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1
**Introduction to Finite Element Methods**

UNIT I Introduction to Finite Element Methods

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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**

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**What is a Numerical Method – An Example**

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**What is a Numerical Method – An Example**

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**What is a Numerical Method – An Example**

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**What is a Numerical Method – An Example**

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**What is a Finite Element Method**

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Discretization 1-D 2-D ?-D 3-D Hybrid

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**Numerical Interpolation Non-exact Boundary Conditions**

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 – 1st order Newton-Cotes integration Trapezoidal rule – multiple application

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**Numerical Integration**

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

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**Numerical Integration**

Calculate: Gaussian Quadrature Trapezoidal Rule: Gaussian Quadrature: Choose according 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 Finite Element Analysis (F.E.A.) of 1-D Problems

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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: Stress: Strain: Strain:**

Deformation: 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**

Unstretched spring Stretched bar x - Axially loaded bar undeformed: deformed: - Elastic body

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**Potential Energy Work Potential (WE) Total Potential Energy**

f P 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**

Example: f P A B Step 1: assume a displacement field f 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**

Example: f P A B Step 3:select ai so that the total potential energy is minimum

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**Galerkin’s Method Example: f P Seek an approximation so**

B 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 Example: f P A B 1 2 3 1 2 3

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**Finite Element Method – Piecewise Approximation**

u x u x

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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 P1 P2 x1 x2

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**Approximation Methods – Finite Element Method**

Example (cont): Step 3: Choosing shape functions - linear shape functions x x x=-1 x=0 x=1 x1 l x2

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**Approximation Methods – Finite Element Method**

Example (cont): Step 4: Forming element equation E,A are constant Let , weak form becomes Let , weak form becomes

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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 Element 1 Element 2 Element 3 1 2 3 4 Approach 2: Element connectivity table local node (i,j) global node index (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 cm2

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**Linear Formulation for Bar Element**

x=x1 x= x2 u1 u2 f(x) L = x2-x1 u x x=x2 1 f2 f1 x=x1

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**Higher Order Formulation for Bar Element**

1 3 u1 u3 u x u2 2 1 4 u1 u4 2 u x u2 u3 3 1 n u1 un 2 u x u2 u3 3 u4 …………… 4

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**Natural Coordinates and Interpolation Functions**

x x=-1 x=1 x x=x1 x= x2 Natural (or Normal) Coordinate: 1 2 x x=-1 x=1 1 3 2 x x=-1 x=1 1 4 2 x x=-1 x=1 3

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**Quadratic Formulation for Bar Element**

x=-1 x=0 x=1 f3 f1 f2

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**Quadratic Formulation for Bar Element**

f(x) P3 P1 P2 x=-1 x=0 x=1

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**Exercises – Quadratic Element**

Example 2: E = 100 GPa, A1 = 1 cm2; A1 = 2 cm2

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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**

Q2 , v2 q P2 , u2 Q1 , v1 P1 , u1

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**Relationship Between Local Coordinates and Global Coordinates**

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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**

Q2 , v2 q P2 , u2 Q1 , v1 P1 , u1

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**Arbitrarily Oriented 1-D Bar Element in 3-D Space**

ax x gx bx y z 2 1 - ax, bx, gx are the Direction Cosines of the bar in the x-y-z coordinate system -

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**Stiffness Matrix of 1-D Bar Element in 3-D Space**

ax x gx bx y z 2 1 -

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**Matrix Assembly of Multiple Bar Elements**

Element I Element I I Element I I I

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**Matrix Assembly of Multiple Bar Elements**

Element I Element I I Element I I I

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**Matrix Assembly of Multiple Bar Elements**

Apply known boundary conditions

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Solution Procedures u2= 4FL/5AE, v1= 0

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**Recovery of Axial Forces**

Element I Element I I Element I I I

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**Stresses inside members**

Element I Element I I Element I I I

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**FEM of 1-D Problems: Applications**

Lecture 5 FEM of 1-D Problems: Applications

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**Torsional Shaft Review Assumption: Circular cross section**

Shear stress: Shear strain: Deformation:

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**Finite Element Equation for Torsional Shaft**

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**Bending Beam y Review x Pure bending problems: Normal strain:**

Normal stress: Normal stress with bending moment: Moment-curvature relationship: Flexure formula:

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**Bending Beam y Review q(x) x**

Relationship between shear force, bending moment and transverse load: Deflection: Sign convention: M + M - M V + - V V

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**Governing Equation and Boundary Condition**

0<x<L Boundary Conditions ----- { Essential BCs – if v or is specified at the boundary. Natural BCs – if or is specified at the boundary.

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**Weak Formulation for Beam Element**

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

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**Weak Form from Integration-by-Parts ----- (2nd time)**

Weak Formulation Weak Form from Integration-by-Parts (2nd time) V(x2) x = x1 M(x2) q(x) y x x = x2 V(x1) M(x1) L = x2-x1

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**Weak Formulation Weak Form y(v) x Q1 q(x) Q3 Q2 Q4 x = x1 L = x2-x1**

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**Ritz Method for Approximation**

q(x) y(v) Q1 Q3 Q2 Q4 x x = x1 L = x2-x1 x = x2 where Let w(x)= fi (x), i = 1, 2, 3, 4

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**Ritz Method for Approximation**

Q3 x = x1 y(v) x x = x2 Q1 Q2 L = x2-x1 Q4

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**Ritz Method for Approximation**

Q3 x = x1 y(v) x x = x2 Q1 Q2 L = x2-x1 Q4

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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 fi??? and where Let Find coefficients to satisfy the interpolation properties.

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**Derivation of Shape Function for Beam Element**

How to select fi??? 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 4th Order 1-D Model**

y(v) u1 q(x) u3 u2 u4 x x = x1 L = x2-x1 x = x2 f4 1 f1 1 f2 f3 x=x2 x=x1

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**Element Equations of 4th Order 1-D Model**

y(v) u1 q(x) u3 u2 u4 x x = x1 L = x2-x1 x = x2

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**Finite Element Analysis of 1-D Problems - Applications**

Example 1. F L Governing equation: Weak form for one element where

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**Finite Element Analysis of 1-D Problems**

Example 1. Approximation function: f1 f4 x=x1 f2 f3 x=x2

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**Finite Element Analysis of 1-D Problems**

Example 1. Finite element model: Discretization: P2 , v2 P3 , v3 P1 , v1 II P4 , v4 I III M1 , q1 M2 , q2 M3 , q3 M4 , q4

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**Matrix Assembly of Multiple Beam Elements**

Element I Element I I

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**Matrix Assembly of Multiple Beam Elements**

Element I I I

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**Apply known boundary conditions**

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 Q1 , v1 E, A, I, L**

P1 , u1 P2 , u2 Q2 , q1 Q4 , q2

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**Finite Element Model of an Arbitrarily Oriented Frame**

q x y q x

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**Finite Element Model of an Arbitrarily Oriented Frame**

local global

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**Plane Frame Analysis - Example**

Rigid Joint Hinge Joint F F F F Beam II Bar Beam I Beam

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Plane Frame Analysis Q3 , v2 Q4 , q2 P2 , u2 P1 , u1 Q2 , q1 Q1 , v1

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Plane Frame Analysis Q1 , v2 Q3 , v3 P1 , u2 P2 , u3 Q2 , q2 Q4 , q3

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Plane Frame Analysis

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

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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 Governing equation: Steady state equation:**

Heat flux q: heat transferred per unit area per unit time (W/m2) Governing equation: Q: heat generated per unit volume per unit time C: mass heat capacity k: thermal conductivity Steady state equation:

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Thermal Convection Newton’s Law of Cooling

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**Thermal Conduction in 1-D**

Boundary conditions: Dirichlet BC: 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<x<L Weighted Integral Formulation ----- Weak Form from Integration-by-Parts -----

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**Formulation for 1-D Linear Element**

x1 x2 1 2 T1 x T2 f1 Let x2 x1 f1T1 f2T2

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**Formulation for 1-D Linear Element**

Let w(x)= fi (x), i = 1, 2

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**Element Equations of 1-D Linear Element**

x1 x2 1 2 T1 x T2 f1

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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 200oC and the outside air temperature is 50oC with a convection coefficient of h = 10 W(m2.K). Find the temperature along the composite wall. t1 t2 t3 x

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**Thermal Conduction and Convection- Fin**

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

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**Fin - Weak Formulation (Steady State Analysis)**

Governing Equation of 1-D Heat Conduction ----- 0<x<L Weighted Integral Formulation ----- Weak Form from Integration-by-Parts -----

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**Formulation for 1-D Linear Element**

Let w(x)= fi (x), i = 1, 2

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**Element Equations of 1-D Linear Element**

x=0 x=L 1 2 T1 x T2 f1

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**Finite Element Analysis of 2-D Problems**

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 W 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 nx and ny are the components of a unit vector, which is normal to the boundary G.

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**Weak Formulation of 2-D Model Problem**

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

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**FEM Implementation of 2-D Heat Conduction – Shape Functions**

Step 1: Discretization – linear triangular element T1 Derivation of linear triangular shape functions: T3 Let T2 Interpolation properties Same

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**FEM Implementation of 2-D Heat Conduction – Shape Functions**

linear triangular element – area coordinates T1 A2 A3 A1 T3 T2 f1 f2 f3

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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 fi 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**

Let 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. f2 f1 f4 f3

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**FEM Implementation of 2-D Heat Conduction – Element Equation**

Weak Form of 2-D Model Problem ----- Assume approximation: and let w(x,y)=fi(x,y) as before, then where

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**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 qi: 3 3 1 1 1 2 2 3 3 1 1 1 2 2

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**Imposing Boundary Conditions**

Consider Equilibrium of flux: FEM implementation:

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**Calculating the q Vector**

Example:

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**2-D Steady-State Heat Conduction - Example**

AB and BC: CD: convection DA: 0.6 m C B 0.4 m y x

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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) Strain-Deformation (Small Deformation) Constitutive Relation (Linear Elasticity)

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**Specification of Boundary Conditions**

EBC: Specify u(x,y) and/or v(x,y) on G NBC: Specify tx and/or ty on G where is the traction on the boundary G at the segment ds.

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UNIT V

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**Weak Formulation for Plane Elasticity**

are components of traction on the boundary G where

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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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**Evaluation of Applied Nodal Forces**

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**Element Assembly for Plane Elasticity**

5 6 B 3 4 3 4 A 1 2

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**Element Assembly for Plane Elasticity**

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 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**

where Ritz method: Weak form where

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**Single-Variable Problem – Heat Transfer**

Weak form where

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**3-Node Axi-symmetric Element**

1 2

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**4-Node Axi-symmetric Element**

h 4 3 b 1 2 x a z r

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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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**Time-Dependent Problems**

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**

and let: 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**

a - family formula: Equation

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**Time Approximation – First Order ODE**

Finite Element Approximation

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**Stability of – Family Approximation**

Example Stability

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**FEA of Transient Heat Conduction**

a - 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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**Transient Heat Conduction - Example**

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**Transient Heat Conduction - Example**

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**Transient Heat Conduction - Example**

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**Transient Heat Conduction - Example**

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**Transient Heat Conduction - Example**

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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**

and let:

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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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**Transverse Motion of Euler-Bernoulli Beam**

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**Transverse Motion of Euler-Bernoulli Beam**

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**Transverse Motion of Euler-Bernoulli Beam**

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MECH4450 Introduction to Finite Element Methods

MECH4450 Introduction to Finite Element Methods

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