plate theory bhadange wo soln

8/8/2019 Plate Theory Bhadange Wo Soln

1/26

MEEN 5330 1

Plate Theory

Name: Swapnil Bhadange

Class: Continuum Mechanics

Date: 15th Nov 04


2/26


3/26

MEEN 5330 3

Definitions [2]

Midplane: It is a plane that dividesthe thickness t into two equal halvesand which is parallel to the face.

Thin plates: The ratio of thethickness to the smaller span lengthshould be less than 1/20.

Thick plates: The above ratio greaterthan 1/20 would be considered asthick plate.


4/26

MEEN 5330 4

Assumptions [2],[3]

The theory that is covered here is valid only for thin plates.These 2D plate theories are obtained from the equations of 3D

elasticity by integrating them through the thickness.

The governing equations of plates are derived by vectormechanics and/or energy and variation principles.


5/26

MEEN 5330 5

General Behavior of Plates [1]

Fundamental Assumptions of Classical theoryfor isotropic, homogenous, elastic thinplates.

Deflection of midsurface is small.

Midplane remains unstrained subsequent tobending.

Plane sections initially normal to the midsurfaceremain plane and normal to that surface afterbending.

The stress normal to the midplane is small

The above hypotheses are also known asKirchhoffs Hypotheses and due to theseassumptions a complex 3D plate problemreduces to 2D problem.


6/26

MEEN 5330 6

Strain-Curvature Relations.[1]

Deformation is measured in terms of strain tensor L.

Strain at any point in the plate is given by

)(

2

1

k

m

j

m

j

k

k

j

jk

X

u

X

u

X

u

X

uL

x

x

x

x

x

x

x

x! Finite Strain Tensor

)(2

1

i

j

j

i

ij

x

u

x

ul

x

x

x

x! Infinitesimal Strain Tensor

yx

wz

y

wz

x

wz xyyx

xx

x!

x

x!

x

x!

2

2

2

2

2

,, KII


7/26

MEEN 5330 7

Strain-Curvature Relations. (cont)

Curvature of a plane curve is defined as the rate of change ofslope angle of the curve wrt the distance along the curve.

The strain curve relation is expressed in the form.

xyyx

y

w

xy

w

yx

w

xOOO !

x

x

x

x!x

x

x

x!x

x

x

x )(,)(,)(

xyxyyyxx zzz OKIOI !!! ,,


8/26

MEEN 5330 8

Stresses [2]

Stress in Elastic bodies on a plane.

jiji nWW !

Where Wij denotes the componentsOf stress vector


9/26

MEEN 5330 9

Stresses [1], [3]

Generalized Hookes Law

When the material is isotropic the no. of elastic co-efficient reduces to 2.Forisotropic material we have E1= E2= E3= E,G12= G13= G23= G,R12=R13=R23=R

The constant E , R , G are modulus of elasticity, Poissons ratio , andshear modulus of elasticity. The connecting equation is

)1(2 R!

EG


10/26

MEEN 5330 10

Stresses [1]

Generalized Hookes Law relates the six components of stress with thesix components of strain

From the assumptions we have Iz

,Kyz,

K

xz

are all negligible.Hence

xyxy

xyy

yxx

G

E

E

KX

RIIR

W

RIIR

W

!

!

!

)(1

)(1

2

2

)]([1

)]([1

)]([1

yxzz

zxyy

zyxx

E

E

E

WWRWI

WWRWI

WWRII

!

!

!

G

G

G

yzyz

xzxz

xyxy

XK

XK

XK

!

!

!


11/26

MEEN 5330 11

Stresses [1]

The stresses distributed over the thickness of the plate produce bendingmoments, twisting moments, and vertical shear forces.

Hence the stresses in bending are found by

yx

wDM

x

w

y

wDM

y

w

x

wDM

xy

y

x

xx

x!

x

x

x

x!

x

x

x

x!

2

2

2

2

2

2

2

2

2

)1(

)(

)(

R

R

R

)1(12 2

3

R!

EtDwhere

is the flexural rigidity of the plate

333

121212

t

zM

t

zM

t

zM xyxy

yy

xx !!! XWW


12/26

MEEN 5330 12

Variation of Stress Within a Plate [1]

These variation are governed by the conditions of equilibrium of

Statics. Fulfillment of these conditions establishes certainrelationships know as equations of equilibrium

Finally the differential equation ofequilibrium for bending of thin plates

dxx

MMdmm

xxyy

x

x!

py

M

yx

M

x

M yxyx!

x

x

xx

x

x

x2

22

2

2

2


13/26

MEEN 5330 13

Variation of Stress Within a Plate(cont)

Expression for vertical shear forces Qx and Qy maynow be written as

)()(2

2

2

2

2

wxDy

w

x

w

xDQx x

x

!x

x

x

x

x

x

!

)()( 22

2

2

2

wy

Dy

w

x

w

yDQy

x

x!

x

x

x

x

x

x!

Where 22

2

22

yx x

x

x

x!

Since equilibrium equation has 3 unknowns, it is internally statically

Indeterminate.


14/26

MEEN 5330 14

The Governing Equation for Deflectionof Plates [1]The basic differential equation for the deflection of plates may be derived

from equilibrium equation

The governing differential equation for deflection of thin plates which isin the concise form and derived by Lagrange is

D

pw ! 4

This equation is reduced to two second order partial differential equations

py

M

x

M!

x

x

x

x2

2

2

2

D

M

y

w

x

w!

x

x

x

x2

2

2

2

D

p

yyxx

yxyx!

x

x

xx

x

x

x2

22

2

2

2OOO

Given the loading and boundary condition w can be found.


15/26

MEEN 5330 15

Boundary Conditions [1]

We can formulate boundary conditions for a variety of commonlyencountered situations. The boundary conditions which applyalong the edge x=a of the rectangular plate with edges parallelto x and y axes are as follows.

Fixed Ends Simply Supported Roller end

w = 0 ,

0!x

x

x

w

w = 0 ,

0)(2

2

2

2

!x

x

x

x!

y

w

x

wDMx R

0!x

x

x

w

0)2(2

3

3

3

!xx

x

x

x

yx

w

x

wR


16/26

MEEN 5330 16

Circular Plates [1]

Introduction:

In practice, members that carry transverse loads, such asend plates and closures of pressure vessels, clutches and

turbine disks etc are usually circular in shape.Thus many of thesignificant applications fall within the scope of the formulasderived ahead.

In all cases the basic relationships in polar co-ordinates

are employed.


17/26

MEEN 5330 17

Basic Relations in Polar Coordinates [1]

In general polar co-ordinates are preferred over Cartesian wherethe degree of axial symmetry exists either in geometry orloading.

The polar coordinate set (r,U) and the Cartesian set (x,y) arerelated by the equations

Ucosrx ! 222 yxr !

Usinry !x

y1tan

!U


18/26

MEEN 5330 18

Basic Relations in Polar Coordinates(cont)

Referring to the above fig,

Ucos!!x

x

r

x

x

r Usin!!x

x

r

y

y

r

rr

y

x

UU sin2

!!x

xrr

x

y

UU cos2!!

x

x


19/26

MEEN 5330 19


Determination of the fundamental equations oflaterally loaded plate in polar coordinatesrequires only that the appropriaterelationships be transformed from Cartesian

to polar coordinates.

Resulting Equations for radial,tangential,twistingmoments are

]11

[)1(

]11

[

)]11

([

2

2

2

2

22

2

2

2

22

2

UUR

UR

UR

U

U

x

x

xx

x!

x

x

x

x

x

x!

x

x

x

x

x

x!

w

rr

w

rDM

w

rr

w

rr

wDM

w

rr

w

rr

wDM

r

r


20/26

MEEN 5330 20


The vertical shearing forces are given by

Similarly,formulas for the plane stress components are written inthe following form

333

12,

12,

12

t

zM

t

zM

t

zM rr

r

r

U

U

U

U XWW !!!

)(

1

)(

2

2

wrDQ

wr

DQ r

xx

!

x

x!

UU


21/26

MEEN 5330 21

Example

1. Determine the deflection in a very long and narrow plate or so-calledinfinite strip(a>>b),if it is simply supported at the edges y=0, and

y=b when the plate carries a non-uniform loading

b

ypyp

Tsin)( 0!

where constant p0represents the load intensity along the line

passing through y=b/2, parallel to the X-axis

Solution:-

From the loading described we have andsubstituting in equation on slide (11) we get

0/ !xx xw 0/2 !xxx yxw

2

2

2

2

dy

wdDM

dy

wdDM yx !! R

(a)

(b)


22/26

MEEN 5330 22

And also we get

Substituting (a) into (c) integrating and satisfying the boundary condition

y=0 and y=b we have

The maximum stresses in the plate are obtained by substituting the abovewith R=1/3 into stress equation

D

p

dy

wd!

4

4 ..(c)

b

y

D

pbw

T

sin4

04

!

!!

!!

2,

26.02.0

2

0max,0max,

2 by

tz

t

bp

tbp yx WW

!!

20max,

tzpzW


23/26

MEEN 5330 23

(continued)

Now consider the following ratio

If, for example ,b=20t , the above quotients are 1/80 and 1/8.For thinplate ,t/b


24/26

MEEN 5330 24

Homework Problem

For the same previous problem consider the plate carrying auniform load p0.


25/26

MEEN 5330 25

References

[1]. Ugural ,A.C. ,McGraw-Hill ,Stresses in Plates and Shells.

[2]. Szilard, Rudolph, Prentice-Hall, INC, Theory and Analysis of

Plates

[3]. Reddy, J.N. , Taylor & Francis, Theory and Analysis of ElasticPlates.


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MEEN 5330 26

To conclude, I would just like to say that the whole topic ofPlate theory has not been covered,as it would be impossibleto give justice to such an interesting topic in just 15-20 min.

I have tried to cover the things which would be easy tocomprehend for all of us and especially me.

Thank-you

plate theory bhadange wo soln

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