Finite Element Method:Beam Element Beam Elementocw.snu.ac.kr/sites/default/files/NOTE/10530.pdf ·...
Transcript of Finite Element Method:Beam Element Beam Elementocw.snu.ac.kr/sites/default/files/NOTE/10530.pdf ·...
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Beam Element
¨ Beam: A long thin structure that is subject to the vertical loads. A beam shows more evident bending deformation than the torsion and/or axial deformation.
¨ Bending strain is measured by the lateral deflection and the rotation -> Lateral deflection and rotation determine the number of DoF (Degree of Freedom).
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
¨ Sign convention
1. Positive bending moment: Anti-clock wise rotation.
2. Positive load: $y-direction.
3. Positive displacement: $y-direction.
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
¨ The governing equation
- + = = -wdx dV or w dVdx
$$
0
V dx dM or V dMdx
$$
+ = =0
ˆ 0 orˆ
ˆ 0 orˆ
dVwdx dV wdx
dMVdx dM Vdx
- - = = -
- = =
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
¨ Relation between beam curvature (k ) and bending moment
k MEI
= =1r or
d vdx
MEI
2
2
$$
=
¨ Curvature for small slope (q = dv dx$ / $ ): kd vdx
=2
2
$$
r: Radius of the deflection curve, $v: Lateral displacement function along the $y-axis direction
E : Stiffness coefficient, I : Moment of inertia along the $z-axis
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Solving the equation with M ,
ddx
EI d vdx
w x2
2
2
2$$$
$FHG
IKJ = - a f
When EI is constant, and force and moment are only applied at nodes,
EI d vdx
4
4 0$$
=
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Step 1: To select beam element type
Step 2: To select displacement function
Assumption of lateral displacement
3 21 2 3 4ˆ ˆ ˆ ˆ ˆ( )v x a x a x a x a= + + +
- Complete 3-order displacement function is suitable, because it has four degree of
freedom (one lateral displacement and one small rotation at each node)
- The function is proper, because it satisfies the fundamental differential equation of a beam.
- The function satisfies continuity of both displacement and slope at each node.
Representing $v with functions of $d y1 , $d y2 , $f1, 2̂f
$ ( ) $v d ay0 1 4= =
dvdv
a$$
$01 3
a f= =f
$ $v L d a L a L a L aya f = = + + +2 13
22
3 4
dv Ldx
a L a L a$$
$a f= = + +f 2 1
22 33 2
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Replacing a1~a4 with $d y1 ,
$d y2 , $f1, 2̂f
$ $ $ $ $ $
$ $ $ $ $ $ $ $
vLd d
Lx
Ld d
Lx x d
y y
y y y
= - + +LNM
OQP
+ - - - +LNM
OQP + +
2 1
3 1 2
3 1 2 2 1 23
2 1 2 1 22
1 1
e j d i
e j d i
f f
f f f
Representing it in matrix form, $ $v N d= o t
$
$$$$
d
d
d
y
yo t =
RS||
T||
UV||
W||
1
1
2
2
f
f N N N N N= 1 2 3 4
NL
x x L L NLx L x L x L
NL
x x L NLx L x L
1 33 2 3
2 33 2 2 3
3 33 2
4 33 2 2
1 2 3 1 2
1 2 3 1
= - + = - +
= - + = -
$ $ $ $ $
$ $ $ $
c h c h
c h c h
where N1, N2, N3, N4 : shape functions of the beam element
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Step 3: To define strain-displacement relation and stress-strain relation
Assume that the equation of strain-displacement relation is valid
e x x ydudx
$, $$$b g = , $ $ $
$u y dv
dx= - => e x x y y d v
dx$, $ $ $
$b g = -2
2
Basic assumption: Cross-section of the beam sustains its shape after deformation by bending,
and generally rotates by degree of dv dx$ / $b g .
Bending moment-lateral displacement relation and shear force-lateral displacement relation
$ $ $$
$ $$
m x EI d vdx
V EI d vdx
b g = =2
2
3
3
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Step 4: To derive an element stiffness matrix and governing equations by direct stiffness
method
¨ Element stiffness matrix and governing equations
$ $ $$
$ $ $ $
$ $$$
$ $ $ $
$ $ $$
$ $ $ $
$ $$$
$
f V EId vdx
EIL
d L d L
m m EId vdx
EIL
Ld L d L
f V EId v Ldx
EIL
d L d L
m m EId v Ldx
EIL
L
y y y
y y
y y y
1
3
3 3 1 1 2 2
1
2
2 3 12
1 22
2
2
3
3 3 1 1 2 2
2
2
2 3
012 6 12 6
06 4 6 2
12 6 12 6
6
= = = + - +
= - = - = + - +
= - = - = - - + -
= = =
b g e jb g e jb g e j
b g
f f
f f
f f
d L Ld Ly y12
1 22
22 6 4+ - +$ $ $f fe j
- Matrix Form
$
$$
$
$$$$
fmfm
EIL
L LL L L L
L LL L L L
d
d
y
y
y
y
1
1
2
2
3
2 2
2 2
1
1
2
2
12 6 12 66 4 6 212 6 12 6
6 2 6 4
RS||
T||
UV||
W||=
--
- - --
L
N
MMMM
O
Q
PPPP
RS||
T||
UV||
W||
f
f
$k EIL
L LL L L L
L LL L L L
=
--
- - --
L
N
MMMM
O
Q
PPPP3
2 2
2 2
12 6 12 66 4 6 212 6 12 6
6 2 6 4
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Step 5: To constitute a global stiffness matrix using boundary conditions
Assemble example
Assume EI of the beam element is constant.
1000 lb load and 1000 lb ft- moment are applied at the center of the beam.
Assume load and moment were only applied at nodes.
Left end of the beam is fixed and right end is pin-connected.
The beam is divided into two elements (nodes 1, 2, and 3 as shown above figure).
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
k EIL
L LL L L L
L LL L L L
d dy y
( )13
2 2
2 2
12 6 12 66 4 6 212 6 12 6
6 2 6 4
1 1 2 2
=
--
- - --
L
N
MMMM
O
Q
PPPP
f f
k EI
L
L LL L L L
L LL L L L
d dy y
( )23
2 2
2 2
12 6 12 66 4 6 212 6 12 6
6 2 6 4
2 2 3 3
=
--
- - --
L
N
MMMM
O
Q
PPPP
f f
FMFMFM
EIL
L LL L L L
L L L LL L L L L L L L
L LL L L L
d
d
d
y
y
y
y
y
y
1
1
2
2
3
3
3
2 2
2 2 2 2
2 2
1
1
2
2
3
3
12 6 12 6 0 06 4 6 2 0 012 6 12 12 6 6 12 6
6 2 6 6 4 4 6 20 0 12 6 12 60 0 6 2 6 4
R
S
|||
T
|||
U
V
|||
W
|||
=
--
- - + - + -- + + -- - -
-
L
N
MMMMMMM
O
Q
PPPPPPP
´
R
S
|||
T
|||
U
V
|||
W
|||
f
f
f
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Boundary conditions and constrains at the node 1(fixed) and node 3(pin-connected) are
f1 1 30 0 0= = =d dy y
-RS|T|
UV|W|=
L
NMMM
O
QPPP
RS|T|
UV|W|
10001000
0
24 0 60 8 2
6 2 43
2 2
2 2
2
2
3
EIL
LL L
L L L
d y
ff
(5.2.5)
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Example: Beam analysis using direct stiffness method
Cantilever beam supported by roller at the center
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Load P is applied at node 1.
Length: 2L ,Stiffness: EI
Constrains: (1) Roller at node 2, (2) Fixed at node 3.
- Global stiffness matrix
K EIL
L LL L L
L L LL L L L
LSymmetry L
d d dy y y
=
--+ - + -
+ --
L
N
MMMMMMM
O
Q
PPPPPPP
3
2 2
2 2 2
2
12 6 12 6 0 04 6 2 0 0
12 12 6 6 12 64 4 6 2
12 64
1 1 2 2 3 3f f f
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
FMFMFM
EIL
L LL L L L
L LL L L L L
L LL L L L
d
d
d
y
y
y
y
y
y
1
1
2
2
3
3
3
2 2
2 2 2
2 2
1
1
2
2
3
3
12 6 12 6 0 06 4 6 2 0 012 6 24 0 12 6
6 2 0 8 6 20 0 12 6 12 60 0 6 2 6 4
R
S
|||
T
|||
U
V
|||
W
|||
=
--
- - --
- - --
L
N
MMMMMMM
O
Q
PPPPPPP
R
S
|||
T
|||
U
V
|||
W
|||
f
f
f
Boundary conditions d dy y2 3 30 0 0= = =f
-RS|T|
UV|W|=
L
NMMM
O
QPPP
RS|T|
UV|W|
PEIL
L LL L LL L L
d y
00
12 6 66 4 26 2 4
32 2
2 2
1
1
2
ff
d PLEIy1
3712
= - f f1
2
2
234 4
= =PLEI
PLEI
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
¨ Substituting the obtained values to the final equation,
FMFMFM
EIL
L LL L L L
L LL L L L L
L LL L L L
PLEI
PLEI
PLEI
y
y
y
1
1
2
2
3
3
3
2 2
2 2 2
2 2
3
2
2
12 6 12 6 0 06 4 6 2 0 012 6 24 0 12 6
6 2 0 8 6 20 0 12 6 12 60 0 6 2 6 4
712
34
0
400
R
S
|||
T
|||
U
V
|||
W
|||
=
--
- - --
- - --
L
N
MMMMMMM
O
Q
PPPPPPP
-R
S
||||
T
||||
U
V
||||
W
||||
F P M F P
M F P M PL
y y
y
1 1 2
2 3 3
0 52
0 32
12
= - = =
= = - =
F Py1 = - : Force at node 1
F y2 , F y3 , M3 : Reacting forces and moment at nodes
M1, M2 : Zero
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
¨ Calculating local nodal loads
Force at the element 1.
When $ $ $f kd=
$
$$
$
fmfm
EIL
L LL L L L
L LL L L L
PLEI
PLEI
PLEI
y
y
1
1
2
2
3
2 2
2 2
3
2
2
12 6 12 66 4 6 212 6 12 6
6 2 6 4
712
34
0
4
RS||
T||
UV||
W||=
--
- - --
L
N
MMMM
O
Q
PPPP
-R
S
|||
T
|||
U
V
|||
W
|||
$ $ $ $f P m f P m PLy y1 1 2 20= - = = = -
Shear moment curve
Mechanics and Design
Finite Element Method: Beam Element
SNU School of Mechanical and Aerospace Engineering
Homework: Distributed load
¨ Equivalent force