Obtaining the modal participation of displacements, stresses, and strain energy in shell finite-element eigen-buckling solutions of thin-walled structural members via Generalized Beam Theory

Thin-Walled Structures 134 (2019) 148–158

Contents lists available at ScienceDirect

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

Full length article

Obtaining the modal participation of displacements, stresses, and strain energy in shell finite-element eigen-buckling solutions of thin-walled structural members via Generalized Beam Theory

T

Junle Cai Department of Civil and Environmental Engineering, Virginia Tech, 200 Patton Hall, Blacksburg, VA 24061, USA

ARTICLE INFO

ABSTRACT

Keywords: Thin-walled structures Shell finite-element Eigen-buckling Modal decomposition Strain energy

This paper presents a method to calculate modal displacement, stress, and strain energy participation in shell finite-element eigen-buckling solutions of thin-walled structural members using Generalized Beam Theory (GBT). The method provides quantitative information that can be used to interpret coupled buckling in structural designs. A finite-element (FE) eigen buckling solution is transformed to a GBT solution, and equivalent GBT modal amplitudes representing the FE solution are retrieved. The modal displacement field, stress tensor and strain energy are retrieved using GBT modal amplitude field and applying GBT constitutive relationships between strain and stress. Theory and examples are provided.

1. Introduction In recent years, the popularity of thin-walled structural sections has increased dramatically. Thin-walled structures save material and labor costs by using lightweight, high strength-to-weight ratio members that are relatively easy to erect. As a trade-off, local stability of the “thinwall” is commonly found coupled with member-wise Euler buckling. Understanding and developing design methods for thin-wall structures considering failure under the influence of more than one buckling mode has been a challenge for researchers and designers. This paper aims to shed light to the topic by employing a method that will enable the calculation of modal displacement, stress, and strain energy participation for thin-walled structural members from an existing shell finite element eigen-buckling solution using the Generalized Beam Theory (GBT). In addition, the method is also meant to facilitate the strength prediction of thin-walled members using the Direct Strength Method [1,2]: buckling modes yielded by finite-element software need to be categorized as local, distortional and global buckling in order to be used by the Direct Strength Method. This paper will provide an automated and quantitative way for that task. A typical GBT stability analysis calculates displacement, stress and strain energy participation for elastic eigen-buckling (e.g., [3]), elastic post-buckling (e.g., [4]), plastic bifurcation (e.g., [5]) and plastic collapse (e.g., [6]) of a thin-walled member. The buckling mode decom-

position method presented herein is different as it works backward from a 3D deformation shape to quantify modal participation. Buckling mode decomposition methods have been developed based on GBT and constrained finite strip method (cFSM) on prismatic members. [7] used cFSM basis function and conducted mode decomposition by reconstructing a displacement field in the cFSM basis space. [8] documented the dominant deformation modes in simulations to collapse using the aforementioned cFSM decomposition method. [9] used the GBT stiffness matrices to calculate the modal participation with success and inspired this manuscript. It is thought that both the methods in [9] and in this manuscript are adequate for the displacement field decomposition. [10] brought out a modal decomposition method by using the GBT mode shapes as basis functions. The extension made in this manuscript's method is that it applies on stress components and strain energy, while the previous work is only applicable to a displacement field decomposition. The modal decomposition method presented is inspired by the existing GBT body of knowledge. GBT modal amplitudes at discrete crosssections are retrieved by using in-plane displacements from a shell finite element solution. The continuous modal amplitude field is reconstructed by assuming that the modal amplitudes can be piece-wisely approximated by Hermitian and Lagrangian polynomials. Stresses and strains then follow using the usual GBT equations. Verification and examples are provided.

E-mail address: [email protected]. https://doi.org/10.1016/j.tws.2018.06.026 Received 8 March 2017; Received in revised form 8 April 2018; Accepted 20 June 2018 0263-8231/ © 2018 Elsevier Ltd. All rights reserved.

Thin-Walled Structures 134 (2019) 148–158

J. Cai

2. Review: GBT eigen-buckling analysis for thin-walled members Generalized Beam Theory describes the cross-sectional displacement field of a prismatic member as a linear combination of the GBT cross-sectional mode shapes. For a thin-walled prismatic member as shown in Fig. 1, with the local coordinate system (x , s , z ) and the corresponding displacement components (u, v, w ) , these mode shapes are found by performing a GBT cross-sectional analysis [11–15]. For the case of open cross-sections and eigen-buckling analyses, the assumptions deemed suitable are: (i) Kicrhhoff-Love hypotheses and Vlasov's assumption ( sxM = 0) ; (ii) the plates of the beam are transversely inextensible ( ssM = 0) , where (·) M denotes the membrane terms; and (iii) the plates of the beam are subject to a plane stress state for the bending terms and an uniaxial stress state for the membrane terms. In this case, in each plate, the vk functions are constant and the warping functions uk are linear or constant. The resulting deformation modes are herein designated as the “conventional modes”. The GBT cross-sectional analysis is a three step process. First, a cross-section is discretized by ‘natural nodes’ and ‘intermediate nodes’, where the ‘natural nodes’ are the corners and ends of the cross-section and the ‘intermediate nodes’ are the end nodes and discretization nodes in between the natural nodes, as illustrated by Fig. 2(a). End nodes are considered both natural nodes and intermediate nodes. Second, the admissible modal space is computed based on the assumptions (i)∼(iii) above. Third, the orthogonal modes are obtained by solving a set of eigenvalue problems [11]. The GBT cross-sectional modes are denoted by continuous functions uk (s ) , vk (s ), and wk (s ), and the displacement field of the mid-line of a cross-section at a distance x is a linear combination of GBT mode shapes

u (s, x ) = uk (s )

Fig. 1. Coordinate systems of a thin-walled member. M xx F xx F ss F xs

Cik

(1)

= u, x = (uk

ss

= v,s =

zwk, ss

zwk )

Xpik =

0

1

2

0

0

0

G

, (4)

(DikI

DikII

DkiII )

k, xx

+ Bik

k

W p0 Xpik

k , xx

= 0,

(5)

b

Etui uk ds +

b

Et 3 12(1

2)

wi wk ds,

b

(6)

b

E

up Cpp

vi vk + wi wk ds.

(7)

where up is the GBT mode shape (warping) corresponding to axial compression ( p = 1), major axis bending ( p = 2), minor axis bending ( p = 3), and torsional warping ( p = 4 ); and Cpp/ E is the cross-sectional property of area ( p = 1), major axis moment of inertia ( p = 2), minor axis moment of inertia ( p = 3), warping moment of inertia ( p = 4 ). Eq. (5) is an ordinary differential equation system with the unknowns being k . With boundary conditions defined, the weak form of this system can be piece-wise approximately solved by the finite element method. The functions k (x ) (conventional modes except for axial extension) are approximated by the Hermite polynomials

k,

i, x wk k, x )/2,

2

1

2

E

pth

k , xx ,

+ wi

1

M xx F xx F ss F xs

where b (·) ds denotes integration along the center line of the crosssection, t is the thickness of the plates. Xpik is a third order tensor defined by:

(2)

2zwk, s k, x , xs = u, s + v , x = NL 2 2 xx = (v , x + w , x )/2 = (vi i, x vk k , x

2

E

0 0

Gt 3 wi, s wk, s ds, 3 Et 3 DikII = b w w ds, 2 ) i k, ss 12 (1 Et 3 Bik = b w w ds, 2 ) i, ss k , ss 12 (1 DikI =

where (U , V , W ) and Uk (s ) , Vk (s ) , Wk (s ) are displacements and GBT cross-sectional modes shapes in the global coordinate system. The first nine (most relevant) conventional mode shapes for a C-section are shown in Fig. 2. The GBT discretization is made identical with the shell finite element models in Section 4 of this paper, following the fashion in [9]. Arguably the fine discretization only slightly increases the computational time, because the governing GBT eigen-buckling equation, Eq. (5), needs not to be solved for in the method in this paper. The GBT strain-displacement relations read xx

E

1

0

k, xxxx

Cik =

k , x (x ),

V (s, x ) = Vk (s ) k (x ), W (s, x ) = Wk (s ) k (x ),

0

E

where the subscripts i, k correspond to the conventional GBT modes; W p0 is the force resultant corresponding to axial compression ( p = 1), major axis bending ( p = 2 ), minor axis bending ( p = 3), and torsional warping ( p = 4 ). Tensors denoted by C, D and B are calculated using the GBT mode shapes, cross-sectional dimensions and material properties:

where standard summation notation applies; ( ), x = ( )/ x ; k (x ) is the modal amplitude of the k th GBT mode at length x along the member; and the derivative k, x in the first term is present due to Vlasov's assumption [10]. At length x along the member; and the derivative k, x in the first term is present due to Vlasov's assumption [11]. The displacement field in Eq. (1) can also be rewritten in the global coordinate system (X , Y , Z ) as

U (s, x ) = Uk (s )

=

0

where E, , G are elastic modulus, Poisson's ratio, and shear modulus, respectively; and ij , ij are stress and strain components, (·) M denotes the membrane terms and (·) F denotes the plate flexural terms. The GBT differential equation system of equilibrium can be derived by using the Principle of Virtual Work and it reads

k , x (x ),

v (s , x ) = vk (s ) k (x ), w (s, x ) = wk (s ) k (x ),

E 0

(3)

k

in which stands for the nonlinear normal strain component essential for forming the geometric stiffness matrix. The stress components are related to the strains by the constitutive law NL xx

=

1 dk.1

+

2 dk.2

+

3 dk.3

+

4 dk.4 .

(8)

The function 1, x (x ) (axial extension mode) is approximated by the Lagrange polynomials 1, x

149

=

1 d1.1

+

2 d1.2

+

3 d1.3

+

4 d1.4 .

(9)

Thin-Walled Structures 134 (2019) 148–158

J. Cai

60 mm

(a)

10 mm

(b)

1

2

3

4

5

(c)

1

2

3

7

8

9

7

6

8

9

E = 2.1e5 MPa ν = 0.3 t = 2 mm Natural node Intermediate node End node (Natural +intermediate)

120 mm

4

5

6

10 mm 60 mm

*Modes not drawn to scale

Fig. 2. GBT mode shapes: (a) cross-section dimensions and discretization; (b) in-plane components of GBT modes; (c) warping components of GBT modes.

In (8) and (9) 1 4 and 1 4 are the shape functions of the Hermite and Lagrange polynomials as shown in Eq. (10) and Fig. 3; and dk . r stands for the r th FE degree of freedom (generalized nodal displacement) in the approximation of the k th modal amplitude, also shown in Fig. 3.

= { Le ( Z=

3

2

2

1 1 2 2

3

1 2

3

+ ) 2

3

1 1

3

2

3

3

+ 1 Le ( 2

2

9 2

3

2)

1

3

2 1

3

3. Calculation of buckling mode participation of displacements, stress components and strain energy 3.1. Reconstruction of discrete GBT modal amplitude The first task of this paper is to decompose the SFEA eigen-buckling displacement field into contributions of GBT modes. Revisiting Eq. (2), the displacement composition is revealed if the GBT modal amplitudes are known. At any given cross-section, the GBT modal amplitudes other than the first mode of uniform compression (Fig. 2), can be found by solving the least squares problem in Eq. (14):

+ 3 2}, 9 2

1

3

T,

= x / Le ,

Vk Wk

(10) In a linear stability analysis, the axial mode can be removed from the analysis and the FE eigen-buckling problem is defined as

where kik , with i , k > 1, reads

k ik = Cik +

dx + DikI

T Le

DkiII

T

dx + Bik

Le

dx + DikII

T Le T Le

dx ,

T Le

dx (12)

In Eq. (12), (·) represents a derivation with respect to x. The geometric stiffness matrix g ij is given by

gik = Wp Xpik

T Le

dx ,

2n × (m 1)

k (x )} (m 1) × 1

least square

=

V W

, 2n × 1

(14)

in which (i) Vk , Wk are displacement components at the discretization nodes of GBT mode k in the global coordinate system, where all the modes generated by the GBT cross-sectional analysis, except for mode 1, are included; (ii) k (x ) is the modal amplitude vector of mode k at location x along the member, (iii) V, W are the in-plane displacements at the GBT discretization nodes in the global coordinate system (Fig. 1) obtained by shell finite element analysis (SFEA) - if the SFEA discretization nodes coincide with the GBT ones, then V and W can be directly read, otherwise, they can be obtained by knowing the SFEA shape functions and the nodal displacements; (iv) the subscripts denote the dimensions of the matrices with n being the number of cross-sectional discretization nodes and m being the number of GBT modes; (v) the warping displacements U are not used. Eq. (14) requires no GBT knowledge other than the GBT mode shapes as shown in Fig. 2. It is applicable to any loading and boundary conditions, and simple to program. Eq. (14) finds the nearest point in the GBT admissible space

(11)

(kik + gik){dk} = {0},

{

(13)

Fig. 3. Shape functions and FE degrees of freedom of (a) Hermite polynomials and (b) Lagrange polynomials.

150

Thin-Walled Structures 134 (2019) 148–158

J. Cai

Fig. 4. Concept of using polynomials to piece-wisely approximate the modal amplitude

k.

warrants knowing continuous GBT modal amplitude. To reconstruct the continuous modal amplitude field, it is assumed that k (x ) can be piecewisely approximated by polynomials like those shown in Eq. (8). The member is divided into several sub-domains each containing four discrete cross-sections, as shown in Fig. 4. Because Hermitian polynomials are used to approximate k 1 (x ), Eq. (8) applies and k (0)

b

u (x )·u1 (s )· tds A

=

b

u (x )·1·tds A

=

b

u (x ) tds A

,

+

2 (0) dk .2

+

3 (0) dk .3

+

Le

4 (0) dk .4 ,

Le

k

that best approximates the displacement field read from SFEA. It is thought the error of the method comes from the transversely-inextensible assumption, as the left side of Eq. (14) satisfies the assumption, and the right side, as the output of SFEA, does not. Eq. (14) retrieves the GBT modal amplitudes of mode 2 to the last one. The first mode amplitude of uniform extension can be retrieved as the average warping of the cross-section, since the GBT modes are decoupled in terms of warping displacements:

=

1 (0) dk .1 Le

Le = 1 dk .1 + 2 dk .2 + 3 dk .3 + 4 dk .4 , 3 3 3 3 3 e e e e 2L 2L 2L 2L 2Le = 1 dk .1 + 2 dk .2 + 3 dk .3 + 4 dk .4, k 3 3 3 3 3 e e e e e k (L ) = 1 (L ) dk .1 + 2 (L ) dk .2 + 3 (L ) dk .3 + 4 (L ) dk .4 ,

Fig. 5. For the torsional mode, the modal amplitude 4 (x ) represents the torsional angle, whilst for the other GBT modes k (x ) represents the maximum inplane displacement.

1, x (x )

=

Le

(16) where denotes the domain under consideration; k ( is the modal amplitude calculated by Eq. (14) at Le ; i ( Le ) corresponds to the value of the ith Hermitian shape function (Eq. (8), Fig. 3) at the location Le ; dk, i denotes the ith generalized nodal displacement of the k th GBT mode (Eq. (8), Fig. 3). dk can be solved for by using Eq. (17) and the continuous distribution of k is approximated. The same operation is performed to retrieve the first modal amplitude 1, x (x ) , only replacing the Hermitian polynomials in Eq. (16) by Lagrangian polynomials as in Eq. (8) and Fig. 3. The concept is depicted in Fig. 4.

Le

(15)

where (i) b means the integration across the cross-section at location x, (ii) u (x ) is warping displacement (Fig. 1) obtained by SFEA, (iii) A is the cross-sectional area.

dk.1 dk.2 = dk.3 dk.4

3.2. Reconstruction of continuous GBT modal amplitude field By retrieving the modal amplitudes at discrete cross-sections, the inplane displacements – (v, w ) or (V , W ) in Fig. 1 – can be decomposed into modal contributions by knowing Eq. (1) and (2). The objectives of this paper include the decomposition of displacements, strain energy and stresses regarding the contribution from GBT modes, which

Le)

1 (0)

3 (0)

4 (0)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1

2 (0)

Le 3

2

2Le 3

2

e 1 (L )

Le 3

3

2Le 3

3

e 2 (L )

Le 3

4

2Le 3

4

e 3 (L )

Le

1

k (0)

( ) ( )

3

k

Le 3

2Le 3

k

2Le 3

e 4 (L )

e k (L )

. (17)

The continuous modal amplitude field is retrieved by using Eq. (8)∼ Eq. (9) rewritten hereafter: k (x )

=

1 (x )· dk.1

+

2 (x )·dk.2

+

3 (x )· dk.3

+

Fig. 6. Example I: dimensions and loading/boundary conditions of a classical ‘pinned-pinned’ column. 151

4 (x )·dk.4 .

(18)

Thin-Walled Structures 134 (2019) 148–158

1.5 1 0.5 (a) φi 0 -0.5 -1 -1.5

(b)

0

Modal Particiaption

J. Cai

180 360 540 720 900 1080 Length (mm)

1 5 0.5

0

7 mode 7 mode 5

0

180 360 540 720 900 1080 Length (mm)

Fig. 7. Example I: GBT buckling mode amplitudes and modal participation: three distortional half-wave mixed with multiple local buckling half-waves.

Fig. 8. Example I: Longitudinal flexural stress and displacement decomposition: (a) GBT reconstruction; (b) contribution from mode 5 and (c) contribution from mode 7: note the distortional mode contributes significantly for the minimum stress but small for the maximum stress.

1, x (x )·= 1 (x )·d1.1

+

2 (x )· d1.2

+

3 (x )·d1.3

+

4 (x )· d1.4 .

(19)

3.3. Decomposition of eigen-buckling displacement field, strain energy and stress components The displacement field of a shell finite element eigen-buckling analysis is decomposed by using the continuous GBT modal amplitude field and the GBT displacement field superposition rule in Eq. (2):

U (s, x ) = Uk (s )

k , x (x ),

V (s, x ) = Vk (s ) k (x ), W (s, x ) = Wk (s ) k (x ),

(20)

where standard summation convention applies, U (s, x ) is the warping displacement and V (s, x ) , W (s , x ) are the in-plane displacements; Uk (s ) , Vk (s ) , Wk (s ) denote GBT modes shapes in the global coordinate system; and k (x ) is the reconstructed GBT modal amplitude field in Eq. (18)∼ Eq. (19). The energy stored in the interaction of GBT modes i and k is

1 di kik dk, 2 1 Eki = dk kki di , 2

Fig. 9. Example I: strain energy decomposition: noticeably most energy is stored in the modes in the diagonal and not in the interaction between modes, indicating that the GBT modes are highly decoupled for this particular problem.

Eik =

(21)

152

Thin-Walled Structures 134 (2019) 148–158

J. Cai

Fig. 10. Example I: Transverse bending stress comparison: (a) GBT reconstruction; (b) SFEA: the modal decomposition process successfully replicates the stress distribution, location and magnitude of the extreme values.

Fig. 11. Example I: Longitudinal bending stress comparison: (a) GBT reconstruction; (b) SFEA: note the stress is retrieved at one side of the plate, if the other side is retrieved, the value will be opposite in sign.

where kik stands for the linear, interactive stiffness matrix of modes i and k as shown in Eq. (12); di , dk are the generalized nodal displacement vectors for the domain where the piece-wise approximation is applied. Obviously, Eii stands for the energy stored in mode i. And the total strain energy is

3.4. Error of the reconstructed displacement field

Etotal =

error =

Eik . i

The error of the reconstructed displacement field, warping displacement and in-plane displacement components can be evaluated respectively by

(22)

k

The strain energy can be used as an unambiguous modal participation indicator as it is independent with the GBT mode normalization. The transverse/longitudinal flexural stresses and longitudinal membrane (warping) stress, decomposed, according to Eq. (1), (3), (4), are F ss

x, s, z =

F xx

x, s, z =

M xx (x ,

E

E 1

s ) = Euk (s )

wk, ss (s )

k (x ) z

w (s ) 2 k , ss

k (x ) z

2

1

k, xx (x ).

errorVW =

E

E 1

wk (s )

k , xx (x ) z ,

w (s ) 2 k

k , xx (x ) z ,

2

1

errorU =

(dFE

[(UFE

(dFE

dreconst )T ·(dFE T (dFE ·dFE )

dreconst )]

,

(26)

Ureconst )2] , 2 (UFE ) inplane

(27) T inplane ) ·(dFE inplane T (dFE inplane ·dFE inplane )

dreconst

dreconst

inplane )]

(23)

(28)

(24)

where means it applies to all the discretization nodes; dFE is the displacement component vector (U , V , W ) of a discretization node read from shell finite element solution; dreconst is the reconstructed displacement vector calculated by using k reconstructed and Eq. (20). dFE inplane and dreconst inplane are the in-plane components of dFE and dreconst . The reason errorU and errorVW are evaluated separately is that the warping

(25) 153

Thin-Walled Structures 134 (2019) 148–158

J. Cai

Fig. 12. Example I: Warping stress comparison: (a) GBT reconstruction; (b) SFEA: the trend and locations of maximum and minimum are closely picked up, though the difference is not neglectable - see the reasoning followed.

Fig. 13. Example II: dimensions and loading/boundary conditions of the member: left-end U = V = W = 0 , right-end pinned-warping free and loaded by uniform compression.

Fig. 14. Example II: GBT buckling mode amplitudes and modal participation: local mode along the member with nine half-waves and magnitude increase from left to right mixed with two distortional buckling half-waves; the local mode dominates the structural response.

displacements U are generally much smaller than V and W.

Pk

3.5. Modal displacement, strain energy and stress participation

displacement (x )

=

|gk k (x )| , n |g (x )| k=1 k k

=

p dx L k displacement n p dx k = 1 L k displacement

,

(30)

where pk displacement is cross-sectional modal displacement participation per Eq. (29).

3.5.1. Modal displacement participation The modal displacement participation measures the modal contribution for the in-plane displacements. The cross-sectional modal displacement participation is calculated as

pk

displacement

3.5.2. Modal strain energy participation The energy participation of GBT mode k is

(29)

Pk

where g4 equals the largest distance from the shear-center to any crosssectional point for the torsional mode, e.g., mode 4 in Fig. 2, and gk = 1 for all the other modes (see Fig. 5). This conversion from the twist angle to displacement produces modal participation factors that are consistent for all modes, including torsion. The member-wise modal inplane displacement participation is calculated as in [16]:

energy

=

Ekk , Etotal

(31)

and the energy participation corresponding to the interaction of the GBT mode k and i is

Pki

154

energy

=

Eki + Eik . Etotal

(32)

Thin-Walled Structures 134 (2019) 148–158

J. Cai

maximum: 264.3 MPa location @ (x,s,z)=(50,130,1) mm

(a) GBT reconstruction

250 200

minimum: -185.6 MPa location @ (x,s,z)=(50,190,1) & (50,70,1) mm

150

q

contribution of mode 7: 252.4 MPa/264.3 MPa (99.2%)

(b) Mode 7 contribution

100

=

contribution of mode 7: -185.6 MPa/-185.6 MPa (100%)

50 q

(c) Remaining modes contribution

0 -50

+

-100

x q

-150 F σss (MPa)

Fig. 15. Example II: transverse bending stress and displacement decomposition: (a) GBT reconstruction; (b) contribution from mode 7 and (c) contribution from the other modes: the 'local' mode 7 contributes over 99% of most extreme stresses. The other stress components can be decomposed as well despite not shown herein.

the discretization is made every 10 mm. The method is applicable on eigen-buckling solutions. In order to show the results, the eigen-buckling solutions are normalized in the following examples such that the maximum displacement equals one, i.e., 2 2 2 max( UFE + VFE + WFE ) = 1.

(34)

4.1. Example I, a pinned-pinned column The first example involves a 1080 mm long member pinned-warping free at both ends and loaded by compression force as shown in Fig. 6. Warping is prevented from the mid-length cross-section. The lowest buckling load is 143.2 kN and the displacement components (U , V , W in Fig. 1) of all the discretization nodes are read from ABAQUS for the modal decomposition analysis. The modal amplitude and displacement participation along the member are presented in Fig. 7. It is shown that the dominant GBT mode is distortional mode 5 mixed with local mode 7. There are three distortional buckling half-waves and multiple local buckling ones. The displacement and the longitudinal flexural stress field can be decomposed into modal contributions as shown in Fig. 8. It can be seen that the maximum stress of 47.4 MPa occurs at the peaks of the first local half-wave near the column ends, and the minimum of 33.7 MPa occurs at the mid-length cross-section. The local mode 7 contributes 74.9% of the maximum stress while the distortional mode 5 contributes 65.0% of the minimum stress. These pieces of information would not be easy to obtain without using the modal decomposition algorithm presented. The displacement field errors by using Eq. (26) – (28) are error = 0.1% , errorU = 3.3% and errorVW = 0.06%. Using Eq. (30), the member-wise displacement participation reads P5 = 88.1% and P7 = 10.0% for mode 5 and 7 respectively. From the strain energy perspective, the total strain energy is computed to be 948.7 N·mm, comparing to the ABAQUS reading of 905.3 N·mm (4.6% difference) as shown in Fig. 9, where strain energy decomposition is conducted per Eq. (21) and the grid (i, j) indicates the strain energy stored in the interaction of mode i and j. Mode 5 is responsible for 89.6% of the total energy per Fig. 9. In order to verify the modal decomposition process, the re-constructed stress fields and those read from SFEA are compared and

Fig. 16. Example II: strain energy decomposition: more than 94.2% of the total elastic strain energy is stored in mode 7, a ‘ local’ mode.

3.5.3. Modal stress participation The stress participation of GBT mode k at any material point is

pk

stress

x , s, z =

k (x ,

s, z ) , (x , s , z )

(33)

where (x , s, z ) is any stress component inspected, for example, M xx (x , s , z ) . k (x , s , z ) is the contribution from the mode k, calculated per Eq. (4). 4. Illustrative examples Model decompositaion is applied on shell finite element analysis (SFEA) results yielded by ABAQUS [17]. S4R elements are used and all the elements are rectangular with an aspect ratio between 1:1 and 2:1. In the cross-sectional directions (Y,Z in Fig. 1), the discretization is identical with the GBT shown in Fig. 2. In the longitudinal direction, 155

Thin-Walled Structures 134 (2019) 148–158

J. Cai

(a) GBT reconstruction

250

min

maximum: 264.3 MPa (2.8% error) location @ (x,s,z)=(50,130,1) mm

s

x

200

max.

min s

s

150

minimum: -185.6 MPa (0.6% error) location @ (x,s,z)=(50,190) & (50,70,1) mm

100 q

(b) SFEA

50

maximum: 257.2 MPa location @ (x,s,z)=(50,130,1) mm

0 -50

minimum: -184.5 MPa location @ (x,s,z)=(50,190,1) & (50,70,1) mm

-100 q

-150 F

σss (MPa) Fig. 17. Example II: Transverse bending stress comparison: (a) GBT reconstruction; (b) SFEA: the minimum stress is retrieved at the intersections of flange and web while maximum stress at the mid-height of the web, both occur at the cross-section corresponding to the peak of mode 7.

maximum: 289.4 MPa (0.6% error) location @ (x,s,z)=(40,130,1) mm

(a) GBT reconstruction x

250 s

max. & min. s s

minimum: -203.8 MPa (2.6% error) location @ (x,s,z)=(140,130,1) mm (b) SFEA

200 150 100

q maximum: 287.6 MPa location @ (x,s,z)=(40,130,1) mm

50 0 -50

minimum: -198.6 MPa location @ (x,s,z)=(140,130,1) mm

q

-100 -150 F

-200 σxx (MPa) Fig. 18. Example II: Longitudinal bending stress comparison: (a) GBT reconstruction; (b) SFEA: the maximum and minimum stress both occur at the mid-height of the web, but at the peaks of the largest and second largest buckling half-waves respectively.

shown in Fig. 10–12. In the comparison, the SFEA warping stress is read from the mid-plane of the plates from ABAQUS, and the SFEA flexural stress of extreme fibers are calculated by subtracting the stress of the mid-plane from that of the extreme fiber. It can be seen that the overall magnitude and distribution of reconstructed flexural stress fields are consistent with the SFEA results. Take Fig. 10 for example, the reconstructed maximum and minimum transverse bending stresses agree with those of SFEA, being 47.4 MPa vs. 45.9 MPa and −33.7 MPa vs. −33.0 MPa (errors 3.3% and 2.1%). The reconstructed warping stress is compared with the SFEA readings in Fig. 12. The values of the reconstructed maximum and minimum stresses are 127.2 MPa and -123.3 MPa respectively, comparing with the SFEA readings of 86.7 MPa and -84.4MPa (error 46.7% and 46.1%). The reason is suspected to be that GBT assumes the warping displacement (u in Fig. 1) varies linearly in each plate, as shown in Fig. 2(c), which may not be accurate enough in this example.

4.2. Example II, a fixed-pinned column The second example is an 870 mm long member as shown in Fig. 13. The member is warping-fixed at the left end and pinned-warping free at the right. It is loaded with a compressive shell edge load q at the pinned-warping free end. The cross-section, longitudinal discretization size and material properties are identical to the previous example. The reconstructed modal amplitudes and participation along the member are retrieved by using Eq. (14) and (17) and shown in Fig. 14. The errors of the reconstructed displacement field are error = 0.4%, errorU = 25.4%, errorVW = 0.003% according to Eqs. (26)–(28). Again, the error of the warping displacement may come from the linear warping assumption of GBT. Local mode 7 is dominant as shown in Fig. 14 and the variation of its amplitude is consistent with the buckling shape in Fig. 15: nine buckling half-waves whose magnitude decrese from the right end to the left is clearly shown by model amplitude in Fig. 14(a). 156

Thin-Walled Structures 134 (2019) 148–158

J. Cai

maximum: 43.8 MPa (52.1%) difference location @ (x,s)=(60,250 & 10) mm

max. s

s (a) GBT reconstruction

s

x minimum: -78.3 MPa (4.2% error) location @ (x,s)=(60,260 & 0) mm

q

40

min. 20

0

-20

maximum: 28.8 MPa location @ (x,s)=(50,250 & 10) mm

-40 (b) SFEA

q

x minimum: -81.7 MPa location @ (x,s)=(50,260 & 0) mm

-60 M

σxx (MPa)

-80

Fig. 19. Example II: Warping stress comparison: (a) GBT reconstruction; (b) SFEA: the error for the minimum stress is acceptable but not the maximum stress.

For this example the transverse bending stress is dominated by mode 7 (a ‘ local’ mode) as shown in Fig. 15, as it is responsible for over 99% of maximum and minimum stresses. For member-wise displacement participation, they read P7 = 70.9% and P5 = 24.9% by using Eq. (30). The total strain energy is computed to be 1518.8 N·mm according to Eq. (22), comparing well to ABAQUS yielding of 1468.4 N·mm (3.4% difference). The strain energy distribution is shown in Fig. 16. It is confirmed that energy-wise mode 7 dominates as it contributes to 94.2% of the strain energy. From Figs. 17–18, the bending stress fields reconstruction is successful and the stress distributions are consistent with the SFEA results. The locations of the reconstructed maximum and minimum bending stresses coincide exactly with those read from SFEA, and the values are acceptably close, as the maximum error is 2.8%. The reconstructed warping stress is compared with the SFEA readings in Fig. 19. The value of the reconstructed minimum stress is −78.3 MPa, while the reading from SFEA is −81.7 MPa (error is 4.2%), but the value of the reconstructed maximum stress has discrepancy with SFEA result being 43.8 MPa vs. 28.8 MPa (52.1% error). The suspected reason is already stated in the last example.

providing buckling mode categorization. In addition, the method can be useful in advancing our understanding of coupled buckling. Acknowledgements The author would like to acknowledge the contribution of this paper by Dr. Cristopher D. Moen, Dr. Ioannis Koutromano, Dr. R.H. Plaut of Virginia Tech, and Dr. Mihai Nedelcu of Universitatea Tehnică ClujNapoca, Romania. The author is grateful for the insights provided by Dr. Sándor Ádány of the Budapest University of Technology and Economics, and last but not least Dr. Dinar Camotim, Dr. Nuno Silvestre and the Generalised Beam Theory Research Group at Lisbon, Instituto Superior Técnico, University of Lisbon, Portugal. References [1] B.W. Schafer, Review: the direct strength method of cold-formed steel member design, J. Constr. Steel Res. 64 (7) (2008) 766–778 〈http://www.sciencedirect. com/science/article/pii/S0143974X08000345 〉. [2] Iron A. Steel Institute. AISI S100-2016. North American Specification for the Design of Cold-formed Steel Structural Members. 2016. [3] D. Camotim, N. Silvestre, R. Gonçalves, P.B. Dinis, GBT analysis of thin-walled members: new formulations and applications, Thin-Walled Struct.: Recent Adv. Future Trends Thin-Walled Struct. Technol. (2004) 137–168. [4] N. Silvestre, D. Camotim, Nonlinear generalized beam theory for cold-formed steel members, Int. J. Struct. Stab. Dyn. 3 (04) (2003) 461–490 〈http://www. worldscientific.com/doi/abs/10.1142/S0219455403001002 〉. [5] R. Gonçalves, D. Camotim, Thin-walled member plastic bifurcation analysis using generalised beam theory, Adv. Eng. Softw. 38 (8) (2007) 637–646 〈http://www. sciencedirect.com/science/article/pii/S0965997806001992 ). [6] R. Gonçalves, D. Camotim, Geometrically non-linear generalised beam theory for elastoplastic thin-walled metal members, Thin-Walled Struct. 51 (2012) 121–129, https://doi.org/10.1016/j.tws.2011.10.006 (ISSN 02638231, URL 〈http:// linkinghub.elsevier.com/retrieve/pii/S026382311100231X 〉). [7] S. Ádány, A.L. Joó, B.W. Schafer, Buckling mode identification of thin-walled members by using cFSM base functions, Thin-Walled Struct. 48 (10–11) (2010) 806–817, https://doi.org/10.1016/j.tws.2010.04.014 (ISSN 02638231, URL 〈http://linkinghub.elsevier.com/retrieve/pii/S0263823110000856 〉). [8] Z. Li, S. Ádány, B. Schafer, Modal identification for shell finite element models of thin-walled members in nonlinear collapse analysis, Thin-Walled Struct. 67 (2013) 15–24, https://doi.org/10.1016/j.tws.2013.01.019 (ISSN02638231, URL 〈http:// linkinghub.elsevier.com/retrieve/pii/S0263823113000207 〉). [9] M. Nedelcu, GBT-based buckling mode decomposition from finite element analysis of thin-walled members, Thin-Walled Struct. 54 (2012) 156–163, https://doi.org/ 10.1016/j.tws.2012.02.009 (ISSN0263-8231, URL 〈http://www.sciencedirect. com/science/article/pii/S0263823112000390 〉). [10] J. Cai, C.D. Moen, Automated Buckling Mode Identification of Thin-walled Structures from 3d Mode Shapes or Point Clouds, in: Proceedings of the Annual Stability Conference, USA, Nashville, TN, 2015. [11] R. Schardt, Verallgemeinerte Technische Biegetheorie, Springer, Berlin, 1989. [12] R. Gonçalves, M. Ritto-Corrêa, D. Camotim, A new approach to the calculation of cross-section deformation modes in the framework of generalized beam theory, Comput. Mech. 46 (5) (2010) 759–781 〈http://link.springer.com/article/10.1007/

5. Conclusions Buckling mode decomposition of displacements, stresses, and the strain energy of thin-walled structural member eigen-buckling solutions are realized by transforming an SFEA solution into a GBT one. The method first reads the in-plane displacements from SFEA and uses a least-square criteria to obtain the approximate GBT modal amplitudes at discrete cross-sections. Then, the method reconstructs the continuous GBT modal amplitude field piece-wisely using polynomials. The normalized displacement field, stresses and strain energy are all decomposed using the reconstructed GBT modal amplitude field. Two illutrative examples are provided for modal decomposition. In order to verify the method, the reconstructed displacement fields and energy are compared against SFEA results. The displacement field and energy errors are small, reading 0.1% and 4.6% (displacement and energy, respectively) for Example I, and 0.4% and 3.4% (displacement and energy, respectively) for Example II. The reconstructed stress fields are compared to SFEA results, showing satisfactory replication in terms of overall stress distribution, extreme stress values (generally less than 5% error) and locations of extreme stress occurrence. However, the discrepancies of 46.1% and 52.1% of the maximum or minimum warping stresses were found in Examples I and II. It is suspected that the linear warping assumption of GBT causes these discrepancies. The method facilitates strength prediction per the Direct Strength Method by 157

Thin-Walled Structures 134 (2019) 148–158

J. Cai s00466-010-0512-2 〉. [13] N. Silvestre, D. Camotim, N.F. Silva, Generalized beam theory revisited: from the kinematical assumptions to the deformation mode determination, Int. J. Struct. Stab. Dyn. 11 (05) (2011) 969–997, https://doi.org/10.1142/S0219455411004427 (ISSN 0219-4554, 1793-6764, URL 〈http://www.worldscientific.com/doi/abs/10. 1142/S0219455411004427 〉). [14] Nuno Silva, Behaviour and Strength of Laminated FRP Composite Structural Elements (Ph.D. thesis), Instituto Superior Técnico, Technical University of Lisbon, 2013.

[15] R. Bebiano, R. Gonçalves, D. Camotim, A cross-section analysis procedure to rationalise and automate the performance of GBT-based structural analyses, ThinWalled Struct. 92 (2015) 29–47, https://doi.org/10.1016/j.tws.2015.02.017 〈http://www.sciencedirect.com/science/article/pii/S0263823115000580 〉. [16] N. Silvestre, D. Camotim, Second-order generalised beam theory for arbitrary orthotropic materials, Thin-Walled Struct. 40 (9) (2002) 791–820 〈http://www. sciencedirect.com/science/article/pii/S0263823102000265 〉. [17] Simulia, Abaqus/CAE, Dassault Systemes Simulia Corp., RI, USA, 2012.

158