HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics The Finite Element Method Introduction

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department 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 Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Geometric nonlinear - Large Deflection - Stress Stiffening Dynamics - Natural Frequency - Forced Vibration - Random Vibration

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Stability - Buckling Field Analysis - Heat Transfer - Magnetics - Fluid Flow - Acoustics

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Evolution of the Finite Element Method 1941HRENIKOFF, MC HENRY, NEWMARK Approximation of a continuum Problem through a Framework 1946 SOUTHWELL Relaxation Methods in theoretical Physics 1954 ARGYRIS, TURNER Energy Theorems and Structural Analysis (general Structural Analysis for Aircraft structures) 1960 CLOUGH FEM in Plane Stress Analysis

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department 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 Element i, j, k = Nodal points (Nodes) of an Element

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics The stiffness relation: [K]{d}= {F} orK d = F K = Total stiffness matrix d = Matrix of nodal displacements F = Matrix of nodal forces

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics K d = F d T = [u 1 v 1 w 1... u n v n w n ] F T = [F x1 F y1... F xn F yn F zn ] K is a n x n matrix K is a sparse matrix and symmetric

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics K d = F Solving the stiffness relation by: - CHOLESKY – Method - WAVE – FRONT – Method

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics 1, 2= Nodes F 1, F 2 = Nodal forces k= Spring rate u 1, u 2 = Nodal displacements u1u1 u2u2 F1F1 F2F2 12 k F 1 = k (u 1 – u 2 ) F 2 = k (u 2 – u 1 ) Spring Element

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Element stiffness matrix

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Spring System u2u2 u3u3 F1F1 F3F3 13 k2k2 k1k1 2 u1u1 F2F2 Element stiffness matrices

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics the stiffness relation by using superposition Total stiffness matrix

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Truss Element x y u2u2 u1u1 F2F2 F1F1 1 2  A Element stiffness matrix c = cosα s = sinα = length A = cross-sectional area E = Young´s modulus Spring rate of a truss element

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics  1 = 45 0  2 = y x F x3 F y  A E Element  : Element  : Node 1  1 Node 1  2 Node 2  3 Stiffness relation

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Beam Element EJ M1M1 M2M2 x y 1 2 Q1Q1 Q2Q2 x y v1v1 v2v2 11 22 1 2 ForcesDisplacements A = Cross – sectional areaE = Young’s modulus I = Moment of inertia = Length

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics the stiffness relation

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Example for practical FEM application Engineering systemPossible finite element model

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Plane stress Triangular Element x y u1u1 v1v1 u3u3 v3v3 u2u2 v2v2 Equilibrium condition: Principal of virtual work Compatibility condition: linear displacement function

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics General displacements (Displacement function) u (x,y) = α 1 + α 2 x + α 2 y v (x,y) = α 4 + α 5 x + α 6 y Nodal displacements u 1 = α 1 + α 2 x 1 + α 3 y 1 v 1 = α 4 + α 5 x 1 + α 6 y 1 similar for node 2 and node 3.

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics u=N dGeneral displacements to nodal displacements ε=B dStrains to nodal displacements σ=D εStresses to strains σ=D B dStresses to nodal displacements

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Other displacement functions quadratic displacement function u (x,y) = α 1 + α 2 x + α 3 y+ α 4 x 2 + α 5 y 2 +α 6 xy v (x,y) = α 7 + α 8 x + α 9 y+ α 10 x 2 + α 11 y 2 +α 12 xy Triangular element with 6 nodes 6

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics cubic displacement function - stress field can be better approximated - more computing time - less numerical accuracy - geometry cannot be good approximated Triangular element with 10 nodes 8 9 3

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics σ = stress matrixp = force matrix ε = strain matrixu = displacement matrix Principal of Virtual Work δU = virtual work done by the applied force δW = stored strain energy δU + δW = 0

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Element stiffness matrix D = Elasticity matrix

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics b 1 = y 2 – y 3 c 1 = x 3 – x 2 b 2 = y 3 – y 1 c 2 = x 1 – x 3 A Δ = Area of element b 3 = y 1 – y 2 c 3 = x 2 – x 1 linear displacement function yields : - linear displacement field - constant strain field - constant stress field

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Dynamics k1k1 k2k2 c1c1 c2c2 m1m1 m2m2 m0m0 u0u0 F0F0 F1F1 F2F2 u1u1 u2u2 Equation of motion

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics M = Mass matrix C = Damping matrix K = Stiffness matrix d = Nodal displacement matrix = Nodal velocity matrix = Nodal acceleration matrix or

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics for a continuum u = N d ε = B d

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics ρ = Mass density μ = Viscosity matrix the element matrices

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics o Modal analysis o Harmonic response analysis - Full harmonic - Reduced harmonic o Transient dynamic analysis - Linear dynamic - Nonlinear dynamic General Equation of Motion Types of dynamic solution

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department 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) Modal Analysis

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Transformation methodsIteration methods JACOBI INVERSE POWER GIVENS INVERSE POWER WITH SHIFTS HOUSEHOLDER SUB – SPACE ITERATION Q – R METHOD for harmonic motion: d = d 0 cos (ωt) (-ω 2 M + K) d 0 = 0 Eigenvalue extraction procedures

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department 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 Harmonic Response Analysis

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Forcing funktion F = F 0 e -iωt Response will be harmonic at input frequency d = d 0 e -iωt (-ω 2 M – iωC + K) d = F 0 is a complex matrix d will be complex (amplitude and phase angle)

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Limiting cases: ω = 0 :K d = F 0 Static solution C = 0 :(-ω 2 M + K) d = F 0 Response in phase C = 0, ω = ω n : (-ω n 2 M + K) d = F 0 infinite amplitudes C = 0, ω = ω n : (-ω n 2 M - iω n C + K) d = F 0 finite amplitudes, large phase shifts

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Transient Dynamic Analysis F (t) = arbitrary time history forcing function periodic forcing function

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics impulsive forcing function Earthquake in El Centro, California

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Two major types of integration: - Modal superposition - Direct numerical integration

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics T0T0 Q0Q0 T1T1 Q1Q1 T2T2 Q2Q  1, A 1 2, A 2  , A  0 0ne-dimensional heat flow principle ,  = conductivity elements  = convection element 0, 1, 2 = temperature elements A = Cross-sectional area = Length λ = Conductivity A α = Convection surface T = TemperatureQ = Heat flow C = Specific heatα = Coefficient of thermal expansion

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Heath flow through a conduction element: Heat stored in a temperature element: c p = specific heat capacity C = specific heat Heat transition for a convection element: Q =  A  (T  – T 2 )

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics Heat balance or

HCMUT 2004 Faculty of Applied Sciences Hochiminh City University of Technology The Finite Element Method PhD. TRUONG Tich Thien Department of Engineering Mechnics C = specific heat matrix K = conductivity matrix Q = heat flow matrix T = temperature matrix = time derivation of T For the stationary state with = 0 KT = Q