Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Plane stress and Plane strain
- Finite element in 2-D: Thin plate element required 2 coordinates
- Plane stress and plane strain problems
- Constant-strain triangular element
- Equilibrium equation in 2-D
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
plane stress: The stress state when normal stress, which is perpendicular to the
plane x-y, and shear stress are both zero.
plane strain: The strain state when normal strain e z , which is perpendicular to the plane x y- , and shear strain g gxz yz, are both zero.
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Stress and strain in 2-D
Stresses in 2-D Principal stress and its direction
{ }ssst
=
UV|W|
RS|
T|x
y
xy
ss s s s
t s
ss s s s
t s
1
22
2
22
2 2
2 2
=+
+-F
HGIKJ + =
=+
--F
HGIKJ + =
x y x yxy
x y x yxy
max
min
tan 22
qt
s spxy
x y
=-
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Displacement and rotation of plane element x y-
e e u g ux y xy
ux y
uy x
=¶¶
=¶¶
=¶¶
+¶¶
{ }eeeg
=
RS|T|
UV|W|
x
y
xy
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
{ } [ ]{ }s e= D
Stress-strain matrix(or material composed matrix) of isotropic material for plane
stress(s t tz xz yz= = = 0 )
[ ]D E
=- -
L
N
MMMM
O
Q
PPPP1
1 01 0
0 0 12
2n
nn
n
Stress-strain matrix(or material composed matrix) of isotropic material for plane deformation
(e g gz xz yz= = = 0 )
[ ]( )( )
D E=
+ -
--
-
L
N
MMMM
O
Q
PPPP1 1 2
1 01 0
0 0 1 22
n n
n nn n
n
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
General steps of formulation process for plane triangular element
Step 1: Determination of element type
Step 2: Determination of displacement function
Step 3: Definition of relations deformation rate – strain and stress-strain
Step 4: Derivation of element stiffness matrix and equation
Step 5: Introduction a combination of element equation and boundary conditions for
obtaining entire equations
Step 6: Calculation of nodal displacement
Step 7: Calculation of force(stress) in an element
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Step 1: Determination of element type
{ }dddd
u
u
u
i
j
m
i
i
j
j
m
m
=RS|T|UV|W|=
R
S
|||
T
|||
U
V
|||
W
|||
u
u
u
Considering a triangular element, the nodes i j m, , are notated in the anti clock wise
direction.
The way to name the nodal numbers in an entire structure must be devised to avoid the
element area comes to be negative.
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Step 2: Determination of displacement function
u x y a a x a yx y a a x a y
( , )( , )
= + += + +
1 2 3
4 5 6u
Linear function gives a guarantee to satisfy the compatibility.
A general displacement function { }y containing function u and u can be expressed as
below.
{ }y =+ ++ +RST
UVW =LNM
OQP
R
S
|||
T
|||
U
V
|||
W
|||
a a x a ya a x a y
x yx y
aaaaaa
1 2 3
4 5 6
1
2
3
4
5
6
1 0 0 00 0 0 1
Substitute nodal coordinates to the equation for obtaining the values of a .
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Calculation of a a a1 2 3, , :
u a a x a yu a a x a yu a a x a y
i i i
j j j
m m m
= + += + +
= + +
1 2 3
1 2 3
1 2 3
or
uuu
x yx yx y
aaa
i
j
m
i i
j j
m m
RS|T|UV|W|=L
NMMM
O
QPPP
RS|T|UV|W|
111
1
2
3
Solving a , { } [ ] { }a x u= -1
Obtaining the inverse matrix of [ ]x , [ ]x
A
i j m
i j m
i j m
- =
L
NMMM
O
QPPP
1 12
a a ab b bg g g
where 2A x y y x y y x y yi j m j m i m i j= - + - + -( ) ( ) ( ) : 2 times of triangle area
a a a
b b b
g g g
i j m j m j i m i m m i j i j
i j m j m i m i j
i m j j i m m j i
x y y x y x x y x y y xy y y y y yx x x x x x
= - = - = -
= - = - = -
= - = - = -
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
After calculation of [ ]x -1, the equation { } [ ] { }a x u= -1
can be expressed as an extended
matrix form.
aaa
A
uuu
i j m
i j m
i j m
i
j
m
1
2
3
12
RS|T|UV|W|=
L
NMMM
O
QPPP
RS|T|UV|W|
a a ab b bg g g
Similarly,
aaa
A
i j m
i j m
i j m
i
j
m
4
5
6
12
RS|T|UV|W|=
L
NMMM
O
QPPP
RS|T|UV|W|
a a ab b bg g g
uuu
Derivation of displacement function u x y( , ) (u can also be derived similarly)
{ } [ ] [ ]u x y
aaa
Ax y
uuu
i j m
i j m
i j m
i
j
m
=RS|T|UV|W|=
L
NMMM
O
QPPP
RS|T|UV|W|
1 12
11
2
3
a a ab b bg g g
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Arranging by depolyment:
u x y x y u x y u
x y u
i i i i j j j j
m m m m
( , ) {( ) ( )
( ) }
= + + + + +
+ + +
124
a b g a b g
a b g
As the same way
u a b g u a b g u
a b g u
( , ) {( ) ( )
( ) }
x y x y x y
x y
i i i i j j j j
m m m m
= + + + + +
+ + +
124
Simple expression of u and u :
u x y N u N u N ux y N N N
i i j j m m
i i j j m m
( , )( , )
= + +
= + +u u u u where
NA
x y
NA
x y
NA
x y
i i i i
j j j j
m m m m
= + +
= + +
= + +
12
121
2
( )
( )
( )
a b g
a b g
a b g
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
{ }( , )( , )
yu u u u
u
u
u
=RST
UVW =+ ++ +
RSTUVW=LNM
OQP
R
S
|||
T
|||
U
V
|||
W
|||
u x yx y
N u N u N uN N N
N N NN N N
u
u
u
i i j j m m
i i j j m m
i j m
i j m
i
i
j
j
m
m
0 0 00 0 0
Making the equation be simple in a form of matrix, { } [ ]{ }y = N d
where [ ]NN N N
N N Ni j m
i j m=LNM
OQP
0 0 00 0 0
The displacement function { }y is represented with shape functions N Ni j, , Nm and
nodal displacement { }d .
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Review of characteristics of shape function:
Ni=1, N j=0, and Nm=0 at nodes ( , )x yi i
A change of Ni of general elements across the surface x y-
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Step 3: Definition of relations deformation rate – strain and stress-strain
Deformation rate:
{ }eeeg
u
u
=
RS|T|
UV|W|=
¶¶¶¶
¶¶
+¶¶
R
S|||
T|||
U
V|||
W|||
x
y
xy
ux
yuy x
Calculation of partial differential terms
¶¶
= =¶¶
+ + = + +ux
uxN u N u N u N u N u N ux i i j j m m i x i j x j m x m, , , ,( )
NA x
x yAi x i i ii
, ( )=¶¶
+ + =1
2 2a b g b
, N AN
Aj xj
m xm
, ,,= =b b2 2
\¶¶
= + +ux A
u u ui i j j m m1
2( )b b b
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Likewise,
¶¶
= + +
¶¶
+¶¶
= + + + + +
u g u g u g u
u g b u g b u g b u
y Auy x A
u u u
i i j j m m
i i i i j j j j m m m m
12
12
( )
( )
Summarizing the deformation rate equation
{ } [ ]{ }eb b b
g g gg b g b g b
u
u
u
=
L
NMMM
O
QPPP
R
S
|||
T
|||
U
V
|||
W
|||
= =RS|T|UV|W|
12
0 0 00 0 0
A
u
u
u
B d B B Bddd
i j m
i j m
i i j j m m
i
i
j
j
m
m
i j m
i
j
m
where
[ ] [ ] [ ]B
AB
AB
Ai
i
i
i i
j
j
j
j j
m
m
m
m m
=L
NMMM
O
QPPP
=
L
NMMM
O
QPPP
=L
NMMM
O
QPPP
12
00 1
2
00 1
2
00
bg
g b
bg
g b
bg
g b
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Strain is constant in an element, for matrix B regardless of x and y coordinates, and
is influenced by only nodal coordinates in an element.
à CST: constant-strain triangle
Relation of stress-strain
sst
eeg
x
y
xy
x
y
xy
DRS|T|
UV|W|=
RS|T|
UV|W|
[ ] ® { } [ ][ ]{ }s = D B d
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Step 4: Derivation of element stiffness matrix and equation
Using minimum potential energy principle.
Total potential energy p p u up p i i j m b p su u U= = + + +( , , ,..., ) W W W
Strain energy U dV D dVV
T
V
T= =zzz zzz1
212
{ } { } { } [ ]{ }e s e e
Potential energy of body force WbV
TX dV= - zzz { } { }y
Potential energy of concentrated load W pTd P= -{ } { }
Potential energy of distributed load (or surface force) WSS
TT dS= -zz { } { }y
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
\ = -
- -
= -
- -
= -
zzz zzzzz
zzz zzzzz
zzz
p pV
T T
V
T T
T
S
T T
T T
V
T
V
T
T T
S
T
T T
V
T
d B D B d dV d N X dV
d P d N T dS
d B D B dV d d N X dV
d P d N T dS
d B D B dV d d f
12
12
12
{ } [ ] [ ][ ]{ } { } [ ] { }
{ } { } { } [ ] { }
{ } [ ] [ ][ ] { } { } [ ] { }
{ } { } { } [ ] { }
{ } [ ] [ ][ ] { } { } { }
where
{ } [ ] { } { } [ ] { }f N X dV P N T dSV
T
S
T= + +zzz zz (7.2.46)
Condition having pole values
¶
¶=LNM
OQP
- =zzzp p T
VdB D B dV d f
{ }[ ] [ ][ ] { } { } 0
® [ ] [ ][ ] { } { }B D B dV d fT
Vzzz =
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
So, the element stiffness matrix is [ ] [ ] [ ][ ]k B D B dVT
V
= zzz
Case of an element having constant thickness t : [ ] [ ] [ ][ ]
[ ] [ ][ ]
k t B D B dx dy
tA B D BA
T
T
=
=
zz
Matrix [ ]k is a 6×6 matrix, and the element equation is as below
ffffff
k k kk k k
k k k
u
u
u
x
y
x
y
x
y
1
1
2
2
3
3
11 12 16
21 22 26
61 62 66
1
1
2
2
3
3
R
S
||||
T
||||
U
V
||||
W
||||
=
L
N
MMMM
O
Q
PPPP
R
S
|||
T
|||
U
V
|||
W
|||
LL
M M ML
u
u
u
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Step 5: Introduction a combination of element equation and boundary conditions for obtaining
a global coordinate system of equation.
[ ] [ ]( )K k e
e
N
==å
1 and { } { }( )F f e
e
N
==å
1
{ } [ ]{ }F K d=
Step 6: Calculation of nodal displacement
Step 7: Calculation of force(stress) in an element
Transformation from the global coordinate system to the local coordinate system: (See Ch. 3)
$ $ $d Td f T f k T kTT= = =
Constant-strain triangle(CST) has 6 degrees of freedom.
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
T
C SS C
C SS C
C SS C
=
-
-
-
L
N
MMMMMMM
O
Q
PPPPPPP
0 0 0 00 0 0 0
0 0 0 00 0 0 00 0 0 00 0 0 0
where
CS==
cossin
A triangular element with local coordinates system not along to the global coordinate system
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Finite Element Method in a plane stress problem
Find nodal displacements and element stresses in the case of the thin plate(see below figure)
under surface force.
thickness t E= = ´ =1 30 10 0 306in. psi, , .n
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
(1) Discretization: Surface tension force is replaced by the following nodal loads.
F TA
F
F
=
= ´
=
1212
1000 1 10
5000
( )( . )psi in in.
lb
The global system of the governing equation is
{ } [ ]{ }F K d= or
FFFFFFFF
RRRR
K
dddddddd
Kdddd
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
1
1
2
2
3
3
4
4
1
1
2
2
1
1
2
2
3
3
4
4
3
3
4
4
50000
50000
0000
R
S
|||||
T
|||||
U
V
|||||
W
|||||
=
R
S
|||||
T
|||||
U
V
|||||
W
|||||
=
R
S
|||||
T
|||||
U
V
|||||
W
|||||
=
R
S
|||||
T
|||||
U
[ ] [ ] V
|||||
W
|||||
where [ ]K is a 8×8 matrix.
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
(2) A combination of stiffness matrix: [ ] [ ] [ ][ ]k tA B D BT=
· Element 1
- Calculation of matrix [ ]B
[ ]B
A
i j m
i j m
i i j j m m
=
L
NMMM
O
QPPP
12
0 0 00 0 0b b b
g g gg b g b g b
where
b
b
b
g
g
g
i j m
j m i
m i j
i m j
j i m
m j i
y yy yy yx xx xx x
= - = - =
= - = - =
= - = - = -
= - = - = -
= - = - =
= - = - =
10 10 010 0 100 10 100 20 200 0 020 0 20
and
A bh=
= FHGIKJ =
1212
20 10 100( )( ) in.2
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Then [ ]B is
[ ]B =-
-- -
L
NMMM
O
QPPP
1200
0 0 10 0 10 00 20 0 0 0 20
20 0 0 10 20 10
- Matrix [ ]D (plane stress)
D E=
- -
L
N
MMMM
O
Q
PPPP=
L
NMMM
O
QPPP( )
( ).
..
.1
1 01 0
0 0 12
30 100 91
1 0 3 00 3 1 00 0 0 35
2
6
n
nn
n
- Calculation of stiffness matrix
i j m
k tA B D BT
= = =
= =
- -- -
- -- -- - -
- - -
L
N
MMMMMMM
O
Q
PPPPPPP
1 3 2
75 0000 91
140 0 0 70 140 700 400 60 0 60 4000 60 100 0 100 60
70 0 0 35 70 35140 60 100 70 240 130
70 400 60 35 130 435
[ ] [ ] [ ][ ] ,.
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
· Element 2
- Calculation of matrix [ ]B
[ ]B
A
i j m
i j m
i i j j m m
=
L
NMMM
O
QPPP
12
0 0 00 0 0b b b
g g gg b g b g b
where
b
b
b
g
g
g
i j m
j m i
m i j
i m j
j i m
m j i
y yy yy yx xx xx x
= - = - = -
= - = - =
= - = - =
= - = - =
= - = - = -
= - = - =
0 10 1010 0 100 0 020 20 00 20 2020 0 20
and
A bh=
= FHGIKJ =
1212
20 10 100( )( ) in.2
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Then matrix [ ]B is
[ ]B =-
-- -
L
NMMM
O
QPPP
1200
10 0 10 0 0 00 0 0 20 0 200 10 20 10 20 0
- Matrix [ ]D (plane stress)
[ ] ( )
.
..
.D =
L
NMMM
O
QPPP
30 100 91
1 0 3 00 3 1 00 0 0 35
6
- Calculation of stiffness matrix
i j m
k
= = =
=
- -- -
- - -- - -- -
- -
L
N
MMMMMMM
O
Q
PPPPPPP
1 4 3
75 0000 91
100 0 100 60 0 600 35 70 35 70 0
100 70 240 130 140 6060 35 130 435 70 400
0 70 140 70 140 060 0 60 400 0 400
[ ] ,.
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Element 1:
1 2 3 4
375 0000 91
28 0 28 14 0 14 0 00 80 12 80 12 0 0 0
28 12 48 26 20 14 0 014 80 26 87 12 7 0 00 12 20 12 20 0 0 0
14 0 14 7 0 7 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0
[ ] ,.
k =
- -- -
- - -- - -- -
- -
L
N
MMMMMMMMMMM
O
Q
PPPPPPPPPPP
Element 2:
1 2 3 4
375 0000 91
20 0 0 0 0 12 20 120 7 0 0 14 0 14 70 0 0 0 0 0 0 00 0 0 0 0 0 0 00 14 0 0 28 0 28 14
12 0 0 0 0 80 12 8020 14 0 0 28 12 48 2612 7 0 0 14 80 26 87
[ ] ,.
k =
- -- -
- -- -- - -
- - -
L
N
MMMMMMMMMMM
O
Q
PPPPPPPPPPP
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
(3) Calculation of displacement: Superpositioning element stiffness matrix, global system of
stiffness matrix is obtained as below.
1 2 3 4
375 0000 91
48 0 28 14 0 26 20 120 87 12 80 26 0 14 7
28 12 48 26 20 14 0 014 80 26 87 12 7 0 00 26 20 12 48 0 28 14
26 0 14 7 0 87 12 8020 14 0 0 28 12 48 2612 7 0 0 14 80 26 87
[ ] ,.
K =
- - -- - -
- - -- - -- - -
- - -- - -
- - -
L
N
MMMMMMMMMMM
O
Q
PPPPPPPPPPP
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Substituting [ ]K to { } [ ]{ }F K d= ,
RRRR
ddd
x
y
x
y
x
y
x
1
1
2
2
3
3
4
50000
50000
375 0000 91
48 0 28 14 0 26 20 120 87 12 80 26 0 14 7
28 12 48 26 20 14 0 014 80 26 87 12 7 0 0
0 26 20 12 48 0 28 1426 0 14 7 0 87 12 8020 14 0 0 28 12 48 2612 7 0 0 14 80 26 87
0000
R
S
|||||
T
|||||
U
V
|||||
W
|||||
L
N
MMMMMMMMMMM
O
Q
PPPPPPPPPPP
=
- - -
- - -
- - -
- - -
- - -
- - -
- - -
- - -
,.
d y4
R
S
|||||
T
|||||
U
V
|||||
W
|||||
Applying given boundary conditions with elimination of columns and rows.
50000
50000
48 0 28 140 87 12 80
28 12 48 2614 80 26 87
375 0000 91
3
3
4
4
RS||
T||
UV||
W||
--
- -- -
L
N
MMMM
O
Q
PPPP
RS||
T||
UV||
W||
=,.
dddd
x
y
x
y
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
Transposing the displacement matrix to the left side
dddd
x
y
x
y
3
3
4
4
1
60 91375 000
10
48 0 28 140 87 12 80
28 12 48 2614 80 26 87
50000
50000
609 64 2
66371041
RS||
T||
UV||
W||
--
- -- -
L
N
MMMM
O
Q
PPPP
RS||
T||
UV||
W||
RS||
T||
UV||
W||
= = ´
-
-.,
.
.
..
in.
The solution of 1-D beam under tension force is
d = =´
= ´ -PLAE
( , )( )
10 000 2010 30 10
670 1066 in.
Therefore, x-component of the displacement at nodes in the equation (7.5.27) of 2-D plane is
quite accurate when the grid is considered as coarse grid.
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
(4) Stresses at each node: { } [ ][ ]{ }s = D B d
- Element 1
{ }( )
sn
n
nn
b b b
g g g
g b g b g b
=- -
´
L
N
MMMM
O
Q
PPPPFHIKL
NMMM
O
QPPP
R
S
|||
T
|||
U
V
|||
W
|||
EA
dddddd
x
y
x
y
x
y
1
1 01 0
0 01
2
12
0 0 00 0 02
1 3 2
1 3 2
1 1 3 3 2 2
1
1
3
3
2
2
Calculating,
sst
x
y
xy
RS|T|
UV|W|=RS|T|
UV|W|
1005301
2 4.psi
Mechanics and Design
Finite Element Method: Plane Stress and Plane Strain
SNU School of Mechanical and Aerospace Engineering
- Element 2
{ }( )
sn
n
nn
b b b
g g g
g b g b g b
=- -
´
L
N
MMMM
O
Q
PPPPFHIKL
NMMM
O
QPPP
R
S
|||
T
|||
U
V
|||
W
|||
EA
dddddd
x
y
x
y
x
y
1
1 01 0
0 01
2
12
0 0 00 0 02
1 4 3
1 4 4
1 1 4 4 3 3
1
1
4
4
3
3
Calculating,
sst
x
y
xy
RS|T|
UV|W|= -
-
RS|T|UV|W|
995122 4..
psi