Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Nguyễn Thanh Nhã
Email: nhanguyen@hcmut.edu.vn
Phone: 0908.56.81.81
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
2.1. Basis equation of plane problem
2.2. Finite element discretization
2.3. Quadrilateral plane element
2.4. Triangular plane element
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
2.1. Basis equation of plane problem
2.1. Basis equation of plane problem Finite Element Method
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artment of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Plane problems
Mechanical problem
- Plane stress state - Plane strain state - Axisymmetric state
Element form
- Triangular element - Quadrilateral element
Polynomial degree of the shape functions
- Linear - Quadratic - Cubic -
Type of the shape functions
- Lagrange shape func. - serendipity shape func. - hierarchical shape func.
2.1. Basis equation of plane problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Geometry
Plane structure / 2D continuum:
- degeneration of the 3D continuum
onto a middle surface with a thickness
parameter
Plane stress model:
- Small thickness relative to other
dimensions of the middle surface
Plane strain model:
- Infinite dimension in an direction
Geometry description
- Position vector of a material point on
the middle surface
1 2
T
X XX or
T
X YX
2.1. Basis equation of plane problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Kinetics and Kinematics
Fundamental assumption for realization of the degeneration of the 3D to 2D
Plane stress VS plane strain
Plane stress model
11 12
21 22
0
0
0 0 0
σ
33 23 13 0
11
22
12
σ
33 0 Note:
2.1. Basis equation of plane problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Kinetics and Kinematics
Fundamental assumption for realization of the degeneration of the 3D to 2D
Plane stress VS plane strain
Plane strain model
11 12
21 22
0
0
0 0 0
ε
33 23 13 0
11 11
22 22
12 122
ε
23 13 0 Note:
33 0
2.1. Basis equation of plane problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Second fundamental assumption about displacement field
1 1 1 2
2 2 1 2
( , )
( , )
u u X X
u u X X
Displacement vector of a body: 1 2 1 1 2 2 1 2( , ) ( , ) ( , )
T
X X u X X u X Xu
Acceleration vector of a body: 1 2 1 1 2 2 1 2( , ) ( , ) ( , )
T
X X u X X u X Xu
Variational of disp. vector: 1 2 1 1 2 2 1 2( , ) ( , ) ( , )
T
X X u X X u X X u
Dirichlet BCs: *, , uu t t X u X X
Neumann BCs: 1 2
T
b bb * * *
1 2
T
t t t
Note: Plane stress: 3 0u Plane strain:3 0u
Kinetics and Kinematics
2.1. Basis equation of plane problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Strain – displacement relationship
sym ε u
In vector mode: ε D u
1
11
1
22
22
12
2 1
0
0
2
X
u
uX
X X
1 2,X X
Strain components are constant along the thickness
Kinetics and Kinematics
2.1. Basis equation of plane problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Constitutive equation
Plane stress state
esσ C ε
2
1 0
1 0
1
1
2
es E
sym
C
Plane strain state
evσ C ε
1 0
1 0
1 1 2
1 2
2
ev E
sym
C
2.1. Basis equation of plane problem Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Principle of virtual work
The weak form for 3D structural problem
*dV dV dV d
u u ε σ u b u t
The principle of virtual work for 2D problem
/2 /2
3 3
/2 /2
/2 /2
*
3 3
/2 /2
h h
A h A h
h h
A h h
dX dA dX dA
dX dA dX d
u u ε σ
u b u t
Because all integrants are functions only of X1 and X2
*
A A A
hdA hdA hdA hd
u u ε σ u b u t
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
2.2. Finite element discretization
2.2. Finite element discretization Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Partitioning into finite elements
Partitioning of 2D continuum Ω into finite domains Ωe
1
NE
e
e
i jwith if i j
The principle of virtual work holds for each finite element e
int
e e e
dyn extW W W
Quadrilateral element
Trianglular element
2.2. Finite element discretization Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Shape function of plane elements
Shape function of plane elements: 2D polynomials
1 2 1 2 1 2
0 0
, ,
p p
i jij
i j
u u
Complete polynomials for Triangular elements
1p
2p
3p
4p
1
1 2
2
1
2
21 2
5
1
4
1 2
3 2
1 2
2 3
1 2
1 4
1 2
5
2
2 2
1 2
3
1 2
3
1
3
1 2
4
2
2
1 2
2
1 2
3
1
3
2
0p
1p
2p
3p
4p
5p
2.2. Finite element discretization Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Shape function of plane elements
Complete Bi-polynomials for Quadrilateral elements
1 2 1 2 1 2
0 0
, ,
p p
i ji j
i j
u u
Complete polynomials of degree p are derived by multiplication of two
1-D polynomials of degree p
1p
2p
3p
4p
1
1 2
2
1
2
21 2
5
1
4
1 2
3 2
1 2
2 3
1 2
1 4
1 2
5
2
2 2
1 2
3
1 2
3
1
3
1 2
4
2
2
1 2
2
1 2
3
1
3
2
0p
1p
2p
3p
4p
5p
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
2.3. Quadrilateral plane element
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
1
2
1
1
Bilinear Lagrange elements
Geometry – Isoparametric mapping
1
2
1
2
1
1
1
2
1
1
1
2
1
1
1 2
34
1X
2X
1eX 2eX
4eX
1
2
3
4 3eX
1 2
T
X X X
Physical coordinates Natural coordinates
An arbitrary materal point within the quadrilateral element is indentifiable by its
physical and natural corrdinaters
1 2
T
ξ
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
The ansatz function of a node i can be derived by multiplication of the
1D Lagrange polynomials
Bilinear Lagrange elements
Ansatz functions
1 1 2 2
i iN and N corresponding to this node
Lagrange polynomials of order p:
1
1 1
1
1 1 1
( )
kp
i
k i
k
k i
N
1
1
( )
0
i k
if i k
N
if i k
1 1 1 1
1 2 1 1 2 2
1 2 1 2 1 2
,
1 1 1
1 1 1
2 2 4
N N N N
ξ
11 1 1
1
1
2
N 12 2 2
1
1
2
N
1D shape functions for node 1:
2D shape function for node 1:
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Bilinear Lagrange elements
Bilinear shape functions
Shape func. for node 1
1
2
4
3
2
1
1
1
1
1 1 2,N 12 2N
11 1N
1
2
4
3
2
1
1 1
1
2 1 2,N
21 1N
12 2N
Shape func. for node 2
1 1 1 1
1 2 1 1 2 2
1 2
1 2 1 2
,
1 1
1 1
2 2
1
1
4
N N N N
ξ
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Bilinear Lagrange elements
Shape func. for node 3 Shape func. for node 4
1
2
4
3
2
1
1
3 1 2,N
1
2
4
3
2
1
1
4 1 2,N
Bilinear shape functions
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Bilinear Lagrange elements
1
2
4
3
2
1
1e
iu
2e
iu
3e
iu
4e
iu
1 2,iu
1
2
4
3
2
1
1
1 1 2,N
1
2
4
3
2
1
1
2 1 2,N
1
2
4
3
2
1
1
3 1 2,N
1
2
4
3
2
1
1
4 1 2,N Bilinear shape functions
1 1 2
1
1 1
4
N ξ
2 1 2
1
1 1
4
N ξ
3 1 2
1
1 1
4
N ξ
4 1 2
1
1 1
4
N ξ
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Bilinear Lagrange elements
Derivations of ansatz functions wrt the natural coords
1
1
,1 2
1
1
1
4
N
N
ξ
ξ
2
2
,1 2
1
1
1
4
N
N
ξ
ξ
3
3
,1 2
1
1
1
4
N
N
ξ
ξ
4
3
,1 2
1
1
1
4
N
N
ξ
ξ
1
1
,2 1
2
1
1
4
N
N
ξ
ξ
2
2
,2 1
2
1
1
4
N
N
ξ
ξ
3
3
,2 1
2
1
1
4
N
N
ξ
ξ
4
4
,2 1
2
1
1
4
N
N
ξ
ξ
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Bilinear Lagrange elements
Geometry
1
2
1
1
1
2
1
2
1
1
1
2
1
1
1
2
1
1
1 2
34
1X
2X
1eX 2eX
4eX
1
2
3
4 3eX
1 1 2 2 3 3 4 4
1 2 1 2 1 2 1 2 1 2
1 2 3 4
,
T
e e e e e e e e e
T
T T T T
e e e e
X X X X X X X X X X
X
X X X X
1
2
ei
ei
ei
X
X
X
4
1 2
1 1
,
NN
ei i ei i
i i
N N
X X ξ X ξ X ξ X ξ
Approximation of geometry
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Bilinear Lagrange elements
1
1
1
2
2
1
1 2 3 4 2
1 2
1 2 3 4 3
2 1
3
2
4
1
4
2
0 0 0 0
0 0 0 0
e
e
e
e
e
e
e
N
e
e
X
X
X
X N N N N X
X N N N N X
X
X
X
ξX ξ
X
ξ ξ ξ ξ ξ
ξ ξ ξ ξ ξ
Approximation of geometry
eNX ξ ξ X
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Bilinear Lagrange elements
1 2
1 2
XX X
X X X
Jacobi transformation
- The calculation of strain vector ε requires that the displacement components
be devivated wrt physical cords X
- Displacement components and approximation of the position vector are
expressed as functions of natural coordinates
Transformation relation between derivatives wrt physical and natural
coords
1,2; 1,2
1 2
1 1 1 1
1 2
2 2 2 2
X X
X
X X
X
J ξ
ξ X
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Bilinear Lagrange elements
Jacobi transformation
The rule for derivating functions in natural coords wrt physical cords is
obtained by inversion of the Jacobi matrix
0J ξ 1
J ξ
X ξ
Inversion of the Jacobi matrix
2 2
2 11
1 1
2 1
1
X X
X X
J ξ
J ξ
1 2 1 2
1 2 2 1
X X X X
J ξDeterminant of the Jacobi matrix
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Jacobi transformation
The derivatives of physical cords wrt natural coords:
,
1
NN
ei i
i
X
X N
ξ
ξ
The derivatives of function in natural coords wrt physical coords
, ,( )
eX
X ξ
ξ N ξ X
1 2
1 2X X X X
Definition of the inverse of the Jacobi matrix
1 2
1 1 1 1
1 2
2 2 2 2
X X X
X
X X
1
J ξ
X ξ
Bilinear Lagrange elements
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Jacobi transformation
Transformation of a surface element dA in natural coords
1 2 1 2dA dX dX d d J ξ
2dX
1dX
1 2dA d d X X
1dξ
2dξ
1 2
1 2
dA d d
d d
ξ ξ
Physical coordinates Natural coordinates
(see more in text book!)
Bilinear Lagrange elements
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of element quantities
Approximations of displacements, variations and second time derivative of
displacements:
e u ξ u ξ N ξ u
e u ξ u ξ N ξ u
e u ξ u ξ N ξ u
1 1 2 2 3 3 4 4
1 2 1 2 1 2 1 2
T
e e e e e e e e eu u u u u u u u u
1 1 2 2 3 3 4 4
1 2 1 2 1 2 1 2
T
e e e e e e e e eu u u u u u u u u
1 1 2 2 3 3 4 4
1 2 1 2 1 2 1 2
T
e e e e e e e e eu u u u u u u u u
Bilinear Lagrange elements
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of element quantities
Approximations of displacements, variations
and second time derivative of displacements:
e u ξ u ξ N ξ u
e u ξ u ξ N ξ u
e u ξ u ξ N ξ u
1 1 2 2 3 3 4 4
1 2 1 2 1 2 1 2
T
e e e e e e e e eu u u u u u u u u
1 1 2 2 3 3 4 4
1 2 1 2 1 2 1 2
T
e e e e e e e e eu u u u u u u u u
1 1 2 2 3 3 4 4
1 2 1 2 1 2 1 2
T
e e e e e e e e eu u u u u u u u u
1
2
3
4
1
1
eu
1
2
eu
4
2
eu
4
1
eu
3
2
eu
3
1
eu
2
2
eu
2
1
eu
2u
1u
Bilinear Lagrange elements
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Strain vector approximation of element
1 2
1 1 1 2
11
11 2
22
22 1 2 2
12
1 2 1 2
2 1 2 2 1 1 1 2
0
0
2
X X
u
uX X
X X X X
D
ε D u
e e ε ξ ε ξ D ξ u D ξ N ξ u B ξ u
Displacements – strain relationship:
Strain vector approximation
Bilinear Lagrange elements
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Strain vector approximation of element
eε ξ B ξ u B ξ D ξ N ξ
:B ξ Differential operator (B-operator)
:eu Linear mapping of element displacement vector
B-operator for element node i
1 2
,1 ,2
1 1
1 2
,1 ,2
2 2
1 2 1 2
,1 ,2 ,1 ,2
2 2 1 1
0i i
i i
i
i i i i
N N
X X
N N
X X
N N N N
X X X X
ξ ξ
B ξ ξ
ξ ξ ξ ξ
Bilinear Lagrange elements
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of internal virtual work
The internal virtual work
Approximation of internal virtual work
1 1
int 1 2
1 1
eW C hd d
ε ξ ε ξ J ξ
1 1
int 1 2
1 1
e e T T eW C hd d
u B ξ B ξ u J ξ
Since ,e eu u do not depend on ξ
1 1
int 1 2
1 1
e e T T e e e eW C hd d
u B ξ B ξ J ξ u u k u
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of internal virtual work
Element stiffness matrix of a bilinear plane quadrilateral element
Using Gaussian integral (2 x 2 integration points)
1 1
1 2
1 1
e T TC hd d
k B ξ B ξ J ξ
2 2
1 2 1 2 1 2 1 2
1 1
, , , ,e i j T i j i j i j i j
i j
h
k B CB J
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of dynamic virtual work
The internal virtual work of inertial forces
Approximation virtual work of inertial forces
1 1
1 2
1 1
e
dynW hd d
u ξ u ξ J ξ
1 1
1 2
1 1
e e T T e e e e
dynW hd d
u N ξ N ξ J ξ u u m u
Element mass matrix of a bilinear plane quadrilateral element
1 1
1 2
1 1
e T T hd d
m N ξ N ξ J ξ
Using Gaussian integral (2 x 2 integration points)
2 2
1 2 1 2 1 2 1 2
1 1
, , , ,e i j T i j i j i j i j
i j
h
m N N J
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of virtual external work
Natural coordinatesPhysical coordinates
1X
2X
1
2
3
4
1 2
T
X X X
*2 1 2,t X X
1
2
3
4 1 2,X Xb
1
2
1
1
1
2
1
2
1
1
1
2
1
1
1
2
1
1
1 2
34
1 2
T
ξ
*2 1 2,t
1 2, b
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of virtual external work
Element vector of volume forces
Volume loads
1 1
1 2
1 1
e e T e e
ext pW hd d
u N ξ b ξ J ξ u r
1 1
1 2
1 1
e T
p hd d
r N ξ b ξ J ξ
1
2
1 2
34
1 2, b
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of virtual external work
Example for element boundary THREE
Boundary loads
4 4
* *
1 1
e ie
ext i ext
i i
W hd hd W
u t u t
3 *1 1 3,1 ,1eextW hd
u t
1X
2X
1
2
3
4
*2 1 2,t X X
1
2
3
4
3d
1dX
2dX
1
2
1
1
1
2
1
2
1
1
1
2
1
1
1
2
1
1
1 2
34
*2 1,1t
3d
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of virtual external work
Using Pythagore theorem
Boundary loads
2 2 2
3 1 2d dX dX
The toal differentials 1 2 1
1 2 1
X X X
dX d d d
The transformation relation of line element
3d
2 2
2 21 2
3 1
1 1
X X
d d
3 3 1 1,1d d J
1
2 2 2
1 2
3
1 1
X X
d
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of virtual external work
Boundary loads
Positon vector of the element edge
3d
4 4
1 1 1
1 3
,1 ,1 ,1ei ei
i i
X N N
X X
Derivative of the position vector wrt natura coord 1
4
1
,1 1 ,1 1
31
,1
,1 ,1ei i e
i
N
X
X N X
Horizontal & Vertical component of the edge element
3d
1 3 3 4 41 ,1 1 ,1 1 1
1
,1
,1 ,1e e
X
dX d X N X N d
3 4
1
1 1
2 2
e eX X d
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of virtual external work
Boundary loads
Jacobi determinant 3 1,1J
1/2
2 2
3 4 3 4
3 1 1 1 2 2
1 1 1 1
,1
2 2 2 2
e e e eX X X X
J
1/2
2 2
3 4 3 4
1 1 2 2
1
2
e e e eX X X X
External virtual work on the boundary 3
3
1
*
1 1 3 1 1
1
,1 ,1 ,1extW h d
u t J
1
*
1 1 3 1 1 3
1
,1 ,1 ,1e T e enh d
u N t J u r
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of virtual external work
Boundary loads
The consistent equivalent load of the boundary load
1
*
3 1 1 3 1 1
1
,1 ,1 ,1e Tn h d
r N t J
External virtual work of an element
4 4 4
1 1 1
i e e e e e e
ext ext ni ni n
i i i
W W
u r u r u r
2.3. Quadrilateral plane element Finite Element Method
Department of Engineering Mechanics – HCMUT 2016Ng. Thanh Nhã
Approximation of virtual external work
Exercise:
Derive the boundary load vectors for the three edges: 1 2 4, ,
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