*Elastic Plate Theory -Basic (Topic 4)

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* Elastic Plate Theory - Basic ( Topic 4 )

Advanced Local Structural Design & Analysis of Marine Structures

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https://sites.google.com/snu.ac.kr/ost

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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 elastic plate theory.

Objectives

http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_II/

http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_I/

Picture from:

http://fgg-web.fgg.uni-lj.si/~/pmoze/esdep/master/wg08/l0100.htm

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

▪ Appreciate the assumptions of elastic plate theory

▪ Be familiar with the bending moment curvature relationship of plate and relationship between twisting moment and twist in a plate.

▪ Establish plate governing equilibrium equation.

▪ Be aware of plate edge conditions.

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Introduction

• Ships are cut, almost entirely, from steel plate. Only a few structural elements are NOT made from plate (e.g. a few castings and some rolled sections).

• An understanding of plate behaviour is therefore Crucial.

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Introduction

• Some components such as deck plate between supports, behave as ideal plates under simple boundary conditions and loads.

• Other components, such as the webs of large frames are subject to more complex loading.

• Still, other components such as deck/stiffener combinations behave somewhat like simple plates, but with special aspects.

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Introduction

• Main hull structures are made up with plating supported by stiffeners.

• For example, deck plates are supported by beams/longitudinals and girders/transverses, bottom plates supported by longitudinals, floors and bottom girders, etc.

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Introduction

• These plates must have sufficient strengths to withstand lateral loads due to water pressures and/or cargoes, and stiffness to resist deformations.

• Now you should think how to relate the bending/twisting moment, curvature and plate flexural rigidity together in order to calculate stresses and strains arisen in the plates.

Weak frame and side impact

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Introduction

• We start by sketching a small portion of plate bending in two directions, but with no in-plane stress.

• In other words, the stress at the mid-plane will be ZERO, and (equivalently) the average stress in the x or y direction will be ZERO.

• There is bending stress and strain in both x and y directions. We assume that σ_z ≃

0.

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Assumptions

It’s usually assumed that in the bending of rectangular plate elements under normal lateral load that:

1. Theboundarysupports(providedbythestiffeners)DoNotDeflectnormaltotheplate.

2. StressandstraininadirectionperpendiculartotheplatemaybeNeglected.

3. Plane sections remain plane,with the middle surface as the neutral surface of the plate in the special case of purebending.

4. DeflectionwisSmall(less thant/2).

5. MaterialremainsElastic.

• The resulting theory based on the above assumptions is known as

“small deflection” or “linear theory”

• The results for a variety of loads are superposable always providing of course that the limit of proportionality of the material isnot exceeded.

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*

Bending Moment-Curvature Relationships

Picture from:

https://physics.stackexchange.com/questions/160497/is-the-moment-curvature-relation-for-an-elastic-beam-general

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Moment-Curvature Relationships

• M_x and M_yare the bending moments per unit length on the edges.

• A plate of thickness t, length a and breadth b is subjected to bending moments on its edges as shown below.

• The subscript indicates the direction of normal of the edge surface on which the bending moment acts.

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Moment-Curvature Relationships

• Consider a small elementdx × dy × t in the plate.

• nn is the neutral plane.

• Due to M_x and M_y, bending stresses s_x and s_y act on lamina abcd of distance z from the neutral plane.

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Moment-Curvature Relationships

• Using assumption 3, we have

[Assumption 3] Plane sections remain plane, with the middle surface as the neutral surface of the plate in the special case of pure bending.

• bending strain in x direction:

e_x = (l₁ – l_o)/l_o = [(R_x + z)q_x – R_xq_x ]/(R_xq_x ) = z / R_x

where R_xandR_yare radii of curvature ofneutral planeinxz andyz plane.

• bending strain in y direction: e_y = z / R_y

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Moment-Curvature Relationships

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• Using assumption 4, we have

2

1 2

x w R_x 

− 

= ₂

1 2

y w R_y 

−  and =

[Assumption 4]DeflectionwisSmall(less than t/2).

http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_I/

Curvature (in x direction) Curvature (in y direction)

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[Source] http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_I/BookSM_Part_I/07_ElasticityApplications/07_Elasticity_Applications_04_Beam_Theory.pdf

Moment-Curvature Relationships

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• Using assumption 5, we have the stress-strain relations for in-plane stresses (i.e. s_z = 0) are:











 





−

= −









y x y

x

E s

s



 e

e

1 1 1

where  is Poisson’s ratio.

[Assumption 5]Material remains Elastic.

• Re-arranging the above equations yields











 





= −









y x y

x E

e e



 s 

s

1 1 1 ²











 



 





− −

 =







2 2

2 /

/ 1

1

1 w y

x Ez w

y x



 s 

s Hence,

(i)

https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson https://en.wikipedia.org/wiki/Poisson%27s_ratio

2 x 2

x

z w

R z x

e = = ^−^ ^ & y ²₂ y

z w

R z y

e = = ^−^ ^

xx y zz

y

e e e

1

zz

xx yy

E s − s + s

 

  =

 

 

 

( )

 

1

zz

yy xx

Es − s( + s )

1

Eszz ( )

1 1 1

1

xx yy

xx yy xx

xx yy yy

v v

E v E v

 s s

s s s

 

 

 

 

 − + 

 

 

− −

    

= −  = −  

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Moment-Curvature Relationships

• The bending moments resulting from the stresses are



−

= ^/²

2 / t

t x

xdy z dydz

M s =



−^/²

2 / t

t y

ydx z dxdz

M s



−

= ^/²

2 / t

t x

x z dz

M s =



−^/²

2 / t

t y

y z dz

M s

and or

and











 



 





− −

 =







2 2

2 /

/ 1

1

1 w y

x Ez w

y x



 s 

s

• Substituting equations (i) into the above equations and integrating give



 





 + 



− 

= ² ₂ ² ₂

y w x

D w

M_x  _



 





 + 



− 

= ² ₂ ² ₂

x w y

D w

M _y 

(

²

)

3

1 12 −

= Et

in which D = plate flexural rigidity Eqn (i)

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Moment-Curvature Relationships

Notes: D = EI

1− 2

 = E E

12

3 / t I =

2 2

dx w EI d

M _x = −

• D is sometimes expressed:

where, is effective elastic modulus for plates

• The moment-curvature relation for a beam is given by in which EI is beam flexural rigidity.

is the second moment of area per unit length



 





 + 



− 

= ² ₂ ² ₂

y w x

D w

M_x  _



 





 + 



− 

= ² ₂ ² ₂

x w y

D w

M_y 

(

²

)

3

1 12 −

= Et D

• Substituting the bending moment-curvature equations into (i) gets

12

3 / t

z M_x

x =

s t³ /12

z M _y

y = s

I M z = 

• These are analogous to s and











 



 





− −

 =







2 2

2 /

/ 1

1

1 w y

x Ez w

y x



 s 

s

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Moment-Curvature Relationships

• Maximum stresses occur on plate surface:

( )

_max

6

₂

t M

_x

x

= 

s

( )

_max

6

₂

t M

_y

y

= 

s

12

3 / t

z M _x

x =

s

t³ /12

z M _y

y = s

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Twisting Moment-Twist of Plates Relationships

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Twisting Moment and Twist of Plates

• At a point in a plate subjected tolateral loadthere will in general actbothbendingandtwistingmoments.

• Bending moments are related to deflection w through curvature. A similar relation is required relating twisting moments to deflection.

Relations between twisting moment and twist in a plane

• M_xy and M_yx are twisting moments per unit length.

• The directions for twisting moments and shear stresses shown above are Positive.

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Twisting Moment and Twist of Plates

y u x

v

xy



+ 



= 



u and v are displacements in x and y direction.

• Shear strain

γ

_xy ^{= angle}âdc ^– ângleâ’d’c’

• Hence,

y x z w

yx

xy

 

− 

=

=  2

²



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Twisting Moment and Twist of Plates

• From stress-strain relations for elastic material:

xy

G 

 =

Thus,

y x zG w

yx

xy  

− 

=

= 2 ²

 (ii)

• The appliedtwisting momentsmustequalthemomentsset up byshear stress:

/ 2 / 2 t

xy xy

M dy t z dydz

= −



− ^and ^{M dx}^yx ⁼



₋^t_t^{/ 2}_{/ 2}^z^ ^yx^dxdz

/ 2 / 2 t

xy xy

M t z dz

= −



− ^{/ 2}/ 2

t

yx yx

M t z dz

=



−

and or

y u x

v

xy



+ 



= 



x y

z w

yx

xy  

− 

=

=  2 ²



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Twisting Moment and Twist of Plates

• Substituting equation (ii) into the foregoing equations and integrating give

y x

w M Gt

M _xy _yx



= 

−

= ³ ²

6

(

⁺^

)

= 2 1

G E

(

²

)

3

1 12 −

= Et

• Putting and D into (iii)

( )

y x D w

M M _xy _yx



− 

=

−

= 1  ²

(iii) y

x zG w

yx

xy  

− 

=

= 2 ²

 (ii) = −



−^/²

2 / t

t xy

xy z dz

M



=



−^/²

2 / t

t yx

yx z dz

M 

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Plate Governing Equilibrium Equation

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Plate Equilibrium Governing Equation

• In beams the lateral load q per unit length is related to the resulting deflection w through

x q

EI w =



4 4

• A similar relation between pressure p on a plate and the resulting deflection w(x,y) can be established.

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Plate Equilibrium Governing Equation

• A small element cut from the plate is subjected to the actions shown below. These actions

(moments, twists, shear forces, etc) vary from point to point, i.e. they are functionsof both x and y.

M_x&M_y bending moments

M_xy&M_yx twisting moments per unit length V_x&V_y shear forces

p pressure

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Plate Equilibrium Governing Equation

• For equilibrium

= 0

 + + 



 p

y V x

Vx y

= 0

 +

− 



y y

xy V

y M x

M

= 0

 − + 



x yx x

x V M y

M

Net force in z direction must be zero

Net moment about x axis must be zero

Net moment about y axis

must be zero M_x&M_y bending moments

M_xy&M_yx twisting moments per unit length V_x&V_y shear forces

p pressure

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Plate Equilibrium Governing Equation

0

2 2

2 2 2

2 + =



− 



 + 



 p

y x

M y

x M y

M x

M x y yx xy

D p y

w y

x w x

w =

 + 



 + 



4 4 2

2 4 4

4

2

p w

D⁴ =

4 4 2

2 4 4

4

4 2

y y

x

x 

+ 



 + 



= 



Eliminating V_x and V_y gives

Substituting the moment-curvature relations into this equation gets or

in which

= 0

 + + 



 p

y V x

Vx y

= 0

 +

− 



y y

xy V

y M x

M − = 0

 + 



x yx x

x V M y

M

2 4

M w

x x

 

 = 

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Plate Equilibrium Governing Equation

Edge conditions

• Simple support: w = 0 and M = 0.

• Edge clamped: w = 0 and slope = 0.

• Elastic rotational restraint: w = 0.

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Plate Equilibrium Governing Equation

• The major problems in applying plate theory to design of ship plates, or to predicting their performance under load, is the uncertainty about the boundary conditions particularly the in-plane ones.

• With large areas of plate under pressure load it is reasonable by symmetry to

assume the edges of plate elements are clamped against rotation at the stiffeners,

and with single loaded plates the rotational conditions are probably nearer those of

simple support.

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• We have investigated the Elastic Plate Theory .

Learning Outcomes (Final Review)

• Now we are able to:

- Appreciate the assumptions of elastic plate theory

- Be familiar with the bending moment curvature relationship of plate and relationship between twisting moment and twist in a plate.

- Establish plate governing equilibrium equation.

- Be aware of plate edge conditions.

• Details can be referred to topics 4 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?