theory anisotropic thick plates

8/10/2019 Theory Anisotropic Thick Plates

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FTD-HT-

2 3-600-69

EDITED

RANSLATION

THEORY

OF

ANISOTROPIC

THICK

PLATES

By:.

G .Lekhnitskiy

English

pages:

Source:N

SSSR.

Izvestiya.

ekhanika

iMashinostroyeniye(Academy

of

Sciences

of

the

USSR.

*ws.

Mechanics

and

MachineBuilding))

No.2 ,1959,pp.

142-1H5.

Translated

by:

.

Koolbeck/TDBRS-3

TH IS

TRANSLAT ION

S

A

Rt

NDIT IONO F

TH R

ORIGI -

NAL

P0USI6N

TIXT

WTHGUT

ANY

ANALYTICAL

O R

EDITORIAL

OMM I N T .

TATEMENTSOR

THRORIIS

ADVOCATED ORIMPLIED ARE THOS E

O P

TH E

SOW

AND 00NO TNECESSAR I LYREFLECT

TNI

POSIT ION

O RPINION

PIC

ORS IONICHNOLOOY

I-

VIS ION.

PREPAREDBY.

TRANSLAT ION

DIVISION

FORE IGN

T E CMNOLO SV DIVIKON

IP-APt,

OH I O .

FTD-

H T

-

3- 6oo -69

Date21

Jan .

19

70


3/12

THEORYOP

ANISOTROPIC

THICK

PLATES

S .G.

Lekhnitskiy

(Saratov)

The

goaloftheseremarks

is

to

show

how

the

classicaltheory

of

thick

plates

isgeneralized

to

thecase

ofa

plate

possessing

anisotropyofa

particular

type,

thatis,

a platemanufactured

from

transversallyIsotropie

materialandconsequently

possessing

five

independent

elasticconstants.

1 .

Generalinformation

. In

the

classical

statement,

the

problem

of

equilibrium

ofan

Isotropie

thick

plateisreduced

to

thedeterminationof six

componentstresseso

a

...,

T nd

x y

xy

threoprojections

of

displacement-

u ,

v ,

w which

strictly

satisfy

allequations

of

elasticity

theoryand

also

theboundary

conditionsonthe

flat

surfaces(faces). Theconditionson

the

lateral

surface,i.e.,

on

the

edge

of

the

plate,

are

satisfied

approximately,"ontheaverage,"a sinthetheoryOfthe

generalized

planestressstateorinthetheoryof

bending

of

thick

plates.

If

the

platei s

deformed

only

by

aload

distributedalong

the

edge,twobasiccasesofequilibrium

aredistinguished:

a )

the

plane

stress

state,and

b )

bending. Each

case

reduces

tothe

determination

inthe

region

of

the

plate

of

a

single

biharmonic

function

of

thevariables

x ,

y

( a

function

of

stresses

or,

correspondingly,of

thedeflectionof

themiddleplane),

satisfying

FTD-HT-23-600-69


4/12

V

theboundaryconditionsonthe

contour;thecorresponding

solutions

are

called

homogeneous

([1],

page

200).

Iftheflatsurfaces

are

alsoloaded

the

solution

is

obtained

by

superimposing

the

homogeneous

solutions

and

thesolution

for

an

infiniteelastic

layer

bounded

bytwo

parallel

planesover

which

given

forces

are

distributed.

he

homogeneous

solutions

foran

Isotropie

plate

are

determined

by

simpleformulas

which

connect

stresses

and

displacements

with

thebiharmonicfunction.

Iti s

easyto

show

that

in

the

caseof

a

transversally

Isotropie

platethehomogeneous

solutions

alsohavea

very

simplestructureandalsoconnect

stresses

and

displacementswiththe

biharmonic

function.

Let

there

be

a

plate

which

is

cutout

of

transversally

Isotropiematerial

in

sucha

way

that

in

it

theplane

ofisotropy

is

parallel

tothemiddlesurface.e

will

take

the

latter

as

plane

xy

and

designate

thethickness

oftheplate

as

h . The

equations

of

the

generalizedHookelaw

are

writtenasfollows

([2],page

28):

T^

+ V

+

A

l

1

(1.1)

2 1 - f v )

xi

Here

Eand

E ,

are

theYoung

moduli

for

tension-compression

indirectionsparallelandperpendicularto

themiddle

plane;

vi s

the

Poissoncoefficient,

characterizing

the

contractioninthe

plane

of

isotropy

during

tension

inthesame

plane;v ,

i sthesame

parameter

for

stretching

in

the

transverse

direction

z ;v

?

i s

the

Poissoncoefficient

which

characterizes

contractioni n

the

directionwhichi s

normal

tothemiddlesurfaceduringtension

in

planes

whichareparallelto

the

middle

surface;

G=E/2(l+

v);

and

G ,i stheshearmodulus

for

planes

whichareparallel

and

normal

to

the

middlesurface, in

all

there

arefive

different

constants,sinceE v ,=E,v

?

.

BTD-HT-23-600-69


5/12

~r

Ten

unknown

functions

must

satisfy

thebasic

systemof

equations

of

equilibrium

for

a

transversallyIsotropie

body;

we

will

obtainthis

system

by

adding

to(1.1)three

equations

of

equilibrium

of

a

continuous

medium:

d ,

4

dt

i> F fh * \* .

d * F

h*

\

*

,

\

V-Vi

0

(2.1)

(2.2)

(2.3)

Here

F

i s

a

function

of

Airystresses,

satisfyingthe

equation

FTD-HT-23-60G-69


6/12

W-? ( V - - J 5 T

+

^-)

2 . 4 )

Thefirst

terms

of

the

expressions

-

u ,v ,

o

o and

T

x

y

x y

will

be

the

average

over

the

thickness

of

the

values

of

displacements

andstresses

andare

connected

by

the

equations

of

theplane

problem

i

a/

(2.5)

a

ii

2< 1

+

v .

* F

B y t r diy

Th e

second

terms,

depending

on

z ,

are

correctionmembers

which

takeinto

account

the

change

inthe

displacements

and

stresses

over

the

thickness.

I n

order

to

obtain

the

distribution

of

stresses

in

athickplate

i t

i s

necessary

firsttosolvetheplaneproblem,

i.e.,

t o

findF ,

u ,

andv .

Inthe

case

ofan

isotropic

material,v~

=

v ;a

e

v/2(l

+

v).

3 .

Bending

,

If

the

plate

is

deformed

by

abending

load

distributed

only

along

the

edge,displacementsand

stresses

are

expressed

through

a

biharmonic

function

a s

follows:

.__

..

,

^...-^v*.

( 3 . 1 )

E

o*n

3 V , \

( *

-

,\*

%h

fop,

z ?

f t \

9

,

Here

*-*+T*I T- **

p-i-i?

3.3)

Stresses

o

ando leadto

bending

momentsM

nd

M

T

leads

x y

x

y

x y

t o

the

torsion

moment

H, :

t ,

T

ead

; o

shearing

forces. The

x y

X i , z

FTD-HT-?

3-600-69


7/12


8/12

7

The

formulas

for

displacements,

stresses,

and

oments

are

obtainedfrom(3.1),(3.2),and

(3.1)

by

adding

toheirright

sides,

respectively,

the

expressions

'-IF*' -ir*

*--*+

^8=5^+

3 . 8 )

v-v-v-

3.9)

The

normal

stress

on

areas

which

are

parallel

to

themiddle

plane

i s

different

from

zero:

kK

| i

- T + i r ) + - f . l T - T r )

3 . 1 0 )

4 .

ending

of

a

round

plate

.

e

shall

consider

in

greater

detail

thecase

of

around

platewithradiusa,

supported

along

the

entire

edge

and

bentbyuniformly

distributed

load

q;

the

weight

of theplatei s

ignored.

Wewill

obtain

theexpressions

fordisplacements

u u

Q

and

r

stresses

c o_,and

T

Q

in

cylindricalcoordinates

from

the

r

re

t f

corresponding

expressionsof

the

theory

ofthin

plates;

wewill

replacew

Q

in

the

latter

by

the

quantity

w.,

whichdepends

on

z ,

and

we

will

add

to

the

right

sides

the

functions

2T

IE

tH ( 3 1 4 i \

(4.1)

I norder

tofindthe

moments

M M andH

n

,i ti s

necessary

r U r

6

to

takethe

formulas

ofthetheory

of

thinplates,to

replacew

Q

in

FTD-HT-23-600-69


9/12

them

b ythe

e x p re s s i on

w ,andtoad dthem o me n t scorresponding

8 '

o

o'

and

o,

i.e.,

(4.2)

It

isclearthatw

n

isafunctiononlyof

di s p e rs i on

r :

0

* W

A

+

D ,%

(4.3)

(termswhich

give

asingularity

in

the

centerarediscarded).

Theboundary

conditions

have

the

form

- 0 ,

/

f

-

o

hen

r

a

(4.4)

Weshall

i n t ro d u ce

the

finalformulasf or

di s p l a ce me n t

wat

anypoint

and

forthestresses:

-*J6(i-v)i>(T+T

a

-TV+ "

v - * [ c ^ * - *

+

-

* - J r ) l

= - 4 F [ 3

+

v ) _ 3 v

,

rfm(

T

-

J

3 * 4 :

N

f r

/

A

(4.5)

(4.6)

Here

the

following

designations

areused:

-r=vfrflr-*S'JBJECTAREA

15, 20

TOPIC

TAGS

modulusof

elasticity,

anisotropic

medium,

elasticity

theory,

circular

plate,Isotropie

property

AUTHOR

CC-AUTHORS

LEKHNITSKIY.

S.

G .

10-OATE

OFNFO

5 9

SOURCE

ANSSSR.IZVESTIYA.MEKHANIKA

I

MASHINOSTROYENIYE

RUSSIAN)

FTT>

6-DOCUMENT

NO.

HT-23-6QO-6Q

PROJECT

NO .

72?01-7fi

7-HEADERCLASN

SECURITYAND

DOWNGRADING

NFORMATION

UMCL.

0

7>-SUPERSEDES

7*-CHANCS

6A-CONTP.OLMARKING

NONE

UNCL

76-

REEL/

FRAME

NO.

1891

785

CONTRACT

NO.

XRE FACC .

NO .

65-

AO-GEOGRAPHICAL

AREA

NO .O FPASES

U

PUBLISHING

OATE

4-

TYPE

PRODUCT

TRANSLATION

REVISION

REQ

NONE

STEP

NO .

02-UR/OI79/59/OOO/OO2/OI42/OI45

ABSTRACT

( )

The

material

ofthe

plate

is

assumed

to

be

transversely

Isotropie,withthe

planes

ofisotropy

parallel

tothemiddle

surface,andthe

material

therefore

possessesfive

independent

elasticconstants.

quationsare

given

without

detailedderi-

vation

forthe

displacements

and

stressescorresponding

to

a

state

of

plane

stress,

and

for

the

displacements,

stressesand

moments

in abentplate,

bothfor

thegeneralcase,andforthe

particularcaseofa

plate

bent b ya

uniformly

distributed

pressureinadditiontoitsownweight.or

a

simply

supported

circular

plate

subjected

to

a

uniformly

distrubuted

pressure

the

equations

for

displacements,

stressesand

moments

are

given,

neglecting

theweight

of

theplate.

or

the

particular

cases

considered,

the

errors

of

thethinplate

approximationare

negligible

if

the

materialisIsotropie,but

areappreciable

ifthematerial

isanisotropic.

the

errorsbecominglargerthe

smaller

the

ratio.here

are

4

Sovietreferences.

AF$C IfoTii F TDOVER PR IN T D EC

8)

AFc lum

theory anisotropic thick plates

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