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Introduction to Finite Element Methods UNIT I

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Numerical Methods – Definition and Advantages Definition: Methods that seek quantitative approximations to the solutions of mathematical problems Advantages:

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What is a Numerical Method – An Example Example 1:

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What is a Numerical Method – An Example Example 1:

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What is a Numerical Method – An Example Example 2:

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What is a Numerical Method – An Example Example 2:

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What is a Numerical Method – An Example Example 3:

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What is a Finite Element Method

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Discretization 1-D2-D 3-D?-D Hybrid

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Approximation Numerical Interpolation Non-exact Boundary Conditions

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Applications of Finite Element Methods Structural & Stress Analysis Thermal Analysis Dynamic Analysis Acoustic Analysis Electro-Magnetic Analysis Manufacturing Processes Fluid Dynamics

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Lecture 2 Review

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Matrix Algebra Row and column vectors Addition and Subtraction – must have the same dimensions Multiplication – with scalar, with vector, with matrix Transposition – Differentiation and Integration

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Matrix Algebra Determinant of a Matrix: Matrix inversion - Important Matrices diagonal matrix identity matrix zero matrix eye matrix

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Numerical Integration Calculate: Newton – Cotes integration Trapezoidal rule – 1 st order Newton-Cotes integration Trapezoidal rule – multiple application

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Numerical Integration Calculate: Newton – Cotes integration Simpson 1/3 rule – 2 nd order Newton-Cotes integration

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Numerical Integration Calculate: Gaussian Quadrature Trapezoidal Rule : Gaussian Quadrature : Chooseaccording to certain criteria

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Numerical Integration Calculate: Gaussian Quadrature 2pt Gaussian Quadrature 3pt Gaussian Quadrature Let:

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Numerical Integration - Example Calculate: Trapezoidal rule Simpson 1/3 rule 2pt Gaussian quadrature Exact solution

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Linear System Solver Solve: Gaussian Elimination: forward elimination + back substitution Example:

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Linear System Solver Solve: Gaussian Elimination: forward elimination + back substitution Pseudo code: Forward elimination: Back substitution: Do k = 1, n-1 Do i = k+1,n Do j = k+1, n Do ii = 1, n-1 i = n – ii sum = 0 Do j = i+1, n sum = sum +

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Finite Element Analysis (F.E.A.) of 1-D Problems UNIT II

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Historical Background Hrenikoff, 1941 – “frame work method” Courant, 1943 – “piecewise polynomial interpolation” Turner, 1956 – derived stiffness matrice for truss, beam, etc Clough, 1960 – coined the term “finite element” Key Ideas: - frame work method piecewise polynomial approximation

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Axially Loaded Bar Review: Stress: Strain: Deformation: Stress: Strain: Deformation:

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Axially Loaded Bar Review: Stress: Strain: Deformation:

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Axially Loaded Bar – Governing Equations and Boundary Conditions Differential Equation Boundary Condition Types prescribed displacement (essential BC) prescribed force/derivative of displacement (natural BC)

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Axially Loaded Bar –Boundary Conditions Examples fixed end simple support free end

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Potential Energy Elastic Potential Energy (PE) - Spring case - Axially loaded bar - Elastic body x Unstretched spring Stretched bar undeformed: deformed:

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Potential Energy Work Potential (WE) P f f: distributed force over a line P: point force u: displacement A B Total Potential Energy Principle of Minimum Potential Energy For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium extremize the total potential energy. If the extremum condition is a minimum, the equilibrium state is stable.

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Potential Energy + Rayleigh-Ritz Approach P f A B Example: Step 1: assume a displacement field is shape function / basis function n is the order of approximation Step 2: calculate total potential energy

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Potential Energy + Rayleigh-Ritz Approach P f A B Example: Step 3:select a i so that the total potential energy is minimum

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Galerkin’s Method P f A B Example: Seek an approximation so In the Galerkin’s method, the weight function is chosen to be the same as the shape function.

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Galerkin’s Method P f A B Example: 1 2 3 1 2 3

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Finite Element Method – Piecewise Approximation x u x u

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FEM Formulation of Axially Loaded Bar – Governing Equations Differential Equation Weighted-Integral Formulation Weak Form

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Approximation Methods – Finite Element Method Example: Step 1: Discretization Step 2: Weak form of one element P2P2 P1P1 x1x1 x2x2

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Approximation Methods – Finite Element Method Example (cont): Step 3: Choosing shape functions - linear shape functions l x1x1 x2x2 x

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Approximation Methods – Finite Element Method Example (cont): Step 4: Forming element equation Let, weak form becomes E,A are constant

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Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Approach 1: Element 1: Element 2: Element 3:

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Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Assembled System:

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Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Approach 2: Element connectivity table Element 1Element 2Element 3 1123 2234 global node index (I,J) local node (i,j)

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Approximation Methods – Finite Element Method Example (cont): Step 6: Imposing boundary conditions and forming condense system Condensed system:

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Approximation Methods – Finite Element Method Example (cont): Step 7: solution Step 8: post calculation

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Summary - Major Steps in FEM Discretization Derivation of element equation weak form construct form of approximation solution over one element derive finite element model Assembling – putting elements together Imposing boundary conditions Solving equations Postcomputation

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Exercises – Linear Element Example 1: E = 100 GPa, A = 1 cm 2

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Linear Formulation for Bar Element x=x 1 x=x 2 2 2 1 1 x=x 1 x= x 2 u1u1 u2u2 f(x) L = x 2 -x 1 u x

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Higher Order Formulation for Bar Element 1 3 u1u1 u3u3 u x u2u2 214 u1u1 u4u4 2 u x u2u2 u3u3 3 1n u1u1 unun 2 u x u2u2 u3u3 3 u4u4 …………… 4

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Natural Coordinates and Interpolation Functions Natural (or Normal) Coordinate: x=x 1 x= x 2 =-1 =1 x 1 3 2 =-1 =1 1 2 =-1 =1 1 4 2 =-1 =1 3

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Quadratic Formulation for Bar Element =-1 =0 =1

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Quadratic Formulation for Bar Element u1u1 u3u3 u2u2 f(x) P3P3 P1P1 P2P2 =-1 =0 =1

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Exercises – Quadratic Element Example 2: E = 100 GPa, A 1 = 1 cm 2 ; A 1 = 2 cm 2

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Some Issues Non-constant cross section: Interior load point: Mixed boundary condition: k

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Finite Element Analysis (F.E.A.) of I-D Problems – Applications

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Plane Truss Problems Example 1: Find forces inside each member. All members have the same length. F

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UNIT II

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Arbitrarily Oriented 1-D Bar Element on 2-D Plane Q 2, v 2 P 2, u 2 Q 1, v 1 P 1, u 1

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Relationship Between Local Coordinates and Global Coordinates

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Stiffness Matrix of 1-D Bar Element on 2-D Plane Q 2, v 2 P 2, u 2 Q 1, v 1 P 1, u 1

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Arbitrarily Oriented 1-D Bar Element in 3-D Space x, x, x are the Direction Cosines of the bar in the x-y-z coordinate system - - - xx x xx xx y z 2 1 - - -

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Stiffness Matrix of 1-D Bar Element in 3-D Space xx x xx xx y z 2 1 - - -

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Matrix Assembly of Multiple Bar Elements Element I I I

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Matrix Assembly of Multiple Bar Elements Element I I I

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Matrix Assembly of Multiple Bar Elements Apply known boundary conditions

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Solution Procedures u 2 = 4FL/5AE, v 1 = 0

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Recovery of Axial Forces Element I I I

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Stresses inside members Element I I I

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Lecture 5 FEM of 1-D Problems: Applications

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Torsional Shaft Review Assumption: Circular cross section Shear stress: Deformation: Shear strain:

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Finite Element Equation for Torsional Shaft

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Bending Beam Review Normal strain: Pure bending problems: Normal stress: Normal stress with bending moment: Moment-curvature relationship: Flexure formula: x y M M

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Bending Beam Review Deflection: Sign convention: Relationship between shear force, bending moment and transverse load: q(x) x y + - M M M + - V V V

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Governing Equation and Boundary Condition Governing Equation Boundary Conditions ----- Essential BCs – if v or is specified at the boundary. Natural BCs – if or is specified at the boundary. { 0

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Weak Formulation for Beam Element Governing Equation Weighted-Integral Formulation for one element Weak Form from Integration-by-Parts ----- (1 st time)

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Weak Formulation Weak Form from Integration-by-Parts ----- (2 nd time) V(x 2 ) x = x 1 M(x 2 ) q(x) y x x = x 2 V(x 1 ) M(x 1 ) L = x 2 -x 1

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Weak Formulation Weak Form Q3Q3 x = x 1 Q4Q4 q(x) y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1

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Ritz Method for Approximation Let w(x)= i (x), i = 1, 2, 3, 4 Q3Q3 x = x 1 Q4Q4 q(x) y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1 where

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Ritz Method for Approximation Q3Q3 x = x 1 y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1 Q4Q4

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Ritz Method for Approximation Q3Q3 x = x 1 y(v) x x = x 2 Q1Q1 Q2Q2 L = x 2 -x 1 Q4Q4

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Selection of Shape Function The best situation is ----- Interpolation Properties

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Derivation of Shape Function for Beam Element – Local Coordinates How to select i ??? and where Let Find coefficients to satisfy the interpolation properties.

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Derivation of Shape Function for Beam Element How to select i ??? e.g. Let Similarly

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Derivation of Shape Function for Beam Element In the global coordinates:

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Element Equations of 4 th Order 1-D Model u3u3 x = x 1 u4u4 q(x) y(v) x x = x 2 u1u1 u2u2 L = x 2 -x 1 x=x 2 x=x 1 1 1 1 1 3 3 2 2 4 4

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Element Equations of 4 th Order 1-D Model u3u3 x = x 1 u4u4 q(x) y(v) x x = x 2 u1u1 u2u2 L = x 2 -x 1

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Finite Element Analysis of 1-D Problems - Applications F L L L Example 1. Governing equation: Weak form for one element where

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Finite Element Analysis of 1-D Problems Example 1. Approximation function: 3 3 2 2 1 1 4 4 x=x 1 x=x 2

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Finite Element Analysis of 1-D Problems Example 1. Finite element model: P 1, v 1 P 2, v 2 P 3, v 3 P 4, v 4 M 1, 1 M 2, 2 M 3, 3 M 4, 4 I II III Discretization:

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Matrix Assembly of Multiple Beam Elements Element I I

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Matrix Assembly of Multiple Beam Elements Element I I

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Solution Procedures Apply known boundary conditions

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Solution Procedures

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Shear Resultant & Bending Moment Diagram

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Plane Flame Frame: combination of bar and beam E, A, I, L Q 1, v 1 Q 3, v 2 Q 2, 1 P 1, u 1 Q 4, 2 P 2, u 2

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Finite Element Model of an Arbitrarily Oriented Frame x y x y

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local global

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Plane Frame Analysis - Example Rigid Joint Hinge Joint Beam II Beam I Beam Bar F FF F

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Plane Frame Analysis P 1, u 1 P 2, u 2 Q 2, 1 Q 4, 2 Q 1, v 1 Q 3, v 2

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Plane Frame Analysis P 1, u 2 Q 3, v 3 Q 2, 2 Q 4, 3 Q 1, v 2 P 2, u 3

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Plane Frame Analysis

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Finite Element Analysis (F.E.A.) of 1-D Problems – Heat Conduction UNIT IV

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Heat Transfer Mechanisms Conduction – heat transfer by molecular agitation within a material without any motion of the material as a whole. Convection – heat transfer by motion of a fluid. Radiation – the exchange of thermal radiation between two or more bodies. Thermal radiation is the energy emitted from hot surfaces as electromagnetic waves.

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Heat Conduction in 1-D Heat flux q: heat transferred per unit area per unit time (W/m 2 ) Q: heat generated per unit volume per unit time C: mass heat capacity Governing equation: Steady state equation: : thermal conductivity

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Thermal Convection Newton’s Law of Cooling

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Thermal Conduction in 1-D Dirichlet BC: Boundary conditions: Natural BC: Mixed BC:

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Weak Formulation of 1-D Heat Conduction ( Steady State Analysis ) Governing Equation of 1-D Heat Conduction ----- 0

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Formulation for 1-D Linear Element Let f2f2 x1x1 x2x2 1 2 T1T1 x T2T2 f1f1 x2 x2 x1 x1 1T1 1T1 2T2 2T2

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Formulation for 1-D Linear Element Let w(x)= i (x), i = 1, 2

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Element Equations of 1-D Linear Element f2f2 x1x1 x2x2 1 2 T1T1 x T2T2 f1f1

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1-D Heat Conduction - Example A composite wall consists of three materials, as shown in the figure below. The inside wall temperature is 200 o C and the outside air temperature is 50 o C with a convection coefficient of h = 10 W(m 2.K). Find the temperature along the composite wall. t1t1 t2t2 t3t3 x

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Thermal Conduction and Convection- Fin Objective: to enhance heat transfer dx t x w Governing equation for 1-D heat transfer in thin fin where

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Fin - Weak Formulation ( Steady State Analysis ) Governing Equation of 1-D Heat Conduction ----- 0

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Formulation for 1-D Linear Element Let w(x)= i (x), i = 1, 2

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Element Equations of 1-D Linear Element f2f2 x=0 x=L 1 2 T1T1 x T2T2 f1f1

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Lecture 7 Finite Element Analysis of 2-D Problems

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2-D Discretization Common 2-D elements:

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2-D Model Problem with Scalar Function - Heat Conduction Governing Equation in Boundary Conditions Dirichlet BC: Natural BC: Mixed BC:

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Weak Formulation of 2-D Model Problem Weighted - Integral of 2-D Problem ----- Weak Form from Integration-by-Parts -----

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Weak Formulation of 2-D Model Problem Green-Gauss Theorem ----- where n x and n y are the components of a unit vector, which is normal to the boundary .

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Weak Formulation of 2-D Model Problem Weak Form of 2-D Model Problem ----- EBC: Specify T(x,y) on NBC: Specify on where is the normal outward flux on the boundary at the segment ds.

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FEM Implementation of 2-D Heat Conduction – Shape Functions Step 1: Discretization – linear triangular element T1T1 T2T2 T3T3 Derivation of linear triangular shape functions: Let Interpolation properties Same

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FEM Implementation of 2-D Heat Conduction – Shape Functions linear triangular element – area coordinates T1T1 T2T2 T3T3 A3A3 A1A1 A2A2 11 22 33

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Interpolation Function - Requirements Interpolation condition Take a unit value at node i, and is zero at all other nodes Local support condition is zero at an edge that doesn’t contain node i. Interelement compatibility condition Satisfies continuity condition between adjacent elements over any element boundary that includes node i Completeness condition The interpolation is able to represent exactly any displacement field which is polynomial in x and y with the order of the interpolation function

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Formulation of 2-D 4-Node Rectangular Element – Bi-linear Element 1 1 2 2 3 3 4 4 Note: The local node numbers should be arranged in a counter-clockwise sense. Otherwise, the area Of the element would be negative and the stiffness matrix can not be formed. Let

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Weak Form of 2-D Model Problem ----- Assume approximation: and let w(x,y)= i (x,y) as before, then where FEM Implementation of 2-D Heat Conduction – Element Equation

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Assembly of Stiffness Matrices

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Imposing Boundary Conditions The meaning of q i : 1 2 3 1 1 2 3 1 2 3 1 1 2 3

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Imposing Boundary Conditions Equilibrium of flux: FEM implementation: Consider

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Calculating the q Vector Example:

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2-D Steady-State Heat Conduction - Example 0.6 m 0.4 m A B C D AB and BC: CD: convection DA: x y

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Finite Element Analysis of Plane Elasticity

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Review of Linear Elasticity Linear Elasticity: A theory to predict mechanical response of an elastic body under a general loading condition. Stress: measurement of force intensity with 2-D

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Review of Linear Elasticity Traction (surface force) : Equilibrium – Newton’s Law

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Review of Linear Elasticity Strain: measurement of intensity of deformation Generalized Hooke’s Law

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Plane Stress and Plane Strain Plane Stress - Thin Plate:

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Plane Stress and Plane Strain Plane Strain - Thick Plate: Plane Stress: Plane Strain: Replace E by and by

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Equations of Plane Elasticity Governing Equations (Static Equilibrium) Constitutive Relation (Linear Elasticity) Strain-Deformation (Small Deformation)

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Specification of Boundary Conditions EBC: Specify u(x,y) and/or v(x,y) on NBC: Specify t x and/or t y on where is the traction on the boundary at the segment ds.

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UNIT V

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Weak Formulation for Plane Elasticity where are components of traction on the boundary

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Finite Element Formulation for Plane Elasticity Let where and

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Constant-Strain Triangular (CST) Element for Plane Stress Analysis Let

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Constant-Strain Triangular (CST) Element for Plane Stress Analysis

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4-Node Rectangular Element for Plane Stress Analysis Let

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4-Node Rectangular Element for Plane Stress Analysis For Plane Strain Analysis: and

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Loading Conditions for Plane Stress Analysis

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Evaluation of Applied Nodal Forces

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Element Assembly for Plane Elasticity A B 1 2 3 4 3 4 6 5

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1 2 3 4 6 5 A B

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Comparison of Applied Nodal Forces

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Discussion on Boundary Conditions Must have sufficient EBCs to suppress rigid body translation and rotation For higher order elements, the mid side nodes cannot be skipped while applying EBCs/NBCs

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Plane Stress – Example 2

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Plane Stress – Example 3

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Evaluation of Strains

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Evaluation of Stresses Plane Stress Analysis Plane Strain Analysis

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Finite Element Analysis of 2-D Problems – Axi- symmetric Problems

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Axi-symmetric Problems Definition: A problem in which geometry, loadings, boundary conditions and materials are symmetric about one axis. Examples:

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Axi-symmetric Analysis Cylindrical coordinates: quantities depend on r and z only 3-D problem 2-D problem

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Axi-symmetric Analysis

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Axi-symmetric Analysis – Single-Variable Problem Weak form: where

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Finite Element Model – Single-Variable Problem Ritz method: where Weak form where

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Single-Variable Problem – Heat Transfer Heat Transfer: Weak form where

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3-Node Axi-symmetric Element 1 2 3

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4-Node Axi-symmetric Element 1 2 3 4 a b r z

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Single-Variable Problem – Example Step 1: Discretization Step 2: Element equation

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Time-Dependent Problems

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In general, Key question: How to choose approximate functions? Two approaches:

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Model Problem I – Transient Heat Conduction Weak form:

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Transient Heat Conduction let: and ODE!

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Time Approximation – First Order ODE Forward difference approximation - explicit Backward difference approximation - implicit

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Time Approximation – First Order ODE - family formula: Equation

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Time Approximation – First Order ODE Finite Element Approximation

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Stability of – Family Approximation Stability Example

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FEA of Transient Heat Conduction - family formula for vector:

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Stability Requirment where Note: One must use the same discretization for solving the eigenvalue problem.

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Transient Heat Conduction - Example

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Model Problem II – Transverse Motion of Euler- Bernoulli Beam Weak form: Where:

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Transverse Motion of Euler-Bernoulli Beam let: and

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Transverse Motion of Euler-Bernoulli Beam

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ODE Solver – Newmark’s Scheme where Stability requirement: where

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ODE Solver – Newmark’s Scheme Constant-average acceleration method (stable) Linear acceleration method (conditional stable) Central difference method (conditional stable) Galerkin method (stable) Backward difference method (stable)

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Fully Discretized Finite Element Equations

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Transverse Motion of Euler-Bernoulli Beam

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