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Introduction to Finite Element MethodsUNIT INumerical Methods Definition and Advantages Definition: Methods that seek quantitative approximations to the solutions of mathematical problems Advantages:

What is a Numerical Method An ExampleExample 1:

What is a Numerical Method An ExampleExample 1:

What is a Numerical Method An ExampleExample 2:

What is a Numerical Method An ExampleExample 2:

What is a Numerical Method An ExampleExample 3:

What is a Finite Element Method

Discretization1-D

2-D3-D

?-D

Hybrid

ApproximationNumerical InterpolationNon-exact Boundary ConditionsApplications of Finite Element Methods Structural & Stress Analysis Thermal Analysis Dynamic Analysis Acoustic Analysis Electro-Magnetic Analysis Manufacturing Processes Fluid DynamicsLecture 2 Review 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

Matrix Algebra Determinant of a Matrix:

Matrix inversion -

Important Matrices diagonal matrix identity matrix zero matrix eye matrix

Numerical Integration Calculate:

Newton Cotes integration Trapezoidal rule 1st order Newton-Cotes integration

Trapezoidal rule multiple application

Numerical Integration Calculate:

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

Numerical Integration Calculate:

Gaussian Quadrature

Trapezoidal Rule:Gaussian Quadrature:Choose according to certain criteria

Numerical Integration Calculate:

Gaussian Quadrature 2pt Gaussian Quadrature

3pt Gaussian Quadrature

Let:

Numerical Integration - Example Calculate:

Trapezoidal rule

Simpson 1/3 rule

2pt Gaussian quadrature

Exact solution

Linear System Solver Solve:

Gaussian Elimination: forward elimination + back substitutionExample:

Linear System Solver Solve:

Gaussian Elimination: forward elimination + back substitutionPseudo code:Forward elimination:Back substitution:Do k = 1, n-1Do i = k+1,n

Do j = k+1, n

Do ii = 1, n-1i = n iisum = 0Do j = i+1, nsum = sum +

Finite Element Analysis (F.E.A.) of 1-D ProblemsUNIT IIHistorical 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

Axially Loaded Bar Review: Stress:Strain:Deformation:

Stress:Strain:Deformation:

Axially Loaded Bar Review: Stress:Strain:Deformation:Axially Loaded Bar Governing Equations and Boundary Conditions Differential Equation

Boundary Condition Types prescribed displacement (essential BC)

prescribed force/derivative of displacement (natural BC)

Axially Loaded Bar Boundary Conditions Examples fixed end

simple support

free end

Potential Energy Elastic Potential Energy (PE)- Spring case- Axially loaded bar - Elastic bodyxUnstretched springStretched bar

undeformed:deformed:

Potential Energy Work Potential (WE)

Pff: distributed force over a lineP: point forceu: displacement AB 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. Potential Energy + Rayleigh-Ritz ApproachPfABExample:Step 1: assume a displacement field

f is shape function / basis functionn is the order of approximationStep 2: calculate total potential energyPotential Energy + Rayleigh-Ritz ApproachPfABExample:Step 3:select ai so that the total potential energy is minimum

Galerkins MethodPfABExample:

Seek an approximation so

In the Galerkins method, the weight function is chosen to be the same as the shape function. Galerkins MethodPfABExample:

123123Finite Element Method Piecewise ApproximationxuxuFEM Formulation of Axially Loaded Bar Governing Equations Differential Equation

Weighted-Integral Formulation

Weak Form

Approximation Methods Finite Element MethodExample:Step 1: DiscretizationStep 2: Weak form of one elementP2P1x1x2

Approximation Methods Finite Element MethodExample (cont):Step 3: Choosing shape functions - linear shape functions

lx1x2xxx=-1x=0x=1

Approximation Methods Finite Element MethodExample (cont):Step 4: Forming element equationLet , weak form becomes

Let , weak form becomes

E,A are constant

Approximation Methods Finite Element MethodExample (cont):Step 5: Assembling to form system equationApproach 1:Element 1:

Element 2:

Element 3:

Approximation Methods Finite Element MethodExample (cont):Step 5: Assembling to form system equationAssembled System:

Approximation Methods Finite Element MethodExample (cont):Step 5: Assembling to form system equationApproach 2: Element connectivity tableElement 1Element 2Element 311232234global node index (I,J)local node (i,j)

Approximation Methods Finite Element MethodExample (cont):Step 6: Imposing boundary conditions and forming condense systemCondensed system:

Approximation Methods Finite Element MethodExample (cont):Step 7: solutionStep 8: post calculation

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

Exercises Linear ElementExample 1:E = 100 GPa, A = 1 cm2Linear Formulation for Bar Element

x=x1 x=x2 1 f2 f1 1

x=x1x= x2u1u2

f(x)L = x2-x1uxHigher Order Formulation for Bar Element

13u1u3uxu2214u1u42uxu2u33

1nu1un2uxu2u33u44Natural Coordinates and Interpolation Functions

Natural (or Normal) Coordinate:x=x1x= x2x=-1x=1xx

132xx=-1x=112xx=-1x=1142xx=-1x=13

Quadratic Formulation for Bar Element

x=-1x=0x=1f3f1f2Quadratic Formulation for Bar Elementu1u3u2f(x)P3P1P2x=-1x=0x=1

Exercises Quadratic ElementExample 2:E = 100 GPa, A1 = 1 cm2; A1 = 2 cm2Some IssuesNon-constant cross section:Interior load point:Mixed boundary condition:kFinite Element Analysis (F.E.A.) of I-D Problems Applications

Plane Truss ProblemsExample 1: Find forces inside each member. All members have the same length. FUNIT IIArbitrarily Oriented 1-D Bar Element on 2-D PlaneQ2 , v2

q

P2 , u2Q1 , v1P1 , u1Relationship Between Local Coordinates and Global Coordinates

Relationship Between Local Coordinates and Global Coordinates

Stiffness Matrix of 1-D Bar Element on 2-D Plane

Q2 , v2

q

P2 , u2Q1 , v1P1 , u1

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

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

ax

xgxbxyz21---

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

ax

xgxbxyz21---

Matrix Assembly of Multiple Bar Elements

Element IElement IIElement II I

Matrix Assembly of Multiple Bar Elements

Element I

Element IIElement II IMatrix Assembly of Multiple Bar Elements

Apply known boundary conditionsSolution Procedures

u2= 4FL/5AE, v1= 0

Recovery of Axial Forces

Element IElement IIElement II I

Stresses inside membersElement IElement IIElement II I

Lecture 5 FEM of 1-D Problems: Applications

Torsional ShaftReview Assumption: Circular cross sectionShear stress: Deformation: Shear strain:

Finite Element Equation for Torsional Shaft

Bending BeamReview Normal strain: Pure bending problems: Normal stress: Normal stress with bending moment: Moment-curvature relationship: Flexure formula: xy

MM

Bending BeamReview Deflection: Sign convention: Relationship between shear force, bending moment and transverse load: q(x)xy

+-MMM+-VVVGoverning 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.

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