*Simply supported plates under Sinusoidal loading (Topic 5)

* Simply supported plates under Sinusoidal loading ( Topic 5)

Advanced Local Structural Design & Analysis of Marine Structures

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[Part I] Plastic Design of Structures

– Plastic theory of bending (Topic 1) – Ultimate loads on beams (Topic 2)

– Collapse of frames and grillage structures (Topic 3)

[Part II] Elastic Plate Theory under Pressure

– Basic (Topic 4)

– Simply supported plates under Sinusoidal Loading (Topic 5) – Long clamped plates (Topic 6)

– Short clamped plates (Topic 7)

– Low aspect ratio plates, strength & permanent set (Topic 7A)

[Part III] Buckling of Stiffened Panels

– Failure modes (Topic 8)

– Tripping (Topic 9) + Post-buckling strength of plate (Topic 9A) – Post-buckling behaviour (Topic 10)

[Theory of Plates and Grillages]

Adv. Local Structural Design & Analysis of Marine Structures (Overview)

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The aim of this lecture is:

• To equip you with the knowledge and understanding of analytic procedures for solving the plate governing equilibrium equation.

Objectives

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Learning Outcome

At the end of this lecture, you should be able to:

▪ Solve the plate governing equilibrium equation for simply supported plate under sinusoidal lateral load and distributed load.

▪ Calculate the deflections, bending moments, and bending stresses in a plate subjected to lateral loads.

▪ Design plates elastically to withstand uniform pressure.

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Introduction

• The linear plate theory developed in the previous lecture leads to the partial differential equation governing the equilibrium.

• The solution of this differential equation depends on loading pattern and edge conditions. Now you should think how to solve this equilibrium equation.

D p y

w y

x

w x

w =

 + 



 + 





4 4 2

2 4 4

4

2

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* How to Plot Sin Function?

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(Basic) How to Plot Sin or Cos Functions?

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* Simply Supported Plates under Sinusoidal Loading

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Simply Supported Plates: Sinusoidal Loading (1/5)

• The basic differential equation governing the small elastic deflection of a flat plate is

• Only in certain simple cases can an exact solution of this equation be found.

• One such case is when the pressure on the plate is distributed sinusoidally:

4 4 4

4

4 2 2 2 4

w w w p

D w p or

x x y y D

  

 = + + =

   

.

b y a

p x

p

_m

 

sin

= sin

in which p

_m

is the maximum pressure and occurs at (a/2, b/2) OST

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• For simply supported edges w = 0 and M = 0

2 0

2 =



 x

w ₂ 0

2 =



 y

w

b y a

w x

w _m  

sin

= sin

and

• One possible solution for deflection w satisfying these edge conditions is

• Substituting the expressions into the plate equilibrium equation for w and p yields

that is,

D p b

b a

w_m a  = ^m



 



 ₄ + _{2 2} + ₄

4 1 2 1

 ²

2 2

4 1 1



 



 +

=

b D a

w_m p^m



(i)

Simply Supported Plates: Sinusoidal Loading (2/5)

4 4 4

4

4 2 2 2 4

w w w p

D w p or

x x y y D

  

 = + + =

   

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• From bending moment-curvature relationships, we have

b y a

w x b

D a y

w x

D w

M_x   1  _m sin  sin

2 2

2 2

2 2

2



 



 +

 =



 





 + 



− 

=

b y a

w x a

D b x

w y

D w

M _y   1  _m sin sin 

2 2

2 2

2 2

2



 



 +

 =



 





 + 



− 

=

Simply Supported Plates: Sinusoidal Loading (3/5)

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( )

_x ^w_m

b D a

M 



 



 +

= ² _{2 2}

max

1 



( )

_y ^w_m

a D b

M 



 



 +

= ² _{2 2}

max

1 



• The maximum bending moments are

at (a/2, b/2) (ii)

Simply Supported Plates: Sinusoidal Loading (4/5)

b y a

w x b

D a y

w x

D w

M _x   1  _m sin  sin 

2 2

2 2

2 2

2



 



 +

 =



 





 + 



− 

=

b y a

w x a

D b x

w y

D w

M _y   1  _m sin  sin 

2 2

2 2

2 2

2



 



 +

 =



 





 + 



− 

=

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( )

_{x m ax} ⁶₂

( )

^M _{x m ax}

 t



=

( )

_{y m ax} ⁶₂

( )

^M _{y m ax}

 t



=

• The maximum stresses are

• Using equations (i) and (ii),

the maximum bending stress can be found for any value of p

_m

, plate dimensions and properties.

Simply Supported Plates: Sinusoidal Loading (5/5)

2 2 2

4 1 1



 



 +

=

b D a

w_m p^m



(i)

^{( )Mx D a b }^wm

 



 +

= ² _{2 2}

max

1 



( )

y wm

a D b

M 



 



 +

= ² _{2 2}

max

1 



at (a/2, b/2) (ii)

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* Simply Supported Plates under Distributed Load

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• Assume that the distributed load can be represented by a Fourier series:

b y n a

x a m

p

m n

mn



 sin sin

1 1



^

=



=

=

• It is known that for simply supported edges the solution for deflection w can be expressed in the form

b y n a

x A m

w

m n

mn



 sin sin

1 1



^

=



=

=

where A_mnis unknown coefficient (=Amplitude of plate deflection),

mis number of half waves in x-direction & n is number of half waves in y-direction.

Simply Supported Plates: Distributed Load (1/3)

4 4 4

4

4 2 2 2 4

w w w p

D w p or

x x y y D

  

 = + + =

   

• Substituting the expressions into the plate equilibrium equationfor w and p yields

0 sin

sin 2

1 1

4 2

2 4

 =









 −











 



 

 +



 



 



 

 + 



 





^ 

=



= b

y m a

x m D

a b

n b

n a

m a

A m

m n

mn mn













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• The previous equation can be rewritten as

0

2 2 2 2

2

4  − =



 



 +

D a b

n a

A_mn m ^mn

• Thus,

2

2 2 2

2

4 

 



 +

=

b n a

D m

A_mn a^mn



(iii)

Simply Supported Plates: Distributed Load (2/3)

• This equation is valid for all values of m and n and so

0 2

4 2

2 4

=

 −









 



 

 +



 



 



 

 + 



 





D a b

n b

n a

m a

A_mn m    ^mn

0 sin

sin 2

1 1

4 2

2 4

 =









 −











 



 

 +



 



 



 

 + 



 





^ 

=



= b

y m a

x m D

a b

n b

n a

m a

A m

m n

mn mn













• From the bending moment-curvature relations, we have



^

=



= 

 



 +

=

1 1

2 2 2

2

2 sin sin

m n

mn

x b

y n a

x A m

b n a

D m

M    



^

=



= 



 



 +

=

1 1

2 2 2

2

2 sin sin

m n

mn

y b

y n a

x A m

a m b

D n

M    

(iv)

• Using equations (iii) and (iv), the bending stresses can be found for any form of p, plate dimensions and properties.

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• The coefficient a_mn in the Fourier series for pressure p can be found by

multiplying both sides of the equation by ^{sin m}a ^{x sin n}b ^x

 

 

   

   

   

and integrating with respect to xfrom 0 to a and with respect to yfrom 0 to b.

( )   

 

^

=



=



= 





1 1 0 0

0 0 , sin sin sin sin sin sin

m n

a b mn

a b

b dxdy y n b

y n a

x m a

x a m

b dxdy y n a

x y m

x

p

( )

 

  

   

 

^

=



=



= 





1 1 0 0

0 0 , sin sin sin sin sin sin

m n

a b mn a b

b dxdy y n b

y n a

x m a

x a m

b dxdy y n a

x y m

x

p      

Simply Supported Plates: Distributed Load (3/3)

b y n a

x a m

p

m n

mn



 sin sin

1 1



^

=



=

=

• Since



=



0 ^{sin sin}  ₀^/2 dx a

a x m a

x m

a  

m m

m m

 

= 



=



0 ^{sin sin}  ₀^/2 dy b

b y n b

y n

b  

n n

n n

 

= 

for for

( )

_mn

a b

aba b dxdy

y n a

x y m

x

p , sin sin 4

0 0 =

 

^{ }

  ( )

= ^{a b}

mn dxdy

b y n a

x y m

x ab p

a 4 0 0 , sin  sin  Then

or

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* Simply Supported Plates under Uniform Pressure

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• When p is constant over the plate surface, say p_o, we have

Simply Supported Plates: Uniform Pressure (1/2)

mand nare odd integers.

mor nis even integer.



= 

= ⁴

 

0 0 ^{sin sin 16 /}₀ ²

mn dxdy p

b y n a

x m ab

a_mn p^{o a b}   ^o 

 for



=

= ⁴

 

0 0 ^{sin sin 16 /}₀ ²

mn dxdy p

b y n a

x m ab

a_mn p^{o a b}   ^o 



= 

= ⁴

 

0 0 ^{sin sin 16 /}₀ ²

mn dxdy p

b y n a

x m ab

a_mn p^{o a b}   ^o 

  ( )

= ^{a b}

mn dxdy

b y n a

x y m

x ab p

a 4 0 0 , sin  sin 

• Hence, based on this value of a_mn, equations (iii) and (iv) for deflection and bending moments can be found and so do the bending stresses.

3 4 1

m ax

Et

b k p

w =

^o

( )

_{m ax 2} ²

2 



 



= 

t p b k ^o

 y

( )

_{m ax 2} ²

2 



 



 

= t

p b k ^o

 x

• The results can conveniently be expressed:

- Maximum bending stress

- Maximum deflection in x & y direction

Page 20/26

Simply Supported Plates: Uniform Pressure (2/2)

• All these maxima occur at the centre of the plate.

• Values of k

₁

, k

₂

and k

₂

 are plotted for  = 0.3

3 4 1

m ax

Et

b k p

w =

^o

( )

_{m ax 2} ²

2 



 



= 

t p b k ^o

 y

( )

_{m ax 2} ²

2 



 



 

= t

p b k ^o

x

- Maximum bending stress

- Maximum deflection in x & y direction

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* Clamped Plates under Uniform Pressure

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• Although the problem of bending stresses and deflections in rectangular plates having clamped edges frequently occurs in design, it is not easy to define a deflection function w which satisfies the plate equilibrium equation and the edge conditions.

• Various approximate solutions have been devised and formulae for maximum stress or deflection have been developed by Inglis, Timoshenko, Grashof, Pounder and Bailey among others.

Clamped Plates: Uniform Pressure (1/2)

• The results can conveniently be expressed:

- Maximum deflection

4

1 3

o m

w k p b

= Et ₂ ²

2 



 



= 

t p b k ^o

m

- Maximum bending stressin y-direction

• w_m occurs at the centre of the plate.

• _m occurs at the centre of the long edge and acts perpendicular to it.

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Clamped Plates: Uniform Pressure (2/2)

• Values of k

₁

and k

₂

according to two reliable theories are plotted on the following page for  = 0.3.

- Maximum deflection

4

1 3

o m

w k p b

= Et

2

2 2 



 



= 

t p b

k ^o

m

- Maximum bending stress in y-direction

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• We have investigated the Elastic Plate Theory .

Learning Outcomes (Review)

• Now we are able to:

- Solve the plate governing equilibrium equation for simple supported plate under sinusoidal lateral load and distributed load.

- Calculate the deflections, bending moments, and bending stresses in a plate subjected to lateral loads.

- Design plates elastically to withstand uniform pressure.

• Details can be referred to topics 5 in the lecture notes.

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[Part I] Plastic Design of Structures

– Plastic theory of bending (Topic 1) – Ultimate loads on beams (Topic 2)

– Collapse of frames and grillage structures (Topic 3)

[Part II] Elastic Plate Theory

– Basic (Topic 4)

– Simply supported plates under Sinusoidal Loading (Topic 5) – Long clamped plates (Topic 6)

– Short Clamped plates (Topic 7)

– Additional (Low aspect ratio plates, strength & permanent set)

[Part III] Buckling of Stiffened Panels

– Failure modes (Topic 8) – Tripping (Topic 9)

– Post-buckling behaviour (Topic 10)

[Theory of Plates and Grillages]

Adv. Marine Structures / Adv. Structural Design & Analysis (Next class)

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Kam Sa Hab Ni Da

Questions?