A finite strip analysis of locally buckled plate structures subject to nonuniform compression

Srinivasan Sridharan

Department of Civil Engineering, Washington University in St Louis, Missouri, USA (Received September 1981; revised December 1981)

The post-local-buckling behaviour of thin plate structures is governed by the mode of loading which can be either one of prescribed load eccentricity or displacement pattern of end sections. A finite strip approach to the analysis of post-local-buckling behaviour under either mode of loading is presented. A particularly simple initial postbuckling analysis based on the perturbation technique is discussed in detail. Examples are presented to illustrate the convergence of the solution and contrast the behaviour of the structure under the two modes of loading. In particular it is demon- strated that a channel-section strut is stiffer and attains a higher collapse load under uniform end shortening than when loaded through the centroid of the cross-section.

Key words: local buckling, post buckling, plate structures, nonuniform compression, modes of loading, perturbation technique

Introduction

It is well known that thin plate structures are susceptible to local buckling, but have significant reserves of strength before they collapse. A study of the post-local-buckling behaviour is therefore of vital importance for the limit state design of these structures.

Prismatic structures built up of rectangular plates constitute one of the most important classes of engineering structures. The use of finite strips in the analysis of post- buckling behaviour of these structures has proved attrac- tivelW4 in view of complexity of the cross-sections that can be analysed with ease and the computational economy that results from the use of the appropriate characteristic functions to describe the variation of displacements in the longitudinal direction. The earlier finite strip analyses were restricted to the case of uniform end-shortening such as would occur if the structure is compressed between the rigid plattens of a testing machine. In the present paper the finite strip method is extended to the case of nonuniform compression with either a prescribed displacement or load eccentricity.

In the case of plate structures which are not symmetric with respect to one of the principal axes, the application of uniform end compression displacement would in general result in a shift of the resultant of the stresses in the cross- section from the centroid after buckling. Conversely, when the structure is loaded through the centroid of the cross-

section, it would undergo some rotation after buckling. This is the result of the redistribution of stresses after buckling which now tend to concentrate near the junctions. Thus in general, it becomes necessary to distinguish between the case of prescribed displacement eccentricity and load eccentricity. The former may be thought of as produced by rigid end plates pivoted at a given point and the latter by the application of a force at a given point on the end plattens. The case of uniform end compression and that of pure bending constitute respectively the limiting cases of these modes of loading, when the pivot or alterna- tively the point of application of load, is at infinity.

The problem of post-local-buckling analysis of plates and plate combinations under prescribed eccentricity of load or pattern of end displacement has been dealt with by Rhodes et al. ‘9’ They employ ed the classical approach of solving von Karman plate equations under the restriction that the buckling mode remains unchanged and the classical approximation7 of boundary conditions along the longi- tudinal junctions which uncouple ‘w’ (the normal displace- ment) and ‘@’ (the Airy’s stress function). The latter approximation appears to be quite justified for the range of practical geometries usually employed in practice.3 The assumption of the fxed buckling mode can lead to serious inaccuracies when the applied load exceeds a certain multiple of the critical load. However, the initial post- buckling stiffness obtained on this basis gives an upper-

0141-0296/82/040249-07/$03.00 0 1982 Butterworth & Co. (Publishers) Ltd Eng. Struct., 1982, Vol. 4, October 249

Finite strip analysis: S. Sridharan

bound to the true values in the postbuckling range and is a useful index in the vicinity of bifurcation.

The mathematical difficulties associated with the creation of a computational scheme which can deal with any given cross-section, however complex, and the consideration of the modification of buckling mode constitute the two major limitations of the classical approach. The finite strip method offers a viable alternative having neither of these lhnitations. However, in the present study, the usual formulation is further simplified by the use of a perturba- tion technique s with the validity of the results produced restricted again to the vicinity of bifurcation. For a given buckling load and mode, the method reduces to solving a set of linear sunultaneous equations giving the inplane stress system associated with the buckling nrode. The extension to the advanced postbuckling range would involve solution of nonlinear equations along the lines of earlier finite strip applications]-3' 9, ~0 The aim of the present paper is to present a very shnple but sufficiently general method of post-local-buckling analysis under non- uniform compression; examples are presented to illustrate the convergence of the method and the serious differences that arise in the behaviour of the structure due to the actual mode of loading. The latter observation has an immediate relevance in the ultimate strength design of thin walled colunms.

T h e o r y

Choice o f displacement functions

The functions characterizing the variation of displace- ments in the longitudinal direction are chosen so as to satisfy the von Karman differential equations of the plate problem. The satisfaction of the end boundary conditions is not vital since the half wave length of local buckling is generally small in comparison to the length of the structure. Further for the local buckling problem, it is possible to uncouple 'w ' and 'v' (Figure lb ) of plates meeting at a junction by assuming:

w = 0 N y y = 0

for each plate at the corners. The former is the result of the out of plane displacements being of a higher order of magnitude than the inplane ones and the latter the result of the plates having much higher extensional than flexural rigidity. Such a simplified modelling of local buckling of plate structures has been suggested by Benthem 7 and has recently been shown to be applicable to a variety of con- figurations of practical interest) ' 3 The relaxation of the displacement compatibility conditions in the cross-sectional plane makes it possible to characterize 'w ' and 'v ' by independent harmonic functions, chosen from the point of view of their compatibility in satisfying the governing differential equations. Figure la and b shows a typical plate structure and a typical plate strip with its local coordinate system. With the notation indicated, it can be shown 3 that for the classical local buckling problem, the displacement functions of a typical strip must take the form:

w = Wm(r?) sin(rnmr~) (la) ( rn= 1,3 . . . . . N)

u = Up(~) sin(pmr~) + Uo ( lb) (p = 2, 4 . . . . . 2N)

v = Vp(r?) cos(pmr~) + Vo (lc) (p = 0, 2, 4, . . . , 2N)

I II /

a tllil" Figure I (a), plate s t ruc tu re unde r l inearly varying end sho r t en ing shown wi th principal axes of sec t ion and fini te strip d iscre t iza t ion; (b), a typical s tr ip (ith) wi th d imens ions and local coo rd ina t e sy s t em

Using the notation: n, number of half waves of buckling; ~, *7 are dimensionless spatial coordinates given by ~ = x/ l and r? = y /b s.

The functions win(r?), Upfr?) and Vp(r?) are taken as cubic and linear functions of r? in terms of the degrees of freedom as below:

Wm (r?) = win, l (1 - 3r? 2 + 2r? :~) + win, 2 (r? -- 2r? 2 + r?3) + w,n,3(3r?z 2r~3) + Wm.4(__r?z+ r?3) (2a)

up(r?) = Up, l (l--r?) + Up,2~ (2b)

Vp(r?) = Vp, 1 ( l --/ '7) -I- Vp,2 ? (2C)

The functions u0 and Vo which are linear across the strip, give rise to a uniaxial state of compression and are to be prescribed appropriately. They take the form:

Uo = [x , ( l - r?) + x2n] l(~ - ~1 ~3a)

[ r?2 ] l 2 7.) 0 = pb 8 Xlr? + - - ( ~ k 2 - - Xl ) - - _ _ ( X 2 - - ~ k , ) ~ ( l - - ~ )

2 2b~ (3b)

For the ith strip, Xl and X2 would take the form:

~k 1 = ~kO q'- O~i, 1 ~,oz q" ~i, IX~3 (4a)

X2 = Xo+ c~i,2X~ +/3i,2X~ (4b)

where Xo is the uniform end compression, X~ and X~ rotations about the c~ and 13 axes of the cross-section of the structure (Figure 1); c~i, l,/3i, l, etc. are constants for each strip depending on its location and must be chosen so as to ensure continuity of displacements along the longitudinal edges of the strip, i.e.

~i, 2 = OQ+I,1 ( 5 a )

13i, z =/3i+t,l (5b)

The condition of given displacement eccentricity is achieved by keeping the ratios between Xo, Xa and X~ constant while prescribing only one of these parameters; on the other hand the condition of given load eccentricity is achieved

250 Eng. Struct., 1982, Vol. 4, October

by prescribing one of these parameters while appropriately varying the ratios with the other two along the equilibrium path.

Energy funct ion

Using the basic assumptions of von Karman plate theory 1'2 an expression for the strain energy of a strip can be obtained in the form:

1 Ustrip =~.~ [aiJ - ()tobii + ~ e i j + Xflil)] w i w l

1 + - - {epq Up Uq + fprUp V r + grsVrVs}

2~

1 + - - [apijU p + brijUr] WjWk

3!

1 + - - ai jk lWiWjWk W l

4!

i , / , k , l = 1 , . . . , 4 M p ,q = 1 , . . . , (4M--2) r , s = 1 . . . . . 4 M

(6)

where M stands for the number of 'w' harmonics; and wi, Up and vr are the abbreviated symbols of the degrees of freedom associated with the various harmonics in the longitudinal direction and aij, . . . , ailkl are coeffi- cients which may be found from integration of appropriate polynomial and trigonometric functions. Because 'w' and 'v' are uncoupled and are described by differing harmonic functions it is neither necessary nor possible to define an arbitrarily oriented global coordinate system for the entire structure. Rather the global degrees of freedom are defined in the same sense as the local degrees of freedom. Along the corners the uncoupling of inplane and out of plane actions is achieved by requiring both 'w' and the inplane normal stress resultant, Nyy to vanish for either strip. The latter is achieved by treating the degrees of freedom corre- sponding to 'v' as independent for each strip at the corner.

On these bases it is possible to generate a potential energy function for the entire structure in the form:

1 Vstma = 2.w {Aii -- (X°Bii + )t°Bii + )taCii

1 + XoDii )} WiW i +~.[ {EpqUpUq + 2FprUpV "

1 + arsVrVs} +31. {ApiIUp +Bri iVr} WiWI

1 + - - AqkIWiWIWkW t

4!

i , j , k , l = l . . . . . Nw

p , q = l . . . . . N u

r , s = l . . . . . N v

where Nw, N u and N v stand for the total number of global degrees of freedom corresponding to 'w', 'u' and 'v' dis- placements, and Aij , . . . , A qkt are appropriate coefficients. Since the parameters )to, )t,~ and )tO which control the displacement of ends are prescribed, the potential energy is the same as the strain energy.

Finite strip analysis: S. Sridharan

Equilibrium equations

The equilibrium equations are generated by invoking the principle of stationary potential energy and take the form:

D Vstruet - - - (Ai l -- (XoBi i + XaCq + XoDi/)} Wij

bWi

+ (Ap iU p + S . f . } Wj

+ W k W f = 0 Aij Wj {i = 1 . . . . ,Ww} (7a)

Vstruct - - - EpqUq + F p r V r + t6 Api lWiW ] = 0

{p = 1 , . . . , N u} (7b)

a - - - r p # p + a,y + 8.lW,.W i = 0

(r = 1 . . . . . N v} (7c)

These three sets of equations describe the nonlinear post- buckling behaviour of the plate structure. Solution of these equations can be achieved either by stipulating or varying )to, ~ and X 0 appropriately to model the desired end condition.

Initial postbuckl ing analysis

In the immediate vicinity of bifurcation, the deflections are infinitesimally small so that it is sufficient to retain only the lowest order terms in equations (7a-c). Thus the equa- tions giving the critical load and the mode of buckling take the form:

{Aii -- (XoBii + XaCii + XoDii )} Wi, = 0 (i = 1 . . . . . Nw)

epqUq, + Fp, V,, = 0

FprUp, + Cr.Vr = 0

(p = 1 , . . . , N u )

( r= 1 . . . . . Nv)

(8a)

(8b)

(8c )

The subscript '1' is attached to the degrees of freedom to indicate that these are the first (or the lowest) order quan- tities in the hierarchy of the perturbation scheme and constitute the buckling mode. In these equations no coupling occurs between 'w' terms on the one hand and u -v terms on the other, so that:

Up, = 0 (p = 1 . . . . . Nu) (9a)

Vr ' = 0 (r = 1 . . . . . Nv) (9b)

and the critical load and the buckling mode (Wi,) are given exclusively by equation (8a). It is assumed that the pre- buckling end displacement pattern is given by the relation- ships:

X~ = ~ Xo

)tO =/~)to

so that the equation (8a) takes the form:

( A i i - )to(Bi/+ + t DD} W/, = 0 (i = 1 . . . . . Nw) (1 O)

Let the critical value of )to be given by Xf). The solution of the buckling problem is the same for the two modes of

Eng. Struct., 1982, Vol. 4, October 251

Finite strip analysis: S. Sridharan

application of load, i.e. whether it is one of prescribed eccentricity of load or pattern of displacement. Note further, because of the orthogonality of the sine functions, the buckling mode is constituted of a single harmonic in the longitudinal direction, given by the integer 'n ' .

If it is assumed that the buckling mode does not undergo any modification, the description of W i takes the form:

Wi = Wi s (11)

where ' s ' is a parameter associated with the geometry of the buckling mode, say the maximum amplitude of the buckle.

Substituting in equation (7b-c) produces the following equations:

S 2 Epq Uq + FprV r + ~v. ApiIWi, WL = 0 (12a)

(p = 1 . . . . ,Nu) $2

FprUp + GrsVs + ~v. Brill¥i'WJz = 0 ( l Z b )

( r = l , . . . , N v )

This gives rise to a solution:

Up =sZ.Up: (p = 1 . . . . . Nu) (13a)

V~=s:.Vr: ( r = l . . . . . Nv) (13b)

It is easily verified that the only harmonic that enters into the description of 'u ' is given by the integer '2n'; and those describing 'v ' are given by 0 and 2n, so that the second order displacement field (proportional to s 2) in any strip takes the form:

u2 = u2(v) sin(2nn~) (14a)

v2 = Vo0?) + v2(r/) cos(Zmr~) (14b)

The use of the procedure presented here thus uncouples the inplane and out of plane degrees of freedom so that the degrees of freedom for any strip is only 4 for the buckling analysis and 6 for the analysis of the second order inplane stress system.

The initially prescribed load eccentricity would be modified as a result of these second order displacements. If the section carries a nonzero axial force, the prescribed load eccentricity can be achieved by prescribing corrective rotations X* and X~. In the special case of pure bending it is necessary to prescribe a corrective axial compression, X* in order to counteract the axial force given rise to by the second order displacement field. Appropriate simpli- fications are possible for a section symmetrical with respect to either one or both of the principal axes. In the following treatment, however, it is assumed that all of the corrections viz. X~, X* and X~ will be assumed to be nonzero. As in equation (13a-b) these are written in the form:

X * - * 2 (15a) - Xo,2S

X * - * 2 (lSb) - - )kc~, 2 S

X~ • z = X ~ y (15c)

Again let :

X*,~ * * = Xo,2a (16a)

~k~, 2 * * = Xo,2/3 (16b)

The second subscript '2 ' is attached to denote the order of magnitude of the prescribed corrective rotations.

It now remains to relate the rate of growth of buckling displacements (given by 's ') to the end compression para- meters (Xo, Xa, ?,~) in order to obtain a complete descrip- tion of the initial postbuckling behaviour. This is achieved by rendering the potential energy stationary with respect to the buckling displacement parameter 's ' . The expression for the potential energy under the total displacement given by Xo + X~, Xc~ + X*, Xt~ + X~, takes the form:

t - [ A i i - ( X o + X * ) B i i - - ( , ~ + k ~ ) C i i Vstruct 2 !

-- ()tfi + ?t~) Di]l Wi Wi s 2 1

9! 2 z

1 +--{ApqUp + BrilV r } WiWIs 4

3! " "

1 + 4v. AiigtWi, Wi, Wk, Wt,s4 (17)

i , j , k , l = 1 . . . . . N w

p , q = l , . . . , N u

r , s = l . . . . ,N v

The condition of stationary potential energy is given by:

[Ai i -- (Xo + X*) B , -- (X, + X*) Cii

- - (X~ + X ~ ) Dill WiW L + 2[EpqUpUq2 + :FprUpVr: + CrsVrVs:ls 2

+ 2{Api/Up2 + BrijVr:} Wi, WI, s2 1

-{-~.Ai]klWi Wl WktWllS2= 0 ( 1 8 )

With the aid of equations (8a) and (12a-b) it can be shown:

AijWiWj, = (Xf)Bii + XcaCij + X~Dij ) WiW L (19a)

EpqUpUq: + 2FprUp Vr: + GrsVrVs2 = --l(ApijUp2 + BrijVr~) WiWj, ( 1 9 b )

Making use of equation (19a-b) in equation (18) there results the relationship:

Xo = Xg + )to, 2s 2 (.20)

where:

[1 {2(Api/Up:+ Brii Vr) W¢ } 17,. - wi, wl, + Aiik'wi, Wi, W~l

--X*,2(Bii + a*Cii + fl*Dq) Wi Wi, } )tO, 2 ~--

(Bq + &Cii + fiDij ) Wi, Wi, (21)

The end rotations can be expressed in the f(mn: X~ = X c + 6Xo,2S 2 + ol*X*,:s 2 (22a)

X~ = X~ + fiXo,2S z+ * * ~2 fl Xo, 2S (22b)

The case of prescribed displacement eccentricity is treated by X* = X* = Xt~ = 0 in equations (17)-(22).

We now summarize the steps involved in the analysis:

(i) Determine the buckling mode and the critical stress using the displacement function for 'w ' in the form of equation (la), taking m = 1. The value of 'n' must be obtained by trial and error to produce the lowest critical load.

252 Eng. Struct., 1982, Vol. 4, October

(ii) Using the displacement functions in equation (14a-b) for u and v, obtain the second order displacement field given by Up ~ and Vr2 using equations (12a-b) and (13a-b). This requires setting up of stiffness matrices for each strip and assembling the same systematically. The third term in equations (12) takes the role of the load vector. ('t/i) If the eccentricity of load is prescribed, determine the corrective end compression and end rotations in the forms ko*2, X~,2 and k~',2 from a knowledge of the second order stress field known in terms of Ups, Vr~ and Wi, ~ to preserve the prescribed eccentricity of the resultant force at the ends. (iv) For any given value of the perturbation parameter, s (say the maximum amplitude of the buckle) values of end compression and rotations are known from equations (20)-(22) as well as the magnitudes of the displacements u, v and w from equations (13) and (11). Thus the stress distribution and the magnitudes of the resultant load are obtained.

Examples In this section a few simple examples are considered to illustrate the convergence of the solution and to contrast the behaviour of plate structures under the two modes of loading.

Convergence study for a square plate problem A square plate, simply supported on all the four edges

with the unloaded edges free to wave in the plane or the plate, is taken as the first example. Figure 2 shows the position of the resultant force on the plate and the variation of longitudinal displacement.across the plate prior to buckling. These remain the same after buckling for the respective cases of prescribed load eccentricity and pre- scribed compression eccentricity. Two different eccen- tricities, given by e/d = 2/3 and 1, are considered.

An index of the response of the plate would be the displacement at the point of action of the load which is the same for both modes of loading prior to buckling. Correspondingly the load required to produce unit displace- ment at this point may be considered the stiffness of the plate. Table I gives the ratio of postbuckling to the pre- buckling stiffnesses of the plate obtained using increasing number of strips across the plate, for tile two eccentricities.

l ~ C o m p r e s s i o n

~- e = 213 d~r~ T ~ " j / Tension

a b F/gum 2 Point of application of load and displacement pattern used in examples

I Xd 2

Finite strip analysis: S. Sridharan

Table I Study of convergence of solution for plate problem

(1) (2) (3)

S*/S for e/d = 2/3 S*/S for e/d = 1.0

Constant Constant Constant Constant No. of load displacement load displacement strips eccentricity eccentricity eccentricity eccentricity

8 0.5335 0.4802 0.6532 0.3140 12 0.5022 0.4392 0.6327 0.2717 16 0.4638 0.3906 0.6274 0.2539 20 0.4623 0.3890 0.6251 0.2506 24 0.4615 0.3882 0.6238 0.2488 30 0.4609 0.3875 0.6228 0.2472 40 0.4603 0.3870 0.6219 0.2461 50 0.4601 0.3867 0.6215 0.2455

From Table 1, it is seen that the convergence is rapid and uniform and a sufficiently accurate result (within an error of 1%) is obtained using 16 strips across the full width of the plate. These values are in good agreement with those obtained from a graphical plot in reference 6. The superior internal consistency of the present solution in comparison to the other numerical solutions is seen from the exact satisfaction of the inplane equilibrium at every cross-section whatever the number of strips employed.

A study of Table I indicates that the plate under pre- scribed load eccentricity exhibits a greater stiffness than that under prescribed compression; and the greater the eccentricity greater the difference. The condition of pre- scribed load eccentricity always involves a rotation of the end cross-sections and, in this case, occurs in such a sense as to compress the lightly loaded regions and reduce the axial displacement at the load point, leading to the greater stiffness of the plate. Stress distribution (not shown) is also seen to be more even in the case of prescribed load eccen- tricity - and this would mean a higher load associated with first yield (generally signalling imminent collapse in locally buckled plate structures). No general conclusion, however, can be drawn from this example as to the greater stiffness or the higher collapse load associated with the case of prescribed loading eccentricity, as such a conclusion is contradicted by the next example.

Comparison of the performance o f channel section struts Channel section struts of constant thickness and of

proportions a/b = 0.25, 0.50 and 0.75 (Figure 4) are studied under the following conditions:

Case (i) compressive load applied through the centroid Case (ii) uniform end compression

The length of the strut is chosen to be an integral multiple of the half wave length corresponding to the minimum critical stress. Attention will be focused on the variation of stresses across the section as it is the concentration of stresses that lead to the plastic yielding and the collapse of the strut.

Figure 3 compares the change in stress distribution from the unbuckled configuration to the equilibrium state with a buckle amplitude of 6/t -- 1.0 at the centre of the flange for a strut with b/a = 0.5 and a/t = 100. Because of the symmetry of the section only one half of the structure is considered. There is a marked difference in the stress distri-

Eng. Struct. , 1982, Vol . 4, October 253

Finite strip analysis: S. Sridharan

n ,ssion

--1.0

- - 2 0

through d

Figure 3 Change in stress distr ibut ion f rom unbuckled configura- t ion to buckled configurat ion corresponding to 8/t = 1.0, for a strut wi th b/a = 0.5, a/t = 100 at middle section of column. (Stresses are given in a dimensionless form, o/E x 103)

8.0

"\b/a : O. 25

6.0

o

~x 4 . o o E

E)

2.0

~t

.75 .50 Case (ii) .25

Case ( i )

0

Figure 4

I 2.0 4.0 6.0

O'av / Oo

Variations of Oma x wi th Oar for struts investigated

I 8 . 0

bution for the two types of loading. For case (i) significant compression is carried by the outstand and the net com- pression carried by the flange plate is a very small fraction of the total. For case (ii) the opposite is true. For a given out of plane displacement, the net compression developed is greater for case (ii) than for case (i). Also the ratio of maximum membrane stress to the average stress carried by the strut, Omax/Oav is smaller for case (ii). A reasonably accurate failure criterion is given simply by Omax = Oy, as the collapse is readily precipitated by the yielding of the

corners) ° Thus the channel strut under uniform end shortening is not only stiffer but would also attain a higher collapse load. A full blown nonlinear analysis would no doubt yield more accurate numerical results, but the general trend indicated by the perturbation solution is expected only to be confirmed.

Figure 4 shows the variation of Oma x with Oar for the three sections investigated. The stresses have been rendered dimensionless by division by Oo, given by:

(:ai E 12(l--v) 2 (23)

In all cases, the channel section under uniform compression would have a higher ultimate strength than the correspond- ing one loaded through the centroid, whatever the value of cry. The differences in behaviour narrow down, however, as the outstand depth becomes smaller. But, the 'effective widths' of the component plates are clearly dependent on the mode of loading.

Figure 5 compares the behaviour of a channel strut loaded through the centroid with that loaded through the flange plate. It is interesting to note that the greatest dis- parity of maximum stress exists prior to buckling; with the occurrence of local buckling the values of maxhnum stresses are about the same, for a given load carried. Thus the eccentricity of the point of application of the load does not influence the collapse load as greatly as the mode of loading does.

8.0

6 .0

oo \

x 4 .0

2.0

S _ I I

i i i

- / / / Looded through / / centroid

/ / I L o o d e d through

I I I ] I 1 0 2 0 3 0 4 0 5.0 6 0

OQV / O 0

Figure 5 Variations of Oma x wi th Oar for channel struts loaded through centroid and through flange

254 Eng. S t ruc t . , 1982, Vo l . 4, O c t o b e r

Conclus ions

A Finite strip analysis of post-local-budding behaviour under nonuniform compression has been presented; the two modes of loading, i.e. one of prescribed load eccen- tricity and the other of prescribed compression eccentricity are considered. The use of the perturbation technique further simplifies the method and is discussed in some detail. The convergence of the solution is illustrated with examples of eccentrically loaded simply supported square plates.

The sample problems illustrate the wide divergence in the behaviour of the plate structures after buckling caused by the mode of loading. In the case of channel struts, uniform compression leads to a greater stiffness and col- lapse load than loading through the centroid. The differ- ences are accentuated for greater depths of outstands. This has an immediate bearing on the current approaches to design of the plate structures which do not distinguish one mode of loading from the other.

N o t a t i o n

The summation convention of indicial notation is employed throughout the text.

Ai D A pi], A i]k Bi D Bri], Ci], Di] Evq , Fv,, G,s E

S

S* ui, wi ui,1, Vi, l, wi,1

U trip Vstruct

a

aft, bi",L ci/' di/' epq , ]pr, grs, api] bri], aijkl bs d

l n $ t

llp,i, Vp,i, Wm,i Ui, Vi, Wi

x , y , z

Coefficients in potential energy function

Young's modulus Membrane stress resultant in transverse direction Stiffness of structure in prebuckling state Postbuckling stiffness Global degrees of freedom First order components of global degrees of freedom Strain energy function for a strip Potential energy function for structure Width of flange plate of channel section strut Coefficients in strain energy function of a strip

Width of strip Side of square plate Distance of point of application from remote edge of square plate Length of structure Number of half waves of buckling Perturbation parameter Thickness of plate element Local degrees of freedom Abbreviated symbols for local degrees of freedom Local coordinate system Principal axes of cross-section of structure

Ol*~ f3*

ai,/, 3i, i

8

X0

~kl, ~k 2

o, '~a, a 3 * * X~, 2 X0,2, Xa,2,

P

O0

grnax O'av

Finite strip analysis: S. Sridharan

Ratios of rotations about respective axes to axial compression parameter in primary path Ratios of corrections of rotation para- meters in respective directions to correction of axial compression parameter in secondary path Geometric parameter applicable for ith strip and ]th edge (] = 1 or 2) giving end compression due to rotations about or/3 axis Amplitude of buckle at centre of flange = y / b s Uniform end compression corresponding to unit length of structure (axial compression parameter) End compression at edges of a strip Rotations about a and 3 axes Values of Xo, ka, X# respectively at bifurcation Corrections to Xo, Xa, X~ respectively Defined by equations (15) Poisson's ratio x / l Defined by equation (23) Maximum membrane stress in x- direction Average stress in x-direction over cross-section

Refe rences

1 Graves Smith, T. R. and Sridharan, S. 'A finite strip method for the post-locaUy-buckled analysis of plate structures', Int. J. Mech. Sci., 1978, 20, 833

2 Sridharan, S. 'Elastic post-buckling behaviour and crinkly collapse of plate structures', PhD Thesis, University of Southampton, UK, 1978

3 Sridharan, S. and Graves Smith, T. R. 'Postbuckling analyses with ffmite strips', J. Eng. Mech. Div. ASCE, 1981, 107, (EM5), 869

4 Hancock, G. J. 'Nonlinear analysis of thin-sections in com- pression', J. Struct. Div., ASCE, 1981, 107 (ST3), 455

5 Rhodes, J. and Harvey, J. M. 'Effects of eccentricity of load or compression on the buckling and post-buckling behaviour of flat plates',Int. J. Mech. Sci., 1971, 13, 867

6 Rhodes, J. and Harvey, J. M. 'Plain channel section struts in compression and bending beyond the local buckling load', Int. J. Mech. Sci., 1976, 8,511

7 Benthem, J. P. 'The reduction in stiffness of combinations of rectangular plates in compression after exceeding the buckling load', National Aeronautical Research Institute, Amsterdam, NLL-TR S.539, 1959

8 Croll, J. G. A. and Walker, A. C. 'Elements of structural stability', Macmillan, London, 1972

9 Graves Smith, T. R. and Sridharan, S. 'Elastic collapse of thin- walled columns', in 'Thin walled structures' (ed. Rhodes, J. and Walker, A. C.), Granada, 1980

10 Graves Smith, T. R. and Sridharan, S. 'The local collapse of elastic thin walled columns', J. Struct. Mech., 1980, 10 (4), 471

11 Walker, A. C. 'The postbuckling behaviour of simply supported square plates', Aeronaut. Quart., 1969, 20, 203

Eng. Struct., 1982, Vol. 4, October 255