Anatolii Isakievich Lur'e.

# Concentration of stresses in the vicinity of an aperture in the surface of a circular cylinder; tr online

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IMM-NYU 280
NOVEMBER I960

NEW YORK UNIVERSITY
INSTITUTE OF
MATHEMATICAL SCIENCES

Concentration of Stresses in the Vicinity of An
Aperture in the Surface of a Circular Cylinder

I. LOURYE

PREPARED UNDER
CONTRACT NO. NONR-285(42)
WITH THE

OFFICE OF NAVAL RESEARCH
UNITED STATES NAVY

h

REPRODUCTION IN WHOLE OR IN PA ^T

IS PERMITTED FOR ANY PURPOSE
OF THE UNITED STATES GOVERNMENT.

IMM-wTU 280
November I960

New York University
Institute of Mathematical Sciences

CONCENTRATION OP STRESSES IN THE VICINITY OF AN
APERTURE IN THE SURFACE OP A CIRCULAR CYLINDER

A. I, Lourye

Translation made by N. Brunswick from Prik. Mat. i. Mekh. ,
Vol. 10, pp. 397-406 (191^.6), under contract no. Nonr-285(l4,2-).

1960

â€˘f.

1. The equations of the theory of cylindrical shells used
in the present paper are of an approximate character. They are
based on the fact that the displacements u and v in the middle
surface are neglected in computing the curvature and twist of the
shell; the latter are expressed only by the radial component w of
the displacement. Under these conditions the problem of
cylindrical shells can be reduced to finding a displacement
function (|) ; if no surface loads are present, then tnis function
^' I

'- A, , r

.d:r Y^

f F-

4 r-

iin

J A.

rr fi ^ -

* J- .cy. ^"^

Therefore, introducing the function

(1,12) C^ = A^ ^;;.

the real part of which represents the radial displacement (this
follows from (lÂ»i|) ) , we obtain for the moments and shear reactions,
the following expressions?

h V in

(1.13)

f' III

1" = -^^ 1^^^^ -2' - ^1^)^- ^^ V^ = - D.5,(^^^,2,^,^2j^^^.

Q

It is clear that oris determined from the same differential
equation (1.11) as - (5) ^
J\. 2_^K) - 2i i\. (^) are linear combinations of the even and odd
components of the functions e" ^ ~ ^ t ^ â€˘

V/e can therefore look at the solution of (1,1[|) in the form

[r^^(Px) - 2i.- .^(ex)])}/(x,y) ,

(2.11)

[iX2(Px) - 2liV^(Px)jH/(x,y)

^hL=i

tA^U - T)

.cri-

\ ',*-;

â– -.S)

;i}ÂŁoe Â«e.Â«d3 lo aoi^si?;

A -irvy ^..-UJ.

â– ;-^*Jiril: cv; J-oJ

.44 :'. :?;

% .;i3

C V c- ^

: X " j. '

â™¦ â– '* -

.-^*rn'> r.-s'-) .-:.*â–  f '" f^ -1,- ^ ;â–  â–

It. iv^ Jfc V^ . V ti '- i, Jj-^ * .*. J - â–  â–  â–  - -

X . f.

e hi J

^-'s -^

, fv,x)^r((x^>

^. v'r .'^ ) 'fl \ "â€˘â– ^

.â€˘,-.'"Â» Of;*'

In the follcwlng we shall investicate "polar" coordinates on
the surface of the cylinder; i.e. we assume

(2.12) X = p cos Xf 7=9 sin X Â«

The curves X = const, are the helical lines: if one unfolds
the cylinder into a plane these lines go over into a system of
straight lines which emit radially from the origin of coordinates.

On this plane the curves p = const, are circles, and on the
cylinder they are curves of the same geodetic distance from the
origin of the coordinates - it is into these curves that the circles
go over when one bends the plane into the surface of a circular
cylinder.

The transformation (2,12) is of course formally identical to
the transf orrration into pol&r coordinates on the plane, iioreover,
formulas (1,13) are also identical with the corresponding formulas
of the plane problem and the problem of bending thin plates.
Therefore the transformation (2,12) to the coordinates p and X
must lead to the same expressions for the shear reactions and
moments as for the plane plate in polar coordinates. We obtain

c _ D a2(n^ ~ 1 ) ,1 > , 1 -^"^ V y^ ^

., c- p 5*p 2 A^2' * â–  ci

\( ra "^ ^ p X

f? T^^ S - ^ / l2(m^ - 1) 3^ T^ ^
(2.13) S - - ^ / ^ :-~^ Im cr,

V m c p

\

^2 p c'jX'

We seek the solution of equation (2,7) in the form
(ZÂ»li|) \J/^ = Rj^cos nX or ij/^ =R^sin. nX*

where n is an integer Â« To determine R (p) we obtain Besael's

c n

differential equation , , â–

(2.15) Rj," + F^n' â– " (2^P^ - ^^^n = Â° â€˘

p - â– ., -

We shall be interested in the solution of this equation which
goes exponentially to zero for large values of the argument Pp ;
this solution is given (as is known) by the first Hankel function of
n-th order

(aa6) h^^(Pp y2l)= Â»i/^(pp) + ix^(Pp) .

Returning to the solution (2.11) of the differential equation
(lÂ»li^.) we shall investigate it in the form

^n "^ f-i'^i^P^) - 2Â±S\^{^x)]i^\l^{M + i-j^(Pp)]cos nX
\2Â»17 )

= (a + ip )cos nX ,

JL .,. .:.. .L ^ â€˘:.k^

Cr - =

V

I J2^ X 1.3t_/2 ..J.

_!,

:i.S)

i ' '- J. â– 'â– ' â€˘*â€˘ -u '^. _ A V

â– :^^ 00 the functions JV.(PX) grow as e^ ' ' â€˘ On the
other hand, the asymptotic representation of the Hankel function
has the form

H

^J^U^P VZI)- â€”^ "^^ [cos (pp-kHgi-l^)

Vupp/vTT

+ 1 sin(Pp - ^" A "^ 7C) J ,

Therefore (^ and x; decrease for pp â€” > oo like
n k '^'^

VPP

and hence for X, different from z.ero or n, go to zero exponentially
and for X = and X = oi not slower than . It is also easily

a/pp

established that the solutions (2,17) - (2,l8) are even with respect
to X and X,

For small values of the argument the functions a. and p. can
be represented by the following series:

a =1 + 1 p^p^[ (2 + cos 2X log ^^^ - 1] + ... ,

t 4 â€˘

Po = I log ^ - ^ P^P^(2 + cos 2X) +. ,.. ,

a^ = cos X(|^ p2p^ - ^ p2p2 cos 2X Â« | P^p^og ^) + . . .,

t ^. -.^

i. ij; - ^ ,U...-..-)^ , Qa)nJ

*"Â» * -^ - â–  - ''V I- : iff'

! â€ž i

^4 V

â€˘ ' - â– - t ''â–  â€˘ â–

â– i.-^jcXe :foa jj

â– "J ^^ii^v Ke-Vi} -;Â»â€˘â–

?/? snoLtiiioB odJ"

^w* ^ ?, Ov.

>!TÂ©t^y - i're eri? O.c eaui-

w O 't t^ o

?-'-^-^- :v:ii"'q:"ii: f.: - 7Si, -^^-^ax "â– ., i.

/ A ao n

10

p^ = COS X {- P^p^ - |) + ... ,

p^ = I cos ax + f pV + ... ,

a^ = cos A[ ^ + ^ P^p^(r - 5Â«os 2X + I cos^X) ] + ... ,

^ 7ip2p2 371 ^ ^^ ij: 2 5

P3 = - ~ cos X (1 - 2coa ZX) + ... ,
a, = - ^ -^ (10 + 6c OS 2X) + ... ,

Here Y stands for the Euler-Masceroni constant log y = 0,5772...

3. Let us investigate a cylindrical shell, with an opening
bounded bj the curve p = p = conatÂ» ; if one unfolds this surface
into a plane, it will have a circular cut-out with radius p^. If
the opening were not there the strained state in the wall of the
cylinder would be given by the equations S, = ph and Sâ€ž = qh Â» V/e
are Interested in the deformation which this strained state is
subjected to in the region close to the opening. This problem
represents a straight generalization, for the case of a cylindrical
surface, of Klrsch's problem for the distribution of stresses in a
plane stress field with a circular opening. It is drubtiul that
one can obtain its exact solution. In the following we give an

assymptotic solution, which is usable under the assumption that the
parameter p /ah is small; this implies that in the follox-jing we
assume that, althought a/h is large, the radius is sufficiently

& > -;

h < c

Â»,. *â–  iU-^^^ ?- â€˘Â»â€˘ .u rtox^jO-i Si'::r iM ^...ij: i

â– >fi !o - c'r; â€˘'-

. i> Ci 8 ^l i3 !â– â– "

+ -â€˘)

\.'

4

AS aoo

i)(p - ci

I

;p + q)

s

.; o L

'rL.a

:^;:( a.

f^X a'

- \Pt. - â€˘â– â– â– I

S.

AS 300 (p - q)i, -" p -^ :? â€˘- j_ ^ '
S

iJO

r V" I

Ay^'

z , ,.\

rm

15

Let us look for example at the case of a cylinder with closed
ends and subject to a constant interior pressure p^ : if no opening
were present, we would have

P ^ Po^

(3.17) P = 2E- ' ^ = h '

and by (3tl6) we obtain

S^ p ^ r

'i^'â€ž = â€ž -'-t

4 + cos 2X +
P o '' L ^

(3.18) rrz 77 TTp/ ^ 1

t

m

2

^ (2.^ cos 2A)J

Vlith zero curvature, the coefficient of the stress

concentration at the opening is in this case equal to (2Â»5)Â»

If curvature is present this coefficient is multiplied by

2

(3.19) 1-^ÂĄh^

Let us consider the stresses which arise as a consequence of
bending of the plate; these stresses become zero together with
the curvature. Let us first of all require that on the contour of
the opening the bending moment G becomes zero. Computing G by
(2,13) and (3*5) we equate to zero the constant term and the
coefficients of cos 2A and cos liX } we obtain three equations
which together with the constants A , C , P,, given above by (3Â»10),
contain five unlmowns , - .

(3.20) B^, D^, E^, H^, M^; ;,

and we have two more constants at our disposal to satisfy the
boundary conditions for the shear reaction Q ", The expression
for Q " we must construct from (2,13) and (3*5) â€˘ The two missing
equations for our unknowns (3.20) we now obtain by setting the
coefficients of cos 2X and cos \\\ in the expression for Q '",

v.- ^;.-

grt.?:n9i:|

A S f,e â– '." ^i-

5 ?â– Â» ^- '> CÂ» 'V T -^ Ca "^f "^

â€˘â– *\: %:i ^v^Hiii !l'>:M:y

U' ^0'

5 J S '. i V" â–  ' ii^ j J i j 4. "<

>:>

-'â–  â– .â€˘

v'3. it p.

16

for p = p , equal to zero. The constant term in this expression
turns out to be, for p = p , equal to

(3.21) V'p = p^ = -2#8(A^-Â°^)=-^^

The resultant shear reacticiis are, therefore, equal to

^^Po (%'"") p = p = - ^Po ^ =â–  ' ^Po Po â€˘

This could be expected since the shear forces must balance
the resultant of the pressure on the surface of the opening.

It remains to determine the normal stresses on the contour

1 - t 2
of the opening produced by the bending moment G, ~ "E'^ X^* where

Cr^ ' is the maximal value of the bending stresses (on the outer
A.

f ibers ) â–ş For this we must first determine the unknowns (3.2C),
Leaving out the computation, we give only the final results:

-' X i^ah m ^^"^^ + 2 "^ 2 P """^ ^^ >t 3m + 1

. ^ i^n-^l + ^!f^ : l^ (3q - p)r')
6 3m + 1 3m + 1 ^ ' y

+ i (p - q) ^^rrj ^^^ h^'^ -

If, for instance, the stresses p and q are determined by (3.17),
and if we take l/m = 0.3 and notice that then

^fah
we obtain

y' = log â€”â€”r - o.ai3

â€˘a :

~j . : : Â» '

( -:ir - k)b â€”

Vis* AJ*-' 'x - 4

r Â«â€˘

P>_j ..- ,â„˘ -..

Of*:

^ in

\X ^.

^.-â– /

17

cr-

I = Â« â€” o_ [3^9 log â€” - + 1.361

ah

vail

(3.^3)

+ cos 2A(0.996 - 1.035 log â€” - } - 0,l5i| cos 1].X] ,

>Sh

The additional normal stresses for non-zero curvature, which
are uniformly distributed along the thickness of the plate by (3.16),
will be given by 2_

(3.24) 0-;' = 1.29 ^ .

Values for these stresses and alsc values for the bending
stresses along the contour of thei opening for some values of the
parameter p /2.a are given in the following table.

Po

-^>

- >

->

^''ah

X =

X = tl/2

X =

X = 7i/2

o>5

^ - 0.055

0..786

1.4^8

- 0.,162

o^k

0Â»065

0>683

0,.932

1 - 0..10i;

o>3

0.112

o..5iii

o,.523

- o,Â»058

0,.2

0.095

0.307

0...233

- 0,,009

0,1

O.OUij.

0.111

0.058

- 0.006

March 25, 19l|6

Institute for Mechanics
of the U.S.S.R* ,

Date Due

fit lr^>/^^>^ - ^Cg^i^J^~

'-'-â™¦-

S^-' - a ^, i

- Â« . â€˘ - sr-j 'i;

^ ^ 'J ml ^-r

IKM-280

c.l

'IÂ»1-230

Lur*i

^â€”-^50

c.^

lair'e

A'_TH03

Concentration of stre sssa ii-

"^^^ the vicinitj of an

^

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