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R&DE (Engineers), DRDO

Theory of Plates

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Ramadas Chennamsetti

Theory of Plates


Introduction

� “When a body is bounded by surfaces, flat in geometry, whose lateral dimensions are large compared to the separation between the surfaces is called a PLATE”

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Ramadas Chennamsetti2

� Plates are initially flat structural elements


Introduction � Plates are subjected to transverse loads – loads

normal to its mid-surface � Transverse loads supported by combined bending

and shear action� Plates may be subjected to in-plane loading also

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� Plates may be subjected to in-plane loading also => uniform stress distribution => membrane

� Membrane action – in-plane loading or pronounced curvature & slope

� Plate bending – plate’s mid-surface doesn’t experience appreciable stretching or contraction

� In-plane loads cause stretching and/or contraction of mid-surface


Introduction

� Plate stretching

NxNxx

zt

Uniform stretching of the plate => uo

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Uniform stretching of the plate => uoq

Axial deformation due to transverse load

Net deformation = Algebraic sum of uniform stretching and axial deformation due to bending load


Introduction

� For plates -� Thin & thick plates –

� Thin plate => b = smallest side � Thick plate =>

bt 20<bt 20>

2000

1

10

1 ≥≥b

t

t≤

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� Small deflections –� Thin plate theory – Kirchoff’s Classical

Plate Theory (KCPT)� Thick plate theory – Reissner – Mindlin

Plate Theory (MPT)

5

tw ≤


KCPT - AssumptionsAssumptions –� Thickness is much smaller than the other physical

dimensions � vertical deflection w(x, y, z) = w(x, y)

� Displacements u, v & w are small compared to plate thickness

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plate thickness � Governing equations are derived based on undeformed

geometry

� In plane strains are small compared to unity –consider only linear strains

� Normal stresses in transverse direction are small compared with other stresses – neglected


KCPT - Assumptions

� Material – linear elastic – Hooke’s law holds good

� Middle surface remains unstrained during bending – neutral surface

� Normals to the middle surface before deformation remain normal to the same surface after

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remain normal to the same surface after deformation => doesn’t imply shear across section is zero – transverse shear strain makes a negligible contribution to deflections.� Transverse shear strains are negligible

� Rotary inertia is neglected


Sign convention

� Following sign convention will be followed

z

Mz

z

ψψψψz

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y

x

Mx

My

Mz

x

ψψψψx

ψψψψy

ψψψψz

y

Positive moments Positive rotations


Bending deformations

y

z

D

Bending takes place in both planes

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x

A

B

z

x

A

B

A’

B’

View ‘D’

Rotation => ψy


Bending deformations

� Deformation in ‘x’ direction

� Deformation in ‘y’ direction

( )yxzzyxu y ,) , ,( ψ−=

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� Deformation in ‘y’ direction

� Vertical deformation

( )yxzzyxv x ,) , ,( ψ−=

) ,( yxww =


Strains

� Assumption – out of plane shear strain -negligible

wwx

w

z

uxz

∂∂

=∂∂+

∂∂=γ 0

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y

w

y

w

y

w

z

vx

w

x

w

xxyz

yz

yyxz

∂∂==>=

∂∂+−==>

=∂∂+

∂∂=

∂∂==>=

∂∂+−==>

ψψγ

γ

ψψγ

0

0

0


Strains

� Non-zero strains

wv

x

wz

xz

x

u yxx

∂∂∂∂∂−=

∂∂

−=∂∂=

2

2

2

ψ

ψε

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yx

wz

x

v

y

u

y

wz

yz

y

v

xy

xyy

∂∂∂−=

∂∂+

∂∂=

∂∂−=

∂∂−=

∂∂=

2

2

2

2γ

ψε


Stresses

� Thin plate – out of plane shear strains vanish – out of plane shear stresses also vanish

0 ,0 == yzxz ττz

τ

z

τxz

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� Out of plane normal stress is also assumed to be zero – logical – thin structure – plane stress conditions

yτyz

τyz

x

τxz

τxz


Stresses

� Non-zero stress components

y

z σxx

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x

σyy

σyy

σxx

τxy

τxy

All three stress components, σxx, σyy, τxy – in-plane


Curvatures

� Curvature – reciprocal of radius of bending

� Rate of change of slope

z

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Rx

dx

x

ww*

dx

ψy

dxx

w

∂∂−

x

wy ∂

∂=== - rotation slope ψ xxy

x

w

xκ

ψ=

∂∂−=

∂∂

=2

2

curvatue


Curvatures

� Similarly bending in yzplane introduces a curvature

2

2

y

wyy ∂

∂−=κ

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� Twisting of plate

y∂

yx

wxy ∂∂

∂−=2

2κ


Constitutive law

� Linear elastic isotropic – Hookean material

� Three stress and strain components

συσε −=

EEyyxx

xx

Writing all three equations in matrix form

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( )υττ

γ

συσ

ε

υε

+==

−=

−=

12EG

EE

EE

xyxyxy

xxyyyy

xx

−−=

+−

−=

xy

yy

xx

xy

yy

xx

xy

yy

xx

xy

yy

xx

E

E

γεε

υυ

υ

υτσσ

τσσ

υυ

υ

γεε

2

100

01

01

1

)1(200

01

011

2


Constitutive law

� Express strains in terms of curvatures

∂∂∂∂

+−−=

y

wx

w

Ezyy

xx

2

2

2

2

21

00

01

01

1 υυ

υ

υτσσ

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∂∂∂

∂

+−

yx

wy

xy 2

22

100

1 υυτ

Variation of stresses across thickness is linear

Basis – thin plates – plane section remain plane after bending – variation of axial deflection is linear across thickness – strains also vary linearly


Equilibrium equations

� Equilibrium of an infinitesimal element

y

z

dx

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x

dx

dy

t

τ*yz

dx

dy

σ*yyσyy

σxx

σ*xx

τxy

τ*xy

τxy

τ*xy

τ*xz

τ*yz

Forces acting on an infinitesimal element dx dy dz

y

x



� Equilibrium in ‘x’ direction

∂

=−−−

++

dzdxdxdydydz

dzdxdxdydydz

yxzxxx

yxzxxx

σ

ττσττσ ***

0

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∂∂

+=

∂∂+=

∂∂+=

dyy

dzz

dxx

yxyxyx

zxzxzx

xxxxxx

τττ

τττ

σσσ

*

*

*



� Substitute – the following equation is obtained

(1) - 0=∂

∂+∂

∂+

∂∂

zyxxzxyxx τττ

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Similarly take equilibrium in ‘y’ and ‘z’ directions

(2) - 0=∂

∂+

∂∂

+∂

∂zyxyzyyxy τττ

(3) - 0=∂

∂+∂

∂+

∂∂

zyxzzyzxz τττ


Shear stresses

� From equation (1) – shear stress τxz can be computed

Use stress deflection/curvature relations

wEzwwEz 222 ∂ ∂∂ ∂∂∂ τ

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( )

( )wx

Ez

z

yx

w

yx

w

x

wEz

z

zyx

wEz

yy

w

x

wEz

x

xz

xz

xz

22

2

3

2

3

3

3

2

2

2

2

2

2

2

1

11

011

∇∂∂

−=

∂∂

∂∂∂−+

∂∂∂+

∂∂

−=

∂∂

=∂

∂+

∂∂∂

+−

∂∂+

∂∂+

∂∂

−−

∂∂

υτ

υυυ

τ

τυ

υυ

Integrate this across thickness to get shear stress


Shear stresses

� Integrate from mid plane to top surface of plate

( ) ( )∂∇∂

−=

∂∇∂

−= ∫∫∫

11

22

2

2 2

2

0

dzzx

wEdz

x

wEzd

hh

xz υυτ

z

z z = h/2

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( )

( )( )

( )( )

−

∂∇∂

−==>

−

∂∇∂

−=−=>

∂∇∂

−=−

∂−∂ − ∫∫∫

412

412

21

11

22

2

2

222

2

222

2

20

2

hz

x

wE

zh

x

wE

z

x

wE

xx

xz

xz

h

z

xz

zxz

υτ

υτ

υτ

υυτ

Parabolic variation


Moments

� Moment wrt ‘y’ axis

∂∂+

∂∂

−−=

===

2

2

2

2

21 x

w

x

wEz

dzzzdFdM

dzdF

xx

xxxxy

xxxx

υυ

σ

σσ

Neutral planez

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( )

∂∂+

∂∂

−−=

∂∂+

∂∂

−−=

∂∂+

∂∂

−−==>

∂∂−

∫+

−

2

2

2

2

2

3

2

2

22

2

2

2

2

2

2

2

2

2

2

112

1

1

1

y

w

x

wEhM

dzzy

w

x

wEM

y

w

x

wEzdM

xx

y

h

h

y

y

υυ

υυ

υυ

υ

x

yσxx

z

dz

dy

dx


Moments

� Moment wrt ‘x’ axis

∂∂+

∂∂

−−=

2

2

2

2

2

3

)1(12 y

w

x

wEhM x υ

υNeutral plane

z

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x

y

z

τxyz

dz

dy

dx

Twisting moment due to shear stress τxy

( )yx

wEhM xy ∂∂

∂−−

−=2

2

3

1)1(12

υυ

)1(12 2

3

υ−= Eh

DCompare with section modulus of beam


Shear forces

� Vertical equilibrium of plate

z

q

dxx

QQQ

dxQdyQqdxdydxQdyQ

xxx

yxyx

∂∂

∂+=

=−−++

*

** 0

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x

y

q

dy

dx

Q*x

Q*y

Qy

Qx

dyy

QQQ y

yy ∂∂

+=*

Substitute these and simplify

qy

Q

x

Q yx −=∂

∂+

∂∂


Shear forces

� Shear forces across thickness can be computed by integrating shear stress across thickness

dzQh

h

yzy = ∫+

−

2

2

τ per unit width

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Ramadas Chennamsetti27y

wD

y

wEhQ

dzh

zy

wEQ

hz

y

wE

y

h

h

y

yz

h

∂∇∂−=

∂∇∂

−−=

−

∂∇∂

−=

−

∂∇∂

−=

∫+

−

−

)()(

)1(12

4

)(

)1(2

4

)(

)1(2

22

2

3

2

2

22

2

2

22

2

2

2

υ

υ

υτ


Governing equation

� Similar expression for Qx

x

wDQ x ∂

∇∂−= )( 2

Substitute Qx and Qy in the following expression – vertical equilibrium

qQQ yx −=

∂+∂

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( ) ( )

( ) ( )

( )D

qw

yx

D

qw

yw

x

qwy

Dy

wx

Dx

qy

Q

x

Q yx

=∇

∂∂+

∂∂=>

=∇∂∂+∇

∂∂=>

−=

∇

∂∂−

∂∂+

∇∂∂−

∂∂

−=∂

∂+

∂∂

22

2

2

2

22

22

2

2

22


Governing equation

� Governing equation

( )D

qw

D

qw

yx

=∇∇=>

=∇

∂∂+

∂∂

22

22

2

2

2

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( )

D

qD

w

=∇=>

=∇∇=>

4

D

q

y

w

yx

w

x

w =∂∂+

∂∂∂+

∂∂

4

4

22

4

4

4

2

Bi-harmonic equationCompare with beam equation


Boundary conditions

� Well posed problem – Governing equations and boundary conditions

� Three basic boundary conditions � Simply supported

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� Simply supported

� Clamped and

� Free edge

� Vertical deflection and their derivatives


Simply supported

� Simply supported – for eg beam => vertical deflection = 0 and moment = 0

( ) 0, =yxw

Simply supportedx = constFor edge, x = const

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02

2

2

2

=

∂∂+

∂∂−=

y

w

x

wDM y υ

Simply supportedx

y

y = const

w = 0 implies second derivative in the direction tangent to this line is zero

02

2

=∂∂

x

w


Simply supported

� Simply supported condition along edge y = const

∂+∂−=

=22

0

wwDM

w

υ

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∂∂+

∂∂−=

22 y

w

x

wDM x υ

w = 0 implies second derivative in the direction tangent to this line is zero

02

2

=∂∂

y

w


Clamped

� Deflection and slope in normal directions vanish

0

0

=∂=

w

wx = constant

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0=∂x

0

0

=∂∂

=

y

w

wy = constant


Free edges

� Free edge – Free from any external loads –Natural boundary conditions

� Bending moment and Shear force vanish

� Bending moment

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� Bending moment

const y at 0

const at x 0

2

2

2

2

2

2

2

2

==

∂∂+

∂∂−=

==

∂∂+

∂∂−=

y

w

x

wDM

y

w

x

wDM

x

y

υ

υ


Free edges

� Moment Mxy and shear forces

Mxy

M

dy

The net force acting on the face

MM

MQ xy +∂

−−='

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dy

Mxy

dxMxy

Mxy

dy

dyy

MM xy

xy ∂∂

+

dyy

MM xy

xy ∂∂

+

y

MQ

My

MMQ

xyx

xyxy

xyx

∂∂

−==>

+∂

∂−−=

'

'

Total shear force,

Vx = Qx + Qx’


Free edges

� Total shear force,

( ) ( )

∂∂

∂∂−−∇

∂∂−==>

+=

x

w

xDw

xDV

QQV

x

xxx

2

22

'

1 υ

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( ) ( )

( )

( )

∂∂∂−+

∂∂−=

∂∂∂−+

∂∂−==>

∂∂

−−∇∂

−==>

yx

w

y

wDV

yx

w

x

wDV

xxDw

xDV

y

x

x

2

3

3

3

2

3

3

3

2

2

2

1

υ

υ

υ


Free edges

� The forces, Vx and Vy => reduced, or Kirchoff’s or effective shear forces

� In case of a free edge,

33 ∂∂

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( )

( ) const y at 02

const at x 02

2

3

3

3

2

3

3

3

==

∂∂∂−+

∂∂−=

==

∂∂∂−+

∂∂−=

yx

w

y

wDV

yx

w

x

wDV

y

x

υ

υ


Bending of plates

� Governing equation of plate rectangular plate bending

( )D

qw =∇∇ 22

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� Solution to this equation – product of two functions – Assume

w = vertical deflection = w(x, y)

external loading, q = q(x, y)

( ) ( ) ( )yGxFyxww == ,


Bending of plates

� Choice of functions – algebraic, trigonometric, hyperbolic etc or combination of these function

� Selection of a function – depends on

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� Selection of a function – depends on boundary conditions

� Simply supported edges – trigonometric function – Navier solution

� Deflection of a plate can be written as sum of infinite trigonometric functions


Bending of plates

� Edges, x = 0 and x = a simply supported

Vertical deflection vanish

w(x=0, y)=0, w(x=a, y)=0

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x

y

x = 0 x = aSSSS

Possible form of solution

( ) ∑∞

=m

m a

xmFxF

sin

π

F1, F2,…….F∞ are coefficients

Symmetric loading wrt x = a/2 Maximum deflection at x = a/2


Bending of plates

� Other edges simply supported

y = b

SSVertical deflection vanish

w(x, y=0)=0, w(x, y=b)=0

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x

y

x = 0, y = 0

x = aSSSS

SS

Possible form of solution

( ) ∑∞

=n

n b

ynGyG

sin

π

G1, G2,…….G∞ are coefficients

Selection of functions based on BCs


Bending of plates

� Final solution,

( )

( )

b

yn

a

xmGFyxw

m nnm

sin

sin,

ππ

=

∑∑

∑∑∞ ∞

∞ ∞

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( )

b

n

a

m

yxwyxw

nm

nm n

mmn

,

sin sin,

πβπα

βα

==

==> ∑∑∞ ∞

Coefficients, Fm, Gn and wmn computed using Fourier Series


Coefficients

� Any periodic function can be expanded into a sine or cosine function using Fourier expansionFunction of one variable

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( )

( )a

xm

a

xmf

a

xmxf

a

xmfxf

m

mm

sin

sin

sin

sin

'' πππ

π

==>

=∑∞

Integrate the above from limits 0 to a


Coefficients

� Integration

( )

( ) ( )a

ma

m

a

m

a

dxmmπx

mmπxf

dxxmxmf

dxa

xm

a

xmfdx

a

xmxf

'''

0

'

0

'

coscos sin sin2

sin sin sin

+−−==>

=

∫∫

∫∫

ππ

πππ

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( ) ( )

( )( )

( )( )

a

m

mm

mmπ

mmπx

mmπ

mmπx

f

dxmmπx

mmπxf

dxa

xm

a

xmf

0

'

'

'

'

0

''

0

a

asin

a

asin

2

a

cosa

cos2

sin sin22

+

+−

−

−=>

+−−==> ∫∫ππ

Lower limit vanishes, evaluate upper limit


Coefficients

� Upper limit

( )( )

( )mma

mma

xf

dxa

xmxf m

a

∫−

−=

'

'

0

' sin

2

sin π

ππ

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( )

( )

( ) dxa

xmxf

af

mm

mmaf

dxa

xmxf

mma

a

m

ma

ax

∫

∫

=

≠=

===>

−=

0

'

'

0

'

sin

2

for 0

for 2

sin

π

π


Coefficients

� Function of variable ‘y’

( )b

yngyg

nn

sin

π∑=

∞

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( )

( ) dyb

ynyg

bg

b

yn

b

yng

b

ynyg

b

n

n

sin

2

sin

sin

sin

0

''

π

πππ

∫=

==>


Coefficients

� Function of two variables

( )

( ) dxa

xm

a

xm

b

ynwdx

a

xmyxw

b

yn

a

xmwyxw

ax

xm nmn

ax

x

m nmn

∫∑∑∫

∑∑=

=

∞ ∞=

=

∞ ∞

=

=>

=

0

''

0

sin

sin

sin

sin,

sin

sin,

ππππ

ππ

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( )

( )

( ) dxdyb

yn

a

xmyxw

abw

dyb

yn

b

ynw

adxdy

b

yn

a

xmyxw

b

ynw

adx

a

xmyxw

aaba

by

y

ax

x

mn

by

ym nmn

by

y

ax

x

m nmn

ax

x

xm nx

∫ ∫

∫∑∑∫ ∫

∑∑∫

=

=

=

=

=

=

∞ ∞=

=

=

=

∞ ∞=

=

==

==>

=

=>

=

0 0

'

00 0

''

'

0

00

sin

sin,

4

sin

sin

2

sin

sin,

sin

2

sin,

ππ

ππππ

ππ


Simply supported plate

� Assume loading over plate

( )b

yn

a

xmqyxqq

m nmn

sin

sin,

ππ∑∑

∞ ∞

==

Solution

rd_

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Solution

( )b

yn

a

xmwyxww

m nmn

sin

sin,

ππ∑∑

∞ ∞

==

Governing equation

D

q

y

w

yx

w

x

w =∂∂+

∂∂∂+

∂∂

4

4

22

4

4

4

2



� Differentiating vertical displacement

=∂∂

∑∑∞ ∞

b

yn

a

xmw

a

m

x

w

m nmn

sin

sin

4

4

4 πππ

rd_

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=∂∂

∂

=∂∂

∑∑

∑∑∞ ∞

∞ ∞

b

yn

a

xmw

b

n

a

m

yx

w

b

yn

a

xmw

b

n

y

w

m nmn

m nmn

sin

sin22

sin

sin

22

22

4

4

4

4

ππππ

πππ



� Plug in governing equation

=

+

+

q

D

qw

b

n

b

n

a

m

a

m mnmn2

4224 ππππ

rd_

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( )

+

==

+

==>

∑∑∞ ∞

b

yn

a

xm

b

n

a

m

q

Dyxww

b

n

a

mD

qw

m n

mn

mnmn

sin

sin

1, 2

2

2

2

24

2

2

2

2

24

πππ

π


UDL

� Uniformly distributed load qo

Computation of coefficients

( )

= ∑ ∑

∞ ∞ ynxmqyxq mn

sin

sin,

ππ

rd_

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( ) dxdyb

yn

a

xmyxq

abq

a b

mn

sin

sin,

4

0 0

ππ∫ ∫=

( )

= ∑ ∑ baqyxq

m nmn sinsin,

q(x, y) = qo


UDL

� Coefficients qmn

0 0

sin

sin

4 ππdxdy

b

ym

a

xmq

abq

a b

omn = ∫ ∫

rd_

mec

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22

00

1644

sin

sin

4

ππ

ππ

mn

q

mn

ab

ab

qq

dyb

ymdx

a

xm

ab

qq

oomn

bao

mn

=

==>

= ∫∫


UDL

� Vertical deflection

( ) ∑∑∞ ∞

+

==m n

mn

nm

b

yn

a

xmq

Dyxww 2224

sin

sin

1,

ππ

π

rd_

mec

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( ) ∑∑∞ ∞

+

==>

+

m n

o

b

n

a

m

b

yn

a

xm

D

qyxw

b

n

a

m

2226

sin

sin

16,

ππ

π


Patch load

� A patch load applied over an area u x v

( )

=∑∑∞ ∞

b

yn

a

xmqyxq mn

sin

sin,

ππvqo

Centroid at (xo, yo)

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==>

∑∑

∑∑∞ ∞

b

yn

a

xmqq

ba

m nmno

m nmn

sin

sin

ππ u

vqo

dxdyb

ym

a

xm

ab

qq

ux

ux

vy

vy

omn

o

o

o

o

sin

sin

4 2

2

2

2

ππ∫ ∫+

−

+

−

=

Patch load


Patch load

� Evaluating integrals

= ∫ ∫+

−

+

−

dxdyb

ym

a

xm

ab

qq

ux

ux

vy

vy

omn

o

o

o

o

sin

sin

4 2 2 ππ

rd_

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==>

=∫+

−

− −

b

vm

b

yn

a

um

a

xmq

mnq

a

um

a

xm

m

adx

a

xm

ooomn

o

ux

ux

x y

o

o

o o

2

sin

sin

2

sin

sin

16

2

sin

sin

2 sin

2

2

2

2 2

πππππ

πππ

π


Patch load

� Deflection

( )

+

= ∑∑∞ ∞

nmmn

b

yn

a

xmS

D

qyxw

m n

mno

sin

sin

16,

226

ππ

π

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=

+

b

vm

b

yn

a

um

a

xmS

b

n

a

mmn

D

oomn

m n

2

sin

sin

2

sin

sin

where,

ππππ

π


Point load

� Point load

uv

Pqo = P

(xo, yo)

Assume the point load acts over an infinitesimal area u x v

Corresponding UDL Pq =

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From the earlier analysis

b

vn

a

um

b

yn

a

xm

uv

P

mnq

b

vn

a

um

b

yn

a

xmq

mnq

oomn

ooomn

2

sin

2

sin

sin

sin

162

sin

2

sin

sin

sin

16

2

2

πππππ

πππππ

==>

=

uvqo =


Point load

� Simplifying

vmb

vn

uma

um

b

yn

a

xm

ab

Pq oo

mn 2

sin

2

sin

sin

sin4

π

π

π

πππ

==>

rd_

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b

yn

a

xm

ab

Pq

b

vm

a

umbaabq

oomn

mn

sin

sin

42

2

sinsin

ππ

ππ

==>

==>


Point load

� Deflection due to point load

( ) b

ym

a

xmS

abD

Pyxw

mn

sin

sin4

, 2

'

4

ππ

π = ∑∑

∞ ∞

rd_

mec

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( )

b

ym

a

xmS

b

n

a

mabDyxw

oomn

m m

sin

sin

,

'

2224

ππ

π

=

+

= ∑∑


Bending & in-plane loading

� Plates are subjected to in-plane loading also – in addition to lateral / transverse loads

� In-plane loading – tensile or compressive

� Large in-plane compressive loads – Buckling takes place

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place

� Buckling – non-linear phenomenon –disproportionate increase of displacement with load

� Critical load – ability to resist axial load ceases –change in deformation shape


Bending & in-plane loading

� Thin walled members – cross-sections like ‘I’, ‘L’, ‘H’, ‘C’ etc – undergo buckling –thin plates of small widths

� Combined loading of a rectangular plate –

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� Combined loading of a rectangular plate –loads � In-plane forces: Nx, Ny, Nxy and Nyx

� Transverse forces / moments: Mx, My, Mxy, Myx, Qx and Qy

� Small deformation and large deformation


In-plane forces

� Infinitesimal element => dA = dx dy

Nx*

dx

Nxy*

θx*

Qx*

θx*

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Nx

dx

x

z θx

Qx

θx

In-plane loading Transverse/out-of-plane loading

For small angles => Sinθ ≈ θ and Cosθ ≈ 1


In-plane forces

� Force equilibrium in x-direction

sincos

cos cos cos

**

*

∂+−

∂

+

+−

∂∂++−

dydxQ

QdxdyN

N

dxNdydxx

NNdyN

xxy

yxyxx

xxx

θθ

θθθ

rd_

mec

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0

0sinsinsin

sincos

*

**

=∂

∂+

∂∂=>

=+

∂∂

+−+

∂∂+−

∂∂

+

y

N

x

N

dxQdxdyy

QQdyQ

dydxx

QQdxdy

y

NN

xyx

yyyy

yxx

xx

xyxy

xy

θθθ

θθ

If there are no in-plane forces – equation vanishes


In-plane forces

� Force equilibrium in y-direction

� Angle is not equal to zero, but, small

0=∂

∂+

∂∂

y

N

x

N yxy

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� Angle is not equal to zero, but, small

y

w

x

wyx ∂

∂=∂∂=

≈≈

θθ

θθθ

,

1cos and sin


In-plane forces

� Force equilibrium in x-direction

2

∂+∂

∂+−

∂

+

+−

∂∂++−

dxww

dydxQ

QdxdyN

N

dxNdydxx

NNdyN

xxy

xyx

xx

rd_

mec

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0

sorder termhigher Neglect

02

2

2

=∂

∂+

∂∂=>

=∂∂+

∂∂+

∂∂

∂∂

+−∂∂+

∂∂+

∂∂

∂∂+−

∂∂

+

y

N

x

N

y

wdxQdy

y

w

y

wdxdy

y

QQ

x

wdyQ

dxx

w

x

wdydx

x

QQdxdy

y

NN

xyx

yy

yx

xx

xyxy

No change in in-plane equilibrium equations


In-plane forces

� Similar expression for force equilibrium in y-direction

� Force equilibrium in z-direction� Contribution from in-plane normal forces, Nx

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� Contribution from in-plane normal forces, Nxand Ny and shear force, Nxy

� Contribution from shear force, Qx and Qy

� Contribution from externally applied load, q


Z-direction

sinsin

sinsin

sinsin

∂∂+∂

∂

++∂−

∂∂

∂∂+

∂∂

∂∂

++∂∂−

∂∂

∂∂+

∂∂

∂∂++

∂∂−

dyww

dxdyN

Nw

dxN

dyy

w

yy

wdxdy

y

NN

y

wdxN

dxx

w

xx

wdydx

x

NN

x

wdyN

xy

yyy

xxx

rd_

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0coscos

coscos

sinsin

sinsin

=+

∂∂

∂∂+

∂∂

∂∂

++∂∂−

∂∂

∂∂+

∂∂

∂∂++

∂∂−

∂∂

∂∂+

∂∂

∂∂

++∂∂−

∂∂

∂∂+

∂∂

∂∂

++∂∂−

qdxdydyy

w

yy

wdy

y

QQ

y

wdxQ

dxx

w

xx

wdydx

x

QQ

x

wdyQ

dxy

w

xy

wdydx

x

NN

y

wdyN

dyx

w

yx

wdxdy

y

NN

x

wdxN

yyy

xxx

xyxyxy

xyxyxy


Z-direction

� If no in-plane forces acting

qy

Q

x

Q yx −=∂

∂+

∂∂

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� Presence of in-plane forces

022

22

2

2

=+∂

∂+

∂∂+

∂∂+

∂∂∂+

∂∂

qy

Q

x

Q

y

wN

yx

wN

x

wN yx

yxyx


Z-direction

� Substituting Qx and Qy

( ) ( )wy

DQwx

DQ yx22 , ∇

∂∂−=∇

∂∂−=

∂+∂+∂ wN

wN

wN

222

2

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( ) ( )

+

∂∂+

∂∂∂+

∂∂=∇=>

=+

∇

∂∂−

∂∂+

∇∂∂−

∂∂+

∂+

∂∂+

∂

qy

wN

yx

wN

x

wN

Dw

qwy

Dy

wx

Dx

yN

yxN

xN

yxyx

yxyx

2

22

2

24

22

22

21

0

2


Plate buckling

� Buckling of thin plate

∂∂+

∂∂∂+

∂∂+=∇

2

22

2

24 2

1

y

wN

xy

wN

x

wNq

Dw yxyx

rd_

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Assume, q = 0, in-plane load, Nx = N1, rest zero

2

2

4

4

22

4

4

4

2x

w

D

N

y

w

yx

w

x

w x

∂∂=

∂∂+

∂∂∂+

∂∂

Assume all four edges are simply supported –Navier solution to plate bending


Plate buckling

� Plate deflection

( )

nm

yxwyxwwm n

nmmn

πβπα

βα

==

== ∑∑∞ ∞

,

sin sin,

Substitute in

rd_

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b

n

a

mnm

πβπα == , Substitute in governing equation

( ) 21222

214224 02

mnm

mnnmm

D

ND

N

αβα

αββαα

=+=>

=−++ Characteristic equation


Plate buckling

� From characteristic equation

( )2222

1

21222

nmmN

D

Nmnm

+

=

=>

=+

πππ

αβα

rd_

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ratioAspect - /

22

2

2

1

222

2

22

1

1

bac

nm

c

c

m

b

DN

b

n

a

m

m

aDN

baaD

=

+==>

+

==>

+

=

=>

π

π


Plate buckling

� Critical load – smallest value

� Increase in N1 with n2 – Minimum value of n is equal to one – buckled shape in y-direction – single half sine wave

rd_

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direction – single half sine wave

2

2

2

1

+=m

c

c

m

b

DN

π

N1 is a function of variable ‘m’ – for minimum value of ‘m’, differentiate N1 wrt ‘m’


Plate buckling

� Differentiate

22

21

1

012

c

m

c

cm

c

c

m

b

D

dm

dN π

=

−

+=

rd_

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2

1

2

4

01

b

DN

b

acm

m

c

c

cr

π=∴

===>

=

−=>

Whole number


Plate buckling

� Number of half sine waves can’t be a whole number – it should be an integer

� Equation for critical loadfor n = 1

2π 2

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2

2

1 b

DKN

π=2

+=m

c

c

mK

Plotting ‘K’ vs aspect ratio = c = a/b for various integer values of ‘m’


Plate buckling

� K vs aspect ratio

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1.414 2.449


Plate buckling

� From figure, value of ‘K’ is same as intersection points of ‘m’ and ‘m+1’N1 critical load at m – when load is increased, buckled form changes from ‘m’ to ‘m+1’

Curves for m = 1 and

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At transition from ‘m’ to ‘m+1’

( )( )1

1

1

1

2

22

+=

+==>

+++=

+

mmc

mmc

m

c

c

m

m

c

c

m

Curves for m = 1 and m = 2meet at c = √2

Aspect ratio less than √2 => m = 1

Aspect ratio from √2to √6, m = 2

C>4 => K ≈ 4


Plate buckling

� Estimation of buckling loadt = 1 cm, a = 2.3 m, b = 1 m, E = 200 GPa, υ = 0.30

Flexural modulus

( )101102002293

=×××==−Et

rd_

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( )( )

( )

0786.42

3.2

3.2

2

23.21

3.2

96.173.0112

10110200

112

22

2

29

2

3

=

+=

+==>

==>===

=−

×××=−

=−

m

c

c

mK

mb

ac

kNmEt

Dυ


Plate buckling

� Minimum critical load

MNbNN

mMNb

D

b

DKN

cr

cr

72.73

/72.73086.4

1

2

2

2

2

1

=×=

=×== ππ

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MNbNN cr 72.731 =×=

In the above expression, for a given width and elastic properties of plate, critical load depends on ‘K’. In turn ‘K’ depends on ‘m’ for a given aspect ratio, c

2890.4 ,3

0786.4 ,2

4790.7 ,1 ,3.2

=======

Km

Km

KmcMinimum value of ‘K’ is considered for estimation of Ncr


Plate and column buckling

� Uniaxial load for plate buckling

( )322

1

2

2

1

ππσ

π

===

=

EtK

DK

Nb

DKN

crp

crp

rd_

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( )

( ) ( )

( ) ( )212

2

11

2

2

21

222

11

112 ,

112

112

υπσ

πυ

σ

υππσ

−===>

−==>

−===

KC

tb

EC

tb

EK

Et

tbK

tb

DK

t

crp

crp

crpcrp



� Buckling of a column

( )2

2

22

2

2

2

2

EAC

EAkCP

l

EICP

cr

cr

ππ

π

===>

=

rd_

mec

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( )

( )

( )22

1

2

2

2

2222

tb

EC

kl

EC

A

P

kl

Cl

CP

crp

crcrc

cr

πσ

πσ

=

===>

===>

Column

Plate



� Critical stress for plate depends on thinness ratio = t/b – not on the length � depends on width

� More thinner plate – lesser bucking load

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� More thinner plate – lesser bucking load

� Critical stress in column depends on slenderness ratio

� Longer columns – lower critical load


Strain energy

� In thin plate theory, out-of-plane shear stresses vanish => τxz, τyz and τzz

� Stress components contributing to strain energy => σxx, σyy and τxy

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energy => σxx, σyy and τxy

� Strain energy,

( ) VdUV

xyxyyyyyxxxx 2

1∫ ++= γτεσεσ

Linear elastic material


Strain energy

� Strain energy,

{ }{ }∫=V

Txyyyxxxyyyxx VdU γεετσσ

2

1

w ∂2

rd_

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{ } [ ]{ }εσ

υυ

υ

υγεε

υυ

υ

υτσσ

C

yx

wy

wx

w

EzEz

xy

yy

xx

xy

yy

xx

==>

∂∂∂∂∂∂∂

+−−=

+−−=

2

2

2

2

2

22

22

100

01

01

1

2

100

01

01

1


Strain energy

� Strain energy,

{ } { } { } [ ]{ }

www

VdCVdUV

T

V

T

222

2

1

2

1∫∫

∂

∂∂

== εεεσ

rd_

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( )

( )

Vdz

yx

w

x

w

y

w

x

w

x

w

y

w

x

w

EU

V

2

22

2

2

2

2

2

2

2

2

2

2

2

2

2

12

12 ∫

∂∂∂−+

∂∂

∂∂+

∂∂+

∂∂

∂∂+

∂∂

−==>

υ

υ

υ

υ


Strain energy

� Infinitesimal volume, dV = dxdydz

� Carry out integration over thickness => dz

322 =∫

+h

h

rd_

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( )2

3

2

2

2

112

12

υ−=

=∫−

EhD

hdzz

h


Strain energy

� Simplify

dxdyx

w

x

w

DU ∫

∂∂+

∂∂

=

2

2

2

2

2

rd_

mec

h@

yah

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.co

.in


( )dxdy

yx

w

x

w

x

wU

A∫

∂∂∂−

∂∂

∂∂−−

=22

2

2

2

2

122

υ

Strain energy – Finite Element Method – Total potential approach


rd_

mec

h@

yah

oo

.co

.in


theory of plates - imechanicaimechanica.org/files/theory of plates.pdf · theory of plates...

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