The elastic stability of a long and slightly bent ...rspa.royalsocietypublishing.org/content/royprsa/162/908/62.full.pdf · 1— The problem of the elastic stability of a plane rectangular

62 H. Jones and N. F. Mott

R eferences

Bethe 1929 Ann. Phys., L pz ., 3, 133.D aunt and Mendelssohn 1937 Proc. Roy. Soc. A , 160, 127.Jones 1937 Proc. Phys. Soc. 49, 50.Jones, Mott and Skinner 1934 Phys. Rev. 45, 378.Keesom and Clark 1935 Physica, ’sGrav., 2, 513.Keesom and Kok 1934 Physica, ’sGrav., 1, 175.

— — 1936 Physica, 'sGrav., 3, 1035.Kronig 1931 ^ P hys . 70, 317.Mott 1935 Proc. Phys. Soc. 47, 571.Mott and Jones 1936 44 The Theory of the Properties of Metals and Alloys. Oxford. Pickard 1936 Nature, Lond., 138, 123.Slater 1936 Phys. Rev. 49, 537.Skinner and Johnston 1937 Proc. Roy. Soc. A (in the Press).

The Elastic Stability of a Long and Slightly Bent Rectangular Plate under Uniform Shear

By D. M. A. L eggett, B.A., Trinity College, Cambridge

(1Communicated by R. V . Southwell, F.R .S .— Received 5 February 1937)

1— The problem of the elastic stability of a plane rectangular plate when subjected to uniform shear has been approximately solved for various conditions (Cox 1933; Timoshenko 1936). In the case of an indefinitely long strip an exact solution has been found (Southwell and Skan 1924), but it appears th a t no attem pt has been made to investigate what happens if the plate is no longer plane. I t is accordingly the object of this paper to consider the stability of a long strip, slightly curved, when its two side edges are subjected to uniform shear.

2— In what follows we assume th a t the thickness and curvature of the plate are constant, and th a t the edges of the plate are two generators and two lines of curvature. I t is, moreover, further assumed th a t the plate is thin as in all similar stability problems, and th a t it is of such length tha t the boundary conditions over the two curved ends can be ignored.

We first consider the equilibrium stresses and displacements caused by applying a uniform shearing force to the two straight edges. Referring the

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Elastic Stability of a Rectangular Plate

system to ordinary cylindrical co-ordinates, it can be shown without difficulty from the equations of equilibrium and the equations connecting up the coefficients of stress and strain (Love 1927, §§ 22, 59, 69) tha t the set of displacements

ur = 0, ug = 0, uz = rd,

where r is constant, can be produced by a stress system in which all the coefficients of stress vanish except 6z which has the value pr/r.

tfttt

4 T

a1II14

^ 9

F ig. 1

3—We now confine our attention to the middle surface of the plate and choose a new system of co-ordinates. We take the generator and line of curvature through the centre of the plate as axes f, y, so th a t the edges of the plate are £ = ± a/2, rj = +6/2, and introduce a th ird axis £, normal to the middle surface in its unstrained condition, and such th a t £ ,y, £ form a right-handed system. Thedisplacement of any point of the middle surface is u, v, w, with reference to these axes. The radius of curvature of the plate is p, its thickness is 26, and the external force system consists of a shear resultant S acting uniformly over the two straight edges.

Since it is supposed th a t the edges of the plate are free to move in the ordinary equilibrium displacement, it follows from § 2 th a t the actual displacements are

S y(l + a)u = — Eh 0,

We shall now use the shell equations obtained by Dean for problems of this nature (1925), when second-order terms which are multiplied by have been om itted, a legitimate approximation since we are assuming th a t 1/p is small. The above set of displacements now satisfies the shell equations exactly, and if the configuration is one of neutral equilibrium, the shell equations can also be satisfied by the displacements

S t](1 + (t)Eh

+ u v',

where u', v ', w' are arbitrarily small but not all zero. Subtracting the shellequations for the two configurations, and ignoring terms of order above the first in u ', v ', w ', we deduce the three fundamental stability equations.



64 D. M. A. Leggett

These are long but can be simplified by making two assumptions. First, th a t 1 Ip is small and of the same order as h\ and second, th a t w' is of larger order than u' or v'. The latter assumption follows a t once from a consideration of the limiting case when the plate is flat, and from the fact th a t we are considering a plate which is only slightly bent. As a result of making these approximations and only retaining the terms of the highest order of magnitude, the equations are

0 f'du ' ,'dv' w'X"I (l — O') 0 /m ^ y p).J 2 07/1

dv'

2 d g \d £ + dri) + dri\\dij p j + Cr0 ^J ° ’(1 - O') 0 t t 8 j~|W tf^

8 dh_ 1 r/dv ' w’\. . . s dhv'i r/dv ' w '\ dur]

(1)

( 2 )

(3)

We will now put the equations (1), (2) and (3) into non-dimensional form by means of the substitutions

b y _ bn 2 '

We getdx\_dx + (T\dy n p ) \ + 2

“ " ~ “ ‘ ‘ ^ ^ ^ d x2 dx\dx + dy) + d y \ \d y 7rpj + Cr0a:J 0,

- V W + H rr2^6\ 2 8 d*W' 63 T ng V ^ + d - O r ) ^ — ^ ¥ _ _ _ j + o. _ J = 0.

(4)

(5)

(6)

We now effect a simplification by introducing a function f (cf. Dean 1926; limoshenko 1934, chap. 1) defined by the equation

(1 — cr) !dur dv'\ d*\Jr2 X d y^ lkc ) = ~ n dxdy‘(7)

From (4), (5), (6) and (7) we deduce the following four equations:

du' (dv' bw'\ d2ilrt e + ° [ ^ - i r p ) - n d y i’ («)

(dv' bw'\ du' d2xlrW ~ ^ ] + ( r 0^ = 77 0 ^ ’ w

V74 / T>d2W'v W - + * a ^ = o. (10)



E la s t ic S ta b i l i t y o f a R e c ta n g u la r P la te 65

(11)

where' - X Y - « - W ‘ - " S ® -

(12)

4— Case 1— Edges Fixed and Sim ply Supported—We shall now find a solution of the above equations which satisfies the boundary conditions*

u' = v' = 0,'j

, d2w’ n f for 0, (13)w = W = 0’\

Since we are om itting the corresponding conditions over the two curved ends, it is possible to express w' and xjr in the form

w' = w[ cos rnx + sin mx,\Jr = x cos mx + sin mx,

whence introducing the two complex quantities W and W defined by the relations

W = w[ + iw'2, W = f x + iilr2.

We are enabled to express w' and rjr more compactly as

w' = {R)W 'e-imx,xjr = (14)

In (14) W' and W are functions of y only, and m is real but otherwise unspecified.

Substituting for w' and >'/ in (10) and (11), we derive the two fourth order differential equations

^ - m ^ W - R m 2W' = 0,

/ /72 \ 2 dW f“ m2 W' + P m 2'F-im 0.

\dy2 / dy

Since they are linear and have constant coefficients, their general solution is of the form

no 00W = 2 areTrV> W' = 2 breTrV,

r = 1 r= 1

where the ar’s and 6r?s are interdependent complex constants and the r r5s are the roots of the algebraic equation

(T2- m 2)2{{T2- m 2)2-im Q T} + PBm^ = 0.

* The expression for the couple in general contains two term s, bu t since w is zero over the two straigh t edges the second term vanishes.

Vol. CLXII—A. F



66 D. M. A. Leggett

From this equation for t it can be seen th a t the algebra involved by this method of solution would be so heavy as to make the method scarcely practicable, but before considering any alternative we make an important observation which we shall use later, namely, th a t the general solution for W and IF' is such th a t W and IF' and all their derivatives are continuous functions of y in (On).

We shall now proceed to develop a method of solving the equations (10) and (11) entirely different from tha t just outlined above. From the last two boundary conditions in (13), we deduce

w's = = 0 , [s = 1, 2] for = 0, (15)

Now consider the fourth derivative of w's 1 or 2) with respect to y in < 0 n > . I t is continuous in th a t range, and hence can be expanded as a

Fourier series of sines throughout (Off) in the form/74 7y / 00

= ^ r * A „ w ir y , (16)

where by a well-known property of Fourier* series r4 | A sr | <K\r, K being some positive number independent of r . Making use of the above inequality and another property of Fourier series, we integrate (16) twice, and deduce tha t in (On)

(]2W' qq(17)

Since each side of (16) is a continuous function of y, it follows tha t the equation is true, not only in the open interval (Off) but in the closed interval < Off > as well. We now put y equal to 0, n, whence it follows from (15) tha t B and C are zero. Integrating (17) twice and adopting the same reasoning as above, we deduce th a t

00w's = 2 A sr sin in < Off > .

r= 1

Reversing this process we obtain the result we want, namely, th a t we can assume a Fourier series in sines for w's in < On > which can be differentiated four times term by term.

Introducing A r defined by the relation

A r — A + A^r)

* For the properties of Fourier series used in this paper see Carslaw (1930, cnaps. i j oj»



and substituting for w 'in (10), we have the following equation from which to determine W :

Elastic Stability of a Rectangular Plate 67

/72iz/ 00

The general solution is

[A cosh my + B sinh my} + y{C cosh my + D sinh my)A r$\nry+ m 2R 2

r= 1 (r2 + w 2)2* (18)

We will now obtain A, B, C, Din term s of A r by virtue of the first two boundary conditions in (13). Since these involve u' and v' it is necessary to solve for them from the equations (7), (8), (9). Assuming a form of solution given by

u ' = u{ cos mx + u 2 sin mx,

v' = v[ cos mx + sin mx,

where u'x, u 2, v[, v2 are independent of x\ and introducing U' and defined by

U' +

so th a t u ’ = (R) U'e~imx,v' = (R) V'e~imx,

we obtain from (7), (8), (9), (19) the three following equations:*

(1 <r)ldU'im V \ _ i7Tmf^_ 2 \ dy 1

(20)

. TT,dV' d2W crbur,- im U + cr —j— = 2 + W ,dy 2 np

(21)

dV ' . TT, 21—-----urm U — —nm lxR + ... .dy np

(22)

M anipulating these equations we haved%w

m U \ 1 — o'2) — in 2 +incrm2W,’ (23)

dU ' 2nm dW (1

(24)

* A t first sight it m ight be thought th a t we had found three equations to be satisfied by the two unknown functions V and b u t it m ust be remembered th a t of the five equations (7) to (11) only four are independent. Since we are treating (10) and (11) as independent, it follows th a t of the three equations (7), (8), (9), and therefore of the three equations (20), (21), (22), only two are independent.



68 D. M. A. Leggett

The first two boundary conditions in (13) can be expressed in the form

U' = V' = 0, for

whence from (23) and (24) we deduced Wdy2

+ crm2lF = 0,

d2W x 2dWW ~ ( + , r )”• ' #

0 ,

for y = 0, (25)

Substituting for W given by (18) in the equation (25), we have the following four equations from which to determine

Am( 1 + cr) + 2D = 0, (26)

im ( l + cr) cosh mn + Bm (l + cr) sinh mir+ <7(2 sinh mn + 1 + cosh mn}+ D{2 cosh mn + mn( 1 + cr) sinh mn} = 0, (27)

- B m { l + o-) + C( l - ( r ) = BX , (28)

— Am{ 1 + cr) sinh mn — Bm( 1 + cr) cosh mn+ (7{( 1 — cr) cosh mn — mn( 1 + cr) sinh mn}+ D{(1 — cr) sinh W77 —ra7r(l + cr) coshm7r} =

wherev ^ A r{rz+{2 + o-)m2r} _ £ A _

r=i (r2 + m2)2 r r ’

(29)

(30)

The second half of the two equations (30) is to be regarded as defining ocr which it will be convenient to use later.

Solving (26) to (29) we deduce

(31)

where A = m2( 1 + a*)2 {(3 —a*)2 sinh2 m77’ — m77'2(l + cr)2}, (32)

A ' = 2RXm (l + <r){(3 — cr) sinh m7y cosh m7r + m77'(l + cr)}— 2R Y m (l + a) {(3— cr) sinhm7r + m77(l + cr) coshm7r}, (33)

B' = i?Xm(l + cr) {m2772(l + cr)2 —2(3—cr) sinh2 m7r}— R Y m 2n(l + a*)2 (1 — cr) sinhm7r, (34)

C' = R X m 2(I + cr)2 (3 — or) sinh2 mn — RYm?n(l + cr)3 sinh mn, (35)

D' = — R X m 2( 1 + o')2 {(3 — 0*) sinh mn cosh mn + mn( 1 + cr)}+ R Y m 2( 1 4- cr2) {(3 — a) sinhm77 + m7r(l + cr) coshm7r}. (36)




After substituting the values of A,B, C, D, obtained from (31) to (36) in the equation (18), the resulting expressions for and given by (14) will satisfy all the fundam ental equations and boundary conditions except the equation (11). We now therefore substitute for w' and ^ in (11) and equate to zero the coefficient of e~imx. We get

2 A r(r2 + m2)2 sin ry — imQ rA r cos ry + PRm* ^ ^

+ P m 2{A cosh my + B sinh my + y(C cosh my + D sinh my)} = 0, (37)

which m ust be satisfied for all y in (O77).Now before we can proceed w ith the equation (37), it is necessary to

express all the term s in y as Fourier series of sines.Expressing cos ny as a sine series of the form

00

cos ny = 2 Ar sin ry> [0 < 77]r= 1

(38)

we deduce th a t the Fourier constants (iT are given by

— cos(r + n )n r + n

1 — cosr — n 1 (39)

Hence from (38) and (39) the coefficient of sin ry in the expansion of 00

2 r A r cos ry isr= 1

1 A p i — cos{r + n)TTi 1 — cos(r—/ n^\. ,n I ■

7T n = \L r + n r ~ n

By straightforw ard evaluation it follows th a t the coefficient of A n in (40), which we shall denote by crn, is given by the adjoining scheme:

(40)

A A 2 A A i

sin y 08 0

16~ 3 n 1577

sin 2 y8

3770

24577

0

sin 3 y 024577 0

48“ 777

sin 4y16

15770

48In

0(41)

I t now remains to expand in a sine series in (On) the expression

(A cosh m y + B sinh my) + y(G cosh my + D sinh my).



70 D. M. A. Leggett

ooWriting it as 2 Kr sinri/ we

r= 1

Denoting the coefficients of A , B , C, D in (42) by 7lr, / 2r? 73r, / 4r, we have

—------ — [1 — cosh mn cos rn],(r2 + m2)

(43)

r(r2 + m2)

sinh mn cos r7r, (44)

- L = - rcosriL [77-(r2 + m2) cosh mn - sinh mn], (45)2 (r2 + m 1)1

tt r- 1. = --------— r7r(r2 + m 2) cos rn sinh mn — 2m(cos rn cosh mn — 1)].2 ( r2 + m2)2

(46)

I t is now essential to get kt expressed in as simple a form as possible, and for this purpose we introduce Lr, Mr, defined by the equation

A Ilr + B I2r+ C I3r + D Iir = (47)l i t

so th a t we have

Kr = ^ \ L r2 A n' a n + M r 2(~ )"A n • (48 )m*\_ n= 1 1 J

From (31) to (36) and (43) to (47), we deduce by straightforward algebratha t

T. - 4 mocrr{(3 — cr) cosh mn sinh mn + mn (1 + cr)}

— cos rn{(3 — cr) sinh mn + mn( 1 + cr) cosh mn}_\l^r — n(\ + a) (3 — cr)2 sinh2 mn — m 2n2( 1 + cr)2

(49)

M - 4 marrcos T7t{(3 — cr) cosh mn sinh mn + mn( 1 + cr)}

— {(3 — cr) sinh mn + mn( 1 + cr) cosh m n}jn( 1 + cr) (3— (r)2sinh2m7r — m2772(l + <x)2

(50)and from (49) and (50) it follows th a t

Lr = ( - Y M r. (51)

On making use of equations (40) and (48), (37) can be expressed in terms of sines of multiples of y only, and since the equation is true for all y in (07r),



71

the coefficient of sin ry(r = 1,2 , . . . , go) must vanish. This gives us an infinite system of equations, linear in the A ’s, namely,

00 PR))!*(r2 + ™2)2 A r - Cm A n + (r2 + m 2)2 A r

+ P x \ L r f o i nA n + Mr f ( - anA ^ \ = 0. (52)L U— 1 71=1 J

In general the only solution of the above equations is th a t in which all the J . ’s are zero. I f however the infinite determ inant formed by eliminating the ^4’s vanishes, this is no longer the case. Since on physical grounds a solution of this kind is only possible when the plate is in limiting equilibrium, it follows th a t by equating the infinite determ inant to zero, we possess an equation from which to determine the critical value of S. This equation, being of indefinitely large order, will have an infinite number of roots, but since for practical purposes we are only interested in the shearing force necessary to produce the most favoured type of distortion, it is only with the smallest of these roots th a t we are concerned.

We now make the substitution

P/?m4(r2 + m2)2 + + PR + ( - )rMr) Z r, (53)

Elastic Stability of a Rectangular Plate

whence the required equation is

£1 l/TYbCC- 2 P R a^L i — -im Q c u

wfiQc^ — imQc23 PRa,i {L2 + Mf)

Pi?a1(L3 — Ms) — imQc$2 z 3 — imQCM

Tt*jo,0II

im Q c^ PjRa2(I>4 + Mf) — imQciS z*

after making use of equation (51), and the fact th a t half the c s vanish. Equation (54) is only formal, however, and it is first necessary to prove tha t the determ inant is convergent before we are a t liberty to s tart evaluating .

Divide the rth row and column by r2. Then the product of the diagonal elements converges absolutely, and it remains to show th a t the sum of the non-diagonal elements also converges absolutely (W hittaker and Watson1927, § 2-81). .

Since from (30), (49), (50), <xr, Lr, Mr, are 0 (l/r), the sum of the modulus otthe term s involving a, L, M is less than

k [ 2 - J 2 (K beins some constant)l r= l )

(55)



72 D. M. A. Leggett

and therefore convergent; but the terms containing the c’s require a more detailed consideration. The sum of the modulus of these terms is

00 00Qm 2 2

r —1n = lK Jr2n2 5

which, on making use of (41), is less than

2Qmi « l ) 2 2Qm ® ® 177 ( j .= l r 2| 71-

00 1 .Now consider —j-------- If is equal ton=in \ r — n \

r— 1 I oo ly -------------p y ----------- .

n=\nnJ?+1

( r ^ n ) (56)

(57)

By the § Cauchy-Maclaurin integral test, we can prove a t once th a t the expression (57) possesses an upper bound independent of r; hence the double series in (56) is convergent, and hence from (55) and (56) the sum of the non-diagonal elements is absolutely convergent— which is w hat we wanted to prove.

Since we cannot obtain the roots of (54) directly, we approxim ate to them by considering the values of S obtained by replacing the left-hand side of (54) by the finite determ inant which contains successively the first 4, 9, 16,... elements of the determ inant in (54).

We will now proceed to evaluate these determ inants, considering the accuracy of the results obtained in this way after we have actually found them.

5—As a first approximation, we consider the determ inant formed by taking the first four elements of the determ inant in (54). We have

— imQc12— imQc21 Z 2

0 ,

giving on evaluation8 Qm (ZxZ 2fi.

Substituting for Q from (12), we deduce finally

where Z l9 Z 2 are given by (53), and m has th a t particular value which makes S a minimum.



For a second approximation we take the first nine elements in (54) giving ultimately, after making use of (41) and (51),

C Dn* f Z ^ Z z - l Z z a l L j P Z R z H2mb2\c |3 Z 1 + c\2Z 3+ '

Proceeding in a similar manner, we obtain from the fourth order determ inant appropriate to the third approximation a quadratic for Q-. This we solve for the smaller of the two roots, bu t as the coefficients in the quadratic are long and complicated, the process is not set down here.

The limiting value of S, as calculated from the third approximation when p is infinite,* is in very close agreement with the value obtained by Southwell and Skan (1924) in their paper. Calling this value S 0, k, in Table I, gives the ratio of the shearing force, calculated for any particular case, to S 0.


First

T a b l e I

Second Thirdapproxim ation approxim ation approxim ation

V R PS’m 2 k m2 k m2 k

0 0-59 1-046 0-63 1-000 0-63 1-0001 0-77 1-199 0-83 1-118 0-83 1-1162 1-05 1-523 1-15 1-342 — —3 1-35 1-914 1-35 1-574 — —4 1-6 2-360 1-55 1-802 1-6 1-7855-5 1-9 3-128 1-75 2-133 — —7 2-3 4-017 1-75 2-458 1-85 2-348

We do not consider the significance of the above resultsf until we have solved the problem for an alternative set of boundary conditions. The results are shown graphically in figs. 2, 3.

6—Case I I — Edges Fixed and Clamped— In this event the boundary conditions to be satisfied are

u' = v'

dw' w — —

0, rr. (59)

The m ethod of solution adopted in this case is similar to th a t explained in § 4, and as in th a t case it is necessary to s tart by considering .to what extent we can assume a differentiable Fourier series for w .

* 52-85 D/52.t The value of cr is taken to be 0-25.



D. M. A. Leggett

Case IFirst Approximation

Second Approximation Third Approximation

Case IISecond Approximation

F ig. 2

Case ICase II

F ig. 3




We express w' and ijr in the form assumed a t the beginning of Case I, but the last two boundary conditions in (59) now give rise to

/ dwi „r „w s = - j - = 0, 0 = 1, 2] for (60)

Expanding the fourth derivative of w's as a sine series in (077-), we have

r= i

and A sr satisfies the inequality

d*w's " , , .2 r (61)

Integrating equation (61) three and four times, we deduce for O ^ y ^ n

dpo*rA sr cos ry + E 2 + 3

w's = f A s r r y + Hs + E + 3,r = l

where /7S, i^, Crg are constants. From (60) these are given by

(62)

H s = 0, ^ s + 7rFs + 7r2(7s = 0,

Es = — 2 f-4sr, F s + 277.FI, + 3n2Gs 2 r cosr = l r = l

leading to o'II Fs =

00Es = - 2 rA sr,

r= 1

II

~ 2 {2 + ( - ) r}^sr>7/ r= 1

- p i { ! + ( - « *-an r= 1

(63)

Hence we can assume for w' a form given by (62) and (63), where the sine series can be differentiated term by term four times. Introducing W', ¥ and A r, as defined in Case I, w’ and >/ can still be expressed in the form given by (14). We now introduce the additional complex quantities E, F and G, defined by the relations

E = E l + iE 2,F = F-y + iF^, G = G + iG^.

Substituting for w' in (10), we have the following differential equation for :

r74U7 ,7271/ / CO 1-t—j- —2m2-7 -j + m4y/ = m 2R \ 2dy dy“ l r = i /



76 D. M. A. Leggett

which has for its general solution

[A cosh my + B sinh my) + y{C cosh my + D sinh my)

i+;MS+K*+ +,v,+<v}]- (64)r “ A rsinry+ m R\ 2 (/.2 + m2)2|_r=

I t is now necessary to obtain the constants A , B, C, D from the first two boundary conditions in (59). The first part of this process is identical with tha t contained in equations (19) to (25) in Case I and so is not repeated here, but on substituting for W given by (64) in (25), we deduce four equations from which to determine A, B, C, D, which are slightly different from the corresponding equations (26) to (29). The left-hand sides of the two sets of equations are the same, but in place of 0, 0, R X , R Y on the right-hand side of (26), (27), (28), (29), we now have Rd , R<fi, R (x + fr), R (y + Q) for the corresponding equations in Case II, where

\]s = —(2+ cr) m2

00I r A r-

r= 1

6(3 + 2o-) 7r2m4

oo

2 {1 + ( - ) r}rr= 1

00

I A y r ,r= 1

Q = - (2 + o’) |TO2

( - Y r A r- 6(3 + 2o-)772 TO4

2 {1 + ( - Y } rr= 1

00

2 ( - M r 7 r .

0 = - - ^ P - l Z V + ( - r ] r A ,7T7YI'

oo

2r— 1

02(l + 2o-)

■nm° 2 [1 + 2( — )r] rA r\ = — 2 { ~ Y A ry r. '=1 ! 1

The two new quantities introduced, y r and /ir, are to be regarded as defined by the second half of the above equations.

Solving the four equations analogous to (26) to (29), A , B, C, D are given by (31), (32), and

A ’ = 2R(X + i/f)m(l + o-){(3 — o-)sinhm77-coshm77- + m77(l + o-)}— 2 R ( Y +Q) m(l + o') {(3— cr) sinh + 1 + cr) cosh+ R6m{ 1 + cr) {(3— cr) (1 — a)sinh2 — 1 + cr)2)— 2 R<fim2n(l + cr)2 sinh mn, (65)

B' = R{X + xJr)m(l + o-){m27r2(l + o-)2- 2(3-0-) sinh2 mn)— R(Y+ {2)m 2n(l + cr)2(1 — o-)sinh.TO7r + R0m( 1 — cr2) {mn( 1 + cr) — (3 — o-) sinh mn cosh mn)+ R<fim( 1 — o-2){(3 — cr) sinh m n — mn{l + a) coshwiTr}, ( 66 )



77Elastic Stability of a Rectangular Plate

C' = R (X + i/r)m2( 1 + cr)2 (3 — cr) sinh2m7r— R (Y + Q )m 3n (\ + cr)3sinhm7r

+ R6m 2( 1 + cr)2 {mn{ 1 + cr) — (3 — crsinh mn cosh mn}+ Rcj>m2{ 1 + cr)2{(3 - cr) sinh + a) cosh mn}, (67)

D' = — R (X + ifr) m2( 1 + cr)2 {(3 — cr) sinh mn cosh mn + mn( 1 + cr)}+ R (Y + Q )m 2{ 1 + cr)2{(3 — cr) sinhm7r + cr) coshw7r}+ Rdm 2( 1 + cr)2 (3 — cr) sinh2m7r + + cr)3 sinh m7r. (68)

Since the expressions obtained for w' and \jr now satisfy all the required conditions except equation (11), it remains to substitute for and in th a t equation and equate to zero the coefficient of e~imx. Corresponding to (37), we get for all y in (07r)

002, A r(r2 + m2)2 sin ry + [m \E y + F y2 + Gy3) — 2m2(2F + 6

r= 1

- im Q 2 rA r cos ry - im Q (E + 2Fy + 3Gy2) + PRmi

+ P m 2{A cosh my + B sinh my + y(C cosh my + D sinh my)}

+ F R \ ^ v ( s : + ^ f ) * V + 0y=) = (l. (69)

I t is now necessary to expand the left-hand side of equation (69) in terms of sines of multiples of y. For the cosine series, and the terms containing A , B, C, D, we make use of the work done in Case I and expressed in equations (38) to (46); bu t for the terms involving E, F, G we have to construct schemes analogous to (41).

Expressing 1, y, y2, y3 as Fourier series of sines in (Orr), we deduce

1 = - f —-—-— — sin ry,rr r=ir

yoo ( _ y

2 £ ^ s i nry,“i r

y* = - 2r2 \ } ~ Y ^ + J 3 r}] sinry>nr3

» In261■2 2 ( - y \ j - r3\ ^ r y .

(70)

Hence from (63) and (70), the coefficients of sin ry in the expansion of Ey + F y2 +• Gy3, F + 3Gy, E + 2 F y + 3G are respectively

A oo

- a s 2 +11 ' n = 1

( — )r 12 n 2r3 2 { \ + { - ) n}

n — 1(71)



78 D. M. A. Leggett

{1 ■-(■- n j l {2 + ( ■->*} n A n + L J p (72)

9 oo / __Yr 9 oo 1 9 co

- := I <*4.+ 4 ^ - 2 (-)“^ „ + -5-5{l-(-)r> ?{l + (-)*M .. (73)' ' • n = l • 7i=l '' • n = 1

Denoting the coefficients of in (71), (72), (73), by ern, f ril, grn, we derive the following schemes for these three sets of quan tities:

A ^ 2 ^ 3 A i

sin £/ 8TT2

024TT2

0

sin 2 y 06

TT20

12TT2

sin 3 y 8 o 80

2 1 n 2u

9rr2

sin 4 y 03

47T2 03

2 n 2(74)

^ 3 Asiny 4

TT20

12TT2

0

sin 2 y 012TT2

024TT2

sin 3 y 43 ^ 0

4TT2

0

sin 4// 06

TT20

12TT2

(75)

A A ^ 3 Asin y 0

8 96---------- 1------ 5TT

016 192

------------- 1--------o 'TT 3

sin 2 y 2TT

06TT

0

sin 3 y 08 32

3 7 t + 9 t t ~30

16 64377 97T3

sin 4// 1TT

03TT

0

(76)



Following the development of Case I, it is necessary to simplify kt, and we accordingly proceed as in th a t case. Two new quantities Ur, Vr are introduced, being defined by the equation


A L r + B I2r + CIor + DI,.

whence K r I/-m , [Lr(X + &) + M r{ Y+

K + 7 n ) \ + ^ r ( w2 ( - + y J

+ u r I A ny n -V r | 1 . (77)n = l n = 1 J

From (31), (32), (43) to (46), (65) to (68), (77), we deduce after considerable algebraic reduction th a t

2 mr r2m77’(lAn( 1 -f <r) [_ ( 2) cos rn sinh mn + 2m3n( 1 + o-)2

(r2 + to2)2 cos rn sinh m n

+ (r a+ m 2) {(3 - o-) (1 - fr) sinh2 tott - to2tt2( 1 + cr)2}

2m2(l + cr)(3 — cr) (r2 + m 2)2 sinh2 TO7T']• (78)

Tr 2mr r 2?n77(l + cr) . . 2m3n (\ + cr)2 . .F' = S ( T T V ) L “ T H ^ sm h^ ‘ ^ T ^ F sln h " ’r

COSrTT {(3 — cr) (1 — <r) sinh2 tott — w 27r2( 1 + <r)2}(r2 + m 2)2m2(l + cr) (3— cr) cos rn sinh2 mn

]■(79)

(r2 + m2)2

where A = (3 — cr)2smh.2mn — to277’2(1 + o')2.

From (78) and (79) it is a t once apparent th a tUr = (-Y+ W r. (80)

We now express the left-hand side of equation (69) in terms of sines of multiples of y, whence equating to zero the coefficient of sin ry we deduce

(r2 + m 2)2A r + m i £ ernA n - 4 m 2 f rnA n - im Q £ crnA nn —1 n — 1 n = l

oo P P yv) A I i 00 ■- im Q J , ?„ .X . + j4*(“‘» + 7 " )) ]+ r ( 2 ( — )UA n fan 2 ^nP'nU=i ; w=i n=i

r oo 4 oo+ Pi? J 2 frn ^ n “

Lw = l m n = l J

an infinite system of equations analogous to (52).



80In order tha t a non-zero solution of these equations should exist, it is

necessary tha t the infinite determinant formed by eliminating the s should vanish. This gives us the following determinantal equation from which to determine the critical value of S :

E x — i m Q ( c 12 + g 12) — i m Q ( C u + y u )

-im Q (c21 + g21) S 2N31 - imQ(c32 + g32)S 3 -im Q {c3i + g3i)

— imQic^ + g^) Ni2 — imQ(ci3 + g i3) “ 4

= 0, (81)

PRrrfi [ 4where A, = (r2 + m -f + m \ r - ±m2f„ + ^ 2 + w2j2 + PR \e" +

+ P R { ( a r + y r )(Lr+ ( - f3 Q + p r<Ur- ( - )rFr)}, (82)

N„ =

+ PR{(ctj+yj) (.Li + ( — yMJ + zijiUi- ( — (83)

I t is now necessary to consider the convergence of the infinite determinant involved in the equation for S, and for this purpose it is convenient to summarize the orders of magnitude of the various quantities involved. We have

D. M. A. Leggett

ocr = Oj(”) ’ 7r=

©II [~)» Pr = 0(r),

©IIoII 1 II O

©II

("r)' ^

Divide the rth row by r,and the rth column by r3, then from (82) to (84) it follows at once that the product of the diagonal terms converges absolutely, and that the sum of the non-diagonal terms not involving the c’s also con-




verges absolutely. The term s involving the c’s we consider separately. The g sum of their modulus is

* which is less than

I c r n2 rn or — 1 71=1 '

2QmTT

I ® l ] f 2Qm y y 1t-iW + 77 [r #

which is convergent by virtue of (57) on interchanging the suffixes r and n. Since the determ inant is convergent, we now approxim ate as in Case I.

7—In order to reduce the algebra and calculation involved to a minimum, we go straight to a second approxim ation formed by taking the first nine elements of the determ inant in question (81). After certain simplification this reduces to

g ________________ ,21.3 2 3 2-^13^31_______________ *~~ 2m b 2 (i \ y c 21 + g21) (c32 + g32) + N 31(c12 + g12) (c23 + g23)

. ~ “ l ( C23 + <723) (C32 + ^32) — “ 3(^12 + ^12) (C21 + 9f2 l)} -

The main results of the subsequent calculation are given in Table II . They are shown graphically in figs. 2 and 3.

T a b l e I I

\ /R P

Second approxim ation.------------A------------s

m2 k0 1-48 1-6641 1-55 1-7172 1-75 1-8553 1-95 2-0354 21 2-2455 2-2 2-471

Here k has the same meaning as in Case I, and it is interesting to note th a t for a plane strip the value of k obtained by Southwell and Skan is 1*68.

8—Before drawing any deductions from the results we have obtained, it is first necessary to consider to w hat extent our th ird approxim ation in Case I, and our second approxim ation in Case II , are likely to differ from the true results.

In Case I we compare the results obtained in this paper for the special case when the strip is flat with those obtained by Southwell and Skan (1924). In the first approxim ation there is an error of some 5% , while in the second and th ird approximations this is reduced to a fraction of 1 %. For the general case in which the plate is curved, no comparison with any

Vol. CLXII—A. G



82 D. M. A. Leggett

known result is possible, but by a consideration of the relative accuracy of the three approximations when the strip is flat, it seems probable th a t for the larger values of B P considered, the error in the third approximation will not exceed 1 %, while for the smaller values it will certainly be less.

In Case I I it is not so easy to ascertain the error in the second approximation, but by a comparison with the results of Southwell and Skan for this case, it is clear th a t the second approximation is not as accurate as in Case I. In spite of this however the results of Case II, as shown in fig. 1, are quite consistent with those of Case I, the curves representing the third approximation in Case I and the second approximation in Case I I tending to become parallel.

W ith reference to the possibility of obtaining further approximations in the two cases, there is no theoretical difficulty, but in view of the calculation which would be necessary and in view of the fact th a t the results wanted have already been obtained with an accuracy sufficient for all practical purposes, the labour involved would not appear to be worth while.

On considering the detailed results of the calculation (not actually given in this paper), it can be seen th a t for a given value of B P , a small change in m2 from its critical value will give rise to a very much smaller change in k. Hence in view of the accuracy to which the results have been evaluated, m2 is only accurate to two figures, and the third figure, where given, is liable to an error of anything up to three.

A final point to which it would appear appropriate to draw attention is the importance and use of Fourier series in solving equations of the type with which we have had to deal in this paper.

In conclusion, the wrriter wishes to thank Mr. W. R. Dean, Trinity College, Cambridge, for his valuable criticism and advice.

Summary

A long plate, slightly curved, is subjected to a shearing force applied uniformly along its two straight edges. A theoretical investigation is carried out to ascertain the value of the shearing force a t which buckling of the plate may be expected to take place.

Two cases are investigated, in one of which the two straight edges are simply supported, and in the other of which they are clamped. The results of the investigation, giving the critical shearing force in terms of the dimensions of the plate in each of the two cases considered, is shown graphically in fig. 2.




R eferen ces

Carslaw, H. S. 1930 “ Introduction to the Theory of Fourier’s Series and Integrals” ,

Cox, H. L. 1933 Rep. Memor. Aero. Res. Comm., No. 1553.Dean, W. R. 1925 Proc. Roy. Soc. A, 107, 734.

— 1926 Proc. Roy. Soc. A, 111, 144.Love, A. E. H. 1927 “ M athem atical Theory of E lasticity” , 4th ed.Southwell, R . V. and Skan, S. W. 1924 Proc. Roy. Soc. A, 105, 582.Timoshenko, S. 1934 “ Theory of E lasticity” , 1st ed. McGraw-Hill.— 1936 “ Theory of E lastic S tab ility” , 1st ed. McGraw-Hill.W hittaker, E . T. and W atson, G. N. 1927 “ Modern Analysis” , 4th ed.

The Absorption Spectra of the Monosulphides of Alkaline Earth Elements and their Latent

Heats of VaporizationB y L. S. Ma th er , M.Sc.

Department of Physics, University of Allahabad, Allahabad, India

{Communicated by M.N. Saha, F .R .S .— Received 6 April 1937)

[Plates 2, 3]

I ntroduction

An im portant extension of Franck’s ideas regarding the photo-dissociation of alkali-halides was made by P. K. Sen-Gupta and others of this laboratory. Sen-Gupta postulated th a t in compounds like ZnO, CdO, HgO, ZnS, CdS and HgS the chemical binding is ionic in nature so th a t the molecule has the structure: M ++X—, where M and X are respectively the electropositive and electro-negative elements. In a number of papers Sen-Gupta emphasizes the view th a t the prim ary processes of absorption of light by this class of compounds results in the simultaneous transition of two electrons from the electro-negative to the electro-positive atom, so th a t the molecule dissociates into two neutral atoms in their ground states. The photo-chemical equation can be represented as given below:

M X + hv1 = M + X(*P), ( 1 )



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