2dsdxdw dxdx dw

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy

W U −

=

∏ ( )

, W = ∫

0^l

qw dx

q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

: strain energy

U

dx dx ds

T T

T

T

dw dwdx

= dx

: the internal work done by on the element

( )

T ⋅ ds − dx T

( )

²

2 2

1 /

ds = dx + dw = dx + dw dx

2 2

let, dw then, 1 dw 1

z z

dx dx

   

=  +  = +

   

( ) 1 1 f z ≅ +2z

Taylor Series

* * * 1 * 2

( ) ( ) ( ) ( ) . . .

f x + ∆ =x f x +f x′ ∆ +x 2 f′′ x ∆ +x 1 2

( ) (0) (0) (0) ...

f x =Maclaurin Seriesf + f′ x+2 f′′ x +

( ) 1 f z = + z

recall,

( )

²

¹

²

1 / 1

2

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy

W U −

=

∏ ( )

, W = ∫

0^l

qw dx

q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

: strain energy

U

dx dx ds

T T

T

T

dw dwdx

= dx

: the internal work done by on the element

( )

T ⋅ ds − dx T

₁ ¹

2 ds dx dw

dx

 

=  + 

2

2

( ) 1 1

2

2

T ds dx T dx dw dx

dx T dw

dx dx

     

 

⋅ − = ⋅      +        −  

 

=    

Considering the stretched string under tension and as the reference state

T q = 0

l ^l

w

T

x z

q ^T

T T

U W Π = −

strain

∴Π = −U

strain

U W

Π = ¹−

strain 2 ext

U W

Π = −

2

strain 2 strain

U U

Π = ³−

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy

W U −

=

∏ ( )

, W = ∫

0^l

qw dx

q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

: strain energy

U

dx dx ds

T T

T

T

dw dwdx

= dx

: the internal work done by on the element

( )

T ⋅ ds − dx T

₁ ¹

2 ds dx dw

dx

 

=  + 

2

( )

2

T dw

T ds dx dx

dx

 

⋅ − =    

( )

²

0

2

0

l

T

l

dw

U T ds dx dx

dx

 

∴ = ∫ − = ∫    

Considering the stretched string under tension and as the reference state

T q = 0

l ^l

w

T

x z

q ^T

T T

U W Π = −

strain

∴Π = −U

strain

U W

Π = ¹−

strain 2 ext

U W

Π = −

2

strain 2 strain

U U

Π = ³−

107/183

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy

W U −

=

∏ ( )

, W = ∫

0^l

qw dx

0 q =

q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

,

0

2

T

l

dw

U dx

dx

 

= ∫    

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∴∏ = ∫     − ∫

variation of :

∏

0 0

l

dw dw

l

T dx q w dx

dx dx

δ ∏ = ∫ δ ^   ^   − ∫ δ

Considering the stretched string under tension and as the reference state

T q = 0

l ^l

w

T

x z

q ^T

T T

U W Π = −

strain

∴Π = −U

strain

U W

Π = ¹−

strain 2 ext

U W

Π = −

2

strain 2 strain

U U

Π = ³−

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy

W U −

=

∏

q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ = ∫     − ∫

variation of :

∏

0 0

l

dw dw

l

T dx q w dx

dx dx

δ ∏ = ∫ δ ^   ^   − ∫ δ

d d

y y

dx δ δ dx

∴ =

recall,

0 0

l

dw dw

l

dw d w

T dx T dx

dx dx dx dx

δ ^   ^   = ^   δ ^  

∫ ∫

Considering the stretched string under tension and as the reference state

T q = 0

l ^l

w

T

x z

q ^T

T T

U W Π = −

strain

∴Π = −U

strain

U W

Π = ¹−

strain 2 ext

U W

Π = −

2

strain 2 strain

U U

Π = ³−

109/183

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy

W U −

=

∏

q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ = ∫     − ∫

variation of :

∏

0 0

l

dw dw

l

T dx q w dx

dx dx

δ ∏ = ∫ δ ^   ^   − ∫ δ

2

0 0 2

0 2 0 2

l l l

l

dw d w dw d w

T dx T w T w dx

dx dx dx dx

T w d w dx dx

δ δ δ

δ

 

=     −

= −

∫ ∫

∫

integrating by part

since

δ w = 0 at x = 0 and x = l

Considering the stretched string under tension and as the reference state

T q = 0

l ^l

w

T

x z

q ^T

T T

U W Π = −

strain

∴Π = −U

strain

U W

Π = ¹−

strain 2 ext

U W

Π = −

2

strain 2 strain

U U

Π = ³−

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy

W U −

=

∏

q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ = ∫     − ∫

variation of :

∏

0 0

l

dw dw

l

T dx q w dx

dx dx

δ ∏ = ∫ δ ^   ^   − ∫ δ

2

0 2 0

l

d w

l

T w dx q w dx

δ ∏ = − ∫ δ dx − ∫ δ

^T

^∫

^{0l dwdx δdwdx dx=T}

^∫

^{0l dw d wdx  dxδ dx} 2

0 0 2

0 2

0 2

l l l

l

dw d w dw d w

T dx T w T w dx

dx dx dx dx

T wd wdx dx

δ δ δ

δ

= −

= −

∫ ∫

∫

2 0 2

l

d w

T q wdx

δ ∏ = − ^  dx + ^  δ

 

∫

Considering the stretched string under tension and as the reference state

T q = 0

l ^l

w

T

x z

q ^T

T T

U W Π = −

strain

∴Π = −U

strain

U W

Π = ¹−

strain 2 ext

U W

Π = −

2

strain 2 strain

U U

Π = ³−

111/183

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ = ∫     − ∫

variation of :

∏

²₂

0

l

d w

T q wdx

δ ∏ = − ^  dx + ^  δ

 

∫

from

δ Π = 0

²₂

0^l

d w 0

T q wdx

dx δ

 

−  +  =

 

∫

since is arbitrary

δ w

2

2

0

T d w q dx + =

l ^l

w

T

x z

q ^T

T T

U W Π = −

strain

∴Π = −U

strain

U W

Π = ¹−

strain 2 ext

U W

Π = −

2

strain 2 strain

U U

Π = ³−

2 0

d dw

T w q

dx^ dx  +

ρω

+ =

recall, differential equation

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

2 0

T d w q dx + =

The principle of potential energy :

Of all the displacement distribution satisfying the conditions of continuity and the prescribed displacement boundary conditions,

the one which actually takes place ( or which satisfies the equilibrium equations) is the one which makes the potential energy assume a stationary(minimum) value)

recall,

We shall now demonstrate that this stationary value is a minimum In order to prove this, we shall show the quantity

( w w ) ( ) w

∆Π = Π + ∆ − Π

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ =

∫

  −

∫

is always positive

l ^l

w

T

x z

q ^T

T T

113/183

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

2 0

T d w q dx + =

We shall now demonstrate that this stationary value is a minimum

In order to prove this, we shall show the quantity

∆Π = Π ( w + ∆ − Π w ) ( ) w

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ =

∫

  −

∫

is always positive

( ) w w x

∆ = ∆ , ∆ w (0 = ∆ )0, w l ( ) = 0

where

( w w ) ( ) w 0

∆Π = Π + ∆ − Π >

means that

if the string is displaced by from its equilibrium position the potential energy is increased

∆ w

l ^l

w

T

x z

q ^T

T T

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

2 0

T d w q dx + =

We shall now demonstrate that this stationary value is a minimum

In order to prove this, we shall show the quantity

∆Π = Π ( w + ∆ − Π w ) ( ) w

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ =

∫

  −

∫

is always positive

( ) w w x

∆ = ∆ , ∆ w (0 = ∆ )0, w l ( ) = 0

where

( w w ) ( ) w 0

∆Π = Π + ∆ − Π >

means that

if the string is displaced by from its equilibrium position the potential energy is increased

∆ w

( ) ( ( ) ) ^{( )}

( ) ( )

2 2

0 0 0 0

2 2

0 0 0 0

( ) ( )

2 2

2 2

l l l l

l l l l

w w w

d w w

T T dw

dx q w w dx dx qw dx

dx dx

T dw d w T dw

dx qw q w dx dx qw dx

dx dx dx

∆Π = Π + ∆ − Π

  + ∆      

 

=   − + ∆ −    − 

 

       

 

  ∆      

=   +  − + ∆   −   − 

   

   

   

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

l ^l

w

T

x z

q ^T

T T

115/183

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

2 0

T d w q dx + =

We shall now demonstrate that this stationary value is a minimum

In order to prove this, we shall show the quantity

∆Π = Π ( w + ∆ − Π w ) ( ) w

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ =

∫

  −

∫

is always positive

( ) w w x

∆ = ∆ , ∆ w (0 = ∆ )0, w l ( ) = 0

where

( w w ) ( ) w 0

∆Π = Π + ∆ − Π >

means that

if the string is displaced by from its equilibrium position the potential energy is increased

∆ w

( ) ( )

( ) ( )

2 2

0 0 0 0

2 2 2

0 0 0 0 0

( ) ( )

2 2

2 2 2

l l l l

l l l l l

w w w

T dw d w T dw

dx qw q w dx dx qw dx

dx dx dx

T d w d wd w d w T d w

dx q w dx qw dx qwdx

dx dx dx dx dx

∆Π = Π + ∆ − Π

  ∆      

=   +  − + ∆   −   − 

   

   

   

     ∆  ∆      ∆ 

=     + +    − ∆ −  −   +

     

     

 

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫

l ^l

w

T

x z

q ^T

T T

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

2 0

T d w q dx + =

We shall now demonstrate that this stationary value is a minimum

In order to prove this, we shall show the quantity

∆Π = Π ( w + ∆ − Π w ) ( ) w

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ =

∫

  −

∫

is always positive

( ) w w x

∆ = ∆ , ∆ w (0 = ∆ )0, w l ( ) = 0

where

( w w ) ( ) w 0

∆Π = Π + ∆ − Π >

means that

if the string is displaced by from its equilibrium position the potential energy is increased

∆ w

( ) ( )

( )

2 2 2

0 0 0 0 0

2

0 0 0

( ) ( )

2 2 2

2 2

2 2 2

l l l l l

l l l

w w w

T d w d wd w d w T d w

dx q w dx qw dx qwdx

dx dx dx dx dx

T d wd w d w T d wd w T d

dx q w dx dx

dx dx dx dx dx

∆Π = Π + ∆ − Π

     ∆  ∆      ∆ 

=           + +        − ∆ −    −     +

   ∆  ∆     ∆ ∆

=       +        − ∆    = +

∫ ∫ ∫ ∫ ∫

∫ ∫ ∫

0 ² 0

l

w

l

dx q wdx dx

  − ∆

 

 

∫ ∫

l ^l

w

T

x z

q ^T

T T

117/183

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

2 0

T d w q dx + =

We shall now demonstrate that this stationary value is a minimum

In order to prove this, we shall show the quantity

∆Π = Π ( w + ∆ − Π w ) ( ) w

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ =

∫

  −

∫

is always positive

( ) w w x

∆ = ∆ , ∆ w (0 = ∆ )0, w l ( ) = 0

where

( w w ) ( ) w 0

∆Π = Π + ∆ − Π >

means that

if the string is displaced by from its equilibrium position the potential energy is increased

∆ w

2

0

2

0 0

l

d wd w T

l

d w

l

T dx dx q wdx

dx dx dx

∆  ∆ 

∆Π = +   − ∆

 

∫ ∫ ∫

2 0 2

0

l l

dw d w

w w dx

dx dx

= ∆ − ∆

∫

integrating by part

0

l

dw d w dx dx dx

∫ ∆

since

∆ = w 0 at x = 0 and x = l

2 0 2

l d w

w dx

= − ∆

∫

dx

2 2

0 2

2

0 0

l

d w T

l

d w

l

T w dx dx q wdx

dx dx

 ∆ 

∆Π = − ∫ ∆ + ∫     − ∫ ∆

l ^l

w

T

x z

q ^T

T T

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

2 0

T d w q dx + =

We shall now demonstrate that this stationary value is a minimum

In order to prove this, we shall show the quantity

∆Π = Π ( w + ∆ − Π w ) ( ) w

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ =

∫

  −

∫

is always positive

( ) w w x

∆ = ∆ , ∆ w (0 = ∆ )0, w l ( ) = 0

where

( w w ) ( ) w 0

∆Π = Π + ∆ − Π >

means that

if the string is displaced by from its equilibrium position the potential energy is increased

∆ w

2 2

0 2

2

0 0

l

d w T

l

d w

l

T w dx dx q wdx

dx dx

 ∆ 

∆Π = − ∫ ∆ + ∫     − ∫ ∆

2 2

0 2

2

0

l

d w T

l

d w

w T q dx dx

dx dx

   ∆ 

∆Π = − ∆ ∫   +   + ∫    

2

2

0

T

l

d w dx dx

 ∆ 

∆Π = ∫    

2

2 0

T d w q dx + = because of equilibrium condition

l ^l

w

T

x z

q ^T

T T

119/183

Illustrative Problem

Uniform Loaded String :

the application of the principle of potential energy q S

q

- initially under a large tensile force - uniform transverse load

- assume that the application of does not change the magnitude of application force - assume that body force is neglected

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2

2 0

T d w q dx + =

We shall now demonstrate that this stationary value is a minimum

In order to prove this, we shall show the quantity

∆Π = Π ( w + ∆ − Π w ) ( ) w

2

( )

0 0

2

l l

T dw

dx qw dx dx

 

∏ =

∫

  −

∫

is always positive

( ) w w x

∆ = ∆ , ∆ w (0 = ∆ )0, w l ( ) = 0

where

( w w ) ( ) w 0

∆Π = Π + ∆ − Π >

means that

if the string is displaced by from its equilibrium position the potential energy is increased

∆ w

2

2

0

T

l

d w dx dx

 ∆ 

∆Π = ∫    

since the integral cannot be negative

∆Π ≥ 0

d w 0 dx

∆ =

the integral vanishes only in the exceptional case when, which only occurs at the equilibrium position

thus we have demonstrated the theorem of minimum potential energy

l ^l

w

T

x z

q ^T

T T

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

Simply supported beam :

the application of the principle of potential energy - uniformly distributed load

- constant cross section

- only consider the strain energy due to pure bending due to - normal stress :

- bending moment :

- moment of inertia of the cross section with respect to the axis :

σ

x

q

x /

M σ = I y

M y I

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

x

y

q w

2 0

d dy

T y p

dx^ dx + ρω + =

2

0^ld dy 0^l 0^l 0

T y dx y y dx p y dx

dx^ dx^δ + ρω δ + δ =

∫ ∫ ∫

Differential Equation

multiply and integrateδy

integrating by part

and two end conditions ₂

2 2

0 0

1 0

2 2

l T dy dy l

y py dx T y

dx dx

δ∫^ ρω + − ^ ^^ +^ δ ^ =

2 2 2

0

1 0

2 2

l T dy

y py dx

δ∫ ρω + − dx = Variational Method

2 2 2

let 1

2 2

T dy

F y py

ρω dx

= + −   

F F 0

x y y

 

∂ ∂ ′−∂ =

∂ ∂  ∂

then the differential equation is obtained by the Euler equation 2

0

l d dy

T y p y dx

dx dx ρω δ

   + + 

   

 

∫

solution of D.E.

Ex.)

solution of Integral Equation

Approximation

-Galerkin /Collocation /Least Square Green Function

Approximation - Rayleigh-Ritz

recall,

this example

121/183

Illustrative Problem

- uniformly distributed load - constant cross section

- only consider the strain energy due to pure bending due to - normal stress :

- bending moment :

- moment of inertia of the cross section with respect to the axis :

σx

q

x / M σ = I y

M y I

⑦ Assume that

( )

dx dx dy

ds≈ ,θ ≈tanθ =

2

2

d d d

s x d

θ y

∴ =

ds d =

⋅ θ

ρ ρ

θ 1 ds = d

y dA E dA

dF =σ = ⋅ρ

dM = −y dAσ ρ

− 1 EI =

M E

d

I ds

θ M

∴ = −

EI M dx

y d

²₂

= −

ds

M M

y θ

ρ d

y x

dx θ d 중립면

dy

dM M+ M

dx V

dx x f )(

dV V+

:분포하중 )

(x f

dV ( )

dx = −f x dM ( )

dx =V x ₄ ( )

4

x dx f

y EI d =

∴

recall,

2 2

M EI d w

= dx

According to the Bernoulli-Euler law in beam theory Simply supported beam :

the application of the principle of potential energy

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

x

y

q w

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

- uniformly distributed load - constant cross section

- only consider the strain energy due to pure bending due to - normal stress :

- bending moment :

- moment of inertia of the cross section with respect to the axis :

σx

q

x / M σ = I y

M y I

2 2

2

0 2

2

E d w

U y

dx

 

∴ =  

 

2 2

M EI d w

= dx

According to the Bernoulli-Euler law in beam theory

The strain energy per unit volume, (the strain energy density)

2 0

1

2 ^x

U = Eσ ₀

2 0

or 1

2 1 2

x x

x

U

U E

σ ε ε

=

=

recall,

2 2 x

/

M I y

y d w EI I dx σ =

=

2 x 2

E d w y σ dx

∴ =

2 0

2 2 2

2 2

2

2 U

x

E

E d w E dx y

= σ

 

=  

 

The strain energy per unit volume, (the strain energy density)

Simply supported beam :

the application of the principle of potential energy

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

x

y

q w

123/183

Illustrative Problem

- uniformly distributed load - constant cross section

- only consider the strain energy due to pure bending due to - normal stress :

- bending moment :

- moment of inertia of the cross section with respect to the axis :

σx

q

x / M σ = I y

M y I

2 2

2

0 2

2

E d w

U y

dx

 

=  

 

2 2

M EI d w

= dx

According to the Bernoulli-Euler law in beam theory The strain energy per unit volume,

(the strain energy density)

2 2 0

2

2

l

EI d w

U dx

dx

 

∴ =  

 

∫

The total Strain Energy absorbed in the beam

∫∫∫

=

V

dz dy dx U

U

₀

2 2

2

2

2

E d w

y dxdydz dx

 

=  

 

∫∫∫

Simply supported beam :

the application of the principle of potential energy

(U W) 0

δ δ

∴ Π = − =

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

since

∫∫ y dydz

²

= I

Neutral Axis

<section view>

dy dz ⋅ y

y x

x

y

q w

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

2 2 0 2

, 2

l

EI d w

U dx

dx

 

=  

 

∫

: the virtual work done by the external (surface and body) forces

δW δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

W U −

=

∏

, W = ∫

0^l

qwdx

recall, uniform loaded string

2 2

0

2

2 0

l

EI d w

l

dx qwdx dx

 

∏ =   −

 

∫ ∫

2 2

0

2

2

l

EI d w

qw dx dx

   

 

∴∏ =   −

   

 

∫

- uniformly distributed load - constant cross section

- only consider the strain energy due to pure bending due to - normal stress :

- bending moment :

- moment of inertia of the cross section with respect to the axis :

σx

q

x / M σ = I y

M y I

Simply supported beam :

the application of the principle of potential energy

(U W) 0

δ δ

∴ Π = − =

x

y

q w

125/183

Illustrative Problem

δW : the virtual work done by the external (surface and body) forces

δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2 2

0

2

2

l

EI d w

qw dx dx

   

 

∏ =   −

   

 

∫

2 2

2 2

0

2

0

2

l l

EI d w d w

dx q wdx dx dx

δ Π = ∫ δ − ∫ δ

2 2

2 2

0 0

l

d w d w

l

EI dx q wdx

dx dx

δ Π = ∫ δ − ∫ δ

integrating by part variation of :

∏

2 2

2 2

0

l d w d w dx dx dx

∫ δ

integrating by part

2 2 2 3

2 2 2 3

0 0

0 3

0 3

4 4

4 0 4

0 4

0 4

l

l l

l

l l

l

d w d w d w d w d w d w

dx dx

dx dx dx dx dx dx

d w d w dx dx dx

d w d w

w wdx

dx dx

d w wdx dx

δ δ δ

δ

δ δ

δ

 

=   −

 

= −

  

 

= −  − 

 

 

 

=

∫ ∫

∫

∫

∫

boundary condition simple support :

2

2 0 0 and

d w at x x l

dx = = =

since

0 0

w at x and x l

δ = = =

4

0 4 0

l

d w

l

EI wdx q wdx

δ Π = ∫ dx δ − ∫ δ

- uniformly distributed load - constant cross section

- only consider the strain energy due to pure bending due to - normal stress :

- bending moment :

- moment of inertia of the cross section with respect to the axis :

σx

q

x / M σ = I y

M y I

Simply supported beam :

the application of the principle of potential energy

(U W) 0

δ δ

∴ Π = − =

x

y

q w

Innovative Ship Design - Elasticity

@

SDALAdvanced Ship Design Automation Lab.

http://asdal.snu.ac.kr Seoul

National Univ.

Illustrative Problem

δW : the virtual work done by the external (surface and body) forces

δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

2 2

0

2

2

l

EI d w

qw dx dx

   

 

∏ =   −

   

 

∫

variation of :

∏

4

0 4 0

l

d w

l

EI wdx q wdx

δ Π = ∫ dx δ − ∫ δ

0 ⁴⁴

l

d w

EI q wdx

δ ^ dx ^ δ

∴ Π =  − 

 

∫

from

δ Π = 0

⁴₄

0^l

d w 0

EI q wdx

dx δ

 

− =

 

 

∫

since is arbitrary

δ w

⁴

4

0

EI d w q dx − =

- uniformly distributed load - constant cross section

- only consider the strain energy due to pure bending due to - normal stress :

- bending moment :

- moment of inertia of the cross section with respect to the axis :

σx

q

x / M σ = I y

M y I

Simply supported beam :

the application of the principle of potential energy

(U W) 0

δ δ

∴ Π = − =

x

y

q w

127/183

Illustrative Problem

δW : the virtual work done by the external (surface and body) forces

δU

(

^{x x y y z z xy xy yz yz zx zx}

)

V

U dx dy dz

δ =

∫∫∫

σ δε +σ δε +σ δε τ δγ+ +τ δγ +τ δγ

(

^{T u T v T w}

)

^dA

(

^{F u F v F w}

)

^dA

W ⁼

∫

A x ^{δ +} y ^{δ +} z ^{δ +}

∫

V x^{δ +} y^{δ +} z^δ

δ ^{µ µ µ}

: the virtual strain energy

or the strain energy absorbed in the body during a virtual displacement

4 0 4

l

d w

EI q wdx

δ Π = ^  dx − ^  δ

 

∫

^from

^{δ Π = 0}

since is arbitrary

δ w

⁴

4

0

EI d w q dx − =

임상전 편저, 재료역학, 2002년 ,문운당 ( Timoshenko S., Young D.H., Elements of strength of materials, 5^thedition, Van Nostrand, 1968)

+

-- -

^comp.

+

+

^tension

+

y or

y κ

ρ

σydA

match

dM M+ M

dx V

dx x f )(

dV V+

y

x

y

x

M

x y

σ2

σ1 dA

2

2

d y M

dx = ±EI ^{− +⇒VdV}_dx^{f x dx( )= −f x( )+(V+dV)=0}

( ) ^{( ) 1 0}

2 M−M+dM +Vdx−f x dx⋅ dx=

) (x dx V dM =

⇒

all sign convention is same

except y-axis in opposite direction

≠

B.M. Κ

=1/ρ y ε σ check dM=

yσdA dM relation btw V, M, f(x)

M =

∫

AdM

modify considering the

2

2

d y M

dx = −EI 4

4 ( )

EI d y f x dx =

- uniformly distributed load - constant cross section

- only consider the strain energy due to pure bending due to - normal stress :

- bending moment :

- moment of inertia of the cross section with respect to the axis :

σx

q

x / M σ = I y

M y I

Simply supported beam :

the application of the principle of potential energy

(U W) 0

δ δ

∴ Π = − =

x

y

q w