The Small Strain TL C1 Plane Beam


 Junior Marshall
 1 years ago
 Views:
Transcription
1 . H The Small Strain TL C Plane Beam H
2 Appendix H: THE SMALL STRAIN TL C PLANE BEAM H H. SUMMARY This Appendix derives the discrete equations of a geometrically nonlinear, C (Hermitian), prismatic, plane beamcolumn in the framework of the Total Lagrangian (TL) description. The formulation is restricted to the three deformational degrees of freedom: d, θ and θ shown in Figure H.. The element rigid body motions have been removed by forcing the transverse deflections at the end nodes to vanish. The strains are assumed to be small while the cross section rotations θ are small but finite. Given the foregoing kinematic limitations, this element is evidently of no use per se in geometrically nonlinear analysis. Its value is in providing the local equations for a TL/CR formulation H. FORMULATION OF GOVERING EQUATIONS H.. Kinematics We consider a geometrically nonlinear, prismatic, homogenous, isotropic elastic, plane beam element that deforms in the x, y plane as shown in Figure H.. The element has cross section area A and moment of inertia I in the reference configuration, and elastic modulus E. y d/ θ d/ C R L L C θ x Figure H. Kinematics of TL Hermitian beam element The plane motion of the beam is described by the two dimensional displacement field {u x (x, y), u y (x, y)} where u x and u y are the axial and transverse displacement components, respectively, of arbitrary points within the element. The rotation of the cross section is θ(x), which is assumed small. The following kinematic assumptions of thin beam theory are used [ ux (x, y) = u a x (x) y ua y (x) ] [ u a ] x u y (x, y) u a y (x) = x (x) yθ(x) u a y (x) (H.) where u a x and ua y denote the displacements of the neutral axes, and θ(x) = ua y / x is the rotation of the cross section. The three degrees of freedom of the beam element are ] u e = [ d θ θ (H.) H
3 H 3 H. FORMULATION OF GOVERING EQUATIONS H.. Strains We introduce the notation ɛ = u x x, = θ x = u a y x. (H.3) for engineering axial strain and beam curvature, respectively. The exact GreenLagrange measure of axial strain is e = u ( ) x x + ux + x ( ) u y = ɛ y + x (ɛ y) + θ (H.4) This can be expressed in terms of the displacement gradients as follows: e = h T g + gt Hg = c T g (H.5) where g = u a x / x u a y / x u a y / x = [ ɛ θ ], h =, H = y [ y y y ] (H.6) We simplify this expression by dropping all y dependent terms form the H matrix: Ĥ = [ ] (H.7) The simplified axial strain is e = h T g + gt Ĥg = ɛ y + ɛ + θ (H.8) The rational for this selective simplification is that e a = ɛ + ɛ is the GL mean axial strain. If the ɛ term is retained, a simpler geometric stiffness is obtained. The term θ is the main nonlinear effect contributed by the section rotations. The vectors that appear in the CCF formulation of TL finite elements discussed in Chapters  are + ɛ + b = h + Hg = θ, c = h + ɛ Hg = θ, (H.9) y y H 3
4 Appendix H: THE SMALL STRAIN TL C PLANE BEAM H 4 N M C M V N V V M M C N V Figure H.. Stress resultants in reference and current configurations. Configurations shown offset for clarity. N H..3 Stresses and Stress Resultants The stress resultants in the reference configuration are N, M and M. The initial shear force is V = (M M )/L. The axial force N and transvese shear force V are constant along the element, whereas the bending moment M (x) is linearly interpolated from M = M ( x/l ) + M x/l. See Figure H. for sign conventions. The initial PK axial stress is computed using beam theory: s = N M y (H.) A I Denote by N, V and M the stress resultants in the current configuration. Whereas N and V are constant along the element, M = M(x) varies linearly along the length because this is a Hermitian model, which relies on cubic transverse displacements. Consequently we will define its variation by the two node values M and M. The shear V is recovered from equilibrium as V = (M M )/L, which is also constant. The PK axial stress in the current state is s = s + Ee = s + Ec T g,or inserting (H.9): s = s + E ( ɛ + ɛ + θ y ) (H.) H..4 Constitutive Equations Integrating (H.) over the cross section one gets the constitutive equations in terms of resultants: N = sda= s A + EA (ɛ + ɛ + θ ) = N + EA (e a + θ ), A M = ys da = M + EI A (H.) H..5 Strain Energy Density H 4
5 H 5 H. FORMULATION OF GOVERING EQUATIONS We shall use the CCF formulation presented in Chapter to derive the stiffness equations. Using α = β = (not a spectral form) one obtains the core energy of a beam particle as U = ( + ɛ) + 4 θ 3 y 3 θ y( + 3 ɛ) = gt E 3 θ 4 (ɛ + θ ) + 3 (ɛ y) 3 yθ + s g y( + 3 ɛ) yθ y 3 (H.3) Integration over this cross section yields the strain energy per unit of beam length: U A = gt A (Ecc T + s H) da g = gt ( + ɛ) A ( + ɛ)θ A E ( + ɛ)θ A 4 θ A + N g I (H.4) To obtain the element energy it is necessary to specify the variation of ɛ, θ and along the beam. At this point shape functions have to be introduced. H..6 Shape Functions Define the isoparametric coordinate ξ = x/l. The displacement interpolation is taken to be the same used for the linear beam element: [ u a x u a = ξ ] d y 8 L ( ξ) ( + ξ) 8 L ( + ξ) θ. (H.5) ( ξ) θ From this the displacement gradients are ɛ g = θ = d L 4 L (ξ )(3ξ + ) 4 L ( + ξ)(3ξ ) θ = Gu e. (H.6) 3ξ 3ξ + The rotation θ varies quadritically and the curvature θ linearly. The node values are obtained on setting ξ =±: ɛ g = θ = ɛ L u e, g L = θ = L u e (H.7) 4 L 4 H..7 Element Energy The strain energy of the element can be now obtained by expressing the gradients g = Gu e and integrating over the length. the result can be expressed as U e = L L U A da= U A L dξ = (ue ) T K U u e (H.8) H 5 θ
6 Appendix H: THE SMALL STRAIN TL C PLANE BEAM H 6 where the energy stiffness is the sum of three contributions: K U = K U a + KU b + KU N. These come from the axial deformations, bending deformations and initial stress, respectively: K U a = EA L K U b = EI L ( + ( + ɛ) ɛ)(4θ θ )L 6 ( + ɛ)(4θ θ )L (θ 3θ θ + θ )L 6 84 ( + ɛ)( θ + 4θ )L ( 3θ + 4θ θ 3θ )L , K U N = N L 4 L /5 L /3. L /3 L /5 ( + ɛ)( θ + 4θ )L 6 ( 3θ + 4θ θ 3θ )L 68, (θ 3θ θ + θ )L 84 (H.9) H.3 INTERNAL FORCE The internal force p is obtained as the derivative ( ) p = U e u = K U + e (ue ) T KU u e = K p u e u e (H.) The internal force stiffness is again the sum of three contributions: K p = K p a + Kp b + Kp N. These come from the axial deformations, bending deformations and initial stress, respectively: K p a = EA L K p b = EI L + 3 ɛ + (3 + ɛ)(4θ ɛ θ )L (3 + ɛ)(4θ θ )L (θ 3θ θ + θ )L 4 (3 + ɛ)( θ + 4θ )L ( 3θ + 4θ θ 3θ )L 84 4, K p N = N L 4 L /5 L /3. L /3 L /5 (3 + ɛ)( θ + 4θ )L ( 3θ + 4θ θ 3θ )L 84, (θ 3θ θ + θ )L 4 (H.) H.4 TANGENT STIFFNESS The tangent stiffness K is obtained as the derivative K = p ) (K u = r + (u e ) T Kr u e e u e (H.) This is again the sum of three contributions: K = K a + K b + K N, which come from the axial H 6
7 H 7 H.4 TANGENT STIFFNESS deformations, bending deformations and current stress, respectively: K a = EA L K b = EI L ( + ɛ) ( + ɛ)(4θ θ )L 3 ( + ɛ)(4θ θ )L (θ 3θ θ + θ )L 3 ( + ɛ)( θ + 4θ )L ( 3θ + 4θ θ 3θ )L 3 4 4, K N = N L 4L L L 4L The material stiffness is K M = K a + K b and the geometric stiffness is K G = K N. ( + ɛ)( θ + 4θ )L 3 ( 3θ + 4θ θ 3θ )L 4, (θ 3θ θ + θ )L (H.3) H 7
Finite Element Formulation for Beams  Handout 2 
Finite Element Formulation for Beams  Handout 2  Dr Fehmi Cirak (fc286@) Completed Version Review of EulerBernoulli Beam Physical beam model midline Beam domain in threedimensions Midline, also called
More informationFinite Element Formulation for Plates  Handout 3 
Finite Element Formulation for Plates  Handout 3  Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the
More informationKeywords: Structural System, Structural Analysis, Discrete Modeling, Matrix Analysis of Structures, Linear Elastic Analysis.
STRUCTURAL ANALYSIS Worsak KanokNukulchai Asian Institute of Technology, Thailand Keywords: Structural System, Structural Analysis, Discrete Modeling, Matrix Analysis of Structures, Linear Elastic Analysis.
More informationIndeterminate Analysis Force Method 1
Indeterminate Analysis Force Method 1 The force (flexibility) method expresses the relationships between displacements and forces that exist in a structure. Primary objective of the force method is to
More informationStiffness Methods for Systematic Analysis of Structures (Ref: Chapters 14, 15, 16)
Stiffness Methods for Systematic Analysis of Structures (Ref: Chapters 14, 15, 16) The Stiffness method provides a very systematic way of analyzing determinate and indeterminate structures. Recall Force
More informationGeometric Stiffness Effects in 2D and 3D Frames
Geometric Stiffness Effects in D and 3D Frames CEE 41. Matrix Structural Analsis Department of Civil and Environmental Engineering Duke Universit Henri Gavin Fall, 1 In situations in which deformations
More informationAN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS
AN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS By: Eduardo DeSantiago, PhD, PE, SE Table of Contents SECTION I INTRODUCTION... 2 SECTION II 1D EXAMPLE... 2 SECTION III DISCUSSION...
More informationChapter 4. Shape Functions
Chapter 4 Shape Functions In the finite element method, continuous models are approximated using information at a finite number of discrete locations. Dividing the structure into discrete elements is called
More informationApplied Finite Element Analysis. M. E. Barkey. Aerospace Engineering and Mechanics. The University of Alabama
Applied Finite Element Analysis M. E. Barkey Aerospace Engineering and Mechanics The University of Alabama M. E. Barkey Applied Finite Element Analysis 1 Course Objectives To introduce the graduate students
More informationThe Basics of FEA Procedure
CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring
More informationMATHEMATICAL MODEL FOR VIBRATIONS OF NONUNIFORM FLEXURAL BEAMS
Engineering MECHANICS, Vol. 15, 2008, No. 1, p. 3 11 3 MATHEMATICAL MODEL FOR VIBRATIONS OF NONUNIFORM FLEXURAL BEAMS Mohamed Hussien Taha, Samir Abohadima* A simplified mathematical model for free vibrations
More informationStrain and deformation
Outline Strain and deformation a global overview Mark van Kraaij Seminar on Continuum Mechanics Outline Continuum mechanics Continuum mechanics Continuum mechanics is a branch of mechanics concerned with
More informationNew approaches in Eurocode 3 efficient global structural design
New approaches in Eurocode 3 efficient global structural design Part 1: 3D model based analysis using general beamcolumn FEM Ferenc Papp* and József Szalai ** * Associate Professor, Department of Structural
More informationTypes of Elements
chapter : Modeling and Simulation 439 142 20 600 Then from the first equation, P 1 = 140(0.0714) = 9.996 kn. 280 = MPa =, psi The structure pushes on the wall with a force of 9.996 kn. (Note: we could
More informationAnalysis of Plane Frames
Plane frames are twodimensional structures constructed with straight elements connected together by rigid and/or hinged connections. rames are subjected to loads and reactions that lie in the plane of
More informationBEAM THEORIES The difference between EulerBernoulli and Timoschenko
BEAM THEORIES The difference between EulerBernoulli and Timoschenko Uemuet Goerguelue Two mathematical models, namely the sheardeformable (Timoshenko) model and the shearindeformable (EulerBernoulli)
More informationStability Of Structures: Basic Concepts
23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for
More informationFinite Element Method (ENGC 6321) Syllabus. Second Semester 20132014
Finite Element Method Finite Element Method (ENGC 6321) Syllabus Second Semester 20132014 Objectives Understand the basic theory of the FEM Know the behaviour and usage of each type of elements covered
More informationEML 5526 FEA Project 1 Alexander, Dylan. Project 1 Finite Element Analysis and Design of a Plane Truss
Problem Statement: Project 1 Finite Element Analysis and Design of a Plane Truss The plane truss in Figure 1 is analyzed using finite element analysis (FEA) for three load cases: A) Axial load: 10,000
More informationStress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t
Stress and Deformation Analysis Material in this lecture was taken from chapter 3 of Representing Stresses on a Stress Element One main goals of stress analysis is to determine the point within a loadcarrying
More information1. a) Discuss how finite element is evolved in engineering field. (8) b) Explain the finite element idealization of structures with examples.
M.TECH. DEGREE EXAMINATION Branch: Civil Engineering Specialization Geomechanics and structures Model Question Paper  I MCEGS 1062 FINITE ELEMENT ANALYSIS Time: 3 hours Maximum: 100 Marks Answer ALL
More information10 Space Truss and Space Frame Analysis
10 Space Truss and Space Frame Analysis 10.1 Introduction One dimensional models can be very accurate and very cost effective in the proper applications. For example, a hollow tube may require many thousands
More informationModule 3. Analysis of Statically Indeterminate Structures by the Displacement Method. Version 2 CE IIT, Kharagpur
odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 21 The oment Distribution ethod: rames with Sidesway Instructional Objectives After reading this chapter the student
More informationThe elements used in commercial codes can be classified in two basic categories:
CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for
More informationBending Stress in Beams
93673600 Bending Stress in Beams Derive a relationship for bending stress in a beam: Basic Assumptions:. Deflections are very small with respect to the depth of the beam. Plane sections before bending
More informationBeam, Plate, and Shell Elements Part I
Topic 19 Beam, Plate, and Shell Elements Part I Contents: Brief review of major formulation approaches The degeneration of a threedimensional continuum to beam and shell behavior Basic kinematic and static
More informationM x (a) (b) (c) Figure 2: Lateral Buckling The positions of the beam shown in Figures 2a and 2b should be considered as two possible equilibrium posit
Lateral Stability of a Slender Cantilever Beam With End Load Erik Thompson Consider the slender cantilever beam with an end load shown in Figure 1. The bending moment at any crosssection is in the xdirection.
More informationElasticity Theory Basics
G22.3033002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationUnit 21 Influence Coefficients
Unit 21 Influence Coefficients Readings: Rivello 6.6, 6.13 (again), 10.5 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Have considered the vibrational behavior of
More informationCHAPTER 9 MULTIDEGREEOFFREEDOM SYSTEMS Equations of Motion, Problem Statement, and Solution Methods
CHAPTER 9 MULTIDEGREEOFFREEDOM SYSTEMS Equations of Motion, Problem Statement, and Solution Methods Twostory shear building A shear building is the building whose floor systems are rigid in flexure
More informationCOMPUTATIONAL ENGINEERING OF FINITE ELEMENT MODELLING FOR AUTOMOTIVE APPLICATION USING ABAQUS
International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 7, Issue 2, MarchApril 2016, pp. 30 52, Article ID: IJARET_07_02_004 Available online at http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=7&itype=2
More informationModule 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur
Module Analysis of Statically Indeterminate Structures by the Direct Stiffness Method Version CE IIT, haragpur esson 7 The Direct Stiffness Method: Beams Version CE IIT, haragpur Instructional Objectives
More informationFUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS
FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS With Mathematica and MATLAB Computations M. ASGHAR BHATTI WILEY JOHN WILEY & SONS, INC. CONTENTS OF THE BOOK WEB SITE PREFACE xi xiii 1 FINITE ELEMENT
More informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
More informationReliable FEModeling with ANSYS
Reliable FEModeling with ANSYS Thomas Nelson, Erke Wang CADFEM GmbH, Munich, Germany Abstract ANSYS is one of the leading commercial finite element programs in the world and can be applied to a large
More informationCAD and Finite Element Analysis
CAD and Finite Element Analysis Most ME CAD applications require a FEA in one or more areas: Stress Analysis Thermal Analysis Structural Dynamics Computational Fluid Dynamics (CFD) Electromagnetics Analysis...
More informationNonlinear analysis and formfinding in GSA Training Course
Nonlinear analysis and formfinding in GSA Training Course Nonlinear analysis and formfinding in GSA 1 of 47 Oasys Ltd Nonlinear analysis and formfinding in GSA 2 of 47 Using the GSA GsRelax Solver
More informationFinite Element Analysis
A Course Material on By Mr. L.VINOTH ME ASSISTANT PROFESSOR DEPARTMENT OF MECHANICAL ENGINEERING SASURIE COLLEGE OF ENGINEERING VIJAYAMANGALAM 638 56 QUALITY CERTIFICATE This is to certify that the ecourse
More informationCHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS
1 CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS Written by: Sophia Hassiotis, January, 2003 Last revision: February, 2015 Modern methods of structural analysis overcome some of the
More informationDeflections. Question: What are Structural Deflections?
Question: What are Structural Deflections? Answer: The deformations or movements of a structure and its components, such as beams and trusses, from their original positions. It is as important for the
More information3 Concepts of Stress Analysis
3 Concepts of Stress Analysis 3.1 Introduction Here the concepts of stress analysis will be stated in a finite element context. That means that the primary unknown will be the (generalized) displacements.
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Analysis of Statically Indeterminate Structures by the Matrix Force Method esson 11 The Force Method of Analysis: Frames Instructional Objectives After reading this chapter the student will be able
More informationCourse in. Nonlinear FEM
Course in Introduction Outline Lecture 1 Introduction Lecture 2 Geometric nonlinearity Lecture 3 Material nonlinearity Lecture 4 Material nonlinearity continued Lecture 5 Geometric nonlinearity revisited
More informationChapter (3) SLOPE DEFLECTION METHOD
Chapter (3) SOPE DEFECTION ETHOD 3.1 Introduction: The methods of three moment equation, and consistent deformation method are represent the FORCE ETHOD of structural analysis, The slope deflection method
More informationMATERIALS SELECTION FOR SPECIFIC USE
MATERIALS SELECTION FOR SPECIFIC USE1 Subtopics 1 Density What determines density and stiffness? Material properties chart Design problems LOADING 2 STRENGTH AND STIFFNESS Stress is applied to a material
More informationThe Bernoulli Euler Beam
8 The ernoulli Euler eam 8 1 Chapter 8: THE ERNOULLI EULER E TLE OF CONTENTS Page 8.1. Introduction 8 3 8.2. The eam odel 8 3 8.2.1. Field Equations................. 8 3 8.2.2. oundary Conditions..............
More informationBack to Elements  Tetrahedra vs. Hexahedra
Back to Elements  Tetrahedra vs. Hexahedra Erke Wang, Thomas Nelson, Rainer Rauch CADFEM GmbH, Munich, Germany Abstract This paper presents some analytical results and some test results for different
More informationEFFICIENT NUMERICAL SIMULATION OF INDUSTRIAL SHEET METAL BENDING PROCESSES
ECCOMAS Congress 06 VII European Congress on Computational Methods in Applied Sciences and Engineering M. Papadrakakis, V. Papadopoulos, G. Stefanou, V. Plevris (eds.) Crete Island, Greece, 5 0 June 06
More informationDESIGN OF BEAMCOLUMNS  I
13 DESIGN OF BEACOLUNS  I INTRODUCTION Columns in practice rarely experience concentric axial compression alone. Since columns are usually parts of a frame, they experience both bending moment and axial
More informationSOME ASPECTS OF FINITE AMPLITUDE TRANSVERSE WAVES IN A COMPRESSIBLE HYPERELASTIC SOLID
SOME ASPECTS OF FINITE AMPLITUDE TRANSVERSE WAVES IN A COMPRESSIBLE HYPERELASTIC SOLID by J. B. HADDOW (Department of Mechanical Engineering, University of Victoria, Victoria, BC, Canada V8W P6) and H.
More informationTUTORIAL FOR RISA EDUCATIONAL
1. INTRODUCTION TUTORIAL FOR RISA EDUCATIONAL C.M. Uang and K.M. Leet The educational version of the software RISA2D, developed by RISA Technologies for the textbook Fundamentals of Structural Analysis,
More informationOn the Free Vibration Behavior of Cylindrical Shell Structures. Burak Ustundag
On the Free Vibration Behavior of Cylindrical Shell Structures by Burak Ustundag B.S., Mechanical Engineering Turkish Naval Academy, 2006 Submitted to the Department of Mechanical Engineering in Partial
More information8.2 Elastic Strain Energy
Section 8. 8. Elastic Strain Energy The strain energy stored in an elastic material upon deformation is calculated below for a number of different geometries and loading conditions. These expressions for
More informationDYNAMIC ANALYSIS OF THICK PLATES SUBJECTED TO EARTQUAKE
DYNAMIC ANALYSIS OF THICK PLATES SUBJECTED TO EARTQUAKE ÖZDEMİR Y. I, AYVAZ Y. Posta Adresi: Department of Civil Engineering, Karadeniz Technical University, 68 Trabzon, TURKEY Eposta: yaprakozdemir@hotmail.com
More informationMatrix Solution of Equations
Contents 8 Matrix Solution of Equations 8.1 Solution by Cramer s Rule 2 8.2 Solution by Inverse Matrix Method 13 8.3 Solution by Gauss Elimination 22 Learning outcomes In this Workbook you will learn to
More informationModeling solids: Finite Element Methods
Chapter 9 Modeling solids: Finite Element Methods NOTE: I found errors in some of the equations for the shear stress. I will correct them later. The static and timedependent modeling of solids is different
More informationLecture 4: Basic Review of Stress and Strain, Mechanics of Beams
MECH 466 Microelectromechanical Sstems Universit of Victoria Dept. of Mechanical Engineering Lecture 4: Basic Review of Stress and Strain, Mechanics of Beams 1 Overview Compliant Mechanisms Basics of Mechanics
More informationThe Finite Element Method for the Analysis of NonLinear and Dynamic Systems. Prof. Dr. Eleni Chatzi Lecture 117 September, 2015
The Finite Element Method for the Analysis of NonLinear and Dynamic Systems Prof. Dr. Eleni Chatzi Lecture 117 September, 215 Institute of Structural Engineering Method of Finite Elements II 1 Course
More informationBending Stress and Strain
Bending Stress and Strain DEFLECTIONS OF BEAMS When a beam with a straight longitudinal ais is loaded by lateral forces, the ais is deformed into a curve, called the deflection curve of the beam. We will
More information820446  ACMSM  Computer Applications in Solids Mechanics
Coordinating unit: 820  EUETIB  Barcelona College of Industrial Engineering Teaching unit: 737  RMEE  Department of Strength of Materials and Structural Engineering Academic year: Degree: 2015 BACHELOR'S
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture  01
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture  01 Welcome to the series of lectures, on finite element analysis. Before I start,
More informationRecommendations for finite element analysis for the design of reinforced concrete slabs
1400 Moment [knm/m] Moment peak over the columns not designed for 1200 Column width 1000 800 20 s120 20 s150 600 20 s180 20 s160 400 20 s230 16 s170 16 s240 16 s190 16 s260 200 12 s240 Lenght [m] 0 0 1
More informationNonlinear behavior and seismic safety of reinforced concrete structures
Nonlinear behavior and seismic safety of reinforced concrete structures K. Girgin & E. Oer Istanbul Technical University, Faculty of Civil Engineering, Istanbul, Turkey Keywords: Reinforced concrete structures,
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationModule 4: Buckling of 2D Simply Supported Beam
Module 4: Buckling of D Simply Supported Beam Table of Contents Page Number Problem Description Theory Geometry 4 Preprocessor 7 Element Type 7 Real Constants and Material Properties 8 Meshing 9 Solution
More informationPhysics 53. Rotational Motion 1. We're going to turn this team around 360 degrees. Jason Kidd
Physics 53 Rotational Motion 1 We're going to turn this team around 360 degrees. Jason Kidd Rigid bodies To a good approximation, a solid object behaves like a perfectly rigid body, in which each particle
More informationBeam Deflections: SecondOrder Method
10 eam Deflections: SecondOrder Method 10 1 Lecture 10: EM DEFLECTIONS: SECONDORDER METHOD TLE OF CONTENTS Page 10.1 Introduction..................... 10 3 10.2 What is a eam?................... 10 3
More informationTutorial for Assignment #2 Gantry Crane Analysis By ANSYS (Mechanical APDL) V.13.0
Tutorial for Assignment #2 Gantry Crane Analysis By ANSYS (Mechanical APDL) V.13.0 1 Problem Description Design a gantry crane meeting the geometry presented in Figure 1 on page #325 of the course textbook
More informationShear Center in ThinWalled Beams Lab
Shear Center in ThinWalled Beams Lab Shear flow is developed in beams with thinwalled cross sections shear flow (q sx ): shear force per unit length along cross section q sx =τ sx t behaves much like
More informationDesign Analysis and Review of Stresses at a Point
Design Analysis and Review of Stresses at a Point Need for Design Analysis: To verify the design for safety of the structure and the users. To understand the results obtained in FEA, it is necessary to
More informationUnit 6 Plane Stress and Plane Strain
Unit 6 Plane Stress and Plane Strain Readings: T & G 8, 9, 10, 11, 12, 14, 15, 16 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems There are many structural configurations
More informationTHREE DIMENSIONAL ACES MODELS FOR BRIDGES
THREE DIMENSIONAL ACES MODELS FOR BRIDGES Noel Wenham, Design Engineer, Wyche Consulting Joe Wyche, Director, Wyche Consulting SYNOPSIS Plane grillage models are widely used for the design of bridges,
More informationA refined shear deformation theory for flexure of thick beams
8(011) 183 195 A refined shear deformation theory for flexure of thick beams Abstract A Hyperbolic Shear Deformation Theory (HPSDT) taking into account transverse shear deformation effects, is used for
More informationCHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES INTRODUCTION
CHAP FINITE EEMENT ANAYSIS OF BEAMS AND FRAMES INTRODUCTION We learned Direct Stiffness Method in Chapter imited to simple elements such as D bars we will learn Energ Method to build beam finite element
More informationDesign of Retraction Mechanism of Aircraft Landing Gear
Mechanics and Mechanical Engineering Vol. 12, No. 4 (2008) 357 373 c Technical University of Lodz Design of Retraction Mechanism of Aircraft Landing Gear Micha l Hać Warsaw University of Technology, Institute
More informationA Simulation Study on Joint Velocities and End Effector Deflection of a Flexible Two Degree Freedom Composite Robotic Arm
International Journal of Advanced Mechatronics and Robotics (IJAMR) Vol. 3, No. 1, JanuaryJune 011; pp. 90; International Science Press, ISSN: 09756108 A Simulation Study on Joint Velocities and End
More informationNonlinear Analysis Using Femap with NX Nastran
Nonlinear Analysis Using Femap with NX Nastran Chip Fricke, Principal Applications Engineer, Agenda Nonlinear Analysis Using Femap with NX Nastran Who am I? Overview of Nonlinear Analysis Comparison of
More informationChapter 5: Indeterminate Structures SlopeDeflection Method
Chapter 5: Indeterminate Structures SlopeDeflection Method 1. Introduction Slopedeflection method is the second of the two classical methods presented in this course. This method considers the deflection
More informationSTRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION
Chapter 11 STRESS AND DEFORMATION ANALYSIS OF LINEAR ELASTIC BARS IN TENSION Figure 11.1: In Chapter10, the equilibrium, kinematic and constitutive equations for a general threedimensional solid deformable
More informationIntroduction to the Finite Element Method (FEM)
Introduction to the Finite Element Method (FEM) Lecture 1 The Direct Stiffness Method and the Global StiffnessMatrix Dr. J. Dean 1 Introduction The finite element method (FEM) is a numerical technique
More informationStresses in Beam (Basic Topics)
Chapter 5 Stresses in Beam (Basic Topics) 5.1 Introduction Beam : loads acting transversely to the longitudinal axis the loads create shear forces and bending moments, stresses and strains due to V and
More informationDevelopment of Membrane, Plate and Flat Shell Elements in Java
Development of Membrane, Plate and Flat Shell Elements in Java by Kaushalkumar Kansara Thesis submitted to the Faculty of the Virginia Polytechnic Institute & State University In partial fulfillment of
More informationShell Elements in ABAQUS/Explicit
ABAQUS/Explicit: Advanced Topics Appendix 2 Shell Elements in ABAQUS/Explicit ABAQUS/Explicit: Advanced Topics A2.2 Overview ABAQUS/Explicit: Advanced Topics ABAQUS/Explicit: Advanced Topics A2.4 Triangular
More information2D Geometric Transformations. COMP 770 Fall 2011
2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrixvector multiplication Matrixmatrix multiplication
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationSimplified Calculation Approach of LoadDeformation Relationships of a BeamColumn Element
G.U. Journal of Science (4): 341350 (009) www.gujs.org Simplified Calculation Approach of oaddeformation Relationships of a BeamColumn Element Ercan YÜKSE 1, Faru KARADOĞA 1 1 Istanbul Technical University,
More informationModeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method
Modeling Beams on Elastic Foundations Using Plate Elements in Finite Element Method Yungang Zhan School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang,
More informationInteraction between plate and column buckling
Delft, University of Technology Engineering office of Public works Rotterdam Interaction between plate and column buckling Master Thesis Name: Alex van Ham Student number: 1306138 Email: vanham.alex@gmail.com
More information330:155g Finite Element Analysis
:55g inite Element Analysis Nageswara Rao Posinasetti Stiffness Matrices Review Matrix Algebra given in App A. Direct stiffness method is sed which simple to nderstand This can be sed for spring, bar and
More informationLaterally Loaded Piles
Laterally Loaded Piles 1 Soil Response Modelled by py Curves In order to properly analyze a laterally loaded pile foundation in soil/rock, a nonlinear relationship needs to be applied that provides soil
More information7 Centrifugal loads and angular accelerations
7 Centrifugal loads and angular accelerations 7.1 Introduction This example will look at essentially planar objects subjected to centrifugal loads. That is, loads due to angular velocity and/or angular
More informationFINITE ELEMENT : MATRIX FORMULATION. Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 7633
FINITE ELEMENT : MATRIX FORMULATION Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 76 FINITE ELEMENT : MATRIX FORMULATION Discrete vs continuous Element type Polynomial approximation
More informationIntroduction to Beam. Area Moments of Inertia, Deflection, and Volumes of Beams
Introduction to Beam Theory Area Moments of Inertia, Deflection, and Volumes of Beams Horizontal structural member used to support horizontal loads such as floors, roofs, and decks. Types of beam loads
More informationAn Integrated Adaptive Environment for Fire and Explosion Analysis of Steel Frames  Part I: Analytical Models
An Integrated Adaptive Environment for Fire and Explosion Analysis of Steel Frames  Part I: Analytical Models L. Song 1, B.A. Izzuddin 2, A.S. Elnashai 3 and P.J. Dowling 4 ABSTRACT This paper presents
More informationPlates and Shells: Theory and Computation  4D9  Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31)
Plates and Shells: Theory and Computation  4D9  Dr Fehmi Cirak (fc286@) Office: Inglis building mezzanine level (INO 31) Outline 1! This part of the module consists of seven lectures and will focus
More informationFINITE ELEMENT METHOD (FEM): AN OVERVIEW. Dr A Chawla
FINITE ELEMENT METHOD (FEM): AN OVERVIEW Dr A Chawla ANALYTICAL / MATHEMATICAL SOLUTIONS RESULTS AT INFINITE LOCATIONS CONTINUOUS SOLUTIONS FOR SIMPLIFIED SITUATIONS ONLY EXACT SOLUTION NUMERICAL (FEM)
More informationTower Cross Arm Numerical Analysis
Chapter 7 Tower Cross Arm Numerical Analysis In this section the structural analysis of the test tower cross arm is done in Prokon and compared to a full finite element analysis using Ansys. This is done
More informationConceptual Approaches to the Principles of Least Action
Conceptual Approaches to the Principles of Least Action Karlstad University Analytical Mechanics FYGB08 January 3, 015 Author: Isabella Danielsson Supervisors: Jürgen Fuchs Igor Buchberger Abstract We
More informationMITES 2010: Physics III Survey of Modern Physics Final Exam Solutions
MITES 2010: Physics III Survey of Modern Physics Final Exam Solutions Exercises 1. Problem 1. Consider a particle with mass m that moves in onedimension. Its position at time t is x(t. As a function of
More informationStructural Analysis  II Prof. P. Banerjee Department of Civil Engineering Indian Institute of Technology, Bombay. Lecture  02
Structural Analysis  II Prof. P. Banerjee Department of Civil Engineering Indian Institute of Technology, Bombay Lecture  02 Good morning. Today is the second lecture in the series of lectures on structural
More information