Edward L Reiss.

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Font size NtW YORK UNIVERSITY
-iNSTITUTE OF MATJ lE.M ATICAL SCIENCES

25 Wavcr/y Place, New York 3, N. Y.

NEW YORK UNIVERSITY
INSTITUTE OF
MATHEMATICAL SCIENCES

IMM-NYU 281
FEBRUARY 1961

Extension of A Thick Infinite Plate With
A Circular Hole

EDWARD L. REISS

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

OFFICE OF NAVAL RESEARCH
UNITED STATES NAVY

REPRODUCTION ^vmOLEOR\; PART

IMM-NYU 281
February I96I

New York University
Institute of Mathematical Sciences

EXTENSION OF A THICK INFINITE PLATE VJITH
A CIRCULAR HOLE

Edward L. Reiss

This report represents results obtained at the
Institute of Mathematical Sciences, New York
University, under the sponsorship of the Office
of Naval Research, United States Navy, Contract
No. Nonr-285(42).

1961

1 Introduction .

The plane stress solution D-] is conventionally employed
to estimate the stress concentration due to a circular hole in
a plate that is uniaxially atretched at infinity. However, this
solution is not a solution of the exact theory. Nevertheless,
we expect it to yield an accurate approximation if e = ^ is
sufficiently "small". We also expect that the accuracy will
Increase as e â€” j- q.

The precise relationship between the plane stress and the
exact theories is given in [j>] for simply connected plates with
"smooth" boundary curves. In addition, a boundary layer
procedure"' is outlined for obtaining increasingly accurate
approximations to the solution of the exact theory. These
approximations are given as "three-dimensional corrections" to
the plane stress solution.

In this paper we extend the method of  to our stress
concentration problem. Results are obtained in the form of a
power series in e. We give here only terms up to and including

We refer to the three-dimensional linear theory of elasticity
for homogeneous and isotropic materials as the exact theory.
There are some special cases for which the plane stress solution
is a solution of the exact theory .

2

Here, h is one half of the plate thickness and R is the radius

of the hole.

'' It is a generalization of the one given by Friedrichs  and
Friedrlchs and Dressier  in a study of the "bending" of plates.

iJ bn^ 38'

second order. Within this approximation we show that the plane
stress theory yields extremely accurate, although non-conservative
predictions of the maximum stress concentration for "small" but
finite values of e. This accuracy depends upon Polsson's ratio v.
For example, with v = I/5 the error in the plane stress solution
is less than 5 9^) if e _^ -3. For "larger " values of v and e,
the error increases. More accurate approximations to the solution
of the exact theory, which may be necessary for these values of e,
can be obtained by determining third and higher terms in the
expansion.

Other approximate solutions of the exact theory for this
problem are given by Sternberg and Sadowsky  using a modifica-
tion of the Ritz method and Green  and Alblas  who
employed infinite series expansions.^ Our results for the stress
concentration compare favorably with those of Alblas if ^ ^ jr >
see Figs. 1 and 2. Agreement is especially good with his
"asymptotic" solution which is closely related to our approxima-
tion method.

2. Formulation .

We introduce a cylindrical coordinate system r,9,z. An
Infinite plate of thickness 2h with a circular hole of radius R
is considered as a three-dimensional elastic body bounded by the
planes (the "faces" of the plate) z = Â± h and the cylindrical
surface r = R. The origin of the coordinates is fixed at the
center of the hole on the mldplane of the plate, z = 0. The plate

4

: ..;ii; ;: t fi'J f '""I

-; '7:^ill0

â–  '. .. â€˘:;;:::r;noo nos.: :.. .'â–
.â– 5 b'-

s. ^ Ti .,â–  .1: i. v. i i O i >"? ii .L O " w .i. -â–  â– ; â– _. .

bC'.' r. .vi-c.s.! X:.- ' S 36

â– â€˘U? LiUr5 f! -*â–  â€˘= ?: {'ÂŁâ€˘'. â–  lo "a30B '

i:vi o'7 . w

is stretched at infinity by a constant tensile force T in a
fixed direction v/hich we take as the x-axis. The boundary of
the hole, i.e. the edge, and the faces of the plate are free of
forces.

If we introduce the dimensionless variables:

i = - ^ , e 1 ; c = f , Id 1 1 ,

and the parameter,

F - h

then the faces of the plate are given by i^ = Â±1 and the boundary
of the hole by | = 0.

Considering the components of stress as functions of 1,9,^
and employing an obvious notation, the stress formulation of the
exact theory is given by:

Equilibrium Equations,

(1) ^0z,c '''^^'^re,e-^^i + ^^"^^^"a,9 +^^re^^=Â°'

"z,C ""^^^rzA^^^^ ^^'^^-^9z,9-'-t gr.

the exact solution occurs only In the direction normal to the
edge, i.e. in the ^ direction. V/e then introduce the "stretched"
boundary layer variable  t] as:

(9) r,=l.

Considering the r],d,t^ coordinate system, boundary layer
stresses indicated by the generic symbol f(T),9,^;e) are defined
as:

(10) f(ri,9,(:;e) =~"(f^ ^a-f^ -2fi ^) = ,

rz ' 'ri^ ^-â€” rz,ri ' â€” rz, 99 rz 9z,9' '

72^n

-^"9 + 11 (fr9,, + nl,^9)-i: Ls^9,99+2^(^r - f9)-^fr9-nl9j = 0,

Jz,Ti"i '9C'

,4,99-1z-^24,9l = >

(14)

f^^(0,9,a = f^(0,9,C) = f2g(O,0,C) = ,

Cjoo,9,a = ,

fS(co,9,a =

T cos 9, n =
, n ^

Here,

r

â€ž (â€˘â€˘ -T/2 sin 29, n =

^0 , n ?^

_1_ (fH n nv ^2 _ ^^ 5^

1 T t / , vJ^JaI tâ€” a^

i+j+l=n

YZ-^Tl (-i)^{j+i)tiJ'a=^ .

i+j-i?=n

In addition we require the boundary layer stresses to
approach or "match" the interior stresses as e â€” > o.
Specifically, we assume that each o''^(4>9,0 has, near ^ = 0,
a Taylor series expansion in |:

I^ t.

to

- \^

2t

i?i;^lJfr

where

(15a) 3n, j.^B-'^lO S,0 n=0,l,... .

Therefore from (5) and (9) we have, In some region about ^ = 0,

00 ^

(16) cr{^,0.^;e) ^rir''(T,9,Oe'' ,

n=0

where

(15b) ^''(^,9,0 = Vâ€” s;;-"^(9,an'"

m=0

are the Interior coefficients near ^ = as functions r\ , 9 and C*
For each n we define the ''redu'-'ed boundary layer stress
coefficients", F^(r],9,0 as:

(17) P'^(n,e,a s f''(n,e,0 - ^""(n^e^C) .

The "matching condition" or the asymptotic form for the f is
obtained from (9), (10) and (l6) by associating each f" with the

corresponding o^ as e â€” > 0. Using (9) and ( ?.? ) we write this

i
condition as: '

(15c) lim F^(t],9,C) = 0, n=0,l,....

Tj â€” >C0

An analysis, simi?-c^r to the preceding, applied to the
boundary at "infinity" yields a corresponding formulation of that
boundary layer problem. We do not exhibit this formulation.

' In obtaining (15g) we assume that the terms in f" which \anish
a-s T] â€” *. 00 do so faster than any negative power of r) .

( â– '. v.^)'^-,

%:1Â±d:

i^r:. no

10

5. A nalysis of the Boundary Layer Problems .

For each n Eqs. (12-15) and (1?) separate into two

distinct systems which we call Problem P^ and Problem T .

Problem P'^ is concerned with the coefficients f^, f"^, f2 and f"

r z y rz

and involves the first two of (12), the first four of (13), (15)
and (1?) for these coefficients and the first two of each of (14).
Problem T is concerned with the remaining two coefficients f g
and Tq and the remaining equations in (12-15) and (17).

We shall associate with p'^ a "stress function", ^ (il,9,C),
which may be the solution of the following boundary value problem
on the sem.i-infinite strip, |^1 _^ 1, t^ >_ 0, and fixed 9:

V^f = ;

i?^^(n,^Â±l) =

where

fcrfs I Of) ^^ -â– '

(Br^)

9"e3riv;

v^-iT " (3.^)0 {ci.^O

\$:ij â€˘ a^Xdoil lo no.WwXoa b ni:B>>cfo oT

vd n^vj" .-â€˘-..-fr- !> - ri rfctxw {8l} mslcfcriq sjjXbv â€˘'^^fim/ocf

iioviig CiOJ:.- â€˘ .â€˘ smxxo'iqqB " srid- YoXqnrs sw iiwoinlny el nieXdCTg

, bS aoo (l-^:S^)(r.d nl8 |+ prd soo)'^Â®"3 p = ^"5

5 / S - , , , S , , s fTB- / d->- iJ , tr Srr

s'lorlw

15

(55b) H = 1^ , a = 2.075 , b = 1.143 .

An Integral representation can be given for the solution

2
of Problem T since Green's function for the boundary value

problem (23) is knovm [l4]. However, we prefer to use the

Infinite series representation which yields for the solution
.2 '

of T'

(36a)

where.

rt oo p

P^^ = >_ K cos nTT? e"'"'^ sin 29 ,
re ^ n

^Q = YH K^ sin nTrC e"'^^^ sin 29 ,

(36b) ,- â€˘ K = ^Â« (-1)"

n ^2 2

TT n

As in  we define the stresses, r , of the N-th
approximation to the exact theorj'- as:

(37) (T^^h^,9,^;E) =rZ[s^{^^,9,r) + F^I,^/e,e,0]e'' â€˘

n=0

.(0)

It follows from this definition, (25), (26) and (33) that j
and (T coincide with the plane stress solution. For the
second approximation to the exact theory we obtain from (37)

fioX^if J.O& â€˘s-dv'- -^col ns-zis sd hbo I'ioiii . -^ Ib':J2

Q

â€˘ ,'oa ^fi!f Tol aftXf'l'r rio-tdv? norti?:': v: ssi'^foe 9.-J-J-r:.!:'>rf-?.

, feS iix3 ^^""s ?-trn eoo ^^^I 2Â»

V' .1

[b'K]

^^'V â– :tÂŁrf,1 i^^). fcn .(eS) ^rioictirrnsfo sln^ mc'.;r c:;.'Oj..L 01 jI

(L)

(38)

16

cr^^^4,e,C;e) = sÂ°(4, 9) + [.71(4,9,0 + F^(^/e,9,0]e^ ,
4i^4,e,^;e) = sO0(4,9) + [cT^e(4,9,C) + P^0(?/e,9,O]e^ ,
3'i2)(4,9,C;e) = F^ {e/ÂŁ,9,ae^ ,

4z^(^>^-JÂŁ) = F^^(|/e,e,C)e^ ,

0-1^^(4, 9, C;e) = f2(4/ÂŁ,0,OÂŁ^ .

,0 2 , â€ž2

Here S , Gr and F are given in (32) and (3^-36).

7. Presentation of Results .

We define the quantity:

(39) ^^^^4,C;ÂŁ) = ^ 5^^^^

1 +-^

(1+?J'^.

In Fig. 1, Oq '(0,0;e) is illustrated as a function of e with

varying Poisson's ratio. For e = we obtain the plane stress

^(2)

result which gives a stress-concentration factor of two for cfg

Although the stress-concentration factor increases with e, for
"small" e, it is only a small percentage of the plane stress
solution. For example, with e = -2 and v = 1/4 our results
Indicate only a 1.25 9^ increase over the plane stress result.
Thus, for "small" e the plane stress solution apparently yields

i^r

, ^sriJ.G-^-^^'^-

.- 1 .. .:

(2)

'T>

, ''3[(3\-e!.-qi o s g*? ^ __- -^ b
.-.CSX

^ . ri rj i M ^ ':c o d'-i A nr. A , f â€˘ .

.'â€˘T-J '"-f^

.'rq ^(31 .lev

Toal ax gy. . .'â– iJ8 iBn

1

Online LibraryEdward L ReissExtension of a thick infinite plate with a circular hole → online text (page 1 of 2)