Effects of eccentricity of load or compression on the buckling and post-buckling behaviour of flat plates

Int. J . mech. Sei. Pergamon Press. 1971. Vol. 13, pp. 867-879. Printed in Great Britain

EFFECTS OF ECCENTRICITY OF LOAD OR COMPRESSION ON THE BUCKLING AND POST-BUCKLING BEHAVIOUR OF FLAT PLATES J. RHODES and J. M. HARVEY D e p a r t m e n t of Mechanics of Materials, University of Strathclyde, Glasgow (Received 9 December 1970, a n d i n revised f o r m 22 J u l y 1971)

Summary--A method of examining the behaviour of plates subjected to either an eccentric compression system or to an eccentrically applied load is outlined. The t e r m compression, as used here, signifies end displacement. The loading in both cases is assumed to be applied through rigid bars so t h a t the plate ends do not warp. I t is shown t h a t the case of eccentric loading is completely different from t h a t of eccentric compression. NOTATION coefficient governing deflexion magnitudes a, b plate length a n d b r e a d t h respectively B 1, B~ constants in stress function D plate flexural rigidity ( - Et3/12(1-v ~) d distance from edge of plate to load point E Young's modulus of elasticity E* apparent modulus of elasticity after buckling e ratio of buckle half wavelength to plate width F Airy stress function P load applied to plate p non-dimensional plate load ( -- Pb/~r ~ D) $ plate thickness x, y Cartesian co-ordinates U~ V in-plane displacements of the plate middle-surface in the directions x and y respectively in-plane displacement system of plate ends under compression U* ~, '~, u L in-plane displacements of edge y -- b, edge y = 0 and load point uc/~ critical value of u* to cause buckling out of plane deflexions of the plate ]7(y) deflexions across the plate o~ coefficient of compression eccentricity direct membrane stresses at a point in the plate in the x and y directions O'CR critical buckling stress for a plate Y Poisson's ratio Other symbols are defined where t h e y first appear. Primes denote differentiation in t h e y-direction, e.g. F ~ = ~ F / ~ y ~.

INTRODUCTION

I~ A RECENTpaper, 1 the authors presented a method whereby the post-buckling behaviour of flat plates uniformly compressed with various boundary conditions o n t h e u n l o a d e d e d g e s c o u l d b e a n a l y s e d . T h e m e t h o d u s e d is e x t e n d e d here to deal with plates which are either compressed eccentrically or loaded eccentrically. 867 59

868

J . RHODES a n d J. M. H ~ v E : g

The buckling of eccentrically loaded plates has been investigated by a number of researchers, including Timoshenko, ~ Johnson and Noel a and Walker, 4 for plates simply supported on their loaded edges and subject to various support conditions on their unloaded edges. While the eccentricity of compression and that of load are identical for an unbuckled plate, the onset of local buckling causes changes in the stress system within the plate which forces the load eccentricity to alter if the compression eccentricity remains constant or, conversely, forces the compression eccentricity to alter if the load eccentricity is constant. To the authors' knowledge, these effects have not been investigated before. Walker examined the post-buckling behaviour of eccentrically loaded plates using Galerkin's method to solve the governing non-linear simultaneous differential equations of Von Karman, and is probably the only researcher who has investigated this problem. His theoretical loading conditions of linearly varying load across the plate ends seem to the authors, however, to be less practical than the condition of linearly varying end compression. In this paper, this end loading condition is considered and the problem is analysed using the methods of ref. 1. END

LOADING

CONDITIONS

Consider the plates shown in Figs. 1 and 2, loaded through rigid bars. The loaded ends are simply supported and the unloaded edges have either of two boundary conditions for deflexion: 1. Both edges restrained against rotation to an equal or unequal degree. 2. One edge rotationally restrained, the other free. Two conditions of loading are examined. (a) The eccentricity of the end displacements remains constant, corresponding to the loading bars rotating about fulcrums as shown in Fig. 1. Loading bars

positions ~ ~_~

-~ r[ I/ Load~

of bars

Plate 1

I/

_

Load

o (=b)

I~E(l-a) ~ ~ -

Fulcrurns

FIG. 1. Plate loaded with constant compression eccentricity.

Effects of eccentricity of load or compression on flat plates

869

(b) The eccentricity of the applied loading remains constant. This corresponds to the loading bar bearing freely on the plate without contact with a n y external con. straints on its movement other t h a n the load, which is applied at a specific point on the bar a~ shown in Fig. 2. I n practice it is to be expected that in general the loading conditions will be somewhere between these two. A plate which is a secondary member of a structure under load will undergo end displacements caused b y loads applied to the structure, which are relatively Loading bars Y

(free from external restraints other than load) r-

-P-~-

I j J I I f I

I J I I I

I I I I t I I

Plate

.=1.__

P __

I

o(I-a)

I I

~

~

c

e

d of bars

Z

positions

FIG. 2. Plate loaded with constant load eccentricity. uninfluenced b y the stress system within the plate itself, approximating to condition (a). I f the plate under consideration is a primary member of a structure then the externally applied loading system and the stress system within the plate are more closely interrelated and condition (b) will apply. This can be illustrated very simply with reference to Fig. 3 which shows diagrammatically a spring system. The two centre springs are considered to represent the plate and the two outer springs the rest of the structure. On the application of a load to the system as shown in the Figure, the behaviour of the two centre springs is very much

[

I

/

/ / / / / /

Stiffness 5"3

si

/ /~/

/ / / / /

/

/

$2

Fro. 3. Diagrammatic illustration of loading conditions.

870

J. RHODES and J . M. HARVEY

influenced b y t h e stiffnesses of the o u t e r springs. I f t h e stiffnesses of the o u t e r springs, $3 a n d $4, are v e r y m u c h greater t h a n those of the inner springs, S1 and $2, t h e displacem e n t s imposed on the centre springs depend almost c o m p l e t e l y on t h e stiffnesses of t h e o u t e r springs a n d are n o t affected b y t h e stiffness of S1 relative to $2. If, on t h e o t h e r hand, the inner springs are m u c h stiffer t h a n t h e o u t e r springs, the displacements imposed on the inner springs are e x t r e m e l y d e p e n d e n t on the relative stiffnesses of S1 and $2. Thus, on one h a n d condition (a) applies a n d on t h e o t h e r h a n d condition (b) applies. THEORETICAL

ANALYSIS

The theoretical analysis used for t h e i n v e s t i g a t i o n of plate post-buckling b e h a v i o u r is similar to t h a t of ref. 1 and so will only be briefly described here. The plates u n d e r consideration are assumed to be free f r o m shear stresses on all boundaries, a n d the u n l o a d e d edges are free f r o m restraints on in-plane waving. The deflexions of the plates can be a p p r o x i m a t e d to b y the expression w

=

~[:F(y)

~rx cos ~e~,

(1)

where ~ is a coefficient defining t h e m a g n i t u d e of the deflexions, a n d :Y is a function of y w h i c h describes t h e deflected f o r m across the plate and satisfies t h e r e l e v a n t b o u n d a r y conditions. I n this i n v e s t i g a t i o n Y is found f r o m an a c c u r a t e buckling analysis and is, therefore, a good a p p r o x i m a t i o n to the actual buckled f o r m at loads n o t g r e a t l y in excess of the critical load. I t can be shown t h a t t h e stress s y s t e m within t h e plate middle-surface corresponding to this deflexion s y s t e m is g i v e n b y 2~rx

F =Fl(y)+F2(y) cos eb '

(2)

where -F1 = ~ z ~ 2 Y 2 + B l Y + B 2 , E~r ~ _ F2 = ~ AS ¢(y). The d e t e r m i n a t i o n of t h e function ~ is shown in ref. 1. T h e constants B 1 a n d B~ fully prescribe t h e end displacement s y s t e m across the plate. T h e end displacement at t h e corner y = 0 is equal to B2(a/2E ), and this varies linearly across t h e plate to (B1 +B~) (a/2E) at y = b. Thus the eccentricity of compression or end displacement is specified b y t h e r e l a t i v e values of B 1 and B 2. This compression eccent r i c i t y can be specified in a clearer fashion b y i n t r o d u c i n g t h e coefficient a, which m a y be defined as t h e coefficient of compression eccentricity, such t h a t

u

I~,=o

~(1--a),

J

(3)

then B~ ----~(1--c~) (2E/a)

(4)

B 1 = ~a(2E/a)

(5)

and a n d t h e end compression system, u*, can be w r i t t e n

u* = ~[1-a+c~(y/b)].

(6}

T h u s t h e eccentricity of end compression is directly specified b y t h e value of a. I f is zero t h e n u n i f o r m compression is applied, if a = 1 t h e n the compression of the edge y -- 0 is zero ff a -- 2 t h e plate is b e n t a b o u t its centre line. A t this stage stresses a n d deformations w i t h i n t h e plate are k n o w n in t e r m s of t h e coefficients ,~. The analysis n o w requires the e v a l u a t i o n of t h e m a g n i t u d e of .~ corresponding to a g i v e n end compression. This m a y be done b y e v a l u a t i n g the strain e n e r g y

Effects of eccentricity of load or compression on flat plates

871

of the plate system in terms of ~r and making this a m i n i m u m with respect to ~I b y setting its derivative to zero, from which .~ is found to be

/4e e be

4 [Jo

+ 2(A){1- v)[Y:Y']ob+ [:Y'Y~']~.

X=

(7)

aE~' 2(~'--"2"n"'32e4 b~t' {~: [~4 + [-~) ~)"~Jdy} Thus for a n y prescribed end displacement system u* greater t h a n UcR, the value of .~ can be obtained. Knowing .~, the stresses and deflexions in the plate can be evaluated. The load on the plate is found from P=t

a~dy = t

dy.

F I+F e

(8)

0

Because of the condition of zero shear stress on the plate edges, F~ integrates to zero across the plate and the load is constant in the x-direction. Substituting for F~ gives

/'b ~E~re ~ r e dy, p = 2tEa ~2 u* d y - tJ0

(9)

i.e. ~b E - e p = 2tEba~(1 - a/2)--tJ0 ~

~e r, dy.

(10)

The stiffness against compression of the plate at loads less t h a n critical is

OP

--

2tEb

= - -

a

(l -~,/2).

(u)

After buckling, the plate stiffness against further compression is influenced b y the deflexions and the method of applying the load. CONSTANT COMPRESSION

ECCENTRICITY

I f the compression eccentricity is kept constant the value of ~ is unchanged after buckling and the post-buckling stiffness can be obtained b y substituting for ~r from equation (7) and obtaining ~ P / ~ , i.e.

fo' f:(

7)

The ratio of post-buckling to pre-buckling stiffnesses is

E* =

1

CONSTANT

f:

:Pedy

f: (

LOAD

:Fe 1 - a + a ~

") dy

(13)

ECCENTRICITY

If the eccentricity of the end loads remains constant the problem of evaluating the post-buckling stiffness is made more complex because a now varies with the increase in load. The variation in a must therefore be investigated. This is done as follows. Consider the plate shown in Fig. 4. The load P, is applied through a point at distance d from the edge y = 0. The relationship between the internal stresses and the load applied to ensure equilibrium of forces is found from equation (8). I f the eccentricity of loading does not change, the moment about the x-axis caused b y the load is

M --- Pd =

a®ydy =

F'~y-t-F'~ycos--~] dy.

(14}

872

J. RHODES and J. M. HARVEY

The conditions of zero shear and normal stresses on the plate edges once more eliminate the periodic terms in equation (14), and this becomes /'b Ew2

Pd = 2tbEa .]o[t~u* y dy - tJo ~

_~2 :~2 y dy,

(15)

E7T2 -4-e-~ A2 Y2 y dy"

(16)

or

Pd = tb2 E~(1-c~3/a)-t

~x

(Tx

d

X

FIG. 4. General stress system in a plate under load Prior to buckling .~ = 0 and the value of a in terms of d and b can be found by dividing equation (16) by equation (9), giving a = b[1 - (a/3)]

211 -

(17)

(~/2)]

from which 2d - b = d-

(b/a)"

(18)

After buckling the division of these two equations is again used to obtain c~. Here the substitution for .~ from equation (7) is employed and, after some algebraic manipulation, a can be obtained in the form a =/~1 +/z2/u,

(19)

where

ae2b't'

24(1 _v~) ~r~

[;

:P~ydy-d

;

Y2dy

] { ; (Y~

\

~Tr]~

--~-~] :P) 2 dy

+ 2(~b)~ ( 1 - v ) [~Y']~ + [Y'~"]~}

(;

Effects of eccentricity of load or compression on flat plates

873

and

Substituting for ~I a n d a in equation (10) gives for the load in this case =

a

pa tJo

+ f ~ Y2 Y dY (/~, + - ~ )

e2abD[fb

Ir 2 (20)

Now

8P

-~=

2the l l a

P'~-2tE (l~:Ys(1-1x,)dy+

~, --~/

Y~-~p, dy

pa ~Jo

)/:

Y2dy.

(21)

I n this case, ~P/~ii represents the increase in load with increase in compression of the edge y = b, and does n o t fully describe the reduced modulus E*. On the edge y = O, the increase in load with increase in compression is obtained as follows. Put U(1-~) = ~(1-/~--~)

= ~,

(22)

then 8P 1 OP ~--~ = (--t/~1) "-~

(23)

or

~P

1

8P

-~ = 1 - ~ 1 ~ "

(24)

This is the increase in load with the increase in compression of the edge y = 0. The displacement of the load point, uL, is related to the load b y a similar reasoning which gives

~P 1 OP Ou~L--- [1-1Xl+l~l(d/b)] ~ "

(25)

I f the compressional displacement is measured at the load point, the apparent reduced modulus E* can therefore be found from equation (25), b y multiplying ~P/~tr. b y a/2tb. Thus the ease of constant load eccentricity can also be examined. TYPICAL

RESULTS

A number of theoretical results are presented here in graphical form. Non-dimensional loads and stresses are used in the graphs and these are defined in the Notation. The results shown are, for the most part, confined to square plates. Fig. 5 shows l o a d - l o a d point displacement paths for square simply supported plates. The loads are applied a t various distances from the plate centre line in the pre-buckling range, and in the post-buckling range the load point displacement p a t h s for constant load eccentricities are shown compared with the corresponding displacement paths which will be encountered if the pre-buekling compression eccentricity is maintained after buckling. I n general compressional stiffness of the load point for this case is greater if the load eccentricity is constant t h a n if the original compression eccentricity is maintained. The variation of E*/E for the load point with variation in initial eccentricity is shown for b o t h cases in Fig. 6. Also shown in this figure is the variation in buckling load with variation in initial eccentricity. A clearer picture of the behaviour of this t y p e of plate under b o t h loading conditions can be obtained from Fig. 7 which compares the end displacements a t both edges and a t the load point f o r b o t h conditions. I t can be seen t h a t for a given load greater t h a n

874

J. RHODES and J. M. HARVEY

buckling the difference between the displacements of the two edges is less for the constant load eccentricity case than for the constant compression eccentricity case. This shows t h a t after buckling the value of a for constant load eccentricity is less than for the 12

~dlb=0.5 {

~-~-~

Load point

2 I////

}-0.6

Const iood s ~ ------ Const. compression ecc

0

4

8

i2

t6

Load pt. displacement X 24

20

24

2E

( I - v 2) b z

FIo. 5, L o a d - e n d displacement curves for simply supported plates. 8--

0.8

7 -

07

6--

0"6

- ~

ET~E-

const. load eoc

8 ¢~

Fie.

LU

4

04

P

3

0.3

2

0.2

05

0.6

07

0.8

d/b

0.9

I

I

l

6. Buckling loads and post-buckling stiffnesses for eccentrically loaded plates.

constant compression eccentricity case, i.e. the compression eccentricity reduces if the load eccentricity remains constant. Also it can be seen t h a t the most highly compressed edge occurs for the constant compression eccentricity case and this would be expected to lead to earlier failure. Fig. 8 shows a comparison of theoretical l o a d - m a x i m u m deflexion curves for an eccentrically loaded, simply supported plate, with experimental results obtained by Walker. l The plates tested by Walker were 10 in. wide and 20 in. long and buckled into two half waves in the direction of loading. The theoretical loa~l position shown in Fig. 8

Effects of eccentricity of l o a d o r compression on fiat plates

875

gives the statically equivalent loading system on the loading bars to t h a t produced in Walker's experiment and, since Walker's loading bars were v e r y strong and rigid and thus h a d negligible bending, produces precisely the same loading conditions on the p l a t e

I

/

/I

~

2

~

-'---*"~

~

/

i

d=06

~ Const. load ecc ---- Const. compression ecc

0

4

Load

/ / / 7 / ~1 p°int

I b ',

I

I

I

I

8

12

16

2(~

J

ePI 24

28

End displocementX24 (l-uZ)b (77r 2f 2

FIO. 7. Comparison of eccentric loading and eccentric compression conditions for simply supported plate. '~

p,

J J

~

~

, d=2/3

b

Plate simply supported on all boundaries

it

× X

2

0

o

~

Theory

Walker's experiments b = loin. o t =0.066 in. a = 20in. x t =0.078 in.

o

I

I

I

t

I

I

0'3

0.6

0-9

1.2

1.5

1.8

2.1

w//

FIo. 8. Comparison of theoretical m a x i m u m deflexions with experiments of W a l k e r for constant load eccentricity case. as were produced in Walker's experiment. As is evident from this Figure, the theoretical analysis predicts the plate deflexions accurately except at loads in the region of buckling where the experimental results are affected to quite a large extent b y the presence of initial imperfections in the plate. As mentioned in ref. 1, the effects of unequal restraints on the rotations of the unloaded edges can cause the load on a uniformly compressed plate to become eccentric after buckling. Conversely, for this t y p e of plate, loading through the centre will produce

876

J. RHOD]~S a n d J. M. HXRVEr

eccentric compression in the post-buckling range. Fig. 9 shows the variation in E*/E with variation in rotational restraint on one edge of a plate whose other edge is simply supported for both loading conditions. F o r the centrally loaded plate, E* is calculated 0.8

Simply supported- rototionolly restrained square plates 0.7

06--

Rotational stiffness of restraining medium = ,Rib D

Uniformly compressed plate

o.5~-

/

Centrally loaded plate 0.4

05

02

I

0

I

4

T

8

I

12

I

16

[

20

24

28

R/b

FIG. 9. P o s t - b u c k l i n g stiffnesses of plates w i t h one u n l o a d e d edge restrained against rotation a n d the other s i m p l y supported.

for the load point. The centrally loaded plate is seen to be slightly less stiff t h a n t h e uniformly compressed plate. Fig. 10 shows the growth in stresses for a plate loaded through the centre with one edge simply supported and the other highly restrained. The growth of compression eccen-

28

1

2(5 24 22

m

2

P:II.763

2O t8

#:9,4r5

~.~ ~

,O 8

~= 7.063

I~

6

5:5"726

o-

2--

Simp[ supported

I

r

I

1

0.2

o4

0.6

0.8

y/b

Restrained almost to clamped condition

FzG. 10. Stresses in a plate w i t h one edge rotational]y restrained and the

other simply supported.

Effects of eccentricity of load or compression on fiat plates

877

trieity in this case causes the simply supported edge of the plate to become v e r y highly stressed after buckling. I n d e e d it is interesting to note t h a t the rate of growth of m a x i m u m membrane stress on the simply supported edge after buckling is almost exactly the same as t h a t for a square plate simply supported on all boundaries, although the buckling load for the plate under consideration is higher. The difference in load-compression behaviour between plates loaded with constant compression eccentricity and plates with constant load eccentricity is strikingly a p p a r e n t when considering plates which have a free edge. Fig. 11 shows the load-compression

3-0

I

2.5

2.0

I

IQ.. 1.5

-

2

1.0

0.5

/ /

I

I

I

2

ly supported free square plate Curve I;- Uniform compression Curve 2:-Centre loading /J for curve 2 token at centre of plate Le. load point I I l I 3

4

5

6

LTX 24 ( I . - ~ 2 ) b 2 a Tr2 t 2

FIG. 11. L o a d - e n d displacement curves for simply supported free plate.

paths for a simply supported free plate under uniform compression and for a similar plato loaded through the centre. F o r the uniform compression case the post-buckling stiffness is a little greater t h a n two-fifths of the initial stiffness, whereas the compressional stiffness of the plate under constant load eccentricity is extremely small. Average membrane stress distributions for the two cases are shown in Fig. 12. This figure shows t h a t t h e highly eccentric stress system which occurs for the uniform compression case is replaced b y a much more symmetrical system when the loading eccentricity is constant. Conversely, the compression system corresponding to the centre loading case is highly eccentric with the free edge being compressed much more t h a n the supported edge. This t y p e of plate under uniform compression will generally fail due to crippling of the supported edge caused b y the high compressive stress on this edge. I f the plate is centrally loaded failure will p r o b a b l y occur closely following buckling due to high bending stresses occurring on the free edge. Fig. 13 shows a comparison of the deflexions of the free edge for both loading conditions. I t can be seen t h a t the deflexions of the centrally loaded plate become v e r y much greater t h a n those of the uniformly compressed plate. Since the bending stresses on this edge are directly proportional to the deflexions this indicates t h a t very high bending stresses occur for the centrally loaded plate immediately after buckling. This is one of the reasons w h y it has been found b y m a n y investigators t h a t short pinended angle struts (constant load eccentricity) fail at or near the local buckling load, whereas Stowell 5 obtained quite large reserves of strength after buckling in cruciform sections which were uniformly compressed.

878

J. RHODES and J. M. HARVEY

~ 94/2~43:72 ~ I

Simply

E Unifolmcompres$ion~ Free

supported

2

J

~5= 1'455 /5= 1'485

I

I

0"2

0.4

l

y/b

0.6

I

0.8

P=1"53

Centreloading FIo. 12. Stress distributions in simply supported free plate.

3.0 2.5 -

/iformiy

compressedplate

2.0 ~ I~. 1.5 I.O

C e n t r a ~

-

0.50

Simplysupported-freesquareplatescomparison of edgedeflectionsof centrallyloadedplateand uniformlycompressedplate !

I

0.5

I'0

,I 1'5

]

I

I

20

2'5

.3.0

w/t

lq'xG. 13. Deflexions of simply supported free plate.

3'5

Effects of eccentricity of load or compression on fiat plates

879

CONCLUSIONS I t has been shown t h a t the behaviour of plates in compression is extremely sensitive to the method of application of loads to the plates. Quite different results are obtained from similar plates subjected to constant compression eccentricity and constant load eccentricity. The failure of plates subjected to both loading systems are at present being investigated both experimentally and analytically taking into account initial imperfections, and the results will be published when they are available. REFERENCES l. J. RHODES and J. M. HARVEY,Int. J. Mech. Sci., 13, 787 (1971). 2. S. P. TIMOSH~NKOand J. M. GERE, Theory of Elastic Stability, 2nd edn. McGraw-Hill, New York (1961). 3. J. H. JOHNSONand R. G. I~OEL,J. Aeronaut. Sci. 20, 535 (1953). 4. A. C. W~LKE~, Thin-waUed Structures. Chatto & Windus, New York (1967). 5. E. Z. ST0WELL,N.A.C.A. Rep. No. 1029 (1951).