Faculty of Applied Sciences Hochiminh City University of

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

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. 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 The Finite Element Method

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

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

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

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics FE = Finite Element i, j, k = Nodal points (Nodes) of an Element - Dividing a solid in Finite Elements Compatibility between the Elements through a displacement function Equilibrium condition through the principal of virtual work HCMUT 2004 The Finite Element Method

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

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics Kd=F d. T = [u 1 v 1 w 1. . . un vn wn] FT = [Fx 1 F y 1. . . F xn F yn F zn] K is a n x n matrix K is a sparse matrix and symmetric HCMUT 2004 The Finite Element Method

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

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

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

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

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics Spring System k 1 1 k 2 2 F 1 u 1 3 F 2 u 3 Element stiffness matrices HCMUT 2004 The Finite Element Method

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

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

Faculty of Applied Sciences Hochiminh City University of Technology y Fy 3 3 1 Element : Node 1 1 Node 2 3 Fx 3 AE 1 = 450 Ph. D. TRUONG Tich Thien Department of Engineering Mechnics AE Element : Node 1 2 Node 2 3 2 = 1350 x 2 Stiffness relation HCMUT 2004 The Finite Element Method

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics Beam Element y y M 1 1 Q 1 M 2 EJ 2 x 1 v 1 1 2 2 v 2 x Q 2 Forces A = Cross – sectional area I = Moment of inertia Displacements E = Young’s modulus = Length HCMUT 2004 The Finite Element Method

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

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

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics Plane stress Triangular Element v 3 y 3 v 1 1 u 3 v 2 2 u 2 x Equilibrium condition: Principal of virtual work Compatibility condition: linear displacement function HCMUT 2004 The Finite Element Method

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. 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 The Finite Element Method

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics u = Nd General displacements to nodal displacements ε = Bd Strains to nodal displacements σ = Dε Stresses to strains σ = DBd Stresses to nodal displacements HCMUT 2004 The Finite Element Method

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics Other displacement functions 3 5 6 Triangular element with 6 nodes 2 1 4 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 HCMUT 2004 The Finite Element Method

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics 3 7 8 6 9 1 10 4 2 Triangular element with 10 nodes 5 cubic displacement function - stress field can be better approximated more computing time less numerical accuracy geometry cannot be good approximated HCMUT 2004 The Finite Element Method

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

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

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

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics Dynamics m 0 k 1 m 1 k 2 c 1 u 0 F 0 m 2 Equation of motion c 2 u 1 F 1 u 2 F 2 HCMUT 2004 The Finite Element Method

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

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics for a continuum u=Nd ε=Bd HCMUT 2004 The Finite Element Method

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

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

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

Faculty of Applied Sciences Hochiminh City University of Technology for harmonic motion: Ph. D. TRUONG Tich Thien Department of Engineering Mechnics d = d 0 cos (ωt) (-ω2 M + K) d 0 = 0 Eigenvalue extraction procedures Transformation methods Iteration methods JACOBI GIVENS HOUSEHOLDER Q – R METHOD INVERSE POWER WITH SHIFTS SUB – SPACE ITERATION HCMUT 2004 The Finite Element Method

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics Harmonic Response Analysis Purpose: To determine the response of a linear system Assumptions: Linear Structure (M, C, K = constant) Harmonic forcing function at known frequency HCMUT 2004 The Finite Element Method

Faculty of Applied Sciences Hochiminh City University of Technology Forcing funktion Ph. D. TRUONG Tich Thien Department of Engineering Mechnics F = F 0 e-iωt Response will be harmonic at input frequency d = d 0 e-iωt (-ω2 M – iωC + K) d = F 0 d is a complex matrix will be complex (amplitude and phase angle) HCMUT 2004 The Finite Element Method

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics Limiting cases: ω=0: K d = F 0 Static solution C=0: (-ω2 M + K) d = F 0 Response in phase C = 0, ω = ωn : (-ωn 2 M + K) d = F 0 infinite amplitudes C = 0, ω = ωn : (-ωn 2 M - iωn. C + K) d = F 0 large phase shifts finite amplitudes, HCMUT 2004 The Finite Element Method

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

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics impulsive forcing function Earthquake in El Centro, California 18. 05. 1940 HCMUT 2004 The Finite Element Method

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

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department of Engineering Mechnics 0 ne-dimensional heat flow principle 1 , A 1 0 1 A λ T C 2 , A 2 1 2 2 , A T 2 Q 2 T 0 Q 0 T 1 Q 1 = = Cross-sectional area Conductivity Temperature Specific heat , = conductivity elements = convection element 0, 1, 2 = temperature elements Aα Q α = = Length Convection surface Heat flow Coefficient of thermal expansion HCMUT 2004 The Finite Element Method

Faculty of Applied Sciences Hochiminh City University of Technology Ph. D. TRUONG Tich Thien Department 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 – T 2) HCMUT 2004 The Finite Element Method

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

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