The document provides an introduction to finite element methods. It discusses how finite element analysis (FEA) is used to solve engineering and science problems involving complex geometries and conditions. FEA works by dividing a body into finite elements and approximating variable fields within each element. This discretization process sets up algebraic equations that approximate the continuous solution. FEA can model problems with complex shapes, loads, materials and include time-dependent effects. It has advantages over closed-form solutions and its accuracy can be improved by refining the mesh. Examples of 1D and 2D elements and approximations are presented.
The document discusses various planar finite elements for structural analysis. It begins by describing the constant strain triangle (CST) element, which assumes constant strain within the element. The document then discusses the linear strain triangle (LST) element and bilinear quadratic (Q4) element, noting issues with modeling bending. An improved bilinear quadratic (Q6) element is presented to better model bending. The document also discusses applying loads via equivalent nodal loads and evaluating stresses in different coordinate systems.
FEM: Introduction and Weighted Residual MethodsMohammad Tawfik
What are weighted residual methods?
How to apply Galerkin Method to the finite element model?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/TopicX+Approximate+Methods+-+Weighted+Residual+Methods
The document discusses the finite element method (FEM) for numerical analysis of structures. It provides the following key points:
1) FEM divides a structure into discrete elements connected at nodes, resulting in a finite number of degrees of freedom and a set of simultaneous algebraic equations to solve.
2) It uses approximate methods like the Rayleigh-Ritz method to obtain solutions for complex geometries and boundary conditions. This involves assuming displacement fields and minimizing the total potential energy.
3) The Galerkin method is presented, which satisfies the governing differential equations in an integral sense by setting the residual equal to zero when multiplied by a weighting function.
4) Applications to 1D problems are discussed,
Finite element method (matlab) milan kumar raiMilan Kumar Rai
The document provides an overview of the finite element method (FEM). It explains that FEM involves dividing a structure into small pieces called finite elements and solving equations over each element. This allows solving complex problems by relating small, simple elements. The document outlines the basic steps of FEM, including discretization, deriving element equations, and provides an example of applying FEM to a 1D bar problem and 2D truss problem.
The document discusses various numerical methods for analyzing mechanical components under applied loads, including the finite element method. It describes weighted residual methods like the Galerkin method and collocation method which approximate solutions by minimizing residuals. The variational or Rayleigh-Ritz method selects displacement fields to minimize total potential energy. The finite element method divides a structure into small elements and applies these methods to obtain approximate solutions for displacements and stresses at discrete points.
The document provides an introduction to the finite element method (FEM). It discusses how FEM can be used to obtain approximate solutions to boundary value problems in engineering. It outlines the general steps involved, including preprocessing (defining the model), solution/processing (computing unknown values), and postprocessing (analyzing results). Examples of FEM applications include structural analysis, fluid flow, heat transfer, and more. The key aspects of FEM include discretizing the domain into simple elements, choosing shape functions to approximate variations within each element, and assembling the element equations into a global system of equations to solve.
This document outlines the contents and concepts of a course on finite element analysis. It covers fundamental concepts like discretization, matrix algebra, and weighted residual methods. It also covers one-dimensional problems involving bars, beams, and trusses. Shape functions, stiffness matrices, and finite element equations are derived for one-dimensional elements. Two-dimensional problems involving plane stress, strain, and heat transfer are also introduced. Numerical integration techniques are discussed. A variety of finite element applications are listed including structural and non-structural problems.
Finite element analysis (FEA) is a numerical technique used to approximate solutions to boundary value problems by dividing the domain into smaller elements. The document discusses the three main stages of FEA: building the model, solving the model, and displaying the results. It provides details on how to create nodes and finite elements to represent an object's geometry, assign material properties and constraints, define the type of analysis, and select parameters to display in the results. Examples of different types of FEA analyses are also listed, such as static, thermal, modal, and buckling analyses.
The document discusses various planar finite elements for structural analysis. It begins by describing the constant strain triangle (CST) element, which assumes constant strain within the element. The document then discusses the linear strain triangle (LST) element and bilinear quadratic (Q4) element, noting issues with modeling bending. An improved bilinear quadratic (Q6) element is presented to better model bending. The document also discusses applying loads via equivalent nodal loads and evaluating stresses in different coordinate systems.
FEM: Introduction and Weighted Residual MethodsMohammad Tawfik
What are weighted residual methods?
How to apply Galerkin Method to the finite element model?
#WikiCourses #Num001
https://wikicourses.wikispaces.com/TopicX+Approximate+Methods+-+Weighted+Residual+Methods
The document discusses the finite element method (FEM) for numerical analysis of structures. It provides the following key points:
1) FEM divides a structure into discrete elements connected at nodes, resulting in a finite number of degrees of freedom and a set of simultaneous algebraic equations to solve.
2) It uses approximate methods like the Rayleigh-Ritz method to obtain solutions for complex geometries and boundary conditions. This involves assuming displacement fields and minimizing the total potential energy.
3) The Galerkin method is presented, which satisfies the governing differential equations in an integral sense by setting the residual equal to zero when multiplied by a weighting function.
4) Applications to 1D problems are discussed,
Finite element method (matlab) milan kumar raiMilan Kumar Rai
The document provides an overview of the finite element method (FEM). It explains that FEM involves dividing a structure into small pieces called finite elements and solving equations over each element. This allows solving complex problems by relating small, simple elements. The document outlines the basic steps of FEM, including discretization, deriving element equations, and provides an example of applying FEM to a 1D bar problem and 2D truss problem.
The document discusses various numerical methods for analyzing mechanical components under applied loads, including the finite element method. It describes weighted residual methods like the Galerkin method and collocation method which approximate solutions by minimizing residuals. The variational or Rayleigh-Ritz method selects displacement fields to minimize total potential energy. The finite element method divides a structure into small elements and applies these methods to obtain approximate solutions for displacements and stresses at discrete points.
The document provides an introduction to the finite element method (FEM). It discusses how FEM can be used to obtain approximate solutions to boundary value problems in engineering. It outlines the general steps involved, including preprocessing (defining the model), solution/processing (computing unknown values), and postprocessing (analyzing results). Examples of FEM applications include structural analysis, fluid flow, heat transfer, and more. The key aspects of FEM include discretizing the domain into simple elements, choosing shape functions to approximate variations within each element, and assembling the element equations into a global system of equations to solve.
This document outlines the contents and concepts of a course on finite element analysis. It covers fundamental concepts like discretization, matrix algebra, and weighted residual methods. It also covers one-dimensional problems involving bars, beams, and trusses. Shape functions, stiffness matrices, and finite element equations are derived for one-dimensional elements. Two-dimensional problems involving plane stress, strain, and heat transfer are also introduced. Numerical integration techniques are discussed. A variety of finite element applications are listed including structural and non-structural problems.
Finite element analysis (FEA) is a numerical technique used to approximate solutions to boundary value problems by dividing the domain into smaller elements. The document discusses the three main stages of FEA: building the model, solving the model, and displaying the results. It provides details on how to create nodes and finite elements to represent an object's geometry, assign material properties and constraints, define the type of analysis, and select parameters to display in the results. Examples of different types of FEA analyses are also listed, such as static, thermal, modal, and buckling analyses.
This document provides an introduction to finite element analysis and stiffness matrices. It discusses modeling a linear spring and elastic bar as finite elements. The key points are:
1. The stiffness matrix contains information about an element's resistance to deformation from applied loads. It relates nodal displacements and forces for the element.
2. A linear spring and elastic bar can each be modeled as a finite element with a 2x2 stiffness matrix. Their matrices are derived from relating nodal displacements to forces based on Hooke's law and the element's geometry.
3. A system of multiple elements is modeled by assembling the individual element stiffness matrices into a global system stiffness matrix, relating total nodal displacements and forces
This document provides an introduction to the course CE 72.52 Advanced Concrete. It discusses the key roles of structural engineers in creating safe built environments. It also outlines some of the main topics that will be covered in the course, including material behavior, section design, member design, ductility, seismic detailing, and prestressed concrete. The document includes several images related to reinforced concrete elements, structural analysis and design processes, and limit state design concepts. It provides an overview of the structural design process from modeling and analysis to detailing and drafting.
This document provides an overview of a course on engineering design and rapid prototyping. It discusses the finite element method (FEM) which will be covered in class. FEM involves cutting a structure into small elements and connecting them at nodes to form algebraic equations that can be solved numerically. This allows for approximate solutions to complex problems. The document outlines the typical FEM procedure of preprocessing, analysis, and postprocessing using software. It also discusses sources of errors in the FEM approach and mistakes users may make.
This document provides an overview of finite element analysis (FEA). It defines FEA as a numerical method for solving governing equations over the domain of a continuous physical system that is discretized into simple shapes. It lists several common structural and non-structural applications of FEA, such as stress analysis, buckling problems, vibration analysis, and heat transfer. Finally, it provides the course outline, textbooks, references, and some common FEA software packages.
This document outlines the agenda for a presentation on finite element analysis. It covers three parts: an introduction and basic concepts, the mathematical formulation, and finite element discretization. In part one, it discusses computational methods, idealization, discretization, and the mechanical approach. Part two covers weighted residual methods, approximating functions, the residual, Galerkin's method, the weak form, and the solution space. Part three discusses finite element discretization, the trial basis, assembling the linear system of equations, and references. The presentation provides an overview of finite element analysis from both physical and mathematical interpretations.
This document provides an overview of a course on the finite element method. The course objectives are for students to learn how to write simple programs to solve problems using FEM. Assessment includes assignments, quizzes, a course project, midterm exam, and final exam. Fundamental agreements include electronic homework submission and using MATLAB or Mathematica. References on FEM are also provided. The document outlines numerical methods for solving boundary value problems and introduces weighted residual methods like the collocation method, subdomain method, and Galerkin method.
The document provides an introduction to the finite element method (FEM). It explains that FEM is a numerical method used to approximate solutions to partial differential equations. It works by dividing a complex problem into smaller, simpler parts called finite elements. This allows for the problem to be solved computationally. The document outlines the basic steps of FEM, including preprocessing (modeling, meshing), solving, and postprocessing (analyzing results). It also discusses applications, history, software, element types, meshing, convergence, and compatibility conditions of FEM.
The document provides an introduction to the finite element method (FEM). It discusses that FEM is a numerical technique used to approximate solutions to boundary value problems defined by partial differential equations. It can handle complex geometries, loadings, and material properties that have no analytical solution. The document outlines the historical development of FEM and describes different numerical methods like the finite difference method, variational method, and weighted residual methods that FEM evolved from. It also discusses key concepts in FEM like discretization into elements, node points, and interpolation functions.
This document discusses the analysis of truss and frame structures using the stiffness method and finite element approach. It provides the derivations of the element stiffness matrices for truss members, beam members, and plane frame members. It expresses the stiffness matrices in both the local and global coordinate systems. The analysis approach can handle arbitrary geometry, loading, material properties and boundary conditions for trusses and frames.
This document discusses finite element analysis for 2D problems. It describes common 2D elements like linear triangles, quadratic triangles, linear quadrilaterals, and quadratic quadrilaterals. It explains how shape functions are used to derive element stiffness matrices and calculate displacements, strains, and stresses within each element from the nodal values. Key elements covered include the constant strain triangle (CST), linear strain triangle (LST), linear quadrilateral (Q4), and quadratic quadrilateral (Q8). The document also discusses calculating von Mises stresses and using contour plots to visualize stress results.
The document provides an introduction to the finite element method (FEM) for numerical analysis of engineering problems. It discusses how FEM can be used to solve complex problems by dividing them into smaller, simpler elements. FEM allows the use of computers to solve problems that cannot be solved through analytical methods. It also describes the different types of FEM formulations including implicit, which tries to obtain structural equilibrium at each time step for faster solutions, and explicit, which does not require iterations but has longer calculation times. Finally, the document gives examples of how FEM can be applied to problems in solid mechanics, fluid mechanics, and thermodynamics.
This document provides an introduction to the theory of plates, which are structural elements that are thin and flat. It defines what is meant by a thin plate and discusses different plate classifications based on thickness. The document derives the basic equations that describe plate behavior by taking advantage of the plate's thin, planar character. It also discusses three-dimensional considerations like stress components, equilibrium, strain and displacement for putting the plate theory into context.
Introduction to finite element analysisTarun Gehlot
The document provides an introduction to finite element analysis (FEA) or the finite element method (FEM). It describes FEA as a numerical method used to solve engineering and mathematical physics problems that cannot be solved through analytical methods due to complex geometries, loadings, or material properties. FEA involves discretizing a complex model into smaller, simpler elements connected at nodes, then applying the governing equations to obtain a numerical solution for the unknown primary variable (usually displacement) at nodes. Secondary variables like stress are then determined from nodal displacements. The process involves preprocessing, solving, and postprocessing steps.
A short introduction presentation about the Basics of Finite Element Analysis. This presentation mainly represents the applications of FEA in the real time world.
The finite element method is used to solve engineering problems involving stress analysis, heat transfer, and other fields. It involves dividing a structure into small pieces called finite elements and deriving the governing equations for each element. The element equations are assembled into a global stiffness matrix and force vector. Boundary conditions are applied and the system is solved for the unknown displacements at nodes. Results like stresses, strains, and temperatures are then determined. Key steps are discretization, deriving the element stiffness matrix, assembling the global matrix, applying boundary conditions, and solving for nodal displacements.
This document discusses finite element analysis and nonlinear finite element analysis. It introduces the finite element method and how it approximates solutions by dividing them into discrete elements. It explains that higher order polynomials and more nodes/elements increase accuracy. The document distinguishes between linear and nonlinear problems, with nonlinear problems not obeying Hooke's law. It lists common finite element methods for nonlinear mechanics like updated Lagrangian formulations. Finally, it discusses solution algorithms for nonlinear problems like Newton's method and line searches.
The document provides an introduction to finite element methods. It discusses how finite element analysis (FEA) is used to solve complex engineering problems that cannot be solved through closed-form analytical methods. The key steps of FEA include discretizing the domain into finite elements, deriving element equations, assembling equations into a global system of equations, and solving the system of equations. Common element types include line, triangular, and brick elements. The accuracy of FEA depends on factors like element size and shape, approximation functions, and numerical integration.
The document provides an introduction to the finite element method (FEM) by comparing it to the finite difference method (FDM) in solving a steady state heat conduction problem. It explains key FEM concepts like weighted residuals, interpolation functions, numerical integration using Gauss quadrature, and applying essential boundary conditions. Examples are presented to illustrate the standard FEM procedure of developing element stiffness matrices, applying nodal connectivity, and assembling the global matrix to obtain a numerical solution.
The document discusses the finite element method (FEM) for analyzing beam structures. FEM involves subdividing a structure into finite elements of simple shape and solving for the whole structure. Elements can be one-, two-, or three-dimensional, with accuracy increasing with more elements. Nodes are points where elements connect, and nodal displacements describe element deformation. FEM allows analyzing complex shapes like plates by treating them as assemblies of beams. A simple bar analysis example demonstrates deriving and solving the stiffness matrix to determine displacements and forces from applied loads.
The document provides an introduction to finite element analysis. It discusses the need for computational methods to solve problems involving complex geometries and boundary conditions that cannot be solved through closed-form analytical methods. The finite element method is introduced as a numerical technique that involves discretizing a continuous domain into discrete subdomains called elements, and approximating variations in dependent variables within each element. This allows setting up algebraic equations that can be solved to approximate the continuous solution. Advantages of the finite element method include its ability to model complex shapes and behaviors, and refine solutions through mesh refinement. Basic concepts such as element types, discretization, and derivation of element equations are described.
This document provides an introduction to finite element analysis and stiffness matrices. It discusses modeling a linear spring and elastic bar as finite elements. The key points are:
1. The stiffness matrix contains information about an element's resistance to deformation from applied loads. It relates nodal displacements and forces for the element.
2. A linear spring and elastic bar can each be modeled as a finite element with a 2x2 stiffness matrix. Their matrices are derived from relating nodal displacements to forces based on Hooke's law and the element's geometry.
3. A system of multiple elements is modeled by assembling the individual element stiffness matrices into a global system stiffness matrix, relating total nodal displacements and forces
This document provides an introduction to the course CE 72.52 Advanced Concrete. It discusses the key roles of structural engineers in creating safe built environments. It also outlines some of the main topics that will be covered in the course, including material behavior, section design, member design, ductility, seismic detailing, and prestressed concrete. The document includes several images related to reinforced concrete elements, structural analysis and design processes, and limit state design concepts. It provides an overview of the structural design process from modeling and analysis to detailing and drafting.
This document provides an overview of a course on engineering design and rapid prototyping. It discusses the finite element method (FEM) which will be covered in class. FEM involves cutting a structure into small elements and connecting them at nodes to form algebraic equations that can be solved numerically. This allows for approximate solutions to complex problems. The document outlines the typical FEM procedure of preprocessing, analysis, and postprocessing using software. It also discusses sources of errors in the FEM approach and mistakes users may make.
This document provides an overview of finite element analysis (FEA). It defines FEA as a numerical method for solving governing equations over the domain of a continuous physical system that is discretized into simple shapes. It lists several common structural and non-structural applications of FEA, such as stress analysis, buckling problems, vibration analysis, and heat transfer. Finally, it provides the course outline, textbooks, references, and some common FEA software packages.
This document outlines the agenda for a presentation on finite element analysis. It covers three parts: an introduction and basic concepts, the mathematical formulation, and finite element discretization. In part one, it discusses computational methods, idealization, discretization, and the mechanical approach. Part two covers weighted residual methods, approximating functions, the residual, Galerkin's method, the weak form, and the solution space. Part three discusses finite element discretization, the trial basis, assembling the linear system of equations, and references. The presentation provides an overview of finite element analysis from both physical and mathematical interpretations.
This document provides an overview of a course on the finite element method. The course objectives are for students to learn how to write simple programs to solve problems using FEM. Assessment includes assignments, quizzes, a course project, midterm exam, and final exam. Fundamental agreements include electronic homework submission and using MATLAB or Mathematica. References on FEM are also provided. The document outlines numerical methods for solving boundary value problems and introduces weighted residual methods like the collocation method, subdomain method, and Galerkin method.
The document provides an introduction to the finite element method (FEM). It explains that FEM is a numerical method used to approximate solutions to partial differential equations. It works by dividing a complex problem into smaller, simpler parts called finite elements. This allows for the problem to be solved computationally. The document outlines the basic steps of FEM, including preprocessing (modeling, meshing), solving, and postprocessing (analyzing results). It also discusses applications, history, software, element types, meshing, convergence, and compatibility conditions of FEM.
The document provides an introduction to the finite element method (FEM). It discusses that FEM is a numerical technique used to approximate solutions to boundary value problems defined by partial differential equations. It can handle complex geometries, loadings, and material properties that have no analytical solution. The document outlines the historical development of FEM and describes different numerical methods like the finite difference method, variational method, and weighted residual methods that FEM evolved from. It also discusses key concepts in FEM like discretization into elements, node points, and interpolation functions.
This document discusses the analysis of truss and frame structures using the stiffness method and finite element approach. It provides the derivations of the element stiffness matrices for truss members, beam members, and plane frame members. It expresses the stiffness matrices in both the local and global coordinate systems. The analysis approach can handle arbitrary geometry, loading, material properties and boundary conditions for trusses and frames.
This document discusses finite element analysis for 2D problems. It describes common 2D elements like linear triangles, quadratic triangles, linear quadrilaterals, and quadratic quadrilaterals. It explains how shape functions are used to derive element stiffness matrices and calculate displacements, strains, and stresses within each element from the nodal values. Key elements covered include the constant strain triangle (CST), linear strain triangle (LST), linear quadrilateral (Q4), and quadratic quadrilateral (Q8). The document also discusses calculating von Mises stresses and using contour plots to visualize stress results.
The document provides an introduction to the finite element method (FEM) for numerical analysis of engineering problems. It discusses how FEM can be used to solve complex problems by dividing them into smaller, simpler elements. FEM allows the use of computers to solve problems that cannot be solved through analytical methods. It also describes the different types of FEM formulations including implicit, which tries to obtain structural equilibrium at each time step for faster solutions, and explicit, which does not require iterations but has longer calculation times. Finally, the document gives examples of how FEM can be applied to problems in solid mechanics, fluid mechanics, and thermodynamics.
This document provides an introduction to the theory of plates, which are structural elements that are thin and flat. It defines what is meant by a thin plate and discusses different plate classifications based on thickness. The document derives the basic equations that describe plate behavior by taking advantage of the plate's thin, planar character. It also discusses three-dimensional considerations like stress components, equilibrium, strain and displacement for putting the plate theory into context.
Introduction to finite element analysisTarun Gehlot
The document provides an introduction to finite element analysis (FEA) or the finite element method (FEM). It describes FEA as a numerical method used to solve engineering and mathematical physics problems that cannot be solved through analytical methods due to complex geometries, loadings, or material properties. FEA involves discretizing a complex model into smaller, simpler elements connected at nodes, then applying the governing equations to obtain a numerical solution for the unknown primary variable (usually displacement) at nodes. Secondary variables like stress are then determined from nodal displacements. The process involves preprocessing, solving, and postprocessing steps.
A short introduction presentation about the Basics of Finite Element Analysis. This presentation mainly represents the applications of FEA in the real time world.
The finite element method is used to solve engineering problems involving stress analysis, heat transfer, and other fields. It involves dividing a structure into small pieces called finite elements and deriving the governing equations for each element. The element equations are assembled into a global stiffness matrix and force vector. Boundary conditions are applied and the system is solved for the unknown displacements at nodes. Results like stresses, strains, and temperatures are then determined. Key steps are discretization, deriving the element stiffness matrix, assembling the global matrix, applying boundary conditions, and solving for nodal displacements.
This document discusses finite element analysis and nonlinear finite element analysis. It introduces the finite element method and how it approximates solutions by dividing them into discrete elements. It explains that higher order polynomials and more nodes/elements increase accuracy. The document distinguishes between linear and nonlinear problems, with nonlinear problems not obeying Hooke's law. It lists common finite element methods for nonlinear mechanics like updated Lagrangian formulations. Finally, it discusses solution algorithms for nonlinear problems like Newton's method and line searches.
The document provides an introduction to finite element methods. It discusses how finite element analysis (FEA) is used to solve complex engineering problems that cannot be solved through closed-form analytical methods. The key steps of FEA include discretizing the domain into finite elements, deriving element equations, assembling equations into a global system of equations, and solving the system of equations. Common element types include line, triangular, and brick elements. The accuracy of FEA depends on factors like element size and shape, approximation functions, and numerical integration.
The document provides an introduction to the finite element method (FEM) by comparing it to the finite difference method (FDM) in solving a steady state heat conduction problem. It explains key FEM concepts like weighted residuals, interpolation functions, numerical integration using Gauss quadrature, and applying essential boundary conditions. Examples are presented to illustrate the standard FEM procedure of developing element stiffness matrices, applying nodal connectivity, and assembling the global matrix to obtain a numerical solution.
The document discusses the finite element method (FEM) for analyzing beam structures. FEM involves subdividing a structure into finite elements of simple shape and solving for the whole structure. Elements can be one-, two-, or three-dimensional, with accuracy increasing with more elements. Nodes are points where elements connect, and nodal displacements describe element deformation. FEM allows analyzing complex shapes like plates by treating them as assemblies of beams. A simple bar analysis example demonstrates deriving and solving the stiffness matrix to determine displacements and forces from applied loads.
The document provides an introduction to finite element analysis. It discusses the need for computational methods to solve problems involving complex geometries and boundary conditions that cannot be solved through closed-form analytical methods. The finite element method is introduced as a numerical technique that involves discretizing a continuous domain into discrete subdomains called elements, and approximating variations in dependent variables within each element. This allows setting up algebraic equations that can be solved to approximate the continuous solution. Advantages of the finite element method include its ability to model complex shapes and behaviors, and refine solutions through mesh refinement. Basic concepts such as element types, discretization, and derivation of element equations are described.
Finite element method have many techniques that are used to design the structural elements like automobiles and building materials as well. we use different design software to get our simulated results at ansys, pro-e and matlab.we use these results for our real value problems.
This document outlines the course objectives and contents for a finite element methods in mechanical design course. The key points are:
1. The course will introduce mathematical modeling concepts and teach how to apply finite element methods (FEM) to solve a range of engineering problems.
2. The content will cover one-dimensional, two-dimensional, and three-dimensional FEM analysis. Solution techniques like inversion methods and dynamic analysis will also be discussed.
3. Applications of FEM include stress analysis, buckling analysis, vibration analysis, heat transfer analysis, and fluid flow analysis for both structural and non-structural problems.
This document provides an introduction to finite element modeling and analysis. It discusses how the finite element method (FEM) can be used to approximate solutions to complex differential equations that describe mechanical systems. The FEM subdivides a system into simple elements that are connected at nodes, allowing the behavior of the overall system to be approximated by the behavior of the discrete elements. The document outlines the basic steps of a finite element analysis and discusses various element types that can be used to model different features.
Topics to be discussed-
Introduction
How Does FEM Works?
Types Of Engineering Analysis
Uses of FEM in different fields
How can the FEM Help the Design Engineer?
How can the FEM Help the Design Organization?
Basic Steps & Phases Involved In FEM
Advantages and disadvantages
The Future Scope
References.
This document outlines the course structure and content for ECE 2408 Theory of Structures V. The course introduces finite element methods for structural analysis. It covers matrix analysis of structures, force and deformation methods, and the use of finite element analysis software. The document compares analytical and finite element analysis methods and explains the key steps in finite element modeling and analysis, including discretization, deriving element stiffness matrices, assembling the global stiffness matrix, applying boundary conditions, and solving for displacements, strains and stresses. The course aims to provide students with skills in using finite element analysis for structural design and problem solving.
Lecture on Introduction to finite element methods & its contentsMesayAlemuTolcha1
The document provides an overview of the Finite Element Method (FEM) course being taught. It discusses:
1. What FEM is and its common application areas like structural analysis, heat transfer, fluid flow.
2. The main steps in FEM including discretization, selecting interpolation functions, developing element matrices, assembling the global matrix, imposing boundary conditions, and solving equations.
3. Different element types like 1D, 2D, and 3D elements and the use of isoparametric formulations.
4. The history of FEM and how it has evolved from being used on mainframe computers to PCs.
The document discusses the finite element method (FEM). FEM is a numerical technique used to find approximate solutions to partial differential equations. It divides a complex problem into small, simpler elements that are solved using relations between each other. There are three phases: pre-processing to mesh the geometry and apply properties/conditions, solution to derive equations and solve for quantities, and post-processing to validate solutions. FEM can model various problem types like static, dynamic, structural, vibrational, and heat transfer analyses. It has advantages like handling complex geometries and loadings but also disadvantages like requiring approximations and computational resources.
Introduction of finite element analysis1ssuser2209b4
The document provides an introduction to the finite element method (FEM). It discusses the different numerical, analytical, and experimental methods used to solve engineering problems. FEM involves discretizing a continuous domain into smaller, finite pieces called elements and using a mathematical representation to approximate the solution. It describes how FEM builds a stiffness matrix and assembles element equations to solve for unknown displacements. The document outlines the general FEM steps and highlights the merits and limitations of FEM, giving examples of its applications in stress analysis of industrial parts, fluid flow analysis, electromagnetic problems, and medical procedures.
The document provides an introduction to finite element analysis (FEA) including:
1) FEA involves discretizing a continuous structure into small substructures called finite elements connected at nodes. This process is called meshing.
2) The finite element method has three main phases - preprocessing (defining geometry, materials, mesh), solution (solving equations to obtain nodal results), and postprocessing (obtaining stresses, strains from nodal results).
3) FEA can model complex problems that cannot be solved analytically due to complicated geometry or materials, and is well-suited for solving problems in solid mechanics, heat transfer, and other fields on computers.
Unit I fdocuments.in_introduction-to-fea-and-applications.pptAdityaChavan99
The document provides an introduction to finite element analysis (FEA) and its applications to engineering problems. It discusses that FEA is a numerical method used to solve problems that cannot be solved analytically due to complex geometry or materials. It involves discretizing a continuous structure into small, well-defined substructures called finite elements. The key steps in FEA include preprocessing such as defining geometry, meshing and material properties, solving the problem, and postprocessing results such as stresses and strains. The document also discusses various element types, assembly of elements, sources of error in FEA, and its advantages such as handling complex geometry, loading and materials.
The document provides an introduction to finite element analysis (FEA) or the finite element method (FEM). It explains that FEA is a numerical method used to solve engineering and mathematical physics problems with complex geometries, loads, and material properties that cannot be solved analytically. FEA discretizes a complex model into smaller, simpler elements and calculates their behavior to obtain approximate solutions. It then describes common FEA applications and outlines the general FEA process of preprocessing, solving, and postprocessing to analyze an object.
A Comprehensive Introduction of the Finite Element Method for Undergraduate C...IJERA Editor
A simple and comprehensive introduction of the Finite Element Method for undergraduate courses is proposed. With very simple mathematics, students can easily understand it. The primary objective is to make students comfortable with the approach and cognizant of its potentials. The technique is based on the general overview of the steps involved in the solution of a typical finite element problem. This is followed by simple examples of mathematical developments which allow developing and demonstrating the major aspects of the finite element approach unencumbered by complicating factors.
The document provides an introduction to finite element analysis (FEA) or the finite element method (FEM). It discusses that FEA is a numerical method used to solve engineering and mathematical physics problems with complex geometries, loadings, and material properties where analytical solutions cannot be obtained. The key steps of FEA include preprocessing such as defining elements and nodes, solving to compute unknown values at nodes, and postprocessing such as plotting and visualization of results. Common applications are structural, fluid, heat transfer, and other analyses. FEA involves discretizing a continuous domain into small, simple elements connected at nodes to approximate the problem.
INTRODUCTION TO FINITE ELEMENT ANALYSISAchyuth Peri
Finite element analysis (FEA) is a numerical technique used to find approximate solutions to partial differential equations. It involves dividing a system into small elements and solving for variables within each element. This allows for analysis of complex geometries, loadings, and materials. The FEM process includes discretizing the system, selecting functions to approximate the solution, assembling element equations into a global system, applying boundary conditions, and calculating displacements, stresses, and strains. FEA offers advantages like analyzing irregular shapes and nonlinear problems, reducing testing costs, and optimizing designs.
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
This document outlines the objectives, course outcomes, and units covered in a finite element analysis course. The objective is to equip students with fundamentals of FEA and introduce the steps involved in discretization, applying boundary conditions, assembling stiffness matrices, and solving problems. The course covers basic FEA concepts, one-dimensional elements, two-dimensional elements, axisymmetric problems, isoparametric elements, and dynamic analysis. Students will learn to formulate and solve structural and heat transfer problems using various finite elements.
The document discusses finite element methods and their applications in microelectromechanical systems (MEMS). It covers the basic formulation of finite element methods, including discretization, selection of displacement functions, derivation of element stiffness matrices, and assembly of global equations. It also discusses specific applications of finite element analysis to problems in MEMS like heat transfer analysis, thermal stress analysis, and static/modal analysis. The finite element method is well-suited for complex geometries and materials and can model irregular shapes, general loads/boundary conditions, and nonlinear behavior.
Applications of artificial Intelligence in Mechanical Engineering.pdfAtif Razi
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1. MCE 565
Wave Motion & Vibration in Continuous Media
Spring 2005
Professor M. H. Sadd
Introduction to Finite Element Methods
2. Need for Computational Methods
• Solutions Using Either Strength of Materials or Theory of
Elasticity Are Normally Accomplished for Regions and
Loadings With Relatively Simple Geometry
• Many Applicaitons Involve Cases with Complex Shape,
Boundary Conditions and Material Behavior
• Therefore a Gap Exists Between What Is Needed in
Applications and What Can Be Solved by Analytical Closed-
form Methods
• This Has Lead to the Development of Several
Numerical/Computational Schemes Including: Finite
Difference, Finite Element and Boundary Element Methods
3. Introduction to Finite Element Analysis
The finite element method is a computational scheme to solve field problems in
engineering and science. The technique has very wide application, and has been used on
problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations,
electrical and magnetic fields, etc. The fundamental concept involves dividing the body
under study into a finite number of pieces (subdomains) called elements (see Figure).
Particular assumptions are then made on the variation of the unknown dependent
variable(s) across each element using so-called interpolation or approximation functions.
This approximated variation is quantified in terms of solution values at special element
locations called nodes. Through this discretization process, the method sets up an
algebraic system of equations for unknown nodal values which approximate the
continuous solution. Because element size, shape and approximating scheme can be
varied to suit the problem, the method can accurately simulate solutions to problems of
complex geometry and loading and thus this technique has become a very useful and
practical tool.
4. Advantages of Finite Element Analysis
- Models Bodies of Complex Shape
- Can Handle General Loading/Boundary Conditions
- Models Bodies Composed of Composite and Multiphase Materials
- Model is Easily Refined for Improved Accuracy by Varying
Element Size and Type (Approximation Scheme)
- Time Dependent and Dynamic Effects Can Be Included
- Can Handle a Variety Nonlinear Effects Including Material
Behavior, Large Deformations, Boundary Conditions, Etc.
5. Basic Concept of the Finite Element Method
Any continuous solution field such as stress, displacement,
temperature, pressure, etc. can be approximated by a
discrete model composed of a set of piecewise continuous
functions defined over a finite number of subdomains.
Exact Analytical Solution
x
T
Approximate Piecewise
Linear Solution
x
T
One-Dimensional Temperature Distribution
7. Discretization Concepts
x
T
Exact Temperature Distribution, T(x)
Finite Element Discretization
Linear Interpolation Model
(Four Elements)
Quadratic Interpolation Model
(Two Elements)
T1
T2
T2
T3 T3
T4 T4
T5
T1
T2
T3
T4 T5
Piecewise Linear Approximation
T
x
T1
T2
T3 T3
T4 T5
T
T1
T2
T3
T4 T5
Piecewise Quadratic Approximation
x
Temperature Continuous but with
Discontinuous Temperature Gradients
Temperature and Temperature Gradients
Continuous
8. Common Types of Elements
One-Dimensional Elements
Line
Rods, Beams, Trusses, Frames
Two-Dimensional Elements
Triangular, Quadrilateral
Plates, Shells, 2-D Continua
Three-Dimensional Elements
Tetrahedral, Rectangular Prism (Brick)
3-D Continua
10. Basic Steps in the Finite Element Method
Time Independent Problems
- Domain Discretization
- Select Element Type (Shape and Approximation)
- Derive Element Equations (Variational and Energy Methods)
- Assemble Element Equations to Form Global System
[K]{U} = {F}
[K] = Stiffness or Property Matrix
{U} = Nodal Displacement Vector
{F} = Nodal Force Vector
- Incorporate Boundary and Initial Conditions
- Solve Assembled System of Equations for Unknown Nodal
Displacements and Secondary Unknowns of Stress and Strain Values
11. Common Sources of Error in FEA
• Domain Approximation
• Element Interpolation/Approximation
• Numerical Integration Errors
(Including Spatial and Time Integration)
• Computer Errors (Round-Off, Etc., )
12. Measures of Accuracy in FEA
Accuracy
Error = |(Exact Solution)-(FEM Solution)|
Convergence
Limit of Error as:
Number of Elements (h-convergence)
or
Approximation Order (p-convergence)
Increases
Ideally, Error 0 as Number of Elements or
Approximation Order
14. One Dimensional Examples
Static Case
1 2
u1 u2
Bar Element
Uniaxial Deformation of Bars
Using Strength of Materials Theory
Beam Element
Deflection of Elastic Beams
Using Euler-Bernouli Theory
1 2
w1
w2
q2
q1
dx
du
a
u
q
cu
au
dx
d
,
:
ion
Specificat
Condtions
Boundary
0
)
(
:
Equation
al
Differenti
)
(
,
,
,
:
ion
Specificat
Condtions
Boundary
)
(
)
(
:
Equation
al
Differenti
2
2
2
2
2
2
2
2
dx
w
d
b
dx
d
dx
w
d
b
dx
dw
w
x
f
dx
w
d
b
dx
d
15. Two Dimensional Examples
u1
u2
1
2
3 u3
v1
v2
v3
1
2
3
f1
f2
f3
Triangular Element
Scalar-Valued, Two-Dimensional
Field Problems
Triangular Element
Vector/Tensor-Valued, Two-
Dimensional Field Problems
y
x n
y
n
x
dn
d
y
x
f
y
x
f
f
f
f
f
f
,
:
ion
Specificat
Condtions
Boundary
)
,
(
:
Equation
ial
Different
Example
2
2
2
2
y
x
y
y
x
x
y
x
n
y
v
C
x
u
C
n
x
v
y
u
C
T
n
x
v
y
u
C
n
y
v
C
x
u
C
T
F
y
v
x
u
y
E
v
F
y
v
x
u
x
E
u
22
12
66
66
12
11
2
2
Conditons
Boundary
0
)
1
(
2
0
)
1
(
2
ents
Displacem
of
Terms
in
Equations
Field
Elasticity
16. Development of Finite Element Equation
• The Finite Element Equation Must Incorporate the Appropriate Physics
of the Problem
• For Problems in Structural Solid Mechanics, the Appropriate Physics
Comes from Either Strength of Materials or Theory of Elasticity
• FEM Equations are Commonly Developed Using Direct, Variational-
Virtual Work or Weighted Residual Methods
Variational-Virtual Work Method
Based on the concept of virtual displacements, leads to relations between internal and
external virtual work and to minimization of system potential energy for equilibrium
Weighted Residual Method
Starting with the governing differential equation, special mathematical operations
develop the “weak form” that can be incorporated into a FEM equation. This
method is particularly suited for problems that have no variational statement. For
stress analysis problems, a Ritz-Galerkin WRM will yield a result identical to that
found by variational methods.
Direct Method
Based on physical reasoning and limited to simple cases, this method is
worth studying because it enhances physical understanding of the process
17. Simple Element Equation Example
Direct Stiffness Derivation
1 2
k
u1 u2
F1 F2
}
{
}
]{
[
rm
Matrix Fo
in
or
2
Node
at
m
Equilibriu
1
Node
at
m
Equilibriu
2
1
2
1
2
1
2
2
1
1
F
u
K
F
F
u
u
k
k
k
k
ku
ku
F
ku
ku
F
Stiffness Matrix Nodal Force Vector
18. Common Approximation Schemes
One-Dimensional Examples
Linear Quadratic Cubic
Polynomial Approximation
Most often polynomials are used to construct approximation
functions for each element. Depending on the order of
approximation, different numbers of element parameters are
needed to construct the appropriate function.
Special Approximation
For some cases (e.g. infinite elements, crack or other singular
elements) the approximation function is chosen to have special
properties as determined from theoretical considerations
19. One-Dimensional Bar Element
udV
f
u
P
u
P
edV j
j
i
i
}
]{
[
:
Law
Strain
-
Stress
}
]{
[
}
{
]
[
)
(
:
Strain
}
{
]
[
)
(
:
ion
Approximat
d
B
d
B
d
N
d
N
E
Ee
dx
d
u
x
dx
d
dx
du
e
u
x
u
k
k
k
k
k
k
L
T
T
j
i
T
L
T
T
fdx
A
P
P
dx
E
A
0
0
]
[
}
{
}
{
}
{
]
[
]
[
}
{ N
δd
δd
d
B
B
δd
L
T
L
T
fdx
A
dx
E
A
0
0
]
[
}
{
}
{
]
[
]
[ N
P
d
B
B
Vector
ent
Displacem
Nodal
}
{
Vector
Loading
]
[
}
{
Matrix
Stiffness
]
[
]
[
]
[
0
0
j
i
L
T
j
i
L
T
u
u
fdx
A
P
P
dx
E
A
K
d
N
F
B
B
}
{
}
]{
[ F
d
K
20. One-Dimensional Bar Element
A = Cross-sectional Area
E = Elastic Modulus
f(x) = Distributed Loading
dV
u
F
dS
u
T
dV
e i
V
i
S
i
n
i
ij
V
ij
t
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces
For One-Dimensional Case
udV
f
u
P
u
P
edV j
j
i
i
(i) (j)
Axial Deformation of an Elastic Bar
Typical Bar Element
dx
du
AE
P i
i
dx
du
AE
P
j
j
i
u j
u
L
x
(Two Degrees of Freedom)
21. Linear Approximation Scheme
Vector
ent
Displacem
Nodal
}
{
Matrix
Function
ion
Approximat
]
[
}
]{
[
1
)
(
)
(
1
2
1
2
1
2
1
2
2
1
1
2
1
1
2
1
2
1
2
1
1
2
1
d
N
d
N
nt
Displaceme
Elastic
e
Approximat
u
u
L
x
L
x
u
u
u
u
x
u
x
u
L
x
u
L
x
x
L
u
u
u
u
L
a
a
u
a
u
x
a
a
u
x (local coordinate system)
(1) (2)
i
u j
u
L
x
(1) (2)
u(x)
x
(1) (2)
1(x) 2(x)
1
k(x) – Lagrange Interpolation Functions
22. Element Equation
Linear Approximation Scheme, Constant Properties
Vector
ent
Displacem
Nodal
}
{
1
1
2
]
[
}
{
1
1
1
1
1
1
1
1
]
[
]
[
]
[
]
[
]
[
2
1
2
1
0
2
1
0
2
1
0
0
u
u
L
Af
P
P
dx
L
x
L
x
Af
P
P
fdx
A
P
P
L
AE
L
L
L
L
L
AE
dx
AE
dx
E
A
K
o
L
o
L
T
L
T
L
T
d
N
F
B
B
B
B
1
1
2
1
1
1
1
}
{
}
]{
[
2
1
2
1 L
Af
P
P
u
u
L
AE o
F
d
K
23. Quadratic Approximation Scheme
}
]{
[
)
(
)
(
)
(
4
2
3
2
1
3
2
1
3
3
2
2
1
1
2
3
2
1
3
2
3
2
1
2
1
1
2
3
2
1
d
N
nt
Displaceme
Elastic
e
Approximat
u
u
u
u
u
x
u
x
u
x
u
L
a
L
a
a
u
L
a
L
a
a
u
a
u
x
a
x
a
a
u
x
(1) (3)
1
u 3
u
(2)
2
u
L
u(x)
x
(1) (3)
(2)
x
(1) (3)
(2)
1
1(x) 3(x)
2(x)
3
2
1
3
2
1
7
8
1
8
16
8
1
8
7
3
F
F
F
u
u
u
L
AE
Equation
Element
25. Simple Example
P
A1,E1,L1 A2,E2,L2
(1) (3)
(2)
1 2
0
0
0
0
0
1
1
0
1
1
1
Element
Equation
Global
)
1
(
2
)
1
(
1
3
2
1
1
1
1
P
P
U
U
U
L
E
A
)
2
(
2
)
2
(
1
3
2
1
2
2
2
0
1
1
0
1
1
0
0
0
0
2
Element
Equation
Global
P
P
U
U
U
L
E
A
3
2
1
)
2
(
2
)
2
(
1
)
1
(
2
)
1
(
1
3
2
1
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
0
0
Equation
System
Global
Assembled
P
P
P
P
P
P
P
U
U
U
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
0
Loading
ed
Distribut
Zero
Take
f
26. Simple Example Continued
P
A1,E1,L1 A2,E2,L2
(1) (3)
(2)
1 2
0
0
Conditions
Boundary
)
2
(
1
)
1
(
2
)
2
(
2
1
P
P
P
P
U
P
P
U
U
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
0
0
0
0
Equation
System
Global
Reduced
)
1
(
1
3
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
P
U
U
L
E
A
L
E
A
L
E
A
L
E
A
L
E
A
0
3
2
2
2
2
2
2
2
2
2
2
2
2
2
1
1
1
L
E
A ,
,
Properties
m
For Unifor
P
U
U
L
AE 0
1
1
1
2
3
2
P
P
AE
PL
U
AE
PL
U
)
1
(
1
3
2 ,
2
,
Solving
27. One-Dimensional Beam Element
Deflection of an Elastic Beam
2
2
4
2
3
1
1
2
1
1
2
2
2
4
2
2
2
3
1
2
2
2
1
2
2
1
,
,
,
,
,
dx
dw
u
w
u
dx
dw
u
w
u
dx
w
d
EI
Q
dx
w
d
EI
dx
d
Q
dx
w
d
EI
Q
dx
w
d
EI
dx
d
Q
q
q
I = Section Moment of Inertia
E = Elastic Modulus
f(x) = Distributed Loading
(1) (2)
Typical Beam Element
1
w
L
2
w
1
q 2
q
1
M 2
M
1
V 2
V
x
Virtual Strain Energy = Virtual Work Done by Surface and Body Forces
wdV
f
w
Q
u
Q
u
Q
u
Q
edV 4
4
3
3
2
2
1
1
L
T
L
dV
f
w
Q
u
Q
u
Q
u
Q
dx
EI
0
4
4
3
3
2
2
1
1
0
]
[
}
{
]
[
]
[ N
d
B
B T
(Four Degrees of Freedom)
28. Beam Approximation Functions
To approximate deflection and slope at each
node requires approximation of the form
3
4
2
3
2
1
)
( x
c
x
c
x
c
c
x
w
Evaluating deflection and slope at each node
allows the determination of ci thus leading to
Functions
ion
Approximat
Cubic
Hermite
the
are
where
,
)
(
)
(
)
(
)
(
)
( 4
4
3
3
2
2
1
1
i
u
x
u
x
u
x
u
x
x
w
f
f
f
f
f
29. Beam Element Equation
L
T
L
dV
f
w
Q
u
Q
u
Q
u
Q
dx
EI
0
4
4
3
3
2
2
1
1
0
]
[
}
{
]
[
]
[ N
d
B
B T
4
3
2
1
}
{
u
u
u
u
d ]
[
]
[
]
[ 4
3
2
1
dx
d
dx
d
dx
d
dx
d
dx
d f
f
f
f
N
B
2
2
2
2
3
0
2
3
3
3
6
3
6
3
2
3
3
6
3
6
2
]
[
]
[
]
[
L
L
L
L
L
L
L
L
L
L
L
L
L
EI
dx
EI
L
B
B
K T
L
L
fL
Q
Q
Q
Q
u
u
u
u
L
L
L
L
L
L
L
L
L
L
L
L
L
EI
6
6
12
2
3
3
3
6
3
6
3
2
3
3
6
3
6
2
4
3
2
1
4
3
2
1
2
2
2
2
3
f
f
f
f
L
L
fL
dx
f
dx
f
L
L
T
6
6
12
]
[
0
4
3
2
1
0
N
30. FEA Beam Problem
f
a b
Uniform EI
0
0
0
0
6
6
12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
/
2
/
3
/
1
/
3
0
0
/
3
/
6
/
3
/
6
0
0
/
1
/
3
/
2
/
3
0
0
/
3
/
6
/
3
/
6
2 )
1
(
4
)
1
(
3
)
1
(
2
)
1
(
1
6
5
4
3
2
1
2
2
2
3
2
3
2
2
2
3
2
3
Q
Q
Q
Q
a
a
fa
U
U
U
U
U
U
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
EI
1
Element
)
2
(
4
)
2
(
3
)
2
(
2
)
2
(
1
6
5
4
3
2
1
2
2
2
3
2
3
2
2
2
3
2
3
0
0
/
2
/
3
/
1
/
3
0
0
/
3
/
6
/
3
/
6
0
0
/
1
/
3
/
2
/
3
0
0
/
3
/
6
/
3
/
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
Q
Q
Q
Q
U
U
U
U
U
U
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
EI
2
Element
(1) (3)
(2)
1 2
32. Special Features of Beam FEA
Analytical Solution Gives
Cubic Deflection Curve
Analytical Solution Gives
Quartic Deflection Curve
FEA Using Hermit Cubic Interpolation
Will Yield Results That Match Exactly
With Cubic Analytical Solutions
34. Frame Element
Generalization of Bar and Beam Element with Arbitrary Orientation
(1) (2)
1
w
L
2
w
1
q 2
q
1
M 2
M
1
V 2
V
2
P
1
P
1
u 2
u
q
q
4
3
2
2
1
1
2
2
2
1
1
1
2
2
2
3
2
3
2
2
2
3
2
3
4
6
0
2
6
0
6
12
0
6
12
0
0
0
0
0
2
6
0
4
6
0
6
12
0
6
12
0
0
0
0
0
Q
Q
P
Q
Q
P
w
u
w
u
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
AE
L
AE
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
EI
L
AE
L
AE
Element Equation Can Then Be Rotated to Accommodate Arbitrary Orientation
35. Some Standard FEA References
Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, 1982, 1995.
Beer, G. and Watson, J.O., Introduction to Finite and Boundary Element Methods for Engineers, John Wiley, 1993
Bickford, W.B., A First Course in the Finite Element Method, Irwin, 1990.
Burnett, D.S., Finite Element Analysis, Addison-Wesley, 1987.
Chandrupatla, T.R. and Belegundu, A.D., Introduction to Finite Elements in Engineering, Prentice-Hall, 2002.
Cook, R.D., Malkus, D.S. and Plesha, M.E., Concepts and Applications of Finite Element Analysis, 3rd Ed., John Wiley,
1989.
Desai, C.S., Elementary Finite Element Method, Prentice-Hall, 1979.
Fung, Y.C. and Tong, P., Classical and Computational Solid Mechanics, World Scientific, 2001.
Grandin, H., Fundamentals of the Finite Element Method, Macmillan, 1986.
Huebner, K.H., Thorton, E.A. and Byrom, T.G., The Finite Element Method for Engineers, 3rd Ed., John Wiley, 1994.
Knight, C.E., The Finite Element Method in Mechanical Design, PWS-KENT, 1993.
Logan, D.L., A First Course in the Finite Element Method, 2nd Ed., PWS Engineering, 1992.
Moaveni, S., Finite Element Analysis – Theory and Application with ANSYS, 2nd Ed., Pearson Education, 2003.
Pepper, D.W. and Heinrich, J.C., The Finite Element Method: Basic Concepts and Applications, Hemisphere, 1992.
Pao, Y.C., A First Course in Finite Element Analysis, Allyn and Bacon, 1986.
Rao, S.S., Finite Element Method in Engineering, 3rd Ed., Butterworth-Heinemann, 1998.
Reddy, J.N., An Introduction to the Finite Element Method, McGraw-Hill, 1993.
Ross, C.T.F., Finite Element Methods in Engineering Science, Prentice-Hall, 1993.
Stasa, F.L., Applied Finite Element Analysis for Engineers, Holt, Rinehart and Winston, 1985.
Zienkiewicz, O.C. and Taylor, R.L., The Finite Element Method, Fourth Edition, McGraw-Hill, 1977, 1989.