a semi-analytical approach to buckling analysis for composite structures

ELSEVIER

Composite Structures 35 (1996) 93-99 Copyright 0 1996 Elsevier Science Ltd

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A semi-analytical approach to buckling analysis for composite structures

James Rhodes Department of Mechanical Engineering, University of Strathclyde, Glasgow, UK

The buckling behaviour of thin-walled structural sections made from specially orthotropic material is studied using a semi-analytical, semi- numerical approach. The approach used combines plate and beam theory in dealing with the out-of-plane and in-plane deformations of the walls of a cross-section, and has common features to both the finite strip approach and the generalised beam theory approach. The particular problems examined here concern the effects of directionality of the material, and for all cases considered material with stiff direction set longitudinally along the section is compared with the same materal set with the stiff direction across the section. It is found that the buckling behaviour in either case can be found directly from an examination of the other case. Copyright 0 1996 Elsevier Science Ltd.

NOTATION

Reference width of plate used to evaluate buckling coefficients Width of a strip Plate flexural rigidity factor, D,=E,it3/12 (1 -V12V21)

Plate flexural rigidity factors for still, 11, and flexible, 22, directions Plate flexural rigidity factors for x and y directions, respectively Torsional rigidity given by D3,=G12t3/12 Elasticity moduli for the subscripted directions Elastic shear modulus Buckling coefficient such that

n21E,E, ccr= 12(1 -v,vJ

(t/b)2 x K

Buckling coefficients for load applied in the subscripted directions Membrane stress at a point on the nth strip due to buckling Reference plate thickness used to evaluate buckling coefficients Strip thickness Potential energy Strain energy

u In-plane displacement of a strip W Out-of-plane displacements in a strip

WlYW2 Out-of-plane displacements of strip boundaries

v12, v21 Poisson’s ratio with respect to the planes 1 and 2

4,@2 Slopes at the strip boundaries

INTRODUCTION

For a number of years now the analysis of thin- walled prismatic members has been best carried out by methods such as the finite strip method,’ in which the number is divided into a number of strips across its cross-section, each strip extending the length of the member. This approach is fundamentally a special case of the finite element method, but because the variation in deformations, etc., along the member are taken out of the direct finite element for- mulation, the resulting numerical calculation requirements are generally an order of magni- tude less than for a normal finite element examination. The variation in deformations, etc., along the member are generally either completely specified a priori or as a limited series of functions which can be optimised using the principle of minimum potential energy. The

93

94 J. Rhodes

finite strip method has been extremely widely used for the examination of buckling problems, see for examples Refs 2-6.

The use of ‘generalised beam theory’ for prismatic members has similarities with the finite strip approach, and this approach has received substantial attention in recent years, e.g. Refs 7 and 8, particularly with regard to steel sections. This method considers that the strips can be analysed using beam analysis, and can generally be set-up using less degrees of freedom than the finite strip approach, although this is offset by the fact that more strips are generally required for a given level of accuracy than would be the case with the finite strip method.

The approach used in this paper has similarities with both the finite strip and the generalised beam theory approaches. Here the in-plane deformations of each strip are examined on the basis of beam theory, while the out-of-plane deformations are examined using plate theory. The approach was developed over 10 years ago for sections with isotropic materials, and has been demonstrated in a number of publications, e.g. Refs 9 and 10 for such materials. As is suggested by the title of Ref. 11 the approach was originally referred to as a finite strip approach, but there are signifi- cant differences in the set-up of this approach in comparison with standard finite strip methods. In the analysis of buckling the deformations along the section have been examined using a number of different deflection forms,” but for a section with simply supported ends the postulation of sinusoidally varying deflections along the section gives an exact representation of the true variation for a large class of problems, and this is used in the present paper. The out-of-plane deflections across a strip can be modelled by polynomial functions of any degree. Linear functions can be used, and if these are used for all strips the classical cubic solution for bending and twisting behaviour of beams is obtained, regardless of the number of strips examined. Cubic, quintic and septic polynomials have also been examined,12 but in the present paper we shall deal only with cubic (and linear in some cases) polynomials.

OUTLINE OF THEORY

Consider the cross-section of arbitrary shape shown in Fig. 1. During buckling the potential

energy change in any strip may be obtained by summing the change in strain energy due to out-of-plane deflections, that due to in-plane deflections and the change in potential energy of the load on the strip. If we consider that the deflections of a strip vary sinusoidally along the member, the out-of-plane displacements of the strip may be approximated by the cubic polynomial:

+ (y” - q2)8,bi} sin ( )

F , (1)

where q=y,/b,. The in-plane displacements of each strip is

assumed constant across the strip, i.e.:

z+,y)=u, sin 7uc ( ) L

(2)

These displacements are illustrated in Fig. 2. Also shown in Fig. 2 are the relationships between the in-plane and out-of-plane displacements of adjacent strips. If we consider the edge displacements of strips i - 1, i and i + 1 we obtain the relationships:

Stress distribution. The stresses may be constant, or may vary linearly as shown.

Strip number ‘7’

Fig. 1. Cross-section of arbitrary shape under load.

v (0

(a) Displacements within a strip (b) Relationship between displacements of adjacent strips

Fig. 2. In-plane and out-of-plane displacements.

Buckling analysis for composite structures 95

Wl(i)=v(i)COt(Pi)-u(i-l)COSeC(Pi)

W2(i)=~(i+l)COSeC(/?~+~)-U~COt(~i+~). (3)

Using these relationships provides a means whereby the problem can be formulated purely in terms of the strip edge rotations and the in- plane displacements.

The strain energy of out-of-plane bending and twisting is:

ss

placement coefficient and setting the derivatives equal to zero. By utilising the relationships given in eqns (3) the number of unknowns which arise with the use of cubic strips is 2N + 3, where N is the total number of strips.

EXAMINATION OF TYPICAL BUCKLING PROBLEMS

The following problems were examined for sections made from specially orthotropic material with the following properties

a2w a2w a2w 2 +2v2J3,---- -

ax2 ay” fD22 - ( 1 aY’

dx dy.

The strain energy of in-plane bending is

Kx I/s2=- =s s2

2 “n’

where n-1

s,= c r$b&- 1) + f: vlbi(Ci + 1) i=l i==n+l

and

5 b,t,-‘i’ b,t, k=i+l k=l

ci= N

k=l

(4)

(5)

The potential lost by the applied loading during buckling is given by the expression

The total potential energy change at buckling may be evaluated in terms of the nodal displacements and slopes by substituting for displacements from (1) and (2) into energy expressions (4)-(6), and applying the principle of minimum potential energy by differentiating the potential energy with respect to each dis-

E,,=lOON mm-’

E,,=20 N mm-’

I!?,,= 10 N mm-’

V 12=@%

For each section examined results are shown for the stiff axis, i.e. EI1, aligned along the member and also for this axis aligned across the member.

Column buckling

Figure 3 shows non-dimensional buckling stress coefficients for a flat strip of material. Here the classical column theory gives the buckling stress for a simply supported column as:

z2E, x2E, t ’ CT =-=- -

0 a (L/r)2 12 b

XK

K 2.5 -

- Kll

-- K22

‘\ <.

0 0 20 40 60 80 loo 120 140 160 180 200

Length

Fig. 3. Buckling of a slender column.

96 J. Rhodes

where

K=(l -v,v,) x L (bTxE.

For the material under examination, taking K,, as the buckling coefficient for loading in the stiff direction and Kz2 as the buckling coefficient for loading perpendicular to the stiff direction we have, from the classical column theory

K,, =2.1958 ; 0

2

K,,=o*4392 ; .

0

2

The above equations are based on column theory which takes no account of Poisson’s ratio effect, and more accurate plate based analysis using the method presented here gives slightly greater buckling coefficients, particularly for very short columns. For example, if L =b the classical results give K,, = 2.1958, K22 = O-4392 compared with 2.218 and 0.441, respectively from the analysis used, i.e. differences of 1 and 0.5%. If L 23b the differences between plate and column analysis are negligible.

From the expressions above it can be seen that K,, =5K22, according to column theory. This is self-evident since El1 =5E22. Aherna- tively, and more usefully in the general situation, we may observe that K22 for a column length L is the same as K,, for a column of length 6L. This may be generalised for columns of any specially orthotropic material as follows: K22 for a column of length L is the same as K,, for a column of length G L.

In the remaining examples considered here the buckling coefficients K,, and K22 are evaluated on the basis that the critical, or buckling stress may be determined from:

rc2a T2

cJcr= 12(1 -v,,v21) B 0 xK,

where T and B are reference values of thickness and cross-sectional breadth, and the values used for each example will be specified in the text.

For the column considered the stresses pro- duced by buckling were bending stresses. If we consider a thin-walled hollow box section column buckling induces substantially membrane stresses within the walls. For a square hollow section with wall mean width, B, of 20 units and thickness, T, of 1 unit the non-dimensional

buckling coefficients determined on the basis of simple column theory are

2

K,,=351*33 + .

0

2

The results obtained using the method described in this paper are shown in Fig. 4 and are extremely close to the values given by these equations. The relationship between KI,, K22 and L described above also holds for this case. The buckling curve for loading in one direction can be used to derive the buckling curve for loading in the orthogonal direction by modify- ing the buckle length as illustrated in Fig. 5. The buckling coefficient K, for a column of length L is equal to the buckling coefficient Km for a column of length ,‘qi x L.

Plate buckling

The situation with regard to plate buckling is illustrated in Figs 6-8. Figure 6 shows the varia-

2000 ,K b=20

17

t=1 b=20 - Kll --K22

Note:- In deriving this graph linear strips were used and local buckling effects have been avoided so that only Euler buckling effects are shown

\ \

0 t 0 20 40 80 80 100 120 140 160 180 200

Length

Fig. 4. Buckling of box section column.

i \

- Kll

K22

1500

Fig. 5. Relationship between K,, and Kz2.


K SO-

25 -

ZcJ- I :\F - Kll

-- Kz?

x K11dWhSdhWlK22

15- \ \ ,/' \

10 - /'

\ /'

\ &xl I=1 //'

1' 5-

\ _-y \ ---

.-___-

0 0 20 40 60 80 loo 120 140 160 180 200 220 240

Length

Fig. 6. Simply supported plate under uniform compression.

tion of buckling coefficients K,, and K..* for plate simply supported on all edges and sub- jected to uniaxial compression. For this example B=b=20 and T=t=l. In this figure the crosses represent the values of Kil obtained by extending the length co-ordinates of specific points on the K..* curve by the factor &,,/E,,. As can be seen the points fall precisely on the K,, curve calculated using computer analysis. The same conclusions can be drawn from Fig. 7 which shows the variation in buckling coefficients with buckle half wavelength for simply supported plates under in-plane bending. Here B and T are as for the previous figure.

The precision with which this simple proce- dure relates buckling coefficients for the different load directions implies that the displacements across the member at buckling for loading in each direction are of the same shape, or at least that the displacement forms at the related half wavelengths are the same. This rai- ses the question as to what happens if the displaced form varies significantly with length. To examine this question Fig. 8 shows the variation in buckling coefficients with buckle half wavelength for a plate with step changes in thickness. Here the plate width is 150 units and the central 50 units is 0.2 units thick while the rest of the plate is 1 unit thick. In specification of the buckling coefficients B=150 and T= 1. For short plates, i.e. length less than about 80 units, the buckling displacements are confined to the central portion, which deflects as if it were fIxed at its junctions with the outer por- tions. For longer half wavelengths buckling occurs across the complete plate width, as illustrated in the figure. Here, as in previous graphs, the curve for K,, is identical to that for KZ2

- Kll

10 - Kz?

x Kll-fm+mKZ!

0 I I I I 4 I I I I I I I

0 20 40 60 80 100 120 140 150 180 200 220 240

Length

Fig. 7. Simply supported plate under pure in-plane bending.

i/-- , / I__;, 0 100 200 300 400 500 600

Length

Fig. 8. Simply supported stepped plate under compression.

stretched along its horizontal axis by a factor of /a. The crosses on the K,, curve were obtained by this method, i.e. plotting the KZ2 value for a buckle half wavelength L at a new buckle half wavelength &,,I,!& L. As may be observed all of these crosses lie precisely on the K,, curve.

Mixed buckling modes

The same effect may be observed even when different types of buckling arise with change in wavelength, as may be observed in Figs 9-12. Figure 9 shows the variation in buckling coefficients (B=20, T=l) with variation in half wavelength for a plain channel section strut. Here the short wave results describe local buckling behaviour while the longer wavelength results describe torsional-flexural buckling

98 J. Rhodes

K 3r

I - K11

2.5 _ Kp t=l

x K11dmhdtmnK22

2 /

20

20

\\\-_ ‘.

u --

I I _-

100 200 300 400 500 600 700 800

Length

Fig. 9. Plain channel section strut under compression.

behaviour. Figure 10 displays the variation in buckling coefficient with variation in buckle half wavelength for a lipped channel column, and shows local buckling behaviour at short wavelengths, distortional buckling, with in-plane movement of the lips at slightly longer buckle half wavelengths, and torsional-flexural buckling at still longer buckle half wavelengths. If the buckle lengths were increased further the torsional-flexural buckling would be replaced by purely flexural buckling about the minor axis. For this member the reference dimensions used were B=lOO, T=l.

Figure 11 examines a T-type section under compression. B= 100 and T= 1 were used as reference dimensions. The geometry of this section is such that under compression buckling will always be local, distortional, torsional-flexural, or some combination of these. As in the case of all previous sections examined, the curve for sections which have the stiff material direction longitudinal is simply an extended version of that for sections with flexible material direction longitudinal.

The final problem examined is that of a Z- section beam under pure bending, taking B=60 and T=2. It is assumed that one flange of the beam is constrained to remain horizontal and prevented from moving laterally. Such restraint conditions are provided, for example, by roof cladding in the case of Z roof pulins. Here again different buckling modes, e.g. local buckling, distortional buckling and lateral-torsional buckling, arise as the wavelength varies, and once more the relationship previously found between the behaviour of members with different material directions applies.

k ‘t

-. --__ ------

01 -

’ ’ 1 1 1 ’ J ’ I 1 ’ ’

0 100 200 300 400 500 600 700 au 990 1000 1100 1200

Length

Fig. 10. Lipped channel section strut under compression.

- Kl,

X K,,*llvadfmmEz l-p:.i; ---

‘.

:: ‘\ (- r

2- _ J q "-....___

OL I I

0 300 600 900 1200 1500 1800 2100 2400 2700 3OW

Length

Fig. 11. Compressed T-type section.

K 60

7m

-Ia, --w x KllddwdfmnKzz 6- _ 3 !_.=;

5-1 I",

i \

4;' / '\

z. 15

/' /

11 /

3- \ \ /

2-T 1

\ '-_- I'

JI 1

0' ,

0 1 2 3 4 5

Length (Thousands)

Fig. 12. Z-section beam in bending.

SUMMARY

The application of an analysis method to the examination of buckling of thin-walled orthotropic plates and structural sections has been


displayed. The method is ideally suited for microcomputer use. To illustrate the method a series of problems were examined in which the specially orthotropic material was examined with the stiff direction set longitudinally along the member, and also with the stiff direction set at right angles to the member longitudinal axis. It was found that regardless of the mode of buckling which arose, and regardless of the combination of modes which occurred for a given member, the buckling coefficient for a buckle length L corresponding to a material with modulus of elasticity E, in the x direction is the same as that corresponding to a buckle length L x $&,/E, if the material is set at right angles to this direction.

REFERENCES

Cheung, Y. K., Finite Strip Method in Structural Analy- sis. Pergamon Press, Oxford, 1976. Mahendran, M. & Murray, N. W., Elastic buckling analysis of ideal thin-walled structures under combined loading using a finite strip method. Thin-Walled Struct., 4 (1986) 329-62. Graves Smith, T. R., The finite strip analysis of structures. In Developments in Thin-Walled Structures - 3,

4.

5.

6.

7.

8.

9.

10.

Il.

12.

3rd Edn. Eds J. Rhodes & A. C. Walker. Elsevier Applied Science, Amsterdam, 1987. Sirdharan, S., A semi-analytical method for the post- local-torsional buckling analysis of prismatic plate structures. Znt. J. Num. Meth. Engng, 18 (1982) 1685-97. Gierlinski, J. T. & Graves Smith, T. R., The geomet- ric nonlinear analysis of thin-walled structures by finite strips. Thin-Walled Struct., 2 (1984) 27-50. Loughlan, J., The buckling behaviour of composite stiffened panel structures subject to combined in- plane compression and shear. Composite Struct., 29 (1994) 197-212. Davies, J. M. & Leach, P., Some observations of generalised beam theory. Proc. 11th Int. Speciality Conj on Cold Formed Steel Structures, St. Louis, Missouri, October 1992. Davies, J. M., Jiang, C. & Leach, P., The analysis of restrained purlins using generalised beam theory. Proc. 12th Int. Speciality Conf on Cold Formed Steel Structures, St. Louis, Missouri, 1994. Rhodes, J., A simple microcomputer finite strip analysis. In Dynamics of Structures, Ed. J. M. Roesset. ASCE, 1987. Rhodes, J. & Khong, P. W., A simple semi-analytical, semi-numerical approach to thin-walled structures sta- bility problems. Proc. 10th Int. Conf on Cold Formed Steel Structures, St. Louis, Missouri, 1990. Khong, P. W., Development of a microcomputer finite strip analysis. PhD Thesis, University of Strath- Clyde, 1988. Chong, S. K., Buckling behaviour of cold-formed steel sections using the finite strip method. MPhil Thesis, University of Strathclyde, 1991.

a semi-analytical approach to buckling analysis for composite structures

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