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**MANE 4240 & CIVL 4240 Introduction to Finite Elements**

Prof. Suvranu De Introduction

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**Office hours: T/F 2:00 pm-3:00 pm Course website:**

Info Instructor: Professor Suvranu De JEC room: 2049 Tel: 6351 Office hours: T/F 2:00 pm-3:00 pm Course website:

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Info TA: Kartik Josyula JEC room: CII 9219 Office hours: M: 5-6pm, R:3:30-4:30pm

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**Course texts and references**

Course text (for HW problems and reading assignments): Title: A First Course in the Finite Element Method Author: Daryl Logan Edition: Fifth Publisher: Cengage Learning ISBN: Relevant reference: Finite Element Procedures, K. J. Bathe, Prentice Hall A First Course in Finite Elements, J. Fish and T. Belytschko Lecture notes posted on the course website

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**Course grades Grades will be based on: Home works (15 %).**

Mini project (10 %) to be handed in by 17th October. Major course project (25 %)to be handed in by December 5th (by noon) Two in-class exams (2x25%) on 10th October, 5th December 1) Form (mini and major) project groups of two by 12th September. 2) All write ups that you present MUST contain your name and RIN 3) There will be reading quizzes (announced AS WELL AS unannounced) on a regular basis and points from these quizzes will be added on to the homework 4) Project grades will be allotted to the group and NOT individually.

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**Collaboration / academic integrity**

Students are encouraged to collaborate in the solution of HW problems, but submit independent solutions that are NOT copies of each other. Funny solutions (that appear similar/same) will be given zero credit. Softwares may be used to verify the HW solutions. But submission of software solution will result in zero credit. 2. Groups of 2 for the projects (no two projects to be the same/similar) A single grade will be assigned to the group and not to the individuals.

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Homeworks (15%) 1. Be as detailed and explicit as possible. For full credit Do NOT omit steps. 2. Only neatly written homeworks will be graded 3. Late homeworks will NOT be accepted. 4. Two lowest grades will be dropped (except HW #1). 5. Solutions will be posted on the course website

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**Mini project (10%) Download the miniproject from the course website.**

You will need to learn the commercial software package ABAQUS (tutorial available at course website + in-class tutorial). Submit a project report by 17th October

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**Mini project (10%)..contd. Project report:**

Must be typed (Text font Times 11pt with single spacing) Must be no more than 5 pages (with figures).Must include the following sections: Problem statement Analysis (with appropriate figures) Results and discussion Attach an appendix containing the program printouts for the different cases you run.

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**Major project (25 %) In this project you will be required to**

Choose an engineering system Develop a simple mathematical model for the system Develop the finite element model Solve the problem using ABAQUS or any other FEM code (you may write one you if you choose to) Discuss whether the mathematical model you chose gives you physically meaningful results. If not, revise your model and perform analysis to improve your results.

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**Major project (25 %)..contd.**

Logistics: Submit 1-page project proposal latest by 26th September (in class). The earlier the better. Projects will go on a first come first served basis. Proceed to work on the project ONLY after it is approved by the course instructor. Submit a one-page progress report on November 4th (this will count as 10% of your project grade) Submit a project report (typed) by noon of 5th December to the instructor.

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**Major project (25 %)..contd.**

Project report: Must be professional (Text font Times 11pt with single spacing) Must include the following sections: Introduction Problem statement Analysis Results and Discussions You must also burn the MS Word version of your project report on a CD with your group number, names, and project title written on it and submit it together with the project report.

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**Major project (25 %)..contd.**

Project examples: (two sample project reports from previous year are provided) 1. Analysis of a rocker arm 2. Analysis of a bicycle crank-pedal assembly 3. Design and analysis of a "portable stair climber" 4. Analysis of a gear train 5.Gear tooth stress in a wind- up clock 6. Analysis of a gear box assembly 7. Analysis of an artificial knee 8. Forces acting on the elbow joint 9. Analysis of a soft tissue tumor system 10. Finite element analysis of a skateboard truck

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**Major project (25 %)..contd.**

Project grade will depend on Originality of the idea Techniques used Critical discussion

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**Finite Element Analysis**

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**Finite Element Analysis**

Cantilever plate in plane strain uniform loading Fixed boundary Problem: Obtain the stresses/strains in the plate Approximate method Geometric model Node Element Mesh Discretization Node Element Finite element model

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**Course content “Direct Stiffness” approach for springs**

Bar elements and truss analysis Introduction to boundary value problems: strong form, principle of minimum potential energy and principle of virtual work. Displacement-based finite element formulation in 1D: formation of stiffness matrix and load vector, numerical integration. Displacement-based finite element formulation in 2D: formation of stiffness matrix and load vector for CST and quadrilateral elements. Discussion on issues in practical FEM modeling Convergence of finite element results Higher order elements Isoparametric formulation Numerical integration in 2D Solution of linear algebraic equations

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For next class Please read Appendix A of Logan for reading quiz next class (10 pts on Hw 1)

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Linear Algebra Recap (at the IEA level)

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What is a matrix? A rectangular array of numbers (we will concentrate on real numbers). A nxm matrix has ‘n’ rows and ‘m’ columns First row Second row Third row First column Second column Third column Fourth column Row number Column number

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**What is a vector? A vector is an array of ‘n’ numbers**

A row vector of length ‘n’ is a 1xn matrix A column vector of length ‘m’ is a mx1 matrix

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**Special matrices Zero matrix: A matrix all of whose entries are zero**

Identity matrix: A square matrix which has ‘1’ s on the diagonal and zeros everywhere else.

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**Matrix operations Equality of matrices**

If A and B are two matrices of the same size, then they are “equal” if each and every entry of one matrix equals the corresponding entry of the other.

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**Addition of two matrices**

Matrix operations Addition of two matrices If A and B are two matrices of the same size, then the sum of the matrices is a matrix C=A+B whose entries are the sums of the corresponding entries of A and B

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**Addition of of matrices**

Matrix operations Properties Properties of matrix addition: Matrix addition is commutative (order of addition does not matter) Matrix addition is associative Addition of the zero matrix

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**Multiplication by a scalar**

Matrix operations Multiplication by a scalar If A is a matrix and c is a scalar, then the product cA is a matrix whose entries are obtained by multiplying each of the entries of A by c

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**Multiplication by a scalar**

Matrix operations Special case If A is a matrix and c =-1 is a scalar, then the product (-1)A =-A is a matrix whose entries are obtained by multiplying each of the entries of A by -1

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Matrix operations Subtraction If A and B are two square matrices of the same size, then A-B is defined as the sum A+(-1)B

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Transpose Special operations If A is a mxn matrix, then the transpose of A is the nxm matrix whose first column is the first row of A, whose second column is the second column of A and so on.

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Transpose Special operations If A is a square matrix (mxm), it is called symmetric if

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**Scalar (dot) product of two vectors**

Matrix operations Scalar (dot) product of two vectors If a and b are two vectors of the same size The scalar (dot) product of a and b is a scalar obtained by adding the products of corresponding entries of the two vectors

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**Matrix multiplication**

Matrix operations Matrix multiplication For a product to be defined, the number of columns of A must be equal to the number of rows of B. A B = AB m x r r x n m x n inside outside

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**Matrix multiplication**

Matrix operations Matrix multiplication If A is a mxr matrix and B is a rxn matrix, then the product C=AB is a mxn matrix whose entries are obtained as follows. The entry corresponding to row ‘i’ and column ‘j’ of C is the dot product of the vectors formed by the row ‘i’ of A and column ‘j’ of B

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**Multiplication of matrices**

Matrix operations Properties Properties of matrix multiplication: Matrix multiplication is noncommutative (order of addition does matter) It may be that the product AB exists but BA does not (e.g. in the previous example C=AB is a 3x2 matrix, but BA does not exist) Even if the product exists, the products AB and BA are not generally the same

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**Multiplication of matrices**

Matrix operations Properties 2. Matrix multiplication is associative 3. Distributive law 4. Multiplication by identity matrix 5. Multiplication by zero matrix 6.

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**Miscellaneous properties**

Matrix operations If A , B and C are square matrices of the same size, and then does not necessarily mean that does not necessarily imply that either A or B is zero

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Definition Inverse of a matrix If A is any square matrix and B is another square matrix satisfying the conditions Then The matrix A is called invertible, and the matrix B is the inverse of A and is denoted as A-1. The inverse of a matrix is unique

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**Inverse of a matrix The inverse of a matrix is unique**

Uniqueness Inverse of a matrix The inverse of a matrix is unique Assume that B and C both are inverses of A Hence a matrix cannot have two or more inverses.

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Some properties Inverse of a matrix Property 1: If A is any invertible square matrix the inverse of its inverse is the matrix A itself Property 2: If A is any invertible square matrix and k is any scalar then

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Properties Inverse of a matrix Property 3: If A and B are invertible square matrices then

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**What is a determinant? The determinant of a square matrix is a number**

obtained in a specific manner from the matrix. For a 1x1 matrix: For a 2x2 matrix: Product along red arrow minus product along blue arrow

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**Example 1 Consider the matrix**

Notice (1) A matrix is an array of numbers (2) A matrix is enclosed by square brackets Notice (1) The determinant of a matrix is a number (2) The symbol for the determinant of a matrix is a pair of parallel lines Computation of larger matrices is more difficult

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**Duplicate column method for 3x3 matrix**

For ONLY a 3x3 matrix write down the first two columns after the third column Sum of products along red arrow minus sum of products along blue arrow This technique works only for 3x3 matrices

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**Example 32 3 -8 8 Sum of red terms = 0 + 32 + 3 = 35**

32 3 -8 8 Sum of red terms = = 35 Sum of blue terms = 0 – = 0 Determinant of matrix A= det(A) = 35 – 0 = 35

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**Finding determinant using inspection**

Special case. If two rows or two columns are proportional (i.e. multiples of each other), then the determinant of the matrix is zero because rows 1 and 3 are proportional to each other If the determinant of a matrix is zero, it is called a singular matrix

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**Cofactor method If A is a square matrix What is a cofactor?**

The minor, Mij, of entry aij is the determinant of the submatrix that remains after the ith row and jth column are deleted from A. The cofactor of entry aij is Cij=(-1)(i+j) Mij

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**What is a cofactor? Sign of cofactor**

Find the minor and cofactor of a33 Minor Cofactor

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**Cofactor method of obtaining the determinant of a matrix**

The determinant of a n x n matrix A can be computed by multiplying ALL the entries in ANY row (or column) by their cofactors and adding the resulting products. That is, for each and Cofactor expansion along the jth column Cofactor expansion along the ith row

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**Example: evaluate det(A) for:**

1 3 -1 4 5 2 -3 -2 A= det(A) = a11C11 +a12C12 + a13C13 +a14C14 det(A)=(1) - (0) + 2 - (-3) = (1)(35)-0+(2)(62)-(-3)(13)=198

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**det(A)=a13C13 +a23C23+a33C33 Example : evaluate det(A)= 5 1 0 3 -1 -3**

1 0 3 -1 -3 2 By a cofactor along the third column det(A)=a13C13 +a23C23+a33C33 det(A)= -3* (-1)4 +2*(-1)5 +2*(-1)6 = det(A)= -3(-1-0)+2(-1)5(-1-15)+2(0-5)=25

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**Quadratic form The scalar Is known as a quadratic form**

If U>0: Matrix k is known as positive definite If U≥0: Matrix k is known as positive semidefinite

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Quadratic form Let Symmetric matrix Then

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**Differentiation of quadratic form**

Differentiate U wrt d1 Differentiate U wrt d2

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**Differentiation of quadratic form**

Hence

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Outline Role of FEM simulation in Engineering Design Course Philosophy

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**Role of simulation in design: Boeing 777**

The Boeing 777 is the first jetliner to be 100 percent digitally designed using three-dimensional solids technology. Throughout the design process, the airplane was "preassembled" on the computer, eliminating the need for a costly, full-scale mock-up. The kg plane is the biggest twin-engine aircraft ever to fly-it can carry 375 passengers 7400 km-and from its first service flight in June 1995, has been certified for extended-range twin-engine operations. Boeing invested more than $4 billion (and insiders say much more) in CAD infrastructure for the design of the Boeing 777 and reaped huge benefits from design automation. The more than 3 million parts were represented in an integrated database that allowed designers to do a complete 3D virtual mock-up of the vehicle. Boeing based its CAD system on CATIA (short for Computer-aided Three-dimensional Interactive Application) and ELFINI (Finite Element Analysis System), both developed by Dassault Systemes of France (Dassault systems acquired ABAQUS in 2005 and ABAQUS+CATIA is known as SIMULIA) and licensed in the United States through IBM. Designers also used EPIC (Electronic Preassembly Integration on CATIA) and other digital preassembly applications developed by Boeing. Much of the same technology was used on the B-2 program. To design the 777, Boeing organized its workers into 238 cross-functional "design build teams" responsible for specific products. The teams used 2200 terminals and the computer-aided three dimensional interactive application (CATIA) system to produce a "paperless" design that allowed engineers to simulate assembly of the 777. The system worked so well that only a nose mockup (to check critical wiring) was built before assembly of the first flight vehicle which was only 0.03 mm out of alignment when the port wing was attached. Boeing also included customers and operators, down to line mechanics, to help tell them how to design the plane. Source: Boeing Web site (http://www.boeing.com/companyoffices/gallery/images/commercial/).

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**Another success ..in failure: Airbus A380**

On Tuesday, Feb. 14, 2006, a resounding crack echoed through the Airbus static test facility in Toulouse, France. The wing of Airbus' mammoth A380 had snapped between the inboard and outboard engines at an applied pressure approximately 1.45 times limit load — about 3.3 percent short of the planned ultimate load. Just before the snap, an extreme wingtip deflection of 24.3 feet had been recorded. Limit load is the highest aerodynamic load expected during an aircraft's lifetime of normal service — often 30 years or more. Ultimate load includes a built-in — and required — safety factor: 1.5 times limit load. Just to be sure. Furthermore, we are talking about the huge, double-decked A380, a brand-new aircraft and the world's largest commercial jetliner, scheduled to enter service at the end of the year with Singapore Airlines. It will seat 555 passengers in three classes — far more in an all-economy configuration. It lists for about $295 million per aircraft, of which 159 have been ordered by 16 airlines around the world. “The failure occurred last Tuesday between 1.45 and 1.5 times the limit load at a point between the inboard and outboard engines,” says Airbus executive vice president engineering Alain Garcia. “This is within 3% of the 1.5 target, which shows the accuracy of the FEM.” But is this sufficient? The European Aviation Safety Agency (EASA) says that the maximum loading conditions are defined in the A380 certification basis. “The aircraft structure is analysed and tested to demonstrate that the structure can withstand the maximum loads, including a factor of safety of 1.5. This process is ongoing and will be completed before type certification.” ABAQUS v6.6 was used to accurately model the failed wing section including every rivet and bolt as well as their interactions. Nonlinear analysis was carried out and finally the problem was fixed and A380 received certification from EASA and FAA on 12 Dec 2006.

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**Drag Force Analysis of Aircraft**

Question What is the drag force distribution on the aircraft? Solve Navier-Stokes Partial Differential Equations. Recent Developments Multigrid Methods for Unstructured Grids

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**Before the 1989 Loma Prieta earthquake**

San Francisco Oakland Bay Bridge Before the 1989 Loma Prieta earthquake

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**San Francisco Oakland Bay Bridge**

After the earthquake

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**A finite element model to analyze the bridge under seismic loads**

San Francisco Oakland Bay Bridge In response to the extensive damage caused by the 1989 Loma Prieta earthquake, the California Department of Transportation (Caltrans) commissioned a seismic-retrofit project for six of the major toll bridges in California, including the famous San Francisco Oakland Bay Bridge. The finite element software company ADINA (located in Watertown, MA) was selected to be used for the seismic analysis of these bridges to determine what retrofitting or new sections of bridges should be constructed. ADINA was used for the linear and highly nonlinear analyses (including large displacements, plasticity, and contact) of the six major toll bridges in California after the 1989 Loma Prieta earthquake. Here, the model of a section of the old Bay Bridge subjected to an earthquake load is shown. Subsequent to this analysis, ADINA was used by the California Department of Transportation and its contractors to design a brand new bridge section spanning from Oakland to Treasure Island. A finite element model to analyze the bridge under seismic loads Courtesy: ADINA R&D

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**Crush Analysis of Ford Windstar**

Question What is the load-deformation relation? Solve Partial Differential Equations of Continuum Mechanics Recent Developments Meshless Methods, Iterative methods, Automatic Error Control Crush analysis of automobiles is the simulation of a slow, dynamic (virtually static) process. In general, only codes based on explicit dynamic analysis procedures have been employed to analyze a crush response. These analyses are computationally expensive, require tuning of the model, and do not accurately represent the actual physics of the crush process. ADINA can be employed to perform crush analyses using the implicit dynamic (practically static) analysis procedures, thereby accurately representing the actual physical process. The solution of the Ford Windstar shown here was solved using ADINA on a Silicon Graphics O2000 machine in about 24 hours of computing time.

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**Engine Thermal Analysis**

Picture from Question What is the temperature distribution in the engine block? Solve Poisson Partial Differential Equation. Recent Developments Fast Integral Equation Solvers, Monte-Carlo Methods

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**Electromagnetic Analysis of Packages**

Thanks to Coventor This is an example of electromagnetic analysis of packages for electronic chips. The problem is that if one of the pins is switching (i.e. conducting) then current may be induced in the neighboring pins due to electromagnetic induction. The purpose of the simulation is to detect such a “glitch” or false signal. Solve Maxwell’s Partial Differential Equations Recent Developments Fast Solvers for Integral Formulations

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**Micromachine Device Performance Analysis**

From Equations Elastomechanics, Electrostatics, Stokes Flow. Recent Developments Fast Integral Equation Solvers, Matrix-Implicit Multi-level Newton Methods for coupled domain problems.

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**Radiation Therapy of Lung Cancer**

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Virtual Surgery

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**Governed by differential equations**

General scenario.. Engineering design Physical Problem Question regarding the problem ...how large are the deformations? ...how much is the heat transfer? Mathematical model Governed by differential equations Assumptions regarding Geometry Kinematics Material law Loading Boundary conditions Etc.

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**Engineering design Example: A bracket Physical problem Questions:**

We consider here a simple example of a bracket supporting a vertical load. We need to choose a mathematical model. The choice of this model clearly depends on what phenomena are to be predicted and on the geometry, material properties, loading and support conditions of the bracket. We notice that The bracket has been fastened to a “very thick steel column”. The term “very thick” is relative to the thickness “t” and height “h” of the bracket. We translate this statement into the assumption that that the bracket is fastened to a (practically) rigid column. We also assume that the load is applied very slowly. The condition of time “very slowly” is relative to the largest natural period of the bracket: i.e., the time span over which the load W is increased from 0 to its full value is much longer than the fundamental period of the bracket. We translate this statement into meaning that we require static analysis (as opposed to a dynamic analysis). Now we are ready to develop a mathematical model of the bracket-depending on the phenomena to be predicted. Let us assume that we want to answer the following questions: What is the bending moment at section AA? What is the deflection at the pin? Questions: What is the bending moment at section AA? What is the deflection at the pin? Finite Element Procedures, K J Bathe

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**Engineering design Example: A bracket Mathematical model 1: beam**

Moment at section AA First we develop a beam mathematical model including shear deformations. We have assumed linear elastic infinitesimal deformation conditions. Hence the load must not be so large as to cause yielding of the material and/or large displacements. Let us now ask whether the mathematical model is reliable and effective. Deflection at load How reliable is this model? How effective is this model?

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**Engineering design Example: A bracket**

Mathematical model 2: plane stress Difficult to solve by hand! To measure the reliability and effectiveness of the simple beam model we consider a slightly more complicated mathematical model. This is a linear elastic two-dimensional plane-stress model. This mathematical model represents the geometry of the bracket more accurately than the beam model and assumes a two-dimensional stress situation. However, note that we have not considered the actual bolt fastening and contact conditions between the steel column and the bracket, nor have we modeled the pin carrying the load onto the bracket.

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**Engineering design ..General scenario.. Physical Problem**

Mathematical model Governed by differential equations The plane stress mathematical model is too complicated to solve by hand. We therefore solve it using a numerical technique- the finite element method. Numerical model e.g., finite element model

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**Engineering design PREPROCESSING 1. Create a geometric model**

..General scenario.. Engineering design Finite element analysis PREPROCESSING 1. Create a geometric model 2. Develop the finite element model How do we perform FEM modeling? We first generate a solid model of the bracket using a commercial package such as Solidworks/ ProE. The next step is to develop the “finite element” model. Solid model Finite element model

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**Engineering design FEM analysis scheme**

..General scenario.. Engineering design Finite element analysis FEM analysis scheme Step 1: Divide the problem domain into non overlapping regions (“elements”) connected to each other through special points (“nodes”) Element Node The FEM analysis scheme can be broken down into 3 steps (we will use these 3 steps throughout the course). Step 1: We divide the solid model up into non-overlapping regions called “elements”. The elements are connected to each other through special points called “nodes”. The solution, in this case the displacement field, is approximated using piece-wise polynomials in each element. This process is called “mesh generation”. This way we discretize a continuous problem, having infinite degrees of freedom to a discrete problem having a finite number of degrees of freedom. Finite element model

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**Engineering design FEM analysis scheme**

..General scenario.. Engineering design Finite element analysis FEM analysis scheme Step 2: Describe the behavior of each element Step 3: Describe the behavior of the entire body by putting together the behavior of each of the elements (this is a process known as “assembly”) Step 2: We then describe the behavior of each element in terms of its nodes Step 3: We then describe the behavior of the entire structure by putting together the behavior of of each of these elements. In this process we are able to compute the displacements at the nodal points. Are we done?

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**Engineering design POSTPROCESSING Compute moment at section AA**

..General scenario.. Engineering design Finite element analysis POSTPROCESSING Compute moment at section AA To compute the moment, we have to do a bit of extra work and this is known as “postprocessing”.

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**Engineering design Preprocessing Analysis Postprocessing**

..General scenario.. Engineering design Finite element analysis Preprocessing Step 1 Analysis Step 2 Step 3 Remember that a complete FEM analysis consists of Preprocessing Analysis Postprocessing The Analysis phase alone has 3 steps (which we will be most interested in the rest of this course) Postprocessing

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**Engineering design Example: A bracket**

Mathematical model 2: plane stress FEM solution to mathematical model 2 (plane stress) Moment at section AA Deflection at load Conclusion: With respect to the questions we posed, the beam model is reliable if the required bending moment is to be predicted within 1% and the deflection is to be predicted within 20%. The beam model is also highly effective since it can be solved easily (by hand). The beam mathematical model completely neglects the stress increase due to the fillets. Hence a plane stress solution including the fillets is necessary. What if we asked: what is the maximum stress in the bracket? would the beam model be of any use?

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Example: A bracket Engineering design Summary The selection of the mathematical model depends on the response to be predicted. The most effective mathematical model is the one that delivers the answers to the questions in reliable manner with least effort. The numerical solution is only as accurate as the mathematical model.

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**Modeling a physical problem**

Example: A bracket ...General scenario Modeling a physical problem Physical Problem Change physical problem Mathematical Model Improve mathematical model Numerical model Does answer make sense? No! Refine analysis YES! Design improvements Structural optimization Happy

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**Verification and validation**

Example: A bracket Verification and validation Modeling a physical problem Physical Problem Validation Mathematical Model Verification Numerical model

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**Critical assessment of the FEM**

Reliability: For a well-posed mathematical problem the numerical technique should always, for a reasonable discretization, give a reasonable solution which must converge to the accurate solution as the discretization is refined. e.g., use of reduced integration in FEM results in an unreliable analysis procedure. Robustness: The performance of the numerical method should not be unduly sensitive to the material data, the boundary conditions, and the loading conditions used. e.g., displacement based formulation for incompressible problems in elasticity Efficiency:

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