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HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
The Finite Element MethodIntroduction
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Linear Structural Analyse
- Truss Structure
- Beam
- Shell
- 3-D Solid
Material nonlinear
- Plasticity (Structure with stresses above yield stress)
- Hyperelasticity (ν = 0.5, i.e. Rubber)
- Creep, Swelling
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Geometric nonlinear
- Large Deflection
- Stress Stiffening
Dynamics
- Natural Frequency
- Forced Vibration
- Random Vibration
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Stability
- Buckling
Field Analysis
- Heat Transfer
- Magnetics
- Fluid Flow
- Acoustics
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Evolution of the Finite Element Method
1941 HRENIKOFF, MC HENRY, NEWMARKApproximation of a continuum Problem through a Framework
1946 SOUTHWELLRelaxation Methods in theoretical Physics
1954 ARGYRIS, TURNEREnergy Theorems and Structural Analysis (general StructuralAnalysis for Aircraft structures)
1960 CLOUGHFEM in Plane Stress Analysis
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
- Dividing a solid in Finite Elements- Compatibility between the Elements through a displacement function- Equilibrium condition through the principal of virtual work
FE = Finite Elementi, j, k = Nodal points (Nodes)
of an Element
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
The stiffness relation:
[K] {d} = {F}
or K d = F
K = Total stiffness matrixd = Matrix of nodal displacementsF = Matrix of nodal forces
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
K d = F
dT = [u1 v1 w1 . . . un vn wn]
FT = [Fx1 F y1 . . . F xn F yn F zn]
K is a n x n matrixK is a sparse matrix and symmetric
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
K d = F
Solving the stiffness relation by:
- CHOLESKY – Method- WAVE – FRONT – Method
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
1, 2 = NodesF1, F2 = Nodal forces
k = Spring rateu1, u2 = Nodal displacements
u1 u2
F1 F21 2k
F1 = k (u1 – u2)
F2 = k (u2 – u1)
Spring Element
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Element stiffness matrix
Fdk
2
1
2
1
F
F
u
u
kk
kk
kk
kkk
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Spring System
u2 u3
F1 F3
1 3k2
k1
2
u1
F2
Element stiffness matrices
11
111 kk
kkk
22
222 kk
kkk
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
the stiffness relation by using superposition
FdK
3
2
1
3
2
1
22
2211
11
F
F
F
u
u
u
kk0
kkkk
0kk
22
2211
11
kk0
kkkk
0kk
K
Total stiffness matrix
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Truss Element
x
yu2
u1
F2
F1 1
2
A
AE
k
Element stiffness matrix c = cosα s = sinα
= lengthA = cross-sectional areaE = Young´s modulus
Spring rate of a truss element
22
22
22
22
kscsscs
csccsc
scsscs
csccsc
AE
k
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
1 = 450 2 = 1350
y
x
Fx3
Fy3
1 2
3
A E A E
Element : Element :Node 1 1 Node 1 2Node 2 3 Node 2 3
3
3
3
3
2
2
1
1
0
0
0
0
200000
021000
001000
000100
000010
000001
2
y
x
F
F
v
u
v
u
v
u
AE
Stiffness relation
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Beam Element
EJM1 M2x
y
1 2Q1 Q2
x
y
v1 v2
1 212
Forces Displacements
A = Cross – sectional area E = Young’s modulusI = Moment of inertia = Length
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
the stiffness relation
d
k
F
2
2
2
1
1
1
22
2323
22
2323
2
2y
2x
1
1y
1x
v
u
v
u
EI4EI60
EI2EI60
EI6EI120
EI6EI120
00EA
00EA
EI2EI60
EI4EI60
EI6EI120
EI6EI120
00EA
00EA
M
Q
Q
M
Q
Q
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Example for practical FEM application
Engineering system Possible finite element model
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Plane stress Triangular Element
1 2
3
x
y
u1
v1
u3
v3
u2
v2
Equilibrium condition: Principal of virtual workCompatibility condition: linear displacement function
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
General displacements (Displacement function)u(x,y) = α1 + α2x + α2y
v(x,y) = α4 + α5x + α6y
Nodal displacementsu1= α1 + α2x1+ α3y1
v1= α4 + α5x1+ α6y1
similar for node 2 and node 3.
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
u = N d General displacements to nodal displacements
ε = B d Strains to nodal displacements
σ = D ε Stresses to strains
σ = D B d Stresses to nodal displacements
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Other displacement functions
1
2
3
4
5
quadratic displacement function
u(x,y) = α1 + α2x + α3y+ α4x2 + α5y
2+α6xy
v(x,y) = α7 + α8x + α9y+ α10x2 + α11y
2+α12xy
Triangular element with 6 nodes6
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
cubic displacement function
- stress field can be better approximated- more computing time- less numerical accuracy- geometry cannot be good approximated
1
2
45
6
7
10
Triangular element with 10 nodes
8
9
3
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
σ = stress matrix p = force matrixε = strain matrix u = displacement matrix
V
T
V
dVδdVUδδU εσ
sA
T
V
Tm dAdVW upuf
Principal of Virtual Work
δU = virtual work done by the applied forceδW = stored strain energy
δU + δW = 0
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Element stiffness matrix
V
T dVBDBk
2
100
01
01
1
ED
2 D = Elasticity matrix
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
b1 = y2 – y3 c1 = x3 – x2
b2 = y3 – y1 c2 = x1 – x3 AΔ = Area of element
b3 = y1 – y2 c3 = x2 – x1
332211
321
321
bcbcbc
c0c0c0
0b0b0b
A2
1B
linear displacement function yields :- linear displacement field- constant strain field- constant stress field
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Dynamics
k1 k2
c1 c2
m1 m2m0
u0
F0 F1 F2
u1 u2
Equation of motion
FdKdCdM
2
1
0
2
1
0
22
2211
11
2
1
0
22
2211
11
2
1
0
2
1
0
F
F
F
u
u
u
kk0
kkkk
0kk
u
u
u
cc0
cccc
0cc
u
u
u
m00
0m0
00m
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
M = Mass matrixC = Damping matrixK = Stiffness matrixd = Nodal displacement matrix
= Nodal velocity matrix= Nodal acceleration matrix
FKddCdM
dd
or
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
for a continuum
u = N dε = B d
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
ρ = Mass densityμ = Viscosity matrix
V
Te dVNNC
V
Te dVB DBk
the element matrices
V
Te dVNNM
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
o Modal analysiso Harmonic response analysis - Full harmonic - Reduced harmonico Transient dynamic analysis - Linear dynamic - Nonlinear dynamic
tFKddCdM
General Equation of Motion
Types of dynamic solution
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Purpose: To determine the natural frequencies and mode shapes for the structure
Assumptions: Linear structure (M, K, = constant)No Damping (c = 0 )Free Vibrations (F = 0)
0 KddM
Modal Analysis
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Transformation methods Iteration methods
JACOBI INVERSE POWERGIVENS INVERSE POWER WITH SHIFTSHOUSEHOLDER SUB – SPACE ITERATIONQ – R METHOD
for harmonic motion: d = d0 cos (ωt)
(-ω2M + K) d0 = 0
Eigenvalue extraction procedures
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Purpose: To determine the response of a linear system
Assumptions: Linear Structure (M, C, K = constant)Harmonic forcing function at known frequency
ti0e FKddCdM
Harmonic Response Analysis
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
K
Forcing funktion F = F0e-iωt
Response will be harmonic at input frequency d = d0 e-iωt
(-ω2M – iωC + K) d = F0
is a complex matrixd will be complex (amplitude and phase angle)
00 FdK
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Limiting cases:
ω = 0 : K d = F0 Static solution
C = 0 : (-ω2M + K) d = F0 Response in phase
C = 0, ω = ωn : (-ωn2M + K) d = F0 infinite amplitudes
C = 0, ω = ωn : (-ωn2M - iωnC + K) d = F0 finite amplitudes,
large phase shifts
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Transient Dynamic Analysis
F(t) = arbitrary time history forcing function
tFKddCdM
periodic forcing function
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
impulsive forcing function
Earthquake in El Centro,
California18.05.1940
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Two major types of integration:
- Modal superposition
- Direct numerical integration
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
T0
Q0
T1
Q1
T2
Q2
1 2
1 2
1, A1 2, A2
, A
0
0ne-dimensional heat flow principle
, = conductivity elements = convection element0, 1, 2 = temperature elements
A = Cross-sectional area = Lengthλ = Conductivity Aα = Convection surface
T = Temperature Q = Heat flow C = Specific heat α = Coefficient of
thermal expansion
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
Heath flow through a conduction element:
Heat stored in a temperature element:
cp = specific heat capacity
C = specific heat
Heat transition for a convection element: Q = A(T – T2)
12 TTA
Q
TCTVcQ p
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
QT
K
TC
TAQ
Q
Q
T
T
T
AAA
0
AAAA
0AA
T
T
T
C00
0C0
00C
2
1
0
2
1
0
2
22
2
22
2
22
2
22
1
11
1
11
1
11
1
11
2
1
0
2
1
0
Heat balance
or QKTTC
HCMUT 2004
Faculty of Applied SciencesHochiminh City University of Technology
The Finite Element Method
PhD. TRUONG Tich ThienDepartment of Engineering Mechnics
C = specific heat matrixK = conductivity matrixQ = heat flow matrix T = temperature matrix = time derivation of TT
For the stationary state with = 0
KT = Q
T
QKTTC