Buckling analysis of thin-walled sections under general loading conditions

ARTICLE IN PRESS Thin-Walled Structures 47 (2009) 730–739

Contents lists available at ScienceDirect

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

Buckling analysis of thin-walled sections under general loading conditions H.C. Bui Construction Faculty, University of Civil Engineering, 5 Giai Phong Street, Ha noi, Vietnam

a r t i c l e in f o

a b s t r a c t

Article history: Received 1 May 2008 Received in revised form 7 December 2008 Accepted 9 December 2008 Available online 4 February 2009

This paper presents an investigation of the buckling behaviour of thin-walled sections subjected to general loading conditions. The semi-analytical finite strip method is used. The existing results are only for sections subjected to a uniform loading, namely: uniform compression, uniform bending and uniform distributed loads, which are applied at the shear centre. For a general loading condition, we proposed the realizing linear analysis first to give longitudinal stresses. The stiffness matrix is provided in the standard manner. Each strip is divided into cells and longitudinal stresses are recorded in these cells. The integrations are performed on each cell domain and the sum of them provides the geometric matrix of the strip. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Finite strip Buckling analysis Local buckling Distortional buckling Lateral buckling General loading condition

1. Introduction The semi-analytical finite strip (FS) method is based on the harmonics functions, and proved to be an efficient tool for analysing prismatic structures. This method was pioneered by Cheung [1], who used the classical plate theory of Kirchhoff. Many authors have adopted and extended this method to analyse thinwalled sections. Hancock [2] presented the buckling curve of a thin-walled I-section. This relates to buckling stress with half wavelengths and enables a better insight into the elastic stability of a section. The distortional buckling phenomenon is also better understood. Schafer [3] utilised the Cheung’s 2-nodal line finite strip to develop CUFSM—a linear buckling analysis programme. The semi-analytical finite strip method is also applied in the nonlinear elastic and elastic–plastic studies [4–6]. However, it is difficult to properly take into account the interaction between local and global behaviour. The semi-analytical finite strip method can also be based on the Mindlin–Reissner plate theory. The latter permits to account for the shear effect through the plate thickness. Hinton [7] highlighted the dependence of the normalised critical stress on the thickness of isolated plates. Dawe et al. [8] and Wang and Dawe [9] analysed the nonlinear elastic response of thin-walled sections. The local and global behaviour is separately examined. Recently, Bui and Rondal [10] studied the buckling behaviour of highly stiffened thin-walled sections, for which the Mindlin– Reissner finite strips are more exact than the Kirchhoff ones.

E-mail addresses: [email protected], [email protected] (H.C. Bui). 0263-8231/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2008.12.003

In the buckling analysis, most of the previous works are limited to the members subjected to uniform compression or uniform bending (UB) or the combination of these two loading cases. Recently, Chu et al. [11] examined a particular case where stresses vary along the longitudinal axis of a channel section under uniform distributed loads (UDL), which are applied at the shear centre. Thus, the general loading condition is not considered yet. The objective of this paper is to propose an approach that permits to study the buckling behaviour of thin-walled sections under a general loading. The 2-nodal line semi-analytical finite strip, which is based on the Kirchhoff plate theory, is developed. Some thin-walled sections are examined to show the potential of the method.

2. Deformation–displacement relation The deformation–displacement relation in the finite strip can be predicted by the combination of the plane elasticity and the Kirchhoff plate theory @u @2 w z 2 , @x @x @v @2 w y ¼  z 2 , @y @y

x ¼

gxy ¼

@u @v @2 w þ  2z , @y @x @x@y

(1)

where u, v and w are the displacements in the x, y and z directions (Fig. 1a).

ARTICLE IN PRESS H.C. Bui / Thin-Walled Structures 47 (2009) 730–739

731

3. Formulation wðx; yÞ ¼

3.1. Displacement functions

uðx; yÞ ¼

r X

f H1

H2 g

m¼1

vðx; yÞ ¼

r X

f H1

m¼1

)

u2m (

H2 g

u1m

v1m v2m

Hy1

f Hw1

Hw2

m¼1

The three displacements of the strip at a point (x,y) can be expressed as [1] (

r X

sin

mpy , a

cos

mpy , a

)

(2)

H y2 g

9 8 w1m > > > > > > = <

y1m

w2m > > > > > > ; :

x x H1 ¼ 1  ; H2 ¼ , b b

(4)

σ1

σ2

y

z, w a

θ1

w2 θ2

u1

v2

x

x, u

u2 σ2

σ1

b

Fig. 1. (a) Finite strip and (b) strip with edge traction.

P M

M

Mmax = M

βM

M

(3)

in which the chosen longitudinal harmonic functions are the classic ‘simply supported’ boundary conditions; r is the number of harmonics considered; Hi, Hwi and Hyi are the Hermite shape functions

z

v1

mpy , a

y2m

y, v

w1

sin

Mmax = PL/4

P

P

−1 ≤ β ≤ 1 Mmax = PL/4

Mmax = M L qz

Mmax = qz L2/8

L Fig. 2. Some types of loading.

ARTICLE IN PRESS 732

H.C. Bui / Thin-Walled Structures 47 (2009) 730–739

3x2 2x3 3x2 2x3 Hw1 ¼ 1  2 þ 3 ; Hw2 ¼ 2  3 , b b b b  2    2x x2 x x þ 2 ; Hy2 ¼ x 2  . Hy1 ¼ x 1  b b b b

The strains can be calculated by substituting Eqs. (2) and (3) into Eq. (1): (5)

r X

fg ¼

(9)

½Bm fdm g,

m¼1

3.2. Stiffness matrix of a strip The stiffness of a strip can be derived from the strain energy Z Z 1 t=2 U¼ fgT ½Dfg dðaireÞ dz, (6) 2 t=2 Ae

in which typical written as 2 dHi s 6 6 dx m 6 ½Bim  ¼ 6 60 6 4 Hi km cm

terms for nodal line i and mth harmonic can be 2

0 Hi km sm dHi cm dx

d Hwi z sm dx2 2 zHwi km sm dH 2z wi km cm dx

3 2 d H yi z s 7 m 7 dx2 7 2 7, zHyi km sm 7 7 5 dHyi km cm 2z dx

where {e} represents the strains (which are given by Eq. (1)): fg ¼ f x

y gxy g,

(10) (7)

and [D] is the elasticity matrix 2 3 1 n 0 7 E 6 6n 1 0 7. ½D ¼ 1  n5 1  n2 4 0 0 2

where km ¼

(8)

mp ; a

sm ¼ sinðkm yÞ;

fdim gT ¼ f uim

vim

wim

yim g.

(12)

7 Buckling curve, m = 1-5 (UB) Critical curve (UB) Critical curve 1 (GM) Critical curve 2 (GM) Critical curve 1 (UDL) Critical curve 2 (UDL)

6

Mcr/My

5

σk1

(11)

The nodal displacements for nodal line i and mth harmonic is written as

y

cell k

cm ¼ cosðkm yÞ.

σk2

4 3

a

2 yk

1

yk-1

0

x 1

L/h = 5 102

103 Beam length (mm)

2 Fig. 3. Division of a strip in cells.

104

Fig. 5. Critical curves of simply supported I-section beam.

M

M

σmax

tw βM

M h

(β = 0)

E =2.05x105 N/mm2 ν = 0.3

qz

b = 200 mm tf = 10 mm tw = 5 mm

tf

fy = 275 N/mm2

b

L

Fig. 4. (a) I-section beam; (b) uniform bending (UB), gradient bending (GB), uniform distributed loads (UDL) and (c) stress distribution at the right end cross-section for the case of GB.

ARTICLE IN PRESS H.C. Bui / Thin-Walled Structures 47 (2009) 730–739

733

Fig. 6. Buckling shapes. (a) UB: L ¼ 3000 mm (local buckling) and L ¼ 10,000 mm (lateral buckling), (b) GM: L ¼ 3000 mm (local buckling) and L ¼ 10,000 mm (lateral buckling) and (c) UDL: L ¼ 3000 mm (local buckling) and L ¼ 10,000 mm (lateral buckling).

in which the mth sub-matrix can be given by Z t=2 Z ½K mm e ¼ ½Bm T ½D½Bm  dðareaÞ dz.

Table 1 Buckling moment of I-section beam. L (mm)

3000 6000 10,000

UM

GM

t=2

FS

FE

Error (%)

FS

FE

Error (%)

1.573 0.537 0.220

1.555 0.535 0.219

1.16 0.37 0.46

1.758 1.024 0.416

1.496 0.936 0.40

17.51 9.40 4.0

The stiffness matrix of the strip can be obtained by substituting Eq. (9) into Eq. (6). Note that for simply supported strips, each harmonic term is uncoupled, thus the global stiffness matrix consists of r sub-stiffness matrices, which have a diagonal location 2 3 ... 0 ½K 11 e 0 6 7 ½K 22 e . . . 0 60 7 7, ½Ke ¼ 6 (13) 6... ... ... ... 7 4 5 0 0 . . . ½K rr e

(14)

Ae

The explicit form of the stiffness matrix in Eq. (14) can be found in Ref. [1]. 3.3. Geometric matrix The geometric matrix can be derived from the potential energy done by the stresses through the nonlinear strains of the buckling displacements. For thin-walled sections, we consider only longitudinal stresses and other types of stresses are supposed to be ignorable. The potential energy can be calculated as "   2  2 # Z Z t a b @u 2 @v @w W¼ gðx; y; sÞ þ þ dx dy, (15) 2 0 0 @y @y @y where g(x,y,s) is a function of co-ordinates (x,y) and the longitudinal stresses. It is assumed that this function can be

ARTICLE IN PRESS 734

H.C. Bui / Thin-Walled Structures 47 (2009) 730–739

expressed for a strip as gðx; y; sÞ ¼ f 1 ðx; sÞ f 2 ðyÞ.

(16)

If the load is applied at the shear centre, the functions f1(x,s) and f2(y) can be explicitly defined. The function f1(x,s) has a common form x f 1 ðx; sÞ ¼ s1  ðs1  s2 Þ , b

(17)

where s1 and s2 are the compressive stresses at nodes 1 and 2, respectively (Fig. 1b). The function f2(y) depends on the type of loading (Fig. 2) and can be determined in some particular cases as follows: (1) uniform bending and/or uniform compression f 2 ðyÞ ¼ 1,

(18)

(2) gradient moments y f 2 ðyÞ ¼ ðb  1Þ þ 1 with  1pbp1, a (3) uniform distributed loads   y y2  2 , f 2 ðyÞ ¼ 4 a a

(19)

stresses are uniform, Chu et al. [11] determined the geometric matrix for the case of uniform distributed loads applied at the shear centre. We also provide the geometric matrix for the case of gradient moment in Appendix A.1. Note, in cases where the stress varies in the longitudinal direction, the element geometric stiffness matrices of different wave numbers are coupled to each other and have the form 2 3 ½K G11 e ½K G12 e . . . ½K G1r e 6 7 6 ½K G21 e ½K G22 e . . . ½K G2r e 7 7, ½K G e ¼ 6 (23) 6 ... ... ... ... 7 4 5 ½K Gr1 e ½K Gr2 e . . . ½K Grr e where [KGmm]e is the geometric matrix associated with wave number m, [KGmn]e ¼ [KGnm]e is the coupled geometric matrix associated with wave number m and n. However, in practice, the loading is more complex; the section can be warped when the load is not applied at the shear centre and a non-uniform torsion moment appears. In these cases, the

5 Central concentrated load Uniform bending Gradient bending Two concentrated loads at the one-fourths Uniform distributed loads

4.5

(20)

4

(4) central concentrated load a if 0pyp ; 2 a if pypa; 2

(5) two concentrated loads at the one-fourths 8 y a > if 0pyp ; 4 > > > a 4 > > < a 3a if pyp ; f 2 ðyÞ ¼ 1 4 4 > > >  > y 3a > > pypa: if :4 1 a 4

3.5 (21)

3 Mcr/My

8 y > <2 a  f 2 ðyÞ ¼ y > :2 1 a

2.5 2 1.5 1

(22)

0.5 0

With the defined functions f1(x,s) and f2(y), the element geometric matrix for each case of loading can be obtained by substituting Eqs. (2) and (3) into Eq. (15), for example: Schafer [3] constructed the geometric matrix for cases where the longitudinal

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 h/L Fig. 8. Relation between critical moment and height.

DOF 3 restrained

DOF 1,3 restrained

DOF 3 restrained Simply supported

DOF 3 restrained 6 5 3

2

DOF 1,3 restrained 1

This node: DOF 1,2,3 restrained DOF 3 restrained

D.O.F. Simply supported Fig. 7. FE model.

4

ARTICLE IN PRESS H.C. Bui / Thin-Walled Structures 47 (2009) 730–739

function f2(y) is much more difficult to define. A procedure described below is proposed to perform approximately the integration of Eq. (15) in a general loading condition; the strip is divided into cells in the longitudinal direction (Fig. 3). In a cell, the longitudinal stress is considered linear varying in the transversal direction and constant in the longitudinal direction. The potential energy (Eq. (15)) can be approximately determined by W¼

N Z yk Z b h tX xi s1k  ðs1k  s2k Þ 2 k¼1 yk1 0 b "   2  2 # @u 2 @v @w  þ þ dxdy, @y @y @y

(24)

where N is the number of cells in the strip; s1k and s2k are the longitudinal stresses obtained from a linear analysis and recorded at the middle of the longitudinal sizes of the kth cell (Fig. 3); yk1 and yk (Eq. (24)) are the ordinates of the cell in the longitudinal direction (Fig. 3). If the strip is uniformly divided into N cells, yk1 and yk are calculated by yk1 ¼ ðk  1Þ

a ; N

yk ¼ k

a . N

(25)

The element geometric matrix can be obtained by substituting Eqs. (2) and (3) into Eq. (24). This geometric matrix has the same form of Eq. (23). The expressions for [KGmm]e and [KGmn]e when the strip is divided into equal cells are given in Appendix A.2.

4. Solution of the problem Firstly, the stiffness matrix and the geometric matrix are transformed into the global axes. Then, an assembly procedure is realised to produce the global stiffness matrix and the global geometric matrix. The buckling problem can be solved by eigenvalue equations ½K þ l½K G  ¼ 0,

(26)

where l is a scaling factor related to the critical load. 5. Numerical applications In order to illustrate the application and to provide a better grasp of the capabilities of the approach presented in the previous sections, three numerical examples are performed. The first example relates to a beam of double symmetry I-section. This beam is subjected to five types of loading, which is given in Fig. 2. The second and third examples examine the buckling behaviour of cold-formed channel sections under uniform distributed loads, a central concentrated load and two concentrated loads, respectively. 5.1. Buckling behaviour of a thin-walled I-section The buckling behaviour of a thin-walled I-section beam is examined in this example. The dimensions of the section and the material properties are shown in Fig. 4a, the height of the crosssection is equal to 600 mm and the flange width is equal to 200 mm. The section is modelled by 16 strips of 2-nodal lines: 4 strips for each flange and 8 strips for the web. Fig. 5 shows the critical curves of the simply supported I-section beam subjected to gradient bending (GM) with b ¼ 0 (Fig. 4b) or uniform distributed loads. For each case of loading, the critical curve is performed with two approaches. In the first approach, the geometric matrix is calculated when the form in the longitudinal direction of the normal stresses is explicitly defined in Eqs. (19) and (20) and the critical curve 1 is obtained. The geometric matrix

735

for the case of gradient bending is given in Appendix A.1. In the case of uniform distributed loads applied at the shear centre, the geometric matrix can be found in Ref. [11]. In the second approach, the geometric matrix is numerically calculated by the procedure described in Section 3.3, in which the longitudinal stresses are obtained by a linear analysis and we have the critical curve 2. For a comparison, typical buckling curves with single harmonic and the critical curve of the I-section subjected to a uniform bending are also plotted in Fig. 5. It can be seen that in the case of uniform distributed loads, the critical curves 1 and 2 are in accordance with L/hX5. These curves diverge when the beam is short (L/ho5). This is due to the fact that in the case of short and very short beams, principal stresses are not only the longitudinal normal stresses. The ignored shear stresses and the vertical normal stresses are also important. For the case of gradient bending, a very good agreement between two critical curves 1 and 2 can be seen in Fig. 5, because we assumed an idealized stress distribution at the right end cross-section of the beam (Fig. 4c). Comparing three cases of loading (Fig. 5), it can be seen that when the beam length is very long, the lateral buckling is dominated. The critical moment of lateral buckling of gradient bending and uniform distributed loads are much higher than that of the uniform bending. This difference is attributed, of course, to the difference in longitudinal stresses in the three loading cases. For the uniform bending, the longitudinal stresses are constant, while it varies linearly for the gradient bending and parabolically for the uniform distributed loads. However, the critical moments of local buckling are close. The gradient bending and uniform distributed loads give slightly higher local critical moments than the uniform bending (8.49% for the gradient bending and 6.67% for the uniform distributed loads). This is attributed to the individual half-wavelengths, which are not equal when the beam buckles locally under the gradient bending and uniform distributed loads. The buckling shapes of local and lateral buckling for the three cases of loading are shown in Fig. 6. To show the influence of shear stresses on the buckling behaviour of beam element, we have performed some numerical examples of buckling analysis of thin-walled sections with 8-node finite element (FE), which is based on the Mindlin–Reissner theory and is implemented in our program FENALYSE [12]. This element has 6 DOF at a node. Table 1 indicates that in case of uniform bending, no shear stresses occur and finite element results are matched with our finite strip results. However, when shear stresses are important, i.e. in case of gradient bending with FE model in Fig. 7, the impact of this type of stresses may be considerable and the FS results diverge significantly from the FE results. Hence, our approach is only useful when shear stresses can be ignored. To examine the buckling behaviour of the thin-walled I-section in this example, the relation between the critical moment and the height of the cross-section can be represented. The length of the beam is constant and equal to 5000 mm. The ratio height to length of the beam is considered in the range 0.03–0.25. The curves relate the critical moment to the height of the beam and are shown in Fig. 8 for five cases of loading, namely: uniform bending, gradient bending, uniform distributed loads, central concentrated load and two concentrated loads at the one-fourth of the beam (Fig. 2). It can be seen that each curve has two distinct regions, which are corresponding to the local and the lateral buckling. The limit between the two regions is different from each other. When the ratio height to length is large, the local buckling is dominated and the critical moments for five loading cases are close. When this ratio is small, the lateral buckling is principal. The curves for the uniform bending and the central concentrated load represent the lower and upper bound, respectively. The

ARTICLE IN PRESS H.C. Bui / Thin-Walled Structures 47 (2009) 730–739

critical moments of lateral loading for these two cases of loading are very different from the rest.

5.2. Buckling behaviour of cold-formed channel sections A channel section with depth h ¼ 250 mm, flange width b ¼ 50 mm, lip length c ¼ 20 mm and thickness t ¼ 1.9 mm is considered here. The channel is subjected to uniform bending and uniform distributed loads, which are applied at the shear centre and at the top flange–web junction. The material properties are given in Fig. 9. The section is modelled by 20 strips: 4 strips are used for each flange, 8 strips for the web and 2 strips for each lip. The typical buckling curve with single half-wave length for the case of uniform bending is given in Fig. 10. It can be seen that it is dominated by local buckling when the beam length is short and intermediate. The critical curves of uniform bending and of uniform distributed loads, which are applied at the shear centre, are also represented in Fig. 10 (critical curve (UB) and critical curve 1 (ULD)). The stress distribution for these cases is shown in Fig. 9b. The geometric matrix of the two loading cases can be found in Refs. [3,11]. As commented by Chu et al., the critical moment of local buckling of uniform distributed loads slightly exceeds that of uniform bending. Now, we examine the buckling behaviour of the channel beam under uniform distributed loads, which are applied at the top flange–web junction (Fig. 9a). The stress distribution is given in Fig. 9c. It is well known that the value of the maximal compression stress in this case is much higher than this value when loads are applied at the shear centre. Thus, with the same intensity of loads, the critical moment of local buckling in the case of loads applied at the flange–web junction is much lower than that in the case of loads applied at the shear centre. The difference can be seen between two critical curves in Fig. 10 (critical curves 1 and 2) when the beam length is intermediate and long. The buckling shapes of local buckling for these two cases of loading are shown in Fig. 11. When the length is very long, the lateral buckling dominates and it can be seen that the results are very close. The difference in the distribution of stresses (Fig. 9a and b) has a minor influence on the lateral buckling. Another channel section of depth h ¼ 200 mm, flange width b ¼ 75 mm, lip length c ¼ 20 mm and thickness t ¼ 2.5 mm is examined. This channel beam is dominated by the distortional buckling as it is shown in Fig. 12 for the case of uniform bending. The critical curves for three cases of loading, namely: uniform bending, uniform distributed loads, which are applied at the shear centre (critical curve 1) and at the top flange–web junction (critical curve 2), are also given in Fig. 12. The value of critical

moment in the case of loads applied at the top flange–web junction is lower than this value of uniform bending and much lower than the case of loads applied at the shear centre. It is noted that the distortional buckling occurs with the uniform bending and with the case of loads applied at the shear centre (Fig. 13a). However, when the loads are applied at the top flange–web junction, the distortional buckling cannot appear because tension stresses reside at the top lip and at the adjacent portion of the top flange (Fig. 9c). These tension stresses prevent the distortional buckling. The local buckling dominates in this loading case when the beam length is intermediate and long (Fig. 13b). When the length is very long, it is dominated by the lateral buckling. The critical moments of lateral buckling are very close for both uniform distributed loads applied at the shear centre and at the top flange–web junction. These critical values are much higher than the ideal loading case of uniform bending.

6. Conclusions The buckling behaviour of thin-walled I-section and channel section subjected to general loading conditions has been investigated using a semi-analytical finite strip method. Comparison of the results have shown that it should apply the present procedure

3

2

1.5

1

0.5

0 102

σmax2 > σmax1

c

σmax1

t

h

103 Beam length (mm)

Fig. 10. Critical curves of simply supported channel-section beam (h ¼ 250 mm, b ¼ 50 mm, c ¼ 20 mm and t ¼ 1.9 mm).

qz

S

Buckling curve, m = 1 (UB) Critical curve (UB) Critical curve 1 (UDL) Critical curve 2 (UDL)

2.5

Mcr/My

736

G gravity centre

shear centre

c

E = 205000 N/mm2 ν = 0.3 fy = 275 N/mm2

b

Fig. 9. (a) Cold-formed channel section; (b) stress distribution when distributed loads are applied at the shear centre and (c) stress distribution when distributed loads are applied at the flange–web junction.

ARTICLE IN PRESS H.C. Bui / Thin-Walled Structures 47 (2009) 730–739

737

Fig. 11. Local buckling shapes of uniform distributed loads when L ¼ 1500 mm: (a) loads applied at the shear centre and (b) loads applied at the top flange–web junction.

6 Buckling curve, m = 1 (UB) Critical curve (UB)

5

Critical curve 1 (UDL) Critical curve 2 (UDL)

Mcr/My

4

3

2

1

0 102

103 Beam length (mm)

Fig. 12. Critical curves of simply supported channel-section beam (h ¼ 200 mm, b ¼ 75 mm, c ¼ 20 mm and t ¼ 2.5 mm).

Fig. 13. Buckling shapes of uniform distributed loads when L ¼ 1500 mm: (a) distortional buckling when loads are applied at the shear centre and (b) local buckling when loads are applied at the top flange–web junction.

for the buckling analysis of thin-walled sections under a general loading in cases where shear stresses can be ignored and/or the thin-walled member is sufficiently long.

In the same type of loading, the critical load of local buckling and distortional buckling depends much on applied position of the load. The difference in applied position of loads gives

ARTICLE IN PRESS 738

H.C. Bui / Thin-Walled Structures 47 (2009) 730–739

the difference in the distribution of longitudinal stresses. The appearance of the distortional buckling depends on this distribution.

Appendix A A.1. The coupled geometric matrix (KGmn) for the case of gradient bending 2

70ð3s1 þ s2 ÞIc

6 6 6 6 6 6 6 ½K Gmn  ¼ C 6 6 6 6 6 6 6 4

0 0

0 0

70ðs1 þ sk ÞIc 0

0 70ðs1 þ s2 ÞIs

0 0

24ð10s1 þ 3sk ÞIc

2bð15s1 þ 7s2 ÞIc

0

0

54ðs1 þ s2 ÞIc

0 70ðs1 þ 3s2 ÞIc

0 0

2bð6s1 þ 7s2 ÞIc 0

70ðs1 þ 3s2 ÞIs

0 24ð3s1 þ 10s2 ÞIc

0 70ð3s1 þ s2 ÞIs

2

b ð5s1 þ 3s2 ÞIc

symmetric

0 0

3

7 7 7 2bð7s1 þ 6s2 ÞIc 7 7 7 2 3b ðs1 þ s2 ÞIc 7 7 7 0 7 7 0 7 7 2bð7s1 þ 15s2 ÞIc 7 5 2

b ð3s1 þ 5s2 ÞIc

where C¼

tbðmnÞp2 , 840a

Z i 1 ah y mpy cos ðb  1Þ þ 1 cos a 0 a a 8 ðb þ 1Þ > > > > > 4 > > < 0 ¼ " # > > ðb  1Þ > 1 1 > > > þ > : p2 ðm  nÞ2 ðm þ nÞ2

Ic ¼

Z i 1 ah y mpy sin ðb  1Þ þ 1 sin a 0 a a 8 ðb þ 1Þ > > > > > 4 > > < 0 ¼ " # > > > ðb  1Þ 1 1 > > >  > : p2 ðm  nÞ2 ðm þ nÞ2

Is ¼

npy dy, a if m ¼ n; if ðm  nÞ is even; if ðm  nÞ is odd;

npy dy, a if m ¼ n; if ðm  nÞ is even; if ðm  nÞis odd:

A.2. The coupled geometric matrix (KGmn) for the case of general loading condition 2 6 6 6 6 6 N 6 X 6 6 ½K Gmn  ¼ C 6 k¼1 6 6 6 6 6 4

70ð3s1k þ s2k ÞIck

0 70ð3s1k þ s2k ÞIsk

0 0 24ð10s1k þ 3s2k ÞIck

0 0 2bð15s1k þ 7s2k ÞIck

70ðs1k þ s2k ÞIck 0 0

0 70ðs1k þ s2k ÞIsk 0

0 0 54ðs1k þ s2k ÞIck

0

0

2bð6s1k þ 7s2k ÞIck

70ðs1k þ 3s2k ÞIck

0 70ðs1k þ 3s2k ÞIsk

0 0 24ð3s1k þ 10sk2 ÞIck

2

b ð5s1k þ 3s2k ÞIck

symmetric

where N is the number of equal cells in the longitudinal direction of the strip, C¼

tbðmnÞp2 , 840a

Z 1 kða=NÞ mpy npy cos dy cos a ðk1Þða=NÞ a a 8   1 1 2mpk 2mpðk  1Þ > > > þ  sin ; sin > < 2N 4mp N N      ¼ > 1 ðm þ nÞpk ðm þ nÞpðk  1Þ 1 ðm  nÞpk ðm  nÞpðk  1Þ > > 1 sin  sin þ sin  sin ; > : 2p m þ n N N mn N N

Ick ¼

if m ¼ n; if man;

0 0 2bð7s1k þ 6s2k ÞIck

3

7 7 7 7 7 7 2 3b ðs1k þ s2k ÞIck 7 7 7 0 7 7 0 7 7 2bð7s1k þ 15s2k ÞIck 7 5 2 b ð3s1k þ 5s2k ÞIck

ARTICLE IN PRESS H.C. Bui / Thin-Walled Structures 47 (2009) 730–739

Z 1 kða=NÞ mpy npy sin dy sin a ðk1Þða=NÞ a a 8   1 1 2mpk 2mpðk  1Þ > > >   sin ; sin > < 2N 4mp N N ¼      > 1 ðm þ nÞpk ðm þ nÞpðk  1Þ 1 ðm  nÞpk ðm  nÞpðk  1Þ > > 1 sin  sin  sin  sin > : 2p m þ n N N mn N N

739

Isk ¼

References [1] Cheung YK. Finite strip method in structural analysis. New York: Pergamon Press; 1976. [2] Hancock GJ. Local distortional and lateral buckling of I-beams. J Struct Div ASCE 1978;104(11):1787–98. [3] Schafer BW. Cold-formed steel behaviour and design: analytical and numerical modelling of elements and members with longitudinal stiffeners. PhD Dissertation, Cornell University, Ithaca, New York, 1997. [4] Bradford MA, Hancock GJ. Elastic interaction of local and lateral buckling in beams. Thin-Walled Struct 1984;2(1):1–25. [5] Key PW, Hancock GJ. A finite strip method for the elastic–plastic large displacement analysis of thin-walled and cold-formed steel sections. ThinWalled Struct 1993;16:3–29. [6] Key PW, Hancock GJ. A theoretical investigation of the column behaviour of cold-formed square hollow sections. Thin-Walled Struct 1993;16(1):31–64.

if m ¼ n; if man:

[7] Hinton E. Buckling of initially stressed Mindlin plates using a finite strip method. Comput Struct 1978;8:99–105. [8] Dawe DJ, Lam SSE, Azizian ZG. Finite strip post-local-buckling analysis of composite prismatic plate structures. Comput Struct 1993;48(6):1011–23. [9] Wang S, Dawe DJ. Finite strip large deflection and post-overall-buckling analysis of diaphragm-supported plate structures. Comput Struct 1996;61(1): 155–70. [10] Bui HC, Rondal J. Buckling analysis of thin-walled sections by semi-analytical Mindlin–Reissner finite strips—A treatment of drilling rotation problem. Thin-Walled Struct 2008;46:646–52. [11] Chu XT, Ye ZM, Kettle R, Li LY. Buckling behaviour of cold-formed channel sections under uniformly distributed loads. Thin-Walled Struct 2005;43: 531–42. [12] Bui HC. Static analysis of thin-walled structures by finite element method and finite strip method based on the shear deformable plate and shallow shell theories. PhD Thesis (in French), University of Liege, Belgium, 2008.