ME8692 Finite Element Analysis PDF

Preview

Summary

Download ME8692 Finite Element Analysis PDF

Description

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

STUDY MA MATERIAL TERIAL FINITE ELEMENT ANALYSIS DEPARTMENT OF MECH

JUNE – 2010

R S

Vel Tech Vel Tech Multi Tech Dr.Rangarajan Dr.Sakunthala Engineering College Vel Tech High Tech Dr. Rangarajan Dr.Sakunthala Engineering College

SEM - VII

VEL TECH

VEL TECH MULTI TECH

1

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

INDEX

UNITS

PAGE NO.

I.

Introduction

06

II.

One Dimensional Problems

46

III. Two Dimensional Continuum

101

IV.

Axisymmetric Continuum

15 150 0

V.

Isoparametric Elements For Two Dimensional Continuum 19 190 0

VEL TECH

VEL TECH MULTI TECH

2

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

# 42 & 60, Avadi – Veltech Road, Avadi, Chennai – 62. Phone : 044 26840603 26841601 26840766

email : [email protected] website : www.vel-tech.org www.veltechuniv.edu.in

R S

            

Student Strength of Vel Tech increased from 413 to 10579, between 1997 and 2010. Our heartfelt gratitude to AICTE for sanctioning highest number of seats and highest number of courses for the academic year 2009 – 2010 in Tamil Nadu, India. Consistent success on academic performance by achieving 97% - 100% in University examination results during the past 4 academic years. Tie-up with Oracle Corporation for conducting training programmes & qualifying our students for International Certifications. Permission obtained to start Cisco Networking Academy Programmes in our College campus. Satyam Ventures R&D Centre located in Vel Tech Engineering College premises. Signed MOU with FL Smidth for placements, Project and Training. Signed MOU with British Council for Promotion of High Proficiency in Business English, of the University of Cambridge, UK (BEC). Signed MOU with NASSCOM. MOU’s currently in process is with Vijay Electrical and One London University. Signed MOU with INVICTUS TECHNOLOGY for projects & Placements. Signed MOU with SUTHERLAND GLOBAL SERVICES for Training & Placements. Signed MOU with Tmi First for Training & Placements.

VELTECH, VEL TECH MULTI TECH engineering colleges Accredited by TCS VEL TECH, VEL TECH MULTI TECH, VEL TECH HIGH TECH, engineering colleges & VEL SRI RANGA SANKU (ARTS & SCIENCE) Accredited by CTS. Companies Such as TCS, INFOSYS TECHNOLOGIES, IBM, WIPRO TECHNOLOGIES, KEANE SOFTWARE & T INFOTECH, ACCENTURE, HCL TECHNOLOGIES, TCE Consulting Engineers, SIEMENS, BIRLASOFT, MPHASIS(EDS), APOLLO HOSPITALS, CLAYTON, ASHOK LEYLAND, IDEA AE & E, SATYAM VENTURES, UNITED ENGINEERS, ETA-ASCON, CARBORANDUM UNIVERSAL, CIPLA, FUTURE GROUP, DELPHI-TVS DIESEL SYSTEMS, ICICI PRULIFE, ICICI LOMBARD, HWASHIN, HYUNDAI, TATA CHEMICAL LTD, RECKITT BENKIZER, MURUGAPPA GROUP, POLARIS, FOXCONN, LIONBRIDGE, USHA FIRE SAFETY, MALCO, YOUTELECOM, HONEYWELL, MANDOBRAKES, DEXTERITY, HEXAWARE, TEMENOS, RBS, NAVIA MARKETS, EUREKHA FORBES, RELIANCE INFOCOMM, NUMERIC POWER SYSTEMS, ORCHID CHEMICALS, JEEVAN DIESEL, AMALGAMATION CLUTCH VALEO, SAINT GOBAIN, SONA GROUP, NOKIA, NICHOLAS PHARIMAL, SKH METALS, ASIA MOTOR WORKS, PEROT, BRITANNIA, YOKAGAWA FED BY, JEEVAN DIESEL visit our campus annually to recruit our final year Engineering, Diploma, Medical and Management Student s.

VEL TECH

VEL TECH MULTI TECH

3

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

Preface to the First Edition

This edition is a sincere and co-ordinated effort which we hope has made a great difference in the quality of the material. ‚Giving the best to the students, making optimum use of available technical facilities & intellectual strength‛ has always been the motto of our institutions. In this edition the best staff across the group of colleges has been chosen to develop specific units. Hence the material, as a whole is the merge of the intellectual capacities of our faculties across the group of Institutions. 45 to 60, two mark questions and 15 to 20, sixteen mark questions for each unit are available in this material.

Prepared By :

Mr. V. Muthuraman. Mr. M.K. Jawahar. Asst. Professor.

VEL TECH

VEL TECH MULTI TECH

4

VEL TECH HIGH TECH

VEL TECH MH1003

VEL TECH MULTI TECH

VEL TECH HIGH TECH

FINITE ELEMENT ANALYSIS

(Common to Mechanical, Automobile, Mechatronics (Elective) and Metallurgical Engineering (Elective)) 1.

INTRODUCTION

9

Historical background – Matrix approach – Application to the continuum – Discretisation – Matrix algebra – Gaussian elimination – Governing equations for continuum – Classical Techniques in FEM – Weighted residual method – Ritz method 2.

ONE DIMENSIONAL PROBLEMS

9

Finite element modeling – Coordinates and shape functions- Potential energy approach – Galarkin approach – Assembly of stiffness matrix and load vector – Finite element equations – Quadratic shape functions – Applications to plane trusses 3.

TWO DIMENSIONAL CONTINUUM

9

Introduction – Finite element modelling – Scalar valued problem – Poisson equation –Laplace equation – Triangular elements – Element stiffness matrix – Force vector – Galarkin approach - Stress calculation – Temperature effects 4.

AXISYMMETRIC CONTINUUM

9

Axisymmetric formulation – Element stiffness matrix and force vector – Galarkin approach – Body forces and temperature effects – Stress calculations – Boundary conditions – Applications to cylinders under internal or external pressures – Rotating discs 5.

ISOPARAMETRIC ELEMENTS FOR TWO DIMENSIONAL CONTINUUM

9

The four node quadrilateral – Shape functions – Element stiffness matrix and force vector – Numerical integration Stiffness integration – Stress calculations – Four node quadrilateral for axisymmetric problems. TEXT BOOKS 1. 2.

Chandrupatla T.R., and Belegundu A.D., ‚Introduction to Finite Elements in Engineering‛, Pearson Education 2002, 3rd Edition. David V Hutton ‚Fundamentals of Finite Element Analysis‛2004. McGraw-Hill Int. Ed.

REFERENCES 1. 2. 3. 4. 5.

Rao S.S., ‚The Finite Element Method in Engineering‛, Pergammon Press, 1989 Logan D.L., ‚A First course in the Finite Element Method‛, Third Edition, Thomson Learning, 2002. Robert D.Cook., David.S, Malkucs Michael E Plesha, ‚Concepts and Applications of Finite Element Analysis‛ 4 Ed. Wiley, 2003. Reddy J.N., ‚An Introduction to Finite Element Method‛, McGraw-Hill International Student Edition, 1985 O.C.Zienkiewicz and R.L.Taylor, ‚The Finite Element Methods, Vol.1‛, ‚The basic formulation and linear problems, Vol.1‛, Butterworth Heineman, 5th Edition, 2000.

VEL TECH

VEL TECH MULTI TECH

5

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

UNIT – I PART – A

1. Define the term finite element. A complex region defining a continuum is discredited into simple geometric shapes called finite elements. 2. Why study FEA? a) To design products that is safe & cost effective. b) To analyze cause of failure in engineering structures 3. Why FEA is so important? FEA is numerical method, which can be used to find location and magnitude of critical stress and reflection in a structure. FEA method can be applied to structure that have no theoretical solution available, and without FEA we will have to use experimental techniques, which can be consuming and expensive.

F

Solid plate –theoretical Solution is possible

f

Plates with notes-No theoretical solution Available.

4. Define nodes: The finite element procedure reduces such unknown to a finite number by dividing the solution region into small parts called elements and by expressing the unknown field variables interms of assumed approximating functions within each element. The approximating functions are defined interms of field variables of specified points called nodes or nodal points.

VEL TECH

VEL TECH MULTI TECH

6

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

5. List out the various steps involved in finite element analysis. i) ii) iii) iv) v) vi) vii) viii)

Select suitable field variables and the elements. Discritise the continuum. Select interpolation functions. Find the element properties. Assemble elements properties to get global properties. Impose the bounding condition. Solve the system equations to get the nodal unknowns. Make the additional calculations to get the required values.

6. Define Matrix: A rectangular array of numbers with a definite number of rows and columns is a matrix.

 a11 a12 ...a1n    a a ... a2n  e.g :  A    21 22 ...................     am1 am2 ... amn  7. How does FEA work? In FEA, an engineering structure is divided into smaller regions, which have simpler geometry and theoretical solution. Collectively the regions represent the entire structure, and the individual element contributes to the solution of the structure. Challenge lies in representing the exact geometry of the structure. Especially, the sharp curves. Generally, a multi-degree polynomial is approximated by a high Number of straight edges. 8. Explain the term transposition in matrix. If A = [aij], then the transpose of A, denoted as AT, is given by AT = [aji]. Thus the rows of a are the columns of AT.

1  5  0 6   then A T  1 0  2 4  e.g : A   5 6 3 2  2 3       4 2

VEL TECH

VEL TECH MULTI TECH

7

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

9. Write the global matrix equation. { F} = [K] {u} Where, {F} = External force matrix. [K] = Global stiffness matrix. {u} = Displacement matrix. 10. List out the major steps used in FEA. i) ii) iii)

Preprocessing or modeling the structure. Analysis Post processing.

11. Briefly explain the term Discretization:Discretization is the process of dividing an engineering structure into small elements. In FEA, Discretization of a structural model is another name for mesh generation. 12. List out the basic elements used in FEA. i)

Line elements: Elements consisting of two nodes.

In computers, a line, connecting two nodes at its ends as shown, represents a line element. The cross, sectional area is assumed constant throughout the elements. e.g: Truss and beam elements.

Line elements ii)

2-D solid elements: Elements that have geometry similar to a flat plate.

2-D solid elements are plane elements, with constant thickness, and have either a triangular or quadrilateral shape, with 3 nodes or 4 nodes. e.g: plane stress, plain strain, plates shells and axisymmetric elements.

VEL TECH

VEL TECH MULTI TECH

8

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

2D solid: Triangular iii)

VEL TECH HIGH TECH

2-D solid: Quadrilateral.

3-D solid elements:

elements that have a 3-D geometry. The basic 3-D solid elements have either a tetrahedral (4 focus) or hexahedral (6 faces) shape. Tetrahedral - 4 nodes.

Hexahedral – 8 nodes.

13. Give the relation ship between matrix & Algebra, algebraic equation:

a11 x1  a12 x2  a13 x3  b1 a21x 1  a 22x 2  a 23x 3  b 2 a31x 1  a 32x 2  a 33x 3  b 3 Matrix form:

a11 a12 a13  x1  b1       a21 a22 a23  x2   b2     a a a    31 32 33  x3  b3 

(or)

[A] {x} = {b}. 14. Write a short note on RA Z-method. (or) Raleigh Ri+z method. The Rayleigh – Ri+z method of expressing field variables by approximate method clubbed with minimization of potential energy has made a big break through in finite element analysis. The Rayleigh – Ri+z method involves the construction of an assumed displacement field,

u   ai i (x,y,z)

i  1 tol

v   a j  j (x,y,z) j  l  1 to m. w   ak k (x,y,z) k  m  1 to n n  m  l. VEL TECH

VEL TECH MULTI TECH

9

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

The functions I are usually taken as polynomials. Displacements u,v,w must be cinematically admissible. That is u,v,w must satisfy specified boundary conditions. Introducing stress-strain and strain – displacement relations, and substituting above equation into the equation 1 T T T v  dv  v u f d v  s u Tds   2 it gives,    (a1,a2 ,...,ar)

 

u i

T

pi

i

Where r = number of independent unknowns. Now, the extremum with respect to ai, (I = 1 to r) yields the set of r equations.  0  ai

i  1,2, ...,r.

from the solutions of r equation, we get these values of all ‘a’. With these values of ai and I satisfying boundary conditions, the displacements are obtained. 15. What is the principally virtual work? A body is in equilibrium if the internal virtual work equals the external virtual work for every kinematically admissible displacement field (,()). 16. What is the principle of minimum potential energy? For conservative systems, of all the kinematically admissible displacements fields, these corresponding to equilibrium extremize the total potential energy. If the extremum condition is a minimum, the equilibrium state is stable. 17. Distinguish between Finite element method is classical methods: i)

ii) iii)

In classifial methods exact equations are formed and exact solutions are obtained where as in finite element analysis exact equations are formed but approximate solutions are obtained . Solutions have been obtained for few standard cases by classical methods, where as solution can be obtained for all problems by finite element analysis. When material property is not isotropic, solutions for the problems become very difficult in classical method. Only few simple cases have been tried successfully by researchers. FEM can handle structures with anisotropic properties also without any difficulty.

VEL TECH

VEL TECH MULTI TECH

10

VEL TECH HIGH TECH

VEL TECH iv) v)

VEL TECH MULTI TECH

VEL TECH HIGH TECH

If structure consists of more than one material, element can be used without any difficulty. Problems with material and geometric non-linearities cannot be handled by classical methods but there is no difficulty in FEM.

18. Distinguish between finite Element method (FEM) vs Finite Difference method (FEM): i)

FDM makes point wise approximation to the governing equations i.e it ensures continuity only at the node points. Continuity along the sides of grid lines are not ensured.

FEM makes piecewise approximation i.e it ensures the continuity at node points as well as along the sides of the element. ii) iii)

FDM needs larger number of nodes to get good results while FEM needs fewer nodes. With FDM fairly complicated problems can be handled where as FEM can handle all complicated problems.

19. Write a short note on plane stress models and plane strain models: Plane stress models: i) ii) iii)

No loading Normal to the plane. No stress Normal to the plane.  z 0, xz  0, yz  0.

iv)

z  0.

v)

If the change in length (T) to the original length is greater, i.e., original length(T) is T  o. small, hence T

Plane strain models. i) ii) iii)

Strain occurs only in the xy – plane. z=0 and shear strains yxz and yyz are also equal to zero.  z 0.

iv)

If the plate with a hole is thick, the change in thickness compared to the original thickness will be small, and therefore, z=0.

VEL TECH

VEL TECH MULTI TECH

11

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

20. Define Aspect ratio. State its significance. Aspect ratio is defined as the ratio of largest to smallest size in an element. Aspect ratio should be as close to unity as possible. For a two dimensional rectangular element, the aspect ratio is conveniently defined as length to breadth ratio. Aspect ratio closer to unity yields better results. 21. What is the post processing in finite element Analysis? This is the last step in a finite element analysis. Results obtained in Analysis after the preprocessing are usually in the form of raw data and difficult to interpret. In post analysis, a CAD program is utilized to manipulate the data for generating deflected shape of the structure, creating stress plots, animation, etc. A graphical representation of the results is very useful in understanding the behaviour of the structure. 22. List out the applications of FEA. FEA can be used in. i) ii) iii) iv) v) vi) vii)

Heat transfer Fluid mechanics (Two dimensional flow). Solid mechanics. Boeing 747 aircraft. Nuclear reaction vessel. Bio-mechanics Reinforced concrete beam.

23. Write some advantages and disadvantages of finite element method. Advantages: i) ii) iii) iv)

VEL TECH

The method can efficiently be applied to cater irregular geometry. It can take care of any type of boundary. Material anisotropy and in homogeneity can be treated without much difficulty. Any type of loading can be handled.

VEL TECH MULTI TECH

12

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

Disadvantages:i) ii) iii)

There are many types of problems where some other method of analysis may probe efficient then the finite element method. Cost involved in the solution of the problem. For vibration and stability problems in many cases the cost of analysis by finite element method may be prohibitive.

24. Use the Gaussian elimination method to solve the simultaneous equations. 2a + b + 2c – 3d = 0 2a – 2b + c – 4d = 5 a +2c – 3d = -4 4a + 4b – 4c + d = -6 Matrix from is

2 1 2  3    2  2 1  4 1 0 2  3    4 4  4 1 

a  0      b  5       c   4  d  6

The upper triangular matrix is

1 0  0  0

0.5 1  3   a 0      1 0.33 0.33  b   1.66     0 1 1.14  c  4.14   0 0 1  d  10.80 

solving the equation, it gives, a = 12, b = -8, c =-8.2, d = 10.2. 25. Define stiffness matrix. The term stiffness matrix originates from structural analysis. The matrix relation between temperature and heat flux is called the stiffness matrix.

VEL TECH

VEL TECH MULTI TECH

13

VEL TECH HIGH TECH

VEL TECH

VEL TECH MULTI TECH

VEL TECH HIGH TECH

PART – B

1.

3 1 4    A=  -1 4 2   -2 2  2

 3 1 4 : 1 0 0  -1 4 2 : 0 1 0  -2 2 -2 : 0 0 1   3 1 4 : 1 0 ...