SlideShare uma empresa Scribd logo

Isoparametric bilinear quadrilateral element _ ppt presentation

•
•
F
Filipe Giesteira

The theoretical formulation of an isoparametric element, from the Lagrange family. In addition, the MATLAB code of the FEM from the bilinear element with four nodes was also implemented. Assignment developed in the scope of the finite element method course, lectured at FEUP (Faculdade de Engenharia da Universidade do Porto).

Engenharia
1 de 40
Baixar para ler offline
2D Plane Elasticity – Isoparametric Bilinear Quadrilateral Lagrange type Element Filipe Giesteira Master student Faculty of Engineering of University of Porto (FEUP) Mechanical Engineering Department (DEMec) Finite Element Method (MEF, Método dos Elementos Finitos) Supervisor: Prof. Francisco Andrade Pires Supervisor: Prof. José Dias Rodrigues Finite Element Method (FEM) - Final Presentation Faculty of Engineering of University of Porto Building L / Room L317 Porto, 24th January 2019 Work Assignment No. 2.3
2 Contents ➢ Introduction – Plane Elasticity Background Theory ➢ Weak & Strong Form – FEM Mathematical Formulation ➢ Finite Element Library – Isoparametric Bilinear Quadrilateral Element ➢ FEM Equations – Element Mechanical Properties ➢ BaPMEF FEM Tool – Script Capabilities and Limitations ➢ Validation Examples – Solutions of the Biharmonic Equation / Airy Functions ➢ Practical Applications 24-01-2019
3 1. Introduction 1.1. Plane Elasticity Background Theory – Assumptions and Simplifications 24-01-2019 y x l t w z 𝑢 Ԧ𝑣 𝑤 σzz x, y, z = ± Τt 2 = τyz x, y, z = ± Τt 2 = τxz(x, y, z = ± Τt 2) = 0 t ≪ l ∧ t ≪ w ⇒ σzz x, y, z = τyz x, y, z = τxz(x, y, z) = 0 t ≪ l ∧ t ≪ w ⇒ ൞ )σxx = fxx(x, y σyy = fyy x, y ൯τxy = fxy(x, y y xl tw z Ԧ𝑣 𝑢 𝑤 t ≫ l ∧ t ≫ w ⇒ σzz x, y, z = C te ቊ 𝑤 𝑥, 𝑦, ± Τ𝑡 2 = 0 𝑤 𝑥, 𝑦, 0 = 0, 𝑏𝑦 𝑠𝑦𝑚. ⇒ 𝑤 𝑥, 𝑦, 𝑧 ≈ 0 𝑤 𝑥, 𝑦, 𝑧 = 0 ⇒ 𝜀 𝑧𝑧(𝑥, 𝑦, 𝑧) = 0 𝛾𝑦𝑧(𝑥, 𝑦, 𝑧) = 0 𝛾𝑥𝑧(𝑥, 𝑦𝑧, 𝑧) = 0 )𝜀 𝑥𝑥 = 𝑓𝑥𝑥(𝑥, 𝑦 ൯𝜀 𝑦𝑦 = 𝑓𝑦𝑦(𝑥, 𝑦 ൯𝛾𝑥𝑦 = 𝑓𝑥𝑦(𝑥, 𝑦 ൞ 𝑓𝑥 = 𝑓𝑥 𝑥, 𝑦 𝑓𝑦 = 𝑓𝑦 𝑥, 𝑦 𝑓𝑧 = 0 ➢ Plane Stress ➢ Plane Strain
4 ➢ Displacement Field ➢ Strain Field ➢ Stress Field 1. Introduction 1.2. Plane Elasticity Background Theory – Solid Mechanics Continuum 24-01-2019 𝑢 = 𝑢 𝑥, 𝑦 = )𝑢(𝑥, 𝑦 )𝑣(𝑥, 𝑦 𝜀 = 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾𝑥𝑦 = 𝜕𝑢 𝜕𝑥 𝜕𝑣 𝜕𝑦 𝜕𝑢 𝜕𝑦 + 𝜕𝑣 𝜕𝑥 𝜀 = L 𝑢 = 𝜕 𝜕𝑥 0 0 𝜕 𝜕𝑦 𝜕 𝜕𝑦 𝜕 𝜕𝑥 )𝑢(𝑥, 𝑦 𝑣 𝑥, 𝑦 𝜎 = 𝐶 𝜀 𝐶 = 𝐸 1 − 𝑣2 𝐸𝑣 1 − 𝑣2 0 𝐸𝑣 1 − 𝑣2 𝐸 1 − 𝑣2 0 0 0 𝐸 2 1 + 𝑣 ; 𝐸 1 − 𝑣 )1 + 𝑣 (1 − 2𝑣 𝐸𝑣 )1 + 𝑣 (1 − 2𝑣 0 𝐸𝑣 )1 + 𝑣 (1 − 2𝑣 𝐸 1 − 𝑣 )1 + 𝑣 (1 − 2𝑣 0 0 0 𝐸 2 1 + 𝑣 Plane Stress Plane Strain Matrix form useful to discretize the Weak-form in Finite Domains! Isotropic Transversely Isotropic Orthotropic Monolithic … …
5 2. Weak & Strong Form 24-01-2019 2.1. Development of the Weak Form ➢ Principle of the Minimum Total Potential Energy ➢ Principle of Virtual Work/Virtual Displacements ➢ Hamilton’s Principle ➢ Galerkin Method of Weighted-Residuals 𝛿𝛱 = 𝜕 𝜕|𝑢| 𝛱 𝑢 𝛿𝑢 = 𝛿𝑉 − 𝛿𝑊 = 0 𝛿𝑊 𝐼𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝐹𝑜𝑟𝑐𝑒𝑠 + 𝛿𝑊 𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝐹𝑜𝑟𝑐𝑒𝑠 + 𝛿𝑊 𝐸𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝐹𝑜𝑟𝑐𝑒𝑠 = 0 න 𝑡1 𝑡2 𝛿 𝑇 − 𝑉 𝑑𝑡 + න 𝑡1 𝑡2 𝛿𝑊𝑛𝑐 𝑑𝑡 = 0 න Ω 𝑤𝑥 − 𝜕 𝜕𝑥 𝜎 𝑥𝑥 − 𝜕 𝜕𝑦 𝜏 𝑥𝑦 − 𝑓𝑥 + 𝜌 𝜕2 𝜕𝑡2 𝑢 𝑥, 𝑦 𝑑𝑥 𝑑𝑦 = 0 න Ω 𝑤 𝑦 − 𝜕 𝜕𝑦 𝜎 𝑦𝑦 − 𝜕 𝜕𝑥 𝜏 𝑥𝑦 − 𝑓𝑦 + 𝜌 𝜕2 𝜕𝑡2 𝑣 𝑥, 𝑦 𝑑𝑥 𝑑𝑦 = 0 Variational Principles Integral Statement
6 ➢ Weak Form – ready to be Discretized in Finite Element Domains ➢ Strong Form – continuously valid in the entire Domain 2. Weak & Strong Form 24-01-2019 2.2. Strong vs. Weak Form 𝑡 ඵ 𝐴 𝐿 𝛿𝑢 𝑇 𝐶 𝐿 𝑢 𝑑𝐴 + ඵ 𝐴 𝛿𝑢 𝑇 𝜌 ሷ𝑢 𝑑𝐴 = 𝑡 ඵ 𝐴 𝛿𝑢 𝑇 𝑏 𝑑𝐴 + 𝑡 ර 𝑙 𝛿𝑢 𝑇 𝑞 𝑑𝑠 + ෍ 𝑝𝑖 𝛿𝑢 𝑝𝑖 𝑇 𝑃𝑝𝑖 𝜕 𝜕𝑥 𝜎𝑥𝑥 + 𝜕 𝜕𝑦 𝜏 𝑥𝑦 + 𝑏 𝑥 = 𝜌 𝜕2 𝜕𝑡2 𝑢 𝑥, 𝑦 𝜕 𝜕𝑦 𝜎 𝑦𝑦 + 𝜕 𝜕𝑥 𝜏 𝑥𝑦 + 𝑏 𝑦 = 𝜌 𝜕2 𝜕𝑡2 𝑣 𝑥, 𝑦 ; 𝜎𝑥𝑥 = 𝐶11 𝜕𝑢 𝜕𝑥 + 𝐶12 𝜕𝑣 𝜕𝑦 𝜎 𝑦𝑦 = 𝐶12 𝜕𝑢 𝜕𝑥 + 𝐶11 𝜕𝑣 𝜕𝑦 ; ൝ 𝜏 𝑥𝑦 = 𝐶11 − 𝐶12 2 𝜕𝑢 𝜕𝑦 + 𝜕𝑣 𝜕𝑥 𝑢 = ො𝑢 , ∀ 𝛤 = 𝛤𝑢 ; 𝑡 = 𝜎 𝑇 ො𝑛 = Ƹ𝑡 , ∀ 𝛤 = 𝛤𝑡 Natural Boundary ConditionsEssential Boundary Conditions 𝛿𝑢 = 𝛿𝑢 𝛿𝑣 𝛿𝑢 𝑝𝑖 = 𝛿𝑢 𝑝𝑖 𝛿𝑣 𝑝𝑖 𝑃𝑝𝑖 = 𝑃𝑝𝑖 𝑥 𝑃𝑝𝑖 𝑦 𝑏 = 𝑏 𝑥 𝑏 𝑦 𝑞 = 𝑞 𝑥 𝑞 𝑦 ሷ𝑢 = ሷ𝑢 ሷ𝑣
7 3. Finite Element Library 24-01-2019 3.1. Isoparametric Bilinear Quadrilateral Element 𝑁𝑖 = 𝑁1 𝑁2 𝑁3 𝑁4 = 1 4 1 − 𝜉 1 − 𝜂 1 4 1 + 𝜉 1 − 𝜂 1 4 1 + 𝜉 1 + 𝜂 1 4 1 − 𝜉 1 + 𝜂 Node 𝝃 𝜼 1 -1 -1 2 1 -1 3 1 1 4 -1 1 𝜂 𝜉 1 4 3 2 Lagrange type Element - 4 nodes Shape / Interpolation Functions
8 3. Finite Element Library 24-01-2019 3.2. Discretization of the Continuum Fields 𝑢 𝑒 ≈ 𝑁𝑖 𝜉, 𝜂 𝑢𝑖 𝑒 = 𝑁1 0 𝑁2 0 𝑁3 0 𝑁4 0 0 𝑁1 0 𝑁2 0 𝑁3 0 𝑁4 𝑢1 𝑣1 𝑢2 𝑣2 𝑢3 𝑣3 𝑢4 𝑣4 𝑢 𝑒 = )𝑢(𝜉, 𝜂 )𝑣(𝜉, 𝜂 ≈ 𝑎 + 𝑏 𝜉 + 𝑐 𝜂 + 𝑑 𝜉𝜂 𝑎 + 𝑏 𝜉 + 𝑐 𝜂 + 𝑑 𝜉𝜂 = 𝑁𝑖 𝑇 𝑢𝑖 𝑁𝑖 𝑇 𝑣𝑖 ε 𝑒 = 𝐿 𝑁 𝑢𝑖 𝑒 = 𝐵 ui e = 𝜕 𝜕𝑥 𝑁1 0 𝜕 𝜕𝑥 𝑁2 0 𝜕 𝜕𝑥 𝑁3 0 𝜕 𝜕𝑥 𝑁4 0 0 𝜕 𝜕𝑦 𝑁1 0 𝜕 𝜕𝑦 𝑁2 0 𝜕 𝜕𝑦 𝑁3 0 𝜕 𝜕𝑦 𝑁4 𝜕 𝜕𝑦 𝑁1 𝜕 𝜕𝑥 𝑁1 𝜕 𝜕𝑦 𝑁2 𝜕 𝜕𝑥 𝑁2 𝜕 𝜕𝑦 𝑁3 𝜕 𝜕𝑥 𝑁3 𝜕 𝜕𝑦 𝑁4 𝜕 𝜕𝑥 𝑁4 𝑢1 𝑣1 𝑢2 𝑣2 𝑢3 𝑣3 𝑢4 𝑣4 ➢ Displacement Field ➢ Strain Field Direct Computation Require Isoparametric Mapping
9 3. Finite Element Library 24-01-2019 3.3. Isoparametric Mapping 𝐽 = 𝐽11 𝐽12 𝐽21 𝐽22 = 𝜕𝑥 𝜕𝜉 𝜕𝑦 𝜕𝜉 𝜕𝑥 𝜕𝜂 𝜕𝑦 𝜕𝜂 = 𝜕𝑁 𝑖 𝜕𝜉 𝑇 𝜕𝑁 𝑖 𝜕𝜂 𝑇 𝑥𝑖 𝑦𝑖 𝑥𝑖 𝑦𝑖 = 𝑥1 𝑦1 𝑥2 𝑦2 𝑥3 𝑦3 𝑥4 𝑦4 𝜕𝑁𝑖 𝜕𝜉 𝑇 = 𝜕𝑁1 𝜕𝜉 𝜕𝑁2 𝜕𝜉 𝜕𝑁3 𝜕𝜉 𝜕𝑁4 𝜕𝜉 = 1 4 𝜂 − 1 1 4 1 − 𝜂 1 4 1 + 𝜂 − 1 4 1 + 𝜂 𝜕𝑁𝑖 𝜕𝜂 𝑇 = 𝜕𝑁1 𝜕𝜂 𝜕𝑁2 𝜕𝜂 𝜕𝑁3 𝜕𝜂 𝜕𝑁4 𝜕𝜂 = 1 4 𝜉 − 1 − 1 4 1 + 𝜉 1 4 1 + 𝜉 1 4 1 − 𝜉 𝜕𝑁𝑖 𝜕𝑥 𝑇 𝜕𝑁𝑖 𝜕𝑦 𝑇 = 𝐽 −1 𝜕𝑁𝑖 𝜕𝜉 𝑇 𝜕𝑁𝑖 𝜕𝜂 𝑇 ➢ Derivatives in respect to the Natural Coordinates ➢ Jacobian Matrix ➢ Derivatives in respect to the Local Coordinates det 𝐽 = 𝐽11 𝐽22 − 𝐽12 𝐽21 > 0
10 3. Finite Element Library 3.3. Isoparametric Mapping ➢ Differential Linear Operators ➢ Differential Area Operator and Determinant of the Jacobian Matrix (Jacobian) ➢ Transformation Matrix – Rotation dA = dx dy =? = det J dξ dη 𝑇 =? = 𝑇 𝜂=𝐶 𝑡𝑒 = cos 𝛼 − sin 𝛼 sin 𝛼 cos 𝛼 = 𝜕𝑥 𝜕𝜉 − 𝜕𝑦 𝜕𝜉 𝜕𝑦 𝜕𝜉 𝜕𝑥 𝜕𝜉 = 𝐽11 −𝐽12 𝐽12 𝐽11 (𝜉,±1) , 𝜂 = ±1 𝑇 𝜉=𝐶 𝑡𝑒 = cos 𝛽 − sin 𝛽 sin 𝛽 cos 𝛽 = 𝜕𝑦 𝜕𝜂 − 𝜕𝑥 𝜕𝜂 𝜕𝑥 𝜕𝜂 𝜕𝑦 𝜕𝜂 = 𝐽22 −𝐽21 𝐽21 𝐽22 (±1,𝜂) , 𝜉 = ±1 dx dy =? = J T dξ dη
11 4. FEM Equations 24-01-2019 4.1. Finite Element Equations – Element Level ➢ Static Equilibrium Equation ➢ Equivalent Load Vector ➢ Stiffness Matrix 𝐾 𝑒 = 𝑡 ඵ 𝐴 𝐵 𝑇 𝐶 𝐵 𝒅𝑨 = 𝑡 ෍ 𝑗 2 ෍ 𝑘 2 𝐵 𝜉𝑗, 𝜂 𝑘 𝑇 𝐶 𝐵 𝜉𝑗, 𝜂 𝑘 det 𝐽(𝜉𝑗, 𝜂 𝑘) 𝑊𝑗 𝑊𝑘 (𝝃𝒋 , 𝜼 𝒌) 𝑾𝒋 𝑾 𝒌 − Τ1 3 , − Τ1 3 1 1 Τ1 3, − Τ1 3 1 1 Τ1 3 , Τ1 3 1 1 − Τ1 3, Τ1 3 1 1 𝐹 𝑒 = 𝑓𝑏 𝑒 + 𝑓𝑞 𝑒 + 𝑓𝑃 𝑒 ෍ 𝑒 𝐸 𝐾 𝑒 𝑢𝑖 𝑒 − 𝐹 𝑒 = 0 Body Forces Vector Surface Tractions Vector Nodal Point Forces Vector Global Equilibrium Equation System of Algebraic Equations 2 𝜂 𝜉 1 4 3 GP2 GP4 GP3 GP1 Illustration of the Gauss Points Table with the Gauss Points and respective Weight for Numerical Integration using Gauss Quadrature
12 4. FEM Equations 24-01-2019 4.2. Equivalent Load Vectors – Part 1 ➢ Body Forces ➢ Nodal Point Loads 𝑓𝑏 𝑒 = 𝑡 ඵ 𝐴 𝑁 𝑇 𝑏 𝑒 𝑑𝐴 = 𝑡 ෍ 𝑗 2 ෍ 𝑘 2 𝑁 𝜉𝑗, 𝜂 𝑘 𝑇 𝑏 𝑥 𝑒 𝑏 𝑦 𝑒 det 𝐽(𝜉𝑗, 𝜂 𝑘) 𝑊𝑗 𝑊𝑘 𝑓𝑃 𝑒 = ෍ 𝑖 𝑃𝑖 = ෍ 𝑖 𝑃𝑥 𝑖 𝑃𝑦 𝑖 𝜂 𝜉 1 4 3 2 𝜂 𝜉 1 4 3 2 Body Forces acting on the quadrilateral element, along the y-direction Nodal Point Loads Body forces acting along the z-direction aren’t considered in plane elasticity (𝝃𝒋 , 𝜼 𝒌) 𝑾𝒋 𝑾 𝒌 − Τ1 3 , − Τ1 3 1 1 Τ1 3, − Τ1 3 1 1 Τ1 3 , Τ1 3 1 1 − Τ1 3, Τ1 3 1 1 Table with the Gauss Points and respective Weight for Numerical Integration using Gauss Quadrature
13 4. FEM Equations 24-01-2019 4.3. Equivalent Load Vectors – Part 2 ➢ Surface Tractions 𝑓𝑞 𝑒 = 𝑡 ර 𝑙 𝑒 𝑁 𝑇 𝒒∗ 𝒅𝒔 = න −1 1 𝑡 𝑁 𝜉, 𝜂 = ±1 𝑇 𝒒∗ 𝑐(𝜉) 𝑑𝜉 , 𝜂 = 𝐶 𝑡𝑒 න −1 1 𝑡 𝑁 𝜉 = ±1, 𝜂 𝑇 𝒒∗ 𝑐 𝜂 𝑑𝜂 , 𝜉 = 𝐶 𝑡𝑒 𝑓𝑞 𝑒 = ෍ 𝑗 2 𝑡 𝑁 𝜉𝑗, 𝜂 = ±1 𝑇 𝒒∗ 𝑐(𝜉𝑗) 𝑊𝑗 , 𝜂 = ±1 ෍ 𝑘 2 𝑡 𝑁 𝜉 = ±1, 𝜂 𝑘 𝑇 𝒒∗ 𝑐 𝜂 𝑘 𝑊𝑘 , 𝜉 = ±1 𝑐 = 𝑐 𝜉 = 𝐽11 𝜉, ±1 2 + 𝐽12 𝜉, ±1 2 , 𝜂 = ±1 𝑐 𝜂 = 𝐽21 ±1, 𝜂 2 + 𝐽22 ±1, 𝜂 2 , 𝜉 = ±1 Division of the Rectangular Boundary in four straight paths
14 4. FEM Equations 24-01-2019 4.3. Equivalent Load Vectors – Part 2 ➢ Surface Tractions Loads in Local Coordinates Loads in Natural Coordinates 𝑞∗ = 𝑞 𝑥 𝑒 𝑞 𝑦 𝑒 𝑞∗ = 𝑇 𝜂=𝐶 𝑡𝑒 𝜏 𝜎 = 𝐽11 −𝐽12 𝐽12 𝐽11 𝜉,±1 𝜏 𝜎 , 𝜂 = ±1 𝑇 𝜉=𝐶 𝑡𝑒 𝜎 𝜏 = 𝐽22 −𝐽21 𝐽21 𝐽22 ±1,𝜂 𝜎 𝜏 , 𝜉 = ±1
15 5. BapMEF FEM Tool 24-01-2019 5.1. Interface – Overview Warnings and Notes Input Boxes Menu Windows Helping graphic output during data input Data Input
16 5. BapMEF FEM Tool 24-01-2019 5.2. Data Input Geometric Variables Body Thickness Definition Material Properties Type of Plane Problem
17 5. BapMEF FEM Tool 24-01-2019 5.3. Special Features ➢ Meshing procedure
18 5. BapMEF FEM Tool 5.3. Special Features ➢ Definition of the Boundary Condition Simple Support Encastre SymmetryFracture Mechanics
19 ➢ Displacements ➢ S𝐭𝐫𝐚𝐢𝐧𝐬 ➢ Stresses ➢ Safety Factor ➢ Von Mises Equivalent Stress 5. BapMEF FEM Tool 24-01-2019 5.4. Output and Post-Processing Variables 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾𝑥𝑦 𝜺 𝒛𝒛 σ 𝑥𝑥 σ 𝑦𝑦 τ 𝑥𝑦 𝝈 𝒛𝒛 𝜎𝑒𝑞 = 1 2 𝜎 𝑥𝑥 − 𝜎 𝑦𝑦 2 𝜎 𝑦𝑦 − 𝜎𝑧𝑧 2 𝜎 𝑥𝑥 − 𝜎𝑧𝑧 2 + 6𝜏 𝑥𝑦 2 𝑆 = 𝜎 𝑦𝑒𝑙𝑑 𝜎 𝑒𝑞 𝑑 = 𝑢2 + 𝑣2 𝑢 𝑣 Plane Stress Plane Strain
20 6. Validation Examples 24-01-2019 6.1. Constant Rectangular cross-section Beam – Pure Bending 𝑙 𝑙 𝒙 𝒛 𝑢 𝑥, 𝑦 = − 𝑀 𝑜 𝐸𝐼 𝑥𝑦 1 𝑢(𝑥, 𝑦) = − 12𝑀 𝑜 𝐸𝑡𝑤3 𝑥 − 𝑙 2 𝑦 − 𝑤 2 Physical Problem Equivalent Problem SimulatedBending Moments ?? Rotation DOF ?? [1] Silva Gomes, Mecânica dos Sólidos e Resistência dos Materiais Change of Coordinate System 𝑦 𝑥 𝑦 𝑥 𝑤 𝑡 𝑤 𝑀 𝑜 = 𝑃 ∙ 𝑎 Original Coordinate System Coordinate System of the BaPMEF tool
21 6. Validation Examples 24-01-2019 6.1. Constant Rectangular cross-section Beam – Pure Bending Comparison of the displacements in the x-direction, determined by the BaPMEF tool against the Analytical solution ➢ Overall good match between the Analytical and Numeric Solution ➢ Null displacements along the Neutral Axis ➢ Small mismatch near the plate’s ends: ▪ Analytical solution badly behaved in plate’s boundaries[1] ▪ Boundary Conditions difficult to model in Plane Elasticity Deformed shape of the mesh
22 6. Validation Examples 24-01-2019 𝑡 𝑤 𝑢(𝑥, 𝑦) = 𝑃 6𝐺𝐼 𝑦 − 𝑤 2 3 − 𝑃𝑥2 2𝐸𝐼 𝑦 − 𝑤 2 − 𝑣 𝑃 6𝐸𝐼 𝑦 − 𝑤 2 3 + 𝑃𝑙2 2𝐸𝐼 − 𝑃𝑤2 8𝐺𝐼 𝑦 − 𝑤 2 𝑣(𝑥, 𝑦) = 𝑣 𝑃𝑥 2𝐸𝐼 𝑦 − 𝑤 2 2 + 𝑃𝑥3 6𝐸𝐼 − 𝑃𝑙2 𝑥 2𝐸𝐼 + 𝑃𝑙3 3𝐸𝐼 𝑢(𝑥, 𝑦) = 𝑃 6𝐺𝐼 𝑦 − 𝑤 2 3 − 𝑃𝑥2 2𝐸𝐼 𝑦 − 𝑤 2 − 𝑣 𝑃 6𝐸𝐼 𝑦 − 𝑤 2 3 + 𝑃𝑙2 2𝐸𝐼 𝑦 − 𝑤 2 𝑣(𝑥, 𝑦) = 𝑣 𝑃𝑥 2𝐸𝐼 𝑦 − 𝑤 2 2 + 𝑃𝑥3 6𝐸𝐼 − 𝑃𝑙2 2𝐸𝐼 + 𝑃𝑤2 8𝐺𝐼 𝑥 + 𝑃𝑙3 3𝐸𝐼 + 𝑃𝑤2 𝑙 8𝐺𝐼 𝜕𝑣 𝜕𝑥 𝑥=𝑙 ; 𝑦=0 = 0 𝜕𝑢 𝜕𝑦 𝑥=𝑙 ; 𝑦=0 = 0 Situation (ii) 1 Situation (i) 1 ➢ Shear distributed Load ➢ There is no approximation in the definition of the BC ➢ It is also necessary to change the coordinate system 6.2. Constant Rectangular cross-section Beam – Cantilever Beam under Tangential Traction at the end [1] Silva Gomes, Mecânica dos Sólidos e Resistência dos Materiais Problem Simulated 𝑦 𝑥 𝑞 𝑙 𝑤𝑃 𝒒 = 𝑷 𝒘 ∙ 𝒕
23 6. Validation Examples 24-01-2019 ➢ Good match between the Numerical and Analytical Solutions – The Numeric Solution is always in-between the two Analytical Solutions ➢ Null displacements along the Neutral Axis ➢ Situation (i) is more “rigid” ➢ Situation (ii) is more “flexible” 6.2. Constant Rectangular cross-section Beam – Cantilever Beam under Tangential Traction at the end Horizontal Displacements along the plate Deformed Shape of the Mesh
24 6. Validation Examples 24-01-2019 6.2. Constant Rectangular cross-section Beam – Cantilever Beam under Tangential Traction at the end ➢ Good match between the Numerical and Analytical Solutions – The Numeric Solution is always in-between the two Analytical Solutions ➢ Null displacements along the Neutral Axis ➢ Situation (i) is more “rigid” ➢ Situation (ii) is more flexible Vertical Displacements along the plate
25 6. Validation Examples 24-01-2019 6.2. Constant Rectangular cross-section Beam – Cantilever Beam under Tangential Traction at the end ➢ H-type Mesh Refinement Elements: Length=1 x Width=2 Elements: Length=2 x Width=2 Elements: Length=8 x Width=2 Elements: Length=25 x Width=10
26 𝑣 𝑥, 𝑦 1 = 𝑸 2𝐸𝐼 ൮ ൲ 𝑦 − 𝑤 2 4 12 − 𝑤2 8 𝑦 − 𝑤 2 2 + 𝑤3 12 𝑦 − 𝑤 2 + 𝑣 𝑙2 4 − 𝑥 − 𝑙 2 2 𝑦 − 𝑤 2 2 + 𝑦 − 𝑤 2 4 6 − 𝑤2 20 𝑦 − 𝑤 2 2 + 𝑙2 8 𝑥 − 𝑙 2 2 − 𝑥 − 𝑙 2 4 12 − 𝑤2 20 𝑥 − 𝑙 2 2 + 1 4 + 𝑣 8 𝑤2 𝑥 − 𝑙 2 2 − 5𝑙4 192 1 + 12𝑤2 5𝑙2 4 5 + 𝑣 2 6. Validation Examples 24-01-2019 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction 𝑙 𝑤 𝑞 𝑦 𝑥 𝑡 𝑤 𝑞 𝑄 ➢ There is no approximation in the definition of the BC [1] Silva Gomes, Mecânica dos Sólidos e Resistência dos Materiais Problem Simulated 𝑸 = 𝒒 ∙ 𝒕
27 6. Validation Examples 24-01-2019 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction ➢ It is also necessary to change the original Coordinate System of the Analytical Solution ➢ Analytical Solutions for the Stresses 𝜎 𝑥𝑥 1 = 𝑸 2𝐼 2 3 𝑦 − 𝑤 2 3 − 𝑤2 10 𝑦 − 𝑤 2 − 𝑸 2𝐼 𝑥 − 𝑙 2 2 − 𝑙2 4 𝑦 − 𝑤 2 𝜎 𝑦𝑦 1 = − 𝑸 2𝐼 𝑦 − 𝑤 2 3 3 − 𝑤2 4 𝑦 − 𝑤 2 − 𝑤3 12 𝜏 𝑥𝑦 1 = 𝑸 2𝐼 𝑦 − 𝑤 2 2 − 𝑤2 4 𝑥 − 𝑙 2 [1] Silva Gomes, Mecânica dos Sólidos e Resistência dos Materiais Deformed Shape of the Mesh
28 6. Validation Examples 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction ➢ Good match between the Numerical and Analytical Solutions for the Vertical Displacement in the y-direction 3D Surface Plot of the vertical displacements along the plate 2D Plot of the vertical displacements along the Neutral Axis
29 6. Validation Examples 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction ➢ Overall good match between the Numerical and Analytical Solutions for the Normal Stress in the x-direction ➢ Effect of the local Supports 3D Surface Plot of the Stress in the x-direction along the plate 2D Plot of the Stress in the x-direction along the Neutral Axis
30 6. Validation Examples 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction ➢ Overall good match between the Numerical and Analytical Solutions for the Normal Stress in the y-direction ➢ Pronounced effect of the local Supports 3D Surface Plot of the Stress in the y-direction along the plate 2D Plot of the Stress in the y-direction along the Neutral Axis
31 6. Validation Examples 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction ➢ Overall good match between the Numerical and Analytical Solutions for the Normal Stress in the x-direction ➢ Effect of the local Supports 3D Surface Plot of the Shear Stress along the plate 2D Plot of the Shear Stress along the Neutral Axis
32 7. Practical Applications 24-01-2019 7.1. Fracture Mechanics – Plate with Centered Crack 2a 𝑊 𝐿 𝑞 𝑞 𝑊 𝑡 a 𝑤 = 𝑊 2 𝑙 = 𝐿 2 Physical Problem Equivalent Problem Simulated ➢ There is no approximation in the definition of the BC ➢ It is also necessary to change the original Coordinate System of the Analytical Solutions ➢ Application of the Mesh Refinement Control Mesh Control mode and Zoom of the Stresses in y-direction near the crack tip
33 7. Practical Applications 24-01-2019 7.1. Fracture Mechanics – Plate with Centered Crack 𝜎 𝑦𝑦 = 𝑞 𝑎 )2(𝑥 − 𝑎 𝜎 𝑦𝑦 = 𝑞 𝑎 )2(𝑥 − 𝑎 sec 𝜋𝑎 𝐿 𝜎 𝑦𝑦 = 𝑞 𝑎 )2(𝑥 − 𝑎 𝐿 𝜋𝑎 tan 𝜋𝑎 𝐿 𝜎 𝑦𝑦 = 𝑞 1 − 𝑎 𝑥 2 4 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 ; 4 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 sec 𝜋𝑎 𝐿 ; 4 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 𝐿 𝜋𝑎 tan 𝜋𝑎 𝐿 ; 2 𝐸 𝑞 𝑎2 − 𝑥2 ; )4(1 − 𝑣2 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 )4(1 − 𝑣2 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 sec 𝜋𝑎 𝐿 )4(1 − 𝑣2 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 𝐿 𝜋𝑎 tan 𝜋𝑎 𝐿 )2(1 − 𝑣2 𝐸 𝑞 𝑎2 − 𝑥2 Plane Stress Plane Strain Solution [2] (i) (ii) (iii) (iv) 𝑣 𝑥, 𝑦 = 𝑣 𝑥, 𝑦 = 𝑣 𝑥, 𝑦 = 𝑣 𝑥, 𝑦 = ➢ Plane Strain Problem [2] David Broke, Elementary Engineering Fracture Mechanics
34 7. Practical Applications 24-01-2019 7.1. Fracture Mechanics – Plate with Centered Crack ➢ Crack Opening Shape ➢ Plate infinitely higher than the crack length – overlap between solution (i), (ii), and (iii) ➢ Solution (i), (ii), and (iii) are only valid near the crack tip – overlap between all solution near the crack tip Vertical Displacements – Plane Stress Vertical Displacements – Plane Strain Overlap Solution (i), (ii), and (iii) Crack tip region
35 7. Practical Applications 24-01-2019 7.1. Fracture Mechanics – Plate with Centered Crack ➢ Stresses along the plate, starting near the crack tip ➢ Plate infinitely higher than the crack length – overlap between solution (i), (ii), and (iii) ➢ Solution (i), (ii), and (iii) are only valid near the crack tip – overlap between all solution near the crack tip Stress in the y-direction near the crack tip Stress in the y-direction far from the crack tip
36 7. Practical Applications 24-01-2019 7.2. BaPMEF vs. Abaqus® 15 Τ𝑁 𝑚𝑚2 𝑷 5 Τ𝑁 𝑚𝑚2 150 [𝑁] 500 [𝑁] 50 [𝑁] 100 [𝑚𝑚] 100 [𝑚𝑚] 30 [𝑚𝑚] 30 [𝑚𝑚] 20 [𝑚𝑚] 20 [𝑚𝑚] 10 Τ𝑁 𝑚𝑚2 BaPMEF ➢ Commercially Available FEA Software – Abaqus ® ➢ Plane Strain Problem ➢ Generic Loading – No Analytical Solution Available ➢ Gravity Body Forces ➢ Surface Tractions acting along the whole boundary ➢ Surface Tractions acting on each Element Problem Simulated in Abaqus Problem Simulated in BaPMEF Gravity Force
37 7. Practical Applications 24-01-2019 7.2. BaPMEF vs. Abaqus® ➢ Horizontal Displacements Results from BaPMEF Tool Results from Abaqus® Software
38 7. Practical Applications 24-01-2019 7.2. BaPMEF vs. Abaqus® ➢ Vertical Displacements Results from BaPMEF Tool Results from Abaqus® Software
39 7. Practical Applications 24-01-2019 7.2. BaPMEF vs. Abaqus® ➢ Displacements Magnitude Results from BaPMEF Tool Results from Abaqus® Software
40 Thank you for your attention! 24-01-2019 Método dos Elementos Finitos EM065 Theory FEM Programming Validation & Practical Applications

Recomendados

Finite Element Analysis - UNIT-2
Finite Element Analysis - UNIT-2Finite Element Analysis - UNIT-2
Finite Element Analysis - UNIT-2
propaul
 
Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5
propaul
 
Constant strain triangular
Constant strain triangular Constant strain triangular
Constant strain triangular
rahul183
 
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANKME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ASHOK KUMAR RAJENDRAN
 
Finite Element Analysis - UNIT-1
Finite Element Analysis - UNIT-1Finite Element Analysis - UNIT-1
Finite Element Analysis - UNIT-1
propaul
 
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANKME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
ASHOK KUMAR RAJENDRAN
 
Finite Element Analysis - UNIT-4
Finite Element Analysis - UNIT-4Finite Element Analysis - UNIT-4
Finite Element Analysis - UNIT-4
propaul
 
Fem lecture
Fem lectureFem lecture
Fem lecture
Muhammad Mohsin Waseem
 

Recomendados

Finite Element Analysis - UNIT-2
Finite Element Analysis - UNIT-2Finite Element Analysis - UNIT-2
Finite Element Analysis - UNIT-2
propaul
 
Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5Finite Element Analysis - UNIT-5
Finite Element Analysis - UNIT-5
propaul
 
Constant strain triangular
Constant strain triangular Constant strain triangular
Constant strain triangular
rahul183
 
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANKME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - III NOTES AND QUESTION BANK
ASHOK KUMAR RAJENDRAN
 
Finite Element Analysis - UNIT-1
Finite Element Analysis - UNIT-1Finite Element Analysis - UNIT-1
Finite Element Analysis - UNIT-1
propaul
 
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANKME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - I NOTES AND QUESTION BANK
ASHOK KUMAR RAJENDRAN
 
Finite Element Analysis - UNIT-4
Finite Element Analysis - UNIT-4Finite Element Analysis - UNIT-4
Finite Element Analysis - UNIT-4
propaul
 
Fem lecture
Fem lectureFem lecture
Fem lecture
Muhammad Mohsin Waseem
 
Finite Element Analysis - UNIT-3
Finite Element Analysis - UNIT-3Finite Element Analysis - UNIT-3
Finite Element Analysis - UNIT-3
propaul
 
Finite elements for 2‐d problems
Finite elements for 2‐d problemsFinite elements for 2‐d problems
Finite elements for 2‐d problems
Tarun Gehlot
 
ME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANKME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANK
ASHOK KUMAR RAJENDRAN
 
ME6603 - FINITE ELEMENT ANALYSIS FORMULA BOOK
ME6603 - FINITE ELEMENT ANALYSIS FORMULA BOOKME6603 - FINITE ELEMENT ANALYSIS FORMULA BOOK
ME6603 - FINITE ELEMENT ANALYSIS FORMULA BOOK
ASHOK KUMAR RAJENDRAN
 
Unit 1 notes-final
Unit 1 notes-finalUnit 1 notes-final
Unit 1 notes-final
jagadish108
 
Point Collocation Method used in the solving of Differential Equations, parti...
Point Collocation Method used in the solving of Differential Equations, parti...Point Collocation Method used in the solving of Differential Equations, parti...
Point Collocation Method used in the solving of Differential Equations, parti...
Suddhasheel GHOSH, PhD
 
Finite element method
Finite element methodFinite element method
Finite element method
Mevada Maulik
 
Basics of finite element method 19.04.2018
Basics of finite element method 19.04.2018Basics of finite element method 19.04.2018
Basics of finite element method 19.04.2018
Dr. Mohd Zameeruddin
 
Finite element - axisymmetric stress and strain
Finite element - axisymmetric stress and strainFinite element - axisymmetric stress and strain
Finite element - axisymmetric stress and strain
أحمد شاكر
 
ME6603 - FINITE ELEMENT ANALYSIS UNIT - V NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - V NOTES AND QUESTION BANKME6603 - FINITE ELEMENT ANALYSIS UNIT - V NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - V NOTES AND QUESTION BANK
ASHOK KUMAR RAJENDRAN
 
Axisymmetric
Axisymmetric Axisymmetric
Axisymmetric
Raj Kumar
 
Fea unit 1
Fea unit 1Fea unit 1
Fea unit 1
s Kumaravel
 
Isoparametric Elements 4.pdf
Isoparametric Elements 4.pdfIsoparametric Elements 4.pdf
Isoparametric Elements 4.pdf
mannimalik
 
Isoparametric mapping
Isoparametric mappingIsoparametric mapping
Isoparametric mapping
Linh Tran
 
Lect14
Lect14Lect14
Lect14
DrASSayyad
 
Finite Element Method
Finite Element MethodFinite Element Method
Finite Element Method
Muhammad Haris
 
Finite element method
Finite element methodFinite element method
Finite element method
Santosh Chavan
 
Rayleigh Ritz Method
Rayleigh Ritz MethodRayleigh Ritz Method
Rayleigh Ritz Method
Sakthivel Murugan
 
Finite Element Method
Finite Element MethodFinite Element Method
Finite Element Method
Tahir Khan
 
L20
L20L20
L20
Sudhir Reddy
 
ITERATIVE METHODS FOR THE SOLUTION OF SADDLE POINT PROBLEM
ITERATIVE METHODS FOR THE SOLUTION OF SADDLE POINT PROBLEMITERATIVE METHODS FOR THE SOLUTION OF SADDLE POINT PROBLEM
ITERATIVE METHODS FOR THE SOLUTION OF SADDLE POINT PROBLEM
IAEME Publication
 
Iterative methods for the solution of saddle point problem
Iterative methods for the solution of saddle point problemIterative methods for the solution of saddle point problem
Iterative methods for the solution of saddle point problem
IAEME Publication
 

Mais conteúdo relacionado

Mais procurados

Isoparametric Elements 4.pdf
Isoparametric Elements 4.pdfIsoparametric Elements 4.pdf
Isoparametric Elements 4.pdf
mannimalik
 
Finite Element Method
Finite Element MethodFinite Element Method
Finite Element Method
Tahir Khan
 

Mais procurados (20)

Finite Element Analysis - UNIT-3
Finite Element Analysis - UNIT-3Finite Element Analysis - UNIT-3
Finite Element Analysis - UNIT-3
 
Finite elements for 2‐d problems
Finite elements for 2‐d problemsFinite elements for 2‐d problems
Finite elements for 2‐d problems
 
ME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANKME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANK
 
ME6603 - FINITE ELEMENT ANALYSIS FORMULA BOOK
ME6603 - FINITE ELEMENT ANALYSIS FORMULA BOOKME6603 - FINITE ELEMENT ANALYSIS FORMULA BOOK
ME6603 - FINITE ELEMENT ANALYSIS FORMULA BOOK
 
Unit 1 notes-final
Unit 1 notes-finalUnit 1 notes-final
Unit 1 notes-final
 
Point Collocation Method used in the solving of Differential Equations, parti...
Point Collocation Method used in the solving of Differential Equations, parti...Point Collocation Method used in the solving of Differential Equations, parti...
Point Collocation Method used in the solving of Differential Equations, parti...
 
Finite element method
Finite element methodFinite element method
Finite element method
 
Basics of finite element method 19.04.2018
Basics of finite element method 19.04.2018Basics of finite element method 19.04.2018
Basics of finite element method 19.04.2018
 
Finite element - axisymmetric stress and strain
Finite element - axisymmetric stress and strainFinite element - axisymmetric stress and strain
Finite element - axisymmetric stress and strain
 
ME6603 - FINITE ELEMENT ANALYSIS UNIT - V NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - V NOTES AND QUESTION BANKME6603 - FINITE ELEMENT ANALYSIS UNIT - V NOTES AND QUESTION BANK
ME6603 - FINITE ELEMENT ANALYSIS UNIT - V NOTES AND QUESTION BANK
 
Axisymmetric
Axisymmetric Axisymmetric
Axisymmetric
 
Fea unit 1
Fea unit 1Fea unit 1
Fea unit 1
 
Isoparametric Elements 4.pdf
Isoparametric Elements 4.pdfIsoparametric Elements 4.pdf
Isoparametric Elements 4.pdf
 
Isoparametric mapping
Isoparametric mappingIsoparametric mapping
Isoparametric mapping
 
Lect14
Lect14Lect14
Lect14
 
Finite Element Method
Finite Element MethodFinite Element Method
Finite Element Method
 
Finite element method
Finite element methodFinite element method
Finite element method
 
Rayleigh Ritz Method
Rayleigh Ritz MethodRayleigh Ritz Method
Rayleigh Ritz Method
 
Finite Element Method
Finite Element MethodFinite Element Method
Finite Element Method
 
L20
L20L20
L20
 

Semelhante a Isoparametric bilinear quadrilateral element _ ppt presentation

Duel of cosmological screening lengths
Duel of cosmological screening lengthsDuel of cosmological screening lengths
Duel of cosmological screening lengths
Maxim Eingorn
 
SIAM-AG21-Topological Persistence Machine of Phase Transition
SIAM-AG21-Topological Persistence Machine of Phase TransitionSIAM-AG21-Topological Persistence Machine of Phase Transition
SIAM-AG21-Topological Persistence Machine of Phase Transition
Ha Phuong
 

Semelhante a Isoparametric bilinear quadrilateral element _ ppt presentation (20)

ITERATIVE METHODS FOR THE SOLUTION OF SADDLE POINT PROBLEM
ITERATIVE METHODS FOR THE SOLUTION OF SADDLE POINT PROBLEMITERATIVE METHODS FOR THE SOLUTION OF SADDLE POINT PROBLEM
ITERATIVE METHODS FOR THE SOLUTION OF SADDLE POINT PROBLEM
 
Iterative methods for the solution of saddle point problem
Iterative methods for the solution of saddle point problemIterative methods for the solution of saddle point problem
Iterative methods for the solution of saddle point problem
 
تطبيقات المعادلات التفاضلية
تطبيقات المعادلات التفاضليةتطبيقات المعادلات التفاضلية
تطبيقات المعادلات التفاضلية
 
Photophysics of dendrimers
Photophysics of dendrimersPhotophysics of dendrimers
Photophysics of dendrimers
 
CHiMaD_pfworkshipXVI_Staublin_lightningtalk.pptx
CHiMaD_pfworkshipXVI_Staublin_lightningtalk.pptxCHiMaD_pfworkshipXVI_Staublin_lightningtalk.pptx
CHiMaD_pfworkshipXVI_Staublin_lightningtalk.pptx
 
Presentation
PresentationPresentation
Presentation
 
Summary of MAST
Summary of MASTSummary of MAST
Summary of MAST
 
document(1).pdf
document(1).pdfdocument(1).pdf
document(1).pdf
 
Lecture 5 backpropagation
Lecture 5 backpropagationLecture 5 backpropagation
Lecture 5 backpropagation
 
Direct solution of sparse network equations by optimally ordered triangular f...
Direct solution of sparse network equations by optimally ordered triangular f...Direct solution of sparse network equations by optimally ordered triangular f...
Direct solution of sparse network equations by optimally ordered triangular f...
 
Exponential decay for the solution of the nonlinear equation induced by the m...
Exponential decay for the solution of the nonlinear equation induced by the m...Exponential decay for the solution of the nonlinear equation induced by the m...
Exponential decay for the solution of the nonlinear equation induced by the m...
 
Basic calculus (ii) recap
Basic calculus (ii) recapBasic calculus (ii) recap
Basic calculus (ii) recap
 
lec27.ppt
lec27.pptlec27.ppt
lec27.ppt
 
Periodic Solutions for Nonlinear Systems of Integro-Differential Equations of...
Periodic Solutions for Nonlinear Systems of Integro-Differential Equations of...Periodic Solutions for Nonlinear Systems of Integro-Differential Equations of...
Periodic Solutions for Nonlinear Systems of Integro-Differential Equations of...
 
lec34.ppt
lec34.pptlec34.ppt
lec34.ppt
 
Duel of cosmological screening lengths
Duel of cosmological screening lengthsDuel of cosmological screening lengths
Duel of cosmological screening lengths
 
Biosight: Quantitative Methods for Policy Analysis: Stochastic Dynamic Progra...
Biosight: Quantitative Methods for Policy Analysis: Stochastic Dynamic Progra...Biosight: Quantitative Methods for Policy Analysis: Stochastic Dynamic Progra...
Biosight: Quantitative Methods for Policy Analysis: Stochastic Dynamic Progra...
 
SIAM-AG21-Topological Persistence Machine of Phase Transition
SIAM-AG21-Topological Persistence Machine of Phase TransitionSIAM-AG21-Topological Persistence Machine of Phase Transition
SIAM-AG21-Topological Persistence Machine of Phase Transition
 
Vectors in mechanics
Vectors in mechanicsVectors in mechanics
Vectors in mechanics
 
E0561719
E0561719E0561719
E0561719
 

Mais de Filipe Giesteira

Fatigue and fracture mechanics _ fadiga e mecânica da fractura
Fatigue and fracture mechanics _ fadiga e mecânica da fracturaFatigue and fracture mechanics _ fadiga e mecânica da fractura
Fatigue and fracture mechanics _ fadiga e mecânica da fractura
Filipe Giesteira
 
Prediction of joint strength and effect of the surface treatment on the singl...
Prediction of joint strength and effect of the surface treatment on the singl...Prediction of joint strength and effect of the surface treatment on the singl...
Prediction of joint strength and effect of the surface treatment on the singl...
Filipe Giesteira
 

Mais de Filipe Giesteira (20)

My Resume
My ResumeMy Resume
My Resume
 
Fatigue and fracture mechanics _ fadiga e mecânica da fractura
Fatigue and fracture mechanics _ fadiga e mecânica da fracturaFatigue and fracture mechanics _ fadiga e mecânica da fractura
Fatigue and fracture mechanics _ fadiga e mecânica da fractura
 
Composite systems - Trace approach
Composite systems - Trace approachComposite systems - Trace approach
Composite systems - Trace approach
 
Composite Systems - Trace Approach _ PPT Presentation
Composite Systems - Trace Approach _ PPT PresentationComposite Systems - Trace Approach _ PPT Presentation
Composite Systems - Trace Approach _ PPT Presentation
 
Isoparametric bilinear quadrilateral element
Isoparametric bilinear quadrilateral elementIsoparametric bilinear quadrilateral element
Isoparametric bilinear quadrilateral element
 
Mechanical speed reducer Design Report _ Motoredutor de Engrenagens cilíndric...
Mechanical speed reducer Design Report _ Motoredutor de Engrenagens cilíndric...Mechanical speed reducer Design Report _ Motoredutor de Engrenagens cilíndric...
Mechanical speed reducer Design Report _ Motoredutor de Engrenagens cilíndric...
 
Involute gear interference _ Interferência em rodas dentadas cilíndricas
Involute gear interference _ Interferência em rodas dentadas cilíndricasInvolute gear interference _ Interferência em rodas dentadas cilíndricas
Involute gear interference _ Interferência em rodas dentadas cilíndricas
 
Relation between contact ratio and interference _ Relação entre razão de Cond...
Relation between contact ratio and interference _ Relação entre razão de Cond...Relation between contact ratio and interference _ Relação entre razão de Cond...
Relation between contact ratio and interference _ Relação entre razão de Cond...
 
Tribology Design - Equation Summary _ Tribologia
Tribology Design - Equation Summary _ TribologiaTribology Design - Equation Summary _ Tribologia
Tribology Design - Equation Summary _ Tribologia
 
Fatigue design - Equation Summary _ Dimensionamento à Fadiga
Fatigue design - Equation Summary _ Dimensionamento à FadigaFatigue design - Equation Summary _ Dimensionamento à Fadiga
Fatigue design - Equation Summary _ Dimensionamento à Fadiga
 
Screw design - Equation summary _ Dimensionamento de Parafusos
Screw design - Equation summary _ Dimensionamento de ParafusosScrew design - Equation summary _ Dimensionamento de Parafusos
Screw design - Equation summary _ Dimensionamento de Parafusos
 
Mechanical vibration - equation summary
Mechanical vibration - equation summaryMechanical vibration - equation summary
Mechanical vibration - equation summary
 
Mechanical vibration mec iii
Mechanical vibration mec iiiMechanical vibration mec iii
Mechanical vibration mec iii
 
Photo-realistic Rendered Images of 3D CAD Model
Photo-realistic Rendered Images of 3D CAD ModelPhoto-realistic Rendered Images of 3D CAD Model
Photo-realistic Rendered Images of 3D CAD Model
 
Shaft - specification drawing
Shaft - specification drawingShaft - specification drawing
Shaft - specification drawing
 
Mechanical speed reducer - 3D Exploded View (SolidWorks)
Mechanical speed reducer - 3D Exploded View (SolidWorks)Mechanical speed reducer - 3D Exploded View (SolidWorks)
Mechanical speed reducer - 3D Exploded View (SolidWorks)
 
Mechanical speed reducer assembly drawing
Mechanical speed reducer assembly drawingMechanical speed reducer assembly drawing
Mechanical speed reducer assembly drawing
 
Prediction of joint strength and effect of the surface treatment on the singl...
Prediction of joint strength and effect of the surface treatment on the singl...Prediction of joint strength and effect of the surface treatment on the singl...
Prediction of joint strength and effect of the surface treatment on the singl...
 
Processos de Fundição do Titânio e suas Ligas
Processos de Fundição do Titânio e suas LigasProcessos de Fundição do Titânio e suas Ligas
Processos de Fundição do Titânio e suas Ligas
 
Incremental sheet Forming (ISF)
Incremental sheet Forming (ISF)Incremental sheet Forming (ISF)
Incremental sheet Forming (ISF)
 

Último

ALCOHOL PRODUCTION- Beer Brewing Process.pdf
ALCOHOL PRODUCTION- Beer Brewing Process.pdfALCOHOL PRODUCTION- Beer Brewing Process.pdf
ALCOHOL PRODUCTION- Beer Brewing Process.pdf
Madan Karki
 
Filters for Electromagnetic Compatibility Applications
Filters for Electromagnetic Compatibility ApplicationsFilters for Electromagnetic Compatibility Applications
Filters for Electromagnetic Compatibility Applications
Mathias Magdowski
 
Electrical shop management system project report.pdf
Electrical shop management system project report.pdfElectrical shop management system project report.pdf
Electrical shop management system project report.pdf
Kamal Acharya
 
Online crime reporting system project.pdf
Online crime reporting system project.pdfOnline crime reporting system project.pdf
Online crime reporting system project.pdf
Kamal Acharya
 
Final DBMS Manual (2).pdf final lab manual
Final DBMS Manual (2).pdf final lab manualFinal DBMS Manual (2).pdf final lab manual
Final DBMS Manual (2).pdf final lab manual
BalamuruganV28
 

Último (20)

ALCOHOL PRODUCTION- Beer Brewing Process.pdf
ALCOHOL PRODUCTION- Beer Brewing Process.pdfALCOHOL PRODUCTION- Beer Brewing Process.pdf
ALCOHOL PRODUCTION- Beer Brewing Process.pdf
 
Fabrication Of Automatic Star Delta Starter Using Relay And GSM Module By Utk...
Fabrication Of Automatic Star Delta Starter Using Relay And GSM Module By Utk...Fabrication Of Automatic Star Delta Starter Using Relay And GSM Module By Utk...
Fabrication Of Automatic Star Delta Starter Using Relay And GSM Module By Utk...
 
5G and 6G refer to generations of mobile network technology, each representin...
5G and 6G refer to generations of mobile network technology, each representin...5G and 6G refer to generations of mobile network technology, each representin...
5G and 6G refer to generations of mobile network technology, each representin...
 
Filters for Electromagnetic Compatibility Applications
Filters for Electromagnetic Compatibility ApplicationsFilters for Electromagnetic Compatibility Applications
Filters for Electromagnetic Compatibility Applications
 
Electrical shop management system project report.pdf
Electrical shop management system project report.pdfElectrical shop management system project report.pdf
Electrical shop management system project report.pdf
 
Quiz application system project report..pdf
Quiz application system project report..pdfQuiz application system project report..pdf
Quiz application system project report..pdf
 
SLIDESHARE PPT-DECISION MAKING METHODS.pptx
SLIDESHARE PPT-DECISION MAKING METHODS.pptxSLIDESHARE PPT-DECISION MAKING METHODS.pptx
SLIDESHARE PPT-DECISION MAKING METHODS.pptx
 
Introduction to Heat Exchangers: Principle, Types and Applications
Introduction to Heat Exchangers: Principle, Types and ApplicationsIntroduction to Heat Exchangers: Principle, Types and Applications
Introduction to Heat Exchangers: Principle, Types and Applications
 
analog-vs-digital-communication (concept of analog and digital).pptx
analog-vs-digital-communication (concept of analog and digital).pptxanalog-vs-digital-communication (concept of analog and digital).pptx
analog-vs-digital-communication (concept of analog and digital).pptx
 
"United Nations Park" Site Visit Report.
"United Nations Park" Site Visit Report."United Nations Park" Site Visit Report.
"United Nations Park" Site Visit Report.
 
Research Methodolgy & Intellectual Property Rights Series 2
Research Methodolgy & Intellectual Property Rights Series 2Research Methodolgy & Intellectual Property Rights Series 2
Research Methodolgy & Intellectual Property Rights Series 2
 
Introduction to Artificial Intelligence and History of AI
Introduction to Artificial Intelligence and History of AIIntroduction to Artificial Intelligence and History of AI
Introduction to Artificial Intelligence and History of AI
 
Research Methodolgy & Intellectual Property Rights Series 1
Research Methodolgy & Intellectual Property Rights Series 1Research Methodolgy & Intellectual Property Rights Series 1
Research Methodolgy & Intellectual Property Rights Series 1
 
Linux Systems Programming: Semaphores, Shared Memory, and Message Queues
Linux Systems Programming: Semaphores, Shared Memory, and Message QueuesLinux Systems Programming: Semaphores, Shared Memory, and Message Queues
Linux Systems Programming: Semaphores, Shared Memory, and Message Queues
 
Instruct Nirmaana 24-Smart and Lean Construction Through Technology.pdf
Instruct Nirmaana 24-Smart and Lean Construction Through Technology.pdfInstruct Nirmaana 24-Smart and Lean Construction Through Technology.pdf
Instruct Nirmaana 24-Smart and Lean Construction Through Technology.pdf
 
Online crime reporting system project.pdf
Online crime reporting system project.pdfOnline crime reporting system project.pdf
Online crime reporting system project.pdf
 
Final DBMS Manual (2).pdf final lab manual
Final DBMS Manual (2).pdf final lab manualFinal DBMS Manual (2).pdf final lab manual
Final DBMS Manual (2).pdf final lab manual
 
Involute of a circle,Square, pentagon,HexagonInvolute_Engineering Drawing.pdf
Involute of a circle,Square, pentagon,HexagonInvolute_Engineering Drawing.pdfInvolute of a circle,Square, pentagon,HexagonInvolute_Engineering Drawing.pdf
Involute of a circle,Square, pentagon,HexagonInvolute_Engineering Drawing.pdf
 
Interfacing Analog to Digital Data Converters ee3404.pdf
Interfacing Analog to Digital Data Converters ee3404.pdfInterfacing Analog to Digital Data Converters ee3404.pdf
Interfacing Analog to Digital Data Converters ee3404.pdf
 
Operating System chapter 9 (Virtual Memory)
Operating System chapter 9 (Virtual Memory)Operating System chapter 9 (Virtual Memory)
Operating System chapter 9 (Virtual Memory)
 

Isoparametric bilinear quadrilateral element _ ppt presentation

  • 1. 2D Plane Elasticity – Isoparametric Bilinear Quadrilateral Lagrange type Element Filipe Giesteira Master student Faculty of Engineering of University of Porto (FEUP) Mechanical Engineering Department (DEMec) Finite Element Method (MEF, Método dos Elementos Finitos) Supervisor: Prof. Francisco Andrade Pires Supervisor: Prof. José Dias Rodrigues Finite Element Method (FEM) - Final Presentation Faculty of Engineering of University of Porto Building L / Room L317 Porto, 24th January 2019 Work Assignment No. 2.3
  • 2. 2 Contents ➢ Introduction – Plane Elasticity Background Theory ➢ Weak & Strong Form – FEM Mathematical Formulation ➢ Finite Element Library – Isoparametric Bilinear Quadrilateral Element ➢ FEM Equations – Element Mechanical Properties ➢ BaPMEF FEM Tool – Script Capabilities and Limitations ➢ Validation Examples – Solutions of the Biharmonic Equation / Airy Functions ➢ Practical Applications 24-01-2019
  • 3. 3 1. Introduction 1.1. Plane Elasticity Background Theory – Assumptions and Simplifications 24-01-2019 y x l t w z 𝑢 Ԧ𝑣 𝑤 σzz x, y, z = ± Τt 2 = τyz x, y, z = ± Τt 2 = τxz(x, y, z = ± Τt 2) = 0 t ≪ l ∧ t ≪ w ⇒ σzz x, y, z = τyz x, y, z = τxz(x, y, z) = 0 t ≪ l ∧ t ≪ w ⇒ ൞ )σxx = fxx(x, y σyy = fyy x, y ൯τxy = fxy(x, y y xl tw z Ԧ𝑣 𝑢 𝑤 t ≫ l ∧ t ≫ w ⇒ σzz x, y, z = C te ቊ 𝑤 𝑥, 𝑦, ± Τ𝑡 2 = 0 𝑤 𝑥, 𝑦, 0 = 0, 𝑏𝑦 𝑠𝑦𝑚. ⇒ 𝑤 𝑥, 𝑦, 𝑧 ≈ 0 𝑤 𝑥, 𝑦, 𝑧 = 0 ⇒ 𝜀 𝑧𝑧(𝑥, 𝑦, 𝑧) = 0 𝛾𝑦𝑧(𝑥, 𝑦, 𝑧) = 0 𝛾𝑥𝑧(𝑥, 𝑦𝑧, 𝑧) = 0 )𝜀 𝑥𝑥 = 𝑓𝑥𝑥(𝑥, 𝑦 ൯𝜀 𝑦𝑦 = 𝑓𝑦𝑦(𝑥, 𝑦 ൯𝛾𝑥𝑦 = 𝑓𝑥𝑦(𝑥, 𝑦 ൞ 𝑓𝑥 = 𝑓𝑥 𝑥, 𝑦 𝑓𝑦 = 𝑓𝑦 𝑥, 𝑦 𝑓𝑧 = 0 ➢ Plane Stress ➢ Plane Strain
  • 4. 4 ➢ Displacement Field ➢ Strain Field ➢ Stress Field 1. Introduction 1.2. Plane Elasticity Background Theory – Solid Mechanics Continuum 24-01-2019 𝑢 = 𝑢 𝑥, 𝑦 = )𝑢(𝑥, 𝑦 )𝑣(𝑥, 𝑦 𝜀 = 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾𝑥𝑦 = 𝜕𝑢 𝜕𝑥 𝜕𝑣 𝜕𝑦 𝜕𝑢 𝜕𝑦 + 𝜕𝑣 𝜕𝑥 𝜀 = L 𝑢 = 𝜕 𝜕𝑥 0 0 𝜕 𝜕𝑦 𝜕 𝜕𝑦 𝜕 𝜕𝑥 )𝑢(𝑥, 𝑦 𝑣 𝑥, 𝑦 𝜎 = 𝐶 𝜀 𝐶 = 𝐸 1 − 𝑣2 𝐸𝑣 1 − 𝑣2 0 𝐸𝑣 1 − 𝑣2 𝐸 1 − 𝑣2 0 0 0 𝐸 2 1 + 𝑣 ; 𝐸 1 − 𝑣 )1 + 𝑣 (1 − 2𝑣 𝐸𝑣 )1 + 𝑣 (1 − 2𝑣 0 𝐸𝑣 )1 + 𝑣 (1 − 2𝑣 𝐸 1 − 𝑣 )1 + 𝑣 (1 − 2𝑣 0 0 0 𝐸 2 1 + 𝑣 Plane Stress Plane Strain Matrix form useful to discretize the Weak-form in Finite Domains! Isotropic Transversely Isotropic Orthotropic Monolithic … …
  • 5. 5 2. Weak & Strong Form 24-01-2019 2.1. Development of the Weak Form ➢ Principle of the Minimum Total Potential Energy ➢ Principle of Virtual Work/Virtual Displacements ➢ Hamilton’s Principle ➢ Galerkin Method of Weighted-Residuals 𝛿𝛱 = 𝜕 𝜕|𝑢| 𝛱 𝑢 𝛿𝑢 = 𝛿𝑉 − 𝛿𝑊 = 0 𝛿𝑊 𝐼𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝐹𝑜𝑟𝑐𝑒𝑠 + 𝛿𝑊 𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝐹𝑜𝑟𝑐𝑒𝑠 + 𝛿𝑊 𝐸𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝐹𝑜𝑟𝑐𝑒𝑠 = 0 න 𝑡1 𝑡2 𝛿 𝑇 − 𝑉 𝑑𝑡 + න 𝑡1 𝑡2 𝛿𝑊𝑛𝑐 𝑑𝑡 = 0 න Ω 𝑤𝑥 − 𝜕 𝜕𝑥 𝜎 𝑥𝑥 − 𝜕 𝜕𝑦 𝜏 𝑥𝑦 − 𝑓𝑥 + 𝜌 𝜕2 𝜕𝑡2 𝑢 𝑥, 𝑦 𝑑𝑥 𝑑𝑦 = 0 න Ω 𝑤 𝑦 − 𝜕 𝜕𝑦 𝜎 𝑦𝑦 − 𝜕 𝜕𝑥 𝜏 𝑥𝑦 − 𝑓𝑦 + 𝜌 𝜕2 𝜕𝑡2 𝑣 𝑥, 𝑦 𝑑𝑥 𝑑𝑦 = 0 Variational Principles Integral Statement
  • 6. 6 ➢ Weak Form – ready to be Discretized in Finite Element Domains ➢ Strong Form – continuously valid in the entire Domain 2. Weak & Strong Form 24-01-2019 2.2. Strong vs. Weak Form 𝑡 ඵ 𝐴 𝐿 𝛿𝑢 𝑇 𝐶 𝐿 𝑢 𝑑𝐴 + ඵ 𝐴 𝛿𝑢 𝑇 𝜌 ሷ𝑢 𝑑𝐴 = 𝑡 ඵ 𝐴 𝛿𝑢 𝑇 𝑏 𝑑𝐴 + 𝑡 ර 𝑙 𝛿𝑢 𝑇 𝑞 𝑑𝑠 + ෍ 𝑝𝑖 𝛿𝑢 𝑝𝑖 𝑇 𝑃𝑝𝑖 𝜕 𝜕𝑥 𝜎𝑥𝑥 + 𝜕 𝜕𝑦 𝜏 𝑥𝑦 + 𝑏 𝑥 = 𝜌 𝜕2 𝜕𝑡2 𝑢 𝑥, 𝑦 𝜕 𝜕𝑦 𝜎 𝑦𝑦 + 𝜕 𝜕𝑥 𝜏 𝑥𝑦 + 𝑏 𝑦 = 𝜌 𝜕2 𝜕𝑡2 𝑣 𝑥, 𝑦 ; 𝜎𝑥𝑥 = 𝐶11 𝜕𝑢 𝜕𝑥 + 𝐶12 𝜕𝑣 𝜕𝑦 𝜎 𝑦𝑦 = 𝐶12 𝜕𝑢 𝜕𝑥 + 𝐶11 𝜕𝑣 𝜕𝑦 ; ൝ 𝜏 𝑥𝑦 = 𝐶11 − 𝐶12 2 𝜕𝑢 𝜕𝑦 + 𝜕𝑣 𝜕𝑥 𝑢 = ො𝑢 , ∀ 𝛤 = 𝛤𝑢 ; 𝑡 = 𝜎 𝑇 ො𝑛 = Ƹ𝑡 , ∀ 𝛤 = 𝛤𝑡 Natural Boundary ConditionsEssential Boundary Conditions 𝛿𝑢 = 𝛿𝑢 𝛿𝑣 𝛿𝑢 𝑝𝑖 = 𝛿𝑢 𝑝𝑖 𝛿𝑣 𝑝𝑖 𝑃𝑝𝑖 = 𝑃𝑝𝑖 𝑥 𝑃𝑝𝑖 𝑦 𝑏 = 𝑏 𝑥 𝑏 𝑦 𝑞 = 𝑞 𝑥 𝑞 𝑦 ሷ𝑢 = ሷ𝑢 ሷ𝑣
  • 7. 7 3. Finite Element Library 24-01-2019 3.1. Isoparametric Bilinear Quadrilateral Element 𝑁𝑖 = 𝑁1 𝑁2 𝑁3 𝑁4 = 1 4 1 − 𝜉 1 − 𝜂 1 4 1 + 𝜉 1 − 𝜂 1 4 1 + 𝜉 1 + 𝜂 1 4 1 − 𝜉 1 + 𝜂 Node 𝝃 𝜼 1 -1 -1 2 1 -1 3 1 1 4 -1 1 𝜂 𝜉 1 4 3 2 Lagrange type Element - 4 nodes Shape / Interpolation Functions
  • 8. 8 3. Finite Element Library 24-01-2019 3.2. Discretization of the Continuum Fields 𝑢 𝑒 ≈ 𝑁𝑖 𝜉, 𝜂 𝑢𝑖 𝑒 = 𝑁1 0 𝑁2 0 𝑁3 0 𝑁4 0 0 𝑁1 0 𝑁2 0 𝑁3 0 𝑁4 𝑢1 𝑣1 𝑢2 𝑣2 𝑢3 𝑣3 𝑢4 𝑣4 𝑢 𝑒 = )𝑢(𝜉, 𝜂 )𝑣(𝜉, 𝜂 ≈ 𝑎 + 𝑏 𝜉 + 𝑐 𝜂 + 𝑑 𝜉𝜂 𝑎 + 𝑏 𝜉 + 𝑐 𝜂 + 𝑑 𝜉𝜂 = 𝑁𝑖 𝑇 𝑢𝑖 𝑁𝑖 𝑇 𝑣𝑖 ε 𝑒 = 𝐿 𝑁 𝑢𝑖 𝑒 = 𝐵 ui e = 𝜕 𝜕𝑥 𝑁1 0 𝜕 𝜕𝑥 𝑁2 0 𝜕 𝜕𝑥 𝑁3 0 𝜕 𝜕𝑥 𝑁4 0 0 𝜕 𝜕𝑦 𝑁1 0 𝜕 𝜕𝑦 𝑁2 0 𝜕 𝜕𝑦 𝑁3 0 𝜕 𝜕𝑦 𝑁4 𝜕 𝜕𝑦 𝑁1 𝜕 𝜕𝑥 𝑁1 𝜕 𝜕𝑦 𝑁2 𝜕 𝜕𝑥 𝑁2 𝜕 𝜕𝑦 𝑁3 𝜕 𝜕𝑥 𝑁3 𝜕 𝜕𝑦 𝑁4 𝜕 𝜕𝑥 𝑁4 𝑢1 𝑣1 𝑢2 𝑣2 𝑢3 𝑣3 𝑢4 𝑣4 ➢ Displacement Field ➢ Strain Field Direct Computation Require Isoparametric Mapping
  • 9. 9 3. Finite Element Library 24-01-2019 3.3. Isoparametric Mapping 𝐽 = 𝐽11 𝐽12 𝐽21 𝐽22 = 𝜕𝑥 𝜕𝜉 𝜕𝑦 𝜕𝜉 𝜕𝑥 𝜕𝜂 𝜕𝑦 𝜕𝜂 = 𝜕𝑁 𝑖 𝜕𝜉 𝑇 𝜕𝑁 𝑖 𝜕𝜂 𝑇 𝑥𝑖 𝑦𝑖 𝑥𝑖 𝑦𝑖 = 𝑥1 𝑦1 𝑥2 𝑦2 𝑥3 𝑦3 𝑥4 𝑦4 𝜕𝑁𝑖 𝜕𝜉 𝑇 = 𝜕𝑁1 𝜕𝜉 𝜕𝑁2 𝜕𝜉 𝜕𝑁3 𝜕𝜉 𝜕𝑁4 𝜕𝜉 = 1 4 𝜂 − 1 1 4 1 − 𝜂 1 4 1 + 𝜂 − 1 4 1 + 𝜂 𝜕𝑁𝑖 𝜕𝜂 𝑇 = 𝜕𝑁1 𝜕𝜂 𝜕𝑁2 𝜕𝜂 𝜕𝑁3 𝜕𝜂 𝜕𝑁4 𝜕𝜂 = 1 4 𝜉 − 1 − 1 4 1 + 𝜉 1 4 1 + 𝜉 1 4 1 − 𝜉 𝜕𝑁𝑖 𝜕𝑥 𝑇 𝜕𝑁𝑖 𝜕𝑦 𝑇 = 𝐽 −1 𝜕𝑁𝑖 𝜕𝜉 𝑇 𝜕𝑁𝑖 𝜕𝜂 𝑇 ➢ Derivatives in respect to the Natural Coordinates ➢ Jacobian Matrix ➢ Derivatives in respect to the Local Coordinates det 𝐽 = 𝐽11 𝐽22 − 𝐽12 𝐽21 > 0
  • 10. 10 3. Finite Element Library 3.3. Isoparametric Mapping ➢ Differential Linear Operators ➢ Differential Area Operator and Determinant of the Jacobian Matrix (Jacobian) ➢ Transformation Matrix – Rotation dA = dx dy =? = det J dξ dη 𝑇 =? = 𝑇 𝜂=𝐶 𝑡𝑒 = cos 𝛼 − sin 𝛼 sin 𝛼 cos 𝛼 = 𝜕𝑥 𝜕𝜉 − 𝜕𝑦 𝜕𝜉 𝜕𝑦 𝜕𝜉 𝜕𝑥 𝜕𝜉 = 𝐽11 −𝐽12 𝐽12 𝐽11 (𝜉,±1) , 𝜂 = ±1 𝑇 𝜉=𝐶 𝑡𝑒 = cos 𝛽 − sin 𝛽 sin 𝛽 cos 𝛽 = 𝜕𝑦 𝜕𝜂 − 𝜕𝑥 𝜕𝜂 𝜕𝑥 𝜕𝜂 𝜕𝑦 𝜕𝜂 = 𝐽22 −𝐽21 𝐽21 𝐽22 (±1,𝜂) , 𝜉 = ±1 dx dy =? = J T dξ dη
  • 11. 11 4. FEM Equations 24-01-2019 4.1. Finite Element Equations – Element Level ➢ Static Equilibrium Equation ➢ Equivalent Load Vector ➢ Stiffness Matrix 𝐾 𝑒 = 𝑡 ඵ 𝐴 𝐵 𝑇 𝐶 𝐵 𝒅𝑨 = 𝑡 ෍ 𝑗 2 ෍ 𝑘 2 𝐵 𝜉𝑗, 𝜂 𝑘 𝑇 𝐶 𝐵 𝜉𝑗, 𝜂 𝑘 det 𝐽(𝜉𝑗, 𝜂 𝑘) 𝑊𝑗 𝑊𝑘 (𝝃𝒋 , 𝜼 𝒌) 𝑾𝒋 𝑾 𝒌 − Τ1 3 , − Τ1 3 1 1 Τ1 3, − Τ1 3 1 1 Τ1 3 , Τ1 3 1 1 − Τ1 3, Τ1 3 1 1 𝐹 𝑒 = 𝑓𝑏 𝑒 + 𝑓𝑞 𝑒 + 𝑓𝑃 𝑒 ෍ 𝑒 𝐸 𝐾 𝑒 𝑢𝑖 𝑒 − 𝐹 𝑒 = 0 Body Forces Vector Surface Tractions Vector Nodal Point Forces Vector Global Equilibrium Equation System of Algebraic Equations 2 𝜂 𝜉 1 4 3 GP2 GP4 GP3 GP1 Illustration of the Gauss Points Table with the Gauss Points and respective Weight for Numerical Integration using Gauss Quadrature
  • 12. 12 4. FEM Equations 24-01-2019 4.2. Equivalent Load Vectors – Part 1 ➢ Body Forces ➢ Nodal Point Loads 𝑓𝑏 𝑒 = 𝑡 ඵ 𝐴 𝑁 𝑇 𝑏 𝑒 𝑑𝐴 = 𝑡 ෍ 𝑗 2 ෍ 𝑘 2 𝑁 𝜉𝑗, 𝜂 𝑘 𝑇 𝑏 𝑥 𝑒 𝑏 𝑦 𝑒 det 𝐽(𝜉𝑗, 𝜂 𝑘) 𝑊𝑗 𝑊𝑘 𝑓𝑃 𝑒 = ෍ 𝑖 𝑃𝑖 = ෍ 𝑖 𝑃𝑥 𝑖 𝑃𝑦 𝑖 𝜂 𝜉 1 4 3 2 𝜂 𝜉 1 4 3 2 Body Forces acting on the quadrilateral element, along the y-direction Nodal Point Loads Body forces acting along the z-direction aren’t considered in plane elasticity (𝝃𝒋 , 𝜼 𝒌) 𝑾𝒋 𝑾 𝒌 − Τ1 3 , − Τ1 3 1 1 Τ1 3, − Τ1 3 1 1 Τ1 3 , Τ1 3 1 1 − Τ1 3, Τ1 3 1 1 Table with the Gauss Points and respective Weight for Numerical Integration using Gauss Quadrature
  • 13. 13 4. FEM Equations 24-01-2019 4.3. Equivalent Load Vectors – Part 2 ➢ Surface Tractions 𝑓𝑞 𝑒 = 𝑡 ර 𝑙 𝑒 𝑁 𝑇 𝒒∗ 𝒅𝒔 = න −1 1 𝑡 𝑁 𝜉, 𝜂 = ±1 𝑇 𝒒∗ 𝑐(𝜉) 𝑑𝜉 , 𝜂 = 𝐶 𝑡𝑒 න −1 1 𝑡 𝑁 𝜉 = ±1, 𝜂 𝑇 𝒒∗ 𝑐 𝜂 𝑑𝜂 , 𝜉 = 𝐶 𝑡𝑒 𝑓𝑞 𝑒 = ෍ 𝑗 2 𝑡 𝑁 𝜉𝑗, 𝜂 = ±1 𝑇 𝒒∗ 𝑐(𝜉𝑗) 𝑊𝑗 , 𝜂 = ±1 ෍ 𝑘 2 𝑡 𝑁 𝜉 = ±1, 𝜂 𝑘 𝑇 𝒒∗ 𝑐 𝜂 𝑘 𝑊𝑘 , 𝜉 = ±1 𝑐 = 𝑐 𝜉 = 𝐽11 𝜉, ±1 2 + 𝐽12 𝜉, ±1 2 , 𝜂 = ±1 𝑐 𝜂 = 𝐽21 ±1, 𝜂 2 + 𝐽22 ±1, 𝜂 2 , 𝜉 = ±1 Division of the Rectangular Boundary in four straight paths
  • 14. 14 4. FEM Equations 24-01-2019 4.3. Equivalent Load Vectors – Part 2 ➢ Surface Tractions Loads in Local Coordinates Loads in Natural Coordinates 𝑞∗ = 𝑞 𝑥 𝑒 𝑞 𝑦 𝑒 𝑞∗ = 𝑇 𝜂=𝐶 𝑡𝑒 𝜏 𝜎 = 𝐽11 −𝐽12 𝐽12 𝐽11 𝜉,±1 𝜏 𝜎 , 𝜂 = ±1 𝑇 𝜉=𝐶 𝑡𝑒 𝜎 𝜏 = 𝐽22 −𝐽21 𝐽21 𝐽22 ±1,𝜂 𝜎 𝜏 , 𝜉 = ±1
  • 15. 15 5. BapMEF FEM Tool 24-01-2019 5.1. Interface – Overview Warnings and Notes Input Boxes Menu Windows Helping graphic output during data input Data Input
  • 16. 16 5. BapMEF FEM Tool 24-01-2019 5.2. Data Input Geometric Variables Body Thickness Definition Material Properties Type of Plane Problem
  • 17. 17 5. BapMEF FEM Tool 24-01-2019 5.3. Special Features ➢ Meshing procedure
  • 18. 18 5. BapMEF FEM Tool 5.3. Special Features ➢ Definition of the Boundary Condition Simple Support Encastre SymmetryFracture Mechanics
  • 19. 19 ➢ Displacements ➢ S𝐭𝐫𝐚𝐢𝐧𝐬 ➢ Stresses ➢ Safety Factor ➢ Von Mises Equivalent Stress 5. BapMEF FEM Tool 24-01-2019 5.4. Output and Post-Processing Variables 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝛾𝑥𝑦 𝜺 𝒛𝒛 σ 𝑥𝑥 σ 𝑦𝑦 τ 𝑥𝑦 𝝈 𝒛𝒛 𝜎𝑒𝑞 = 1 2 𝜎 𝑥𝑥 − 𝜎 𝑦𝑦 2 𝜎 𝑦𝑦 − 𝜎𝑧𝑧 2 𝜎 𝑥𝑥 − 𝜎𝑧𝑧 2 + 6𝜏 𝑥𝑦 2 𝑆 = 𝜎 𝑦𝑒𝑙𝑑 𝜎 𝑒𝑞 𝑑 = 𝑢2 + 𝑣2 𝑢 𝑣 Plane Stress Plane Strain
  • 20. 20 6. Validation Examples 24-01-2019 6.1. Constant Rectangular cross-section Beam – Pure Bending 𝑙 𝑙 𝒙 𝒛 𝑢 𝑥, 𝑦 = − 𝑀 𝑜 𝐸𝐼 𝑥𝑦 1 𝑢(𝑥, 𝑦) = − 12𝑀 𝑜 𝐸𝑡𝑤3 𝑥 − 𝑙 2 𝑦 − 𝑤 2 Physical Problem Equivalent Problem SimulatedBending Moments ?? Rotation DOF ?? [1] Silva Gomes, Mecânica dos Sólidos e Resistência dos Materiais Change of Coordinate System 𝑦 𝑥 𝑦 𝑥 𝑤 𝑡 𝑤 𝑀 𝑜 = 𝑃 ∙ 𝑎 Original Coordinate System Coordinate System of the BaPMEF tool
  • 21. 21 6. Validation Examples 24-01-2019 6.1. Constant Rectangular cross-section Beam – Pure Bending Comparison of the displacements in the x-direction, determined by the BaPMEF tool against the Analytical solution ➢ Overall good match between the Analytical and Numeric Solution ➢ Null displacements along the Neutral Axis ➢ Small mismatch near the plate’s ends: ▪ Analytical solution badly behaved in plate’s boundaries[1] ▪ Boundary Conditions difficult to model in Plane Elasticity Deformed shape of the mesh
  • 22. 22 6. Validation Examples 24-01-2019 𝑡 𝑤 𝑢(𝑥, 𝑦) = 𝑃 6𝐺𝐼 𝑦 − 𝑤 2 3 − 𝑃𝑥2 2𝐸𝐼 𝑦 − 𝑤 2 − 𝑣 𝑃 6𝐸𝐼 𝑦 − 𝑤 2 3 + 𝑃𝑙2 2𝐸𝐼 − 𝑃𝑤2 8𝐺𝐼 𝑦 − 𝑤 2 𝑣(𝑥, 𝑦) = 𝑣 𝑃𝑥 2𝐸𝐼 𝑦 − 𝑤 2 2 + 𝑃𝑥3 6𝐸𝐼 − 𝑃𝑙2 𝑥 2𝐸𝐼 + 𝑃𝑙3 3𝐸𝐼 𝑢(𝑥, 𝑦) = 𝑃 6𝐺𝐼 𝑦 − 𝑤 2 3 − 𝑃𝑥2 2𝐸𝐼 𝑦 − 𝑤 2 − 𝑣 𝑃 6𝐸𝐼 𝑦 − 𝑤 2 3 + 𝑃𝑙2 2𝐸𝐼 𝑦 − 𝑤 2 𝑣(𝑥, 𝑦) = 𝑣 𝑃𝑥 2𝐸𝐼 𝑦 − 𝑤 2 2 + 𝑃𝑥3 6𝐸𝐼 − 𝑃𝑙2 2𝐸𝐼 + 𝑃𝑤2 8𝐺𝐼 𝑥 + 𝑃𝑙3 3𝐸𝐼 + 𝑃𝑤2 𝑙 8𝐺𝐼 𝜕𝑣 𝜕𝑥 𝑥=𝑙 ; 𝑦=0 = 0 𝜕𝑢 𝜕𝑦 𝑥=𝑙 ; 𝑦=0 = 0 Situation (ii) 1 Situation (i) 1 ➢ Shear distributed Load ➢ There is no approximation in the definition of the BC ➢ It is also necessary to change the coordinate system 6.2. Constant Rectangular cross-section Beam – Cantilever Beam under Tangential Traction at the end [1] Silva Gomes, Mecânica dos Sólidos e Resistência dos Materiais Problem Simulated 𝑦 𝑥 𝑞 𝑙 𝑤𝑃 𝒒 = 𝑷 𝒘 ∙ 𝒕
  • 23. 23 6. Validation Examples 24-01-2019 ➢ Good match between the Numerical and Analytical Solutions – The Numeric Solution is always in-between the two Analytical Solutions ➢ Null displacements along the Neutral Axis ➢ Situation (i) is more “rigid” ➢ Situation (ii) is more “flexible” 6.2. Constant Rectangular cross-section Beam – Cantilever Beam under Tangential Traction at the end Horizontal Displacements along the plate Deformed Shape of the Mesh
  • 24. 24 6. Validation Examples 24-01-2019 6.2. Constant Rectangular cross-section Beam – Cantilever Beam under Tangential Traction at the end ➢ Good match between the Numerical and Analytical Solutions – The Numeric Solution is always in-between the two Analytical Solutions ➢ Null displacements along the Neutral Axis ➢ Situation (i) is more “rigid” ➢ Situation (ii) is more flexible Vertical Displacements along the plate
  • 25. 25 6. Validation Examples 24-01-2019 6.2. Constant Rectangular cross-section Beam – Cantilever Beam under Tangential Traction at the end ➢ H-type Mesh Refinement Elements: Length=1 x Width=2 Elements: Length=2 x Width=2 Elements: Length=8 x Width=2 Elements: Length=25 x Width=10
  • 26. 26 𝑣 𝑥, 𝑦 1 = 𝑸 2𝐸𝐼 ൮ ൲ 𝑦 − 𝑤 2 4 12 − 𝑤2 8 𝑦 − 𝑤 2 2 + 𝑤3 12 𝑦 − 𝑤 2 + 𝑣 𝑙2 4 − 𝑥 − 𝑙 2 2 𝑦 − 𝑤 2 2 + 𝑦 − 𝑤 2 4 6 − 𝑤2 20 𝑦 − 𝑤 2 2 + 𝑙2 8 𝑥 − 𝑙 2 2 − 𝑥 − 𝑙 2 4 12 − 𝑤2 20 𝑥 − 𝑙 2 2 + 1 4 + 𝑣 8 𝑤2 𝑥 − 𝑙 2 2 − 5𝑙4 192 1 + 12𝑤2 5𝑙2 4 5 + 𝑣 2 6. Validation Examples 24-01-2019 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction 𝑙 𝑤 𝑞 𝑦 𝑥 𝑡 𝑤 𝑞 𝑄 ➢ There is no approximation in the definition of the BC [1] Silva Gomes, Mecânica dos Sólidos e Resistência dos Materiais Problem Simulated 𝑸 = 𝒒 ∙ 𝒕
  • 27. 27 6. Validation Examples 24-01-2019 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction ➢ It is also necessary to change the original Coordinate System of the Analytical Solution ➢ Analytical Solutions for the Stresses 𝜎 𝑥𝑥 1 = 𝑸 2𝐼 2 3 𝑦 − 𝑤 2 3 − 𝑤2 10 𝑦 − 𝑤 2 − 𝑸 2𝐼 𝑥 − 𝑙 2 2 − 𝑙2 4 𝑦 − 𝑤 2 𝜎 𝑦𝑦 1 = − 𝑸 2𝐼 𝑦 − 𝑤 2 3 3 − 𝑤2 4 𝑦 − 𝑤 2 − 𝑤3 12 𝜏 𝑥𝑦 1 = 𝑸 2𝐼 𝑦 − 𝑤 2 2 − 𝑤2 4 𝑥 − 𝑙 2 [1] Silva Gomes, Mecânica dos Sólidos e Resistência dos Materiais Deformed Shape of the Mesh
  • 28. 28 6. Validation Examples 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction ➢ Good match between the Numerical and Analytical Solutions for the Vertical Displacement in the y-direction 3D Surface Plot of the vertical displacements along the plate 2D Plot of the vertical displacements along the Neutral Axis
  • 29. 29 6. Validation Examples 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction ➢ Overall good match between the Numerical and Analytical Solutions for the Normal Stress in the x-direction ➢ Effect of the local Supports 3D Surface Plot of the Stress in the x-direction along the plate 2D Plot of the Stress in the x-direction along the Neutral Axis
  • 30. 30 6. Validation Examples 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction ➢ Overall good match between the Numerical and Analytical Solutions for the Normal Stress in the y-direction ➢ Pronounced effect of the local Supports 3D Surface Plot of the Stress in the y-direction along the plate 2D Plot of the Stress in the y-direction along the Neutral Axis
  • 31. 31 6. Validation Examples 6.3. Constant Rectangular cross-section Beam – Simple Supported Beam under Uniform Surface Traction ➢ Overall good match between the Numerical and Analytical Solutions for the Normal Stress in the x-direction ➢ Effect of the local Supports 3D Surface Plot of the Shear Stress along the plate 2D Plot of the Shear Stress along the Neutral Axis
  • 32. 32 7. Practical Applications 24-01-2019 7.1. Fracture Mechanics – Plate with Centered Crack 2a 𝑊 𝐿 𝑞 𝑞 𝑊 𝑡 a 𝑤 = 𝑊 2 𝑙 = 𝐿 2 Physical Problem Equivalent Problem Simulated ➢ There is no approximation in the definition of the BC ➢ It is also necessary to change the original Coordinate System of the Analytical Solutions ➢ Application of the Mesh Refinement Control Mesh Control mode and Zoom of the Stresses in y-direction near the crack tip
  • 33. 33 7. Practical Applications 24-01-2019 7.1. Fracture Mechanics – Plate with Centered Crack 𝜎 𝑦𝑦 = 𝑞 𝑎 )2(𝑥 − 𝑎 𝜎 𝑦𝑦 = 𝑞 𝑎 )2(𝑥 − 𝑎 sec 𝜋𝑎 𝐿 𝜎 𝑦𝑦 = 𝑞 𝑎 )2(𝑥 − 𝑎 𝐿 𝜋𝑎 tan 𝜋𝑎 𝐿 𝜎 𝑦𝑦 = 𝑞 1 − 𝑎 𝑥 2 4 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 ; 4 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 sec 𝜋𝑎 𝐿 ; 4 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 𝐿 𝜋𝑎 tan 𝜋𝑎 𝐿 ; 2 𝐸 𝑞 𝑎2 − 𝑥2 ; )4(1 − 𝑣2 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 )4(1 − 𝑣2 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 sec 𝜋𝑎 𝐿 )4(1 − 𝑣2 𝐸 𝑞 𝑎2 − 𝑎𝑥 2 𝐿 𝜋𝑎 tan 𝜋𝑎 𝐿 )2(1 − 𝑣2 𝐸 𝑞 𝑎2 − 𝑥2 Plane Stress Plane Strain Solution [2] (i) (ii) (iii) (iv) 𝑣 𝑥, 𝑦 = 𝑣 𝑥, 𝑦 = 𝑣 𝑥, 𝑦 = 𝑣 𝑥, 𝑦 = ➢ Plane Strain Problem [2] David Broke, Elementary Engineering Fracture Mechanics
  • 34. 34 7. Practical Applications 24-01-2019 7.1. Fracture Mechanics – Plate with Centered Crack ➢ Crack Opening Shape ➢ Plate infinitely higher than the crack length – overlap between solution (i), (ii), and (iii) ➢ Solution (i), (ii), and (iii) are only valid near the crack tip – overlap between all solution near the crack tip Vertical Displacements – Plane Stress Vertical Displacements – Plane Strain Overlap Solution (i), (ii), and (iii) Crack tip region
  • 35. 35 7. Practical Applications 24-01-2019 7.1. Fracture Mechanics – Plate with Centered Crack ➢ Stresses along the plate, starting near the crack tip ➢ Plate infinitely higher than the crack length – overlap between solution (i), (ii), and (iii) ➢ Solution (i), (ii), and (iii) are only valid near the crack tip – overlap between all solution near the crack tip Stress in the y-direction near the crack tip Stress in the y-direction far from the crack tip
  • 36. 36 7. Practical Applications 24-01-2019 7.2. BaPMEF vs. Abaqus® 15 Τ𝑁 𝑚𝑚2 𝑷 5 Τ𝑁 𝑚𝑚2 150 [𝑁] 500 [𝑁] 50 [𝑁] 100 [𝑚𝑚] 100 [𝑚𝑚] 30 [𝑚𝑚] 30 [𝑚𝑚] 20 [𝑚𝑚] 20 [𝑚𝑚] 10 Τ𝑁 𝑚𝑚2 BaPMEF ➢ Commercially Available FEA Software – Abaqus ® ➢ Plane Strain Problem ➢ Generic Loading – No Analytical Solution Available ➢ Gravity Body Forces ➢ Surface Tractions acting along the whole boundary ➢ Surface Tractions acting on each Element Problem Simulated in Abaqus Problem Simulated in BaPMEF Gravity Force
  • 37. 37 7. Practical Applications 24-01-2019 7.2. BaPMEF vs. Abaqus® ➢ Horizontal Displacements Results from BaPMEF Tool Results from Abaqus® Software
  • 38. 38 7. Practical Applications 24-01-2019 7.2. BaPMEF vs. Abaqus® ➢ Vertical Displacements Results from BaPMEF Tool Results from Abaqus® Software
  • 39. 39 7. Practical Applications 24-01-2019 7.2. BaPMEF vs. Abaqus® ➢ Displacements Magnitude Results from BaPMEF Tool Results from Abaqus® Software
  • 40. 40 Thank you for your attention! 24-01-2019 Método dos Elementos Finitos EM065 Theory FEM Programming Validation & Practical Applications