Buckling analysis of trapezoidally corrugated panels using spline finite strip method

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Thin-Walled Structures 18 (1994) 209-224 © 1994 Elsevier Science Limited

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ELSEVIER

Buckling Analysis of Trapezoidally Corrugated Panels Using Spline Finite Strip Method

R. Luo & B. Edlund Division of Steel and Timber Structures, Chalmers University of Technology, S-412 96 Gothenburg, Sweden (Received 7 March 1993; accepted 19 April 1993)

ABSTRACT Buckling of trapezoidally corrugated panels under in-plane loading is analyzed by a spline finite strip method. The influence on the elastic buckling load of various parameters, such as geometry, loading forms and boundary conditions, etc., is studied. It is found that: (1) for longitudinal compression the buckling load increases with the corrugation angle ~, and for a given ~ the highest buckling load is achieved when the 'proportion parameter' ~ = 1; (2) for shear loading the buckling load increases as increases, and for a given ~ the highest buckling load is obtained when = 2; and (3) for a combined loading of compression and shear, interactive curves can be approximated by unit circles when • = 15 °, 30 °, 45 °, 60 ° and 90 °. However, when • = 75 ° a parabola seems to be a better approximation. Based on the numerical experiments, simplified formulae and interactive curves are suggested for practical design.

INTRODUCTION

The use of trapezoidal profiles (Fig. 5) has increased considerably in recent years. The profiled panel is widely used for girders, roofing, decking, wall cladding, and in offshore structures. The corrugation provides a continuous stiffening which permits the use of thinner plates, and fabrication costs for elements with corrugated panels are normally considered to be lower than those for stiffened plates. In the design of elements with 209

R. Luo, B. Edlund

210

trapezoidally corrugated profiles, stability problems are of prime importance because of the high slenderness ratio of the element. Many design codes have been published for the design of profiled sheeting, for example the Swedish Code for Light-Gauge Metal Structures, the European Recommendations and ISO Standards, etc. However, there is still not enough knowledge of how different geometrical parameters and imperfections influence the bearing capacity of this kind of structures. At Chalmers University of Technology experimental research on stability problems of this kind of panels has been carried out. In this paper, the elastic buckling of such panels under in-plane loading is analyzed by a spline finite strip method and numerical results are compared with the experimental results. A parametric study is performed and suggestions for the practical design of such panels are given. The finite element method 1 is regarded as being a most powerful and versatile tool for structural analysis. However, in practice its application is often restricted because it demands a large computer and long computation time. Because of this, a simple and economical approach known as the finite strip method 2 was successfully developed for structures with regular shape. Nevertheless, the use of conventional finite strips experienced difficulties in dealing with concentrated loads, complicated boundary conditions, etc. In order to overcome these difficulties and retain the advantages of the finite strip method, a linear combination of local B3splines will be used here to form spline finite strips. 3 By using spline functions 4 instead of trigonometric series, which have been used in the semi-analytical finite strip method, while retaining the transverse interpolation polynomials, the method can easily be used in cases with complicated boundary and loading conditions. For the analysis of the elastic and inelastic buckling of thin-walled structures the method has earlier proven to be both accurate and efficient, s 7

SPLINE FINITE STRIP METHOD AND BUCKLING ANALYSIS The spline function chosen here for the approximation of the displacement is the B3-spline of equal section length, namely: m + I

fO') = Z i

o~i~,O')

(1)

--I

where ~iO') is a local B3-spline as shown in Fig. 1; and ~i is a coefficient to be determined. A standard B3-spline function is defined by:

Buckling of trapezoidally corrugated pane&

1 ~i(Y) = 6h 3

211

0 (Y -- Yi- 2)3 h3 q- 3h2(y - Yi- l)

Y ~Yi-2 Yi- 2 <~Y <~ Yi-1

+ 3h(y - Yi- 1) 2 - 3(y - Yi- 1) 3 h3 + 3h2(yi+ l _ Y)

Yi- 1 <~ Y <~Yi

-+- 3h(yi+ 1 - y)2 _ 3(yi+ 1 - y)3

Yi <<-Y <~ Yi+ 1

(Yi+2 - y)3 0

(2) Yi+l <~Y <~ Yi+2 Yi+2 <~Y

Modified splines are used near the boundary points. 3 A trapezoidally corrugated panel is subdivided transversely into a n u m b e r of strips using n nodal lines and subdivided longitudinally into m sections using (m + 3) section knots. The nodal lines and section knots for one strip are shown in Fig. 2. Each section knot has four degrees of freed o m corresponding to two out-of-plane displacements, w and 0, and two in-plane displacements, u and v. The displacement functions {f} of a strip are expressed as the product of conventional transverse shape functions Ni and longitudinal B3-splines: {f}

v

=

=

w

[ Nt [~./] 0 0

-{u,} {v,} {w,}

0 0 0 N2 [~9./] 0 0 0 ] Nt [~b,,i] 0 0 0 N2 [ff~./] 0 0 ] 0 N3[~,,.i] N4[@,i] 0 0 Ns[~'wi] N6[~oi]

{0,} { ui} {vl} {% } _

{0j}. (3)

where: Nl = 1 - x , N2 = x, N3 = 1-3Yc 2 + 2 2 3 ,N4 = x ( 1 - 2 2 + . 9 2 N5 = 3Sc2 - 2.v 3, N6 = x ( x 2 - CY),x = x / b

) (4)

T h e t e r m s [~ui], [~vi], [~wi], [lltoi], [~/uj], [~u], [ff/,,j], [~lOj]a r e r o w m a t r i c e s in terms of B3-splines and {u,}, {v,}, {w,}, {0~}, {UJ}, {vi}, {w/}, {Oi} are the corresponding displacement parameter vectors for the two adjacent nodal lines i and j, respectively. Define a generalized displacement vector for a strip by:

{a} =

(S)

The strain energy of a strip resulting from buckling deformation is given by:

212

R. Luo, B. Edlund

(a)

~i

. ~ e c t i o n

knots

Yi-2 Yi-I Yi Yio! Yi*2

YO Yl Y2

Ym-2Ym-I Ym m section$

~'~

,'~XX •

-

X X X X k-L,*r. •

~

•

i

Yo Yl h

,

•

-~1

Ym..2 Ym.4Ym

Fig. 1. (a) Typical B3-spline function, eqn (2). (b) Basis of B3-spline expression. (c) Amended local spline functions. Nodallinei

t

~

~

~

/

m

Nodallinej

secti°ns

Fig. 2. A spline finite strip.

ui

= ~

{o'}T{e}dxdy

(6)

in which b is the strip width and L is the strip length. The stress matrix {a} and strain matrix {5} can be related to the generalized displacement vector {6} by: {e} = [B]{6}

(7)

and: {a} = [D]{~} = [D][B]{6}

(8)

Bucklingof trapezoidallycorrugatedpanels

213

Substituting eqns (7) and (8) into eqn (6), and summing for all strips, the strain energy U for the structural system is:

U = Eui = E 21 {~)T Jo [L

f0 ~[B]T[D][B]dxdy{6}

(9)

where E means the sum for all strips. The basic state of stress assumed in a strip is shown in Fig. 3. It consists of a uniform shear stress -c, a longitudinal compressive stress O'L, and a transverse compressive stress aT. The increase in potential energy of the membrane forces resulting from the bending deformation was derived 8 as:

owow}

WF =--E~

aL\--~y/l +~rT\Ox] +2r~X--~-fy tdxdy (10)

The increase in potential energy of the membrane forces resulting from inplane buckling deformation was derived 8 to be:

WM = --E

aL ~ y ) +~-~y

}

tdxdy

(11)

The total increase in potential energy of the membrane forces resulting from buckling deformation is: W -

WF+WM

(12)

By applying the Ritz principle 9 to the total potential energy we have:

O(U + W)

-

0

(13)

t3

L

Fig. 3. Basic stresses of a strip.

214

R. Luo, B. Edlund

which gives: ([K] - 2[G]){6} = {0}

(14)

where [K] is the stiffness matrix; 2 is the buckling load factor; and [G] is the stability matrix. For the details of the matrices [K] and [G], we refer to Luo. 10 This is an eigenvalue problem. By solving for the smallest eigenvalue of eqn (14) we obtain the first bifurcation load. The stiffness and stability matrices described in the previous sections are derived in a local coordinate system. Transformation to a global coordinate system is necessary when two adjoining plate strips have different orientations. Let a plate strip be inclined at an a n g l e / / t o the global x-axis as shown in Fig. 4. The displacements in the local axes {6'} are related to those in the global axes {6} by: {6'}

(15)

=

JR] {6}

=

[4 < o', u; v; w.; 0;], {a}

where: {a'}

= [ui vi w~ 0i u/v1 wj 0i]

and by: [R] -where: cos/~ 0 -sinfl 0

0 sin/~ 0 1 0 0 0 cos B 0 0 0 l

(17)

The stiffness matrix [K] and the stability matrix [G] in the global coordinate system become: [K] =

[R]T[K'][R],

[G] =

[R]T[G'][R]

(18)

where [K'] and [G'] are the corresponding matrices determined in the local coordinate system. The stiffness and stability matrices are highly banded; therefore, an eigenvalue routine for banded matrices can be applied. In this paper, the subspace iteration m e t h o d introduced by Bathe and Wilson 1~ has been used. When only shear load is present a 'shifting' procedure introduced by Bathe 12 is applied to avoid zero eigenvalues. In the solution of eqn (14), we perform a shift p on [G] by calculating: [G*] = [G] +p[K]

(19)

Buckling of trapezoidally corrugatedpanels

215

y (v) y' ( v ' ) ~

x' (u')

)

~z'

(w') z (w)

Fig. 4. Transformation from local to global coordinates. and we then consider the eigenvalue problem: [K]{6} = #[G*]{6}

(20)

In order to identify how the eigenvalues and eigenvectors of [K]{6} = 2[G]{6} are related to those of the problem [K]{6} = #[G*]{6}, we rewrite eqn (19) in the form: =

(21)

where ~/= # / ( 1 - p#). Equation (21) is equivalent to [K]{6} = 2[G]{6}. Hence, we have: 2 = #/(1 - p#), {6i} = {~i}

(22)

N U M E R I C A L RESULTS Several numerical examples are presented in this section. Loading cases of longitudinally pure compression and pure shear, and a combined loading case of axial compression and shear, are considered.

Loading case I: corrugated panels under longitudinal compression In the analysis, each panel, which is subjected to longitudinal compression as shown in Fig. 5(a), is subdivided into longitudinal strips. The two vertical edges are hinged supports, the bottom edge is simply supported and the upper edge is a sliding support (in the longitudinal direction). At the two latter edges the original shape of the corrugated profile is preserved after

216

R. Luo, B. Edlund

a) Y

t

d

b2 "t=bl/b 2 I~

B=1997

v"

...1

"1

x

Fig. 5. (a) Panel under longitudinal compression. (b) Notation of corrugation geometry. buckling. Young's modulus E = 1-98 x l05 MPa, Poisson's ratio # = 0.3, and the thickness of the web t = 2 mm. The overall panel width is always constant B = 1997 mm, the overall panel height is also constant H = 1270 mm. The following factors (see definition in Fig. 5(b)) which influence the buckling load are studied: (1) angle o f the corrugation at, with values of 15 °, 30 °, 45 °, 60 °, 75 °, and 90°; (2) ratio 7 = bl/b2, with values of 34/34, 68/34, 102/34, 135/34, and 171/34. The results obtained are shown in Fig. 6. Buckling modes o f the panels with ratio ? -- 171/34, ~ -- 30 ° and ~ = 75 ° are shown in Figs 7 and 8, respectively. For the panel with ~ = 61.57°, V = 171/34, o-or(experiment) = 135.7 MPa, o-cr (calculated) = 127.3 MPa, o-cr (ABAQUS) = 125-8 MPa. This corresponds to (o./E)x 103= 0.68 (experiment), 0.64 (calculated), and 0-63 (ABAQUS), respectively. It can be seen that the result using the spline finite strip m e t h o d gives a good agreement with the experimental result ~3 and the result using the finite element program ABAQUS. 14 It is known that for thin-walled structures two main types of buckling modes must be considered, one being local plate buckling and the other overall buckling. The buckling stress associated with the local buckling mode of a component plate is: 15 O'loc,cr

=

k 0-o

(23)

where: rc2E

0.0- 12(]-

1,'2)

(b)!

(24)

and k is a buckling coefficient which depends on the type and distribution of the applied stress on the plate, the ratio between the plate sides and the boundary conditions of the plate.

Buckling of trapezoidally corrugated panels

217

x

3-

a

,.:j.:"':"/'m......" .- ...... .

0 J if) z

9

/ ,,~'f" . . . . . . . . . . • . . . . . . . -.-0 . . . . . .

--'--B'---

171/34

---'-t--

135/34

.... IB---

102/34

-- -,t

--.

68/34

.... • -'-

34/34

•

z w

z_ o

Angle 1

o

300

45 °

600

75 °

90 °

Fig. 6. Influence of angle ~t and ratio y = bl/b2 on the buckling stresses.

Hllllll Ittnllllllllll I I llll111111

IIII I I I

Fig. 7. Buckling mode of panel with y = 171/34, ct = 30°. The buckling stress associated with the global buckling m o d e o f a wide panel is:

n2 E agl,cr -- (L/r)2

(25)

where r is the radius o f gyration for the considered cross-section; and L is a reduced c o l u m n length which depends on the edge conditions. In Fig. 6 some interesting observations may be made. In the cases w h e n the ratio ~ = 171/34, ~, = 135/34 and y = 102/34, respectively, the buckling stresses increase almost linearly until the corrugation angle 0t reaches a value around 30 °, 45 °, and 60 °, respectively. After 0~ has reached these values, the buckling stress remains constant as 0t increases. This can be

218

R. Luo, B. Edlund

Fig. 8. Buckling mode of panel with 7 = 171/34, ~ = 75°.

explained by the fact that when c~ increases above these values, the buckling mode changes from global buckling to local buckling. In the case when the ratio 7 = 171/34, and when ~ is between 15 ° to 30 °, the global buckling mode (see Fig. 7) is governing. When ~ ~> 45 ° the local buckling mode is governing (see Fig. 8). In the cases when the ratio ~/-- 68/34 and 34/34, the global buckling mode dominates for all angles ~. These results correspond well to the experimental results, ~3 and show that for the panels with ratio 7 - - 171/34 and angle ~ = 61-57 °, local buckling takes place before global buckling. For the local plate buckling mode (corresponding to the flat part of the curves in Fig. 6), using eqn (24), for the flat panels with width b = 171 mm, o0 = 24-48 MPa; and for the panels with width b = 135 mm, a0 = 39-3 MPa. Using eqn (23) and buckling stresses in Fig. 6, average results for 'flat part of curves', for panels with bl = 171 mm, k = 126.11/24.48 ~ 5.16 (when ~ varies from 30 ° to 90°); and for the panels with b~ = 135 mm, k = 203.81/39-3 ~ 5.18 (when ~ varies from 45 ° to 90°).

Buckling of trapezoidally corrugatedpanels

219

Taking the average value of k for these two cases, a simplified formula for calculating local buckling stress can be given as: trloc,cr = 5" 17 tr0

(26)

For the panels with b = 102 mm, using the same steps as above, a simplified formula for calculating local buckling stress can be given as: O'loc,cr

=

6-02 a0

(27)

which shows that the plate is elastically restrained along the longitudinal edges.

Loading case 2: corrugated panels under shear loading Buckling of corrugated panels under shear loading (Fig. 9) is studied. These •panels are also subdivided into longitudinal strips. The boundary conditions, material properties and the factors which influence the buckling load are the same as in loading case 1. The computational results are shown in Fig. 10. The buckling mode of the panel with ~ = 45 ° and ratio 7 = 135/34 is shown in Fig. 11.

Loading case 3: panel under combined longitudinal compression and shear Buckling of corrugated panels under combined longitudinal compression and shear (Fig. 12) is studied. The material properties are the same as above. The influence o f the angle ~ on the buckling load is studied for the ratio ? = 171/34. The two vertical edges are hinged supports, the bottom and the upper edges are free in the direction of the x-axis. The results computed are shown as interaction curves in Fig. 13. Two types o f interaction curve are used here for comparison with the results computed. One is called the parabolic curve as proposed by Batdorf and Stein:16

t,, Ix

T I

tl

b, t kb ;

H = 1270

l

q

~ "f "L"Z

" f "J / = 1997

V

"J " f

~×

~-

Fig. 9. P a n e l u n d e r s h e a r l o a d i n g .

220

R. Luo. B. Edlund

o !

, 15"

Fig. 10. Influence

.

, 30"

.

, 45"

.

, 60"

.

, 75"

.

,

1 Angle a

90"

of angle u and ratio y on the buckling stresses under shear.

Fig. 11. Buckling mode of panel with angle z = 45”, ratio y = 135/34.

(3) The other is the unit circle curve proposed by Timoshenko:‘7 (29)

Buckling of trapezoidally corrugatedpanels

221

U

T l ~,

H =1270

B=1997 ~

x

Fig. 12. Panel under combined longitudinal compression and shear. 200

1 O0

'

O' 0

~

100

[]

15 °

•

30*

•

45*

&

60*

•

75 °

D

90*

200

Fig. 13. Interaction curves for compression and shear. The stresses O'cr and "~cr are the compression and shear stresses which 0 and l:cr 0 are produce buckling when acting simultaneously. The values Gcr the critical values o f the panel when subjected to compression stress alone or to shear stress alone. In Fig. 14 these two curves are plotted together with the interaction curve computed for the angle ~t = 60 °. It shows that the unit circle curve gives a better correspondence with the computed results than the parabolic interaction curve. The latter one gives conservative results. F o r the panel with angle ct = 75 ° (Fig. 15) the parabolic interaction curve gives a better correspondence with the calculated results than the unit circle curve. The latter one gives overestimated results. F r o m Fig. 13 we can see that for panels with ~ -- 15 °, 30 °, 45 °, 60 ° and 90 °, the interaction curves have similar shapes, so that the use of a unit circle curve gives better correspondence, while for panels with ~ = 75 °, the parabolic curve gives better results.

222

R. Luo, B. Edlund 120

100

•,iri'icI

80

60

40

20

(Y

0 0

20

40

60

80

100

120

Fig. 14. Interaction curves for compression and shear compared with results for the panel with ~ = 60°.

1;

120

80 60

Parabolic--,P"~

40, 20, 0

(Y

=

100

200

Fig. 15. Interaction curves for compression and shear, compared with the results for the panel with 2 = 75'.

CONCLUSIONS We have shown that the numerical results given by the spline finite strip m e t h o d agree well with the experimental results. F r o m the numerical results, we have observed the following: (1) F o r longitudinal compression the buckling load increases with the c o r r u g a t i o n angle ~, a n d for a given ~ the highest buckling load is achieved when the ' p r o p o r t i o n p a r a m e t e r ' 7 = 1. It is also noticed

Buckling of trapezoidally corrugatedpanels

223

that as the angle a increases the buckling mode changes from global buckling to local buckling for 7 = 171/34, 135/34 and 102/34. (2) For shear loading, the buckling load increases as a increases provided that local buckling is prevented, and for a given ~ the highest buckling load is obtained when 7 = 68/34. For a given 7 the highest buckling load is obtained when ~ = 90 °. (3) For a combined loading of compression and shear, interactive curves can be approximated by unit circles when a = 15 °, 30 °, 45 °, 60 ° and 90 °. However, when ~ = 75 ° a parabola seems to be a better approximation for the interactive curve. Based on the numerical experiments, the authors have suggested formulae (26) and (27) to determine approximately the local buckling loads for cases with pure compression. The interactive curves given in Figs 14 and 15 should be useful in the practical design. It should be remarked here that the buckling of panels under patch loading is more complicated than what has been considered in this paper and the authors shall report such cases separately. 18

REFERENCES 1. Zienkiewicz, O. C. & Taylor, R. L., The Finite Element Method. McGrawHill, London, 1989. 2. Cheung, Y. K., The Finite Strip Method in Structural Analysis. Pergamon Press, Oxford, 1976. 3. Cheung, Y. K. & Fan, S. C., Static analysis of right box girder bridges by spline finite strip method. Proc. Instn Cir. Engrs, Part 2, 75 (1983) 311-23. 4. Schoenberg, I. J., Contributions to the problem of approximation of equidistant data by analytic functions. Q. Appl. Math., 4 (1946) 45-99, 112-14. 5. Lau, S. C. W. & Hancock, G. J., Buckling of thin-walled structures by a spline finite strip method. Thin-Walled Structures, 4 (1986) 269-94. 6. Lau, S. C. W. & Hancock, G. J., Inelastic buckling analysis of beams, columns and plates using the spline finite strip method. Thin-Walled Structures, 7 (1989) 213-38. 7. Van Erp, G. M. & Menken, C. M., The spline finite strip method in the buckling analysis of thin-walled structures. Commun. Appl. Numer. Methods, 6 (1990) 477-84. 8. Plank, R. J. & Wittrick, W. H., Buckling under combined loading of thin flat-walled structures by a complex finite strip method. Int. J. Numer. Methods Engg 8(2) (1974) 323-39. 9. Ritz, W., Ober eine neue Methode zur L6sung gewisser Variationsprobleme der mathematischen Physik. J. Reine Angew. Math., 135 (1908) 1 61. 10. Luo, R., Buckling analysis of trapezoidally corrugated panels under in-plane loading using spline finite strip method. Thesis presented to the Chalmers University of Technology at Gothenburg, Sweden in partial fulfillment of the degree of Licentiate of Engineering, 1991.

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R. Luo, B. Edlund

11. Bathe, K. J. & Wilson, E. L., Large eigenvalue problems in dynamic analysis. Proc. ASCE, EM6, 98 (1972) 1471-85. 12. Bathe, K. J., Finite Element Procedures in Engineering Analysis. Prentice Hall, Englewood Cliffs, NJ, 1982. 13. Leiva-Aravena, L., Trapezoidally corrugated panels - - Buckling behaviour under axial compression and shear. Chalmers University of Technology, Division of Steel and Timber Structures, Gothenburg, Sweden, PuN. 87:1, 1987. 14. ABAQUS Manual, Hibbitt, Karlsson & Sorensen, Inc., Providence, Rhode Island, USA, 1989. 15. Timoshenko, S. & Gere, J. M., Theory of Elastic Stability. McGraw-Hill, New York, 1961. 16. Batdorf, S. B. & Stein, M., Critical combinations of shear and direct stress for simply supported rectangular flat plates. NACA Tech. Note 1223, National Advisory Committee for Aeronautics, US Government Printing Office, Washington D.C., 1947. 17. Timoshenko, S., Stability of the webs of plate girders. Engineering, 238 (1935) 207. 18. Luo, R. & Edlund, B., Elastic buckling of trapezoidally corrugated panels under in-plane patch loading. Division of Steel and Timber Structures, Chalmers University of Technology, (in preparation).