Lecture 3 Plate Bending

(1)

OPen INteractive Structural Lab

Topics in Ship Structural Design

(Hull Buckling and Ultimate Strength)

Lecture 3 Plate Bending

Reference : Ship Structural Design Ch.09

NAOE

Jang, Beom Seon

(2)

"Long" Plates(Cylindrical Bending)

 An equation relating the deflections to the loading can be developed as for the beam.

 Cylindrical Bending:

 A plate which is bent about one axis only(a>>b)

 Transverse deformation does not occur,

since such a deformation would required a saddle shape deformation.

9.1 Small Deflection Theory

2

=

^x ^y

x

E E

 

  

(1-

2

)

=

^x

x

E

 



= 2 =E

x 1- x x

 E  

 ^

ε_x -ε_x ε_y

-ε_y

E

y y

x

 



  



 0



 E E

y x y

 

  

^y

 

^x

Plate in cylindrical bending

Saddle shape

(3)

Poisson Ratio

Reference



_x



_x



_y



_z

 E 1  

( )



_y



_y

 

_z _x

 E 1  

( )



_z

 

_z _x



_y

 E 1  

( )

axial trans



   

E

y y

x

 



  



Strain Δε

_y

induced by ε

_y

(4)

Formulas between a beam and a long plate

 E΄ : effective modulus of elastictiy, always > E.

→ A plate is always “stiffer” than a row of beams

 For a long prismatically loaded plate, the extra stiffness may be fully taken into account for by using E΄ in place of E.

 For simply supported plates:

 For clamped plates:

4

4 4 2

max 3

5 5 (1- )

= =

384 32

pb pb

E I Et

 



4 4 2

max 3

(1- )

= =

384 32

pb pb

E I Et

 



) 1

(  

²

 

E E

(5)

Formulas between a beam and a long plate

 Moment curvature relation may be obtained from External bending moment = moment of stress force

where

 D : the flexural rigidity of the plate

the constant of proportionality between moment and curavature analogous with EI in beam theory

5

2 3

/ 2 / 2

2 2

/ 2 / 2

t 1

= ( ) =

1 12(1- )

t t

t x t

x x

E z E

M zdz dz

r r

  

 

 

       

x

= / z r

x



3 2

t 1

= =

12(1- )

_x _x

E D

M  r r

 

 

 

3

12(1- 2) D Et





x x

E 

 

) 1

( 

²



 M  EI

(6)

Formulas between a beam and a long plate

 The radius of curvature of the plate can be expressed in terms of the deflection w of the plate

 Also for a unit strip in a long plate the max. stress is where the section modulus

 For a uniform pressure p, the maximum bending moment is proportional to pb². The stress is expressed in

 The coefficient k depends on the boundary conditions:

k=3/4 for simply supported edges k=1/2 for clamped edges

6

max

= M

max

c I /



3 2

/ c=(t /12)/(t/2)=t /6 I

2

max

= b

kp t

 ^{ } _{ }

 

rx

x

w 1

2

2 



  M

x D w 



 ² ₂

EI v  M



(7)

Beam Theory

Reference

1) strain-curvature Geometric relation

    ^y

 

  E   ^Ey

2)strain-stress relation

M

_x

ydA

  

A

^

3) moment-curvature relation

4) Stress-moment relation

 

_x

  ^My I

   



1 M

EI stress – strain - curvature - Moment

Ey



 ^ ^ 1

1

2

3

4

(8)

Beam theory

1) Strain-Curvature Relation

Reference

L y d dx y

1

 (   )    dx



Initial length of Line e-f = dx Length after bending = L

₁

d  dx

 

compression

Tension

Natural axis

No change in length

  

   



_x

^L ^dx   dx

y y

1

(9)

Beam theory

3) moment-curvature relation

Reference

dM   

_x

ydA

M

_x

ydA

  

A

^

M Ey dA E y dA EI EI

A

 

A

^

²

 ^ 

²

 ^  _ ¹

   



1 M

EI





A

y dA

I

² : Moment of Inertia

: EI = bending rigidity

 Ey

  

(10)

 Force equilibrium

→ Relation between shear force (V) and load (q)

 Moment equilibrium

→ Relation between Moment (M) and shear force (V)

 Combining two relations

 Relation between deflection and distributed load

Beam Theory : Relation between load, shear force, Moment

Reference

dx q dV dV

V qdx V

F_vert        



⁰ ^: ⁽ ⁾ ⁰

dx V dM dM

M dx

dV dx V

qdx M

M       



 



 





 ⁰ ^: ₂ ⁽ ⁾ ⁽ ⁾ ⁰

dx q M

d² ₂  

 d  dx

M EI

2 2



EI q dx

v

d

⁴₄

 

Homework #1 Derive in detail

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 Force equilibrium

→ Relation between shear force (V) and load (q)

 Moment equilibrium

 Combining two relations

 Relation between deflection and distributed load

Forces and moments in a plate element

x q d

M

d²₂   + ^y + =0

x q

q p

x y

 

  q :shear force per unit length

m : moment per unit length

dx q dV  

dx V - +q =0 dM 

xy y

y

m m

x y

 

  m^yx + m^x -q =0_x

y x

 

2 2

2

2^x -2 m^xy + m2^y +p=0 m

x x y y

 



   



 







 



 

 ²₂ ²₂

x v w y D w m_y

y x v w D

m_xy



 

 (1 ) ²



 







 



 

 ²₂ ²₂

y v w x D w m_x

D p y

w y

x w x

w 



 



 



4 4 2 2

4 4

4

2 EI

q dx

v d⁴₄  

(12)

Derivation of the Plate Bending Equation

 The previous theory is only applicable if :

 Plane cross sections remain plane.

 The deflections of the plate are small(w_max not exceeding 3/4 t)

 The maximum stress nowhere exceeds the plate yield stress(i.e., the material remains elastic

)

 A panel of plating will have curvature in two directions

=

^x ^y

x

E E

 

 

_y

=

^y ^x

E E

 

 

1-

2

= +

x

x y

E

 ^（  ^）  

E v E

v

_y

v 

^y



^x

  

²

Differential element of plating

(13)

Derivation of the Plate Bending Equation

 Assumptions stated previously give rise to the strain-curvature relations 9.1 Small Deflection Theory

1-

2

= +

x

x y

E

 ^（  ^）  

3

12(1- 2) D Et







 







 



 

 ²₂ ² ₂

x v w y

D w m_y



 







 



 

  ₂ ( ) ² ₂ ² ₂

1 y

v w x

z w v

E

x



 







 

( ) ² ₂ x z w

x _



 







 

 ( ) ² ₂ y z w

y



 







 



 

 ² ₂ ² ₂

y v w x

D w m_x



  



 







 



 



 ^/²

2 /

2 2 2 2

2 2 2

/ 2

/ 1

t t t

t x

x z dz

y v w x

w v

zdz E

m 

(14)

Beam Theory : Deflection curve

 Relation between deflection and distributed load

Reference – Mechanics of Material

 

d



ds

 

 1 

d



ds

d dx

  tan 

ds  dx

 

 1  d  dx

tan   dv   dx

   





1

²

2

d dx

 ^  ^

1 M

EI

 d  dx

M EI

2 2



From Geometric relation

EI q dx

v

d

⁴₄

 

dx q M

d² ₂  

(15)

OPen INteractive Structural Lab OPen INteractive Structural Lab

Forces and moments in a plate element

 The lateral load pdxdy is carried by the distributed shear forces q acting on the four edges of the element.

 In the general case of bending, twisting moments will also be generated on all the four faces

Forces and moments in a plate element

Without the lateral load pdxdy With the lateral load pdxdy

(16)

Forces and moments in a plate element

 Sine convention in beam theory

– Determined how deform the material

– Shear force : rotate an element clockwise (+) – Bending moment : shorten the upper skin (+)

Sine convention in beam theory Sine convention in plate bending

(17)

Forces and moments in a plate element

 From equilibrium of vertical forces :

 Taking moments parallel to the x-axis, at Point A

+

^y

+ =0

x

q

q p

x y

 

- +q =0

xy y

y

m m

x y

 

+ -q =0

yx x

x

m m

y x

 

0 )

( ) (





 

 



 

 

pdxdy dx

q dx y dy

q q

dy q dy x dx

q q

y y

y

x x

x

0 )

(

) (



 

 





 

 

dxdy q

dx y dy

m m

dx m dy m dy x dx

m m

y y

y

y xy

xy xy

q :shear force per unit length m : moment per unit length

Point A

 Taking moments parallel to the y-axis

(18)

Forces and moments in a plate element

 Because of the principle of complementary shear stress, it follows that m

_yx

=-m

_xy

, so that

 Substituting for q

_x

and q

_y

- m

^xy

+ m

^x

-q =0

_x

y x

 

2 2

2

2^x -2 m^xy + m2^y +p=0 m

x x y y

 



   

+ -q =0

yx x

x

m m

y x

 

m^xy

-

m^y

+q =0

_y

x y

 

+

^y

+ =0

x q

q p

x y

 

0 )

( )

(  



 



 



 



 



p

x m y

m y

x m y

m x

xy x y

xy

(19)

 If a point A in the plate a distance z from the neutral surface is displaced a distance v in the y direction,

 The change in slop of the line AB will be

 The change in slop of the line AD

 The shear strain is

 The shear stress

Forces and moments in a plate element

= + u

x y



^



^

 

=G + u

x y

 ^ _ ^  ^ ^ _

 

 

Shear strain and plate deflection

x v dx

v x dx

v v



 

 

 

x u



Shear strain is induced by deflection. That is, the vertical deflection generates shear strain.

(20)

 from the geometry of the slope of w in each direction (x and y):

 Making these substitutions gives

 The twisting moment

Forces and moments in a plate element

y x v w y D

x w v

v v Et

y x

w v

m_xy Et



 

 







 



 ³ ² ³ ² (1 ) ²

) 1 )(

1 ( 12

) 1 ( )

1 ( 12

y x

w dz Gt

y z x G w

m ^t

xy t  

 



 









2 3 2

2 2 /

2

/ 2 6

) 1 (

2 v

G E

 

=-z

u x





 ⁽ ^), ⁽ ^), ^ ⁽^^),

 

 x

z w u

x z w

u 

 

 y

z w

v 

 



y x Gz w



 

 2 ²



τ

부호 정의 필요 (+) convention

τ

-τ +τ

(21)

Forces and moments in a plate element

 Substitute m_x, m_y, m_xy

 Finally

 Equation of equilibrium for the plate

 Assumption

• small deflection, max deflection < ¾ t

• no stretching of the middle plane ( no membrane effect)

x p v w y D w y

y x v w y D

x y

v w x D w

x  



 



 



 







 



 



 



 







 



 



 



 



 







 



 



2 2 2

2 2

2

) 1 ( 2



 







 



 

 ²₂ ²₂

x v w y D w m_y

2 2

2

2^x -2 m^xy + m2^y +p=0 m

x x y y

 



   

y x v w D

m_xy



 

 (1 ) ²

D p w /

4 

D  p y

w y

x w x

w 



 



 



4 4 2 2

4 4

4

2



 







 



 

 ²₂ ²₂

y v w x D w m_x

(22)

Boundary Conditions

 The types of restraint around the boundary of a plate can be idealized as follows:

1. Simply supported: edges free to rotate and to move in the plane 2. Pinned: edges free to rotate but not free to move in the plane

3. Clamped but free to slide: edges not free to rotate but free to move in the plane of the plate;

4. Rigidly clamped: edges not free to rotate or move in the plane

 In most plated frame structures (particularly in vehicles and free- standing structure) : little restraint against edge pull-in.

→ BC. 1 and 3 are generally applicable 9.1 Small Deflection Theory

(23)

Simply Supprted Plates

 The general expression for the load

 The coefficient A_mn can be obtained by Fourier analysis for any particular load condition. For a uniform pressure p₀.

 For the biharmonic equation, a sinusoidal load distributi on produces a sinusoidal deflection

1 1

( , )= _mn sin sin

m n

m y n x

p x y A

a b

 

 

 

 

0 2

= 16

mn

A p

 mn

= =

1 1

= sin sin

m n

mn

m n

m y n x

B a b

 

 ^ ^

 

 

0

2 2 2

6

2 2

= 16

+

mn

B p

m n

Dmn a b



^_ ^_

 

Dmn p b

n b

a n m a

B_mn m ₄ ₆ ⁰

4 4 2

2 2 2 4 4

4

4 2 16



 _



 



  

D p y

w y

x w x

w 



 



 



4 4 2 2

4 4

4

2

Homework #2 Prove it.

(24)

Simply Supprted Plates

 Due to the symmetry of the problem, M and n only take odd values.

 Effect of boundary condition Clamped < Simply supported

about 4~5 times.

 Effect of aspect ratio

when a=b, the deflection is minimum 9.1 Small Deflection Theory



^^







 



 



 

 ^m

m n

n b

x m a

y m

b n a

Dmn m w p

1 1

2 2 2

2 6

0 sin sin

16  



Maximum deflection of rectangular plates under uniform pressure

Homework #3 Plot the cur ve using any kinds of soft ware.

(25)

Simply Supported Plates

 To determine the bending stress in the plate, calculate the bending stress.

 The bending moment will always be greater across the shorter span.

Homework #4 Prove this.

 The symbol b is used for shorter dimension, then a/b> 1

 m_xis the larger of the two moments and has its maximum value in the center of the plate.

2 2

=- +

mx D

x y

  

   

   

 

2 2

= =

0

2 2 2

2 2

1 1 4

2 2

= 16 + sin sin

+

m n

p n m m y n x

b a a b

m n

mn a b

 





 

 

 

 

   

 

 

 



^^















 

 



 







 



 



 

 ^m

m n

n m

m n

n b

x n a

y m

b n a Dmb m

n p b

x n a

y m b

n

b n a Dmn m

p x

w

1 1

2

2 2 2

2 2 4

0 1 1

2 2 2 2

2 2 2

2 6

0 2

2

sin 16 sin

sin

16 sin  





^^







 

 



 



 

 ^m

m n

n b

x n a

y m

b n a Dna m

m p y

w

1 1

2

2 2 2

2 2 4

0 2

2

sin

16 sin  



(26)

Clamped Plates

 The usual methods are the energy(or Ritz) method and the method of Levy

 The energy method, while giving only approximate results

 The Levy type solution is achieved by the superposition of three loading systems applied to a simply supported plate :

(1) uniformly distributed along the short edges : deflection w₁ (2) moments distributed along the short edges : deflection w₂ (3) moments distributed along the long edges : deflection w₃

short edges : long edges :

(

where x=0, y=0 is at the center of the plate) 9.1 Small Deflection Theory

3

1 2

/ 2 / 2

+ + 0

y a y a

y y y



 

 

     

      

   

3

1 2

/ 2 / 2

+ + 0

x b x b

x x x



 

 



  

  

      

   

(27)

Clamped Plates

 k depends on :

 plate edge conditions

 Plate side ratio a/b

 Position of point considered The effect of aspect ratio is

smaller for clamped plates, and beyond about a/b=2

such a plate behaves essentially as a clamped strip and the

influence of aspect aspect ratio is negligible.

Stresses in rectangular plates under uniform lateral pressure

2

Stress 



 



 

t kp b

Homework #5 Plot Max. S tress curve in simply sup ported plates.

(28)

Large-Deflection Plate Theory

 Membrane stress arises when the deflection becomes large and/or when the edges are prevented from pulling in.

 Small-deflection theory fails to allow for membrane stresses.

 As the deflection increases, an increasing prortion of the load is carried by this membrane action.

 The lateral load is supported by both bending and membrane action.

 Amore comprehensive plate theory, usually referred to as "large- deflection" plate theory.

9.2 Combined Bending and Membrane Stresses-Elastic Range

(29)

Membrane Tension(Edges Restrained Against Pull-in)

 What is membrane tension?

 Membrane stress : Stresses that act tangentially to the curved surface of a shell

 Pressure vessel : closed structures containing liquids or gases under pressure

 Gage pressure (Internal – external pressure) is resisted by membrane tension

Spherical Pressure vessel Cylindrical Pressure

vessel

(30)

Membrane stress

 Spherical Pressure vessel

Reference

 F

_horiz

 0

  ( 2 r t

_m

)  p r ( 

²

)

r r t

m

 

2    ^pr  r t

pr

m

t

2

2 2

(31)

 Cylindrical Pressure vessel

Membrane stress

Reference

σ

₁

: circumferential stress hoop stress



₁

( 2 bt )  2 pbr  0

 

₁

 ^pr t



₂

( 2  rt )  p r 

²

 0

 

₂

 2 pr

t



₁

 2 

₂

σ

₂

: longitudinal stress

(32)

Membrane stress

Reference

Hull Girder Bending

Double Bottom Bending

Local Bending of Longitudinal

Plate Bending

Global Stress

Local Stress

Membrane stress on plating

Bending stress on plating

# of elements should be more than thee (3) between boundaries.

 Ship & Offshore Structure

Membrane stress Bending stress

(33)

 N_x, N_y : membrane tension per unit length

 N_xy : membrane shear force per unit length

 Vertical component of the tension force at side (1)

 Net force in the x-direction

 Net force in the y-direction

Large-Deflection Plate Theory by von Karman

+N dy_x dx

x x x

 

     

      

 

) / ( )

tan(

)

sin(

slope N dy slope N dy w x dy

N_x

 

_x

 

_x

 



) ( x dy w N_x



 

Vertical component Vertical component

) ( ₂

2

x dy w N_x



 

) ( ₂

2

y dx w N_y



 

(34)

 The vertical component of the shear force along sides (3) and (4) is

Large-Deflection Plate Theory by von Karman

) ( x dx w N_xy



Vertical component



 



 



 







 



 dy

x w y x dx w N_xy

xy xy

N dx dy N dx

x y x x

  

       

         

 

(35)

Large-Deflection Plate Theory by von Karman

 Total vertical forces

 New terms are added

 N

_x

, N

_y

, N

_xy

are functions of x and y.

 Except for a few simple cases, precise mathematical solutions are very difficult.

y dxdy N w

y x N w x

N_x w _xy _y 



 







 



 



2 2 2

2 2

2



 







 



 



 

 

 



 



2 2 2

2 2 4

4 2 2

4 4

4

1 2

2 y

N w y x N w x

N w D p

y w y

x w x

w

y xy

x



 







 



 



 



⁴ 1 ² ₂ 2 ² ² ₂

y N w y x N w

x N w D p

w _x _xy _y

D p y

w y

x w x

w 







 



4 4 2 2

4 4

4

2

(36)

Membrane Tension(Edges Restrained Against Pull-in)

 The relative magnitude of membrane effects depends on

: the degree of lateral deflection or "curvature" of the plate surface the degree to which the edges are restrained form pulling in.

 If the edges are restrained and large deflection (w>1.5 t)

the contribution of membrane tension > bending, vice versa.

 In ship plating : little restraint against edge pull-in

for large deflections, supporting stiffeners or beams fails earlier.

 Nevertheless, some situations in which large deflections can be permitted,

→ the use of membrane tension can give substantial weight savings.

(37)

Membrane Tension(Edges Restrained Against Pull-in)

 The case of a unit-width strip

 Laterally loaded

 Edges are prevented from approaching

 the difference between the arc length of the deflected strip and the original straight length

 Membrane force is unidirectional (N_y=N_xy=0) and constant over the length.

 Extension due to the tension

2

= 0^b 1 d 1

dx dx





^^ ^  ^  ^^ ^ ^



₀^b ¹₂ ^^_ ^d_dx^ ^^_² ^dx

2

2 0

Et b d

T dx

b dx











 

(38)

Membrane Tension(Edges Restrained Against Pull-in)

 w

₀

initial deflection, w

₁

due to the load

 The total change in length is

 The change in length due to the initial deflection

0 1

( ) ( ) sin x

x b

     

2

2 2

0 1 2

0

1 ( ) cos 2

b

x

b b dx

 

     

2

0 1

( )

   4b

 

b w 4

2 0



2