CIDECT 2- Structural Stability of Hollow Sections

Construction with Hollow Steel Sections - Structural stability of hollow sectionsDiscuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t p] WITH CONSTRUCTION HOLLOW STEEL SECTIONS Edited by: Comite International pour le Developpement et I’Etude Authors: J acques Rondal, University of Liege de la Construction Tubulaire Karl-Gerd Wurker, Consulting engineer Dipak Dutta, Chairman of the Technical Commission CIDECT J aap Wardenier, Delft University of Technology Noel Yeomans, Chairman of the Cidect Working Group “J oints behaviour and Fatigue-resistance’’ Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t J . Rondal, K.-G. Wurker, D. Dutta, J . Wardenier, N. Yeomans Verlag TUV Rheinland Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t I Die Deutsche Bibliothek - CIP Einheitsaufnahme Structural stability of hollow sections / [Comite International pour le Developpement et I’Etude de la Construction Tubulaire]. J . Rondal . . . - Koln: Verl. TUV Rheinland, 1992 (Construction with hollow steel sections) Dt. Ausg. u.d.T.: Knick- und Beulverhalten von Hohlprofilen (rund und rechteckig). - Franz. Ausg. u.d.T.: Stabilite des structures en profils creux ISBN 3-8249-0075-0, Reprinted edition NE: Rondal, J asques; Comite International pour le Developpement et I’Etude de la Construction Tubulaire ISBN 3-8249-0075-0 0 by Verlag TUV Rheinland GmbH, Cologne Entirely made by: Verlag TUV Rheinland GmbH, Cologne Printed in Germany First edition 1992 Reprinted with corrections 1996 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t The objective of this design manual is to present the guide lines for the design and calculation of steel structures consisting of circular and rectangular hollow sections dealing in particular with the stability of these structural elements. This book describes in a condensed form the global, local and lateral-torsional buckling behaviour of hollow sections as well as the methods to determine effective buckling lengths of chords and bracings in lattice girders built with them. Nearly all design rules and procedures recommended here are based on the results of the analytical investigations and practical tests, which were initiated and sponsored by CIDECT. These research works were carried out in the universities and institutes in various parts of the world. The technical data evolving from these research projects, the results of their evaluation and the conclusions derived were used to establish the “European buckling curves” for circular and rectangular hollow sections. This was the outcome of a cooperation between ECCS (European Convention for Constructional Steelwork) and CIDECT. These buckling curves have now been incorporated in a number of national standards. They have also been proposed for the buckling design by Eurocode 3, Part 1: “General Rules and Rules for Buildings” (ENV 1993-1-1). Extensive research works on effective buckling lengths of structural elements of hollow sections in lattice girders in the late seventies led in 1981 to the publication of Monograph No. 4 “Effective lengths of lattice girder members” by CIDECT. A recent statistical evaluation of all data from this research programme resulted in a recommendation for the calculation of the said buckling length which Eurocode 3, Annex K “Hollow section lattice girder connections” also contains. This design guide is the second of a series, which CIDECT has already published and also will publish in the coming years: 1. Design guide for circular hollow section (CHS) joints under predominantly static loading 2. Structural stability of hollow sections (reprinted edition) 3. Design guide for rectangular hollow section joints under predominantly static loading 4. Design guide for hollow section columns exposed to fire (already published) 5. Design guide for concrete filled hollow section columns under static and seismic loading 6. Design guide for structural hollow sections for mechanical applications (already published) 7. Design guide for fabrication, assembly and erection of hollow section structures (in 8. Design guide for circular and rectangular hollow section joints under fatigue loading (in All these publications are intended to make architects, engineers and constructors familiar with the simplified design procedures of hollow section structures. Worked-out examples make them easy to understand and show how to come to a safe and economic design. Our sincere thanks go to the authors of this book, who belong to the group of wellknown specialists in the field of structural applications of hollow sections. We express our special thanks to Prof. J acques Ronda1 of the University of Liege, Belgium as the main author of this book. We thank further Dr. D. Grotmann of the Technical University of Aix-la-Chapelle for numerous stimulating suggestions. Finally we thank all CIDECT members, whose support made this book possible. (already published) (already published) (already published) preparation) preparation) Dipak Dutta Technical Commission CIDECT 5 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 7 Quadrangular vierendeel columns 6 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Page . . 9 1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1 Limit states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Limit state design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Steel grades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Increase in yield strength due to cold working . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Cross section classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Members in axial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Design method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Design aids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Membersi n bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1 Design for lateral-torsional buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Members in combined compression and bending . . . . . . . . . . . . . . . . . . . . . 28 5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2 Design method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2.1 Design for stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2.2 Design based on stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2.2.1 Stress design without considering shear load . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2.2.2 Stress design considering shear load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6 Thin-walled sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.2 Rectangular hollow sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.2.1 Effective geometrical properties of class 4 cross sections . . . . . . . . . . . . . . . . . 34 6.2.2 Design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.2.3 Design aids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.3 Circular hollow sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7 Buckling length of members in lattice girders . . . . . . . . . . . . . . . . . . . . . . . . 40 7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 7.2 Effective buckling length of chord and bracing members with lateral support . . 40 7.3 Chords of lattice girders, whose joints are not supported laterally . . . . . . . . . . . 40 8 Desi gnexampl es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 8.1 Design of a rectangular hollow section column in compression . . . . . . . . . . . . . 43 8.2 Design of a rectangular hollow section column in combined compression and 8.3 Design of a rectangular hollow section column in combined compression and uni-axialbending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 bi-axialbending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 8.4 Design of a thin-walled rectangular hollow section column in compression . . . 47 8.5 Design of a thin-walled rectangular hollow section column in concentric compression and bi-axial bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 9 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 References 53 CIDECT . International Committee for the Development and Study of Tubular Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Introduction It is very often considered that the problems to be solved while designing a steel structure are only related to the calculation and construction of the members and their connections. They concern mainly the static or fatigue strength and the stability of the structural members as well as the load bearing capacity of the joints. This point of view is certainly not correct as one cannot ignore the important areas dealing with fabrication, erection and when necessary, protection against fire. It is very important to bear in mind that the application of hollow sections, circular and rectangular, necessitates special knowledge in all of the above mentioned areas extending beyond that for the open profiles in conventional structural engineering. This book deals with the aspect of buckling of circular and rectangular hollow sections, their calculations and the solutions to the stability problems. The aim of this design guide is to provide architects and structural engineers with design aids based on the most recent research results in the field of application technology of hollow sections. It is mainly based on the rules given in Eurocode 3 “Design of Steel Structures, Part 1: General Rules and Rules for Buildings” and its annexes [l , 21. Small differences can be found when compared to some national standards. The reader will find in reference [3] a review of the main differences existing between Eurocode 3 and the codes used in other countries. However, when it is possible, some indications are given on the rules and recommendations in the codes used in Australia, Canada, J apan and United States of America as well as in some european countries. Lift shaft with tubular frames 9 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 1 General 1 . l Limit states Most design codes for see1 structures are, at the present time, based on limit state design. Limit states are those beyond which the structure no longer satisfies the design performance requirements. Limit state conditions are classified into - ultimate limit state - serviceability limit state Ultimate limit states are those associated with collapse of a structure or with other failure modes, which endanger the safety of human life. For the sake of simplicity, states prior to structural collapse are classified and treated as ultimate limit states in place of the collapse itself. Ultimate limit states, which may require consideration, include: - Loss of equilibrium of a structure or a part of it, considered as a rigid body - Loss of load bearing capacity, as for example, rupture, instability, fatigue or other agreed Serviceability limit states correspond to states beyond which specified service criteria are no longer met. They include: - Deformations or deflections which affect the appearance or effective use of the structure (including the malfunction of machines or services) or cause damage to finishes or non- structural elements - Vibration which causes discomfort to people, damage to the building or its contents or which limits its functional effectiveness Recent national and international design standards recommend procedures proving limit state resistance. This implies, in particular for stability analysis, that the imperfections, mechanical and geometrical, which influence the behaviour of a structure significantly, must be taken into account. Mechanical imperfections are, for example, residual stresses in structural members and connections. Geometrical imperfections are possible pre- deformations in members and cross sections as well as tolerances. limiting states, such as excessive deformations and stresses 1.2 Limit state design In the Eucrocode 3 format, when considering a limit state, it shall be verified that: where yF =Partial safety factor for the action F yU =Partial safety factor for the resistance R F =Value of an action R =Value of a resistance for a relevant limit state yF F =Fd is called the design load while RlrM =R, is designated as the design resistance. It is not within the scope of this book to discuss in detail these general provisions. They can be taken from Eurocode 3 and other national codes, which can sometimes show small deviations from one another. As for example, the calculations in the recent US-codes are made with 6 =? h M . 10 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 1.3 Steel grades Table 1 gives the grades of the generally used structural steels with the nominal minimum values of the yield strength f,, range of the ultimate tensile strength f, and elongations. The steel grades correspond to the hot-rolled hollow sections as well as to the basic materials for cold-formed hollow sections. The designations of the steel grades in Table 1 are in accordance with EN 10 025 1311. They can be different in other standards. For hot-rolled hollow sections (circular and rectangular), the european code EN 10 210, Part 1 1201, 1994 is available. Table 1 - Steel grades for structural steels min. yield strength f, (N/mm2) f, (Nlmrn') tensile strength min. percentage elongation steel grade L, =5.65 4, longitudinal transverse S 235 15 17 550. . .720 460 S 460 20 22 490.. ,630 355 S 355 20 22 410.. ,560 275 S 275 24 26 340. . ,470 235 Table 2 contains the recommended physical properties valid for all structural steels. Table 2 - Physical properties of structural steels modulus of elasticity: E =210 000 N/mm2 shear modulus: G =- =81 000 N/mm2 poi son co-efficient: U =0.3 co-efficient of linear expansion: Q =12 . 10 ~ 6/ oC density: e =7850 kg/m3 E 2(1 +U) 1.4 Increase in yield strength due to cold working Cold rolling of profiles provides an increase in the yield strength due to strain hardening, which may be used in the design by means of the rules given in Table 3. However, this increase can be used only for RHS in tension or compression elements and cannot be taken into account if the members are subjected to bending (see Annex A of Eurocode 3 121). For cold rolled square and rectangular hollow sections, eq. (1.2) can be simplified (k =7 for all cold-forming of hollow sections and n =4) resulting in: 14t fya =fyb +b+h (fu-fyb) I f, 5 1.2 . fyb Fig. 1 allows a quick estimation of the average yield strength after cold-forming, for square and rectangular hollow sections for the four basic structural steels. 11 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Table 3 - Increase of yield strength due to cold-forming of RHS profiles Average yield strength: The average yield strength f , , may be determined from full size section tests or as follows [19, 321: f , , =f , , +(k . n . tz/A) . (f, - f , , ) (1 4 where f , , , f , =specified tensile yield strength and ultimate tensile strength of the basic material (N/mm2) t =material thickness (mm) A =gross cross-sectional area (mm2) k =co-efficient depending on the type of forming (k =7 for cold rolling) n =number of 90° bends in the section with an internal radius <5t (fractions of 90' fya =should not exceed f, or 1.2 f , , bends should be counted as fractions of n) The increase in yield strength due to cold working should not be utilised for members which are annealed. or subject to heating over a long length with a high heat input after forming, which may produce softening. Basic material: Basic material is the flat hot rolled sheet material out of which sections are made by cold forming. * Stress relief annealing at more than 58OoC or for over one hour may lead to deterioration of the mechanical properties [29] Increase In wel d st r engt h f va/ f yb 1 20 - 1 15 -. 1 10 ~. 1.05 - 1.004 , 1 i , ~ , ~ , 0 10 20 30 40 50 60 70 80 90 100 b + h 2 t Fig. 1 - Increase in yield strength for cold-formed square and rectangular hollow sections 12 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 2 Cross section classification Different models can be used for the analysis of steel structures and for the calculation of the stress resultants (normal force, shear force, bending moment and torsional moment in the members of a structure). For an ultimate limit state design, the designer is faced mainly with three design methods (see Fig. 2). The cross section classes 3 and 4 with the procedure “elastic-elastic” differ from each other only by the requirement for local buckling for class 4. Procedure “ plastic-plastic’’ Cross section class 1 This procedure deals with the plastic design and the formation of plastic hinges and moment redistribution in the structure. Full plasticity is developed in the cross section (bi-rectangular stress blocks). .The cross section can form a plastic hinge with the rotation capacity required for plastic analysis. The ultimate limit state is reached when the number of plastic hinges is sufficient to produce a mechanism. The system must remain in static equilibrium. Procedure “ elastic-plastic’’ Cross section class 2 In this procedure the stress resultants are determined following an elastic analysis and they are compared to the plastic resistance capacities of the member cross sections. Cross sections can develop their plastic resistance, but have limited rotation capacity. Ultimate limit state is achieved by the formation of the first plastic hinge. Procedure “ elastic-elastic” Cross section class 3 This procedure consists of pure elastic calculation of the stress resultants and the resistance capacities of the member cross sections. Ultimate limit state is achieved by yielding of the extreme fibres of a cross section. The calculated stress in the extreme compression fibre of the member cross section can reach its yield strength, but local buckling is liable to prevent the development of the plastic moment resistance. Procedure “ elastic-elastic” Cross section class 4 The cross section is composed of thinner walls than those of class 3. It is necessary to make explicit allowances for the effects of local buckling while determining the ultimate moment or compression resistance capacity of the cross section. The application of the first three above mentioned procedures is based on the presumption that the cross sections or their parts do not buckle locally before achieving their ultimate limit loads; that means, the cross sections must not be thin-walled. In order to fulfil this condition, the blt-ratio for rectangular hollow sections or the dlt-ratio for circular hollow sections must not exceed certain maximum values. They are different for the cross section classes 1 through 3 as given in Tables 4, 5 and 6. A cross section must be classified according to the least favourable (highest) class of the elements under compression andlor bending. Tables 4 through 6 give the slenderness limits blt or d/t for different cross section classes based on Eurocode 3 [l , 21. Other design codes show slightly different values (compare Tables 8 and 9). 13 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t cross section class 1 classes load resistance capacity stress distribution and rotation capacity full plasticity in the cross section full rotation capacity procedure for the determination of the stress resultants plastic procedure for the determination of the ultimate resistance capacity of a section plastic class 2 full plasticity in the cross section restricted rotation capacity - f V + f Y plastic Fig. 2 - Cross section classification and design methods class 3 elastic cross section yield stress in the extreme fibre elastic elastic class 4 elastic cross section local buckling to be taken into account & - f v 1’ +f y elastic elastic Tabl e 4 - Limiting d/t ratios for circular hollow sections 82 cross section class 2 dlt S 50t2 1 compression andlor bending dlt S 90e2 3 dlt 5 70e2 f, (Nlmm2) 460 355 275 235 e 0.51 0.66 0.85 1 €2 0.72 0.81 0.92 1 14 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Table 5 - Limiting h,/t-ratios for webs of rectangular hollow sections webs: (internal element perpendicular to the axis of bending) h, =h - 3t class compression bending web subject to web subject to stress distribution in element (compression positive) @h #3 / h - -1 f V - -2 + 1 h,/t 5 72c h,lt S 33 e 2 h,lt 5 83c h,lt 5 386 I I stress distribution in element positive) (compression DTh h1'2y f v - ha -i -i, fv 3 h,/t I 4 2 ~ h,lt 5 124~ h, =h - 31 web subject to bending and compression f V - when CY >0.5 h,/t 5 396e/(13~~- 1) when CY <0.5 h,/t S 36cla when CY >0.5 h,lt S 456~1(13~~- 1) when CY <0.5 h,lt S 41.5cla PT! h * t v - when 11. >- 1 h,/t 5 42d (0.67 +0.3311.) when 11. <- 1 h,/t 5 62t(l - 11.) m f Y 235 460 355 275 € 0.72 0.81 0.92 1 15 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Tabl e 6 - Li mi ti ng b,/t-rati os for flanges of rectangular hollow sections flanges: (internal elements parallel to the axis of bending) b, =b ~ 3t class I I section in bending 1 section in compression stress distribution in element and cross section (compression positive) +n f~ ’ I I b,/t 5 33e I b,/t I 42e 2 1 I b,/t I 38c I b,/t 5 42 c stress distribution in element and cross section (compression positive) 3 1 I b,/t 5 42 e I b,lt S 42 t f, (N/mm2) 460 355 275 235 t 0.72 0.81 0.92 1 In Table 7 the blt, hlt and dlt limiting values for the different cross section classes, cross section types and stress distributions are given for a quick determination of the cross section class of a hollow section. The values for width b and height h of a rectangular hollow section are calculated by using the relationship blt =b,lt +3 and hlt =h,lt +3. For the application of the procedures “plastic-plastic’’ (class 1) and “elastic-plastic’’ (class 2), the ratio of the specified minimum tensile strength f, to yield strength f, must be not less than 1.2. Further, according to Eurocode 3 [l , 21, the minimum elongation at failure on a gauge length l, =5.65 no (where A, i s the original cross section area) is,not to be less than 15%. For the application of the procedure “plastic-plastic’’ (full rotation), the strain E” corrres- ponding to the ultimate tensile strength f, must be at least 20 times the yield strain E , corresponding to the yield strength f,. The steel grades in Table 1 for hot formed RHS and hot or cold formed CHS may be accepted as satisfying these requirements. Tables 8 and 9 give, for circular hollow sections and for square or rectangular hollow sections respectively, the limiting blt and hlt ratios, which are recommended in various national codes around the world [3]. Table 8 shows that there are significant differences in dlt limits recommended by the national codes, when a circular hollow section is under bending. In particular, this is clear in the case of the recent american code Al SC 86. For the concentrically loaded circular hollow sections, the deviations are significantly smaller (less than about 10%). Table 9 shows that the differences in blt limits for rectangular hollow sections between the national codes are, in general, not as large as those for circular hollow sections. 16 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Table 7 - blt- , hlt- and dlt limits for the cross section classes 1, 2 and 3 with blt =b,lt +3 and hlt =h,lt +3 r r I 3 r class 1 2 - 235 45 - - 36 - !75 - 355 - !75 - 235 45 - - 41 - 86.0 - 70.0 355 460 !35 275 355 - 36.6 - 36.6 - 103.3 - 59.6 460 __ 32.2 - 32.2 - 90.8 - 46.0 cross section I element 41.6 36.6 32.2 41.6 36.6 32.2 45 41.6 RHP compression' compression 08 33.3 29.3 25.7 37.9 __ 79.5 33.4 29.3 45 41.6 RHP m bending compression 75 69.3 61 .l 53.6 70.0 61.5 127 1 1 7.3 - 76.9 RHP bending bending m 50 42.7 33.1 - 25.5 - 59.8 - 46.3 - 35.8 - 90.0 - CHS - compression andlor bending There is no difference between blt and hlt limits for the classes 1, 2 and 3, when the whole cross section is only under compression. Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Table 8 - Max. d/t limits for circular hollow sections by country and code (t =e ; f, in N/mm*) axial compression Tabelle 9 - Max. b,/t limits for rectangular hollow sections by country and code - (E =dy ; f, in N/mm2) Y country code I Australia ASDR 87 164 Belgium I NBN 851-002 (08.88) Canada I CANICSA-S16.1 -M89 Germany I DIN 18800, Part 1 (1 1.90) J apan I AIJ 80 Netherlands I NEN 6770, publ. draft (08.89) United BS 5950 Part 1 (1985) U.S.A. AlSClLRFD (1986) European Eurocode 3 [l ] Community bending axial compression plastic limit yield limit (class 2) (class 3) 40.2~' 40.2~' 29.9 t 45.4t" 45.4€" 42 t 42.2 t 34.6t 42.2 t 42 t 34 t 42 t 47.8 t - 47.8 t 37.8 t 37 t 37.8 t 43.6 t 34. 2~ 37.66 42 c 34 t 40.8~ 42 c 38 t 42 t 4 0 . 8 ~ - for cold formed non-stress relieved hollow sections * * for hot-formed and cold-formed stress relieved hollow sections 18 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 3 Members in axial compression 3.1 General This chapter of the book is devoted to the buckling of compressed hollow section members belonging to the cross section classes 1, 2 and 3. Thinwalled cross sections (class 4) will be dealt with in chapter 6. The buckling of a concentrically compressed column is, historically speaking, the oldest problem of stability and was already investigated by Euler and later by many other researchers 151. At the present time, the buckling design of a steel element under compression is performed by using the so called “European buckling curves” in most european countries. They are based on many extensive experimental and theoretical investigations, which, in particular, take mechanical (as for example residual stress, yield stress distribution) and geometrical (as for example, linear deviation) imperfections in the members into account. 00 0 0 5 1.0 1 5 x Fig. 3 - European buckling curves [l ] 0 A detailed discussion on the differences between buckling curves used in codes around the world is given in reference [3]. Both design methods, allowable stress design and limit state design, have been covered. For ultimate limit state design, multiple buckling curves are mostly used (as for example, Eurocode 3 with a,, a, b, c curves, similarly in Australia and Canada). Other standards adopt a single buckling curve, presumably due to the fact that emphasis is placed on simplicity. Differences up to 15% can be observed between the various buckling curves in the region of medium slenderness (X). 3.2 Design method At present, a large number of design codes exist and the recommended procedures are often very similar. Eurocode 3 [l , 21 is referred to in the following. For hollow sections, the only buckling mode to be considered is flexural buckling. It is not required to take account of lateral-torsional buckling, since very large torsional rigidity of a hollow section prevents any torsional buckling. 19 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t The design buckling load of a compression member is given by the condition; Nd Nb,Rd where Nd =Design load of the compressed member (7 times working load) =Design buckling resistance capacity of the member f YM Nb,Rd =X . A . (3.1) A is the area of the cross section; X is the reduction factor of the relevant buckling curve (Fig. 3, Tables 1 1 through 14) fy is the yield strength of the material used; -yM is the partial safety factor on the resistance side (in U.S.A.: l/y, =6) The reduction factor X is the ratio of the buckling resistance Nb,Rd to the axial plastic resistance N, , , , , : dependent on the non-dimensional slenderness 5; of a column; x = - - - Nb.Rd fb,Rd Npl.Rd fy,d - fb,,d =design buckling stress =- Nb,Rd A fY,d =design yield strength =- YM The non-dimensional slenderness 5; is determined by f Y with X =- (Ib =effective buckling length; i =radius of gyration) Ib I X, =P . fi (“Eulerian” slenderness) E =210 000 N/mm2 Tabl e 10 a - Eulerian slenderness for various structural steels steelgrade 1, (N/rnmz) S 460 S 355 S 275 S 235 460 355 275 235 I I X, 67.1 76.4 86.8 93.9 The selection of the buckling curve (a through c in Fig. 3) depends on the cross section type. This is mainly based on the various levels of residual stresses occurring due to different manufacturing processes. Table 10b shows the curves for hollow sections. 20 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Table 10 b - Buckling curves according to manufacturing process f,, =Yield strength of the basic (not cold-formed) material f,, =Yield strength of the material after cold-forming I cross section 1 manufacturing process I buckling curves I Table 11 - Reduction factor X - buckling curve a, - - h 0.00 .l0 .20 .30 .40 .50 .60 .70 .80 .90 1 .oo 1.10 1.20 1.30 1.40 1.50 1.60 l .70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 - - 0 1 .oooo 1 .oooo 1 .oooo 0.9859 0.9701 0.9513 0.9276 0.8961 0.8533 0.7961 0.7253 0.6482 0.5732 0.5053 0.4461 0.3953 0.3520 0.3150 0.2833 0.2559 0.2323 0.21 17 0.1 937 0.1 779 0.1639 0.1515 0.1404 0.1305 0.1216 0.1136 0.1063 0.0997 0.0937 0.0882 0.0832 0.0786 0.0744 1 1 .oooo 1 .oooo 0.9986 0.9845 0.9684 0.9492 0.9248 0.8924 0.8483 0.7895 0.7178 0.6405 0.5660 0.4990 0.4407 0.3907 0.3480 0.31 16 0.2804 0.2534 0.2301 0.2098 0.1920 0.1764 0.1626 0.1503 0.1394 0.1296 0.1207 0.1128 0.1056 0.0991 0.0931 0.0877 0.0828 0.0782 0.0740 2 3 1 .oooo 1 .oooo 1 .oooo 1 .oooo 0.9973 0.9959 0.9829 0.9814 0.9667 0.9649 0.9470 0.9448 0.9220 0.9191 0.8886 0.8847 0.8431 0.8377 0.7828 0.7760 0.7101 0.7025 0.6329 0.6252 0.5590 0.5520 0.4927 0.4866 0.4353 0.4300 0.3861 0.3816 0.3441 0.3403 0.3083 0.3050 0.2775 0.2746 0.2509 0.2485 0.2280 0.2258 0.2079 0.2061 0.1904 0.1887 0.1749 0.1735 0.1613 0.1600 0.1491 0.1480 0.1383 0.1373 0.1286 0.1277 0.1199 0.1191 0.1120 0.1113 0.1049 0.1043 0.0985 0.0979 0.0926 0.0920 0.0872 0.0867 0.0823 0.0818 0.0778 0.0773 0.0736 0.0732 4 1 .oooo 1 .oooo 0.9945 0.9799 0.9631 0.9425 0.9161 0.8806 0.8322 0.7691 0.6948 0.6176 0.5450 0.4806 0.4248 0.3772 0.3365 0.3017 0.2719 0.2461 0.2237 0.2042 0.1871 0.1721 0.1587 0.1469 0.1363 0.1268 0.1183 0.1106 0.1036 0.0972 0.0915 0.0862 0.0814 0.0769 0.0728 5 1 .oooo 1 .oooo 0.9931 0.9783 0.9612 0.9402 0.9130 0.8764 0.8266 0.7620 0.6870 0.6101 0.5382 0.4746 0.4197 0.3728 0.3328 0.2985 0.2691 0.2437 0.221 7 0.2024 0.1855 0.1 707 0.1575 0.1458 0.1353 0.1259 0.1 175 0.1098 0.1029 0.0966 0.0909 0.0857 0.0809 0.0765 0.0724 6 l .oooo 1 .oooo 0.991 7 0.9767 0.9593 0.9378 0.9099 0.8721 0.8208 0.7549 0.6793 0.6026 0.5314 0.4687 0.4147 0.3685 0.3291 0.2954 0.2664 0.2414 0.2196 0.2006 0.1840 0.1693 0.1563 0.1447 0.1343 0.1250 0.1167 0.1091 0.1023 0.0960 0.0904 0.0852 0.0804 0.0761 0.0720 -- 7 1 .oooo 1 .oooo 0.9903 0.9751 0.9574 0.9354 0.9066 0.8676 0.8148 0.7476 0.6715 0.5951 0.5248 0.4629 0.4097 0.3643 0.3255 0.2923 0.2637 0.2390 0.2176 0.1989 0.1824 0.1 679 0.1 550 0.1436 0.1333 0.1242 0.1159 0.1084 0.1016 0.0955 0.0898 0.0847 0.0800 0.0756 0.071 7 8 1 .oooo 1 .oooo 0.9889 0.9735 0.9554 0.9328 0.9032 0.8630 0.8087 0.7403 0.6637 0.5877 0.5182 0.4572 0.4049 0.3601 0.3219 0.2892 0.261 1 0.2368 0.21 56 0.1971 0.1809 0.1665 0.1538 0.1425 0.1324 0.1233 0.1151 0.1077 0.1010 0.0949 0.0893 0.0842 0.0795 0.0752 0.0713 9 1.0000 1 .oooo 0.9874 0.9718 0.9534 0.9302 0.8997 0.8582 0.8025 0.7329 0.6560 0.5804 0.51 17 0.4516 0.4001 0.3560 0.3184 0.2862 0.2585 0.2345 0.2136 0.1954 0.1794 0.1652 0.1526 0.1414 0.1314 0.1224 0.1143 0.1070 0.1003 0.0943 0.0888 0.0837 0.0791 0.0748 0.0709 21 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Tabl e 12 - Reducti on factor X - buckl i ng curve “ a” - - x 1.00 .l0 .20 .30 .40 .50 .60 .70 .80 .90 1 .oo 1.10 1.20 1.30 1.40 1.50 1.60 1.70 l B O 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 - - 0 1 .oooo 1 .oooo 1 .oooo 0.9775 0.9528 0.9243 0.8900 0.8477 0.7957 0.7339 0.6656 0.5960 0.5300 0.4703 0.4179 0.3724 0.3332 0.2994 0.2702 0.2449 0.2229 0.2036 0.1867 0.1717 0.1 585 0.1467 0.1362 0.1267 0.1 182 0.1 105 0.1036 0.0972 0.0915 0.0862 0.0814 0.0769 0.0728 1 1 .oooo 1 .oooo 0.9978 0.9751 0.9501 0.921 1 0.8862 0.8430 0.7899 0.7273 0.6586 0.5892 0.5237 0.4648 0.4130 0.3682 0.3296 0.2963 0.2675 0.2426 0.2209 0.2018 0.1851 0.1 704 0.1 573 0.1456 0.1352 0.1258 0.1 174 0.1098 0.1029 0.0966 0.0909 0.0857 0.0809 0.0765 0.0724 2 1 .oooo 1 .oooo 0.9956 0.9728 0.9474 0.9179 0.8823 0.8382 0.7841 0.7206 0.6516 0.5824 0.5175 0.4593 0.4083 0.3641 0.3261 0.2933 0.2649 0.2403 0.2188 0.2001 0.1 836 0.1690 0.1560 0.1445 0.1342 0.1 250 0.1166 0.1091 0.1022 0.0960 0.0904 0.0852 0.0804 0.0761 0.0721 3 1 .oooo 1 .oooo 0.9934 0.9704 0.9447 0.9147 0.8783 0.8332 0.7781 0.7139 0.6446 0.5757 0.51 14 0.4538 0.4036 0.3601 0.3226 0.2902 0.2623 0.2380 0.2168 0.1983 0.1820 0.1676 0.1548 0.1434 0.1332 0.1241 0.1158 0.1084 0.1016 0.0954 0.0898 0.0847 0.0800 0.0757 0.071 7 4 1 .oooo 1 .oooo 0.9912 0.9680 0.9419 0.91 14 0.8742 0.8282 0.7721 0.7071 0.6376 0.5690 0.5053 0.4485 0.3989 0.3561 0.3191 0.2872 0.2597 0.2358 0.2149 0.1966 0.1 805 0.1663 0.1536 0.1424 0.1323 0.1232 0.1150 0.1077 0.1010 0.0949 0.0893 0.0842 0.0795 0.0752 0.0713 5 1 .oooo 1 .oooo 0.9889 0.9655 0.9391 0.9080 0.8700 0.8230 0.7659 0.7003 0.6306 0.5623 0.4993 0.4432 ,03943 0.3521 0.3157 0.2843 0.2571 0.2335 0.2129 0.1949 0.1790 0.1649 0.1524 0.1413 0.1313 0.1224 0.1143 0.1070 0.1003 0.0943 0.0888 0.0837 0.0791 0.0748 0.0709 6 1 .oooo 1 .oooo 0.9867 0.9630 0.9363 0.9045 0.8657 0.8178 0.7597 0.6934 0.6236 0.5557 0.4934 0.4380 0.3898 0.3482 0.3124 0.2814 0.2546 0.2314 0.21 10 0.1932 0.1775 0.1636 0.1513 0.1403 0.1304 0.1215 0.1135 0.1063 0.0997 0.0937 0.0882 0.0832 0.0786 0.0744 0.0705 7 1 .oooo 1 .oooo 0.9844 0.9605 0.9333 0.9010 0.8614 0.8124 0.7534 0.6865 0.6167 0.5492 0.4875 0.4329 0.3854 0.3444 0.3091 0.2786 0.2522 0.2292 0.2091 0.1915 0.1 760 0.1623 0.1501 0.1392 0.1295 0.1207 0.1 128 0.1056 0.099 1 0.0931 0.0877 0.0828 0.0782 0.0740 0.0702 8 1 .oooo 1 .oooo 0.9821 0.9580 0.9304 0.8974 0.8569 0.8069 0.7470 0.6796 0.6098 0.5427 0.4817 0.4278 0.3810 0.3406 0.3058 0.2757 0.2497 0.2271 0.2073 0.1899 0.1746 0.1610 0.1490 0.1 382 0.1285 0.1198 0.1120 0.1049 0.0985 0.0926 0.0872 0.0823 0.0778 0.0736 0.0698 9 1 .oooo 1 .oooo 0.9798 0.9554 0.9273 0.8937 0.8524 0.8014 0.7405 0.6726 0.6029 0.5363 0.4760 0.4228 0.3767 0.3369 0.3026 0.2730 0.2473 0.2250 0.2054 0.1 883 0.1732 0.1598 0.1478 0.1372 0.1276 0.1190 0.1113 0.1042 0.0978 0.0920 0.0867 0.081 8 0.0773 0.0732 0.0694 The buckling curves can be described analytically (for computer calculations) by the equation: 1 x =b + ( P ’ but x S 1 with 4 =0,5 [l +a (x - 0,2) +x*] The imperfection factor a (in equation 3.4) for the corresponding buckling curve can be obtained from the following table: 22 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Table 13 - Reduction factor X - buckling curve “ b” - - x 0.00 .l0 .20 .30 .40 .50 .60 .70 .80 .90 1 .oo 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1 .80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 - - 0 1 .oooo 1 .oooo 1 .oooo 0.9641 0.9261 0.8842 0.8371 0.7837 0.7245 0.6612 0.5970 0.5352 0.4781 0.4269 0.3817 0.3422 0.3079 0.2781 0.2521 0.2294 0.2095 0.1920 0.1 765 0.1 628 0.1506 0.1397 0.1299 0.1211 0.1132 0.1060 3.0994 0.0935 0.0880 0.0831 0.0785 0.0743 D.0704 1 1 .oooo 1 .oooo 0.9965 0.9604 0.9221 0.8798 0.8320 0.7780 0.7183 0.6547 0.5907 0.5293 0.4727 0.4221 0.3775 0.3386 0.3047 0.2753 0.2496 0.2272 0.2076 0.1903 0.1751 0.1615 0.1494 0.1387 0.1290 0.1203 0.1124 0.1053 0.0988 0.0929 0.0875 0.0826 0.0781 0.0739 0.0700 2 1 .oooo 1 . 0000 0.9929 0.9567 0.91 81 0.8752 0.8269 0.7723 0.7120 0.6483 0.5844 0.5234 0.4674 0.41 74 0.3734 0.3350 0.301 6 0.2726 0.2473 0.2252 0.2058 0.1887 0.1 736 0.1 602 0.1483 0.1 376 0.1281 0.1195 0.1117 0.1046 0.0982 0.0924 0.0870 0.0821 0.0776 0.0735 0.0697 3 1 .oooo 1 .oooo 0.9894 0.9530 0.9140 0.8707 0.8217 0.7665 0.7058 0.6419 0.5781 0.51 75 0.4621 0.4127 0.3693 0.3314 0.2985 0.2699 0.2449 0.2231 0.2040 0.1871 0.1722 0.1590 0.1472 0.1366 0.1272 0.1186 0.1109 0.1039 0.0976 0.0918 0.0865 0.0816 0.0772 0.0731 0.0693 4 5 1 .oooo 1 .oooo 1 .oooo 1 . 0000 0.9858 0.9822 0.9492 0.9455 0.9099 0.9057 0.8661 0.8614 0.8165 0.8112 0.7606 0.7547 0.6995 0.6931 0.6354 0.6290 0.571 9 0.5657 0.51 17 0.5060 0.4569 0.4517 0.3653 0.3613 0.3279 0.3245 0.2955 0.2925 0.2672 0.2646 0.2426 0.2403 0.221 1 0.2191 0.2022 0.2004 0.1855 0.1840 0.1708 0.1694 0.1577 0.1565 0.1461 0.1450 0.1356 0.1347 0.1263 0.1254 0.1178 0.1170 0.1102 0.1095 0.1033 0.1026 0.0970 0.0964 0.0912 0.0907 0.0860 0.0855 0.0812 0.0807 0.0768 0.0763 0.0727 0.0723 0.0689 0.0686 0.4081 -0.4035 6 1 .oooo 1 .oooo 0.9786 0.9417 0.901 5 0.8566 0.8058 0.7488 0.6868 0.6226 0.5595 0.5003 0.4466 0.3991 0.3574 0.321 1 0.2895 0.2620 0.2381 0.21 71 0.1987 0.1825 0.1681 0.1553 0.1439 0.1 337 0.1 245 0.1 162 0.1 088 0.1020 0.0958 0.0902 0.0850 0.0803 0.0759 0.0719 0.0682 7 1 .oooo 1 .oooo 0.9750 0.9378 0.8973 0.8518 0.8004 0.7428 0.6804 0.6162 0.5534 0.4947 0.4416 0.3946 0.3535 0.3177 0.2866 0.2595 0.2359 0.2152 0.1970 0.1809 0.1667 0.1541 0.1428 0.1327 0.1237 0.1155 0.1081 0.1013 0.0952 0.0896 0.0845 0.0798 0.0755 0.0715 0.0679 8 1 .oooo 1 . 0000 0.9714 0.9339 0.8930 0.8470 0.7949 0.7367 0.6740 0.6098 0.5473 0.4891 0.4366 0.3903 0.3497 0.3144 0.2837 0.2570 0.2337 0.2132 0.1953 0.1794 0.1654 0.1 529 0.1418 0.1318 0.1 228 0.1 147 0.1074 0.1007 0.0946 0.0891 0.0840 0.0794 0.0751 0.0712 0.0675 9 1 .oooo 1 .oooo 0.9678 0.9300 0.8886 0.8420 0.7893 0.7306 0.6676 0.6034 0.5412 0.4836 0.431 7 0.3860 0.3459 0.31 1 1 0.2809 0.2545 0.2315 0.21 13 0.1936 0.1 780 0.1641 0.1517 0.1407 0.1308 0.1219 0.1139 0.1067 0.1001 0.0940 0.0886 0.0835 0.0789 0.0747 0.0708 0.0672 Eurocode 3, Annex D allows the use of the higher buckling curve “ao” instead of “a” for compressed members of I-sections of certain dimensions and steel grade S 460 [6]. This is based on the fact that, in case of high strength steel, the imperfections (geometrical and structural) play a less detrimental role on the buckling behaviour, as shown by numerical calculations and experimental tests on I-section columns of S460. As a consequence hot formed hollow sections using S 460 steel grade may be designed with respect to buckling curve “a,” instead of “a”. 23 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Table 14 - Reduction factor X - buckling curve “ c” - - h 0.00 .l0 .20 .30 .40 .50 .60 .70 .80 .90 1 .oo 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 - - 0 1 .oooo 1 .oooo 1 .oooo 0.9491 0.8973 0.8430 0.7854 0.7247 0.6622 0.5998 0.5399 0.4842 0.4338 0.3888 0.3492 0.3145 0.2842 0.2577 0.2345 0.2141 0.1962 0.1803 0.1662 0.1537 0.1425 0.1325 0.1234 0.1153 0.1079 0.1012 0.0951 0.0895 0.0844 0.0797 0.0754 0.0715 0.0678 1 1 .oooo 1 .oooo 0.9949 0.9440 0.8920 0.8374 0.7794 0.7185 0.6559 0.5937 0.5342 0.4790 0.4290 0.3846 0.3455 0.31 13 0.2814 0.2553 0.2324 0.2122 0.1945 0.1 788 0.1649 0.1525 0.1415 0.1315 0.1226 0.1145 0.1072 0.1006 0.0945 0.0890 0.0839 0.0793 0.0750 0.071 1 0.0675 2 1 .oooo 1 .oooo 0.9898 0.9389 0.8867 0.831 7 0.7735 0.7123 0.6496 0.5876 0.5284 0.4737 0.4243 0.3805 0.3419 0.3081 0.2786 0.2528 0.2302 0.2104 0.1929 0.1774 0.1636 0.1514 0.1404 0.1306 0.1 21 7 0.1 137 0.1065 0.0999 0.0939 0.0885 0.0835 0.0789 0.0746 0.0707 0.0671 3 1 .oooo 1 .oooo 0.9847 0.9338 0.881 3 0.8261 0.7675 0.7060 0.6433 0.5815 0.5227 0.4685 0.41 97 0.3764 0.3383 0.3050 0.2759 0.2504 0.2281 0.2085 0.1912 0.1 759 0.1623 0.1502 0.1394 0.1297 0.1209 0.1 130 0.1058 0.0993 0.0934 0.0879 0.0830 0.0784 0.0742 0.0703 0.0668 4 1 .oooo 1 .oooo 0.9797 0.9286 0.8760 0.8204 0.7614 0.6998 0.6371 0.5755 0.5171 0.4634 0.4151 0.3724 0.3348 0.3019 0.2732 0.2481 0.2260 0.2067 0.1896 0.1745 0.1611 0.1491 0.1384 0.1287 0.1201 0.1 122 0.1051 0.0987 0.0928 0.0874 0.0825 0.0780 0.0738 0.0700 0.0664 5 1 .oooo 1 .oooo 0.9746 0.9235 0.8705 0.8146 0.7554 0.6935 0.6308 0.5695 0.51 15 0.4583 0.41 06 0.3684 0.3313 0.2989 0.2705 0.2457 0.2240 0.2049 0.1880 0.1731 0.1598 0.1480 0.1374 0.1278 0.1 193 0.1 115 0.1045 0.0981 0.0922 0.0869 0.0820 0.0775 0.0734 0.0696 0.0661 6 1 .oooo 1 .oooo 0.9695 0.9183 0.8651 0.8088 0.7493 0.6873 0.6246 0.5635 0.5059 0.4533 0.4061 0.3644 0.3279 0.2959 0.2679 0.2434 0.2220 0.2031 0.1864 0.1717 0.1585 0.1468 0.1364 0.1269 0.1184 0.1108 0.1038 0.0975 0.0917 0.0864 0.0816 0.0771 0.0730 ‘0.0692 0.0657 7 1 .oooo 1 .oooo 0.9644 0.9131 0.8596 0.8030 0.7432 0.6810 0.6184 0.5575 0.5004 0.4483 0.4017 0.3606 0.3245 0.2929 0.2653 0.241 2 0.2200 0.201 3 0.1849 0.1 703 0.1573 0.1457 0.1354 0.1260 0.1176 0.1 100 0.1031 0.0969 0.091 1 0.0859 0.081 1 0.0767 0.0726 0.0689 0.0654 8 1 .oooo 1 .oooo 0.9593 0.9078 0.8541 0.7972 0.7370 0.6747 0.6122 0 551 6 0.4950 0.4434 0.3974 0.3567 0.321 1 0.2900 0.2627 0.2389 0.2180 0.1996 0.1833 0.1689 0.1561 0.1446 0.1344 0.1252 0.1168 0.1093 0.1025 0.0963 0.0906 0.0854 0.0806 0.0763 0.0722 0.0685 0.0651 9 1 .oooo 1 .oooo 0.9542 0.9026 0.8486 0.7913 0.7309 0.6684 0.6060 0.5458 0.4896 0.4386 0.3931 0.3529 0.31 78 0.2871 0.2602 0.2367 0.21 61 0.1979 0.1818 0.1676 0.1549 0.1436 0.1334 0.1243 0.1161 0.1086 0.1018 0.0957 0.0901 0.0849 0.0802 0.0759 0.0719 0.0682 0.0647 24 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 3.3 Design aids The reduction factor for buckling X is equal to 1 .O for X S 0.2. When this limit is exceeded, the design resistance must take the buckling reduction factor X into acount. For identical X, X is independent of the steel grade (yield strength fv) Figures 4 through 7 allow a quick determination of buckling resistance. The diagrams give the buckling strength as a function of X =T (buckling lengthlradius of gyration) with yield strength of the material as a parameter. Ib Bucklmg strength (N, ’ Vu/A) Nlmm2 l l 400 350 300 250 200 150 100 50 0 0 50 100 150 200 250 I 0 20 40 60 80 Il l d-1) 0 20 40 M] 80 100 l / ( b - l j I I , m I I , , # I , Fig. 4 - Buckling curve for hot-formed hollow sections of S460, basis “a,” (see Table 11) Buckling strength (Nb ’ YM/A) Nlmm’ 450 4M) 350 300 250 200 150 100 50 0 0 50 100 150 200 25 b 20 40 60 80 2 I / ( d- t ) 0 ’ ’20 40 6h 80 100 I / ( b- t l Fig. 6 - Buckling curves for hollow sections of various steel grades, basis “b” (see Table 13) Bucklmg strength (N, m . TM/A) Nlmm’ 450m--- 400 35 f , =460 N/mmz ~ , f y =275 N/mmZ fk =355 N/mm2 f.. =77‘1 Ni mmZ 0 50 100 150 200 250 I I # # I , ~ I # I 0 20 40 60 80 l / l d - t l 0 20 40 60 80 100 Il l b-11 r Fig. 5 - Buckling curves for hollow sections of various steel grades, basis ”a” (see Table 12) Buckling strength (N, ’ ” I A ) Nlmmz ‘5 . 0 20 40 60 80 Il l d-11 OA=f 0 20 40 60 80 ’ I d0 I/l b-t 1 , , , , , , Fig. 7 - Buckling curves for hollow section of various steel grades, basis “c” (see Table 14) For circular and square hollow sections the abscissa values I/(d - 1) or I/(b- t) can approximately replace the slenderness X. This is precisely valid fort e d or t 4 b. 25 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Tubular triangular arched truss for the roof structure of a stadium 26 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 4 Members in bending In general, lateral-torsional buckling resistance need not be checked for circular hollow sections and rectangular hollow sections normally used in practice (b/h 2 0.5). This is due to the fact that their polar moment of intertia I, is very large in comparison with that of open profiles. 4.1 Design for laterial-torsional buckling The critical lateral-torsional moment decreases with increasing length of a beam. Table 15 shows the length of a beam (of various steel grades) exceeding which lateral- torsional failure occurs. The values are based on the relation: I 113 400 -<-.- h- t - f, 1 +3Y y f, =Yield strength in N/mm2 YY = Equation (4.1) has been established on the basis of the non-dimensional slenderness limit XLT =0.4' (see Eurocode 3 [l]), which is defined by the relation: b- t where fCr,LT is the critical elastic stress for lateral-torsional buckling. Equation 4.1 is based on pure bending of a beam (most conservative loading case) for elastic stress distribution (cross section class 3). However, it is also valid for plastic stress distribution (cross section classes 1 and 2). The lowest value for I/(h-t) is 37.7 (S460) according to Table 15. Assuming a size of 100 x 200 mm, the critical length, for which lateral-torsional buckling can be expected, is: I , , =37.7. 0.2 =7.54 m, This span length can be regarded as quite large for the given size (and full utilization of yield strength for yF times load). Tabl e 15 - Limiting I/(h - t) ratios for a rectangular hollow section, below which no lateral-torsional buckling check is necessary - * X , I 0.4 i s also recommended by some other codes 13,211 27 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 5 Members in combined compression and bending 5.1 General Besides concentrically compressed columns, structural elements are most often loaded simultaneously by axial compression and bending moments. This chapter is devoted to classes 1,2 and 3 beam-columns. Thin-walled members (class 4) are considered in chapter 6. 5.2 Design method 5.2.1 Design for stability Lateral-torsional buckling is not a potential failure mode for hollow sections (see chapter 4). According to Eurocode 3 [l ] the relation is based on the following linear interaction formulae: N,, My.Sd +K - Mz,Sd +K,- Mz,Rd 5 1 Nb.Rd where: NSd =Design value of axial compression (yF times load) X =min ( X, , X, ) =Reduction factor (smaller of x, and xz), see chapter 3.2 A =Cross sectional area f, =Yield strength yM =Partial safety factor for resistance My,Sd, Mz,Sd =Maximum absolute design value of the bending moment about y-y or z-z axis according to the first order theory’) M, , , , =W,,,, . f, by elastic utilization of a cross section (class 3) YM or M, , , , =W,,,, . f, by plastic utilization of a cross section (class 1 and 2) YM Mz,Rd =W,,,, . f, by elastic utilization of a cross section (class 3). YM or =W,,,, f, by plastic utilization of a cross section (class 1 and 2) YM ’) increment of bending moments according to the second order theory is considered by determining $ and X, by buckling lengths of whole structural system 28 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t K, =1 -~ x , . Np, p z , however K, I 1.5 (5.6) (2:'; - l), however p, I 0.9 (5.7) For elastic sections (class 3) the value - In the equations (5.5) and (5.7) is taken to be ~ L Z = ( 2@M, z - 4) + - - WPIZ . W€ll,Z equal to 1. PM,, and PM,, are equivalent uniform moment factors according to Table 16, column 2, in order to determine the form of the bending moment distribution M, and M,. Remark 1 : For uni-axial bending with axial force, the reduction factor X is related to the loaded bending axis, as for example, X , for the applied M, with M, =0. Then the following additional requirement has to be fulfilled: Table 16 - Equivalent uniform moment factors pp and p , 1 moment diagram end moments 1 <* <l moment from lateral load v 7 Ma Ma moment due to combined lateral load plus end moments 2 equivalent uniform moment factor pu M, = I max M 1 due to lateral load only AM = I max M I for moment diagram without change of sign I maxM( +I mi nMI where sign of moment changes 3 equivalent uniform moment factor p, Dm,+ =0.66 +0.44 4, however p,,+ B 1 - - and p,,+ B 0.44 N NKi P,,, =1.0 29 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Remark 2: A further design method for the loading case of bending moment and axial compression is available in the literature [21, 22, 231, which is called “substituting member method” [24, 251. It is based on the formula for uni-axial bending moment and axial force’), which is used frequently: NSd P m ’ 1 +-- . 5 1 ‘y ’ Npl ,Rd Sd 1 -~ N, , ’ ’ Y where, besides the definitions already described, (5.9) r2. El N, , =- - - ‘g - Np l (Eulerian buckling load) x2 P, =Equivalent uniform moment factor from Table 16, column 3. P, < 1, allowed only for fixed ends of a member and constant compression without lateral load M, , , , according to equation (5.3) (elastic or plastic) Equation (5.9) can be written conservatively in a simplified manner: NSd P m ’ My.Sd ‘y ‘ Npl ,Rd + 5 0.9 (5.9a) 5.2.2 Design based on stress A compressed member has to be designed on the basis of the most stressed cross section in addition to stability. Axial force, bending moments M, and M, and shear force have to be considered simultaneously. According to Eurocode 3 [l], an applied shear force V,, can be neglected, when the following condition is fulfilled: ‘Sd 0.5 ‘pI,Rd (5.10) where v,,,,, =Design plastic shear resistance of a cross section =2t.d;- f i . h fi . YM f, for CHS (5.1 1) f =2t. h,. 2- (5.12) for RHS (b, instead of h, when shear force is parallel to b) A, =21 dm or 2t. h, ’) Corresponding formulae for uni- or bi-axial bending and axial force are given in [21, 231 30 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Equation (5.10) is satisfied in nearly all practical cases. In some other codes [21] the limiting values for ~ , up to which the shear force can be disregarded, is significantly lower than 0.5. ' Sd Vpl,Rd 5.2.2.1 Stress design without Considering shear load [l] The following relationship is valid for plastic design (cross section classes 1 and 2): where cy =p =2 for CHS c y = @ = - , however 5 6 1 - 1.13n2 (5.13) (5.14) (5.15) YM M, , , , , and MNz,Rd are the reduced plastic resistance moments taking axial forces into account. These reduced moments are described by the relations given below. For rectangular hollow sections: = Mpl,y,Rd (l - n), however Mpl,y,Rd (5.16) MNz.Rd =Mpl.z.Rd 0.5(+h, 1 however Mpl,z,Rd l - n For square hollow sections: M, , , , =1,26 Mpl,Rd (1 - n), however 5 MP,,,, For circular hollow sections: M, , , , =1,04. MP, (1 - n'.'), however 5 MP, (5.1 7) (5.18) (5.19) For circular hollow sections, the following exact and simple equation [23] is also valid instead of the equation (5.19): M Sd Mpl,Rd (5.20) where MSd =i - (5.21) But the shear force must be limited to ~ 5 0.25 For elastic design the following simple linear equation can be applied instead of the equation (5.13): 'Sd 'pl.Rd Sd My Sd Sd + L + C 1 1 A . fyd wel.y . fyd fyd (5.22) where fyd =fJ yM This equation can also be used, as a lower bound, but more simple to use, for plastic design of cross section classes 1 and 2 instead of the equation (5.13). 31 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 5.2.2.2 Stress design considering shear load [l] If the shear load Vs, exceeds 50% of the plastic design resistance of the cross section the design resistance of the cross section to combinations of moment and axial force shall be calculated using a reduced yield strength for the shear area, where: red. f, =(1 - e ) f, e =6%-l>’ (5.23) (5.24) Vpl,Rd is according to equation (5.1 1) or (5.12). For circular hollow section: A, =- 2A For rectangular hollow section: - shear load parallel to depth: A, =- Ah b +h - shear load parallel to width: A, =- Ab b +h For circular hollow section, the following exact but simple equation can be given taking also the shear force into account (231: a M Sd Mpl,Rd where 7 = (5.25) (5.26) ‘Sd =i-i (5.27) V, , , , , is according to the equation (5.1 1). M, , is according to the equation (5.21). No reduction for f, as shown in the equation (5.23) has to be made. 32 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Uni-planar tubular broken-off truss Tubular supports for a canvas roof construction 33 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 6 Thin-walled sections 6.1 General The optimisation of the buckling behaviour of hollow sections leads, for a constant value of cross sectional area, to profiles of large dimensions and small thicknesses (large moment of inertia). Small thicknesses (relative to outer dimensions) can cause failure, before reaching yield strength in the outer fibres, by local buckling. The unavoidable imperfections of the profiles involve an interaction between local buckling in the cross section and flexural buckling in the column. This decreases the resistance to both types of buckling. By keeping within the dlt or blt limits for the respective cross section classes given in Tables 4, 5 and 6, it is not required to check local buckling. Only when exceeding the dlt or blt limits for class 3 sections, does the influence of local buckling on the load bearing capacityof the structural members have to be taken into account. The cross section thus involved shall be classified as class 4 (see Fig. 2). It should be noted that the phenomenon of local buckling can become more critical by applying and utilizing higher yield strength, so that smaller b/t ratios have to be selected (see Tables 4 and 5, last line). Eurocode 3 [l] takes account of local buckling by the determining the load bearing capacity using effective cross section dimensions, which are smaller than the real ones. In the structures, which are dealt with in this book, circular hollow sections with a dlt ratio higher than the limiting values given in Table 4 are seldom used; in general, dlt values are 50 at the highest. In consequence, this chapter is mainly devoted to class 4 square and rectangular hollow sections. 6.2 Rectangular hollow sections 6.2.1 Effective geometrical properties of class 4 cross sections The effective cross section properties of class 4 cross sections are based on the effective widths of the compression elements. The effective widths of flat compression elements shall be obtained using Table 17. The plate buckling reduction facor 4 shall be calculated by means of the relations given in Table 18. For the sake of simple calculation, the equation (6.2) and (6.1) are described in Fig. 8 ( e =f(X)) and Fig. 9 (k, =f(4)). In order to determine the effective width of a flange element, the stress ratio 4 used in Table 17 shall be based on the properties of the gross (not reduced) cross section. To calculate the effective depth (h,”) of web elements, the effective area of the compressed flange (be,, . t) but the gross area of the webs (h . t) has to be used. This simplification allows a direct calculation of effective widths. Strictly speaking, an exact calculation of the effective width of a web element requires an iterative procedure. Under bending moment loading it is possible that the effective (reduced) width becomes valid only for one flange. This results in a mono-symmetrical cross section with a corresponding shift of the neutral axis. As a consquence, the effective section modulas has to be calculated with reference to the new neutral axis. Note: Eurocode 3 [ l , 21 is not consistent regarding the definition of a so-called “thin-walled profile”. 34 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Table 17 - Effecti ve wi dths and buckl i ng factors for thin-walled rectangular hollow sections stress distribution (compression positive) b , = h - 3 t o r b - 3 t effective width ben ben =e . bt bel =0.5 be, be, =0.5 ben $ =a2/a, 0 + 1 > $ > 0 + l buckling factor Alternatively: for 1 2 $ z - 1 3.2 4.0 km 1.05 - $ 7.81 k, = 16 h1+$)2+0. 112(1 -$)2+(1 +$) ben =e . bc bel =0.4ben be, =0.6 be, 0 > $ > - 1 1 - 1 I - 1 > $ > - 2 7.81 - 6.29$ +9.78$? 5.98 ( 1 - 4)' 23.9 Plate buckllng reducton tactor p Buckling factor K, 1. 0, , , r , 1 60 I Nondomensional slenderness x p Fig. 8 - Plate buckling reduction factor p 35 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Tabl e 18 - Plate buckling reduction factor e I where xp, the non-dimensional slenderness of the flat compression element, is given by: where f, is the critical plate buckling stress and k, is the plate buckling factor (see Table 17 and Fig. 9) 1 with t =e and f, =yield strength in N/mm2 Reference [2] considers that the influence of the internal corner radius need not to be taken into account provided that: r s 5 t These conditions are fulfilled by practically all actually produced square and rectangular hollow sections. The b,/t limit above which local buckling needs to be taken into account according to Tables 8 and 9 is blt > 42 E for a uniformly compressed flange. However equation (6.2) in Table 18 for an identically loaded flange gives X, > 0.673; this results in b,/t > 38.2 E , some what smaller than the 42 E above. It is well known, that the equation (6.3) for plate buckling gives conservative results. On account of this, possible local buckling of thin-walled sections has to be considered first, when the b,/t limits given in Tables 5 through 7 are exceeded. 6.2.2 Design procedure When the effective geometrical properties of a class 4 cross section, e. g. effective area A, , , effective radius of gyration ieff, effective section modulus Wef f , have been calculated, it is easy to check the stability and the resistance. Indeed, it is just necessary to use these effective properties in place of the geometrical properties of the gross section in class 3 calculations. For dimensioning thin-walled cross section, equation (5.22) is replaced by the relation: NSd + + Mz.Sd Aeff ’ Iyd ’ fyd weff,z ’ fyc! 5 1 4 with fyd = YM Hollow sections have two axes of symmetry and therefore there is no shift of the neutral axis when the cross section is subject to uniform compression. This leads to an important simplification of class 4 beam-column equations, because additional bending moments due to this shift do not exist in the case of structural hollow section. The use of effective geometrical properties of thin-walled sections is recommended in the codes of the most countries around the world. Only in the japanese code, the load bearing capacity of a thin-walled rectangular hollow section is given by the smaller of the maximum plate buckling load and global buckling load. At last, as shown in reference [ 101, the lateral-torsional buckling can also be disregarded for thin-walled hollow sections of class 4. 36 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 6.2.3 Design aids For practical application, the transition from the cross section class 3 to class 4 is of special imporance showing the blt limits, below which local buckling can be disregarded. With e =1, the equation (6.2) leads to the limit x,, S 0.673. Fig. 10 gives -on base of the depth or width-to-thickness ratio and of the k, coefficient (Table 17) as well as of the yield strength f, - the possibility of a quick check of the zone where no allowance for local buckling is necessary. The area to the left of the curves belongs to cross section class 3, while that to the right covers class 4, all of them lying in the elastic range. When blt limits given by the curves are exceeded (local buckling), the plate buckling reduction factor e according to the equation (6.2) has to be determined. KO fv (N/mm21 = 460 355 275 235 50 40 30 23.9 20 10 4 0 25 50 75 100 125 150 - bl or . ! ! L Fig. 10 - b,/t or h,/t limits, below which local buckling can be disregarded P 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0 0 10 20 30 40 50 60 70 I bl i t - W Fig. 11 - Plate buckling curves Fig. 12 - Effective RHS cross section under axial force N and bending moments M,, M, 37 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Plate buckling reduction factor e vs. - for various structural steel grades is drawn in Fig. 11 (see equation 6.3). Effective geometrical values for the cross sections of class 4 can be calculated by means of the formulae given in Table 19. The notations in Table 19 are explained in Fig. 12. b,/t f i o Table 19 - Effective geometrical properties axial force: Aen 2t (ben +h," +41) bending moments: W,,, =t I (be,, +2t) (>- 6, ) - 2 (+- 6> (h,,, +ben +2t) 6.3 Circular hollow sections For thin-walled circular hollow sections, it is more difficult to judge the local buckling behaviour, especially the interaction between global and local buckling, than in the case of plates. This is due to the local instability behaviour of cylindrical shells, their high susceptibility to imperfections and sudden reduction of load bearing capacity without reserve [23]. Local buckling has also to be considered for CHS, when the d/t limits for the cross section 3 are exceeded (see Tables 4 and 7) . 38 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Circular hollow sections, which are applied in practice, do not or seldom, possess dlt ratios exceeding those given in Tables 4 and 7; in general dlt S 50. In cases, where thin-walled circular hollow sections are applied, the procedure of substituting the yield strength f, in the already mentioned formulae by the real buckling stresses. for a short cylinder, can be used. These buckling stresses can be calculated by the procedure shown in [26] or [27]. The procedures in both cases are simple; however, there is no equation describing the buckling stress explicitly. * U" in [26]; uXS,RK in [ 27] 39 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 7 Buckling length of members in lattice girders 7.1 General Chord and bracing members of a welded lattice girder are partially fixed at the nodes, although the static calculation of the forces in the members is carried out assuming the joints to be hinged. As a consequence of this partial restraint, a reduction of the system length I is made to obtain the effective buckling length l b. 7.2 Effective buckling length of chord and bracing members with lateral support The buckling of hollow sections in lattice girders has been treated in [14, 15, 281. Based on this, Eurocode 3 [ l , 2 - Annex K] recommends the buckling lengths for hollow sections in lattice girders as follows: Chords: - in-plane: I, =0.9 x system length between joints - out-of-plane: I , =0.9 x system length between the lateral supports Bracings: - in- and out-of-plane: I b =0.75 x system length between joints. When the ratio of the outer diameter or width of a bracing to that of a chord is smaller than 0.6, the buckling length of the bracing member can be determined in accord with Table 20. The equations given are only valid for bracing members, which are welded on the chords along the full perimeter length without cropping or flattening of the ends of the members. Due to the fact that no test results are, at present time, available on fully overlapped joints, the equation given in Table 20 cannot be applied to this type of joint. Fully overlapped pmls In both of the last cases, a buckling length equal to the system length of the bracing member has to be used. 7.3 Chords of lattice girders, whose joints are not supported laterally The calculation is difficult and lengthy. Therefore, it is convenient to use a computer. For laterally unsupported truss chords the effective buckling length can be considerably smaller than the actual unsupported length. References (12, 151 give two calculation methods for the case of compression chords in lattice girders without lateral support. Both methods are based on an iterative method and require the use of a computer. However, in order to facilitate the application for commonly encountered cases (laterally restrained in direction), 64 design charts have been drawn and appear as appendices in CIDECT Monograph no. 4 [15]. The effective buckling length of a bottom chord loaded in compression (as for example, by uplift loading) depends on the loading in the chord, the torsional rigidity of the truss, the 40 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t bending rigidity of the purlins and the purlin to truss connections. For detailed information, reference is given to [ 12, 151. For the example given in the following figure, the buckling length of the unsupported bottom chord can be reduced to 0.32 times the chord length L. buckllng length bottom chord l t ~ =- 0.32 L IPE 140 ' Q 1 3 9 . 7 ~ 4 Q 60x 3 Q 1397x 4 Lateral buckllng of laterally UnSUpPOrIed chords Tabl e 20 - Buckling length of a bracing member in a lattice girder do: outer diameter of a circular chord member dl: outer diameter of a circular bracing member bo: external width of a square chord member dl dl bl b,: external width of square bracing member for all P: Ib/1 S 0.75 when p <0.6, in general 0.5 5 - S 0.75 calculate with: Ib I 41 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Lattice girder of square hollow sections supported by a cable construction General view of a RHS roof structure 42 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 8 Design examples 8.1 Design of a rectangular hollow section column in compression Nqd =1150 k N 1150 kN rl /2 1 =4m _I Y - V v @m l I t Fig. 13 - Column under concentric compression A column is to be designed using a rectangular hollow section 300 x 200 x 7.1 mm, hot- formed with a yield strength of 235 N/mm2 (steel grade S 235). The length of the column is 8 m. It has hinged support at both ends. An intermediate support at the middle of the column length exists against buckling about the weak axis 2-2. Given: Concentric compression (design load) NSd =1150 kN buckling length: =8 m l b,z =4 m steel grade: S235; f , =235 N/mm2 geometric properties: A =67.7 cm2; iy =11.3 cm; iz =8.24 cm b, max. - =300 - ‘ 7’1 =39.25 < 42 (compare with Tab. 5 and 6) t 7.1 X =- 8oo =70.8; X, =- =48.6 <Xy 400 Y 11.3 8.24 - x =- - L - 70 - 0.754 (see Tab. loa) y 93.9 xY =0.821 (Tab. 12, buckling curve “a”) Acc. to equation (3.1): N, , , , =0.821 .6770. - . =1187 kN > 1150 kN. Therefore column okay. 235 1.1 8.2 Design of a rectangular hollow section column in combined compression and uni- axial bending Fig. 14 - NSd =800 k N 2 - 2 %,rd Column under combined compression and uni-axial bending 43 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t given: hot-formed rectangular hollow section column 300 x 200 x 8 mm compression Nsd =800 kN bending moment My,Sd =60 kNm or 18 kNm at both ends buckling length l b., = =8.0 m steel grade S275; f, =275 Nlmm' W, =634 cm3; W, =510 cm3 WPl,, =765 cm3; Wpl,z =580 cm3 geometric properties: A =75.8 cm'; i, =11.2 cm; iz =8.20 cm - - b, 200 - 3.8 =22 t 8 - - h, 300 - 3.8 =34.5 (Tables 5 and 6) - < 38.0.92 =35 for class 2 cross section of S275 - t 8 a) Calculation for flexural buckling: / A =~ =71.4; 800 y 11.2 A, =~ =97.6 800 8.2 - A =- 71'4 =0.823 (see Tab. loa); - 97.6 86.8 Y 86.8 X, =- =1.124 i cy =0.782 (see Tab. 12, buckling curve "a"); X , =0.580 Acc.toTable16:~,,,=1.8-0.7~0.3=1.59 Acc. to equation (5.5): p, =0.823 (2 . 1.59 - 4) +765 - 634 = - 0.468 < 0.9 634 Acc. to equation (5.4): K, =1 - (- 0'468) . ' O3 =1.23 1.5 0.782.7580.275 Calculation for the stability about y-y axis acc. to equation (5.1): 800.103.1.1 + '23 ' 6o ' lo6 ' =0.540 +0.386 =0.926 < 1 .O 0.782 ' 7580.275 765. lo3. 275 Calculation for buckling about z-z axis: NSd Nb.z.Rd 800 < 0.580 .7580 . 275 ' =1099.1 kN. Therefore column okay. b) Calculation for the load bearing capacity Shear load V: 'y.Sd =~ - - - 5.25 kN 8 Acc. to equation (5.1 1): =2 . 8 (300 - 8) . 275 ' =674 kN 4 3 . 1 . 1 ---- 5'25 - 0.008 < 0.5 "Pl.y,Rd - 674 The shear load can be disregarded. Acc. to equation (5.13): My,Sd I " (MNy,Rd) M, , , , =60 kNm (max) ff= 1.66 NSd 800. 1.1 =o,422 n =- 1 - 1.13 n2 Npl Rd 75.8 ' 27.5 - - 1.66 1 - 1.13. 0.422' =2.07 44 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t ACC. to equation (5.16): M,,,,, =1.33 765 lo3 1.1 275 ( 1 - 800-103-1.1) 7580 275 =147. lo6 Nmm =147kNm ( % ) m =(~)'"' =0.156 c 1.0. Therefore column okay. MNy,Rd 8.3 Design of a rectangular hollow section column i n combined compression and bi-axial bending Nld =1000 kN 1000 kN 2 - 2 My,rd v - v Mz. rd Fig. 15 - Column under combined compression and bi-axial bending Given: Hot formed rectangular hollow section column 300 x 200 x 8.8 mm The length of the column is 8 m. Both ends of the columns have hinged support about the strong axis y-y and fixed support at the foot end about the weak axis z-z. Compression N, , =1000 kN Bending moment M,,, =60 kNm Mz,Sd =50 kNm Steel grade: S355; f, =355 Nlrnm' Buckling length: lb., =8 m lb., =0.7.8.0 =5.6 m Geometric properties: A =82.9 cm3 W, =689 cm3; W, =553 cm3 Wp,,y =834 cm3; W,,,, =632 cm3 iy =11.2 cm; i, =8.16 cm The cross section just satisfies the requirements for the class 2 of S355 (Tables 5 and 6). 45 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t a) Calculation for the global buckling acc. to equation (5.1) X, =- =71.4 X, =- =68.6 560 11.2 8.16 - 71.4 X, =- =0.935 X, =- =0.898 - 68 6 76.4 76.4 X , =0.71 1 (= X,,,) x Z =0.735 (buckling curve “a”) Acc. to equation (5.2): N, , , , , , =0.71 1 . 8290. - . =1902 kN (= min Nb,R,,) 355 1.1 Nb,,,,d =0.735 ’ 8290. F . i o-3 =1966 kN ACC. to equation (5.3): MP,,,,,, =834. lo3. - - =269 kNm =632. lo3. =204 kNm 355 1 .l Acc. to Tab. 16: OM,, =1.8 Acc. to equation (5.5): py =0.935 (2. 1.8 - 4) +(g - l) =- 0.164 < 0.9 Acc. to equation (5.4): K, =1 - (- 0.164) 1000. lo3 0.71 1 .a290 * 355 =1.078 < 1.5 Acc. to Tab. 16: OM,, =1.8 - 0.7(- 0.5) =2.15 Acc. to equation (5.7): =0.898 (2. 2.15 - 4) +( g: : - - l) =0.412 < 0.9 Acc. to equation (5.6): K, =1 - 0.412 1000 . lo3 =o.809 < ,5 0.735.8290 * 355 Finally, acc. to equation (5.1): - 1902 269 ’Oo0 + ’ 6o + * 50 =0.526 +0.240 +0.198 204 =0.964<1.0 b) Calculation for load bearing capacity In order to obtain sufficient load bearing capacity of the cross section the “elastic” equation (5.22) is applied conservatively (all values in kN and mm): 1000~1:1 + 60.103.1.1 + 8290 .0.355 689 . l O3 .0.355 553 . l O3 .0.355 50 * lo3 . =0.374 +0.270 +0.280 =0.924 < 1 .O If this calculation would not have led to a satisfactory result (that means 1 .O), then the calculation must be carried out using equation (5.1 3). The assumption to neglect shear load in equations (5.13) and (5.22) is V,, S 0.5 V,,,,,, see equation (5.10) [l , 21. The shear resistance acc. to equation (5.12) is decisive in this case: Vpl,z,Rd =2 ’ 8.8 (200 - 8.8) ~ 355 d3- 1. 1 =627 kN ‘Sd vpl,Rd -- - 0.015 < 0.5. Therefore shear is not critical. 46 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 8.4 Design of a thin-walled rectangular hollow section column in compression 2 - 2 V - V Fig. 16 - Thin-walled column under concentric compression Given: Cold-formed rectangular hollow section column 400 x 200 x 4 mm (acc. to IS0 4019 11 71) The length of the column is 10 m. Both ends of the column have hinged support about the strong axis y-y and fixed supports at both ends about the weak axis z-z. Steel grade: S275, f, =275 N/mm2 (basic hot rolled strip) Buckling length: lb., =10 m =2 =5 m 10 N, , =500 kN Cross sectional area A =46.8 cm2 1. Calculation of average increased yield strength after cold-forming Acc. to equation (1.2): f , , =275 + 400 +200 14’4 (410 - 275) =287.6 N/mm2 .: 1.2.275 =330 Nlmrn’ 2. Cross section classification Long side: - - Short side: r - h, 400 - 3. 4 t - 4 =g71 > 42 c$ =38.8 (Tables 5 and 6) b, 200 - 3. 4 =47 - 4 The cross section is thin-walled (class 4) and the calculation shall be made using effective width. According to Fig. 8, the limit for plate buckling: Xp, ,lml, =0.673 (xp acc. to equation (6.2) with e =1.0). Non-dimensional slenderness taking yield strength of the basic material f , b acc. to equation (6.3): - 97 =28.4 . dTi- - - 1.85 > 0.673 - Ap,z = 28.4 ’ 47 - =0.90 > 0.673 47 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Non-dimensional slenderness taking average increased yield strength f , , (287.6 N/mm2) after cold-forming: - 97 hp. , = xp,, = 28.4. izi- =1.89 > 0.673 - 47 28.4. dTd235/287.6 =0.92 > 0.673 In all cases, the cross section belongs to class 4. 3. Effective geometric values a) With yield strength of the basic material f , , (275 N/mm2) and K, =4 (simple compression): e, =0.476 e, =0.840 1 acc. to equation (6.2) heft =0.476 (400 - 3.4) =184.7 mm be, =0.840 (200 - 3.4) =157.7 mm A,, =28.69 cm2 ieff,y =17.50 cm acc. to Tab. 19 i ,,,, = 8.32 cm ace. to Tab. 17 1 b) With average increased yield strength after cold forming (fya =287.6 N/mm2) e , = 0.468 e , = o.827 ] acc. to equation (6.2) h, , =0.468 (400 - 3.4) =181.6 mm be, =0.827 (200 - 3. 4) =155.5 mm Aetf =28.25 cm2 ie+,y =17.60 cm ief,., = 8.33 cm 1 acc. to Tab. 17 4. Design for global buckling a) With yield strength of the basic material (fyb =275 N/mm2): 0 Strong axis x, - 17.5 ’Oo0 - 57.1 - X, -- - 86.8 = 57’1 0.66 (seeTab. loa) x , =0.806 (acc. to Tab. 13, curve “b”) Nb, , , , =0.806.2869 . 1.1=578 kN (see equation (3.1)) 0 275 0 Weak axis x, =- - 8.32 500 - 60.1 - 60.1 X, =86.8 =0.69 x , =0.7893 (acc. to Tab. 13, curve “b”) Nb,Rd =0.7893.2869. - =566 kN 0.275 1.1 48 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t b) With average increased yield strength after cold-forming (289 N/mm2): XE =93.9 J235/287.6 =84.9 (see Tab. loa) 0 Strong axis X, --- - 17.6 ’Oo0 - 56.8 - X, -- - 84.9 = 56‘8 0.67 > 0.2 xY =0.743 (acc. to Tab. 14, curve “c”) N, , , , =0.743 * 2825 * ~ =549 kN 0.2876 1.1 0 Weak axis X, - 8.33 500 - 60.0 - 60 X, =84.9 =0.71 xZ =0.719 (acc. to Tab. 14, curve “c”) N, , , , =0.719.2825 * L =531 kN 0 2876 1 .l Conclusion: Assuming both criteria (basic and average increased yield strength, the design com- pressive load (= 500 kN) lies lower than the calculated lead bearing capacity. The calculated values for the strong and weak axis differ by a small margin from each other. An economic selection of the cross section has been made. 8.5 Design of a thin-walled rectangular hollow section column in concentric com- pression and bi-axial bending N,d 250 kN 250 kN 2 - 2 My.rd v - v M2, Sd Fig. 17 - Thin-walled column under combined compression and bi-axial bending Given: Cold-formed rectangular hollow section column 400 x 200 x 4 mm. Bending moments: My,,, =25 kNm and 12.5 kNm at the ends of the column Concentric compression N, , =250 kN M, , , , =12.5 kNm and - 12.5 kNm at the ends of the column 49 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Under bending moment the yield strength of the basic material is always to be assumed even for cold-formed profiles. The strain hardening of cold-formed section is desregarded. Steel grade: S275; f, =f,,b =275 N/mmz Column system length I =10 m Buckling lengths: lb., =10 m 10 l b., =- =5 m From design example 8.4: X, =0.806 --t x, =0.66 xZ =K,,,," =0.7893 + X, =0.69 - heft =184.7 mm be,, =157.9 mm Aeff =28.69 cm2 i,,,,, =17.5 cm i ef f ,z =8.32 cm Ratio of the end moments: 4, =g =0.5 PM,, =1.45 acc. to Tab. 16, - 12.5 second column $2 =m = - 1.0 =2.50 Further effective geometric values acc. to Tab. 19: 6, = 5.2 mm 6, =20.2 mm W,,,,, =482.2 cm3 =219.9 cm3 - Ace. to equation (5.5): p, =X, (2p,,, - 4) =0.66 (2. 1.45 - 4) = - 0.726 <0.9 Acc. to equation (5.4): K, =1 - - 0.726 1 250 . lo3 =, ,256 < ,5 0.806.2869.275 Acc. to equation (5.7): F, =0.69 (2 . 2.50 - 4) =0.69 0.9 Acc. to equation (5.6): K, =1 - 0.69 250. lo3 =0.722 1.5 0.7893.2869.275 Calculation to check stability acc. to equation (5.1): 250000. 1.1 + 1.256.25.106.1.1 + 0.722.12.5-106.1.1 0.806.2869.275 482.2. l o3. 275 21 9.9 . l O3 .275 =0.432 +0.260 +0.164 =0.856 < 1.0 Calculation to check maximum stress at the foot end acc. to equation (5.22): 250.103.1 .I + 25. lo6. 1.1 + 12.5. lo6. 1 .l 2869.275 482.2. lo3. 275 219.9 - lo3. 275 =0.348 +0.207 +0.227 =0.782 <1.0 Conclusion: The cross section 400 x 200 x 4 mm satisfies the requirements 50 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 9 Symbols Gross area of the cross section Effective area of the cross section Circular hollow section Modulus of elasticity Calculated value of an action Shear modulus Moment of inertia Effective moment of inertia Amplification co-efficient for a beam-column (see equations 5.1, 5.4, 5.6) Reduced design plastic resistance moment allowing for the axial force Design value of the bending moment Design value of the buckling resistance of a compression member Plastic design value of the resistance of a compression member Design value of the axial force Resistance Rectangular hollow section Plastic design shear resistance Design value of the shear force Section modulus Effective section modulus Plastic section modulus External width of RHS Width of a flat element (see Tab. 6) Average width of RHS (b - t ) Average width of RHS (h - t ) External diameter of CHS Critical plate buckling stress Ultimate tensile strength of the basic material of a hollow section Tensile yield strength Average design yield strength of a cold-formed section Tensile yield strength of the basic material of a hollow section Design yield strength (=4> Critical stress (elastic) for lateral buckling External depth of RHS Radius of gyration Effective radius of gyration Buckling factor (see Tab. 18) Length Effective buckling length Internal corner radius for RHS Wall thickness 51 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Y Z Py P2 e e V X II. Strong axis of the cross section Weak axis of the cross section CO-efficient of linear expansion (see Tab. 2) Imperfection co-efficient of the buckling curves Exponents of the criterion for the resistance of a beam-column Equivalent uniform moment factor (see Tab. 16) Ratio of the width minus thickness to depth minus thickness of RHS Partial safety factor for the resistance Shift of the neutral axis of a thin-walled section Ultimate strain Yield strain Slenderness of a column Eulerian slenderness Non-dimensional slenderness of a column Non-dimensional slenderness of a flat plate for lateral-torsional buckling Non-dimensional slenderness of a flat plate Co-efficient used for a beam-column (see equations 5.5 and 5.7) Poisson’s ratio Density Reduction factor of the yield strength to take account of the shear force and effective width Reduction factor for buckling curves (see Fig. 3) Stress or moment ratio (see Tab. 17) 52 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 10 References [ l ] EC3: Eurocode no. 3, Design of Steel Structures, Part I General Rules and Rules for Buildings. Commission of the European Communities, chapters 1 to 9, EN 1993-1-1: 1992. [2] EC3: Eurocode no. 3, Design of Steel Structures, Part 1 General Rules and Rules for Buildings. Commission of the European Communities - annexes, EN 1993-1-1: 1992. [3] SSRC: Stability of Metal Structures - A World View. Structural Stability Research Council, 2nd Edition, 1991. [4] Sherman, D. R.: Inelastic Flexural Buckling of Cylinders. Steel Structures - Recent Research Advances and their Application to Design, International Conference, Budva, M. N. Pavlovic editor, Elsevier, London, 1986. [5] J ohnston, B. G.: Column Buckling Theory - Historic Highlights. A. S. C. E., J ournal of the Structural Division, Vol. 109, no. 9, September 1983. [6] EC3: Eurocode no. 3, Design of Steel Structures, Part 1 - General Rules and Rules for Buildings. Annex D - The Use of Steel Grades S460 and S420, Commission of the European Communities, ENV 1993-1-1 : 1992/A1: 1994. [7] Beer, H., and Schulz, G.: The European Buckling Curves, International Association for Bridge and Structural Engineering, Proceedings of the International Colloqium on Column Strength, Paris, November 1972. [8] Austin, W. J .: Strength and Design of Metal Beam-Columns, A. S. C. E. J ournal of the Structural Division, Vol. 87, no. 4, April 1961. 191 Chen, W. F., and Atsuta, T.: Theory of Beam-Columns, Volume 1 : In-Plane Behaviour and Design. Mc.Graw Hill, New-York, 1976. 1101 Rondal, J ., and Maquoi, R.: Stabilite des poteaux en profils creux en acier, Soditube, Notice 1 1 17, Paris, Mai 1986. [ l l ] Ellinas, C. P., and Croll, J . G. A.: Design Loads for Elastic-Plastic Buckling of Cylinders under Combined Axial and Pressure Loading, Proceedings of the BOSS '82 Confe- rence, Boston, August 1982. 1121 CIDECT: Construction with Hollow Steel Sections, ISBN 0-9510062-07, December 1 984. [l31 Grimault, J . P.: Longueur de flambement des treillis en profils creux soudes sur membrures en profils creux, Cidect report 3E-3G-8013, J anuary 1980. [l41 Rondal, J.: Effective Lengths of Tubular Lattice Girder Members, Statistical Tests, Cidect report 3K - 8819, August 1988. [l51 Mouty, J .: Effective Lengthsof Lattice Girder Members, Cidect, Monograph no. 4, 1980. [l61 lSO1DlS 657-14: Hot-rolled steel Sections; Part 14: Hot formed structural hollow sections - Dimensions and sectional properties, Draft Revision of Second edition IS0 657: 14 - 1982. [l71 IS0 4019: Cold-finished steel structural hollow sections - Dimensions and sectional properties, 1 st edition, 1982. [l81 IS0 630: Structural Steels, 1st edition, 1980. 53 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 11 91 I201 1221 1231 1241 1251 1271 1281 1291 1301 131 I 1321 IIW XV - 701/89: Design Recommendations for hollow section joints - Predominantly statically loaded, 2nd Edition, 1989, International Institute of Welding. EN 10 21 0: Hot finished structural hollow section of non-alloy and fine grained structural steels Part 1 : Technical delivery requirements, 1994. Part 2: Tolerances, dimensions and sectional properties (in preparation). DIN 18 800, Teil 1: Stahlbauten, Bemessung und Konstruktion, November 1990. Teil2: Stahlbauten, Stabilitatsfalle, Knicken von Staben und Stabwerken, November 1990. ECCS-CECM-EKS: European Recommendation for Steel Structures - 2E, March 1978 Dutta, D. and Wijrker K.-G.: Handbuch Hohlprofile in Stahlkonstruktionen, Verlag TUV Rheinland GmbH, Koln 1988. Roik, K. and Kindmann, R.: Das Ersatzstabverfahren - Tragsicherheitsnachweise fur Stabwerke bei einachsiger Biegung und Normalkraft, Der Stahlbau 91982. Roik, K. and Kindmann, R.: Das Ersatzstabverfahren - eine Nachweisform fur den einfeldrigen Stab bei planmal3ig einachsiger Biegung mit Druckkraft, Der Stahlbau 12/1981. European Convention for Constructional Steelwork (ECCS-EKS): Buckling of Steel shells, European Recommendations (section 4.6 als selbstandige Schrift), 4th Edition, 1988. DIN 18 800, Teil4: Stahlbeton, Stabilitatsfalle, Schalenbeulen, November 1990. Sedlacek, G., Wardenier, J ., Dutta. D. and Grotmann, D.: Eurocode 3 (draft), Annex K - Hollow section lattice girder connections, October 1991. prEN 10 21 9-1, 1991 : Cold formed structural hollow section of non-alloy and fine grain structural steels, Part 1 -Technical delivery conditions, EClSS/TC 1O/SC 1, Structural Steels: Hollow Sections. Boeraeve, P., Maquoi, R. and Rondal, J .: Influence of imperfections on the ultimate carrying capacity of centrically loaded columns, 1st International Correspondence Conference "Design Limit States of Steel Structures", Technical University of Brno, Czechoslovakia, Brno, 1983. EN 10 025: Hot-rolled products of non-alloy structural steels, Technical delivery conditions, 1993. European Convention for Constructional Steelwork: ECCS-E6-76, Appendix no. 5: Thin walled cold formed members. Acknowledgements for photographs: The authors express their appreciation to the following firms for making available the photographs used in this Design Guide: British Steel plc. Mannesmannrohren-Werke A.G. Mannhardt Stahlbau llva Form Valexy 54 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t [@l Comite International pour le Developpement 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 major hollow steel section manufacturers to create a major force in the research and application of hollow steel sections worldwide. The objectives of CIDECT are: 0 3 0 0 to increase knowledge of hollow steel sections and their potential application by initiating and participating in appropriate researches and studles to establish and maintain contacts and exchanges between the producers of the 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 for good engineering practice and suitable architecture, in general by disseminating information, organizing congresses etc. to co-operate with organizations concerned with practical design recommen- dations, regulations or standards at national and international level. Technical activities The technical activities of CIDECT have centred on the following research aspects of hollow steel section design: G 0 0 0 0 0 0 0 0 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 joints Aerodynamic properties Bending strength Corrosion resistance Workshop fabrication The results of CIDECT research form the basis of many national and international design requirements for hollow steel sections. 55 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t CIDECT, the future Current work is chiefly aimed at filling up the gaps in the knowledge regarding the structural behaviour of hollow steel sections and the interpretation and imple- mentation of the completed fundamental research. As this proceeds, a new complementary phase is opening that will be directly concerned with practical, economical and labour saving design. CIDECT Publications The current situation relating to CIDECT publications reflects the ever increasing emphasis on the dissemination of research results. Apart from the final reports of the CIDECT sponsored research programmes, which are available at the Technical Secretariat on demand at nominal price, CIDECT has published a number of monographs concerning various aspects of design with hollow steel sections. These are available in English, French and German as indicated. Monograph No. 3 - Windloads for Lattice Structures (G) Monograph No. 4 - Effective Lengths of Lattice Girder Members (E, F, G) Monograph No. 5 - Concrete-filled Hollow Section Columns (E, F) Monograph No. 6 - The Strength and Behaviour of Statically Loaded Welded Connections in Structural Hollow Sections (E) Monograph No. 7 - Fatigue Behaviour of Hollow Section J oints (E, G) A book “Construction with Hollow Steel Sections”, prepared under the direction of CIDECT in English, French, German and Spanish, was published with the sponsor- ship of the European Community presenting the actual state of the knowledge acquired throughout the world with regard to hollow steel sections and the design methods and application technologies related to them. In addition, copies of these publications can be obtained from the individual members given below to whom technical questions relating to CIDECT work or the design using hollow steel sections should be addressed. The organization of CIDECT comprises: 0 0 0 56 President: J . Chabanier (France) Vice-president: C. L. Bijl (The Netherlands) A General Assembly of all members meeting once a year and appointing an €xecutive Committee responsible for adiministration and executing of esta- bished policy Technical Commission and Working Groups meeting at least once a year and directly responsible for the research and technical promotion work Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t 0 Secretariat in Paris responsible for the day to day running of the organization. Present members of CIDECT are: (1 995) 0 British Steel PLC, United Kingdom 0 CS1 Transformados S.A., Spain 0 EXMA, France 0 ILVA Form, Italy 0 IPSCO Inc., Canada 0 Laminoirs de Longtain, Belgium 0 Mannesmannrohr, Federal Republic of Germany 0 Mannstadt Werke GmbH, Federal Republic of Germany 0 Nippon Steel Metal Products Co. Ltd., J apan 0 Rautaruukki Oy, Finland 0 Sonnichsen A/S, Norway 0 Tubemakers of Australia, Australia 0 Tubeurop, France 0 VOEST Alpine Krems, Austria Cidect Research Reports can be obtained through: Mr. E. Bollinger Office of the Chairman of the CIDECT Technical Commission c/o Tubeurop France lmmeuble Pacific TSA 20002 92070 La Defense Cedex Telephone: (33) 1 /41258265 Telefax: (33) 2/41 258783 Mr. D. Dutta Marggrafstrasse 13 40878 Ratingen Germany Telephone: (49) 21021842578 Telefax: (49) 21 02/842578 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 misinterpretation of the information contained in this book or in its use. 57 Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t Construction with Hollow Steel Sections - Structural stability of hollow sections Discuss me ... C r e a t e d o n 2 4 M a y 2 0 0 8 T h i s m a t e r i a l i s c o p y r i g h t - a l l r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e S t e e l b i z L i c e n c e A g r e e m e n t