Steps in the Finite Element Method. Chung Hua University Department of Mechanical Engineering Dr. Ching I Chen
|
|
- Curtis Virgil Preston
- 5 years ago
- Views:
Transcription
1 Steps in the Finite Element Method Chung Hua University Department of Mechanical Engineering Dr. Ching I Chen
2 General Idea Engineers are interested in evaluating effects such as deformations, stresses, temperature, fluid pressure, and fluid velocities caused by forces, such as applied loads or pressures and thermal and fluid fluxes. The nature of distribution of the effects in a body depends on the characteristics of the force system and of the body itself. Our aim is to find this distribution of the effects.
3 Solid Mechanics: Example 1: Truss deformation under the external forces y x
4 Solid Mechanics: Example 1 : deformation results
5 Solid Mechanics : Example 2 : plate stress distribution under the external forces
6 Solid Mechanics : Example 2 : stress distribution results
7 Heat Transfer : Example 3 : temperature distribution in a short solid cylinder T Top = 40 T wall =T bot = 0
8 Heat Transfer : Example 3 : temperature distribution results
9 Fluid Mechanics : Example 4 : acceleration of a tank fluid a
10 Fluid Mechanics : Example 4 : height distribution
11 Fluid Mechanics : Example 5 : plane poiseuille flow P 1 =0.1 P 2 = 0
12 Fluid Mechanics : Example 5 : velocity distribution
13 Electrical Analysis : Example 6 : electric current flowing in a network
14 Electrical Analysis : Example 6 : results comparison V 1, volts V 2, volts V 3, volts V 4, volts I 2-1, amps I 3-1, amps I 2-3, amps I 4-2, amps I 4-3, amps I 1-4, amps Target ANSYS
15 From previous examples we know the effects may be deformation, pressure, velocity, temperature, voltage, current, etc. depending on the type of the problem. Let s called these effects distribution as a function of u(x,y,z). For 2-D illustration: y Distribution of u u(x,y) (T,p) for entire body 2-D Body to be considered x
16 For convenience, we temporality use displacements or deformations in place of effects. We assume that it is difficult to find the distribution of u(x,y,z) by using conventional methods and decide to use the concept of discretization -- finite element method. We divide the body into a number of smaller regions called finite elements.
17 The consequence of such subdivision is that the distribution of displacements (effects) is also discretized into corresponding subzones. The subdivided elements are now easier to examine as compared to the entire body and distribution of u over it. y u (T,p) Distribution of u e (x,y) over a generic element e Finite element x
18 Steps of Finite Element Method Formulation and application of the finite element method are considered to consist of eight basic steps : Discretize and Select Element Configuration Select Approximation Models or Functions Define Gradient Unknown and Constitution Relations Derive Element Equations Assemble Element to Obtain Global System Solve for the Primary Unknowns Solve for Derived or Secondary Quantities Interpretation of Results
19 Step 1 : Discretize and Select Element Configuration How to discretize the body? Node 1. A point processes coordinate and number to describe its location in the structure body. N3 N2 N4 N1 N1 = (10, 0) N2 = ( 5,10) N3 = ( 0, 5) N4 = ( 8, 2)
20 Totally, there are 71 nodes
21 Totally, there are 723 nodes
22 Node (continued) 2. Having degree of freedom (DOF) to specify the load effects. - A degree of freedom can be defined as an independent (unknown) displacement that can occur at a point. - The solution is solved only for those nodes that we created.
23 Example 1: Truss deformation under the external forces dof = displacements at x, y direction (UX, UY) y UY x 3 UX
24 Example 2 : plate stress distribution under the external forces dof = displacements at x, y direction (UX, UY) UY UX
25 Example 3 : temperature distribution in a short solid cylinder dof = temperature (T) T
26 Example 4 : acceleration of a tank fluid dof = displacements at x, y,z direction (UX, UY, UZ)
27 Example 5 : plane poiseuille flow dof = velocity at x, y direction (VX, VY) P 1 =0.1 P 2 = 0 VY VX
28 Example 6 : electric current flowing in a network dof = voltage (Volt) V
29 Node (continued) 3. Using to create the elements - The creation of element should be based on node to node connection.
30 Element 1. Elements are created by node to node connection. 2. The connection between node to node is straight line. Therefore the actual outside contour is approximation for curve line and surface boundary. How to approach the actual contour?
31 3-D course element to fine element
32 Element (continued) 3. What type of element should be used? Depending on the problem defined, However, geometrically, 1 D element 2 D element 3 D element
33 1 D element Cantilever Beam T1 T2 T T Heat transfer
34 2 D element The shape of 2-D plane element quadrilateral triangle
35 2 D element P P h This is plane stress problem, neglect the thickness (a unit).
36 2 D element t Real Beam 2-D beam 2-D beam finite element
37 2 D element The effects of x-y plane are to be found. Y X This is plane strain problem, neglect z direction length.
38 3 D element The shape of 3-D solid element hexahedron prism tetrahedron
39 3 D element The shape of 3-D solid element tetrahedron hexahedron
40 3 D element The shape of 3-D solid element hexahedron
41 Element Configuration Globally, the configuration (location) of each element is not same y x x element #1 at 0 to 5 element #2 at 5 to 10 element #3 at 10 to 15 element #1 in the range 0 x 5, 0 y 5 element #6 in the range 5 x 10, 6 y 9
42 Element Configuration (continued) In order to generally describe the displacements of each element, the local coordinate system should be used s s s each element has its own coordinate system, called local coordinate system, such that 0 s x global coordinate system
43 Element Configuration (continued) Based on the local coordinate system, the generalization to the governing equation of each element cab be reached. No matter how the element domain is!! 1-D element L 1 L 2 s L 3 s s Each element can be different length. The domain will be 0 s Li
44 Element Configuration (continued) 2-D element t y t s x The local coordinate system can be selected by your convenience. s
45 Step 2 : Select Approximation Models or Functions 1. Choose a pattern or shape for the distribution of the unknown quantity u e in the element. 2. The nodal points of the element provide strategic for writing mathematical functions to describe the shape of the distribution of the unknown quantity over the domain of the element. distribution pattern or shape of u e (s,t) over a generic element e e
46 Step 2 : Select Approximation Models or Functions (continued) 3. A number of mathematical functions such as polynomials and trigonometric series can be used for this purpose, especially polynomials because of the ease and simplification they provide in the finite element formulation. What is f(x) x Interpolation function To find a function f(x) to pass the given points
47 Step 2 : Select Approximation Models or Functions (continued) 4. If an plane element has 4 unknown (u 1, u 2, u 3, u 4 ), the polynomial interpolation function can be expressed as u 3 u 4 e u 2 t u 1 s ust (, ) = N( stu, ) + N( stu, ) N ( s, t) u + N ( s, t) u where u i are the values of the unknowns at the nodal points
48 Step 2 : Select Approximation Models or Functions (continued) 5. Generally, if we denote u as the unknown, the polynomial interpolation function can be expressed as u = Nu + N u + N u + L + N u m m where u 1,u 2,u 3,,u m are the values of the unknowns at the nodal points N 1,N 2,N 3,,N m are the interpolation functions
49 Step 2 : Select Approximation Models or Functions (continued) 6. Remark One should realize that the solution in each step will be in terms of the unknowns only at the nodal points. In the finite element method, one should find the solution as the values of the unknown u at all the nodes. To initiate action toward obtaining the solution, one has assumed a priori or in advance a shape or pattern that one hope will satisfy the conditions, laws, and principles of the problem at hand.
50 Step 3 : Define Gradient - Unknown and Constitution Relations 1. This step depends on the type of the problem. Various disciple processes different unknown attribution. Solid mechanics Fluid mechanics Heat transfer (Thermal)
51 Solid mechanics x direction of displacement du Gradient Unknown ε = dx x strain-displacement σ = E ε x x x Constitution Relations Hook s law stress-strain
52 Fluid mechanics Fluid flow (poiseuille flow) in one dimension φ : fluid head or potential = p/γ + z k x : coefficient of permeability d φ g = Gradient Unknown dx x gradient-fluid head v=-kg x x x Constitution Relations Darcy s law velocity-gradient
53 Heat transfer Heat flow in one dimension k x : thermal conductivity q x : rate of heat flow (Btu/W or W) dt T = Gradient Unknown dx x gradient-temperature q = -kat x x x Constitution Relations Newton s cooling law Rate of heat flow-gradient
54 Step 4 : Derive Element Equations A number of alternatives are available for the derivation of element equations. Two two most commonly used are : The energy methods Principle of stationary potential energies Principle of stationary complementary energies Reissner s mixed principle Hybrid formulations Weighted residual methods Collocation Method Subdomain Method Galerkin s method Least-squares Method
55 The energy methods - These procedures are based on the ideas of finding consistent states of bodies or structures associated values of a scalar quantity assumed by the loaded bodies. In engineering, usually this quantity is a measure of energy or work. - In case of stress-deformation analysis, we define the function F to be the potential energy in a body under load. - The potential energy is defined as the sum of the internal strain energy U and the potential of the external loads W p. Π = U W p
56 Principle of minimum potential energy - Stationary value : the term stationary can imply a maximum, minimum point of a function F(x). F local maximum F(x) local minimum x - to find the point of stationary value, we have df dx = 0
57 Principle of minimum potential energy - The potential energy is function of unknown nodes values Π=Π ( u u Lu ),, n to find the point of stationary value, we have Π Π Π = 0 = 0 L = 0 u1 u2 u n
58 Weighted residual methods - The weighted residual methods are based on assuming an approximate solution for the governing differential equations. - The assumed solution must satisfy the initial and boundary conditions of the given problem. - Because the assumed solution is not exact, substitution of the solution into the differential equations will lead to some residuals or errors. - Each residual method requires the error to vanish over some selected interval or at some points.
59 Weighted residual methods Define 1-D column problem with A 1 = 0.25 in, A 2 = in, L = 10 in, thickness = in, P = 1000 lb, E = lb/in 2 y A 1 L The governing differential equations P du EA( y) = 0 dy u(y) A 2 P Assume an approximate solution uy ( ) = Cy+ Cy + Cy
60 Weighted residual methods (continued) Since A A = + L du dy = Ay ( ) A1 y t 2 ( C1 2C2y 3 C3y ) The residual error function will be 2 Re s/ E= ( y)( C1+ 2Cy Cy 3 )
61 Collocation Method The error function is force to be zero at as many points as there are unknowns coefficients. We select y = L/3, y = 2L/3, y = L C1+ 2C2 + 3C = C1+ 2C2 + 3C = ( ) ( ( )) C C ( ) C ( ) = 0 which leads to uy ( ) = (423.08y y y )
62 Subdomain Method The integral of the error function over some selected subintervals is forced to be be zero. The number of subintervals chosen must equal the number of unknowns coefficients. In this case we assumed three integrals: L /3 0 2 L /3 L /3 L 2 L /3 Res dy = 0 Res dy = 0 Res dy = 0
63 Subdomain Method (continued) After manipulation L /3 0 2 L /3 L /3 L 2 L /3 ( )( ) = y C 2Cy 3Cy dy 0 ( )( ) = y C 2C y 3C y dy 0 ( )( ) = y C 2Cy 3Cy dy 0 which leads to uy ( ) = (391.35y y y )
64 Galerkin Method This Method requires the error to be orthogonal to some weighting functions Φ i, according to the integral of the domain b a Φ i Res dy = 0 i=1,2,3...n The weighting functions are chosen to be members of the approximation solution. uy ( ) = Cy+ Cy + Cy The weighting function Φ ( y) = y Φ ( y) = y Φ ( y) = y
65 0 Galerkin Method (continued) This leads to L y ( y )( C Cy Cy ) dy = L y 2 ( y )( C Cy Cy 2) 6 dy = 0 ( )( ) L y 3 y C C y C y 2 6 dy = 0 which leads to uy ( ) = (400.64y y y )
66 Least-Squares Method This method requires the error to be minimized with respect to the unknowns coefficients in the assumed solution of the domain, according to the relationship minimize ( b 2 ) Res dy a which leads to b a Res Res dy = 0 C i
67 Least-Squares Method (continued) Since, the residual error function is Re s/ E= ( y)( C+ 2Cy+ 3 Cy ) L ( y)( C1 Cy 2 Cy 3 ) ( ) y dy = 0 L 0 0 L ( y)( C ) 1 C2y C3y ( y)( C ) 1 Cy 2 Cy 3 ( y) ydy = ( y) y dy 0 =
68 Least-Squares Method (continued) which leads to uy ( ) = (389.73y y y )
69 The Exact Solution The differential equation is P du A2 A1 EA( y) = 0 and ( ) 1 dy L The exact solution in obtained by integrating A y = A + y t u y Pdy du = 0 0 EA( y) or y Pdy PL A A uy ( ) = = ln A+ y ln A 0 EA( y) Et ( A2 A 1) L
70 Comparison of weighted residual solution Location along the bar Exact method Collect method Subdomain method Galerkin method Least square method y = y = y = y = y =
71 The element equation Use of either of the forgoing methods will leads to equation describing the behavior of an element, which are commonly expressed as [ k]{ u } = { f} [k] = element property matrix, stiffness matrix {u} = vector of unknowns at the element nodes, node displacements {f} = vector of element forcing parameters, nodal forces
72 Step 5 : Assemble Element to Obtain Global System Once the element equations are established for a generic element in step 4, it will recursively add them together to find the global equations. This assembling process is based on the law of compatibility continuity
73 Compatibility : plane problem Slope may not be equal i j Equal displacement The displacements of two adjacent or consecutive points must have identical values
74 Compatibility : bending problem (more severely) Equal slopes or gradients i j Equal displacement Not only the continuity of displacements but also the slopes or the first derivative of displacements are also continuity at adjacent or consecutive points.
75 The assemblage equations are or = [K]{U}={F} [K] = assemblage property matrix {U} = assemblage vector of nodal unknowns {F} = assemblage vector of nodal forcing parameters
76 Boundary conditions The global equations [K] {U} = {F} represent the properties of a body or structure. It tell us about the capabilities of the body to withstand applied forces.
77 How the body or structure will perform its engineering duties will depend on the surrounding and the problems it faces. These aspects can be called constrains or boundary conditions. Boundary conditions are the physical constraints or supports that must exist so that the structure or body can stand in space uniquely.
78 Boundary conditions are categorized as Essential, forced or geometric boundary conditions: This type of condition is commonly specified in terms of known values on the unknowns on a part of the body or structure. Such as the fixed point without slope or displacement in the body. Natural boundary conditions: This type of condition is commonly specified in terms of the first or second derivative of displacement. Such as the free or simply support end of the beam imply the moment is zero.
79 Example fluid head Is specified essential B.C. temperature Is specified cylinder Impervious to fluid fluid flux is proportion to the first derivative of fluid head natural B.C. heat flux is proportion to the first derivative of temperature Insulated against heat
80 The final assemblage equations are = or [ K ]{ U } = { F }
81 Step 6 : Solve for the Primary Unknowns Once the global system equation is derived, this is a set of linear (or nonlinear) simultaneous algebraic equations, which can be written in a standard familiar form as K U + K U + L+ K U = F n n 1 K U + K U + L+ K U = F n n 2 M M K U + K U + L+ K U = F n1 1 n2 2 nn n n
82 Step 6 : Solve for the Primary Unknowns These equations can be solved using any numerical method. At the end of this step, we have solved for the unknowns (displacements) U 1, U 2, U 3,,U n. These are called primary unknowns because they appear as the first quantities sought in the basic. Also, they are the degree of freedom defined in each node at the beginning - in solid mechanics displacements - in heat transfer temperature - one dimensional flow fluid head
83 Step 7 : Solve for Derived or Secondary Quantities Very often additional or secondary quantities must be computed from the primary quantities Stress-deformation problems - strains, stresses, moments and shear forces Flow problem - velocities, flow flux Thermal problem - heat flux
84 Step 8 : Interpretation of Results The results are usually obtained in the form of printed output from the computer. - Plot the values of displacement and stresses along the domain. - Tabulate the results.
Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I
Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix
More informationInstitute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I
Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix
More informationME FINITE ELEMENT ANALYSIS FORMULAS
ME 2353 - FINITE ELEMENT ANALYSIS FORMULAS UNIT I FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS 01. Global Equation for Force Vector, {F} = [K] {u} {F} = Global Force Vector [K] = Global Stiffness
More information12. Stresses and Strains
12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)
More informationBHAR AT HID AS AN ENGIN E ERI N G C O L L E G E NATTR A MPA LL I
BHAR AT HID AS AN ENGIN E ERI N G C O L L E G E NATTR A MPA LL I 635 8 54. Third Year M E C H A NICAL VI S E M ES TER QUE S T I ON B ANK Subject: ME 6 603 FIN I T E E LE ME N T A N A L YSIS UNI T - I INTRODUCTION
More informationChapter 3 Variational Formulation & the Galerkin Method
Institute of Structural Engineering Page 1 Chapter 3 Variational Formulation & the Galerkin Method Institute of Structural Engineering Page 2 Today s Lecture Contents: Introduction Differential formulation
More informationStress analysis of a stepped bar
Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.
More informationBilinear Quadrilateral (Q4): CQUAD4 in GENESIS
Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS The Q4 element has four nodes and eight nodal dof. The shape can be any quadrilateral; we ll concentrate on a rectangle now. The displacement field in terms
More informationDue Monday, September 14 th, 12:00 midnight
Due Monday, September 14 th, 1: midnight This homework is considering the analysis of plane and space (3D) trusses as discussed in class. A list of MatLab programs that were discussed in class is provided
More informationFLEXIBILITY METHOD FOR INDETERMINATE FRAMES
UNIT - I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 1. What is meant by indeterminate structures? Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These
More informationFinite Element Method in Geotechnical Engineering
Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps
More informationChapter 5 Structural Elements: The truss & beam elements
Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations
More informationME 1401 FINITE ELEMENT ANALYSIS UNIT I PART -A. 2. Why polynomial type of interpolation functions is mostly used in FEM?
SHRI ANGALAMMAN COLLEGE OF ENGINEERING AND TECHNOLOGY (An ISO 9001:2008 Certified Institution) SIRUGANOOR, TIRUCHIRAPPALLI 621 105 Department of Mechanical Engineering ME 1401 FINITE ELEMENT ANALYSIS 1.
More informationTruss Structures: The Direct Stiffness Method
. Truss Structures: The Companies, CHAPTER Truss Structures: The Direct Stiffness Method. INTRODUCTION The simple line elements discussed in Chapter introduced the concepts of nodes, nodal displacements,
More informationMethods of Analysis. Force or Flexibility Method
INTRODUCTION: The structural analysis is a mathematical process by which the response of a structure to specified loads is determined. This response is measured by determining the internal forces or stresses
More informationDHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI DEPARTMENT OF MECHANICAL ENGINEERING ME 6603 FINITE ELEMENT ANALYSIS PART A (2 MARKS)
DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI DEPARTMENT OF MECHANICAL ENGINEERING ME 6603 FINITE ELEMENT ANALYSIS UNIT I : FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PART A (2 MARKS) 1. Write the types
More informationDr. D. Dinev, Department of Structural Mechanics, UACEG
Lecture 6 Energy principles Energy methods and variational principles Print version Lecture on Theory of Elasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACEG 6.1 Contents
More informationJEPPIAAR ENGINEERING COLLEGE
JEPPIAAR ENGINEERING COLLEGE Jeppiaar Nagar, Rajiv Gandhi Salai 600 119 DEPARTMENT OFMECHANICAL ENGINEERING QUESTION BANK VI SEMESTER ME6603 FINITE ELEMENT ANALYSIS Regulation 013 SUBJECT YEAR /SEM: III
More informationChapter 12 Plate Bending Elements. Chapter 12 Plate Bending Elements
CIVL 7/8117 Chapter 12 - Plate Bending Elements 1/34 Chapter 12 Plate Bending Elements Learning Objectives To introduce basic concepts of plate bending. To derive a common plate bending element stiffness
More informationUsing MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup,
Introduction to Finite Element Analysis Using MATLAB and Abaqus Amar Khennane Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business
More informationReview of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis
uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods
More informationLecture 7: The Beam Element Equations.
4.1 Beam Stiffness. A Beam: A long slender structural component generally subjected to transverse loading that produces significant bending effects as opposed to twisting or axial effects. MECH 40: Finite
More informationIntroduction to Finite Element Method. Dr. Aamer Haque
Introduction to Finite Element Method 4 th Order Beam Equation Dr. Aamer Haque http://math.iit.edu/~ahaque6 ahaque7@iit.edu Illinois Institute of Technology July 1, 009 Outline Euler-Bernoulli Beams Assumptions
More informationCRITERIA FOR SELECTION OF FEM MODELS.
CRITERIA FOR SELECTION OF FEM MODELS. Prof. P. C.Vasani,Applied Mechanics Department, L. D. College of Engineering,Ahmedabad- 380015 Ph.(079) 7486320 [R] E-mail:pcv-im@eth.net 1. Criteria for Convergence.
More informationBack Matter Index The McGraw Hill Companies, 2004
INDEX A Absolute viscosity, 294 Active zone, 468 Adjoint, 452 Admissible functions, 132 Air, 294 ALGOR, 12 Amplitude, 389, 391 Amplitude ratio, 396 ANSYS, 12 Applications fluid mechanics, 293 326. See
More informationTopic 5: Finite Element Method
Topic 5: Finite Element Method 1 Finite Element Method (1) Main problem of classical variational methods (Ritz method etc.) difficult (op impossible) definition of approximation function ϕ for non-trivial
More informationLevel 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method
9210-203 Level 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method You should have the following for this examination one answer book No additional data is attached
More informationVirtual Work and Variational Principles
Virtual Work and Principles Mathematically, the structural analysis problem is a boundary value problem (BVP). Forces, displacements, stresses, and strains are connected and computed within the framework
More information3D Elasticity Theory
3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.
More informationLecture 8: Assembly of beam elements.
ecture 8: Assembly of beam elements. 4. Example of Assemblage of Beam Stiffness Matrices. Place nodes at the load application points. Assembling the two sets of element equations (note the common elemental
More informationThe Finite Element Method for Mechonics of Solids with ANSYS Applicotions
The Finite Element Method for Mechonics of Solids with ANSYS Applicotions ELLIS H. DILL 0~~F~~~~"P Boca Raton London New Vork CRC Press is an imprint 01 the Taylor & Francis Group, an Informa business
More information4 NON-LINEAR ANALYSIS
4 NON-INEAR ANAYSIS arge displacement elasticity theory, principle of virtual work arge displacement FEA with solid, thin slab, and bar models Virtual work density of internal forces revisited 4-1 SOURCES
More informationInterpolation Functions for General Element Formulation
CHPTER 6 Interpolation Functions 6.1 INTRODUCTION The structural elements introduced in the previous chapters were formulated on the basis of known principles from elementary strength of materials theory.
More informationIntroduction. Finite and Spectral Element Methods Using MATLAB. Second Edition. C. Pozrikidis. University of Massachusetts Amherst, USA
Introduction to Finite and Spectral Element Methods Using MATLAB Second Edition C. Pozrikidis University of Massachusetts Amherst, USA (g) CRC Press Taylor & Francis Group Boca Raton London New York CRC
More informationStiffness Matrices, Spring and Bar Elements
CHAPTER Stiffness Matrices, Spring and Bar Elements. INTRODUCTION The primary characteristics of a finite element are embodied in the element stiffness matrix. For a structural finite element, the stiffness
More informationEML4507 Finite Element Analysis and Design EXAM 1
2-17-15 Name (underline last name): EML4507 Finite Element Analysis and Design EXAM 1 In this exam you may not use any materials except a pencil or a pen, an 8.5x11 formula sheet, and a calculator. Whenever
More informationStructural Analysis of Truss Structures using Stiffness Matrix. Dr. Nasrellah Hassan Ahmed
Structural Analysis of Truss Structures using Stiffness Matrix Dr. Nasrellah Hassan Ahmed FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS In general, there are three types of relationships: Equilibrium
More informationCIVL 7/8117 Chapter 4 - Development of Beam Equations - Part 2 1/34. Chapter 4b Development of Beam Equations. Learning Objectives
CIV 7/87 Chapter 4 - Development of Beam Equations - Part /4 Chapter 4b Development of Beam Equations earning Objectives To introduce the work-equivalence method for replacing distributed loading by a
More informationPLAXIS. Scientific Manual
PLAXIS Scientific Manual 2016 Build 8122 TABLE OF CONTENTS TABLE OF CONTENTS 1 Introduction 5 2 Deformation theory 7 2.1 Basic equations of continuum deformation 7 2.2 Finite element discretisation 8 2.3
More informationCIV-E1060 Engineering Computation and Simulation Examination, December 12, 2017 / Niiranen
CIV-E16 Engineering Computation and Simulation Examination, December 12, 217 / Niiranen This examination consists of 3 problems rated by the standard scale 1...6. Problem 1 Let us consider a long and tall
More informationPrepared by M. GUNASHANKAR AP/MECH DEPARTMENT OF MECHANICAL ENGINEERING
CHETTINAD COLLEGE OF ENGINEERING AND TECHNOLOGY-KARUR FINITE ELEMENT ANALYSIS 2 MARKS QUESTIONS WITH ANSWER Prepared by M. GUNASHANKAR AP/MECH DEPARTMENT OF MECHANICAL ENGINEERING FINITE ELEMENT ANALYSIS
More informationENGN2340 Final Project: Implementation of a Euler-Bernuolli Beam Element Michael Monn
ENGN234 Final Project: Implementation of a Euler-Bernuolli Beam Element Michael Monn 12/11/13 Problem Definition and Shape Functions Although there exist many analytical solutions to the Euler-Bernuolli
More informationDevelopment of Truss Equations
CIVL 7/87 Chapter 3 - Truss Equations - Part /53 Chapter 3a Development of Truss Equations Learning Objectives To derive the stiffness matri for a bar element. To illustrate how to solve a bar assemblage
More informationLecture 21: Isoparametric Formulation of Plane Elements.
6.6. Rectangular Plane Stress/Strain Element. he CS element uses a triangular shape. he 3 nodes of the CS element allow us to employ linear multivariate approximations to the u and v displacements. he
More informationCIVL4332 L1 Introduction to Finite Element Method
CIVL L Introduction to Finite Element Method CIVL L Introduction to Finite Element Method by Joe Gattas, Faris Albermani Introduction The FEM is a numerical technique for solving physical problems such
More informationApplications in Solid Mechanics
Companies, 4 CHAPTER 9 Applications in Solid 9. INTRODUCTION The bar and beam elements discussed in Chapters 4 are line elements, as only a single coordinate axis is required to define the element reference
More informationThe Finite Element Method for Solid and Structural Mechanics
The Finite Element Method for Solid and Structural Mechanics Sixth edition O.C. Zienkiewicz, CBE, FRS UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in
More informationCHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES
CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES 14.1 GENERAL REMARKS In structures where dominant loading is usually static, the most common cause of the collapse is a buckling failure. Buckling may
More information3. Numerical integration
3. Numerical integration... 3. One-dimensional quadratures... 3. Two- and three-dimensional quadratures... 3.3 Exact Integrals for Straight Sided Triangles... 5 3.4 Reduced and Selected Integration...
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationAdvanced Vibrations. Distributed-Parameter Systems: Approximate Methods Lecture 20. By: H. Ahmadian
Advanced Vibrations Distributed-Parameter Systems: Approximate Methods Lecture 20 By: H. Ahmadian ahmadian@iust.ac.ir Distributed-Parameter Systems: Approximate Methods Rayleigh's Principle The Rayleigh-Ritz
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 06
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 06 In the last lecture, we have seen a boundary value problem, using the formal
More informationFinite element modelling of structural mechanics problems
1 Finite element modelling of structural mechanics problems Kjell Magne Mathisen Department of Structural Engineering Norwegian University of Science and Technology Lecture 10: Geilo Winter School - January,
More informationNonconservative Loading: Overview
35 Nonconservative Loading: Overview 35 Chapter 35: NONCONSERVATIVE LOADING: OVERVIEW TABLE OF CONTENTS Page 35. Introduction..................... 35 3 35.2 Sources...................... 35 3 35.3 Three
More informationFinite Element Method
Finite Element Method Finite Element Method (ENGC 6321) Syllabus Objectives Understand the basic theory of the FEM Know the behaviour and usage of each type of elements covered in this course one dimensional
More informationCOPYRIGHTED MATERIAL. Index
Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,
More information14. *14.8 CASTIGLIANO S THEOREM
*14.8 CASTIGLIANO S THEOREM Consider a body of arbitrary shape subjected to a series of n forces P 1, P 2, P n. Since external work done by forces is equal to internal strain energy stored in body, by
More informationThe Kinematic Equations
The Kinematic Equations David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 0139 September 19, 000 Introduction The kinematic or strain-displacement
More information7. Hierarchical modeling examples
7. Hierarchical modeling examples The objective of this chapter is to apply the hierarchical modeling approach discussed in Chapter 1 to three selected problems using the mathematical models studied in
More informationChapter 2: Deflections of Structures
Chapter 2: Deflections of Structures Fig. 4.1. (Fig. 2.1.) ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 1 (2.1) (4.1) (2.2) Fig.4.2 Fig.2.2 ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 2
More informationIntroduction to Finite Element Method
Introduction to Finite Element Method Dr. Rakesh K Kapania Aerospace and Ocean Engineering Department Virginia Polytechnic Institute and State University, Blacksburg, VA AOE 524, Vehicle Structures Summer,
More informationQuintic beam closed form matrices (revised 2/21, 2/23/12) General elastic beam with an elastic foundation
General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation
More informationAdvanced Vibrations. Distributed-Parameter Systems: Exact Solutions (Lecture 10) By: H. Ahmadian
Advanced Vibrations Distributed-Parameter Systems: Exact Solutions (Lecture 10) By: H. Ahmadian ahmadian@iust.ac.ir Distributed-Parameter Systems: Exact Solutions Relation between Discrete and Distributed
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras (Refer Slide Time: 00:25) Module - 01 Lecture - 13 In the last class, we have seen how
More informationLecture 27: Structural Dynamics - Beams.
Chapter #16: Structural Dynamics and Time Dependent Heat Transfer. Lectures #1-6 have discussed only steady systems. There has been no time dependence in any problems. We will investigate beam dynamics
More informationChapter 1 General Introduction Instructor: Dr. Mürüde Çelikağ Office : CE Building Room CE230 and GE241
CIVL222 STRENGTH OF MATERIALS Chapter 1 General Introduction Instructor: Dr. Mürüde Çelikağ Office : CE Building Room CE230 and GE241 E-mail : murude.celikag@emu.edu.tr 1. INTRODUCTION There are three
More informationMEC-E8001 FINITE ELEMENT ANALYSIS
MEC-E800 FINIE EEMEN ANAYSIS 07 - WHY FINIE EEMENS AND IS HEORY? Design of machines and structures: Solution to stress or displacement by analytical method is often impossible due to complex geometry,
More informationAircraft Structures Kirchhoff-Love Plates
University of Liège erospace & Mechanical Engineering ircraft Structures Kirchhoff-Love Plates Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/ Chemin
More informationChapter 6 2D Elements Plate Elements
Institute of Structural Engineering Page 1 Chapter 6 2D Elements Plate Elements Method of Finite Elements I Institute of Structural Engineering Page 2 Continuum Elements Plane Stress Plane Strain Toda
More informationChapter 4 Deflection and Stiffness
Chapter 4 Deflection and Stiffness Asst. Prof. Dr. Supakit Rooppakhun Chapter Outline Deflection and Stiffness 4-1 Spring Rates 4-2 Tension, Compression, and Torsion 4-3 Deflection Due to Bending 4-4 Beam
More informationCIVL 8/7117 Chapter 12 - Structural Dynamics 1/75. To discuss the dynamics of a single-degree-of freedom springmass
CIV 8/77 Chapter - /75 Introduction To discuss the dynamics of a single-degree-of freedom springmass system. To derive the finite element equations for the time-dependent stress analysis of the one-dimensional
More informationD : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each.
GTE 2016 Q. 1 Q. 9 carry one mark each. D : SOLID MECHNICS Q.1 single degree of freedom vibrating system has mass of 5 kg, stiffness of 500 N/m and damping coefficient of 100 N-s/m. To make the system
More informationPart D: Frames and Plates
Part D: Frames and Plates Plane Frames and Thin Plates A Beam with General Boundary Conditions The Stiffness Method Thin Plates Initial Imperfections The Ritz and Finite Element Approaches A Beam with
More informationCode No: RT41033 R13 Set No. 1 IV B.Tech I Semester Regular Examinations, November - 2016 FINITE ELEMENT METHODS (Common to Mechanical Engineering, Aeronautical Engineering and Automobile Engineering)
More informationGeneral elastic beam with an elastic foundation
General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation
More informationMulti-Point Constraints
Multi-Point Constraints Multi-Point Constraints Multi-Point Constraints Single point constraint examples Multi-Point constraint examples linear, homogeneous linear, non-homogeneous linear, homogeneous
More informationProcedure for Performing Stress Analysis by Means of Finite Element Method (FEM)
Procedure for Performing Stress Analysis by Means of Finite Element Method (FEM) Colaboração dos engºs Patrício e Ediberto da Petrobras 1. Objective This Technical Specification sets forth the minimum
More informationComputational Stiffness Method
Computational Stiffness Method Hand calculations are central in the classical stiffness method. In that approach, the stiffness matrix is established column-by-column by setting the degrees of freedom
More informationDiscrete Analysis for Plate Bending Problems by Using Hybrid-type Penalty Method
131 Bulletin of Research Center for Computing and Multimedia Studies, Hosei University, 21 (2008) Published online (http://hdl.handle.net/10114/1532) Discrete Analysis for Plate Bending Problems by Using
More informationEsben Byskov. Elementary Continuum. Mechanics for Everyone. With Applications to Structural Mechanics. Springer
Esben Byskov Elementary Continuum Mechanics for Everyone With Applications to Structural Mechanics Springer Contents Preface v Contents ix Introduction What Is Continuum Mechanics? "I Need Continuum Mechanics
More informationChapter 1: The Finite Element Method
Chapter 1: The Finite Element Method Michael Hanke Read: Strang, p 428 436 A Model Problem Mathematical Models, Analysis and Simulation, Part Applications: u = fx), < x < 1 u) = u1) = D) axial deformation
More informationTable of Contents. Preface...xvii. Part 1. Level
Preface...xvii Part 1. Level 1... 1 Chapter 1. The Basics of Linear Elastic Behavior... 3 1.1. Cohesion forces... 4 1.2. The notion of stress... 6 1.2.1. Definition... 6 1.2.2. Graphical representation...
More informationME 475 Modal Analysis of a Tapered Beam
ME 475 Modal Analysis of a Tapered Beam Objectives: 1. To find the natural frequencies and mode shapes of a tapered beam using FEA.. To compare the FE solution to analytical solutions of the vibratory
More information4. Mathematical models used in engineering structural analysis
4. Mathematical models used in engineering structural analysis In this chapter we pursue a formidable task to present the most important mathematical models in structural mechanics. In order to best situate
More informationLinear Elasticity ( ) Objectives. Equipment. Introduction. ε is then
Linear Elasticity Objectives In this lab you will measure the Young s Modulus of a steel wire. In the process, you will gain an understanding of the concepts of stress and strain. Equipment Young s Modulus
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More information2 marks Questions and Answers
1. Define the term strain energy. A: Strain Energy of the elastic body is defined as the internal work done by the external load in deforming or straining the body. 2. Define the terms: Resilience and
More informationUNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich
UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST
More informationBending Load & Calibration Module
Bending Load & Calibration Module Objectives After completing this module, students shall be able to: 1) Conduct laboratory work to validate beam bending stress equations. 2) Develop an understanding of
More informationName (Print) ME Mechanics of Materials Exam # 1 Date: October 5, 2016 Time: 8:00 10:00 PM
Name (Print) (Last) (First) Instructions: ME 323 - Mechanics of Materials Exam # 1 Date: October 5, 2016 Time: 8:00 10:00 PM Circle your lecturer s name and your class meeting time. Gonzalez Krousgrill
More informationPIEZOELECTRIC TECHNOLOGY PRIMER
PIEZOELECTRIC TECHNOLOGY PRIMER James R. Phillips Sr. Member of Technical Staff CTS Wireless Components 4800 Alameda Blvd. N.E. Albuquerque, New Mexico 87113 Piezoelectricity The piezoelectric effect is
More informationUNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES
UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES A Thesis by WOORAM KIM Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the
More information2D Kirchhoff Thin Beam Elements
2D Kirchhoff Thin Beam Elements General Element Name Y,v X,u BM3 2 3 1 Element Group Element Subgroup Element Description Number Of Nodes 3 Freedoms Node Coordinates Geometric Properties Kirchhoff Parabolically
More informationPLAT DAN CANGKANG (TKS 4219)
PLAT DAN CANGKANG (TKS 4219) SESI I: PLATES Dr.Eng. Achfas Zacoeb Dept. of Civil Engineering Brawijaya University INTRODUCTION Plates are straight, plane, two-dimensional structural components of which
More informationFINITE-VOLUME SOLUTION OF DIFFUSION EQUATION AND APPLICATION TO MODEL PROBLEMS
IJRET: International Journal of Research in Engineering and Technology eissn: 39-63 pissn: 3-738 FINITE-VOLUME SOLUTION OF DIFFUSION EQUATION AND APPLICATION TO MODEL PROBLEMS Asish Mitra Reviewer: Heat
More informationContents as of 12/8/2017. Preface. 1. Overview...1
Contents as of 12/8/2017 Preface 1. Overview...1 1.1 Introduction...1 1.2 Finite element data...1 1.3 Matrix notation...3 1.4 Matrix partitions...8 1.5 Special finite element matrix notations...9 1.6 Finite
More informationFinite-Elements Method 2
Finite-Elements Method 2 January 29, 2014 2 From Applied Numerical Analysis Gerald-Wheatley (2004), Chapter 9. Finite-Elements Method 3 Introduction Finite-element methods (FEM) are based on some mathematical
More informationPh.D. Preliminary Examination Analysis
UNIVERSITY OF CALIFORNIA, BERKELEY Spring Semester 2014 Dept. of Civil and Environmental Engineering Structural Engineering, Mechanics and Materials Name:......................................... Ph.D.
More information2 Introduction to mechanics
21 Motivation Thermodynamic bodies are being characterized by two competing opposite phenomena, energy and entropy which some researchers in thermodynamics would classify as cause and chance or determinancy
More informationMITOCW MITRES2_002S10linear_lec07_300k-mp4
MITOCW MITRES2_002S10linear_lec07_300k-mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources
More information