lecture 1 overview of the fem

Finite Element Method

Instructor:

Wong Foek Tjong, Ph.D.

TS 4466 2 Credits

22023.04.22

Course description

The course aims to enable the students to understand the basic concepts and procedures of the finite element method (FEM) and to apply the FEM by using a commercial softwareTeaches understanding of how finite element

methods work rather than how to use a software

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Graduated from Universitas Parahyangan, Bandung in March1994

Final project: Dynamic Analysis of Multi-degree-of-Freedom Structures Subjected to Ergodic Random Excitation

Graduated from Institut Teknologi Bandung in April1998Master thesis: Active Vibration Control of Structures by Using Artificial Neural Network Observer

Graduated from Asian Institute of Technology, Thailand in May 2009

Dissertation: Kriging-based Finite Element Method for Plates and Shells

Contact: [email protected] Building, Room P402BTel. 62-31-298-3391

The instructor

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Course outline1. Overview of the FEM2. The direct stiffness method

Spring and bar systems Truss structures

3. One-dimensional elements Bar, beam, torsional bar elements Frame element in 3D space

4. Two-dimensional elements for plane-strain/plane-stress problems

Constant strain triangle element Bilinear isoparametric quadrilateral element

5. Introduction to plate and shell elements 6. Applications of the FEM using SAP2000

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References D.L. Logan (2007)

A First Course in the Finite Element Methodthe 4th Ed., Toronto, Nelson

D.V. Hutton (2004)Fundamentals of Finite Element AnalysisNew York, McGraw-Hill

R. D. Cook, D.S. Malkus, M.E. Plesha and R.J. Witt (2002) Concepts and Applications of Finite Element Analysis 4th Ed., John Wiley and Sons

W. Weaver, Jr. and P.R. Johnston (1984)Finite Elements for Structural AnalysisNew Jersey, Prentice-Hall

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References (cont’d) Computers and Structures, Inc. (2006)

CSI Analysis Reference Manual, Berkeley, CSI C. Felippa (2008) Introduction to Finite Element Methods

http://www.colorado.edu/engineering/cas/courses.d/IFEM.d/

R. Krisnakumar (2010) Introduction to Finite Element Methods

http://www.youtube.com(Video of lecture series on FEMs)

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Softwares

MATLAB Ver. 6.5Strongly recommended software for matrix computation and programming

SAP 2000 Ver. 11.0.0

For applications

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Grading weights

Homework assignments 15% Mid-semester exam 35% Take home test 15% Final exam-- project 35%

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Late coming to the class The tolerance for coming late to the class is 20

minutes. Those who come late more than 20 minutes are

NOT allowed to attend the class. Please refer to the “FEM Lecture Plan” for more

academic norms

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Any question about the course before we begin with the Overview of the FEM?

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Discussion: the task of a structural engineer Let take a look on a typical job vacancy announcement

that you may read once you graduate from your study

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Why do you think a design engineer is required to master a structural analysis and design software?

An engineer needs to understand the behavior of a structure so that he/she can make judicious decisions in design, retrofitting, or rehabilitation of the structure

Discussion (cont’d)

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Behavior of a Real Structure

Experiment

Replicate conditions of the structure (possibly on a smaller scale) and observe the behavior of the model

Simulation

Simplifications and assumptions of the real structure

MathematicalModel Physical

Model

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An example of the FEM applications

Real experiment FE simulationIt is often expensive or dangerous

It replicates conditions of the real experiment

Source: W.J. Barry (2003), “FEM Lecture Slides”, AIT Thailand

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The need for modeling

A real structure cannot be analyzed, it can only be “load tested” to determine the responses

We can only analyze a “model” of the structure (perform simulation)

We need to model the structure as close as possible to represent the behavior of the real structure

Source: W. Kanok-Nukulchai

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The idealization process for a simple structure

Source: C. Felippa

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MathematicalModels

Analytical Solution Techniques

Numerical Solution Techniques

Closed-form Solutions

Only possible for simple geometries and boundary conditions

•Finite difference methods•Finite element methods•Boundary element methods•Mesh-free methods•etc.

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It is a computational technique used to obtain approximate solutions of engineering problems.

In the context of structural analyzes, it may be regarded as a generalized direct stiffness method. The direct stiffness method you studied in MK 4215 Structural

Analysis III is actually the application of the FEM to frame structures

It is originated as a method of structural analysis but is now widely used in various disciplines such as heat transfer, fluid flow, seepage, electricity and magnetism, and others.

Finite element method (1)

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Modern FEM were first developed and applied by aeronautical engineers, i.e. M.J. Turner et al., at Boeing company in the period 1950s. 1956: The first engineering FEM paper


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The name “finite element method” was coined by R.W. Clough in 1960. It is called “finite” in order to distinguish with “infinitesimal element” in Calculus.

1967: First FEM book by O.C. Zienkiewicz


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The computation is carried out automatically using a computer or a network of computers.

The results are generally not exact.

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Example of applications in structural engineering

(a) Truss

1. Framed structures

(b) Grid

Source: Weaver and Johnston, 1984

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Example of applications in structural engineering (cont’d)

1. Framed structures (cont’d)

(c) Frame (d) Arch


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(a) Plane stress

2. Two-dimensional continua

(b) Plane strain


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(a) General solid

3. Three-dimensional continua

(b) Axisymmetric solid


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4. Plate in bending


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5. Shells


(a) General shell

(b) Axis symmetric shell

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Source: http://gid.cimne.upc.es/gidinpractice/gp01.html

The analysis and design of buildings

The analysis of a double curvature dam taking into account soil-structure interactions effects

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Source: http://gid.cimne.upc.es/gidinpractice/gp01.html

The structural analysis of an F-16 aircraft The analysis of the Cathedral of

Barcelona using 3D solid elements. (courtesy of Barcelona Cathedral)

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Fundamental concept is discretization, i.e. dividing a continuum (continuous body, structural system) into a finite number of smaller and simple elements whose union approximates the geometry of the continuum.

Mesh generation programs, called preprocessors, help the user in doing this work GiD, a software for pre and post processor

Discretization (1)

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Some basic element shapes

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Some 1st order (linear) elements

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Some 2nd order (quadratic) elements

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One suggestion on performing discretization is to divide structural regions with high stress concentration into finer division e.g. in the vicinity of the support and around the hole(s).

The accuracy of the results can be improved by using a finer mesh (h-refinement) or using a higher order elements (p-refinement).

Discretization (2)

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Examples of discretization (1)

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x

yz

h

D

Clamped

D=100, D/h =100E = 2 x 106 ; ν = 0.3; k = 5/6Load: uniform q = -1E-6

76 nodes, 119 elements172 active DOF

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Cooling Tower– Nuclear Power Plant (taken from a FEM Course Project of Doddy and Andre, Dec 2008)

150 m

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Structural Model and Its Example of the Analysis Results

The structure is divided into smaller parts called “element”

Membrane force contour in the circumferential direction

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The FE Model with a Finer Mesh

The structure is modeled with a finer mesh

The result is now better

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For General purposes:NASTRAN, ANSYS, ADINA, ABAQUS, etc.

For structural analysis, particularly in Civil Engineering:SANS, SAP, STAAD, GT STRUDL, etc.

For building structures:ETABS, BATS etc.

For geotechnical design:PLAXIS

For conducting researches on earthquake engineering:DRAIN-2D, DRAIN-3D, RUAOMOKO, OpenSees etc.

Examples of FEM software

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Typical capabilities of a FE program

Data generation Automatic generation of nodes, elements, and restraints

Element types E.g. SAP2000: Frame, Cable, Shell, Plane, Asolid, Solid, etc.

Material behavior Linear-elastic, nonlinear

Load types Force, displacement, thermal, time-varying excitation

Plotting results Original and deformed geometry, stress contours

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Cook, Malkus, and Plesha (1989, pp.6)Concepts and assumptions behind the computer codes (FEM software) should be mastered. Engineers are expected to be able to use the software to gain better advantages and will less likely misuse them.

SAP2000 disclaimer The user accepts and understands that no warranty is

expressed or implied by the developers or the distributors on the accuracy or reliability of the program. The user must explicitly understand the assumptions of the program and must independently verify the results.

Why do we need to study the basic theory of FEM?

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Any question before we proceed to computational steps of the FEM?

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Discretize the structure (problem domain) Divide the structure or continuum into finite elements

Once the structure has been discretized, the computational steps faithfully follow the steps in the direct stiffness method.

The direct stiffness method: The global stiffness matrix of the discrete structure are obtained

by superimposing (assembling) the stiffness matrices of the element in a direct manner.

Computational steps of the FEM- the direct stiffness method

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Generate element stiffness matrix and element force matrix for each element.

Assemble the element matrices to obtain the global stiffness equation of the structure.

Apply the known nodal loads. Specify how the structure is supported:

Set several nodal displacements to known values.

Computational steps… (cont’d)

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Solve simultaneous linear algebraic equation.

The nodal parameters (displacements) are obtained.

Calculate element stresses or stress resultants (internal forces).

General steps of the FEM (cont’d)

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Any question before we continue to a brief introduction to MATLAB?

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Suppose you want to calculate the natural frequency (Hz) of a SDOF system with the mass m=100 kg and stiffness k=5 KN/m

The formula is

Type in the Command Window:>>m=100>>k=5*1000>> f=1/(2*pi)*sqrt(k/m)

Example

1

2 2

kf

m

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Edit/Debug Window

Click this icon to open a new Edit/Debug

window

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Array operations are operations performed between arrays on an element-by-element basis, e.g.

>> A=[ 1 2; 3 4], B=[-1 3; -2 1]>> A+B, A+4 Common array operations:

Array multiplication (A .* B) Array right division (A ./ B) Array left division (A .\ B) B in the numerator Array exponential (A .^ B)

Array and matrix operations

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Matrix operations follow the normal rules of linear algebra, e.g.

>> A=[ 1 2; 3 4]>> B=[-1 3; -2 1]>> A*B What is the different between A.^3 and A^3?

Be careful to distinguish between array operations and matrix operations in your MATLAB code

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Solving a set of linear algebraic equations

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

n n nn n n

a x a x a x b

a x a x a x b

a x a x a x b

The equations can be written in matrix form as follows:

Ax b

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MATLAB command to solve the equations:

>> x= A \ B (left division operator) Other commands related to linear

algebra:>> det (A)>> rank (A)>> inv (A)

Solving a set of linear algebraic equations (cont’d)

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Please look at the Matlab Tutorial folder to learn more about Matlab

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Homework (due date next class)

1. Write an essay explaining (approx. 500 words):

What is finite element method? Why do you interested to take this course

(TS4466 Finite Element Method)? What do you expect?

2. Divide the following continuum into finite elements:

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Homework (2)

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Homework (3)

3. Divide this equilateral triangle into several quadrilateral elements. You are not allowed to use a triangular element.

4. Solve the following simultaneous algebraic equations using Matlab.

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Explain why the second equation (No. b) does not have a unique solution? (Connect this fact with the determinant and the rank of the coefficient matrix)

500 250 250 88

250 800 150 66

400 150 400 44

x y z

x y z

x y z

500 250 250 88

250 800 150 66

125 400 75 44

x y z

x y z

x y z

a.

b.

lecture 1 overview of the fem

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