Shear Def Plate Buckling

8/10/2019 Shear Def Plate Buckling

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NORTH DAKOTA STATE UNIVERSITYDEPARTMENT OF MECHANICAL ENGINEERING

MEAM 722ADVANCED MECHANICS OF DEFORMABLE SOLIDS

INDEPENDENT PROJECT

EFFECTS OF SHEAR DEFORMATIONSON PLATE BUCKLING

Author: Sait MekicInstructor: Dr. Mohammad H. Alimi

Fargo, January, 2001


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C O N T E N T S

Page

1. INTRODUCTION .2

2. CLASICAL PLATE THEORY 3

2.1 Classifications and definitions ..3

2.2 Kinematic equations ..5

2.3 Constitutive relations and bending moment resultants .6

2.4 Equilibrium of the plate element ..8

3. BUCKLING OF PLATES ...12

3.1 Differential equation of buckling 12

3.2 Buckling analysis of plates in compression and shear ..16

3.3 Critical load of a plate uniformly compressed in one direction ..17

3.4 Critical load of a plate in shear by Galerkin method ...19

4. REFINED PLATE THEORIES .21

4.1 Governing differential equation .21

4.2 Numerical methods .23

Conclusions and recommendations ...24

REFERENCES 25


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1. INTRODUCTION

Before any discussion of buckling of plates can take place, it is necessary to establish the theory of bendingof plates leading to a differential equation connecting applied loads flexural rigidity and curvature.

Therefore the first task of this project was to analyze these aspects of the problem.

In Section 2 is given an overview of classification of plates and their theories. Then, a comparison of theKirchhoff and the Reissner-Mindlin kinematics is given. Further, using Kirchhoff assumptions, constitutiverelations, and equilibrium of plate element, differential equation of plate bending under transverse load isderived.

Analysis of effects of shear deformations on plate buckling, which was main concern of this project, can gointo two directions. One direction is to analyze in-plane shear deformations caused by in-plane loads, andthe other is to analyze what happens when transverse shear stresses are not negligible and have to be takeninto account.

The analysis of in-plane shear deformations by in-plane shear loads is given in section 3, where thedifferential equation of plate buckling is derived. In this section is presented one problem that considerscritical load of a plate uniformly compressed in one direction, and one problem that considers critical loadof a plate in shear by Galerkin method.

The analysis of transverse shear effects is given in Section 4, where the differential equations governingdisplacement field and applied loads is presented. Some observations concerning to numerical methods ofsolution of these equations, also, are given in this section.

Although there were presented two examples of solutions of plate problems, the main task of this projectwas to analyze some theoretical aspects of the given problem, rather then to solve any particular problem

connected with the subject.


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3

2 CLASICAL PLATE THEORY

2.1 Classifications and definitions

Consider an elastic body, as shown in Fig. 2.1, comprising the region 0 x a , 0 y b, and -h/2 z h/2 , such that h


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2.1 Assumptions of the Kirchhoff and the Reissner-Mindlin plate theories

The assumptions of the Kirchhoff plate theory are:

1. strains and deflections are small and geometrically linear2. material of the plate is homogeneous, isotropic, and obeys Hooke's law.3. plate is thin

The latter assumption leads to the two Bernoulli hypotheses (Fig. 2.1):

a) straight lines perpendicular to the mid-surface before deformation remain straight after deformation b) transverse normals rotate such that they remain perpendicular to the mid-surface after deformation

and to a further assumption:

c) the transverse normals do not experience elongation, i.e. they are in-extensible:

e zx = 0(2.1)

zx = 0

z,w n x

P ( x,y) t

h/2 w/xh/ 2

w( x, y)

x

Fig.2.1. Bernoulli hypothesis

If Bernoulli hypothesis a), and b) hold we have Kirchhoff theory, for which

xw x

=

(2.2)

yw

yx=


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If Bernoulli hypothesis b) is released we have Reissner-Mindlin theory, for which (see Fig. 2.2)

x x

w z x

= +

(2.3)

y y

w

y= +

z

As a consequence of these hypotheses, the local effects of load applications cannot be described by thetheory.

The assumptions given by (2.1) are in contradiction to the materials with Poisson effect ( 0), but thiscontradiction does not cause further difficulties for the analysis.

Neglecting the transverse shear strains xz , and yz is inconsistent with the material law. Shear stresses exist because of equilibrium of forces and stresses, i.e. xz , and yz are not equal to zero, while from equations

xz = G xz , yz = G yz (2.4)

it follows that these stresses are equal to zero. Both inconsistencies are similar to those encountered in theBernoulli-Euler beam model.

The Reissner-Mindlin theory does not have these inconsistencies. Kinematics of this theory is shown inFig. 2.2.

z,w

P ( x, y)

w/x

w(x,y) xy

- w/ xx x

Fig. 2.2. Comparison of Kirchhoff and Reissner-Mindlin kinematics

2.2 Kinematic equations

The rotations of the material normal about x and y-axes are denoted by y and x respectively. For smalldeflections and rotations it follows


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x y

w x

w y

= = , (2.5)

The displacements { u , v , w} of a plate particle P( x, y, z ) not necessarily located on the mid surface aregiven by

u z w x

z v z w y

z w w x x y= = = = =

, , y( , ) (2.6)

Using assumption a) from section 2.1 and elasticity kinematics we can write

eu

x z

w

x zk xx xx= = =

2

2

ev

y z

w

y

zk yy yy= = =

2

2

ew z

z w

z zk zz zz = = =

2

2

(2.7)

eu

yv

x z

w x y

zk xy xy= +

= = 12

2

eu z

w x xz

= +

=12

0

e v z

w y yz = + =12 0

where

k (2.8)w

xk

w

yk

w x y xx xy xy

= = =

2

2

2

2

2

, ,

In (2.7), and (2.8) k xx , and k yy represent curvatures of the deflected mid-surface, while k xy representswarping of the plate. From (2.7) we see that entire displacement field is defined when w( x, y) is defined.

2.2 Constitutive relations and bending moment resultants

To establish the plate constitutive equations further assumptions are necessary:

a) plate is homogenous and isotropic with elastic modules E b) each plate lamina z = constant is in plane stressc) the plate material obeys Hooke's law for plane stress, which can be represented in the form


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xx 1 0 e xx

yy = E

1 2 1 0 e yy (2.9)

xy 0 0 (12

1 2e zy )

Them bending moments M xx , M yy , M xz ,are stress resultants with dimensions of moment per unit length, i.e.force. Action of moments upon the plate is shown on Fig. 2.3.The moments are calculated by integratingthe elementary stress couples through the thickness.

y M yx

M yy M xx

M xy M xy

M xx x

M yy

M yx

Fig. 2.3. Action of bending moments

M dy zdydz M zdz xx xx xx xxh

h

h

h

= =

2

2

2

2

,

M dx zdxdz M zdz yy yy yy yyh

h

h

h

= =

2

2

2

2

,

(2.10)

M dy zdydz M zdz xy xy xy xyh

h

h

h

= =

2

2

2

2

,

M dx zdxdz M zdz yx yx yx yxh

h

h

h

= =

2

2

2

2

,


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Since bending moments are in equilibrium it follows that M xy = M yx . Using (2.8) and (2.9) we can write

M xx 1 0 k xx

M yy = D 0 0 k yy (2.11)

M xy 0 0 (12

1 + 2k zy ) where

D Eh=

3

212 1( ) (2.12)

represents the bending or isotropic flexural plate rigidity.

If the bending moments M xx , M yy , M xz ,are known, the maximum values of in plane stresses can be obtainedusing (2.9) and (2.11):

xx xx M

h) max,min =

62

yy yy M

h) max,min =

62 (2.13)

xy xy M

h) max,min =

62

These maximum and minimum values occur for z = h/2, i.e. on the plate surfaces.

2.3 Equilibrium of the plate element

To derive the interior equilibrium equations we consider differential mid-surface element dxdy aligned with x, y-axe s as shown in Fig. 2.4. Consideration of force equilibrium along the z direction requires the presence of transverse shear forces. The components of these forces In the { x, y} system ate called Q x andQ y and are defined as shown in Fig. 2.4(a). These are forces per unit length.

With these shear forces are associated shear stresses xz and yz . For a homogeneous plate and usingequilibrium similar to Euler-Bernoulli beams, the stresses vary parabolically over the thickness:

xz xz z

h=

) max 14 2

2 (2.14a)


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yz yz z

h=

) max 14 2

2 (2.14b)

in which the maximum values xz

)max

and yz

)max

, which occur on the mid-surface z = 0, are only functionof x and y. Integrating over the thickness gives.

z y QQ

ydy y

y+

q

(a) Q x

dy Q x + Q x

dx x

x

Q y dx

dx

M yx + ( M yx / y)dy

M yy + ( M yy / y)dy (b) M xx

M xy + ( M xy / x)dx M xy dy

M xx + ( M xx / x)dx x

M yy

M yx

Fig.2.4 External and internal forces on the element of the mid-surface

in which the maximum values xz )max and yz )max , which occur on the mid-surface z = 0, are only functionof x and y. Integrating over the thickness yields


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Q d (2.15) z h Q dz x xz h

h

xz y yz h

h

yz = = = =

/

/

max/

/

max) , )2

2

2

223

23

h

If Q x and Q y are known then

xz x

yz yQ

h

Q

h) , )max max=

23

23

= (2.16)

as in the case of Bernoulli-Euler beams, stresses (2.16) come entirely from eqilibrium analysis. TheKirchhoff plate model ignores the transverse shear energy, and in fact xz = yz = 0.from cinematicequations (2.7).Practically this means that stresses (2.16) should be significantly smaller then (2.13). If theyare not, the Kirchhoff model does not apply.

Considering force equilibrium along the z direction in Fig. 2.4 (a) yields the shear equilibrium equation:

Q x

Q

yq x

y+ = (2.17)

where q is applied transverse force per unit area. Force equilibrium along the x and y-axes is in this caseautomatically satisfied and does not give additional equilibrium equations. Force equilibrium along the xand y-axes will be considered in section (2.XX) where the differential equation of plate buckling is derived.

Considering moment equilibrium about x and y yields two moments differential equations:

M

x

M

yQ xx

xy x+ = ,

M

x

M

yQ

yx yy y+ = (2.18)

Moment equilibrium about z gives

M xy = M yx (2.19)

The four equilibrium equations (2.17) - (2.19) relate six Fields M xx , M xy , M yx , M yy , Q x and Q y .Hence plate problem is statically indeterminate.

Eliminating Q x and Q y from (2.18)gives The following Moment equilibrium equation in terms of the load:

2

2

2 2

22 M

x

M

x y

M

y q xx xy yy

+ + = (2.20)

Elimination of bending moments from equation (2.20) using equations (2.8) and (2.11) gives equation for bending of thin plates:

(2.21) D w q =4

where


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= (2.22)+ +44

4

4

2 2

4

42

x x y y is the biharmonic operator.

Equation (2.21) is the analog of the Bernoulli-Euler beam equation

EI w

xq

4

4 = (2.23)


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3 BUCKLING OF PLATES

3.1 Differential equation of buckling

Plate structures are often subject to normal and shearing forces acting in the plane of the plate. If these in- plane forces are sufficiently small, the equilibrium is stable and the resulting deformations are characterized by the absence of lateral displacements ( u 0, v 0, w = 0). As the magnitude of these in-plane forcesincreases, at certain load intensity, a marked change in the character of deformation pattern takes place.That is, simultaneously with the in-plane deformations, lateral displacements are introduced. In thiscondition, the originally stable equilibrium becomes unstable and the plate is said to have buckled.

Classical buckling problems of plates can be formulated using the differential equation of staticequilibrium, various energy methods, and dynamic approaches.

In most general form in-plane displacements can be written as

u x y z u x y z x y x( , , ) ( , ) ( , )= +0

(3.1)v x y z v x y z x y y( , , ) ( , ) ( , )= +0

w( x, y, z ) = w0( x, y)

where { u0 , v0 , w0 } are displacement components of a point along the ( x, y, z ) coordinates, and x , and y are rotation about y and x axes respectively. These rotations are given by equation (2.5), if transverse shear

strains are zero, as it is the case that we are considering now.

We here consider the in plane forces acting on a plate element, in which the forces are assumed to befunctions of the mid-surface coordinates x and y, as shown in Fig. 2.5. The in-plane force resultants aredefined to be:

(3.2) N x xh

h

=

/

/

2

2

dz

dz

dz

dz

(3.3) N y yh

h

=

/

/

2

2

(3.4) N z z h

h

=

/

/

2

2

(3.5) N xy xyh

h

=

/

/

2

2


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(3.6) N dz yx yxh

h

xy= =

/

/

2

2

N

=

=

z

N xdy

N xydy y

N yxdx N ydx

( N y + ( N y / y)dy)dx)dx

( N yx + ( N yx / y)dy)dx)

dy

( N x + ( N x / x)dx)dy) ( N xy + ( N xy / x)dx)dy) x

Fig. 2.5. In-plane forces on a plate element

Using relation (2.24) end equilibrium of the in-plane forces it can be shown that, for the case of no surfaceshear stresses

(3.7)40

0u x y( , )

(3.8)4 0 0v x y( , ) In Fig. 2.6 is shown the relationship between forces and displacements when the plate is subject to bothlateral and in-plane forces, so that there is lateral deflection w. The z component of the loading per unit areais, for small slopes:

1 2

2dxdy N

N

xdx dy

w x

w

xdx N dy

w x x

x x+

+

(3.9)

Neglecting terms of higher order, the component of N x force per unit area in the z direction is

N w

x

N

xw x x

x

2

2 + (3.10)

Similarly, the z component of the N y force per unit area is


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N w

x

N

xw x x

x

2

2 + (3.11)

z ( w/ x + ( 2w/ x2)dx) ( N x + ( N x / x)dx)dy)

dx

N xdy x w/ x

Fig. 2.6. Relationship between forces and displacements

The z component of the in plane shear resultants N xy and N yx can be investigated using Fig. 2.7:

dy

N xydy

dx N xy ( N yx + ( N yx / y)dy)dx)

( N xy + ( N xy / x)dx)dy)

Fig. 2.7. In-plane shear resultants

1 2 2

dxdy N

N

xdx

w y

w x y

dx dy N N

ydy

w x

w x y

dy dx xy xy

yx yx+

+

+ +

+


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N w y

dy N w x

dx xy yx

(3.12)

If higher order terms are neglected in the above expression reduces to

N w

x y

N

xw y

N w

x y

N

yw x xy

xy yx

yx

2 2

+ + + (3.13)

With all the above z components of forces per unit area evaluated, the governing plate equation (2.21) can be modified to include their effect:

D w q x y N w

x N

w

y N

w x y x y xy

= + + +42

2

2

2

2

2( , )

+

N

xw x

N

yw

yx x y+ (3.14)

+

N

xw y

N

yw

yx xy yx+

Having in mind the in plane force equilibrium and the assumption that there are no applied surface shearstresses, it follows that

N

x

N

y

x yx+ =0 (3.15)

N

x

N

y xy y+ =0 (3.16)

Substituting (3.15) and (3.16) into (3.14) the final form of the differential equation of plate buckling becomes

D w q x y N w

x N

w

y N

w x y x y xy

= + + +42

2

2

2

2

2( , )

(3.17)

This equation is analogous to the Beam-column equation, which can be obtained by multiplying equation(3.17) by the width b of the beam, letting ( )/ y = 0, = 0, P = -bN x , and bq( x ,y) = q( x):

d w

xk

w

x

q x EI

4

42

2

2

+ = ( ) (3.18)

Where k 2 = P/EI.


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3. 2 Buckling analysis of plates in compression and shear

Classical buckling problems of plates can be formulated using a) the differential equation of staticequilibrium, b) various energy methods, and c) dynamic approaches [1].

a) Equilibrium method

In the equilibrium method, we assume that the plate has buckled slightly. When the in plane edge loads areslightly above the critical load, the lateral deflections, w, are approaching very large values, regardless howsmall the increase in the edge loads becomes. The smallest load corresponding to this condition is thecritical load.

b) Energy methods

In the state of neutral equilibrium, the plate changes its original flat shape to a curved shape withoutgaining or losing energy. The corresponding energy equation is

(3.19) W W i e* *+ =0 where W i

* and where W e* represent the work of the internal forces, in the form of potential energy due to

bending ( U b*), and the work done by the external compressive forces due to the in-plane displacements

produced by bending, respectively.

c) Dynamic methods

The stable system will return to its original position after the introduction of small oscillations. If the stateof equilibrium is unstable, the system will not return to its initial position, since the small disturbance will

be followed by increasingly large deflections.

In setting up the differential equation of transverse vibration the effect of in-plane forces is considered.

When plates are subject to the simultaneous action of in plane compressive and shear forces, combinedwith lateral bending, buckling occurs at lower load intensities then when these forces act individually. Theeffect of the combined loading can be approximated by the so-called interaction equation, which has thefollowing form:

(3.20) R R R1 2 3 1 + + + .. .

where Ri is the load ratio defined by

Ri th edgeload acting alone

corresponding i thcritical load

N

N ii

cr i

=

=,

(3.21)

Equation (3.3) represented graphically gives the interaction curve (Fig. 3.1). Buckling take place when the plot of the forces is on, or above these limited bounds.

For simply supported plate subject to biaxial compression:

R x + R y 1 (3.22)


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R2

1.0 R2 + R2 (Bounds of Safe

Zone )SafeZone

R2

0 R1 1.0 R1

Fig.3.1. Interaction curve

For simply supported plate with a/b 1, and long plates with elastically restrained edges, subject tolongitudinal compression and shear:

Rcomp + R2 shear 1 (3.23)

For bending and shear:

R2bend + R2 shear 1 (3.24)

3.3 Critical load of a plate uniformly compressed in one direction

Here we consider a simply supported rectangular plate with sides a and b units long (Fig. 3.2). the plate isacted on by a compression force, N x.

y

a

N xb

x

Fig. 3.2. Plate in compression

The differential equation of the plate buckling reduces to:


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Dw

x

w

x y

w

y N

w

x x

4

4

4

2 2

4

4

2

22 0+ +

+ = (3.25)

Since all four edges of the plate are simply supported, the lateral deflection as well as the bending momentvanishes along each edge. The boundary conditions are:

w = 0, at x = 0, and x = a(3.26)

w = 0, at x = 0, and y = b

Assume that the solution is of the form:

w A (3.27)m x

am y

bm nmn

nm= =

11

1 2 3sin sin , , , , ...

Then after derivation of (3.9) and substitution into (3.7)

Am

a

m n

a b

n

b

N

Dm

a

m xa

n ybmnnm

x

+ + =11

4 4

4

2 2 2

2 2

4 4

4

2 2

22 0

sin sin (3.28)

This yields

N D

b

mba

n amb x

= +

2

2

2 2

(3.29)

N x depends on the dimensions and the physical properties of the plate and on m and n, the number of half-waves that the plate buckles into. Since the critical value of N x is the smallest value that satisfies equation(3.11), the value of m and n that minimize N x must be determined. It is obvious that N x increases as nincreases and that n = 1. The number of half-waves in the x direction that correspond to a minimum valueof N x is found by minimizing N x with respect to m.

Thus

dN

dm D

b

mba

amb

ba

a

m b x = +

=

20

2

2

2

(3.30)

from which

m (3.31)ab

=

This yields

N D

b x cr , =

4 2

2

(3.32)


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3.4 Critical load of a plate in shear by Galerkin method

Here we consider the simply supported square plate loaded by uniform shearing forces N xy applied alongthe four edges (Fig3.3).

ya

N yx

N xy N xy a

x

N yx

Fig. 3.3. Plate in shear

The differential equation of the plate buckling reduces to:

Dw

x

w

x y

w

y

N w

x y xy

4

4

4

2 2

4

4

2

2 2+ +

+ 0= (3.33)

Assume that

w x y Ai xa

i ya

iii

( , ) sin sin , ,==

1

2

1 2= (3.34)

For this plate subject to a pure shear, whose deflection is assumed by (3.16), weighted residuals are of theform

= (3.35)= j w x y dxdy j jaa

( ) ( , ) , ,00 1 2 where

( )ww

x

w

x y

w

y

N

Dw

x y xy= + + +

4

4

4

2 2

4

4

2

22

(3.36)


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4 REFINED PLATE THEORIES

4.1 Governing Differential equations

The extension of Timoshenko beam theory to plates is Reissner-Mindlin, or "thick" plate theory. Reissner-Mindlin theory relaxes assumption of negligible shear deformation, and accounts for an average shearstrain in the plate.

From (2.9), (2.10), and (2.11) it follows

x x Ez w

x

w

y

M z

h=

+

=1

122

2

2

2

2 3

y y Ez w

y

w

x

M z

h=

+

=1

122

2

2

2

2 3 (4.1)

xy

xy Ez w x y

M z

h= + =1

122

3

and from the equilibrium equation

x xy xz

x y z + + =0 (4.2)

it follows

xz x xy

z z

h

M

x

M

y= +

123 (4.3)

Integration of (4.3) with respect to z yields

xz x xy z

h

M

x

M

y f x y= +

+62

3( , ) (4.3)

Applying boundary conditions xz = 0 at z = h/2, function f ( x, y) is determined:

f x yh

M

x

M

x x xy( , ) = +

32

(4.4)

Therefore


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xz x xy

h

M

x

M

x z h

= +

32

1 42

(4.5)

Similarly we find

yz

xy y

h M

x M

x z h

= +

3

21 4

2

(4.6)

The transverse normal stress can be obtained from the equilibrium in z direction, and it is equal:

z qh

z h

z h

= + 2

1 3 43

(4.6)

As it was mentioned, for thin plates the transverse normal and shear stresses are negligible in comparison tothe remaining stresses. For thick plates the contribution of transverse shear stresses can be large.

Using the total energy principle, equilibrium equation can be derived. Further, from the equilibriumequations, next five differential equations, that relate displacement field and apply load can be derived [2],[3], and [4]:

x A

u x

Av y

A y

u y

v x11

012

066

0 0 0+

+ +

= (4.7)

A x

u y

v x yx

Au x

Av y66

0 012

022

0 0

+

+ +

= (4.8)

A x

w x

A y

w y

q x y550

440

0

+

+ +

+ = (4.9)

x

D x

D y

D y y x

Aw x

x y x y x11 12 66 55

00+

+ +

+

= (4.10)

D x y x y

D x

D y

Aw y

x y x y yx66 12 22 44

00

+

+ +

+

= (4.11)

where

A c dz D z c d i jij ijh

h

ij ijh

h

= = /

/

/

/

, , ( ,2

22

2

2

1 2 6= , , )

= )

(4.12) A K K c dz i jij i j ijh

h

=

/

/

, ( , ,2

2

4 5


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23

c E

c E

c c c c111

12 2122

2

12 2112 12 22 21 11 66 121 1

= = = = = , , , G

c G c c G44 23 45 55 130= = =, ,

For an isotropic material, equation (4.9), (4.10), and (4.11) can be written as:

= +2 02 1

w x y

q Eh

x y

( )

+ (4.14)+

+

=

22 2

2 201

26 1

0

x

y x x x y y h

w x

( ) ( )

+ +

+

=2

2 2

2 201

26 1

0

y x y

y x y x h

w y

( ) ( )

where represent shear correction factor, which is usually chosen as 5/6.

Equation (4.14) is uncoupled from equation (4.7) and (4.8) and can be solved for w0, x, and x independently from displacements u0, and v0.

In the Table 4.1 is given a comparison of the results obtained by the classical plate theory and the resultsobtained by the refined plate theory, of nondimensionalized center deflections of square, simply supportedisotropic plates under uniform loading [2].

Table 4.1

R e f I n e d P l a t e T h e o r y C l a s I c a lP l a t e

T h e o r y

a/h 5 10 12.5 20 50 100

Deflection 5.3556 4.6660 4.5832 4.4936 4.4453 4.4438 4.4436

Differences are much more pronounced for othotropic and composite plates.

4.2 Numerical methods

Equations (4.14), in general cannot be solved exactly. Although the Reissner-Mindlin is much moredifficult then classical plate theory plate theory, the resulting set of equations is much easier too solvenumerically. Finite elements based on Reissner-Mindlin assumptions have one important advantage overelements based on classical thin plate theory. Reissner-Mindlin plate elements require only C 0 continuity of


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the lateral displacement w0, and independent normal rotations x, and y. However, elements based onclassical thin plate Kirchhoff theory require C 1 continuity. In other words, w0 / x and w0 / y as well as w0 should ideally be continuous across element interface although this condition is relaxed in non conforming

plate elements. Thus it would appear that Reissner-Mindlin plate elements are simpler to formulate.

However, when exact numerical integration is used with standard Reissner-Mindlin finite elements, very

disappointing results are obtained in application to thin plates. This Phenomenon which is called shearlocking, is caused by the imposition of the constraints xz = yz = 0 by the shear strain energy terms in thetotal potential energy when limiting thin plate situation are approached.

Conclusions and recommendations

Study of a subject as Stability of plates under shear requires deep insight into Theory of plates and, as a prerequisite, knowledge of Theory of elasticity and FEM.

In the next phase of research on this project, I would make a finite element model, based on Reisner-Mindlin theory, for a certain types of plates and certain type of loads, and do numerical experiments toobtain a valid comparative analysis.


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REFERENCES

1. Szilard R., Theory and Analysis of Plates, Classical and Numerical Methods, Prentice-Hall, Inc.,Englewood Cliffs, New Jersey, 1974.

2. Reddy J. N., Energy and Variational Methods in Applied Mechanics, John Wiley & Sons, New York,1984.

3. Mindlin R. D., Influence of Rotary Inertia and Shear on Flexural Motion of Isotropic Elastic Plates,Journal of Applied Mechanics, vol 18, pg 31-38, 1951.

4. Reissner E., The Effect of Transverse Shear Deformation on the Bending of Elastic Plates, Journal ofApplied Mechanics, vol 12 (2), pg 69-77, 1949.

5. Timoshenko S., Woinowsky-Kriger S., Theory of Plates and Shells, Mc Graw-Hill, New York, 1959.

6. Timoshenko S.,Gere J M., Theory of elastic Stability, Mc Graw-Hill, New York, 1961.

7. Langhaar H. L., Energy Methods in Applied Mechanics, Mc Graw-Hill, New York, 1959

8. Chuen-Yuan C., Nonlinear Analysis of Plates, Mc Graw-Hill, New York, 1980.

9. Reismann H., Elastic Plates, Theory and Applications, John Wiley & Sons, New York, 1988.

10. Bulson P. S., The Stability of Plates, Elsevier, New York, 1969.

11. Bletzinger K. U., Theory of Plates, Part I and Part II, http://www.statik.bauwesen.tu-muenchen.de/

12. Deduced Thick Plate Solutions From Thin Plate Solutions,

http://www.eng.edu.sg./EResnews/9808/p8.html

13. Felippa C. A., Introduction to Finite Element Methods,http://caswww.colorado.edu/courses.d/IFEM.d/Home.html

14. http://www.umr.edu/~umreec/web-courses/me334/section05/page176-180.html
http://www.statik.bauwesen.tu-muenchen.de/http://www.eng.edu.sg./EResnews/9808/p8.htmlhttp://caswww.colorado.edu/courses.d/IFEM.d/Home.htmlhttp://www.umr.edu/~umreec/web-courses/me334/section05/page176-180.htmlhttp://www.umr.edu/~umreec/web-courses/me334/section05/page176-180.htmlhttp://caswww.colorado.edu/courses.d/IFEM.d/Home.htmlhttp://www.eng.edu.sg./EResnews/9808/p8.htmlhttp://www.statik.bauwesen.tu-muenchen.de/

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