BUCKLING STUDIES OF COLD-FORMED CHANNELS IN SHEAR USING THE SEMIANALYTICAL FINITE STRIP AND SPLINE FINITE STRIP METHODS CAO HUNG PHAM GREGORY J. HANCOCK RESEARCH REPORT R932 FEBRUARY 2013 ISSN 1833-2781

SCHOOL OF CIVIL ENGINEERING

SCHOOL OF CIVIL ENGINEERING

BUCKLING STUDIES OF COLD-FORMED CHANNELS IN SHEAR USING THE SEMI-ANALYTICAL FINITE STRIP AND SPLINE FINITE STRIP METHODS RESEARCH REPORT R932

CAO HUNG PHAM GREGORY J. HANCOCK February 2013 ISSN 1833-2781

Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods Copyright Notice School of Civil Engineering, Research Report R932 Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods Cao Hung Pham Gregory J. Hancock February 2013 ISSN 1833-2781 This publication may be redistributed freely in its entirety and in its original form without the consent of the copyright owner. Use of material contained in this publication in any other published works must be appropriately referenced, and, if necessary, permission sought from the author. Published by: School of Civil Engineering The University of Sydney Sydney NSW 2006 Australia This report and other Research Reports published by the School of Civil Engineering are available at http://sydney.edu.au/civil

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Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods

ABSTRACT The finite strip method is computationally efficient for the static, stability, post-buckling and vibration analyses of thin-walled structures. The finite strip employs simple polynomial functions to describe the transverse variations of the displacements and continuous harmonic series functions or discontinuous spline functions to describe the longitudinal variation of the strip displacements. While the Semi-Analytical Finite Strip Method (SAFSM) generally uses the longitudinal harmonic series to satisfy the boundary conditions at the longitudinal ends and to give compatibility between strips, the Spline Finite Strip Method (SFSM) employs local spline functions in the longitudinal direction to account for different boundary conditions. The Semi-Analytical Finite Strip Method (SAFSM) has been widely used in computer software (THIN-WALL, CUFSM) to develop the signature curves of the buckling stress versus buckling half-wavelength for a thinwalled section under compression or bending to allow identification of buckling modes. Recently, a complex mathematical technique has been applied in the SAFSM theory to allow for the case of shear. The shear buckling modes produced include local, distortional and overall with phase shifts along the member. This report provides the analysis and comparison between the new SAFSM development for shear and the SFSM for whole plain channel sections including flanges and lips where the sections are loaded in pure shear parallel to the web. The main variables are the flange widths and lip sizes. For the longitudinal direction, the SAFSM determines the shear signature curves versus buckling half-wavelength and the SFSM determines the elastic shear buckling stresses versus the member lengths of the whole channel section. The SAFSM is limited to a single half-wavelength whereas the SFSM can include multiple buckles as seen in the wellknown garland curve. The report demonstrates the potential for coupling between multiple short halfwavelength modes in shear and longer single half-wavelength as may occur in distortional buckling.

KEYWORDS Cold-formed channel sections; Semi-Analytical Finite Strip; Spline Finite Strip; Shear buckling analysis; Signature curve; Coupling buckling modes.

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Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods

TABLE OF CONTENTS ABSTRACT ...................................................................................................................................................... 3 KEYWORDS .................................................................................................................................................... 3 TABLE OF CONTENTS ................................................................................................................................... 4 INTRODUCTION .............................................................................................................................................. 5 ELASTIC SHEAR BUCKLING OF FULL CHANNEL SECTIONS .................................................................... 5 SEMI-ANALYTICAL FINITE STRIP METHOD (SAFSM).......................................................................................... 5 SPLINE FINITE STRIP METHOD (SFSM) ............................................................................................................ 5 MODELLING LIPPED CHANNEL SECTIONS IN SHEAR ............................................................................... 6 RESULTS OF BUCKLING ANALYSES AND COMPARISONS OF SAFSM AND SFSM ................................ 7 CONCLUSIONS ............................................................................................................................................. 12 ACKNOWLEDGEMENT ................................................................................................................................. 12 REFERENCES ............................................................................................................................................... 12

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Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods

INTRODUCTION The Finite Strip Method (FSM) is a computationally efficient method for the static, stability, post-buckling and vibration analyses of thin-walled structures. The FSM has various advantages compared with the finite element method in the analysis of thin-walled structures. Firstly, the FSM requires a greatly reduced number of degrees of freedom for a given structural problem. Further, the FSM generally shows no compatibility problems between membrane and flexural displacements at plate junctions. The FSM eases the definition of the geometry of the structural problem and its numerical discretization. The FSM employs simple polynomial functions to describe the transverse variations of the displacements and continuous harmonic series functions or spline functions to describe the longitudinal variation of the strip displacements. This approach differs from the finite element method in which polynomial functions are normally used in both the longitudinal and transverse directions to describe the element displacements. For compression and bending, the longitudinal harmonic series for the FSM are generally chosen to satisfy simply supported boundary conditions at the longitudinal ends and to give compatibility between strips. This is also known as the Semi-Analytical Finite Strip Method (SAFSM). However, for shear, the longitudinal harmonic series are chosen such that they are free at the ends and allow unrestrained shear buckling. When local spline functions are used instead of the harmonic series functions in the longitudinal direction, a more widely applicable FSM is developed to account for different boundary conditions. This is known as the Spline Finite Strip Method (SFSM).

ELASTIC SHEAR BUCKLING OF FULL CHANNEL SECTIONS SEMI-ANALYTICAL FINITE STRIP METHOD (SAFSM) The Semi-Analytical Finite Strip Method was originally developed by Cheung (1968, 1976) for the stress analysis of simply supported isotropic and orthotropic variable thickness plates in bending and later applied to the buckling analysis of plate assemblies under biaxial compression by Przmieniecki (1973). This method was further extended by Plank and Wittrick (1974) to analyse the buckling behaviour of thin-walled crosssections under the various loading conditions such as longitudinal and transverse compression, longitudinal in-plane bending and shear. Hancock (1978) applied the SAFSM developed by Plank and Wittrick (1974) to beams and identified local, distortional and lateral-torsional modes. The SAFSM has the advantage that it includes strips in bending and so could study beams as well as compression members. The paper by Hancock (1978) clearly identified the signature curve for a beam being the buckling stress versus the buckle half-wavelength for a single halfwavelength. Recent developments by Adany and Schafer (2006) have included the Constrained Finite Strip Method (cFSM) which has allowed the buckling mode decomposition into pure local, distortional and overall modes. In the earlier papers, the modes tended to be a combination of the basic modes although normally dominated by one at a particular half-wavelength. The application of the SAFSM for pure shear buckling analysis of section has not been studied until recently, although the methodology was available in the Plank and Wittrick (1974) paper. Recently, Hancock and Pham (2011, 2012) applied Plank and Wittrick’s methodology using complex mathematical techniques and developed a computer program “bfinst7.cpp” employing the SAFSM to study pure shear buckling. Recent research by Pham SH, Pham CH and Hancock (2012a, b) using the SAFSM has investigated the elastic shear buckling of lipped channel sections with complex web stiffeners. They proved that the presence of web stiffeners mainly improves the shear buckling stress of sections by increasing the local buckling stress. The web stiffeners slightly improve the distortional buckling stress. The SAFSM analysis in this report is also based on this program to further investigate the effect of flange widths and lip sizes of the whole channel sections on the elastic buckling stresses in shear. SPLINE FINITE STRIP METHOD (SFSM)

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B Buccklin ng Stu S udies of Co old--Fo orme ed Cha ann nelss in Shearr us sing g the S Sem mi-A Anallytic cal Finite Stri rip a and d Sp pline e Fiinite e Strip p Me etho odss Th he Spli S ine Fin nite e Sttrip Me etho od (SF FSM M) is i a de eve elop pme ent of the e SA AFS SM orrigin nally y deve elop ped byy Cheu ung g (19 968 8, 1976 6). It uses u s sp pline fu unc ction ns in the t lon ngitu udin nal dire ection in pla ace of the e sin ngle e ha alf sine ew wave e over o r the e le engtth of o th he secction n, and a d ha as b been prov p ven to be an n effficie ent too ol fo or ana a lyziing structure es with w co onsttantt ge eom metriic prop p perttiess in a part p ticu ular direction, ge ene erallly th he longitu udin nal one e. T The ad dvan ntag ge of the t SF FSM M iss tha at itt allowss more m e co omp plex x tyypes s off loa adin ng a and d bo oundarry cond c ditio onss oth her tha an sim s ple suppo orts s to be easilyy invvesttiga ated d an nd buc b cklin ng in n sh hea ar iss alsso eas e sily acccounte ed fo or. FSM M was w fully d developed d fo or th he line l ear elasticc strucctura al ana a lysis of o fo olde ed plat p te stru s ctures byy Initially, tthe SF Fa an and a Ch heung (19 982)). The T SF FSM M wa as the t n extende ed to t buck b kling and a nonlin nearr an nalyyses off fla at p plate es and a d folded d-pllate e sttruc ctures by La au and d Han H cocck (19 ( 86)) an nd Kw won an nd Hanco ock (1991 1, 199 1 93). Th he SFS SM M invvolv ves sub bdivvidiing a tthin-wa alled dm mem mbe er in nto longitu udin nal striips wh here e ea ach strrip is i a assu ume ed to t b be free f e to o de eform m b both in n itts plan p ne (me embra ane dis spla ace eme entss) and a ou ut o of itts plan p ne (fle exural dis spla acem men nts)). The T e fun nctionss ussed in the e longittudinal direction n arre norm n mally B3 B spli s iness. The T en nds of the t sectio on u und der sstud dy are e no orma allyy fre ee to o de eform lon ngitu udin nally but are a pre evented frrom m de eforrmin ng in a cro oss-se ectio onal plane e. he SFS S SM wa as rece r ently emp e ploy yed by Ph ham m an nd Hanco ock (20 009 9a, 201 12b b) to o study th he elastic c bu uckkling g off Th thin-w walle ed plain cha c anne el sect s tion ns a and thin-w walle ed cha ann nel sec s ction ns with w h in nterm med diatte w web b sttiffe enerrs in n pure e Pham m and a Ha anco ock k, 20 009 9b). Th hese e sttudiies pro ovid ded an efffecttive too ol to o de eterrmine the e sh hearr bu ucklling g shearr (P s w which was w ussed to devvelo op the e prropo ose ed sshear des d sign n cu urve e in n th he D Dire ect Streng gth Me etho od (DS ( SM)) strress (Pham ma and Hanco ock, 20 012a).

MOD DEL LLING G LIP L PPE ED D CHA ANN NE EL SE SECT TIO ONS S IN SH SHEA AR he geo g metry of the t lipp ped d ch hannel sectio on sstud died d in thiis re epo ort is show wn in F Fig. 1(a a). The e ch han nnel co onsistss Th of a web w b ch hann nel of 200 0mm m (b1),, a flan nge e width of 0.0 01 mm m to 160 0 mm m (b2), a lip p siz ze o of 0.01 0 1 mm m tto 20 2 mm mm a with h th he thicckness s off 2 mm m. The T e ve ery sm mall va alue e off 0.01 mm m in n b both h fla ang ge widt w th and a d lip p siize,, (d1), all altthou ugh h im mpra actic cal,, ha as bee b en used u d to o allow w lim mitin ng con c ndition of zerro flang f ge wid dth and d un nlip pped d chan nne el to o e ap ppro oach hed d. Thes T se d dim menssion ns a are all cen ntre eline e an nd not overa all. be

1. (a a) Geo G ome etryy, Shea ar Stre S ess Dis strib butio on a and d (b b) Boun nda ary Con C ndittions of o Lippe ed Cha C ann nel S Secction ns Figure 1

Schoo ol off Civvil Engi E nee ering g The Unive ersitty of o Sy ydne ey

Re esea arch h Re eporrt R932 R 2

Pag ge 6

Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods

Table 1: Boundary Conditions of Lipped Channel-SAFSM vs SFSM SAFSM SFSM Edges u v w u v w 0 0 1 0 0 0 3478 1 0 0 0 0 0 1 2 5 6 9 10 0 0 0 0 0 0 11 12 Note: u, v and w are translations in the x,y and z directions respectively. 0 denotes free and 1 denotes restraint DOF The member is subdivided into 40 longitudinal strips which include 16 strips in the web, 10 strips in each flange and 2 strips in each lip. For the longitudinal direction, the half-wavelength in the SAFSM or the member length in the SFSM of the channel sections varies from 30 mm to 10000 mm. The shear flow distribution resulting from a shear force parallel with the web through the shear centre, shown in Fig. 1(a), is incorporated in the lipped channel section. This case of shear stress distribution is based on Case D in Pham and Hancock (2009a) which is the most representative of practice. The shear flow distribution is not in equilibrium longitudinally as this can only be achieved by way of a moment gradient in the section. However, it has been used in these studies to isolate the shear from the bending for the purpose of identifying pure shear buckling loads and modes. The finite strip buckling analysis allows the uniform shear stresses in each strip to be used to assemble the stability matrix of each strip then the system stability matrix. To simulate the variation in shear stress, each strip in the cross-section is assumed to be subjected to a uniform shear stress which varies from one strip to the other strip. The more the cross-section is subdivided into strips, the more accurately the shear stress is represented in order to match the practical shear flow distribution. Fig. 1(b) and Table 1 shows the boundary conditions of the lipped channel. In the SAFSM, the channel is free to distort at the two end sections, whereas all edges of the two end cross-sections in the SFSM are simply supported. These latter boundary conditions prevent cross-section distortion. For the longitudinal direction, both finite strip methods (SAFSM and SFSM) permit the plates to undergo free deformation longitudinally including warping deformations along the channel member are also permitted.

RESULTS OF BUCKLING ANALYSES AND COMPARISONS OF SAFSM AND SFSM Fig. 2 shows the relationship curves between the shear buckling stresses and the half-wavelengths or lengths of lipped channel sections where the lip size (d1) is 20 mm and the flange width (b2) varies from 5 mm to 160 mm. In particular, the shear signature curves are plotted against the half-wavelengths (HWL) in Fig. 2(a) for the SAFSM and the elastic shear buckling stress versus the member length curves are graphically reproduced in Fig. 2(b) for the SFSM. Fig. 3 shows the corresponding shear buckling modes of the lipped channel with b2=5 mm and d1=20 mm at the typical half-wavelengths or lengths of 100 mm, 500 mm and 1000 mm respectively. As can be seen in Fig. 2, for the same small flange width of 5 mm (b2/b1=0.025 mm), the SAFSM and SFSM curves have similar shapes. At the very short half-wavelength or length of 100 mm, the shear buckling stress is 141.44 MPa for the SAFSM and is significantly lower than 378.45 MPa of the SFSM. This fact can be explained due to the differences in boundary conditions between the two methods. It can be seen in Fig. 3(a) that the SFSM analysis assumes no cross-section distortion at both section ends under analysis and this restraint increases the buckling stress above that of the SAFSM which is free to distort at the ends. When the half-wavelength or length increases to 500 mm, the shear buckling stresses drop dramatically to 19.48 MPa and 42.11 MPa for the SAFSM and SFSM respectively. The buckling mode for the SAFSM in Fig. 3(b) at HWL=500 mm shows twisting and relative rotation between the two ends, whereas the channel buckles in a twisting mode between the restrained section ends for the SFSM. At a longer half-wavelength or length of 1000 mm, the shear bucking stresses reduce to 4.37 MPa and 13.72 MPa for the SAFSM and SFSM respectively. The corresponding buckling mode for both methods is a single half-wave and the channels buckle sideways as shown in Fig. 3(c). At lengths greater than 1000 mm, the effect of the end restraints is small and so the SFSM curve drops close to that of the SAFSM. When School of Civil Engineering The University of Sydney

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Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods the flange width increases to 20 mm (b2/b1=0.1 mm), the SAFSM and SFSM curves have the same trend and behave similarly to those with small flange width of 5 mm. The shapes of the two relationship curves are shifted up relatively with flange width increments as shown in Fig. 2. 250 b 2 =5 b 2 =10

200

b 2 =15 b 2 =20 b 2 =40 b 2 =60

150

b 2 =80 b 2 =100 b 2 =120

100

b 2 =140 b 2 =160

50

0 100

1000

Maximum Stress in Section at Buckling (MPa)

Maximum Stress in Section at Buckling (MPa)

250

b2 =5 b2 =10 b2 =15

200

b2 =20 b2 =40 b2 =60

150

b2 =80 b2 =100 b2 =120 b2 =140

100

b2 =160

50

0

10000

100

Buckle Half-Wavelength (mm)

(a). Semi-Analytical Finite Strip Method

1000

10000

Member Length (mm)

(b). Spline Finite Strip Method

Figure 2. Shear Buckling Stress vs Buckling Half-Wavelength/Length of Lipped Channel Sections for Lip Size (d1) of 20 mm

SAFSM HWL=100mm

HWL=500mm

HWL=1000mm

a=500mm (b)

a=1000mm (c)

SFSM

a=100mm (a)

Figure 3. Shear Buckling Modes of Lipped Channel for Flange Width (b2) of 5mm and Lip Size (d1) of 20 mm It is interesting to note at the wider flange width of 40 mm (b2/b1=0.2 mm) that the SAFSM graph reduces rapidly to 115.05 MPa at a half-wavelength of 200 mm. It then remains approximately unchanged up to a half-wavelength of 300 mm and slightly decreases to 111.13 MPa at 400 mm. Fig. 4 shows the corresponding shear buckling mode shapes at typical half-wavelengths or lengths for both SAFSM and SFSM. Interestingly, for the SAFSM, although the signature curve has not clearly shown a minimum point,

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Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods there is a trend for the lipped channel to buckle in pure local buckling where the junction lines between web, flanges and lips remain almost straight as shown in Fig. 4(a) for half-wavelength of 200 mm. At the half-wavelength of 400 mm, the buckling mode shown in Fig. 4(b) turns slightly to the distortional buckling mode as the junction lines between flanges and lips start to distort. When the half-wavelength increases further to 600 mm and 800 mm, the shear buckling stress then decreases as a result of switching from the local buckling mode to the distortional buckling mode as shown in Figs 4(c) and 4(d) respectively.

SAFSM HWL=200mm

HWL=400mm

HWL=600mm

HWL=800mm

SFSM a=200mm (a)

a=400mm (b)

a=600mm (c)

a=800mm (d)

Figure 4. Shear Buckling Modes of Lipped Channel for Flange Width (b2) of 40mm and Lip Size (d1) of 20 mm For the SFSM at the flange width of 40 mm (b2/b1=0.2 mm), the graph drops dramatically to 125.01 MPa at the length of 400 mm. In the length range from 400 mm to 800 mm, the shear buckling stress then reduces slightly from 125.01 MPa to 112.59 MPa. The explanation for this fact is that the flange width of 40 mm is just wide enough to provide lateral restraints to the web. Along with the restraint at the two section ends, these lateral restraints enforce the channel to buckle locally and turn to two buckles when the length increases from 200 mm to 400 mm as shown in Figs. 4(a) and 4(b). Especially, Fig. 4(b) for SFSM shows the interesting shear buckling mode where there is a coupling between the local buckling mode in the web at short half-wavelength and a distortional buckling mode in the flanges and lips at a longer half-wavelength. At the length of 600 mm in Fig. 4(c), the buckling mode occurs similarly to that of 400 mm except there are three local buckles in the web. As can be seen in Fig. 2(b), for the SFSM, the shear buckling stress starts to drop dramatically from the length of 800 mm. The reason for this fact is due to the switching of buckling modes from local to distortional. At the particular length of 800 mm in Fig. 4(d) for the SFSM, the coupling still occurs between two local buckles but at longer half-wavelength with distortional buckles in the flanges and lips. As the flange width increases above 40 mm, all the signature curves shown in Fig. 2(a) for SAFSM form minimum points at a half-wavelength of approximately 200 mm. Fig. 5 shows the corresponding shear buckling mode shapes at typical half-wavelengths or lengths of the channel with a flange width of 80 mm (b2) and a lip size of 20 mm (d1) for both SAFSM and SFSM. It is worth noting that the shear buckling stresses at all minimum points converge roughly at 116.5 MPa irrespective of the flange widths. The channel buckles in a pure local buckling mode since the junction lines between the web, flanges and lips remain straight. Fig. 5(a) for the SAFSM shows this type of local buckling at a half-wavelength of 200 mm. When the halfwavelength increases above 200 mm, the shear signature curves raise relatively with flange width increments. The maximum shear stresses are shifted up to 162.74 MPa at a flange width of 100 mm. When the flange width increases from 120 mm to 160 mm, all the signature curves seem to coincide with each other no matter how big the flange width. At a longer half-wavelength of 800 mm, the signature curves start to drop and the buckling mode switches to shear distortional buckling. Fig. 5(b) shows typical shear distortional buckling mode at a half-wavelength of 1400 mm for the SAFSM. School of Civil Engineering The University of Sydney

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Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods

SAFSM HWL=200mm

SAFSM

SFSM

HWL=1400mm

a=600mm

SFSM

a=1000mm

SFSM a=1800mm

SFSM a=1400mm (b) Distortional Buckling

(a) Local Buckling

Figure 5. Shear Local and Distortional Buckling Modes of Lipped Channel for Flange Width (b2) of 80mm and Lip Size (d1) of 20 mm For the SFSM, when the flange width increases above 40 mm, all the SFSM curves of different flange widths almost coincide with each other and behave identically. They drop dramatically to approximately 133 MPa at a length of 400 mm then slowly reduce to around 123 MPa at 200 mm. Above the length of 600 mm, they almost stay unchanged at 117.5 MPa. This value is very close to the minimum points on the SAFSM curves of 200 mm half-wavelength (approximately 116.5 MPa). At this particular shear stress for the SFSM, local buckling occurs irrespective of the channel member length. The longer the channel, the more multi local buckles can be added and the effect of the two restrained end sections is very small. In Fig. 5(a), the local buckling turns from three buckles at 600 mm length to five and seven at 1000 mm and 1400 mm respectively. Depending on the flange width, the SFSM curves start to drop at different lengths, and the buckling mode switches from local to distortional. Fig. 5(b) shows the shear distortional mode for the SFSM with 80 mm flange width at 1800 mm length where the SFSM curve starts to drop and the buckling modes. The wider the flange width, the longer the channel will stay until the shear local buckling mode changes to distortional buckling mode. As the flange width increases further from 120 mm to 160 mm, the SFSM curves are almost unchanged and coincide with each other. The effect of flange width in this range is therefore small for both shear local and distortional buckling. This fact is similar to that of the SAFSM. As discussed above, the lips also play an important role in the shear buckling capacity of lipped channel section. Figs. 6 and 7 show similar relationship curves between the shear buckling stresses and the halfwavelengths or lengths of lipped channel sections where the lip sizes (d1) are reduced to 10 mm and 0.01 mm (unlipped channel) respectively. Each figure also includes comparison between the SAFSM and SFSM. Each relation curve also represents a flange width which is in the range from 0.01 mm to 160 mm. By comparison between the various lip sizes (20 mm, 10 mm and 0.01 mm in Figs 2, 6 and 7 respectively), the relationship curves behave similarly. For the SAFSM, the minimum point for the lip size of 10 mm is unchanged at approximately 115.5 MPa, whereas it slightly reduces to 108.5 MPa for the lip size of 0.01 mm (unlipped channel). Similarly, for the SFSM, shear buckling stresses are comparable to those of the SAFSM at different lip sizes (10 mm and 0.01 mm) where all the curves stay unchanged. It is clear from these values of the minimum points for local buckling in shear that the effect of the lip sizes is very small. However, when reducing the lip sizes, the flange width has to be wider to get to the same shear buckling stress. For example, it can be seen in Fig. 2, for a lip size of 20 mm, the shear signature curve can form a minimum point with the flange width of 40 mm. From Figs. 6 and 7, the minimum points appear and give almost the same values of shear buckling stresses when the flange widths are wider at 60 mm and 80 mm for the lip sizes of 10 mm and 0.01 mm respectively. School of Civil Engineering The University of Sydney

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Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods

250 b 2=5 b 2=10

200

b 2=15 b 2=20 b 2=40 b 2=60

150

b 2=80 b 2=100 b 2=120

100

b 2=140 b 2=160

50

0 100

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Maximum Stress in Section at Buckling (MPa)

Maximum Stress in Section at Buckling (MPa)

250

b 2 =5 b 2 =10

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b 2 =80 b 2 =100 b 2 =120

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b 2 =140 b 2 =160

50

0

10000

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Member Length (mm)

Buckle Half-Wavelength (mm)

(b). Spline Finite Strip Method

(a). Semi-Analytical Finite Strip Method

Figure 6. Shear Buckling Stress vs Buckling Half-Wavelength/Length of Lipped Channel Sections for Lip Size (d1) of 10 mm

250 b 2 =5 b 2 =10

200

b 2 =15 b =20 2

b 2 =40

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b 2 =60 b =80 2

b 2 =100 b 2 =120

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Maximum Stress in Section at Buckling (MPa)

Maximum Stress in Section at Buckling (MPa)

250

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b =80 2

b 2=100 b 2=120

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50

0 100

Buckle Half-Wavelength (mm)

(a). Semi-Analytical Finite Strip Method

1000

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Member Length (mm)

(b). Spline Finite Strip Method

Figure 7. Shear Buckling Stress vs Buckling Half-Wavelength/Length of Lipped Channel Sections for Lip Size (d1) of 0.01 mm (Unlipped Channel)

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Buckling Studies of Cold-Formed Channels in Shear using the Semi-Analytical Finite Strip and Spline Finite Strip Methods

CONCLUSIONS This report has outlined the elastic buckling analyses of channel sections subject to pure shear stresses using Semi-Analytical Finite Strip Method (SAFSM) and Spline Finite Strip Method (SFSM). The SAFSM differs from the SFSM in which local spline functions are used instead of harmonic series in the longitudinal direction. For boundary conditions, while the channel in the SAFSM is free to distort at the two section ends, all edges of the two end cross-sections in the SFSM are simply supported. The SAFSM is therefore limited to a single half-wavelength whereas the SFSM can include multiple buckles. By varying the flange widths and lip sizes, the analysis results show that the flanges with lips can improve significantly the shear buckling stress of the channel sections. When the flange width is small, the behaviours of the two relationship curves of the two methods are similar. The shear buckling stresses reduce dramatically as the half-wavelengths or lengths of the channel increase. The values of the shear buckling stresses for the two methods are significantly different at short half-wavelengths or lengths due to different boundary conditions at the two end sections. At longer half-wavelengths or lengths, the SFSM curves drop closer to those of the SAFSM due to the small effect of the end boundary conditions. Channel sections with a very narrow flange may buckle in a twisting mode. When the flange is wide enough to provide elastic torsional restraint to the web, the SAFSM curves start to form minimum points where the pure shear local buckling can be determined. Meanwhile, at approximately the same shear stresses, the SFSM curves stayed unchanged, the local buckling modes can be observed as a result of the multi local buckles in which the half-wavelength of each local buckle is almost similar to that of the SAFSM. Depending on the flange width, the two relationship curves start to drop at different lengths where the buckling mode switches from local to distortional buckling. The potential coupling between multiple local buckles in the web and longer half-wavelength distortional buckles in the flanges and lips may occur at a particular flange width where it is just wide enough to form the minimum point in the SAFSM.

ACKNOWLEDGEMENT Funding provided by the Australian Research Council Discovery Project Grant DP110103948 has been used to perform this project. The SFSM program used was developed by Gabriele Eccher. The graphics program used to draw the 3D buckling modes was developed by Song Hong Pham under an Australian Government AusAid Scholarship.

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