computer aided ship design part.3 grillage analysis of midship...
TRANSCRIPT
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Naval Arc
hitectu
re &
Ocean E
ngin
eering
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
Computer Aided Ship DesignPart.3 Grillage Analysis of Midship
Cargo Hold
2009 Fall
Prof. Kyu-Yeul Lee
Department of Naval Architecture and Ocean Engineering,Seoul National University of College of Engineering
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Application
Bar
Beam
Shaft
Summary
Element Behavior
Tension
Bending
Torsion
Midship Cargo Hold
2
2
( )( ) 0
d u xEA f x
dx
4
4
( )( ) 0
w xEI f x
x
Structure
•Superposition of Stiffness Matrix•Coordinates Transformation
Truss
Frame
Grillage2
2
( )( ) 0
xGJ f x
x
Beam Theory : Sign Convention, Deflection of Beam
Elasticity : Displacement, Strain, Stress, Force Equilibrium, Compatibility, Constitutive Equation
:A Sectional Area :
:
E
I
Young’s Modulus
Moment of Inertia
:G Shear Modulus
:J Polar Moment of Inertia:l Length
Differential Equation
Mx F , 0where x
VariationalMethod
2
0( ) 0
2
l EA duf u dx
dx
22
20( ) 0
2
l EA d wf w dx
dx
2
0( ) 0
2
l GJ df dx
dx
1 1
2 2
1 1
1 1
u fEA
u fl
0 1( )u x a a x
11
2 2
11
3
2 2
2 2
2 2
6 3 6 3
3 2 32
6 3 6 3
3 3 2
ful l
Ml l l lEI
ul l fl
l l l l M
2 3
0 1 2 3( )w x b b x b x b x
1 1
2 2
1 1
1 1
MGJ
Ml
0 1( )x c c x
Kd F
Finite Element Method
•Discretization•Approximation
Grillage Modeling
•Equivalent Force & Moment•3D 2D
•Boundary condition
Engineering Concept !
Solution
•programming•visulaization
:u
Vertical Displacement
:G Shear Modulus
: Angle of Twist:w
Axial Displacement
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Chapter 1. Element : Bar
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Differential Eqn.
Rao,S.S.,Mechanical Vibrations, Fourth Edition, Prentice Hall, 2004,
z
xo
ac
bdx dx
( , )f x t
l
a
b
c
d
f dx
P dP
dx
Equilibrium position a
b
c
d u
u du
Displacedposition
P
( )u x
x
l
( )f x
( )u x
x0 l
an example of displacement in x-direction at x
:density, : Young s Modulus, :sectionalareaE A
( , ):external force per unit lengthf x tif is constantAand is time invariantf
P : the axial forces acting on the cross sections of a small element of the bar of length dx
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Differential Equ.
A Bar in Axial Vibration
2
2
( , )( ) ( )
u x tP dP f dx P A x dx
t
F ma
2
2
( , )( )
u x tdP f dx A x dx
t
2
2
( , ) ( , )( ) ( , ) ( )
u x t u x tEA x dx f x t dx A x dx
x x t
( , ):external force per unit lengthf x t
2
2
( )( ) 0
d u xEA f x
dx 0
u
t
2 2
2 2
( , ) ( , )( , )
u x t u x tEA f x t A
x t
dynamics (vibration)
( ) : .if A x A const
( , )( ) ( ) ( )
( , )( )
u x tP A x EA x EA x
x
P u x tdP dx EA x dx
x x x
‘constitutive equation’
“Longitudinal Vibration of a Bar or Rod : Rao,S.S.,Mechanical Vibrations, Fourth Edition, Prentice Hall, 2004, pp597-560
subjected to
0
0
( , 0) ( ), 0
( , 0) ( ), 0
u x t u x x l
ux t u x x l
t
I.V.P
(0, ) 0, 0
( , ) 0 or ( , ) 0, 0
u t t
u uAE l t l t t
x x
at the free end , axial force
B.V.P
statics
:density, : Young s Modulus, :sectionalareaE A
z
xo
ac
bdx dx
( , )f x t
l
a
b
c
d
a
b
c
d
f dx
P dP
u
u du
dx
Equilibrium position
Displacedposition
P
( )u x
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Variational MethodDifferential Equation
Boundary condition
2
2
( )( ) 0
d u xEA f x
dx
2
200
l d uEA f u dx
dx
2
0( ) 0
2
l EA duf u dx
dx
00 , 0
xx l
duu EA
dx
multiply by and integrateu
2
20
l d uEA u f u dx
dx
L.H.S:
0
0
ll d udu du
EA u EA f u dxdx dx dx
0
l du duEA f u dx
dx dx
2
0( )
2
l EA duf u dx
dx
integration by part
d du u
dx dx
21
2u u u
f u fu
21
2
u u u
x x x
( ) ( )b b
a ah x dx h x dx
operation
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Rayleigh-Ritz methodDifferential Equation
Boundary condition
2
2
( )( ) 0
d u xEA f x
dx
2
0( ) 0
2
l EA duf u dx
dx
00 , 0
xx l
duu EA
dx
Variational Method2
1 2u a x a x
2
2
1 2 1 20
2 02
l EAa a x f a x f a x dx
1
n k
kku a x
Rayleigh-Ritz method
assume,
2
0( ) 0
2
l EA duf u dx
dx
solution
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
(solution)
2 3 32 2
1 2 1 1 2 2
4
2 3 3
f l l f lEAl a EAl a a EAl a EA a a
2
2
1 2 1 20
22
l EAa a x f a x f a x dx
2 2 2 21 1 2 2 1 20
4 42
l EAa a a x a x f a x f a x dx
3 2 32 2 2
1 1 2 2 1 2
0
42
2 3 2 3
l
EA x x xa x a a x a f a f a
3 2 32 2 2
1 1 2 2 1 2
42
2 3 2 3
EA l f l f ll a l a a a a a
3 2 32 2
1 1 1 2 1 2 2 2 1 2
82 2 2
2 3 2 3
EA l f l f ll a a l a a l a a a a a a
3 2 32 2
1 1 1 2 1 2 2 2 1 2
4
3 2 3
l f l f lEAl a a EAl a a EAl a a EA a a a a
8/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
(solution)
2 3 32 2
1 2 1 1 2 2
4
2 3 3
f l l f lEAl a EAl a a EAl a EA a a
2
2
1 2 1 20
22
l EAa a x f a x f a x dx
22
1 2
3 32
1 2
02
40
3 3
f lEAl a EAl a
l f lEAl a EA a
2
2
1 2 1 20
2 02
l EAa a x f a x f a x dx
since
22
13
322
24
33
f ll l
aEA l
a f ll
9/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
(solution)
1
2
2
f l
a EA
fa
EA
5 5
1
4 4 42
2
3 3 3
2 3
f l f l
a
EAla f l f l
23
21
4 322
43 2
3
3
f ll
a l
EAla f ll l
5
4 4
3 3
26
f l f l
EA
fEAl f l
EA
22
13
322
24
33
f ll l
aEA l
a f ll
10/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Rayleigh-Ritz methodDifferential Equation
Boundary condition
2
2
( )( ) 0
d u xEA f x
dx
2
0( ) 0
2
l EA duf u dx
dx
00 , 0
xx l
duu EA
dx
Variational Method
1
n k
kku a x
2
1 2u a x a x
Rayleigh-Ritz method
2
2
1 2 1 20
2 02
l EAa a x f a x f a x dx
2( )2
f l fu x x x
EA EA
assume,
2
0( ) 0
2
l EA duf u dx
dx
1
2
2
f l
a EA
fa
EA
11/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Finite Element Method
2
0( ) 0
2
l EA duf u dx
dx
Variational Method
x
( )u xl
1u 2u( )u x
1 2( )u x c c x assume: 1 2, (0) , ( )u u u l u
discretization
1 element , 2 nodesfinite element method
12/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Finite Element Method
1 2( )u x c c x assume:
1 2( )u l c c l
1 2, ( ) 1x x
or u x u ul l
1 2, (0) , ( )u u u l u
1(0)u c 1 1c u
2 12
u uc
l
2 11( )
u uu x u x
l
1u 2u( )u x 2
0( ) 0
2
l EA duf u dx
dx
Variational Method
13/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Finite Element Method
1 2( ) 1x x
u x u ul l
1
2
( ) 1 1 udu x
udx l l
( ) ,u x Nd( )du x
dx Bd
1
2
( ) 1ux x
u xul l
differentiation with respect to x
1
2
1 11 , ,
ux xwhere
ul l l l
N B d
1u 2u( )u x2
0( ) 0
2
l EA duf u dx
dx
Variational Method
14/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Finite Element Method
1 2( ) 1x x
u x u ul l
( )u x Nd
( )du x
dx Bd
2
0( ) 0
2
l EA duf u dx
dx
0 0
( ) 02
l lT TEA dx f dx
d B Bd Nd
derivation
1u 2u( )u x
1
2
1 11 , ,
ux xwhere
ul l l l
N B d
2
0( ) 0
2
l EA duf u dx
dx
Variational Method
15/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
(derivation)2
0( )
2
l EA duf u dx
dx
T T0 0
( )2
l lT TEA dx f dx
d B Bd d N
T T T0 0
1( )
2
l lTEA dx f dx
d B B d d N
T T1
2
d Kd d F
T T
d Kd d F
T
d Kd F
0
0
2 0
1
1 1
1
1 1
1 1
1 1
1 1
lT
l
l
EA dx
lEA dx
l l
l
EAdx
l
EA
l
K B B
T TT1 1
2 2 d Kd d K d d F
T
0
1
2
( )l
f dx
u
u
F N
d
1
1 1
x x
l l
l l
N
B
T T T1 1
2 2 d Kd d Kd d F
T T d Kd d K d
TK Ksymmetry
0 0
( )2
l lT TEA dx f dx
d B Bd Nd
T T T :f f f scalar Nd Nd d N Nd
16/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
* * * ** *1 2 1 2
1 1 2 2
1 2
( , ) ( , )( ) ( )
f x x f x xx x x x
x x
(cf. Taylor Series for a Function of Two Variables)
* *
1 2 1 2( , ) ( , )f x x f x x
* *( ) ( ) ( )Tf f f x x x d
*1 * *1
*2 2
, ,x x
x x
x x d x x
* * * ** *1 2 1 2
1 1 2 2
1 2
( , ) ( , )( ) ( )
f x x f x xx x x x
x x
** * * *1 11 2 1 2
*
1 2 2 2
( , ) ( , ) x xf x x f x x
x x x x
* *( ) ( )Tf x x x
* *
1 2
*1 1 1
** *2 21 2
2
( , )
( , )
f x x
x x x
x xf x x
x
T
2 * * 2 * * 2 * ** 2 * * * 21 2 1 2 1 2
1 1 1 1 2 2 2 22 2
1 1 2 2
( , ) ( , ) ( , )1( ) 2 ( )( ) ( )
2
f x x f x x f x xx x x x x x x x R
x x x x
Taylor Series for a Function at ),( 21 xxf ),(*
2
*
1 xx
17/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
* * * ** *1 2 1 2
1 1 2 2
1 2
( , ) ( , )( ) ( )
f x x f x xx x x x
x x
(cf. Taylor Series for a Function of Two Variables)
* *
1 2 1 2( , ) ( , )f x x f x x
*1 * *1
*2 2
, ,x x
x x
x x d x x
2 * * 2 * * 2 * ** 2 * * * 21 2 1 2 1 2
1 1 1 1 2 2 2 22 2
1 1 2 2
( , ) ( , ) ( , )1( ) 2 ( )( ) ( )
2
f x x f x x f x xx x x x x x x x R
x x x x
2 * * 2 * * 2 * ** 2 * * * 21 2 1 2 1 2
1 1 1 1 2 2 2 22 2
1 1 2 2
( , ) ( , ) ( , )1( ) 2 ( )( ) ( )
2
f x x f x x f x xx x x x x x x x
x x x x
2 * * 2 * * 2 * * 2 * ** 2 * * * * * 21 2 1 2 1 2 1 2
1 1 1 1 2 2 1 1 2 2 2 22 2
1 1 2 1 2 2
( , ) ( , ) ( , ) ( , )1( ) ( )( ) ( )( ) ( )
2
f x x f x x f x x f x xx x x x x x x x x x x x
x x x x x x
2 * * 2 * * 2 * * 2 * *
* * * *1 2 1 * *2 1 2 1 21 1 2 21 1 2 2 1 1 2 22 2
1 1 2 1 2 2
( , ) ( , )( ) (
( , ) ( , )1( ) ( ) ( ) ( )
2)
f x x f x x f x x f x xx x x x x x x x
x x x xx x x
xx
x
* * * *
1 1 2 2 1 1 2 2
2 2 2 2
2 2
1 2 1 1 2 2
*
1 1
*
2 2
( ) ( ) ( )2
(1
)x x x x xf f f f
x
x
x x x xx x x
x
x
x x
Taylor Series for a Function at ),( 21 xxf ),(*
2
*
1 xx
18/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
* * * ** *1 2 1 2
1 1 2 2
1 2
( , ) ( , )( ) ( )
f x x f x xx x x x
x x
(cf. Taylor Series for a Function of Two Variables)
* *
1 2 1 2( , ) ( , )f x x f x x
*1 * *1
*2 2
, ,x x
x x
x x d x x
2 * * 2 * * 2 * ** 2 * * * 21 2 1 2 1 2
1 1 1 1 2 2 2 22 2
1 1 2 2
( , ) ( , ) ( , )1( ) 2 ( )( ) ( )
2
f x x f x x f x xx x x x x x x x R
x x x x
2 * * 2 * * 2 * ** 2 * * * 21 2 1 2 1 2
1 1 1 1 2 2 2 22 2
1 1 2 2
( , ) ( , ) ( , )1( ) 2 ( )( ) ( )
2
f x x f x x f x xx x x x x x x x
x x x x
* * * *
1 1 2 2 1 1 2 2
2 2 2 2
2 2
1 2 1 1 2 2
*
1 1
*
2 2
( ) ( ) ( )2
(1
)x x x x xf f f f
x
x
x x x xx x x
x
x
x x
2 2
2 *1 1 2 1 1
*2 22 2
2
2
* *
1
1 2
1 2 2
1
2
f f
x x x x x
x xf f
x x x
x x x x
* * *1( ) ( ) ( ) ( )2
T Tf f f R x x x d d H x d
Taylor Series for a Function at ),( 21 xxf ),(*
2
*
1 xx
19/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
* * * ** *1 2 1 2
1 1 2 2
1 2
( , ) ( , )( ) ( )
f x x f x xx x x x
x x
(cf. Taylor Series for a Function of Two Variables)
* *
1 2 1 2( , ) ( , )f x x f x x
*1 * *1
*2 2
, ,x x
x x
x x d x x
2 * * 2 * * 2 * ** 2 * * * 21 2 1 2 1 2
1 1 1 1 2 2 2 22 2
1 1 2 2
( , ) ( , ) ( , )1( ) 2 ( )( ) ( )
2
f x x f x x f x xx x x x x x x x R
x x x x
2 2
2 *1 1 2 1 1
*2 22 2
2
2 1
* *
1 1 2
2
2
1
2
f f
x x x x x
x xf fx
x
x x
x x
x
** * * *1 11 2 1 2
*
1 2 2 2
( , ) ( , ) x xf x x f x x
x x x x
* * *1( ) ( ) ( ) ( )2
T Tf f f R x x x d d H x d
Taylor Series for a Function at ),( 21 xxf ),(*
2
*
1 xx
20/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Finite Element Method
1 2( ) 1x x
u x u ul l
( )u x Nd
( )du x
dx Bd
2
0( ) 0
2
l EA duf u dx
dx
0 0
( ) 02
l lT TEA dx f dx
d B Bd Nd
derivation
1u 2u( )u x
1
2
1 11 , ,
ux xwhere
ul l l l
N B d
T
0 d Kd F
-Chapter 1. Element : Bar
1 1,
1 1
EAwhere
l
K
T
0, ( )
l
f dx F N Kd F
2
0( ) 0
2
l EA duf u dx
dx
Variational Method
21/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
x
( )u x
Element : Bar - Finite Element Method
1u 2u( )u x
1
2
1 1
1 1
uEA
ul
F
T
0, ( )
l
f dx F N
Kd F
( ) :f x f const
constant external force per unit lengthequivalent nodal forces
equivalent nodal forces
x
( )u xl1 2( )u x c c x assume:
1 2, (0) , ( )u u u l u
T
0( )
l
f dx N 0
1l
x
lf dx
x
l
2
2
0
2
2
l
xx
lf
x
l
1
2
f
f
F
1u 2u( )u x
1
1
2f f 1
1
2f f
1
2
1
2
f l
f l
2
0( ) 0
2
l EA duf u dx
dx
Variational Method
22/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
x
( )u x
Element : Bar - Finite Element Method
1u 2u( )u x
1
2
1 1
1 1
uEA
ul
F
T
0, ( )
l
f dx F N
Kd F
( ) :f x f const
constant external force per unit length
equivalent nodal forces
x
( )u xl1 2( )u x c c x assume:
1 2, (0) , ( )u u u l u
x
( )u x
( ) :f x f const
boundary conditionequivalent nodal forces free body diagram
( )u x
( ) :f x f const
f l
1u 2u( )u x
1
1
2f f 1
1
2f f
1
2
1
2
f l
f l
1
2
f
f
F
1
2
1
2
f l
f l
1
2
f lf
f
F
2
1
1 1 2
1 1 1
2
0f l
EA
ulf l
and 1 0u
1 0u 2u( )u x
1
1
2f f 1
1
2f f
2
0( ) 0
2
l EA duf u dx
dx
Variational Method
23/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Finite Element Method
x
( )u x
( ) :f x f const
1
2f l
free body diagram
f l
1
2f lf l 1
4f lequivalent
nodal forces
f l
1
4f l
f l
( ) :f x f const ( ) :f x f const
2
4f l
Kd F2
1
01 1 2
1 1 1
2
f lEA
ulf l
1 0u 2u( )u x
1
1
2f f
1
1
2f f
1 element , 2 nodes
1 0u 3u( )u x
1
3
4f f
3
1
4f f
2 element , 3 nodes
2u
2
2
4f f
l
2
l
4
l 2
4 4
l
/ 2ll
F.E.M
2
3
3
41 1 0 02
1 1 1 1/ 2 4
0 1 11
4
f l
EAu f l
lu
f l
superposition of stiffness matrix
2
0( ) 0
2
l EA duf u dx
dx
Variational Method
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Finite Element Method
x
( )u x
( ) :f x f const
Kd F2
1
01 1 2
1 1 1
2
f lEA
ulf l
1 0u 2u( )u x
1
1
2f f
1
1
2f f
1 element , 2 nodes
1 0u 3u( )u x
1
3
4f f
3
1
4f f
2 element , 3 nodes
2u
2
2
4f f
l
F.E.M
2
3
3
41 1 0 02
1 1 1 1/ 2 4
0 1 11
4
f l
EAu f l
lu
f l
superposition of stiffness matrix
2
1 2
10,
2
f lu u
EA
1 2( ) 1x x
u x u ul l
sol)2
2
3
8
f lu
EA
sol)
2
3
4,
8
f lu
EA
,0 x l
1 0,u
givenfind
displacement solution
2
0( ) 0
2
l EA duf u dx
dx
Variational Method
25/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
(solution)1 0u 3u
( )u x
2 element , 3 nodes
2u
1 2
2 3
1 ,0/ 2 / 2 2
( )
2 1 ,/ 2 / 2 2
x x lu u x
l lu x
x x lu u x l
l l
1 1
1 2( )u x c c x
1 2, (0) , ( / 2)u u u l u 2 3, (0) , ( )u u u l u
2 2
1 2( )u x c c x
) 02
li x ) 2
lii x l
1 2, ( ) 1/ 2 / 2
x xor u x u u
l l
1 2( / 2) / 2u l c c l
1
1(0)u c1
1 1c u
2 12
/ 2
u uc
l
2 11( )
/ 2
u uu x u x
l
0x / 2x l x l
2 2
1 2 2( / 2) / 2u l c c l u 2
1 2 32c u u
2 3 22
/ 2
u uc
l
2 2
1 2 3( )u l c c l u
3 22 3( ) 2
/ 2
u uu x u u x
l
2 3, ( ) 2 1/ 2 / 2
x xor u x u u
l l
26/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Finite Element Method
x
( )u x
( ) :f x f const
Kd F2
1
01 1 2
1 1 1
2
f lEA
ulf l
1 0u 2u( )u x
1
1
2f f
1
1
2f f
1 element , 2 nodes
1 0u 3u( )u x
1
3
4f f
3
1
4f f
2 element , 3 nodes
2u
2
2
4f f
l
F.E.M
2
3
3
41 1 0 02
1 1 1 1/ 2 4
0 1 11
4
f l
EAu f l
lu
f l
superposition of stiffness matrix
2
1 2
10,
2
f lu u
EA
1 2( ) 1x x
u x u ul l
sol)2
2
3
8
f lu
EA
sol)
2
3
4,
8
f lu
EA
,0 x l
1 0,u
1 2
2 3
1 ,0/ 2 / 2 2
( )
2 1 ,/ 2 / 2 2
x x lu u x
l lu x
x x lu u x l
l l
givenfind
displacement
given : x
find : ( )u x
2
0( ) 0
2
l EA duf u dx
dx
Variational Method
27/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - ComparisonDifferential Equation
Boundary Condition
2
2
( )( ) 0
d u xEA f x
dx
2
0( ) 0
2
l EA duf u dx
dx
00 , 0
xx l
duu EA
dx
VariationalMethod
Rayleigh-Ritz Method
2( )2
f l fu x x x
EA EA
x
l ( )u x
( ) :f x f const
2 23
2 2 2 2 8
l f l l f l f lu
EA EA EA
for example,
2
2 1
2 2
f l f f lu l l l
EA EA EA
Finite Element Method
mathematical modelingby using Newton’s second law
1 2( ) 1x x
u x u ul l
1 2
2 3
1 ,0/ 2 / 2 2
( )
2 1 ,/ 2 / 2 2
x x lu u x
l lu x
x x lu u x l
l l
1 0u 2u( )u x
1
1
2f f
1
1
2f f
1 element , 2 nodes
1 0u 3u( )u x
1
3
4f f
3
1
4f f
2 element , 3 nodes
2u
2
2
4f f
2
1 2
10,
2
f lu u
EA
2 2
1 2 3
3 40, ,
8 8
f l f lu u u
EA EA
2 2/ 2 1 1
2 2 4
l l f l f lu
l EA EA
2
2
1
2
f lu l u
EA
2
2
3
2 8
l f lu u
EA
2 2
3
4 1
8 2
f l f lu l u
EA EA
28/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Classification
ref. Logan D.L., A First Course in the Finite Element Method, Third edition, Brooks/Cole, p.116
We developed the bar finite element equations by the direct method in Section 3.1 and by the potential energy
method (one or a number of variational methods) in Section 3.10.
In fields other than structural/solid mechanics, it is quite probable that a variational principle, analogous to the
principle or minimum potential energy, for instance, may not be known or even exist. In some flow problems in
fluid mechanics and in mass transport problems (Chapter 13), we often have only the differential equation and
boundary conditions available. However, the finite element method can still be applied.
The methods or weighted residuals applied directly to the differential equation can be used to develop the finite
element equations. In this section, we describe Galerkin's residual method in general and then apply it to the bar
element. This development provides the basis for later applications of Galerkin's method to the nonstructural
heat-transfer element (specifically, the one-dimensional combined conduction, convection, and mass transport
element described in Chapter 13). Because of the mass transport phenomena, the variational formulation is not
known (or certainly is difficult to obtain), so Galerkin's method is necessarily applied to develop the finite element
equations.
Virtual Work
Potential Energy Method
Weighted Residual
Differential Equation Variational Method sol) Rayleigh-Ritz
applicable only to structural problem
sol) F.E.M
Collocation
Least Square
Galerkin
Integral Form
discretize
Algebraic Equation
29/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
-Chapter 1. Element : Bar
Galerkin’s Residual Method
V
R dV
1
2
1 ,ux x
whereul l
N d
Differential Equation2
2
( )0
d u xEA
dx
( )u x Nd
test function
since it is approximated solution
2
2
( )d u xEA
dx 0 Rresidual
Thus substituting the approximated solution into
the differential equation results in a residual over
the whole region of the problem as follows
In the residual method, we require that a weighted value of
the residual be a minimum over the whole region. The
weighting functions allow the weighted integral of
residuals to go to zero
0V
R W dV weighting function or
the basis functions are chosen to play the role of
the weighting functions W
Galerkin Method
1x x
l l
N
1N 2N
iN
0 ,( 1,2)iV
R N dV i
1
0.) ( ) 0ref u v uv xv dx
basis function
30/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Galerkin’s Residual Method
the test functions are chosen to play the role of
the weighting functions W
Galerkin Method
iN
0 ,( 1,2)iV
R N dV i weighting function
residual (test function used)N
Differential Equation
2
2
( )0
d u xEA
dx
2
20
( )0 ,( 1,2)
l
i
d u xAE N dx i
dx
Bar - Galerkin’s Residual Method
integration by parts
00
0
ll
ii
dNdu duN AE AE dx
dx dx dx
1 1 2 2, ( )where u x N u N u Nd 1 2, 1 ,x x
N Nl l
1
2
1 1 u
ul l
1
002
1 1,( 1,2)
ll
ii
udN duAE dx N AE i
udx l l dx
1 21 2
dN dNduu u
dx dx dx since
111
002
1 1l
l udN duAE dx N AE
udx l l dx
122
002
1 1l
l udN duAE dx N AE
udx l l dx
1:i
2 :i
1 1
0x l x
du duN AE N AE
dx dx
2 2
0x l x
du duN AE N AE
dx dx
1 2(0) 1, ( ) 0N N l
2 2(0) 0, ( ) 1N N l
sincedu
AE AE A fdx
since
1 0xN f
2 x lN f
1f
2f
1
0.) ( ) 0ref u v uv xv dx
31/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Galerkin’s Residual Method
the test functions are chosen to play the role of
the weighting functions W
Galerkin Method
iN
0 ,( 1,2)iV
R N dV i weighting function
residual (test function used)N
Differential Equation
2
2
( )0
d u xEA
dx
2
20
( )0 ,( 1,2)
l
i
d u xAE N dx i
dx
Bar - Galerkin’s Residual Method
integration by parts
1 1 2 2, ( )where u x N u N u Nd 1 2, 1 ,x x
N Nl l
111
02
1 1l udNAE dx f
udx l l
122
02
1 1l udNAE dx f
udx l l
1 1 1 12 20 02 2 2 2
1 1 1 1 11 1 1 1
l lu u u uAEAE dx AE dx AE l l
u u u ul l l l l l
1 1 1 12 20 02 2 2 2
1 1 1 1 11 1 1 1
l lu u u uAEAE dx AE dx AE l l
u u u ul l l l l l
1 1
2
1 1uAE
ful
1 22
1 1uAE
ful
1 1
2 2
1 1, , ,
1 1
u fEAwhere
u fl
K d F Kd F
1
0.) ( ) 0ref u v uv xv dx
32/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Potential Energy Approach
the principle of minimum potential energy
Of all the geometrically possible shapes that a body can assume,the true one, corresponding to the safisfaction of stable equilibrium of the body, is identified by a minimum value of the total potential energy
the total potential energy Π is defined as the
sum of the internal strain energy Πin and the potential energy of the external forces Πext
in ext
33/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Potential Energy Approach the total potential energy Π is defined as the
sum of the internal strain energy Πin and the potential energy of the external forces Πext
in ext
1
2in in x x
V V
d dV the strain energy for one-dimensional stress.
x
x
Linear-elastic
(Hooke’s law)material
x xE
0
x
in xd d dxdydz
xd
0 x
0
x
x xE d dxdydz
21
2x xE d dxdydz
1
2xdxdydz
To evaluate the strain energy for a bar,
we consider only the work done by the internal forces during
deformation.
34/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Potential Energy Approach
the total potential energy Π is defined as the
sum of the internal strain energy Πin and the potential energy of the external forces Πext
in ext
The potential energy of the external forces, being opposite in sign from the extenal work
expression because the potential energy of external forces is lost when the work is done by
the external forces, is given by
11
M
ext b x s ix i
iV S
X udV T u dS f u
l
1xf2xf
bX
xT1S
x
ubXbody forces typically from the self-weight of the bar (in units of force per unit volume) moving
through displacement function
susurface loading or traction typically from distributed loading acting along the surface of
the element (in units of force per unit surface area) moving through displacements
where are the displacements occurring over surface
xT
su 1S
nodal concentrated force moving through nodal displacements iuixf
35/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Potential Energy Approach the principle of minimum potential energy
Of all the geometrically possible shapes that a body can assume,the true one, corresponding to the safisfaction of stable equilibrium of the body, is identified by a minimum value of the total potential energy
the total potential energy Π is defined as the sum of the internal strain energy Πin and the potential energy of the
external forces Πext
in ext 1
2in x x
V
dV 1
1
,M
ext b x s ix i
iV S
X udV T u dS f u
1. Formulate an expression for the total potential energy.
2. Assume the displacement pattern to vary with a finite set of undetermined parameters (here
these are the nodal displacements ), which are substituted into the expression for total
potential energy.
3. Obtain a set of simultaneous equations minimizing the total potential energy with respect to
these nodal parameters. These resulting equations represent the element equations.
Apply the following steps when using the principle of minimum potential energy
to derive the finite element equations.
iu
36/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Potential Energy Approach the principle of minimum potential energy
Of all the geometrically possible shapes that a body can assume,the true one, corresponding to the safisfaction of stable equilibrium of the body, is identified by a minimum value of the total potential energy
the total potential energy Π is defined as the sum of the internal strain energy Πin and the potential energy of the
external forces Πext
in ext 1
2in x x
V
dV 1
1
,M
ext b x s ix i
iV S
X udV T u dS f u
l
1xf2xf x
1 1 2 202
l
x x x x
Adx f u f u
1. Formulate an expression for the total potential energy.
Apply the following steps when using the principle of minimum potential
energy to derive the finite element equations.dV Adx
2. Assume the displacement pattern to vary with a finite set of undetermined parameters (here these are the nodal
displacements ) ,which are substituted into the expression for total potential energy.
we have the axial displacement function expressed in terms of the shape functions and nodal displacements by
1
2
, 1 ,ux x
whereul l
N d( )u x Nd
assume that there is no surface traction and body force and the
sectional area is constantA
iu
1u2u
37/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Potential Energy Approach the principle of minimum potential energy
Of all the geometrically possible shapes that a body can assume,the true one, corresponding to the safisfaction of stable equilibrium of the body, is identified by a minimum value of the total potential energy
the total potential energy Π is defined as the sum of the internal strain energy Πin and the potential energy of the
external forces Πext
in ext 1
2in x x
V
dV 1
1
,M
ext b x s ix i
iV S
X udV T u dS f u
1 1 2 202
l
x x x x
Adx f u f u
1
2
, 1 ,ux x
whereul l
N d( )u x Nd
assume that there is no surface traction and body force and the
sectional area is constantA
1
2
1 1x
udu
udx l l
x xE E Bd
Bd
T T
02
lAE dx Bd Bd d F
1 T1 1 2 2 1 22
x
x x
x
ff u f u u u
f
d F
T T T
02
LEAdx d B Bd d F
l
1xf2xf x
dV Adx1u2u
38/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Potential Energy Approach the principle of minimum potential energy
Of all the geometrically possible shapes that a body can assume,the true one, corresponding to the safisfaction of stable equilibrium of the body, is identified by a minimum value of the total potential energy
the total potential energy Π is defined as the sum of the internal strain energy Πin and the potential energy of the
external forces Πext
in ext 1
2in x x
V
dV 1
1
,M
ext b x s ix i
iV S
X udV T u dS f u
1 1 2 202
l
x x x x
Adx f u f u
assume that there is no surface traction and body force and the
sectional area is constantA
1
2
, 1 ,ux x
whereul l
N d( )u x Nd
1
2
1 1x
udu
udx l l
x xE E Bd
Bd
1 T1 1 2 2 1 22
x
x x
x
ff u f u u u
f
d F
T T T
02
lEAdx d B Bd d F
3. Obtain a set of simultaneous equations minimizing the total potential
energy with respect to these nodal parameters. These resulting
equations represent the element equations.
The minimization of Π with respect to each nodal displacement requires that
1
0u
and
2
0u
l
1xf2xf x
dV Adx1u2u
39/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar - Potential Energy Approach the principle of minimum potential energy
Of all the geometrically possible shapes that a body can assume,the true one, corresponding to the safisfaction of stable equilibrium of the body, is identified by a minimum value of the total potential energy
the total potential energy Π is defined as the sum of the internal strain energy Πin and the potential energy of the
external forces Πext
in ext 1
2in x x
V
dV 1
1
,M
ext b x s ix i
iV S
X udV T u dS f u
1
2
1 1 u
ul l
Bd
1T 1 22
1 1 2 2
x
x
x x
fu u
f
f u f u
d F
T T T
02
lEAdx d B Bd d F
The minimization of Π with respect to each nodal displacement requires that
1
0u
and
2
0u
11 22
1
1 1
1
T T Tul
u uul l
l
d B D Bd 1
1 22
2
2 2
1 1 2 22
1 11
1 1
1( 2 )
uu u
ul
u u u ul
2 21 1 2 2 1 1 2 22 0
( 2 )2
l
x x
EAu u u u dx f u f u
l
1 2 1 1 2 12 01
(2 2 ) ( )2
l
x x
EA EAu u dx f u u f
u l l
1 2 2 1 2 22 02
( 2 2 ) ( )2
l
x x
EA EAu u dx f u u f
u l l
1 2 1
1 2 2
( ) 0
( ) 0
x
x
EAu u f
l
EAu u f
l
In matrix form, we express
11
2 2
1 1
1 1
x
x
fuEA
u fl
11
22
1 1, , ,
1 1
x
x
fuEAwhere
ful
K d F Kd F
40/41
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SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.krSeoul NationalUniv.
2009 Fall, Computer Aided Ship Design, Part3 Grillage Analysis of Midship Cargo Hold 01- Bar Element
Element : Bar
Bar
Element
2
2
( )( ) 0
d u xEA f x
dx
Differential Equation
Mx F , 0where x
VariationalMethod
2
0( ) 0
2
l EA duf u dx
dx
1 1
2 2
1 1
1 1
u fEA
u fl
0 1( )u x a a x
Kd F
Finite Element Method
•Discretization•Approximation
f1 f2
kNode1 Node 2
δ1 δ2
! Notation
2
1
2
1
kk
kk
f
f
]][[][ Kf
EAk
l
: , :E Young s Modulus A sectional area
stiffness matrix
Galerkin’sWeighted Residual
Energy Method
from now on,we will use this!
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