Modal identification for shell finite element models of thin-walled members in nonlinear collapse analysis

Thin-Walled Structures 67 (2013) 15–24

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Modal identification for shell finite element models of thin-walled members in nonlinear collapse analysis Z. Li a,n, S. A´da´ny b, B.W. Schafer c a

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

a r t i c l e i n f o

abstract

Article history: Received 15 June 2012 Received in revised form 14 January 2013 Accepted 24 January 2013

The objective of this paper is to provide a method for applying modal identification (i.e. the separation of general deformations into fundamental modal deformation classes: local, distortional, global, shear, and transverse extension) to the collapse analysis of thin-walled members modeled using material and geometric nonlinear shell finite element analysis. The advantage of such a modal identification is the ability to categorize and reduce the complicated deformations that occur in a shell finite element model—and ultimately to (a) quantitatively associate failures with particular classes, e.g. state a model as a local failure, and (b) track the evolution of the classes, e.g., mixed local and distortional buckling leading to a distortional failure in a given model. Ultimately, this capability will aid Specification development, which must simplify complicated behavior down to strength predictions in isolated buckling-induced limit states. The modal identification method is enabled by creating a series of base vectors, consistent with the fundamental deformation classes, that are used to categorize the general finite element displacements. The base vectors are constructed using the constrained finite strip method for general end boundary conditions, previously developed by the authors. A fairly sizeable minimization problem is required for assigning the contributions to the fundamental deformation classes. The procedure is illustrated with shell finite element examples of cold-formed steel members modeled to collapse with geometric or/and material nonlinearity. The failure modes of the member are tracked (i.e., identified as a function of displacement), and the collapse mechanism is investigated. The provided examples provide both proof of concept for the modal identification and demonstrate the potential of using such information to better understand the behavior of thin-walled members. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Modal identification Constrained finite strip method Finite element method Collapse analysis Post-buckling analysis

1. Introduction The strength of thin-walled members is often governed by its buckling behavior due to its high cross-sectional slenderness. In terms of design procedures, current design specifications (e.g., for cold-formed steel [1–3]) all tackle the strength prediction in member design by investigating the elastic buckling based on rational analyses (numerical or analytical), then using empirical design equations calibrated from experimental or even numerical simulation data to predict the design strength of the member. However, as commonly acknowledged, instabilities of thinwalled members can be rather complicated and are generally categorized as: local (local-plate), distortional, and global (Euler) buckling. As currently reflected in design specifications, such as AISIS100 for cold-formed steel [3], appropriate separation and identification of the buckling modes are necessary because of the different

n

Corresponding author. Tel.: þ1 410 9490843. E-mail addresses: [email protected], [email protected] (Z. Li), [email protected] (S. A´da´ny), [email protected] (B.W. Schafer). 0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.01.019

post-buckling strength and interactions between the modes. Moreover, as in the Direct Strength Method [4] for cold-formed steel member design as provided in [3], the formulation of the empirical design equations is calibrated based on the appropriate categorization of the failure modes into the three buckling mode classes (local, distortional, and global), this categorization both in experimental tests and numerical simulations is still completed largely by subjective visualization and engineering judgment. Given the many advances in computation, the Finite Element (FE) Method employing shell finite elements has become popular in analyzing thin-walled structures both for elastic (eigen) buckling and nonlinear collapse analyses. FE’s unique applicability to handle complex geometry and boundary conditions makes it a natural choice in many situations. However, FE itself provides no means of modal identification, and instead requires a laborious and subjective procedure employing visual investigation to classify the buckling modes in elastic buckling analysis or categorize the failure mode in nonlinear collapse analysis. For elastic buckling analysis, the newly developed constrained finite strip method (cFSM) provides the ability to decompose buckling modes as well as categorize (identify) arbitrary buckling

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modes into the fundamental deformation classes of global (G), distortional (D), local (L), and/or shear and transverse extension (ST) as presented in [5–7]. The cFSM modal identification method was extended to general end boundary conditions in [5,7]. Extension of the modal identification method for elastic buckling of shell FE models with general end boundary conditions has also recently been completed [8,9]. In this paper, the cFSM-based modal identification procedure is applied to the nonlinear collapse analysis of shell finite element models with both geometric and material nonlinearity. Generalized Beam Theory has been utilized in nonlinear elastic analysis for a similar modal identification purpose [10].

2. Formulation: modal identification for shell FE method The goal for the proposed modal identification is to take a general FE displacement vector (deformed shape or buckling mode) based on a shell element model of a thin-walled member, and accurately assign quantitative modal participations in terms of the G, D, L, and ST classes. For nonlinear collapse analysis, equilibrium in the incremental FEM solution may be expressed as ðK e þK g þ K p ÞDFE0 ¼ F

ð1Þ

where, Ke is the conventional elastic stiffness matrix, Kg is the geometric stiffness matrix and depends upon the internal forces in the model, Kp is the plastic reduction matrix and accounts for material yielding, F is the consistent nodal forces, and DFE’ is the vector of displacements to be identified. For the special case considering only geometric nonlinearity on the perfect structure an eigen-value problem associated with elastic buckling may be defined as ðK e lK g ÞFFE ¼ 0

ð2Þ

where, l is the eigenvalue (load factor is compression positive), and FFE is the eigenmode (buckling mode) vector, which is nothing more than a special case of DFE. Note, for nonlinear analysis, Ke and Kg are formulated on the deformed shape in each increment, and thus change throughout the analysis. Note, the cFSM base vectors used to identify the deformation in the nonlinear collapse analysis (as illustrated later) are based on the perfect geometry. However, for nonlinear collapse analysis the deformation vectors are normally based on a model of the member with initial geometric imperfections. Consequently, the deformation vector to be identified is DFE ¼ DFE0 þ Dimp

Table 1 Criteria of mode classification. Mechanical criteria

G

D

L

ST

gxy ¼ 0, ex ¼ 0, v is linear (Vlasov’s hypotheses) ey a 0 (longitudinal warping) kx ¼ 0 (undistorted section)

Yes

Yes

Yes

No

Yes Yes

Yes No

No –

– –

into four fundamental deformation classes: global (G), distortional (D), local (L), and shear and transverse extension (ST). The separation is based on formal mechanical criteria [5–7,11–13] as summarized in Table 1, and is accomplished through constraint matrix RM (where M¼G, D, L, or ST). Combined, the constraint matrices of each subspace (RM) span and form an alternative basis of the standard FS space, usually termed the natural cFSM basis. Two primary capabilities are enabled by cFSM: modal decomposition and modal identification. Modal decomposition is a model reduction technique whereby the general displacement field is constrained to only a desired subset, e.g. focusing only on L deformations and hence local buckling. Modal identification is a model classification technique whereby a general displacement field is quantitatively categorized into G, D, L, and/or ST participations, e.g. quantifying the participation of deformations consistent with distortional buckling in a visually mixed-mode eigen-buckling solution. The cFSM derived RM constraint matrices are the key toward application of both modal decomposition and modal identification. The column vectors of RM define a set of basis vectors within the G, D, L, and/or ST subspaces. Transformation inside each subspace by performing an auxiliary eigen problem within each subspace either for a unit axial stress, or for the actual applied stresses on the member being studied, may be used to form the axial or applied modal basis, respectively. For general end boundary conditions, further distinctions are also available for coupled and uncoupled modal bases. Depending on the desired analysis and desired numerical efficiency certain bases hold advantages in some problems; see [5,6,14,15] for complete details. The base vectors (columns of RM) must be appropriately normalized with vector norm, strain energy norm, and work norm available [12–14]. With the defined basis and selected normalization scheme, any nodal displacement vector in FS method, DFS, (deformed shape or buckling mode) may be transformed into the basis spanned by the four subspaces, via

ð3Þ

where, DFE is the deformation vector to be identified in terms of G, D, L, and ST participations, and Dimp is the imperfection deviating from the perfect model. This is consistent with the fact that the cFSM base vectors are defined on the perfect structure and furthermore only influences the identification solution for small DFE. The modal identification of the FE deformation vector, DFE, is completed by utilizing a linear combination of the FE base vectors, RFE, which are interpolated from the cFSM base vectors, RFS, and consist of predefined column vectors defining G, D, L, and ST deformations. In the following sections, cFSM is briefly summarized along with the RFE and RFS base vectors and the basic modal identification methodology.

3. Constrained finite strip method and generalized base functions The cFSM was originally derived from the semi-analytical Finite Strip (FS) Method by separating the FS general displacement field

DFS ¼ RFS C FS

ð4Þ

where RFS ¼[RG RD RL RST] and CFS ¼[CG CD CL CST]T. The column vectors of CFS include the contribution coefficients for each base vector inside each deformation class and may be summed to determine the participation of DFS in a given class. The solution is sensitive to the choice of basis, normalization of the base vectors, and summation method employed for the participation calculation as detailed in [14,15]. Based on existing work the cFSM basis used in this paper is the uncoupled axial modal basis, normalization is by the vector norm, and the ST subspace is employed. Participation is calculated via the L2 norm, as discussed further in Section 4.3. The final step in defining the cFSM solution is the selection of the FS longitudinal shape (or base) function. The modal identification employed here interpolates the FS base functions to any FE degrees of freedom (DOF). In cases where the end boundary conditions of the FS base functions matches the end boundary conditions in the shell FE model these are termed associated base functions and agreement between modal identification in the FS

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17

Fig. 1. Longitudinal field diagram: (a) C–F&F–C m ¼1; (b) S–S m¼1,2,y,10, note only the first 10 terms are shown in (b) m may be any positive integer and does not have to be consecutive.

Fig. 2. Illustration of interpolation of FS DOF to FE DOF (FS local displacement is illustrated for longitudinal term m¼ 1 of S–S).

and FE model is essentially complete [5–7]. However, when end boundary conditions in the shell FE model, particularly with respect to translations at the end, do not match the underlying FS base functions errors can be large [8]. Having to generate associated base functions for all cases is both cumbersome and in general impossible. Hence, a set of approximate, but accurate, generalized FS base functions that meets all the essential boundary conditions has been proposed and validated [9,14]. In the longitudinal direction the generalized base function utilizes the analytically attractive sine series expansion (for simply supported ends) augmented with the first (m¼1 longitudinal term) for clamp-free and free-clamp ends as shown in Fig. 1. All the modal identification results in this paper are performed using the generalized base function.

4. Methodology for modal identification of shell FE models Modal identification of a shell FE model requires interpolating the cFSM basis (RFS) to the FE DOFs (RFE). Minimization of the error between the general FE displacement, DFE, and the same deformation transformed into the cFSM basis in FE DOF, provides the modal contributions. These contributions are summed to determine the modal participations utilized in modal identification and the error is monitored as detailed in the following sections.

4.1. cFSM base function interpolation to FE DOF General shell finite elements typically have 6 DOF, i.e., 3 translational and 3 rotational displacements, at each node. Plate bending finite strips typically have 4 DOF, i.e., 3 translational displacements (U, V, W) and 1 rotational displacement about the longitudinal axis (Y). Fig. 2 illustrates the essential concept and steps of expanding the FSM DOF into the FEM DOF (note, the FEM mesh is not required to be regular as shown in Fig. 2). First, any FEM node (e.g., node i in Fig. 2) is mapped into a strip of the corresponding FSM model consisting of two strip lines (e.g., line j and k in Fig. 2). Second, the global nodal displacement in the

FSM model is transferred to local nodal displacements via coordinate transformation GFS, e.g., for strip line j, dFSi ¼ GFS DFSi

ð5Þ T

T

where dFSj ¼ [uj vj wj yj] , and DFSj ¼[Uj Vj Wj Yj] . Note, this is a generalized notation for the FS displacement vector. In reality, the FS displacement is differentiated into longitudinal terms, m (e.g., m¼1 in Fig. 2) and FS shape functions vary with m [5,7]. Third, the local nodal displacement of the FE model is interpolated by using the two (j and k) local FS displacement vectors, the FS shape functions, and a transformation matrix T, and evaluated at the FE node coordinates, e.g., for FE node i, dFEi ¼ TNFS djkFS 9@i

ð6Þ

where dFEi, the local nodal displacement vector of the FE model, is [u v w yx yy yz]T at node i, djkFS , the local displacement vectors of the FS model consisting of two strip lines (i.e., j and k), is [dFSj dFSk]T, NFS is the FS shape function (see [5,7] for explicit expression), T is a transformation matrix expanding the 4 FS DOF to 6 FE DOF given as below and assuming that the drilling DOF is zero: 2 3 1 0 0 0 60 1 0 07 6 7 6 7 60 0 1 07 6 7 T ¼6 ð7Þ 7 6 0 0 @=@y 0 7 6 7 40 0 0 15 0

0

0

0

Fourth, the global nodal displacements of the FE model are then obtained via coordinate transformation GFE, e.g., for FE node i, DFEi ¼ GTFE dFEi

ð8Þ

where DFEi is [U V W Yx Yy Yz]T at node i. It is worth noting here that even though the drilling DOF is assumed zero in local coordinate, YZ may be nonzero in global coordinates because of the transformation from local to global coordinates. Moreover, though not necessary for the case illustrated in Fig. 2, special transformations between coordinate systems is required if they

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are different (traditionally FS uses a left-handed coordinate system and FE a right-handed coordinate system). Finally, with a known FS displacement vector (including RFS), the FE displacement vector can be assembled by the preceding steps for all nodes of an FE model. Particularly, the FE base vectors RFE are formulated by replacing the FS displacement vector with the base vectors of the FS generalized base function (i.e., column vector of RFS that define G, D, L, and ST deformation modes).

4.4. Error estimation To assess the error in the modal identification the L2 norm of the residual error, Derr, relative to the L2 norm of the displacement vector, DFE, is calculated: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð16Þ e ¼ DTerr Derr = DTFE DFE

5. Modeling parameters for numerical examples

4.2. Minimization With RFE known, any shell FE displacement vector can be approximated by a linear combination of the cFSM basis vectors interpolated to FE DOF. Minimization of the residual error is pursued since the shape functions employed in cFSM are only an approximation to the FE shape functions, specifically Derr ¼ DFE RFE C FE

ð9Þ

where, DFE is the FE displacement vector, Derr is the residual error, RFE is the cFSM base vectors transformed to FE DOF, and CFE is the contribution coefficients (for G, D, L, and ST). This is treated as a linear least squares problem where the best approximation is defined as that which minimizes the sum of the squared residual error, i.e.: min DTerr Derr

ð10Þ

Substituting Eq. (9) into Eq. (10), the least squares problem can be rewritten as minðDFE RFE C FE ÞT ðDFE RFE C FE Þ

ð11Þ

Provided that the base vectors in RFE are linearly independent, the problem has a unique solution based on the minimum condition d½ðDFE RFE C FE ÞT ðDFE RFE C FE Þ dC FE

ð12Þ

Therefore, the optimal solution C nFE that provides the minimal sum squared error is C nFE ¼ ðRTFE RFE Þ1 RTFE DFE

ð13Þ

4.3. Participation evaluation Typically modal identification results sum the contributions in a particular class (e.g., L) to form the predicted modal participation. Two options are available for the participation (p) of each mode class M (G, D, L, and ST):

Shell FE modeling of thin-walled members for ultimate strength prediction and investigation of collapse behavior necessitates the inclusion of both geometric and material nonlinearity. The solutions of these models are highly sensitive to model inputs, such as geometric imperfections, residual stresses, plastic strain, yield criteria, material model, boundary conditions, and also the fundamental mechanics, particularly with regard to element selection and solution schemes [16,17]. All the analyses performed herein utilize the commercial finite element package ABAQUS [18]. In this study, residual stresses and initial plastic strains are not included. Additional model assumptions are discussed in detail as follows. Three cold-formed steel members, as shown in Fig. 3, are selected for study. All the members are modeled as columns. The members are classified based on the expected dominant buckling mode as determined from modal identification of an eigen buckling analysis. Member (a) with a length of 600 mm is local dominated, i.e., the first eigen buckling mode is local, see Table 2. Member (b) with a length of 1200 mm is distortional dominated. Member (c) with a length of 700 mm is local-distortional interacted. Table 2 provides the G, D, L, and ST participations for the eigen buckling analysis used to classify the members. The relatively short length for each section is intended to exclude global buckling. The material is assumed to be homogeneous and isotropic and modeled as elastic-perfectly plastic (von Mises yield criteria with isotropic hardening) with Young’s modulus E¼ 210,000 MPa, Poisson’s ratio v¼0.3, and a yield stress of 345 MPa. Prediction of the peak strength and failure behavior of thinwalled structural members requires a fine mesh in nonlinear FE models to provide reliable results. The shell element used here is the S4 shell element from the ABAQUS library of elements [18], the S4 is a 4-node linear element (fully integrated). In addition, the element aspect ratio is controlled between 1/2 and 2 to avoid element distortion under large deformations.

pM ¼

M X

4 X

M X

M ¼ G,D,L,ST

i

9C nFEðiÞ 9=

i

45.0

50.8

 L1 norm (used in [13]) 9C nFEðiÞ 9

4 X

i

M ¼ G,D,L,ST

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M uX n t ðC FEðiÞ Þ2

60 10

 L2 norm (used in [7]) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M uX n ðC FEðiÞ Þ2 = pM ¼ t

12.7

15.9

ð14Þ

152.5 ð15Þ

Thickness: t=1.1455

100

Thickness: t=2.000

203.2

Thickness: t=3.300

i

Modal identification displays near independence in choice of bases when using the L2 norm in cFSM [14,15]. Therefore, the FE modal identification in this study is based on the L2 norm, unless otherwise specified.

Fig. 3. Dimensions of cross sections (mm) and mesh: (a) local dominated model; (b) distortional dominated model; (c) local-distortional interacted model (nodes indicate the mesh density within the cross-section).

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Table 2 Participations of eigenmode and nonlinear collapse models (no imperfection). Model

Local dominant model

Distortional dominant model

Local–distortional interacted model

Participation (%)

1st eigenmode at peak at post 1st eigenmode at peak at post 1st eigenmode at peak at post

Load (kN)

G

D

L

ST

0.4 0.7 0.4 6.0 0.1 0.1 2.9 0.2 18.4

4.8 12.8 55.9 89.2 93.8 82.6 38.5 50.5 52.0

94.6 85.4 41.0 4.7 4.6 14.8 58.1 47.1 27.5

0.1 1.1 2.8 0.1 1.6 2.5 0.4 2.2 2.1

Pcr

Pu

20.8 72.9 173.4 163.8 283.8 296.4

Note: at peak refers to the peak of the load–displacement response in a nonlinear collapse analysis; at post refers to collapse mechanisms at force levels less than 80% postpeak deformation.

The end boundary conditions simulate local-plate simply supported conditions, which imply warping fixity at the member ends. Specifically, all translational degrees of freedom are fixed at one end of the model, at the opposite end the transverse translational degrees of freedom are fixed, while the longitudinal translational degrees of freedom are tied to a single reference node, where the end shortening (loading) is performed. Careful treatment of geometric imperfections is of significant importance in modeling thin-walled members because the ultimate strength and failure mechanism are both imperfection sensitive. In this study, for modeling convenience, the distribution of the imperfection is seeded from local and/or distortional buckling mode shapes generated from an FS analysis of S-S boundary conditions [7], and the magnitude is a function of the member’s thickness t. It is worth noting here that if one wants to simulate tests or provide strength predictions of cold-formed steel members in a more reliable and accurate manner, the distribution and magnitude of imperfections should be more closely tied to measured data [16,17]. The solution scheme employed in the study is the arc-length method (modified Riks method [19] in ABAQUS).

6. Numerical examples Numerical examples of the proposed modal identification for shell element FE models are provided in this section. First, a simple linear static analysis is shown to demonstrate the applicability of modal identification on a general FE displacement vector. Then, modal identification of elastic post-buckling (models with geometric nonlinearity but no material nonlinearity) and collapse analyses are demonstrated. The evolution of the modal participation during post-buckling and nonlinear collapse analyses are utilized to explore imperfection sensitivity and the relative importance of the deformation classes (G, D, L, and ST) in describing the pre- and post-peak deformations. 6.1. Modal identification: a linear elastic static analysis Before delving into the more complicated cases, a simple problem—modal identification for DFE of a linear static analysis is considered. This is a simplified version of Eq. (1) with Kg ¼Kp ¼0 and no updating of Ke. In this particular study, a special set of linear elastic static analyses are performed to illustrate a key separation in the cFSM formulation. According to cFSM, for global and distortional deformation classes (the GD space, see Table 1) if Vlasov’s hypothesis is enforced the full deformation field (U, W, and Y) can be determined once the warping (longitudinal) displacements, V, are

Fig. 4. Warping distributions: (a) major-axis bending; (b) symmetric distortional.

known, see [5,12] for complete details. End warping displacements (longitudinal deformations), as illustrated in Fig. 4, corresponding to major-axis bending and symmetric distortional buckling are prescribed on member (b) of Fig. 3. The warping displacements along the member length are extrapolated from the end warping displacements by the cFSM longitudinal shape function: V¼

q X

V ½m Y 0½m a=m½m

ð17Þ

m¼1

where, V[m] is the warping distributions given in Fig. 4 associated with longitudinal term m, see [5,7] for the complete shape functions. Linear elastic static shell FE analysis is performed on the member with the two prescribed warping displacements. The deformed shapes from the static analyses for both simple (S– S) and clamped (C–C) end boundary conditions are provided in Figs. 5 and 6, respectively. As shown in Fig. 5, major-axis warping deformations result in appropriate transverse bending and modal identification indicates 99.9% G participation. For distortional buckling, Fig. 6, modal identification indicates 98% D participation. The predicted participations are consistent with visual identification and expectations based on the applied warping displacements. 6.2. Modal identification: nonlinear analysis In this section modal identification of the cold-formed steel columns of Fig. 3 is performed on: (1) elastic post-buckling analysis, with only geometric nonlinearity (i.e., Kp ¼0 in Eq. (1)); and (2) nonlinear collapse analysis, with both geometric and material nonlinearity. In both cases the analysis is based on

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Fig. 5. Deformed shapes of major-axis bending warping functions for S–S and C–C. (a) S–S, major-axis bending and (b) C–C, major-axis bending.

Fig. 6. Deformed shapes of symmetric distortional warping functions for S–S and C–C. (a) S–S, symmetric distortional with longitudinal term m ¼6 and (b) C–C, symmetric distortional with multiple longitudinal terms.

incremental load-deflection response and equilibrium is established in the deformed shape. For modal identification, determining the displacements to consider in the identification requires some judgment. In this study, the applied displacement of column end shortening is highly correlated with the fourth global buckling mode (G4, the axial compression or extension [13,15]) of the cFSM base vectors. Modal identification for elastic post-buckling analysis is provided for the three models in Figs. 7e–9e. In all cases G has significant participation throughout the analysis, but this deformation is nearly entirely due to G4. Although this deformation is important for understanding the response comprehensively, the buckling behavior is made clearer when this primary displacement is removed. Therefore, modal identification results without this correlated global mode are provided and termed as the adjusted participations (as illustrated in Figs. 7–9). For simple loading cases, as studied here, the primary displacement is typically coincident with a global mode and readily removed. However, for more complicated loading the separation is less clear. It is proposed that the linear elastic deformations associated with the applied loading on the perfect structure (DLE) should be removed prior to identification, thus the displacement vector to be identified, DFE*, is DFEn ¼ DFE DLE

ð18Þ

where, DLE is DG4 in the problem studied here. 6.2.1. Local dominated model Modal identification of the elastic post-buckling and nonlinear collapse behavior of a lipped channel dominated by local buckling (local dominated model of Fig. 3a, and Table 2) is explored in this section. The member exhibits significant post-buckling reserve with an elastic local buckling load of 20.8 kN and a peak strength of approximately 73 kN, depending on imperfections. Five imperfection cases, as listed in Table 3, are considered. Fig. 7a provides the load–displacement response and Fig. 7b–d the deformations. Careful study of the overall response and evolution of the modal participations as a function of end shortening, as provided in Fig. 7, reveals significant insights about the behavior of this member.

First, the local dominated member exhibits only mild imperfection sensitivity with respect to strength for both elastic postbuckling and collapse analyses, as indicated in Fig. 7a and Table 3. Table 3 also illustrates that not all imperfections are detrimental with respect to strength: at small magnitudes the distortional imperfection (case III, Table 3) actually increases the strength above the no imperfection result (case I). Second, both the elastic post-buckling and collapse analyses demonstrate the importance of distortional (D) deformations in the post-buckling and post-peak collapse regimes. The importance of D deformations is independent of magnitude and distribution of imperfections, as shown in Fig. 7f–j. Material nonlinearity does trigger a more dramatic transition from L to D deformations in the post-peak regime (see Fig. 7f–j) and the yielding creates localization in the deformation response (see Fig. 7c and d), but the final mechanisms are similar at large deflections. The importance of distortional deformations in collapse mechanisms triggered from local buckling modes has been observed in experiments [20]. Third, imperfection magnitude and distribution can alter the predicted pre-peak modal identification (L vs. D) and the peak failure mode, but in the end the same post-peak collapse behavior is triggered. For example, with a large enough distortional imperfection the pre-peak identification emphasizes D participation (Fig. 7j, imperfection case V) even though the model is fundamentally dominated by local buckling. However, in all models L participations are greatest at peak load, but drop precipitously in post-peak collapse. Inward and outward distortional imperfections (cases III and IV) trigger subtly different responses even in a member dominated by local buckling. In the post-peak collapse regime distortional deformations dominate. Finally, shear and transverse deformations (ST), which are not obvious in either the eigen-buckling mode or in visual analysis, has growing importance in the post-buckling/post-peak regimes. Participations are typically greater than 2%, consistent with observations from GBT analysis [10]. Also, for the no imperfection model (case I) in Fig. 7(f), the initial deformations (primarily associated with localized deformations due to the end boundary conditions) have a noticeable ST contribution.

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Fig. 7. Nonlinear analyses of the local dominated models. (a) Load-displacement responses with typical deformation shapes (scale factor: 2), (b) Near peak, (c) (d) Near 2 mm, (e) I: no imperfection with G4 (elastic), (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).

6.2.2. Distortional dominated model Modal identification of the elastic post-buckling and nonlinear collapse behavior of a cross-section dominated by distortional buckling (distortional dominated model of Fig. 3b, and Table 2) is explored in this section. For the distortional dominated model, three imperfection cases are studied: (I) no imperfection; (II) local imperfections with 8 half-waves and a magnitude of 0.1 t; and (III) distortional imperfections with 2 half-waves and a magnitude of 0.1 t. Load–displacement responses for elastic postbuckling and collapse analyses are shown in Fig. 8a with typical deformation shapes in Fig. 8b–d. The evolution of the modal participation for both elastic post-buckling and collapse analyses and for the three imperfection cases is provided in Fig. 8f–h. Predicted strength in the distortional dominated member is more imperfection sensitive than in the local dominated member. As shown in Fig. 8a, introduction of even a 0.1 t imperfection results in peak strength decreases from the no imperfection (case I)

model of 9% for the local imperfection (case II) and 14% loss for the distortional imperfection (case III). While the local dominated member had significant post-buckling reserve, the selected distortional dominated member is in the inelastic buckling regime (i.e., with strength of 163.8 kN and elastic buckling of 173.4 kN, as summarized in Table 2). The dominance of distortional deformations in the post-buckling/post-peak collapse regimes and the role of material nonlinearity are generally the same as in the local dominated model. For the distortional dominated model the initial imperfection is more influential—as evidenced by the strongly local–distortional interacted failure that results from the no imperfection (case I) and local imperfection (case II) results of Fig. 8f and g, respectively. If the imperfection matches the collapse, as in the distortional imperfection of case III, then the matching participation persists throughout the analysis. Such a ‘‘sympathetic’’ imperfection is likely to be the most detrimental with respect to strength.

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Fig. 8. Nonlinear analyses of the distortional dominated model. (a) load-displacement responses with typical deformation shapes (scale factor: 1), (b) near peak, (c) near 4mm, (d) near 4mm, (e) I: no imperfection with G4 (elastic), (f) I: no imperfection, (g) II: local imperfection (0.1t) and (h) III: distortional imperfection (0.1t).

Contrary to the local dominated member, the distortional dominated member exhibits some global participation (minoraxis bending) in the post-buckling regime. However, for post-peak collapse global only participates in the case of the local (case II) imperfection.

6.2.3. Local/distortional interacted model Modal identification of the elastic post-buckling and nonlinear collapse behavior of a cross-section with interacted local and distortional buckling (Fig. 3c, and Table 2) is explored in this section. Similar to the distortional dominant model, three imperfection cases are studied for the local–distortional interacted model: (I) no imperfection; (II) local imperfections with 5 halfwaves and a magnitude of 0.1 t; and (III) distortional imperfections with 2 half-waves and a magnitude of 0.1 t. Load– displacement responses of elastic post-buckling and collapse analyses are shown in Fig. 9a with typical deformation shapes in Fig. 9b–d. The member is not as sensitive to imperfections as the distortional dominated member. Slight post-buckling reserve is observed. The modal participation for both elastic post-buckling and collapse analyses for the three imperfection cases are shown in Fig. 9f–h. If the imperfection is local the pre-peak identification is primarily local with increasing distortional participation; conversely for distortional imperfections D dominates with increasing L participation. The mode is clearly ‘‘interacted’’ in the sense that the deformations introduce both L and D participations

independent of the imperfections in the pre-peak range. Regardless, the dominance of distortional deformations in the post-buckling/ post-peak collapse regimes and the role of material nonlinearity still follow the earlier results: D deformations dominate as the collapse mechanism forms. For local/distortional interacted models, the noticeable global mode in the collapse analysis is largely weakaxis bending.

7. Discussions The examples provided herein provide proof-of-concept for the extension of modal identification, driven by cFSM base vectors, to nonlinear collapse analysis. The proposed modal identification has limitations. The cFSM base vectors, as currently formed, require a straight, open cross-section, prismatic member consisting of plane plates. Further, cFSM (and GBT) cannot provide a meaningful distinction between local and distortional buckling if neighboring plates have small relative angle changes (e.g. due to small longitudinal stiffeners, or when modeling corners as a series of straight segments). Identification herein shares this problem. Work is underway to relax these limitations. Limitations with regard to the displacement field to be identified are also possible, mathematically the proposed method can be applied to any displacement field, but the question remains: are the results practically meaningful? To date our studies have shown meaningful identification results for elastic buckling with a variety of end restraints (including semi-rigid ends), intermediate elastic restraints, member with

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23

Fig. 9. Nonlinear analyses of the local/distortional interacted models. (a) load-displacement responses (Deformation scale factor: 2), (b) near peak, (c) near 4mm, (d) near 4mm, (e) I: no imperfection with G4 (elastic), (f) I: no imperfection, (g) II: local imperfection (0.1t) and (h) III: distortional imperfection (0.1t).

Table 3 Imperfection cases and their peak loads. Case Waves along member

Distribution

Magnitude (t)

Peak load (kN)

I II III

n/a 5 1

0 0.1 0.1

72.9 72.1 73.1

IV V

1 1

n/a Local Outward distortional Inward distortional Outward distortional

0.1 0.94

72.6 68.3

Note: (1) t is the thickness of the cross-section. (2) Wave numbers are determined from the characteristic half-wavelengths of local and distortional buckling from CUFSM analysis [7].

holes, and in collapse analysis of members with imperfections. As long as the member meets the limitations of cFSM the displacement field to be identified is not severely limited. Although the proposed method is general questions remain, most importantly, how to handle the primary displacements associated with a given loading particularly when it is complex, as discussed in Section 7. Error in the modal identification is generally quite small. Error in the nonlinear collapse analyses of the three studied cases is provided in Fig. 10. Error is greatest when plasticity and localization dominate the response. A larger number of cFSM base vectors could be included to reduce the error, but is currently not felt to

be worth the computational cost. Overall the error is read to be more than satisfactory for successful modal identification. The modal identification performed herein demonstrates some unique challenges associated with assigning buckling-induced limit states to lipped channels. In particular, models with imperfections demonstrate that the failure mode also relies on the prescribed imperfections. Moreover, from a behavioral standpoint the dominance of distortional buckling in the post-buckling/ collapse regime of lipped channels, independent of the prebuckling or pre-peak deformations is an intriguing result. Local failures, whereby local buckling dominates the response and significant post-buckling is developed is observed, but the collapse mechanisms are typically better captured by distortional deformations. In general plasticity triggers the development of the distortional post-peak mechanism earlier and may also bring some other modes in due to localization (e.g., global, shear and transverse extension deformations). Moreover, when material nonlinearity is present, this extension of elastic base vectors to describe plastic deformation fields is practical, and appears potentially useful in an engineering sense, but further work is needed before such identification could be considered rigorous. A long-term goal of this work is to connect collapse mechanisms and their related energy dissipation to the G, D, L, and ST deformations. Currently, no simple means exists for predicting energy dissipation in members, and modal identification and categorization provides new information to make these connections. Future work in this direction is anticipated.

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Z. Li et al. / Thin-Walled Structures 67 (2013) 15–24

15 Local dominated model Dist. dominated model

10 e, %

Local/Dist. interacted model

5

0 0

1

2

3

4 5 Displacement, mm

6

7

8

Fig. 10. Error versa end shorten displacement of all three models for nonlinear collapse analysis.

8. Conclusions Modal identification of a thin-walled member modeled by shell finite elements and analyzed to collapse is successfully demonstrated. The method extends the applicability of the constrained finite strip method to the domain of general finite element analysis of thin-walled members. The utility of the approach is demonstrated through a series of numerical examples of geometrically perfect and/or imperfect cold-formed steel lipped channel members under uniform end shortening. The results indicate how the participation of a given bucklingassociated deformation space (global, distortional, local, and/or shear and transverse extension) evolves under load. The results challenge current conventions as in all the studied lipped channel members, regardless of the eigen-buckling, or pre-peak classification, the post-peak collapse mechanisms have large distortional participations. Thus, even in a member entirely dominated by local buckling up to peak load, final failure is typically in a distortional dominated mechanism—a fact strongly accentuated when material nonlinearity is included in addition to geometric nonlinearity. Insights of this nature are only available with the type of quantitative modal identification explored here. Significant future work remains to further utilize this capability and determine how best to inform design methods based on these findings.

Acknowledgments This paper is based in part upon work supported by the U.S. National Science Foundation under Grant no. 0448707. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. References [1] Eurocode 3: design of steel structures, part 1.3: general rules. ENV 1993-1-3, European Committee for Standardisation (CEN); 1996. [2] AS/NZS, Cold-formed steel structures, AS/NZS 4600, Standards Australia/ Standards New Zealand; 1996.

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