SHEAR BUCKLING OF CHANNEL SECTIONS WITH SIMPLY SUPPORTED ENDS USING THE SEMI-ANALYTICAL FINITE STRIP METHOD CAO HUNG PHAM GREGORY J.

Transcription

1 SHEAR BUCKLNG OF CHANNEL SECTONS WTH SMPLY SUPPORTED ENDS USNG THE SEM-ANALYTCAL FNTE STRP METHOD CAO HUNG PHAM GREGORY J. HANCOCK RESEARCH REPORT R931 FEBRUARY 2013 SSN SCHOOL OF CVL ENGNEERNG

2 SCHOOL OF CVL ENGNEERNG SHEAR BUCKLNG OF CHANNEL SECTONS WTH SMPLY SUPPORTED ENDS USNG THE SEM-ANALYTCAL FNTE STRP METHOD RESEARCH REPORT R931 GREGORY J. HANCOCK CAO HUNG PHAM February 2013 SSN

3 Copyright Notice School of Civil Engineering, Research Report R931 Shear Buckling of Channel Sections with Simply Supported Ends Gregory J. Hancock Cao Hung Pham February 2013 SSN This publication may be redistributed freely in its entirety in its original form without consent of copyright owner. Use of material contained in this publication in any or published works must be appropriately referenced,, if necessary, permission sought from author. Published by: School of Civil Engineering Sydney NSW 2006 Australia This report or Research Reports published by School of Civil Engineering are available at School of Civil Engineering Research Report R931 Page 2

4 ABSTRACT Buckling of thin-walled sections in pure shear has been recently investigated using Semi-Analytical Finite Strip Method (SAFSM) to develop signature curve for sections in shear. The method assumes that buckle is part of an infinitely long section unrestrained against distortion at its ends. For sections restrained at finite lengths by transverse stiffeners or or similar constraints, Spline Finite Strip Method (SFSM) has been used to determine elastic buckling loads in pure shear. These loads are higher than those from SAFSM due to constraints. The SFSM requires considerable computation to achieve buckling loads due to large numbers of degrees of freedom of system. n 1980 s, Anderson Williams developed a shear buckling analysis for sections in shear where ends are simply supported based on exact finite strip method. The current report furr develops SAFSM buckling ory of YK Cheung for sections in pure shear accounting for simply supported ends using methodology of Anderson Williams. The ory is applied to buckling of plates of increasing length channel sections in pure shear also for increasing length. The method requires increasing numbers of series terms as sections become longer. Convergence studies with strip subdivision number of series terms is provided in report. KEYWORDS Cold-formed channel sections; Simply supported ends; Shear buckling analysis; Finite strip method; Semianalytical finite strip method; Complex mamatics. School of Civil Engineering Research Report R931 Page 3

5 TABLE OF CONTENTS ABSTRACT... 3 KEYWORDS... 3 TABLE OF CONTENTS... 4 NTRODUCTON... 5 THEORY... 5 Strip Axes Nodal Line Deformations... 5 Plate Buckling Deformations with Simply Supported Ends... 6 Forces along The Nodal Lines... 6 Total Energy... 7 Stiffness Stability Matrices... 8 Assembly of Stiffness And Stability Matrices... 8 EGENVALUE ROUTNES... 9 Sturm Sequence Property... 9 Direct Computation of Sign Count of ([A] - λ [B])... 9 Eigenvector Calculation... 9 Computer Program bfinst8.cpp... 9 SOLUTONS TO PLATES AND SECTONS N SHEAR... 9 Plate Simply Supported on Both Longitudinal Edges... 9 Lipped Channel Section in Pure Shear Web-Stiffened Channel in Pure Shear CONCLUSONS ACKNOWLEDGEMENTS REFERENCES Appendix A: Buckling Stress Shear Buckling Coefficient for Plain Lipped Channel Appendix B: Buckling Stress Shear Buckling Coefficient for Stiffened Web Channel Appendix C: Flexural Stiffness Stability Matrices For Mth Series Term Appendix D: Membrane Stiffness Stability Matrices For Mth Series Term School of Civil Engineering Research Report R931 Page 4

6 Shear Buckling of Cha annel Sections with Simply Supported Ends using Semi-Analytical Fini ite Strip Method NTRODUCTON The recent development of Dire ect Strength Met thod (DSM) of design, as specified in North American Specification NAS S100 (AS, 2007) for design of cold-formed steel structural mem mbers Australian/New Zeal Stard AS/NZS 4600:2005 (Stards Australia, 2005), has required need to compute local, distortional overall elastic buckling loads of full sections. The usual approach is to compute buckling signature curv ves for sect tions using a finite strip buc ckling ana alysis method such as in programs CUF FSM (Adany Schafer, 2006) THN-WALL (CASE, 2006). A recent development has been DSM for sections in pure shear (Ph am Hancock, 2012a) as currently balloted for acceptance in 2012 Edition of NAS S100. The method requires elastic buckling load of full sections in pure shear to be computed. The complex sem mi-analytical finite strip method (SAFSM) of Plank Wittrick (197 74) has been used by Hancock Pham (2011, 2012) to com mpute signature curves for channel sections in pure shear. The method assumes ends of half-wavelength under consideration are free to disto ort buc kle is part of a very long length witho out restraint from end conditions. Detailed studies of sec tions with rectangular triangular intermediate stiffeners in web have been performed by Pham SH, Pham CH Hancock (2012a, 2012b) using this analysis. n practice, sections may be restrained at ir ends or by transverse stiffeners so that shear buckling modes are changed buckling loads increased by end effects. Pham Hancock (2009a, 2012b) have used Spline Finite Strip Met hod (SFSM) developed for elastic buckling by Lau Hancock (1986) to study channel sections with simply supported ends in shear. Although it is muc ch mo re effic cient than Finite Element Method (FEM), it still requires substantial computer resources to obta ain a result. Anderson Willi iams ( 1985) have pro posed a version of exact finite strip buckling analysis developed by Wittrick (1968), Williams Wittrick (1969), accounting for simply supported end conditions for sections in shear. This report furr dev velops evaluates this method for SAFSM of Plank Wittrick ( 1974). Comparisons with SFSM are provided to establish accuracy. THEORY STRP AXES AND NODAL LNE DEFORMATONS The x-ax xis is in longitudinal dire ection in plane of strip, y-axis is in transversee direction in plane of strip, w is in z-direction perpendicular to strip as shown in Fig. 1. Figure 1. Strip Axes Nodal Line Deformations School of Civil Engineering Research Report R931 Page 5

7 The strip nodal line flexural deformations in Fig. 1 are given in vector format by: {δ F } = (w 1, θ x1, w 2, θ x2 ) T (1) Similarly, strip nodal line membrane deformations in Fig. 1 are given in vector format by: {δ M } = (u 1, v 1, u 2, v 2 ) T (2) n finite strip method of Cheung (1968, 1976), longitudinal variations of displacement are described by harmonic functions transverse variations are described by polynomial functions. PLATE BUCKLNG DEFORMATONS WTH SMPLY SUPPORTED ENDS For analysis of plates sections with simply supported ends, strip nodal line deformations are regrouped into those associated with variation according to sine function, those associated with variation according to cosine function as {δ s } = (v 1, w 1, θ x1, v 2, w 2, θ x2 ) T {δ c } = (u 1, u 2 ) T respectively. The deformations {δ} of a strip for μ series terms assuming simply supported ends can refore be expressed by: µ δ Re iδ δ X x δ sin µ L δ cos (3) L where {δ sm } T = (v 1, w 1, θ x1, v 2, w 2, θ x2 ) m T {δ cm } T = (u 1, u 2 ) m T X x cos i sin (4) L The symbol Re means real part, m is number of series term, elements of {δ sm } {δ cm } are real. The displacements in Equation 3 satisfy simply supported boundary conditions. L FORCES ALONG THE NODAL LNES There are N nodal lines corresponding to intersection of strips both at plate junctions within plates subdivided into strips. The forces at nodal lines can be computed at a given load factor λ E from unrestrained stiffness stability matrices for mth series term derived in Hancock Pham ( ) using: where [H m ] = [K m ] λ E [G m ] F H δ (5) The stiffness matrix [K m ] is real stability matrix [G m ] is real when shear stresses shown in Fig. 2 are zero but complex Hermitian when shear stresses are non-zero. The load factor λ E applies to stresses shown for each strip as in Fig. 2. Equation 5 for nth series term can be partitioned according to sequence in Equation 3 so that: F if H H H H iδ iδ (6) The forces are out-of-phase with displacements so that multiple i on {F c } {δ c } accounts for 90 degree phase difference between se two components when shear is zero, so that [H n ] is real symmetric for such cases. School of Civil Engineering Research Report R931 Page 6

8 Shear Buckling of Cha annel Sections with Simply Supported Ends using Semi-Analytical Fini ite Strip Method Figure 2. Membrane Stressess The forces along nodal lines corresponding to assumed displacements in Equation 3 are derived as follows: F Re µ F F X x µ H R δ H R δ sin H R δ H R δ cos L wheree superscripts R denote real imaginary parts respectively. L H δ H δ cos H δ H δ sin L L (7) TOTAL ENERGY n order to compute stiffness stability matrices of strip according to conventional finite strip ory (Cheung 1968, 1976) buckling ory (Plank Wittrick (1974)), it is necessary to dete ermine total energy in strip under action of membrane forces. The total energy E is given by: E L δ T F dx L µ δ T H R δ L µ µ c δ T H δ H δ c δ T H δ H δ (8) wheree c L L sin L cos L dx (9) m m+n odd 0 m+n even School of Civil Engineering Research Report R931 Page 7

9 STFFNESS AND STABLTY MATRCES For equilibrium, orem of minimum total potential energy with respect to each of elements {δ m } is: The result is: where E 0 (10) H R µ δ Q δ 0 m 1, 2,., µ (11) Q c H c H c H c H c H c H c H (12) c H The matrices [H m R ] [Q mn ], vectors {δ m }, {δ n }, are real. n Equation 11, when shear stress τ is zero, Q mn is zero hence individual series terms are uncoupled in m = 1, 2,..., μ each can be solved independently. Furr, [Q mn ] is transpose of [Q nm ] since [H m ] is Hermitian so that resulting stiffness stability matrices are symmetric. For example, when μ = 3 (3 series terms), full stiffness stability matrix derived from Equation 11 can be represented as follows: H R Q 0 H R Q 0 H Q H R Q Q T H R Q (13) 0 Q H R 0 Q T H R The matrix [H] has 4 * N * μ degrees of freedom. f rows columns in matrix [H] are organised so that each degree of freedom is taken over μ series terms, n half-bwidth of matrix is simply μ times half-bwidth of problem with one series term. This speeds computation of eigenvalues eigenvectors considerably. ASSEMBLY OF STFFNESS AND STABLTY MATRCES The component flexural stiffness stability matrices used to compute [K m ] [G m ] in Eq. 5 ( hence [H m ]) was derived in Hancock Pham (2011, 2012) to be: k F λ E g F δ F 0 (14) The flexural stiffness matrix [k Fm ] is real flexural stability matrix [g Fm ] is real if shear stress τ is zero. However, stability matrix [g Fm ] is complex Hermitian if shear stress is non-zero. They are given for mth series term in Appendix C. The component membrane stiffness stability matrices used to compute [K m ] [G m ] in Eq. 5 were derived by Hancock Pham (2011, 2012) to be: k M λ E g M δ M 0 (15) The membrane stiffness matrix [k M ] is real membrane stability matrix [g M ] is real. They are given for mth series term in Appendix D. The term λ E is load factor. For folded plate assemblies including thin-walled sections such as channels, (14) (15) must be transformed to a global co-ordinate system to assemble stiffness [K m ] stability [G m ] matrices of folded plate assembly or section for mth series term. School of Civil Engineering Research Report R931 Page 8

10 t is clear that only [g F ] has complex terms. So only [H 11n ] [H 11m ] in Equation 12 are non-zero. Equation 12 refore simplifies to: Q c H c H 0 (16) 0 0 EGENVALUE ROUTNES STURM SEQUENCE PROPERTY From ory of equations (Turnbull, 1946), leading principal minors of [C]-λ[] (where [] is a unit matrix) form a Sturm sequence. The leading principal minor of order r is given by det ([C r ] λ[]) where [C r ] is leading principal sub-matrix of order r of [C]. The first term of Sturm sequence is leading principal minor of order r=0 is defined to be unity. The number of eigenvalues greater than λ is equal to number of agreements in sign between consecutive members of Sturm sequence from r = 0 to r = n where n is dimension of matrix [C]. This property is very useful in isolating range of λ in which a particular eigenvalue is located. The eigenvalue corresponding to a particular mode number can be isolated by bisection between values of λ which bound eigenvalue. DRECT COMPUTATON OF SGN COUNT OF ([A] - λ [B]) Peters Wilkinson (1969) have shown that sign of det([a r ] λ[b r ]) is same as that of det([c r ] λ[]). Consequently, it is possible to apply Sturm sequence directly to ([A] - λ[b]) without need to transform to stard eigenvalue problem det([c] - λ[]) = 0. For finite strip buckling analysis given by (11), [G] component of [H] is chosen as [A] [K] component of [H] is chosen as [B] so that computed eigenvalues λ of ([A] - λ[b]) are reciprocals of load factors λ E EGENVECTOR CALCULATON Wilkinson (1958) has produced a method for computing eigenvector {δ} of (11) by solving equations at value of λ E for a unit right h side vector {1} replacing {0} in (11). The process is usually repeated once to purify eigenvector with unit vector {1} replaced by {δ} from first iteration. This method has been used in calculations in this report. COMPUTER PROGRAM bfinst8.cpp A computer program bfinst8.cpp has been written in Visual Studio C++ to assemble stiffness stability matrices given by Equation 11 to solve for eigenvalues using Sturm sequence property described in Sections above, to compute corresponding eigenvectors as per Section above. The program stores real [K] matrix real [G] complex [G] components of stability matrices in order to extract eigenvalues eigenvectors. The results from this analysis are given notation resafsm (restrained SAFSM) in this report to distinguish it from unrestrained SAFSM in Hancock Pham (2011, 2012). SOLUTONS TO PLATES AND SECTONS N SHEAR PLATE SMPLY SUPPORTED ON BOTH LONGTUDNAL EDGES School of Civil Engineering Research Report R931 Page 9

11 The solution for a plate simply supported along both longitudinal edges determined using resafsm analysis is compared with classical solution of Timoshenko Gere (1961) tem 9.7 Buckling of rectangular plates under action of shearing stresses. The equation for elastic buckling of a rectangular plate is given (Timoshenko Gere (1961)) as: τ D (17) where D is plate flexural rigidity, b is width of plate which may consist of multiple strips k v is plate buckling coefficient in shear. The analysis is carried out for 8 equal width strips an increasing number of series terms. Aspect ratios of 1:1, 2:1, 3:1 4:1 have been investigated solutions for buckling coefficient k v are compared in Table 1 with those of Timoshenko Gere Table t is clear that solutions become more accurate with increasing numbers of series terms, that more terms are required for higher aspect ratios. The solutions converge to values slightly lower than those of Timoshenko Gere (1961) presumably because y used less series terms. These k v values can be compared with that for a buckle in an infinitely long section of given in Plank Wittrick (1974) Hancock Pham (2011, 2012). Anderson Williams (1985) achieved values of 9.37, for 6 strips 3, 4 6 series terms respectively using exact strip formulation for a square plate simply supported along all four longitudinal edges. Aspect ratio Timoshenko 3 series terms 4 series terms 6 series terms Gere (1961) 1: : : : Table 1 Buckling coefficients k v for simply supported square rectangular plates The buckling modes for an aspect ratio of 2:1 6 series terms determined from bfinst8.cpp are plotted as a contour plot using Mathcad in Fig. 3. The buckling mode has one elongated local buckle of web satisfying simply supported boundary conditions on all four edges. The contour line finishing at centres of ends is zero displacement contour. Figure 3. Shear Buckling Mode of Plate of Aspect Ratio 2:1 with Four Edges Simply Supported (6 Series Terms) School of Civil Engineering Research Report R931 Page 10

12 LPPED CHANNEL SECTON N PURE SHEAR n order to extend study to lipped channel sections, a 200mm deep lipped channel with flange width 80mm, lip length 20mm thickness 2mm as studied by Pham Hancock (2009a, 2012b) has been used. These dimensions are all centreline not overall. n Pham Hancock (2009a), three different shear stress distributions have been investigated. These are uniform shear in web alone (called Cases A/B), uniform shear in web flanges (called Case C), a shear stress equivalent to a shear flow as occurs in a channel section under a shear force parallel with web through shear centre (Case D as shown in Fig. 4). n this report, only Case D is studied as it is 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 section. However, it has been used in se studies to isolate shear from bending for purpose of identifying pure shear buckling loads modes. The finite strip buckling analysis allows uniform shear stresses in each strip, as shown in Fig. 2, to be used to assemble stability matrix [k g ] of each strip n system stability matrix [G]. Fig. 4 demonstrates that shear flow in each strip is uniform. n studies of plain channels in this report, web is divided into sixteen equal width strips, flanges into ten each lips into two each making 40 strips 41 nodal line with a total of 164 degrees of freedom. Adequate accuracy can be achieved for engineering purposes with 18 strips, 19 nodal lines 76 degrees of freedom. However, 40 strips have been used in this case for accurate benchmarking against SFSM analyses. Figure 4. Shear Flow Distribution Assumed (Case D) The SAFSM resafsm curves of buckling stress versus half-wavelength/length are compared in Fig. 5. The resafsm graph (circles) of buckling stress versus length (as opposed to half-wavelength for SAFSM) was computed using bfinst8.cpp program described above with 8 series terms. The resafsm analysis assumes no cross-section distortion at both ends of section under analysis (Z = 0, L). For a section of length 200mm, resafsm analysis gives a buckling coefficient k v of for web which is higher than that for a square panel in shear at 9.34 due to flange restraint. The buckling mode is shown in Fig. 6 encompasses a single buckle half-wavelength. At L = 600mm, 1000mm 1600mm, buckling modes are shown in Fig. 7, 8 9 respectively involve 3, 5 6 local buckle half-waves respectively. At L = 2000mm, buckling mode involves one half-wavelength is a type of distortional buckle is shown in Fig. 10. For comparison, SAFSM graph (squares) of buckling stress versus buckle half-wavelength (signature curve) derived in Hancock Pham (2011, 2012) using program bfinst7.cpp is also shown in Fig. 5 for lipped channel for a range of buckle half-wavelengths from 30mm to 10000mm. The graph reaches a minimum at approximately 200mm half-wavelength n rises starts to drop at about 800mm. The buckling coefficient k v corresponding to minimum point is based on average stress in web (τ av = V/A w ) computed from shear load V on section divided by area of web A w. This compares with an asymptotic value of 6.60 for multiple local buckles using resafsm analysis. School of Civil Engineering Research Report R931 Page 11

13 Maximum Stress in Section at Buckling (MPa) Shear Buckling of Channel Sections with Simply Supported Ends resafsm (bfinst8.cpp) SAFSM (bfinst7.cpp) Buckle Half-Wavelength/Length (mm) Figure 5. Buckling Stress versus Length/Half-Wavelength from SAFSM (bfinst7.cpp) resafsm (bfinst8.cpp) for Simple Lipped Channel Figure 6. Simple Lipped Channel Shear Buckling Mode at L = 200mm Figure 7. Simple Lipped Channel Shear Buckling Mode at L = 600mm School of Civil Engineering Research Report R931 Page 12

14 Shear Buckling of Channel Sections with Simply Supported Ends Figure 8. Simple Lipped Channel Shear Buckling Mode at L = 1000mm Figure 9. Simple Lipped Channel Shear Buckling Mode at L = 1600mm Figure 10. Simple Lipped Channel Shear Buckling Mode at L = 2000mm School of Civil Engineering Research Report R931 Page 13

15 To validate accuracy of resafsm analysis, buckling stresses coefficients are compared in Appendix A with Spline Finite Strip Method (SFSM) reported in Pham Hancock (2009a, 2012b). The SFSM values have also been computed for a section with same strip subdivision of 40 strips can be regarded as an accurate solution for benchmarking. They were given previously in Hancock Pham (2011, 2012). n general, resafsm values are higher than SFSM values by less than 1%. However, at L=1600mm, error increases to approximately 4% as use of only 8 series terms does not accurately predict behaviour when at least 6 buckle half-waves occur as shown in Fig. 9. WEB-STFFENED CHANNEL N PURE SHEAR Shear buckling of thin-walled channel sections with intermediate web stiffeners have been studied using SFSM by Pham Hancock (2009b) using SAFSM by Hancock Pham (2011, 2012) Pham SH, Pham CH Hancock (2012a, 2012b). n this report, particular web stiffener used is same as that in Hancock Pham (2011, 2012) has a rectangular indent of 5mm over a depth of 80mm located symmetrically about centre of web. Swage stiffeners of this type are common in practice. The web is divided into 6 equal width strips for each of 2 outer vertical elements in web, 8 for inner vertical element of web, stiffeners in web into 2 each, flanges into 6 each lips into two each making 40 strips 41 nodal line with a total of 164 degrees of freedom, same number as for simple lipped channel. The SAFSM resfsm curves of buckling stress versus half-wavelength/length are compared in Fig.11. The effect of different end conditions (Z = 0,L) between SAFSM (squares) resafsm (circles) are clear from this comparison. The shape of two curves is remarkably similar except that SFSM shows several plateau points at approximately 200mm 1000mm lengths corresponding to switch in buckling modes. The differences in modes at se switch points are shown in Figs. 12, Maximum Stress in Section at Buckling (MPa) 1000 resafsm (bfinst8.cpp) 800 SAFSM (bfinst7.cpp) Buckle Half-Wavelength/Length (mm) Figure 11. Buckling Stress versus Length/Half-Wavelength from SAFSM (bfinst7.cpp) resafsm (bfinst8.cpp) for Stiffened web Channel School of Civil Engineering Research Report R931 Page 14

16 Figure 12. Stiffened web channel shear buckling mode at L = 200mm Figure 13. Stiffened web channel shear buckling mode at L = 600mm Figure 14. Stiffened web channel shear buckling mode at L = 1000mm Figure 15. Stiffened web channel shear buckling mode at L = 1600mm School of Civil Engineering Research Report R931 Page 15

17 t is interesting to observe that stiffened web channel has only two buckle half-waves at a length of 1000mm compared with five for plain channel. The effect of stiffener is to decrease number of buckle half-waves with a corresponding increase in buckling stress. To validate accuracy of resafsm analysis for web stiffened channel, buckling stresses coefficients are compared in Appendix B with Spline Finite Strip Method (SFSM) reported in Hancock Pham (2011). The SFSM values have also been computed for a section with 40 strips can be regarded as an accurate solution for benchmarking. n general, resafsm values are higher than SFSM values by less than 1% for lengths greater than 600mm. At shorter lengths, error is slightly higher decreasing from 3.8% at 100mm to 1.8% at 200mm for mode shown in Fig. 12 n falling rapidly to 0.5% at 400mm. CONCLUSONS A new version of semi-analytical finite strip buckling analysis of thin-flat-walled structures under combined loading with simply supported end conditions has been derived programmed in Visual Studio C++. The method has been called resafsm to reflect restrained ends. The method uses multiple series terms to allow for increasing numbers of buckle half-waves as sections become longer. The method includes extraction of eigenvalues eigenvectors from matrices produced when thin-walled sections are subjected to shear in addition to compression bending. The method is based on ory of Anderson Williams applied to general complex matrices developed by Plank Wittrick. The method using up to 6 series terms has been checked against solutions of Timoshenko Gere for simply supported rectangular plate in pure shear found to produce accurate results. The solutions are slightly more accurate than Timoshenko Gere as more series terms have been used. The buckling stress versus length curve for a plain lipped channel in pure shear has been produced using resafsm with 8 series terms compared with signature curve from semi-analytical finte strip method with unrestrained ends (SAFSM). The resfsm curve has also been compared with spline finite strip buckling analysis (SFSM) where ends are fixed against distortion. For subdivision of section into 40 strips, method is accurate to generally better than 1% except in at L=1600mm where an error of 4% occurs more series terms would be required at this length before mode switches to a single distortional buckle at longer half-wavelengths. The buckling stress versus length for a web-stiffened lipped channel with a 5mm swage stiffener has also been investigated. The modes have changed significantly from those of simple lipped channel. Accuracies generally better than 1% have been achieved compared with SFSM analysis except at lengths less than 200mm where an error of 1.8% occurs. ACKNOWLEDGEMENTS Funding provided by Australian Research Council Discovery Project Grant DP has been used to perform this project. The SFSM program used was developed by Gabriele Eccher. The graphics program used to draw 3D buckling modes was developed by Song Hong Pham under an Australian Government AusAid Scholarship. REFERENCES Adany, S. Schafer, B.W. (2006), Buckling mode decomposition of thin-walled, single branched open cross-section members via a constrained finite strip method, Thin-Walled Structures, 64(1): p Anderson, M.S. Williams, F.W. (1985), Buckling of simply supported plate assemblies subject to shear loading, Aspects of analysis of plate structures, (Eds. Dawe et al), Clarendon Press, Oxford, pp School of Civil Engineering Research Report R931 Page 16

18 American ron Steel nstitute (AS). (2007). North American Specification for Design of Cold- Formed Steel Structural Members Edition, AS S Centre for Advanced Structural Engineering (2006), THN-WALL A computer program for cross-section analysis finite strip buckling analysis direct strength design of thin-walled structures, Version 2.1, School of Civil Engineering, University of Sydney, Sydney, Australia. Cheung, Y. K. (1968) Finite Strip Method Analysis of Elastic Slabs. ASCE J. Eng. Mech. Div., Vol. 94, EM6, pp Cheung, Y.K. (1976), Finite Strip Method in Structural Analysis, Pergamon Press, nc. New York, N.Y. Hancock, G. J. Pham, C. H. (2011), A Signature Curve for Cold-Formed Channel Sections in Pure Shear, Research Report R919, School of Civil Engineering, University of Sydney, July. Hancock, G. J. Pham, C. H. (2012), Direct Method of Design for Shear of Cold-Formed Channel Sections based on a Shear Signature Curve, Proceedings, 21 st nternational Specialty Conference on Cold-Formed Steel Structures (Eds. LaBoube Yu), St Louis, Missouri, USA, Oct, pp Lau, S. C. W. Hancock, G. J. (1986). Buckling of Thin Flat-Walled Structures by a Spline Finite Strip Method. Thin-Walled Structures, Vol. 4, pp Peters, G. Wilkinson, J.H. (1969), Eigenvalues of Ax = λbx with B Symmetric A B, Computer Journal, Vol. 12, pp Pham, C. H. Hancock, G. J. (2009a). Shear Buckling of Thin-Walled Channel Sections. Journal of Constructional Steel Research, Vol. 65, No. 3, pp Pham, C. H. Hancock, G. J. (2009b). Shear Buckling of Thin-Walled Channel Sections with ntermediate Web Stiffener. Proceedings, Sixth nternational Conference on Advances in Steel Structures, Hong Kong, pp Pham, C. H. Hancock, G. J. (2012a). Direct Strength Design of Cold-Formed C-Sections for Shear Combined Actions Journal of Structural Engineering, ASCE, Vol. 138, No. 6, pp Pham, C. H. Hancock, G. J. (2012b) Elastic Buckling of Cold-Formed Channel Sections in Shear. Thin- Walled Structures, Vol. 61, pp Pham, S.H., Pham, C.H. Hancock, G.J. (2012a), Shear Buckling of Thin-Walled Channel Sections with Complex Stiffened Webs, Research Report R924, School of Civil Engineering, University of Sydney, January. Pham, S.H., Pham, C.H. Hancock, G.J.(2012b), Shear Buckling of Thin-Walled Channel Sections with Complex Stiffened Webs, Proceedings, 21 st nternational Specialty Conference on Cold-Formed Steel Structures (Eds. LaBoube Yu), St Louis, Missouri, USA, Oct 2012, pp Plank, RJ Wittrick, WH. (1974), "Buckling Under Combined Loading of Thin, Flat-Walled Structures by a Complex Finite Strip Method", nternational Journal for Numerical Methods in Engineering, Vol. 8, No. 2, pp Stards Australia. (2005). AS/NZS 4600:2005, Cold-Formed Steel Structures. Stards Australia/ Stards New Zeal. Timoshenko, S.P. Gere, JM. (1961),"Theory of Elastic Stability", McGraw-Hill Book Co. nc., New York, N.Y. Turnbull, H.W. (1946), Theory of Equations, Oliver Boyd, Edinburgh London. Wilkinson, J.H. (1958), The calculation of eigenvectors of co-diagonal matrices, Computer Journal, Vol. 1, pp Williams, F.W. Wittrick, W.H. (1969), Computational procedures for a matrix analysis f stability vibration of thin flat-walled structures in compression, nt. Journal of Mechanical Sciences, 11, Wittrick, W.H. (1968), A unified approach to initial buckling of stiffened panels in compression, Aeronautical Quarterly, 19, School of Civil Engineering Research Report R931 Page 17

19 APPENDX A: BUCKLNG STRESS AND SHEAR BUCKLNG COEFFCENT FOR PLAN LPPED CHANNEL Table A. Buckling Stress Buckling Coefficient for Plain Lipped Channel Length (mm) resafsm (8 terms) Buckling Stress (MPa) Shear Buckling Coefficient k v Buckling Stress (MPa) SFSM Shear Buckling Coefficient k v School of Civil Engineering Research Report R931 Page 18

20 APPENDX B: BUCKLNG STRESS AND SHEAR BUCKLNG COEFFCENT FOR STFFENED WEB CHANNEL Table B. Buckling Stress Buckling Coefficient for Stiffened Web Channel Half-Wavelength/ Length (mm) resafsm (8 terms) Buckling Stress (MPa) Shear Buckling Coefficient k v Buckling Stress (MPa) SFSM Shear Buckling Coefficient k v School of Civil Engineering Research Report R931 Page 19

21 APPENDX C: FLEXURAL STFFNESS AND STABLTY MATRCES FOR MTH SERES TERM k F C F T k F C F g F C F T g F ig S C F (C-1) (C-2) where 1 0 C F 0 b 3 2b 2 b b 2 b k F = DX 6 DX 8 DX DX 2 4 DX DX 4 6 2DXY D1 D1 DX 8 2 2DXY 3D1 DX 2 10 D12DXY DX 8 DX 6 DX 2D1 10 DX 12 3D1 D1 DX 8 2 D1 2DXY DX 2 10 D12DXY 3 2DY8DXY DX D1 3DY 3DXY D1 D1 3DY 3DXY DX DY18DXY 5 where DX D mπ L A DY D b A D1 ν D mπ bl A DXY G t 12 mπ bl A Ab L E t D 12 1 ν f f 2 f 2 f 3 f g F 2 f 3 f 3 f 4 f f 3 f 4 f 4 f 5 f f 4 f 5 f 5 f 6 f f 3 f 4 f 4 f 5 f 4 f 5 f f 5 f 6 f f 5 f 6 4f 3 f 6 f 7 6f 4 f 6 f 7 6f 4 f 7 f 8 9f 5 where f mπ L V 2 σ f mπ L V 2 σ σ f 1 V b 2 σ T Vb t L School of Civil Engineering Research Report R931 Page 20

22 g S The stresses σ 1, σ 2, σ T τ are shown in Fig. 2 0 f f 0 f f 3 f f 2 f mπ bl V 2 τ f f f f 3 2 f f 5 School of Civil Engineering Research Report R931 Page 21

23 APPENDX D: MEMBRANE STFFNESS AND STABLTY MATRCES FOR MTH SERES TERM k M C M T k M C M g M C M T g M C M (D-1) (D-2) where 1 0 C M k M = G1 2 G1 4 G1 4 G1 6 0 E12 2 E1 2 G1 G1 E12 2bk 4bk 4 0 G1 2bk E12 G1 E12 2 4bk 4 E2 E2 2 4 E2 G1 E2 4 2bk 6 where G1 G mπ L V E1 E b V E2 E mπ L V E12 νe mπ bl V Vb t L k mπ L E E 1 ν g M f f 2 f 2 f 3 f 2 f 3 f 3 f f f 2 f 2 f 3 f 2 f 3 f 3 f 4 where f mπ L V 2 σ f mπ L V 2 σ σ Vb t L School of Civil Engineering Research Report R931 Page 22