On stress concentrations in thick plates

ON STRESS CONCENTRATIONSIN THICK PLATES (0 KONTSENTRATSII PMM

NAPRIAZHENII

V TOLSTYKH

PLITAKH)

Vol. 30, No. 5, 1966 pp. 963-970 O.K. AKSENTIAN (Rostov on the Don) (ReceivedOctaber30,

Investigation

is continued

the Kirchhoff

hypothesis,

method developed

1. Consider

of applied

of stress

plate bending

concentration

theory,

problems

based

by means

on of the

in [ 1 and 21. the problem of axisymmetric

2h with an opening Introduce

on the applicability to the solution

1965)

which is bounded

dimensionless

plate are traction-free and is free of shear.

coordinates

bending

by the circular s, n and [(Fig.

while the cylindrical

surface

Here, m is an odd integer,

of an infinite cylindrical 1). Assume

is subjected

k is a constant

plate of a thickness

surface

r with radius

that the flat faces

to a normal load N=k of proportionality

a.

of the gm,

and

h= h/a. Applying

the method described

the boundary r,

in [l

and 21, we obtain

expressions

for the stresses

on

which are of the form

(1.1)

On stress

+ (v -

+ ; [(v- 1) sp(5)

concentrations

1147

in thick plates

$q,,=,

1)

1 $4

-

r,n,, (s;)cpa-

I+... cp2 J I

:!npK)

p=1

ZnzIr=-2tlLh

-g

rprp(5) cp2 +

t'tlh+

aA$o 5') 7

(1 -

I7I=O

p=1

; rp (5) p=1

-

(Tp%3 ++ cpPj} + *. .

p=1

p=1

z,, = 0,

0,

7,, =

(1.3)

1

v=m

(1.5)

Here ~1 is the shear ratio rp (0,

in [ 1 and 31, 2y

The

while

In (1.1) circle,

the even-numbered

to (1.4),

with $0 (n) being

opening

as given

the constants

quadrant.

system

are obtained

Pi of equations.

The boundary

It is readily bending

theory

From (1.6).

conditions

seen based

for I/Y, (n) are given

that the above

we have

The boundary

from the boundary

on the Kirchhoff

Table

in the fourth

of the above.

The roots

in the order of increasing roots

being

1 con-

are located

those

magnitude,

in the first

are in the fourth quadrant.

for the bending

theory.

part.

which

The roots

are conjugates

tii (n) are biharmonic

the solution

by the applied

c

roots

the quantities

roots

computer.

of forty roots

the odd-numbered quadrant

out over

were obtained

are numbered

1

is carried

first 80 of these

in the first FIG.

of 4 of function

real

the values

quadrant

‘i, (&J,

which have a positive

with aid of an electronic tains

(T is Poisson’s

are the roots

P 1. Summation

sin (2) -

the roots

modulus,

d = l/s); n,, (Q,

are known functions

and rp (5)

given z-t

calculations

(in

coincide

functions

of an infinite values

conditions

for the exterior plate

of a

with a circular

of the functions on r by means

$ri (n) and of an infinite

by

with the conditions

of the applied

plate

hypothesis.

k +o=Bo-~qv+l)(nz_r_)

:j

ln(~~?-l)

(1.7)

1148

0. K. Aksentian

P

5

7

Rerp

3.7488381

6.9499798

IO. 119259

K3.271274

Imrp

1.3843390

1.6761046

1.8583834

1.9915705

P

11

1

3

Re yp 19.579408 Imrp P

13

15

22.727036

25.873384

2.2573196

2.1833970 21

38.451800

Im 7,

2.476&020

2.5188989

P

31

Re rP 51.02’1838 fm rP P

2.6599693 41 2.7939639

P

51

Im r, P

2.8995’tZl 61 98.159562

49 2.8801345

57

59

91.875516

2.9182217

2.9362345

2.9536?82

65

67

104.44351

107.58546

3.0744514

79.307064

2.8599 Ii33

55

117.01119

2.7CQ8KW

76.164856

88.733453

3.0176753

73

39 sn. 595487

2.7445856

85.591359

3.0024137

2.6282535

37 60 .L,j2Q7:;

63

71 3.0~~0~

2*5W3901

101.30155

2.9866716

Re y, 113.86930 Imrp

2.8389026

53

29 47.881869

47

73.0226~

2.8169378

2.4299576

44.738731

45

69.880291

Re rp 82.449231

P

2.7179394

43

Im r,

Re 7,

57.310371

2.6897936

Re yp 66.737923

Imrp

2.5580671) 35

54.167664

32.163617

2.3’787569 27

41.595390

33

19

29.018831

25

35.307902

2.0966252

17

2.3217134

23

Re y,

9 16.429870

2.9704179 69 110.72739

3.0324849

75

77

120.15307

123.29494

3 -0876925

95.017552

3.0468686 79 t26.43680

3 *f~59~

3.1131672

In order to exclude rigid body motion of the plate, we set B0 = 0. To determine cpz , we have an infinite set of linear algebraic equations &ST? (c=a 7t W-7

P “)“(r

toss rp) t

--r

P)

Iv (rt i r,P -

(r, -

1,Fl cpI -

+

(&se

yt -1)

ct*=Ft*

P+t

(La) (t = 1, 2, 3, . . .) --I

which may be written in the form IIM II- Iha II = II Ff, II Here llMl[ is the complex matrix of the left-hand side of (1.8) and llc&

(1.9) and llF,,[[

On stress

are, respectively,

concentrations

in thick

the column matrix of the unknowns

1149

plates

and that of the right-hand

side of

(1.8). To transform

(1.8)

into real form, set Cur =

Noting that yzn_t

is the conjugate

%-1.2

=

corresponding

order of the system

f 211-1,~--

-

is halved.

to first twenty roots

(1.10)

1, 2, 3,.

(n =

van,2

shown that

If we limit ourselves

Introducing

P --V x2nk

27+1,2'

where \lW,ll is th e matrix of the transformed

20, 18, . . .

(1.12)

by truncation.

of the loading

or geometry

of the first fourteen

mation being insufficient m = 5. The results

1

i

real system. With the aid of a computer,

i

11

x j=

-

i

0.89.10-6

i m=5

xj=

6 -

i

0.318.10-4 11

xj=-o.12.10-5

To illustrate

of rank

so that the results

problem.

The inverted

Calculations

matrices

we’re carried

i.e.

it is

of the matrix inpermit the deof the approxi-

out for m = 3 and

2.

2 -0.12745.10-3

2 3 0.1993.10-4

7 -0.400~10-5 12 0.30.10-8

1

xj=0.18609.10-2

matrices

the matrix is universal,

of the plate,

for the remainder.

6

xj=-On.9967.10-5

(1.14)

out of the twenty unknown x., the accuracy

are shown in Table

xf=0.180170.10-2

m=3

(1.13)

II“j II = IIf jz II

TABLE i

. .)

inverted.

may be used for any plate bending

termination

2, 3,.

2n-1,2

I‘t was shown that, for a given material,

In [ll, version

is solved

, 4 were successively

independent

the

the notation

in the form IIMl II.

System (1.14)

boundary

side semi-plane,

(IIX1,

we can write (1.9)

(1.11)

fm, 2 = -$ Im F2,_l,2

Re F2,+1,2,

IL --U xm-lk

. .)

to twenty

yp lying in the right-hand

will be equal to twenty.

CLk

ivn,

of yzn, it can be easily

van-1,2 =

u2n,29

so that the order of the real system layers

uPa -

2 -0.689.10-4 7 -0.18.10-5 12 -0.7.10-a

the rate of convergence

8 -0.119.10-5 13 -0.42.10-6

4

9

8 -0.842.10-b 13 -0.7.10-0

of the process

-0.797.10-5 10

-0.177.10-5

0.15.10-6

14 0.23.10+

3 0.1785.10-3

5

-0.7332.10”

4 -0.1152.10-a 9 -0.20.10-5

5 0.119.10-4 10 -0.24.10-S

14 -0.2.10”

used in the determination

of xi,

1150

O.K. Aksentian

we list various superscript

approximations

indicates

x2, zll and x1, as the most typical (Table

of zI,

the order the system

from which the particular

3). The

determination

was

made. TABLE

W

m-5

(n) “13

-ri

n

m=3

G 8 10 12 14 lG 18 20

0.1803134.10-” 0.1801734. IO-2 0.1801719~10-2 0.1801708.10-2 0.1801705~10-2 0.1801703.10-3 0.1801702. IO-2 0.1801702~10-*

G 8 10 12 1’1 IG 18 20

0.187135.10-” 0.186330.20-’ 0.186181. IO-” 0.186139.10-2 0.186119.10-” 0.18G109.10-2 (~.18GlO1.10-” 0.186098.10-2

-0.128097.10-3

-0.127338.

IO-3 _ 0.127381.10-3

-0.127$10.10-3 -0.127429.10-3 --0.1274339.10-3 --0.127143.10-3

-0.458.10-6 -0.436.10-G -mO. 4.28. IO-6

0.28.10-6 O.25.1O-6 0.24.10-e

-0:69875. -0 731 $5. IO--’ lo-‘& -0. ti8719. IO-4 -o.m7;2.10-1 -0.fi8802. IO--0.128.10-5 -0.74.10-” -0.70.2OP

4. KS’,‘, .1OW ---0 . G&l . II I- 1 -0.G8919.10~~

-0.83.10-7 -0.13. IO-” --0.16.10-”

with which the xi were obtained

is sufficient

for

purposes.

Utilizing

(1.1)

to (1.4),

of the state

of stress

formulation.

Let us examine

points.

‘i . IO-3

-0.127J0

We will see later that the accuracy practical

3

Here and hereinafter

for the stresses

(1.7),

we obtain, %I,

These

a first approximation

contain

the infinite

the rate of convergence we denote

series

of these series

the coefficients

of all components of the boundary

layer

at the moat typical

of hz in the series

expressions

os and Tnz on r by ani, uSi and Tnzi, respectively.

us,

Substituting and (1.3),

we may now obtain

in the plate.

(l.lO),

(1.13)

and the values

previously

obtained

for xi into (1.1)

for m = 3,

I r.c=*1 = +- k{O.GOOO + 149.3082 -

- I.102 - 0.69 - 0.45 -...].lO-2} Tnzllr,t=O = -k[7.4879-10.885+4.919

z

1.924 -

+ k (0.6000

-2.179+

Z-

2.792 +

0.405)

-

1.806 =

+ k-l.005

0.919-0.31+0.06-.

. .].10-2s

k.O.0001

For m = 5, we obtain

qr

cc+1 = J- k (0.4286 + (52.492 -i- 13.056 - 0.044-1.80-1.68-

0.9 -:..]:AO-2) _, + k (0.4286 -t 0.598) = tkl.027 = - k [5.38G - 11.684 -j- 10.42 - 6.63 -+ 4.0 %*, Ir,+l z - k0.004

1.28 -

The boundary

conditions

Comparing [=

the immediately

f 1, where

1.4 -

. . .] 10-2~

yield

c,,i I’,c=:*1 points

2.5 +

one would

=

+ h,h,

preceding expect

(1.15)

q,z /I’,<_ 0 = 0

results

with (1.15),

convergence

to be the

we see that even at the slowest,

the series

results

On stress

concentrations

in thick plates

qr -QJ

-94

-a;

-a,

-a

FIG. 2 using

seven

vergence

terms do not differ significantly

Fig. 2 shows

curves

of successive

Kirchhoff

solution;

solutions

taking into account

the broken line is the exact

Now let us calcnlate is usually

calculated a,lr

c=

dete;ination

ones.

the basis

the first approximation for determining

from formula (1.2).

of c

At other points,

con-

(~,,r[ f(c)

u,I

of

curves

one, two and to the

1, 2 and 3 correspond

at the points

r

shown that’a

[=

factor.

first approximation

solving

the infinite

f 1. This us/ r

may be

of

system

for the

.

1) RP (+I)

and (1.2) +

yr,$)

and taking into account (+I)

=

*

to

layer terms, respectively.

concentration

without

using

in the figure correspond

solution;

the stress

but it is easily

(1.27) into (1.1) (Y -

for

lines

one, two and three boundary

* 1 for arbitrary m may be obtained

Substituting

we obtain

approximations

layer terms with m = 3. The straight

three boundary

stress

from the exact

is even better.

W,,”

=

(1.11)

2*s,

(&I)

as well as

1152

O.K.

%I1

be1

I r,

I r,c=*1= --fk

I;-+1

f k (-

Aksentian

I“-+svp m

+

2

2

p=1, 3,

1

ReVrp2cp2) . ..

&+4(v-l)+pzI,3,

I

Re(THICK,)

Ix

But cn1I

r, c=c1=

f

k

Hence

p-1,

k

2

Re (rp2cp,)

no-1

8pv

3, . . .

m-12

Whereupon, we have

QelJr.&-+l=fk In (1.16),

tion factor is not obtained

asymptotically

theory, in the first approximation,

m-l

+ --

2v

the first term in the braces corresponds

From (1.16) it is clear that in this case,

theory is 22%

v-l

{-&

from the Kirchhoff as m increases.

(1.16)

1

to the solution

form/= 1 of course,

increases

m+2

of applied theory.

the exact stress

concentra-

theory. The error of this For m = 3, the error of applied

; for m = 5 it is 44% ; for m = 7 it is 66%, etc.

At other points on the surface r the error will also not be small. Fig. 2 shows the curves of the function

cSl\ p (5). as calculated

by means of the Kirchhoff

theory (straight

line) as well as those using one, two and three boundary layer terms (1, 2 and 3). 2. As discussed solution

in [l],

the next step in constructing

to the problem is the determination

The boundary conditions

From (2.1).

the asymptotic

expansion

of the

of I+$~(n) and c

P3’

for $r (n) are given by

we have

(2.2) As before,

we set B, = 0. The system of equations II M II

. II~13II = IIF

for cp3 has the form

t3 II

(2.3) P+1

(L==l, Here llfnll is the same matrix asin system (1.8). same manner as system (1.9),

2, :i,.

The system (2.3) is solved

noting that cZn 3 is the conjugate .

of czn_r

.) in the

3 and setting

I

On stress

C2~-l,3-

Thereupon, as

it is clear

yi system

$

it is possible

from (2.3)

concentrations

in thick

(n=i,

(Yzn_1-

iY*)

to obtain

the values

that with fourteen

plates

. .)

2, 3,.

(Table

known values

1153

(2.4)

4) of the first ten

unknown

yi,

of xi the order of the truncated

will also be fourteen. TABLE j=

1

2

~j=-0.36003.10-3 m=3

j=

i -=

1

rn-= 5 j=

6

yj=

TO illustrate approximations

! 10 12 14

m=3

m = 5

4 6 8 10 12 14

-0.37288.10-3 -0.37072.10-3 -0.37037.10-3 -0.37036.10-3 -0.370354.10-3 -0.370352~10-3

We may now determine vergence previously

of the series determined

determining

approximations

of y,. into (1.1)

-0.49.10-7 -0.46.10-’

O.lO8.1O-e 0.1075.10-a

0.23.10-7 0.27.10-7

0.190.10-a 0.191~10-a

of the stress

in this step will now be checked. vafnes

yi, successive

5

-0.61675.10-a -0.62075.10-8 -0.62262.10-3 -0.62258.10-a -0.62255.10-a -0.62253. 1O-3

the second

0.3.10-7

5)

-0.4938. IO-* -0.50775.10-4 -0.56738.10-4 -0.50732.10-4 -0.50726.10-4 -0.50724.10-4

-0.36050.10-3 -0.35991~10-3 -0.35998.iO-3 -0.36000.10-a -0.360014.10-3 -0.360020.10-3

10

0.19.10-a

of the process

TABLE

0.38.10-a

9

0.33.10”

the rate of convergence

5

0.1155.10-4

8

0.48.10-a

of yI , ya, yp and yIO are shown (Table

4

-0.4*10-7 4

-0.1661.10-4

7

0.221.10-5

10

0.11.10-6 3

-0.6225.10-4

0.200*10-6

9

-0.5.10-7

2

5

0.9050.10-5

8

0.21.10-6

0.37035.10-3

4

0.190.10-5

7

0.26.10-G

yj== -

3

-O.5O72.1O-4

6

yj=

4

and (1.3).

components.

Substituting we obtain,

(2.2),

(2.4)

The conand the

for m = 3,

u,~I~,~=~~= +k{[~1.7504+0.714+0.498+0.080+ 0.062+. . .]+ [10.71918- 0.1446 - 0.1832 - 0.1518 - 0.0456 - 0.0242 - 0.0138 - . . .]}1O-~z z + k (- 0.1040 + 0.1016) = T kO.0024 + 1.504 - 0.38: %rza Ir,c=o = - k (if.3572 + [- 2.03747 - 0.6441 + 0.2492 - 0.9939

= -k

(0.0249

-

-j- 0.03 - 0.02 + + 0.0359 - 0.013 0.0251) =

k 0.0002

. . .] + -

0.004

+

. . .]} 10-g =

1154

while

0. K. Aksentian

for m = 5, we shall have

@Ttz Ir, L-3 1 =: -4; k {[+ (11.3542 + 1.4340 +

12.3362 0.0244 z

%z2

=

I'. c-0

1.i650 0.0850

+ k (-0.1285

+

0.290 +

-

0.0664

+

0.1259)

0.242 +

-

9.0432 = T

0.126 -

-;-

0.0254

. . .] + -

. . .])

10-a z

kO.0026

k{(1.774+ 1.221--0.894+0.40-0.15+. . .]+ - 0.261 + 0.419 - 0.236 + 0.125 - 0.06 + 0.03 + . . .]} 10-Y z - k (0.0235 - 0.0237) = k0.0002

-

+ [-2.384 Comparing

-

the remits

thus obtained

on and ~na on the boundary

f,

with the values

we find that convergence

given in (1.15) of the second

for the stresses approximation

is

also satisfactory. Now consider

the stress l=

ing the infinite

system

Gs2

i

of equations

utilizing =

r,c=+1

In a manner similar to that of section

* t may also be obtained

ahown *ataalr,

Calculations

aair.

to determine

(2.5) yield,

k $

f

for the second c pJ.

1, it is readily

approximation

withont solv-

Thus we obtain

for m = 3, 0.0723

j5.35959 --a0228

-

-

0.0916

0.0121

-

0.0069

-

0.0759

-

-...].2O_ZzJt

k-0.135

For m = 5, we obtain OS2

I

r, Z=ctl = $ k $

[5.6771 -

+

0.0332

0.7170

3. The third step of the construction determination

0.0226

-

+ -

0.0122 0.0127

of the asymptotic

of r+!~a (n) and cp4. The boundary

conditions

-

0.0425

--...].fO-a expansion

-u’F: deals

k-O.168 with the

for $a (s) ara given by (3.1)

From (3.1),

we have

(3.2) To determine given matrix, tion Of aa\r

c

, we P4

have the infinite

but, as previously

discussed,

system

of equations

it is easily

* <= f 1 may be obtained without the determination

Calculations

utilizing

(3.3) yield

for m = 3,

with the previonsly

shown that the third approximaof c

P-

Thns, we obtain (3.3)

On stress

c&,

+ ‘/,[1.64360

1.2423+

k”/s{[-

=i

<=,l

+

0.00538

concentrations

-

0.0335+

0.0178

-

1155

in thick plates

0.0170+

0.0034

-

0.0021

0.0014

-

+

0.0013+.

0.00065

-

. .]+

. . .]} 1O-2 zz

=: _+ k.O.O1OO Form 5s31

r,

= 5, we obtain

+*1

=

1.3005

ks,h{[-

T

+ l/z 11.7137 -to.1335

$0.00629

-

0.06460

-

0.0022

+ -

0.0090 0.0018

+ -

0.0061

+

0.0010 -

0.0026 0.0006

+ -

. . .] + . . .], 10-2~

=: + k 0.0113 We now present q?,

c=

approximation

of the asymptotic

+,k [- 0.4667 f k [-. 0.2381

=

cI II’, ++-1

From (3.4)

h -

0.135 112+

0.0100h3 +

h -

0.168 k2 +

0.0113

and (3.5) it is clear

the diameter

of the opening

of

determination

of. the stress

in powers

It is readily

h3 +

that even for x = 2 (i.e.

concentration

when the plate thickness represents

we can recommend

factor be baaed

m = 5

for

. . .]

(3.4)

m = 3

for

. . .I

the third term in the expansion

8% of the sum of the first two terms. As a result, expansion

expansions

f 1

(J, I r,i=_+1 =

twice

a three-term

t3 . 5) is

only 5% to

for x < 2 that the

on the first two terms of the

of h.

seen

from (3.4)

and (3.5)

that, for x = 0.1, the sum of the next two terms

is equal to 3% to 7% of the first term. Thus, for cS may be obtained

from the following

for x < 0.1,

v-l

I r,I;=,1- -fkh[-_++T-

6,

the stress

concentration

factor

expression, m-l

(3.6)

m+2

3

BIBLIOGRAPHY 1.

Aksentian,

O.K. and Vorovich,

(The state of stress 2.

Aksentisn, osnove theory).

3.

Lnr’e,

O.K. and Vorovich,

prikladnoi

PMhf, Vol.

K teorii

sostoianie

plity maloi

tolshchiny

27, No. 6, 1963.

I.I., Ob opredelenii

teorii (On the determination

PMM, Vol. AI.,

I.I., Napriazhennoe

in a thin plate).

kontsentratsii

of stress

napriazhenii

concentrations

using

na applied

28, No. 3, 1964. tolstykh

plit (On a theory of thick plates).

PMM, Vol. 6, Nos.

and 3, 1942.

Translated

by H.H.

2