Review: Constrained finite strip method developments and applications in cold-formed steel design

Thin-Walled Structures 81 (2014) 2–18

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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws

Review

Review: Constrained finite strip method developments and applications in cold-formed steel design Zhanjie Li a, Jean C. Batista Abreu a, Jiazhen Leng a, Sándor Ádány b, Benjamin W. Schafer a,n a b

Department of Civil Engineering, Johns Hopkins University, Baltimore, Maryland, USA Department of Structural Mechanics, Budapest University of Technology and Economics, Budapest, Hungary

art ic l e i nf o

a b s t r a c t

Available online 6 November 2013

The stability of thin-walled members is decidedly complex. The recently developed constrained Finite Strip Method (cFSM) provides a means to simplify thin-walled member stability solutions through its ability to identify and decompose mechanically meaningful stability behavior, notably the formal separation of local, distortional, and global deformation modes. The objective of this paper is to provide a review of the most recent developments in cFSM. This review includes: fundamental advances in the development of cFSM; applications of cFSM in design and optimization; identifying buckling modes and collapse mechanisms in shell finite element models; and, additional stability research initiated by the cFSM methodology. A brief summary of the cFSM method, in its entirety, is provided to explain the method and highlight areas where research remains active in the fundamental development. The application of cFSM to cold-formed steel member design and optimization is highlighted as the method has the potential to automate generalized strength prediction of thin-walled cold-formed steel members. Extensions of cFSM to shell finite element models is also highlighted, as this provides one path to bring the useful identification features of cFSM to general purpose finite element models. A number of alternative methods, including initial works on a constrained finite element method, initiated by cFSM methods, are also detailed as they provide insights on potential future work in this area. Research continues on fundamentals such as methods for generalizing cFSM to arbitrary cross-sections, improved design and optimization methods, and new ideas in the context of shell finite element method applications. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Constrained finite strip method Modal identification Local buckling Distortional buckling Buckling mode interaction

Contents 1. 2.

3.

4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The constrained finite strip method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1. Classic FSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Signature curve stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 cFSM formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1. Modal decomposition and identification for simply supported ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2. General boundary conditions stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3. Modal identification for general end boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.4. Stiffness matrix options in FSM and cFSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.5. New shear modes in cFSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 cFSM in cold-formed steel research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1. cFSM in design of cold-formed steel members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2. ‘FSM at cFSM buckling length’ approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3. ‘cFSM with correction factors’ approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.4. cFSM in shape optimization of cold-formed members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.4.1. Unconstrained optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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). 0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.09.004

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

3

4.4.2. Optimization with manufacturability constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 cFSM in modal identification of FEM elastic buckling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.5.1. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.5.2. Buckling mode identification of regular members (with general BC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.5.3. Members with holes and irregular FEM mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.5.4. Members undergoing thermal gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.6. cFSM in modal identification of FEM nonlinear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.6.1. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.6.2. Lipped channel column examples (GNIA and GMNIA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.6.3. Lipped channel parametric study of L vs. D instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5. Other works initiated by cFSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.1. Imperfection identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2. Alternative cFSM and constrained spline FSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.3. Constrained FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.4. cFSM in analytical solutions for global buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.5. Alternative FEM modal identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.5.

1. Introduction Thin-walled cold-formed steel members enjoy a relatively complicated stability response for typical geometries and loading. As a result, specialized tools for studying this stability response have been developed and advanced. One of the most successful of these tools has been the Finite Strip Method (FSM). In particular, the signature curve for member stability analysis popularized by Hancock [1] has provided 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 in a modal nature that mechanically separates global, distortional, local, and other modes [2]. This formal separation is integral to GBT, and allows measurement of modal participation. By extracting the mechanical assumptions that lead to the separation one may extend the definitions to other methods. In particular, this insight lead to the development of the constrained FSM (cFSM), which imbues FSM with the same ability as GBT, in terms of the separation of the deformations. In fact, the methods have been compared and shown to be nearly coincident in their end result [3–5]. This paper is a modified and significantly extended version of the paper presented at the CIMS2012 conference [6]. The paper provides a review of fundamental developments in cFSM as well as research results that are closely related and/or made possible by cFSM. This review focuses on the last three years, though older results are referenced and briefly presented if germane to understanding the latest results. The paper begins, in Section 2, with a summary of the constrained finite strip method (cFSM). The method is built-up from the simplest case (simply supported ends) then extended to general end boundary conditions. Ongoing research in the basic assumptions and the definition of the modes is also summarized. Section 3 of the paper provides a summary of efforts to apply cFSM in a variety of design, optimization, and modal identification problems. The design efforts focus on the use of cFSM to automate the identification of modes for use in cold-formed steel member design. This process is further generalized in the examination of shape optimization of cold-formed steel members. The last topic in Section 3 focuses on the use of cFSM base functions for modal identification in shell finite element method (FEM) models. Specifically local, distortional, and global classifications are 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 not directly linked to, but unquestionably initiated by, the constraining technique of cFSM, including nascent efforts in the constrained finite element method (cFEM).

2. The constrained finite strip method 2.1. Classic FSM The finite 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 a strip, local displacement fields u, v, and w are discretized as follows: ( ) i u1½m q h x x ð1  Þ Y ½m ; u¼ ∑ b b u2½m m¼1 ( ) i v1½m q h a x x v ¼ ∑ ð1  bÞ b ð1Þ Y 0½m v2½m μ½m m¼1

q

w¼ ∑

m¼1



2

3

1  3x2 þ 2x3 b

b

  2 x 1  2x þ x2 b b

3x2 2 b

3

 2x3 b

x



x2 2 b

9 8 > > > w1½m > > >  > = < θ1½m >  bx Y w > ½m > > > > 2½m > > > : θ2½m ;

ð2Þ where the longitudinal shape function is Y ½m ¼ sinðmπ y=aÞ

ð3Þ

the strip degrees of freedom (DOF): ui[m], vi[m], wi[m], θi[m] are indicated for the first term (m¼1) of the simply supported (SS) end boundary condition in Fig. 1. Unlike FEM DOF, FSM DOF always occur at the same

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

4

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

local x location (the strip lines) but longitudinal (y) location varies depending on the boundary condition and the [m] term in the series. The strain–displacement relation that defines the strips is a combination of a plane stress (subscript PS) condition for the membrane DOF (u, v) and Kirchhoff thin plate theory for the bending (subscript B) DOF (w), namely 9 9 9 8 9 8 8 8 2 2 ε = > ε = > ∂u=∂x > > > = < = < x > < x > <  z∂ w=∂x > ∂v=∂y fεM g ¼ εy þ εy ¼ þ  z∂2 w=∂y2 > > > > > > > : ∂u=∂y þ ∂v=∂x ; ; :γ ; :γ ; : 2z∂2 w=∂x∂y > xy xy PS ps

B

B

ð4Þ The strain–displacement relation is combined with the shape functions in the conventional manner. First, the internal strain energy can be formed Z  Z Z  1 1 1 U¼ ½BT ½D½BdV d fsgT fεgdV ¼ fεgT ½DfεgdV ¼ fdgT 2 2 2 ð5Þ where [D] is the matrix representation of the generalized (2D) Hooke's law, and [B] defines the relationship between the strain vector {ε} and the displacement vector {d}, the latter of which is constructed from the nodal displacements for each m, {d}[m] ¼ [u1[m], v1[m], u2[m], v2[m], w1[m], θ1[m] w2[m], θ2[m]]T. From the strain energy expression the local stiffness matrix [ke] is Z ½ke  ¼ ½BT ½D½BdV ð6Þ traditionally 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 taken due to the use of a left-handed coordinate system for W and a right-handed coordinate system for Θ, a convention that traces back to the pioneering work of [7] and adopted in [8] and in the CUFSM software [9]. Using the strip connectivity the strip stiffness matrix is assembled into the global stiffness: [Ke]. Where the final size is (4nsq  4nsq), where ns is the number of strip lines. The local strip geometric stiffness matrix is formed from the work done by the edge tractions (Fig. 1) on the second-order strains, namely

Z  Z Z  1  T T  1 W¼ T εIIy dV ¼ T d ½G ½G d dV ¼ fdgT T½GT ½GdV d 2 2 ð7Þ where T is the distributed traction over the cross-section, εIIy is the second-order strain (see Eq. (20) ad latter discussion), and [G] is a matrix that describes the relationship between the second-order strain components and the displacement vector. From the work expression the strip's geometric stiffness matrix [kg] is Z ½kg  ¼ T½GT ½GdV ð8Þ the local geometric stiffness matrix [kg] is also typically expressed in terms of membrane and bending terms. Transformation to global coordinates and assembly proceeds identically to the elastic stiffness matrix and results in the (4nsq  4nsq) global geometric stiffness matrix [Kg]. For a given distribution of edge tractions on a member the geometric stiffness matrix scales linearly, resulting in the classic eigen-buckling problem, namely ½K e ½Φ  ½Λ½K g ½Φ ¼ 0 ½Λ ¼ diag½ λ1

λ2 ::: λ4ns q ;

ð9Þ

½Φ ¼ ½ fϕg1

fϕg2

:::

fϕg4ns q ; ð10Þ

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

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 a series of q separate solutions. In this case, if the member (strip) length a is varied and m ¼ 1, and the first mode (λ1) is plotted as a function of a, the classic signature curve for the stability of a thinwalled member results, as shown in Fig. 2 for a 600S200-43 [10,11] cold-formed steel lipped channel cross-section under compression. The buckling modes identified in the signature curve (Fig. 2) are typically known as local (-plate), (flange-) distortional, and global (flexural) buckling.

3. cFSM formulation The constrained FSM (cFSM) is an extension to FSM that uses mechanical assumptions to constrain [Ke] and [Kg] down to those deformations that are consistent with a desired set of criteria, e.g., those consistent with local (-plate) buckling. The method is presented in [12–16], and also implemented in the FSM software CUFSM [9]. The cFSM constraints are defined in Table 1 and are utilized to formally categorize deformations into global (G), distortional (D), local (L), and shear and transverse extension (ST) deformation spaces. Specifically, any FSM displacement field {d} (e.g. an eigenbuckling mode {ϕ} is an important special case) may be constrained to any deformation space M (where M ¼G, D, L, and/or ST) via fdg ¼ ½RM fdM g

ð11Þ

explicit determination of RM is lengthy, but not overly complicated. The shape functions of Eqs. (1)–(3) are utilized with the mechanical 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 completed by introducing the desired constraint matrix [RM], that constrain the deformations to the desired space (M ¼G, D, L, and/or ST) and thus Eq. (9) becomes ½RM T ½K e ½RM ½ΦM   ΛM ½RM T ½K g ½RM ½ΦM  ¼ 0

ð12Þ

Eq. (12) reduces the problem size to the size of the M space, and constrains the deformations to be consistent with that same M space, effectively decomposing the solution.

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

Modal identification, i.e. categorization of a deformation into the M spaces, of any FSM solution is also possible. First, we must recognize that G, D, L, ST spans the entire FSM deformation space, as such, taken as a whole the developed RM constraint matrices represent an alternative basis in the FSM space, one in which deformations are categorized. This basis transformation of a general displacement vector {d} may be expressed as  d ¼ ½ ½RG  ½RD  ½RL  ½RST  fcg ð13Þ where c now provides the deformations within each class: cG, cD, cL, cST. The values of c are dependent on the normalization of the base vectors within R. A full discussion of the normalization selection for R is provided in [17]. Here, a simple vector norm is used throughout – this norm provides practically reasonable results, is most consistent with norms used in GBT, and is most compatible with the application of cFSM in FEM modal identification. Once the c is determined participation in each mode must also be determined, a variety of options exist as discussed in [17], here the vector norm is used, namely P M ¼ jjfC M gjj=jjfCgjj

ð14Þ

the ability to quantitatively define the participation in a given deformation space; provides a unique measure of coupling amongst deformations. If the deformations are coincident with buckling modes – as they often are – then the participations are a direct measure of the degree of coupling in the given instability. 3.1. Modal decomposition and identification for simply supported ends The application of modal decomposition and identification on the signature curve analysis of the simply supported lipped channel of Fig. 2 is provided in Fig. 3. The modal decomposition (Fig. 3a) provides pure mode (deformation space) solutions and indicates that only D separates significantly from the full solution. The modal identification (Fig. 3b) is scaled by the load factor so that the identification may be visualized along with the signature curve. The results indicate a measurable L–D coupling even at the traditional distortional minima and further indicate how L, D, and G couple as a function of buckling half-wavelength. Table 1 Mechanical constraint criteria for mode classification.

5

3.2. General boundary conditions stability analysis By properly selecting longitudinal shape functions for Eqs. (1) and (2), FSM can be extended to cover various end boundary conditions. The extension is presented in [18–20] and implemented into CUFSM [21]. Altogether five boundary conditions 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 sin ðmπ y=aÞ sin ðπ y=aÞ

ð15Þ

SC : Y ½m ¼ sin ½ðm þ 1Þπ y=a þðm 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 sin ðπ y=2=aÞ

ð18Þ

for all cases, except simply supported ends, the longitudinal [m] terms are not orthogonal and potentially [Ke] and [Kg] are fully populated. As a result, the signature curve loses its meaning, and rather than perform analysis at a variety of lengths it is more logical to choose the physical length and investigate the higher modes, as is typical in FEM linear buckling analyses. An advantage of the FSM stability analysis for general end boundary conditions is the ability to identify the [m] longitudinal terms that participate in the solution for a given buckling mode {ϕ}. A simple vector norm may be applied to find the participation (p) of term [m] p½m ¼ ‖fϕM g‖=‖fϕg‖

ð19Þ

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

Mechanical criteria

G

D

L

ST

3.3. Modal identification for general end boundary conditions

Vlasov's hypotheses: (γxy)PS ¼ 0, (εx)PS ¼ 0, v is linear longitudinal warping: (εy)PS a0 undistorted section: κx ¼ 0

Yes Yes Yes

Yes Yes No

Yes No –

No – –

The end restraint conditions, i.e., the selected longitudinal shape function, only minimally influences the modal decomposition/ identification process, and can be done similarly as described

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

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

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

Fig. 5. Modal identification of a fixed ended (600S200-43, L ¼ 2540 mm) column.

above for any length and boundary condition. This is attributed to the special nature of the selected longitudinal shape functions. However, for general end boundary conditions the signature curve is no longer practically appropriate and the modal identification is instead performed on the higher modes, at a given physical length. Fig. 5 provides modal identification of the first 50 modes of a fixed ended (600S200-43) column. While FSM (or FEM) provides only the load factors and buckling mode shapes, cFSM can identify the buckled shapes by assigning participation percentages to any buckled shape. As shown in the second column of Fig. 5, local dominant, distortional dominant, and global dominant modes are all readily identifiable. 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 identifiable.

3.4. Stiffness matrix options in FSM and cFSM The derivation of the stiffness matrices, as summarized in Section 2.1, can be realized in a number of subtly different ways. Specifically, decisions are necessary in (at least) three steps: (i) definition of the second-order strain, εIIy for [kg], (ii) integration of the external energy for determining [kg], and (iii) integration of the internal strain energy for determining [ke]. For the second-order (longitudinal) strain (εIIy ) two formulations are in typical use

aÞ

εIIy ¼

1 2

"

"   

2 #

2 # ∂u 2 ∂w 1 ∂u 2 ∂v 2 ∂w þ þ þ bÞ εIIy ¼ ∂y ∂y 2 ∂y ∂y ∂y ð20Þ

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

Eq. (20a) is popular in engineering beam-model-based stability studies (e.g., Euler buckling solutions), while Eq. (20b) is primarily used in shell models (e.g. shell FEM or FSM) and is the formal application of the Green–Lagrange strain definition. In performing the integration to find [kg] two options may be used Z aZ b Tðx; yÞ½Gðx; yÞT ½Gðx; yÞdxdy aÞ ½kg  ¼ t Z bÞ

0 t

½kg  ¼

0

Z

0

a

Z

0

b

Tðx; y; zÞ½Gðx; y; zÞT ½Gðx; y; zÞdxdydz

bÞ

0 t

½kg  ¼ 0

0

Z

a 0

Thus, given these options the finite strip stiffness matrices can be formulated in altogether 2  2  2 ¼8 different combinations (as discussed in [22]). Classical FSM [7,8] is implemented with Eqs. (20b), (21a) and (22b). In many practical cases the stability results are hardly influenced 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 influence the response, particularly at short or long lengths.

ð21Þ

0

note, in Eq. (21b) the variation through the thickness is directly considered in both T and [G], while in Eq. (21a) both T and [G] should be taken at z¼ 0 (i.e. the mid-plane of the plate/strip). In classical finite strip derivations [7,8] the first, simpler, formula (Eq. (21a)) is used. The second expression, Eq. (21b), is mathematically more precise and for thicker strips can make a nonnegligible difference. Finally, in calculating the strain energy, two options might be established similarly to those of the external work. The variation of strains and stresses through the thickness can be considered or disregarded, which latter case corresponds to neglecting the membrane energy. The corresponding two formulae; therefore, are as follows: Z aZ b aÞ ½kg  ¼ t ½Bðx; yÞT ½D½Bðx; yÞdxdy Z

7

Z

b

½Bðx; y; zÞT ½D½Bðx; y; zÞdxdydz

ð22Þ

0

note, in Eq. (22b) the variation through the thickness is directly considered in [B], while in Eq. (21a) [B] should be taken at z ¼0.

3.5. New shear modes in cFSM In cFSM global modes satisfy Vlasov's null-strain hypotheses (i.e., εx,PS ¼ γxy,PS ¼0) and cross-sections are not distorted. For distortional modes Vlasov's hypotheses are still satisfied, but the cross-sections are distorted. Shear modes are not specifically defined (Table 1), but obviously must involve in-plane shear strains. 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 material, etc. Recently, cFSM mode definitions have been generalized, and shear modes have been defined on a more rigorous mechanical basis [23]. Examples of the new shear modes are provided in Fig. 7. Utilizing the newly defined novel shear modes [23], cFSM can be extended from open cross-sections to handle closed cross-sections or 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, E¼ 210 GPa) is provided. Furthermore, the new shear mode definition makes it easy to simulate

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

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

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

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

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

4. cFSM in cold-formed steel research 4.1. cFSM in design of cold-formed steel members A beneficial feature of cFSM is that pure (i.e., global, distortional, and/or local) modes can easily be calculated. This feature can 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 Strength Method (DSM), see [24,25]. However, pure mode cFSM elastic buckling loads are slightly different than conventional FSM (e.g., see Fig. 3a) and cannot be fully used for cross-section models with rounded corners. This is of importance since DSM is calibrated to the conventional FSM, thus the advantages of cFSM cannot be immediately employed. Two approaches have recently been proposed to address this limitation.

half-wavelength's identified in a conventional FSM analysis. This observation is powerful, because the pure mode cFSM solution always has a critical half-wavelength (a minimum in the signature curve) but conventional FSM often has indistinct minima, particularly for distortional buckling. In addition, this observation holds true even when the cFSM pure mode calculation uses a model based on sharp corners and the conventional FSM analysis uses a model with round corners. Thus, as illustrated in Fig. 9, the recommended approach is to use a cross-section model with sharp corners and perform pure mode cFSM analysis to find Lcr and then employ that Lcr in a cross-section model with round corners using conventional FSM to find the elastic buckling load (Pcr or Mcr). These elastic buckling loads are then utilized in DSM for strength prediction. This approach was validated against all CFS lipped channels commercially available in the United States [10,11] under compression, as well as major- and minor-axis bending. The use of a cross-section model with rounded corners is recommended, since although corners have only a modest influence on the local and distortional buckling solutions, the decrease in gross properties and hence global buckling as well as yield loads (and moments) can be significant. The method can readily be extended to any cross-section; however, the accuracy of the approach has only been validated for lipped channel sections. 4.3. ‘cFSM with correction factors’ approach

4.2. ‘FSM at cFSM buckling length’ approach The ‘FSM at cFSM buckling length’ approach [26,27] takes advantage of the fact that the cFSM calculated pure mode critical buckling half-wavelength (Lcr) is nearly equal to the critical

A different strategy is taken in the ‘cFSM with correction factors' approach [28–30]. Parametric studies are completed on the DSM pre-qualified cross-sections (see Appendix 1 of [25]) as illustrated in Fig. 10. Column and beam members with cross-section models

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having both sharp and round corners were calculated by FSM and pure mode cFSM, and the results are analyzed (altogether approximately 15,000 cases). Based on the studies a design approach is proposed which uses the elastic buckling results from pure mode cFSM employing a cross-section model with straight corners, and the effect of round corners and the FSM-cFSM difference is handled through empirical correction factors. The corrected elastic buckling values 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 distortional buckling and found to be as accurate as the current DSM using conventional FSM. 4.4. cFSM in shape optimization of cold-formed members One of the desirable features of cold-formed steel members is that they may be shaped (cold-bent) to nearly any open crosssection that provides an efficient and economical solution. As a result, finding optimal shapes for CFS members from the vast geometry of possible designs is a problem of great interest, which is reflected by the recent research activity in this field [31–38]. Shape optimization requires the ability to efficiently perform a large number of stability solutions. Further, for formal optimization the stability calculation needs to be automatic (no user intervention) and general. CFS member design based on FSM and DSM is reasonably fast, and among the available design methods it is certainly the most general and most convenient for automation; all these feature make it a promising candidate in optimization. Further, cFSM is potentially able to make the FSM–DSM design process fully automatic (as demonstrated e.g., in Section 3.1), though the calculation of pure local and distortional critical load can still be challenging if the cross-section has unusual shape.

9

width, thus providing the ability to form nearly any possible shape. Three optimization algorithms are considered: a gradient-based steepest descent method; and two stochastic search methods, genetic algorithm and simulated annealing. The final optimal shapes (see Figs. 11 and 12) are length-dependent and non-conventional, but compared with a standard cold-formed steel lipped channel have capacities 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 with a different objective function. Here, cross-section shape with minimal cross-sectional area is searched for a given value of load-bearing capacity under varying column length. The optimal shapes are also non-conventional, though symmetric. Final results are similar in spirit to [31], but with notable differences suggesting formulation of the optimization problem has an important influence on the results. 4.4.2. Optimization with manufacturability constraints In [32] manufacturability constraints are introduced into the shape optimization. The constraints are as follows: (i) the crosssection must be symmetric or point-symmetric, (ii) the crosssection must have some flanges and lips which are no shorter than 25 mm or 12.5 mm, respectively, and (iii) there must be a clearance of a minimum of 25 mm between the two lips. Typical results are presented in Fig. 13. The results show that constrained optimization has only a modest decrease in capacity compared with unconstrained optimization and leads to practical optimal shapes. 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 increased capacity for a complete range of axial load and major-axis bending. 4.5. cFSM in modal identification of FEM elastic buckling analysis

4.4.1. Unconstrained optimization In [31] results from a cFSM–DSM-based shape optimization are presented. The cross-section shape is not limited by predetermined elements (flanges, webs, stiffeners, etc.), instead the full solution space of cold-formed steel shapes is explored. Columns with various length, but made of a given width of steel sheet, are optimized for strength. Bending of the sheet is allowed at 20 locations along its

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

4.5.1. Formulation FSM's power is also its weakness – the lack of generality along the length prohibits efficient application to tapered members, members with holes, members with unusual loading along the length, etc. Further, the penetration of general purpose FEM models into engineering is so complete that a means to apply the concepts of modal decomposition and identification to general FEM models is needed for successful dissemination. Here we examine the extension of cFSM to FEM models, where thin-walled (cold-formed steel) members have been modeled with shell (plate) finite elements, for the purpose of modal identification. The method is presented in [39–45] and briefly summarized here. Given that the cFSM base vectors are easily constructed and available the approach that has been taken is to extend these vectors into the FEM space only for the purposes of defining a deformation basis that is already categorized into G, D, L, and ST deformations.

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

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Fig. 12. Maximal capacity column shapes from unconstrained optimization, L¼ 4.9 m. (a) Pn ¼ 13.39 kN (3.01 kips) and (b) Pn ¼ 14.33 kN (3.22 kips).

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. 14. Illustration of generalized shape functions utilizing (aþ b). (a) m ¼1 CF and FC of Eq. (6) and (b) m¼ 1,2,…,10 SS of Eq. (3).

Construction of the cFSM base vectors is completed in the FSM space. A thin-walled member is defined and discretized into strips. As discussed in [42] to make the end conditions general a special set of longitudinal (Y[m]) shape functions are utilized, namely the first term for CF and FC and the full SS series

 π y mπ y q q π ð  y þaÞ þ1  cos þ ∑ sin ∑ Y ½m ¼ 1  cos 2a 2a a m¼1 m¼1 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 conditions for identification across all general end boundary condition cases. With the longitudinal shape function selected the cFSM base vectors in [R] are constructed as before. At the heart of the FE modal identification is the transformation of [R] in the cFSM basis to the standard FEM nodal basis. This is more than just a simple transformation matrix, because the location of the FEM DOF are different than the FSM DOF. Fig. 15 provides a conceptual view of the FSM to FEM transformation, and [41,45] provides the complete details. Construction of a typical FEM DOF at a node is illustrated in Fig. 15, the node location is mapped from the global FEM coordinate system into the global FSM coordinate system, transformation from FSM global to FSM local is performed to place the node within (or on) a strip, the strip shape functions are used to interpolate the desired DOF, the value is then transformed into the FEM local coordinate system, then into the FEM global coordinate system. The end result of the process is that [R] is transformed into

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11

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

Fig. 16. Modal identification of column with semi-rigid ends, (a) model, (b)–(d) modes, (e) identification (lipped channel: h ¼ 100 mm, b ¼60 mm, d ¼ 10 mm, t ¼2 mm, E¼ 203500 GPa, ν¼ 0.3. (a) End model, (b) #1, (c) #4, (d) #8 and (e) modal identification.

[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 (deformation space) contributions can be determined through transformation utilizing [RFE], similar to Eq. (12). However, one must recognize that the cFSM basis vectors used to form [RFE] are only an approximate basis and thus error may exist, namely fderr g ¼ fdFE g  ½RFE fcFE g

ð24Þ T

if the sum squared error ({dFE} {dFE}) is minimized one finds that the solution for {cFE} is fcFE g ¼ ð½RFE T ½RFE Þ  1 ½RFE T fdFE g

ð25Þ

Modal 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 identification 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 ½ΦFE  ¼ ½ fϕFE g1

fϕFE g2

:::

fϕFE gnFE  DOF 

ð26Þ

any {ϕFE} may be used as {dFE} of Eq. (25); therefore, after construction of a companion FSM model, creation of [R] and interpolation and transformation to [RFE], Eq. (25) may be solved for the {cFE} contribution coefficients. Eq. (14) is then used to get the participation coefficients. The results are similar to FSM modal identification with general end boundary conditions (Fig. 5), but now also the error in the approximation is monitored. An FEM model of a column with semi-rigid end conditions is explored to demonstrate, in part, the efficacy of the selected base function system, see Eq. (23). The error (last column of Fig. 16e) is negligible for the first 43 modes. Global, distortional, and local dominant modes are identified along with higher modes that are severely interacted. Error is non-negligible when attempts to identify local modes at short wavelengths, that are not included in the cFSM basis, are performed.

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4.5.3. Members with holes and irregular FEM mesh The introduction of holes into a thin-walled (cold-formed) steel member is generally regarded to greatly complicate the analysis. Recently spline finite strip models have been developed for this special case [46,47], but in general shell FEM models provide the most complete (if numerically costly) solution [48–50]. The problem is, how to identify the resulting output? Here, as shown in Fig. 17, cFSM-based modal identification is performed on a FEM model of a member with holes to illustrate the potential. The error is negligible across the first 50 modes, indicating excellent success with the identification procedure. With the exception of the first two global modes the modal identification results ably demonstrates the challenge of members with holes: L and D are at least partially interacted across essentially all modes. Even in modes with a visual dominance, e.g., L in mode 17, the other mode (D in mode 17) has 20% or greater participation. It is possible to identify the mode with the largest G, D, or L contributions, or the first 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 quantified, currently it is not utilized in design. Clearly, despite significant practical progress on this problem, theoretical issues remain.

4.5.4. Members undergoing thermal gradients Another problem of significant interest is the evaluation of the stability of thin-walled members under temperature gradients. Of particular interest is coupling and mode switching that is triggered by the thermal gradient. An FEM shell element model of a column is constructed and exposed on one side (Fig. 18a) to the time– temperature profile of Fig. 18b. The modulus is assumed to be temperature dependent based on the results of [51,52]. Elastic buckling analysis is performed at different times throughout the analysis and the buckling load and modes generated. As Fig. 18e indicates the elastic buckling load decreases steadily with time, but as the buckling mode shapes of (Fig. 18c, d, f and g) show the nature of the buckled shape evolves with time. Modal identification indicates a growth in distortional deformations even though the initial (t¼ 0) buckling mode is local dominant. For the 7th mode the dominant mode switches at t  25 min. 4.6. cFSM in modal identification of FEM nonlinear analysis 4.6.1. Formulation The application of the cFSM basis to FEM eigen-buckling helps solve a longstanding problem in the identification of buckled shapes.

Fig. 17. Modal identification of (a) member with holes, (b) full results, (c)–(i) mode shapes. (a) Geometry (mm), (b) identification results, (c) 1st, (d) 3rd, (e) 5th, (f) 17th, (g) 22nd, (h) 22nd zoom and (i) 5th global feature.

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13

Fig. 18. Modal identification of column under thermal gradients. (a) Thermal gradient at t¼ 23 min, (b) temperature vs. time curve (T-t), (c) 1st, t ¼0 min, (d) 1st, t¼ 32 min, (e) elastic buckling vs. time, (f) 7th, t=0 min (g) 7th, t=32 min and (h) modal identification vs. time.

However, the general nature of the procedure makes it possible to identify any deformation – thus a natural extension is to explore modal identification of geometrically nonlinear analysis on the imperfection structure (GNIA) and geometric and material nonlinear analysis on the imperfection structure (GMNIA) models. For the general case of the GMNIA solution equilibrium (incrementally) is represented by fFg ¼ ð½K e   ½K g   ½K p ÞfdFE g

ð27Þ

direct modal identification of {dFE} of Eq. (27) has two problems: first, in a model with initial imperfections the comparison to the undeformed cFSM basis is problematic; second, in the pre-buckling stage the linear elastic deformations (e.g. axial shortening in a column) dominate and tend to cloud the evolution of the buckling-associated deformations. As a result, the deformation vector for performing the identification {dFE}ID is modified to fdFE gID ¼ fdFE g þ fδimp g

removing this contribution from the participations, per pM ¼

‖fcFE  M g‖  ‖fcLE  M g‖ ‖fcFE g‖  ‖fcLE g‖

ð29Þ

in practical implementations it has been enough to remove only the global (M ¼G) linear elastic deformations; and this is what is illustrated in the following. 4.6.2. Lipped channel column examples (GNIA and GMNIA) Modal identification of GNIA and GMNIA analysis of a lipped channel with varying imperfections is provided in Fig. 19. In all cases the collapse mechanism is identified as interacted with L and D, but dominated by D deformations. The pre-buckling identification is shown to be imperfection sensitive. Although the perfect (no imperfection) model is dominated by local (L) buckling, depending on the imperfection magnitude and shape it is possible to trigger D dominance. See [44,45] for further studies of this nature on additional members.

ð28Þ

where {δimp} is the imperfection. Further, the participation is modified to remove linear elastic deformations {dLE} resulting from {F}. This is completed by finding the contribution for the undesired deformations {cLE} per Eq. (25) with {dLE} replacing {dFE} and then

4.6.3. Lipped channel parametric study of L vs. D instability A 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

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Fig. 19. Modal identification 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 imperfection, (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).

this is solved by current design codes that provide local and distortional buckling as separate limit states. However, it is still surprising to see just how different the actual member response is from elastic and/or elastic buckling assumptions. Consider the lipped channel column with varying lip length as depicted in Fig. 20a. As the lip length increases the elastic distortional buckling load (Pcrd) increases while the elastic local buckling load (Pcrl) stays largely constant. The resulting ratio of distortional to local elastic buckling is provided in Fig. 20b, and indicates a Pcrd/Pcrl  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% exceedance probability for Type I and Type II imperfections. Modal identification of the imperfection shows approximately 80% of distortional and 20% local participation. Modal identification of the deformations, at peak load and in the collapse regime (Fig. 20d and k), indicates the interacted nature of the deformations and the extent to which lip length influences 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 local buckling steadily increases with lip length, but only for extremely long lips is local buckling the dominant predicted deformation. One might think of coupled local and distortional buckling occurring with Pcrl ¼ Pcrd, but coupled inelastic deformations occur with Pcrd  (10–20)Pcrl. In this example, one can see how modal identification of GMNIA analysis opens up new avenues for exploring interacting modes.

5. Other works initiated by cFSM 5.1. Imperfection identification Geometric imperfections that are the result of the manufacturing process are the focus of [53–55], where a summary of the available imperfection measurements for cold-formed steel members is presented and three methods to simulate imperfection fields are introduced. The first is the classical approach employing a superposition of eigenmode imperfections, but scaled to match peaks in

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15

Fig. 20. Lip length study: (e) d ¼10 mm at peak; (f) d ¼ 10 mm at post; (g) d ¼ 20 mm at peak; (h) d ¼20 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 identification at peak and (k) modal identification at post.

the measured physical measurements. The second is a method based on the multi-dimensional spectral representation method, in which imperfections are considered as a two-dimensional random field and simulations are performed taking a spectra-based approach. The third, is termed as the 1D Modal Approach, and is a novel combination of modal approaches and spectral representation that directly considers the frequency content of the imperfection field, but employs a spectral representation method driven by the crosssectional eigenmode shapes to generate the imperfection fields. Though the eigenmode shapes can be determined by various numerical methods, cFSM is beneficial to use and has been applied in the actual study. Based on the analysis of available imperfection measurement data, design power spectrums are proposed for GMNIA analysis. From GMNIA-based finite element parametric studies the 1D Modal Approach is found to be the most powerful method. 5.2. Alternative cFSM and constrained spline FSM In [56–58] an alternative cFSM formulation is presented. The method is based on the same mechanical criteria as presented herein, but the practical implementation of these criteria into the FSM is slightly different from that used in the original cFSM [13–16]. The practical advantage of the new implementation is that closed cross-sections, which are not considered in the original cFSM derivations, are now treated. It is worth noting; however, that the shear modes are not fully incorporated; therefore, torsional modes (e.g., lateral-torsional buckling) of closed cross-sections are not fully solved. Recently, the constraining technique has also been applied to the spline finite strip method by the same research group [59]. The method is not fully developed yet, but first results are promising.

5.3. Constrained FEM In [60–62] the constraining technique of cFSM is applied within the finite element method. The method is realized in a commercial FEM code: ANSYS [63], an approach that has both advantages and disadvantages. The advantage is that ANSYS is widely available and reliable, while the disadvantage is that its application introduces certain limitations for the constraining method. The major limitations are as follows: (i) only the modal decomposition problem has been solved, i.e., the presented method is able to provide critical loads and buckled shapes in pure modes, but model identification is not possible; (ii) the modal decomposition is not full in the sense that it is not possible to transform the whole displacement field into a modal basis; and (iii) the introduction of constraints increases the problem size unlike in cFSM where constraining decreases the effective degrees of freedom. In spite of the disadvantages, the method provides useful and unique practical results, e.g., for perforated cold-formed steel members such as rack uprights that cannot be properly handled by any other methods to date. Moreover, the presented constrained method can be considered as an important step toward to a more general cFEM. 5.4. cFSM in analytical solutions for global buckling cFSM buckling solutions, when reduced to only a small number of modes, can readily produce analytical solutions. Given the various options available in the implementation, e.g., Eqs. (20)–(22), cFSM can be utilized to explore the effect of different mechanical assumptions 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]

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the classical pin-ended column stability problem is derived, and analytical solutions for flexural, torsional, and flexural-torsional buckling are generated that follow Euler–Bernoulli beam theory or the mechanics consistent with Kirchoff plate theory. In [67] additional analytical solutions for the column problem are derived, now with shear deformations included. In [68] the classical pin-ended beam stability problem is explored and the case of a doublysymmetric section under uniform moment is also analytically treated with the various mechanical definitions. In all cases, the dependence of the derived analytical solutions on the assumed mechanics is fully discussed, allowing one to understand specifically how classical assumptions impact commonly used formulae for global buckling. 5.5. Alternative FEM modal identification Modal identification for shell FEM solutions is a much needed tool for assessment of thin-walled member stability. As an alternative to the use of the full cFSM basis vectors (Sections 3.3 and 3.4), in [69,70] it is proposed to only use the cross-section deformation modes. The advantage of such an approach is to make the identification independent of the boundary conditions. The resulting identification, provided along the member length, is similar in spirit to GBT-based modal identification.

6. Discussion A significant number of extensions to cFSM are possible and needed. Within FSM/cFSM 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 cFSM basis to FEM, modal decomposition (instead of just modal identification) should be explored. In particular, the modal decomposition could be implemented in a manner such that the reduced DOF can be utilized in a beam element formulation. Here again the pioneering work of GBT provides some potential avenues for exploration. Also, on the practical side, further automation of cFSM particularly its application to FEM modal identification is needed. A number of issues in the behavior and design of thin-walled members deserve further study with cFSM. The identification of limit state dependency on imperfections instead of the cross-section geometry alone needs to be quantified and further explored. As energy dissipation becomes a more important property for thinwalled members (in disproportionate collapse, seismic, blast, etc.) connecting the impact of the buckling deformations to the collapse mechanics and the energy dissipation within the different mechanisms is needed. For cFSM, probably the most important next step is to determine meaningful ways that the quantification of the modal participations can be used in design. Utilizing the participations as a measure of modal coupling/interaction and then using weighted strength formulas is one possible approach, but currently no method utilizes cFSM (or GBT for that matter) participations in a manner that informs design. A number of issues remain in the context of modal identification extensions to shell FEM models and GMNIA analysis. The practice of removing the linear elastic participations needs further study. As the FEM models become more complex the ability to generate the base vectors in FSM may be compromised. Further, the use of elastic base vectors for assessing plastic deformations in a GMNIA analysis is practically interesting, but lacks theoretical rigor. Strategies for error reduction in the FEM modal identification are also needed. While today the field may lack perfect clarity on the notions of coupled instabilities versus interacted modes versus other similarly labeled phenomenon, cFSM does provide a manner to categorize instabilities into a workable small number of classes and then quantify the coupling/interaction across those classes. Thus, the

definition of the deformation classes (Table 1) is the key to the entire method; these definitions work well for simple shapes with sharp corners, but the separation between local and distortional deformations, in particular, becomes challenged in shapes with many corrugations and round corners. As a result, further effort into formalizing the deformation classes would still be useful.

7. Conclusions The finite strip method (FSM) is an efficient numerical tool that has provided a unique ability to improve our understanding of thin-walled member stability. The constrained FSM (cFSM) extends the capabilities of FSM to include modal identification and modal decomposition. Modal identification provides a means to 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 to only desired deformations; allowing an analyst to isolate a given buckling mode, or even study the impact of differing mechanical theories (beam theory vs. plate theory, impact of shear, etc.) on a buckling solution. This paper provides a summary of recent research applications and fundamental developments in cFSM. A summary of cFSM, built-up from the case of the semi-analytical solution for strips with simply-supported ends; then generalized to the case of varying end boundary conditions as well as different assumptions regarding the calculation of the internal strain energy, external work, and shear is provided. The application of cFSM to automate the calculation of member stability modes (local, distortional, or global) is demonstrated. This automation, combined with the Direct Strength Method of design creates the ability to study optimal member cross-sections; and several examples of recent work in this area are presented. The application of cFSM to provide modal identification in shell finite element models is one of the richest areas of current study. The general nature of shell finite element models allows for situations largely outside of the realm of traditional cFSM, i.e., members with holes, under longitudinal loading or temperature gradients, etc. Further, the application of cFSM-based modal identification to FEM models opens up the extension of the identification to general deformations such as those from geometric and material nonlinear analysis on the imperfect structure (GMNIA). Modal identification of deformations from FEM GMNIA models demonstrates the evolution of coupled and competing modes in pre-buckling, post-buckling, and collapse regimes. Other research initiated by cFSM is also ongoing, including work on imperfection modeling that employs cFSM modes, alternative approaches to extend cFSM to finite element models, and analytical investigations of global stability solution employing the cFSM framework. Significant work remains to advance the theory and continue to generalize the cFSM approach, but today significant new capabilities in the analysis of thin-walled members are now available thanks to the modal decomposition and modal identification capabilities of cFSM.

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