review constrained fi nite strip method developments and

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ReviewReview: Constrained nite strip method developments andapplications in cold-formed steel designZhanjie Lia , Jean C. Batista Abreua , Jiazhen Lenga , Sándor Ádányb , Benjamin W. Schafera ,na Department of Civil Engineering, Johns Hopkins University, Baltimore, Maryland, USAb Department of Structural Mechanics, Budapest University of Technology and Economics, Budapest, Hungary

a r t i c l e i n f o

Available online 6 November 2013

Keywords:Constrained nite strip methodModal identi cationLocal bucklingDistortional bucklingBuckling mode interaction

a b s t r a c t

The stability of thin-walled members is decidedly complex. The recently developed constrained FiniteStrip Method (c FSM) provides a means to simplify thin-walled member stability solutions through itsability to identify and decompose mechanically meaningful stability behavior, notably the formalseparation of local, distortional, and global deformation modes. The objective of this paper is to providea review of the most recent developments in c FSM. This review includes: fundamental advances in thedevelopment of c FSM; applications of c FSM in design and optimization; identifying buckling modesand collapse mechanisms in shell nite element models; and, additional stability research initiated bythe c FSM methodology. A brief summary of the c FSM method, in its entirety, is provided to explain themethod and highlight areas where research remains active in the fundamental development. Theapplication of c FSM to cold-formed steel member design and optimization is highlighted as the methodhas the potential to automate generalized strength prediction of thin-walled cold-formed steel members.Extensions of c FSM to shell nite element models is also highlighted, as this provides one path to bringthe useful identi cation features of c FSM to general purpose nite element models. A number of alternative methods, including initial works on a constrained nite element method, initiated by c FSMmethods, are also detailed as they provide insights on potential future work in this area. Research continueson fundamentals such as methods for generalizing c FSM to arbitrary cross-sections, improved design and

optimization methods, and new ideas in the context of shell

nite element method applications.& 2013 Elsevier Ltd. All rights reserved.

Contents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. The constrained nite strip method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1. Classic FSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2. Signature curve stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3. cFSM formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1. Modal decomposition and identication for simply supported ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2. General boundary conditions stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Modal identi cation for general end boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Stiffness matrix options in FSM and cFSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5. New shear modes in cFSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. c FSM in cold-formed steel research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1. c FSM in design of cold-formed steel members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. ‘FSM at c FSM buckling length’ approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.3. ‘cFSM with correction factors’ approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. cFSM in shape optimization of cold-formed members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.1. Unconstrained optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/tws

Thin-Walled Structures

0263-8231/$ -see front matter & 2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.tws.2013.09.004

n Corresponding author. Tel.: þ 1 410 16 6265; fax: þ 1 410 516 7473.E-mail addresses: [email protected] (Z. Li), [email protected] (J.C. Batista Abreu),

[email protected] (J. Leng), [email protected] (S. Ádány),[email protected] (B.W. Schafer).

Thin-Walled Structures 81 (2014) 2– 18

http://www.sciencedirect.com/science/journal/02638231http://www.elsevier.com/locate/twshttp://dx.doi.org/10.1016/j.tws.2013.09.004mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.tws.2013.09.004http://dx.doi.org/10.1016/j.tws.2013.09.004http://dx.doi.org/10.1016/j.tws.2013.09.004http://dx.doi.org/10.1016/j.tws.2013.09.004mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.tws.2013.09.004&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.tws.2013.09.004&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.tws.2013.09.004&domain=pdfhttp://dx.doi.org/10.1016/j.tws.2013.09.004http://dx.doi.org/10.1016/j.tws.2013.09.004http://dx.doi.org/10.1016/j.tws.2013.09.004http://www.elsevier.com/locate/twshttp://www.sciencedirect.com/science/journal/02638231


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4.4.2. Optimization with manufacturability constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.5. cFSM in modal identication of FEM elastic buckling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.1. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2. Buckling mode identication of regular members (with general BC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3. Members with holes and irregular FEM mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.5.4. Members undergoing thermal gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6. cFSM in modal identication of FEM nonlinear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2. Lipped channel column examples (GNIA and GMNIA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6.3. Lipped channel parametric study of L vs. D instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Other works initiated by cFSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Imperfection identi cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2. Alternative cFSM and constrained spline FSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Constrained FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. c FSM in analytical solutions for global buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.5. Alternative FEM modal identication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction

Thin-walled cold-formed steel members enjoy a relatively com-plicated stability response for typical geometries and loading. As aresult, specialized tools for studying this stability response have beendeveloped and advanced. One of the most successful of these toolshas been the Finite Strip Method (FSM). In particular, the signaturecurve for member stability analysis popularized by Hancock [1] hasprovided the organizing thrust of today's member design: global,distortional, and local(-plate) buckling based on the signature curve.

In recent years, an additional tool: Generalized Beam Theory (GBT)has shown that the buckling deformations may be formally treated ina modal nature that mechanically separates global, distortional, local,and other modes [2]. This formal separation is integral to GBT, andallows measurement of modal participation. By extracting the

mechanical assumptions that lead to the separation one may extendthe de nitions to other methods. In particular, this insight lead to thedevelopment of the constrained FSM (c FSM), which imbues FSM withthesame ability as GBT, in terms of the separation of the deformations.In fact, the methods have been compared and shown to be nearlycoincident in their end result [3– 5].

This paper is a modi ed and signi cantly extended version of the paper presented at the CIMS2012 conference [6]. The paperprovides a review of fundamental developments in c FSM as wellas research results that are closely related and/or made possibleby c FSM. This review focuses on the last three years, though olderresults are referenced and brie y presented if germane to under-standing the latest results.

The paper begins, in Section 2, with a summary of theconstrained nite strip method (c FSM). The method is built-upfrom the simplest case (simply supported ends) then extended togeneral end boundary conditions. Ongoing research in the basicassumptions and the de nition of the modes is also summarized.Section 3 of the paper provides a summary of efforts to apply c FSMin a variety of design, optimization, and modal identicationproblems. The design efforts focus on the use of c FSM to automatethe identi cation of modes for use in cold-formed steel memberdesign. This process is further generalized in the examinationof shape optimization of cold-formed steel members. The last topicin Section 3 focuses on the use of cFSM base functions formodal identi cation in shell nite element method (FEM)models. Speci cally local, distortional, and global classicationsare provided for elastic buckling, geometrically nonlinear, and

full nonlinear collapse analysis of shell FEM models. Finally, in

Section 4 a series of research results are discussed that are notdirectly linked to, but unquestionably initiated by, the constrainingtechnique of c FSM, including nascent efforts in the constrained

nite element method (c FEM).

2. The constrained nite strip method

2.1. Classic FSM

The nite strip method leverages the longitudinal regularity of many thin-walled members to dramatically decrease the problem size.Members are discretized into longitudinal strips per Fig. 1. Within astrip, local displacement elds u, v , and w are discretized as follows:

u ¼ ∑q

m ¼ 1ð1 xbÞ xbh i u

1½mu2½m( )Y ½m ;

v ¼ ∑q

m ¼ 1ð1 xbÞ xbh i

v1½mv2½m( )Y 0½m a μ m½ ð1Þ

w ¼ ∑q

m ¼ 11 3x2b2 þ

2 x3b3

x 1 2xb þ x2

b2 3x2

b2 2 x3

b3 x x

2

b2 xb

w 1½mθ 1½mw 2½mθ 2½m

8>>>>>>>:

9>>>>=>>>>;Y ½m

ð2Þ

where the longitudinal shape function is

Y ½m

¼ sin ðmπ y=aÞ ð3Þ

the strip degrees of freedom (DOF): ui[m ], v i[m ], wi[m ], θ i[m ] are indicatedfor the rst term (m ¼ 1) of the simply supported (SS) end boundarycondition in Fig.1. Unlike FEM DOF, FSM DOF always occur at the same

Fig. 1. Finite strip discretization, strip DOF, and notation.

Z. Li et al. / Thin-Walled Structures 81 (2014) 2 – 18 3


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local x location (the strip lines) but longitudinal ( y) location variesdepending on the boundary condition and the [m] term in the series.

The strain– displacement relation that de nes the strips is acombination of a plane stress (subscript PS ) condition for themembrane DOF (u , v ) and Kirchhoff thin plate theory for thebending (subscript B) DOF (w), namely

εM f g ¼

ε xε yγ xy8>: 9>=>; ps

þ

ε xε yγ xy8>: 9>=>;B

¼

∂u=∂ x∂v=∂ y

∂u=∂ yþ ∂v=∂ x8>: 9>=>;PS þ

z ∂2w =∂ x2

z ∂2w =∂ y2

2 z ∂2w =∂ x∂ y8>: 9>=>;Bð4ÞThe strain– displacement relation is combined with the shape

functions in the conventional manner. First, the internal strainenergy can be formed

U ¼ 12Z fs gT fεgdV ¼ 12Z fεgT ½D f εgdV ¼ 12 fdgT Z ½B T ½D ½B dV d ð5Þ

where [D] is the matrix representation of the generalized (2D)Hooke's law, and [B] de nes the relationship between the strainvector {ε} and the displacement vector {d}, the latter of which isconstructed from the nodal displacements for each m, {d}[m ] ¼[u1[ m ], v 1[ m ], u 2[ m ], v 2[ m ], w 1[ m ], θ 1[ m ] w 2[ m ], θ 2[ m ]]T . From the strainenergy expression the local stiffness matrix [ke] is

½ke ¼ Z ½B T ½D ½B dV ð6Þtraditionally this matrix is broken into membrane (plane stress)and bending terms. The size of the strip stiffness matrix is(8q 8q). The local stiffness matrix is transferred to global DOF(U , V , W , > Θ) per coordinate transformation. Care must be takendue to the use of a left-handed coordinate system for W and aright-handed coordinate system for Θ, a convention that tracesback to the pioneering work of [7] and adopted in [8] and in theCUFSM software [9]. Using the strip connectivity the strip stiffnessmatrix is assembled into the global stiffness: [K e]. Where the nalsize is (4n sq 4n sq), where n s is the number of strip lines.

The local strip geometric stiffness matrix is formed from the workdone by the edge tractions (Fig.1) on the second-order strains, namely

W ¼ Z T ε II y dV ¼ Z 12 T d T

½G T ½G d dV ¼ 12 fdg

T Z T ½G T ½G dV d ð7Þ

where T is the distributed traction over the cross-section, εII y is thesecond-order strain (see Eq. (20) ad latter discussion), and [G] is amatrix that describes the relationship between the second-orderstrain components and the displacement vector. From the workexpression the strip's geometric stiffness matrix [k g ] is

½k g ¼

Z T ½G T ½G dV ð8Þ

the local geometric stiffness matrix [k g ] is also typically expressed interms of membrane and bending terms. Transformation to globalcoordinates and assembly proceeds identically to the elastic stiffnessmatrix and results in the (4nsq 4nsq) global geometric stiffnessmatrix [K g ].

For a given distribution of edge tractions on a member thegeometric stiffness matrix scales linearly, resulting in the classiceigen-buckling problem, namely

½K e ½Φ ½ Λ ½K g ½Φ ¼ 0 ð9Þ

½ Λ ¼ diag½ λ1 λ2 ::: λ4ns q ; ½Φ ¼ ½ f g1 f g2 ::: f g4nsq ;

ð10Þ

2.2. Signature curve stability analysis

For the special case of simply supported end conditions the [m]longitudinal shape function terms are orthogonal and thus each[m] term is separable and the problem may be approached as aseries of q separate solutions. In this case, if the member (strip)length a is varied and m ¼ 1, and the rst mode ( λ1) is plotted as afunction of a , the classic signature curve for the stability of a thin-walled member results, as shown in Fig. 2 for a 600S200-43 [10,11]cold-formed steel lipped channel cross-section under compres-sion. The buckling modes identi ed in the signature curve (Fig. 2)are typically known as local (-plate), (ange-) distortional, andglobal ( exural) buckling.

3. cFSM formulation

The constrained FSM (c FSM) is an extension to FSM that usesmechanical assumptions to constrain [K e] and [K g ] down to thosedeformations that are consistent with a desired set of criteria,e.g., those consistent with local (-plate) buckling. The method ispresented in [12– 16], and also implemented in the FSM softwareCUFSM [9]. The c FSM constraints are de ned in Table 1 and areutilized to formally categorize deformations into global (G), dis-tortional (D), local (L), and shear and transverse extension (ST )deformation spaces. Speci cally, any FSM displacement eld {d}(e.g. an eigenbuckling mode { } is an important special case) maybe constrained to any deformation space M (where M ¼ G, D, L,and/or ST ) via

fdg ¼ ½RM f dM g ð11Þexplicit determination of RM is lengthy, but not overly complicated.The shape functions of Eqs. (1)– (3) are utilized with the mechan-ical criteria of Table 1 to achieve the desired constraint matrices.Full derivations are provided in [13,14,16].

Modal decomposition of the eigen-buckling solution is com-pleted by introducing the desired constraint matrix [RM ], thatconstrain the deformations to the desired space (M ¼ G, D, L, and/or ST )and thus Eq. (9) becomes

½RM T ½K e ½RM ½Φ M ΛM ½RM T ½K g ½RM ½Φ M ¼ 0 ð12Þ

Eq. (12) reduces the problem size to the size of the M space, andconstrains the deformations to be consistent with that same M space,

effectively decomposing the solution.

Fig. 2. Classic signature curve for simply supported column.

Z. Li et al. / Thin-Walled Structures 81 (2014) 2 – 184


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Modal identi cation, i.e. categorization of a deformation intothe M spaces, of any FSM solution is also possible. First, we mustrecognize that G, D, L, ST spans the entire FSM deformation space,as such, taken as a whole the developed RM constraint matricesrepresent an alternative basis in the FSM space, one in whichdeformations are categorized. This basis transformation of a generaldisplacement vector {d} may be expressed as

d

¼ ½ ½RG ½RD ½RL ½RST fc g ð13Þ

where c now provides the deformations within each class: c G, c D, c L,c ST . The values of c are dependent on the normalization of the base vectors within R. A full discussion of the normalizationselection for R is provided in [17]. Here, a simple vector normis used throughout – this norm provides practically reasonableresults, is most consistent with norms used in GBT, and is mostcompatible with the application of c FSM in FEM modal identica-tion. Once the c is determined participation in each mode must alsobe determined, a variety of options exist as discussed in [17], herethe vector norm is used, namelyP M ¼ jjf C M gjj=jjfC gjj ð14Þthe ability to quantitatively de ne the participation in a givendeformation space; provides a unique measure of coupling amongstdeformations. If the deformations are coincident with bucklingmodes – as they often are – then the participations are a directmeasure of the degree of coupling in the given instability.

3.1. Modal decomposition and identi cation for simplysupported ends

The application of modal decomposition and identicationon the signature curve analysis of the simply supported lippedchannel of Fig. 2 is provided in Fig. 3. The modal decomposition(Fig. 3a) provides pure mode (deformation space) solutions andindicates that only D separates signi cantly from the full solution.The modal identi cation (Fig. 3b) is scaled by the load factor so

that the identi cation may be visualized along with the signaturecurve. The results indicate a measurable L– D coupling even at thetraditional distortional minima and further indicate how L, D, andG couple as a function of buckling half-wavelength.

3.2. General boundary conditions stability analysis

By properly selecting longitudinal shape functions forEqs. (1) and (2), FSM can be extended to cover various endboundary conditions. The extension is presented in [18– 20] andimplemented into CUFSM [21]. Altogether ve boundary condi-tions are considered based on simply supported (S), clamped (C),free (F), and guided (G). The shape function for SS is given by

Eq. (3) while the other (series) shape functions are as follows:CC : Y ½m ¼ sinðmπ y=aÞ sinðπ y=aÞ ð15Þ

SC : Y ½m ¼ sin ½ðm þ 1Þπ y=a þð m þ 1=mÞ sin ðmπ y=aÞ ð16Þ

CF : Y ½m ¼ 1 cos½ðm 1=2Þπ y=a ð17Þ

CG : Y ½m ¼ sin ½ðm 1=2Þπ y=a sinðπ y=2=aÞ ð18Þ

for all cases, except simply supported ends, the longitudinal [m]terms are not orthogonal and potentially [K e] and [K g ] are fully

populated. As a result, the signature curve loses its meaning, andrather than perform analysis at a variety of lengths it is morelogical to choose the physical length and investigate the highermodes, as is typical in FEM linear buckling analyses. An advantageof the FSM stability analysis for general end boundary conditions isthe ability to identify the [m] longitudinal terms that participate inthe solution for a given buckling mode {}. A simple vector normmay be applied to nd the participation ( p) of term [m]

p½m ¼ f M g = f g ð19Þ

for the problem of Fig. 1, but now with clamped– clamped ends and aphysical length (Lb) of 2450 mm, the 1st and 25th modes at mid-length and thepredicted [ m] term participation are provided in Fig.4.Fig. 4a identi es a local mode with a dominant half-wave of Lb/22 forthe 1st mode, while Fig. 4b, for the 25th buckling mode, shows adistortional mode with dominant half-waves near Lb/4 and Lb/6, butnon-negligible participation across a large magnitude of [m] terms.Fig. 4 provides the basic FSM solution, while the next section extendsc FSM to the case of general end boundary conditions.

3.3. Modal identi cation for general end boundary conditions

The end restraint conditions, i.e., the selected longitudinal shapefunction, only minimally inuences the modal decomposition/identi cation process, and can be done similarly as described

Fig. 3. Modal decomposition and identi cation of simply supported column (600S200-43). (a) decomposition and (b) identication.

Table 1Mechanical constraint criteria for mode classi cation.

Mechanical criteria G D L ST

Vlasov's hypotheses: (γ xy)PS ¼ 0, (ε x)PS ¼ 0, v is linear Yes Yes Yes Nolongitudinal warping: (ε y)PS a 0 Yes Yes No –undistorted section: κ x ¼ 0 Yes No – –



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above for any length and boundary condition. This is attributedto the special nature of the selected longitudinal shape functions.However, for general end boundary conditions the signaturecurve is no longer practically appropriate and the modal identi ca-tion is instead performed on the higher modes, at a given physicallength.

Fig. 5 provides modal identi cation of the rst 50 modes of axed ended (600S200-43) column. While FSM (or FEM) provides

only the load factors and buckling mode shapes, c FSM can identifythe buckled shapes by assigning participation percentages to anybuckled shape. As shown in the second column of Fig. 5, localdominant, distortional dominant, and global dominant modes areall readily identi able. In addition, weak coupling (e.g., mode #1)as well as strong coupling across two modes (mode #8) or three

modes (mode #31) are all identi able.

3.4. Stiffness matrix options in FSM and cFSM

The derivation of the stiffness matrices, as summarized inSection 2.1, can be realized in a number of subtly different ways.Speci cally, decisions are necessary in (at least) three steps:(i) de nition of the second-order strain, ε II y for [k g ], (ii) integrationof the external energy for determining [k g ], and (iii) integration of the internal strain energy for determining [ke].

For the second-order (longitudinal) strain (ε II y ) two formula-tions are in typical use

aÞ εII y ¼ 12

∂u∂ y

2þ

∂w∂ y

2" # bÞ εII y ¼ 12 ∂u∂ y 2

þ ∂v

∂ y 2

þ ∂w

∂ y 2" #

ð20Þ

Fig. 5. Modal identi cation of a xed ended (600S200-43, L¼ 2540 mm) column.

Fig. 4. Participation of [m] terms in buckling of a xed ended (600S200-43) column. (a) 1st mode longitudinal participation and (b) 25th mode longitudinal participation.



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Eq. (20a) is popular in engineering beam-model-based stabilitystudies (e.g., Euler buckling solutions), while Eq. (20b) is primarilyused in shell models (e.g. shell FEM or FSM) and is the formalapplication of the Green– Lagrange strain de nition.

In performing the integration to nd [k g ] two options may beused

aÞ ½k g ¼ t

Z a

0

Z b

0T ð x; yÞ½Gð x; yÞ T ½Gð x; yÞdxdy

bÞ ½k g ¼ Z t

0 Z a

0 Z b

0T ð x; y; z Þ½Gð x; y; z ÞT ½Gð x; y; z Þdxdydz ð21Þ

note, in Eq. (21b) the variation through the thickness is directlyconsidered in both T and [G], while in Eq. (21a) both T and [G]should be taken at z ¼ 0 (i.e. the mid-plane of the plate/strip).In classical nite strip derivations [7,8] the rst, simpler, formula(Eq. (21a)) is used. The second expression, Eq. (21b), is mathema-tically more precise and for thicker strips can make a non-negligible difference.

Finally, in calculating the strain energy, two options might beestablished similarly to those of the external work. The variationof strains and stresses through the thickness can be consideredor disregarded, which latter case corresponds to neglecting themembrane energy. The corresponding two formulae; therefore,are as follows:

aÞ ½k g ¼ t Z a

0 Z b

0½Bð x; yÞ T ½D ½Bð x; yÞdxdy

bÞ ½k g ¼ Z t

0 Z a

0 Z b

0½Bð x; y; z Þ T ½D ½Bð x; y; z Þdxdydz ð22Þ

note, in Eq. (22b) the variation through the thickness is directlyconsidered in [B], while in Eq. (21a) [B] should be taken at z ¼ 0.

Thus, given these options the nite strip stiffness matrices can beformulated in altogether 2 2 2¼ 8 different combinations (asdiscussed in [22]). Classical FSM [7,8] is implemented with Eqs. (20b),(21a) and (22b).

In many practical cases the stability results are hardly in u-enced by the details of choosing expression (a) or (b) in Eqs. (20),(21) and (22). However, in some instances, as illustrated in Fig. 6,the decisions greatly in uence the response, particularly at short

or long lengths.

3.5. New shear modes in cFSM

In c FSM global modes satisfy Vlasov's null-strain hypotheses(i.e., ε x,PS ¼ γ xy,PS ¼ 0) and cross-sections are not distorted. Fordistortional modes Vlasov's hypotheses are still satis ed, but thecross-sections are distorted. Shear modes are not speci callyde ned (Table 1), but obviously must involve in-plane shearstrains. In many applications shear has negligible effect; however,in some cases shear is crucially important: torsional behavior of closed cross-sections, members made of low shear rigidity mate-rial, etc. Recently, c FSM mode de nitions have been generalized,and shear modes have been de ned on a more rigorous mechan-ical basis [23]. Examples of the new shear modes are provided inFig. 7.

Utilizing the newly de ned novel shear modes [23], c FSM can beextended from open cross-sections to handle closed cross-sectionsor even general cross-sections with one or multiple closed parts.In Fig. 8a lateral-torsional buckling of a rectangular hollow section(RHS) beam calculated using the novel shear modes (depth: 100 mm,width: 20 mm, thickness: 1 mm, E ¼ 210 GPa) is provided. Further-more, the new shear mode de nition makes it easy to simulate

Fig. 7. Illustration of shear modes. (a) Lipped channelshear-bending mode (major-axis), (b) lipped channel shear-distortional mode (symmetrical) and (c) RHS shear-

distortional mode.

Fig. 6. Effect of stiffness matrix options (600S200-43 column with SS). (a) Minor axis exural buckling and (b) distortional buckling (symmetrical).



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shear-deformable beams with only a few modal degrees of freedom.In (Fig. 8b) global minor-axis exural buckling of a 600S200-43lipped channel column with and without shear deformation, i.e., inaccordance with Euler– Bernoulli and Timoshenko beam theory,respectively, is provided. The shear deformability adds only oneadditional mode (i.e., one additional DOF) to the problem.

4. c FSM in cold-formed steel research

4.1. cFSM in design of cold-formed steel members

A bene cial feature of c FSM is that pure (i.e., global, distor-tional, and/or local) modes can easily be calculated. This featurecan potentially be utilized in cold-formed steel (CFS) design, since

capacity prediction is typically based on elastic buckling loads of the modes, as is most evident in the case of the Direct StrengthMethod (DSM), see [24,25]. However, pure mode c FSM elasticbuckling loads are slightly different than conventional FSM (e.g.,see Fig. 3a) and cannot be fully used for cross-section models withrounded corners. This is of importance since DSM is calibratedto the conventional FSM, thus the advantages of c FSM cannot beimmediately employed. Two approaches have recently been pro-posed to address this limitation.

4.2. ‘ FSM at cFSM buckling length ’ approach

The ‘FSM at c FSM buckling length’ approach [26,27] takesadvantage of the fact that the c FSM calculated pure mode critical

buckling half-wavelength (Lcr ) is nearly equal to the critical

half-wavelength's identi ed in a conventional FSM analysis. Thisobservation is powerful, because the pure mode c FSM solutionalways has a critical half-wavelength (a minimum in the signaturecurve) but conventional FSM often has indistinct minima, parti-cularly for distortional buckling. In addition, this observation holdstrue even when the c FSM pure mode calculation uses a modelbased on sharp corners and the conventional FSM analysis usesa model with round corners. Thus, as illustrated in Fig. 9, therecommended approach is to use a cross-section model with sharpcorners and perform pure mode cFSM analysis to nd Lcr and thenemploy that Lcr in a cross-section model with round corners usingconventional FSM to nd the elastic buckling load (P cr or M cr ).

These elastic buckling loads are then utilized in DSM forstrength prediction. This approach was validated against all CFSlipped channels commercially available in the United States [10,11]under compression, as well as major- and minor-axis bending. Theuse of a cross-section model with rounded corners is recom-mended, since although corners have only a modest in uence onthe local and distortional buckling solutions, the decrease in grossproperties and hence global buckling as well as yield loads (andmoments) can be signi cant. The method can readily be extendedto any cross-section; however, the accuracy of the approach hasonly been validated for lipped channel sections.

4.3. ‘ cFSM with correction factors ’ approach

A different strategy is taken in the ‘c FSM with correction factors'approach [28– 30]. Parametric studies are completed on the DSMpre-quali ed cross-sections (see Appendix 1 of [25]) as illustrated

in Fig. 10. Column and beam members with cross-section models

Fig. 9. FSM at cFSM buckling length approach illustration. (a) Sharp-corner and rounded-corner model and (b) critical load calculation.

Fig. 8. Application of new shear modes. (a) Lateral-torsional buckling of RHS and (b) exural buckling of 600S200-43.



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having both sharp and round corners were calculated by FSM andpure mode c FSM, and the results are analyzed (altogether approxi-mately 15,000 cases). Based on the studies a design approach isproposed which uses the elastic buckling results from pure modec FSM employing a cross-section model with straight corners, and theeffect of round corners and the FSM-c FSM difference is handledthrough empirical correction factors. The corrected elastic bucklingvalues are then employed in the DSM formulation for strength

prediction. The results are validated against experimental results(on Z and lipped channel sections) for local and distortionalbuckling and found to be as accurate as the current DSM usingconventional FSM.

4.4. cFSM in shape optimization of cold-formed members

One of the desirable features of cold-formed steel members isthat they may be shaped (cold-bent) to nearly any open cross-section that provides an ef cient and economical solution. As aresult, nding optimal shapes for CFS members from the vastgeometry of possible designs is a problem of great interest, whichis re ected by the recent research activity in this eld [31– 38].Shape optimization requires the ability to ef ciently perform

a large number of stability solutions. Further, for formal optimiza-tion the stability calculation needs to be automatic (no userintervention) and general. CFS member design based on FSM andDSM is reasonably fast, and among the available design methods itis certainly the most general and most convenient for automation;all these feature make it a promising candidate in optimization.Further, c FSM is potentially able to make the FSM– DSM designprocess fully automatic (as demonstrated e.g., in Section 3.1),though the calculation of pure local and distortional critical loadcan still be challenging if the cross-section has unusual shape.

4.4.1. Unconstrained optimizationIn [31] results from a c FSM– DSM-based shape optimization

are presented. The cross-section shape is not limited by pre-

determined elements ( anges, webs, stiffeners, etc.), instead the fullsolution space of cold-formed steel shapes is explored. Columns withvarious length, but made of a given width of steel sheet, are optimizedfor strength. Bending of the sheet is allowed at 20 locations along its

width, thus providing the ability to form nearly any possible shape.Three optimization algorithms are considered: a gradient-basedsteepest descent method; and two stochastic search methods, geneticalgorithm and simulated annealing. The nal optimal shapes (seeFigs. 11 and 12) are length-dependent and non-conventional, butcompared with a standard cold-formed steel lipped channel havecapacities more than double the original design.

It is worth noting that in [36,37] a similar optimization problem is

solved and based on FSM–

DSM for functional evaluation, but witha differentobjective function.Here, cross-section shape with minimalcross-sectional area is searched for a given value of load-bearingcapacity under varying column length. The optimal shapes are alsonon-conventional, though symmetric. Final results are similar inspirit to [31], but with notable differences suggesting formulation of the optimization problem has an important in uence on the results.

4.4.2. Optimization with manufacturability constraintsIn [32] manufacturability constraints are introduced into the

shape optimization. The constraints are as follows: (i) the cross-section must be symmetric or point-symmetric, (ii) the cross-section must have some anges and lips which are no shorterthan 25 mm or 12.5 mm, respectively, and (iii) there must be aclearance of a minimum of 25 mm between the two lips. Typicalresults are presented in Fig. 13. The results show that constrainedoptimization has only a modest decrease in capacity comparedwith unconstrained optimization and leads to practical optimalshapes. Comparison of the optimal shapes to the full family of conventional lipped channel sections with the same amount of material shows that the developed optimal shapes have increasedcapacity for a complete range of axial load and major-axis bending.

4.5. cFSM in modal identi cation of FEM elastic buckling analysis

4.5.1. FormulationFSM's power is also its weakness – the lack of generality along

the length prohibits ef cient application to tapered members,members with holes, members with unusual loading along thelength, etc. Further, the penetration of general purpose FEM modelsinto engineering is so complete that a means to apply the conceptsof modal decomposition and identi cation to general FEM models isneeded for successful dissemination.

Here we examine the extension of c FSM to FEM models, wherethin-walled (cold-formed steel) members have been modeled withshell (plate) nite elements, for the purpose of modal identi ca-tion. The method is presented in [39– 45] and brie y summarizedhere. Given that the c FSM base vectors are easily constructedand available the approach that has been taken is to extendthese vectors into the FEM space only for the purposes of deninga deformation basis that is already categorized into G, D, L, and ST deformations.

Fig. 11. Maximal capacity column shapes from unconstrained optimization, L¼ 1.2 m. (a) Pn ¼ 56.03 kN (12.60 kips) and (b) Pn ¼ 55.34 kN (12.44 kips).

Fig. 10. Cross-section topologies in studies [28– 30].



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Construction of the c FSM base vectors is completed in the FSMspace. A thin-walled member is de ned and discretized into strips.As discussed in [42] to make the end conditions general a specialset of longitudinal (Y [m ]) shape functions are utilized, namely the

rst term for CF and FC and the full SS series

∑q

m ¼ 1Y ½m ¼ 1 cos

π y2a þ 1 cos

π ð yþ aÞ2a þ ∑

q

m ¼ 1sin

mπ ya

ð23Þ

this set of shape functions is illustrated for [m] up to 10 in Fig. 14,and as detailed in [42] provides the essential boundary conditionsfor identi cation across all general end boundary condition cases.With the longitudinal shape function selected the c FSM basevectors in [R] are constructed as before.

At the heart of the FE modal identi cation is the transformation of [R] in the c FSM basis to the standard FEM nodal basis. This is morethan just a simple transformation matrix, because the location of the FEM DOF are different than the FSM DOF. Fig. 15 provides aconceptual view of the FSM to FEM transformation, and [41,45]provides the complete details. Construction of a typical FEM DOF ata node is illustrated in Fig. 15, the node location is mapped from theglobal FEM coordinate system into the global FSM coordinate system,transformation from FSM global to FSM local is performed to placethe node within (or on) a strip, the strip shape functions are used tointerpolate the desired DOF, the value is then transformed into theFEM local coordinate system, then into the FEM global coordinate

system. The end result of the process is that [R] is transformed into

Fig.14. Illustration of generalized shape functions utilizing (aþ b). (a) m ¼ 1 CF andFC of Eq. (6) and (b) m ¼ 1,2,… ,10 SS of Eq. (3).

Fig. 13. Maximal capacity column shapes from constrained optimization. (a) L ¼ 1.2 m, Pn ¼ 54.49 kN (12.25 kips) and (b) L ¼ 4.9 m, Pn ¼ 13.16 kN (2.96 kips).

Fig. 12. Maximal capacity column shapes from unconstrained optimization, L ¼ 4.9 m. (a) Pn ¼ 13.39 kN (3.01 kips) and (b) Pn ¼ 14.33 kN (3.22 kips).



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[RFE ] a set of FEM basis vectors that are consistent interpolations of the cFSM basis vectors at the FEM nodal locations and DOF.

Given an FEM displacement vector {dFE } the modal (deformationspace) contributions can be determined through transformationutilizing [RFE ], similar to Eq. (12). However, one must recognize thatthe c FSM basis vectors used to form [RFE ] are only an approximatebasis and thus error may exist, namelyfderr g ¼ fdFE g ½RFE f c FE g ð24Þif the sum squared error ({dFE }T {dFE }) is minimized one nds thatthe solution for {c FE } is

fc FE g ¼ ð½RFE T ½RFE Þ 1½RFE T fdFE g ð25ÞModal participation factors are formulated the same as Eq. (14),

since within [RFE ] the [RGFE ], [RDFE ], [RLFE ], and [RSTFE ] are known.

4.5.2. Buckling mode identi cation of regular members(with general BC)

Consider the eigenbuckling problem of Eq. (9), but now fully

conducted with a shell FEM model. The resulting FEM buckling

mode shapes are

½Φ FE ¼ ½ f FE g1 f FE g2 ::: f FE gnFE DOF ð26Þ

any { FE } may be used as {dFE } of Eq. (25); therefore, after construc-tion of a companion FSM model, creation of [R] and interpolation andtransformation to [RFE ], Eq. (25) may be solved for the {c FE }contribution coef cients. Eq. (14) is then used to get the participationcoef cients. The results are similar to FSM modal identication withgeneral end boundary conditions (Fig. 5), but now also the error inthe approximation is monitored.

An FEM model of a column with semi-rigid end conditions isexplored to demonstrate, in part, the ef cacy of the selected basefunction system, see Eq. (23). The error (last column of Fig. 16e)is negligible for the rst 43 modes. Global, distortional, and localdominant modes are identi ed along with higher modes thatare severely interacted. Error is non-negligible when attempts toidentify local modes at short wavelengths, that are not included in

the c FSM basis, are performed.

Fig. 16. Modal identi cation of column with semi-rigid ends, (a) model, (b)– (d) modes, (e) identi cation (lipped channel: h ¼ 100 mm, b ¼ 60 mm, d ¼ 10 mm, t ¼ 2 mm,E ¼ 203500 GPa, ν ¼ 0.3. (a) End model, (b) #1, (c) #4, (d) #8 and (e) modal identication.

Fig. 15. Illustration of interpolation between FE and FSM displacement elds.



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4.5.3. Members with holes and irregular FEM meshThe introduction of holes into a thin-walled (cold-formed) steel

member is generally regarded to greatly complicate the analysis.Recently spline nite strip models have been developed for thisspecial case [46,47], but in general shell FEM models providethe most complete (if numerically costly) solution [48– 50]. Theproblem is, how to identify the resulting output? Here, as shownin Fig. 17, c FSM-based modal identi cation is performed on a FEM

model of a member with holes to illustrate the potential. The erroris negligible across the rst 50 modes, indicating excellent successwith the identi cation procedure.

With the exception of the rst two global modes the modalidenti cation results ably demonstrates the challenge of memberswithholes: Land D areat least partially interacted acrossessentially allmodes. Even in modes with a visual dominance, e.g., L in mode 17, theother mode (D in mode 17) has 20% or greater participation. It ispossible to identify the mode with the largest G, D, or L contributions,or the rst mode where G, D, or L has greater than 50% participation,but which would be appropriate for design is unknown. Further,although the interaction is quanti ed, currently it is not utilized indesign. Clearly, despite signicant practical progress on this problem,theoretical issues remain.

4.5.4. Members undergoing thermal gradientsAnother problem of signi cant interest is the evaluation of the

stability of thin-walled members under temperature gradients. Of particular interest is coupling and mode switching that is triggeredby the thermal gradient. An FEM shell element model of a columnis constructed and exposed on one side (Fig. 18a) to the time–temperature pro le of Fig. 18b. The modulus is assumed to betemperature dependent based on the results of [51,52]. Elastic

buckling analysis is performed at different times throughout theanalysis and the buckling load and modes generated. As Fig. 18eindicates the elastic buckling load decreases steadily with time,but as the buckling mode shapes of (Fig. 18c, d, f and g) show thenature of the buckled shape evolves with time. Modal identi ca-tion indicates a growth in distortional deformations even thoughthe initial (t ¼ 0) buckling mode is local dominant. For the 7thmode the dominant mode switches at t 25 min.

4.6. cFSM in modal identi cation of FEM nonlinear analysis

4.6.1. FormulationThe application of the c FSM basis to FEM eigen-buckling helps

solve a longstanding problem in the identi cation of buckled shapes.

Fig. 17. Modal identi cation of (a) member with holes, (b) full results, (c)– (i) mode shapes. (a) Geometry (mm), (b) identication results, (c) 1st, (d) 3rd, (e) 5th, (f) 17th,

(g) 22nd, (h) 22nd zoom and (i) 5th global feature.



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However, the general nature of the procedure makes it possible toidentify any deformation – thus a natural extension is to exploremodal identi cation of geometrically nonlinear analysis on theimperfection structure (GNIA) and geometric and material nonlinearanalysis on the imperfection structure (GMNIA) models. For thegeneral case of the GMNIA solution equilibrium (incrementally) isrepresented by

fF g ¼ ð½K e ½ K g ½ K p ÞfdFE g ð27Þ

direct modal identi cation of {dFE } of Eq. (27) has two problems: rst,in a model with initial imperfections the comparison to the unde-formed c FSM basis is problematic; second, in the pre-buckling stagethe linear elastic deformations (e.g. axial shortening in a column)dominate and tend to cloud the evolution of the buckling-associateddeformations. As a result, the deformation vector for performing theidenti cation {dFE }ID is modi ed to

fdFE gID ¼ f dFE gþf δ imp g ð28Þ

where {δ imp } is the imperfection. Further, the participation is mod-i ed to remove linear elastic deformations {dLE } resulting from {F }.This is completed by nding the contribution for the undesired

deformations {c LE } per Eq. (25) with {dLE } replacing {dFE } and then

removing this contribution from the participations, per

pM ¼ fc FE M g fc LE M g

fc FE g fc LE g ð29Þ

in practical implementations it has been enough to remove onlythe global (M ¼ G) linear elastic deformations; and this is what isillustrated in the following.

4.6.2. Lipped channel column examples (GNIA and GMNIA)

Modal identi cation of GNIA and GMNIA analysis of a lippedchannel with varying imperfections is provided in Fig. 19. In allcases the collapse mechanism is identi ed as interacted with L and D,but dominated by D deformations. The pre-buckling identi cationis shown to be imperfection sensitive. Although the perfect (noimperfection) model is dominated by local (L) buckling, dependingon the imperfection magnitude and shape it is possible to trigger Ddominance. See [44,45] for further studies of this nature on addi-tional members.

4.6.3. Lipped channel parametric study of L vs. D instabilityA classic problem with the cold-formed steel lipped channel

column, which has been the focus of the examples provided here,is: how long does the lip stiffener need to be so that local buckling

instead of distortional buckling controls? From a practical standpoint

Fig.18. Modal identi cation of column under thermal gradients. (a) Thermal gradient at t ¼ 23 min, (b) temperature vs. time curve (T -t ), (c) 1st, t ¼ 0 min, (d) 1st, t ¼ 32 min,(e) elastic buckling vs. time, (f) 7th, t = 0 min (g) 7th, t = 32 min and (h) modal identi cation vs. time.



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this is solved by current design codes that provide local anddistortional buckling as separate limit states. However, it is stillsurprising to see just how different the actual member response isfrom elastic and/or elastic buckling assumptions.

Consider the lipped channel column with varying lip lengthas depicted in Fig. 20a. As the lip length increases the elasticdistortional buckling load (P crd ) increases while the elastic localbuckling load (P crl ) stays largely constant. The resulting ratio of distortional to local elastic buckling is provided in Fig. 20b, andindicates a P crd /P crl 10 at a lip length of 25 mm. GMNIA analysis of the section indicates increasing strength with lip length (Fig. 12c).Note, in the GMNIA model the imperfection has a 50% exceedanceprobability for Type I and Type II imperfections. Modal identica-tion of the imperfection shows approximately 80% of distortionaland 20% local participation. Modal identication of the deforma-tions, at peak load and in the collapse regime (Fig. 20d and k),indicates the interacted nature of the deformations and the extentto which lip length in uences the character of the deformations.

For a large regime of lip lengths distortional buckling dominates

even though its elastic buckling load is 10 or more times greater than

local buckling. Transition towards participations associated with localbuckling steadily increases with lip length, but only for extremelylong lips is local buckling the dominant predicted deformation. Onemight think of coupled local and distortional buckling occurringwith P crl ¼ P crd , but coupled inelastic deformations occur withP crd (10– 20)P crl . In this example, one can see how modal identi -cation of GMNIA analysis opens up new avenues for exploringinteracting modes.

5. Other works initiated by cFSM

5.1. Imperfection identi cation

Geometric imperfections that are the result of the manufacturingprocess are the focus of [53– 55], where a summary of the availableimperfection measurements for cold-formed steel members is pre-sented and three methods to simulate imperfection elds areintroduced. The rst is the classical approach employing a super-

position of eigenmode imperfections, but scaled to match peaks in

Fig. 19. Modal identi cation of column for GNIA and GMNIA. (a) Load-disp. Response (scale: 2), (b) peak, (c) 2 mm, (d) 2 mm, (e) GNIA I: with GLE, (f) I: no imperfectio(g) II: local imperfection (0.1t ), (h) III: outward dist. Imperfection (0.1t ), (i) IV: inward dist. Imperfection (0.1t ) and (j) V: large dist. Imperfection (0.94t ).



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the measured physical measurements. The second is a method basedon the multi-dimensional spectral representation method, in whichimperfections are considered as a two-dimensional random eldand simulations are performed taking a spectra-based approach.The third, is termed as the 1D Modal Approach, and is a novelcombination of modal approaches and spectral representation thatdirectlyconsiders the frequency contentof the imperfection eld, butemploys a spectral representation method driven by the cross-sectional eigenmode shapes to generate the imperfection elds.Though the eigenmode shapes can be determined by variousnumerical methods, c FSM is benecial to use and has been appliedin the actual study. Based on the analysis of available imperfectionmeasurementdata, design power spectrums areproposed for GMNIAanalysis. From GMNIA-based nite element parametric studies the1D Modal Approach is found to be the most powerful method.

5.2. Alternative cFSM and constrained spline FSM

In [56– 58] an alternative c FSM formulation is presented. Themethod is based on the same mechanical criteria as presentedherein, but the practical implementation of these criteria into theFSM is slightly different from that used in the original c FSM [13– 16].The practical advantage of the new implementation is that closedcross-sections, which are not considered in the original c FSMderivations, are now treated. It is worth noting; however, that theshear modes are not fully incorporated; therefore, torsional modes(e.g., lateral-torsional buckling) of closed cross-sections are not fullysolved. Recently, the constraining technique has also been appliedto the spline nite strip method by the same research group [59].The method is not fully developed yet, but rst results are

promising.

5.3. Constrained FEM

In [60– 62] the constraining technique of c FSM is applied withinthe nite element method. The method is realized in a commercialFEM code: ANSYS [63], an approach that has both advantages anddisadvantages. The advantage is that ANSYS is widely available andreliable, while the disadvantage is that its application introducescertain limitations for the constraining method. The major limita-tions are as follows: (i) only the modal decomposition problem hasbeen solved, i.e., the presented method is able to provide criticalloads and buckled shapes in pure modes, but model identi cation isnot possible; (ii) the modal decomposition is not full in the sensethat it is not possible to transform the whole displacement eld intoa modal basis; and (iii) the introduction of constraints increasesthe problem size unlike in c FSM where constraining decreasesthe effective degrees of freedom. In spite of the disadvantages,the method provides useful and unique practical results, e.g., forperforated cold-formed steel members such as rack uprights thatcannot be properly handled by any other methods to date. More-over, the presented constrained method can be considered as animportant step toward to a more general c FEM.

5.4. cFSM in analytical solutions for global buckling

c FSM buckling solutions, when reduced to only a small number of modes, can readily produce analytical solutions. Given the variousoptions available in the implementation, e.g., Eqs. (20)– (22), c FSMcan be utilized to explore the effect of different mechanical assump-tions on analytical stability solutions. Given the widespread use of analytical solutions for global buckling, these comparisons are of

particular interest, as explored in [64–

68]. For example, in [64–

66]

Fig. 20. Lip length study: (e) d ¼ 10 mm at peak; (f) d ¼ 10 mm at post; (g) d ¼ 20 mm at peak; (h) d ¼ 20 mm at post; (i) d ¼ 40 mm at peak; (j) d ¼ 40 mm at post (post:5 peak displacement; scale: 2), (a) cross sections and lip variations, (b) ratio of distortional/local critical load, (c) load-disp. response, (d) modal identication at peak and(k) modal identi cation at post.



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the classical pin-ended column stability problem is derived, andanalytical solutions for exural, torsional, and exural-torsionalbuckling are generated that follow Euler– Bernoulli beam theory orthe mechanics consistent with Kirchoff plate theory. In [67] addi-tional analytical solutions for the column problem are derived, nowwith shear deformations included. In [68] the classical pin-endedbeam stability problem is explored and the case of a doubly-symmetric section under uniform moment is also analytically treated

with the various mechanical de nitions. In all cases, the dependenceof the derived analytical solutions on the assumed mechanics is fullydiscussed, allowing one to understand speci cally how classicalassumptions impact commonly used formulae for global buckling.

5.5. Alternative FEM modal identi cation

Modal identi cation for shell FEM solutions is a much neededtool for assessment of thin-walled member stability. As an alter-native to the use of the full c FSM basis vectors (Sections 3.3and 3.4), in [69,70] it is proposed to only use the cross-sectiondeformation modes. The advantage of such an approach is to makethe identi cation independent of the boundary conditions. Theresulting identi cation, provided along the member length, is

similar in spirit to GBT-based modal identication.

6. Discussion

A signi cant number of extensions to c FSM are possible andneeded. Within FSM/c FSM closed sections, multi-branched sections,tapered sections, and shear are all worthy next topics of study;indeed work is underway. For the extension of the c FSM basis toFEM, modal decomposition (instead of just modal identication)should be explored. In particular, the modal decomposition could beimplemented in a manner such that the reduced DOF can be utilizedin a beam element formulation. Here again the pioneering work of GBT provides some potential avenues for exploration. Also, on thepractical side, further automation of c FSM particularly its applicationto FEM modal identication is needed.

A number of issues in the behavior and design of thin-walledmembers deserve further study with c FSM. The identication of limit state dependency on imperfections instead of the cross-sectiongeometry alone needs to be quanti ed and further explored.As energy dissipation becomes a more important property for thin-walled members (in disproportionate collapse, seismic, blast, etc.)connecting the impact of the buckling deformations to the collapsemechanics and the energy dissipation within the different mechan-isms is needed. For c FSM, probably the most important next step is todetermine meaningful ways that the quanti cation of the modalparticipations can be used in design. Utilizing the participations as ameasure of modal coupling/interaction and then using weightedstrength formulas is one possible approach, but currently no methodutilizes c FSM (or GBT for that matter) participations in a manner thatinforms design.

A number of issues remain in the context of modal identi cationextensions to shell FEM models and GMNIA analysis. The practice of removing the linear elastic participations needs further study. Asthe FEM models become more complex the ability to generate thebase vectors in FSM may be compromised. Further, the use of elasticbase vectors for assessing plastic deformations in a GMNIA analysisis practically interesting, but lacks theoretical rigor. Strategies forerror reduction in the FEM modal identi cation are also needed.

While today the eld may lack perfect clarity on the notions of coupled instabilities versus interacted modes versus other similarlylabeled phenomenon, c FSM does provide a manner to categorizeinstabilities into a workable small number of classes and then

quantify the coupling/interaction across those classes. Thus, the

de nition of the deformation classes (Table 1) is the key to theentire method; these de nitions work well for simple shapes withsharp corners, but the separation between local and distortionaldeformations, in particular, becomes challenged in shapes withmany corrugations and round corners. As a result, further effort intoformalizing the deformation classes would still be useful.

7. Conclusions

The nite strip method (FSM) is an ef cient numerical tool thathas provided a unique ability to improve our understandingof thin-walled member stability. The constrained FSM (c FSM)extends the capabilities of FSM to include modal identicationand modal decomposition. Modal identi cation provides a meansto quantify the participation of a given deformation in terms of global, distortional, local (-plate), shear, and transverse extension.Modal decomposition allows a general solution to be reduced toonly desired deformations; allowing an analyst to isolate a givenbuckling mode, or even study the impact of differing mechanicaltheories (beam theory vs. plate theory, impact of shear, etc.) on abuckling solution.

This paper provides a summary of recent research applicationsand fundamental developments in c FSM. A summary of c FSM,built-up from the case of the semi-analytical solution for stripswith simply-supported ends; then generalized to the case of varying end boundary conditions as well as different assumptionsregarding the calculation of the internal strain energy, externalwork, and shear is provided. The application of c FSM to automatethe calculation of member stability modes (local, distortional,or global) is demonstrated. This automation, combined with theDirect Strength Method of design creates the ability to studyoptimal member cross-sections; and several examples of recentwork in this area are presented.

The application of c FSM to provide modal identication in shellnite element models is one of the richest areas of current study. The

general nature of shell

nite element models allows for situationslargely outside of the realm of traditional c FSM, i.e., members withholes, under longitudinal loading or temperature gradients, etc.Further, the application of c FSM-based modal identi cation to FEMmodels opens up the extension of the identi cation to generaldeformations such as those from geometric and material nonlinearanalysis on the imperfect structure (GMNIA). Modal identication of deformations from FEM GMNIA models demonstrates the evolutionof coupled and competing modes in pre-buckling, post-buckling, andcollapse regimes. Other research initiated by c FSM is also ongoing,including work on imperfection modeling that employs c FSM modes,alternative approaches to extend c FSM to nite element models, andanalytical investigations of global stability solution employing thec FSM framework. Signicant work remains to advance the theoryand continue to generalize the c FSM approach, but today signicantnew capabilities in the analysis of thin-walled members are nowavailable thanks to the modal decomposition and modal identi ca-tion capabilities of c FSM.

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review constrained fi nite strip method developments and

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