 1.12 Induction welding process

6

Fig. 1.13 Manufacturing of rectangular hollow sections

Fig. 1.14 Forming of large CHS

Fig. 1.15 Spirally welded CHS

7

8

2. PROPERTIES OF HOLLOW SECTIONS

f yd fu  1,25 or  0,8 f yd fu

2.1 MECHANICAL PROPERTIES

This is only one aspect for ductility. In the case of impact loading, the steel and members should also behave in a ductile manner. Hence, Tables 2.1a and 2.2a also give requirements based on the standard Charpy test to ensure adequate notch toughness.

Hollow sections are made of similar steel as used for other steel sections, thus in principle there is no difference in mechanical properties.

Nowadays, more refined characterisation methods exist to describe the ductility of cracked bodies, e.g. the CTOD (Crack Tip Opening Displacement) method.

These characterisation methods are generally used for pressure vessels, transport line pipes and offshore applications, which are beyond the scope of this book.

Tables 2.1a and 2.2a show, as an example, the mechanical properties according to the European standard EN 10210-1 (2006) for hot finished structural hollow sections of non-alloy and fine grain structural steels. The cold formed sections are given in EN 10219-1 (2006): Cold formed welded structural hollow sections of non-alloy and fine grain structural steels (see Tables 2.1b and 2.2b). As shown, the requirements of EN 10210-1 and EN 10219-1 are almost identical.

Another characterisation is sometimes required for thick walled sections which are loaded in the thickness direction. In this case, the strength and ductility in the thickness direction should be sufficient to avoid cracking, called lamellar tearing, see Fig. 2.1.

This type of cracking is caused by non metallic manganese-sulphide inclusions. Thus, if the sulphur content is very low or the sulphur is joined with other elements such as calcium (Ca), such a failure can be avoided. Indirectly this is obtained by requiring a certain reduction of area RAZ in the tensile test. For example, RAZ = 35 means that in the tensile test the cross sectional area at failure has been reduced by 35% compared to the original cross sectional area.

Hollow sections can also be produced in special steels, e.g. high strength steel with yield strengths up to 690 N/mm2 or higher, weathering steels and steel with improved or special chemical compositions, etc.

Generally, the design of members is based on the yield strength. In this chapter the recommended γM0 and γM1 factors of 1,0 are adopted for the design yield strength fyd.

In statically indeterminate structures, sufficient deformation capacity or rotation capacity is required for redistribution of loads. In this case, yielding of members or yielding in the joints may provide the required rotation capacity. A tensile member made of ductile steel can be brittle if a particular cross section is weakened, e.g. by holes, in such a way that this cross section fails before the whole member yields. It is therefore required that yielding occurs first. This shows that the yield-to-ultimate tensile strength ratio is also important, especially for structures with very non-uniform stress distributions, which is a situation that occurs in tubular joints. Some codes, such as Eurocode 3 (EN 1993-1-1, 2005), specify the following requirement for the minimum ratios:

fu  1,1 f yd

(2.1b)

In most structural steel specifications the minimum required yield strength, ultimate tensile strength, elongation and the Charpy V-notch values are specified. Design standards or specifications give further limitations for the fu/fy ratio, whereas depending on the application, more restrictive requirements may be given related to CTOD values or the properties in the thickness direction (Z quality).

Another aspect is the effect of cold forming on the mechanical properties of the parent steel. In the case of cold forming of hollow sections, the yield strength and to a lesser extent the ultimate tensile strength are increased, especially in the corners, as shown in Fig. 2.2. Further, the yield-to-ultimate tensile ratio is increased and the elongation slightly decreased.

(2.1a)

If the standards, e.g. EN 10210-1 and EN 10219-1, specify the properties at a particular cross section location based on the finished product, these properties have been already partly taken into account. Thus, this generally applies in Europe.

The IIW (2009) recommendations and many offshore codes require a higher ratio between fu and fyd:

9

However, some standards outside Europe specify the material properties of the parent material. In this case, the increased yield strength can be taken into account for design. A small corner radius produces a small cold formed area with a large cold forming effect and consequently a large increase in yield strength, while a large corner radius does just the opposite.

According to research work of Lind & Shroff (1971), the product of area and increase in yield strength can approximately be taken as constant. Lind & Shroff assumed that in every corner of 90 the yield strength of the parent material fyb is increased over a length of 7t to the ultimate tensile strength of the parent material fu. The total increase over the section 4(7t)t(fu - fyb) can be averaged over the section, resulting in a design yield strength fya, as shown in Fig. 2.2.

2.2 STRUCTURAL HOLLOW SECTION DIMENSIONS AND DIMENSIONAL TOLERANCES The dimensions and sectional properties of structural hollow sections have been standardised in EN (EN 10210-2, 2006; EN 10219-2, 2006) and ISO standards (ISO 657-14, 2000; ISO 4019, 2001) for hot finished and cold formed structural hollow sections respectively.

The two applicable standards in Europe are EN 10210-2 (2006) "Hot finished structural hollow sections of non-alloy and fine grain steels – Part 2: Tolerances, dimensions and sectional properties" and EN 10219-2 (2006) "Cold formed welded structural hollow sections of non-alloy and fine grain steels – Part 2: Tolerances, dimensions and sectional properties". However, the majority of manufacturers of structural hollow sections do not produce all the sizes shown in these standards. It should be further noted that other sizes, not included in these standards, may be produced by some manufacturers.

It is noted that the cold formed sections should satisfy the requirements for minimum inside corner radius to guarantee sufficient ductility, see Table 2.3 for fully aluminum killed steel (steel with limited Si content).

Part 10 of Eurocode 3 (EN 1993-1-10, 2005) specifies the material selection. Here, a permissible thickness can be determined based on a reference temperature, the steel grade and quality and the stress level. The reference temperature covers, besides the air temperature, also cold forming effects, strain rate, etc.

However, the current rules cannot be adopted to cold formed hollow sections because the determination of the effect of cold forming for cold formed hollow sections is not yet clearly specified. Based on the data obtained by Soininen (1996), Kosteski et al. (2003), Björk (2005), Kühn (2005), Puthli & Herion (2005) and Sedlacek et al. (2008), presently a proposal is being worked out for an amendment of EN 1993-1-10. In this proposal of CEN/TC 250/SC 3-N 1729 (2010), it is recommended that for cold formed hollow sections according to EN 10219, the procedure for hot formed material can be used provided that for the cold forming effects the reference temperature is reduced by Tcf. For CHS, Tcf varies from 0 C to 20 C depending on the thickness and the d/t ratio. For RHS with steel qualities according to EN 10219, Tcf varies from 35 C to 45 C depending on the thickness and the ratio between the inside corner radius and the thickness. For cold formed hollow sections with Charpy impact strengths significantly exceeding the requirements of EN 10219, a lower value of Tcf is allowed.

The majority of the tolerances given in EN 10219-2 are the same as those in EN 10210-2, see Tables 2.4a and 2.4b.

Internationally, the delivery standards in various countries deviate considerably with respect to the thickness and mass tolerances (Packer, 1993). In most countries besides the thickness tolerance, a mass tolerance is given, which limits extreme deviations. However, in some production standards, e.g. in the USA, the thickness tolerance is not always compensated by a mass tolerance. This has resulted in associated design specifications which account for this, by designating a lower "design wall thickness" of 0,9 or 0,93 times the nominal thickness t. In Eurocode 3, where design is based on nominal thicknesses, the thickness tolerances in EN 10210-2 and EN 10219-2 are (partly) compensated by the mass tolerance. It is foreseen that in the next revision these tolerances will be tightened.

Although the circular, square and rectangular hollow sections are the generally-used shapes; other shapes are sometimes available. For example, some tube manufacturers deliver the shapes given in Table 2.5.

Of these, the elliptical hollow sections have become more popular for architectural designs. These shapes are not dealt with further in this book. However, more information about elliptical hollow sections can be found in Bortolotti et al. (2003), Chan & Gardner (2008), Choo et al. (2003), Martinez-Saucedo et al.

10

(2008), Packer et al. (2009b), Pietrapertosa & Jaspart (2003), Theofanous et al. (2009), Willibald et al.

(2006) and Zhao & Packer (2009).

The buckling behaviour is influenced by initial eccentricities, straightness and geometrical tolerances as well as residual stresses, non-homogeneity of the steel and the stress-strain relationship.

2.3 GEOMETRIC PROPERTIES

Based on extensive investigations by the European Convention for Constructional Steelwork (ECCS) and CIDECT, "European buckling curves" (Fig. 2.3 and Table 2.7) have been established for various steel sections including hollow sections. They are incorporated in Eurocode 3 (EN 1993-1-1, 2005).

2.3.1 Tension The design capacity Nt,Rd of a member under tensile loading depends on the cross sectional area and the design yield strength, and is independent of the sectional shape. In principle, there is no advantage or disadvantage in using hollow sections from the point of view of the amount of material required. The design capacity is given by:

Nt,Rd  Afyd

The reduction factor  shown in Fig. 2.3 is the ratio of the design buckling capacity Nb,Rd to the axial plastic capacity.



(2.2)

If the cross section is weakened by bolt holes or slots, the net cross section should also be checked, in a similar way as for other sections, e.g. according to Eurocode 3 (EN 1993-1-8, 2005): Nt,Rd 

A net fu 0,9  M2

fb,Rd =

(2.3)

Nb,Rd A

(2.6)

The non-dimensional slenderness  is determined by: 

The factor 0,9 may vary from country to country depending on the partial factor M used. Where ductile behaviour is required (e.g. under seismic loading), the yield resistance shall be less than the ultimate resistance at the net section of fastener holes.

 E

(2.7)

where:

E  

2.3.2 Compression

E (Euler slenderness) fy

(2.8)

The buckling curves for the hollow sections are classified according to Table 2.6. Most open sections fall under curves "b" and "c". Consequently, for the case of buckling, the use of hot formed hollow sections generally provides a considerable saving in material.

For centrally loaded members in compression, the critical buckling load depends on the slenderness λ and the section shape.

The slenderness λ is given by the ratio of the buckling length ℓb and the radius of gyration i. b i

(2.5)

where:

where the partial safety factor M2 = 1,25.



Nb,Rd f  b,Rd Npl,Rd fyd

Fig. 2.4 illustrates, for a buckling length of 3 m, a comparison between the required mass of open and hollow sections for a given load. It shows that in those cases in which loads are small, leading to relatively slender sections, hollow sections provide a great advantage (considerably lower use of material).

However, if loads are higher, resulting in low slenderness, the advantage (in %) will be less.

(2.4)

The radius of gyration of a hollow section (in relation to the member mass) is generally much higher than that for the weak axis of an open section. For a given length, this difference results in a lower slenderness for hollow sections and thus a lower mass when compared with open sections.

The overall buckling behaviour of hollow sections improves with increasing diameter- or width-to-wall 11

these limits can be taken considerably larger.

thickness ratio. However, this improvement is limited by local buckling. To prevent local buckling, d/t or b/t limits are given e.g. in Eurocode 3 (EN 1993-1-1, 2005), see Table 2.7. In the case of thin walled sections, interaction between global and local buckling should be considered.

It is apparent that hollow sections are especially favourable compared to other sections if bending about both axes is present.

Hollow sections used for elements subjected to bending can be more economically designed by using plastic design. However, then the sections have to satisfy more restricted conditions to avoid premature local buckling. Like other steel sections loaded in bending, different moment-rotation behaviour can be observed.

In addition to the improved buckling behaviour due to the high radius of gyration and the enhanced design buckling curve, hollow sections can offer other advantages in lattice girders. Due to the torsional and bending stiffness of the members in combination with joint stiffness, the effective buckling length of compression members in lattice girders can be reduced (Fig. 2.5). Eurocode 3 (EN 1993-1-1) recommends an effective buckling length for hollow section brace members in welded lattice girders equal to or less than 0,75ℓ, in which ℓ represents the system length, see also Rondal et al. (1992).

Fig. 2.7 shows various moment-rotation diagrams for a member loaded by bending moments.

The moment-rotation curve "1" shows a moment exceeding the plastic moment Mpl and a considerable rotation capacity. Moment-rotation curve "2" shows a moment exceeding the plastic moment capacity Mpl, but after the maximum, the moment drops immediately, so that little moment-rotation capacity exists. Moment-rotation curve "3" represents a capacity lower than the plastic moment capacity, which, however, exceeds the yield moment capacity Mel. In the moment-rotation curve "4" the capacity is even lower than the yield moment capacity Mel and failure is by elastic buckling. The effect of the moment-rotation behaviour is reflected in the classification of cross sections as shown in Fig. 2.8 and Table 2.7. The cross section classification is given in limits for the diameter- or flat width-to-thickness ratio.

For chords, 0,9 times the system length for in-plane buckling or 0,9 times the length between the supports for out-of-plane buckling, is taken as the effective buckling length.

These reductions are also based on the fact that the chord and brace members are generally not fully optimised. If for example the chord would be fully utilized with different members for every panel then these reductions would not be allowed.

Laterally unsupported compression chords of lattice girders (see Fig. 2.6) have a reduced buckling length due to the improved torsional and bending stiffness of the tubular members (Baar, 1968; Mouty, 1981).

These factors make the use of hollow sections in girders or trusses even more favourable.

The limits are based on experiments and can be expressed as:

2.3.3 Bending In general, I and H sections are more economical under bending about the major axis (Imax larger than for hollow sections). Only in those cases in which the design resistance in open sections is largely reduced by lateral buckling, hollow sections offer an advantage.

It can be shown by calculations that lateral instability is not critical for circular hollow sections, square hollow sections and for the most commonly used rectangular hollow sections with bending about the strong axis. Table 2.8 shows allowable span-to-depth ratios for the most commonly used sections (EN 1993-1-1, 2005). According to a study of Kaim (2006)

d 235 c t fyd

for CHS

(2.9)

b 235 3  c t f yd

for RHS

(2.10a)

h 235 3  c t f yd

for RHS

(2.10b)

with fyd in N/mm2 and c depending on the section class, the cross section and the loading. For RHS, it is conservatively assumed that the width of the "flat" is equal to the external width b or depth h of the RHS minus 3t.

12

The cross section classes 1 and 2 can develop the plastic moment capacity up to the given b/t or d/t limits with bi-linear stress blocks, whereas the moment capacity of the cross section classes 3 and 4 is based on an elastic stress distribution (see Fig. 2.8). The difference between the cross section classes 1 and 2 is reflected in the rotation capacity. After reaching the plastic moment capacity, the cross section class 1 can keep this capacity after further rotation, whereas the capacity of the cross section class 2 drops after reaching this capacity. As a consequence, the moment distribution in the structure or structural component should be determined by elastic analysis for structures made of sections with cross section classes 2, 3 or 4. For structures made of sections with cross sections in class 1 a plastic moment distribution can be adopted, but an elastic moment distribution is still permissible (and in some countries more common).

do not exceed 50% of the shear capacity Vpl,Rd, the effect of shear may be neglected and the bending moment capacity about one axis is given by:

Mc,Rd  Wpl fyd for cross section classes 1 or 2 (2.12) Mc,Rd  Wel fyd

for cross section class 3

(2.13)

Mc,Rd  Weff fyd for cross section class 4

(2.14)

When the shear force exceeds 50% of the shear capacity, combined loading has to be considered, see Eurocode 3 (EN 1993-1-1).

2.3.4 Shear The elastic shear stress in circular and rectangular hollow sections can be determined with simple mechanics by:

For a class 1 beam fully clamped at both ends and subjected to a uniformly distributed loading q, the plastic moment distribution implies that after reaching the plastic moment capacity at the ends, the beam can be loaded until a further plastic hinge occurs at mid span (see Fig. 2.9).



VEd S f yd  2 It 3

(2.15)

Fig. 2.10 shows the elastic stress distribution. The design capacity based on plastic design can be easily determined based on the Huber-Hencky-Von Mises criterion by assuming the shear yield strength in those parts of the cross section active for shear.

For the class 4 cross section, the maximum stress is determined by local buckling and the stress in the outer fibre is lower than the yield strength fy.

Alternatively, an effective cross sectional area based on the yield strength may be determined.

Vpl,Rd  A v

Detailed information about the cross sectional classification is given by Rondal et al. (1992).

fyd

(2.16)

3

where: Research by Wilkinson & Hancock (1998) showed that especially the limits for the side wall slenderness of RHS need to be reduced considerably. E.g. for class 1 sections, they suggest reducing the Eurocode 3 limits (EN 1993-1-1) for the side wall slenderness to: 5(b  2t  2r ) (h  2t  2r)  70  6t t

with

Av  A

for RHS

(2.17)

(or just 2 h t) with V in the direction of h.

(2.11)

Av 

b  2t  2r  30 t

2 A 

for CHS

(2.18)

2.3.5 Torsion

For r = t, this can be simplified to:

b h b  34 with  77  0 ,83 t t t

h bh

Hollow sections, especially CHS, have the most effective cross section for resisting torsional moments, because the material is uniformly distributed about the polar axis. A comparison of open and hollow sections of nearly identical mass in Table 2.9 shows that the torsional constant of hollow sections is about 200 times that of open sections.

(2.11a)

In the absence of shear forces or if the shear forces

13

The design capacity for torsional moments is described by:

Mt,Rd  Wt

2.11, is given by:

p  f yd

f yd

(2.19)

3

2 It   (d  t ) t dt 2

(2.20)

where: It 

 d  t 3 t 4

(2.21)

2.3.7 Combined loadings

For rectangular hollow sections (Marshall, 1971): Wt 

It t2

Various combinations of loadings are possible, e.g. tension, compression, bending, shear and torsion.

(2.22)

Am A

Depending on the cross sectional classification, various interaction formulae should be applied.

Reference can be made to Eurocode 3 (EN 1993-1-1).

It is beyond the scope of this book to deal with all possible formulae; however, the interaction between the various loads in the cross section can be based on the Huber-Hencky-Von Mises stress criterion (Roik & Wagenknecht, 1977). For the member checks, other interaction formulae apply, see e.g. EN1993-1-1.

where: 2

It 

t 3  A 4A m t  A 3

(2.23)

 A  2 hm  b m   2 rm 4   

(2.24)

A m  b m hm  rm2 4   

(2.25)

2.4 DRAG COEFFICIENTS Hollow sections, especially circular hollow sections, have a striking advantage for use in structures exposed to fluid currents, i.e. air or water. The drag coefficients are much lower than those of open sections with sharp edges, see Fig. 2.12 (Schulz, 1970; CIDECT, 1984; Dutta, 2002).

For thin walled rectangular hollow sections, eq. (2.22) can be approximated by: Wt  2 hm b m t

(2.27)

In eq. (2.27), M0 = 1,0, but for transport pipelines, the M0 value may be considerably larger than for other cases, depending on the hazard of the product, the effect of failure on the environment and inspectability.

The design capacities for RHS sections subjected to internal pressure are much more complicated; reference can be made to the Deutscher Dampfkesselausschuβ (1975).

For circular hollow sections:

Wt 

2t d  2t

(2.26)

The first term in eq. (2.23) is generally only used for open sections. However, research by Marshall (1971) showed that the given formula provides the best fit with the test results.

2.5 CORROSION PROTECTION Structures made of hollow sections offer advantages with regard to corrosion protection. Hollow sections have rounded corners (Fig. 2.13) resulting in a better protection than that for sections with sharp corners.

This is especially true for the joints in circular hollow sections where there is a smooth transition from one section to another. This better protection increases the protection period of coatings against corrosion.

The exact, more complicated equations for the cross sectional properties are given in EN 10210-2 (2006) and EN 10219-2 (2006).

2.3.6 Internal pressure The circular hollow section is most suitable to resist an internal pressure p.

Structures designed in hollow sections have a 20 to 50% smaller surface to be protected than comparable structures made of open sections.

Many

The design capacity per unit length, shown in Fig.

14

investigations, summarized by Tissier (1978), have been conducted to assess the likelihood of internal corrosion. These investigations, carried out in various countries, show that internal corrosion does not occur in sealed hollow sections.

section ratio of the steel and concrete, reinforcement percentage of the concrete and the applied load, see Fig. 2.14 (Twilt et al., 1994).

2.6.2 Fire protection by water circulation

Even in hollow sections which are not perfectly sealed, internal corrosion is limited. If there is concern about condensation in an imperfectly sealed hollow section, a drainage hole can be made at a point where water can drain by gravity.

Another method for fire protection of buildings is to use water filled hollow section columns.

The columns are interconnected with a water storage tank. Under fire conditions, the water circulates by convection, keeping the steel temperature below the critical value of 450 ď‚°C. This system has economical advantages when applied to buildings with more than about 8 storeys. If the water flow is adequate, the resulting fire resistance time is virtually unlimited.

2.6 USE OF INTERNAL VOID The internal void in hollow sections can be used in various ways, e.g. to increase the compressive resistance by filling with concrete, or to provide fire protection. In addition, heating or ventilation systems are sometimes incorporated into hollow section columns.

In order to prevent freezing, potassium carbonate (K2CO3) is added to the water. Potassium nitrate is used as an inhibitor against corrosion.

The possibilities of using the internal space are briefly described below.

2.6.3 Heating and ventilation The inner voids of hollow sections are sometimes used for air and water circulation for heating and ventilation of buildings. Many examples in offices and schools show the excellent combination of the strength function of hollow section columns with the integration of heating or ventilation systems. This system offers maximization of floor area through elimination of heat exchangers, a uniform provision of warmth and a combined protection against fire.

2.6.1 Concrete filling If the commonly-available wall thicknesses are not sufficient to meet the required load bearing resistance, the hollow section can be filled with concrete. For example, it may be preferable in buildings to have the same external dimensions for the columns on every floor. At the top floor, the smallest wall thickness can be chosen, and the wall thickness can be increased with increasing load for lower floors. If the hollow section with the largest available wall thickness is not sufficient for the ground floor, the hollow section can be filled with concrete to increase the load bearing resistance.

2.6.4 Other possibilities Sometimes hollow section chords of lattice girder bridges are used for conveying fluids (pipe bridge). In buildings, the rain water downpipes may go through the hollow section columns or in other cases electrical wiring is located in the columns. The internal space can also be used for prestressing a hollow section.

A very important reason for using concrete filled hollow sections is that the columns can be relatively slender. Design rules are given in e.g. Eurocode 4 (EN 1994-1-1, 2004).

2.7 AESTHETICS

Concrete filling of hollow sections contributes not only to an increase in load bearing resistance, but it also improves the fire resistance duration. Extensive test projects carried out by CIDECT and the European Coal and Steel Community (ECSC) showed that reinforced concrete filled hollow section columns without any external fire protection like plaster, vermiculite panels or intumescent paint, can attain a fire life of even 2 hours depending on the cross

A rational use of hollow sections leads in general to structures which are cleaner and more spacious.

Hollow sections can provide slender aesthetic columns, with variable section properties but flush external dimensions. Due to their torsional rigidity, hollow sections have specific advantages in folded structures, V-type girders, etc.

15

Lattice construction, which is often made of hollow sections directly connected to one another without any stiffener or gusset plate, is often preferred by architects for structures with visible steel elements.

However, it is difficult to express aesthetic features in economic comparisons. Sometimes hollow sections are used only because of aesthetic appeal, see e.g.

Fig. 2.15, whilst at other times appearance is less important.

16

Table 2.1a Hot finished structural hollow sections – Non-alloy steel properties (EN 10210-1, 2006) (2)

Minimum yield strength (1) 2 (N/mm )

Steel designation

Minimum tensile strength 2 (N/mm )

Longitudinal minimum elongation (%) on gauge Lo  5,65 So

Charpy impact strength (10 x 10 mm)

t  16 mm

16 < t  40 mm

40< t  63 mm

t<3 mm

3  t  100 mm

3 < t  40 mm

40 < t  63 mm

Temp. C

J

S235JRH

235

225

215

360-510

360-510

26

25

20

27

S275J0H S275J2H

275

265

255

430-580

410-560

23

22

0 -20

27 27

S355J0H S355J2H S355K2H

355

345

335

510-680

470-630

22

21

0 -20 -20

27 27 (3) 40

(1) (2) (3)

For thicknesses above 63 mm, these values are further reduced.

In transverse direction 2% lower.

Corresponds to 27 J at -30 C.

Table 2.1b Cold formed welded structural hollow sections – Non-alloy steel (EN 10219-1, 2006) – Steel properties different from EN 10210-1 (2006) Steel designation

Minimum longitudinal elongation (%), all thicknesses, tmax = 40 mm

S235JRH

24 (1)

S275J0H S275J2H

20

(2)

S355J0H S355J2H S355K2H

20

(2)

(1)

(2)

For t > 3 mm and d/t < 15 or

bh  12,5 the minimum elongation is reduced by 2 to 22%; for t  3 mm the minimum 2t

elongation is 17%. bh For d/t < 15 or  12,5 the minimum elongation is reduced by 2 to 18%.

2t

17

Table 2.2a Hot finished structural hollow sections – Fine grain steel properties (EN 10210-1, 2006) Minimum tensile strength 2 (N/mm )

Minimum yield strength 2 (N/mm )

Steel designation

Minimum elongation (%) on gauge

Charpy impact strength (10 x 10 mm)

Lo  5,65 So t  65 mm

t  16 mm

16 < t  40 mm

40 < t  65 mm

t  65 mm

Long.

Trans.

Temp. C

S275NH S275NLH

265

255

370-510

24

22

-20 -50

40 27

(1)

275

S355NH S355NLH

345

335

470-630

22

20

-20 -50

40 27

(1)

355

S420NH S420NLH

400

390

520-680

19

17

-20 -50

40 27

(1)

420

S460NH S460NLH

440

430

540-720

17

15

-20 -50

40 27

(1)

460

(1)

J

Corresponds to 27 J at -30 C.

Table 2.2b Cold formed welded structural hollow sections – Fine grain steel (EN 10219-1, 2006) – Steel properties different from EN 10210-1 (2006) Feed stock condition M (1) Steel designation

Minimum tensile strength 2 (N/mm )

Minimum longitudinal elongation (%) (2)

S275MH S275MLH

360 - 510

24

S355MH S355MLH

450 - 610

22

S420NH S420NLH

520 - 660

19

S460NH S460NLH

530 - 720

17

(1) (2)

M refers to thermal mechanical rolled steels. bh For d/t < 15 or  12,5 the minimum elongation is reduced by 2, e.g. from 24% to 22% for S275MH and S275MLH. 2t

Table 2.3 Minimum inner corner radii of cold formed RHS according to EN 1993-1-8 (2005) Maximum wall thickness t (mm) r/t

 25  10  3,0  2,0  1,5  1,0

Strain due to cold forming (%)

2 5  14  20  25  33

General Predominantly static loading

Fatigue dominating

Aluminium-killed steel (Al  0,02%)

any any 24 12 8 4

any 16 12 10 8 4

any any 24 12 10 6

18

Table 2.4a Hot finished structural hollow sections – Tolerances (EN 10210-2, 2006) Section type

Square/rectangular

Mass

the greater of ± 0,5 mm and ± 1% but not more than 10 mm

the greater of ± 0,5 mm and ± 1% (1)

Outside dimension Thickness

Circular

Welded

-10%

Seamless

-10% and -12,5% at maximum 25% cross section

Welded

± 6% on individual lengths

Seamless

-6%; +8%

Straightness

0,2% of the total length and 3 mm over any 1 m length

Length (exact)

+10 mm, -0 mm, but only for exact lengths of 2000 to 6000 mm

Out of roundness

-

2% for d/t  100

Squareness of sides

90 ± 1

-

Corner radii

3,0t maximum

-

Outside

Concavity/convexity

± 1% of the side

Twist (1)

2 mm + 0,5 mm/m

(1)

-

For elliptical hollow sections with h  250 mm, the tolerances are twice the values given in this table.

Table 2.4b Cold formed welded structural hollow sections (EN 10219-2, 2006) – Tolerances different from EN 10210-2 (2006) Section type

Square/rectangular

Circular

b < 100 mm: the greater of ± 0,5 mm and ± 1% 100 mm  h, b  200 mm: ± 0,8% b > 200 mm: ± 0,6%

± 1%, minimum ± 0,5 mm maximum ± 10 mm

t  5 mm: ± 10% t > 5 mm: ± 0,5 mm

for d  406,4 mm: t  5 mm: ± 10% t > 5 mm: ± 0,5 mm for d > 406,4 mm: ± 10% with maximum ± 2,0 mm

Mass

± 6%

± 6%

Straightness

0,15% of the total length and 3 mm over any 1 m length

Outside corner radii (profile)

t  6 mm: 1,6 to 2,4t 6 mm < t  10 mm: 2,0 to 3,0t t > 10 mm: 2,4 to 3,6t

-

Concavity/convexity

maximum 0,8% with a minimum of 0,5 mm

-

Outside dimension

Thickness

Welded

Table 2.5 Special shapes available Triangular

Hexagonal

Octagonal

Shape

19

Flat - oval

Elliptical

Half-elliptical

Table 2.6 European buckling curves according to manufacturing processes (EN 1993-1-1, 2005)

Cross section

bb t

Manufacturing process

Buckling curves

Hot finished 420 N/mm2 < fy  460 N/mm2

a0

Hot finished fy  420 N/mm2

a

Cold formed

c

Flange

t

d

d

Web

h

tt

h

Table 2.7 Limits for b/t, h/t and d/t for cross section classes 1, 2 and 3 (EN 1993-1-1, 2005) Class

1

2 2

fyd (N/mm ) Cross section

RHS (1) b/t

Load type

Considered element

235

Compression

Top face

(2)

235

275

355

fyd (N/mm ) 460

235

275

355

460

Bending

Side wall

Compression and/or bending

33,5

29,8

c = 38 26,6

41,0

38,1

33,9

c = 42 30,2

45,0

41,8

37,2

33,0

h 235 c 3 t fyd

(2)

c = 72 69,6

61,6

c = 83 51,8

86,0

79,7

70,5

c = 124 62,3 127,0 117,6 103,9 91,6

d 235 c t fyd

tt

c = 50 50,0

(1)

460

c = 33

75,0

CHS d/t

355

2

fyd (N/mm )

b 235 c 3 t fyd

36,0

RHS (1) h/t

275

3 2

42,7

33,1

c = 70 25,5

70,0

59,8

46,3

c = 90 35,8

90,0

76,9

59,6

46,0

For all hot finished and cold formed RHS, it is conservative to assume that the width-to-thickness ratio of the "flat" is b - 2t - 2r b h - 2t - 2r h   3 or  3 . t t t t Wilkinson & Hancock (1998) suggested reducing the Eurocode limits (EN 1993-1-1) for the side wall slenderness of RHS b h b considerably, e.g. for class 1 in a simplified form to: with  34 .  77  0 ,83 t t t

20

Table 2.8 Allowable span-to-depth ratios L/(h-t) to avoid lateral buckling based on EN 1993-1-1 (2005) L  ht

bt ht

bb Flange b

tt

h

h

S235

S275

S355

S460

0,5

73,7

63,0

48,8

37,7

0,6

93,1

79,5

61,6

47,5

0,7

112,5

96,2

74,5

57,5

0,8

132,0

112,8

87,4

67,4

0,9

151,3

129,3

100,2

77,3

1,0

170,6

145,8

112,9

87,2

Table 2.9 Torsional strength of various sections Mass (kg/m)

Torsion constant It 4 4 4 (10 mm ) or (cm )

UPN 200

25,3

11,9

INP 200

26,2

13,5

HEB 120

26,7

13,8

HEA 140

24,7

8,1

140 x 140 x 6

24,9

1475

168.3 x 6

24,0

2017

Section

21

Fig. 2.1 Lamellar tearing

Actual fy mean after cold forming

Fig. 2.2 Influence of cold forming on the yield strength for a square hollow section of 100 x 100 x 4 mm

1,00

0

0,75

0,50

0,25

0

0

0,5

1,0

1,5

2,0

Fig. 2.3 Eurocode 3 buckling curves (EN 1993-1-1, 2005)

22

Buckling stress (N/mm2) Fig. 2.4 Comparison of the masses of hollow and open sections under compression in relation to the loading

Fig. 2.5 Restraints for the buckling of a brace member

Fig. 2.6 Bottom chord laterally spring supported by the stiffness of the members, joints and purlins

23

Mpl Mel Me

Fig. 2.7 Moment-rotation curves

Fig. 2.8 Stress distribution for bending

Fig. 2.9 Moment distribution in relation to the cross section classification

24

Fig. 2.10 Elastic shear stress distribution

t

ttffyd yd

d - 2t

t fyd

Fig. 2.11 Internal pressure

Fig. 2.12 Wind flow for open and circular hollow sections

25

paint layers

steel

steel

corner protection for RHS and open sections

Fig. 2.13 Painted corners of RHS vs. open sections

fire life (min.)

RHS 304,8x304,8x9,5 nonreinforced concrete filling

120.

60.

only RHS

111 min. steel fibre reinforced concrete filling

50min.

14 min.

1650.

3150.

3150.

working load (kN) Fig. 2.14 Fire resistance of concrete filled hollow sections

26

Fig. 2.15 Aesthetically appealing structures

27

28

Very attractive applications can be found in the halls and buildings for the Olympic Games in Athens, e.g.

Fig. 3.8.

3. APPLICATIONS The applications of structural hollow sections nearly cover all fields. Hollow sections may be used because of the beauty of their shape or to express lightness, while in other cases their geometrical properties determine their application. In this chapter, examples are given for the various fields and to show the possibilities of constructing with hollow sections.

Elliptical hollow sections are becoming more and more popular among architects and already several examples exist, see for example Fig. 3.9, the airport building in Madrid.

Nowadays, many examples of tubular structures are found in railway stations (Figs. 3.10 and 3.11) and roofs of stadia and halls (Figs. 3.12 to 3.14).

3.1 BUILDINGS AND HALLS

Indeed, as stated by one of the former CIDECT vice presidents, Jim Cran, at the Tubular Structures Symposium in Delft (1977) "The sky is the limit", whilst presenting beautiful applications of structural hollow sections.

In buildings and halls, hollow sections are mainly used for columns and lattice girders or space frames for roofs. In modern architecture, they are also used for other structural or architectural reasons, e.g. facades.

Fig. 3.1 shows a 10-storey building in Karlsruhe, Germany with rectangular hollow section columns 180 x 100. Special aspects are that the columns are made of weathering steel and are water filled to ensure the required fire protection. The columns are connected with water reservoirs to ensure circulation.

Besides the fire protection, a further advantage is that due to the water circulation in the columns, the deformation of the building due to temperature differences by sunshine is limited.

3.2 BRIDGES As mentioned in the introduction, the Firth of Forth Bridge is an excellent example of using the hollow section shape for structural applications in bridges.

Nowadays, many modern examples exist (IISI, 1997).

Figs. 1.4, 3.15 to 3.17 and 3.20 show various examples of pedestrian bridges; some of these are movable bridges.

Fig. 3.2 shows an example of lattice girder trusses used in a roof of an industrial building. For an optimal cost effective design, it is essential that the truss joints are made without any stiffening plates.

Circular hollow sections can also be used as a flange for plate girders, as shown in Fig. 3.17 for a triangular box girder.

An especially appealing application is given in Fig. 3.3, showing a tree-type support of the airport departure hall in Stuttgart, Germany. For the joints, streamlined steel castings are used.

A very nice example of a road-pedestrian bridge is illustrated in Fig. 3.18, being a composite steel-concrete bridge with hollow sections for the arch and braces and a concrete deck.

Fig. 3.4 shows the roof of the terminal of Kansai International Airport in Osaka, Japan with curved triangular girders of circular hollow sections.

Fig. 3.19 shows a railway bridge near Rotterdam, The Netherlands with circular hollow section arches.

3.3 BARRIERS

Fig. 3.5 shows a dome under construction, whereas Fig. 3.6 illustrates a special application using columns and beams in the faรงade for ventilation assuring clean windows in the swimming pool.

There are a few aspects which make hollow sections increasingly suitable for hydraulic structures, such as barriers. Due to environmental restrictions, the maintenance of hydraulic structures requires severe precautions, making durability an important issue.

Structures of hollow sections are less susceptible to corrosion due to the rounded corners. Furthermore, especially circular hollow sections have lower drag coefficients, leading to lower forces due to wave loading. Fig. 3.21 shows a barrier with a support

Fig. 3.7 shows a very nice architectural application in Bush Lane House in the city of London, UK. The external circular hollow section lattice transfers the faรงade loads and the loads on the floors to the main columns. The hollow sections are filled with water for fire protection.

29

structure of circular hollow sections. Fig. 3.22 shows the storm surge barrier near Hook of Holland with triangular arms made of circular hollow sections and a length (250 m) equal to the height of the Eiffel Tower in Paris.

In the agricultural field, glass houses (Fig. 3.33) and agricultural machinery are typical examples. Also in transport, many examples exist but these are outside the scope of this book. Indeed, the sky is the limit.

3.4 OFFSHORE STRUCTURES Offshore, many application examples are available; most of them in circular hollow sections. For the support structure, the jacket or tower, not only is the wave loading important, but also other aspects are leading to the use of circular hollow sections. E.g. in jackets, the circular hollow section piles are often driven through the circular hollow section legs of the jacket, thus the pile is guided through the leg.

Sometimes the internal void is used for buoyancy.

Further, the durability and easy maintenance in severe environments are extremely important.

Hollow section members are used in jackets, towers, the legs and diagonals in topside structures, cranes, microwave towers, flare supports, bridges, support structures of helicopter decks and further in various secondary structures, such as staircases, ladders, etc.

Figs. 3.23 and 3.24 show two examples.

3.5 TOWERS AND MASTS Considering wind loading, corrosion protection and architectural appearance, there is no doubt that hollow sections are to be preferred. However, in many countries, electric power transmission towers are made of angle sections with simple bolted joints.

Nowadays, architectural appearance becomes more important, while stringent environmental restrictions make protection and maintenance increasingly expensive. These factors stimulate designs made of hollow sections (Figs. 3.25 and 3.26).

3.6 SPECIAL APPLICATIONS The field of special applications is large, e.g. along the roads, petrol stations (Fig. 3.27), sound barriers (Fig. 3.28), traffic information gantries (Fig. 3.29), guard rails, parapets and sign posts.

Further, excellent application examples are found in radio telescopes (Fig. 3.30), in mechanical engineering, cranes (Fig. 3.31) and roller coasters (Fig. 3.32).

30

Fig. 3.2 Roof with lattice girders

Fig. 3.1 Faรงade of the Institute for Environment in Karlsruhe, Germany

Fig. 3.3 Airport departure hall in Stuttgart, Germany 31

Fig. 3.5 Dome structure in Gothenburg, Sweden Fig. 3.4 Roof of Kansai International Airport in Osaka, Japan

Fig. 3.7 Bush Lane House in London, UK

Fig. 3.6 Faรงade with ventilation through the RHS columns and beams, Borkum, Germany

Fig. 3.8 Hall for the 2004 Olympic Games, Athens, Greece

Fig. 3.9 Airport Madrid with EHS sections, Spain

32

Fig. 3.10 Railway station in Rotterdam, The Netherlands

Fig. 3.11 TGV railway station at Charles de Gaulle Airport, France

Fig. 3.12 Barrel dome grid for the Trade Fair building in Leipzig, Germany

Fig. 3.13 Retractable roof for the Rogers Centre in Toronto, Canada

Fig. 3.14 Stadium Australia for the 2000 Olympic Games, Sydney, Australia

33

Fig. 3.15 Movable pedestrian bridge in RHS, The Netherlands

Fig. 3.16 Movable pedestrian bridge in RHS near Delft, The Netherlands

Fig. 3.18 Composite road bridge in Marvejols, France Fig. 3.17 Pedestrian bridge in Houdan, France

Fig. 3.20 Movable pedestrian bridge in CHS near Delft, The Netherlands

Fig. 3.19 Railway bridge with CHS arches, The Netherlands

Fig. 3.21 Eastern Scheldt barrier, The Netherlands

Fig. 3.22 Storm surge barrier, The Netherlands 34

Fig. 3.24 Amoco P15 offshore platform with jack-up, North Sea

Fig. 3.23 Bullwinkle offshore structure, Gulf of Mexico

Fig. 3.25 Electric power transmission tower

Fig. 3.26 Mast, The Netherlands

Fig. 3.27 Petrol station, The Netherlands

Fig. 3.28 Sound barrier, Delft, The Netherlands

35

Fig. 3.29 Traffic information gantry, The Netherlands

Fig. 3.30 Radio telescope

Fig. 3.31 Cranes

Fig. 3.33 Green house, The Netherlands

Fig. 3.32 Roller coaster

36

4. COMPOSITE STRUCTURES

4.3 AXIALLY LOADED COLUMNS

4.1 INTRODUCTION

From the work of Roik et al. (1975), a simplified design method is given in Eurocode 4 (EN 1994-1-1), similar to the design method adopted for steel columns, i.e.:

Concrete filled hollow sections (Fig. 4.1) are mainly used for columns. The concrete filling gives a higher load bearing capacity without increasing the outer dimensions. The fire resistance can be considerably increased by concrete filling, in particular if proper reinforcement is used.

NEd   Npl,Rd

(4.1)

where: design normal force (including load factors) NEd χ reduction factor for the relevant buckling curve, i.e. curve "a" for ρs  3% and curve "b" for 3% < ρs  6% (see Fig. 2.3) Npl,Rd resistance of the cross section to normal force according to eq. (4.2)

Due to the fact that the steel structure is visible, it allows a slender, architecturally-appealing design. The hollow section acts not only as the formwork for the concrete, but also ensures that the assembly and erection in the building process are not delayed by the hardening process of the concrete.

Npl,Rd = Aa fyd + Ac fcd + As fsd

CIDECT research on composite columns started already in the sixties, resulting in monographs and design rules, adopted by Eurocode 4 (EN 1994-1-1, 2004). CIDECT Design Guide No. 5 (Bergmann et al., 1995) provides detailed information for the static design of concrete filled columns.

(4.2)

To a large extent, this chapter follows the information given in Design Guide No. 5, but updated with the latest revisions to Eurocode 4 (EN 1994-1-1).

where: Aa, Ac, As cross sectional areas of structural steel, concrete and reinforcement fyd, fcd, fsd design strengths of steel, concrete (see Table 4.1) and reinforcement using the recommended γM factors according to Eurocode 2 (EN 1992-1-1, 2004) and Eurocode 3 (EN 1993-1-1, 2005) being γa = 1,0 for fy, γc = 1,5 for fc, and γs = 1,15 for fs

4.2 DESIGN METHODS

The load factors for the actions F have to be determined from EN 1990 (2002).

In the last decades, several design methods for composite columns were developed, e.g. in Europe by Guiaux & Janss (1970), Roik et al. (1975) and Virdi & Dowling (1976), finally resulting in the design rules given in Eurocode 4 (EN 1994-1-1, 2004).

Concrete classes higher than C50/60 should not be used without further investigation and classes lower than C20/25 are not allowed for composite construction.

In concrete filled hollow sections, the concrete is confined by the hollow section. Therefore, the concrete strength reduction factor of 0,85 does not have to be considered.

In this chapter, the design method given is based on the approach presented in Eurocode 4 (EN 1994-1-1).

The design of composite columns has to be carried out at the ultimate limit state, i.e. the effect of the most unfavourable combination of actions should not exceed the resistance of the composite member.

The χ reduction factor follows from the relative slenderness 

An exact calculation of the load bearing capacity considering the effect of imperfections and deflections (second order analysis), the effect of plastification of the section, cracking of the concrete, etc. can only be carried out by means of a computer program. With such a program, the resistance interaction curves as shown in Fig. 4.2, can be calculated. Based on these calculated capacities, the following simplified design methods have been developed.



  E

Npl,Rk Ncr,eff

(4.3)

where: Npl,Rk resistance of the cross section to axial load according to eq. (4.2), however, with fyd, fcd and fsd replaced by fyk, fck and fsk Ncr,eff elastic buckling capacity of the member (Euler 37

critical load)





Ncr,eff =

 (EI)eff  2b

If the parameter δ is less than 0,2, the column shall be designed as a concrete column following Eurocode 2 (EN 1992-1-1). On the other hand, when δ exceeds 0,9, the column shall be designed as a steel column according to Eurocode 3 (EN 1993-1-1).

The buckling (effective) length of the column can be determined by following the rules of Eurocode 3 (EN 1993-1-1).

Ec,eff =

Ecm NG,Ed 1 t NEd

(4.8)

Npl,Rd

(4.4)

where: ℓb buckling length of the column (EI)eff effective stiffness of the composite section

(EI)eff = Ea Ia + 0,6 Ec,eff Ic + Es Is

A a f yd

To avoid local buckling, the following limits should be observed for bending and compression loading (EN 1994-1-1, 2004): - For concrete filled rectangular hollow sections (with h being the greater overall dimension of the section):

(4.5) (4.6)

h/t  52ε

(4.9)

- For concrete filled circular hollow sections:

where: Ia, Ic, Is

d/t  90ε2

moments of inertia of the cross sectional areas of structural steel, concrete (with the area in tension assumed to be uncracked) and reinforcement, respectively Ea, Ecm, Es moduli of elasticity of structural steel, concrete and reinforcement Ec,eff modulus of elasticity of concrete corrected for creep with Ecm according to Table 4.1 NEd acting design normal force NG,Ed permanent part of NEd φt creep factor according to Clause 3.1 of Eurocode 2 (EN 1992-1-1)

(4.10)

The factor ε accounts for different yield strengths: ε=

235 fyd

(4.11)

with fyd in N/mm2.

Although the d/t and h/t values given in Table 4.2 are equal (for CHS) or higher (for RHS) than those of class 3 for unfilled sections, the plastic resistance of the section can be used. However, for the analysis of the internal forces in a structure, an elastic analysis should be performed. Further discussions on slenderness limits for unfilled CHS and RHS and the effect of concrete filling can be found in Zhao et al.

(2005).

The calibration factor 0,6 in eq. (4.5) is incorporated to consider, for example, the effect of cracking of concrete under moment action due to second order effects.

4.3.1 Limitations 4.3.2 Effect of long term loading

The reinforcement to be included in the design calculations should not exceed 6% of the concrete area. There is no minimum requirement.

The influence of the long-term behaviour of the concrete on the load bearing capacity of the column is included by a modification of the concrete modulus of elasticity, since the load bearing capacity of the columns may be reduced by creep and shrinkage. As shown in eq. (4.6) for a load which is fully permanent, the modulus of elasticity of the concrete will be considerably reduced.

The composite column is considered as "composite" if: 0,2  δ  0,9

(4.7)

where:

38

Comparing the stress distribution of point B, where the normal force is zero, and that of point C with the same moment as in point B and axial force NC,Rd (Fig. 4.10), the neutral axis moves over a distance 2hn. Hence, the normal force NC,Rd can be calculated by the additional compressed parts of the section with depth 2hn. Because the force NC,Rd does not contribute to the moment MC,Rd = MB,Rd.

4.3.3 Effect of confinement For concrete filled circular hollow section columns with a small relative slenderness   0,5 (for CHS, this is approximately ℓ/d  12) and e/d  0,1, the bearing capacity is increased due to the impeded transverse strains. This results in radial compression in the concrete and a higher resistance to normal stresses, see Fig. 4.3. Above these values, the confinement effect is very small.

Furthermore, the normal force at point C is twice the value of that at point D: NC,Rd = 2ND,Rd.

For concrete filled rectangular hollow sections, any confinement effect is neglected.

4.6 INFLUENCE OF SHEAR FORCES

Detailed information can be found in Eurocode 4 (EN 1994-1-1).

The influence of the shear stresses on the normal stresses does not need to be considered if: VEd  0,5 Vpl,Rd

4.4 RESISTANCE OF A SECTION TO BENDING

(4.12)

The shear force on a composite column may either be assigned to the steel profile alone or be divided into a steel and a reinforced concrete component. The component for the structural steel can be considered by reducing the axial stresses in those parts of the steel profile which are effective for shear (Fig. 4.11).

For the determination of the resistance of a concrete filled section to bending moments, a full plastic stress distribution in the section is assumed (Fig. 4.4). The concrete in the tension zone of the section is assumed to be cracked and is therefore neglected. The internal bending moment resulting from the stresses and depending on the position of the neutral axis is the resistance of the section to bending moments Mpl,Rd.

The reduction of the axial stresses due to shear stresses may be carried out according to the Huber-Hencky-Von Mises criterion or according to Eurocode 4 (EN 1994-1-1). For the determination of the cross-section interaction, it is easier to transform the reduction of the axial stresses into a reduction of the relevant cross sectional areas equal to that used for hollow sections without concrete filling:

4.5 RESISTANCE OF A SECTION TO BENDING AND COMPRESSION The resistance of a concrete filled cross section to bending and compression can be shown by the interaction curve between the normal force and the internal bending moment.

2   2V   reduced Av = Av 1   Ed  1      Vpl,Rd   

Figs. 4.5 to 4.8 show the interaction curves for RHS and CHS columns in relation to the cross section parameter δ. These curves have been determined without any reinforcement, but they may also be used for reinforced sections if the reinforcement is considered in the δ values and in Npl,Rd and Mpl,Rd respectively.

Vpl,Rd  A v

f yd 3

(4.13)

(4.14)

For Av, see Chapter 2.

4.7 RESISTANCE OF A MEMBER TO BENDING AND COMPRESSION

The interaction curve has some significant points, shown in Fig. 4.9. These points represent the stress distributions given in Fig. 4.10. The internal moments and axial loads belonging to these stress distributions can be easily calculated if effects of the corner radius are excluded.

4.7.1 Uniaxial bending and compression Fig. 4.12 shows the principle of the method for the design of a composite member under combined compression and uniaxial bending using the 39

cross-section interaction curve. Due to imperfections, the resistance of an axially loaded member is given by eq. (4.1) or χ on the vertical axis in Fig. 4.12.

by:

The moment capacity factor at the level of χ is defined as the imperfection moment. Having reached the load bearing capacity for axial compression, the column cannot resist any additional bending moment.

where:

MEd,|| = k MEd

k

The value of χd resulting from the actual design normal force NEd (χd = NEd/Npl,Rd) determines the moment capacity factor μd for the capacity of the member. This factor μd gives the moment capacity including the imperfection moment, thus the imperfection moment should be added to the external moment including second order effects.

(4.16)

1 N 1  Ed Ncr,eff

(4.17)

k is the amplification factor to incorporate the second order effects.

Ncr,eff can be determined with eq. (4.4), however, with a modified (EI)eff,|| due to the simplifications mentioned before:

The capacity for the combined compression and bending of the member can now be checked by:

(EI)eff,|| = 0,9 (Ea Ia + 0,5 Ec,eff Ic + Es Is)

M||,max  αM μd Mpl,Rd

The total moment including the imperfection moment is:

(4.15)

where: M||,max design bending moment of the column, including the imperfection moment and second order effects αM 0,9 for S235 to S335 and 0,8 for S420 and S460 μd to be obtained from the interaction diagrams in Figs. 4.5 to 4.8

M||,max =

1 N 1  Ed Ncr,eff

(MEd + NEd e0)

(4.18)

(4.19)

The capacity can now be checked with eq. (4.15).

Columns with different end moments If the end moments are not equal (see Fig. 4.13), then the k factor in eq. (4.17) has to be corrected for the external moment by a factor β:

The additional reduction by the factor αM accounts for the assumptions of this simplified design method, e.g. the interaction curve of the section is determined assuming full plastic behaviour of the materials with no strain limitation.

k

Note: Interaction curves of the composite sections always show an increase in the bending capacity higher than Mpl,Rd. The bending resistance increases with an increasing normal force, because former regions in tension are compressed by the normal force. This positive effect may only be taken into account if it is ensured that the bending moment and the axial force always act together. If this is not ensured, and the bending moment and the axial force result from different loading situations, the related moment capacity μd has to be limited to 1,0.

 N 1  Ed Ncr,eff

(4.20)

where: β = 0,66 + 0,44r but β  0,44

(4.21)

with r being the ratio between the smallest and largest end moment (Fig. 4.13).

The total moment including the imperfection moment is now: M||,max 

Columns with equal end moments The verification procedure for columns with the same end moments given in Eurocode 4 (EN 1994-1-1) is as follows:

 MEd N e  Ed 0 N NEd 1  Ed 1 Ncr,eff Ncr,eff

(4.22)

This moment has to be used in eq. (4.15). If the first order moment is larger than MEd,||, then this value

The second order moment MEd,|| can be approximated 40

should be used.

4.9 SPECIAL COMPOSITE MEMBERS WITH HOLLOW SECTIONS

4.7.2 Biaxial bending and compression

The previous sections consider composite members consisting of a hollow section at the outer side and concrete inside. The concrete may be reinforced or not. However, an alternative is to reinforce the concrete with steel fibres instead of reinforcing bars which provide advantages in the extension of the fire resistance.

A composite member under biaxial bending and compression has first to be examined for both axes under uniaxial bending and compression, see Section 4.7.1. Additionally the combined situation has to be verified. The influence of the imperfection is only taken into account for the buckling axis which is most critical.

Other types of reinforcement used are solid sections or another hollow section inside a circular or rectangular hollow section with concrete in between.

Fig. 4.16 shows an example of a CHS with another CHS member inside. Although many combinations are possible, the design is in principle similar to that for the reinforced concrete hollow section columns described in the previous sections (Zhao et al., 2010).

The check can be expressed by the following condition: M y,Ed  dy Mpl,y,Rd



Mz,Ed  dz Mpl,z,Rd

 1,0

(4.23)

The values μdy and μdz are determined at the level of χd.

4.8 LOAD INTRODUCTION In the design of composite columns, a full composite action of the cross section is assumed. This means that in the bond area no significant slip can occur between the steel and the concrete. At locations of load introduction, e.g. at beam-column connections, this has to be verified. If no calculation is carried out, the length of load introduction should be assumed to be the minimum of 2b, 2d or ℓ/3, where b or d is the minimum transverse dimension of the column, and ℓ is the column length.

If the steel is not painted and is free of oil and rust, the maximum bond stress, based on friction is (EN 1994-1-1, 2004): - τRd = 0,55 N/mm2 for CHS columns - τRd = 0,40 N/mm2 for RHS columns The shear load transfer can be considerably by shear connectors components, see Fig. 4.14.

increased or steel

For concentrated loads, a load distribution according to Fig. 4.15 can be assumed. For such locally loaded parts of encased concrete, higher design values for the concrete strength can be used.

41

Table 4.1 Strength classes of concrete, characteristic cylinder strength and modulus of elasticity for normal weight concrete

Strength class of concrete fck,cyl/fck,cub Cylinder strength fck (N/mm2) 2

Modulus of elasticity Ecm (N/mm )

C20/25

C25/30

C30/37

C35/45

C40/50

C45/55

C50/60

20

25

30

35

40

45

50

30000

31000

33000

34000

35000

36000

37000

Note: The recommended values γa = 1,0, γc = 1,5 and γs = 1,15 should be used to determine the design values.

Table 4.2 Limits for wall thickness ratios of concrete filled hollow sections for preventing local buckling under axial compression (EN 1994-1-1, 2004)

Steel grade

S235

S275

S355

S460

Rectangular hollow sections – eq. (4.9)

h/t

52,0

48,1

42,3

37,2

Circular hollow sections – eq. (4.10)

d/t

90,0

76,9

59,6

46,0

42

Fig. 4.1 Concrete filled hollow sections with notations

S235 / C45 d = 500 mm t = 10 mm

NEd/Npl,Rd 1,00

0,75

0,50

0,25

0 0

0,25

0,50

0,75

1,00 NEde/Mpl,Rd

Fig. 4.2 Bearing capacity of a composite hollow section column

43

Fig. 4.3 Three dimensional confinement effect in concrete filled hollow sections

Fig. 4.4 Stress distribution for the bending resistance of a section

NEd/Npl,Rd

,

parameter  

1,0

A a fyd Npl,Rd

 = 0,45 0,8

0,40

0,6

0,35

0,30 0,275 0,25 0,225 0,20

0,4 0,9 0,8

0,2

0,7 0,6 0,5

0 0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

MEd/Mpl,Rd ,

Fig. 4.5 Interaction curve for rectangular hollow sections with bending about the weak axis, b/h = 0,5

44

NEd/Npl,Rd

,

A a fyd

parameter  

1,0

N, pl,Rd  = 0,45

0,8

0,40

0,35

0,6

0,4

0,30 0,275 0,25 0,225 0,20

0,9 0,8 0,7

0,2

0,6 0,5 0,5

0 0,2

0

0,4

0,6

0,8

1,0

1,2

MEd/Mpl,Rd

1,4

Fig. 4.6 Interaction curve for square hollow sections with b/h = 1,0

NEd/Npl,Rd

parameter  

1,0

A a fyd Npl,R, d  = 0,45

0,40

0,35

0,30

0,8 0,6

0,275 0,25 0,225 0,20

0,9

0,4

0,8 0,7

0,2

0,6 0,5

0 0

0,2

0,4

0,6

0,8

1,0

1,2

MEd/Mpl,Rd

1,4

Fig. 4.7 Interaction curve for rectangular hollow sections with bending about the strong axis, h/b = 2,0

NEd/Npl,Rd 1,0

parameter  

,

A a fyd Npl,Rd ,  = 0,45

0,8

0,40

0,35

0,30 0,275

0,6

0,4

0,25 0,225 0,20

0,9 0,8 0,7

0,2

0,6 0,5

0 0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

Fig. 4.8 Interaction curve for circular hollow sections 45

MEd/Mpl,Rd

NA,Rd NE,Rd NC,Rd

ND,Rd NB,Rd MA,Rd

MB,Rd MD,Rd MC,Rd

Fig. 4.9 Interaction curve approached by a polygonal connection of the points A to E

Npl,Rd

MB,Rd = Mpl,Rd

MC,Rd = Mpl,Rd NNC,Rd C,Rd

MD,Rd = Mmax,Rd ND,Rd = 0,5NC,Rd D,Rd= 0,5NC,Rd

ME,Rd NE,Rd -

Fig. 4.10 Stress distributions of selected positions of the neutral axis (points A to E) 46

Fig. 4.11 Reduction of the normal stresses due to shear

,

NEd Npl,Rd

MEd Mpl,Rd

d ,

Fig. 4.12 Design for compression and uni-axial bending

MEd

r MEd

Fig. 4.13 Relation between the end moments (-1  r  +1)

47

Fig. 4.14 Load introduction into hollow sections by inserted plates

1:2,5

1:2,5

Fig. 4.15 Load introduction in a composite column outer tube t0 ti inner tube di

d0

concrete

Fig. 4.16 Tube-in-tube composite column concept

48

swell up to a multiple of their original thickness) - Suspended ceilings (mainly protecting roofs, trusses) - Heat radiation shielding

5. FIRE RESISTANCE OF HOLLOW SECTION COLUMNS 5.1 INTRODUCTION

In some countries, intumescent coatings are restricted to a fire resistance of 30 or 60 minutes, but this technology is rapidly developing and nowadays considerably larger protection times are possible.

This chapter is a reduced version of CIDECT Design Guide No. 4 (Twilt et al., 1994), however, updated with the latest revisions to Eurocodes 3 and 4 on structural fire design (EN 1993-1-2, 2005; EN 1994-1-2, 2005).

5.1.2 Concrete filling of the section

Unprotected structural hollow sections have an inherent fire resistance of approximately 15 to 30 minutes. Traditionally, it was assumed that unprotected steel members fail when they reach temperatures of about 450 to 550 ď&#x201A;°C. However, the temperature at which a steel member reaches its ultimate limit state depends on the massivity of the section and the actual load level. If the service load level of a column is less than 50% of its resistance, the critical temperature rises to over 650 ď&#x201A;°C, which, for bare steel, means an increase in failure time of more than 20%.

Usually, fire protection through concrete filling of the section is applied to columns only. Filling hollow sections with concrete is a very simple and attractive way of enhancing fire resistance. The temperature in the unprotected outer steel shell increases rapidly.

However, as the steel shell gradually loses strength and stiffness, the load is transferred to the concrete core.

Apart from the structural function, the hollow section also acts as a radiation shield to the concrete core, in combination with a steam layer between the steel and the concrete core.

When hollow steel sections are required to withstand extended amounts of time in fire, additional measures have to be considered to delay the rise in steel temperature.

Depending on the fire resistance requirements, the concrete in the hollow section can be plain concrete (fire resistance 30 minutes up to 60 minutes) or concrete with reinforcing bars or steel fibres. New research aimed at increasing the fire resistance of concrete filled hollow sections is focused on the use of high strength concrete.

5.1.1 External insulation of the steel section External insulation of the steel section is a type of fire protection that can be applied to all kinds of structural elements (columns, beams and trusses). The temperature development in a protected hollow steel section depends on the thermal properties of the insulation material (conductivity), on the thickness of the insulation material and on the section factor (massivity) of the steel profile.

5.1.3 Water cooling Water cooling is a type of fire protection that can be applied to all kinds of hollow sections, but is mostly used for columns. The hollow section acts both as the load bearing structure and as the water container.

This protection system is quite sophisticated; it needs a thorough design and proper hydraulic installations.

External fire protection materials can be grouped as follows: - Insulating boards (based mainly on gypsum or mineral fibre or lightweight aggregates such as perlite and vermiculite). If board protection is to be used, care must be taken to ensure the integrity of joints between the boards.

- Spray coating or plaster (based mainly on mineral fibre or lightweight aggregates such as perlite and vermiculite) - Intumescent coatings (paint-like mixtures applied directly to the steel surface which, in case of fire,

The cooling effect consists of the absorption of heat by water, the removal of heat by water circulation and its consumption in the vaporization of water. In practical applications, these effects are combined. A suitably designed water filled system will limit the average steel temperature to less than 200 ď&#x201A;°C. Two different systems can be used: permanently filled elements or elements filled only when a fire breaks out. In the latter case, protection depends on a fire

49

detection system and a short water filling time. In unreplenished systems, the attainable fire resistance time depends on the total water content (including any reservoir tank) and on the shape of the heated structure. In systems where the water is constantly renewed, the fire resistance is unlimited. Water cooling by natural flow is mainly used for vertical or inclined elements in order to ensure the circulation of the water.

element during fire. For more details, see Twilt et al.

(1994).

5.2 FIRE RESISTANCE

The experimental approach, i.e. the determination of the fire resistance of columns based on standard fire tests, is the traditional approach. Although employing different national testing procedures, the concept of fire testing is, by and large, the same in the various countries.

For building components such as columns, with a load bearing function, the only relevant performance criterion is "stability".

As far as the determination of the fire resistance is concerned, there are basically two possibilities: an experimental approach and an analytical approach.

5.2.1 Concept Fire safety precautions are specified with the intent of avoiding any casualties and reducing economic fire damage to an acceptable level. As far as building construction is concerned, it is important that the construction elements can withstand a fire for a specified amount of time. In this respect, one should bear in mind that the strength and deformation properties of the commonly used building materials deteriorate significantly at the temperatures that may be expected under fire conditions. Moreover, the thermal expansion of most building materials is considerable. As a result, the structural elements and assemblies may deform or even collapse when exposed to fire conditions.

The analytical approach is the modern approach and has become possible by the development of computer technology. On an international level, calculation rules for the fire resistance of both steel and composite steel concrete columns, including concrete filled hollow section columns, are available. The analytical approach offers significant advantages, when compared with the experimental one.

Important factors influencing the fire resistance of columns are: - Load level - Shape and size of the cross section - Buckling length - Concrete filling and reinforcement

The amount of time that a construction element can resist a fire largely depends on the anticipated temperature development of the fire itself. This temperature development depends, among other aspects, on the type and amount of combustible materials present, expressed in terms of kg of wood per m2 floor surface and called the fire load density, see Fig. 5.1, and on the fire ventilation conditions.

Bare steel columns (i.e. hollow section columns without external protection or concrete filling) possess only a limited fire resistance. Depending on the load level and the section factor (massivity), a fire resistance of 15 to 20 minutes is usually attainable. A 30 minutes fire resistance can only be achieved in more exceptional cases. This situation may be dramatically improved by applying thermal insulation to the column. Depending on the type and thickness of the insulation material, fire resistances of many hours can be achieved, although most requirements today are limited to 120 minutes.

In practical fire safety design, however, it is conventional to use a so-called "standard fire curve", defined in ISO 834-1 (1999), which is more or less representative for post flash-over fires in buildings with relatively small compartments, such as apartment buildings and offices. Alternative standard fire curves, with small differences from the ISO-curve, are in use in the USA.

Hollow section columns filled with concrete have a much higher load bearing capacity and a higher fire resistance than unprotected, empty hollow section columns. Provided the concrete is of good quality (over, say, a crushing strength of 20 N/mm2) and the cross sectional dimensions are not too small (not less than 150 x 150 mm), a fire resistance of at least 30 minutes will be achieved. Sections with larger

The amount of time a building component is able to withstand heat exposure according to the standard fire curve, is called the "fire resistance". In order to be able to determine the fire resistance of a building component, proper performance criteria have to be determined. These criteria are defined in relation to the anticipated function of the respective building

50

fire curve (ISO 834-1, 1999) is common practice in Europe and elsewhere. The standard fire test is not intended to reflect the temperatures and stresses that would be experienced in real fires, but provides a measure of the relative performance of elements of structures and materials within the capabilities and dimensions of the standard furnaces. In general, uncertainties about structural behaviour in real fires are taken into account by making conservative fire resistance requirements.

dimensions will have a higher fire resistance and by adding additional reinforcement to the concrete, the fire resistance may be increased to over 120 minutes.

Infinite fire resistance can be achieved by water filling, provided an adequate water supply is available.

Improved fire performance of hollow section columns can also be achieved by placing the columns outside the building envelope â&#x20AC;&#x201C; an expedient sometimes used for architectural purposes. By preventing direct flame impingement on the member, the need for additional fire protection measures can be significantly reduced or even become unnecessary.

Required safety levels are specified in Codes and normally depend on factors like: - Type of occupancy - Height and size of the building - Effectiveness of fire brigade action - Active measures, such as vents and sprinklers

Since fire safety requirements for columns are normally expressed in terms of the fire resistance to be attained, this emphasizes the need to consider the fire resistance requirements from the beginning in a structural design project.

An overview of fire resistance requirements as a function of the number of storeys and representative for European countries is given in Table 5.1.

5.2.2 Requirements

The following general features may be identified: - No specified fire resistance requirements for buildings with limited fire load density (say, 15-20 kg/m2) or where the consequences of collapse of the structure are acceptable.

- Fire resistance for a specified but limited amount of time, where the time requirement is mainly intended to allow for safe evacuation of the occupants and intervention by rescue teams.

- Extended fire resistance of the main structure to ensure that the structure can sustain a full burn out of combustible materials in the buildings or a specified part of it.

Fire safety in buildings is based on achieving two fundamental objectives: - Reducing the loss of life - Reducing the property or financial loss in, or in the neighbourhood of, a building fire In most countries, the responsibility for achieving these objectives is divided between the government or civic authorities who have the responsibility for life safety via building regulations, and the insurance companies dealing with property loss through their fire insurance policies.

Sometimes unprotected steel may be sufficient, for example for situations where safety is satisfied by other means (e.g. sprinklers) and/or if requirements with respect to fire resistance are low (i.e. not over, say, 30 minutes).

The objectives of fire safety may be achieved in various ways. For example: - By eliminating or protecting possible ignition sources (fire prevention).

- By installing an automatic extinguishing device, in order to prevent the fire from growing into a severe fire (operational or active measures, e.g. sprinklers).

- By providing adequate fire resistance to the building components using passive measures to prevent fire spreading from one fire compartment to adjacent compartments.

A full fire engineering approach (Natural Fire Concept), in which compartment and steel temperature are calculated from a consideration of the combustible material present, compartment geometry and ventilation, is nowadays more accepted and has shown considerable savings in fire protection costs in specific cases.

Often a combination of the above measures is applied.

5.2.3 Performance criteria

Requirements with regard to fire resistance clearly belong to the passive measures. To date, the use of a conventional fire scenario employing the ISO standard

The fundamental concept behind all methods designed to predict structural stability in fire is that

51

construction materials gradually lose strength and stiffness at elevated temperatures. The reduction in the yield strength of structural steel and the compression strength of concrete with increasing temperature according to Eurocode 3 (EN 1993-1-2) and Eurocode 4 (EN 1994-1-2) is given in Fig. 5.2. It shows that there is not much difference in the relative reduction in strength of concrete and steel under high temperatures. The reason for the difference in the structural behaviour of steel and concrete elements under fire conditions is that heat propagates about 10 to 12 times faster in a steel structure than in a concrete structure of the same massivity, because the thermal conductivity of steel is higher than that of concrete.

However, the handling of the necessary computer programs is quite time consuming and requires expert knowledge.

Normally, the fire resistance design of structures is based on a similar design approach as used for design under ambient temperature. In a multi-storey braced frame, the buckling length of each column at room temperature is usually assumed to be the column length between floors. However, such structures are usually compartmented and any fire is likely to be limited to one storey. Therefore, any column affected by fire will lose its stiffness, while adjacent members will remain relatively cold.

Accordingly, if the column is rigidly connected to the adjacent members, built-in end conditions can be assumed in the event of fire. Investigations by Twilt & Both (1991) showed that in the case of fire on one storey the buckling length of columns in braced frames is reduced to between 0,5 and 0,7 times the column length, depending on the boundary conditions, see Fig. 5.3.

In the following chapters, only the principles of the design tables and the simple calculation procedures are shown. In all equations, the recommended γM and γM,fi factors in the Eurocodes for steel and concrete are 1,0.

There is an increasing tendency toward assessing the fire resistance of individual members or sub-assembles by analytical fire engineering. The Eurocodes on structural fire design (EN 1993-1-2, 2005; EN 1994-1-2, 2005) define three levels of assessments: - Level 1: Tabulated data - Level 2: Simple calculation models - Level 3: Advanced calculation models

The critical temperature of an axially loaded steel column depends on the load Nfi,Ed which is present during a fire and the buckling resistance NRd at room temperature (Nfi,Ed = ηfi,t NRd).

"Advanced calculation models" is the most sophisticated level. Such calculation procedures include a complete thermal and mechanical analysis of the structure and use the values for the material properties given in the Eurocodes. General calculation methods enable real boundary conditions to be considered and take account of the influence of non-uniform temperature distribution over the section and therefore lead to more realistic failure times and, consequently, to the most competitive design.

As a simplification for the calculation, the load Nfi,Ed which is present during a fire may also be related by the ratio ηfi to the design load NEd at room temperature. This ratio depends on the load ratio between variable and permanent loading, but as a simplification, generally a ratio ηfi = 0,65 may be assumed. Only in cases of areas susceptible to accumulation of goods, the recommended value is 0,70.

For practising engineers and architects not accustomed to handling specialised computer programs, "Simple calculation models" have been developed, which lead to a comprehensive design, but are limited in application range. They use conventional calculation procedures and normally provide adequate accuracy.

"Tabulated data", which provide solutions on the safe side and allow fast design for restricted application ranges, forms the lowest level of assessment.

5.3 UNFILLED HOLLOW SECTION COLUMNS The simple calculation rules for the critical temperature of steel columns discussed hereafter, hold for classes 1, 2 and 3 cross sections only (as defined in Eurocode 3 (EN 1993-1-1, 2005)) and can be applied both to protected and unprotected columns. For columns with a class 4 cross section, a default value for the critical temperature of 350 C is to be used.

Note: Instead of the symbols Efi,d,t and Rfi,d,t used in Eurocode 3 (EN 1993-1-2, 2005), for consistency the symbols Nfi,Ed and Nfi,Rd refer here to the design load and the design resistance in the fire situation.

52

with t being the thickness of the steel hollow section.

Hence, generally:

Nfi,Ed  fi NEd  0,65 NEd

For a practical range of the hollow section thickness of 20 to 2.5 mm, this also leads to section factors varying between 50 and 400 m-1. However, for equivalent cross sections, hollow sections have an Am/V ratio which is about 60% of that of comparable open sections.

(5.1)

where: Nfi,Ed design load in the fire situation NEd design load at room temperature ηfi reduction factor for NEd to obtain Nfi,Ed

For any given critical steel temperature, the fire resistance of an unprotected steel element – assuming standard fire conditions – depends only on its section factor as illustrated in Fig. 5.7. In many practical situations, the critical temperature of a steel member will be approximately 550 C. This figure shows that an unprotected steel member with a section factor smaller than approximately 40 m-1 may have a fire resistance of 30 minutes or beyond.

Now, the following verification has to be carried out:

Nfi,Ed  Nfi,Rd   fi k y, Npl,Rd

(5.2)

where: Nfi,Ed design load in the fire situation Nfi,Rd buckling resistance in the fire situation Npl,Rd compression resistance of gross cross section at room temperature χfi reduction factor for flexural buckling in the fire design situation, see Fig. 5.4 ky,θ reduction factor for the yield strength of steel at a steel temperature θa, see Fig. 5.5

If external fire insulation is provided, the steel temperature development depends not only on the section factor, but also on the type and thickness of the insulation material.

This check is, in principle, similar to that for members in compression at room temperature. Only the buckling coefficient χ at room temperature has to be replaced by χfi at temperature θa in the fire situation and the yield strength fyd at room temperature by ky,θ fyd in the fire situation. The load factors are usually 1,0.

5.4 CONCRETE FILLED HOLLOW SECTION COLUMNS 5.4.1 Unprotected columns – thermal and mechanical response Because of their different locations in the cross section, the various components of a concrete filled hollow section column will each have different time-dependent strength reduction characteristics.

The unprotected, directly exposed steel shell will be rapidly heated and will show a significant strength reduction within a short time.

For unprotected steel sections, it can be shown that – for standard fire exposure – the temperature development of a steel section depends only on the relative geometry of the profile. This effect is taken into account by means of the shape factor, Am/V, where: Am exposed surface area of the member per unit length [m2/m] V volume of the member per unit length [m3/m] This ratio is equal to the exposed steel perimeter/steel cross section.

The concrete core with its high massivity and low thermal conductivity will, for some time, maintain a significantly high proportion of its strength, mostly in the core area rather than near the surface.

The curves presented in Fig. 5.6 illustrate the effect of the section factor on the temperature development of an unprotected steel section when exposed to standard fire conditions. For commonly used I sections, shape factors are within a range of, say, 50 to 400 m-1. For hollow sections exposed to heat from all sides, the section factor may be approximated by:

Reinforcement, if used, is normally placed near the surface, but is protected by typically 20-50 mm of concrete cover. For this reason, it will have a retarded strength reduction. Fig. 5.8 demonstrates this characteristic behaviour and describes the fire performance of the various components of concrete filled hollow section columns.

Am/V = Perimeter/(Perimeter x thickness t) = 1/t [m-1]

The load bearing capacity R of a cross section is the sum of the load bearing capacities of each of its

53

components rj. Under fire conditions, all component capacities are dependent on the fire endurance time t.

R(t) = Σrj(t)

- The minimum cover of the reinforcing bars (us) The slenderness relation is indirectly incorporated in the load level ηfi,t which is given by:

(5.3)

In room temperature design, the steel shell is likely to be the dominant load bearing component because of the high strength of the steel and the location of the profile. However, after a fire time t1, only a small percentage of the original load bearing capacity of the steel shell can still be activated. This means that in the case of fire the main part of the load carried by the steel section will be redistributed to the concrete core, which loses strength and stiffness more slowly than the steel section. Therefore: - The load bearing capacity of the steel shell should be minimised, which means thin shell thickness and low steel grade.

- The load bearing capacity of the concrete core should be optimised, which means higher concrete strength and reinforcement.

ηfi,t = Nfi,Ed /NRd

(5.4)

where: Nfi,Ed design load in the fire situation NRd buckling resistance at room temperature NRd is calculated according to the room temperature procedures given in Eurocode 4 (EN 1994-1-1, 2004).

However, the following limitations apply: - Irrespective of the actual steel grade, the yield strength of the hollow sections for fire calculations is limited to a maximum of 235 N/mm2.

- The wall thickness of the steel is limited to a maximum of 1/25 of the cross sectional dimension d, b or h.

- Reinforcement ratios higher than 3% are not taken into account.

- The values given in Table 5.2 are valid for steel grade S500 used for the reinforcement As.

Since the strength reduction of the components is directly affected by the heating characteristic of the cross section, a minimum column cross sectional dimension is often necessary to fulfill a required fire resistance.

Level 2: Simple calculation models At level 2, a computer program has been developed to model the fire performance of concrete filled hollow section columns (Grandjean et al., 1980; Twilt & Van de Haar, 1986). This software tool is based on Annex H (informative) of Eurocode 4 (EN 1994-1-2) for concrete filled hollow section columns at elevated temperatures. However, this program will be modified, because comparisons with other programs revealed discrepancies on the non-conservative side.

With increasing temperature, the strength and Young's modulus decrease. Thus, the load bearing capacity of a structural member decreases with time, while its deformation increases. In practical fire design, the influence of the column slenderness has also to be taken into account.

5.4.2 Assessment methods for unprotected columns

Design charts have been developed in which, for a standard fire exposure of 30, 60, 90 and 120 minutes, the design axial buckling load Nfi,Rd of concrete filled hollow section columns is given as a function of the buckling length ℓθ and the sectional parameters. Figs.

5.9 and 5.10 illustrate typical charts of Annex H (status 2010) of Eurocode 4 (EN 1994-1-2).

Levels of assessment As already explained in Section 5.2.3, Eurocodes 3 and 4 on structural fire design (EN 1993-1-2, 2005; EN 1994-1-2, 2005) define three different levels of assessment. This chapter deals with design information at levels 1 and 2, i.e. "Tabulated data" and "Simple calculation models". For more general calculation models, see Twilt et al. (1994).

These charts should only be used if the following conditions are satisfied: - Buckling length in the fire condition ℓθ  4,5m - Width (b or h) or diameter (d): 140  (b, h or d)  400 mm - Concrete strength minimum C20/25 and maximum C40/50 - Reinforcement ratio  5% - Fire resistance R  120 minutes

Level 1: Tabulated data The fire rating of unprotected concrete filled hollow section columns may be classified according to Table 5.2 as a function of: - The load level ηfi,t - The cross section size (b, h or d) - The amount (%) of reinforcement (As /(Ac + As) 100)

For a given buckling length and loading, the fire 54

of hollow sections with other steel sections. The advantages of such special cross section types are an increased load bearing capacity without the need to increase the outer cross sectional dimensions, or reduced dimensions for a given load capacity.

resistance of concrete filled hollow section columns mainly depends on the cross sectional dimensions, the concrete quality and the reinforcement, if any. By a proper choice of these parameters, practically any fire resistance can be achieved. If no reinforcement is used, a fire resistance of 30 minutes can normally be achieved; 60 minutes, however, is not attainable unless the load level is significantly decreased. As a result, the design charts focus on reinforced hollow section columns with fire resistances of 60 minutes and more.

To fulfill architectural requirements, special steels, such as weathering steel, can also be used for the hollow sections of the columns.

Careful design of the top and bottom of a single column or at the connection of a continuous column is necessary to ensure that the loading is introduced into the composite cross section in a proper way.

The effect of small eccentricities (M/N  0,5(d or b)) can be taken into account by increasing the axial load Nfi,Ed to an equivalent axial load Nequ:

Nequ 

Nfi,Ed s e

5.4.4 Externally protected concrete filled hollow section columns

(5.5)

If an extended fire resistance is desired, in combination with a high load level and/or a minimised column cross section, it may be necessary to apply conventional external protection to a concrete filled hollow section column.

where: φs correction coefficient related to the reinforcement, see Fig. 5.11. φe correction coefficient related to the eccentricity e, see Fig. 5.12.

Note: There are some concerns about the Annex H method. Wang & Orton (2008) pointed out that this method is rather antiquated. An alternative method, developed by Wang & Orton (2008), is based on the well established cold design method for composite columns in the main part of Eurocode 4 (EN 1994-1-1, 2004), but modified to take into account strength and stiffness degradations of steel and concrete at high temperatures. A design software package named "Firesoft" is now available to assist designers, which has been verified by Wang & Orton (2008).

5.5 WATER FILLED HOLLOW SECTION COLUMNS 5.5.1 Basic principles Water filling using natural circulation provides a safe and reliable fire protection method for hollow section columns, provided that two conditions are satisfied (Hönig et al., 1985): - The system is self activating in fire.

- The system is self controlling.

In a properly designed system, the natural circulation will be activated when the columns are locally heated by a fire. The density of warm water is lower than that of cold water, which produces the pressure differentials that activate the natural circulation. The effect will be intensified when localised boiling commences and steam is formed. As the fire develops, the rate of steam production will also increase, thus forcing the cooling effect obtained by naturally activated circulation.

5.4.3 Technological aspects Small vent holes (10 to 15 mm diameter) are required in the hollow section walls, usually in pairs. Such holes must be provided for each storey length at each floor level, with a maximum distance of 5,0 m. between pairs. They must be placed between 100 and 120 mm from each column end. Those holes are intended to prevent the bursting of the column under steam pressure from the heating of entrapped water in the enclosed concrete.

The following methods of permanent water filling are available:

Besides the standard hollow section cross sections, in the past, a variety of different cross section designs were developed and successfully applied in building projects. They are all based either on combinations of hollow sections (tube inside tube) or on combinations

1. Unreplenished columns Simply filling a column with water, with no provision for replacing any water lost through steam production, 55

It is not advisable to use any electro-mechanical installation, such as pumps, acting against the naturally produced circulation. This may lead to a failure of the cooling system and thus to a collapse of the water filled structure.

will lead to an increased, but limited fire resistance compared to that of the empty column. In multi-storey columns the water in the top-storey-columns will be first evaporated, but the fire resistance can be increased by externally protecting the top storey length and using it as a reservoir for the lower storeys.

Heavy steam production may lead to an additional critical loss of water. Therefore, unreplenished columns should be used only for lower fire resistance requirements, up to, say, 60 minutes.

5.5.2 Assessment methods A careful design is necessary to ensure the positive behaviour of a water filled hollow section column system. Two main criteria must be fulfilled to ensure the cooling effect: - Natural circulation of the water is maintained.

- Water losses due to steam production are replaced.

2. Columns with external pipe This system has a connecting down pipe between the bottom and top of the columns. The lighter, upward flowing water-steam mixture must be separated at the top, so that the water can return down through the pipe to the bottom. In this manner, an external naturally forced circulation will be activated. In addition, the pipe can be connected with a water storage tank at the top of the building to replace the water lost from steam production and possibly act as a common water/steam separating chamber. A group of individual columns can be connected at their bottom to a shared connecting pipe as well as with a connecting pipe at the top. For such a group of columns, only one down pipe is necessary, connecting top and bottom of the whole group, see Fig. 5.13a.

The mass of the water cooled steel structure as well as the water within the system can be taken into account when calculating the time of commencement of boiling. The loss of water mass by evaporation has to be estimated only for the time difference between the start of boiling and the required fire resistance time. For the characteristic thermal behaviour, refer to Fig. 5.14.

The maximum temperature reached by the steel can be estimated from the boiling temperature of the water filling. The boiling temperature itself depends on the hydraulic water pressure, i.e. the static head. In addition, there will be a temperature gradient across the wall of the hollow section, which will lead to a slight increase of the temperature of the steel surface directly exposed to the fire. However, the maximum external steel surface temperature will normally not reach a value high enough to significantly affect the mechanical properties of the steel.

3. Columns with internal pipe In this system, an internal down tube is used within each column to provide a supply of cool water to the bottom of each column. This promotes the internal, naturally activated circulation of the upward flowing water-steam mixture and the down flowing water after steam separation. Thus, each column acts as an individual member without any connection to the other columns.

5.6 JOINTS

To minimise the number of water storage tanks, the tops of several columns can be connected by a common pipe leading to one storage tank for the whole group, see Fig. 5.13b.

5.6.1 Unfilled hollow section columns Normally, the joints of both protected and unprotected steel structures have a lower local section factor than the adjacent members and will therefore attain lower steel temperatures. However, if the section factors are higher than for the connected members, they may be the critical elements and the behaviour under fire conditions has to be considered, especially if catenary action may occur (Wang & Ding, 2009). When bolted joints are used for insulated steel members, care must be taken to ensure that the bolt heads and nuts are as well protected as the cleat. Normally, this will lead to a local increase of insulation thickness.

4. Mixed systems The above mentioned systems can be mixed within a building and they can be connected to act as a mixed integrated system. This can be advantageous for structures containing not only columns, but also water filled diagonals for bracing, etc.

In the naturally circulating systems described above, a minimum declination of the diagonals of about 45ď&#x201A;° is recommended.

56

5.6.2 Concrete filled hollow section columns The joints should be designed based on the principle that the loads can be transferred from the beams to the columns in such a way that all structural components â&#x20AC;&#x201C; structural steel, reinforcement and concrete â&#x20AC;&#x201C; contribute to the load bearing capacity according to their strength. This can be done with a through plate shown in Fig. 4.14 or with fin plates in combination with an additional connection between the hollow section and the concrete e.g. by a through pin.

57

Table 5.1 Variations in required fire resistance Type of building

Requirements

Fire class

One storey

None or low

Possibly up to R30

2 or 3 storeys

None up to medium

Possibly up to R60

More than 3 storeys

Medium

R60 to R120

Tall buildings

High

R90 and more

Table 5.2 Minimum cross-sectional dimensions, reinforcement ratios and axis distances of the reinforcing bars for fire resistance classification for various degrees of utilisation levels ηfi,t

As

Ac

h

Standard fire resistance

us t

b

us

t d

us

steel section: b/t  25 and d/t  25

1

Minimum cross-sectional dimensions for load level ηfi,t  0,28

1.1 1.2 1.3

Minimum dimensions h and b or minimum diameter d (mm) Minimum ratio of reinforcement As/(Ac+As) in % Minimum axis distance of reinforcing bars us (mm)

2

Minimum cross-sectional dimensions for load level ηfi,t  0,47

2.1 2.2 2.3

Minimum dimensions h and b or minimum diameter d (mm) Minimum ratio of reinforcement As/(Ac+As) in % Minimum axis distance of reinforcing bars us (mm)

3

Minimum cross-sectional dimensions for load level ηfi,t  0,66

3.1 3.2 3.3

Minimum dimensions h and b or minimum diameter d (mm) Minimum ratio of reinforcement As/(Ac+As) in % Minimum axis distance of reinforcing bars us (mm)

R30

R60

R90

R120

R180

160 0 -

200 1,5 30

220 3,0 40

260 6,0 50

400 6,0 60

260 0 -

260 3,0 30

400 6,0 40

450 6,0 50

500 6,0 60

260 3,0 25

450 6,0 30

550 6,0 40

-

-

Note: In Eurocode 4 (EN1994-1-2), the thickness of the hollow section "t" is called "e".

58

1,0 0,8 0,6 0,4 0,2 0 Time (min.)

Fig. 5.1 Natural fire curves and the ISO standard fire curve (ISO 834-1, 1999)

Fig. 5.2 Schematic material strength reduction for structural steel and concrete

Mode of deformation

Rigid core

For ffire ire conditions in Europe Europe e.g.: e.g. : in

Fire exposed column

(a) Section through the building

topfloor f loor : : ℓkθ =0.7 0,7ℓ top 0,5ℓ otherfloors f loors: : ℓkθ =0.5 other

(b) Room temperature

(c) Elevated temperature

Fig. 5.3 Schematic structural behaviour of columns in braced frames

59

1.0 1,0 0.9

S500 S500 S355 S355 S235 S235

0,8 0.8

fifi

0.7

0,6 0.6

ENV 1993

0.5

0,4 0.4 0.3

0,2 0.2 0.1

0,0 0.0

0,0 0.0

0,5 0.5

1,0 1.0

1,5 1.5

2,0 2.0

2,5 2.5

3,0 3.0

 ( rela)(a) Fig. 5.4 Reduction factor χfi for flexural buckling in the fire situation for a particular critical temperature θa (for comparison, the previous ENV 1993 lower bound curve is also shown)

Reduction factor ky,

reduction factor ky,

1 1,0 0,8 0,8 0,6 0,6 0,4 0,4 0,2 0,2 00 0

200

400 600 800 steel temperature a

1000

1200

Fig. 5.5 Reduction factor ky,θ for the yield strength of steel at a steel temperature θa

60

Time to reach a (min.)

Steel temperature a (oC)

-1) Section factor Amm/V (m-1

Time (min.)

Fig. 5.6 Calculated temperature development in an unprotected steel section as a function of the shape factor

a

Fig. 5.7 Time for an unprotected steel section to reach a given mean temperature under standard conditions as a function of the section factor

Fig. 5.8 Typical strength reduction characteristics of the various components of a concrete filled hollow section column

61

t

Fig. 5.9 Examples of buckling curves for CHS exposed to fire

t

t

Fig. 5.10 Examples of buckling curves for RHS exposed to fire

62

1,0 1,0 0,9 0,8 0,7 0,6 0,5

0,4 0,0

0,5

1,0

1,5

2,0

2,5 3,0

3,5

4,0 4,5 4,5 5,0

Fig. 5.11 Correction coefficient φs as a function of the reinforcement (in %)

e

1,0 0,9 0,8 0,7

ℓ /b or ℓθ/d ℓθθ/b or ℓθ/d

0,6 0,5 0,4 0,3 0,0

e/b or e/d

0,10

0,20

0,30

0,40

0,50

Note: In Eurocode 4 (EN 1994-1-2), the eccentricity "e" is called "δ".

Fig. 5.12 Correction coefficient φe as a function of the eccentricity e

63

Fig. 5.13 Options for columns with external and internal pipes

Time (min.)

Fig. 5.14 Typical temperature development in a water filled hollow section column, exposed to standard fire conditions

64

equilibrated by brace member(s) on the opposite side, the joint is classified as an X joint.

6. DESIGN OF HOLLOW SECTION TRUSSES

When brace members transmit part of their load as K joints and part of their load as T, Y, or X joints, the adequacy of each brace needs to be determined by linear interaction of the proportion of the brace load involved in each type of load transfer. However, the effect of the chord loading should be added to the joint type with the most unfavourable chord load function.

6.1 TRUSS CONFIGURATIONS Various types of trusses are used in practice, see Fig. 6.1. Trusses made of hollow sections should be designed in such a way that the number of joints and thus fabrication is minimised. This means that due to the lower number of joints, a Warren type truss with K joints (Fig. 6.1a) is preferred to a Pratt type truss with N joints (Fig. 6.1b).

One K joint, in Fig. 6.3b, illustrates that the brace force components normal to the chord member may differ by as much as 20% and still be deemed to exhibit K joint behaviour. This is to accommodate slight variations in brace member forces along a typical truss, caused by a series of panel point loads.

Vierendeel girders (Fig. 6.1c) are mainly used in those cases where architectural or functional aspects require that no diagonals are used.

Trusses are characterised by their length L, depth h, geometry and the distance between the joints.

The depth is normally related to the span, being about 1/10 to 1/16 L. Considering total economy of a hall with all costs, a depth of 1/15 L is a common choice.

Whenever feasible, the joints are located at the load application points, e.g. at the purlin locations.

The N joint in Fig. 6.3c, however, has a ratio of brace force components normal to the chord member of 2:1.

In this case, that particular joint needs to be analysed as both a "pure" K joint (with balanced brace forces) and an X joint (because the remainder of the diagonal brace load is being transferred through the joint), as shown in Fig. 6.4. For the diagonal tension brace in that particular joint, one would need to check that:

6.2 JOINT CONFIGURATIONS

0,5 N 0,5 N ď&#x20AC;Ť ď&#x201A;Ł 1,0 K joint resistance X joint resistance

6.2.1 Joint classification

If the gap size in a gapped K (or N) joint becomes large and exceeds the value permitted by the gap limit, then the "K joint" should also be checked as two independent Y joints.

Depending on the type of truss, various types of joints are used (Fig. 6.2), i.e. X, T, Y, N, K or KT. Although here the designation X, T, Y, etc. is related to the configuration, the classification is determined by the loading, see Fig. 6.3. The basic joint types can be defined as follows:

In X joints such as Fig. 6.3e where the braces are close or overlapping, the joint should be checked as an X joint, considering both brace load components perpendicular to the chord.

(a) When the force component normal to the chord in a brace member is equilibrated by beam shear (and bending) in the chord member, the joint is classified as a T joint when the brace is perpendicular to the chord, otherwise it is classified as a Y joint. (b) When the force component normal to the chord in a brace member is essentially equilibrated (within 20%) by loads in other brace member(s) on the same side of the joint, the joint is classified as a K joint. The relevant gap is, in principle, between the primary brace members whose loads equilibrate.

An N joint is to be considered as a type of K joint with one brace at 90o. (c) When the force component normal to the chord is transmitted through the chord member and is

In K joints such as Fig. 6.3d, where a brace has very little or no loading, the joint can be treated as a Y joint, as shown.

6.2.2 Terminology and notation In principle, the terminology adopted by CIDECT and IIW to define joint parameters, is used wherever possible. However, to be consistent with the notation in Chapters 2 to 5, for the resistance, the Eurocode 3 notations Ni,Rd and Mi,Rd are used instead of N* or M* in the CIDECT Design Guides.

65

eccentricity e represents an offset from the chord centreline towards the outside of the truss.

The term "joint" is used to represent the zone where two or more members are interconnected, whereas "connection" is used to represent the location at which two or more elements meet.

6.3 LIMIT STATES AND LIMITATIONS ON MATERIALS

The "through member" of a joint is termed the "chord" and attached members are termed "braces" (although the latter are also often termed bracings, branch members or web members).

6.3.1 Limit states As stated in Chapter 1, this book is written in a limit states design format already including appropriate material and joint partial safety factors (γM) or joint resistance (or capacity) factors (). This means that the effect of the factored loads (the specified or unfactored loads multiplied by the appropriate load factors) should not exceed the factored resistance of the joint.

Fig. 6.5 shows some of the common joint notation for gapped uniplanar K joints. The numerical subscripts (i = 0, 1, 2) to symbols shown in Fig. 6.5 are used to denote the member of a hollow section joint. The subscript i = 0 designates the chord; i = 1 refers in general to the brace for T, Y and X joints, or it refers to the compression brace member for K and N joints; i = 2 refers to the tension brace member for K and N joints. For K and N overlap joints, the subscript i is used to denote the overlapping brace member and j is used to denote the overlapped brace member (see Fig. 6.6).

Some connection elements, which are not specific to hollow sections, such as plate material, bolts and welds, need to be designed in accordance with local or regional structural steel specifications. Thus, additional safety or resistance factors should only be used where indicated.

6.2.3 Limitations on geometric parameters

If allowable stress design (ASD) or working stress design is used, the joint factored resistance expressions provided herein should, in addition, be divided by an appropriate load factor. A value of 1,5 is recommended by the American Institute of Steel Construction (AISC, 2005).

Most of the joint resistance formulae are subject to a particular "range of validity". This often represents the range of the parameters or variables for which the formulae have been validated, by either experimental or numerical research. In some cases, it represents the bounds within which a particular failure mode will control, thereby making the design process simpler.

Joints with parameters outside these specified ranges of validity are sometimes allowed, but they may result in lower joint efficiencies and generally require considerable engineering judgement and verification.

6.3.2 Limitations on materials The recommendations given are applicable to both hot-finished and cold-formed steel hollow sections, as well as cold-formed stress-relieved hollow sections.

The nominal specified yield strength of hollow sections should not exceed 460 N/mm2 (MPa). This nominal yield strength refers to the finished product and should not be taken larger than 0,8fu.

With reference to Figs. 6.6 and 6.7 for RHS sections, the gap g or overlap q, as well as the eccentricity e, may be calculated by eqs. (6.1) and (6.2) (Packer et al., 1992; Packer & Henderson, 1997):

h  sinθ1  θ 2  h1 h2  g  e  0    2  sin θ1 sin θ 2 2 sin θ1 2 sin θ 2 

(6.1)

 h1  sin θ1 sin θ 2 h0 h2   g   e    2 sin θ1 2 sin θ 2  sinθ1  θ 2  2

(6.2)

The joint resistances given are for hollow sections with a nominal yield strength up to 355 N/mm2. For nominal yield strengths greater than this value, the joint resistances should be multiplied by 0,9. On one hand, this provision considers the relatively larger deformations that take place in joints with nominal yield strengths around 450 to 460 N/mm2, when plastification of the chord face or chord cross section occurs (for large brace-to-chord diameter or width ratios β, it may be conservative); on the other hand, for other joints the deformation or rotation capacity may be lower with yield strengths exceeding 355

For CHS sections replace hi by di.

Note that a negative value of the gap g in eq. (6.1) corresponds to an overlap q. A positive value of the 66

N/mm2. Furthermore, for any formula, the "design yield stress" used for computations should not be taken higher than 0,8 of the nominal ultimate tensile strength. This provision allows for ample connection ductility in cases where punching shear failure or failure due to local yielding of a brace or plate govern, since strength formulae for these failure modes are based on the yield stress. For S460 steel hollow sections, the reduction factor of 0,9, combined with the limitation on fy to 0,8fu, results in a total reduction in joint resistance of about 15%, relative to just directly using a yield stress of 460 N/mm2 (Liu & Wardenier, 2004).

validity given in the following chapters.

- It is common practice to design the members with the centre lines noding. However, for ease of fabrication it is sometimes required to have a certain noding eccentricity. The effect should be considered for member and joint design, see Section 6.5.

The gap g (Fig. 6.5) is defined as the distance measured along the length of the connecting face of the chord, between the toes of the adjacent brace members (ignoring welds). In good designs, the minimum gap between adjacent brace members should be g  t1 + t2, to ensure that there is adequate clearance to form satisfactory welds.

Some codes, e.g. Eurocode 3 (EN 1993-1-12, 2007) give additional rules for the use of steel S690. These rules prescribe an elastic global analysis for structures with partial-strength joints. Further, a reduction factor of 0,8 to the joint capacity equations has to be used instead of the 0,9 factor which is used for S460.

- In overlapped K joints, the in-plane overlap should be large enough to ascertain that the interconnection of the brace members is sufficient for adequate shear transfer from one brace to the other. This can be achieved by ensuring that the overlap, which is defined in Fig. 6.6, is at least 25%.

Where overlapping brace members are of different widths, the narrower member should overlap the wider one. Where overlapping brace members with the same diameter have different thicknesses and/or different strength grades, the member with the lowest fyi ti value should overlap the other member.

6.4 GENERAL DESIGN CONSIDERATIONS In designing hollow section trusses it is important that the designer considers the joint behaviour right from the beginning. Designing members of a truss based on member loads only may result in undesirable stiffening of joints afterwards. This does not mean that the joints have to be designed in detail in the conceptual design phase. It only means that chord and brace members have to be chosen in such a way that the main governing joint parameters provide an adequate joint strength and an economical fabrication.

- Gap joints are preferred to partial overlap joints, since the fabrication is easier with regard to end cutting, fitting and welding.

- An angle of less than 30 between a brace member and a chord creates serious welding difficulties at the crown heel location and is not covered in the recommendations. However, angles less than 30 may be possible if the design is based on an angle of 30 and it is shown by the fabricator that a satisfactory weld can be made.

Since the design is always a compromise between various requirements, such as static strength, stability, economy in fabrication and maintenance, which are sometimes in conflict with each other, the designer should be aware of the implications of a particular choice.

- In common lattice structures (e.g. trusses), about 50% of the material weight is used for the chords in compression, roughly 30% for the chord in tension and about 20% for the web members or braces.

This means that with respect to material weight, the chords in compression should likely be optimised to result in thin walled sections. However, for corrosion protection (painting), the outer surface area should be minimised. If both chords have to be designed for compression loads, for example due to uplift wind loading, the above values may change depending on the lateral supports.

The following guidance is given to arrive at an optimum design: - Lattice structures can usually be designed assuming pin jointed members. Secondary bending moments due to the actual joint stiffness can be neglected for static design if the joints have sufficient rotation capacity. This can be achieved by limiting the wall slenderness of certain members, particularly the compression members, which is the basis for some of the geometric limits of validity. This will be the case if the joint parameters are within the range of

- Joint strength increases with decreasing chord 67

between axial load and bending moment, for each brace member.

diameter- or width-to-thickness ratio. As a result, the final diameter- or width-to-thickness ratio for the chord in compression will be a compromise between joint strength and buckling strength of the member and relatively stocky sections will usually be chosen.

For the chord in tension, the diameter- or width-to-thickness ratio should be chosen to be as small as possible.

A rigid joint frame analysis is not recommended for most planar, triangulated, single-chord, directly welded trusses, as it generally tends to exaggerate brace member moments, while the axial force distribution will still be similar to that for a pin jointed analysis.

- Since the joint strength efficiency (i.e. joint strength divided by the brace yield load Ai fyi) increases with increasing chord-to-brace thickness t0/ti this ratio should be chosen as high as possible, preferably above 2. Furthermore, the weld volume required for a thin walled brace is smaller than that for a thick walled brace with the same cross section, if the welds are to develop the capacity of the connected brace member.

Transverse loads applied to either chord away from the panel points produce primary moments which must always be taken into account when designing the chords.

Computer plane frame programs are regularly used for truss analysis. In this case, the truss can be modelled by considering a continuous chord with brace members pin connected to it at distances of +e or -e from it (e being the distance from the chord centreline to the intersection of the brace member centrelines). The links to the pins are treated as being extremely stiff as indicated in Fig. 6.8. The advantage of this model is that a sensible distribution of bending moments is automatically generated throughout the truss, for cases in which bending moments need to be taken into account in the design of the chords.

- The joint strength also depends on the yield stress ratio between chord and brace, thus the use of higher strength steel for chords (if available and practical) may offer economical advantages.

- In principle, multiplanar trusses can be approached in a similar way as uniplanar trusses, although the depth can usually be smaller, between 1/15 and 1/18 L.

Secondary moments, resulting from end fixity of the brace members to a flexible chord wall, can generally be ignored for both members and joints, provided that there is deformation and rotation capacity adequate to redistribute stresses after some local yielding at the connections. This is the case when the prescribed geometric limits of validity for design formulae given in Chapters 8 to 11 are followed. Welds in particular need to have potential for adequate stress redistribution without premature failure, and this will be achieved with the recommendations given in Section 6.5.2. Table 6.1 summarizes when moments need to be considered for designing CHS or RHS trusses.

6.5 TRUSS ANALYSIS Elastic analysis of hollow section trusses is frequently performed by assuming that all members are pin connected. Nodal eccentricities between the centre lines of intersecting members at panel points should preferably be kept to e ď&#x201A;Ł 0.25d0 or 0.25h0. These eccentricities produce primary bending moments which, for a pinned joint analysis, need to be taken into account in chord member design, e.g. by treating it as a beam-column. This is done by distributing the panel point moment (sum of the horizontal components of the brace member forces multiplied by the nodal eccentricity) to the chord on the basis of relative chord stiffness on either side of the joint (i.e. in proportion to the values of moment of inertia divided by chord length to the next panel point, on either side of the joint).

6.5.1 Truss deflections For the purpose of checking the serviceability condition of overall truss deflection under specified (unfactored) loads, an analysis with all members being pin jointed will provide a conservative (over)estimate of truss deflections when all the joints are overlapped. A better assumption for overlap conditions is to assume continuous chord members and pin jointed brace members.

If these eccentricity limits are violated, the eccentricity moment may have a detrimental effect on joint strength and the eccentricity moment must be distributed between all members of a joint. If moments are distributed to the brace members, the joint capacity must then be checked for the interaction

68

However, for gap-connected trusses, a pin jointed analysis still generally underestimates overall truss deflections, because of the flexibility of the joints. At the service load level, gap-connected hollow section truss deflections are underestimated by around 5-10%. Thus, a conservative approach for gap-connected trusses is to estimate the maximum truss deflection by 1,1 times that calculated from a pin jointed analysis.

6.5.2 Weld design To avoid weld failure it is recommended to design the welds to be stronger than the connected brace members, wherever possible. Designing fillet welds in this way, and adopting Eurocode 3 (EN 1993-1-8, 2005), results in the following minimum throat thickness "a" for fillet welds around brace members, assuming matched electrodes: a  0,92t, for S235 (fyi = 235 N/mm2) a  0,96t, for S275 (fyi = 275 N/mm2) a  1,10t, for S355 (fyi = 355 N/mm2) a  1,42t, for S420 (fyi = 420 N/mm2) a  1,48t, for S460 (fyi = 460 N/mm2) For very lightly loaded structures, smaller welds are allowed, provided care has been taken of the effective weld lengths around the perimeter (Frater & Packer, 1990; Packer & Wardenier, 1992).

69

Table 6.1 Moments to be considered for CHS or RHS truss design Type of moment

Primary

Primary

Secondary

Moments due to

Nodal eccentricity (e ď&#x201A;Ł 0.25d0 or 0.25h0)

Transverse member loading

Secondary effects such as local deformations

Chord design

Yes

Yes

No

Design of braces

No

Yes

No

Design of joints

Yes, for Qf only

Yes, influences Qf

No, provided parametric limits of validity are met

Note: For structures subjected to fatigue loading, all primary and secondary bending moments should be considered, see Chapter 14.

70

a. Warren truss

b. Pratt truss

c. Vierendeel truss

d. truss with cross braces Fig. 6.1 Various types of trusses

X joints

T and Y joints

N and K joints

KT joints

Fig. 6.2 Basic types of joints

71

N

1,2N 1.2N

N

100%

within tolerance within tolerance for: for:

K

K 







0,2N sin sin  0.2N

gap (a) 0.5Nsin sin 0,5N  100%

N

100%

(b)

50% K 50% X

K

N

N

0

100%

Y 



+e

0.5N sin sin 0,5N 

(c)

(d)

0.5 N/sin 0,5N / sin 

N 100%

K

0.5 N/sin 0,5N / sin 

X 





+e

gap

(e)

(f)

N 100%

X

 N (g)

Fig. 6.3 Examples of hollow section joint classification

72

N

K



N



0 100%

100%

0,5N sin 0.5N sin

0,5N sin 0.5N sin

0,5N 0.5N

N

+

= 

0,5N 0.5N



N cos N cos 

 0.5N 0,5Ncos cos 

0,5N coscos  0.5N

0.5N 0,5Nsin sin 

0.5N sin sin 0,5N

Figure 6.4 Checking of a K joint with imbalanced brace loads

N1

N2 b1

d1

b2

1

h1

t1

t2

h2

2

g

1

d2

b0

t0

2

0

N0 +e

h0

Fig. 6.5 Symbols used for K gap joints

i

j -e

i = 1 or 2 (overlapping member) j = overlapped member

q p

Overlap =

q p

x 100%

Fig. 6.6 Definition of overlap

73

g

g

e>0

e=0

e e or ď&#x201A;Ł 0,25 d0 h0

ee << 00

e<0

Fig. 6.7 Noding eccentricity

For most overlap joints

Extremely stiff members

Extremely stiff members

Pin

For most gap joints

Fig. 6.8 Plane frame joint modelling assumptions to obtain realistic forces for member design

74

7. BEHAVIOUR OF JOINTS

stiffness of the hollow section side walls.

For a proper understanding of the behaviour of welded joints between hollow sections it is important to consider the load path, the internal stiffness distribution in a joint and the material properties.

Consequently, the stiffness for a q2 load is considerably smaller than for a q1 load. This is graphically shown in Fig. 7.3. For loads on the top face at locations between q1 and q2 the behaviour is in between that for q1 and q2.

7.1 GENERAL INTRODUCTION

The resulting elastic stress pattern in the plate can now be determined in two ways.

7.1.1 Load path

1. Consider the deformations under a uniform stress For a uniform stress, the plate and hollow section faces do not have the same deformed shape. To ensure that the plate and the hollow section face have the same deformation, the stresses at the centre should be lower and at the sides higher. Thus, additional stresses have to be added to the uniformly distributed loading, as shown in Fig. 7.4b. This increases the stresses at the sides and reduces the stresses at the plate centre. Hence, the highest stresses occur at the stiff parts.

The load path shows the elements through which the loads have to pass and where failure may occur. For example, Fig. 7.1 illustrates a welded joint between plates and a hollow section. The load has to pass via the following elements: - Plate - Weld - Hollow section face (through thickness) - Hollow section side wall In principle, failures can occur in any of these parts. If the width of the plate b1 is small compared to the chord width b0, more types of failure can occur in the chord face. This will be discussed later.

As shown in Fig. 7.4b, the plate remains almost straight due to the much higher axial stiffness of the plate compared to the bending stiffness of the chord top face. Therefore, the plate could have been assumed to be nearly rigid compared with the stiffness of the hollow section top face.

7.1.2 Internal stiffness distribution The stiffness distribution in the joint determines the elastic stress distribution. Here, the plate to RHS chord joint of Fig. 7.1 will be examined again.

2. Assume rigid plate If the plate is assumed to be rigid, the stress pattern can be directly determined with Fig. 7.3. For a deformation δ1, the stress for q1 is much higher than for q2, resulting in the stress pattern of Fig. 7.4c.

Consider the stiffness of the plate and the connected face of the hollow section.

From this evaluation, it is clear that the non-uniformity largely depends on the b0/t0 ratio. If b0/t0 is very small, approaching a solid profile, the stress distribution is uniform if contraction is not considered. If b0/t0 is large, it may even be that the stress at the centre has the opposite sign of that at the sides.

1. Plate The plate end remains straight if loaded by a uniform loading q per unit length. The deformation is determined by the plate stiffness for axial stresses, which is high.

2. Hollow section face First consider a unit load q1 at a small unit length at the sides (Fig. 7.2b). The load q1 can flow directly into the hollow section side walls. Thus, the deformation is determined by the stiffness of the hollow section side wall for axial stresses.

7.1.3 Effect of material properties Fig. 7.5 shows the Ď&#x192;-Îľ diagram of two materials: (a) Steel with a yield strength fy and a strain hardening part with an ultimate tensile strength fu. (b) A fictitious steel without any deformation capacity, i.e. it fails immediately after reaching the maximum stress fu,b.

Now consider a unit load q2 at the centre of the hollow section face (Fig. 7.2c). The load has to be transmitted to the side walls by bending. Thus, the deformation is determined by the bending stiffness of the top face of the hollow section and the axial

Suppose that failure of the plate-to-RHS joint is governed by failure of the plate just before the weld.

75

This means that the stress pattern in the plate (Fig. 7.4c) has to be considered in relation to the material behaviour.

7.1.4 Failure modes Following the load path (see Section 7.1.1) shows the possible failure locations, whereas the stiffness distribution (Section 7.1.2) in combination with the material behaviour (Section 7.1.3) determine the failure mode for the various locations. The lowest failure load for all possible failure modes gives the governing strength. Now the possible failure modes for the plate-to-RHS joint of Fig. 7.1 will be evaluated.

The stress pattern in Fig. 7.4c is based on an elastic material behaviour, thus equivalent to material "b". As soon as the maximum stress at the side (location 1) reaches the ultimate stress fu,b, the material will start cracking.

If material "a" of Fig. 7.5 had been used, the maximum stress would first reach the yield stress fy. With increasing load, the material at location (1) yields i.e. the stress remains constant fy and the strain ε increases. With further increasing load, the material just beside location (1) in Fig. 7.4c will yield, etc. At a certain strain, the material at location (1) will reach the strain hardening part in the σ-ε diagram of Fig. 7.5.

After a further increase of the loading, the stress will increase until the ultimate stress fu, after which the "actual stress" will still increase, although the "engineering stress", based on the original cross section will decrease. At a certain ultimate strain εu cracking will occur at location (1).

1. Plate Fig. 7.6a shows the possible stress distribution in the plate after yielding and after reaching the ultimate strain at the sides (location 1). If the chord width-tothickness ratio b0/t0 is low and the material has sufficient ductility, the yield capacity of the plate can be attained. In most cases the capacity is lower.

2. Welds If the strength of the fillet welds (Fig. 7.6b) is lower than that of the plate, the welds may fail. If plastic deformation occurs in the welds only, the total deformation for the joint is small, resulting in a joint with no deformation capacity (which is generally not allowed). Therefore, it is recommended that the welds should preferably be designed to be stronger than the connected brace members.

Sometimes cracking occurs at the very stiff locations and still the loading can be increased due to a more uniform stress distribution in the remaining cross section.

Only for very lightly loaded structures, e.g. where members have been selected based on aesthetical aspects, are smaller welds allowed, provided the secondary effects and the effective perimeter are considered (Frater & Packer, 1990; Packer & Wardenier, 1992).

The above example shows the importance of yielding for the load capacity of hollow section joints.

Another aspect which is also extremely important for static design is the deformation capacity. The deformation capacity determines if secondary moments can be redistributed in structures.

3. Chord face The loading and hence the stresses have to pass via the top face to the side walls. Especially for thick material, cracking can occur due to manganese-sulphide (MnS) inclusions, called lamellar tearing (Fig. 7.6c). To avoid this material problem, material with good through thickness properties (TTP) should be used, i.e. steels with low sulphur contents.

For example in a truss, secondary bending moments exist due to the joint stiffness of the welded joints.

However, these moments are not necessary for equilibrium of the structure. If the truss is loaded up to failure and the joint strength is governing instead of the member strength, at a certain moment, yielding occurs due to the combination of axial loading and the (secondary) bending moments. If the deformation capacity is sufficient, the axial forces in the members can increase with a decrease of the (secondary) bending moments due to plastic rotation of the joint. In the failure stage, the secondary bending moments may have totally disappeared.

If b1 < b0, other failure modes can be obtained for the chord, i.e. chord face plastification or chord punching shear.

For the joint with b1 = b0 the connecting chord face is held in position by the plate and the stiff connection to the side wall. Therefore, chord face yielding with a distinct yield line pattern can only develop after excessive yielding of the plate at the sides and/or excessive yielding of the hollow section side walls

76

under the plate.

7.3 GENERAL FAILURE MODES

Punching shear of the hollow section face can only occur if the plate width b1 is smaller than b0 - 2t0 (see Fig. 7.6d).

Similar to the plate-to-hollow section chord joint in Fig. 7.1, hollow section joints exhibit, depending on the loading, joint type and geometric parameters, various modes of failure.

4. Chord side wall All the stresses have to be transmitted through the side walls over a limited width, thus this may be a critical failure mode (chord side wall yielding shown in Fig. 7.6e).

As an example, Fig. 7.8 illustrates typical failure modes for a K joint of rectangular hollow sections, i.e.: - Plastification of the chord face - Chord punching shear - Local brace failure (effective width) - Chord shear failure - Local buckling of the compression brace - Local buckling of the chord

If the loading is compression instead of tension, the stability of the side wall may be critical.

7.2 GENERAL FAILURE CRITERIA

If the welds are not strong enough, weld failure can also occur, or if the material does not have sufficient through thickness properties (TTP) lamellar tearing is possible.

In general, the static strength can be characterized by various criteria, i.e.: - Ultimate load resistance - Deformation limit - (Visually observed) crack initiation

The failure modes and associated analytical models for determination of the strength formulae are described in detail in the following chapters.

The ultimate load capacity is well defined for those joints which show a maximum in the load deformation diagram, e.g. for selected joints loaded in compression. Other joints show an increasing load capacity with increasing deformation such that the maximum is obtained at excessive deformation.

7.4 JOINT PARAMETERS The geometry of a particular joint is generally defined by the dimensions given in Fig. 6.5 and by the joint parameters , , ,  and g’ shown in Fig. 7.9.

Originally the parameters were related to the radius of a circular section. Nowadays the diameter, width or depth are used which explains the factor of 2 in the definition of the and ratios.

Besides the ultimate capacity criterion, a deformation limit has been defined (Lu et al., 1994) to avoid deformations which are too large. This limit, being 0,03d0 or 0,03b0, as shown in Fig. 7.7, is based on the fact that the deformation at serviceability should not be governing and that crack initiation should not occur at serviceability either. The deformation limit considers the local displacement of the chord wall at the connection of the brace to the chord.

Thus, the ultimate load capacity is defined by the criterion that is met first, i.e. the maximum capacity or the load at the deformation limit.

For serviceability, an arbitrary limit of 0,01d0 or 0,01b0 is adopted. This 1% limit is the same as the out-of-roundness limit and has shown to give acceptable deformations. However, it must be mentioned that many formulae in the codes and design guides were initially developed based on ultimate load or end-of-test data and were later on evaluated for the 1% d0 or 1% b0 rule for serviceability.

77

t1

b1

b0

t0

h0

a

ℓ1

q

q1

q1

q1

b. RHS loaded at the sides

a. plate

A1 = b1 x t1

q2

q2 c. RHS loaded at the centre

Fig. 7.2 Plate-to-RHS chord joint – stiffness

Fig. 7.1 Plate-to-RHS joint

q

q1

q1

stress

deformation

plate

q2

RHS

1



Fig. 7.3 Load-deformation diagram

Fig. 7.4a Stress and resulting deformation

stress actual stress

fu,b b fu

a

σ fy

deformation

plate

engineering stress

RHS

strain hardening

Fig. 7.4b Compatibility

yield



u 1

Fig. 7.5 σ-ε diagram steel

2

1

Fig. 7.4c Resulting stress pattern

78

0,5be

max.

0,5be fu

N

a

fy

3% 3%dd 0 0or 3% b0 1

2

1

Fig. 7.6a Plastic stress pattern and ultimate situation at failure

(b)

(c)

ď ¤ Fig. 7.7 Deformation limit

(d)

Fig. 7.6b Weld failure Fig. 7.6c Lamellar tearing Fig. 7.6d Punching shear

plate

chord

difficult to make a proper weld

t1

2,5:1 elastic plastic ultimate

fy bw

fu

Fig. 7.6e Chord side wall failure

79

Fig. 7.8 Failure modes for a K joint of rectangular hollow sections



2 0 2 or 0 d0 b0



b d b d1 or 1 or 1 or 1 d0 d0 b0 b0

for T, Y and X joints



d1  d2 d  d2 b  b2 or 1 or 1 2 d0 2b0 2b0

for K and N joints

2  

d0 b or 0 t0 t0

N1

t1

ti t0

1

g g'  t0

n

0 fy0

N2

N1

t0

N0p

ℓ0

Fig. 7.9 Parameters used for defining the joint geometry 80

1

2

N0

As indicated in Chapter 6, to avoid weld failure it is recommended to design the welds to be stronger than the connected braces. Prequalified full penetration welds can be considered always to be stronger than the connected braces. Partial penetration plus fillet, or fillet welds alone, can also usually provide a weld connection as strong as the connected braces.

8. WELDED JOINTS BETWEEN CIRCULAR HOLLOW SECTIONS 8.1 INTRODUCTION Circular hollow sections can be connected in various ways, e.g.: - With special prefabricated connectors (Fig. 8.1) - With end pieces which allow a bolted joint (Fig. 8.2) - Welded to a plate (Fig. 8.3) - Welded directly to the through member (chord) (Fig. 8.4)

The material should not be susceptible to lamellar tearing, i.e. especially for the larger thicknesses, the sulphur content should be low (TTP quality).

Furthermore, in the current design recommendations, the d/t ratios have been limited to avoid local buckling.

In addition, limiting the d/t ratio has the effect that local brace failure is no longer a governing failure mode for T, Y, X and K gap joints.

For transport or erection it may be that bolted joints are preferred or required, whereas for space structures prefabricated connectors are generally used. However, the simplest solution is to profile the ends of the members which have to be connected to the through member (chord) and weld the members directly to each other. Nowadays, end profiling does not give any problem and the end profiling can be combined with the required bevelling for the welds.

Furthermore, within the range of validity of the design recommendations, it has been shown that the chord shear criterion can be covered by the formula for chord plastification.

As a result, the governing modes of failure to be considered for uniplanar joints have been reduced to: - Chord plastification - Chord punching shear

Although the directly welded joint (Fig. 8.4) is the simplest and cleanest solution, the load transfer is rather complex due to the non-linear stiffness distribution along the perimeter of the connected braces. The design rules have been based on simplified analytical models in combination with experimental evidence, resulting in semi-empirical design formulae.

8.3 ANALYTICAL MODELS For the determination of the influencing joint parameters, three models are used, i.e.: - Ring model (for chord plastification) - Punching shear model (for chord punching shear) - Chord shear model

8.2 MODES OF FAILURE In Chapter 7 it was already indicated that the ultimate load capacity is based on either the maximum in the load deformation diagram (if the chord displacement is smaller than 0,03d0) or the load at a chord deformation of 0,03d0.

8.3.1 Ring model The ring model, originally developed by Togo (1967), is based on the assumption that, for example in an X joint, most of the loading is transferred at the saddles of the brace, since the chord is stiffest at these parts of the connection perimeter (see the elastic stress distribution in Fig. 8.6).

In accordance with the procedure described in Chapter 7, i.e. following the loads, various possible failure modes (Fig. 8.5) can be expected: - Local brace failure (yielding, local buckling) - Weld failure - Lamellar tearing - Chord plastification (face/wall, or cross section) - Chord punching shear failure - Chord local buckling - Chord shear failure

Consequently, the load N1 in the brace can be divided into two loads of 0,5 N1 sinθ1 at the saddles of the brace perpendicular to the chord and at a distance c1d1 with c1 < 1,0. These loads will be transferred by an effective length Be of the chord. In the model, the load 0,5 N1 sinθ1 is now considered as a line load over the length Be, see Fig. 8.7.

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unequal stiffness distribution, the stress distribution will be non uniform, even after yielding. However, tests on CHS joints have shown that within the range of validity given, the full perimeter can be considered to be effective.

At failure, the plastic moment capacity will be reached at the locations A and B (in Fig. 8.8). Neglecting the influence of axial and shear stresses the plastic moment capacity per unit length mp results in:

mp 

1 2 t 0 fy0 4

(8.1)

For joints with θ1 = 90 the punching shear area is  d1 t 0 and the limiting value for the punching shear

Assuming d0 - t0  d0 gives for equilibrium: 2 mp B e 

N1 sin 1  d0 c 1 d1     2 2   2

stress is f y 0 / 3 . Thus, the punching shear capacity

is given by: (8.2)

N1   d1 t 0

fy0

or:

N1 

2 2 B e / d0 t 0 f y 0 (1  c 1 ) sin 1

For angles θ1 < 90 the component perpendicular to the chord has to be considered and the joint perimeter will increase. Projecting the connection perimeter to a flat surface through the chord crown gives an ellipse and the ratio between the perimeter of this ellipse and 1  sin 1 the circle for θ1 = 90 is given by , resulting 2 sin 1

(8.3)

The effective width Be is determined experimentally and depends on the β ratio, e.g. for β = 1,0 the width Be is smaller than for β = 0,5 due to the direct load transfer through the chord. The average value is: Be = 2,5d0 to 3d0.

in:

This ring model only considers chord plastification which is caused by the brace load components perpendicular to the chord. It may be clear that the loads in the chord also have an influence on the load capacity of the joint. The effect of chord stress is given by a function Qf. As a result, the strength equation has the form:

N1 

t 02 f y 0 c0 Qf (1  c 1 ) sin 1

(8.5)

3

N1  0,58  d1 t 0 f y 0

1  sin 1 2 sin 2 1

(8.6)

Tests have shown that the chord stresses have a minor effect, thus, the chord stress function Qf is not included in eq. (8.6).

8.3.3 Chord shear model

(8.4)

In T and Y joints, failure is governed by a combination of local failure of the chord cross section due to the brace load component perpendicular to the chord and chord failure due to shear, bending and, if present, axial loading of the chord. This has been worked out in detail by Van der Vegte & Makino (2006).

where c0, c1 and Qf are determined using results from experiments and numerical analyses; see Section 8.4.

For X joints, this model gives good agreement with the test results, but the formula needs further adjustments for the more complicated joints, such as K and N joints.

K joints with a large β value may fail by a shear failure in the gap location, see Fig. 8.10. The failure mode is a chord cross section plastification due to shear load, axial load and, if present, bending.

8.3.2 Punching shear model

For compact chords, the chord shear capacity can be derived from plastic analysis (see Section 2.3.4):

Originally, the punching shear model served as a basis for many design recommendations (Marshall, 1992). The punching shear failure mode is also caused by the brace load component perpendicular to the chord, i.e. N1 sinθ1. The joint resistance is obtained by multiplying the effective punching shear area and the punching shear yield stress (Fig. 8.9). Due to the

Vpl,0  A v

82

fy0 3



2 A 0 (0,58 f y 0 ) 

(8.7)

The axial load capacity of a chord member is given by:

8.5 BASIC JOINT STRENGTH FORMULAE

Npl,0 = A0 fy0 = π (d0 - t0) t0 fy0

8.5.1 T, Y, X joints and K, N gap joints

(8.8)

If the bending moments are small, only the interaction between axial load and shear load has to be considered, i.e.: 2

 Ni sin i   Ngap,0     Vpl,0   Npl,0   

The analytical ring model approach has served as a basis for the determination of the joint strength formulae. Based on the available numerical and test results for X joints and using eq. (8.4), the values for c0 and c1 have been determined to obtain the function for the mean strength, see Fig. 8.14. Further, in the recent IIW (2009) recommendations a minor correction in the function for β and γ has been included to reduce the capacity for very low β ratios to be more consistent with the capacity for T joints.

2

   1,0  

(8.9)

or:

Ngap,0  A 0 f y 0

 Ni sin i 1   0,58 f y 0 A v 

   

2

(8.10)

Since for T, Y, K and N joints the load transfer is more complicated, semi-empirical functions for β and  as well as for the gap g of K joints have been derived, resulting in the following, general format:

If the chord is only loaded by the brace load components, i.e. N0p = 0, then Ngap,0 = Ni cos θi, which is shown in Fig. 8.11.

N1,Rd  f () f (  ) f (g' )

8.4 EXPERIMENTAL AND NUMERICAL VERIFICATION

f y 0 t 02 sin 1

Qf

(8.11)

In the new recommendations of IIW (2009) and CIDECT Design Guide No. 1 (Wardenier et al., 2008a), this equation has been presented as:

Nowadays, not only does a lot of experimental evidence exist, but also many results derived with numerical analyses are available. Makino et al. (1996) give a good survey of the available experimental data.

Ni,Rd  Qu Q f

The experimental work has been mainly carried out in Germany, Japan, USA, The Netherlands, UK and Norway. The joints have been tested in various testing set-ups (e.g. Fig. 8.12), primarily on isolated joints.

Only a few tests have been carried out on joints in complete girders. For example, in the framework of an offshore programme, Bolt & Billington (2000) have conducted scale tests on complete frames of jackets to simulate the interaction between joint and member behaviour and to calibrate numerical models for frame behaviour; see also Choo et al. (2005a).

fy0 t 02 sin θi

(8.11a)

with Qu = f(β) f() f(g’) and Qf = f(n), where n is the ratio between the maximum chord stress at the connecting face and the chord yield stress: n

N0 M  0 Npl,0 Mpl,0

(8.12)

In the comparison of the test results with the resulting formulae for the basic uniplanar joints, it was concluded that within the range of validity given, the results can be described by a primary joint strength function for chord plastification and an additional check for punching shear. The chord shear criterion does not have to be checked separately.

For experiments, attention should be given to the support and loading conditions to avoid constraint effects (Liu et al., 1998). For numerical results, it is important that the FE models used have been calibrated against experimental data (Van der Vegte et al., 2010a). The elements and the mesh should be considered properly (see Fig. 8.13); details are given

by Van der Vegte (1995).

In the previous recommendations of IIW (1989) and CIDECT Design Guide No. 1 (Wardenier et al., 1991), the chord stress function, called f(n’) was based on the chord prestress which was in contradiction with the function for joints with square hollow sections, which used the maximum chord stress as the governing parameter. This inconsistency has been

83

corrected in the current design recommendations of IIW (2009) and the 2nd edition of CIDECT Design Guide No. 1 (Wardenier et al., 2008a) adopted in this book.

More detailed information regarding the design equations for overlap joints is given in Chapter 11.

8.6 EVALUATION TO DESIGN RULES 8.5.2 K, N overlap joints

In the analysis for the basic joint strength formulae, at first, functions have been derived which predict the mean strength with the lowest coefficient of variation.

For overlap joints, the same approach is adopted for all types of overlap joints, regardless whether circular or rectangular braces are used in combination with a circular, rectangular or open section chord (Wardenier, 2007; Qian et al., 2007). Only the effective width parameters depend on the type of section. The resistance of overlap joints between circular hollow sections with 25%  Ov  100% overlap is based on the following criteria: (1) Local failure of the overlapping brace (2) Local chord member yielding at the joint location based on interaction between axial load and bending moment (3) Shear of the connection between the brace(s) and the chord

Considering the scatter in test results, typical tolerances in dimensions and workmanship and the variation in yield stress, in the second step, characteristic joint strength formulae have been determined with a 5% probability of lower strength (Kurobane, 1981; Wardenier, 1982; Van der Vegte et al., 2008).

In the analyses by Van der Vegte et al. (2007, 2008, 2010b) and Zhao et al. (2008) for the new recommendations of IIW (2009) and CIDECT Design Guide No. 1 (Wardenier et al., 2008a), these characteristic formulae have been further divided by a partial factor M of 1,1 for chord plastification and 1,0 for punching shear and 1,0 for chord shear.

Figure 8.16 shows the overlap joint configuration with the cross sections to be examined for these criteria.

Finally, the equations have been slightly simplified to derive the design resistance formulae.

Local failure of the overlapping brace (criterion 1) should always be checked, while shear between the braces and the chord (criterion 3) may only become critical for larger overlaps, i.e. larger than 60% or 80%, depending on whether or not the hidden toe location of the overlapped brace is welded to the chord. The check for local chord member yielding (criterion 2) is, in principle, a member check and may become critical for larger overlaps and/or larger β ratios.

Table 8.1 summarizes the design resistance formulae for the basic uniplanar joints. As shown, only the chord plastification criterion and the general punching shear criterion should be checked.

Fig. 8.15 shows that a compression stress n < 0 in the chord may give a considerable reduction of the joint resistance, whereas for chord tension, the influence depends on the γ ratio. The influence functions given are fully based on experimental and numerical evidence. For chord compression, a function that provided the best fit with the test results is used whereas for chord tension, a lower bound is adopted for the joints with high γ ratios. Recent results (Qian et al., 2008) for low γ ratios show that the chord stress function acts more as a mean function for these data.

For 100% overlap joints, similar criteria have to be checked. Only here, as shown by Qian et al. (2007), shear of the overlapped brace and chord member yielding will generally be the governing criteria.

Although an overlap of 100% is given in the recommendations, in general, the overlap will be slightly larger to allow proper welding of the overlapping brace to the overlapped brace.

In the previous recommendations of IIW (1989) and CIDECT (Wardenier et al., 1991), no reduction for chord tensile loading was used since at that time, the available experiments did not show a reduction for chord tension loads up to about 80% of the chord yield capacity (Wardenier, 1982).

Joints with overlaps between 0% and 25% should be avoided because in those cases, the stiffness of the connection between the overlapping brace and the overlapped brace is much larger than that of the overlapping brace to chord connection, which may lead to premature cracking and lower capacities (Wardenier, 2007).

The joint resistances given in Table 8.1 are for hollow sections with a nominal yield strength not exceeding

84

355 N/mm2. As discussed in Section 6.3, for nominal yield strengths greater than this value (up to S460), the joint resistances given have to be multiplied by 0,9 (Liu & Wardenier, 2004).

strength function (Wardenier et al., 2009): N1  f () f (  ) f () f y0 t 02 Q f

(8.13)

or presented as eq. (8.11a) with:

8.7 OTHER TYPES OF JOINTS

Qu  f () f (  ) f ()

8.7.1 Related types of joints

(8.14)

where for X joints:

Large diameter CHS sections are built up from cans with a maximum length equal to the maximum plate width from which they are made. In structures with these large size tubular sections, it is easy to use a larger can thickness at the joint locations. This is, for example, commonly used in offshore structures.

However, it can also be applied to other heavily loaded structures, such as bridges and large span structures.

 1    1  0,4η  0,15 Qu  2,2   1  0,7 

(8.14a)

and for T joints:





Qu  2,2 1  6,82 1  0,4η 0,2

(8.14b)

The database for plate to CHS joints mainly includes tests carried out in Japan and is summarised by Kurobane (1981), Makino et al. (1991) and Wardenier (1982).

X Joints with short can lengths have been numerically investigated by Van der Vegte (1995). For X joints, it was shown that the can should have a minimum length of 2,5d0 in order to obtain a joint resistance based on the can thickness. For smaller can lengths, a linear interpolation can be made between the resistance of the joint with and without a can.

The capacities of joints between transverse plates, I sections or RHS sections as braces and a CHS chord, as illustrated in Table 8.3, are directly related to those of joints between CHS braces and chords. However, a detailed analysis (Wardenier et al., 2009) revealed that for selected cases, large discrepancies exist between the data sets. Hence, until more evidence is available, e.g. from Voth (2010), the constant in eqs.

(8.14a) and (8.14b) is taken as 2,2 instead of 2,6 derived for joints between CHS braces and chords. As a result, the current design capacities are considerably smaller than those in the previous IIW (1989) recommendations.

The joint design resistance for the special types of joints shown in Table 8.2 can be directly related to that for the basic type of joints in Table 8.1. In all cases, the brace load components perpendicular to the chord have to be considered, since these affect the chord plastification.

The first and second joints in Table 8.2 have a loading effect similar to that for an X joint and the design resistance is therefore related to that for X joints.

Care has to be taken with regard to the effective perimeter for punching shear. If, for example, the flanges of I sections are close to each other, thus η is small, e.g. η  2, then the inner part between the flanges cannot resist forces and only the outer perimeter should be considered as effective, similar to CHS or RHS braces.

The third and fourth joints have a loading comparable to that for a K joint and the design resistance should therefore be related to that for K joints.

It is also clear that in the latter case the shear of the chord is higher than that for a K joint and should be checked seperately.

8.7.3 Multiplanar joints 8.7.2 Plate to CHS joints

In multiplanar joints, two additional effects influence the joint capacity compared to that for uniplanar joints, i.e.: - The geometric effect (stiffening by the braces) - The loading effect

Table 8.3 shows T and X joints with a circular hollow section chord and various configurations for the brace.

The design resistance for these types of joints can be related to each other by the following, general

85

For example, consider the XX joint in Table 8.4. If the out-of-plane braces are very small in diameter, and unloaded, they have hardly any effect on the deformation of the chord. However, if the diameter increases (e.g. β = 0,6), the chord cross section is stiffened considerably. As a consequence, for chord plastification, the geometric effect on the joint capacity will be minor for small β values and significant for larger β values.

8.7.5 Interaction between axial loads and bending moments Joints with brace members subjected to combined loading should satisfy: Ni,Ed Ni,Rd

The deformation capacity further decreases for β ratios close to 0,7 since the load transfer is concentrated at the out-of-plane gap locations between the braces. Below β = 0,7, the braces are not overlapping.

2

 Mip,i,Ed  M   op,i,Ed  1,0   Mip,i,Rd  Mop,i,Rd  

(8.15)

This interaction equation is based on the work of Hoadly & Yura (1985), although the exponents have been rounded off. Eq (8.15) shows directly that the influence of axial load and brace out-of-plane bending, both with the loading concentrated at the saddles, is comparable, whereas brace in-plane bending, with the maximum loading more concentrated at the crown, has a smaller effect in the interaction.

For the load effects in XX joints, it is clear that, for chord plastification, loads in the brace planes in the opposite sense will decrease the joint capacity, whereas loads in the same sense will increase the joint capacity. However, for chord punching shear, the factor  should not exceed 1,0.

8.8 DESIGN CHARTS In the design process, it is important that the designer knows how to design and that he or she can quickly check if a particular design will be appropriate. Hence, design graphs have been established (Wardenier et al., 1991, 2008a) in which the joint resistance is expressed as an efficiency, i.e. the joint resistance is given as a fraction of the yield capacity Ai fyi of the connected brace. This results in the following efficiency formula:

Although the effects depend on the joint parameters (Paul, 1992; Van der Vegte, 1995) the influence function in Table 8.4 for XX joints can be considered to be a lower bound, especially for loading in both planes in the same sense.

8.7.4 Joints loaded by brace bending moments

Efficiency =

In principle, the design resistance formulae for joints loaded by brace in-plane or out-of-plane bending moments have been determined in a similar way to that for axially loaded joints, also resulting in two strength criteria, i.e. chord plastification and chord punching shear. The design resistance formulae are based on the analyses by Wardenier (1982) and Van der Vegte et al. (2010b) and are given in Table 8.5.

Ni,Rd A i f yi

 Ce

fy0 t 0 Q f f yi t i sin i

(8.16)

In the case of d1  d 2 for K joints, eq. (8.16) has to be

d1  d 2 , where di is the diameter of the 2d i brace considered.

multiplied by

The efficiency parameter Ce (CT for T joints, CX for X joints and CK for K gap joints), see Figs 8.17 to 8.19, gives the efficiency for a joint with: - θi = 90 - fy0 t0 = fyi ti (identical thickness and yield stress for brace and chord) - Qf = 1,0

For out-of-plane bending, the loading is concentrated at the saddles, similar to X joints in the ring model approach. This explains why the same function is used for f(β).

In the case of Vierendeel girders, it is recommended to choose joints with width ratios close to β =1,0 to provide sufficient stiffness and strength.

As an example, Fig. 8.19b shows that for a K joint with g = 2t0, 2  30 and β  0,5, a value of CK  0,4 applies. Thus, for an angle θi = 45, a 100% efficiency can be obtained if:

For plate, I, H or RHS to CHS chord joints the moment resistance is given in Tables 8.3a and 8.3b.

86

fy0 t 0 f yi t i

previous recommendations, a reduction factor was only given for chord compression loading and was based on the chord preload.

 1,77

Note that this ratio should be slightly larger because the chord stress effect Qf has not been included.

For brace in-plane bending, only the constant has been changed resulting in about 10% lower strength.

This reduction is also a result of newly derived data with larger dimensions and relatively smaller welds.

This example shows that for a 100% efficiency, fy0 t0 should always be considerably larger than fyi ti.

For brace out-of-plane bending, the capacity in the recommendations of IIW (2009) and the 2nd edition of CIDECT Design Guide No. 1 is again related to the axial load capacity of X joints. On average, for medium β and γ ratios the joint capacities are about the same as those in Eurocode 3 (EN 1993-1-8, 2005).

The chord stress function Qf for CHS joints is given as a function of the parameter n, thus related to the maximum stress to yield stress ratio in the chord, see Figs. 8.11 and 8.15.

For simply supported lattice girders, the influence of the chord stress function is small (small chord loads) near the supports, whereas it has a pronounced influence at the centre of the girder where the chord loads are high and, in general, the brace loads are small. The chord stress function is especially important in the case of continuous or cantilevered lattice girders.

For the plate, I or RHS to CHS chord joints, selected data of Akiyama et al. (1974) were originally excluded from the database due to unexplained deviations from other data. In the new analyses these data are included since no satisfactory explanation could be obtained for the discrepancies observed. However, this resulted in a considerable reduction of the joint capacities.

8.9 RELATION TO THE PREVIOUS RECOMMENDATIONS OF IIW (1989) AND CIDECT (1991)

Currently, a research programme, including both experiments and numerical analyses, is being carried out at the University of Toronto (Voth, 2010) to investigate this type of configuration more in detail.

These results confirm that Akiyama’s data are far too conservative.

Recently all data have been reanalysed (Van der Vegte et al., 2007, 2008, 2010b) for the revision of the recommendations of IIW (2009) and CIDECT Design Guide No. 1 (Wardenier et al., 2008a). Zhao et al.

(2008) give a detailed overview of the modifications made for the IIW (2009) recommendations. These new recommendations are also to be included in the new ISO standard in this field and will be the basis for the next revision of the Eurocode 3 recommendations.

In most cases, the updated recommendations of IIW (2009) and CIDECT (Wardenier et al., 2008a) give slightly lower capacities than the previous equations, although there are areas where they give the same or higher capacities. A comprehensive comparison between these new recommendations and the previous equations is given by Wardenier et al. (2008a, 2008b). These references further give a comparison with the latest update of the API (2007); see also Pecknold et al. (2007).

For these reanalyses, extensive numerical data have been obtained. One of the main reasons for the analyses was to define the chord stress function by the maximum chord stress, in accord with that for RHS joints. However, this also influences the basic joint strength functions, while the same analytical models apply. Further, the current database now includes more joints with larger sections where the size of the welds has a smaller influence on the joint capacity.

Detailed information about the background of American codes is given by Marshall & Toprac (1974), Marshall (1984, 1992, 2004, 2006) and Packer et al.

(2010). The background for the ISO 19902 (2007) offshore code is given by Dier & Lalani (1998) and Dier (2005).

In the new recommendations of IIW (2009) and CIDECT (Wardenier et al., 2008a), for both chord compression and tension loading, the maximum chord load has to be considered for the chord load function and both give a reduction in joint resistance. In the

8.10 CONCLUDING REMARKS For more detailed information about special types of

87

joints such as cropped end joints, flattened end joints, stiffened joints, slotted CHS gusset plate joints, etc., reference is made to the appropriate literature; see Choo et al. (2004, 2005b), Dutta (2002), Kurobane (1981), Packer (2006), Packer & Henderson (1997), Rondal (1990), Thiensiripipat (1979), Wardenier (1982) and Wardenier et al. (2008a).

88

Table 8.1 Design axial resistances of welded joints between circular hollow sections Type of joint

Design limit state

T and Y joints

Chord plastification t1

N1,Rd 

N1

d0

d1

t0

1 M0

N0

X joints

 2,6  17,7 β   2

sin 1

0,2

Qf (for d1  d0 - 2t0)

Chord punching shear

N1,Rd = 0,58 fy0  d1 t0

1  sin 1 2 sin2 θ1

Chord plastification

N1,Rd 

t1 N1

d0

d1

t0

1

N1,Rd  0,58 fy 0

K and N gap joints

t2

t1 1

(for X joints, if cos θ1 > β) ( 2 / ) A 0 sin 1

  N1,Ed sin 1  Ngap,0,Rd  A 0 fy 0 1    0,58 fy 0 ( 2 / ) A 0   

2

Chord plastification

N1,Rd  d1

(for d1  d0 - 2t0)

Chord punching shear

Chord shear

N1

N1

fy 0 t 02  2,6  2,6   0,15   Qf sin 1  1 - 0,7β 

See chord punching shear equation for T and Y joints

N0

d0

fy0t 02

g

N2

d2 2

+e

N2,Rd 

fy0 t02 sin 1

  1 (1,65  13,2 1,6 )  0,3 1  Q 0,8  f 1 , 2 ( g / t )  0  

sin 1 N1,Rd sin 2

(for di  d0 - 2t0)

Chord punching shear N0

Ni,Rd = 0,58 fy0  di t0



Qf  1  n

Function Qf

C

1

1  sin i 2 sin2 θi

with n 

N0,Ed Npl,0,Rd



Chord compression stress (n < 0) T, Y and X joints

C1 = 0,45 - 0,25β

K gap joints

C1 = 0,25

M0,Ed Mpl,0,Rd

in connecting face

Chord tension stress (n  0) C1 = 0,20

Range of validity

ti  t 0

e  0,25 d0

i  30

g  t1  t 2

fyi  fy0 fy  0,8fu fy  460 N/mm2

Compression

class 1 or 2 (2) and d0 /t 0  50 (for X joints : d0 /t 0  40 )

0,2 

General

Chord

Braces (1) (2)

di  1,0 d0

d0 /t 0  50 (for X joints : d0 /t 0  40 )

Tension Compression

class 1 or 2

(2)

and di /ti  50 di /ti  50

Tension

For 355 N/mm2 < fy0  460 N/mm2, use a reduction factor of 0,9 for the design resistances.

Class 1 and 2 limits for di/ti are given in Table 2.7.

89

(1)

Table 8.2 Design resistance of related types of joints Type of joint

Relationship to the formulae of Table 8.1

All brace member forces act in the same sense (compression or tension).

N1,Ed  N1,Rd with N1,Rd for X joint given in Table 8.1

All brace member forces act in the same sense (compression or tension).

N1,Ed sinθ1 + N2,Ed sinθ2  NX,Rd sinθX with NX,Rd sinθX from X joint given in Table 8.1, being the larger of the values for brace 1 or 2

Forces in members 1 are in compression and members 2 in tension.

Ni,Ed  Ni,Rd (i = 1 or 2) with Ni,Rd from K joint given in Table 8.1, but with the actual chord force

Forces in members 1 are in compression and members 2 in tension.

Ni,Ed  Ni,Rd (i = 1 or 2) with Ni,Rd from K joint given in Table 8.1 Note: In a gap joint, the chord cross section in the gap has to be checked for shear failure: 2

2

 Ngap,0,Ed   Vgap,0,Ed       1,0  Npl,0,Rd   Vpl,0,Rd     

NN22

N N11

where: Ngap,0,Ed = design axial force in gap

Npl,0,Rd  A 0 fy 0

Vgap,0,Ed = design shear force in gap

Vpl,0,Rd  0,58fy 0

90

2A 0 

Table 8.3a Design resistances of welded T joints connecting plates or open sections to CHS chords Type of T joint

Design limit state Chord plastification

N1,Rd

Mip,1,Rd

Mop,1,Rd

N1,Rd  fy0 t 02 2,2  15 β2  0,2 Qf

Mip,1,Rd  0

Mop,1,Rd  0,5 b1 N1,Rd

N1,Rd  5fy 0 t 02 (1  0,4) Q f

Mip,1,Rd  h1 N1,Rd

Mop,1,Rd  0

N1

b1



t1 to



do N1 h1

t1 t0 d0

N1

h1

1 t0

d0



h1



N1,Rd  fy0 t 02 2,2  15 β2 (1 0,4)  0,2 Qf

N1

Mip,1,Rd 

h1 N1,Rd (1  0,4)

Mop,1,Rd  0,5 b1 N1,Rd

b1

1

t0 d0

(for b1  d0 - 2t0)

Chord punching shear

I section brace with η  2 (axial loading and brace out-of-plane bending) and RHS brace

N1,Ed A1 N1,Ed

All other cases

A1







Mip,1,Ed Wel,ip,1 Mip,1,Ed



Wel,ip,1

Qf  1  n

Function Qf



C

1

Mop,1,Ed Wel,op,1 Mop,1,Ed Wel,op,1

with n 

 0,58 fy0

t0 t1

 0,58 fy 0

2t 0 t1

N0,Ed Npl,0,Rd



M0,Ed Mpl,0,Rd

(t1 = flange thickness for I section brace)

in connecting face

Brace axial load, brace in-plane bending and brace out-of-plane bending Chord compression stress (n < 0): C1 = 0,25

Chord tension stress (n  0): C1 = 0,20

Range of validity 0,2 

General CHS chord

RHS braces I section braces Plates (1) (2)

Compression

b1  1,0 d0

fy1  fy0

fy  0,8fu

fy  460 N/mm2

class 1 or 2 (2) and d0 /t 0  50 (for X joints : d0 /t 0  40 ) d0 /t 0  50 (for X joints : d0 /t 0  40 )

Tension class 1 or 2

(2)

Compression

class 1 or 2

(2)

Tension

none

Compression

1  90

and b1/t1  40 and h1/t1  40 b1/t1  40 and h1/t1  40

Tension

Transverse plate:  

b1  0,4 d0

Longitudinal plate: 1   

For 355 N/mm2 < fy0  460 N/mm2, use a reduction factor of 0,9 for the design resistances.

Section class limitations are given in Table 2.7.

91

h1 4 d0

(1)

Table 8.3b Design resistances of welded X joints connecting plates or open sections to CHS chords Type of X joint

Design limit state Chord plastification

N1,Rd

Mip,1,Rd

Mop,1,Rd

 2,2  2,2   0,15  Qf N1,Rd  fy 0 t 02   1 - 0,7β 

Mip,1,Rd  0

Mop,1,Rd  0,5 b1 N1,Rd

N1,Rd  5fy 0 t 02 (1  0,4) Q f

Mip,1,Rd  h1 N1,Rd

Mop,1,Rd  0

 2,2  2,2    (1  0,4)  0,15 Q f N1,Rd  fy 0 t 02  1 0,7β  

Mip,1,Rd 

N1

b1 t1 t0 d0

N1

N1 h1

t1 t0 d0

N1 N1

h1

1 t0

d0

N1

N1 h1

b1

1

h1 N1,Rd (1  0,4)

t0 d0

N1

Chord punching shear

See chord punching shear equations for T joints in Table 8.3a Function Qf

Same as in Table 8.3a

Validity range

Same as in Table 8.3a (1)

(1)

For 355 N/mm2 < fy0  460 N/mm2, use a reduction factor of 0,9 for the design resistances.

92

Mop,1,Rd  0,5 b1 N1,Rd

Table 8.4 Correction factors for the design resistance of multiplanar joints Type of joint

Correction factor μ to uniplanar joint resistance

TT joints

General

Members 1 may be either in tension or compression N1

N1

N1

N1

gt



μ = 1,0

XX joints

Chord plastification

Members 1 and 2 can be either in compression or tension

  1 0,35

N1 1 N

N

N1 1

NN22

  1  0,35

N1

N1

General

Members 1: compression Members 2: tension

μ = 1,0

N1

A A

A A

Validity range

N2

N2

gt



N2,Ed N1,Ed

but  1,0

Notes: - Take account of the sign of N1,Ed and N2,Ed, with |N2,Ed |  |N1,Ed| - N2,Ed/N1,Ed is negative if the members in one plane are in tension and in the other plane in compression

KK gap joints

N1

N1,Ed

Chord punching shear

N N 22

N1 N 1

N2,Ed

Note: In a KK gap joint, the chord cross section in the gap has to be checked for shear failure: 2

2

 Ngap,0,Ed   Vgap,0,Ed       1,0  Npl,0,Rd   Vpl,0,Rd     

where: Ngap,0,Ed = design axial force in gap

Npl,0,Rd  A 0 fy 0

Vgap,0,Ed = design shear force in gap

Vpl,0,Rd  0,58fy 0

Same as in Table 8.1 and 60°    90°

93

2A 0 

Table 8.5 Design moment resistances of welded joints between circular hollow sections Type of joint – brace loading

Design limit state

T, Y and X joints – in-plane bending

Chord plastification

Mip,1,Rd  4,3

fy0 t02 d1 sin 1

  0,5 Qf (for d1  d0 - 2t0)

Chord punching shear

Mip,1,Rd  0,58 fy0 t0 d12 T, Y and X joints – out-of-plane bending

1  3 sin θ1 4 sin2 θ1

Chord plastification

Mop,1,Rd 

fy 0 t 02 d1  1,3  1,3   0,15   Qf sin 1  1 - 0,7β  (for d1  d0 - 2t0)

Chord punching shear

Mop,1,Rd  0,58 fy0 t0 d12

Function Qf



Qf  1  n

C

1

3  sin θ1 4 sin2 θ1

with n 

N0,Ed Npl,0,Rd



M0,Ed Mpl,0,Rd

in connecting face

Chord compression stress (n < 0)

Chord tension stress (n  0)

T, Y and X joints

C1 = 0,45 - 0,25β

C1 = 0,20

Validity range

Same as in Table 8.1 (1) (2)

(1) (2)

For 355 N/mm2 < fy0  460 N/mm2, use a reduction factor of 0,9 for the design resistances.

The equations in Table 8.5 may also be used for K gap joints, if brace moments have to be considered, by checking that the brace utilization due to bending plus the brace utilization due to axial load  0,8. For K overlap joints, no evidence exists.

94

Fig. 8.1 Example of a prefabricated connector (this Nodus type is out of production)

Fig. 8.2 Joints with end pieces for bolted joints

Fig. 8.3 Welded CHS slotted gusset plate joints

95

Y joint

T joint d1

d1

N1 t1

N1

t1

1

t0

1

d0

X joint

d0

t0

N0

K joint with gap

d1 t1

d1

d2

N1

N2

t1

t2

N1

g

1 t0

N0

d0

N0

2

1 t0

d0

N1 N joint with overlap

KT joint with gap

di dj Nj

d3 d2

Ni tj

N2

ti

t2

t3 g2

N0

j

i

t0

d0

N0

Fig. 8.4 Welded CHS joints

96

2

d1

N3 t1

N1

g1 1 t0

d0

(a) Brace failure (yielding, local buckling)

(e) Chord punching shear failure

(b) Weld failure

(f) Chord local buckling

as (a) but failure in the weld

(g) Chord shear failure

(c) Lamellar tearing

see e.g. Fig. 7.6c

(d) Chord plastification (face/wall, thus cross section)

or

Fig. 8.5 Failure modes for joints between circular hollow sections

N1 σ1,Ed σjoint

σjoint

N1 Fig. 8.6 Elastic stress distribution in an X joint

97

N1

c1d1

N1 sin 1 2B e

N1 sin 1 2

Be ≈ 2,5 to 3d0

N1

Fig. 8.7 Ring model

Fig. 8.8 Plastic hinges in the ring model at failure

N1

1 Vp N0

Fig. 8.9 Punching shear model

98

N1 sin 1 2

N1 sin 1 2

N2

N1 A

N0

A

2

1

V

A Fig. 8.10 Chord shear model

N2

Ngap,0

A

N1 2

N0 

A N0

2

N cos   N i

i

0p

i 1

1 N0p

A Fig. 8.11 Chord load N0 and chord preload N0p

Fig. 8.12 Test rig for isolated joint tests

99

Ngap,0  N1 cos 1  N0p

N1/(fy0t02)

 = 0,60 2 = 40,0

δ (mm) Fig. 8.13 The effect of the element type on numerical results

8

1  1  0,7

f(N1u) X joints

6

4

2

0 0,0

0,2

0,4

0,6

0,8

1,0

 Fig. 8.14 Comparison of experiments with the mean joint resistance function (X joints)

100

1,0

2

N1u sin 1 / (fy0 t0 Qu)

1,2

0,8 0,6 2γ = 63,5 0,4

2γ = 25,4 2γ = 63,5 2γ = 50,8

0,2

2γ = 25,4 0,0 -1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

N0 / Npl,0

Fig. 8.15 Chord stress function Qf

di

dj

ti

tj Nj

Ni (1) 4.8

j

(3) 4.11

Nop

4.11

i

t0

N0o

d0

4.10 4.10 (2) (2) brace i =overlapping member; brace jj = overlapped brace i = overlapping overlapped member member brace i = overlapping member; brace j = overlapped member

Fig. 8.16 Checks for overlap joints

101

T joint efficiency

X joint efficiency 1,0

0,9

N1,Rd

0,8

A1 fy1

 CT

fy 0 t 0



Qf fy1 t1 sin 1



0,7 0,6 0,5



0,4



0,3

effciency C X

effciency C T

1,0



0,2 0,1 0,0

 0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

0,9

N1,Rd

0,8

A1 fy1

 

0,5 0,4 0,3



0,2 0,1



0,0 0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

1



Fig. 8.18 Design charts for CHS X joints

K gap joint efficiency g'=2

K gap joint efficiency g'=1 1,0



Qd N* fy0 t 0 fQ t d   CK1  C y 0f 0 1 f 2 A i fyi A1 fy1 fyi t i Ksin 2di1 fy1ti1 sin

Ni,Rd

0,7



0,6



0,5 0,4



0,3



0,2 0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9



0,7 0,6



0,5 0,4



0,3



0,2 0,1 0,0

 0



f t d Qd N* fy0 t 0 Q  CK1  C y 0f 0 1 f 2 A i fyi A1 fy1fyi t i Ksin 2di 1 fy1ti1 sin

Ni,Rd

0,9 0,8

effciency C K

0,8

effciency C K



Qf fy1 t1 sin 1

0,6

1

Fig. 8.17 Design charts for CHS T and Y joints

 0

1

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

1





Fig. 8.19a Design charts for CHS K joints with gap (g’ = g/t0 = 1)

Fig. 8.19b Design charts for CHS K joints with gap (g’ = g/t0 = 2) K gap joint efficiency g'=10

K gap joint efficiency g'=5 1,0

1,0 

Qd N* fy0 t 0 fQ t d   CK1  CK y 0f 0 1 f 2 A i fyi A1 fy1 fyi t i sin 2di1 fy1ti1 sin

Ni,Rd



0,7 0,6



0,5 0,4



0,3



0,2 0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9



0,5 0,4



0,3



0,1 0,0

 0

1

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

1





Fig. 8.19c Design charts for CHS K joints with gap (g’ = g/t0 = 5)



0,7 0,6

0,2

 0



f t d Qd N* fy0 t 0 Q  CK1  CK y 0f 0 1 f 2 A i fyi A1 fy1fyi t i sin 2di 1 fy1ti1 sin

Ni,Rd

0,9 0,8

effciency C K

0,9 0,8

effciency C K

fy 0 t 0

0,7



1,0 0,9

 CX

Fig. 8.19d Design chart for CHS K joints with gap (g’ = g/t0 = 10)

102

9. WELDED JOINTS BETWEEN RECTANGULAR HOLLOW SECTIONS 9.1 INTRODUCTION The most economical and common way to connect rectangular hollow sections is by direct connection without any intersecting plates or gussets, as shown in Fig. 9.1. This also gives the most efficient way for protection and maintenance.

Joints between rectangular hollow sections can be easily made, since the connecting members have to be provided with straight end cuts only.

Although the fabrication is simple, the load transfer is more complex due to the non-uniform stiffness distribution in the joints. Due to the flat sides, the difference in stiffness at the side and at the centre of a face is even greater than for circular hollow sections.

The general philosophy to identify the various failure modes and to observe the load transfer has been described in Chapter 7, but will be discussed here in more detail for the joints between rectangular and square hollow sections.

Most failure modes can be related to analytical models to study the impact of the various influencing parameters. Based on the analytical models and the tests carried out, design rules have been established.

9.2 MODES OF FAILURE Similar to circular hollow section joints, the ultimate load capacity is based on the maximum in the load deformation diagram (if the chord deformation is less than 0,03b0) or the load at a deformation of 0,03b0 of the chord.

As already indicated in Chapter 7 and shown in Fig. 9.2, the following modes of failure can occur: - (Local) brace failure (cracking or local buckling) - Weld failure - Lamellar tearing - Chord face plastification - Chord punching shear failure - Chord side wall yielding or buckling - Chord local buckling - Chord shear failure

avoid weld failure, the welds should be stronger than the connected braces and the throat thickness should satisfy the same requirements as given in Section 6.5.2. Prequalified full penetration welds can always be considered to be stronger than the connected brace members. Partial penetration plus fillet, or fillet welds alone, can also usually provide a weld connection as strong as the connected braces.

Also here, the steel should not be susceptible to lamellar tearing. Especially for larger thicknesses (t > 25 mm), a TTP quality with a low sulphur content should be used for the chords.

In the design recommendations the width-to-wall thickness ratios b/t have been limited to avoid local buckling and/or to limit deformations.

As a result, the following failure modes still have to be considered in design: - Local brace failure (yielding, local buckling) - Chord face plastification - Chord punching shear - Chord side wall failure - Chord shear failure Due to the fact that rectangular hollow sections can be connected with various orientations and in various combinations, several failure modes have to be considered, which makes the checking procedure more complicated.

In the case of joints between square hollow sections and within a smaller validity range, the failure modes to be checked can be limited to one or two.

Local brace failure generally occurs for joints with relatively thin walled braces and is a general failure mode for overlap joints.

Chord face plastification is the most common type of failure for T, Y, X and K and N gap joints with width ratios β < 0,85.

Chord punching shear may occur in joints with relatively low or high β ratios. However, to allow shear of the chord face, the brace width bi should satisfy: bi < b0 - 2 t0 - 2(1,4a) where a is the weld throat thickness.

Chord side wall failure is a common failure mode for T, Y and X joints with a β ratio close or equal to 1,0.

Similar to joints between circular hollow sections, to 103

Chord shear may occur in K gap joints with a high β ratio or K gap joints with chords with a low h0/b0 ratio.

The energy dissipated in the various yield lines is also indicated in Fig. 9.3. Equating the sum to the external work gives:

9.3 ANALYTICAL MODELS

2 f y 0 t 02    (1  )  tan     N1 sin 1  1   tan  sin 1 

Similar to circular hollow section joints, analytical models are used to describe the joint behaviour and to determine the governing parameters for the joint strength. Sometimes the joint behaviour is too complicated to cover all influencing parameters and in combination with the test results semi-empirical formulae have been developed to describe the joint strength.

This is a minimum for:

dN1 0 d

tan   1  

In principle, the yield line method is an upper bound approach. Therefore, various yield line patterns have to be examined in order to obtain the lowest capacity.

However, the difference in capacity between the various yield line patterns is relatively small.

Furthermore, local strain hardening effects and membrane action are ignored. Therefore, the simplified yield line pattern shown in Fig. 9.3 (model a) is generally used for T, Y and X joints instead of the more complicated pattern shown in Fig. 9.3 (model b).

The principle of the yield line method is based on equating the work done by the external force N1 over a deflection δ and the internal energy dissipated by the plastic hinge system with yield lines of length ℓi and rotation angles i.

N1 sin 1     i i mp

(9.1)

where:

mp 

1 2 t 0 f y 0 per unit length 4

(9.2)

(9.4)

or:

9.3.1 Yield line model The yield line model, originally developed by the Danish researcher Johansen for plates, is widely used for joints between rectangular hollow sections. For joints with medium β ratios, the yield line model gives a good estimate of the chord face plastification capacity (Packer, 1978; Wardenier & Giddings, 1986; Yu, 1997; Zhao, 1992), even under low temperatures (Björk et al., 2003). For very small β ratios, the deformation to realise the yield line pattern may be too high. For high β ratios, the model predicts infinite strengths and other failure modes will be governing, e.g. chord side wall failure.

(9.3)

(9.5)

Substitution of eq. (9.5) in eq. (9.3) gives the capacity: N1 

f y 0 t 02  2  1   4 1    1    sin 1  sin 1

(9.6)

In this model, some simplifications have been incorporated, i.e. the thickness of the sections has been neglected (b0 - 2t0  b0). The same applies to the weld sizes, which have not been considered. Further, the effect of the chord load has to be included, which will be done by a chord load function Qf. For K joints, yield line models can also be used.

However, the load transfer is more complicated since in the gap area, the stress situation in the yield hinge is largely influenced by membrane stresses, shear stresses and work hardening. These effects complicate the analytical models to such an extent that semi-empirical formulae are used for design.

9.3.2 Punching shear model Similar to joints between circular hollow sections, the brace can be pulled out of the chord, resulting in cracking in the chord by shear around the brace connection perimeter. Since the stiffness along the perimeter is non uniform, the deformation capacity of certain parts may not be sufficient to obtain a full effective perimeter for punching shear; i.e. only certain parts can be assumed to be effective for resisting the punching shear. For example, for a T or Y joint (Fig. 9.4) the sides along the chord walls are the stiffest part. Depending on the b0/t0 ratio of the chord, a larger or smaller part along the cross walls will be effective, designated as be,p.

104

Chord punching shear is caused by the brace load component perpendicular to the chord face, thus the punching shear criterion is given by: N1 

 2 h1  1 t 0   2b e,p  3  sin 1  sin 1

fy0

(9.7)

It will be clear that be,p is a function of b0/t0. The smaller b0/t0, the larger be,p. The value for be,p is determined experimentally (Wardenier, 1982).

9.3.3 Local brace failure model The model to describe local brace failure (Fig. 9.6) has a certain relationship with the chord punching shear model. Due to the non uniform stiffness along the connection perimeter, both models consider an effective part, although due to the different deformation capacities for failure in the brace and chord punching shear, the values for be and be,p are different.

For K gap joints, the gap size is extremely important for the effective punching shear length. For example, if the gap size is close to zero and the β value is low to medium (Fig. 9.5a), the gap part is relatively very stiff compared to the other perimeter parts, resulting in:

Furthermore, chord punching shear is caused by the brace load component perpendicular to the chord, whereas for local brace failure, the brace load is taken. The effect of the angle θ has not yet been defined clearly and has been conservatively excluded up to now.

 h  t 0  b i  2c i  with c << 1 sin i  3 

For a T, Y and X joint, the criterion for local brace failure can thus be given by:

Ni sin i 

fy0

(9.8)

For a large gap (Fig. 9.5c) a similar situation occurs as for T, Y and X joints, thus: Ni sin i 

 2hi  t 0   2b e,p  3  sin i 

fy0

(9.9)

For a gap where the stiffness is about the same as at the sides of the braces (Fig. 9.5b), the punching shear criterion becomes: Ni sin i 

 2hi  t 0   b i  b e,p  3  sin i 

fy0

(9.10)

or

g  1  b0

g  1,5 1   b0

For K joints with gap, the same applies as for the punching shear criterion, i.e. the gap size should satisfy eq. (9.11) for having a full effective cross wall of the brace at the gap: (9.13)

This criterion is also directly applicable to overlap joints for the overlapping brace, see Chapter 11.

9.3.4 Chord side wall bearing or buckling model

Due to the deformation capacity of the material, which has been proved experimentally, this limit can be extended to: 0,5 1    

(9.12)

The term 4t1 has to be included to avoid the corners being counted twice. Similar to the punching shear criterion, the effective width be is determined experimentally (Wardenier, 1982) and becomes larger if b0/t0 decreases.

Ni = fyi ti (2hi + bi + be - 4ti)

Neglecting the thickness and the weld sizes, the gap value here has to satisfy: g b 0  bi  2 2

N1 = fy1 t1 (2h1 + 2be - 4t1)

(9.11)

T, Y and X joints with a high β ratio generally fail by yielding or buckling of the chord side walls, shown in Fig. 9.8. The model used is similar to that used for beam-to-column joints between I sections. For joints with β = 1,0, the capacity can be easily determined by:  h  1 N1  2f y 0 t 0  1  5t 0  sin  1   sin 1

For  values close to 1,0 these limits are impractical and a minimum gap g = t1 + t2 for welding is required.

(9.14)

For slender walls, the yield stress fy0 is replaced by a buckling stress fk which depends on the chord web

105

slenderness h0/t0. This model is simple and gives a larger margin of safety for high slendernesses.

A model which is in better agreement with the test results for all slendernesses is based on the "4 hinge yield line" mechanism, shown in Fig. 9.9 dating back to the 1970s (Packer, 1978). In the case of compression of the chord side walls, Yu (1997) also used a buckling stress, but with a buckling length of (h0 - 2t0)/2.

9.3.5 Chord shear model Similar to circular hollow section joints, this model, shown in Fig. 9.10, can be described by the basic formula for plastic design. The plastic shear load capacity is given by:

Vpl,0 

fy0 3

(9.15)

Av

Wardenier, 1982), whereas for K joints with gap, a semi-empirical formula is used. As an example, Fig. 9.11 shows a comparison between the experiments and the formula for the strength of K gap joints, established by Wardenier (1982), which was included in the previous recommendations of IIW (1989) and CIDECT Design Guide No. 3 (Packer et al., 1992).

In the last 20 years, the results of many numerical investigations became available, e.g. Lu (1997), Yu (1997), Liu & Wardenier (2001, 2003, 2006), Kosteski & Packer (2003), Wardenier et al. (2007a, 2007b).

Furthermore the recommendations had to be extended to steel S460, which made a revision of the recommendations and the validity ranges necessary (Liu & Wardenier, 2004).

9.5 BASIC JOINT STRENGTH FORMULAE 9.5.1 T, Y, X joints and K, N gap joints

In principle, the webs are effective for shear, but if the gap is small, a part of the top flange may also be effective, thus: A v  2 h0   b 0  t 0

(9.16)

The coefficient α depends on the g/t0 ratio and can be easily determined based on plastic analysis (Wardenier, 1982).

The remaining cross section has to transmit the axial force. Using the Huber-Hencky-Von Mises criterion, the following interaction formula can be derived: Ngap,0  A 0  A v  f y 0  A v f y 0

 Vgap,0 1   Vpl,0 

   

2

(9.17)

This formula is comparable to that for circular hollow section joints, eq. (8.10).

9.4 EXPERIMENTAL AND NUMERICAL VERIFICATION

For T, Y and X joints up to β = 0,85, the yield line model for chord face plastification is used as a basic lower bound formula for the joint resistance (Fig. 9.12).

Above β = 0,85, the joint resistance is governed either by chord side wall failure, brace failure or by chord punching shear, for b1 < b0 - 2 t0 - 2(1,4a).

For K gap joints, initially the semi-empirical formula of Wardenier (1982) based on chord face plastification was used as the primary criterion. However, the ultimate strength data which served as a basis for this equation did not consider the deformation limit of 3% b0 and only included chord sections of b0/t0  35.

Hence, a modification has recently been carried out (see Section 9.6).

Depending on the joint parameters, various other criteria may become critical for K gap joints, e.g. local brace failure, chord punching shear or chord shear.

9.5.2 K, N overlap joints

Based on the models and test evidence, either analytical or semi-empirical formulae were developed.

For example, for T, Y and X joints, the yield line model is used as a lower bound for the test results and also incorporated in the recommendations (EN 1993-1-8, 2005; IIW, 1989, 2009; Packer, 1978; Packer & Henderson, 1997; Packer et al., 1992, 2009a;

As mentioned before, for overlap joints the same approach is adopted for all types of overlap joints, regardless of whether circular or rectangular braces are used in combination with a circular, rectangular or an open section chord (see Chapter 11).

The resistance of overlap joints between rectangular

106

hollow sections with 25%  Ov  100% overlap is based on the following criteria (Chen et al., 2005; Liu et al., 2005; Wardenier & Choo, 2006): (1) Local failure of the overlapping brace (2) Local chord member yielding at the joint location based on interaction between axial load and bending moment (3) Shear of the connection between the brace(s) and the chord Fig. 9.13 shows the overlap joint configuration with the cross sections to be examined for these criteria. For K and N overlap joints, the subscript i is used to denote the overlapping brace member, while the subscript j refers to the overlapped brace member.

Local failure of the overlapping brace (criterion 1) should always be checked, while shear between the braces and the chord (criterion 3) may only become critical for larger overlaps, i.e. larger than 60% or 80%, depending on whether or not the hidden toe location of the overlapped brace is welded to the chord. The check for local chord member yielding (criterion 2) is, in principle, a member check and may become critical for larger overlaps and/or larger β ratios.

capacity for large b0/t0 ratios would be reduced.

Changing the original function 8,9βγ0,5, used in the previous editions of IIW (1989) and CIDECT Design Guide No. 3 (Packer et al., 1992), into 14βγ0,3 gives the same capacity for b0/t0 = 20 and a reduction for large b0/t0 ratios. The new expression, included in IIW (2009) and CIDECT Design Guide No. 3 (Packer et al., 2009a) is a reasonable compromise between covering the data determined with the 3% b0 deformation limit, extension of the validity range and backup by previous analyses (Packer & Haleem, 1981; Wardenier, 1982).

Since in joints of rectangular hollow sections the sections can have different orientations and depth-to-width ratios, many configurations are possible, resulting in a large number of failure modes and related strength formulae, see Table 9.1.

However, with a smaller range of validity, the design formulae for joints between square hollow sections can be reduced to such an extent that only a single check has to be carried out (see Table 9.2).

Fleischer & Puthli (2008) initiated research on how to deal with RHS joints with gaps and/or chord width-to-thickness ratios outside the current range of validity.

Joints with overlaps between 0% and 25% should be avoided because in those cases, the stiffness of the connection between the overlapping brace and the overlapped brace is much larger than that of the overlapping brace to chord connection, which may lead to premature cracking and lower capacities (Wardenier, 2007).

9.7 OTHER TYPES OF JOINTS OR OTHER LOAD CONDITIONS

More detailed information regarding the design equations for overlap joints is given in Chapter 11.

As for circular hollow section joints, various joint configurations exist for which the resistance can be directly related to the basic types presented in Tables 9.1 and 9.2.

9.6 EVALUATION TO DESIGN RULES In principle, the evaluation to design rules, e.g. for K joints with gap, is similar to that described for joints between circular hollow sections.

For T, Y and X joints, a lower bound analytical yield line criterion is used for chord face plastification.

Therefore, no statistical evaluation has been carried out.

Considering Lu’s deformation limit of 3% b0 (Lu et al., 1994), and extending the validity range to class 1 and 2 sections with b0/t0  40 and h0/t0  40 instead of an upper limit of 35, for K gap joints, the functions for β and γ needed to be modified in such a way that the

9.7.1 Related types of joints

Table 9.3 gives the design resistance for some special types of RHS uniplanar joints with braces directly welded to the chord; notice the similarity with Table 8.2 for circular hollow section joints.

9.7.2 Joints between circular braces and a rectangular chord As far as chord face plastification is concerned, the strength of a joint with a circular hollow section brace with diameter di is about π/4 times that of a joint with a square hollow section brace with a width bi = di, see Fig. 9.14 (Wardenier, 1982; Packer et al., 2007). As a consequence, the same formulae can be used as for square hollow section joints, but the resistances have

107

to be multiplied by π/4. This also means that the joints have the same efficiency, i.e. the joint strength divided by the squash load of the brace.

9.7.3 Joints between plates or I sections and RHS chords Joints between plates or I sections and RHS chords are approached in a similar manner as the rectangular hollow section joints and in principle the same modes of failure have to be considered. Within the scope of this book, these joints are not further discussed, but reference is made to Lu (1997), Packer et al. (2009a) and Chapter 12. The design resistances are given in Table 9.4.

Tee joints to the ends of RHS members When an axial force is applied to an RHS member, via a welded Tee joint as shown in Fig. 9.15, the capacity is determined by local failure of the RHS walls or the Tee web.

For a commonly used distribution slope of 2.5:1 from each face of the Tee web (Kitipornchai & Traves, 1989), the dispersed load width is (5tp + tw). A conservative assumption is to use this effective width at two sides of the RHS member. Thus, the resistance of the RHS can be computed by summing the contributions of the parts of the RHS cross sectional area into which the load is distributed: N1,Rd  2 f y1 t1 (t w  5t p )  A 1 fy1

(9.18)

A similar load dispersion can be assumed for the capacity of the Tee web. If the Tee web has the same width as the width of the cap plate, i.e. (h1 + 2s), the capacity of the Tee web is: N1,Rd  2 f yw t w (t 1  2,5t p  s)

(9.19a)

 2 f yw t w (t 1  5t p )

(9.19b)

In eqs. (9.18) and (9.19), the size of any weld legs to the Tee web has been conservatively ignored.

Gusset plate-to-slotted RHS joints Single gusset plates, slotted into the ends of hollow section members and concentrically aligned with the axis of the member, as shown in Fig. 9.16, are commonly found in diagonal brace members of steel framed buildings.

As a consequence of only part of the RHS cross section being connected, an uneven stress distribution

around the RHS perimeter occurs during load transfer at the connection. This phenomenon, known as shear lag, is illustrated in Fig. 9.16.

The possible failure modes for the gusset plate-to-slotted RHS joints loaded in tension are circumferential failure of the RHS and tear out or "block shear" failure of the RHS. Shear lag is principally influenced by the weld length, Lw in relation to the dimension w which is the distance between the welds measured from plate face-to-plate face, around the perimeter of the RHS. For long weld lengths, shear lag effects become negligible, while for short weld lengths, tear out governs over circumferential fracture of the RHS. However, if Lw = 1,65b for square braces and 1,3d for circular braces, it can be assumed that the capacity is equal to that of the connected hollow section or plate.

Detailed design rules are given by Packer et al.

(2009a).

9.7.4 Multiplanar joints Compared to uniplanar joints, multiplanar joints have a geometric effect and a loading effect to be considered.

It is plausible that a multiplanar joint has a geometric influence only if the β value is large, because then the chord side wall is stiffened, see e.g. Fig. 9.17 for an XX joint.

For multiplanar joints of rectangular hollow sections, the tendency of the loading effect is similar but less pronounced compared to that of joints of circular hollow sections, see Fig. 9.18.

Extensive analytical and numerical research by Liu & Wardenier (2001, 2003) showed that the differences in capacity between uniplanar K gap and multiplanar KK gap joints are caused by the larger chord force acting in multiplanar joints. Based on this work, the following design recommendations are given for multiplanar KK joints (see Table 9.5).

Multiplanar KK gap joints (Fig. 9.19) - For chord face plastification (small or medium β), the strength of the joint can be based on the joint resistance formulae for uniplanar joints given in Tables 9.1 and 9.2, and no further multiplanar correction is necessary, provided that the actual, total chord force is used for the chord stress function Qf.

108

- For large  ratios or rectangular chord sections, the strength of a KK gap joint is governed by chord shear and chord axial force interaction, presented in Table 9.5. The KK gap joint (with  = 90) is subjected to a shear force of 0,5 2 Vgap,0,Ed in each

expressed in terms of the efficiency of the connected braces in a similar way to that for circular hollow section joints, i.e. the joint resistance is given as a fraction of the yield capacity Ai fyi of the connected brace. This results in the following efficiency formula:

plane, where Vgap,0,Ed is the total "vertical" shear force. The shear force in each plane is resisted by the two walls of the RHS chord.

Multiplanar overlap KK joints - For multiplanar overlap KK joints, the strength of the joint is similar to that for uniplanar overlap joints given in Chapter 11. Thus, compared to the previous IIW (1989) recommendations, a brace shear criterion and a local chord yielding criterion have been added.

The design resistances for joints loaded by brace bending moments are derived in a similar way to that for axially loaded joints. To simplify the design, limitations are also given here for the range of validity to reduce the criteria to be checked.

For Vierendeel girders it is recommended to choose joints with  = 1,0 to provide sufficient stiffness and strength.

The design resistance formulae are based on the analyses of Wardenier (1982), Mang et al. (1983), Yu (1997) and Packer et al. (2009a), and are given in Table 9.6.

9.7.6 Interaction between axial loads and bending moments For joints with brace members subjected to combined loading, the effect of axial load on the joint moment capacity depends on the critical failure mode, and hence a complex set of interactions exists.

Consequently, it is conservatively proposed to use a linear interaction relationship:

N1,Rd



Mip,1,Ed Mip,1,Rd

 1,0

Ni,Rd A i f yi

 Ce

fy0 t 0 Q f f yi t i sin i

(9.21)

In the case of b1  b2 for K joints, eq. (9.21) has to be

b1  b2 , where bi is the width of the 2bi brace considered.

multiplied by

For a detailed explanation, see Section 8.8.

Using the chart of Fig. 9.23 shows that e.g. a K gap joint with 2  20 and b1 = b2 gives an efficiency parameter CK  0,37. Thus, for an angle θi = 45, a 100% efficiency can be obtained if:

9.7.5 Joints loaded by brace bending moments

N1,Ed

Efficiency 

fy0 t 0 f yi t i

 1,9 (for Qf = 1,0)

If the chord load effect Qf is included, this ratio should be slightly larger. Figs. 9.24 and 9.25 show the chord load effect Qf as a function of the parameter n, defined as the ratio between the maximum stress in the connecting chord face and the chord yield stress.

9.9 CONCLUDING REMARKS For more detailed information about joints loaded by bending moments as well as special types of joints, reference is made to the appropriate literature, see Dutta (2002), Korol et al. (1977), Packer & Henderson (1997), Packer et al. (2009a), Ono et al. (1991), Syam & Chapman (1996), Wardenier (1982) and Wardenier & Giddings (1986).

(9.20)

9.8 DESIGN CHARTS In Figs. 9.20 to 9.23, the joint resistances are 109

Table 9.1 Design axial resistances of welded joints between RHS or CHS braces and RHS chord Type of joint

Design limit state

T, Y and X joints

Chord face plastification

N1,Rd  d1

(for β  0,85)

fy 0 t 02  2 4  Qf  sin 1  (1  ) sin θ1 1    (general check)

Local brace failure N1

N1,Rd  fy1 t1 ( 2h1  2b e  4 t1 )

b1

t1 1

(for b1  b0 - 2t0)

Chord punching shear

h1

b0

t0

N1,Rd 

h0

0,58 fy0 t 0  2h1    2be,p   sin θ1  sin 1  (for X joints, if cos θ1 > h1/h0)

Chord shear

See chord shear equations for K gap joints, but with V0,Ed instead of Vgap,0,Ed (for  = 1,0) (1)

Chord side wall failure

N1,Rd  K gap joints

 fk t 0  2h1   10 t 0  Qf sin θ1  sin1  (general check)

Chord face plastification

Ni,Rd  14

2 0,3 fy0 t 0

sin θi

Qf (general check)

Local brace failure N1

N2 b1

d1 1

d2

h2

h1 t1

1 0

Ni,Rd  fyi t i (2hi  bi  be  4t i )

b2

t2

2

g

b0

2 t 0 h0

N0 +e

(for bi  b0 - 2t0)

Chord punching shear

Ni,Rd 

0,58 fy0 t 0  2hi    bi  be,p  sin θi  sin i  (general check)

Chord shear

Ni,Rd 

0,58 fy0 A v sin θi

Av and Vpl,0,Rd

Vpl,0,Rd  0,58 fy0 A v

T, Y and X joints

A v  2h0 t 0

K gap joints

A v  2h0 t 0   b0 t 0

Function Qf

Qf  1  n



and

Ngap,0,Rd  (A 0  A v ) fy0  A v fy0

RHS braces:  

C

1

with n 

N0,Ed Npl,0,Rd



T, Y and X joints

C1 = 0,6 – 0,5β

K gap joints

C1 = 0,5 – 0,5β but  0,10

be and be,p

 10   fy 0 t 0   bi but  bi  be      b0 /t 0   fyi ti 

fk

2

1 1  ( 4g2 ) /(3t 02 )

CHS braces:   0 M0,Ed Mpl,0,Rd

Chord compression stress (n < 0)

Tension: fk  fy0

 Vgap,0,Ed   1   Vpl,0,Rd   

in connecting face Chord tension stress (n  0) C1 = 0,10  10   bi but  bi be,p     b0 /t 0 

Compression: fk   fy0 for T and Y joints, and fk  0,8  fy0 sin θ1 for X joints

where  = reduction factor for column buckling according to e.g. Eurocode 3 (EN 1993-1-1, 2005) using h  1 the relevant buckling curve and a slenderness   3,46  0  2  t sin θ1  0 

(1)

For 0,85 <  < 1,0 use linear interpolation between the resistance for chord face plastification at  = 0,85 and the resistance for chord side wall failure at  = 1,0.

110

Table 9.1 Design axial resistances of welded joints between RHS or CHS braces and RHS chord (continued) T, Y, X and K gap joints with CHS brace

For CHS braces, multiply the above resistances by /4 (except for chord shear criterion) and replace bi and hi by di (i = 1 or 2)

Range of validity T, Y or X joints Brace-to-chord ratio RHS chord

RHS braces

bi /b0  0,1  0,01b0 /t 0 but  0,25

CHS braces

di /b0  0,1  0,01b0 /t 0 and 0,25  di /b0  0,80

Compression

class 1 or 2

(2)

class 1 or 2

(2)

class 1 or 2

(2)

and b0 /t 0  40 and h0 /t 0  40 b0 /t 0  40 and h0 /t 0  40

Tension RHS braces

Compression

and bi /ti  40 and hi /ti  40 bi /t i  40 and hi /t i  40

Tension CHS braces

Compression

and di /ti  50 di /ti  50

Tension

0,5 (1  )  g/b0  1,5 (1  )

Gap

N/A

Eccentricity

N/A

Aspect ratio

0,5  hi /bi  2,0

Brace angle

i  30

Yield stress

fyi  fy0

(2)

K gap joints

and g  t1  t 2 e  0,25h0

fy  0,8fu

fy  460 N/mm2

(4)

(3)

Section class limitations are given in Table 2.7.

For g/b0  1,5 (1  ) , check the joint also as two separate T or Y joints.

(4)

For 355 N/mm2 < fy  460 N/mm2, use a reduction factor of 0,9 for the design resistances.

111

(3)

Table 9.2 Design axial resistances of welded joints between square or circular braces and a square hollow section chord Joints between square hollow sections

Design limit state

T, Y and X joints

Chord face plastification d1 N1

b1 h1 t1

1

N1,Rd 

b0

t0 h0

K gap joints

fy 0 t 02  2 4   Qf sin 1  (1  ) sin θ1 1   

Chord face plastification

N1

N2 b1

d1

b2

1

1 0

d2

h2

h1 t1

t2

2

g

b0

2 t 0

Ni,Rd  14  0,3

fy0 t 02 sin θi

Qf

h0

N0 +e

Function Qf

Same as in Table 9.1

T, Y, X and K gap joints with CHS brace

For CHS braces, multiply the above resistances by /4 and replace bi by di (i = 1 or 2)

Range of validity

Same as in Table 9.1 with additional limits given below

General SHS braces CHS braces

T, Y and X joints

b1/b0  0,85

K gap joints

0,6  (b1  b2 )/(2bi )  1,3

b0 /t 0  15

K gap joints

0,6  (d1  d2 )/(2di )  1,3

b0 /t 0  15

112

Table 9.3 Design resistance of related types of joints Type of joint

Relationship to the formulae of Tables 9.1 and 9.2

All brace member forces act in the same sense (compression or tension).

N1,Ed  N1,Rd with N1,Rd from X joint given in Table 9.1 or 9.2

N1

1

1

N1

All brace member forces act in the same sense (compression or tension).

N1

N2

1

2

N2

N1,Ed sinθ1 + N2,Ed sinθ2  NX,Rd sinθX with NX,Rd sinθX from X joint given in Table 9.1 or 9.2, being the larger of the values for brace 1 or 2

N1

Forces in members 1 are in compression and members 2 in tension.

N1

1

N2

Ni,Ed  Ni,Rd (i = 1 or 2)

2

1

with Ni,Rd from K joint given in Table 9.1 or 9.2, but with the actual chord force

N2

N1 1

Forces in members 1 are in compression and members 2 in tension.

N1

1

1

Ni,Ed  Ni,Rd (i = 1 or 2) with Ni,Rd from K gap joint given in Table 9.1 or 9.2

N2 2

Note: In a gap joint, the chord cross section in the gap has to be checked for shear failure: Vgap,0,Ed  Vpl,0,Rd  0,58 fy0 A v

N1

N2 1

 Vgap,0,Ed   Ngap,0,Ed  Ngap,0,Rd  (A 0  A v ) fy0  A v fy0 1    Vpl,0,Rd   

113

2

Table 9.4 Design resistances of uniplanar plate-to-RHS joints Type of joint

Design limit state

T and X joints – transverse plate

Chord face plastification

N1

 2  2,8   N1,Rd  fy0 t 02   Qf  1  0,9 

b1

t1 t0

h0

Chord punching shear

b1

Chord side wall failure

t1 t0

h0

(for β ≈ 1,0) (1)

N1,Rd  2 fy0 t 0 ( t1  5t 0 ) Q f

Local plate failure

N1

(for 0,85b0  b1  b0 - 2t0)

N1,Rd  0,58 fy0 t 0 ( 2t1  2b e,p )

b0 N1

(for 0,4  β  0,85)

(for all β)

N1,Rd  fy1 t1 be

b0

T and X joints – longitudinal plate

Chord face plastification N1

h1

t1 t0

h0 b0 h1

 t  N1,Rd  2 fy 0 t 02   2 1  1  Q f b 0   

N1 t1 t0

h0

b0 N1 (1)

For 0,85 <  < 1,0, use linear interpolation between the resistance for chord face plastification at  = 0,85 and the resistance for chord side wall failure at  = 1,0.

114

Table 9.4 Design resistances of uniplanar plate-to-RHS joints (continued) Type of joint

Design limit state

T joints – longitudinal through-plate

Chord face plastification

N1

h1

t1 t0

h0

 t  N1,Rd  4 fy 0 t 02   2 1  1  Qf b0  

b0

T stub joints – stiffened longitudinal plate N1

h1

t sp  0,5 t0 e3

t1

t sp

bsp  t1

with: * 

b0  t 0

t0

h0 bsp b0

If tsp fulfills the above requirement, the joint can be regarded as an RHS-to-RHS T joint. In the design equations for RHS-to-RHS T joints, the stiffening plate width bsp is then used for the brace width b1.



Qf  1  n

Function Qf

*

C

1

with n 

N0,Ed Npl,0,Rd



M0,Ed Mpl,0,Rd

Chord compression stress (n < 0) Transverse plate

C1 = 0,03γ but  0,10

Longitudinal plate

C1 = 0,20

be and be,p

 10   fy 0 t 0   bi but  bi  be      b0 /t 0   fyi ti 

in connecting face

Chord tension stress (n  0) C1 = 0,10  10   bi but  bi be,p     b0 /t 0 

Range of validity

Compression RHS chord

class 1 or 2

(2)

and b0 /t 0  40 and h0 /t 0  40 b0 /t 0  40 and h0 /t 0  40

Tension Aspect ratio

0,5  h0 /b0  2,0

Transverse plate

  b1/b0  0,4

Longitudinal plate

1    h1/b0  4

Plate angle

1  90o

Yield stress

fy1  fy0

(2) (3)

fy  0,8fu

fy  460 N/mm2

(3)

Section class limitations are given in Table 2.7.

For 355 N/mm2 < fy  460 N/mm2, use a reduction factor of 0,9 for the design resistances.

115

Table 9.5 Correction factors for the design resistance of multiplanar joints Type of joint

Correction factor μ to uniplanar joint strength

TT joints

General

Members 1 may be either in tension or compression

2N1

N1

N1

μ = 1,0



XX joints

Chord face plastification

Members 1 and 2 can be either in compression or tension

  1 0,35

N1

(for   0,85)

N2,Ed N1,Ed

N1

N2

N2

Notes: - Take account of the sign of N1,Ed and N2,Ed, with |N2,Ed |  |N1,Ed| - N2,Ed/N1,Ed is negative if the members in one plane are in tension and in the other plane in compression.

Other failure modes

N1

μ = 1,0

N1

KK gap and overlap joints

General

Members 1: compression Members 2: tension

μ =1,0

N2

N1 A

A

N1

N1 

but in a KK gap joint, the chord cross section in the gap has to be checked for shear failure according to: 2

2

 Ngap,0,Ed   0,71Vgap,0,Ed       1,0  Npl,0,Rd   Vpl,0,Rd      where: Ngap,0,Ed  design axial force in gap Npl,0,Rd

 A 0 fy0

Vgap,0,Ed  design shear force in gap

Vpl,0,Rd Range of validity

 0,58 fy 0 (0,5 A 0 ) for a square hollow section chord

Same as in Tables 9.1 and 9.2  ≈ 90°

116

Table 9.6 Design moment resistances of welded joints between rectangular hollow sections Type of joint – brace loading

Design limit state

T and X joints – in-plane bending

Chord face plastification

Mip,1

(for β  0,85)

 1 2   Mip,1,Rd  fy0 t 02 h1   Qf  2 1   1    

b1

(for 0,85 <   1,0)

Local brace failure h1 t1 1

b0

t0

  b Mip,1,Rd  fy1  Wpl,1  (1  e ) b1(h1  t1) t1 b1   (for  = 1,0) (1)

Chord side wall failure

h0

Mip,1,Rd  0,5fk t 0 h1  5t 0  Qf 2

T and X joints – out-of-plane bending (2) Chord face plastification

Mop,1

 h (1   )  Mop,1,Rd  fy0 t 02 b1 1  2b1(1   ) 

b1

2(1   )  Qf (1  )  (for 0,85 <   1,0)

Local brace failure

h1 t1 1

(for β  0,85)

Mop,1,Rd  fy1 [ Wpl,1  0,5t1b1  be  ] 2

b0

t0 h0

(for  = 1,0) (1)

Chord side wall failure Mop,1,Rd  fk t 0 (b0  t 0 ) (h1  5t 0 ) Qf

Function Qf

Same as in Table 9.1

be

 10   fy0 t 0   be   b but  b1   1  b0 /t 0   fy1 t1 

fk

Brace in-plane bending

Brace out-of-plane bending

T and Y joints

X joints

T and Y joints

X joints

fk  fy 0

fk  0,8  fy 0

fk   fy 0

fk  0,8  fy 0

where  = reduction factor for column buckling according to e.g. Eurocode 3 (EN 1993-1-1, 2005) using the relevant buckling curve and a slenderness  h   3,46  0  2  t   0 Range of validity (1)

(2) (3)

Same as in Table 9.1, but with θ1 ≈ 90°

(3)

For 0,85 <  < 1,0, use linear interpolation between the resistance for chord face plastification at  = 0,85 and the resistance for chord side wall failure at  = 1,0.

Chord distortion to be prevented for brace out-of-plane bending.

The equations are conservative for θ1 < 90°.

117

Y joint

T joint h1 b1

h1

N1 t1

t1

1

b0

t0

N1

1

b0

t0

N0

h0

b1

h0

X joint

K joint with gap h1

h1

h2

t1

N1

b1

1

t0

b0

t2

b1

t1

g

2

N0

N0

N1

N2

b2

1

t0

b0 h0

h0

N1

N joint with overlap

KT joint with gap h3

hi bi

hj bj

Ni

Nj tj

b3

h2

N2

b2

ti

t2

N0

i

t0

b0

2

N0

h0

Fig. 9.1 Welded RHS joints

118

t1

t3 g2

j

h1

N3

N1

b1

g1 1

t0

b0 h0

(a) Brace failure

(e) Chord punching shear failure

(b) Weld failure

(f) Chord side wall yielding or buckling

as (a) but failure in the weld

(c) Lamellar tearing

(g) Chord local buckling

see e.g. Fig. 7.6c

(d) Chord face plastification

(h) Chord shear failure

Fig. 9.2 Failure modes for welded RHS joints

119

The total energy dissipated in the yield lines 1 to 5 is as follows: Yield lines 1:

2b 0

2 mp (b0  b1 ) cot 

=

4 tan  mp (1  )

Yield lines 2:

2b1

2 mp (b0  b1 ) cot 

=

4 tan  mp (1  )

Yield lines 3:

2(

b  b1 h1 2 2 0 cot  ) mp sin 1 2 (b 0  b1 )

  4 =   4 cot  mp  (1  ) sin 1 

Yield lines 4:

2(

h1 2 ) mp sin 1 (b 0  b1 )

  4 mp =  ( 1   ) sin  1 

Yield lines 5:

    mp 4 5     5 tan   5 cot  

with:

mp 

= 4(tan   cot  )mp

fy20 t 0 4

Total energy Ed

=

Fig. 9.3 Yield line model for a T, Y and X joint

120

8mp   (1  )     tan   (1  )  tan  sin 1 

0,5be,p

h1 sin 1

L eff  2 (

h1  2 be,p ) sin 1

0,5be,p

Fig. 9.4 Chord punching shear model for a T, Y or X joint

, 0,5b e,p

, 0,5b

e,p

h2 sin 2

Fig. 9.5 Chord punching shear model for a K gap joint (chord face)

121

0,5be 0,5be

Fig. 9.6 Local brace failure model for a T, Y or X joint

0,5be,ov

hi

0,5be,ov

Nj

Ni bi

Fig. 9.7 Local brace failure model for a 100% overlap joint

122

,

,

Fig. 9.8 Chord side wall failure model

Fig. 9.9 Four hinge yield line model for chord side wall failure (Packer, 1978)

Fig. 9.10 Chord shear failure model 123

500

400

300

200

100

0

0

100

200

300

400

500

600

Fig. 9.11 Comparison between experiments and the mean ultimate joint strength equation for chord plastification for K gap joints with β  0,85 (Wardenier, 1982)

250

200

150

100

50

0 0

50

100

150

200

250

Fig. 9.12 Comparison between experiments and the analytical yield line criterion for chord plastification for T, Y and X joints with β  0,85 (Wardenier, 1982)

124

hj

tj

bj

hi

bi

Nj

Ni

ti

(1) ď ąj

ď ąi (3)

b0 t0

N0p

N0

(2)

h0

(2)

i = overlapping member and j = overlapped member Fig. 9.13 Overlap joint configuration with cross sections to be checked

Fig. 9.14 Comparison of a K joint with a circular brace and an equivalent joint with a square brace

N1

N1

tw

tp

2.5

1

5tp+tw t1 s

h1

s

b1

Fig. 9.15 Load dispersion for a Tee joint on the end of an RHS member

125

TOP

SIDE

Stress trajectory

Fig. 9.16 Shear lag in gusset plate-to-slotted RHS joints

1,8

N1u (JAA = 0) / N1u

1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0,0 0,0

0,2

0,4

0,6

0,8

1,0

Note: The vertical axis plots the ratio of the ultimate strength N1u(JAA = 0) of a multiplanar joint with unloaded out-of-plane braces and the ultimate strength N1u of the corresponding uniplanar joint.

Fig. 9.17 Multiplanar geometry effect for an XX joint of square hollow sections (Yu, 1997)

1,5

For  = 0,2  = 0,4  = 0,6  = 0,8

1,0

0,5 -1,0

N1u (JAA ) / N1u (J AA = 0)

N1u (JAA ) / N1u (J AA = 0)

1,5

0,0

1,0

For  = 1,0 1,0

0,5 -1,0

0,0

1,0

Note: The vertical axis plots the ratio of the ultimate strength N1u(JAA) of a multiplanar joint and the ultimate strength N1u(JAA = 0) of the multiplanar joint with unloaded out-of-plane braces.

Fig. 9.18 Multiplanar loading effect for an XX joint of square hollow sections (Yu, 1997)

126

Fig. 9.19 Multiplanar KK joint of square hollow sections

127

Efficiency X joints in compression

Efficiency T and Y joints in compression 1,0 0,9

N1,Rd

0,8

A1 fy1

 CT

fy 0 t 0

Qf fy1 t1 sin1

0,7

1,0



0,9

N1,Rd



0,8

A1 fy1



0,7



0,6



0,5





0,5

CX

CT

0,6

only for  1 = 90o and fy = 355N/mm2



0,4

0,4

0,3

0,3

0,2

0,2

0,1

0,1



fy 0 t 0

Qf fy1 t1 sin1



0,0

0,0 0

0,1

0,2 0,3

0,4

0,5 

0,6

0,7 0,8

0,9

0

1

Fig. 9.20 Efficiency for T and Y joints of square hollow sections (brace in compression)

0,9

N1,Rd A 1 f y1

0,8

 CT

Qf fy1 t1 sin1

0,7

0,5 0,6 

0,7

0,8 0,9

1

0,9



0,8





0,7



0,6 0,5

Qf



0,5

0,4

 

0,6

0,3

X and T joints: chord axial stress functions T joints: chord bending stress function 1



fy 0 t 0

0,1 0,2

Fig. 9.21 Efficiency for X joints of square hollow sections (braces in compression)

Efficiency T, Y and X joints in tension 1,0

CX or CT

 CX



0,4 0,3

0,4

0,2

0,3

0,1 0

0,2 -1

0,1

-0,8

-0,6

-0,4

-0,2

0 n

0,0 0

0,1 0,2

0,3 0,4

0,5 

0,6 0,7

0,8 0,9

1

0,2

0,4

0,6

0,8

1

X joints: chord bending stress function 1

Fig. 9.22 Efficiency for T, Y and X joints of square hollow sections (brace(s) in tension)



0,9 0,8



0,7 0,6

 

Qf

0,5 0,4 0,3 0,2 0,1 0 -1

-0,8

-0,6

-0,4

-0,2

0 n

0,2

0,4

0,6

0,8

1

Fig. 9.24 Chord load function for T and X joints of square hollow sections

Efficiency K gap joints 1,0 0,9

Ni,Rd

0,8

A i fyi

 CK

K gap joints: chord axial stress functions

fy 0 t 0 Q f b1  b2 fyi t i sin i 2bi

1,0

0,7 CK

0,6

0,9



0,8



0,7

0,5 Qf

0,4 0,3



0,6 0,5



0,4 0,3

0,2

0,2

0,1

0,1

0,0

0,0

15

20

25



30

35

40

-1

Fig. 9.23 Efficiency for K gap joints of square hollow sections

-0,8

-0,6

-0,4

-0,2

0 n

0,2

0,4

0,6

0,8

1

Fig. 9.25 Chord load function for K gap joints of square hollow sections

128

As a result, the governing failure modes to be considered are: - Brace failure - Chord web failure - Chord shear failure

10. WELDED JOINTS BETWEEN HOLLOW SECTIONS AND OPEN SECTIONS 10.1 INTRODUCTION Hollow sections and open sections are used in various ways, for example: - Hollow section braces and open section chords as shown in Figs. 10.1 and 10.2a - Open section braces and rectangular hollow section chords, shown in Fig. 10.2b - I section beams connected to hollow section columns, dealt with in Chapter 12 Other combinations, but with bolted connections, are considered in Chapter 13.

In this chapter, it is shown that in many respects the behaviour of joints between open sections and hollow sections is comparable to that of joints between rectangular hollow sections.

10.2 MODES OF FAILURE Following the same procedure as described in Chapter 7, the following failure modes (Fig. 10.3) can be expected and observed for joints between hollow section braces and I section chords: - Brace failure (yielding, local buckling) - Weld failure - Lamellar tearing - Chord web failure (yielding, local buckling) - Chord shear failure - Chord local buckling Chord face plastification will not occur here, since this can only take place after excessive yielding of the chord web.

As indicated before, the welds should ideally be stronger than the connected brace members, assuming the latter are loaded to their limit, thus the welds should satisfy certain requirements, see Section 6.5.2.

For joints between hollow section braces and channel section chords it can be easily shown that with the same limitations as discussed above, the failure modes (Fig. 10.4) to be considered in more detail are: - Brace failure - Chord face plastification - Chord punching shear failure - Chord side wall failure - Chord shear failure Since hot rolled channel sections (UNP) have thick flanges, which act as walls here, chord side wall failure will not be critical for these sections. In principle, the other failure modes can be approached in a similar way to that for joints between rectangular hollow sections (Wardenier & Mouty, 1979; Wardenier, 1982; Aribert et al., 1988). If chords of cold formed channel sections are used (Fig. 10.1), the situation will be different, since the side walls may deform when the top face deforms, resulting in lower strengths. For detailed information about joints with channel section chords, reference can be made to Wardenier (1982).

The joints of Fig. 10.2 with welded angles or channels at the sides of a rectangular hollow section are not different from other joints in open sections. Besides weld failure, chord shear failure has to be considered.

10.3 ANALYTICAL MODELS 10.3.1 Local brace failure model The most effective part of the brace is located at the crossing with the chord web, shown in Fig. 10.5. Here, the same model can be used as for beam-to-column joints of open sections, i.e. for a T, Y or X joint: N1 = 2 fy1 t1 be

(10.1)

where:

fy0

Similar to other hollow section joints, lamellar tearing should be avoided by choosing proper material.

b e  t w  2r  7

Local buckling of the members can be avoided by choosing proper diameter-, width- and depth-tothickness ratios, thus by limiting the range of validity.

If be > b1, it is conservatively proposed to follow the perimeter, as shown in Fig. 10.5.

f y1

t0

(10.2)

Similar to joints of circular and rectangular hollow 129

sections, the criterion for local brace failure is also used for overlap joints. For overlaps Ov > Ovlimit, the connection between the braces and the chord has to be checked for shear.

Mpl,f 

b 0 t 02 fy0 4

Vpl,f = b0 t0

10.3.2 Chord web failure model

(10.4)

Vf  Vpl,f

(10.11)

b 0 t 0

b w  2b wf 

2t i  10( t 0  r ) sin  i

(10.5)

10.3.3 Chord shear model Similar to joints with a rectangular hollow section chord, an interaction formula can be determined for the combined effect of shear and axial load at the gap location in the chord, see Fig. 10.7. When the web is yielding, a part of the flange may also be effective for shear if the gap is small.

By equilibrium, the moment in the flange is: Vf g 2

(10.6)

The interaction formula for the flange with a rectangular cross section is: 2

  Vf     Vpl,f  



where:

1 4g 2 1 2 3t 0

(10.13)

For deep sections the effectiveness of the bottom flange will be restricted to the area below the web and corners, thus the following effective shear area is used for joints with RHS braces: A v  A 0  2 b 0 t 0    b 0 t 0  t w  2 r t 0

(10.14)

If circular hollow section braces are used, the chord flange is less stiffened at the gap location, and consequently, α is lower. Although here the same area could be used as in the bottom flange, assuming α = 0 in eq. (10.14) gives a better correlation (lower bound) with the test results.

Thus, shear of the chord cross section at the gap location has to be checked by:

Vgap,0  Ni sin i  Vpl,0  A v

2

   1,0  

(10.12)

where:

For beam-to-column joints, a stress dispersion of 2,5:1 is used, which was shown also to be valid for these joints (Wardenier, 1982).

 Mf   Mpl,f 

1 4g 2 1 2 3t 0

Thus, for an I or H section, the active part of the top flange is:

but:

Mf 

(10.10)

Substitution in eq. (10.7) gives:

(10.3)

hi  5( t 0  r ) sin  i

(10.9)

3

Mf V 2g  f Mpl,f Vpl,f t 0 3

where:

bw 

fy 0

Combination of eqs. (10.6), (10.8) and (10.9) results in:

The load from the brace has to be transferred by an effective area of the chord web, see Fig. 10.6. The effective areas are located in the chord web at the location where the brace walls cross the chord web.

Similar to the formulae used for beam-to-column joints of open sections: Ni sinθi = fy0 tw bw

(10.8)

(10.7)

fy0 3

(10.15)

with Av according to eq. (10.14).

For the interaction between the axial load and the 130

shear load at the gap location, based on the Huber-Hencky-Von Mises criterion, the following formula applies (see Section 9.3.5): Ngap,0  A 0  A v f y 0  A v f y 0

 Vgap,0 1   Vpl,0 

   

2

(10.16)

10.4 EXPERIMENTAL VERIFICATION The formulae developed in Section 10.3 for joints with an I section chord have been verified by experiments (Wardenier & Mouty, 1979; Wardenier, 1982), e.g. the comparison of eq. (10.15) with test results is given in Fig. 10.8.

The design criteria for joints with a hot finished channel chord section have also been checked by tests. It was shown that for medium to high width ratios β, chord shear failure was the most common failure mode for K gap joints, whereas chord punching shear or chord face plastification was governing for low β values.

10.5 EVALUATION TO DESIGN RULES In principle, the same approach has been used as for the joints discussed in Chapters 8 and 9. However, due to the limited number of tests, no thorough statistical analysis has been carried out. The resulting design resistances are given in Table 10.1.

depending on whether or not the hidden toe location of the overlapped brace is welded to the chord. The check for local chord member yielding (criterion 2) is, in principle, a member check and may become critical for larger overlaps and/or larger β ratios.

For 100% overlap joints, similar criteria have to be checked. Only here, as shown by Chen et al. (2005) and Qian et al. (2007), shear of the overlapped brace and chord member yielding will generally be the governing criteria. Although an overlap of 100% is given in the recommendations, in general, the overlap will be slightly larger to allow proper welding of the overlapping brace to the overlapped brace.

More detailed information regarding the design equations for overlap joints is given in Chapter 11.

10.6 JOINTS PREDOMINANTLY LOADED BY BENDING MOMENTS Here, only beam-to-column joints with an RHS brace (or beam) and an I section column are of practical interest.

The design resistance in Table 10.2 is governed by local failure of the RHS brace with formulae based on eqs. (10.1) and (10.2) (see Fig. 10.9) and column web failure based on eqs. (10.3) to (10.5) (see Fig. 10.10).

For overlap joints, the same approach is adopted for all types of overlap joints, regardless whether circular or rectangular braces are used in combination with a circular, rectangular or open section chord. Only the effective width parameters depend on the type of section. The resistance of overlap joints between circular hollow sections with 25%  Ov  100% overlap is based on the following criteria: (1) Local failure of the overlapping brace (2) Local chord member yielding at the joint location based on interaction between axial load and bending moment (3) Shear of the connection between the brace(s) and the chord The cross sections to be examined for these criteria are the same are those shown in Figures 8.16 and 9.13.

Local failure of the overlapping brace (criterion 1) should always be checked, while shear between the braces and the chord (criterion 3) may only become critical for larger overlaps, i.e. larger than 60% or 80%,

131

Table 10.1 Design resistances of welded joints between RHS or CHS braces and I or H section chords Type of joint

Design limit state

T, Y, X and K gap joints

Local brace failure t1 t1

Ni,Rd  2fyi ti be

h1

N1

1

d1 t0

b1

Chord web failure

b0 tw

r

h0

Ni,Rd 

d1 h1

t2

b1

h2 d2

N2

t1 N1 1

g

2

t0

b2

sin i (for K gap joints; for T joints in member check; for X joints with cos 1 > h1/h0)

Chord shear

t2

t1

fy 0 t w b w

Ni,Rd 

0,58 fy 0 A v

b0

r

tw

h0

sin i

 Vgap,0,Ed   Ngap,0,Rd  ( A 0  A v ) fy 0  A v fy 0 1    Vpl,0,Rd   

2

Factors RHS braces be

be  t w  2r  7t 0

bw

bw 

CHS braces fy 0 fyi

but  bi  hi  2ti

2t i hi  5( t 0  r ) but   10( t 0  r ) sin i sin i

be  t w  2r  7t 0

bw 

fy 0 fyi

but  0,5 (di  ti )

2t i di  5( t 0  r ) but   10( t 0  r ) sin i sin i

A v  A 0  (2   ) b0 t 0  ( t w  2r ) t 0 AV

Vpl,0,Rd



1 1  ( 4g2 ) /(3t 02 )

0

Vpl,0,Rd  0,58 fy 0 A v

Range of validity X joints I or H section chord

CHS braces

RHS braces

Compression

T and Y joints

K gap joints

Flange

class 1 or 2

Web

class 1 and dw  400 mm class 1 or 2 and dw  400 mm

Tension

none

Compression

class 1 or 2 (1)

Tension

di /ti  50

Compression

class 1 or 2 (1)

Tension

bi /ti  40 and hi /ti  40

Aspect ratio

0,5  hi /bi  2,0

Gap

N/A

g  t1  t 2

Eccentricity

N/A

e  0,25h0

Brace angle

i  30

Yield stress

fyi  fy0

(1) (2)

fy  0,8fu

fy  460 N/mm2

Section class limitations are given in Table 2.7.

For 355 N/mm2 < fy  460 N/mm2, use a reduction factor of 0,9 for the design resistances.

132

(2)

Table 10.2 Design moment resistance of uniplanar RHS braces (beams) to I or H section chord joints Type of joint

Design limit state

T joints

Local brace failure

I or H chord

'I' or 'H' Column

Mip,1,Rd  fy1 t1 be hz

RHS brace (beam) RHS Beam

1

where hz is the distance between the centres of gravity of the effective parts of the RHS brace (beam)

be hz h1

Mip,1

be

Chord web failure

Mip,1,Rd  0,5 fy 0 t w b w (h1  t1 )

Factors fy 0

be

be  t w  2r  7t 0

bw

b w  h1  5( t 0  r ) but  2t1  10( t 0  r )

Range of validity

Same as in Table 10.1, but θ1 ≈ 90°

133

fy1

but  b1  h1  2t1

T joint

X joint

h1 b1 t1

h1 t1

t1

b1 t1

d1

d1

b0

t0

t0 tw

h0 r

h0 r

b0

K gap joints d2

t1

t2

h2

d11

b1

b2

N2

h1

N1

2

1

b0

t0

b1

b2 h2

tw

r

h0

t1

t2

h1

N1

N2 g = 0,1b0

1

2

t0 tw b0

r

Fig. 10.1 Welded truss joints between hollow section braces and open section chords

134

h0

tw

Fig. 10.2 Welded truss joints between hollow sections and open sections (Packer & Henderson, 1997)

135

(a) Brace failure

(d) Chord web failure

(b) Weld failure

(e) Chord shear failure

as (a) but failure in the weld

(c) Lamellar tearing

(f) Chord local buckling

see e.g. Fig. 7.6c

Fig. 10.3 Governing failure modes for joints between hollow section braces and I section chords

136

(a) Brace failure

(d) Chord side wall failure

(b) Chord face plastification

(e) Chord shear failure

(c) Chord punching shear failure

Fig. 10.4 Governing failure modes of joints between hollow section braces and a channel section chord

137

(a)

(b)

Fig. 10.5 Model for local brace failure

bwf

bwf

1 : 2,5

bw Fig. 10.6 Model for chord web failure

Fig. 10.7 Chord shear model 138

1 : 2,5

700 RI joints 600

500

400

300

200

100

0

0

100

200

300

400

500

600

700

800

(a) Joints between RHS braces and an I section chord (RI joints)

600 CI joints 500

400

300

200

100

0

0

100

200

300

400

500

600

700

800

(b) Joints between CHS braces and an I section chord (CI joints) Fig. 10.8 Comparison between test results and eq. (10.15) for joints between hollow section braces and an I section chord (Wardenier, 1982)

139

1

Fig. 10.9 Local brace failure criterion for moment loading of RHS beam-to-column joints

bwf

1

Fig. 10.10 Column web failure criterion for moment loading

140

thickness ratios b/t and diameter-to-wall thickness ratios d/t have been limited to avoid local buckling.

11. WELDED OVERLAP JOINTS 11.1 INTRODUCTION In the latest revision of the IIW recommendations for the design of hollow section joints (IIW, 2009), the design strength formulae for overlap joints have also been reanalysed based on the research results of Chen et al. (2005), Liu et al. (2005), Wardenier & Choo (2006), Wardenier (2007) and Qian et al.

(2007).

The objective was to present design strength formulae which are, as far as possible, based on a physical model and which have a good agreement with available experimental and/or numerical data.

Further, the formulae should be logical, as simple as possible, consistent for similar types of joints and easy to understand for designers. Therefore, in the case of overlap joints, it was preferred that the approach for CHS and RHS overlap joints would be consistent and based on the same philosophy.

As a result, for overlap joints, the same approach is now adopted for all types of overlap joints, regardless whether circular or rectangular braces are used in combination with a circular, rectangular or an open section chord. Only the effective width parameters depend on the type of section.

As a result, the following failure modes still have to be considered in design: (1) Local brace failure (2) Local chord member yielding (3) Shear failure at the connection between the brace(s) with the chord The locations of these failure modes are given in Fig. 11.1. Note that for K and N overlap joints, the subscript i is used to denote the overlapping brace member, while the subscript j refers to the overlapped brace member.

In the previous IIW recommendations (IIW, 1989), local chord member yielding was considered as a member failure. However, because designers sometimes forgot to check this criterion, it has now been included in the joint strength equations.

Although considered as a possible failure mode, in the experiments of the seventies and eighties, brace shear failure was not observed. More recently, numerical studies showed that especially for large overlaps or rectangular braces with hj < bj, brace shear failure could govern. A concentrated local brace shear may further, in case of chords loaded in compression, initiate local buckling of the connecting face of the chord.

11.2 MODES OF FAILURE In overlap joints the following modes of failure can occur: - (Local) brace failure (cracking or local buckling) - Weld failure - Lamellar tearing - Local chord member yielding or local buckling - Shear failure at the connection between the brace(s) with the chord Similar to other welded hollow section joints, to avoid weld failure, the welds should ideally be stronger than the connected braces. For fillet welds, the throat thickness should satisfy the same requirements as those given in Section 6.5.2.

Also here, the steel should not be susceptible to lamellar tearing. Especially for larger thicknesses (t > 25 mm), a TTP quality with a low sulphur content should be used for the overlapped brace and chord members.

11.3 ANALYTICAL MODELS FOR RHS OVERLAP JOINTS In the past, various models for overlap joints have been studied in detail (e.g. Dexter & Lee, 1998; Kurobane, 1981; Marshall, 1992; Packer, 1978; Wardenier, 1982). However, in these studies, overlap joints with CHS, RHS or open section chords were studied separately, leading to different design approaches. As shown in this chapter, these joints can be described by similar models.

11.3.1 Local brace failure model The local brace failure model, shown in Fig. 11.2 for 50% and 100% overlap joints, is consistent with that used for other RHS joints, see Chapter 9. Local brace failure, also referred to as "brace effective width criterion" is especially critical for overlap joints with relatively thin walled overlapping braces.

In the design recommendations, the width-to-wall 141

For 100% overlap joints in which the brace widths of the overlapped and overlapping brace are about the same, three walls of the overlapping brace are fully effective, while the overlapping brace cross wall on the overlapped brace is only partly effective by be,ov.

Hence, the capacity of the overlapping brace i can be given by eq. (11.1): Ni = fyi ti (2hi + bi + be,ov - 4ti)

(11.1)

For overlaps which still sufficiently stiffen the connection with the chord wall, e.g. for 50%  Ov < 100% (the lower limit of 50% has been verified by tests), the sides of the overlapping brace are still assumed to be fully effective and the capacity can be given by eq. (11.2): Ni = fyi ti (2hi + bei + be,ov - 4ti)

(11.2)

 N0   Npl,0 

 f y 0 t 0   f t  yi i

  b i but  b i  

(11.3)

 10  f yj t j    b but  b i b e,ov    b j /t j  f yi t i  i   

(11.4)

For overlaps smaller than 50%, the effective parts of the side walls are taken in relation to the overlap, i.e.

2hi(Ov/50). Overlaps smaller than 25% are not recommended due to the possible large difference in stiffness between the overlapping connection and the connection to the chord, which may lead to premature cracking and lower capacities (Wardenier, 2007).

11.3.3 Shear failure of the connection between brace(s) and chord Shear failure at the connection between the brace(s) and the chord may occur for large overlaps or in cases where hj < bj. As shown in Fig. 11.3 for 100% overlap joints, three sides of the overlapped brace are fully effective while the cross wall at the heel is only effective for a part bej. Thus, the yield capacity can be given by eq. (11.6): Ni cos i  N j cos  j 

f yj (2h j  b j  b ej ) t j 3

sin  j

(11.6)

For joints with overlaps Ov < 100%, the effective parts of both braces with relevant thicknesses and steel grade have to be considered, see Wardenier & Choo (2006). Further, the effective parts depend on whether or not the hidden location of the overlapped brace has been welded to the chord.

For partially overlapped joints with the hidden toe of the overlapped brace not welded, the parts effective for shear are shown in Fig. 11.3 and the yield capacity is given by eq. (11.7): Ni cos i  N j cos  j   100  Ov     2h  b ei  t i f yi  100  i   f yj (2h j  b ej ) t j sin i sin  j 3 3

11.3.2 Local chord member failure At the joint location, the chord should always be checked for local chord member failure based on an interaction between axial load and bending moment.

Various interaction formulae exist, e.g. for RHS eq.

(11.5) with c = 1,5 which is based on Roik & Wagenknecht (1977). In the experiments of Sopha et al. (2006), the capacity according to eq. (11.5) with c = 1,5 could just be reached. However, the numerical results showed that at the joint location it would be better to use a linear interaction, i.e. with c = 1,0:

(11.5)

This criterion becomes critical for joints with large overlaps, thus with a large eccentricity.

where bei and be,ov are the effective width parts for the connected cross walls, being consistent with those for other joints:  10 b ei    b 0 /t 0

c

   M0  1,0  Mpl,0 

(11.7)

where: bei effective width at the connection between the overlapping brace cross wall and the chord according to eq. (11.3) bej effective width at the connection between the overlapped brace cross wall and the chord according to eq. (11.8):  10 b ej    b 0 /t 0

 f y 0 t 0   f t  yj j

  b j but  b j  

(11.8)

The overlap limit for which this criterion may become decisive depends on various geometrical parameters.

142

effective width terms bei and bej at the connection of the braces with the chord should again be based on eq. (10.2).

11.4 ANALYTICAL MODELS FOR CHS OVERLAP JOINTS For CHS joints, the same principles are followed as for RHS joints and the criteria are directly related to those of RHS joints (Wardenier, 2007; Qian et al., 2007).

For local failure of the overlapping brace, the criteria for overlap joints with RHS braces are multiplied by π/4 to obtain the capacity of CHS brace joints, since π/4 is the ratio of cross sectional areas of thin walled CHS and RHS braces with d = b = h. Further all "b" and "h" dimensions in the formulae are replaced by "d".

Since the local stiffness of the joints between CHS-toCHS members is more uniform than that for RHS-toRHS members, the constant in the effective width terms of eqs. (11.3), (11.4) and (11.8) is increased by 20%, i.e. changed from 10 to 12. This increase is also found when comparing the efficiency of CHS X joints to that of RHS X joints. Adopting these modifications in eqs. (11.1) and (11.2) results in the functions given in Tables 11.1 and 11.4. Numerical data showed that due to the more uniform stiffness distribution in CHS overlap joints, a single expression, related to the expression for RHS joints with 50%  Ov < 100%, can be used to describe the local brace failure of CHS joints with overlaps 25%  Ov < 100%.

For the local chord yield criterion, eq. (11.5) with c = 1,7 for CHS sections is given, based on Roik & Wagenknecht (1977).

11.5 ANALYTICAL MODELS FOR OVERLAP JOINTS WITH AN OPEN SECTION CHORD Overlap joints with open section chords behave in principle similar to those with an RHS chord section.

Only the effective width terms bei and bej at the connection of the braces with the chord differ.

However, these terms are in agreement with the expression used for T, X and K gap joints, see eq.

(10.2).

For local chord member failure, eq. (11.5) is used with c = 1,0, similar to the equation for overlap joints with an RHS chord.

The brace shear criterion is also consistent with overlap joints with an RHS chord, although the

11.6 EXPERIMENTAL AND NUMERICAL VERIFICATION The equations in Sections 11.3 to 11.5 for local brace failure have been verified with experiments and/or numerical data; for RHS overlap joints in Wardenier (1982), Chen et al. (2005), Liu et al. (2005) and Wardenier & Choo (2006) and for CHS joints in Wardenier (2007) and Qian et al. (2007). For joints with open section chords, the local brace failure criterion was previously evaluated in Wardenier (1982).

As an example, Fig. 11.4 shows, for all experimental and numerical data for CHS overlap joints with 25%  Ov < 100% in the screened database of Makino et al.

(1996), the ratio of the actual capacity and the predicted value using the local brace failure criterion, see Table 11.1. This evaluation gives a mean value of 1,11 with a coefficient of variation (COV) of 6,0%.

Due to the non-uniform stiffness distribution in joints with small overlaps, and depending on the geometric parameters, the joint capacity may drop for small overlaps, leading to a large scatter. Therefore, it is proposed to limit the validity of equations for local brace failure to 25%  Ov < 100%.

As already mentioned in Section 11.3.2, for local chord failure, the experiments by Sopha et al. (2006) just reached the capacity according to eq. (11.5) with c = 1,5. However, the numerical results showed that at the joint location it would be better to use a linear interaction, i.e. with c = 1,0, which is also adopted for open section chords.

The available experimental results and numerical data (Chen et al., 2005; Qian et al., 2007) showed that the brace shear criteria, i.e. eqs. (11.6) and (11.7) are too conservative if based on the yield strength fy. Since fracture does not occur but the shear deformation has to be limited to avoid local chord deformation leading to premature chord local buckling, it was proposed to base this strength criterion on the ultimate strength fu of the parts effective for shear.

In 1994, Davies & Crocket (1994) already investigated the effect of the welded or non-welded hidden toe of the overlapped brace for RHS joints.

They found, for their numerically investigated joints,

143

no effect. However, Dexter & Lee (1998) observed for CHS joints with the hidden location welded, a capacity which was about 10% higher than that for joints with no weld. The recent reanalysis for the IIW (2009) recommendations showed that the effect depends on the overlap and the governing criterion. If local brace failure or the local chord yielding is decisive, then welding of the hidden toe of the overlapped brace has no effect on the joint strength.

However, if the brace shear criterion is governing, the capacity of the joints with the hidden location welded is about 10% higher than that of the joints with no weld, which agrees with the above results found in the literature.

Fig. 11.5 shows, for CHS joints, that the local brace failure criterion and the brace shear criterion with the hidden location welded or unwelded give about the same capacity for an overlap of 60% if the hidden toe location is unwelded and for an overlap of 80% if the hidden toe location is welded. From this comparison, the limits decisive for checking the brace shear criterion are determined. This also explains why, in general, no brace shear failures were observed in the experiments: in most cases the hidden location was welded and the overlap was smaller.

For joints with braces with hj < bj, the brace shear criterion can become critical for smaller overlaps.

11.7 JOINT STRENGTH FORMULAE Using the equations developed in Sections 11.3 to 11.5 with modifications based on the experimental and/or numerical data discussed in Section 11.6, the final joint strength equations shown in Table 11.1 have been developed.

The range of validity is given in Table 11.2, while details of the parameters used in Table 11.1 are provided in Tables 11.3 and 11.4.

Designers who feel the equations for overlap joints with RHS braces are too complicated could, as a (conservative) simplification, neglect the effective width terms (bei + be,ov - 4ti) for the local brace failure criterion and bei and bej for the brace shear criterion, thus reducing many parameters in Table 11.4 to zero or to a certain conservative value. Similar comments apply for dei, de,ov and dej for overlap joints with CHS braces.

144

Table 11.1 Design axial resistance of uniplanar overlap joints with a CHS, RHS, I or H section chord Type of joint

Design limit state

Axially loaded overlap joints

Local failure of overlapping brace

di ti

tj

Ni,Rd  fyi ti  b,eff .

Nj Ni i

t0

j

dj

Local chord member yielding c

hi

tj

bi

ti

 N0,Ed     M0,Ed  1,0  Npl,0,Rd  Mpl,0,Rd   hj

Nj Ni i

j

c = 1,7 for CHS chord c = 1,0 for RHS or I section chord (for Ovlimit < Ov  100%) (1)

Brace shear t0

bj

Ni,Ed cos i  Nj,Ed cos  j  Ns,Rd (see Table 11.3)

ℓb,eff.

(2)

CHS braces

25%  Ov < 50%  b,eff .

50%  Ov < 100%

Ov = 100%

RHS braces

  (2di  dei  de,ov  4ti ) 4

 b,eff . 

 (2di  2de,ov  4ti ) 4

 b,eff .  (

Ov )2hi  bei  be,ov  4ti 50

 b,eff .  2hi  bei  be,ov  4ti  b,eff .  2hi  bi  be,ov  4ti

The efficiency (i.e. design resistance divided by the yield load) of the overlapped General note

(1)

(2)

 A j fyj   brace j shall not exceed that of the overlapping brace i, i.e. Nj,Rd  Ni,Rd   A i fyi   

Ovlimit = 60% if hidden toe of the overlapped brace is not welded.

Ovlimit = 80% if hidden toe of the overlapped brace is welded.

The expressions for dei, de,ov, bei and be,ov are given in Table 11.4.

145

Table 11.2 Design axial resistance of uniplanar overlap joints with a CHS, RHS, I or H section chord Range of validity

di /d0 and d j /d0  0,20 bi /b0 and b j /b0  0,25

General

di /b0 and d j /b0  0,25 CHS

Chord

RHS

I or H section

di /d j  0,75

ti and t j  t 0

bi /b j  0,75

ti  t j

i and  j  30

fy  0,8fu

Ov  25%

Compression

class 1 or 2 (1) and d0 /t 0  50

Tension

d0 /t 0  50

Compression

class 1 or 2 (1) and b0 /t 0  40 and h0 /t 0  40

Tension

b0 /t 0  40 and h0 /t 0  40

Aspect ratio

0,5  h0 /b0  2,0

Compression

Flange

class 1 or 2

Web

class 1 or 2 and dw  400 mm

Tension

Braces RHS

(1) (2)

fy  460 N/mm2

(2)

none CHS or RHS chord

CHS

fyi and fyj  fy 0

(1)

I or H section chord

and d1/t1  50

class 1

Tension

d2 /t 2  50

Compression

class 1 or 2 (1) and b1/t1  40 and h1/t1  40

class 1

Tension

b2 /t 2  40 and h2 /t 2  40

Aspect ratio

0,5  hi /bi  2,0 and 0,5  h j /b j  2,0

Compression

class 1 or 2

hi /bi  1,0 and h j/b j  1,0

Section class limitations are given in Table 2.7.

For 355 N/mm2 < fy  460 N/mm2, use a reduction factor of 0,9 for the design resistances.

Table 11.3 Design brace shear resistance of uniplanar overlap joints with a CHS, RHS or I section chord Ns,Rd for brace shear criterion (1)

Ovlimit < Ov < 100% CHS braces

Ov = 100%

Ovlimit < Ov < 100% RHS braces

Ov = 100% (1) (2)

(only to be checked for Ovlimit < Ov  100%) (2)

Ns,Rd

 100  Ov    2di  dei  ti    100     0,58 f (2d j  c s dej ) t j ]  [0,58 fui uj 4 sin i sin  j

Ns,Rd  0,58fuj

Ns,Rd

 (3d j  dej ) t j 4 sin  j

 100  Ov    2hi  bei  ti   100     0,58 f (2h j  c s bej ) t j  0,58fui uj sin i sin  j

Ns,Rd  0,58fuj

(2h j  b j  bej ) t j sin  j

The expressions for dei, dej, bei and bej are given in Table 11.4.

Ovlimit = 60% and cs = 1 if hidden toe the overlapped brace is not welded.

Ovlimit = 80% and cs = 2 if hidden toe of the overlapped brace is welded.

In the case of overlap joints with hi < bi and/or hj < bj, the brace shear criterion shall always be checked.

146

Table 11.4 Effective width factors (be and de) used in Tables 11.1 and 11.3 Factors for CHS braces to CHS chords CHS braces

Overlapping CHS brace to CHS chord  12  fy 0 t 0   di but  di  dei      d0 /t 0  fyiti  Overlapped CHS brace to CHS chord  12  fy 0 t 0   d j but  d j  dej      d0 /t 0  fyjt j  Overlapping CHS brace to overlapped CHS brace  12  fyjt j   d but  di  de,ov    d j /t j  fyiti  i    Factors for CHS or RHS braces to RHS chords CHS braces

RHS braces

Overlapping CHS brace to RHS chord  10  fy 0 t 0   di but  di  dei      b0 /t 0  fyiti  Overlapped CHS brace to RHS chord  10  fy 0 t 0   d j but  d j  dej      b0 /t 0  fyjt j  Overlapping CHS brace to overlapped CHS brace  12  fyjt j   d but  di  de,ov    d j /t j  fyiti  i   

Overlapping RHS brace to RHS chord  10  fy 0 t 0   bi but  bi  bei      b0 /t 0  fyiti  Overlapped RHS brace to RHS chord  10  fy 0 t 0   b j but  b j  bej      b0 /t 0  fyjt j  Overlapping RHS brace to overlapped RHS brace  10  fyjt j   b but  bi  be,ov    b j /t j  fyiti  i   

Factors for CHS or RHS braces to I section chords CHS braces

RHS braces

Overlapping CHS brace to I section chord fy 0 dei  t w  2r  7t 0 but  di fyi

Overlapping RHS brace to I section chord fy 0 bei  t w  2r  7t 0 but  bi fyi

Overlapped CHS brace to I section chord fy 0 dej  t w  2r  7t 0 but  d j fyj

Overlapped RHS brace to I section chord fy 0 bej  t w  2r  7t 0 but  b j fyj

Overlapping CHS brace to overlapped CHS brace  12  fyjt j   d but  di  de,ov    d j /t j  fyiti  i   

Overlapping RHS brace to overlapped RHS brace  10   fyjt j   b but  bi  be,ov    b j /t j   fyiti  i   

147

hj

bj Nj

tj

hi

bi Ni

ti

(1) j

i (3)

b0 t0

N0p

N0

(2)

h0

(2)

i = overlapping member and j = overlapped member Fig. 11.1 Overlap joint with cross sections to be checked (for other sections, the same locations apply)

0,5be,ov

0,5be,ov

0,5bei

i

i

j

j

b0

t0

h0

Fig. 11.2 Local failure of the overlapping brace for RHS joints with 50% and 100% overlap

hi

bi

hi

hj

bj

ti

tj

ti

j

i

tj j

i

hj / sinj

0,5hi / sini 0,5bei

hj / sinj bj

0,5bej ti

hj

bj

bi

tj

0,5bej tj

Fig. 11.3 Effective shear area for RHS joints with 50% (no hidden weld) and 100% overlap

148

CHS 25-100% Overlap joints (excl. brace yielding)

N exp. / N local brace failure

1,4

Washio '63

1,2

Kurobane '64

1,0

Togo '67 Wardenier-de Koning '77

0,8

Kurobane '80

0,6

Ochi '81

0,4

De Koning-Wardenier '81

0,2

Kurobane '82 Dexter '94

0,0 0,0

0,2

0,4



0,6

0,8

1,0

Fig. 11.4 Ratio of the experimental joint capacity (database Makino et al., 1996) and the capacity based on the local brace failure criterion (25%  Ov < 100%)

Local brace failure vs. brace shear criteria 1,2 Local brace failure

Efficiency

1,0 Brace shear; hidden toe welded; Ov = 80%

0,8 0,6

Brace shear; hidden toe not welded; Ov = 60%

0,4 0,2 0,0 0

20

d0/t0

40

60

Fig. 11.5 Comparison between the criteria for local brace failure and brace shear with the hidden location welded or unwelded (for di/ti = 25, di = dj, ti = tj, fyi = fyj, fui = fuj = 1,25fyi, θi = θj = 45°)

149

150

12.3 ANALYTICAL MODELS

12. WELDED I BEAM-TO-CHS OR RHS COLUMN MOMENT JOINTS

12.3.1 Local failure of the beam flange

12.1 INTRODUCTION Connections in beam-to-column joints can be welded or bolted. This chapter will focus mainly on unstiffened, welded joints between CHS or RHS columns and I section beams, as shown in Fig. 12.1. Examples of some stiffened joints, especially used in earthquake prone regions, e.g. Japan, are shown in Fig. 12.2.

The effective width for the beam flange be shown in Fig. 12.4 can be determined from plate-to-CHS or plate-to-RHS joints (see Sections 8.7.2 and 9.7.3 with more detailed information in Section 9.3) because the flange connections are governing.

The moment capacity can be expressed by: M1,Rd = N1,Rd (h1 - t1)

(12.1)

In Chapters 8 and 9 it is already stated that the design formulae for hollow section joints loaded by bending moments can be derived in a similar way to that for axially loaded joints. This also applies to beam-tocolumn joints. For more detailed information, reference can be made to Dutta (2002), Kamba & Tabuchi (1994), Kurobane et al. (2004), Lu (1997), Packer & Henderson (1997) and De Winkel (1998).

where N1,Rd is the flange capacity for axial loading based on local failure of the beam flange determined in a similar way to that for the joints between hollow sections.

12.2 MODES OF FAILURE

For plate-to-RHS column joints, the capacity for local failure of the plate is given by:

In a similar way to that described in Chapter 7, various modes of failure (Fig. 12.3) can be identified by following the load transfer through the joint: - Local beam flange failure (yielding, local buckling) - Weld failure - Lamellar tearing - Column plastification (face, wall or cross section) - Column punching shear - Column local buckling - Column shear failure As indicated in Chapter 7, several modes of failure can be avoided, e.g.: - Weld failure, by making the welds stronger than the connected beam, i.e. for double fillet welds the throat thickness "a" should meet at least 0,5 times the value given in Section 6.5.2.

- Lamellar tearing, by choosing a material quality which is not susceptible to lamellar tearing (TTP quality).

- Local buckling, by limiting the width-to-thickness and/or the diameter-to-thickness ratio.

As a result, the following modes of failure have to be considered for design: - Local failure of the beam flange - Column plastification (face, wall or cross section) - Column punching shear - Column shear failure

Tests on plate-to-CHS column joints showed that, within the range of validity of the formulae, local failure of the plate was not critical compared to the other criteria.

M1,Rd = fy1 t1 be (h1 - t1)

(12.2)

where be is similar to that for joints between rectangular hollow sections, see Table 9.1.

12.3.2 Column plastification The plastification of I beam-to-CHS or RHS column joints is not only determined by the connection at the flanges but also by the column depth. The web of the I beam forces the chord face of an RHS column into a different yield line pattern than that which would be observed by two flanges at a certain distance apart, see Fig. 12.5.

If the web of the beam was not present, the capacity of the flange could be given according to eq. (12.1) where N1,Rd is the flange capacity based on the column plastification criterion. For example, for an RHS column (with θ1 = 90 and η = t1/b0 which is very small), eq. (9.6) would result in:

 4   (h  t ) M1,Rd  f y 0 t 02   1   1 1  

151

(12.3)

If the influence of the web is incorporated, the equation becomes considerably more complicated and reference can be made to Lu (1997).

For the chord side wall plastification criterion, similar rules can be used to those for beam-to-column joints between I sections and those used for RHS joints with β =1,0, i.e.

M1,Rd  2 f y 0 t 0 b wf (h1  t 1 )

(12.4)

where: b wf  t 1  5t 0 but 

h1  5t 0 2

(12.5)

For I beam-to-CHS column joints, the strength of the flange plate connection can be based on the ring model (see Chapter 8) with the resistances given in Table 8.3. However, for moment loading with the beam web included, the formulae become rather complicated and have to be calibrated with test results, resulting in semi-empirical formulae (De Winkel, 1998).

The column punching shear strength of I beam-toCHS or RHS joints can be directly determined from plate-to-CHS or plate-to-RHS joints (Wardenier et al., 2008a; Packer et al., 2009a). For more detailed information, see Wardenier (1982) and Voth (2010).

Here, similar to the criterion for local failure of the beam flange, the flanges are governing because the webs are located at the softest part of the column face and are generally not effective.

As shown in Fig. 12.6, the capacity is given by:

fy0 3

t 0 (2b e,p  2t 1 )(h1  t 1 )

If the beam-to-column joints only have a moment loaded beam on one side, or alternatively the beam moments on either side of the joint do not balance each other, shear forces will act in the column, which may cause shear failure of the column. The cross section of the column has to be checked here for the combined actions of axial load, shear load and bending moment. For class 1 and class 2 sections, the interaction can be based on the Huber-Hencky-Von Mises criterion (Wardenier, 1982) or a suitable, possible stress distribution can be assumed, as e.g. shown in Fig. 12.7b. According to the Huber-Hencky-Von Mises criterion, the following condition applies for a side wall of an RHS column: f y2   2  3 2

(12.7)

or: 2

      1      fy   f / 3     y 

2

(12.7a)

or:

12.3.3 Column punching shear

M1,Rd 

12.3.4 Column shear failure

2

2

 M0,Ed   V0,Ed      1  Mpl,0,Rd   Vpl,0,Rd      2

(12.7b)

2

 N0,Ed   V0,Ed      1  Npl,0,Rd   Vpl,0,Rd     

(12.7c)

or:

Mpl,V,0,Rd  Mpl,0,Rd

 V  1   0,Ed   Vpl,0,Rd   

Npl,V,0,Rd  Npl,0,Rd

  V 1   0,Ed   Vpl,0,Rd   

2

(12.8)

(12.6)

For plate-to-CHS column joints, it was shown that, within the range of application given in Table 8.3, be,p can be taken as b1.

For plate-to-RHS column joints, the same be,p can be taken as that given in Table 9.4.

2

(12.9)

Adding the flange parts (which are not reduced by shear) and conservatively assuming that the effective shear area is 2 hm t0 results in:

Mpl,V,0,Rd  b m hm t 0 f y 0  0,5 h t f

2 m 0 y0

  V 1   0,Ed   Vpl,0,Rd   

2

(12.10)

152

  V Npl,V,0,Rd  2 b m t 0 f y 0  2 hm t 0 f y 0 1   0,Ed   Vpl,0,Rd   

2

(12.11) The formulae (12.10) and (12.11) show the plastic capacities for axial loading and moment, reduced by the effect of shear.

In a similar way, the interaction between axial load and bending moment can be derived (Wardenier, 1982). By introducing Npl,V,0,Rd and Mpl,V,0,Rd instead of Npl,0,Rd and Mpl,0,Rd, the full interaction can be obtained.

In the standards (EN 1993-1-1, 2005) these formulae are approximated by simpler formulae. Also, the effect of small shear loads has been neglected, e.g. for V0,Ed  0,5 Vpl,0,Rd.

12.4 EXPERIMENTAL AND NUMERICAL VERIFICATION Most of the initial tests on plate-to-CHS and I beam-to-CHS joints were carried out in Japan (Makino et al., 1991; Kamba & Tabuchi, 1994). A good survey of all existing evidence on beam-to-CHS column joints, including many tests on stiffened joints, is given

by Kamba & Tabuchi (1994). Later work by De Winkel (1998) concentrated on a numerical parameter study with experiments carried out for validation of the numerical models. This study not only dealt with simple, unstiffened joints, but also with I beam-to-CHS column joints in which the column was filled with concrete and/or combined with a composite steel-concrete floor.

A similar investigation to that carried out by De Winkel (1998) was done by Lu (1997) for plate and I beam-to-RHS column joints.

Stiffened joints with strips welded at the sides of the flanges were investigated by Shanmugam et al.

(1993) i.e. the cross section of the flange at the connection with the RHS column also had an I shape.

12.5 BASIC JOINT STRENGTH FORMULAE In the studies mentioned in Section 12.4 on unstiffened RHS joints, strength formulae have been determined using analytical models as a basis. These formulae have been modified to fit the numerical data.

Since the strength data have been based on a

deformation criterion of 3% of the chord width, the resulting formulae for plate-to-RHS column joints give, for low β ratios, lower strengths than those based on the yield line method.

At present, only strength functions are established for uniplanar and multiplanar joints, but no formulae have been developed for the stiffness, although the stiffness is extremely important for the determination of the moment distribution in unbraced frames.

However, a large number of moment-rotation diagrams is available for a wide range of parameters.

From these moment-rotation diagrams (Lu, 1997; De Winkel, 1998) indications can be obtained for the stiffness.

The effect of the joint stiffness on the elastic moment distribution is represented in Figs. 12.8 and 12.9. It is shown that with semi-rigid joints the elastic moment distribution can be influenced considerably.

If a rigid-plastic analysis is used, not only is the joint moment resistance of primary importance, but also the rotation capacity. For example, if the stiffness of the beam-to-column joints in Fig. 12.8 is very low, the plastic moment capacity of the beam at mid-span Mpl,Rd may be reached first. The moment capacity of the end joints Mj,Rd can only be attained if the beam has sufficient rotation capacity at the location of the plastic moment. In the case of joints with a very low stiffness, this might not be the case, e.g. see curve "e" in Fig. 12.10.

If the stiffness of the joint is high, the (partial) strength capacity of the beam-to-column joints (e.g. curve "b" in Fig. 12.10) may be reached first. Now, these joints should have sufficient deformation capacity to allow the plastic moment capacity of the beam at mid-span to be attained. The joints with behaviour "a" or "c" are stronger than the connected beam, thus the beam should have sufficient deformation capacity if the plastic moment is reached first at the ends.

Hence, for an accurate analysis of frames with semi-rigid joints, an appropriate description of the moment-rotation behaviour is required as well as evidence regarding: - Stiffness (at serviceability and at the ultimate limit state) - Strength (ultimate limit state) - Rotation capacity However, all this information is not yet generally available for beam-to-tubular column joints, although research has been initiated by Weynand et al. (2006).

153

Alternative options are to use joints with such a stiffness that the joints can be classified as (nearly) rigid or (nearly) pinned. For both cases, limits can be defined. However, the deflections can only be determined properly if the joint stiffnesses are known (Fig. 12.11).

Part 8 of Eurocode 3 (EN 1993-1-8, 2005) provides stiffness classifications, see Fig. 12.12. A possible joint modelling approach is given in Fig. 12.13.

For I beam-to-tubular column joints, the factors c1 and c2 still have to be defined. Another complication is that the axial loads and moments in the column do not only affect the strength but also the stiffness, as shown by the curve in Fig. 12.13 for N0 â&#x2030; 0.

12.6 CONCLUDING REMARKS The design recommendations in CIDECT Design Guide No. 9 (Kurobane et al., 2004) give strength functions for beam-to-CHS and beam-to-RHS column joints.

This chapter intends to give basic background information without going too much into detail in the resulting design formulae. The design of frames with semi-rigid joints is not typical for tubular structures and is therefore not described here in detail.

154

Fig. 12.1 Unstiffened I beam-to-CHS and I beam-to-RHS column joints

tp

tp

t0

t0 d0

b0

bf Bf

Fig. 12.2 Flange diaphragm I beam-to-CHS and I beam-to-RHS column joints

155

b1 Bf

a. Beam flange failure

b. Weld failure

Joint configuration

c. Lamellar tearing

d1. Column face plastification

d2. Column wall plastification (side view)

f. Column local wall buckling (side view)

e. Column punching shear

g. Column shear failure

Fig. 12.3 Modes of failure for I beam-to-RHS column joints

156

Fig. 12.4 Stress distribution in beam flange

Fig. 12.5 RHS column face plastification

0,5be,p t1

Fig. 12.6 RHS column punching shear 157

Fig. 12.7 Column shear failure

Fig. 12.8 Beam with various end conditions

158

0,67 0,50 0,33

Fig. 12.9 Variation of elastic moment distribution with joint stiffness

Fig. 12.10 Various M-ď Ś characteristics

159

1,0 0,8 0,6 0,4 0,2 0

Fig. 12.11 Variation of midspan deflection with joint stiffness

,

,

Fig. 12.12 Boundaries for stiffness classification of beam-to-column joints according to Eurocode 3 (EN 1993-1-8)

Ed

Fig. 12.13 M-ď Ś modelling 160

13. BOLTED JOINTS The calculation methods used for bolted joints between, or to, hollow sections are basically not different from those used for any other type of joint in conventional steel construction.

Most details given in this chapter are presented without (detailed) design formulae.

13.1 FLANGE PLATE JOINTS 13.1.1 Flange plate joints to CHS under axial tension load For the flange plate joints shown in Fig. 13.1, various investigations were carried out (Kato & Hirose, 1984; Igarashi et al., 1985; Cao & Packer, 1997).

Economical joints under tension load can be obtained if prying force is permitted at the ultimate limit state, with the connection proportioned on the basis of a yielding failure mechanism of the flange plates. In CIDECT Design Guide No. 1 (Wardenier et al., 2008a) formulae and tables are given, based on the work of Igarashi et al. (1985). In the context of this book, only the failure modes are presented (Fig. 13.2). It is preferable to design primary structural joints on the basis of the yield resistance of the circular hollow section.

13.1.2 Flange plate joints to RHS under axial tension load Research by Birkemoe & Packer (1986) and Packer et al. (1989) on bolted RHS flange plate joints with bolts on two sides of the RHS only, see Fig. 13.3, showed that in principle the strength of these joints can be analysed on the basis of the traditional prying model developed for T-stubs by Struik & De Back (1969).

The location of the plastic hinge lines may be adjusted for greater accuracy, i.e. the distance b in Fig. 13.4 is adjusted to b' according to: b'  b 

d  ti 2

(13.1)

Detailed formulae are given by Packer & Henderson (1997) and Packer et al. (2009a).

Many tests have been carried out on RHS flange plate joints with bolts on 4 sides of the RHS, as shown in Fig. 13.3. A thorough study of this type of bolted joint has been undertaken by Willibald et al. (2002, 2003a).

It was revealed that RHS flange plate joints bolted on all four sides could still be proportioned on the basis of the two-dimensional T-stub prying model of Struik & De Back (1969), with some minor modifications.

Following the procedure for bolted RHS flange plate joints with bolts on two sides, the inner yield lines in the flange plate can now be expected adjacent to the RHS outer face and hence the term ti should be deleted from eq. (13.1). The bolt pitch to be used is the minimum of p from both sides. The dimension p, the plate width or depth divided by the number of bolts in that direction, is illustrated in Fig. 13.3. This "minimum p" value is then used in the joint analysis based of a two-dimensional prying model. In order for this design model to be valid, the centres of the bolt holes should not be positioned beyond the corners of the RHS (as illustrated in Fig. 13.3).

Detailed information can be found in CIDECT Design Guide No. 3 (Packer et al., 2009a).

13.1.3 Flange plate joints to CHS or RHS under axial tension load and moment loading Design methods for bolted flange plate joints to date have generally been developed for axial tension loading. Frequently, however, hollow sections are subjected to both axial tension load (Ni) and bending moment (Mi). In such cases, a hypothetical "effective" axial load can be computed (Kurobane et al., 2004) for use with the flange plate joint design procedures given in Sections 13.1.1 and 13.1.2: N Mi   A i Effective axial   i  A W i   i

(13.2)

where: Ai cross sectional area of the CHS or RHS Wi elastic (or plastic) section modulus of the CHS or RHS This procedure will be conservative, especially for CHS, as it computes the maximum tensile normal stress in the CHS or RHS and then applies this to the whole member cross section.

13.2 END JOINTS Some bolted end joints are shown in Fig. 13.5. The flange of the tee in Fig. 13.5d, as well as the other flange plates perpendicular to the CHS or RHS

161

section, must be sufficiently thick to effectively distribute the load to the cross section (Wardenier et al., 2008a; Packer et al., 2009a), see also Section 9.7.3.

13.3 GUSSET PLATE JOINTS For bolted gusset plate joints, the design can be based on the various possible failure modes, e.g. for a tension member: - Yielding of the cross section - Rupture of the net area - Rupture of the effective net area reduced for shear lag Similar to other bolted joints, the total net area is the sum of individual net areas along a potential critical section of a member or gusset plate, see Fig. 13.6. If such a critical section comprises net areas loaded in tension and segments loaded in shear, the shear segments should be multiplied by the shear strength and the tension areas by the ultimate strength.

Eurocode 3 (EN 1993-1-1, 2005) specifies a γM factor of 1,0 for yielding and 1,25 for ultimate strength (rupture).

A failure mode of the gusset plate which also must be checked is yielding across an effective dispersion width of the plate, which can be calculated using the Whitmore (1952) effective width concept illustrated in Fig. 13.7. For this failure mode (for one gusset plate), the strength is given by:



N i,Rd  f yp t p g  2 (tan 30 o )  p

 1

(13.3)

M

where the term  p represents the sum of the bolt pitches in a bolted connection or the length of the weld in a welded connection, and M =1,1.

If the member is in compression, buckling of the gusset plate must also be prevented.

Fig. 13.8 shows some examples of bolted gusset plate joints. It must be borne in mind that fitting of these connections is very sensitive with regard to dimensional tolerances and to deformations of the welded gusset due to weld-induced distortions. Thus, care has to be taken to ensure fitting at site.

When a member is connected by some, but not all parts of its cross section elements and if the net section includes elements which are not connected, the net area perpendicular to the load has to be

multiplied by a shear lag factor which depends on the shape of the section, the number of connected faces and the number of transverse rows of fasteners.

Such a case is illustrated in Fig. 13.8b where bolting plates are welded to the sides of the RHS brace member. For welds parallel to the direction of load (as the four flare groove welds would be in Fig. 13.8b, along the four corners of the RHS), the shear lag factor is a function of the weld lengths and the distance between them. For the RHS, the shear lag reduction factors can be applied to each of the four sides (two of width w = bi - ti, and two of width w = hi ti), to produce a total effective net area of the RHS reduced by shear lag. Suggested shear lag reduction factors for these four element areas, in terms of the weld length Lw, are (CSA, 2009): - 1,00 when the weld Iengths (Lw) along the RHS corners are  2bi (or 2hi as applicable) - (0,5 + 0,25Lw/bi) when the weld lengths along the RHS corners are bi  Lw < 2bi, or - (0,5 + 0,25Lw/hi) when the weld lengths along the RHS corners are hi  Lw < 2hi - 0,75Lw/bi when the weld lengths along the RHS corners are Lw < bi (or hi as applicable)

13.4 SPLICE JOINTS Fig. 13.9 shows a splice joint for circular hollow sections. This type of connection can, for example, be executed with four, six or eight strips welded longitudinally on the periphery of the hollow sections and connected by double lap plates, one on each side.

Lightly loaded splice joints in tension can be made as shown in Fig. 13.10 and for architectural appearance the bolts can be hidden. Using one plate on each side, instead of the solution in Fig. 13.10, provides a more fabrication-friendly solution. Such an eccentric joint, however, may have little stiffness and resistance to out-of-plane flexure under compression loading, thus the designer should be confident that such a condition has been considered. Experimental and numerical research on this RHS joint type, under tension loading, has been conducted by Willibald et al. (2003b).

13.5 BEAM-TO-COLUMN JOINTS Bolted beam-to-column joints can be designed in various ways, mainly depending on the type of load that has to be transmitted. In general, shear joints are simpler to fabricate than moment joints. Typical joints

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are given in Figs. 13.11 to 13.15 without detailed description.

13.6 BRACKET JOINTS Some typical joints for lightly loaded beams are shown in Fig. 13.16.

13.7 BOLTED SUBASSEMBLIES Lattice structures are often connected to columns by bolted flanges, plates or Tee profiles. Some examples are shown in Fig. 13.17.

13.8 PURLIN JOINTS Fig. 13.18 shows some examples of purlin joints for trusses with CHS or RHS chords.

13.9 BLIND BOLTING SYSTEMS Due to the closed nature of hollow sections, in many cases additional welded plates are used for bolted joints. However, solutions are then not aesthetically appealing. Nowadays, bolting systems are available which can be used when only one side of the connection is accessible. Blind bolting systems make use of either special types of bolts or inserts or special drilling systems.

13.9.1 Systems using bolts and inserts Special types of bolts and systems allow one to bolt from one side of a hollow section. A number of patented blind bolting systems is available, e.g. Huck "Ultra Twist Blind Bolt" and Lindapter "HolloFast" and "HolloBolt". The latter, which uses a special insert and a standard bolt, has been investigated by CIDECT (Sidercad & British Steel, 1996; Yeomans, 1998) with regard to its axial, shear and bending capacity (see Fig. 13.19).

The systems are based on the principle that after bringing them in from one side, the bolts are torqued and a "bolt head" forms on the inside of the connected plies.

The design rules for blind bolting systems are based on typical failure modes, i.e.

- Punching shear of the fastener through the column face

- Yielding of the column face (yield line pattern around the bolts) - Bolt failure in shear, tension or a combination of both

13.9.2 Drilling system The Flowdrill system, see Fig. 13.20, is a special patented method for extruded holes. CIDECT has carried out research (Yeomans, 1994; British Steel, 1996) to assess the load bearing capacity of this type of joint in structural hollow sections.

Flowdrilling is a thermal drilling process (Fig. 13.21) to make a hole through the wall of a hollow section by bringing a tungsten carbide bit into contact with the hollow section wall and generating sufficient heat by friction to soften the steel. As the bit moves through the wall, the metal flows to form an internal bush. In the next step, the bush is threaded using a roll tap.

Conventional bolts are then used in this tapped hole.

Bolting to hollow sections with wall thicknesses up to 12,5 mm can be recommended by using the Flowdrill method, see Yeomans (1994).

13.10 NAILED JOINTS As an alternative to bolting or welding, steel circular hollow sections can be nailed together to form reliable structural joints. Up to now, this method of connection has only been verified for splice joints between two co-axial tubes (see Fig. 13.22). In such a joint, one tube can fit snugly inside the other, in such a way that the outside diameter of the smaller equals the inside diameter of the larger. Nails are then shot fired and driven through the two wall thicknesses and arranged symmetrically around the tube perimeter.

As an alternative, two tubes of the same outside diameter can be joined by means of a tubular collar over both tube ends; in this case nails are again inserted by driving them through the two tube walls.

Research to date has covered a range of tube sizes with various diameter-to-thickness ratios, tube wall thickness and lack of fit (Packer, 1996). The observed failure modes were nail shear failure, tube bearing failure, and net section fracture of the tube. These failure modes have been identified for both static and fatigue loading. Simple design formulae, derived from bolted and riveted joints, have been verified for both these load cases.

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Fig. 13.1 Bolted CHS flange plate joint

Fig. 13.2 Failure modes for bolted CHS flange plate joints

164

p

p

p

p

p

Fig. 13.3 Bolted RHS flange plate joints

Fig. 13.4 RHS flange plate joint with bolts at two sides of the RHS

165

Fig. 13.5 Bolted end joints

Tension segment

Bolt hole diameter d’

Inclined segments

Shear segments

Total net area for critical section A-A is the sum of the individual segments: For tension segment : An = (g1 - d’/2) t For shear segment : Agv = L t 2 For each inclined segment : An = (g2 - d’) t + (s /4g2) t

Fig. 13.6 Calculation of total net area for a gusset plate

166

,

Fig. 13.7 Whitmore criterion for gusset plate yielding

Fig. 13.8 Some examples of bolted gusset plate joints

Fig. 13.9 Bolted splice joint for CHS

167

Fig. 13.10 Hidden bolted splice joint

IPE or HE cut off

Fig. 13.11 I section beam-to-CHS column joints 168

a

b

c

d

e

f

Fig. 13.12 I section beam-to-RHS column simple shear joints 169

a

b

c

d

Fig. 13.13 Moment joints between open section beams and CHS or RHS columns

170

Fig. 13.14 RHS sections connected to I section columns

Fig. 13.15 Knee joint assemblies for portal frames

Fig. 13.16 Bracket joints

171

a

b

c

d

e

f

Fig. 13.17 Bolted joints for lattice girder supports

172

a

b

c

d

e

f

Fig. 13.18 Purlin joints

Fig. 13.19 Lindapter "HolloFast" connection

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Fig. 13.20 Flowdrill connection for joining end plates or angles to RHS

Fig. 13.21 Flowdrill process

Fig. 13.22 Nailed CHS joint

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as the endurance or fatigue life.

14. FATIGUE BEHAVIOUR OF HOLLOW SECTION JOINTS Fatigue is a mechanism whereby cracks grow in a structure under fluctuating stress. Final failure generally occurs when the reduced cross section becomes insufficient to carry the load without rupture.

Generally the fatigue cracks start at locations with high stress peaks.

High stress peaks may occur at local notches, e.g. at welds (Fig. 14.1). However, geometric peak stresses may also occur due to the geometry, e.g. at holes or in hollow section joints due to the non-uniform stiffness distribution at the perimeter of the connection (Fig. 14.2).

Thus the fatigue behaviour is largely influenced by the loading and the way the members are connected. In a fatigue analysis the loading and the loading effects should be carefully evaluated and compared with the fatigue resistance.

Sometimes it is difficult to determine the loading effects accurately, e.g. the secondary bending moments in lattice girders. In such cases, simplified approaches can be used.

14.1 DEFINITIONS Stress range Δσ The stress range Δσ (shown in Fig. 14.3) is the difference between the maximum and the minimum stress in a constant amplitude loading regime.

Stress ratio R The stress ratio R is defined as the ratio between the minimum stress and the maximum stress in a stress cycle of constant amplitude loading (Fig. 14.3). σ-N or Wöhler line The relation (on a log-log scale) between the stress range Δσ and the number of cycles N to failure is presented in a so-called σ-N or Wöhler line (Fig. 14.4).

Fatigue strength The fatigue strength of a welded component is defined as the stress range Δσ, which causes failure of the component after a specified number of cycles N. Fatigue life The number of cycles N to a defined failure is known

Fatigue limit The fatigue limit is defined as the stress range below which it is assumed that no fatigue failure occurs for a constant amplitude loading, see Fig. 14.4. For Eurocode 3 (EN 1993-1-9, 2005) and the IIW recommendations for hollow section joints (IIW, 1999), for example, this occurs at N = 5 x 106 cycles. Note: IIW (2008) for fatigue design of welded joints and components has recently changed this limit to 107 cycles.

Cut off limit The cut off limit is defined as the stress range below which it is assumed that the stress ranges of a variable amplitude loading do not contribute to the fatigue damage. For Eurocode 3 (EN 1993-1-9), for example, this occurs at N = 108 cycles (see Fig. 14.5).

Note: The recently revised recommendations of IIW (2008) no longer give a cut off limit for variable amplitude loading.

Geometric stress The geometric stress, also called hot spot stress, is defined as the extrapolated principal stress at a specified location at the weld toe. The extrapolation must be carried out from the region outside the influence of the effects of the weld geometry and discontinuities at the weld toe, but close enough to fall inside the zone of the stress gradient caused by the global geometrical effects of the joint. The extrapolation is to be carried out on the brace side and the chord side of each weld (see Fig. 14.6).

Generally the geometric stress (or the hot spot stress) can be determined by considering the stress normal to the weld toe since the orientation of the maximum principal stress is normal or almost normal to the weld toe.

Stress concentration factor The stress concentration factor (SCF) is the ratio between the geometric peak stress, or hot spot stress, excluding local effects, at a particular location in a joint and the nominal stress in the member due to a basic member load which causes this geometric stress.

14.2 INFLUENCING FACTORS The fatigue behaviour can be determined either by Δσ-N methods or with a fracture mechanics approach.

The various Δσ-N methods are based on experiments resulting in Δσ-N graphs (Fig. 14.4) with a defined

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stress range Δσ on the vertical axis and the number of cycles N to a specified failure criterion on the horizontal axis.

The relation between the number of cycles to failure N and the stress range Δσ can be given by:

N  C (  m )

(14.1)

or:

log N  log C  m log  

(14.2)

On a log-log scale, this gives a straight line with a slope of -m, see Fig. 14.4.

The fracture mechanics approach is based on a fatigue crack growth model and is not further discussed in the context of this book.

Due to the appearance of residual stresses, the stress ratio R = σmin /σmax (see Fig. 14.3) is not taken into account in modern fatigue design. Only if the structure is fully stress relieved, it might be advantageous to take the stress ratio into account.

Fig. 14.7 shows an example illustrating the influence of residual stresses on a plate with fy = 240 N/mm2. A nominal stress range Δσnom = 120 N/mm2 is applied on the plate, resulting in a nominal average stress range in the net cross section at the hole of:  nom 

80 120  160 N/mm 2 60

With a geometric stress concentration factor SCF = 3 for this detail, this nominal stress range results in a theoretical geometric stress range:

The example in Fig. 14.7 with a tensile loading results in a compressive residual stress at the hole, which is favourable. However, it also shows that a compressive loading would have resulted in a tensile residual stress at the hole, which is unfavourable. In this latter case, it would result in fatigue failure under an external nominal compression stress range Δσ. In practical situations, various stress ranges occur and the residual stress ranges are not known. That is why, in general, no difference is made between tensile and compressive loading.

For simple details, e.g. butt welded, end-to-end connections, cover plates, attachments, etc., Δσ-N lines can be determined. These Δσ-N lines (Figs. 14.4 and 14.5) already take into account the effect of the welds and the geometry of the connection.

For more complicated geometries, such as the directly welded joints between hollow sections, the peak stresses and thus the peak stress ranges depend on the geometrical parameters. This means that for every joint, a separate Δσ-N line exists. Hence, in modern design, the fatigue behaviour of hollow section joints is related to basic Δσ-N curves, in which Δσ represents a geometrical or hot spot stress range taking into account the effect of the geometry of the joint. This geometrical stress range can also be calculated by multiplying the nominal stress range with a relevant stress concentration factor SCF. As a consequence of this method, stress concentration factors should be available for joints with various geometries. The effect of the weld is included in the basic Δσ-N lines. As shown later, the thickness of the sections is also an influencing parameter.

14.3 LOADING EFFECTS As stated before, the stress range Δσ is a governing parameter for fatigue.

2

Δσgeom = SCF Δσnom = 3 x 160 = 480 N/mm

However, at 240 N/mm2 the material is yielding, resulting in stress pattern "b". Unloading to zero (loading in the opposite sense) means that the elastic stress pattern "a" has to be subtracted from "b" resulting in residual stress pattern "c". The next cycle will thus fluctuate between patterns "c" and "b", i.e. starting from a residual stress equal to the yield stress.

In welded structures, residual stresses and discontinuities exist at welds too. Therefore, it is assumed that locally the stress will fluctuate between the yield stress and a lower limit (fy - Δσgeom).

For constant amplitude loading, the Δσ-N line is generally cut off between N = 2 x 106 to 107 cycles depending on the code. CIDECT (Zhao et al., 2001) and IIW (1999) follow Eurocode 3 (EN 1993-1-9, 2005) and use 5 x 106 cycles (see Fig. 14.4).

For random loading, also called spectrum loading, the smaller cycles may also have an effect and the cut off is taken at 108 to 2 x 108 cycles, depending on the code. Eurocode 3 (EN 1993-1-9) uses a cut off at N = 108 cycles. Between 5 x 106 to 108 cycles, a Δσ-N line is used with a shallower slope of m = 5.

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Although the sequence of the stress ranges affects the damage, the simplest and best rule available up to now to determine the cumulative damage, is the Palmgren-Miner linear damage rule, i.e. 

ni  1,0 Ni

(14.3)

in which ni is the number of cycles of a particular stress range Δσi and Ni is the number of cycles to failure for that particular stress range.

If vibration of the members occurs, the nominal stress ranges may be considerably increased. This will be the case if the natural frequency of a structural element is close to the frequency of the loading. It is therefore essential to avoid this.

For very high stress ranges and consequently a low number of cycles, low cycle fatigue may occur. In this case, no Δσ-N curves are given in the codes since the fatigue is mainly governed by the strains. Research by Van der Vegte et al. (1989) showed that the various Δσ-N curves can be used if translated into strain, i.e. Δε-N curves. In this case, the loading should be evaluated based on the resulting strain range Δε.

14.4 FATIGUE STRENGTH In IIW (1999), CIDECT (Zhao et al., 2001) and Eurocode 3 (EN 1993-1-9, 2005) the fatigue strength for members with butt welded or fillet welded end-to-end connections, with plates or members with attachments, etc. is given as the stress range at 2 x 106 cycles. This classification is in line with the classifications given for other members such as I sections, etc. Table 14.1 shows some of these classifications, adopted by Eurocode 3 (EN 1993-1-9, 2005), which are based on the analyses of Noordhoek et al. (1980) and Wardenier (1982).

Table 14.1 shows that the classification of 160 or 140 N/mm2 for plain sections without attachments or connections is much higher than the classification for details.

Table 14.1 further makes clear that a butt welded end-to-end connection has a better fatigue behaviour than a fillet welded end-to-end connection with an intersecting plate.

For welded details, the classification is independent of the steel grade. Higher strength steels are more sensitive to notches, which reduces the favourable

effect of the higher strength. However, if improved welding methods are used or post weld improvements (grinding, TIG dressing, plasma dressing, ultrasonic impact treatment (UIT), etc.) are carried out, higher fatigue strengths can be obtained, especially for higher strength steels (e.g. for S460 and S690).

14.5 PARTIAL FACTORS If fatigue actions are included in a design, it should be noted that the stress ranges produced by the (unfactored) loading have to be multiplied by a partial (safety) factor which depends on the type of structure (fail-safe or non fail-safe) and the possibility for inspection and maintenance. The recommended partial (safety) factors according to CIDECT (Zhao et al., 2001) are given in Table 14.2. For example, a partial factor of M = 1,25 and m = 3 reduces the design fatigue life by about a factor of (1,25)-3  0,5.

14.6 FATIGUE CAPACITY OF WELDED JOINTS In Section 14.4 the basic parameters influencing the strength of members with attachments or with end-to-end connections are discussed. It has already been stated that in welded joints between hollow sections the stiffness around the intersection is non-uniform, resulting in a geometrical non-uniform stress distribution as shown in Fig. 14.2 for an X joint of circular hollow sections. This non-uniform stress distribution depends on the type of brace loading (axial, in-plane bending, out-of-plane bending), the joint type and the geometry. Therefore, fatigue design of hollow section joints is, in general, different from, for example, that of simple welded connections between plates.

14.6.1 Geometrical stress approach Since peak stresses determine the fatigue behaviour, in modern design methods, the fatigue design is related to the geometrical stress range of a connection. This geometrical stress range includes the geometrical influences but excludes the effects related to fabrication such as the configuration of the weld (flat, convex, concave) and the local condition at the weld toe (radius of weld toe, undercut, etc.). Since the geometrical peak stress (also called the hot spot stress) can only be determined with finite element methods or by measurements on actual specimens, stress concentration factors have been developed for

177

the basic types of joints and basic loadings. These stress concentration factors are defined as the ratio between the geometrical (peak) stress and the nominal stress causing the geometrical stress, for example for an axially loaded X joint without chord loading:

SCFi, j,k 

geometrica l stress i, j,k nominal stress in the brace

(14.4)

one set of curves with a single, general thickness correction factor can be conservatively used for all joint types of CHS and RHS. For a designer it is important to have an insight into the parameters which determine the stress concentration factors. Optimal design requires the stress concentration factors to be as low as possible.

where: i chord or brace j location, e.g. crown, saddle or in between for CHS joints k type of loading

As an indication, the stress concentration factors are given for various joint configurations. Fig. 14.9 shows the stress concentration factors for axially loaded X joints of circular hollow sections at four locations, i.e. for the chord and brace, at the crown and saddle.

The following conclusions can be made:

Thus, several SCFs have to be determined for various likely critical locations (Fig. 14.6). The maximum SCF and the location depend on the geometry and loading.

This is particularly important for combined loadings.

Thus, for determining the fatigue life the geometrical (peak) stress range has to be calculated for the various locations:

For the chord: - Generally the highest SCF occurs at the saddle position - The highest SCFs at the saddle are obtained for medium β ratios - SCFs decrease with decreasing τ value - SCFs decrease with decreasing 2 value

 geomi, j,k  (  nomi, j,k SCFi, j,k )

For the brace: - Generally the same applies as for the chord; however, a decrease in τ value results in an increase in SCF at the crown position (in some cases the graphs are only given here for τ = 1,0).

- SCF in the brace may become critical compared to that in the chord for small τ values; however, the brace thickness is then smaller than the chord thickness. Considering the thickness effect, it is most likely that the chord location is still the critical position.

(14.5)

The Δσ-N line to be considered is also based on the geometrical stress range.

Initially, it was expected that one Δσ-N curve could be used. However, for the same geometrical stress the strain gradient may be different, resulting in a different fatigue life. Since the gradient is steeper for thin walled members than for thick walled members, this effect is accounted for by a thickness correction.

The geometrical Δσ-N curves in IIW (1999) and CIDECT (Zhao et al., 2001) are related to fatigue class 114 (at 2 x 106 cycles) for a wall thickness of 16 mm. For joints with wall thicknesses other than 16 mm, a wall thickness function should be applied.

The Δσ-N curves with the thickness correction included are given in Fig. 14.8. It should be noted that no further thickness effect should be used for thicknesses less than 4 mm, since the weld performance may overrule the geometrical influence, which may sometimes result in lower fatigue strengths (Wardenier, 1982; Mashiri et al., 2007).

The curves presented in Fig.14.8 slightly deviate from the classifications previously given in CIDECT Design Guide No. 6 (Wardenier et al., 1995). However, the reanalysis by Van Wingerde et al. (1997) showed that

For X joints of square hollow sections (Van Wingerde, 1992), the SCFs are given for various locations in Fig. 14.10. As mentioned before, similar observations can be made as for circular hollow section X joints: - For τ = 1,0 the highest SCFs generally occur in the chord at locations B and C - The highest SCFs are found for medium β ratios - The lower the 2γ ratio, the lower the SCF - The lower the τ ratio, the lower the SCF in the chord, whereas τ has a small influence on the SCF in the brace.

Fig. 14.11 shows the SCFs for an axially loaded K gap joint of circular hollow sections with g = 0,1d0. Here, the same observations apply as for X joints, but the SCFs are considerably lower due to the stiffening effect of both braces with opposite loadings connected to the chord face. Here, only maximum SCF values for chord and brace are given (i.e. no differentiation is

178

made between the saddle and crown positions).

Although the effect of the brace angle is not included in the figures, a decrease in the angle between brace and chord results in a considerable decrease in SCF. If the chord is loaded, the geometrical or hot spot stress range at the chord locations at the crown (CHS) or at the locations C and D (RHS) (see Fig. 14.6) has to be increased by the chord nominal stress range multiplied by the stress concentration factor produced by the chord stresses. As shown in Fig. 14.12 for RHS T and X joints, the SCF for chord loading varies between 1 and 3, depending on the loading, joint type and geometrical parameters (Van Wingerde, 1992).

Another aspect to be considered is multiplanar loading. For the same type of joint and the same geometry, different SCFs can be obtained for different loading conditions (see Fig. 14.13).

All the SCFs are based on measurements taken at the toe of the weld, since this location is generally critical.

However, for very low SCFs, crack initiation can start at the weld root. Therefore a minimum SCF = 2,0 is recommended.

Further, the SCF values have been determined for butt or groove welded joints. Fillet welds cause slightly lower SCFs in the chord, but due to local wall bending effects, result in considerably higher SCFs in the braces. It is therefore recommended to increase the SCFs in RHS braces of T and X joints by a factor of 1,4 when fillet welds are used (Van Wingerde, 1992).

Considering all these aspects, it can be concluded that optimal design can be achieved if the SCFs are as low as possible. Thus the following guidelines can be given: - Avoid medium β ratios. Ratios close to β = 1,0 give the lowest SCFs. - Make the wall thickness of the brace as low as possible (low τ ratio).

- Take relatively thick walled chords (low 2γ ratio).

In this way, SCFs of about 2 to 4 are possible, resulting in an economic design.

If the bending moments in girders have not been determined by finite element analyses or other methods, e.g. a rigid frame analysis, the effect of the bending moments can be incorporated in a simplified manner as discussed in Section 14.6.2.

More detailed information about stress concentration

factors can be obtained for CHS joints from Efthymiou (1988) and Romeijn (1994) and for RHS joints from Herion (1994) and Van Wingerde (1992).

14.6.2 Classification method based on nominal stress ranges To simplify design methods, it would be easier if the SCFs could already be incorporated in the design class, taking account of the main influencing parameters. However, this is impossible for T, Y and X joints, since the variation in SCF is considerable.

For K joints, classification is possible to a certain extent. Table 14.3 shows the detail categories for lattice girder joints given in Eurocode 3 (EN 1993-1-9, 2005), based on the analyses of Noordhoek et al.

(1980) and Wardenier (1982). The thickness effect is indirectly included. The method can only be used for thin walled sections (see recommended range of validity). As shown in Table 14.3, the classification only depends on the gap, overlap and τ ratio. It must be stated that the design classes are based on test results and on an independent analysis, and do not comply with the geometric stress method. Further, the design classes given clearly show the advantage of using a low τ ratio or high t0/ti.

Since it might be difficult for designers to determine the secondary bending moments, the CIDECT and IIW recommendations allow a calculation based on the assumption that the members are pin connected.

However, the stress ranges in chord and braces caused by the axial loading in the braces have to be multiplied by the factors given in Tables 14.4 and 14.5 to account for the secondary bending moments due to joint stiffness. These multiplication factors are based on measurements in actual girders (De Koning & Wardenier, 1979). In the case of eccentricities or loads in between the joints, a calculation based on the model in Fig. 6.8 is recommended. Although the effects of the eccentricities and loads in between the joints are then covered, the effects of joint stiffness still have to be added in a similar way as described here before.

14.7 FATIGUE CAPACITY OF BOLTED JOINTS Joints with pretensioned high strength friction grip (HSFG) bolts have a favourable fatigue behaviour compared to joints with non-pretensioned bolts. Joints loaded in tension can, for example, be designed in

179

such a way that the fatigue load does not critically affect the joints. Joints with pretensioned high strength bolts subjected to shear or friction loads can bear higher fatigue loads than welded joints, which means that, in general, these bolted joints in shear do not represent the most critical elements. Bolted joints under fatigue loading should be designed in such a way that there is no play and no slip between the faying surfaces.

14.7.1 Bolted joints under tensile load Theoretical investigations and experimental results (Bouwman, 1982) show that the fatigue behaviour of joints with pretensioned bolts is significantly influenced by the way in which the load is transmitted.

A few examples of bolted joints in tension are shown in Fig. 14.14.

For configurations a1 and a2 (Fig. 14.14) the contact face pressure is, by design, co-axial with the external load. In this case, as the load increases, at first there is a significant decrease in the contact pressure. Only when the applied load exceeds the contact pressure, the load in the high strength bolts begins to increase appreciably. For configurations c1 and c2, the load in the high strength bolts increases from the very beginning with growing applied load due to prying.

Thus, connection types a1 and a2 exhibit a much better fatigue performance than the other types.

For the a1 and a2 configurations, the stiffness of the flanges is distinctly higher than that for the axial load transmission through the bolts accompanied by flange bending. As a result, for configurations a1 and a2 under fatigue load, only a small stress increase in the bolts is observed. This change in stresses due to fatigue loading can generally be neglected, when the applied load per bolt is smaller than the bolt preload.

Should the load transmitting parts not be located on the same plane, the connections must then be designed in such a manner that the load transmission is mainly effected through a reduction of contact loads. The arrangements shown in Fig. 14.15 are recommended for bolted ring flange joints between hollow sections. These proposed flange connections require expensive machining and finishing. However, the same level of load transmission can be achieved simply with the help of packing plates (shims). It is further recommended that the high strength bolts be located as closely as possible to the load carrying structural parts.

Snug-tight bolted connections without pretensioning should be avoided, since the fatigue behaviour is bad.

14.7.2 Bolted joints under shear load The fatigue behaviour of high strength pretensioned bolted joints under shear load is generally better than that of welds connecting hollow sections to end plates.

As shown in Fig. 14.16, the stress distribution in joints with pretensioned bolts is significantly better than that in bolted joints without preload. The reason is that part of the external load is transmitted by friction around the bolt hole. After a high strength pretensioned bolted joint has slipped, a more unfavourable stress distribution develops than before, since part of the force is now transmitted by pressure on the face of the hole (see Fig. 14.16c).

Non-pretensioned, non-fitted bolts should be avoided for structural parts subjected to fatigue loading.

Recommendations for the design class of high strength bolted joints are, for example, given in Eurocode 3 (EN 1993-1-9, 2005), see Table 14.6.

It is noted that, depending on the conditions, partial (safety) factors have to be used; see e.g. Table 14.2.

14.8 FATIGUE DESIGN In the previous sections the fatigue resistance of hollow section joints has been discussed in relation to the geometry. However, when a designer starts with the design process, the geometry is not known and he or she has first to determine the geometry, e.g. for a truss. Here, several steps have to be followed: Step A: For the determination of the geometry, use has to be made of the knowledge obtained in the previous sections. For a good performance of joints, the following points have to be considered to obtain the best fatigue behaviour: - Select joints with high or low β values, and avoid intermediate values. Considering fabrication, β = 0,8 is preferred to β = 1,0.

- Choose the chord diameter- or chord width-tothickness ratio 2γ to be as low as possible; e.g. for tension chords about 20 or lower and for compression chords about 25 or lower.

- Design the braces to be as thin as possible to achieve preferable brace-to-chord thickness ratios τ  0,5 and to minimise welding.

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- Design girders in such a way that the angles θ between braces and chord are preferably about 40.

Considering the above, joints with relatively low stress concentration factors can be achieved. For example, for a K joint of circular hollow sections (Fig. 14.11) with β = 0,8; 2γ = 20; τ = 0,5 and θ = 40: SCF chord  2,1 SCF brace  2,0 Step B: Based on the required lifetime, the number of cycles N has to be determined.

Step C: For the number of cycles N (from B), the geometric stress range Δσgeom can be obtained from the Δσ-N line assuming a certain thickness, see Fig. 14.8.

Step D: Depending on the inspection frequency and the type of structure ("fail-safe" or "non fail-safe") the partial factor M can be determined (Table 14.2).

Step E: Determine the additional multiplication factors C to account for secondary bending moments in chord and braces (Tables 14.4 and 14.5).

Step F: Based on steps A, C, D and E, the allowable nominal stress range for the braces can be calculated:

 nom,brace 

 geom  M C (SCF)

 min  max

and

Step G: The configuration now determined should be checked for ease of fabrication, inspection and the validity range for joints.

If the design satisfies the requirements, the final check can be carried out, now starting from the loading and the known geometry.

The procedure is then as follows: (1) Determine loads and moments in members and from these, the stresses in braces and chord, e.g. assuming pin ended braces to continuous chords if nodal eccentricities between the centrelines of intersecting members at the joints should be included, see Fig. 14.17.

(2) Determine from (1) the nominal stress ranges Δσnom,brace in the brace and Δσnom,chord in the chord.

(3) Multiply the nominal stress ranges by the partial factor γM, the factors C to account for secondary bending moments (step E) and the SCFs to obtain the maximum geometric stress ranges in chord and braces.

(4) Determine the fatigue life (number of cycles) from the Δσ-N line for geometric stress for the relevant thickness.

If the calculated fatigue life meets or exceeds the required fatigue life, the design satisfies the requirements. Otherwise modifications have to be made.

If a spectrum (variable amplitude) loading is acting, the spectrum can be divided into stress blocks and for each stress range the number of cycles to failure can be determined. Using the Palmgren-Miner rule, given in eq. (14.3), will result in the cumulative damage which should not exceed 1,0. It may sometimes be easier to determine first an equivalent constant amplitude stress range with the corresponding number of cycles.

For a particular R ratio, σmax can be determined with: R

If the chord is also subjected to large loads, a larger cross section should be taken to account for these chord loads.

   max   min

   max (1  R) Thus:

 max,nom,brace 

 geom  M C (1  R ) SCF

With this maximum brace stress, the cross section of the braces can be determined and, with the parameters selected under step A, the chord dimensions and the joint layout selected.

181

Table 14.1 Detail categories for hollow sections and simple joints according to Eurocode 3 (EN 1993-1-9, 2005), IIW (1999) and CIDECT (Zhao et al., 2001) Details loaded by nominal normal stresses Detail category m=3

Constructional detail

Description

160

Rolled and extruded products Non-welded elements Sharp edges and surface flaws to be improved by grinding

140

Continuous longitudinal welds Automatic longitudinal welds with no stop-start positions, proven free of detectable discontinuities

71

Transverse butt welds Butt welded end-to-end connection of circular hollow sections Requirements: - Height of the weld reinforcement less than 10% of weld with smooth transitions to the plate surface - Welds made in flat position and proven free of detectable discontinuities - Details with wall thickness greater than 8 mm may be classified two detail categories higher ( 90)

56

Transverse butt welds Butt welded end-to-end connection of rectangular hollow sections Requirements: - Height of the weld reinforcement less than 10% of weld with smooth transitions to the plate surface - Welds made in flat position and proven free of detectable discontinuities - Details with wall thicknesses greater than 8 mm may be classified two detail categories higher ( 71)

71

Welded attachments (non load-carrying welds) Circular or rectangular hollow section, fillet welded to another section Section width parallel to stress direction  100 mm

50

Welded connections (load-carrying welds) Circular hollow sections, end-to-end butt welded with intermediate plate Requirements: - Welds proven free of detectable discontinuities - Details with wall thicknesses greater than 8 mm may be classified one detail category higher ( 56)

45

Welded connections (load-carrying welds) Rectangular hollow sections, end-to-end butt welded with intermediate plate Requirements: - Welds proven free of detectable discontinuities - Details with wall thicknesses greater than 8 mm may be classified one detail category higher ( 50)

40

Welded connections (load-carrying welds) Circular hollow sections, end-to-end fillet welded with intermediate plate Requirements: - Wall thickness less than 8 mm

36

Welded connections (load-carrying welds) Rectangular hollow sections, end-to-end fillet welded with intermediate plate Requirements: - Wall thickness less than 8 mm

182

Table 14.2 Partial factors M according to IIW (1999) and CIDECT (Zhao et al., 2001)

"Fail-safe" structures (redundant)

Non "fail-safe" structures (non-redundant)

Periodic inspection and maintenance Accessible joint detail

M = 1,00

M = 1,25 (1)

Periodic inspection and maintenance Poor accessibility

M = 1,15

M = 1,35

Inspection and access

(1)

In Eurocode 3 (EN 1993-1-9): M = 1,15

Table 14.3 See next page

Table 14.4 Multiplication factors to account for secondary bending moments in CHS lattice girder joints (EN 1993-1-9, 2005), IIW (1999) and CIDECT (Zhao et al., 2001)

Type of joint Gap joints Overlap joints

Chords

Verticals

Diagonals

K

1,5

-

1,3

N

1,5

1,8

1,4

K

1,5

-

1,2

N

1,5

1,65

1,25

Table 14.5 Multiplication factors to account for secondary bending moments in RHS lattice girder joints (EN 1993-1-9, 2005), IIW (1999) and CIDECT (Zhao et al., 2001)

Type of joint Gap joints Overlap joints

Chords

Verticals

Diagonals

K

1,5

-

1,5

N

1,5

2,2

1,6

K

1,5

-

1,3

N

1,5

2,0

1,4

Table 14.6 Fatigue classes for various pretensioned bolted joints according to Eurocode 3 (EN 1993-1-9)

Constructional detail

Detail category

m

Bolts (d  30 mm) loaded in tension based on tensile stress area

50

3

One-sided, slip-resistant joints, e.g. splices or cover plates (based on stress in gross section)

90

3

Double-sided, slip-resistant joints, e.g. splices or cover plates (based on stress in gross section)

112

3

183

Table 14.3 Detail categories for lattice girder joints based on nominal stresses according to Eurocode 3 (EN 1993-1-9, 2005) and CIDECT (Zhao et al., 2001)

Detail categories for lattice girder joints based on nominal stresses Detail category (1) (2) m=5 90

Constructional details

Description

t0/ti  2,0 Circular hollow sections K and N gap joints

45

t0/ti = 1,0

71

t0/ti  2,0

Rectangular hollow sections K and N gap joints

36

t0/ti = 1,0

Requirements:  -0,5(b0 - bi)  g  1,1(b0 - bi)  g  2t0

71

t0/ti  1,4

56

t0/ti = 1,0

71

t0/ti  1,4

50

K overlap joints Requirements:  30%  Ov  100%

N overlap joints Requirements:  30%  Ov  100%

t0/ti = 1,0

General requirements 35  θi  50 4  t0  8 mm 4  ti  8 mm

b0  200 mm d0  300 mm

0,40  bi/b0  1,0 0,25  di/d0  1,0

(b0/t0)(t0/ti)  25 (d0/t0)(t0/ti)  25

(3) (3)

-0,5h0  e  0,25h0 -0,5d0  e  0,25d0

Out-of-plane eccentricity:  0,02b0 or  0,02d0 Fillet welds are permitted for braces with wall thicknesses  8 mm (1) (2) (3)

Note that the detail category is based on the stress range in the braces.

For intermediate t0/ti values, use linear interpolation between nearest detail categories.

This formulation is based on test data and slightly deviates from the IIW (1999) recommendations.

184

Stress increase due to weld toe effects Stress increase due to weld geometry

Brace wall

Maximum geometrical peak stress

Geometrical stress weld notch

a

b Chord wall

Fig. 14.1 Peak stress due to weld discontinuity

nominal peak in brace

peak in chord

Fig. 14.2 Geometrical stress distribution in an axially loaded X joint of circular hollow sections

Stress R>0



R=0 R = -1

 

Fig. 14.3 Stress range σ and stress ratio R 185

1000 Detail category

Stress range  (N/mm2)

500

Constant amplitude fatigue limit

100 50 m=3

10 104

5

105

5

106

107

5

5

108

Number of stress cycles N

Fig. 14.4 σ – N curves for classified details and constant amplitude loading (IIW, 1999; CIDECT, 2001; EN 1993-1-9, 2005)

1000 Detail category

Stress range  (N/mm2)

500

Constant amplitude fatigue limit Cut-off limit 100 50

m=3

m=5 10 104

5

105

5

106 2

5

107

5

108

Number of stress cycles N

Fig. 14.5 σ – N curves for classified details and variable amplitude loading (IIW, 1999; CIDECT, 2001; EN 1993-1-9, 2005)

186

Brace

Brace

Saddle

Crown Chord

Chord

Fig. 14.6 Locations of extrapolation of geometric peak stresses for a T joint

fy =240 N/mm2

b

a 30

20

A

30

A

B

c B

A

-240

B 240

average = 160 N/mm2 theoretical stress at max

 = 120 N/mm2

residual stress at 

actual stress at max

Fig. 14.7 Plate with a hole

For 103 < N < 5 x 106 , , log(N) log(16/t) log(geom) = 1/3 [(12,476 - log(N)] + 0,06

 geom ( t )

For 5 x 106 < N < 108 , log(geom) = 1/5 [(16,327 - log(N)] + 0,402 , log(16/t)

 geom ( t  16 mm )

 16     t 

0,402

Geometrical stress range  (N/mm2)

1000 500

100 t=4 t=8 t = 12,5 t = 16 t = 25

50

(t in mm)

10 104

105

106

5 107

108

109

Number of stress cycles N

Fig. 14.8 Basic σgeom - N design curves for the geometrical stress method for hollow section joints (IIW, 1999; CIDECT, 2001)

187

2 = 15

24

2 = 30

2 = 50

= 0,5 = 1,0

Brace saddle 20 = 1,0

SCF

16 12

= 1,0

8

= 1,0

4 0

0

0,2

0,4

0,6

0,8

1,0

5



Brace crown

SCF

4

40

3 2

Chord saddle

36

1 = 1,0

0

32

0

0,6

0,8

1,0

Brace crown independent of  = 1,0

24

SCF

0,4



28 8 7

20

Chord crown = 1,0

6 16

SCF

= 1,0

12

= 1,0

5

= 1,0

4 3

8

2

4 0

0,2

1 0

0,2

0,4

0,6

0,8

0

1,0

0

0,2

0,4

0,6

0,8





Fig. 14.9 SCFs for axially loaded circular hollow section X joints (IIW, 1999; CIDECT, 2001)

188

1,0

32 Brace

28 24

Chord 2 = 30

20

2 = 50

SCF/0,75

2 = 15

= 0,25 – 1,0 Symbol size ~ 

16

12 20

8

SCF

16

4

12

0

0 Line B

8

0,2

0,4

0,2

0,4



0,6

0,8

1,0

0,6

0,8

1,0

32 4 28 0

0 Line A, E

0,2

0,4



0,6

0,8

1,0 24

20

SCF/0,75

20

SCF/0,75

16

16

12

12

8

8

4

4

0

0 Line D

0,2

0,4



0,6

0,8

1,0

0

0 Line C



Notes: - For a T joint, the effect of chord bending due to the axial brace load should be separately included in the analysis.

- For fillet welded joints: multiply SCFs for the brace by 1,4.

- A minimum SCF = 2,0 is recommended to avoid crack initiation from the root.

Fig. 14.10 SCFs for butt welded T and X joints of square hollow sections, loaded by an axial force on the brace (parametric formulae compared with FE calculations (Van Wingerde, 1992))

189

2 = 15

2 = 30

2 = 50

= 0,5 = 1,0 8 7 6

6

5

5

4

4

3

3

2

2

1

1

0

0 0

0,2

0,4

0,6

0,8

Brace saddle / crown

7

SCF

SCF

8

Chord saddle / crown

1,0

0

0,2

0,4

0,6

0,8

1,0





4

4

3

3

SCF/0,24

SCF/0,19

Fig. 14.11 Maximum SCFs for axially loaded K joints of circular hollow sections with gap g = 0,1d0

2

1 0

2

1

0

0,2

0,4

0,6

0,8

0

1,0



Line C

0

0,2

Line D

0,4

0,6

0,8

1,0



Fig. 14.12 SCFs for T and X joints of rectangular hollow sections (chord locations C and D of Fig. 14.6 only), loaded by an in-plane bending moment or an axial force on the chord

=0,5

Max. SCF

4,7

2= 24 = 0,5

7,0

10,6

Fig. 14.13 Effect of multiplanar loading on the SCF

190

F

F

F

F

F

F

a1

b1

c1

Good

a2

Bad

b2

ď Ś circular hollow section

Fig. 14.14 Examples of bolted joints (with deformed flanges) in tension

Fig. 14.15 Recommended bolted ring flange joint for fatigue loading

191

c2

a. No preload

b. Preloaded

c. Preloaded with slip

Fig. 14.16 Possible stress distributions in bolted shear joints

Noding condition for most overlap joints

Pin

Extremely stiff members

Extremely stiff members

Noding condition for most gap joints

Fig. 14.17 Plane frame joint modelling assumptions

192

15. DESIGN EXAMPLES

Maximum chord force N0 = -1148 kN (compression).

15.1 UNIPLANAR TRUSS OF CIRCULAR HOLLOW SECTIONS

Possible section sizes are shown in Table 15.1, along with their compressive resistances.

In this example (Wardenier et al., 2008a), the design principles of Chapter 6 are illustrated as well as the joint design methods.

• Truss layout and member loads A Warren type truss with low brace member angles is chosen to limit the number of joints, see Fig. 15.1.

The trusses are spaced at 12 m intervals and the top chord is considered to be laterally supported at each purlin position at 6 m centre-to-centre. The spanto-depth ratio is 15, which is approximately the optimal limit considering service load deflections and overall costs.

For this example, hot finished members are selected and the member resistances are calculated according to Eurocode 3 (EN 1993-1-1, 2005), assuming a partial factor M = 1,0 (this factor may be different for various countries).

From a material point, the sections Ø244,5 x 5,6 and Ø219,1 x 7,1 are most efficient. However, for the supplier considered in this example, these two dimensions are not available from stock (only deliverable from the mill). These dimensions can only be used if a large quantity is required, which is assumed in this example.

Bottom chord For the capacities of the joints, it is best to keep the tension chord as compact and stocky as possible.

However, to allow gap joints and to keep the eccentricity within the limits, a larger diameter may be needed. Possible section sizes are given in Table 15.2.

Diagonals Try to select members (see Chapters 6 and 8) which satisfy:

fy 0 t 0 fyi t i

 2,0

The factored design load P from the purlins including the weight of the truss has been calculated as P = 108 kN. A pin jointed analysis of the truss gives the member forces shown in Fig. 15.2.

i.e.

• Design of members

For the braces loaded in compression, use an effective buckling length ℓb (see Chapter 2) of:

In this example, the chords are made from steel S355 with a yield stress of 355 N/mm2 and the braces from steel S275 with a yield stress of 275 N/mm2.

 b  0,75   0,75 2,4 2  3,0 2  2,88 m

For member selection, use can be made of either member resistance tables with the applicable effective length or the applicable strut buckling curve. The availability of the member sizes selected has to be checked. Since the joints at the truss ends are generally decisive, the chords should not be too thin walled. As a consequence, a continuous chord with the same wall thickness over the whole truss length is often the best choice.

Top chord Use a continuous chord with an effective in-plane and out-of-plane buckling length ℓb (see Chapter 2) of: ℓb = 0,9 x 6000 = 5400 mm

355  7,1  2,0 or ti  4,5 mm 275t i

The possible sizes for the compression diagonals are given in Table 15.3 and for the tension diagonals in Table 15.4.

Member selection The number of sectional dimensions depends on the total tonnage to be ordered. In this example, for the braces only two different dimensions will be selected.

Comparison of the members suitable for the tension members and those suitable for the compression members shows that the following sections (see Fig. 15.3) are most convenient:

- Braces:

Ø139,7 x 4,5 Ø 88,9 x 3,6 - Top chord: Ø219,1 x 7,1 - Bottom chord: Ø193,7 x 6,3

193

(These chord sizes allow gap joints; no eccentricity is required.)

Right side:

It is recognized that the d0/t0 ratios of the chords selected are high. This may give joint strength problems in joints 2 and 5. The check for joint resistance is given in Table 15.5.

n

• Joint strength checks, commentary and revision General In Table 15.5, all joints are treated as K joints, initially neglecting the additional X joint action in the joints 2, 3 and 4. In this case, there should be a larger margin between the design and the acting efficiency as will be shown in the more detailed, accurate evaluation given below.

Joint 1 If, in joint 1, a gap g = 2t0 is chosen between the cap plate and the brace, nearly no eccentricity exists for the bolted joint of the cap plate. This joint is checked as a K(N) joint (see Table 15.5) because the load transfer is similar as in an N joint (the reaction in the cap plate is upwards and the diagonal loading is downwards). In the example, β is conservatively based on the diameter of the brace.

Joint 2 Table 15.5 shows that the strength of joint 2 with g = 12,8t0 (and eccentricity e = 0 mm) is not sufficient.

The easiest way to obtain adequate joint strength will be to decrease the gap from 12,8t0 to 3t0, resulting in a larger CK = 0,39 and a slightly lower Qf. However, this means that a (negative) eccentricity of e = 28 mm is introduced resulting in an eccentricity moment of: M0 = (878 - 338) x 28 x 10-3 = 15,12 kNm.

Since the length and the stiffness El of the top chord members between joints 1-2 and 2-3 are the same (see Fig. 15.3), this moment can be equally distributed over both members, i.e. both members have to be designed additionally for M0,Ed = 7,56 kNm. Including the chord bending moment effect gives the following values for the chord stress parameter n in the connecting face at the left and right side of the joint.

Left side:

n

N0,Ed A 0 f y0



M0,Ed Mpl,0,Rd



N0,Ed A 0 f y0



M0,Ed Mpl,0,Rd



 878 - 7,56  4728  0,355 113,3

 -0,52  0,067  -0,59 The right side with n = -0,59 is decisive, resulting in Qf = 0,80. In combination with CK, this gives a joint strength efficiency (see Table 15.5):

N1,Rd A 1 f y1

 0,84 

N1,Ed A 1 f y1

 0,82

(o.k.)

The chord members between joints 1-2 and 2-3, which are in compression, should also be checked as a beam-column for buckling. From these, chord member 2-3 is most critical. This check depends on the national code to be used. In general, the criterion to be verified has the following format: N0,Ed  A 0 f y0

k

M0,Ed Mpl,0,Rd

 1,0

where: χ reduction factor for column buckling (see Table 15.1 for the values of χ for the possible chord sections) k amplification factor for second order effects depending on slenderness, section classification and moment diagram (in this case use a triangular shape) Mpl,0,Rd plastic moment resistance (Wpl,0 fy0) of the chord (class 1 or 2 sections)

N0,Ed  A 0 f y0

k

M0,Ed Mpl,0,Rd



878 7,56 k 1189 113,3

 0,74  0,067k  1,0 Independent of the code used, this will not be critical.

More accurate calculation based on combined K and X joint actions As already mentioned, this joint actually has a combination of K joint and X joint actions and should be substituted by a K joint and an X joint, see Chapter 6.2.1.

 338 7,56  4728  0,355 113,3

 -0,20  0,067  -0,13 194

-108 -108

0

-338

=

-878

-338

n = -0,28 - 0,067 = -0,35 -474

+ -259

-432

-404 259

Thus, for

-173

259

Q f  (1  0,35 )0,45 -0,250,64   0,88

Joint 2 – K joint action:

n

N0,Ed A 0 fy0



 404  0,24 compression 4728  0,355

For

Q f  (1  0,24 )0,25  0,93

CK = 0,39 (see Fig. 8.19)

N1,Rd 0,88  0,20  2,04   0,57 A 1 fy1 0,625 Due to acting load:

For brace 1:

N1,Ed 173   0,33 A 1 fy1 525,5

N1,Rd 0,93  0,39  2,04   0,82  0,97 A 1 fy1 0,625

Hence, the utilization ratio for X joint action is:

Due to acting load:

N1,Ed



N1,Ed N1,Rd

259  0,49 525,5

A 2 f y2

 0,39  2,55 

Note: Based on the check as a K joint only (see Table 15.5 for the evaluation of joint 2 with g/t0 = 3,0), this N 0 ,82 utilization ratio was 1,Ed   0 ,98 which would N1,Rd 0 ,84

0,93  1,29  1,0 0,625

have been about 10% over-optimistic.

The actual efficiency is: N2,Ed A 2 f y2

 0,98  1,0

(o.k.)

Joint 2 – X joint action:

n

N0,Ed A 0 fy0



(not o.k.)

Thus, the joint is still not o.k.

For brace 2: N2,Rd

0,33  0,58 0,57

0,50 + 0,58 = 1,08 > 1,0

0,49   0,50 0,97

N1,Rd



The combined acting efficiencies for brace 1 due to K joint and X joint action are:

Hence, the utilization ratio for K joint action is:

N1,Ed

d1 139,7   0,64 and 2γ = 30,9: d0 219,1

CX = 0,20 (see Fig. 8.18)

For the modified configuration with g = 3t0, β = 0,52 and 2γ = 30,9:

A 1 f y1

d1 139,7   0,64 : d0 219,1

 474  0,28 compression 4728  0,355

Including the above mentioned bending moment in the chord M0,Ed gives:

Further decreasing the gap will not help because the design efficiency as a K joint is already close to 1,0.

Hence, the effect of the X joint action should be decreased. This can be done as follows: (1) By using a section for brace 1 with about the same cross sectional area but a lower thickness (e.g. Ø 168,3 x 3,6), which increases the design efficiency. However, this increases the number of section types for the braces to three.

(2) By increasing the thickness of the top chord and choosing Ø 219,1 x 8,0.

An additional type of section can increase the costs, but increasing the chord thickness also increases

195

material costs. The choice will be made after checking the other joints.

Thus, for

Joint 3

Q f  (1  0,60 )0,45 -0,250,64   0,77

-108 -108 -878

0 -1148

-878

-1014

+ -86

-259 -259

-134

= 86

For

-173

86

n

A 0 fy0

N1,Rd 0,77  0,20  2,04   0,50 A 1 fy1 0,625

 134   0,08 compression 4728  0,355

Due to acting load:

Q f  (1  0,08 )0,25  0,98

N1,Ed 173   0,33 A 1 fy1 525,5

For g = 12,8t0 with β = 0,52 and 2γ = 30,9: CK = 0,34 (see Fig. 8.19) N1,Rd

 0,34  2,04 

A 1 f y1

d1 139,7   0,64 and 2γ = 30,9: d0 219,1

CX = 0,20 (see Fig. 8.18)

Joint 3 – K joint action:

N0,Ed

d1 139,7   0,64 : d0 219,1

Hence, the utilization ratio for X joint action is:

0,98  0,82  0,89 0,625

N1,Ed N1,Rd

Due to acting load:



0,33  0,66 0,50

The combined acting efficiencies due to K joint and X joint actions are:

N1,Ed 86   0,16 A 1 f y1 525,5

0,18 + 0,66 = 0,84 < 1,0

(o.k.)

Hence, the utilization ratio for K joint action is:

Thus, the criteria are satisfied.

N1,Ed

Note: Based on the check as a K joint only (see Table 15.5 for the evaluation of joint 3), this utilization ratio N 0 ,49  0 ,72 which would have been about was 1,Ed  N1,Rd 0 ,68



N1,Rd

0,16  0,18 0,89

For brace 2: N2,Rd A 2 f y2

14% over-optimistic.

 0,34  2,55 

0,98  1,29  1,0 0,625

Joint 4 -108

The actual efficiency is: N2,Ed A 2 f y2

-1148 -1148

-1148

= -86

-86

 0,32  1,0

Joint 3 – X joint action:

n

-1148

N0,Ed A 0 fy0



 1014  0,60 compression 4728  0,355

-86

(o.k.)

-86

54 54

At joint 4, a site joint will be made consisting of two plates which also transfer the purlin load to the chord.

This means that joint 4 behaves as two N joints.

Assuming no eccentricity at the bolted joint and cap plates of 15 mm, the gap between the toe of the brace and the cap plate will be (see eq. (6.1)):

196

resulting in a joint according to Fig. 13.1 with 13 bolts Ø 22 – 10,9 with an end plate thickness of 20 mm (fy = 355 N/mm2) for the bottom tensile chord joint. To avoid displacements in the joint it is recommended to pretension the bolts. For fatigue loaded joints, the bolts have to be pretensioned.

 219,1 sin(2  38,7) 88,9  g  0,5    15 2 sin 38,7  sin 38,7  2

 50,6 mm  7,1t 0

The check in Table 15.5 shows that the joint is o.k.

Evaluation The joint checks showed that joint 2 is not o.k.

Considering the options mentioned, in this example, the top chord section will be changed from Ø 219,1 x 7,1 to Ø 219,1 x 8,0. Recalculating joint 2 for e = 0 (with g = 12,8t0) gives a utilization ratio of 0,49 for K joint action and 0,45 for X joint action, thus a combined utilization of 0,94 < 1,0.

Compared to the selected members in Fig. 15.3, only the top chord is changed to Ø 219,1 x 8,0 and all joints can be made without any eccentricity.

For the top chord joint, the compression loading is transferred through contact pressure. The number of bolts required depends on the erection loads which can be tensile, and the national code requirements with regard to the minimum joint strength related to the member tensile strength.

15.2 UNIPLANAR TRUSS OF SQUARE HOLLOW SECTIONS

The above extensively worked-out example shows that checking as a K joint only is much faster than using the combined K and X joint actions.

In Packer et al. (2009a), a truss with the same configuration and loading has been designed with square hollow sections, all with a yield stress of 355 N/mm2. In principle, the approach is similar, resulting in the member dimensions shown in Fig. 15.7.

A fast alternative would be to check as a K joint with C and Qf factors selected between the values for K and X joints depending on the contributions.

15.3 MULTIPLANAR TRUSS (TRIANGULAR GIRDER)

• Purlin joints

Depending on the type of purlins, various purlin joints are possible. If corrosion will not occur, a cut-out of a channel section welded on top of the chord at the purlin support location and provided with bolt stubs gives an easy support, see Fig. 15.6. Table 8.3 provides evidence for the design of plate-to-tube joints.

The joints in Table 8.3 are not exactly similar to those between open U sections and a CHS chord but the capacity may be based on the design resistance for an RHS-to-CHS joint. Since no cross plates are present, only the sides are effective; therefore a very conservative reduction factor to be applied is h1/(h1+b1).

For the purlin joint at the centre, another alternative has to be used to allow a site bolted truss joint. If the top chord parts are provided with cap plates, a T-stub for purlin support can be fitted in between the cap plates.

• Site bolted flange joints

This book does not give complete design procedures for bolted flange joints. However, in Wardenier et al.

(2008a) this example is worked out further, e.g.

For an easy comparison, for this example a multiplanar truss (Fig. 15.8) is chosen with the side elevation dimensions equivalent to the uniplanar truss discussed in Section 15.1.

• Member loads

The member loads can be determined in a similar way as for the uniplanar truss, assuming pin ended members.

The load in the bottom chord follows by dividing the relevant moment by the girder depth. Since two top chords are used, the load at the top has to be divided by 2. The loads in the braces follow from the shear forces V in the girder (Fig. 15.9).

The top chords should be connected in the top plane for equilibrium of loading, see Fig. 15.10. This can be achieved by a bracing system which connects the loading points. Connection of the loading points only in the horizontal plane results in a triangular truss which has no torsional rigidity. A combination with diagonals in the horizontal plane, thereby completing a Pratt type truss in the horizontal plane, gives torsional resistance. It is also possible to use the purlins or the roof structure as the connecting parts

197

between the loading points. Once the loads in one plane are known, the design can be treated in a similar way as for uniplanar trusses.

• Joints

The joints can also be treated in a similar way as for uniplanar joints, however, taking account of the larger chord loads. This means a larger Qf reduction factor for the joints with the bottom chord. From a fabrication point of view, it is better to avoid overlaps of the intersecting braces from both planes. Sometimes this may result in an eccentricity in the two planes, also called an offset (see Fig. 15.11).

The offset has to be incorporated in member design and joint capacity verification. For the chords, the moments due to this offset have to be distributed over the chord members, affecting the chord stress function Qf and hence, the joint capacity.

• Design calculation

the purlins as connection between the top chords. A simple bolted connection, as given in Fig. 15.6, can easily be designed to transfer the shear load of 54 kN. However, in this way the truss has no torsional rigidity and cannot act as horizontal wind bracing for the roof.

If this is required, braces between the top chords should be used.

• Joint strength check

As mentioned, the initial difference with the joint strength checks for the uniplanar truss in Section 15.1 is that the effect of noding eccentricity has to be incorporated. A joint without any eccentricity would result in an overlap of the braces in the two planes, see Fig 15.14a. To allow welding, an out-of-plane gap of 22,5 mm is chosen which results in an eccentricity of 50 mm (in-plane eccentricity = 43 mm  0,25d0). As a consequence, the in-plane gap increases, resulting in slightly lower CK values.

Besides the joint capacity checks carried out in Section 15.1, the multiplanar joint has to be checked for chord shear, see Table 8.4. The joint with maximum shear in the gap is joint 5 with:

Assume P = 187 kN (at ultimate limit state).

This means that the loads acting in the side planes of the triangular truss (see Fig. 15.12) are:

Vgap,0,Ed = 2,5 P = 2,5 x 187 = 467,5 kN Further (see Fig. 15.2):

P  108 kN 2 cos 30 o

Ngap,0,Ed = 0,5 x (2 x 675) = 675 kN

This is equal to the purlin loads used in the design example for the uniplanar truss in Section 15.1. As a consequence, the top chord and the diagonals can be the same to those for the uniplanar truss, provided the same steel grades are used (see Fig. 15.13).

Mgap,0,Ed = 675 x 0,05 = 33,75 kNm

For the bottom chord, the required cross section should be twice that required for the uniplanar truss, i.e. Ø 219,1 x 11,0 with A0 = 7191 mm2 and Wpl,0 = 476,8 x 103 mm3. (This section may have a longer delivery time.)

A detailed check of the members is already given in Section 15.1 and is the same here. However, the eccentricity moment should be taken into account both for member design and joint strength verification (i.e. the effect of the chord moment on the chord stress factor Qf).

The braces between the top chords are determined by the horizontal loads of 54 kN at each purlin support or by loads resulting from unequally distributed loading on the roof. Since transport is simpler for V-trusses than for triangular trusses, it is also possible to use

198

Npl,0,Rd  A 0 f y0  7191 0,355  2552 kN Ngap,0,Ed Npl,0,Rd



675  0,26 2552

Vpl,0,Rd  0,58 f y 0

2A 0 2  7191  0,58  0,355   

 943 kN Vgap,0,Ed Vpl,0,Rd



467,5  0,50 943

Mpl,0,Rd  Wpl,0 f y0  476,8  10 3  0,355  10 3  169,3 kNm

Mgap,0,Ed Mpl,0,Rd



33,75  0,20 169,3

A conservative, linear interaction gives:

Ngap,0,Ed Npl,0,Rd



Vgap,0,Ed Vpl,0,Rd



Mgap,0,Ed Mpl,0,Rd

Thus, for b0 = 150 mm and  



b1  b 2 2  120   0,8 : 2b 0 2  150

0,5(1  0,8)  g/150  1,5(1  0,8)

0,26  0,50  0,20  0,96  1,0

or:

The exact interaction is more complicated (Wardenier, 1982). In general, this chord shear check becomes critical for larger β ratios.

15.4 MULTIPLANAR TRUSS OF SQUARE HOLLOW SECTIONS

15  g  45 The eccentricity (e) corresponding to the minimum gap of 15 mm, giving the minimum value for e, can be calculated with:  h1  sin θ1 sin θ 2 h 0 h2   g   e   2 sinθ 2 sin θ 1 2   sin θ1  θ 2  2

The approach for a multiplanar truss of square hollow sections is similar to that in Section 15.3. Generally, the braces in the two side planes are connected at different faces of the bottom chord, giving no problems with out-of-plane overlaps as would be possible for circular hollow sections.

 120  sin 38,7 sin 38,7 150   15    8 mm sin 38,7 2   sin 38,7  38,7  b i 120 150  0,34   0,8  0,1  0,01 6,3 b 0 150

Working out the example used in Section 15.3 for square hollow sections (all members with fy = 355 N/mm2), results in the same dimensions for the top chords and braces as those given in Fig. 15.7. For the bottom chord, a section with twice the cross sectional area has to be selected.

b1 120   24  class 2 limit (= 33,9) and  40 (o.k.) t1 5 b 0 150   23,8 t0 6,3

15  23,8  40

(o.k.)

b1  b 2 2  120   1,0 0,6  1,0  1,3 2b i 2  120

(o.k.)

θi = 38,7  30

(o.k.)

15.5 JOINT CHECK USING THE JOINT RESISTANCE FORMULAE The joints in the previous examples have been checked using the efficiency parameters Ce from the design graphs. However, the joints can also be checked, using the formulae given in Chapters 8 and 9. Here, as an example, only joint 5 of the uniplanar (RHS) truss shown in Fig. 15.7 will be checked using the resistance formulae.

The dimensions of the sections and the yield stresses are presented in Fig. 15.7. All other information remains similar to that given in Figs. 15.1 and 15.2.

Since all members are square hollow sections, Table 9.2 applies.

(o.k.)

• Check for chord plastification

Ni,Rd  14   0,3

f y0 t 02 sin θ i

Qf

For tension:

• Check of validity range

Q f  (1  n ) 0,10



b1  b 2 2  120   0,8 2b 0 2  150



150  11,9 2  6,3

According to Table 9.1, the gap g has to satisfy:

0,5(1  )  g/b 0  1,5(1  )

199

n

N0,Ed Npl,0,Rd



M0,Ed Mpl,0,Rd



675 675  8  3480  0,355 192000  0,355

As (16 Ø25) = 16  Ac =

 0,54  0,08  0,46

7854

s 

0,355  6,3 2  14  0,8  (11,9)0,3   0,94 sin 38,7 = 498,7 kN > Ni,Ed = 432 kN

  406,4 2  10013  7854  111850 mm 2 4

• Reinforcement ratio

Q f  (1  0,46 ) 0,10  0,94

Ni,Rd

  25 2  7854 mm 2 4

(o.k.)

The approach using design charts, given in an example in CIDECT Design Guide No. 3 (Packer et al., 2009a), results in approximately the same efficiency.

If rectangular hollow sections had been used, the joint resistance check would have been considerably more complicated, which is evident if Table 9.1 is compared to Table 9.2. The extra checks involve: - Chord shear - Local brace failure - Chord punching shear

Concrete C20 with c = 1,5 CHS S275 with a = 1,0 Reinforcement S500 with s = 1,15

= 6,6% > 6%

The ratio of reinforcement ρs has to be limited to 6% for the calculation (see Section 4.3.1). This may be achieved by: - Using 14 Ø25 - Considering only reinforcing bars which lie in the most favourable position of the section for bending so that ρs 6% (neglect two centre bars). In the current design example, this option is selected.

- Reducing the diameters of the reinforcing bars to such an extent that ρs = 6%

15.6 CONCRETE FILLED COLUMN WITH REINFORCEMENT Here, the axial compressive capacity is calculated for a concrete filled circular hollow section with a cross section and reinforcement as shown in Fig. 15.15 and with the factors according to Table 4.1.

  406,4 2  10013 4

As 

6  7854  7140 mm 2 6,6

Ac 

  406,4 2  10013  7140  112564 mm 2 4

• Check of concrete filled column

Npl,Rd  A a f yd  A c fcd  A s fsd  10013  275  7140  435  112564  13,3  7357 kN 0,2 <  

A a f yd Npl,Rd



10013  0,275  0,37 < 0,9 7357

• Assumptions for the analysis ℓb NEd NG,Ed φt

= 3,6 m = 6000 kN = 0,5NEd = 3,0



Npl,Rk Ncr,eff

Npl,Rk  A a f yk  A c fck  A s fsk  10013  275  7140  500  112564  20

• Strength

 8575  10 3 N  8575 kN

fyd = 275/1,0 = 275 N/mm2 fsd = 500/1,15 = 435 N/mm2 fcd = 20/1,5 = 13,3 N/mm2 • Cross sectional areas

  E

Ncr,eff 

  (EI)eff  2b

(EI)eff = Ea Ia + 0,6 Ec,eff Ic + Es Is

Aa (Ø 406,4 x 8,0) =  406,4  8,0   8,0  10013 mm 2

200

(o.k.)

E c,eff 

E cm 30000   12000 N/mm 2 NG,Ed 1  0,5  3 1 t NEd

E a Ia  2,1 10 5  19870  10 4  41727  10 9 Nmm 2

 406,4  2  84 64  8210  10 9 Nmm 2

0,6 E c,eff Ic  0,6  12000 

Neglecting the two centre bars gives: 7854 ( 4  59,3 2  4  109,6 2 16  4  143,2 2  2  155,0 2 )  19812  10 9 Nmm 2

E s Is  2,1 10 5 

(EI)eff = 41727 + 8210 + 19812 = 69749 kNm2 Ncr,eff 



   69749  53117 kN 3,6 

Npl,Rk Ncr,eff



8575  0,40 53117

 The reinforcement ratio ρs> 3%, thus use curve "b" (see Fig. 2.3): χ = 0,93

NEd  6000 kN   pl,Rd  0,93  7357  6842 kN (o.k.)

• Check for local buckling

235 d 406,4   50,8  90 2  90   76,9 275 t 8,0

(o.k.)

Note: The increase in compression capacity caused by confinement effects is here neglected, but would give an increase in capacity due to the low  ratio.

201

Table 15.1 Possible section sizes for top (compression) chord

fy0 (N/mm2)

355

(1)

N0 (kN)

ℓb (mm)

Possible sections (mm)

A0 (mm2)

d0/t0



5400

Ø 193,7 x 10,0 Ø 219,1 x 7,1 Ø 219,1 x 8,0 Ø 244,5 x 5,6 Ø 244,5 x 6,3

5771 4728 5305 4202 4714

19,4 30,9 27,4 43,7 38,8

1,09 0,94 0,95 0,84 0,84

-1148

(1)

χ (1)

χ fy0 A0 (kN)

0,61 0,71 0,71 0,78 0,78

1245 1189 1329 1159 1298

χ (1)

χ fyi Ai (kN)

Buckling curve "a" of Eurocode 3 (EN 1993-1-1, 2005).

Table 15.2 Possible section sizes for bottom (tension) chord

fy0 (N/mm2)

N0 (kN)

Possible sections (mm)

A0 (mm2)

d0/t0

fy0 A0 (kN)

355

1215

Ø 168,3 x 7,1 Ø 177,8 x 7,1 Ø 193,7 x 6,3

3595 3807 3709

23,7 25,0 30,7

1276 1351 1317

Table 15.3 Possible section sizes for compression diagonals

fyi (N/mm2)

275

(2)

Ai (mm2)



2,881

Ø 168,3 x 3,6 Ø 139,7 x 4,5

1862 1911

0,57 0,69

0,90 0,85

462 448

2,881

Ø 114,6 x 3,6

1252

0,85

0,77

266

546

1,08

0,61

92

ℓb (m)

-432 -259 -86 (1)

Possible sections (mm)

Ni (kN)

2,881

Ø 88,9 x 2,0

(2)

(1)

Buckling curve "a" of Eurocode 3 (EN 1993-1-1, 2005).

Wall thickness is rather small for welding, outside the validity range.

Table 15.4 Possible section sizes for tension diagonals

fyi (N/mm2) 275

Ni (kN)

Possible sections (mm)

Ai (mm2)

fyi Ai (kN)

432

Ø 133,3 x 4,0

1621

445

259

Ø 88,9 x 3,6

964

265

86

Ø 48,3 x 2,3

332

91

202

Table 15.5 Joint strength check, assuming K joint action only

Member sizes Joint

Joint parameters

Chord load

Chord (mm)

Braces (mm)

β

d0/t0

g/t0

n

1

Ø 219,1 x 7,1

Plate Ø 139,7 x 4,5

0,64

30,9

2,0

-0,20

2

Ø 219,1 x 7,1

Ø 139,7 x 4,5 Ø 88,9 x 3,6

0,52

30,9

12,8

-0,52

2a

Additional analysis of joint 2 with g/t0 = 3,0 and e = -28 mm

0,52

30,9

3,0

-0,59

3

Ø 219,1 x 7,1

Ø 139,7 x 4,5 Ø 88,9 x 3,6

0,52

30,9

12,8

-0,68

4

Ø 219,1 x 7,1

Ø 88,9 x 3,6 Ø 88,9 x 3,6

0,41

30,9

7,1

-0,68

5

Ø 193,7 x 6,3

Ø 139,7 x 4,5 Ø 139,7 x 4,5

0,72

30,7

2,9

0,51

6

Ø 193,7 x 6,3

Ø 88,9 x 3,6 Ø 139,7 x 4,5

0,59

30,7

9,4

0,82

7

Ø 193,7 x 6,3

Ø 88,9 x 3,6 Ø 88,9 x 3,6

0,46

30,7

15,8

0,92

Actual efficiency Joint

Ni,Ed A i f yi

Joint strength efficiency CK

fy0 t 0 f yi t i 2,04

Check

Qf

1 sin i

d1  d2 2 di

A i f yi

0,95

1,60

> 1,0

> 1,0

o.k.

Ni,Rd

Ni,Rd  Ni,Ed

1

-0,82

2

0,82 0,98

0,34

2,04 2,55

0,83

1,60

0,82 1,29

0,76 > 1,0

not o.k. o.k.

2a

0,82 0,98

0,39

2,04 2,55

0,80

1,60

0,82 1,29

0,84 > 1,0

o.k. o.k.

3

0,49 0,32

0,34

2,04 2,55

0,75

1,60

0,82 1,29

0,68 > 1,0

o.k. o.k.

4

0,32 0,32

0,35

2,55 2,55

0,75

1,60

1,0 1,0

> 1,0 > 1,0

o.k. o.k.

5

0,82 0,82

0,41

1,81 1,81

0,87

1,60

1,0 1,0

> 1,0 > 1,0

o.k. o.k.

6

0,98 0,49

0,37

2,26 1,81

0,71

1,60

1,29 0,82

> 1,0 0,62

o.k. o.k.

7

0,32 0,32

0,32

2,26 2,26

0,60

1,60

1,0 1,0

0,69 0,69

o.k. o.k.

0,41

Note: Joints 1-4 discussed in detail in text.

203

Bolted joint

L = 6 x 6000 = 36000 mm tan θ = 2,4 / 3 = 0,8 θ = 38,7

Fig. 15.1 Truss layout

-

-

-

-

-

-

Fig. 15.2 Truss member axial loads

Ф 219,1 x 7,1

Ф 88,9 x 3,6

Ф 139,7 x 4,5

Ф 193,7 x 6,3

Fig. 15.3 Initially selected member dimensions and joint numbers for CHS truss.

(In the final design, the top chord is changed to Ø 219,1 x 8,0)

t0

2t0

Fig. 15.4 Joint 1 204

t0 -338 kN

-878 kN

M0,Ed = 7,56 kNm

Ф 219,7 x 7,1 Ф 139,7 x 4,5

Ф 88,9 x 3,6

3t0

M0,Ed 1

2

Fig. 15.5 Joint 2 (with M0,Ed = 7,56 kNm in chord on both sides of the joint)

Fig. 15.6 Purlin joint

180 x 180 x 8,0

120 x 120 x 5,0

80 x 80 x 3,0

150 x 150 x 6,3

Site bolted joint

Fig. 15.7 Member dimensions and joint numbers for RHS truss (fy0 = fyi = 355 N/mm2)

Fig. 15.8 Triangular truss

205

3

Ni 

 2

P 2

Vi  2 cos  sin i 2

Fig. 15.9 Shear force

P 2

Fig. 15.10 Horizontal loads

 ti

Offset  0,25d0

24 00

Fig. 15.11 Gap and offset

93,5 kN

108 kN

54 kN

Fig. 15.12 Cross section of the triangular truss with circular hollow sections

206

Ф 219,1 x 8,0

Ф 88,9 x 3,6

Ф 219,1 x 11,0

Ф 139,7 x 4,5

Chord: fy0 = 355 N/mm2 Diagonals: fyi = 355 N/mm2

Fig. 15.13 Member dimensions and steel grades

Diagonals: Ф 139,7 x 4,5 Chord:

22,5

Fig. 15.14 Connection of the diagonals to the bottom chord

207

Ф 219,1 x 11

406,4

59,3

155,0 143,2 109,6

8,8

C20 16 Ф 25, S500 S275

Fig. 15.15 Concrete filled column

208

16. REFERENCES AISC, 2005: Specification for structural steel buildings. ANSI/AISC 360-05, American Institute of Steel Construction, Chicago, USA. AISC, 2010: Specification for structural steel buildings. ANSI/AISC 360-10, American Institute of Steel Construction, Chicago, USA. Akiyama, N., Yayima, M., Akiyama, H., & Otake, F., 1974: Experimental study on strength of joints in steel tubular structures. Journal of Society of Steel Construction, JSSC, Vol. 10, No. 102, pp. 37-68, (in Japanese).

API, 2007: Recommended practice for planning, designing and constructing fixed offshore platforms – Working st stress design. API RP 2A, 21 Edition, Suppl 3, American Petroleum Institute, Dallas, USA. Aribert, J.M., Ammari, F., & Lachal, A., 1988: Influence du mode d'application d'une charge de compression locale sur la résistance plastique de l'âme d'un profilé cas des assemblages tubulaires. Construction Métallique No. 2, pp. 3-30.

Baar, S., 1968: Etude théorique et expérimentale du déversement des poutres à membrures tubulaires.

Collection des publications de la Faculté des Sciences Appliquées de Université de Liège, No. 10, Liège, Belgium.

Bergmann, R., Matsui, C., Meinsma, C., & Dutta, D., 1995: Design guide for concrete filled hollow section columns under static and seismic loading. CIDECT Series "Construction with hollow steel sections" No. 5, Verlag TÜV Rheinland, Köln, Germany, ISBN 3-8249-0298-2.

Birkemoe, P.C., & Packer, J.A., 1986: Ultimate strength design of bolted tubular tension connections.

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220

SYMBOLS Abbreviations of organisations AISC AWS CEN CIDECT CSA IIW ISO

American Institute of Steel Construction American Welding Society European Committee for Standardization Comité International pour le Développement et l'Etude de la Construction Tubulaire Canadian Standards Association International Institute of Welding International Organization for Standardization

Other abbreviations CHS CTOD FE RHS SCF SHS TTP

circular hollow section crack tip opening displacement finite element rectangular or square hollow section stress concentration factor square hollow section through thickness properties

General symbols A Aa Ac Agv Ai Am An Anet As Av Be Bf C Ce CK CT CX C1 E Ea Ecm Ec,eff Ed Efi,d,t Es (EI)eff

cross sectional area cross sectional area of structural steel in a composite column cross sectional area of concrete in a composite column gross area in shear for block failure cross sectional area of member i (i = 0, 1, 2, 3) exposed surface area of a steel member/unit length under fire conditions; cross section parameter for torsion net section in a bolted joint net cross sectional area cross sectional area of reinforcement in a composite column shear area effective chord length in ring model width parameter of diaphragm plate in beam-to-column joints constant; constant in Δσ-N relationship; multiplication factor to account for secondary bending moments efficiency parameter for joints (general) efficiency parameter for K joints efficiency parameter for T joints efficiency parameter for X joints constant used in chord stress function modulus of elasticity modulus of elasticity of steel section in a composite column modulus of elasticity of concrete in a composite column modulus of elasticity of concrete in a composite column, corrected for creep total energy dissipated in yield line model design load under fire conditions, at fire exposure time t modulus of elasticity of reinforcement in a composite column effective stiffness of a composite column 221

(EI)eff,|| F I Ia Ib Ic Is It JAA Kb L Lb Leff Lo Lw M Mb Mc,Rd Me Mel MEd MEd,|| Mf Mgap,0,Ed Mi Mi,Rd Mip,i Mip,i,Ed Mip,i,Rd Mj Mj,Ed Mj,Rd Mmax,Rd Mop,i Mop,i,Ed Mop,i,Rd Mpl Mpl,f Mpl,V,0,Rd Mpl,Rd Mpl,y,Rd Mpl,z,Rd Mpl,0 Mpl,0,Rd MRd Mt,Rd My,Ed Mz,Ed M0 M0,Ed M1 M||,max N

stiffness of a composite column used for a second order analysis force moment of inertia moment of inertia of steel section in a composite column moment of inertia of a beam moment of inertia of concrete in a composite column moment of inertia of reinforcement in a composite column torsional moment of inertia ratio of out-of-plane axial load and in-plane axial load in a multiplanar joint stiffness of a beam length; span; length in a bolted (gusset plate) joint length of a beam effective length measured length of a tensile test specimen weld length in a gusset plate joint moment bending moment in a beam at mid-span design resistance of a cross section to bending moment elastic moment capacity elastic yield moment capacity design bending moment second order bending moment bending moment in a flange design bending moment in the cross section of a chord at the gap applied bending moment in member i (i = 0, 1, 2, 3) design resistance of a joint, expressed in terms of bending moment in member i (i = 0, 1, 2, 3) applied in-plane bending moment in member i (i = 0, 1, 2, 3) design applied in-plane bending moment in member i (i = 0, 1, 2, 3) design resistance of a joint, expressed in terms of in-plane bending moment in member i (i = 0, 1, 2, 3) applied bending moment in a beam-to-column joint applied design bending moment in a beam-to-column joint design moment resistance of a beam-to-column joint maximum design resistance of a composite cross section to bending moment applied out-of-plane bending moment in member i (i = 0, 1, 2, 3) design applied out-of-plane bending moment in member i (i = 0, 1, 2, 3) design resistance of a joint, expressed in terms of out-of-plane bending moment in member i (i = 0, 1, 2, 3) plastic moment capacity plastic moment capacity of a flange design value of the plastic moment capacity of a chord cross section, reduced by shear design value of the plastic moment capacity of a (composite) cross section design resistance of a (composite) cross section to bending moment about the y-y axis design resistance of a (composite) cross section to bending moment about the z-z axis plastic moment capacity of a chord cross section design value of the plastic moment capacity of a chord cross section design resistance of a (composite) cross section to bending moment design resistance of a cross section to torsional moment design bending moment about the y-y axis design bending moment about the z-z axis bending moment in a chord design bending moment in a chord brace or beam bending moment design bending moment of a member including imperfection and second order effects axial load; number of cycles (to failure) 222

Nb,Rd Ncr,eff NEd Nequ Nfi,Ed Nfi,Rd NG,Ed Ngap,0 Ngap,0,Ed Ngap,0,Rd Ni Ni Ni,Ed Ni,Rd Nj Nj,Ed Npl Npl,V,0,Rd Npl,Rd Npl,Rk Npl,0 Npl,0,Rd NRd Ns,Rd Nt,Rd N0 N0,Ed N0p N1u N1u(JAA) N1u(JAA = 0) P Ov Ovlimit Qf Qu R RAZ Rfi,d,t R(t) S Sj Sj,ini So V VEd Vf Vgap,0,Ed Vi Vp Vpl,f Vpl,Rd Vpl,0 Vpl,0,Rd

design buckling resistance of a member buckling capacity used to determine the second order effects (acting) design axial load (at room temperature) equivalent design axial load under fire conditions including effect of eccentricities design axial load under fire conditions buckling resistance of a member under fire conditions permanent part of the acting design force in a composite column axial load in the cross section of a chord at the gap design axial load in the cross section of a chord at the gap reduced resistance to axial load, due to shear, in the cross section of a chord at the gap applied axial force in brace i (i = 1, 2, 3); applied axial force in overlapping brace number of cycles to failure (in Palmgren-Miner rule) design applied axial load in member i (i = 0, 1, 2, 3) design resistance of a joint, expressed in terms of axial load in member i (i = 0, 1, 2, 3) applied axial load in overlapped brace design applied axial load in overlapped brace axial load capacity of a cross section design value of the axial load capacity of a chord cross section, reduced by shear design resistance of a (composite) cross section to axial load (at room temperature) characteristic resistance of a (composite) cross section to axial load axial yield capacity of a chord cross section design axial yield capacity of a chord cross section design resistance of a (composite) cross section to axial load; buckling resistance of a member at room temperature design value of the shear resistance of the brace(s) at the connection with the chord design tensile capacity of a cross section chord load design value of the chord load chord preload ultimate axial load capacity of a joint based on the load in member 1 ultimate strength of a multiplanar joint with load ratio JAA ultimate strength of a multiplanar joint with unloaded out-of-plane braces factored design load overlap, Ov = q/p x 100% limit for overlap, decisive for brace shear check chord stress function function in the design strength equations accounting for the effect of geometric parameters stress or load ratio reduction of area in a tensile test (in %) design resistance under fire conditions, at fire exposure time t resistance under fire conditions static moment to neutral axis rotational stiffness of a joint initial rotational stiffness of a joint cross sectional area of a standard tensile test specimen (in mm2) volume of a steel member/unit length; shear load (acting) design shear load shear load in a flange design shear load in the cross section of a chord at the gap shear load in brace i (i = 1, 2, 3) punching shear stress plastic shear capacity of a flange design plastic shear yield capacity plastic shear yield capacity of a chord cross section design plastic shear yield capacity of a chord cross section 223

V0 V0,Ed Weff Wel Wel,ip,i Wel,op,i Wi Wpl Wpl,i Wt

shear load in a chord design shear load in a chord effective section modulus elastic section modulus elastic section modulus for in-plane bending moment in member i elastic section modulus for out-of-plane bending moment in member i elastic section modulus of member i (i = 0, 1, 2, 3) plastic section modulus plastic section modulus of member i (i = 0, 1, 2, 3) section modulus for torsion

a aâ&#x20AC;&#x2122; b be bei bej be,ov

throat thickness of a weld; edge distance of bolt edge distance of plastic hinge line in bolted end plate connection external width of a rectangular hollow section; width of a plate; distance from bolt to RHS face effective width of a plate, a beam flange or a brace member effective width of an overlapping RHS brace member at the chord connection effective width of an overlapped RHS brace member at the chord connection effective width of an overlapping RHS brace member at the connection to the overlapped brace effective punching shear width minimum width of a diaphragm plate in beam-to-column joints external width of brace i (i = 1, 2, 3) width of overlapped brace average width of an RHS (b-t) width of a stiffening plate effective width of a web effective width of a web under a brace wall, a plate or a beam flange distance between plastic hinge lines in bolted end plate connection external width of a chord width of a plate or width of an I section or RHS brace constant in chord M-N interaction equation coefficients factor considering the condition (welded or unwelded) at the hidden toe of the overlapped brace external diameter of a hollow section; bolt diameter effective width of an overlapping CHS brace member at the chord connection effective width of an overlapped CHS brace member at the chord connection effective width of an overlapping CHS brace member at the connection to the overlapped brace external diameter of brace i (i = 1, 2, 3); diameter of overlapping brace; external diameter of inner tube in composite column diameter of overlapped brace depth of the web of an I section chord (dw = h0 - 2t0 - 2r) bolt hole diameter external diameter of a chord eccentricity distance from bolt to CHS face edge distance of bolt design buckling stress strength of concrete in a composite column design strength of concrete in a composite column characteristic concrete cylinder strength (in N/mm2) characteristic concrete cube strength (in N/mm2) characteristic concrete cylinder strength (in N/mm2) buckling stress for chord side wall failure (general)

be,p bf bi bj bm bsp bw bwf bâ&#x20AC;&#x2122; b0 b1 c c, c0, c1, c2 cs d dei dej de,ov di dj dw dâ&#x20AC;&#x2122; d0 e, e0 e1 e2 fb,Rd fc fcd fck fck,cub fck,cyl fk

224

fs fsd fsk fu fu,b fui fuj fu0 fy fya fyb fyd fyi fyj fyk fyp fyw fy0 g, g1, g2 g g1 g2 g’ h hi hj hm hn hz h0 h1 i k ky,θ l, ℓ ℓA ℓb ℓb,eff. ℓi ℓθ lx m mp n n’ ni p q q, q1, q2 r rj rm

yield strength of reinforcement in a composite column design strength of reinforcement in a composite column characteristic strength of reinforcement in a composite column specified ultimate tensile strength ultimate tensile strength of material "b" ultimate tensile strength of overlapping brace ultimate tensile strength of overlapped brace ultimate tensile strength of a chord yield strength; design yield strength in joint strength equations design yield strength based on the average yield strength of an RHS section yield strength of parent material specified design yield strength design yield strength of brace i (i = 1, 2, 3); design yield strength of overlapping brace design yield strength of overlapped brace characteristic yield strength of hollow steel section in a composite column design yield strength of a plate design yield strength of a web design yield strength of a chord gap between the braces (ignoring welds) of a K, N or KT joint at the face of the chord bolt gauge bolt edge distance bolt distance gap divided by chord thickness: g’ = g/t0 depth of a rectangular hollow section external depth of brace i (i = 1, 2, 3) depth of overlapped brace average depth of an RHS (h-t) parameter in a composite column moment arm external depth of a chord depth of a plate or depth of an I section or RHS brace radius of gyration amplification factor to incorporate second order effects reduction factor for the yield strength of steel at steel temperature θa length circumferential parameter for torsion buckling length effective perimeter for local brace failure length of yield line i; length of member i (i = 0, 1, 2, 3) buckling length under fire conditions length used in the "4 hinge yield line" model slope of Δσ-N curve plastic moment per unit length chord stress divided by chord yield stress ratio of chord stress due to chord preload and chord yield stress applied number of cycles (in Palmgren-Miner rule) length of the projected contact area of the overlapping brace onto the face of the chord, without the presence of the overlapped brace; internal pressure; bolt pitch projected length of overlap between braces of a K or N joint at the chord face; uniformly distributed loading loadings inside corner radius of a rectangular or square hollow section; radius of an I or U section; ratio of the smaller to the larger end moment (-1  r  +1) load bearing capacity in fire of a single component of a composite cross section mean corner radius of a rectangular or square hollow section 225

s t tf ti tj tp tsp tw t0 t1 us w

α α αM β β

β* χ χd χfi δ, δ1 δ ε εu  , j γ γa γc γF γM, γM0, γM1, γM2 γs η ηfi ηfi,t φ

extension of a Tee from a brace member; bolt distance (fire exposure) time, thickness; wall thickness flange plate thickness wall thickness of brace i (i = 1, 2, 3); thickness of overlapping brace; wall thickness of inner tube in composite column; plate thickness (i = 1, 2) in a bolted splice joint wall thickness of overlapped brace thickness of a plate thickness of a stiffening plate thickness of a web flange thickness of an I section chord or wall thickness of a CHS or RHS chord or “through member” (e.g. column) thickness of a plate or flange thickness of an I section brace or wall thickness of a CHS or RHS brace concrete cover of reinforcement distance between the welds measured from plate face-to-plate face around the perimeter of a CHS or RHS member non-dimensional factor for the effectiveness of the chord flange in shear; angle used in yield line model chord length parameter: α = 2ℓ0/d0 or 2ℓ0/b0 reduction factor for the plastic moment capacity of a composite column parameter for the effect of the end moments on member buckling diameter or width ratio between braces and chord: d d b b   1 or 1 or 1 or 1 (T, Y or X joints); d0 b0 d0 b0 

d1  d2 d  d2 b  b2 (K or N joints); or 1 or 1 2d0 2b 0 2b 0



d1  d2  d3 b  b2  b3 (KT joints) or 1 3d0 3b 0

width parameter used for T stub joints reduction factor for buckling according to EN 1993-1-1 (2005) or any equivalent national buckling curve ratio between actual design force and the axial capacity of a composite section reduction factor for buckling in the fire design situation deformation section parameter (contribution ratio of steel section in a composite column) section class parameter accounting for different steel grades; engineering strain; logarithmic strain ultimate strain joint resistance (or capacity) factor; angle between the planes in a multiplanar joint rotation of a beam-to-column joint half diameter or half width-to-thickness ratio of the chord: γ = d0/2t0 or b0/2t0 partial (safety) factor for steel section in a composite column partial (safety) factor for concrete in a composite column load factor for the action material or joint partial (safety) factor partial (safety) factor for reinforcement in a composite column brace member depth-to-chord diameter or width ratio: η = hi/d0 or hi/b0 ratio between the design load present under fire conditions and the design load at room temperature ratio between the design load present under fire conditions and the buckling resistance at room temperature angle used in ring model

226

φe φi φs φt  E 

μ μd μdy μdz θ θa θi θj ρs σ σa σc σjoint σmax σmin σnom σpeak σr σ σ0 σ1,Ed τ τ τRd Δ ΔTcf Δε Δσ Δσgeom Δσnom

correction coefficient related to the eccentricity rotation in a yield line i correction coefficient related to the reinforcement creep factor slenderness of a member under compression Euler slenderness relative slenderness ratio correction factor to uniplanar joint resistance to account for multiplanar effect related bending capacity in a composite column related bending capacity about the y axis related bending capacity about the z axis temperature; angle between brace and chord steel temperature acute angle between brace member i (i = 1, 2, 3) and the chord; acute angle between overlapping brace and the chord acute angle between overlapped brace and the chord reinforcement ratio of a composite column (in %) axial or bending stress; engineering stress; logarithmic stress stress in steel section stress in concrete stress at the intersection of brace and chord maximum stress minimum stress nominal stress in a member peak stress at the weld toe of a joint stress in radial direction stress in tangential direction chord stress nominal stress in brace shear stress brace-to-chord wall thickness ratio: τ = ti/t0 design bond stress midspan deflection of a beam under a uniformly distributed load temperature shift in the notch impact energy - temperature diagram due to cold forming (cf) strain range stress range geometrical stress range nominal stress range

Subscripts a c e, eff el fi i

ip

referring to structural steel in a composite column referring to concrete in a composite column effective elastic referring to fire conditions subscript used to denote the member of a hollow section joint. Subscript i = 0 designates the chord (or “through member”); i = 1 refers in general to the brace for T, Y and X joints, or it refers to the compression brace for K, N and KT joints; i = 2 refers to the tension brace for K, N and KT joints; i = 3 refers to the vertical brace for KT joints. For K and N overlap joints, the subscript i is used to denote the overlapping brace member; subscript used to indicate the member for which the SCF is given in-plane

227

j k max min n nom op p pl s t u v w y Ed Rd Rk

subscript used to denote the overlapped brace member for K and N overlap joints; subscript used to indicate the location where the SCF is given subscript used to indicate the type of loading for which the SCF is given maximum minimum net nominal out-of-plane plate; preload plastic referring to reinforcement in a composite column tension; torsion; time ultimate shear web; weld yield design value of action design value of resistance characteristic value of resistance

Symbols not shown here are specifically defined at the location where they are used.

In all calculations, the nominal (guaranteed minimum) mechanical properties should be used.

228

Comité International pour le Développement et I’Etude de la Construction Tubulaire International Committee for the Development and Study of Tubular Structures CIDECT, founded in 1962 as an international association, joins together the research resources of the principal hollow steel section manufacturers to create a major force in the research and application of hollow steel sections world-wide.

The CIDECT website is http://www.cidect.com

The objectives of CIDECT are: • To increase the knowledge of hollow steel sections and their potential application by initiating and participating in appropriate research and studies.

• To establish and maintain contacts and exchanges between producers of hollow steel sections and the ever increasing number of architects and engineers using hollow steel sections throughout the world.

• To promote hollow steel section usage wherever this makes good engineering practice and suitable architecture, in general by disseminating information, organising congresses, etc.

• To co-operate with organisations concerned with specifications, practical design recommendations, regulations or standards at national and international levels.

Technical activities The technical activities of CIDECT have centred on the following research aspects of hollow steel section design: • • • • • • • • • •

Buckling behaviour of empty and concrete filled columns Effective buckling lengths of members in trusses Fire resistance of concrete filled columns Static strength of welded and bolted joints Fatigue resistance of welded joints Aerodynamic properties Bending strength of hollow steel section beams Corrosion resistance Workshop fabrication, including section bending Material properties

The results of CIDECT research form the basis of many national and international design requirements for hollow steel sections.

229

CIDECT Publications The current situation relating to CIDECT publications reflects the ever increasing emphasis on the dissemination of research results.

The list of CIDECT Design Guides, in the series "Construction with Hollow Steel Sections", already published, is given below. These Design Guides are available in English, French, German and Spanish.

1. Design guide for circular hollow section (CHS) joints under predominantly static loading (1st edition 1991 and 2nd edition 2008) 2. Structural stability of hollow sections (1992, reprinted 1996) 3. Design guide for rectangular hollow section (RHS) joints under predominantly static loading (1st edition 1992 and 2nd edition 2009) 4. Design guide for structural hollow section columns exposed to fire (1994, reprinted 1996) 5. Design guide for concrete filled hollow section columns under static and seismic loading (1995) 6. Design guide for structural hollow sections in mechanical applications (1995) 7. Design guide for fabrication, assembly and erection of hollow section structures (1998) 8. Design guide for circular and rectangular hollow section welded joints under fatigue loading (2001) 9. Design guide for structural hollow section column connections (2004) In addition, as a result of the ever-increasing interest in steel hollow sections in internationally acclaimed structures, two books have been published, i.e. "Tubular Structures in Architecture" by Prof. Mick Eekhout (1st edition 1996 and 2nd edition 2010) and "Hollow Sections in Structural Applications" by Prof. Jaap Wardenier (1st edition 2002) and this 2nd edition by Prof. Jaap Wardenier et al. (2010).

Copies of the Design Guides, the architectural book and research papers may be obtained through the CIDECT website: http://www.cidect.com "Hollow Sections in Structural Applications" by Prof. Jaap Wardenier et al. (2010) is available in hard copy colour print from the publisher: Bouwen met Staal Boerhaavelaan 40 2713 HX Zoetermeer, The Netherlands P.O. Box 190 2700 AD Zoetermeer, The Netherlands Tel. +31(0)79 353 1277 Fax +31(0)79 353 1278 E-mail info@bouwenmetstaal.nl

CIDECT Organisation (2010) • President: P. Ritakallio, Finland • Treasurer/Secretary: R. Murmann, United Kingdom • A General Assembly of all members meeting once a year and appointing an Executive Committee responsible for administration and execution of established policy.

• A Technical Commission and a Promotion Committee meeting at least once a year and directly responsible for the research work and technical promotion work.

Chairman Technical Commission: G. Iglesias, Spain Chairman Promotion Committee: J. Krampen, Germany

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Present members of CIDECT are: • • • • • • • • • • •

Atlas Tube, Canada/USA OneSteel Australian Tube Mills, Australia Borusan Mannesmann Boru, Turkey Corus Tubes, United Kingdom Grupo Condesa, Spain Industrias Unicon, Venezuela Rautaruukki Oyj, Finland Robor Steel Services, South Africa Sidenor SA, Greece Vallourec & Mannesmann Tubes, Germany Voest-Alpine Krems, Austria

Acknowledgements for photographs The authors express their appreciation to the following firms and persons for making available some of the photographs used in this book: Bouwdienst, Rijkswaterstaat, The Netherlands CORUS Tubes, UK Delft University of Technology, The Netherlands HGG Profiling Equipment BV, Wieringerwerf, The Netherlands Instituto para la Construcción Tubular (ICT), Spain University of Toronto, Canada Vallourec & Mannesmann Tubes, Germany Prof. Y.S. Choo, National University of Singapore, Singapore Mr. D. Dutta, Germany Mr. Félix Escrig, Spain Mr. José Sánchez, Spain

Disclaimer Care has been taken to ensure that all data and information herein is factual and that numerical values are accurate. To the best of our knowledge, all information in this book is accurate at the time of publication.

CIDECT, its members and the authors assume no responsibility for errors or misinterpretations of information contained in this book or in its use.

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