2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Naval Arc
hitectu
re &
Ocean E
ngin
eering
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul 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
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
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
2/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Chapter 2. Element : Beam
3/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Differential Eqn.
x ε i
① strain at in x-direction : y
,x x σ i ε i , , y θ k y j E
ds
d ds
1d
ds
( )y d d d yy
ds ds
, : initial length
, :elongated length
ds
y d
ds
x
y
neutralsurface y
dx
d
ds
xσ* neutral surface : Longitudinal lines on the lower part of the beam are elongated, whereas those on the upper part are shortened. Thus the lower part of the beam is in tension and the upper part is in compression. Somewhere between the top and bottom of the beam is a surface in which longitudinal lines do not change in length. This surface is called neutral surface
4/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Differential Eqn.
x ε i
,x
yE where
σ i i
( )x xd dA dA dA F σ i i③ force acting on in x-direction : dA
④ moment about z-axis :
2
( ) ( )y y
d d y E dA E dA
M y F j i k
② stress at in x-direction :y
① strain at in x-direction : y
,x x σ i ε i , , y θ k y j E
ds
d ds
1d
ds
x
yE
σ i i
yd E dA
F i
( )y d d d yy
ds ds
, : initial length
, :elongated length
ds
y d
neutralsurface
y
dx
d
ds
ds
x
y
xσ
5/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Differential Eqn.
ds
d ds
1d
ds
① strain at in x-direction :
x ε i
,x x σ i ε i
④ moment about z-axis :
2
( ) ( )y y
d d y E dA E dA
M y F j i k
y
, , y θ k y j
( )y d d dy
ds ds
E
② stress at in x-direction :y x
yE
σ i i
③ force acting on in x-direction : dAy
d E dA
F i
2
A A
yd E dA
M M k
⑤ assume dx
dydxds tan,
2
2
d d y
ds dx
2
2
d d dy d dy d y
ds ds dx dx dx dx
AdAyI 2Define then, ,
EI EIM
M k EI
M k
, : initiallength, :elongatedlengthd y d
2
2,
d yM EI
dx
2
2
d yEI
dxM k
dEI
ds
M k
neutralsurface
y
dx
d
ds
ds
x
y
xσ
6/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Differential Eqn.1d
ds
① strain at in x-direction :
x ε i
,x x σ i ε i
④ moment about z-axis :2
( ) ( )y y
d d y E dA E dA
M y F j i k
y
, , y θ k y j
( )y d d dy
ds ds
E
② stress at in x-direction :y x
yE
σ i i
③ force acting on in x-direction : dAy
d E dA
F i
2
A A
yd E dA
M M k
⑤ assume dx
dydxds tan,
2
2
d d y
ds dx
2,A
I y dA EI
k
, : initiallength, :elongatedlengthd y d
2
2
d yEI
dxM k
x
y
( )f x
neutralsurface
y
dx
d
ds
ds
x
y
xσ
EI dEI
ds
M k k
⑥ relationships between loads, shear forces, and bending moments
1 2 1 2, , ,V M
V V dx M M dxx x
V j V j M k M k
•force equilibrium1 2 ( ) 0y x F V V f
11 1
11 1
( ) 0
( ) 0
VV V dx f x dx
x
VV V dx f x dx
x
j j j
j ( )dV
f xdx
2
2,
d yM EI
dx
7/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Differential Eqn.1d
ds
① strain at in x-direction :
x ε i
,x x σ i ε i
④ moment about z-axis :2
( ) ( )y y
d d y E dA E dA
M y F j i k
y
, , y θ k y j
( )y d d dy
ds ds
E
② stress at in x-direction :y x
yE
σ i i
③ force acting on in x-direction : dAy
d E dA
F i
2
A A
yd E dA
M M k
⑤ assume dx
dydxds tan,
2
2
d d y
ds dx
2,A
I y dA EI
k
, : initiallength, :elongatedlengthd y d
x
y
( )f x
neutralsurface
y
dx
d
ds
ds
x
y
xσ
⑥ relationships between loads, shear forces, and bending moments
1 2 1 2, , ,V M
V V dx M M dxx x
V j V j M k M k
•force equilibrium ( )dV
f xdx
( )dM
V xdx
•moment equilibrium 1 2 2
1( ) 0
2z d d x dx M M M x V x f
1
( ) 02
M VM M dx dx V dx dx f x dx
x x
k k i j i j
2
2
d yEI
dxM k
EI dEI
ds
M k k
2
2,
d yM EI
dx
8/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Differential Eqn.1d
ds
① strain at in x-direction :
x ε i
,x x σ i ε i
④ moment about z-axis :2
( ) ( )y y
d d y E dA E dA
M y F j i k
y
, , y θ k y j
( )y d d dy
ds ds
E
② stress at in x-direction :y x
yE
σ i i
③ force acting on in x-direction : dAy
d E dA
F i
2
A A
yd E dA
M M k
⑤ assume dx
dydxds tan,
2
2
d d y
ds dx
2,A
I y dA EI
k
, : initiallength, :elongatedlengthd y d
x
y
( )f x
neutralsurface
y
dx
d
ds
ds
x
y
xσ
⑥ relationships between loads, shear forces, and bending moments
1 2 1 2, , ,V M
V V dx M M dxx x
V j V j M k M k
•force equilibrium ( )dV
f xdx
( )dM
V xdx
•moment equilibrium
2
2
d yEI
dxM k
EI dEI
ds
M k k
2
2,
d yM EI
dx
2
2
d y M
dx EI
3
3
1 1( )
d y dMV x
dx EI dx EI
4
4
1 1( )
d y dVf x
dx EI dx EI
4
4( )
d yEI f x
dx
9/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Variational MethodDifferential Equation
Boundary condition
4
4( ) 0
d vEI f x
dx
4
400
l d vEI f v dx
dx
multiply by and integrateu
4
40
l d vEI v f v dx
dx
L.H.S:
3 3
3 30 00
ll ld vd v d v
EI v EI dx f v dxdx dx dx
3
30 0
l ld vd vEI dx f v dx
dx dx
2 2 2
2 2 20 00
ll ld v d v d v d v
EI EI dx f v dxdx dx dx dx
integration by part
d dv v
dx dx
21
2v v v
f v fv
( ) ( )b b
a ah x dx h x dx
operation
0
2 2
2 2
0
0 , 0
, 0, 0
x x l
x x l
v v
d v d vEI EI
dx dx
2 2
2 20 0
l ld v d vEI dx f v dx
dx dx
22 2 2
2 2 2
1
2
v v v
x x x
10/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Variational MethodDifferential Equation
Boundary Condition
4
4( ) 0
d vEI f x
dx
4
400
l d vEI f v dx
dx
multiply by and integrateu
4
40
l d vEI v f v dx
dx
L.H.S:
integration by part
d dv v
dx dx
21
2v v v
f v fv
22 2 2
2 2 2
1
2
v v v
x x x
( ) ( )b b
a ah x dx h x dx
operation
0
2 2
2 2
0
0 , 0
, 0, 0
x x l
x x l
v v
d v d vEI EI
dx dx
2 2
2 20 0
l ld v d vEI dx f v dx
dx dx
2 2
2 20 0
1
2
l ld v d vEI dx fv dx
dx dx
2
20 2
l EI d vfv dx
dx
11/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Finite Element Method
Variational Method
x
( )v xl
1v 2v( )v x
12
2 3
0 1 2 3( )v x c c x c x c x assume: 1 2, (0) , ( )v v v l v
1 2, (0) , ( )dv dv
ldx dx
discretization
1 element , 2 nodesfinite element method
2
2
20 2
l EI d vfv dx
dx
12/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Finite Element Method
2 3
0 1 2 3 2( )v l c c l c l c l v
0(0)v c0 1c v
2 1 2 1 22
3 12c v v
l l
Variational Method1v 2v
( )v x
12
22
20 2
l EI d vfv dx
dx
2 3
0 1 2 3( )v x c c x c x c x assume: 1 2, (0) , ( )v v v l v
1 2, (0) , ( )dv dv
ldx dx
1(0)dv
cdx
2
1 2 3 2( ) 2 3dv
l c c l c ldx
1 1c 3 1 2 1 23 2
2 1c v v
l l
13/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Finite Element Method
0 1c v 2 1 2 1 22
3 1, 2c v v
l l
Variational Method1v 2v
( )v x
12
22
20 2
l EI d vfv dx
dx
2 3
0 1 2 3( )v x c c x c x c x assume: 1 2, (0) , ( )v v v l v
1 2, (0) , ( )dv dv
ldx dx
1 1,c
3 1 2 1 23 2
2 1c v v
l l
2 3
1 1 1 2 1 2 1 2 1 22 3 2
3 1 2 1( ) 2v x v x v v x v v x
l l l l
3 2 3 3 2 2 3 3 2 3 2 2
1 1 2 23 3 3 3
1 1 1 1( ) (2 3 ) ( 2 ) ( 2 3 ) ( )or v x x x l l v x l x l xl x x l v x l x l
l l l l
14/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Finite Element Method
1
1
1 2 3 4
2
2
( )
v
v x N N N Nv
Variational Method1v 2v
( )v x
12
22
20 2
l EI d vfv dx
dx
3 2 3 3 2 2 3 3 2 3 2 2
1 1 2 23 3 3 3
1 1 1 1( ) (2 3 ) ( 2 ) ( 2 3 ) ( )v x x x l l v x l x l xl x x l v x l x l
l l l l
3 2 3
1 3
1(2 3 )N x x l l
l
3 2 2 3
2 3
1( 2 )N x l x l xl
l
3 2
3 3
1( 2 3 )N x x l
l
3 2 2
4 3
1( )N x l x l
l
15/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Finite Element Method
1
21
1 2 3 42
2
2
( )
v
d v xB B B B
vdx
( ) ,v x Nd
2
2
( )d v x
dx Bd
1
1
1 2 3 4
2
2
( )
v
v x N N N Nv
differentiation with respect to twice x
Variational Method1v 2v
( )v x
12
22
20 2
l EI d vfv dx
dx
3 2 3 3 2 2 3 3 2 3 2 2
1 1 2 23 3 3 3
1 1 1 1( ) (2 3 ) ( 2 ) ( 2 3 ) ( )v x x x l l v x l x l xl x x l v x l x l
l l l l
1 3
1(12 6 )B x l
l
2
2 3
1(6 4 )B xl l
l
3 3
1( 12 6 )B x l
l 2
4 3
1(6 2 )B xl l
l
3 2 3
1 3
1(2 3 )N x x l l
l
3 2 2 3
2 3
1( 2 )N x l x l xl
l
3 2
3 3
1( 2 3 )N x x l
l 3 2 2
4 3
1( )N x l x l
l
16/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Finite Element Method
( )v x Nd
2
2
( )d v x
dx Bd
22
20( ) 0
2
l EI d vfv dx
dx
0 0
( ) 02
l lT TEI
dx f dx
d B Bd Nd
derivation
1
1
1 2 3 4 1 2 3 4
2
2
, ,
v
where N N N N B B B Bv
N B d
Variational Method1v 2v( )v x
12
2
2
20 2
l EI d vfv dx
dx
2 2
3 3 3 3
1 1 1 1(12 6 ) (6 4 ) ( 12 6 ) (6 2 )x l xl l x l xl l
l l l l
T B B
3
2
3
3
2
3
1(12 6 )
1(6 4 )
1( 12 6 )
1(6 2 )
x ll
xl ll
x ll
xl ll
17/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
(derivation)2
2
20( )
2
l EA d vfv dx
dx
T T
0 0( )
2
l lT TEI
dx f dx
d B Bd d N
T T T
0 0
1( )
2
l lTEI 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
lTEI dx K B B
T TT1 1
2 2 d Kd d K d d F
1 1 2 2[ ]Td v v
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 TEI
dx f dx
d B Bd Nd
T T T :f f f scalar Nd Nd d N Nd
2 2
3
2 2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
l l
l l l lEI
l ll
l l l l
T
0( )
l
f dx F N
18/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
(derivation)
2 2
3 3 3 3 3 3 3 3
2 2 2 2 2 2
3 3 3 3 3 3 3 3
3
1 1 1 1 1 1 1 1(12 6 ) (12 6 ) (12 6 ) (6 4 ) (12 6 ) ( 12 6 ) (12 6 ) (6 2 )
1 1 1 1 1 1 1 1(6 4 ) (12 6 ) (6 4 ) (6 4 ) (6 4 ) ( 12 6 ) (6 4 ) (6 2 )
1( 12 6
x l x l x l xl l x l x l x l xl ll l l l l l l l
xl l x l xl l xl l xl l x l xl l xl ll l l l l l l l
EI
x ll
2 2
3 3 3 3 3 3 3
2 2 2 2 2 2
3 3 3 3 3 3 3 3
1 1 1 1 1 1 1) (12 6 ) ( 12 6 ) (6 4 ) ( 12 6 ) ( 12 6 ) ( 12 6 ) (6 2 )
1 1 1 1 1 1 1 1(6 2 ) (12 6 ) (6 2 ) (6 4 ) (6 2 ) ( 12 6 ) (6 2 ) (6 2 )
x l x l xl l x l x l x l xl ll l l l l l l
xl l x l xl l xl l xl l x l xl l xl ll l l l l l l l
0
l
dx
2 2
3
2 2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
l l
l l l lEI
l ll
l l l l
2 2 2 2 3 2 2 2 2 3
6 6 6 6
2 2 3 2 2 3 4 2 2 3 2 2 3 4
6 6 6 6
2 2
6 6
1 1 1 1(144 144 36 ) (72 84 24 ) ( 144 144 36 ) (72 60 12 )
1 1 1 1(72 84 24 ) (36 48 16 ) ( 72 84 24 ) (36 36 8 )
1 1( 144 144 36 ) (
x xl l x l xl l x xl l x l xl ll l l l
x l xl l x l xl l x l xl l x l xl ll l l l
EI
x xl ll l
0
2 2 3 2 2 2 2 3
6 6
2 2 3 2 2 3 4 2 2 3 2 2 3 4
6 6 6 3
1 172 84 24 ) (144 144 36 ) ( 72 60 12 )
1 1 1 1(72 60 12 ) (36 36 8 ) ( 72 60 12 ) (36 24 4 )
l
dx
x l xl l x xl l x l xl ll l
x l xl l x l xl l x l xl l x l xl ll l l l
0
lTEI dx K B B
19/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Finite Element Method
T
0 d Kd F
-Chapter 2. Element : Beam
T
0, ( )
l
f dx F N Kd F
( )v x Nd
2
2
( )d v x
dx Bd
22
20( ) 0
2
l EI d vfv dx
dx
0 0
( ) 02
l lT TEI
dx f dx
d B Bd Nd
derivation
1
1
1 2 3 4 1 2 3 4
2
2
, ,
v
where N N N N B B B Bv
N B d
Variational Method1v 2v( )v x
12
2
2
20 2
l EI d vfv dx
dx
2 2
3
2 2
12 6 12 6
6 4 6 2,
12 6 12 6
6 2 6 4
l l
l l l lEIwhere
l ll
l l l l
K
20/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Finite Element Method
1
2 2
1
3
2
2 2
2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
vl l
l l l lEI
l l vl
l l l l
FT
0, ( )
l
f dx F N
Kd F
constant external force per unit length
equivalent nodal forces
equivalent nodal forces
2 3
0 1 2 3( )v x c c x c x c x assume:
1 2
1 2
, (0) , ( ) ,
(0) , ( )
v v v l v
v v l
T
0( )
l
f dx N
3 2 3
3
3 2 2 3
3
03 2
3
3 2 2
3
1(2 3 )
1( 2 )
1( 2 3 )
1( )
l
x x l ll
x l x l xll
f dx
x x ll
x l x ll
43 3
4 23 2 3
3 43
4 32
0
2
2
1 4 3 2
2
4 3
l
xx l xl
x xl x l l
fl x
x l
x xl l
1
1
2
2
f
m
f
m
F
1v2v( )v x
1
2fl
1
2fl
2
2
20 2
l EI d vfv dx
dx
Variational Method
1v 2v( )v x
12
x
( )v xl
x
( )v x
( ) :f x f const
2
2
2
12
2
12
lf
lf
lf
lf
21
12fl
21
12fl
21/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Finite Element Method
constant external force per unit length
equivalent nodal forces
boundary conditionequivalent nodal forces free body diagram
2
f l
2
2
0
12
0
12
lf
lf
2
2
2
2
012
2 2
0
12
lf
lf
lf
l lf f
lf
F
Variational Method
Kd F
2 3
0 1 2 3( )v x c c x c x c x assume:
1 2
1 2
, (0) , ( ) ,
(0) , ( )
v v v l v
v v l
1v 2v( )v x
12
x
( )v xl
1v2v( )v x
1 0f
21
12fl21
12fl
x
( )v x
( ) :f x f const
1v2v( )v x
1
2fl
1
2fl
21
12fl21
12fl
2 0f
x
( )v x
( ) :f x f const
x
( )v x
( ) :f x f const
2
f l
1
2 2
1
3
2
2 2
2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
vl l
l l l lEI
l l vl
l l l l
F T
0, ( )
l
f dx F N
1
1
2
2
f
m
f
m
F
2
2
2
12
2
12
lf
lf
lf
lf
2
2 2
1
3
2 2 22
0
012 6 12 6
6 4 6 2 12
12 6 12 6 0 0
6 2 6 4
12
l l lf
l l l lEI
l ll
l l l l lf
2
2
20 2
l EI d vfv dx
dx
22/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Finite Element Methodequivalent nodal forces
Variational Method
1v2v( )v x
1 0f
21
12fl21
12fl
2 0f
x
( )v x
( ) :f x f const
2
2 2
1
3
2 2 22
0
012 6 12 6
6 4 6 2 12
12 6 12 6 0 0
6 2 6 4
12
l l lf
l l l lEI
l ll
l l l l lf
Kd F
givenfind
3 3
1 1 2 20, , 0,24 24
fl flv v
EI EI
,0 x l
displacement
given : x
find : ( )v x
3 2 2 3 3 2 2
1 23
1( ) ( 2 ) ( )
24
fv x x l x l xl x l x l
EI l 2 2 3( ) (2 )
24
fv x x l xl
EI
2
2
20 2
l EI d vfv dx
dx
23/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Galerkin’s Residual Method
V
R dV
1
1
1 2 3 4
2
2
,
v
where N N N Nv
N d
Differential Equation4
4
( )0
d v xEI f
dx
( )u x Nd
test function
since it is approximated solution
4
4
( )d v xEA f
dx 0 R
residual
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
iN
0 ,( 1,2)i
V
R N dV i
1
0.) ( ) 0ref u v uv xv dx
basis function
24/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Galerkin’s Residual Method
the test functions are chosen to play the role of
the weighting functions W
Galerkin Method
iN
0 ,( 1,2)i
V
R N dV i weighting function
residual (test function used)N
Differential Equation
4
4
( )0
d v xEI f
dx
4
40
( )0 , ( 1,2,3,4)
l
i
d v xEI f N dx i
dx
Beam - Galerkin’s Residual Method
integration by parts
3 3
3 30 00
0 ,( 1,2,3,4)
ll l
ii i
d v d v dNN EI EI dx f N dx i
dx dx dx
2 2 3 2
2 2 3 20 00
0 ,( 1,2,3,4)
ll l
i ii i
d N d v d v dN d vEI dx EI N f N dx i
dx dx dx dx dx
1
0.) ( ) 0ref u v uv xv dx
, ( )where v x Nd
3 2 3
1 3
1(2 3 )N x x l l
l
3 2 2 3
2 3
1( 2 )N x l x l xl
l
3 2
3 3
1( 2 3 )N x x l
l
3 2 2
4 3
1( )N x l x l
l
integration by parts again
2
20 00
0 ,( 1,2,3,4)
ll l
i ii i
d N dNEI dx EI N V m f N dx i
dx dx
B d
3 2
3 2( ), ( )
d v d vV x m x
dx dx
Recall,
0 00
0
lT
l lT T Td
EI dx EI m V f dxdx
NB B d N N
In matrix form,
25/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Galerkin’s Residual Method
the test functions are chosen to play the role of
the weighting functions W
Galerkin Method
iN
0 ,( 1,2)i
V
R N dV i weighting function
residual (test function used)N
Differential Equation
4
4
( )0
d v xEI f
dx
4
40
( )0 ,( 1,2)
l
i
d v xEI f N dx i
dx
Beam - Galerkin’s Residual Method
integration by parts
1
0.) ( ) 0ref u v uv xv dx
, ( )where v x Nd
3 2 3
1 3
1(2 3 )N x x l l
l
3 2 2 3
2 3
1( 2 )N x l x l xl
l
3 2
3 3
1( 2 3 )N x x l
l
3 2 2
4 3
1( )N x l x l
l
2 2
3
2 2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
l l
l l l lEI
l ll
l l l l
0
lTEI dx K B B2 2 2 3 2 2 2
3
( ) 16 6 3 4 6 6 3 2
d xx xl x l xl l x xl x l xl
dx l
N
3 2 3 3 2 2 3 3 2 3 2 2
3
1( ) 2 3 2 2 3x x x l l x l x l xl x x l x l x l
l N
(0) 1 0 0 0N
( )
, 0 0 0 1x l
d x
dx
N
0
( ), 0 1 0 0
x
d x
dx
N
, ( ) 0 0 1 0l N
0 00
( ) ( ) ( ) ( ) (0) (0) (0) (0)
lT T
l lT T T Td d d
EI m V f dx m l l V l l m V f dxdx dx dx
N N NN N N N N
R.H.S
0 00
0
lT
l lT T Td
EI dx EI m V f dxdx
NB B d N N
K
26/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Galerkin’s Residual Method
the test functions are chosen to play the role of
the weighting functions W
Galerkin Method
iN
0 ,( 1,2)i
V
R N dV i weighting function
residual (test function used)N
Differential Equation
4
4
( )0
d v xEI f
dx
4
40
( )0 ,( 1,2)
l
i
d v xEI f N dx i
dx
Beam - Galerkin’s Residual Method
integration by parts
1
0.) ( ) 0ref u v uv xv dx
, ( )where v x Nd
3 2 3
1 3
1(2 3 )N x x l l
l
3 2 2 3
2 3
1( 2 )N x l x l xl
l
3 2
3 3
1( 2 3 )N x x l
l
3 2 2
4 3
1( )N x l x l
l
0 00
ll l
T dEI dx EI m V f dx
dx
NB B d N N
2 2
3
2 2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
l l
l l l lEI
l ll
l l l l
0
lTEI dx K B B
R.H.S
0 00
2
2
( ) ( ) ( ) ( ) (0) (0) (0) (0)
20 0 0 1
0 0 1 0 12( ) ( ) (0) (0)
0 1 0 0
21 0 0 0
12
lT T
l lT T T Td d d
EI m V f dx m l l V l l m V f dxdx dx dx
lf
lf
m l V l m Vl
f
lf
N N N
N N N N N
2
2
(0)2
(0)12
( )2
( )12
lV f
lm f
lV l f
lm l f
Kd F2 2
3
2 2
12 6 12 6
6 4 6 2,
12 6 12 6
6 2 6 4
l l
l l l lEIwhere
l ll
l l l l
K
K
F (0) 1 0 0 0N
( )
, 0 0 0 1x l
d x
dx
N
0
( ), 0 1 0 0
x
d x
dx
N
, ( ) 0 0 1 0l N
2
2
(0)2
(0)12
,
( )2
( )12
lV f
lm f
lV l f
lm l f
F
27/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - Galerkin’s Residual Method
the test functions are chosen to play the role of
the weighting functions W
Galerkin Method
iN
0 ,( 1,2)i
V
R N dV i weighting function
residual (test function used)N
Differential Equation
4
4
( )0
d v xEI f
dx
4
40
( )0 ,( 1,2)
l
i
d v xEI f N dx i
dx
1
0.) ( ) 0ref u v uv xv dx
, ( )where v x Nd
Beam - Galerkin’s Residual Method
For simple support beam,
x
( )v x
( ) :f x f const
(0) , (0) 0, ( ) , ( ) 02 2
l lV f m V l f m l 2
1 2 3 4
2
0
12[ ]
0
12
T
lf
f f f f
lf
F
Kd F2 2
3
2 2
12 6 12 6
6 4 6 2,
12 6 12 6
6 2 6 4
l l
l l l lEIwhere
l ll
l l l l
K1 2 3 4, [ ]Tf f f fF
2
1 2
2
3 4
(0) , (0)2 12
( ) , ( )2 12
l lf V f f m f
l lf V l f f m l f
recall
1m 2m
28/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - 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
29/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - 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.
30/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - 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
1
2 2
1 1
ext y s iy i i i
i iS
T v dS f v m
vbXbody forces typically from the self-weight of the bar (in units of force per unit volume) moving
through displacement function
svsurface 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
yT
sv1S
nodal concentrated force moving through nodal displacements iviyf
l
1yf 2m
yT
x
,y v
2 yf
1m
31/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - 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. 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.
iv
1
2 2
1 1
, ext y s iy i i i
i iS
T v dS f v m
32/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
l
1yf 2m
yT
x
,y v
2 yf
1m
Element : Beam - 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
Apply the following steps when using the principle of minimum potential
energy to derive the finite element equations.
assume that there is no surface traction and body force and the
sectional area is constantA
1
2 2
1 1
, ext y s iy i i i
i iS
T v dS f v m
dV dAdx
dS bdx
1
2 2
1 1
2 2
01 1
1
2
1
2
in ext
x x y s iy i i i
i iV S
l
x x y s iy i i i
i ix A
dV T v dS f v m
dAdx bT v dx f v m
The differential volume for the beam element
Width of beam, :b
The differential area over which the surface loading acts is
33/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - 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
2 2
01 1
1
2
l
x x y s iy i i i
i ix A
dAdx bT v dx f v m
assume that there is no surface traction and body force and the
sectional area is constantA
2
2x
du d vy
dx dx
1
2 2
1 1
, ext y s iy i i i
i iS
T v dS f v m
dvu y
dx
dv
dx
dv
dx
y
y Bd
x xE yE Bd
1
2 21
1 1 2 20 0
1 1 2
2
[ ]
y
l lT T T T T
y iy i i i y
i i y
f
mf dx f v m f dx v v
f
m
d N d N d F
T
02
lT TEI
dx d B Bd d F
0,
lT T
y s ybT v dx bT dx d N
l
1yf 2m
yT
x
,y v
2 yf
1m
y ybT f
34/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - 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
assume that there is no surface traction and body force and the
sectional area is constantA
T T T
02
lEIdx 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
0v
and
1
, 0
2 2
01 1
1
2
l
x x y s iy i i i
i ix A
dAdx bT v dx f v m
2
, 0v
2
, 0
1
2 2
1 1
, ext y s iy i i i
i iS
T v dS f v m
l
1yf 2m
yT
x
,y v
2 yf
1m
35/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - 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
T T T
02
lEIdx d B Bd d F
1
2 2
1
1 1 2 23
2
2 2
2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
T T
vl l
l l l lEIv v u
l l vl
l l l l
d B Bd
00
lTEI dx B B d F
0d
1
1
02
2
,
y
l
y
y
f
mwhere f dx
f
m
F N
Kd F2 2
3
2 2
12 6 12 6
6 4 6 2,
12 6 12 6
6 2 6 4
l l
l l l lEIwhere
l ll
l l l l
K
1
1
02
2
,
y
l
y
y
f
mf dx
f
m
F Nx
( )v x
( ) :f x f const
36/37
2009 Fall, Computer Aided Ship Design, Part 3. Grillage 02-Beam Element
SDAL@Advanced Ship Design Automation Lab.http://asdal.snu.ac.kr
Seoul NationalUniv.
Element : Beam - 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
Kd F2 2
3
2 2
12 6 12 6
6 4 6 2,
12 6 12 6
6 2 6 4
l l
l l l lEIwhere
l ll
l l l l
K
1
1
02
2
,
y
l
y
y
f
mf dx
f
m
F N
x
( )v x
( ) :f x f const
For simple support beam with uniform load f
,yf f 1
1
2
2
2
0
2
0
y
y
ff
m
ff
m
2 2
0
2
2
02 2
20
0 12 12,
0
2 2 2
0 12012
l
l lf f
f
l lf f
f dxf l l
f fl
fl
f
F N
37/37