Scholarly article on topic 'Structural design of elliptical hollow sections: a review'

Structural design of elliptical hollow sections: a review Academic research paper on "Civil engineering"

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Academic research paper on topic "Structural design of elliptical hollow sections: a review"

Proceedings of the Institution of Civil Engineers

Structures and Buildings 163

December 2010 Issue SB6 Pages 391-402

doi: I0.I680/stbu.20I0.I63.6.39l

Paper 900095

Received 30/11/2009 Accepted 22/07/2010

Keywords: codes of practice & standards/reviews/steel structures

Tak Ming Chan Leroy Gardner

Assistant Professor, School of Reader, Department of Civil Engineering, University of and Environmental Warwick, UK Engineering, Imperial College

London, UK

Kwan Ho Law

PhD student, Department of Civil and Environmental Engineering, Imperial College London, UK

Structural design of elliptical hollow sections: a review

T. M. Chan MSc, DIC, PhD, L. Gardner MSc, DIG, PhD, CEng, MICE, MIStructE and K. H. Law MSc, DIC, CEng, MIStructE

Tubular construction is synonymous with modern architecture. The familiar range of tubular sections -square, rectangular and circular hollow sections - has been recently extended to include elliptical hollow sections (EHSs). Due to differing flexural rigidities about the two principal axes, these new sections combine the elegance of circular hollow sections with the improved structural efficiency in bending of rectangular hollow sections. Following the introduction of structural steel EHSs, a number of investigations into their structural response have been carried out. This paper presents a state-of-the-art review of recent research on EHSs together with a sample of practical applications. The paper addresses fundamental research on elastic local buckling and post-buckling, cross-section classification, response in shear, member instabilities, connections and the behaviour of concrete-filled EHSs. Details of full-scale testing and numerical modelling studies are described, and the generation of statistically validated structural design rules, suitable for incorporation into international design codes, is outlined.

NOTATION

A gross cross-section area

Ac cross-section area of the concrete within a concrete-

filled steel tube

Aeff effective cross-section area

As cross-section area of a steel tube

Av shear area

a half of the larger outer diameter of an EHS

b half of the smaller outer diameter of an EHS

De equivalent diameter

De1 equivalent diameter (Kempner, 1962)

De2 equivalent diameter (Ruiz-Teran and Gardner, 2008)

De3 equivalent diameter (Zhao and Packer, 2009)

E Young's modulus

f coefficient dependent on thickness and larger outer

diameter of an EHS

fck compressive concrete strength

fy material yield stress

L0 perimeter

Mel,Rd elastic moment resistance

Mel,z,Rd elastic moment resistance about the minor (z-z) axis

Mpl,Rd plastic moment resistance

Mpl,y,Rd plastic moment resistance about the major (y-y) axis

M y,Ed M z ,Ed Nb,Rd Nc,Rd NCFT

Ncr NEd

rmax rmin

Vpl,Rd Vu

Weff Wei

Ol, O2

ultimate bending moment

design bending moment about the major (y-y) axis design bending moment about the minor (z-z) axis member buckling resistance cross-section compressive resistance cross-section compression resistance of a concrete-filled EHS

elastic flexural buckling load design axial force ultimate axial load plastic yield load rotation capacity radius of curvature

radius of a circular section with the same perimeter

as the corresponding oval

critical radius of curvature

maximum radius of curvature

minimum radius of curvature

coordinate along the curved length of an oval

thickness of shell

plastic shear resistance

ultimate shear force

effective section modulus

elastic section modulus

coordinate along the major (y-y) axis

cross-section major axis

coordinate along the minor (z-z) axis

cross-section minor axis

coefficient dependent on the material yield stress

non-dimensional member slenderness

Poisson's ratio

eccentricity of an oval

end stresses

elastic buckling stress

yield stress in shear

ratio of end stresses

1. INTRODUCTION

The opening of Britannia Bridge in the UK in 1850 (Collins, 1983; Ryall, 1999) heralded a new era for structural hollow sections. It was the first major civil engineering application to adopt rectangular hollow sections (RHSs) in the main structural skeleton. Behind the scenes, viable design options involving circular hollow sections (CHSs) and elliptical hollow sections (EHSs) were also considered during the conceptual design

stage. Nine years later, the engineer Isambard Kingdom Brunei adopted EHSs as the primary arched compression elements in one of his masterpieces - the Royal Albert Bridge (Binding, 1997). Subsequently, in 1890, the Forth Railway Bridge (Paxton, 1990) was completed, displaying extensive use of CHSs. The hollow sections employed in these early structures had to be fabricated from plates connected by rivets. As the construction industry continued to evolve, new design and production techniques were developed, and hollow sections are now manufactured as hot-finished structural products with square, rectangular and circular geometries.

More than a century after their initial use by Brunel, EHSs have emerged as a new addition to the hot-finished product range for tubular construction, and have already been utilised as the primary elements in a number of structural applications. Examples include the Zeeman Building at the University of Warwick completed in 2003 (Figure 1), Society Bridge in Scotland (Corus, 2006) completed in 2005 (Figure 2) and the main airport terminal buildings in Madrid (Viñuela-Rueda and Martinez-Salcedo, 2006) completed in 2004 (Figure 3), Cork completed in 2006 (Figure 4) and London Heathrow completed in 2007 (Figures 5 and 6).

Early analytical research into the structural characteristics of non-circular cylindrical shells initially centred on oval hollow sections (OHSs), after which attention turned to sections of elliptical geometry. The primary focus of these early studies was the elastic buckling and post-buckling response of slender oval and elliptical shells. More recently, following the introduction of hot-finished elliptical tubes of structural proportions, attention has shifted towards the generation of

Figure 1. Zeeman Building, University of Warwick (2003)

Figure 4. Cork Airport, Ireland (2006)

structural performance data through physical testing and numerical simulations and to the subsequent development of structural design rules. The structural scenarios investigated to date include axial compression, bending and shear at both cross-sectional level and member level, concrete-filled tubular construction and connections. This paper presents a state-of-the art review of previous research and current provisions for all aspects of the design of structural steel EHSs.

2. GEOMETRY

The recent addition to the family of hot-finished tubular sections is marketed as OHSs. An oval may be described generally as a curve with a smooth, convex, closed 'egg-like' shape, but with no single mathematical definition. Hence, a range of geometric properties, depending on the degree of elongation and asymmetry of ovals, exists. The recently introduced sections are, in fact, elliptical in geometry (an ellipse being a special case of an oval), as described later. In early investigations, a number of formulations were examined by Marguerre (1951) to describe the geometry of an oval and the simplified expression given by Equation 1 was adopted by a number of researchers to describe a doubly symmetric oval cross-section.

where r is the radius of curvature at point s along the curved length of the section, £ is the eccentricity of the section (£ = 0 represents a circle while, for £ = 1, the minimum curvature is zero at the narrow part of the shell cross-section), L0 is the perimeter of the section and r0 is the radius of a circular section with the same perimeter.

An ellipse is a special case of an oval and can be described mathematically as

3'* (;)'■1

1 _ 1 r ro

where y and z are the Cartesian coordinates, a is half of the larger outer diameter and b is half of the smaller outer diameter, as shown in Figure 7. The aspect ratio of an ellipse is defined as a/b, while the maximum and minimum radii of curvature may be shown to be rmax = a2/b and rmin = b2/a. The ratio between the maximum radius of curvature and the minimum radius of curvature characterises the shape of the ellipse and is given by (a/b)3.

Romano and Kempner (1958) derived a relationship between the eccentricity £ of an oval and the aspect ratio a/b of an ellipse and concluded that the two shapes, defined by Equations 1 and 2, were comparable provided 0 < £ < 1. It is worth noting that for £ = 0, Equation 1 exactly represents a circle (i.e. an ellipse with a/b = 1); for £ = 1, the corresponding aspect ratio is 2.06.

= b_ max a

Figure 7. Geometry of an ellipse

3. ELASTIC LOCAL BUCKLING AND POST-BUCKLING

Extensive analytical work on the elastic buckling and post-buckling of OHSs and EHSs under axial compression was conducted in the 1950s and 1960s, with the earliest study being performed by Marguerre (1951). Following on from this critical work, Kempner (1962) concluded that the elastic buckling stress of an OHS could be accurately predicted by the buckling stress of a CHS with a radius equal to the maximum radius of curvature of the OHS and that the solution was a lower bound. The post-buckling behaviour of OHSs was first studied by Kempner and Chen (1964), who observed that the higher the aspect ratio of the OHS, the more stable the post-buckling behaviour (approaching that of a flat plate) and, the lower the aspect ratio, the more unstable the post-buckling behaviour (approaching that of a circular shell). The stable post-buckling response of sections with high aspect ratios, enabling loads beyond the elastic buckling load to be sustained, was attributed to the ability of the sections to redistribute stresses to their stiffer regions of high curvature upon buckling (Kempner and Chen, 1966).

applied to EHS. Tennyson et al. (1971) carried out physical tests to assess the buckling behaviour of EHSs with aspect ratios between 1 and 2. The tests confirmed that elliptical shells with aspect ratios close to unity exhibit unstable post-buckling behaviour and high imperfection sensitivity, resulting in collapse loads below the elastic buckling load. Conversely, while the elliptical sections with an aspect ratio of 2 exhibited initially unstable post-buckling behaviour, the response quickly restabilised, resulting in attainment of collapse loads in excess of the initial buckling loads. These findings were corroborated by Feinstein et al. (1971).

The recent introduction of hot-finished EHSs has prompted further research, including a re-evaluation of the fundamental elastic buckling and post-buckling characteristics of elliptical shells, principally by means of numerical analysis techniques. While the findings of the previous researchers have been largely confirmed, detailed numerical modelling (Ruiz-Teran and Gardner, 2008; Silvestre, 2008; Zhu and Wilkinson, 2006) has revealed that use of the maximum radius of curvature in the prediction of the elastic buckling stress of an EHS in compression becomes increasingly inaccurate for higher aspect ratios and thicker tubes, and revised expressions have thus been devised (Ruiz-Teran and Gardner, 2008; Silvestre, 2008). Most recently, the post-buckling stability and imperfection sensitivity of EHSs were systematically quantified (Silvestre and Gardner, 2010) in terms of bifurcation angle and slope of ascending post-buckling equilibrium path. This study provides insight for the future development of effective area formulae for local buckling of slender EHSs.

4. HOT-FINISHED EHSS

Hot-finished structural sections of standardised geometries are the staple products employed within the steel construction industry. Such sections are now available in elliptical profiles with outer dimensions ranging from 150 X 75 mm to 500 X 250 mm; thicknesses range from 4 to 16 mm and all sections have an aspect ratio of 2. Approximate formulae for the determination of geometric properties of EHSs are provided in the European product standard EN 10210-2 (CEN, 2006). The following sections summarise the latest research findings and design proposals for EHSs in a range of structural scenarios. Extensive laboratory testing and numerical modelling studies have been conducted on EHSs over the past few years, and a summary of the physical tests that have been performed is given in Table 1. These include stub column tests, in-plane bending tests, combined bending and shear tests, combined axial load and bending tests, column flexural buckling tests, connection tests and tests on concrete-filled tubes. These tests, supplemented by numerically generated structural performance data, have been employed in the development and verification of design rules. A series of tests has also been carried out on cold-formed stainless steel EHSs and corresponding design guidance has been developed (Lam et al., 2010; Theofanous et al., 2009a, 2009b).

The buckling and initial post-buckling behaviour of EHSs was first studied by Hutchinson (1968). Hutchinson concluded that Kempner's proposal (Kempner, 1962), whereby the elastic buckling stress of an OHS could be accurately predicted using the classical CHS formulation with an equivalent radius equal to the maximum radius of curvature of the OHS, may also be

5. CROSS-SECTION BEHAVIOUR 5.1. Compression

Axial compression represents one of the fundamental loading arrangements for structural members. For cross-section classification under pure compression, of primary concern is

Structural configuration Structural carbon steel Stainless steel

No. Reference No. Reference

of tests of tests

Cross-section tests Compression Unfilled 33 Chan and Gardner, 6 Theofanous et al.,

2008a; Zhao and Packer, 2009a

Concrete filled 42 Yang et al., 2008; Zhao 6 Lam et al., 2010

and Packer, 2009

Bending and Minor axis 23 Chan and Gardner, 3 Theofanous et al.,

combined bending + 2008b; Gardner et al., 2009b

shear 2008

Major axis 19 Chan and Gardner, 3 Theofanous et al.,

2008b; Gardner et al., 2009b

Member buckling Compression Minor axis 12 Chan and Gardner, 4 Theofanous et al.,

tests 2009a 2009a

Major axis 12 Chan and Gardner, 2 Theofanous et al.,

2009a 2009a

Connection tests Fully welded truss- 7 Bortolotti et al., 2003;

type connections Pietrapertosa and

Jaspart, 2003

Gusset plate Branch and through- 6 Willibald et al, 2006a

connections plate connections

End connections 5 Willibald et al, 2006b

Total number of tests performed 159 24

Table 1. Summary of experiments performed on elliptical hollow sections

the occurrence of local buckling before yielding. Cross-sections that reach the yield load are considered class 1-3 (fully effective), while those where local buckling prevents attainment of the yield load are class 4 (slender). For uniform compression, a cross-section slenderness parameter has been determined by consideration of the elastic critical buckling stress.

The elastic critical buckling stress acr of a uniformly compressed OHS/EHS may be reasonably approximated by substituting the expression for the maximum radius of curvature rmax into the classical buckling stress of a circular cylinder (Hutchinson, 1968; Kempner, 1962) given by

[3(1 - V2)]1/2(rmax/t)

Dei (a2/b)

f£2 f£2

where De1 is the equivalent diameter based on Kempner's (1962) proposal for acr and e2 — 235/fy to allow for a range of yield strengths.

Further research on the elastic buckling of elliptical tubes (Ruiz-Teran and Gardner, 2008) revealed inaccuracies in Kempner's predictive formula (Equation 3) for EHSs with higher aspect ratios and tube thicknesses. Following analytical and numerical studies, an improved expression for the elastic buckling stress of a uniformly compressed EHS was derived and hence a revised expression for the equivalent diameter was proposed

De2 — 2a 1+f(a - V

where f — 1 - 1 t \ -2-3 —

where E is Young's modulus, v is Poisson's ratio and t is the thickness of the shell. This assumes that buckling initiates at the point of maximum radius of curvature and ignores the restraining effect of the surrounding material of lower radius of curvature and the influence of the boundary conditions. For an elliptical section, rmax may be shown to be equal to a2/b. It has therefore been proposed (Chan and Gardner, 2008a) that under pure compression, the cross-section slenderness of an EHS is defined as

The corresponding cross-section slenderness of a compressed EHS may therefore be defined as

6 De2 , D2 — 2a [1+f(a - 01 ! t£2

where De2 is the equivalent diameter proposed by Ruiz-Teran and Gardner (2008).

The above slenderness measures apply over the full range of practical aspect ratio of EHSs (say 1 < a/b < 4) and are comparable with the current treatment of CHSs in the sense that, for the case of a/b — 1, both give an equivalent diameter equal to the actual diameter of a CHS. A comparison of CHS and EHS test data (Chan and Gardner, 2008a; Giakoumelis and Lam, 2004; Sakino et a/., 2004; Teng and Hu, 2007; Tutuncu and O'Rourke, 2006; Zhao and Packer, 2009) in compression is shown in Figure 8, while a typical experimental failure mode for a compressed EHS is shown in Figure 9. For EHS, the

t EHS (Equation 4) □ EHS (Equation 6)

-CHS regression

EHS (Equation 4) regression EHS (Equation 6) regression

120 150 180 210 240

Figure 8. Comparison of different equivalent diameters employed in EHS slenderness parameters

yields closer agreement between the two section types and increases the number of sections from the current range of EHSs being fully effective; it is thus more accurate and appropriate for design. On this basis, it was recommended that EHSs may be classified in compression using the current CHS slenderness limit of 90 in EN 1993-1-1 (CEN, 2005) and the equivalent diameter and slenderness parameter defined by Equations 5 and 6. The more straightforward measure of slenderness based on De1 (Equation 4) has been adopted in the design tables published by the Steel Construction Institute (SCI) and British Constructional Steelwork Association (BSCA), commonly referred to as 'the blue book' (SCI/BSCA, 2009).

An alternative approach to the cross-section classification of EHSs was proposed by Zhao and Packer (2009). The structural response was likened to that of an RHS comprising flat plates rather than a circular tube, and the degree of curvature in the section ignored. The proposed slenderness measure, based on an equivalent diameter De3 = (2a - 2t) was given by

De3 _ 2(fl - t)

and it was recommended that the class 3 slenderness limit for flat internal elements in compression of 42 (EN 1993-1-1 (CEN, 2005)) should apply. It is worth nothing that, for an aspect ratio a/b = 2, assuming the thickness of the section to be small, De3 is approximately half De1 or De2 and the slenderness limit for flat elements in compression is approximately half that for a CHS. Hence, both approaches will typically yield similar results. However, for lower aspect ratios, use of De3 with the RHS slenderness limit will be increasingly conservative. A further interesting difference between the two approaches lies in the use of e, which is employed to modify the section slenderness based on material strength fy. Assuming shell-like behaviour, De1 and De2 are normalised by e2 while, based on plate-like behaviour, De3 is normalised by e. The reality is likely to be intermediate between these two extremes, and will clearly be dependent on the aspect ratio of the section.

Failure to reach the yield load in compression due to the occurrence of local buckling is generally treated in design using either an effective stress or an effective area approach, with recent trends favouring the latter. A preliminary effective area formula (Equation 8) for class 4 (slender) EHSs was proposed by Chan and Gardner (2008a) and found to be suitable for design for the current practical range of EHSs

" 90 235" 0-5

8 Aeff = A A/1 /y

This proposal, taking De = 2a2/b, has been adopted in the SCI/ BCSA design tables (SCI/BSCA, 2009).

results are plotted on the basis of the two equivalent diameters De1 (Equation 4 (Chan and Gardner, 2008a)) and De2 (Equation 6 (Ruiz-Teran and Gardner, 2008)). Regression curves have also been added for the three datasets in Figure 8. The results demonstrate that both slenderness parameters for EHSs are conservative in comparison to CHSs; however, Equation 6

5.2. Bending

For minor (z-z) axis bending, similar to axial compression, local buckling initiates at the point of greatest radius of curvature, which coincides with the most heavily compressed part of the cross-section. It was therefore proposed that the same cross-section slenderness parameter given by Equation 4 can also be adopted for EHSs in minor axis bending; this

proposal was supported by available test data and adopted in the blue book (SCI/BSCA, 2009). For bending about the major (y-y) axis, local buckling initiates, in general, neither at the point of maximum radius of curvature (located now at the neutral axis of the cross-section with negligible bending stress) nor at the extreme compressive fibre, since this is where the section is of greatest stiffness (i.e. minimum radius of curvature). A critical radius of curvature rcr was therefore sought to locate the point of initiation of local buckling (Gerard and Becker, 1957). This was achieved by optimising (i.e. finding the maximum value of) the function composed of the varying curvature expression and an elastic bending stress distribution. The theoretical point of initial of buckling, assuming a linear elastic stress distribution was found at rcr = 0.65a2/b. This result was modified (Chan and Gardner, 2008b) to provide better prediction of observed physical behaviour, to yield rcr = 0.4a2/b (see Figure 10). As the aspect ratio of the section reduces (i.e. the section becomes more circular), the point of initiation of buckling tends towards the extreme compressive fibre of the section. This is reflected by a transition in r, to the local radius of curvature at the extreme fibre where r = b2/a, at an aspect ratio a/b — 1.357. The slenderness parameters for major axis bending proposed by Chan and Gardner (2008b) and adopted in the SCI/BSCA design tables (SCI/BSCA, 2009) are given by

9 De (a2/b) —= 0-8—'— for a/b > 1-357 18 2 ts2

10 De (b2/a) ^ ! = 2 V for a/b < 1-357 ts2 ts2

Based on their proposed measures of slenderness (Equations 4, 9 and 10), Chan and Gardner (2008b) assessed the applicability of current CHS slenderness limits to EHSs, with the following criteria to demark the classes of cross-section

(a) class 1 sections were required to reach the plastic moment capacity MpljRd and possess a minimum rotation capacity R of 3

(b) class 2 sections were required only to reach MplRd

(c) class 3 sections were required to reach the elastic moment capacity Mel,Rd

(d) otherwise, the sections were class 4.

By means of comparison with available test and finite-element (FE) data, the current CHS slenderness limits given in EN 19931-1 (CEN, 2005) of De/fc2 = 50 for class 1, 70 for class 2 and 90 for class 3 were found to be suitable for EHSs. It was further recommended that the class 3 limit of 90 could be relaxed to 140 for both CHSs and EHSs.

An interim effective section modulus formula, Weff for class 4 (slender) EHSs was also proposed and found to be safely applicable when compared with test and FE results

■ T„ / 140 235\0'25

Weff = H D=t 2/W

where Wel is the elastic section modulus of the EHS.

5.3. Combined compression and bending

For cross-section classification under combined compression and bending (Gardner and Chan, 2007), designers may initially simply check the cross-section against the most severe loading case of pure compression. If the classification is class 1, then there is no benefit to be gained from checking against the actual stress distribution. Similarly, if plastic design is not being utilised, there would be no benefit in reclassifying a class 2 cross-section under the actual stress distribution. Under combined compression and minor axis bending, clearly local buckling will initiate in the region of the maximum radius of curvature, similar to the cases of pure compression or pure minor axis bending. Under combined compression and major axis bending, the critical radius of curvature (i.e. the point of initiation of local buckling) will shift towards the centroidal axis as the compressive part of the loading increases. This effect is shown in Figure 11, where z/a is the normalised distance of rcr from the centroid of the section and ^ = a2/o1 is

Figure 10. Modified location of critical radius of curvature for EHS with a/b — 2 in major axis bending

a/b = 1001

a/b = 11

-. „ \

a/b = 125 - K s \

a/b = 1-5ss - i \

a/b = 2 0. - "S, . \ \

-1 0 -0-5 0 05 10

Figure 11. Theoretical variation in position of rcr with aspect ratio a/b and stress distribution ^

the ratio of the end stresses between which a linear gradient is considered. Note that Figure 11 shows the theoretical elastic buckling response, which has not be adjusted in the light of experimental observations, as described for the case of pure major axis bending in the previous section. For ^ = 1, which corresponds to pure compression, buckling initiates at z/a — 0 (i.e. the centroid of the section) for all aspect ratios. As ^ decreases, the position of initiation of buckling migrates up the section where the greater stresses exist. This migration is more rapid in sections of low aspect ratio where there is less variation in radius curvature around the section.

For class 3 sections under combined loading, EN 1993-1-1 (CEN, 2005) provides a linear interaction formula, given by Equation 12. When compared with test results, this interaction formula was shown (Chan and Gardner, 2009b) to be applicable to EHSs.

(NEd/Nc,Rd) + (Mz3d/MeURd) < 1-0

where Mz,Ed is the design bending moment about the minor (z-z) axis and Mel,z,Rd is the design elastic bending resistance about the minor (z-z) axis, NEd is the design axial force and Nc,Rd is the design cross-section resistance under uniform compression.

In the plastic regime, Nowzartash and Mohareb (2009) derived interaction surfaces for EHSs under combined compression and bending about the two principal axes. Their proposed interaction expression is

(My,Ed/Mpi>y,Rd)2 + 2(NEd/NCjRd)1-75 + (NEd/Nc>Rd)3'5 < 1-0

where My Ed is the design bending moment about the major (y-y) axis and Mpl, y,Rd is the design plastic bending resistance about the major (y-y) axis.

The two interaction formulations (Equation 12 for class 3 sections and Equation 13 for class 1 and 2 sections) are plotted, together with available test data, in Figure 12, which may be seen to provide safe-side predictions of the resistance of EHSs to combined bending and axial compression.

Equation 13

Concentric tests Eccentric tests (major axis, classl) Eccentric tests (minor axis, class3)

5.4. Shear and combined bending and shear

The plastic shear resistance of an EHS was derived by Gardner et al. (2008) based on the assumption that shear stresses at yield are distributed uniformly around the section and act tangentially to the surface (see Figure 13). For transverse loading in the y- y direction, this yielded a plastic shear resistance Vpl>Rd equal to twice the product of the vertical projection of the elliptical section (measured to the centreline of the thickness) and the thickness (i.e. 2(2b — t)t) multiplied by the yield stress in shear ry. Likewise, for transverse load in the z-z direction (see Figure 13), the corresponding projected area is equal to 2(2a — t)t. Therefore, for an EHS of constant thickness, the shear area Av may be defined by Equations 14 and 15. These proposed shear areas have been adopted in the SCI/BCSA design tables (SCI/BSCA, 2009)

Av = (4b - 21 ) t

for loading in the y - y direction and

Av = (4fl - 21) t

for loading in the z-z direction.

Test results on an EHS under combined bending and shear are plotted in Figure 14. The results demonstrate that where the shear force Vu is less than half the plastic shear resistance Vpl,Rd, the effect of shear on the bending moment resistance is small. Conversely, for high shear force (greater than 50% of Vpl,Rd), there is a degradation of the bending moment resistance. The proposed moment-shear interaction (Gardner et al., 2008) derived from EN 1993-1-1 (CEN, 2005) is plotted in Figure 14 and shows good agreement with the experimental data.

6. MEMBER BEHAVIOUR 6.1. Flexural column buckling

Flexural buckling of EHS columns has been studied by Chan and Gardner (2009a). A total of 24 experiments were performed, 12 buckling about the major axis and 12 about the minor axis. The experimental data were supplemented with additional structural performance data generated from validated numerical models. The combined results are shown in

For shear along y-y

For shear along z-z

Av = (4b - 2t)t

Av = (4a - 2t)t

Figure 12. Test results and interaction curves for combined bending and axial compression

Figure 13. Derivation of plastic shear area for EHS

Figure 14. Test results and interaction diagram for EHS under combined bending and shear

Figure 15 in which, on the vertical axis, the buckling load has been normalised by the cross-section resistance; the horizontal axis is the member slenderness 1 — (A/y/Ncr)0'5, Ncr being the elastic buckling load of the column. The results were found to follow a similar trend to buckling data for hot-finished CHS columns (also shown in Figure 15). Supported by statistical analysis, it was therefore proposed that the buckling curve (curve 'a' in EN 1993-1-1 (CEN, 2005)) for hot-finished CHSs could also be safely applied to hot-finished EHSs. This proposal has been adopted in the SCI/BCSA design tables (SCI/BSCA, 2009).

6.2. Lateral torsional buckling

Lateral torsional buckling is the member buckling mode associated with laterally unrestrained beams loaded about their major axis. The closed nature of tubular sections results in high torsional stiffness, making them inherently resistant to buckling modes featuring torsional deformations. For circular sections, lateral torsional buckling is not possible, while for EHSs with low aspect ratios, it is of no practical concern. However, for higher aspect ratios, the disparity in major and minor axis flexural stiffness grows and susceptibility to lateral torsional buckling increases. Initial studies have indicated that lateral torsional buckling should be considered in EHSs with aspect ratios a/b higher than about 3.

7. CONNECTIONS

A number of recent studies have been performed to examine the behaviour of connections to and between EHS members. Two general connection types have been considered

Figure 15. EHS column buckling test results and proposed design curve

(a) fully welded connections between elliptical tubes in truss-type applications

(b) gusset plate connections, which might be employed in trusses or for diagonal bracing members in steel-framed buildings.

The following sections describe the latest research findings in these areas.

7.1. Welded truss-type connections

The first full-scale experimental studies on connections between structural steel EHSs were performed by Bortolotti et al. (2003) and Pietrapertosa and Jaspart (2003). The test configurations mimicked fully welded brace-to-chord connections typically found in trusses. The experimental data were utilised to validate numerical models, which were subsequently employed to perform parametric analyses. Existing design rules for equivalent RHS and CHS connections were reviewed and preliminary observations and recommendations on their suitability for EHS connections were made. Additional numerical analyses, covering a wider range of variables, were carried out by Choo et al. (2003), who concluded that the ring model originally devised for CHS joints (Togo, 1967) may also be applied to describe the behaviour of EHS joints and that, with appropriate orientation of brace and chord member, axially loaded EHS connections can achieve higher capacities than equivalent CHS connections. Further research on welded truss-type connections featuring EHSs is under way.

7.2. Gusset plate connections

The behaviour of gusset plate connections to EHS members has been investigated experimentally and numerically. Two general configurations have been studied

(a) gusset plates welded to the sides of EHS members, representing, for example, connections to chord members in trusses

(b) gusset plates employed in end connections, for example to bracing members in frames or web members in trusses.

For the first configuration, six full-scale tests were conducted by Willibald et al. (2006a), exploring different orientations and different connection details, covering both branch and through-plate connections, orientated longitudinally and transversely, and connected to both the wider and narrower sides of the EHS. Comparisons of the test results with existing RHS and CHS design formulae revealed that neither fully represented the behaviour of EHS connections, but that the RHS provisions could be conservatively adopted.

For the second configuration, five full-scale tests on gusset plate connections to the ends of EHS members were reported by Willibald et al. (2006b), together with eight similar tests on end connections to CHS members. Both slotted tube and slotted plate connection details were examined, with plates orientated to span either the smaller or larger EHS diameter (see Figure 16). Failure of all specimens was either by circumferential fracture of the tube or tear-out of the base material of the tube along the weld. The five test results were utilised by Martinez-Saucedo et al. (2008) for the validation of numerical models, which were subsequently used to perform parametric studies to

Figure 16. Slotted end EHS connections

enable a wider range of variables (including weld length, connection eccentricity and EHS diameter) to be examined. Based on the findings, design recommendations for slotted end EHS connections, accounting for a number of possible failure modes, were proposed.

8. CONCRETE-FILLED EHSS

An increasingly popular means of improving structural efficiency in tubular construction is through concrete infilling (Shanmugam and Lakshmi, 2001). Concrete infilling of steel tubes provides enhanced strength and stiffness, greater resistance to local buckling and improved performance in fire.

The behaviour of filled elliptical tubes has been investigated analytically (Bradford and Roufegarinejad, 2007), experimentally (Yang et al., 2008; Zhao and Packer, 2009) and numerically (Jamaluddin et al., 2009). The studies examined composite load-carrying capacity, ductility, level of concrete confinement afforded by the elliptical tube and the simulated effect of concrete shrinkage. A model for predicting the strength of the confined concrete was also proposed by Dai and Lam (2010). The response of concrete-filled EHSs was found, in general, to be intermediate between that of concrete-filled square hollow sections (SHSs) or RHSs and CHSs (Yang et al., 2008; Zhao and Packer, 2009). An analytical model to predict the strength of confined concrete in elliptical tubes, based on a modification to a previously devised model for concrete columns with elliptical reinforcement hoops (Campione and Fossetti, 2007), was proposed and verified (Yang et al., 2008). As anticipated, thicker tubes were found to provide greater confinement and improved ductility. Existing design rules for concrete-filled SHSs and RHSs, including those provided in EN 1994-1-1 (CEN, 2004), were shown to be safely applicable to concrete-filled EHSs, while the corresponding rules for CHSs generally resulted in an overprediction of capacity. It was concluded (Yang et al., 2008; Zhao and Packer, 2009) that the cross-section compression resistance of a concrete-filled EHS Ncft could be most accurately predicted by a simple summation of the steel and concrete resistances (Equation 16), as recommended for SHSs and RHSs in EN 1994-1-1 (CEN, 2004).

Ncft = Asfy + Acf ck

where As is the cross-sectional area of the steel tube, fy is the

yield strength of the steel, Ac is the cross-sectional area of the concrete and fck is the compressive concrete strength.

9. CONCLUSIONS

A series of research programmes have been recently conducted around the world following the introduction of EHSs as hot-finished structural steel construction products. These studies have included fundamental analytical investigations of the buckling and post-buckling response of elliptical tubes building on earlier studies performed in the 1960s, full-scale experimental programmes on members and connections, and detailed numerical simulations. A total of over 150 tests have been performed, supplemented by a multiplicity of numerically generated structural performance data. On the basis of the findings of these studies, a number of proposals for structural design rules have been made. Many of these design rules have been incorporated into industry design guidance (SCI/BSCA, 2009). In this paper, a state-of-the-art review of this research has been presented; it is concluded that the design recommendations made are suitable for incorporation into international structural design standards.

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