The document contains a sixth semester examination question paper for the subject Modeling and Finite Element Analysis. It has two parts with a total of 8 questions. Some of the key questions asked are:
1) Derive an expression for maximum deflection of a simply supported beam with a point load at the center using Rayleigh-Ritz method and trigonometric functions.
2) Explain the basic steps involved in the finite element method.
3) Define a shape function and discuss the properties that shape functions should satisfy.
4) Derive the stiffness matrix for a 2D truss element and the strain-displacement matrix for a 1D linear element.
5) Discuss the various
This document provides a question bank for the Finite Element Analysis course ME6603 taught at R.M.K College of Engineering and Technology. It contains 180 questions divided into two parts - Part A (short questions) and Part B (long questions). The questions cover the main topics of the course including the basic concepts and procedure of finite element analysis, discretization, element types, weighted residual methods, potential energy approach, and boundary conditions. Commercial FEA software packages and steps to use them are also discussed. The document aims to help students prepare for exams by providing a variety of questions related to the finite element method and its applications in engineering problems.
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.
The document discusses various topics related to stress and strain including: principal stresses and strains, Mohr's stress circle theory of failure, 3D stress and strain, equilibrium equations, and impact loading. It provides examples of stresses acting in different planes including normal, shear, oblique, and principal planes. It also gives examples of calculating normal and tangential stresses on an oblique plane subjected to stresses in one, two, or multiple directions with and without shear stresses.
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 discusses finite element analysis using axisymmetric elements. It begins by introducing axisymmetric elements, which reduce 3D axisymmetric problems to 2D by assuming symmetry around a central axis. It then derives the strain-displacement matrix [B] and stress-strain matrix [D] for an axisymmetric triangular element. It shows how to assemble the element stiffness matrix [K] and accounts for temperature effects. An example problem of a thick-walled pressure vessel is presented to illustrate the axisymmetric element method. Practical applications of axisymmetric elements include pipes, tanks, and engine parts that have cylindrical symmetry.
This document provides a question bank for the Finite Element Analysis course ME6603 taught at R.M.K College of Engineering and Technology. It contains 180 questions divided into two parts - Part A (short questions) and Part B (long questions). The questions cover the main topics of the course including the basic concepts and procedure of finite element analysis, discretization, element types, weighted residual methods, potential energy approach, and boundary conditions. Commercial FEA software packages and steps to use them are also discussed. The document aims to help students prepare for exams by providing a variety of questions related to the finite element method and its applications in engineering problems.
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.
The document discusses various topics related to stress and strain including: principal stresses and strains, Mohr's stress circle theory of failure, 3D stress and strain, equilibrium equations, and impact loading. It provides examples of stresses acting in different planes including normal, shear, oblique, and principal planes. It also gives examples of calculating normal and tangential stresses on an oblique plane subjected to stresses in one, two, or multiple directions with and without shear stresses.
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 discusses finite element analysis using axisymmetric elements. It begins by introducing axisymmetric elements, which reduce 3D axisymmetric problems to 2D by assuming symmetry around a central axis. It then derives the strain-displacement matrix [B] and stress-strain matrix [D] for an axisymmetric triangular element. It shows how to assemble the element stiffness matrix [K] and accounts for temperature effects. An example problem of a thick-walled pressure vessel is presented to illustrate the axisymmetric element method. Practical applications of axisymmetric elements include pipes, tanks, and engine parts that have cylindrical symmetry.
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
This document discusses two-dimensional vector variable problems in structural mechanics. It describes plane stress, plane strain and axisymmetric problems, and provides the stress-strain relations for materials under these conditions. It also discusses thin structures like disks and long prismatic shafts. Additionally, it covers dynamic analysis and vibration of structures, describing free vibration, forced vibration and types of vibration. Equations of motion are developed using Lagrange's approach and the weak form method. Mass and stiffness matrices for axial rod and beam elements are also presented.
ME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANKASHOK KUMAR RAJENDRAN
This document contains a collection of practice problems related to finite element analysis of two-dimensional vector variable problems, including axisymmetric problems. The problems cover derivation of element stiffness matrices and strain-displacement matrices for various element types under different conditions, calculation of element stresses and displacements, modeling of cylinders under pressure, and determination of global stiffness matrices for structures. The elements and conditions include constant strain triangles, linear strain triangles, axisymmetric triangles, plane stress, plane strain, and shells.
General steps of finite element analysisSasi Kumar
The document outlines the 10 general steps of the finite element method (FEM) for analyzing structures: 1) Discretize the structure into elements and nodes, 2) Number the nodes and elements, 3) Select displacement functions, 4) Define material behavior, 5) Derive the element stiffness matrix, 6) Assemble the global stiffness matrix, 7) Apply boundary conditions to remove singularities, 8) Solve the equations for unknown displacements, 9) Compute element strains and stresses, and 10) Interpret the results. The 10 steps provide the overall process for using FEM to model a structure and calculate its response to loading.
- The document discusses one-dimensional finite element analysis.
- It describes the derivation of shape functions for linear one-dimensional elements like a bar element. Shape functions define the variation of displacement within the element.
- The stiffness matrix, which represents the element's resistance to deformation, is also derived for a basic linear bar element. It is shown to be symmetric and its properties are discussed.
- Examples are provided to demonstrate calculating displacements at points within a one-dimensional element using the shape functions.
The document provides an introduction to the finite element method (FEM) through lecture notes. It discusses the basic concepts of dividing a complex problem into smaller, simpler pieces called finite elements. A brief history is given of the FEM from its origins in the 1940s to its widespread use today in engineering fields. The typical procedure of FEM for structural analysis is outlined as dividing a structure into finite elements connected at nodes.
This document contains formulas and equations related to finite element analysis (FEA) for one-dimensional structural and heat transfer problems. It includes formulas for weighted residual methods, Ritz method, beam deflection and stress, springs, one-dimensional bars and frames, and one-dimensional heat transfer through walls and fins. Displacement functions, stiffness matrices, thermal loads, and conduction/convection equations are provided for linear and quadratic elements undergoing static structural and thermal analysis.
The document discusses two methods for mesh refinement - the p-method and h-method. The p-method increases the order of the polynomial used in the finite element model, allowing for more accurate results without changing the mesh. The h-method reduces the size of elements to create a finer mesh, better approximating the real solution in areas of high stress gradients. Both methods aim to improve the accuracy of finite element analysis results, with the p-method doing so without requiring changes to the mesh.
This document discusses sources of error in finite element analysis, including modeling errors due to simplifying assumptions, discretization errors from approximating solutions, and numerical errors from limited computer precision. It provides examples of common mistakes that can cause incorrect results, such as incorrect material properties or insufficient boundary constraints. It also discusses best practices for verifying models, such as element testing, mesh refinement studies, and checking results against analytical solutions or boundary conditions.
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.
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 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.
Application of Boundary Conditions to Obtain Better FEA ResultsKee H. Lee, P.Eng.
This document discusses applying proper boundary conditions in finite element analysis to obtain better results. It covers:
1) Typical boundary conditions like supports, connections, and structural symmetry to model structures accurately
2) Examples show boundary conditions significantly affect results like displacements and moments
3) Nonlinear behaviors from large deformations, materials, and contact require special boundary conditions in analysis
This document discusses analysis of statically indeterminate structures using the force method. It begins by introducing statically and kinematically indeterminate structures. It then discusses the degree of static indeterminacy for different types of structures like beams, trusses, frames, and grids. It also discusses the different types of deformations that can occur in these structures. The document then covers the concepts of equilibrium, compatibility, and the force method of analysis using the method of consistent deformation. Several examples are provided to illustrate the calculation of degree of static indeterminacy for beams, trusses and frames. It also discusses kinematic indeterminacy and provides examples of its calculation for different structures.
Finite Element Analysis Rajendra M.pdfRaviSekhar35
This document provides an overview of a presentation on applied finite element analysis for civil and mechanical engineering applications. It discusses the general procedure of the finite element method, including dividing a structure into elements, formulating element properties, assembling elements, applying loads and supports, solving equations, and calculating element outputs. It also covers various element types such as bars, trusses, and beams, and provides examples of how to derive the stiffness matrix for different elements.
1. The stiffness method is used to analyze the beam by determining its degree of kinematic indeterminacy, selecting unknown displacements, restraining the structure, and generating a stiffness matrix.
2. A 4m beam with supports at 1.5m and 3m is analyzed using a stiffness matrix approach. The displacements selected are the rotations at joints B and C.
3. The stiffness matrix is generated by applying unit rotations at each joint and calculating the actions. This matrix is then used along with the applied loads in a superposition equation to solve for the unknown displacements.
General steps of the finite element methodmahesh gaikwad
General Steps used to solve FEA/ FEM Problems. Steps Involves involves dividing the body into a finite elements with associated nodes and choosing the most appropriate element type for the model.
The document discusses machine foundations used in the oil and gas industry. It begins by introducing the different types of machines, such as centrifugal and reciprocating machines, and how they are classified based on speed. It then discusses the various types of foundations used to support these machines, including block foundations and frame foundations. The document outlines the inputs needed for foundation design, which include project specifications, soil parameters, and machine details from the vendor. It describes the process of analyzing machine foundations, including dynamic and static analyses. Key aspects like natural frequencies, displacements, and strength are evaluated.
STIFFNESS MATRIX FOR 3-NODE TRIANGULAR ELEMENT murali mohan
The document discusses the stiffness matrix for a 3-node triangular element. It provides the stiffness matrix that defines the relationship between forces and displacements for a basic 3-node triangular element used in finite element analysis. The matrix captures the element's geometric and material properties to model how it resists deformation when external forces are applied.
This document contains the questions from a Third Semester B.E. Degree Examination in Network Analysis. It consists of 5 questions with 3 sub-questions each, selecting at least 2 questions from each part A and B.
Part A questions focus on network analysis techniques like star-delta transformation, mesh analysis, node voltage method, graph theory concepts and tie set scheduling. Sample circuits are provided to solve using these techniques.
Part B questions discuss dual networks, matrix representation of networks using tie-sets, network theorems and two-port networks. Definitions and explanations are provided along with examples where needed.
The document tests the examinee's knowledge of various network analysis concepts, theorems and problem solving
This document contains the questions from a third semester B.E. degree examination on Network Analysis. It has 8 questions divided into two parts - Part A and Part B.
The questions assess concepts related to network analysis including Fourier series expansion, Fourier transforms, Laplace transforms, solution of differential equations using separation of variables, curve fitting, eigen analysis, and more. Methods like Newton-Raphson, simplex method, relaxation method, and power method are also tested. Circuit analysis concepts involving RC circuits, transfer functions, and network theorems are covered.
The questions require deriving equations, solving problems numerically and graphically, explaining concepts, and designing circuits to assess the candidate's understanding of core topics in network analysis
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
This document discusses two-dimensional vector variable problems in structural mechanics. It describes plane stress, plane strain and axisymmetric problems, and provides the stress-strain relations for materials under these conditions. It also discusses thin structures like disks and long prismatic shafts. Additionally, it covers dynamic analysis and vibration of structures, describing free vibration, forced vibration and types of vibration. Equations of motion are developed using Lagrange's approach and the weak form method. Mass and stiffness matrices for axial rod and beam elements are also presented.
ME6603 - FINITE ELEMENT ANALYSIS UNIT - IV NOTES AND QUESTION BANKASHOK KUMAR RAJENDRAN
This document contains a collection of practice problems related to finite element analysis of two-dimensional vector variable problems, including axisymmetric problems. The problems cover derivation of element stiffness matrices and strain-displacement matrices for various element types under different conditions, calculation of element stresses and displacements, modeling of cylinders under pressure, and determination of global stiffness matrices for structures. The elements and conditions include constant strain triangles, linear strain triangles, axisymmetric triangles, plane stress, plane strain, and shells.
General steps of finite element analysisSasi Kumar
The document outlines the 10 general steps of the finite element method (FEM) for analyzing structures: 1) Discretize the structure into elements and nodes, 2) Number the nodes and elements, 3) Select displacement functions, 4) Define material behavior, 5) Derive the element stiffness matrix, 6) Assemble the global stiffness matrix, 7) Apply boundary conditions to remove singularities, 8) Solve the equations for unknown displacements, 9) Compute element strains and stresses, and 10) Interpret the results. The 10 steps provide the overall process for using FEM to model a structure and calculate its response to loading.
- The document discusses one-dimensional finite element analysis.
- It describes the derivation of shape functions for linear one-dimensional elements like a bar element. Shape functions define the variation of displacement within the element.
- The stiffness matrix, which represents the element's resistance to deformation, is also derived for a basic linear bar element. It is shown to be symmetric and its properties are discussed.
- Examples are provided to demonstrate calculating displacements at points within a one-dimensional element using the shape functions.
The document provides an introduction to the finite element method (FEM) through lecture notes. It discusses the basic concepts of dividing a complex problem into smaller, simpler pieces called finite elements. A brief history is given of the FEM from its origins in the 1940s to its widespread use today in engineering fields. The typical procedure of FEM for structural analysis is outlined as dividing a structure into finite elements connected at nodes.
This document contains formulas and equations related to finite element analysis (FEA) for one-dimensional structural and heat transfer problems. It includes formulas for weighted residual methods, Ritz method, beam deflection and stress, springs, one-dimensional bars and frames, and one-dimensional heat transfer through walls and fins. Displacement functions, stiffness matrices, thermal loads, and conduction/convection equations are provided for linear and quadratic elements undergoing static structural and thermal analysis.
The document discusses two methods for mesh refinement - the p-method and h-method. The p-method increases the order of the polynomial used in the finite element model, allowing for more accurate results without changing the mesh. The h-method reduces the size of elements to create a finer mesh, better approximating the real solution in areas of high stress gradients. Both methods aim to improve the accuracy of finite element analysis results, with the p-method doing so without requiring changes to the mesh.
This document discusses sources of error in finite element analysis, including modeling errors due to simplifying assumptions, discretization errors from approximating solutions, and numerical errors from limited computer precision. It provides examples of common mistakes that can cause incorrect results, such as incorrect material properties or insufficient boundary constraints. It also discusses best practices for verifying models, such as element testing, mesh refinement studies, and checking results against analytical solutions or boundary conditions.
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.
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 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.
Application of Boundary Conditions to Obtain Better FEA ResultsKee H. Lee, P.Eng.
This document discusses applying proper boundary conditions in finite element analysis to obtain better results. It covers:
1) Typical boundary conditions like supports, connections, and structural symmetry to model structures accurately
2) Examples show boundary conditions significantly affect results like displacements and moments
3) Nonlinear behaviors from large deformations, materials, and contact require special boundary conditions in analysis
This document discusses analysis of statically indeterminate structures using the force method. It begins by introducing statically and kinematically indeterminate structures. It then discusses the degree of static indeterminacy for different types of structures like beams, trusses, frames, and grids. It also discusses the different types of deformations that can occur in these structures. The document then covers the concepts of equilibrium, compatibility, and the force method of analysis using the method of consistent deformation. Several examples are provided to illustrate the calculation of degree of static indeterminacy for beams, trusses and frames. It also discusses kinematic indeterminacy and provides examples of its calculation for different structures.
Finite Element Analysis Rajendra M.pdfRaviSekhar35
This document provides an overview of a presentation on applied finite element analysis for civil and mechanical engineering applications. It discusses the general procedure of the finite element method, including dividing a structure into elements, formulating element properties, assembling elements, applying loads and supports, solving equations, and calculating element outputs. It also covers various element types such as bars, trusses, and beams, and provides examples of how to derive the stiffness matrix for different elements.
1. The stiffness method is used to analyze the beam by determining its degree of kinematic indeterminacy, selecting unknown displacements, restraining the structure, and generating a stiffness matrix.
2. A 4m beam with supports at 1.5m and 3m is analyzed using a stiffness matrix approach. The displacements selected are the rotations at joints B and C.
3. The stiffness matrix is generated by applying unit rotations at each joint and calculating the actions. This matrix is then used along with the applied loads in a superposition equation to solve for the unknown displacements.
General steps of the finite element methodmahesh gaikwad
General Steps used to solve FEA/ FEM Problems. Steps Involves involves dividing the body into a finite elements with associated nodes and choosing the most appropriate element type for the model.
The document discusses machine foundations used in the oil and gas industry. It begins by introducing the different types of machines, such as centrifugal and reciprocating machines, and how they are classified based on speed. It then discusses the various types of foundations used to support these machines, including block foundations and frame foundations. The document outlines the inputs needed for foundation design, which include project specifications, soil parameters, and machine details from the vendor. It describes the process of analyzing machine foundations, including dynamic and static analyses. Key aspects like natural frequencies, displacements, and strength are evaluated.
STIFFNESS MATRIX FOR 3-NODE TRIANGULAR ELEMENT murali mohan
The document discusses the stiffness matrix for a 3-node triangular element. It provides the stiffness matrix that defines the relationship between forces and displacements for a basic 3-node triangular element used in finite element analysis. The matrix captures the element's geometric and material properties to model how it resists deformation when external forces are applied.
This document contains the questions from a Third Semester B.E. Degree Examination in Network Analysis. It consists of 5 questions with 3 sub-questions each, selecting at least 2 questions from each part A and B.
Part A questions focus on network analysis techniques like star-delta transformation, mesh analysis, node voltage method, graph theory concepts and tie set scheduling. Sample circuits are provided to solve using these techniques.
Part B questions discuss dual networks, matrix representation of networks using tie-sets, network theorems and two-port networks. Definitions and explanations are provided along with examples where needed.
The document tests the examinee's knowledge of various network analysis concepts, theorems and problem solving
This document contains the questions from a third semester B.E. degree examination on Network Analysis. It has 8 questions divided into two parts - Part A and Part B.
The questions assess concepts related to network analysis including Fourier series expansion, Fourier transforms, Laplace transforms, solution of differential equations using separation of variables, curve fitting, eigen analysis, and more. Methods like Newton-Raphson, simplex method, relaxation method, and power method are also tested. Circuit analysis concepts involving RC circuits, transfer functions, and network theorems are covered.
The questions require deriving equations, solving problems numerically and graphically, explaining concepts, and designing circuits to assess the candidate's understanding of core topics in network analysis
1. The document contains questions from a third semester B.E. degree examination in discrete mathematical structures.
2. It asks students to define sets, prove properties of sets, solve problems involving sets and functions, write symbolic logic statements, and determine if logic arguments are valid or not.
3. Several questions also involve topics like tautologies, propositional logic, and predicate logic.
1. The document contains questions from a third semester B.E. degree examination in discrete mathematical structures.
2. It asks students to define sets, prove properties of sets, solve problems involving sets and functions, write symbolic logic statements, and determine if logic arguments are valid or not.
3. Several questions also involve topics like tautologies, propositional logic, and predicate logic.
b.
(08 Marks)
, 10, 12, 15)
(10 Marks)
Design a 4-bit binary adder using half adders and full adders.
(08 Marks)
c. Design a 4-bit binary subtractor using half subtractors and full subtractors.
(08 Marks)
3 a.
Design a 4-bit magnitude comparator using basic gates.
(10 Marks)
b.
Design a 4-bit binary comparator using basic gates.
(10 Marks)
4 a.
Design a 4-bit binary multiplier using AND gates and half adders.
(10
Metrology is the science of measurement. Some key points:
1) A wavelength standard has advantages over line and end standards as it provides a stable reference without endpoints.
2) Limit gauges are used to check if a part's dimensions fall within the acceptable tolerance range. They are classified based on their application as go, no-go, adjustable, and ring gauges.
3) Measurement systems involve accuracy, precision, calibration, and other factors. Primary transducers directly measure physical quantities while secondary transducers convert one form of energy to another.
This document appears to be an examination paper for Engineering Mathematics from a third semester B.E. degree program. It contains 10 questions across two parts - Part A and Part B. The questions cover a range of topics including Fourier series, differential equations, matrix eigenvalues, interpolation, and numerical methods. Students are instructed to answer any 5 full questions, selecting at least 2 from each part. The questions vary in marks from 4 to 10 marks each.
1) The document discusses various questions related to the finite element method and analysis. It includes questions on determining stresses and displacements in a bar element, principle of minimum potential energy, derivation of shape functions, consistent nodal load vector, Gauss quadrature, advantages and disadvantages of finite element method, Hermite shape functions, and short notes on topics like C0, C1 and C2 functions.
2) The questions cover various fundamental concepts in finite element method like discretization, shape functions, potential energy approach, assembly of global stiffness matrix, consistent loading, numerical integration and analysis of trusses, beams and frames.
3) The document tests the candidate's understanding of basic finite element method concepts and their ability to
1) The document discusses various questions related to the finite element method and analysis. It includes questions on determining stresses and displacements in a bar element, principle of minimum potential energy, derivation of shape functions, consistent nodal load vector, Gauss quadrature, advantages and disadvantages of finite element method, Hermite shape functions, and short notes on C0, C1 and C2 functions, node numbering, serendipity elements, and patch tests.
2) The questions cover topics like derivation of shape functions, evaluation of shape functions, consistent nodal load vector, Gauss quadrature, Hermite shape functions, and flexibility and stiffness methods. Mathematical relations involving [K], [f], [a] are also given
This document contains a question paper for the Sixth Semester Mechanical Engineering examination covering the topic of Finite Element Analysis. It includes 20 short answer questions in Part A and 5 long answer questions in Part B. Some example questions are on distinguishing between analysis methods, defining terms, providing examples of different problem types, deriving element characteristics, and solving physical problems using various finite element techniques. Students are required to answer all questions in the allotted time of three hours.
This document appears to be an exam question paper for a structural engineering course focused on earthquake engineering and seismic analysis. It contains 10 questions related to topics like lessons learned from past earthquakes, seismic waves, response spectra, seismic analysis of buildings, retrofitting structures, and base isolation systems. It also includes 4 figures showing building plans and mode shapes for dynamic analysis. The questions range from explaining concepts to calculating total base shear and performing vibration analysis of buildings.
1) The document discusses topics related to digital communication systems including sampling theory, PCM, delta modulation, line coding techniques, and spread spectrum.
2) It asks questions about deriving expressions, sketching spectra, block diagrams, and analyzing digital modulation techniques.
3) The exam covers two parts - Part A focuses on digital modulation concepts while Part B covers advanced topics like DPSK, channel coding, and adaptive equalization.
This document contains a summary of an engineering mathematics exam with questions covering various topics including:
1) Solving differential equations using Taylor series, Runge-Kutta, and Picard's methods.
2) Computing values for functions that satisfy given differential equations using Runge-Kutta and Milne's methods.
3) Analyzing functions in complex plane including Cauchy-Riemann equations and conformal mappings.
4) Solving problems involving Legendre polynomials, addition theorems of probability, and Poisson and normal distributions.
5) Testing hypotheses using statistical methods and fitting distributions to data.
This document contains exam questions for the subject Digital Communication. It has two parts - Part A and Part B. Part A focuses on digital communication systems, sampling, PCM, delta modulation, line coding techniques and adaptive equalization. Part B covers passband transmission schemes, modulation techniques like BPSK, MSK, spread spectrum techniques and correlation receivers. The questions test concepts like block diagrams, derivations, explanations and comparisons related to digital communication topics.
The document is a question paper for the Strength of Materials subject. It contains 10 questions divided into 5 modules. The questions are on topics like stresses and strains in materials, bending moments, shear forces, torsion, columns and thin-walled pressure vessels. Some questions ask students to derive expressions, draw shear force and bending moment diagrams, calculate stresses and pressures in different scenarios, and design columns. The document provides instructions to answer any 5 full questions with 1 from each module. Missing data can be suitably assumed.
This document appears to be an exam paper for the subject Logic Design. It contains 10 questions divided into two parts - Part A and Part B. The questions cover various topics related to logic design including canonical forms, minimization of logic functions, multiplexers, decoders, adders and code converters. Students are instructed to answer any 5 full questions selecting at least 2 questions from each part. The exam is worth a total of 100 marks and is meant to evaluate students' understanding of fundamental concepts in logic design.
This document contains the details of an examination for a third semester engineering degree. It includes instructions to answer any five full questions selecting at least two from each part. The document then lists 14 questions across two parts (A and B) related to topics in logic design and electronic circuits. The questions cover various concepts including universal gates, Boolean functions, amplifiers, feedback, operational amplifiers, timers and voltage regulators. Diagrams and calculations are included in some of the questions.
This document contains questions from a third semester Bachelor of Engineering degree examination in Mechanics of Materials. It includes two parts, Part A and Part B.
Part A contains three questions. Question 1 has sub-parts asking students to analyze data from a tensile test on mild steel and calculate properties like Young's modulus, proportional limit, true breaking stress and percentage elongation. Question 2 has sub-parts asking students to calculate total elongation of a brass bar under axial forces and find Poisson's ratio and elastic constants from tensile test data.
Part B likely contains similar analysis questions related to mechanics of materials, though the specific questions are not included in the document provided. The document provides the framework and context for the examination,
This document contains questions from a third semester engineering examination in Basic Thermodynamics. It is divided into two parts, with Part A containing definitions and concepts in thermodynamics, and Part B containing applications of thermodynamic principles. Some of the questions ask students to define terms, derive equations, and solve problems related to processes like expansion, throttling, heating and mixing of substances. Maximizing and minimizing entropy, and determining properties like enthalpy are also examined. Diagrams including P-V, T-S and H-S are discussed.
This document contains questions from a third semester Bachelor of Engineering degree examination in Mechanics of Materials. It includes two parts, Part A and Part B.
Part A contains three questions. Question 1 has sub-parts asking students to analyze data from a tensile test on mild steel and calculate properties like Young's modulus, proportional limit, true breaking stress and percentage elongation. Question 2 has sub-parts asking students to calculate total elongation of a brass bar under axial forces and find Poisson's ratio and elastic constants from tensile test data.
Part B likely contains similar analysis questions related to mechanics of materials, though the specific questions are not included in the document provided. The exam covers a range of topics testing students' understanding
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This document contains instructions and questions for an exam in Analog and Digital Electronics. It is divided into 5 modules. For each module, there are 2 full questions with multiple parts to choose from. Students must answer 5 full questions, choosing 1 from each module. They must write the same question numbers and answers should be specific to the questions asked. Writing must be legible. The questions cover topics like operational amplifiers, logic gates, multiplexers, flip-flops, counters, and more. Diagrams and explanations are often required.
Unix and Shell Programming,
Q P Code: 60305.
Additional Mathematics I
Q P Code: 60306
Computer Organization and Architecture
Q P Code: 62303
Data Structures Using C
Q P Code: 60303
Discrete Mathematical Structures
Q P Code: 60304
Engineering Mathematics - III
Q P Code: 60301
Soft Skill Development
Q P Code: 60307
Unix and Shell Programming,
Q P Code: 60305.
Additional Mathematics I
Q P Code: 60306
Computer Organization and Architecture
Q P Code: 62303
Data Structures Using C
Q P Code: 60303
Discrete Mathematical Structures
Q P Code: 60304
Engineering Mathematics - III
Q P Code: 60301
Soft Skill Development
Q P Code: 60307
Unix and Shell Programming,
Q P Code: 60305.
Additional Mathematics I
Q P Code: 60306
Computer Organization and Architecture
Q P Code: 62303
Data Structures Using C
Q P Code: 60303
Discrete Mathematical Structures
Q P Code: 60304
Engineering Mathematics - III
Q P Code: 60301
Soft Skill Development
Q P Code: 60307
Unix and Shell Programming,
Q P Code: 60305.
Additional Mathematics I
Q P Code: 60306
Computer Organization and Architecture
Q P Code: 62303
Data Structures Using C
Q P Code: 60303
Discrete Mathematical Structures
Q P Code: 60304
Engineering Mathematics - III
Q P Code: 60301
Soft Skill Development
Q P Code: 60307
Unix and Shell Programming,
Q P Code: 60305.
Additional Mathematics I
Q P Code: 60306
Computer Organization and Architecture
Q P Code: 62303
Data Structures Using C
Q P Code: 60303
Discrete Mathematical Structures
Q P Code: 60304
Engineering Mathematics - III
Q P Code: 60301
Soft Skill Development
Q P Code: 60307
This document appears to be an examination for an advanced concrete design course. It includes questions related to concrete mix design, properties of concrete, testing methods, durability, special concretes, and more. Specifically, Part A asks about Bogue's compounds in cement, concrete rheology, porosity calculations, superplasticizers, fly ash, and mix design. Part B covers permeability, alkali-aggregate reaction, sulfate attack, fiber reinforced concrete, ferrocement, lightweight/high density concrete, and concrete properties. Part C asks about non-destructive testing methods, high performance concrete, and special topics like self-consolidating concrete. The document provides an examination covering a wide range of advanced concrete topics.
- The document discusses design considerations for prestressed concrete structures. It begins with explaining the principle of prestressing and differentiating between bonded and unbonded prestressed concrete members.
- It then explains Hoyer's Long Line system of pre-tensioning with a neat sketch. Further, it provides examples to calculate extreme fiber stresses in prestressed concrete beams using the load balancing method, and designs the required area of prestressing tendons.
- In short, the document focuses on fundamental concepts of prestressing like various systems, design of prestressed beams, and calculation of stresses in the extreme fibers using the load balancing method.
This document contains questions for a highway engineering exam, organized into 5 modules. It includes questions on various topics relating to highway planning, design, construction and management. Some example questions are on road alignment factors, preliminary survey details for highway alignment, flexible pavement design, properties and tests for road aggregates, subgrade soil properties, construction of water bound macadam base and granular sub-base. Students are required to answer 5 full questions, choosing one from each module. Formulas, tables and design standards can be referred to solve numerical problems.
This document contains questions from a B.E. Degree Examination in Design of RCC Structural Elements. The exam has 5 modules.
Module 1 asks questions about the difference between working stress and limit state methods, definitions related to partial safety factors and characteristic values, and checking a simply supported beam for serviceability limit state of cracking.
Module 2 contains questions on determining moment of resistance for T-beams, central point loads for simply reinforced beams, and ultimate moment capacity for doubly reinforced beams.
Module 3 involves designing a rectangular reinforced concrete beam and a T-beam slab floor system.
Module 4 distinguishes one-way and two-way slabs and asks about bond, anchorage length,
1. The document provides a series of problems from an Engineering Mathematics examination. It includes problems across four modules involving calculus, differential equations, linear algebra, and probability.
2. Students are asked to solve problems using various mathematical techniques like Taylor's series, Runge-Kutta method, Euler's method, linear transformations, and the Laplace transform.
3. Questions involve finding derivatives, solving differential equations, evaluating integrals, finding eigenvectors and eigenvalues, and solving problems involving probability.
The document provides information about an examination for Operations Management. It includes 10 questions across two parts (A and B) assessing various topics related to operations management. Part A questions cover topics like defining operation management, service vs goods production differences, decision making frameworks, capacity analysis, forecasting methods, breakeven analysis and aggregate planning models. Part B questions assess topics such as inventory management, manufacturing models, supply chain components, and capacity planning strategies. The document provides context and questions for an exam, assessing students' understanding of key operations management concepts.
This document outlines the structure and content of an examination for Engineering Economy. It contains 10 questions split into 2 parts with at least 2 questions to be answered from each part. The duration of the exam is 3 hours. Questions cover various topics related to Engineering Economy including present worth analysis, annual equivalent cost, rate of return, depreciation, etc. Students are permitted to use discrete interest factor tables.
This document contains the questions and solutions for a Computer Integrated Manufacturing exam. It includes multiple choice and descriptive questions across two parts (A and B) that cover topics like automated manufacturing systems, flow line analysis, computer aided process planning, computer numerical control, industrial robots, and design of manufacturing systems. The questions require calculations, explanations, and diagrams to demonstrate understanding of key concepts in computer integrated manufacturing.
This document contains exam questions for Management and Engineering subjects. It is divided into two parts (PART A and PART B) with multiple choice and long answer questions covering various topics:
- PART A includes questions on management functions, planning, communication, recruitment processes, and engineering topics like stresses in bodies, stress concentration factors, and shaft design.
- PART B includes questions on entrepreneurship, types of businesses, barriers to entrepreneurship, location selection for small industries, sources of finance, project selection criteria, network techniques, and engineering topics like dynamics of machines, balancing of rotating masses, and engine dynamics.
The document provides detailed exam questions to test students' understanding of management and engineering concepts. Students must answer 5
This document contains questions from a Material Science and Metallurgy exam. It covers various topics:
- Crystal structures of BCC, FCC and HCP lattices and their properties. Diffusion of iron atoms in BCC lattice.
- Mechanical properties in the plastic region from stress-strain diagrams. True and conventional strain expressions. Twinning mechanism of plastic deformation.
- Fracture mechanisms based on Griffith's theory of brittle fracture. Factors affecting creep. Fatigue testing and S-N curves for materials.
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The document appears to contain exam questions for an Operations Management course. It includes questions related to topics like linear trend analysis, capacity planning, aggregate planning, inventory management, MRP, and supply chain management. It also contains exam questions for other engineering courses on subjects like control systems, power plants, and nuclear power. The questions generally provide relevant context and ask students to define terms, explain concepts, calculate values, or discuss strategies related to the various operations, engineering, and management topics.
More from BGS Institute of Technology, Adichunchanagiri University (ACU) (20)
Modelling and finite element analysis: Question Papers
1. USN 06ME63
Sixth Semester B.E. f)egree Examination, December 2Ol2
Modeling and Finite Element Analysis
Tirne: 3 hrs. Max. Marks:100
Note: Answer FIVE full questions, selecting
at leust TWO questions from eoch part.
!
.! PART _ A
a
u
e la. Using Rayleigh-Ritz method, derive an expression for maximum det.lection of the simply
= b. supported beam with point load P at centre. Use trigonometric function. (08 Marks)
a Solve the following system of simultaneous equations by Gauss elimination method.
O X-l Y 'l Z:9
aX, x-2y+32:8
=D- 2x+ Y - z:3 (08 Marks)
c. Explain the principle of minimum potential energy and principle of virtual work. (04 Marks)
3
otll
=co
.= a.l
2a. Explain the basic steps involved is FEM. (10 Marks)
b. Explain the concepts of iso, sub and super parametric elements. (05 Marhs)
Etf c. Define a shape function. What are the properlies that the shape functions should satisly?
-O (05 Marks)
=ts
a2
3a. What are the convergence requirements? Discuss three conditions of
convergence i
6= requirements. (05 Marks)
1
OO b. What are the considerations for choosing the order of the polynomial functions? (05 Marks) I
Derive the shape functions for CST element.
rl
-1 c. (10 Marks) {
boi
:'
2G 4a. Derive the Hermite shape function tbr a 2-noded beam element. (10 Marks)
b. Derive the shape functions fbr a four noded quadrilateral element in natural coordinates.
3u (10 Marks)
AE
6X I
:
o --: PART _ B
,i .9.
6E
oLE 5a. Derive an expression for stifthess matrix for a2-D truss element. (10 Marks)
a,-
b Derive the strain displacement matrix tbr 1-D linear element and show that o: E[B]{u}
>(k (10 Marks)
cno bIr
=
0=
so
F>
6a. Discuss the various steps involved in the finite element analysis of a one dirnensional heat
o transfbr problem with refbrence to a straight unifbrm fin. (10 Marks)
(-) <
b. Derive the element matrices, using Galerkin for heat conduction in one dimensional element
with heat generation Q. (l0 Marks)
-N
o
Z 7 a. A bar is having uniform cross sectional area of 300 mm2 and is subjected to a load
6 P : 600kN as shown in Fig.Q7(a). Determine the displacement field, stress and support
o reaction in the bar. Consider two element and rise elimination method to handle boundary
conditions. Take E :200 GPa. (10 Marks)
I of 2
2. a
I
,
,
,
/.
t
Fig.Q7(a)
b. For the two bar truss shown in Fig.Q7(b), determine the nodal displacements and stress in
each number. Also find the support reaction. Take E :200 GPa. (10 Marks)
, SotlN
Fig.Q7(b)
8a. For the beam shown in Fig.Q8(a), determine the end reaction and deflection at mid span.
Take E :200 GPa,I:4x106 mma. (10 Marks)
TYc
24hNlm
h,,|tl,lfi
Fig.Q8(a) Fie.Q8(b)
Determine the temperature distribution through the composite wall subjected to convection
heat loss on the right side surface with convection heat transfer coefficient shown in
Fig.Q8(b). The ambient temperature is *5oC. (10 Marks)
**+*8
2 of2
3. 06ME63
sixth semester B.E. Degree Examinatlon, December 2011
Modelling and Fisrite Elememt Analysis
Time:3 hrs" Max. Marks:100
Note: Answer uny FIYE full questions, selecting
at least TWO questions from each part'
PART-A
equatio, for ffrtate of stress and state the terms involved"
(04 Marks)
oi
I a. write the equilibrium
o
o b. solve the following system of equations by Gaussian elimination rnethod :
(d
a x1*x2*x:=6
(6
Xr-Xz*2x3=5
Marksi (08
rd x1* 2x2-x3=2.
{)
c. Determine the displacements of holes of the spring system shown in the figure using
iE
e) (08 Marks)
principle of minimum potential en?rg{;
_o? o t{" trln"*
Srtcll.t"r
Sorf
(!u
!.,
Fig.Q.1(c). 6-s rr lx't
ll
ao
traP
.=N
d+
i. 60 number and
otr 2a. Explain the discreti zationprocess of a given domain based on element shapes
-o (06 Marks)
slze.
a structural
o= b. Explain basic steps involved in FEM with the heip of an example involving
Es
member subjected to axial loads.
(08 Marks)
od
vd
0. Why FEA is widely accepted in engineering? List various appiications of FEA in
(06 Marks)
6o engineering
o'o
boc
3a. Derive interpolation model for 2-D simplex element in global co - ordinate system'
"o! (10 Marks)
26
!s=
'd(g b. What is an interpolation function? Write the interpolation functions for:
.a
EO
o€
(,
i) 1 -Dlinearelement ; ii) 1 -Dquadraticelement'
2O
iiU 2-D linearelement ; iv) 2-Dquadraticelement'
tro. v) 3-Dlinearelement. (06 Marks)
(04 Marks)
oj c. Explain "complete" and "conforming" elements'
AE
Derive shape function for 1 - D quadratic bar element in neutral co-ordinate tVttelm
5L)
olE 4a. Marks)
LO
o.E
>.9 b. Derive shape functions for CST element in NCS. (08 Marks)
on-
troo c. What ur. rhup. functions and write their properties. (any two). (04 Marks)
qo
:a)
EE PART -B
-h
U< 5a. Derive the body force load vector for I - D linear bar etrement.
(04 Marks)
(06 Marks)
--.; c'i b. Derive the Jacobian matrix for CST element starting from shape function'
(10 Marks)
o c. Derive stiffness matrix for a beam element starting from shape function'
o
z
d 6a, Explain the various boundary conditions in steady state heat transfer problems with simple
o
o, sketches. (06 Marks)
b. Derive stiffness matrix for 1 - D heat conduction problem using either functional approach
or Galerkin's approach
l
(08 Marks) .j
l'
I of Z ij
ii
4. 06M863
r
I c. For the composite wall shown in the figure, derive the global stifftress matrix. (06Marts)
Take
Ar:Az=A3:A
Fie.Q.6(c)
7 a. The structured member shown in figure consists of two bars. An axial load of P:200 kN is
loaded as shown. Determine the following :
i) Element stiffness matricies.
ii) Global stiffness matrix.
iii) Global load vector.
iv) Nodaldisplacements.
i) Steel Ar = 1000 mm2
Er :200 GPa
ii) Bronze Az:2000 mm2
Ez: 83 GPa' (08 Marls)
b. For the truss system shown, determine the nodal displacements. Assume E: 210 GPa and A
= 500 mm2 for both elements. (I2 Marks)
;f
=loovl.rt
Fie.Q.7(b)
8 a. Determine the temperature distribution in 1 - D rectangular cross - section fin as shown in
figure. Assume that convection heat loss occurs from the end of the fin. Take
'3w
K=-.
CmoC'
- = 0.1w , T*:20oC. Consider two elements
h "" (10 Marks)
=
Cm'oC
fo v 5 E*fr.,r
Fr z reg,il
.f.y-tol nnll
Fie.Q.8(a)
b. For the cantilever beam subjected to UDL as shown in Fig.Q.8(b), determine the deflections
of the free end. Consider one element. (10 Marks)
fo;5s1.
Fs z rD qlt
.t-tot$fltt
Fie.Q.8(b)
,*****
2 of?
5. r-
I
I
06ME63
USN
SixthSemesterB.E.DegreeExamination,December2010
Modeling and Finite Element Analysis
Max. Marks:100
Time:3 hrs. selecting
Note: Answer ony FIVE futl questions'
at least TWO questions from each part'
PART _ A
,9
o
ffi'"-" tt for two dimensions' (06 Marks)
H
I a. Explain, with a sketch, plain stress
p*"'ii"f energy' Explain the potential energy' with usual
"in
a
(g b. State the principles of minimu* (06 Marks)
notations.
c. t^hTT: the steps invotved in Ravleigh-Ritz methog?
DeTnTl:,'X ul?l]":::'*'
d
shown in Fig.l(c). use second degree
::#::
()
d
0) ffii il'r'ffi:i:.1il;J*_d#;,I;J;;g-^
approximation, for the displacement'
(08 Marks)
39 iolynomial
d9
-o ,,
ao"
Fm
.=+
'E-f
b?p
Fig.l (c).
Pfr
method with finite element methocl'
o> (04 Marks)
!1 a 2a. Bring out the four differences in continuum with example'
b. What do you underrtuod FEM?
eri"ny e*piain the steps involved in FEM'
acd
the generai node numbering *d t"l-ff:;Tl
5(J
do
Write properries of stiffness matrix K. Show - (06 Marks)
the half bandwidth-
6d
{tz Marks)
}E 3a. What is an interpolation function? ,..r - of convergence ,
-r ^^--.^- requirements'
tr5 b. what are convergence requirements? Discuss three
conditions
(08 Marks)
Write a shot notes on :
!O
oe c. -
o- gt
Eo. i1 C.o*etrical isotropy for 2D Passal triangle (CST) elernent' with a sketch' (lG Marks)
ii) Shapg function for constant strain triangrilar
si ^9
bar eiernent, in natural co-ordinates"
'@q 4a. Derive the shape functions for the one-dimensional (08 Marlcs)
quadrilateral eler'rent, in natural co-'rdinates'
Derive the shape functions for a four-node
L0
b. (08 Marks)
>.k
mo (04 Marksi
c. Write four properties of shape functions'
g0
=(6
tr> PART - B
59
o-
U<
5a. Derive the following :
1) Element stiffness matrix (K")'
il Element load vector (f)
c.i
-i
() (12 Manlis)
o Uy aire"t method for one-dimensional
bar etrement'
Z (l-1) for constant strain triangle (csr)'
b. K:ff:"Iffi::f the Jocabian transformation matrix (08 Marks)
(08
d
o
a (06 Marks)
6a.Explainwithasketch,one-dimensionalheatconduction. for heat conduction in one
b.Derivetheelementmatrices,usingGalerkinapproach, (10 Marks)
dimensional element' (04 Marks)
dimension'
c. Explain heat flux boundary condition in one
6. I
06M863
7 a. Solve for nodal displacements and elemental stresses for the following. Fig.Q,7(a), shows a
thin plate of uniform lmm thickness, Young's modulus E : 200 Gfa, weighr aensity of the
plate : 76.6 x 10-6 N/mm2. In addition to its weight, it is subjected to a point load of 1 kN at
its mid point and model the plate with 2 bar elements. (10 Marks)
r
Fig.Q.7(a). I
I
+
t
b. For the pin-jointed configuration shown in Fig.e.7(b), formulate the stiffness matrix. Also
determine the nodal displacements. (10 Marks)
IKN
fiIom'rJ.
{.
Es = E2=26r6$Pr.
Fig-Q.7(b).
8a. Solve for vertical deflection and slopes, at points 2 and,3, using beam elements, for the
structure shown in Fig.Q.8(a). Also determine the deflection at the centre of the
the beam carrying UDL.
ir"rtffi
E:z.o06P(
Fig.Q.8(a).
J = 4x lob**ti
b. Determine the temperature distribution through the composite wall, subjected to convection
heat transfer on the right side surface, with convective heat transfer co-efficient shown in
Fig.Q.8(b). The ambient temperature is -5oC. (10 Marks)
a-
t*
[2 looo $ly*tt
Fig.Q.8(b).
Kz=***
****,r
2 of2
7. 06M863
USN
May/June 20L0
Sixth semester B.E. Degree Examination,
Modeling and Finite Element Analysis
Max' Marks:100
Time: 3 hrs.
d questions, seleeting atleast TIYO from each part'
o
o
Note: Answer any FIVE futl
a PART - A
of a simply supported beam with
(g
i
! a. Using Rayleigh Ritz method, find tt. ,il*l*--a"flection (10 Marks)
'o
() point load at center' . . -- ,--- ^^-.^+r^-- L., I],,,oci;rn eli method'
by Gaussian elimination
b. Solve the following system of simultaneous equations
(B
o
B9
qp- 4xr f 2W+ 3x3:4
2xr * 3x2* 5x3:2 (10 Marks)
'=h
Zxr * 7xz: 4
aoll
t-6
.= e'l 2a.Explainthedescretizationprocess'sketchthedifferenttypesofelementslD'2D'3D
(06 M'::Y)
cdS
c^ bI)
!i {) elements used in the finite element
analysis' -..,-- L-- A:-^
!'a
otr
rA b. considerinil;gton"rrt,.ou.oio it. "tl*.rrt
stiffness matrix by direct stiffiress
fir;111*;
eE
Comment on its characteristics' tudti9,}*rr.ut
o7 the properties that the shape flrnction should
8z a. De{ine
"
J# #ffi;. irh;;*"
oid
?d
6o criteria with suitable examples and compatibil*
*o*T#H:i;
do 3 a. Explain the convergence
boc
.dd
b.
FEM.
Explain simplex, complex and multiplex
elements using element shapes' (06 Marks)
rk coordinates for one dimensionai
}E
!o= c. Explain linear interpolatiorr, potyrro*ials
in terms
"igilu"r - (06 Marks)
!rg
-2" ts
simPlex element'
irO
oe
E3 ."t,,rte'Pr vr revr*r*.
4a,Explaintheconceptofisoparametric,subparametricandsuperparametricelementsanrj
(06Marks)
o9'
tro theirurrr'to a ,1 r:--ri^^^*a
o-i
b.DerivetheshapefunctionsforaCsTelementandalsothedisplacementmatrix.(08Marks)
for abeam element'
(06 Marks)
9E
A,E
c. Derive the Hermite shape nn.ti*
=9
LO PART. B
shown in fig' Qs(a)'
s a. Find the shape tunctions forpgintp ruiffi"lement
o.<
>.(I at
g";o ^t11r11r*;
(10 Marks)
6E area and Jacobian the eiement'
matrix
AE 61 8)
tr>
=6J
Ek
'P*
_h C6rsJ
o< I
..I e.i Fig.Qs(a) I
(trt{
C' C$,
o
z (10 l![arks)
(l
b.Derivethestiffrressmatrixfota2_dimensionaltrusselement.
o
+ analysis of a one dimensional heat
6 a. Discuss the various steps involvedain the finite element
unjform fin' (10 Marks)
transfer problem with reference to straight for linear interpolation of
b. Explain the finite element *oa"rirrg La rrrrp" functions (10 Marks)
tieat trunsfeielement)'
temperature field (one - dimensional
1of?
8. 7 a. Determine the nodal displacement and stresses in the erement shown in
fig. e7(a).1r0 Marks)
Ar = 500 mm2
Fie.Q7(a)
lokN Az = 2000mm2
:
Er 100 GPa
E2:200 GPa
300mm 300mm
3::::_1"":::rl-::tg::r^ *"gx oferementr (1);;
b.
truss etements shown in fig.
of 200mmz and .the e7(b). Au the
elements have an area
irt;; sil,- f;*.-;:';fi'bil;:
(10 Marks)
go l.^1,
Fie.Q7(b)
f
6o*t
A composite wall consists of three materials as
shown in fig. eg. The outer temperature
To = 200c' convective heat transfe, tuk., place
on the inner surface of the wall with Too =
8000c and h :25 wrmz o.[""ir.
it . ,.rp.rature distribution on the wall.
'C. (20 Marks)
f*: I, otc
^*Jli kr:20 WmoC
kz:30 WmoC
k3 = 50 WimoC
Fig.Q8 ' h-25WlmzoC
T*:8000C
*****
2 of2
9. 06M863
USN
sixth semester B.E. Degree Examination, June-July 2009
Modeling and Finite Element Analysis
Time: 3 hrs. Max. Marks:100
Note: Answer any FIVE full questions, selecting
at least TWO questions from each part.
PART _ A
L a. Explain the principle of minimum potential energy and principle of virtual work. (06 Marks)
+l
b. Evaluate the integral 1= J{fE' +2z ++2F by using 2 point and 3 point Gauss
-t
(06 Marks)
quadrature.
c. Sotve the following system of simultaneous equations by Gauss Elimination method:
x, -2x, * 6x, = Q
Zxr+Zxr*3x, =l
- Xr * 3x, = 0 (08 Marks)
2a. Explain briefly about node location system' (06 Marks)
Explain preprocessing and preprocessing in FEM. (06 Marks)
b.
Explain the basic steps involved in FEM. (08 Marks)
c.
3a. What are the considerations for choosing the order of polynomial functions? (06 Marks)
b. Explain convergence requirements of a polynomial displacement model. (06 Marks)
c. Derive the linear interpolation polynomial in terms of natural co-ordinate for 2-D triangular
elements. (08 Marks)
4a. What are Hermite shape functions of beam element? (06 Marks)
b. Derive the shape function for a quadratic bar element using Lagrangian method. (06 Marks)
c. Derive the shape function for a nine noded quadrilateral element. (08 Marks)
PART _ B
5a. Derive the element stiffness matrix for truss element. (10 Marks)
b. Derive the Jacobian matrix for 2D triangular element. (I0 Marks)
6a. Explain the types of boundary conditions in heat transfer problems. (r0 Marks)
b. Discuss the Galerkin approach for l-D heat conduction problem. (10 Marks)
la. Using the direct stiffness method, determine the nodal displacements of stepped bar shown
in figure Q7 (a). (lo Marks)
Er :200 GPa
Ez:70 GPa
Ar : 150 mm2
Az: i00 mm2
Fr:l0kW
Fz:5 kW
Fie. Q7 (a)
I ofZ
10. 06M863
7 b. For the truss shown in figure Q7 (b), find the assembled stiffness matrix. (10 Marks)
lkN
T
5oo ' E1 : E2:200 GPa
I
L
Fig. Q7 (b)
8 a. Determine the temperature distribution through the composite wall subjected to convection
heat loss on the right side surface with convective heat transfer coefficient shown if figure
Q8 (a). The ambient temperature is -5"c. (r0 Marks)
t:rdc
k= 6 =j_
Kr: )*cr.l/*.r-lt
, v+K
prq6.b- -ri._o,94
F
Fig. Q8 (a)
b' Determine the maximum deflection in the uniform cross section of Cantilever beam shown
in figure Q8 (b) by assuming the beam as a single element. (10 Marks)
loe i< Fj
E:7x10e N/m2
i I:4x10-a ma
----* l
Fig. Q8 (b)
**{.**
2 of2
11. u
ME6Fl
-
USN
OLD SCI{EME -;?
l--/--
sixth semester B.E. Degree Examination, July 20A6
Mechanical Engineering
Finite Element Methods
Time:3 hrs.l [Max. Marks:100
Note: Answer any FIVE full questions'
(03 Marks)
Define functional.
(10 Marks)
Derive Euler's Langranges's equation'
(07 Marks)
Expiain principle of minimum potential energy'
Briefly explain the steps involved in FEM'
(10 Marks)
(10 Marks)
Derive shape functions for CST triangular element in local co-ordinater.
Explain Banded matrix. Write an algor'ithm for Guass elimination technique'
(10 Marks)
Explain Raieigh's Ritz method in detail' (10 Marks)
4 What do you understand by weak form of differential equation. (05 Marks)
a*,..'u_-,
. 3j!.j
ft, ="lS$*
tY,,Y.'d.,c
:1
u
'.j
'-"F,
'-lt
ffi':gnr*:.et*r-** bar whose cross
i) For the above problem compute [B] and [c] matrix. It is^tapered
- section area decreases linearly from t-000 m*2 to 500 Take E:2x 10s N/mm2'
mm2.
ii) Use two elements and findthe nodal displacements. (15 Marks)
a. Derive shape functions and stiffness matrix for beam element' (15 Marks)
b. Explain the need of Jacobian transformation matrix. (05 Marks)
a. Explain in detail ISO - parametric, sub - parametric and Super - parametric
(10 Marks)
elements.
b. Explain "penalty approach" for handling the boundary conditions' (10 Marks)
a. Discuss the requirements to be fulfilled for the convergence of FEM solution' Marks)
(10
b. Derive FEM equation by variational principle' (10 Marks)
Write short notes on anY four :
a. Pascal's triangle d. Truss element
b. Local - co - ordinate sYstem e. Shell element
c. Patch test. f. EliminationaPProach'
:t:t:k* *
12. Poge No,,. I ME6FI
Reg. No.
Sixth Semesler B.E. Degree Exominolion, Jonuory/Februory 2006
Mechonicol Engineering
(Old Scheme)
Finite Elemenl Methods
Time: 3 hrs.) (Mox.Morks: 100
NOtg: Answer any FVE lull questions.
l. (o) Find the inverse of
[lt] (5 Morks)
ror a:
[3 ]]
,:l; {l
Find : i) AB ii1 gT 4T (5 Morks)
Solve by Gouss eliminotion
2q * * rJ: -7
3x'2
5r1 * n2 * a3: Q (10 Morks)
321 *2x214x3:11
2. @t Whot is finite element method? Whot ore the odvontoges of FEM over finite
difference method? (4 Morks)
(b) Exploin boundory volue ond initiol volue problems using suitoble exomples.
' (8 Morks)
(c) Exploin the steps involved in the finite element onolysis of solids ond structures.
(8 Morks)
.
3. (o) whot is meont by 'Bclnd width' of o motrix? Give on exomple. Exploin why it
should be minimized. (6 Morks)
(b) Stote the principle of minimum potentiol energy, ond derive on expression for totol
potentiol energy of o solid bor under compression. (6 Morks)
(c) Exploin the Royleigh-Rit method with on exomple, (8 Morks)
4. (o) Exploin the Golerkin's opprooch for obtoining stiffness motrix of o bor element,
(10 Morks)
(b) TwopointsPl(10,5)ondP2(80,10)onosolidbodydisplocesto PlOO.z,b.4)ond
P;(80.5,10.2) ofter looding. Determine normol ond sheor stroins. (10 Morks)
Confd.,.. 2
13. Poge No,,, 2 ME6FI
5. A solid stepped bor os shown in fig.l is subjected to on oxiol force. Determine the
following
D Element ond ossembled stiffness motrix
iD Displocement of eoch node
iii) Reoction force ot fixed end (20 Morks)
2-
A,=t0O mm.
*r=1-Oo mm' Lku
E = 200G Pa
I t'r= ro Q Po
6. (o) Whot is Jocobion Motrix? Derive o Jocobion motrix for Two-Dimensionol element.
(10 Morks)
,, l]
(b) Derive shope function CST triongulor element. (10 Morks)
7. @) Derive shope functions for o l-D quodrotic element with 3 nodes. (10 Mofts)
. (b) Exploin convergence criterio ond potch test in brief, (10 Morks)
8. Write short note on ony FOUR:
o) Voriotionol opprooch
'b) Hermition shope functions
c) Penolty opprooch for hondling boundory conditions
d) Logronge ond serendipity fomily of elements
e) ISO porometric: elements (5x4 Morks)
14. Page N0... 1
ME6F1
USN
Sixth Semester B.E. Degree Examination, July/August 2005
Mechanical En gineering
Finite Element Methods
Time: 3 hrs.I [Max.Marks : 100
Note: 1. Answer any FIVE full questions.
2. Missing data may be suitable assumed.
1. (a) Define positive definite matrix. (2 Marks)
(b) Solve the system of simultaneous equations given below by Gaussian elimination method.
2c1 * 2n2 * ns :9
n1*n2+fry:6 (10 Marks) '
2a1 * a2: 4
(c) Determine the inverse and eigen values of the given matrix A
. I 4 -2.286 (8 Marks)
^: -z.zJG
L
8
2. (a) Explain basic steps in FEM. (10 Marks)
(b) Explain potential energy of an elastic body. (5 Marks)
(c) Explain isoparametric, subparametric and superparametric element concept. (5 Marks)
(a) Derive the shape functions for a three node 1-D element in natural coordinate. (EMarks)
(b) Determine the displacemenl field, stress and support reactions in the body shown in
fis.Q3(b).
, P :60k/f E : 2O0kN lrrlnlZ : Et : Ez At :2000mm2 Az : l00Omrn2
(12 Marks)
F tS , a.z ir.
4. (a) Explain steps involved in Galerkin method. (10 Marks)
(b) Determine the defleetioh of canlilever beam of length I and loaded with a vertical load P
at the free end by Rayleigh-Ritz method. (10 Marks)
Contd.... 2
15. Page N0... 2 ME6Fi
5. (a) For the one dimensional truss element, develop the element stiffness matrix in the global
coordinate system. (10 Marks)
(b) Determine the nodal displacement and stress by using truss element. (10 Marks)
(a) Derive the stiffness matrix for a two node beam element. (10 Marks)
(b) For the beam shown in fig,Q.No.6(b). Determine the maximum deflectlon and the reaction
at the support. El is constant throughout the beam. (10 Marks)
7. (a) What is the significance of the band width? lllustrate best method of node numbering with
an example. (5 Marks)
(b) Evaluate the following by Gaussian quadrature
i) /: /]i (s"* + *, + #)da by one point and two point formula. (3 Marks)
ii) I : I: * OV 3-point formula. (8 Marks)
8. Write short nole on the following :
(a) Coordinate systems
(b) Convergence criteria
(c) Variational method
(d) Plane stress and plane strain conditions
(e) Penalty approach for handling boundary conditions. (5x4=20 Marks)
*****
16. Page N0,. 1 ME6F1
USN
Sixth Semester B.E, Degree Examination, January/February 2005
Mechanical Engineering
Finite Element Methods
Time: 3 hrs.l [Max.Marks : lO0
Note: Answer any FIVE full questions.
1. (a) Distinguish between :
Symmetric and skew symmetric matrix, transpose and inverse of a matrix. (4 Marks)
(b) What is a banded matrix? What are its merits? (4 Marks)
(c) Solve the following system of simultanegus equations :
11l2t2lrt:4
3*t-4xz-2r3-2
5r1l3r2*5r3- -7
either by Gaussian elimination method or malrix inversion method. (6 Marks)
(d) Find the eigen values of the matrix A
lz B
A- lr 4 -21
-2lr (6 Marks)
Lz 10 ,r j
2. (a) What is the basis of the Finite Element Method? Explain the basic steps involved in the
finite element method. (10 Marks)
(b) Determine the true displacement field for a two noded one dimensional tapered elemenl
shown in Fig.1. Also compute the stiffness matrix for this elemerit.
o c.n^--*J
I"t-eJ
At= loo
n;r'o[, "
q2&) , Ftq' t'
At :700rnz
-t12 :900mm2
A
. '2 (10 Marks)
An : ('* #)
Contd.... 2
17. Pase N0... 2 MEOF1
3. (a) What are the principles of continuum method? Compare this method with finite element
method clearly bringing out their relative merits. (6 Marks)
(b) Stale the variational principle of minimum potential energy. (4 Marks)
A uniform cantilever is subjected to a uniformly distributed load of W kN/m over ils
span 'l'. The displacement function is given as
y: 4(*, +612a,2 - lrs)where A is fhe displacement at the free end, Compute
the v"a'lue of the deflection A by the principle of minimum polential energy. Compare
this with the exact value. (r0 Marks)
4. (a) Derive the strain displacement relations. (2 Marks)
(b) b<plain the concepts of plane stress and plane strain with suitable examples, Also derive
the corresponding equations. (8 Marks)
(c) A uniform rod of lengh I fixed al both ends is subjected to a constant axial load of w
kN/m. Establish the displacement field and compute the stresses at the fixed ends and
rnidspan. What are lhe nalure and magnitude of the reaclions at the lwo ends? Use
Rayleigh-Bitz method. (10 Marks)
5. (a) What are interpolation rnodels? Give reasons for choosing polynomial funclions for such
npdels. (5 Marks)
(b) Explain briefly the penalty approach for handling displacement boundary conditions. '
(5 Marks)
Using the penalty approach, determine the nodal displacements and lhe stresses in
each material in the axially loaded bar shown in Fig.2
A l,^v, i*1,^r"
3oo t'tt'T 4 OO x^1^4
Area of (1):2400mm2
Area of (2) :6A0mm2
(10 Marks)
EAL:o'7 xTosNfrnrnz
Esteel:2x705Nlmrnz
6. (a) Explain the concept of isoparametric formulation. (5 Marks)
(b) Derive an elemenl stiffness matrix of a constant strain triangular element using the above
concept. (15 Marks)
Contd.... 3
18. Pase N0... 3 MEOF1
7, (a) what is a higher order element? what is its importance? (4 Marks)
(b) Derive the stiffness matrix for an element in the form
K: IW)r t"l tBl d,a
Show that the above matrix is symmetric. (10 Marks)
(c) A beam element carries a concentrated load P af { from one end. Obtain nodal loads
using the formulae of fixed beam. (6 Marks)
8. Write brief explanatory notes on any FOUR: (5x4=!Q [irs*s;
D Advantages and disadvantages of finite element methods
ii) Types of Finite Elements
iii) Boundarycondifions
iv) Principle of virtual work
v) Cohvergence criteria ** * **
19. Page No., 1
ME6Fl
USN
Sixth Semester B,E. Degree Examinatlon, July/August 2004
Mechanical Engineerlng
Finite Element Methods
Time: 3 hrs.I lMax.Marks : lOO
Note: 1, Answer any FIVE futt questions.
2. Assume suitable dak if necessiry.
1. (a) Explain with example.
i) Symmetric matarix
ii) Determinant of a matrix
iii) Positive definite matrix
iv) Half band width
v) Partitioning of matrices. (10 Marks)
(b) Give the aigorithm for fonruard elimination and back substitution of Gauss elimination for
a general matrix, (10 Marks)
2, (a) With suitable examples explain.
i) Essential (geometric) boundary condition
ii) Natural (force) boundary condition. (5 Marks)
(b) Outline the steps in finite element analysis. (5 Marks)
(c) State the. principle. of minimum potential energy. Obtain the equilibrium equation of the
system shown in fig 2.c using the principle of minimum potential energy. (10 Marks)
3. (a) Derive the equilibrium equation of 3D elastic body occupying a volume V and having a
: surface S, subjected to body force and a concentiated lodd. (10 Marks)
(b) An elastic bar of length L, modulus of elasticity E, area of cross section A, which is fixed
at one end and is subjected to axial load at the other end. Obtain the Euler equation
governing the bar, and natural boundary conditions. (10 Marks)
4. (a) For a two noded one dimensional element, show that the strain and stress are constant
with in the element. (10 Marks)
(b) Explain the criteria for monotonic convergence. (10 Marks)
5. (a) A component shown in fig. 5(a) is subjected to a load 5 kN. Detennine the lollowoing.
ii Element stiffness matrices
iD B - matrices
iii) Dispiacemerrts and strains
iv) Stresses and reactions.
Obtain the stiffness matrix and load vector assuming two elements, (12 Marks)
(b) What are characteristics of stiffness matrix ? (8 Marks)
6. (a) For a pin jointed configuration shown in Fig 6.a determine the stilfness matrix. Also
determine g, interms of g,. (10 Marks)
(b) Derive the Hermite shape functions of a beam. (r0 Marks)
Contd.... 2
20. :
Fage No... 2 ME6FT
7. (a) Evaluate
I
I [r,,* ;r*ffif*
-1
Using two point Gauss quadrature. (5 Marks)
(b) Derive the expression for shape functions of eight noded isoparametric element. (15Marks)
8, (a) Determine the Jacobian for the triangular element shown in lig Q8.a. (5 Marks)
(b) Give the element number and mode numbers for the structure shown in Fig Q 8.b, so as
matrix.
to minimize the half band width of the resulting stilfness (5 Marks)
(c) For the beam shown in fig Q.8c. obtain the global stiffness matrix. (10 Marks)
F
fi?. qL. c
vf, ts*' +'ol
)I o'. g-- 7oxto3 ^'/t",'ol
-/-
I fZ A= l3oo ss +ozn'
I J CLo,P) L= S m
*
fi3. Q6.o
''l s0
ooo
c+1) mm
. / A.: 5oo ms ,
gnz Qoo
c z 3.5) ri , too 6Pa
Ct.gr.l L'; zoo a'oo-
63' QB'o- FS' E(")
+R
=l qe. b
ng.
7* l'rD ---+L t -o ----l
|[---6--G-re-Z-
/,r-'----=---'---i{--a----v
-7.,
I ,=2-ooePd
Fs, Ee .c ?=- +^iie *'"-4
; - r -,r^O
*****
21. Page No... 1 ME6Fl
Heg. No.
Sixth Semester B.E. Degree Examination, January/February 2009
Mechanical Engineering
Finite Elembnt Methods
Tirne: 3 hrs.I lMax.Marks : IOO
Note: Answer any FIVE questions. l
1. (a) Solve the following system of simultaneous equations by Gaussian Elimination Method.
t1 -2n2 f 613 - 0 l
l
2a1*2c2*3n3-3
-rr*3r2-2 (10 Marks)
(b) Find the inverse of the following matrices
l0 1 21
f1 2
', Ll?il ilL;:, ll -21
(5+5 Marks)
2. (a) Explain the theorem,of minimum potential energy. Distinguish between minimum potential
-
theorem and principle of virtual displacement" (10 Marks)
(b) Explain the basic steps in the formulation of finite element analysis. (10 Marks)
3. (a) ,Flnd_thg.s!re!9 al w.:0 and displacement at the mid - point of the rod shown below.
Use Raleigh Ritz method
A)
I
Tq-Ke E = L1q6J- (Yo*,,.3'S ^".U.*[-y
A' luniF (A tea
fl(r.,1-r a-h.orr
(10 Marks)
(b) Explain plane stress and plane strain methods with rerevant equations. (10 Marks)
4. (a) Explain the penalty approach for handling the specified displacement boundary conditions.
(10 Marks)
Contd.... 2
22. Page N0... 2 ME6F1
(b) For the {ollowing figure (bar), find the nodal displacements. The cross sectional area
decreases linearly from 1000rnm2 lo 500mm2. Use two elements.
Take E :2x1O5MPa,7:0.3
,t 5ooss
lbbo -, looo A1
k- J$o''twr 4
- (10 Marks)
(a) Explain convergence criteria in detail, (10 Marks)
to) Derive shape functions for 'CST' element from
generalized co-ordinates. (10 Marks)
(a) Derive the stiffness matrix for a two noded beam element (12 Marks)
i
(b) Distinguish between isoparametric, sub-parametric and super-parametric elements.la uarxsl"
l
[^ 7. Consider the 4 -bur truss shown below, Determine.
i) Element stiffness matrix for each element
I
I
ii) Using eliminations approach to solve for the nodal displacements.
(iiD Calculate stresses in each element. (20 Marks)
+v 2-gooor..t (n.+i5 Ja-svr*,; ll
Qg Ar
t
3otv t"t
@
I 20,0001.; >1
rQrC)
4-4O
I
* *r
I
Write shorl notes on any FOUR of the following.
a) Eliminationapproach
b) Patch test
c) Galerkin's approach
d) Geometric isotropy
e) Post Processing
f) LST triangular element ** * ** (5x&20 Marks)
23. a
Poge No.,. I ME6FI
Reg. No.
Sixth Semesler B.E. Degree Exominolion, Jonuory/Februory 2006
Mechonicol Engineeilng
(Old Scheme)
Finiie Elemenl Methods
'1.
Time: 3 hrs.) ':.
(Mox.Morks: 100
NOle: Answer ony FIVE tuil queslions.
I. (o) Find the inverse of
[r ol
lo rl (5 Morks)
,o, a:
[3 1] ,:l; {l
Find : i) AB ii1 BT ar (5 Morks)
(c) Solve by Gouss eliminotion
2*t+3a2*nJ:-1
541*e2*rs:0 (10 Morks)
3rr + 2a2l4a3 -']".1
2. @, Whot is finite element method? Whot ore the odvontoges of FEM over finite
difference method? (4 Morks)
(b) Exploin boundory volue ond initiol volue problems using suitoble exomples.
' (8 Morks)
(c) Exploin the steps involved in the finite element onolysis of solids ond structures.
(S Morks)
: .
3. tol whot is meont by 'Bcind width' of o motrix? Give on exomple, Exploin why it
should be minimized, (6 Morks)
(b) Stote the principle of minimum potentiol energy, ond derive on expression for totol
potentiol energy of o solid bor under compression. (6 Morks)
(c) Exploin the Royleigh-Ritz method with on exompte. (8 Morks)
4. (o) Exploin the Golerkin's opprooch for obtoining stiffness motrix of o bor element,
(10 Morks)
(b) TwopointsPl(10,8)ondP2(80,10)onosotidbodydisptocesto pl(Lo.z,b.4)ond
Pj(80.5,10.2) ofter looding. Determine normol ond sheor stroins, (t0 Mql1s)
Confd.... 2
24. Poge No,,, 2 ME6FI
5. A solid stepped bor os shown in fig.l is subjected to on oxiol force. Determine the
following
i) Element ond ossembled stiffness motrix
iD Displocement of eoch''node I
iii) Reoction force of fixed end (20 Morks)
2-
A,= tOo hm ,
*r=LOo mhn- h-k u
g = 2,00G Pa
rt"= lo q Pq
6. (o) Whot is Jocobion Motrix? Derive o Jocobion motrix for Two-Dimensionol element.
(10 Morks)
(b) Derive shope function CST triongulor element, (10 Morks)
7. @| Derive shope functions for o l-D quodrotic element with 3 nodes. (t0 Morks)
(b) Exploin convergence criterio ond potch test in brief. (10 Morks)
8. Write short note on ony FOUR:
o) Voriotionol opprooch
6) 'Hermition shope functions
c) Penolty opprooch for hondling boundory conditions
d) Logronge ond serendipity fomily of elements
e) ISO porometric elements (5x4 Mqrks)
25. Page No.., 1
ME6Fl
USN
$ixth sernester B"E. Degree Examination, July/August 2004
Mechanical Engineering
Finite Element Methods
3 hrs.l [Max.Marks : 10O
Note: 1. Answer any F|VE full questions.
2. Assume suitable data if necessary.
1. (a) Explain with example,
i) Syrnmetric matarix
ii) Determinant of a matrix
iii) Pcsitive definite matrix
iv) Half band width
v) Partitioning of matrices. (10 Marks)
(b) Give the algorithm for forurard elimination and back substitution of Gauss elimination for
a general matrix. (io Marks)
2. (a) With suitable examples explain.
i) Essential (geometric) boundary mndition
ii) Ndtural (force) boundary condition. (5 Marks)
(b) outline the steps in finite element analysis. (5 Marks)
(c) State the principle of minimum potential energy. Obtain the equilibrium equation ol the
system shown in fig 2.c using the principle of-minimum potentidl energy. (10 Marks)
3. (a) Derive the equilibriqm equation ol 3D.elastic body oc.cypyt"ng a volume V and having a
surface s, subjected to body force and a concentrated lddd. (r0 Marks)
(b) ry elastic bar of length.L, modulusof elasticity E, area of cross section A, which is fixed
at one end and is subjected to axial load at-the other end. Obtain the'Euler equation
governing the bar, and natural boundary conditions. t10 Marks)
4. (a) Fo1 a two noded one dimensional element, show that the strain and stress are constant
with in the element" (ro Marks)
(b) Explain the criteria for monotonic convergence. (,l0 Marks)
5. (a) A component shown in fig. 5(a) is subjected to a load 5 kN. Determine the foilowoing.
i) Element stiffness matrices
ii) B - matrices
iii) Displaeements and strains
iv) Stresses and reactions.
Obtain the stiffness matrix and load vector assurning two eiements. ('t2 Marks)
(b) What are characteristics of stiffness matrix ? (8 Marks)
(a) For a.pin jointed configuration shown in Fig 6.a detennine the stiffness matrix. Also
determine qt interms of g,. (10 Marks)
(b) Derive the Hermite shape functions of a beam. (10 Marks)
Contd.... 2
26. Page Nor, 2 illE6F1
7. (a) Evaluate
1
-1
Using two point Gauss quadrature. (5 Marks)
{b) Derive the expression for shape lunctions of eight noded isoparametric element. (15 Marks)
8. (a) Determine the Jacobian for the triangular element shown in fig eg.a, (5 Marks)
(b) Give thp element number and mode numbers for the structure shown in Fig Q 8.b, so as
to minimize the half band width of the resulting stiffness matrix.
(5 Marks)
(c) For the beam shown in fig Q.8c. obtain the global stiffness matrix. (10 Marks)
i
I
fi?. qL. e-
r'
t
vf, ct'o'+c>
)t o', -/" Et 7oxto3^l/t''ol
I {/Clo,rs) A= l3oo ss m"n'
I J.
* V-- S n
t
fr3. Q6.a
le
ooo
, )',+;') mm
/ A.; 5oo mw ,
gn: QOO
C z s's) c : 0o GPa'
L1.51) L'; r-oo aOo.
63' Qe'o- F3 5(a1
s
+R
ft. q8. b
,@
h-
/l
I'ro nD I .tlo -,
^^
,-2oold.
.qc.c i= "^lo6+nYo*
t ,
*****
27. a
Page No... 1 ME6F1
USN
Sixth Semester B.E. Degree Examination, January/February 2004
Mechanical Engineering
Finite Element Methods
Time: 3 hrs.l [Max.Marks : IO0
Note: 1. Answer any FIVE full questions.
2. Missing data may be suitably assumed,
1. (a) Find the eigen values of
A- 4 -{51 (5 Marks)
-,/3 a l
(b) Solve the following system of simultaneous equations by Gaussian elimination method.
2e1*12!3rs:t$
4r1*r21.a3:$
3n1*2r2 * rs:3 (10 Marks)
(c) Define the following with example
i) Skew matrix
ii) Symmetric banded matrix. (5 Marks)
(a) Explain difference between continuum method and finite element method, (5 Marks)
(b) Explain basic steps involved in FEM. (10 Marks)
(c) Explain principle of minimum potential energy and virlual work. (5 Marks)
(a) Expain steps involved in Rayleigh - Ritz method. (B Marks)
(b) Determine the deflection at the free end of a cantilever beam of length '1, carrying a
vertical load 'P' at its free end by Rayleigh Ritzmethod (i0 Marks)
(c) List the demerits of cantinuum methods. (2 Marks)
4' (a) Derive strain displacement matrix, stiffness matrix for one dimentional bar element.
(8 Marks)
(b) Solve for stresses and strains for the following problem by using bar element.
(12 Marks)
? = loco l.J
/t<_
E:2.7xlA5Nfrrurnz
At :5Omm2
Az :25mm2
P : 100011
Contd.... 2
28. Page N0... 2
ME6F1
5. (a) Derive stiffness matrix for a truss element. (8 Marks)
(b) For a pin jointed configuration shown in figure, determine nodal
displacements and stress
by using truss elemenls.
f : looo;?
T
5oo r
Ar : LAAmmz
t :lSovnr'
Az:125Amm,2
E:200GPa (12 Marks)
6. (a) Compute.the deflection of simply supported beam carrying concentrated load at its centre,
Use two beam elments.
(16 Marks)
(b) ls FEM analysis applicable for highly elastic materials? Explain. (4 Marks)
7. Find the displacement of node 1 in the triangurar element shown using one triangular
element. Also find stress and strain in the elefient.
. 1+----- 3o n
(-3o,o ) loo l,/
l r.-__ 5o
I
2o
I.(,2,o )
I
E:70GPa L
7:0.3 c 3o,
Le : lAmm (20 Marks)
Write short notes on any FOUR of the following :
a) Static condensation
b) lsoparametric, super parametric and subparametrlc element
c) Static and kinematic boundary condition
d) Lagrangian and Hermite shape functions
e) Convergencecriterion (4x5=2Q fYl2Y[s)
*****
29. a
-----
'
-'-t/'
Page N0,,. I ME6F1
USN
Sixth Semester B.E. Degree Examination, July/August 2000
Mechanical Engineering
Finite Element Methods
Time: 3 hrs.I [Max.Marks : 10O
Note: Answer any FIVE futt questions.
1. (a) Given o:l; i], ort.,*in.
i) Inverse of matrix ii) Eigen values. (10 Marks)
(b) lf ,7"r: [€, 1-(2], evaluate /, wT Nag (5 Marks)
(c) Explain symmetric banded matrix. (5 Marks)
2. (a) With an example explain Rayleigh -Ritz method. (10 Marks)
(b) State the principle of minimum potential energy. (4 Marks)
(c) Sketch the quadratic and Hermite shape functions. (6 Marks)
3. (a) Derive the following characteristics of three noded l-D element.
i) Strain displacement matrix [B]
ii) Stiffness matrix [frr] (10 Marks)
(b) Solve for nodal displacements and stresses for the structure shown in fig 1. Use penality
approach to apply boundary csnditions. (10 Marks)
h t"laao n{' 2"17o frrn*
.,€ r 2lo$ pa
*1,€=zo$fo"
?JaoN
4. (a) Derive an expression for
i) Jacobian matrix
ii) Stiffness matrix for axisymmetric element. (10 Marks)
Contd.... 2
30. _ _
, ___:_
Page N0... 2 ME6F1
(b) 0onsider a rectangular element as shown in Fig.2. Evaluate J and B matrices at
(=0, =0, (10 Markr)
+
+
C1i,o,{)
cv>-
t A,>
L -t a)
(0, ,)
5. (a) Explain with neat sketches the library of elements used in FEM. (10 Marks)
(b) Using Gaussian quadrature, evaluate the following integral by two point formula
d, /], (€2 + zrt€ + rf) dt drt (10 Marks)
6, (a) For the pin jointed_ configuration shown in Fig.3 determine the stiflness values of
'' kn, l*e and,-k2, of global stiffness matrix. (10 Marks)
O hra'tgroivl"nL'
/L
L
I
I
"l/
b MvY'
vjup ln7
>}lac?", ,
E-
(b) Derive an expression lor stiffness matrix ol a two noded beam element. (10 Marks)
7. (a) Explain in detail the leatures of any one commercial FEA software package. (l0Marks)
(b) Bring out the differences between continuum methods and FEM. (10 Marks)
Write short notes on any FOUR :
a) State functions
b) Galerkin methods
c) Elimination method of handling boundary conditions.
d) Temperature effects
e) Convergence criteria. ** * **
(4x5=20 Marks)
31. I
Page No... l ME6Fl
Reg. No.
sixth serrester B.E. Degree Examflnatlon, Februar5r zooz
Mechanical Englneering
Ftntte Element Methods
Time: 3 hrs.l [Max.Marks : I0O
Note: Answer any FIVE full questions,
1. (a) What is a banded matrix and state its advantage?
(b) Calculate the eigen values of the matrix A.
o:lt ?,1
lz 0 1l
(c) Evaluate .4.-1 when -d. : lo 4 ol
fr o 2l
(d) Drptain Gauss-elimination method to solve a set of simultaneous equations.
(4X6=20 Marks)
2. (a) What is finite element method? Drplain the basic steps in the formulation of
finite element analysis. (12 Marks)
(b) Differentiate between continuum method and finite element mettrod. (8 Marks)
3. (a) A rectangular bar in subjected to an axial load P as shown in fig.l. Derive an
expression for potential energr and hence determine the extreme value of the
potential 9le-1ry forthe-following data. Modutus of elasticity E :200Gpa,
load P - SkNr length of the bar I : L00mm, width of the ba; b :20mm arrd
thickness of the bar t : Llmm. Also state its equilibrium stability. . ,
l_
{
T
-+
'L
Fta, I
iff
(b) use Rayleigh-Ritz method to find the disptacement and the stress .,tilIill
point of the rod as shown in fig.2. The area of cross section of the bar is 4OO
mmz and. the modulus of elasticity of the material is 7O GPa. Assume the
displacement to be second degree polynomial. (to Marks)
4. (a)
-Explain the elimination approach for handling the specified displacement
boundary conditions (5 Marks)
Contd.... 2