FORMULATION OF THE
DISPLACEMENT-BASED
FINITE ELEMENT
METHOD
LECTURE 3
58 MINUTES
3·1
Formulation of the displacement-based finite element method
LECTURE 3 General effective formulation of the displace­
ment-based finite element method
Principle of virtual displacements
Discussion of various interpolation and element
matrices
Physical explanation of derivations and equa­
tions
Direct stiffness method
Static and dynamic conditions
Imposition of boundary conditions
Example analysis of a nonuniform bar. detailed
discussion of element matrices
TEXTBOOK: Sections: 4.1. 4.2.1. 4.2.2
Examples: 4.1. 4.2. 4.3. 4.4
3·2
Formulation of the displacement-based finite element method
FORMULATION OF
THE DISPLACEMENT ­
BASED FINITE
ELEMENT METHOD
- A very general
formu lation
-Provides the basis of
almost all finite ele­
ment analyses per­
formed in practice
-The formulation is
really a modern appli ­
cation of the Ritz/
Gelerkin procedures
discussed in lecture 2
-Consider static and
dynamic conditions, but
linear analysis
Fig. 4.2. General three-dimensional body.
3·3
FOl'Dlulation of the displaceDlent·based finite e1mnent lDethod
The external forces are
fB
X
fB
= fBy
fS
fB
f~
i
FX
= fSy
Fi = Fyi
fS
i
FZ
Z
Z
(4.1)
The displacements of the body from
the unloaded configuration are
denoted by U, where
uT = [u
V
w]
(4.2)
The strains corresponding to U are,
~T = [E XX Eyy EZZ YXy YyZ YZX]
and the stresses corresponding to
are
3·4
€
(4.3)
Formulation of the displacement-based finite element method
Principle of virtual displacements
where
ITT = [IT If w]
(4.6 )
Fig. 4.2. General three-dimensional body.
3·5
Formulation of the displaceaenl-based filile e1eDlenl .ethod
,,
x,u
""
Finite element
For element (m) we use:
!!(m) (x, y, z)
"T
!!
= [U, V, W,
z)
0
(4.8)
U2V2W2 ••• UNVNW N]
!!"T = [U,U 2U3
... Un]
§.(m) (x, y, z)
=~(m) (x, y,
!.(m)
3·&
=!:!.(m) (x, y,
=f(m)~(m)
+ -rI(m)
(4.9)
z)
!!
(4.'0)
(4.'1)
'OI'IIalation of the displaceDlenl-based filile eleDlenl method
Rewrite (4.5) as a sum of integrations
over the elements
(4.12)
Substitute into (4.12) for the element
displacements, strains, and stresses,
using (4.8), to (4.10),
____---..ll.c=~~-----I
j-
'iTl
If
~
~
1
B(m) Tc(m)B(m)dv(m)j U=
v(m) l-
j [I
T
I 1
_"
-(m) T
--£
£1
L l(m)
T (m)
!!(m) 1.B
dV(m)
j
---- ~(m) = f.(m) ~(m)
-
m
V
I
, .
( )T
B(m)TTI(m) dv(m)j
m
-___.r__.........1
"<I:::
--
_~m
:,. L f. ) !!sCm)Ti m)dScmlj
m
_m_JV...:........<,==~I______
El;rm) -
= B(m) l..u·)
(-£ )(m)
y:(m)
=!!(m)
~
(m) T
-US
-(m) T
------.
... ~
(4.13)
3·7
Formulation of the displacement-based finite element .ethod
We obtain
KU
=R
(4.14)
where
R=.Ba + Rs - R1 + ~
~f
K=
J
-
( 4. 15)
B(m)Tc(m)B(m)dV(m)
(4.16)
m V(m)- -
R =
~
"'1.
~ lm) -
H(m)TfB(m)dV(m)
(4.17)
-
"'1
= "'1
-1
R
=
-S
R
R
~
HS (m)Tfs(m)dS(m) (4.18)
~ ~m) B(m)TT1(m)dV(m)
~ V(m) -
(4.19)
=-F
(4.20)
In dynam ic analysis we have
~B = ~
f
(m)T -B(m)
V(m) .!:!.
[1.
_ p(m).!:!.(m)~]dV(m)
MD+KU= R
3·8
B(m)
1.
(4.21 )
(4.22)
-B(m)
=
1.
•• (m)
- p!!
Formulation of the displacement-based finite element method
To impose the boundary conditions,
we use
~a ~b
~a ~b
+
~a
t!t>b
-~
=
(4.38)
..
..
~a~+~a~=~-~b~-~b~
(4.39)
~=~a~+~b~+~a~+~b~
(4.40)
i
!
I
!
•
A
V
-
Global degrees
of freedom
;~
V
I
~
(restrained\
L
COS
T=
[
'/..
Transformed
C,-:~:e)
rl'f
u
a -sin
a]
sin a
U= T
\eedom
degrees of
f
cos a
IT
Fig. 4.10. Transformation to skew
boundary conditions
3·9
Formulation of the displacement-based finite element method
.th
1
column
For the transformation on the
total degrees of freedom we use
1. ••
i th row
(4.41 )
so that
j
1
cos a.
T=
..
Mu+Ku=R
(4.42)
}h
sin a.
where
L
Fig. 4.11. Skew boundary condition
imposed using spring element.
We can now also use this procedure
(penalty method)
Say Ui = b, then the constraint
equation is
,
k U. = k b
where
k » k ..
"
___ 3·10
(4.44)
.th
J
!
-s ina.
1
cos a.
1
FormDlation of the displacement·based finite element method
Example analysis
area
z
=1
100
x
y
area
=9
100
80
element
®
Finite elements
J~
I"
-I
100
80
~I
3·11
Formulation of the displacement·based finite element method
Element
interpolation functions
1.0
I ...
--I
L
Displacement and strain
interpolation matrices:
H(l} = [(l-L)
-
100
!:!.(2} = [
a
!!(l)=[
!!(2) = [
3·12
y
100
(1- L)
a]
v(m} = H(m}U
80
:0]
1
100
1
100
a]
a
1
80
1
80]
av = B(m}U
ay -
FOI'IDDlation of the displacement·based finite element method
stiffness matrix
- 1
100
5.= (1 HEllO
a
l~O [-l~O l~O
o}Y
a
U
1
- 80
1
80
Hence
E
=240
[ 2.4
-2.4
-2.4
15.4
a
-13
Similarly for M '.!!B ' and so on.
Boundary conditions must still be
imposed.
3·13
MIT OpenCourseWare
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Resource: Finite Element Procedures for Solids and Structures
Klaus-Jürgen Bathe
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