Hw 4

Weak form

∴

∫

L

L

L

w′Tu ′dx + ∫ wpu dx − ∫ wfdx = 0

0

0

0

Potential energy 2  L L 1  L  dv  Π ( v ) =  ∫ T   dx + ∫ pv2 dx  − ∫ v f dx 0 2  0  dx   0

Galerkin method and N

N

u ≈ u h = ∑ NA ( x) dA = °N ⋅d%

° w ≈ wh = ∑ NB ( x) CB = °N ⋅C

A =1

Substituting

and

u L

∫0 ( w′)

T

into the weak form,

w L

( w) 0

Tu′dx + ∫

where

L

0

T

pu dx − ∫

L

0

( w)

T

f dx = 0

and

° ⋅C ° w′ = B ⇒∫

B =1

° ⋅ d% u′ = B

( B° ⋅ C° ) T ( B° ⋅ d%) dx + ∫ ( °N ⋅ C° ) p ( °N ⋅ d%) dx − ∫ ( °N ⋅ C° ) T

T

L

T

L

0

f dx = 0

0

L ° T⋅B ° T ⋅T ⋅ B ° ⋅ d%dx + L C ° T⋅ N ° T ⋅ p⋅ N ° ⋅ d%dx − L C ° T⋅ N ° T⋅ f dx = 0 ⇒ ∫ C ∫ ∫      0  0  0 

{ ∫ ( B° ⋅ T ⋅ B° dx ) + ∫ ( N°

°T ⇒C 

L

L

T

0

0

T

)}

)

(

T  ° dx ⋅ d%− L N ⋅ p⋅ N ∫0 ° ⋅ f dx  = 0

Since the coefficients are arbitrary,

(

) (

)

)

(

 L B ° T ⋅T ⋅ B ° dx + L N ° T ⋅ p⋅ N ° dx  ⋅ d%= L N ° T ⋅ f dx ∫0 ∫10 44 2 4 43  ∫0  1 4 4 4 4 4 44 2 4 4 4 4 4 4 43 ° ° F K Rayleigh-Ritz method

1

and N

v = ∑ NI dI = °N ⋅d% I =1

Substituting

and

N dN I ° ⋅d% dI = ∑ BI dI =B dx I =1 I =1

into the potential energy,

v

Π ( v) =

N

v′ = ∑

v′

(

)

(

)

(

)

(

)

(

)

1  L ° %T ° ⋅ d% dx  + 1  L °N ⋅ d%T ⋅ p ⋅ °N ⋅ d% dx  − L °N ⋅ d%T f dx B ⋅ d ⋅ T ⋅ B  2  ∫0  ∫0 2  ∫0

1 T L° T ° dx + L °N T ⋅ p ⋅ °N dx  ⋅ d%− d%T L °NT f dx ⇒ Π ( v ) = d%  ∫ B ⋅T ⋅ B ∫0 ∫0  2  0 ∂Π  L ° T ° dx + L N ° T ⋅ p⋅ N ° dx ⋅ d%− L N ° T f dx = 0 = ∫ B ⋅T⋅ B ∫ ∫ 0 0  ∂ d%  0 L ° T ⋅T ⋅ B ° dx + L N ° T ⋅ p⋅ N ° dx ⋅ d%= L N ° T f dx ⇒ ∫ B ∫ ∫  0  0 0 1 4 2 43 1 4 4 4 4 44 2 4 4 4 4 4 43 ° ° F K

2