Mathematics. - On the general theory of elastic stability. (Second communieation). By C. B. BIEZENO and H. HENCKY. (Communi­cated by Prof. W. VAN DER WOUDE).

(Communicated at the meeting of January 28, 1928.)

111. Applications.

9. /ntroduction. In the following paragraphs we give some applieations of the general

equations, ' deducted in a previous paper (see vol. 31. p. 569). Already in the simplest cases great difficulties arise when the problem is treated in an exact manner, and therefore we shall have to look af ter admissible simplifications. The most evident one is to restriet by a suitable "kinematie scheme" the mobility of the different points of the body. By such a kinema tie scheme, not only the displacements u, v, w but also their variations bu, bv, bw are affected, so that the integral:

.fffdV(LdU +Mbv+Ndw}=O

whieh is to be looked upon as the representant ofthe equations (6a, b, c) may increase in usefulness. In the following examples th is integral will always be our starting point.

10. The flat plate, supported elastically and subjected to volumetrie forces.

We suppose the plate lying in a vertieal plane. the z-axis being directed normally to this plane. A deflection w in the direction of the axis of z involves alocal elastie reaction of the magnitude - kw. Except t6 volumetrie-forces the plate is only subjected to edge-stresses.

Both edge-stresses and volumetrie forces are supposed to act in the xy plane.

The kinematie scheme is characterised by the statement: 1. that points. originally lying on a normal of the plate surface. af ter

deformation find themselves again on a normal of the deformed middle­plane of the plate.

2. that strain in the direction of z will be excluded; 3. th at for points of the plane z = O. u and v will be zero.

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Consequently the displacements of any point (x. y. z) must be put equal to:

ow u=- OX.Z

ow ........ (16) v=- ay .z

w= w

where w. indicating the displacement of a point in the plane z = O. is a function of x and y only.

By the relations (16) we conclude from 3a and . 3b :

and from (4)

aw w x =+ oy

Ow W y =- ax

W z = 0

av 2 . rx = - Ty fz - Tx ay

au 2 . ry = + Tx fz + Ty ax

Substituting the variations

in equation (11). we find:

o!5w !5u=- ox .z

o!5w !5v= - oy . z

!5w = lJw

which may be transformed in:

fJdX .dy l.f Ndz + ~rz. ~:. dz+J~. a:; . dzl dw

. . . . • (168)

. . . . (16b)

- JdS [cos (nX)fLz dz + cos (ny)f Mz dz] dw = O.

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This equation holds for all possible variations of wand for every part of the plate. Putting öw = 0 for all edge~points. we find:

J'~r f'( àL aM) Jdx.dy.Ö~ N+z·ax-+z. dy dz=O.

Therefore

and in consequence

Jds . öw [ cos (nx).[Lz dz + cos (ny) JMz dz ] = O.

Application of the same reasoning to a part of the plate with contour C. leads to:

JdS . ÖW [cos (nx) , [Lzdz + cos (ny)f Mzdz ] = 0

so that for every point Q. of this contour

cos (nx)f Lzdz + cos (ny) f Mzdz = O.

As for such a point Q the factors cos(nx) and cos(ny) vary with varying contour C. it follows. that for every point of the plane z = 0:

I. .[LZ dz =0

Ir. fMzdz=O

and consequently

fz.~~ .dz=O J' aM z. ay . dz=O.

[( àL dM) From . N + z . àx + z. ay dz = 0 it follows that:

lIl. JNdz=O.

11. Restricting ourselves to the case Tx = Ty = Sz = Z = 0 and Sx. Sy and T. independent of z, we find:

L = asx + ~ + aty _ ar. ax ay az ay

M=~+ asy + atx + arz

ax ay az dx

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N = atg + atx + asz + Tz . awx + S . awx àx ag az dx g ag

_ Sx . a:; _ Tz • a;; + XWg - YWx .

Substituting these expressions in 1 and 11. and integrating with respect to z. we find. with the abbreviations

Mx JSx. z.dz

Mil --JSg . Z • dz

D Jtz .z. dz:

r. d aMx aD J'orz d .) tg Z = a.x:- + ay - . diJ' Z z . . . . . (I')

f d _oD aMg Jorz d tx z - ox + Oy + ax' z Z (11')

from which the following equation can be deduced:

JOtg d JOtx d _ a2M x a2D +02M g

ox' z+ diJ' z- ox2 +2' oxog fiii2' Combining this relation with 111. we find:

o2Mx a2D o2Mg a2w a2w a2w (h2 + 2. ox og + ----aij2 + Sx h . ox2 + 2 . Tz h. dx dg + Sg h. ag2

aw aw - Xh . ox - Yh . àg - kw = O.

is obtained.

1) A specialisation of this equation can be found in "The math. theory of elasticity" by A. E. H. LOVE. 3rd edition 1920. p. 541.

583

12. The girder, without volumetric-forces, elastically supported. When in the faregaing equatian the differentiaI-quatients with respect

ta y, as weIl as X and Y are put equaI ta zera, and if - Sx h be repIaced by the tataI campressive farce p, we fall back an the well­knawn equatian

d4w d2w El· dx4 +P. dx2 +kw=O

far the e1astically supparted girder.

13. Vertical strat, subjected to its own weight. -G

Again specialising and putting : Xh = - 1- (G= tataI weight af strut),

G (I-x) and Sx h = - 1 (x ta be measured from the cIamped end) we

find:

EJ diW +G(I-x) d2w _ G dw -0 1) . dx4 1 . dx2 1· dx - .

14. Buckling of a girder, clamped at one side. We examine a girder with a rectanguIar crass-sectian (height = h,

breadth = b, h » b). The system af caardinates is pIaced in such a manner, that the x-axis caincides with the axis af the girder, and that the dimensians hand b af the crass-sectian are measured aIang the axis y and z respectiveIy. The arigin af the system af caardinates coincides with the centre af the cIamped crass-sectian. The Iaading farce p , acting in the centre af the free-end sectian, is parallel ta the y-axis..

Then we have :

12 S x = - b . h3 • Py (1 - x)

Tz = + b ~~3 • ~ ( ; y- y2 } (17)

T x = Ty=Sy=Sz =O.

The kinematic scheme af the girder will be described with the aid af twa quantities Wo and t'. The first af them gives the deflectian af the axis af the beam in the z-directian; the secand the angIe af ratatian af any crass-sectian abaut the x-axis. The dispIacements u, v, w af any paint af the girder are then determined by the relatians:

dwo at' ~ u = - ax . z - yz . ax (17a )

v=-t'.z

w=+wo+t'·y

1) Compare A. E . H. LaVE, p. 431-432.

584

as can easily be seen by using the kinematic scheme of the flat plate, and by remembering that the deflection of the axis of the girder in the direction y is negligible with respect to the other one, in the direction z. The quantities Wo and 1:, as a matter of fact, only depend on x.

The equations (3a), (3b), and (4) give:

2.rx =0

w%=o

The variations Öa, öv, Öw are

(Mwo aÖ1: öu = - ----a.x . z - y . z . ax

öv=-z. Ör

ÖW=+ öWo + y. Ör.

Hence, the equation (11), (af ter partial integration) can be written as follows:

ÖW 1dxlöw1J(z.~;+N)dy.dz+örjJ( Ny+yz.~:-Mz)dy.dZ(

- [öw~jJLZdydz1:- [ör jjLyzdy dz 1:=0. Putting Ör = 0 everywhere, and ÖWo = 0 at the ends of the girder

we find:

JJ( N + z . ~:) dy dz = O.

Putting ÖWo = 0 everywhere, and Ö1: = 0 at the ends of the girder:

JJ( Ny + yz . ~; - Mz) dy dz = o.

Moreover the expressions between the [ ] brackets must vanish, so that:

JJL . z dy dz = const.

JJL yz dy dz = const.

585

and therefore

zdy dz=O JJ' ·OdXL .

. . . . . . (17d )

(J'OL .J OX· yzdydz=O

Consequently

.JJN dy dz = 0 . . . . . . . (I)

.Û(Ny - Mz) dy dz = 0 (11)

15. N ow we have to . determine the functions L, Mand N, and find

L = oSx + ~ + 0 tg _ or. dx dy OZ dy

M=~+ OSg + otx + arz

ax oy dz ax N = a tg + 0 tx + oSz + 2 Tz . dr: + Sx (d

2 Wo + . d 2

r: ) . ax oy az dx dx2 y dx2

so that equation 1 becomes:

Partial integration, leads, with due regard to the conditions : tx = Sz = 0 h b

for y = ± 2' resp. z = ± 2' to

jJatg dr: d 2r: - (I , ax . dy dz + 2 . dx' P - dx 2 • (1 - x) P - 0 . .. )

Equation 11 takes the form:

In order to determine the integrals

1J' at JJot y. d~ . z dy dz and a~ . dy dz

we use equation (17d ).

586

In consequence of the facto that

oL _ 02sx + o2tz + o2ty a2Tz

ax - ax2 oxay axaz - axay

we find by transforming in the usual manner IJ ~; . z dy dz = 0 :

Jj'ot c fJ" a2 JJ' a

2

o~ . dg dz =J. z. o;~ . dy dz - z . ax ~~ dy dz. . (111)

JJ' 'oL From Ox' yz dy dz = 0 we derive:

J y. ox . dydz~ z. ih .dydz=JJ yz. ox2 .dydz- (IV) J"r. aty D' (Uz = r' a2sx ~

J/'r. iF - J yz . ox~~ . dy dz

At the other hand the torsional moment of the cross~section is equal to:

Mw JJtz. z. dy dz -.J]~y . y dy dz.

so that

jJ~~ .Zdydz-jJ~~ . ydydz=d~w . ... (V)

Substituting (111) in (I). we find:

jJ a2sx - rJ' a2Tz de d2

-,; . z. ax2 .dydz-

J .... z'oxay.dy dz+2' dx .P-P(1-x)'dx2 =O . (1/)

Combining (IV) and (Il). we have:

j(' r.' a2 J' at J" a2

J yz. o;~ .dy dz-2j a~ .zdydz-J yz. ox~~·dydz-(11/)

JJ'a J' d2

- ;~ . z dy dz + J ySx. d:20 . dy dz = O.

From (V) and (IV) it follows that:

Jj' at, d d - tj' (a2Sx a2

Tz ) d d + dMw 2. • z. ax' y z =J. yz ax2 - axay y z ---;IX .

If now Tz is replaced by the value. determined byequation (17b).

the equations (I) and (11) become:

(I")

(Il")

587

According to equation (17b). we have

or on account of (17)

2 JJ arz d d - d3wo 6. P JJ(h2 2 ) 2 d d . z. ax' Y z - dx3 . bh3 . 4 - y . z . y z

resp. :

so that (11) can be reduced to

dMw + P(I- ) d2wO+ Pb2

d3wO=O

dx x . dx2 24' dr .

Finally we have to make use of the re1ations

au (d2w d2l)

Sx = + E. àx = - Ez dx20 + y. dr

and

(where C is a constant. depending on the dimensions of the cross~ section).

Doing so. and putting b i;3 = I. we find the simultaneous differential

equations:

d 4w d 2l dl El· dx40 + P (1 - x) . dx2 - 2 . P . dx = 0

Pb2 d3w which. with the exception of the term 24 . dx30. ag ree with those found

by other investigators. Along other lines TIMOSCHENKO 1) introduced a similar term. when he made experiments with I~girders and stated a discrepancy between the usual theory and his results.

1) TIMOSCHENKO. Sur la stabilité des systèmes êlastiques. Paris 1913. or Annales des ponts et ehaussées. fase. 3. -4 and 5 1913.

588

16. The two-dimensional problem in polar coordinates. If the state of stress of the body under consideration is independent

of z (sa that W x = W g = o. {x = (y = 0). and if the volumetrie farces are negligible. the problem of elastic stability is governed by the equations:

L = oSx + otz _ T oWz _ S dwz _ orz - 0 - àx iJy Z' dx g' rly oy - . (I)

M=~+àsg +S oWz + T oWz +orz - 0 - ox ày X' dx Z' oy ox - . (11)

(I1I)

Though. as a matter of facto it would be possible to transform these equations in polar coordinates in a formal manner. we shall give here the direct deduction of the main equations. The polar coordinates of a point being (e. q;). the radial and tangential displacements u and u. we have:

(1 Ou u ou) 2 . r z = (So - Sr) fz + T z -. -0 + - - ~ . , e q; e ve

The relations (5) have to be replaced by

- T z W z

x? = T z + tz - r z - Sr W z

Y:o = T z + tz + rz + Sp W z

Yr = Sr + Si' + T • . W z

where obviously Xp. X~ .. " signify stress-components in the directions e and q;. related to surface-elements in state I. but acting on the corresponding surface-elements in state Ir.

The equations of equilibrium of an element in state II. originally lying between the two angles q; and q; + dq;. and the two circles (e) and (e + de). take the form

l o(Xp~+l àXi' _ Y,,-O e' oe e . oq; e-

l. o(Y:o e) +l. 0 Y? +~I =0 e oe e oq; e

589

or

axp +J.-. ax\, +J.-. (Xo _ Y~) = 0 ae e arp e ' ,

avp +J.- àY\, + J.- (X. + Y) - 0 àe e . arp e ' ? ,e - •

Substituting in these equations the values of X p, X 'f ' Yp• Yr we obtain. with regard to the equations of equilibrium. holding for the element in state I:

as our Bnal equations:

They reduee. with T z = O. to

L asp +J.-. atz +J.-. (so _ s,,) _ Sf awz _J.-. arz = O. (I) ae e arp e ' , e arp e arp

M--~+J.-. as? +2tz + s awz +~~= O ae e drp e p (\; ae

17. The circular ring under external pressure. The kinematic seheme be eharaeterised by

u = uo

e 1 auo v = Vo - - -. -a . (e - a) ;

a a rp

. . (II)

a is the average ring-radius. u. vare the displaeements of an arbitrary point (e. rp). uo. Vo the displaeements of the eorresponding point (a. rp).

We find eonseeutively:

öu = tSuo

e 1 aöuo öv = - öVo - - . -a- . (e - a)

a a rp fz = 0 r z = 0

aw. _ 1 (avo a2uo)

arp - --; . arp - arp2

Proceedings Royal Acad. Amsterdam. Vol. XXXI.

39

590

and therefore:

The equation

whïch now governs the prohJem, therefore can he written as follows:

• [dep [buo.JLe de + bV{ ~e2 . de - : ~~o .J Me (e - a) de] = 0

.fiep[buo lJLe de + J:~ e (e -a) de t + bVJ'~e2 de] = o.

Consequently

The Jatter of these equations hecomes

Now we have

SI' = E .1 Uo + ~. dvo _ a2 Uo • e - a I

, 0 e a aep Oep2 ae \

~L-E lOl oUo +l 02vo_ <PUo e- a I aep - . e· dep a· Oep2 aep3· ae \

J' aSt - OUo a2vo e . aq; de - Eh . (Jep + Eh . aep2

f' a h f f 2 tz _ 2 a+ 2 _ e . a de - (tz e) _.!!.-- 2 tz e de - - 2 tz e de L e a 2 ' 0

h h as tz =0 for e=a+ 2 and a- 2·

Therefore the equation reduces to

OUo + a2vo_ 0 or dep aep2-

(11)

591

It follows that the centre-line of the ring does not change its length. Equation (I) passes into:

where

The houndary conditions are

which may he written as

lP,? =0

h e=a+- h 2 sp = - Pp . 2a (a + h/2) .

in consequence of the facto that

h ( CFu) = - 2a (a + h/2)' Uo + aq;g .

Transforming equation (I). we find successively:

f oSp J - a+h/2 e - . de + Sp. de - (e . sp) . e a-h/2

h ( (Puo) = (a + h/2) . sa+h/2 = - Pp • 2a' Uo + aq;2

39*

592

and putting

fJ= In a+h(~ a-h/2

dvo aep = - ua

j ' e2-ea a2t. _ \ at. e2- ae la+hJ2 Jat. 2e - a -a- ' aeaep . de - lag;' -a-\ a-hJ2 - aep' --a- . de

--2f~ e-a_J~ d - aep' a aep . e ~

• a2 a2 ' ()2 ai j" ( )2

J ~ e- a d =-E __ ua JI _e-~d -E ~ e- a d a 2' ·e 'a2' 2 e 'a'" 2 e ep a ep ~ ,ae ep. ae

= - E (fJ - hfa) c;:~+ a;;~).

The flnal result therefore is:

or

or

with _ Pp _ Pp • a 3

1'-- E(fJ- hfa) ---W-The solutions cos ep and sin ep of th is equation are non-essential. as

they relate to displacements of the ring as a whole. The remaining solutions cos VT+r cp. sin VT+;; ep have to be periodical with the period 2 n. The smallest value for I' therefore follows from

Vl+I'=2 . 1'=3. which is in accordance with results already known.