Elastic and elasto-plastic buckling of thin-walled columns subjected to uniform compression

Thin-Walled Structures 3 (1985) 93-108

Elastic and Elasto-plastic Buckling of Thin-wailed Columns Subjected to Uniform Compression

R. Gr~dzki and K. Kowal-Michalska Institute of Applied Mechanics, Technical University of,E6d~, Poland

ABSTRACT The problem of local stability loss in the elasto-plastic range of a thinwalled column, loaded by uniform compressive stresses, is examined on the basis of the J2 deformation theory and the J2 incremental theory of plasticity. The problem is solved in two different ways. Several types of closed and open cross-sections are considered. The results of numerical calculations are presented in graphical form, showing the relationship between the critical stress and the slenderness ratio for the column section. All possible buckling modes in the elastic range and local buckling in the elasto-plastic range are demonstrated in the case of a column of channel form cross-section.

NOTATION bi E Es E, hi i m Wi xi, Yi Eij

Width of wall 'i' of the column. Y o u n g ' s modulus of elasticity. Secant modulus for uniaxial compression. Tangent modulus for uniaxial compression. Thickness of wall 'i' of the column. Length of the column section. N u m b e r of half-waves in the direction of an axis xi. Deflection function of wall 'i' of the column. Coordinate system for wall 'i' of the column. Strain increment tensor i,] = 1,2.

93 Thin-Walled Structures 0263-8231/85/$03.30 © Elsevier Applied Science Publishers Ltd, England, 1985. Printed in Great Britain

94

o" O'cr

R. Gr#dzki, K. Kowal-Michaiska

Poisson's ratio. Compressive stresses applied to the column. Critical stress. Stress increment tensor.

1 INTRODUCTION The inelastic buckling of plates has been investigated among others by Ilyushin, ~Stowell: and Bijlaard 3 on the basis of the constitutive relations of the J2 deformation theory of plasticity and by Handelman and Prage# on the basis of the constitutive relations of the J2 incremental theory of plasticity. It has been well known that calculations based on the less respectable deformation theories of plasticity gave reasonably good agreement with test results. 5 The major objections to stability predictions based on these theories are due to the fact that deformation theory does not represent a physically acceptable theory of plasticity. Saunders, 6 and later Sewell, 7 have constructed incremental theories with vertices on the yield surface for which the J2 deformation theory predictions of bifurcations are valid as long as the total loading condition is met. In 1979, Christoffersen and Hutchinson s proposed a phenomenological c o m e r theory of plasticity, called J~ corner theory, in which for a restricted range of deformations, such as are found in the usual buckling problems, the bifurcation loads are equivalent to those obtained on the basis of the J2 deformation theory of plasticity. In recent years work concerning the inelastic buckling of compressed plates or columns has been done by Shirivastava, 9 Graves-Smith ~°'1l and Marinetti and Olivetto. 12The plastic postbuckling behaviour of structures has been a point of interest in the fundamental work of Hutchinson 5 and recently in the extended papers of Needleman and Tvergaard. ~3.t4 In the present paper the problem of stability in the elastic and elastoplastic range of thin flat-walled columns compressed by uniform stresses is examined on the basis of the conventional theory of plates. The constitutive relations of the J2 deformation theory and the J2 incremental theory of plasticity are used. The results of calculations in the elasto-plastic range are presented, showing the relationship between the critical stress and slenderness of a column section. All possible buckling

Buckling of thin-walled columns

95

Pig. 1. Columnsof closed and open cross-sections.

modes in the elastic range---e.g. Euler buckling, bending-torsional buckling and local buckling in the elastic and elasto-plastic range---are demonstrated for a column of a channel-form cross-section. The calculations are carded out for box cross-sections of rectangular and trapezoidal form, and for open cross-sections such as cruciform, channel and omega forms.

2 CONSTITUTIVE RELATIONS The strains are considered to be small and the assumptions of isotropy and compressibility in the elasto-plastic state are made. The material is considered to obey the Huber-Hencky-Mises yield condition for both theories. According to the Shanley concept of continued loading during buckling, no unloading takes place; this should provide the lowest critical load. ~5 Due to buckling, the perfectly flat plate hardens under uniform compression in the longitudinal direction. The relationships between the stress and strain increments are 0"11 =

E(Alleu + A i 2 e 2 2 )

0"22 = E ( A I 2 ~ I I +A22~22)

0"33 = 2 EA33E12

(1)

96

R. Grqdzki, K. Kowal-Michalska

where A 11

=

39~+~o, 3 ~,~, + 2(1 - 2 v ) ~ , - (1 - 2v) 2

2~,-2(1-2v) A~2 = 3~p~,+ 2(1 - 2v)~o,- (1

-

2v) 2

(2) 4~t

A22 = 3~o~ot+2(l_2v)~ _(1_2v)2

1 A3 3 -

3~0,- (1 - 2v)

and ~o, = E / E ,

~

= E/E,

(3)

are obtained from the uniaxial stress-strain curve of the material of the column. The relationships above correspond to the deformation theory of plasticity and have the same form as in the paper of Shirivastava. 9 The relationships for the incremental theory are obtained by putting ~ps = 1. L2 Thus, while the analysis is carried out on the basis of the deformation theory alone, the results for the incremental theory may be obtained simply by taking ~s = 1 in the results for the deformation theory. Substituting in eqns (1) and (2), ~, = ~, = 1, the relationships concerning the elastic range are obtained.

3 MAIN ASSUMPTIONS AND BOUNDARY CONDITIONS It is assumed that the column consists of i flat plates. Next, the following assumptions are made: the critical load is larger than the load corresponding to the proportional limit of the material of the column; the column section of length l, contained between consecutive modal lines, is considered;

Buckling of thin-walled columns

97

the junction lines of column flanges remain straight during buckling. This means that along each junction line the following conditions must be fulfilled (a) deflections w, are equal to zero ( W i ) yi

=

O.bi=0

(4)

(b) slopes of neighbouring walls are equal to each other, for example ay,

y,= b,

(5)

\ ay~+, ]y,+,= o

(c) the equality of bending moments must be provided ( M y , ) y i = bi = (Myi+l)ri+t = o

(6)

Additionally, in the case of an open cross-section the conditions along the free edge have to be taken into account as follows:

03Wi

( g y ) y i = b i "~"

03Wi

A22~/3 "4"(A 12+ 4 A 3 3 ) ~ i

a2wi (M,i)y~ = b, = A22 ~-Z:T + A

oyi

o2wi 12T~xt

: 0

(7)

(8)

= 0

1

I 2'h2

4 , h ,

'h2

~bb h,

J

l~g. 2. T y p e s of cross-section considered.

R. Gr#dzki, K. Kowal-Michalska

98

Several types of cross-sections of the columns that are considered are presented in Fig. 2.

4 METHODS OF SOLUTION The values of parameters of the critical load are determined in two different ways: I. on the basis of a set of equations derived from homogeneous boundary conditions (the so-called L6vy method) and II. by means of the variational method. Some remarks about each of the methods used are given below. In the case of the L6vy method the solution of the differential equation of equilibrium for the deformed middle plane of a plate is used. In the inelastic range this differential equation has the form 04Wi

0 4W i

04Wi

Or 02Wi

m , - ~ + 2 ( A , 2 + 2 A 3 3 ) ~ x ~ , + a z z - - ~ y + 12~-~ ~

= 0

(9)

The deflection function wi of a flange may be expressed as wi = F(yi)sin

mTrx i

l

(10)

Substituting eqn (10) into eqn (9), an ordinary differential equation is obtained. The general solution of this equation is

F(yi) = Aisinh aiyi + Bisinfliyi + C~coshaiyi + D~cosBiyi

(11)

where Ot 2 =

+

fl/2 = _

m2,;r 2 AlL+A33

l2

A22

/mazr4[[A,,+a33~z_al,]

m2rrz lZor

(12)

Next, for each of the columns considered the appropriate boundary conditions are taken into account. A set of homogeneous algebraic equations in terms of the unknown (independent) parameters Ai-Di is written. The main determinant of this set must vanish, and from this condition the parameter of the critical load can be found.

Bucklingof thin-walledcolumns

99

axis of symmetry~/

x~

/x,

Y / / ~ / / / / /~-,/ / / . 7 / o,_.._ / -

/

/

/.

/

o,

/ -,-

/

7--

/

/

//.

/

/ /

b,/

/

.///

7 /

Fig. 3. A multi-spanplate loadedby uniformcompressivestresses. In the second method of investigation the approach proposed by Protte 16is used. According to this method the problem of stability analysis of the column is reduced to the investigation of the buckling of a multispan plate with different widths and thicknesse~ see Fig. 3. It is possible to formulate the problem in this way due to the kinematic and dynamic boundary conditions which must be fulfilled along the junction lines (eqns (4)-(8)). The deflection function for a single plate is assumed in the same form as before, eqn (10). For the coordinate systems x~,y~ shown in Fig. 3, the functions F~(y,) are assumed to be polynomials of an arbitrary degree having the form y7 F(yi) = A i

~

a..,---~i + Bi n=l

n=l

y7

B.,, ,, bi

(13)

The coefficients ~t.,~and/3.,i are determined as follows. For beams of unit length, loaded by unit forces (see Fig. 4) deflection curves are found. The coefficients in these polynomials are taken as coefficients ot.,~and /3,,i in the functions F~. Parameters A~ and B; are unknown. Taking into account conditions of deformation compatibility for neighbouring walls, the relationships between parameters A~ and B~ are obtained. However, two parameters always remain independent. The total energy, E, of the structure is expressed as

E = ~ U,- Z W ' i

i

(14)

100

R. Grqdzki, K. Kowal-Michalska

2-2

c2/

= ?- 2~÷~~

.

lb =i

{V Fig. 4. Beams of unit length loaded by unit forces, and their deflection functions.

where the strain energy of a single plate in the elasto-plastic range has the form U,-

;o au~ -~xi ] ~2Wi~2Wi " 1102W-'-)~2 +2A12 Ox2 Oy~ +A22\ OY2 24

+4A33 (~ °2w' ]

]=ldx,dy, J

(15)

The work of external forces applied to the plate i is

h~ Sot sb~ 'aw~ `2 Wi=--2 o tr~--~xi ) dx,dy,

(16)

By applying the variational Timoshenko-Ritz method to the problem, the minimum energy of the system considered is found. The critical load is determined from the resulting second-degree equation.

5 RESULTS OF NUMERICAL CALCULATIONS Numerical calculations have been carried out on the basis of both methods described earlier. Closed cross-sections of rectangular and trapezoidal forms, and open cross-sections such as cruciform, channel

Buckling of thin-walled columns

101

and o m e g a forms of different geometrical parameters were examined (Fig. 2). In these calculations the material properties of mild steel are used, obtained from uniaxial compression. The parameters E,, Es and v have been established for the range of stresses tr varying from 200 MPa to 280 MPa (Table 1). It may be noticed that for values ~0,--->~ eqns (2) become indeterminate. This difficulty is avoided by putting in this case an arbitrarily large value (for instance one million) or when ¢,goes to infinity, the coefficients in eqns (2) take the form 1

limAlt = 3~o~+2(1-2v) ~Ot ---->

limA22 =

4 3~0~+ 2(1 - 2 v )

~Ot ----~

lim A t2 = 3~os+2(1-2v) ~ t ----.> 0o

1 A33 =

3~o,- (1 - 2v)

For the cases considered here, it is assumed that in the elasto-plastic range the n u m b e r of half-waves, m, is unity. In Fig. 5 the curves of stability loss in the elasto-plastic range are presented for rectangular cross-section of dimensions given in the figure. Curves 1 and 3 are based on the deformation theory of plasticity---curve 1 for compressible material and curve 3 for incompressible material with v = 0-5. Curves 2 and 4 are obtained on the grounds of the incremental theory---curve 2 for compressible material and curve 4 for v = 0.5, respectively. As is well known from the literature, the deformation theory of plasticity for a compressible material gives results in good agreement with test results. Thus, further calculations are carried out according to this theory. Comparing the two methods of solution, it has been established that in the case of closed cross-sections the results of calculations are in complete agreement. For open cross-sections the results obtained from the variational m e t h o d are greater by several (up to five) per cent. This difference m a y be explained by the fact that in the variational m e t h o d the boundary

R. Gr~lzki, K. Kowal-Michalska

102

TABLE 1 Parameters for Mild Steel at Different Stresses o'(MPa)

~o,

~p,

v

200.0 210.0 220.0 228.0 234.0 238.0 239.0 240.0 240"0 240"0 240"0 240"0 240"0 240"0 241"0 242"0 247"0 252"0 257"0 263"0 268"0 274"0 280"0

1.000 00 1.470 59 2.127 66 3.125 00 4.545 40 8.333 33 16.129 00 34.482 60 ~ oo ~ o0 oo o0 100"100 00 52"631 60 41"666 67 41"666 67 41"666 67 41"666 67 41 "666 67 41"666 67 41"666 67

1-000 00 1.010 10 1-052 63 1.111 11 1.162 79 1.234 57 1.315 79 1.408 45 1"587 30 1"754 39 2"173 91 2"631 58 3"000 00 3"448 28 3"846 15 4"347 83 5"000 00 5"882 35 6"666 67 7"142 85 7"692 30 8"333 33 9"090 91

0.270 0.272 0.274 0.280 0.285 0-290 0.296 0.304 0"318 0"330 0"358 0"375 0"390 0"400 0"405 0"408 0"414 0"417 0"420 0"424 0"426 0"428 0"430

conditions along the free edges of the column are not fulfilled. Since this is a small difference, it is not shown in the figures. The curves of critical stresses O'er versus b/h ratio are presented for a column section whose geometrical parameters are determined by the ratios of thicknesses and widths of c o m p o n e n t plates and by the ratio of length of a section to width of a chosen plate. In Fig. 6 the curves O'er = f(b/h) for rectangular cross-sections are shown. It is easy to see that a column of a cubic form with equal wall thicknesses buckles as a square plate simply supported ~ee curve 1 which is in agreement with a curve given in the literature. 17 Results for trapezoidal cross-sections are presented in Fig. 7. After an analysis it may be found that in cases 1 and 2 the side plates which have the smallest stiffness provoke the stability loss of the column considered. The characteristics of inelastic stability for channel cross-sections are

Buckling of thin-walled columns

103

[MPa )

286 L._=I b, i I

260

b_tht b-2 ~=2

"2

220

20(

I0

20

30

1.0

50

60

70

Fig. 5. Stability curves obtained on the basis of the deformation and incremental theory of plasticity.

~,IMPo] Icurve

286

I I

L ._~

11,1

•

2

I

3

122

2

1

260

240

226

& 200

I0

20

30

40

50

60

IFig. 6. Stability curves for rectangular cross-section.

70

104

R. Grqdzki, K. Kowal-Michalska

~,[IVlPaJ L bfb, h,h~ curve bz b2 ~ hz hz

280

250 ,

r

2 3

I~11! IT!

4

I

l

~- -~ll

:

,\

240

220

h~

220

20

40

50

80

lBlg.7. Stability curves for trapezoidal cross-section. ~,/MPa] L b~ h, curve -E2 b2 h2

280

~2

25O

] 240

1

1

2 3 4

1 I 1 I ~- I 1 I ~

4

I 1 h

220i 4

206

tO

20

3O

40

Fill. 8. Stability curves for channel form cross-section.

I

2

Buckling of thin-walled columns

G,,IMPa/ 286

105

± b, ~ h,

curve b2: bR b2 h-2 h~

~ ~

I ,lflit 2 il I 1 12

~

3

4 5

26O 4

3

5

1711~1 1~- I I 121

I !

3

21,0

220

200

10

2O

30

40

50

6O

Fig. 9. Stability curves for omega form cross-section.

demonstrated in Fig. 8. It may be seen that in case 1 the flanges of a channel buckle, approximately, like plates clamped along one of the unloaded edges, and in case 4 they buckle like simply supported plates. Curves 2 and 3 correspond to the buckling of plates with both unloaded edges elastically fixed. The relations between the critical stresses and the width to thickness ratios of the lips of top-hat type cross-sections are shown in Fig. 9. In cases 1, 3, 4 and 5 the lips initiate local stability loss of the column. Only in case 2 do the elements which are supported along all edges initiate buckling. The results for cruciform cross-sections are presented in Fig. 10. The same p h e n o m e n o n as for a cubic column may be noticed--since a cruciform consists of four identical plates, the stability loss of the column corresponds to the buckling of a single simply supported free plate. In Fig. 11 the relationship between critical stresses and the ratio of length of the column section to the thickness of the column wall is presented. This relationship covers all possible buckling modes for a column with a channel cross-section, if its geometrical parameters are fixed. For the mild steel the local buckling in the elasto-plastic range occurs up to the value l/h = 40 and when the proportional limit is reached elastic local buckling appears. For values of l/h higher than 680, i.e. for very long and slender columns, the overall buckling is of the bending

106

R. Grqdzki, K. Kowal-Michalska ~, [rePel f ....

286 1

1

1

2{i 3 Z, 5

250

2 1 1 2 1 1

240

220

200

b~ h,. 10

20

30

40

50

60

F i g . 10. Stability curves for cruciform cross-section.

I

28O

elas~o-plos~c /ocal buc,
6~

\

24O

200 ffo,op

~.~.,,,,'~

cr

l~

•~

bending- ~orslor~71

IOCdl buckhng

160

~

.... \

120

~ 8o

I I l

I

~0

4-

I

I i

/

i

± h

I

5

~

50

100

I

i

500

1000

5000

10000

Fig. 11. All possible buckling modes in the elastic and elasto-plastic range for channelform cross-section.

Bucklingof thin-walledcolumns

107

torsional mode. The Euler buckling occurs when l/h surpasses the value 4300. The chain-broken line shows the Euler buckling m o d e found on the basis of a so-called tangent modulus theory.

6 FINAL REMARKS Comparing the relations o-, = f(b/h) presented for different crosssections it may be noticed that for the material with explicitly m a r k e d yield platform the character of curves in all cases is the same. It should be emphasized that while the calculations have been carried out only for one material, it is possible, on the basis of equations discussed earlier, to obtain the relations o-~ = f(b/h) for an arbitrary material of known properties in the elasto-plastic range. T h e results of calculations are in good agreement with the results found in the literature in the elastic range, and---where it is possible to c o m p a r e - - i n the elasto-plastic range. The solutions given here are valid in the cases of the uniform compression of the structure. Other types of loading would n e e d further consideration.

ACKNOWLEDGEMENT T h e authors are grateful to Prof. Dr J. Leyko for his scientific support.

REFERENCES 1. Ilyushin, A. A., The elasticplastic stability ofplates, Tech. Note, N A C A - 1188, 1947. 2. Stowell, E. Z., A unified theory of plastic buckling of columns and plates, Tech. Note, NACA--1556, 1948. 3. Bijlaard, P. P., Theory and tests on the plastic stability of plates and shells, J. Aeronaut. Sci. 9 (1948) 529--41. 4. Handelman, G. H. and Prager, W., Plastic buckling of rectangular plates under edge thrusts, Tech. Note, NACA--1530, 1948. 5. Hutchinson, J. W., Plastic buckling, Adv. Appl. Mech., 14 (1974) 67-144. 6. Saunders, J. L., Plastic stress-strain relation based on linear loading functions, Proc. 2nd US Natn. Congr. Appl. Mech., University of Michigan, Ann Arbor, 1954, pp. 55-460.

108

R. Grqdzki, K. Kowal-Michalska

7. Sewell, M. J., A plastic flow rule at a yield vertex, J. Mech. Phys. Solids, 22 (1972) 469-90. 8. Christoffersen, J. and Hutchinson, J. W., A class of phenomenological corner theories of plasticity, J. Mech. Phys. Solids, 27 (1979) 465-87. 9. Shirivastava, S. C., Inelastic buckling of plates including shear effects, J. Solid Structures, 15 (1979) 567-75. 10. Graves-Smith, T. R., The ultimate strength of locally buckled columns of arbitrary length, Thin-walled Steel Structures, pp. 35-60, London, Crosby Lockwood, 1969. 11. Graves-Smith, T. R., The post-buckled behaviour of a thin-walled boxbeam in pure bending, Int. J. Mech. Sci. (1972) 711-22. 12. Marinetti, A. and Olivetto, G., Inelastic buckling of thin, prismatic flatwalled structures, Stability in the Mechanics of Continua (F. H. Schroeder, Ed.), Berlin, Springer-Verlag, 1982. 13. Tvergaard, V. and Needleman, A., On the buckling of elastic-plastic columns with asymmetric cross-sections, Int. J. Mech. Sci., 17 (1975) 41924. 14. Needleman, A. and Tvergaard, V., Aspects of plastic postbuckling behaviour, Mechanics of Solids, pp. 453-98, Oxford, Pergamon Press, 1982. 15. Hill, R., A general theory of uniqueness and stability in elastic plastic solid, J. Mech. Phys. Solids, 6 (1958) 236-49. 16. Protte, W., Zur Beulung versteifler Kastentr/iger mit symmetrischem Trapez-Querschnitt unter Biegemomentetr--Normalkraft und Querkraftbeanspruchung, Tech. Mitt. Krupp-Forsch. Ber. 34 (Band 11.2)(1976). 17. Volmir, A. S., Biegsame Platten und Schalen, Berlin, VEB Verlag for Bauwesen, 1962.