theelastic stability of a corrugated...

144

The Elastic Stability of a Corrugated Plate.

By W. E. D e a n , M.A., Fellow of Trinity College, Cambridge.

(Communicated by R. V. Southwell, F.R.S.— Received January 1, 1926.)

1. Introduction and Summary— This paper deals with the elastic stability of a corrugated plate under thrust along its generators. Besides the assumption, common to all such problems, that the plate is thin, it is supposed that the depth of a bay (d, fig. 1) is a small multiple of h, the semi-thickness, and that the transverse expansion that is the usual accompaniment of a longitudinal thrust, is prevented by a thrust in a perpendicular direction. The equations derived in §§ 2-7 are then soluble, and the critical stress in any case can be found from an equation expressing that an infinite determinant is zero.

The numerical work that has been done has been limited to the two cases in which d = lOh and d = 5 h,respectively. As a preliminary, in §§ 8-11, it has been supposed that a, Poisson’s ratio, is zero. The equations are greatly simplified by this supposition, and results can easily be obtained which are a valuable guide to the more complicated arithmetic of the normal case in which a does not vanish. In particular, this preliminary work is used to find the “ favourite type of distortion” (that possible under the least stress) in the other case : it can be seen that the favourite types are the same whether a is or is not zero. The necessary exploration is therefore done for a = 0. and the arithmetic in the more practical case, in which we suppose that a — \ , is confined to the calculation of stresses causing definite modes of distortion.

For a = numerical results are given in §§ 12-14, and shown in figs. 3 and 4 It is found in the cases considered that the ratio of length of bay (l) to height (H) should not exceed if the critical stress is to be as large as possible. The critical stress in a corrugated plate can be conveniently stated by giving its ratio to that in a plane plate of the same thickness and height. If < J, this ratio is about 30 when d — 5 h,and over 130 when d = 10A. This may be stated in different terms in a manner that has an application more direct to practical problems. Two plates of equal thickness and height will weigh the same if they are of the same length and of the same material. If one is plane and the other corrugated so that d — 10/i and l/H < the latter though weighing the

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Elastic Stability o f a Corrugated Plate. 145

same will take over 130 times the total load of the other. The possible application of this work is to the use in aeroplane construction of corrugated plates, which must be light and yet take considerable loads. The possibility of extending this work to plates with deeper bays is considered in §15; this extension is necessary in deciding what is the greatest value of djh to which the approximate solution of this paper will apply. I t does not appear worth while to attempt this until some experimental results are available. For the same reason the form of the surface of the plate after collapse has taken place has not been considered numerically, though in § 16 the general nature of the favourite types of distortion is stated.

The writer is greatly indebted to Prof. G. I. Taylor at whose suggestion this problem was attacked, and to Mr. R. Y. Southwell for his valuable criticism and advice.

§ 2. The position of P, any point of the middle surface of the plate, is specified by a, the distance of P measured along a generator from an arbitrary line of

curvature, and by (3, the distance of P measured along a line of curvature from a generator, p, the radius of curvature of the section of the surface by a plane perpendicular to the generators, is a function of (3 only, and we write

Fig. 1. 1/p = c sin wp, (1)

where c and n are constants, c being the maximum curvature.Given c and n equation (1) defines the middle surface, but a more convenient

description of a corrugated plate can be given in terms of four lengths : H, the height; h, the semi-thickness of the material of the plate ; l, the length of a bay of the middle surface ; and d, the depth of a bay (fig. 1).

The fundamental assumptions made in what follows are that the ratio H is small, and that d is comparable with h. The first of these is, of course, inevitable : instability is only of practical interest, and is only calculable, if the plate considered is thin. Little progress has been made without some such assumption as the second, whose limitations are discussed later in § 15. The numerical work has been confined to the two cases in which d — 10 hand d — oh respectively, so that we take d to be of the order of h throughout.

VOL. CXI.— A. L



146 W . R. Dean.

The four lengths and the quantities c and of (1) are naturally not independent. Clearly, l — re/n, while

if it is assumed that cjn, which is approximately equal to rcdjl, is small. This is supposed here, but the restriction can probably be removed in most cases at the cost of slightly complicating the arithmetic ; it is necessary, however, that the plate should not be so rapidly corrugated that the maximum curvature is large in spite of the fact that d is small.

§ 3. The Shell Equations.—We investigate the stability of the plate under thrust parallel to its generators, it being supposed that the aggregate lateral expansion that would normally take place is prevented by suitable thrusts in a perpendicular direction. Since d is not more than 10A in the cases to be considered numerically, it may be assumed that the critical thrust is the same as that of a plane plate of the same height, and of semi-thickness some small multiple of h.

The displacements of any point of the middle surface are written v, w ; u being the displacement along the generator, v the perpendicular tangential displacement, and w the normal displacement. From the plane-plate comparison above it follows that in the critical configuration u is of the order h2,* while it can be seen a 'posteriori that v is also of order h2, and w is of order h.

The reason for the higher order of w is fairly clear physically. To prevent lateral expansion a transverse thrust must be applied, which, in the case of a corrugated plate with a shallow bay, may be expected to be, roughly, the longitudinal thrust multiplied by Poisson’s ratio. In the critical configuration there is, accordingly, a transverse thrust about one-quarter of the high collapsing thrust: evidently a corrugated plate can offer but little resistance to such a stress, and relatively high normal displacements ensue.

Lastly, from (2), 1/p is of the order of h. Shell equations that are applicable with slight modification to this problem have been given by the writer in a recent paper.f They need not be set down in full here, as we immediately

* It is convenient to make some statements of this type in this section. The meaning is merely that u contains h*as a factor. In the critical equilibrium of a plane plate dul(‘M is of the order (fc/H)2.

t ‘ Roy. Soc. Proc.,’ A, vol. 107, p. 734 (1925). By shell equations is meant equations in terms of the displacements of points of the middle surface only.

This can be writtenc = dn2, (2)




approximate to them by retaining in the first two equations, the principal terms of which involve uand v linearly, no terms of order higher than A2, and in the third equation, whose most important term is of order h2w, no terms of order higher than A3. The equations are considerably simplified by this procedure, and become

8aCM , ! / 8 e e \ 2 , f 8 y

8a l0£ p 1 — 8

0(3

+ / M 2Y18(3/ J.

+ 0v cu civ u------Lea

00p L0(3

03 ' 0a 03

™ 4 - JL ( d w ) 2 . a ! 1-03 p 2 '03 / 1 10a ' ^0a' J

1 — (7 0

- 0 ,

8aand

d v . du 8w_0a 8(3 8a 8p_ = 0,

A2 4 13 V l W “ p

"0̂ , 4 id w f , f0W , i /8w\2V

0W J0W , /dv___ W \ \ | _ _8_0a [0 a ' \0(3 p/J J 8 3 L0^ \0 (3

_0_0a

l - a 0 r t 2 0a

IV , 0w)P + 0 a f i -

cw cv o«\~l03 \0« _r 0S/J

a _003

'dw /cv , dir ,0a \8a d$i~

( 1 )

The equations from which (3), (4), and (5) have been deduced ignore all terms of the first order in u,v, w if multiplied by A3 or higher power of A, and all terms of the third and higher orders in u, iv. This approximation is sufficient so far as deducing terms of order A2 in (3) and (4) is concerned, but, clearly, there is no reason for excluding terms in 3 from (5). The importance of such terms in the case of a plane plate when w is of the order A has been pointed out by J. Prescott,* and it happens that the terms that must be added to the left side of (5) are exactly the same as those shown to be necessary by him in the corresponding plane-plate equation. The reason is this: the complete third- order terms in (5) must contain those of the plane-plate case, which is reached by putting 1 /p = 0 ; the other third-order terms must contain a positive integral power of 1/p as a factor,'!* and hence must be of order higher than A3.£

* ‘ Phil. Mag.,’ vol. 43, p. 97 (1922). t P only appears in this form in the equations.t An independent evaluation of such third-order terms as are required in this problem

has been made on the lines of Section 5 of the paper by the writer quoted above ; that is to say, without appealing to results of the theory of thin shells. There appears to be no point in setting out this work : the necessary terms may be found in equations (5) to (14) of Prescott’s paper.



148 W . E . Dean.

The complete equations for this problem are now (3), (4) and

A2 4 Irdv wi i , f0w , , /0wA2l"]t v * * Lap - p- + M a p ) + 1 k + * k ) }J

0 [dw f B u , 1 , /0V, - 1 ^ - 1 ^ + i f — ) + « (5 p

w\ , a + 2 \0j3 } ]

0 f0w f0v w , i fdw'y , 0m . a/0w;\2) lip Lap lap ~ p + a W + ffa i + 2 [t J j J0(3

1 — (7 0 2 0^— (7 0~ 0 ( 3

dwfdv | 0-m , 0W 0w) '_0(3 i0a 0(3 0a 0(3J ,'dw f0v , 0M , 0w 0w).0a 10a 0(3 0a 0(3j"_

§ 4. The Equilibrium CoWriting

"0VTx du0a

0V0(3

! 1 /3w+ S \0a m 8(3

fr 4- il 2 .ap +• {

pdu

4- i 20wr |0(3/ J ’

0«V\2]0 a / / ’

andq 1 — a jdv , du . dw 0«v'|

2 \0 a 0(3 0a 0(3 J ’

equations (3), (4), and (6) become

0Tx

and

Vi T2P

0S _ 0a 0(3

0T2 0S _ 0(3 1 0a

0 /m dw Tl 0^

0,

0,

0a

0a

0 /m dw'0(3 V 2 0p

0 /o 0Wdw\ ^ q0(3/ 0(3 \ 0a

The last of these can be simplified by the use of (7) and (8) to the form

d2W n a d2W nh2 — 4 3 Vl "■

T,P

m d2WT l d ?

The equations can be solved by

T° a s ? - 2S

o,

0a0(3




where and C2 are constants, provided that

/r „ 4 C2 - Vi4w - - o pn 02W n 02<C _ AU 0a2 2 0S2 ’

0M I 1 (dw\2 I f dv5« + % J + a i r P + 4 ,'0-mA2 \

0(3/ i Cl,

0V3B

. , l d w \2, I 0W , 4 /0ttA2) n+ 4(5-0) + <r{ ^ + 4 (—) ! = c=.and

0(3 0a/ J

0V , 0 W , 0W 0W _ ~0a 0(3 0a 0(3

( 10)

( 11)

( 12)

(13)

Setting u = Aa, A constant, and supposing that v and w are independent of a, equations (10), (11) and (12) become

and

h2 diw C2 p d2iv _ 03" d(34 p “ d{32 ~~ ’ ’

(14)

a Ci — aC2A ” 1 - c r 2 ’

(15)

dv w, 1 / dw\2 '_ C2 — cr Cid(3 p i 1 - <r ’ (16)

while (13) is satisfied identically.Substituting the value of p given by (1), it is seen that

w = kc sin 3,k constant, will satisfy (14) provided

hWlc/'d- (l - h i2) C2 = 0 .From (16)

dv C2 — crCi | jo •. o o 9 _— = - ^ — .r + sin- n(3------— cos- 3,

_ Co - oCi

(17)

0 8 )

+ &c2/2 — k2c2n2j4 — (kc2/2 -f- cos 2n(3.

As it is supposed that there is no aggregate lateral expansion of the plate, dv/dQ must be periodic, so that we must have

(C2 - <tC1)/(1 - a2) + - W / 4 = 0. (19)The expression for v is not wanted.*

* This expression contains an arbitrary constant which properly should correspond to a rigid body displacement. That v = const, does not exactly represent such a displacement is due to the fact that the shell equations are only approximate. As the plate is assumed shallow, the discrepancy is of minor importance. In any case, the oonstant can naturally be ignored.



150 W . R. Dean.

Cx and C2 are proportional to the applied thrusts along the generators and transversely, respectively. Given Cl5 C2 and can be found from (18) and (19), and A from (15); hence the values of u, v and w. That C2 is known when Cj is given is, of course, a consequence of the restriction that there should be no lateral expansion. The first part of the problem, the determination of the equilibrium configuration of the plate under thrust, is therefore completed.

§ 5. The Stability Equations.—If the equilibrium configuration, u, w, investigated above is critical, there will be a neighbouring configuration of equilibrium which we write u + u', v -f- v', w + . The stability equations are derived by substituting first u, v, w in the shell equations, then u + u', v -f- v', w -j- w', and finally subtracting the two sets of equations. In the

equations thus produced powers of u', v' and w' above the first can be ignored. There results

d_"0a

_00P

du' ,3 i + a {ap

u/ . dwdv£\ "I7 + d p W /J

\ *h/L Qw'~\2 0pL0a 0a J 0, (20)

dv . dw , du0p p dp 0p 0a ]

1 — <7 0 Vdv , du/ , _ 02 0aL0a 0p dp 0a J

and'n j t r 1 i drw~\rdv w' . dwdw'ViW - h + w m ~ ' i + T v w

+ a

d2w' rT du, wm d i + a \ d & ~ i

w + i (dw

d^u/ f dv0p*Ldp

w . , /dw\/ , du+ a r * .

(21)

Using (15) and (16) the last of these becomes

[- +0V

h/ y-y 4 ,3 VlW

d~w -P

d v '_«/ , dw , du!_0p p ~ r dp 0p dx-

- < v -C< d2w' ( 22)

At this point it is convenient to state the mathematical problem in full, repeating some equations that have already been given. We have to find




the value of Cj that leads to a non-zero solution in a', and w' of (20), (21) and (22), where

1/p = wjk — c sin (23)

J M /S - (1 - C2 = 0, (24)and

(C2 - gCJKI- <r“) + kc2/2 - J c W /i = 0. (25)

The next consideration is to put these equations in a form suitable for numerical work.

§ 6.—The first stage in the manipulation is effectively equivalent to the elimination of one of the variables u', v', Setting

ou , i bv _ w _ 0'“ 68a \0jS p d fid fi) 0[32 ’

d v ' w', dw d w ' , / 82̂0{3 p d$8(3 0a 8a2’

1 — o/dv' , du_ , dw dw'\ _ __ 82(j>2 \8a 0(3 1 dfi 0a / 8a0^’

equations (20) and (21) are satisfied,* while (22) becomes

(26)

(27)

(28)

(29)

u and v' can be eliminated from (26), (27) and (28), the result being

V,‘ 4, + ( l - < r ) ( i + | p ) | / f ' = 0. (30)♦

A non-zero solution in w' and ^ of (29) and (30) evidently implies a nonzero solution in u', v' and iv' of the stability equations. As stated what has been done is effectively to eliminate one of the original variables : the method adopted gives a more symmetrical form than would result from a direct attempt at elimination, and entails less labour.

It may be noticed that if1 , drw? W

= 0,

equation (29) reduces to the stability equation of a plane plate ; this relation, in fact, evidently implies that in the equilibrium configuration the middle

* This procedure is analogous to that commonly followed in two-dimensional problems. Of. A. E. H. Love, 4 Mathematical Theory of Elasticity * (3rd edition), § 144 ; also Prescott, loc. cit.,p. 101.



152 W . R. Dean.

surface is straightened out. However, from equation (24) an infinite transverse tension* is required for this ; it is a familiar enough experience that it is exceedingly difficult to straighten by tension any naturally curved body, as, for instance, a coil of wire.

§ 7.—The equations must now be put in non-dimensional form; (29) and (30) may be satisfied by

andw' = W sin (xa,

<]> = <]q sin [ix,

(31)

(32)

and

where ;x is a constant, and W and are functions of (3 only. Let the variable P be changed to 0 (= n$), and let

Ci - c a v , c 2 = c a'A y , = H ,

djh — df, — (a'.

Equations (29) and (30) then become

— - 2(x'2 ^ + [i/4 W + 3 Ci' [x'4 W - 3 C2y 2d04 do2

and

d02+ 3(1 - for) d'fx'2 sin 0 </> = 0 ,

d ^d64 2[x'2 + (x'4̂ — (1 — a2) (1 — d'rx'2 sin 0 W = 0.

and

dO*Finally, let Jen2 — ci,

F = 3 (1 - a)d'[x'2 , G = (1 - a2) (1 - a) d'ft'2

y = 3 C1y * 2 = 3 C2'[x'2.

The resulting equations are

d4Wd04

2+ 2 d2W d2W + F sin 00 = 0,and

d^>dO4

2 ^ 2 _j_ . . '4d02 + [x'4+ - G s i n 0 W = O,

2 = a/(l — a),

(33)

(34)

(35)

(36)

(37)while from (24) and (25)

and

a [1 + 3 (1 - a2) d'2/2] = 3u C / + 2 + 3 (1 - a2) 8/4 - a2/(l - a). (38)

* A tension because k must be positive in this case ; Jc is always negative in the conditions considered here— a transverse thrust being, of course, necessary to prevent lateral expansion.




In the first case considered numerically we take cr = 1 /4, d' = 10. The last equation is then

a = (0*0052957) C hy2 + (0-49647) a2 - (0-0070609) -3 ^ . (39)

In the second case, the same value of a is used, and d' 5, so that

a = (0-020743) C/fx'2 + (0-48617) a2 - (0-027657) (40)JL &

Equations (33) to (39) contain a complete statement of the problem, are non-dimensional, and are in the form that has been found most suitable for numerical work.

§ 8.-—The Assumption a — 0.—By analogy with other problems in elastic stability, we can suppose that the collapsing thrust in a corrugated plate is not importantly affected by the value of Poisson’s ratio for the material. For instance, in the problem most resembling the present one, that of the stability of a cylindrical shell under axial thrust, a appears in the formula for the collapsing thrust solely in the usual factor (1 — a2). Since for most materials a is in the neighbourhood of J, the supposition that it is zero leads to an error in the determination of the critical thrust usually less than 10 per cent.

The numerical discussion of a problem of stability is necessarily lengthy, since the object is not merely to find the stress that will cause a given type of collapse, but also to find that type of collapse (the “ favourite type of distortion ”) that takes place under the least stress. Inevitably, a considerable exploration is necessary. By assuming that <7 = 0, a striking simplification of the equations results, and it is relatively easy to find the favourite type of distortion. We can then find the stress that will cause collapse of the type when a is not zero, and if there is a reasonable agreement between the two values it is natural to conclude that the favourite modes are the same in the two cases.* At the least, the numerical work in one case is an obviously valuable guide to that in the other.

The physical interpretation of the assumption cr = 0 is a simple one. It means that the sole effect of a longitudinal thrust is to compress the parallel filaments of the plate ; the transverse expansion that normally accompanies such a compression does not take place, and no perpendicular thrust is necessary to prevent lateral movement. As a consequence, in the equilibrium configura-

* The argument as stated here in general terms is not, of course, conclusive, but it is quite clear from numerical and other considerations that in the work below the favourite type when <r* 0 has, in fact, been obtained.



154 W . R Dean.

tion w — 0, and there is no need for the third order terms that arise in thegeneral case.

§ 9. The Simplified Equations.—Putting = 0 in (38), we see that = 0 ;from (37) z = 0 ; and

F = 3 G! = 3d' pi'2.

If x = d'jx'2, equations (35) and (36) become

+ (/‘w + + 3*sin 0 ̂= °> («)and

+ W = 0. (42)

d! (= d/h)is known ; assuming a value of (x', we have that of x, and have to find the least value of y that permits a non-zero solution in W and <f> of (41) and (42).

§ 10. Type Solutions.—The expressions

W = W\ cos X0 + tv-2 cos (2 — X) 0 + w$ cos (2 + X) 0-j- Wi cos (4 — X) 0 + ...

<f> — (f>isin (1 — X) 0 + (f>2 sin (1 + X) 0 + sin (3 — X) 0~n (f>± sin (3 ~i~ X) 0 -J- ...

(43)

where X, wv w2, ..., <f>v <j>2, ... are constants, will satisfy (41) and (42) if certain relations hold between the coefficients. There is an infinity of such relations, which can be satisfied by non-zero constants if an infinite determinant vanishes. This condition is

F ( l - X ) ,X

2 ’ 0, X

2 ’ 0,

3x¥ ’ F (X) + y, 3x

¥ ’ o, 0,

o, X

2 ’ F( 1 + X), o, 0,

3x ~2 5 o, 0, F (2--x )+ y ,

3.x2"’

o, o, 0, F(3 - x )

— 0, (44)

where F (z) = (z2 + jx'2)2.




That the determinant is convergent is evident upon dividing rows andcolumns by the appropriate factors. I t can be written

..., F (2n — 1 + X), 0, 0, X

2 ’

0, F (2n — X) + y, 3 x

2 ’o,

i p

io\

H F(2» + l - X ) , 0,

— — 0 2 ’ 0, F (2m + x) + y,

Divide the first row and column written down by (2 — 1 -j- X)2, the second row and column by (2n — X)2, and so on. Typical diagonal terms are then

[1 + [i/2/(2>» — X)2]2 -j- yj{2n — X)4 and [1 + |x/2/(2 — 1 + X)2]2,

and typical non-diagonal terms,

3z/2 (2n - X)2 (2n + 1 - X)2 and - (2 - X)2 (2 n + 1 - X)2.

Conditions sufficient for convergence are that the product of the diagonal terms should converge absolutely, and that the sum of the non-diagonal terms should converge absolutely.* The product of the diagonal terms converges like the product II (1 -j- 1/w2), the sum of the non-diagonal terms like the series 2 1 /m4, and consequently the determinant converges. There is, of course, no need to divide the rows and columns in numerical work, and they are left as they stand in the case considered below, § 11.

Given x, y can be determined from equation (44) for a given value of X. It is not necessary to consider all values of X when finding the minimum value of y. Writing X — 2 -j- X' in (43) the forms of W and are not altered, so that values of X between 0 and 2 alone need be considered. Again, putting first X = 1 -f- X', and then X = 1 — X', a pair of identical expressions for W and (f> are obtained. For given x, yis a function of Xf ; this function is periodic with period 2, and is symmetrical about the value X = 1. It already appears probable that the minimum value of y is to be found when X — 0 or when

* Whittaker and Watson, ‘ Modern Analysis,’ § 2-81.t It is clear from the form of (44) that, given x and X, infinitely many values of satisfy

the equation. Only the numerically least of these is of importance.



156 W . B. Dean.

X = 1. It is convenient to treat the solutions when X is integral separately. There are four, and no more, distinct cases, Types I-IV below.

Type 1:—W = Wi + cos 26 + w-z cos 46 + ... , d

<f> — (f)i sin 6 + $2 sin 36 -f- <̂3 sin 56 + . . . , J

giving the equationnr

= 0 .F (0) + 3a?,~ 2’ 0 , o,

- X , F (1), a?2’ 0 ,

o, 3a?J ’ F (2) + y, 3x

2 ’

o, o, X

2’ F (3),

Type 11 :—

W = ivi sin 6 + «’2 sin 36 + ws sin 56 -j-

(f> — </>i + ^2 cos 26 + 3 cos 46 -h ... ,

Type 111W — W\ cos 6 -j- W2cos 36 -J- cos 56 -f- ..

== <f>i sin 26 -(- <f>2 sin 46 -f- <̂3 sin 66 —J— ..

F(1 ) + y ,

X

~ 2 ’

0,

3a?2 ’

F (2),

_ 3a?2" ’

0,

x~T

F (3) + y,

0,

0,

3a?2*’

F (0), X

2’ 0, 0, = 0 .

3a?, F (l) + 2/, 3a?I f ’ 0,

o ,a?2’ F(2),

' 2’

0, o, 3a?*2’ F(3) + y, ...




Type I V :—W — Wi sin 20 + Wo sin 40 + w3 sin 60 + ••• ,

(b — (f>icos 0 ~r ^2 cos 30 -j- 3 cos 50 -f- ... ,

F ( l) , X

~ 2* 0 , 0 ,

3x 2 ’ F (2) +

Sx 2 ’ 0 ,

0 ,X

2’F(3), X

2’

0 , 0 ,3x2 ’ F(4 ) + y,

If the first term of W is taken to be the sine or cosine of any integral multiple of 0, one or other of these four forms is reached. Types I and IV correspond to X = 0 in the general solution, Types II and III to X = 1. I t appears that on putting X = 0 the determinant of (44) becomes the product of the determinants corresponding to Types I and IV, and similarly in case X = 1, but this has not been verified algebraically.

§ 11. Numerical Results.—Theoretically, values of y are required for all values of p', but the sequel will show that when p,' is large the values of y' are of no importance.

Table I gives the relation obtained by computation between y and p,' for the values 10 and 5 of d!.

Table I.—Relation between y and p/ when a = 0.

(a) d' = 10. . (6) d' = 5.

p- - - C /m'2- Type. /*'• - y- - c/ m'2 Type.

0 1 0-0148 0-493 I 0-1 0-00378 0-126 I0-2 0-222 1-85 I 0-2 0-0570 0-475 I0-3 0-999 3-70 1 0-3 0-262 0-970 I0-32 1-25 4-07 1 0-4 0-724 1-51 I0-3 1-22 4-52 III 0-5 1-50 2-00 I0-4 1-46 3-04 III 0-6 2-56 2-37 I0-5 1-82 2-43 III 0-5 1-63 2-17 III0-6 2-36 2-19 III 0-6 1-98 1-83 III0-7 3-10 2-11 III 0-7 2-44 1-66 III0-8 4-09 2-13 III 0-8 3-04 1-58 III0-9 5-35 2-20 III 0-9 3-80 1-56 III1-0 6-91 2-30 III 1-0 4-74 1-58 III1-5 19-5 2-89 III 1-1 5-89 1-62 III



158 W . R. Dean.

The numerical work for the single case d' 10, = 1*5, which gives thelargest value of x, will illustrate the method, and show how many rows and columns of the determinant need be considered. In this case x — 22*5

F (1) = 10-563, F (2) = 39-063,F (3) = 126-59, F (4) = 333-09.

If — y == 19, the determinant of Type III is

- 8-437, 33-75, o, 0,

- 11-25, 39-063, 11-25, o,

o, - 33-75, 107-59, 33-75,

o3 o, — 11-25, 333-09,

while if — y — 20, it is

- 9-437, 33-75, o, 0.— 11-25, 39-063, 11-25, o,

o, - 33-75, 106-59, 33-75,

0, o, — 11-25, 333-09,

The value of the first is

( - 8-437) ( - 5-94) (43-7) (341-8) ...,

and that of the second

( - 9-437) ( - 1-17) ( - 218) (331-4).......

Considering only three rows and columns we have to take only three factors ; by linear interpolation it is found that the determinant vanishes when — =19 • 5, to three figures. This result is not altered by including the fourth factors. Consequently in all cases considered at most only four rows and columns are needed, and the position is the same when a is not zero.

The values of — yin Table I are (with some obvious exceptions) the least that lead to non-zero solutions of equations (41) and (42), the type of solution giving them being shown. The proof that there are no smaller values is mainly numerical, but some general arguments are available.

With the solution of Type IV the determinant is positive unless — y exceeds F (2), that is (4 + p/2)2. This expression is greater in every case than the value of — y given.



Elastic Stability o f a Corrugated Plate.

The determinant corresponding to Type II can be reduced to

F(0) F ^ + 2F (0) + V’3a;! ’ o ,

X F(2), X

2’

0 ,2x¥ ’ F(3)+ ;

159

which is positive unless — y exceeds 3jc2/2 F (0). This expression is 3 2,and therefore is 150 when d' — 10, and 37-5 when d' = 5. Types II and IV are consequently excluded from consideration in the range of values of Y taken.

With Type III — y must exceed F (1), and therefore 1, and need not be considered till \x >0-3 , when d' = 10, nor till Y >0*4, when d' = 5.

As stated, there is some reason to suppose that the general solution for values of X other than 0 or 1 cannot provide a minimum value, but it seemed advisable to verify this numerically in some particular cases. When d' 10 this work has been confined to the values 0*3, 0-5, 1*0, and 1*5 of f . In all four cases it has been shown that with the value of —y* given in Table I, and with each one of the values 0-2, 0-5, and 0-8 of X, the relevant determinant is positive. It was clear that there was no possibility of a smaller root, and further that in case \x — 0-3 the determinant increased with X, while with the other values of Y the contrary was the case. Similar work has been done for the values 0-4 and 1 - 0 of Y when d' — 5. These considerations appear adequate to show that the least values of y have actually been found.

The results are shown in fig. 2. The physical significance of y and CY is explained later after the more practical results for J have been obtained.

§ 12. Results when a = —If a is not zero, but, as is here supposed, J, theprocedure of the last paragraph is not essentially altered. Assuming a value of GY Y2, those of a, y, z, F and G can be calculated from equations (33) to (40), and the value of GY Y2 f°r which the appropriate determinant vanishes is found by linear interpolation as before. The type solutions of § 10 may be used without alteration, the determinants corresponding to Types I and III being now

F (0) + y, 17 2, 0, 0,— G, F (l), 0/2, 0,0, - F / 2 , F (2) + y + 4 z, F/2,0, 0, - 0/2, F(3),

* That corresponding to Type I was, of course, used for Y = 0-3.



160 W . R. Dean.

F (1) + «/ + 2, F/2, 0, 0,— a/2, F(2), G/2, o,

o, - F / 2 , F (3) + y4- 9*,F/2,o, 0, — G/2,

respectively.

d=5 ^Type ZZZ

1 -2 *3 4 -5

Fig. 2.—Values of C* p'2, when 0.

The results of calculation are given in the following table.Table II.—Relation between CV ; /2 and p/ when a J.

(a) d' = 10. . (6) d' = 5.

/• - c xy 2.

0-1 0-4650-2 1-770-3 3-600-325 4-090-3 4-450-4 2-990-5 2-400-6 2-150-7 2-080-8 2-100-9 2-161-0 2-261-2 2-491-5 2-84

Type.

IIII

IIIIIIIIIIIIIIIIIIIIIIIIIIIIII

/•

0-10-20-30-40-50-60-50-60-70-80- 9 1 0 1 11 - 2 1-5

- w * . Type.

0-1190-4540- 9451- 502- 03- 2-44 2-12 1-80 1-64 1-57 1-55 1-57 1-61 1-67 1-92

IIIIII

IIIIIIIIIIIIIIIIIIIIIIIIIII



Elastic Stability of a Corrugated Plate. 161

The values of C / f 2 in this table are in all cases within 7 per cent, of the corresponding values when a = 0. This being so there appeared to be no need to repeat the numerical work of verifying that the favourite types have actually been obtained, especially as the general remarks of § 11 apply with little alteration to this case. The results are shown in figs. 3 and 4.

§ 13. The collapsing thrust* is — 2 E hTj/( 1 — a2), and the corresponding stress is therefore proportional to Tx or Cl5 while

Cx = Cj h2 y2 = (t (CY (F2), (45)being the length of a bay.Before proceeding further it must be pointed out that in this, as in most other

problems of elastic stability, it is impracticable to consider in detail how the plate is held at its edges. For instance, in the investigation of the stability under longitudinal thrust of a cylindrical strut, the condition for the collapse of a strut of length X under given end conditions has not been found ; what has been done is to find what thrust will maintain a distortion of wave-length X in an indefinitely long tube.f

From (31) X (— 7c/p,) is the distance between successive zeros of the normal displacement in the distortion, and the best way to bring H, the height of the

•1 -2 -3 -4 -5 10rFig. 3.—Values of Cx' /x'2, when d/h = 10, o* =

* The notation of the theory of thin shells differs in writing the thrust — Tr f R. V. Southwell, “ On the General Theory of Elastic Stability,” ‘ Phil. Trans.,’ A,

vol. 213, pp. 235, 236. The effect of the manner of constraint at the ends in the similar problem of the boiler flue is discussed on pp. 226, 227.

VOL. CXI. — A. M



r62 W. R. Dean.

plate, into the analysis would appear to be to decide that X must not exceed H. Then \i (— ]xjn) must not be less than Z/H. The appropriate value of C/p,'2 to be substituted in (45) is the least value the expression can take for p,'>: Z/H ; this is merely to say that the wave length of distortion will be that possible under

Fig. 4.—Values of CYm a> when djh = = £.

the least stress. Consequently when d' 10, for values of Z/H between 0-22 and 0-7 the wave-length is to be calculated from p,' = 0-7, and the value of — C iV 2 is 2*08. This least value can be read off from the continuous line

1 -2 -3 4 5

Fig. 5.—Values of C.

in fig. 3 for p/ = Z/H ; those parts of the curves shown dotted are of no physical significance. Similarly for d' — 5 the continuous curve of fig. 4 must be used. The value of the stress can now be calculated from (45), but it is simpler to



find the relation between a new function C' and —C 'f 2 being represented by the continuous lines in the two cases. This relation is given in the following table, and shown in fig. 5.

Elastic Stability of a Corrugated Plate. 163

Table III.—Relation between (7 and f .

(a) d' = 10. (b) d' = 5.

O'. P- O'.

0-1 46-5 0-1 11-90-2 44-2 0-2 11-30-225 41-1 0-3 10-50-3 23-1 0-4 9-370-4 13-0 0-425 8-580-5 8-32 0-5 6-200-6 5-78 0-6 4-310-7 4-25 0-7 3-160-8 3-28 0-8 2-420-9 2-67 0-9 1-911-0 2-26 1-0 1-571-2 1-73 1-1 1-331-5 1-26 1-2 1 -16

1-5 0-853

The collapsing stress (a thrust) is now

W (h\* ,— * la;c

the value of Of being determined for f = H.§ 14.—The critical stress of a long plane plate of height H and semi-thickness

is*

Ett2 { h f 3(1 - a2) 'H / ’

so that the ratio of the critical stress of a corrugated plate to that of a plane plate of the same thickness and height is 3(7. From fig. 5 if the depth of the bay of a corrugated plate is 10A it will withstand over 130 times the stress that can be put on a plane plate of the same thickness and of the same height, provided that the ratio of length of bay to height does not greatly exceed 1/5 ; but with the increase of IjH beyond this point, the strength of the plate diminishes rapidly. In the other case, d' 5, it is not so important that H should be small; a critical stress 30 times that of a similar plane plate can. be obtained if 1/H is less than O'35. Even such shallow plates as are considered

* Love, op. cit., § 332 A.



164 W . R. Dean.

here are therefore remarkably strong : in the first case, the critical stress (though not, of course, the total thrust per unit length of edge line) is actually greater than that in a plane plate of the same over-all thickness, 22

A rough expression for the maximum value of C/ is 2. There is some theoretical basis for this ; when a — 0 it can be seen from the condition for a solution of Type I that as [i' decreases, — yapproaches the value (1 -f- 3d'2/2) fjt/4, whence —C/ = 1/3 -f- d'2/2. By means of this approximate rule and fig. 5, the values of C' for all values of d' less than 10 and all \x can be obtained probably to within 5 per cent.

The limiting value of C' as [/ is increased is seen immediately : by increasing the length of bay and keeping the height constant the case of the plane plate is approached. Hence C' tends to the value 1/3. The distortion of a plane plate corresponds to the solution of Type I in the corrugated plate. Consequently it may be expected that for large values of f the curves of Type I in figs. 3 and 4 will again intersect those of Type III, and will then remain below them. This is borne out to some extent by the point of inflection on the Type I curve when d' — 5 ; when d' = 10 the point of inflection is on a part of the curve not drawn.

§ 15. The method of reduction of § 6 is not available if the assumption that the depth of bay is comparable with the semi-thickness is abandoned. An extension of the theory to plates with deeper bays, though certainly possible if it is supposed that a = 0, must therefore entail considerable labour, and it is only by this extension that it is possible to estimate within what limits d' must lie if the approximations of the preceding work are to be valid. I t does not appear worth while to attempt this until it is known from experiment what features of the problem are missed by the approximations. Again the rough rule that the ratio of the critical stress in a corrugated plate to that in a similar plane plate is, in the most favourable conditions, 3d'2/2, indicates that by increasing d' a point is rapidly reached, even with the thinnest plates, beyond which Hooke’s law will no longer hold, and the possibility of elastic failure (as opposed to failure by instability) must be taken into account. Suppose that E = 2*0 X 1012 (C.G.S.), and that the limit of proportionality is 4*6 X 109,* then if A/H = 0-01, the critical stress given by the rough approximation exceeds the limit of proportionality as soon as — 2 • 1 , if A/H = 0-005, the corresponding value of d' is twice that above, 4-2. Now the smaller is A/H, the greater is the range of values of d' in which the work

* The writer is indebted to Mr. H. J. Gough of the National Physical Laboratory for these figures, which are average values for some types of steel used in aeroplane construction.



Elastic Stability o f a Corrugated Plate.165

of this paper will be valid, so that there is some hope that with values of A/H likely to occur in practice the solution will hold in all cases in which breakdown is due to instability alone, or at any rate in all cases wherein the critical stress can be calculated by a theory based on Hooke’s law. When elastic failure must be considered, theory can at present do no more than indicate a stress that cannot in any circumstances be exceeded ; theoretical values in such of these cases as have been worked out are usually far in excess of the stresses that actually cause failure.* The rapidly increasing maximum stress given by the ratio 3d'2/2 is not likely to hold for large values of d' (it is 4 per cent, too high when d' — 5, and 6 per cent, too high when = 10), and there is some reason therefore to conclude that this stress is one that cannot be exceeded in any case. If this is so, where extension of this work is needed is in finding whether the theory can set a lower limit to the maximum stress for larger values of d'.

It may be pointed out again that it has not been found possible to allow for the way in which the plate is held at its edges ; there is, however, no evidence that on this account the theoretical values of the stress are likely to be too low. I t is true that by clamping a plane plate the critical stress is increased to four times that given by the above formula, because the wavelength of distortion is halved. But the effect of clamping, or any method of support, in any other case is that high local stresses, and therefore conditions likely to lead to instability, result in the neighbourhood of the edges, due to what is usually known as the “ edge effect. ”f I t is true that if u = 0 the way in which the plate is held at the edges can be considered ; but it is only in a certain type of problem that the value of a is of minor importance, and those that concern the edge effect, which does not take place at all if 0, are certainly not in this class.

§ 16. The utility of a theoretical discussion of the stability of a body is not, however, confined to the deduction of a formula for the critical stress ; there is indeed, for the reason mentioned, but rarely agreement with experiment on this point. Qualitative information, with usually a wider range of validity, can often be obtained. Thus it may be concluded by extrapolation from fig. 5 that for values of d' up to 15 or 20, any value of Z/H not exceeding 1/5 will lead to roughly the maximum stress possible in a plate of given depth of bay, and this is a result likely to be true when the theoretical value of the critical stress is unreliable.

* Southwell, loc. cit., footnote to p. 241.



166 W . R. Dean.

Again, theoretical information in respect of the favourite type of distortion is usually in good agreement with practice. This matter has not been considered here in detail, as there is little point in elaborating the above work before experimental results are available, but the general nature of the distortion can .be stated. The highest stress is obtained when the ratio of length of bay to height is small, and then the distortion of Type I takes place. This resembles that of a long plane plate, or, what is perhaps a closer analogy, the primary flexure of a cylindrical strut under thrust, in which the strut as a whole bends like a thin solid rod. The normal displacement w' of a point of the plate is of the form

7T0Cw' = (wx -f- w2 cos 2nfi -f- ... ) sin ——,

where wv w2, . . . . are constants whose ratios can be determined, for when the ratio of length of bay to height is small, the wave-length of distortion is as long as possible ; moreover, in this case, the most important term in the bracket is wx, so that the normal displacement is practically dependent only on on* The tangential movement of points of the plate is of minor importance.

The distortion of Type III is not of the same practical importance, for it only appears when the plate is not so corrugated as to give nearly the maximum stress for given depth of bay and height. In this,

w' — ( wicos + w 2 cos 3w(U -f- . . . . ) sin p.a,

where p, is not now necessarily equal to tc/H, and exceeds this value over a certain range of values of 1/H, so that a distortion of short wave-length takes place. Again there is a parallel to this in the stability of a cylindrical strut. The distortion cannot be simply described in this case as certainly two terms of the series are needed, and they are likely to be of equal importance, but it is of interest to notice that the middle generator n(3 = (2m -j- 1) 7u/2, m integral), of any bay remains fixed in the displacement.

§ 17. Notation.—It may be useful to set down again here the notation for convenience in reading the figures.

(V — djh, where d is the depth of a bay, and h the semi-thickness of the plate.

* The number of terms that must be taken in the series for the displacements is indicated by the number of rows of columns of the determinant necessary to ensure a given accuracy. In the work above four rows and columns are in all cases found sufficient, so that two terms of the series for w' give a result of the same accuracy ; but when l/H is small, two rows and columns are generally enough, and w' is therefore roughly a function of a only.



Elastic Stability o f a Corrugated Plate.167

M = 7i/Z, [x = 7t/X and p /= p/w; l is the length of a bay of the plate, and A the distance between consecutive zeros of w', the normal displacement in the distortion. X, the semi-wave length of distortion, will be that possible under the least stress, but we may suppose that X must not exceed H, the height of the plate. The proper value of p' is then the value > Z/H that gives the least stress ; the fact that p' is not in every case equal to H is allowed for by the continuous straight lines in figs. 3 and 4.

The critical stress is

the appropriate value of C /p '2 is that given by the continuous lines in figs. 3 and 4 for the value Z/H of p'.

Fig. 5 gives the relation between a new variable C' and p/. The critical stress is

Ejt2 /h\2

Ett1 —

the value of C' being taken from fig. 5 for the value Z/H of p/. 3C' is, therefore,the ratio of the critical stress in the corrugated plate to that in a plane plate of the same thickness and height.



theelastic stability of a corrugated...

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