Nonlinear FEM. Critical Points. NFEM Ch 5 Slide 1
|
|
- Elijah Cain
- 6 years ago
- Views:
Transcription
1 5 Critical Points NFEM Ch 5 Slide
2 Assumptions for this Chapter System is conservative: total residual is the gradient of a total potential energy function r(u,λ) = (u,λ) u Consequence: the tangent stiffness matrix (Hessian of Π) K = r(u,λ) u 2 = (u,λ) u u is real symmetric NFEM Ch 5 Slide 2
3 Spectral Properties of Tangent Stiffness Matrix Nonlinear FEM The algebraic eigenproblem for K has the form Kz i = κ i z i, i =, 2,...N Since K is real symmetric, it enjoys two important spectral properties:. All eigenvalues of K are real. K has a complete set of real eigenvectors, which can be orthonormalized so that z T i z j = δ ij in which δ ij denotes the Kronecker delta NFEM Ch 5 Slide 3
4 Regular Versus Critical Regular point: K is nonsingular Critical point: K is singular (also called singular or nonregular points) Useful criterion for small systems with small # of DOF: The determinant of K vanishes at a critical point NFEM Ch 5 Slide 4
5 Why Are Critical Points Important? Along a static equilibrium path of a conservative system, the transition from stability to instability (or vice-versa) always occurs at a critical point Notes: () This does not mean that the transition will happen (2) Property does not extend to nonconservative systems Consequence: in designing against instability, the analyst is primarily interested in locating critical points NFEM Ch 5 Slide 5
6 Critical Point Classification as per Number of Zero Eigenvalues (= Rank Deficiency) Nonlinear FEM Isolated critical point: K has one zero eigenvalue (= rank deficiency of K is one) Multiple critical point: K has two or more zero eigenvalues (= rank deficiency of K is two or more) We will focus on the isolated type for now, since multiple critical points are more difficult to deal with NFEM Ch 5 Slide 6
7 Classification of Isolated Critical Points Nonlinear FEM Limit point: path continues with no branching, but tangent is normal to λ (control parameter) axis Bifurcation point: two or more paths cross and there is no unique tangent A limit point may be a maximum, minimum or inflexion point. If a maximum or minimum, its occurrence is also called snap-through or snap-buckling by structural engineers NFEM Ch 5 Slide 7
8 Limit or Bifurcation? Denote the values of state vector, staging control parameter, tangent stiffness matrix and incremental load vector at the critical point as u λ K q cr cr Denote the null eigenvector of K by z so K z = 0 T Compute the indicator z cr q cr (a scalar). Then (proven in the Notes) cr cr cr cr cr cr zcr T q cr 0 : limit point z T q cr cr = 0 : bifurcation point NFEM Ch 5 Slide 8
9 Response Example (2 DOF, pictured in 2D control-state space) λ L B λ L B L 2 L 2 B 2 u Limit point before bifurcation u Bifurcation before limit point NFEM Ch 5 Slide 9
10 Response Example (2 DOF, pictured in 3D control-state space) λ L B L 2 B 2 u 2 u Bifurcation before limit point NFEM Ch 5 Slide 0
11 A Simple Example (from Notes) Nonlinear FEM Find the critical points of the scalar residual equation 2 3 r = u 2 u + u λ = 0 To be worked out on whiteboard NFEM Ch 5 Slide
12 The Circle Game Example Nonlinear FEM L Control parameter λ T 2 B 2 S 2 R S B T Total residual: L State parameter µ r µ, λ = λ µ λ 2 + µ 2 = 0 (Artificial, not associated with a real structure) NFEM Ch 5 Slide 2
13 9 Special Points in Circle Game Example L Control parameter λ T 2 B 2 S 2 R S B T L State parameter µ One reference point R, atλ = µ = 0 Two bifurcation points B and B 2,atλ =±/ 2, µ =±/ 2 Two limit points L and L 2,atλ =±, µ = 0 Two turning points T and T 2,atλ = 0, µ =± Two non-equilibrium stationary points ("vortex points") S and S 2,atλ =±/ 6, µ = / 6 NFEM Ch 5 Slide 3
14 Incremental Flow for Circle Game Example Nonlinear FEM Control parameter λ State parameter µ NFEM Ch 5 Slide 4
15 Tangent Stiffness, Incremental Load, and Incremental Velocity for Circle Game Example For DOF these matrix and vector quantities reduce to scalars: K = r µ = λ2 + 2λµ 3µ 2 q = r λ = 3λ2 + 2λµ µ 2 v = q/k NFEM Ch 5 Slide 5
16 Stiffness Sign Regions for Circle Game Example Nonlinear FEM Control parameter λ K<0 Unstable T 2 B 2 L K >0 Stable R L State parameter µ B T K = 0 Neutral NFEM Ch 5 Slide 6
17 Incremental Load Sign Regions for Circle Game Example Nonlinear FEM Control parameter λ T 2 q< 0 L B R B 2 L 2 q > 0 q = State parameter µ T NFEM Ch 5 Slide 7
18 Incremental Velocity Sign Regions for Circle Game Example Nonlinear FEM Control parameter λ v=q/k=0/0 S T 2 B 2 L B R L 2 T S State parameter µ q=0 & v=0 K = 0 NFEM Ch 5 Slide 8
19 Propped Rigid Cantilevered (PRC) Column - Perfect Structure Nonlinear FEM ; C k ; A rigid L ;; B ;; C' ;; u A= L sin θ A θ spring stays A' horizontal as column tilts L L cos θ ;; B P = λ kl NFEM Ch 5 Slide 9
20 Perfect PRC Column: Response Control parameter λ Unstable 0.4 Unstable 0.2 T 2 R T State parameter µ B Unstable Stable λ cr Stiffness coefficient K State parameter µ B Control versus state parameter response for perfect column Tangent stiffness versus state parameter response for perfect column NFEM Ch 5 Slide 20
21 Propped Rigid Cantilevered (PRC) Column - Imperfect Structure Nonlinear FEM ; k C ; Initial Imperfection εl A θ 0 A 0 L ; C' ; spring stays horizontal as column tilts u A= L sin θ A θ 0 θ A 0 L P = λ kl A' L cos θ rigid B B NFEM Ch 5 Slide 2
22 Imperfect PRC Column: Response Control parameter λ State parameter µ State parameter µ Stiffness coefficient K Control versus state parameter response for varying imperfection Tangent stiffness versus state parameter response for varying imperfection NFEM Ch 5 Slide 22
23 Imperfect PRC Column: Response (cont'd) Control parameter λ λ(µ)=( µ 2 3/2 ) Unstable Stable Unstable State parameter µ Imperfection parameter ε Critical load parameter λ λ(ε)=( ε 2/3 ) 3/2 Critical point locus as stableunstable region separator Imperfection sensitivity diagram NFEM Ch 5 Slide 23
Residual Force Equations
3 Residual Force Equations NFEM Ch 3 Slide 1 Total Force Residual Equation Vector form r(u,λ) = 0 r = total force residual vector u = state vector with displacement DOF Λ = array of control parameters
More informationNonlinear Bifurcation Analysis
33 Nonlinear ifurcation Analysis 33 1 Chapter 33: NONLINEAR IFURCATION ANALYSIS TALE OF CONTENTS Page 33.1 Introduction..................... 33 3 33.2 ifurcation Analysis Levels............... 33 3 33.3
More information34 Imperfections 34 1
34 Imperfections 34 1 Chapter 34: IMPERFECTIONS TABLE OF CONTENTS Page 34.1 No Body is Perfect................... 34 3 34.2 Imperfection Sources................. 34 3 34.2.1 Physical Imperfections..............
More informationOverview of Solution Methods
20 Overview of Solution Methods NFEM Ch 20 Slide 1 Nonlinear Structural Analysis is a Multilevel Continuation Process Stages Increments Iterations Individual stage Increments Iterations NFEM Ch 20 Slide
More information4 Second-Order Systems
4 Second-Order Systems Second-order autonomous systems occupy an important place in the study of nonlinear systems because solution trajectories can be represented in the plane. This allows for easy visualization
More informationNonconservative Loading: Overview
35 Nonconservative Loading: Overview 35 Chapter 35: NONCONSERVATIVE LOADING: OVERVIEW TABLE OF CONTENTS Page 35. Introduction..................... 35 3 35.2 Sources...................... 35 3 35.3 Three
More informationOne-Parameter Residual Equations
4 One-Parameter Residual Equations NFEM Ch 4 Slide 1 Total Force Residual Equation with One Control Parameter r(u, ) = 0 total force residual state vector staging parameter = single control parameter Derivatives
More informationOn Nonlinear Buckling and Collapse Analysis using Riks Method
Visit the SIMULIA Resource Center for more customer examples. On Nonlinear Buckling and Collapse Analysis using Riks Method Mingxin Zhao, Ph.D. UOP, A Honeywell Company, 50 East Algonquin Road, Des Plaines,
More informationStability Analysis of a Single-Degree-of-Freedom Mechanical Model with Distinct Critical Points: I. Bifurcation Theory Approach
World Journal of Mechanics, 013, 3, 6-81 doi:10.436/wjm.013.31005 Published Online February 013 (http://www.scirp.org/journal/wjm) Stability Analysis of a Single-Degree-of-Freedom Mechanical Model with
More informationChapter 12 Elastic Stability of Columns
Chapter 12 Elastic Stability of Columns Axial compressive loads can cause a sudden lateral deflection (Buckling) For columns made of elastic-perfectly plastic materials, P cr Depends primarily on E and
More informationDesign Sensitivity Analysis and Optimization for Nonlinear Buckling of Finite-Dimensional Elastic Conservative Structures 1
Design Sensitivity Analysis and Optimization for Nonlinear Buckling of Finite-Dimensional Elastic Conservative Structures 1 M. Ohsaki Department of Architecture and Architectural Engineering, Kyoto University
More informationDirect calculation of critical points in parameter sensitive systems
Direct calculation of critical points in parameter sensitive systems Behrang Moghaddasie a, Ilinca Stanciulescu b, a Department of Civil Engineering, Ferdowsi University of Mashhad, P.O. Box 91775-1111,
More informationUnit 18 Other Issues In Buckling/Structural Instability
Unit 18 Other Issues In Buckling/Structural Instability Readings: Rivello Timoshenko Jones 14.3, 14.5, 14.6, 14.7 (read these at least, others at your leisure ) Ch. 15, Ch. 16 Theory of Elastic Stability
More informationErrors in FE Modelling (Section 5.10)
Errors in FE Modelling (Section 5.10) Modelling error : arises because physical reality is replaced by a mathematical model. Example: A beam that can resist both axial and transverse loads being modelled
More informationClearly the passage of an eigenvalue through to the positive real half plane leads to a qualitative change in the phase portrait, i.e.
Bifurcations We have already seen how the loss of stiffness in a linear oscillator leads to instability. In a practical situation the stiffness may not degrade in a linear fashion, and instability may
More informationAdaptive Analysis of Bifurcation Points of Shell Structures
First published in: Adaptive Analysis of Bifurcation Points of Shell Structures E. Ewert and K. Schweizerhof Institut für Mechanik, Universität Karlsruhe (TH), Kaiserstraße 12, D-76131 Karlsruhe, Germany
More information. D CR Nomenclature D 1
. D CR Nomenclature D 1 Appendix D: CR NOMENCLATURE D 2 The notation used by different investigators working in CR formulations has not coalesced, since the topic is in flux. This Appendix identifies the
More informationInstitute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I
Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix
More informationHomework Exercises for Chapter 10 TL Bar Elements: Truss Analysis
Solutions to Exercises omework Exercises for Chapter TL Bar Elements: Truss Analysis Note: not all solutions fully worked out. EXECISE. The strain measures (engineering strain versus GL strain) are different.
More informationConstitutive models: Incremental plasticity Drücker s postulate
Constitutive models: Incremental plasticity Drücker s postulate if consistency condition associated plastic law, associated plasticity - plastic flow law associated with the limit (loading) surface Prager
More informationOPTIMAL CONTROL AND ESTIMATION
OPTIMAL CONTROL AND ESTIMATION Robert F. Stengel Department of Mechanical and Aerospace Engineering Princeton University, Princeton, New Jersey DOVER PUBLICATIONS, INC. New York CONTENTS 1. INTRODUCTION
More informationCourse in. Geometric nonlinearity. Nonlinear FEM. Computational Mechanics, AAU, Esbjerg
Course in Nonlinear FEM Geometric nonlinearity Nonlinear FEM Outline Lecture 1 Introduction Lecture 2 Geometric nonlinearity Lecture 3 Material nonlinearity Lecture 4 Material nonlinearity it continued
More informationMatrices and Vectors. Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A =
30 MATHEMATICS REVIEW G A.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A = a 11 a 12... a 1N a 21 a 22... a 2N...... a M1 a M2... a MN A matrix can
More informationApplications of Eigenvalues & Eigenvectors
Applications of Eigenvalues & Eigenvectors Louie L. Yaw Walla Walla University Engineering Department For Linear Algebra Class November 17, 214 Outline 1 The eigenvalue/eigenvector problem 2 Principal
More informationIntroduction to Finite Element Method
Introduction to Finite Element Method Dr. Rakesh K Kapania Aerospace and Ocean Engineering Department Virginia Polytechnic Institute and State University, Blacksburg, VA AOE 524, Vehicle Structures Summer,
More informationStructural Stability: Basic Concepts
28 Structural Stability: asic oncepts 28 1 hapter 28: STRUTUR STIITY: SI ONETS TE OF ONTENTS age 28.1 Introduction..................... 28 3 28.2 Terminology..................... 28 3 28.3 Testing Stability....................
More information1. Background. is usually significantly lower than it is in uniaxial tension
NOTES ON QUANTIFYING MODES OF A SECOND- ORDER TENSOR. The mechanical behavior of rocks and rock-like materials (concrete, ceramics, etc.) strongly depends on the loading mode, defined by the values and
More informationPart E: Nondestructive Testing
Part E: Nondestructive Testing Non-destructive Testing General Concepts The Southwell Plot Examples Some Background Underlying General Theory Snap-Through Revisited Effect of Damping Range of Prediction
More informationME 680- Spring Geometrical Analysis of 1-D Dynamical Systems
ME 680- Spring 2014 Geometrical Analysis of 1-D Dynamical Systems 1 Geometrical Analysis of 1-D Dynamical Systems Logistic equation: n = rn(1 n) velocity function Equilibria or fied points : initial conditions
More informationNonlinear Control Lecture 2:Phase Plane Analysis
Nonlinear Control Lecture 2:Phase Plane Analysis Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2010 r. Farzaneh Abdollahi Nonlinear Control Lecture 2 1/53
More informationSTRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS
1 UNIT I STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define: Stress When an external force acts on a body, it undergoes deformation. At the same time the body resists deformation. The
More informationAutomatic Control Systems theory overview (discrete time systems)
Automatic Control Systems theory overview (discrete time systems) Prof. Luca Bascetta (luca.bascetta@polimi.it) Politecnico di Milano Dipartimento di Elettronica, Informazione e Bioingegneria Motivations
More informationNewton Method: General Control and Variants
23 Newton Method: General Control and Variants 23 1 Chapter 23: NEWTON METHOD: GENERAL CONTROL AND VARIANTS TABLE OF CONTENTS Page 23.1 Introduction..................... 23 3 23.2 Newton Iteration as Dynamical
More informationMechanical Design in Optical Engineering
OPTI Buckling Buckling and Stability: As we learned in the previous lectures, structures may fail in a variety of ways, depending on the materials, load and support conditions. We had two primary concerns:
More informationLecture 6 Positive Definite Matrices
Linear Algebra Lecture 6 Positive Definite Matrices Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices
More information9.1 Introduction to bifurcation of equilibrium and structural
Module 9 Stability and Buckling Readings: BC Ch 14 earning Objectives Understand the basic concept of structural instability and bifurcation of equilibrium. Derive the basic buckling load of beams subject
More informationBIFURCATION theory is the commonly used tool to analyze
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 51, NO. 8, AUGUST 2004 1525 Computation of Singular and Singularity Induced Bifurcation Points of Differential-Algebraic Power System Model
More informationProblem Set Number 2, j/2.036j MIT (Fall 2014)
Problem Set Number 2, 18.385j/2.036j MIT (Fall 2014) Rodolfo R. Rosales (MIT, Math. Dept.,Cambridge, MA 02139) Due Mon., September 29, 2014. 1 Inverse function problem #01. Statement: Inverse function
More informationPOST-BUCKLING BEHAVIOUR OF IMPERFECT SLENDER WEB
Engineering MECHANICS, Vol. 14, 007, No. 6, p. 43 49 43 POST-BUCKLING BEHAVIOUR OF IMPERFECT SLENDER WEB Martin Psotný, Ján Ravinger* The stability analysis of slender web loaded in compression is presented.
More informationTangent spaces, normals and extrema
Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent
More informationPositive Definite Matrix
1/29 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Positive Definite, Negative Definite, Indefinite 2/29 Pure Quadratic Function
More informationOn optimisation of structures under stability constraints - a simple example
Rakenteiden Mekaniikka Journal of Structural Mechanics Vol. 49, No 2, 26, pp. 69 77 rmseura.tkk.fi/rmlehti/ c The authors 26. Open access under CC BY-SA 4. license. On optimisation of structures under
More information1 Nonlinear deformation
NONLINEAR TRUSS 1 Nonlinear deformation When deformation and/or rotation of the truss are large, various strains and stresses can be defined and related by material laws. The material behavior can be expected
More informationReduction in number of dofs
Reduction in number of dofs Reduction in the number of dof to represent a structure reduces the size of matrices and, hence, computational cost. Because a subset of the original dof represent the whole
More informationLinear Algebra using Dirac Notation: Pt. 2
Linear Algebra using Dirac Notation: Pt. 2 PHYS 476Q - Southern Illinois University February 6, 2018 PHYS 476Q - Southern Illinois University Linear Algebra using Dirac Notation: Pt. 2 February 6, 2018
More informationEsben Byskov. Elementary Continuum. Mechanics for Everyone. With Applications to Structural Mechanics. Springer
Esben Byskov Elementary Continuum Mechanics for Everyone With Applications to Structural Mechanics Springer Contents Preface v Contents ix Introduction What Is Continuum Mechanics? "I Need Continuum Mechanics
More informationControl Systems. Internal Stability - LTI systems. L. Lanari
Control Systems Internal Stability - LTI systems L. Lanari outline LTI systems: definitions conditions South stability criterion equilibrium points Nonlinear systems: equilibrium points examples stable
More informationSolutions of Semilinear Elliptic PDEs on Manifolds
Solutions of Semilinear Elliptic PDEs on Manifolds Jeff, Northern Arizona University 1 2 3 4 5 What is a PDE? Definition A Partial Differential Equation (PDE) is a relation involving an unknown function
More informationOPTIMUM PRE-STRESS DESIGN FOR FREQUENCY REQUIREMENT OF TENSEGRITY STRUCTURES
Blucher Mechanical Engineering Proceedings May 2014, vol. 1, num. 1 www.proceedings.blucher.com.br/evento/10wccm OPTIMUM PRE-STRESS DESIGN FOR FREQUENCY REQUIREMENT OF TENSEGRITY STRUCTURES Seif Dalil
More informationCHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES
CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES 14.1 GENERAL REMARKS In structures where dominant loading is usually static, the most common cause of the collapse is a buckling failure. Buckling may
More informationReview of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
More informationImperfection sensitivity analysis of hill-top branching with many symmetric bifurcation points 1
Imperfection sensitivity analysis of hill-top branching with many symmetric bifurcation points 1 M. Ohsaki Department of Architecture and Architectural Engineering, Kyoto University Kyotodaigaku-Katsura,
More informationSupplementary Information. for. Origami based Mechanical Metamaterials
Supplementary Information for Origami based Mechanical Metamaterials By Cheng Lv, Deepakshyam Krishnaraju, Goran Konjevod, Hongyu Yu, and Hanqing Jiang* [*] Prof. H. Jiang, C. Lv, D. Krishnaraju, Dr. G.
More informationSimon D. GUEST Reader University of Cambridge Cambridge, UK.
Multi-stable Star-shaped Tensegrity Structures Jingyao ZHANG Lecturer Ritsumeikan University Kusatsu, Shiga, JAPAN zhang@fc.ritsumei.ac.jp Robert CONNELLY Professor Cornell University Ithaca, NY, USA rc46@cornell.edu
More information5 Linear Algebra and Inverse Problem
5 Linear Algebra and Inverse Problem 5.1 Introduction Direct problem ( Forward problem) is to find field quantities satisfying Governing equations, Boundary conditions, Initial conditions. The direct problem
More informationNonlinear analysis in ADINA Structures
Nonlinear analysis in ADINA Structures Theodore Sussman, Ph.D. ADINA R&D, Inc, 2016 1 Topics presented Types of nonlinearities Materially nonlinear only Geometrically nonlinear analysis Deformation-dependent
More informationParametrically Excited Vibration in Rolling Element Bearings
Parametrically Ecited Vibration in Rolling Element Bearings R. Srinath ; A. Sarkar ; A. S. Sekhar 3,,3 Indian Institute of Technology Madras, India, 636 ABSTRACT A defect-free rolling element bearing has
More informationSborník vědeckých prací Vysoké školy báňské - Technické univerzity Ostrava číslo 1, rok 2014, ročník XIV, řada stavební článek č. 18.
Sborník vědeckých prací Vysoké školy báňské - Technické univerzity Ostrava číslo, rok, ročník XIV, řada stavební článek č. 8 Martin PSOTNÝ NONLINER NLYSIS OF BUCKLING & POSTBUCKLING bstract The stability
More informationCE-570 Advanced Structural Mechanics - Arun Prakash
Ch1-Intro Page 1 CE-570 Advanced Structural Mechanics - Arun Prakash The BIG Picture What is Mechanics? Mechanics is study of how things work: how anything works, how the world works! People ask: "Do you
More informationLinear Algebra & Geometry why is linear algebra useful in computer vision?
Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia
More informationEE16B, Spring 2018 UC Berkeley EECS. Maharbiz and Roychowdhury. Lectures 4B & 5A: Overview Slides. Linearization and Stability
EE16B, Spring 2018 UC Berkeley EECS Maharbiz and Roychowdhury Lectures 4B & 5A: Overview Slides Linearization and Stability Slide 1 Linearization Approximate a nonlinear system by a linear one (unless
More informationOne Dimensional Dynamical Systems
16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with one-dimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar
More informationMore Examples Of Generalized Coordinates
Slides of ecture 8 Today s Class: Review Of Homework From ecture 7 Hamilton s Principle More Examples Of Generalized Coordinates Calculating Generalized Forces Via Virtual Work /3/98 /home/djsegal/unm/vibcourse/slides/ecture8.frm
More informationA BIFURCATION ANALYSIS OF SPACE STRUCTURES BY USING 3D BEAM-COLUMN ELEMENT CONSIDERING FINITE DEFORMATIONS AND BOWING EFFECT
Advanced Steel Construction Vol. 8, No. 3, pp. 56-8 (0) 56 A BIFURCATION ANALYSIS OF SPACE STRUCTURES BY USING 3D BEAM-COLUMN ELEMENT CONSIDERING FINITE DEFORMATIONS AND BOWING EFFECT K.S. Lee,* and S.E.
More information16.20 Techniques of Structural Analysis and Design Spring Instructor: Raúl Radovitzky Aeronautics & Astronautics M.I.T
16.20 Techniques of Structural Analysis and Design Spring 2013 Instructor: Raúl Radovitzky Aeronautics & Astronautics M.I.T February 15, 2013 2 Contents 1 Stress and equilibrium 5 1.1 Internal forces and
More informationLecture Slides. Chapter 4. Deflection and Stiffness. The McGraw-Hill Companies 2012
Lecture Slides Chapter 4 Deflection and Stiffness The McGraw-Hill Companies 2012 Chapter Outline Force vs Deflection Elasticity property of a material that enables it to regain its original configuration
More informationMatrices A brief introduction
Matrices A brief introduction Basilio Bona DAUIN Politecnico di Torino Semester 1, 2014-15 B. Bona (DAUIN) Matrices Semester 1, 2014-15 1 / 41 Definitions Definition A matrix is a set of N real or complex
More informationContinuation methods for non-linear analysis
Continuation methods for non-linear analysis FR : Méthodes de pilotage du chargement Code_Aster, Salome-Meca course material GNU FDL licence (http://www.gnu.org/copyleft/fdl.html) Outline Definition of
More informationLinear Algebra & Geometry why is linear algebra useful in computer vision?
Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia
More information202 Index. failure, 26 field equation, 122 force, 1
Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic
More informationUniversity of Houston, Department of Mathematics Numerical Analysis, Fall 2005
3 Numerical Solution of Nonlinear Equations and Systems 3.1 Fixed point iteration Reamrk 3.1 Problem Given a function F : lr n lr n, compute x lr n such that ( ) F(x ) = 0. In this chapter, we consider
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationPlane Trusses Trusses
TRUSSES Plane Trusses Trusses- It is a system of uniform bars or members (of various circular section, angle section, channel section etc.) joined together at their ends by riveting or welding and constructed
More informationA = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].
Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes
More informationPart D: Frames and Plates
Part D: Frames and Plates Plane Frames and Thin Plates A Beam with General Boundary Conditions The Stiffness Method Thin Plates Initial Imperfections The Ritz and Finite Element Approaches A Beam with
More informationMath 322. Spring 2015 Review Problems for Midterm 2
Linear Algebra: Topic: Linear Independence of vectors. Question. Math 3. Spring Review Problems for Midterm Explain why if A is not square, then either the row vectors or the column vectors of A are linearly
More informationCOMPUTATION OF KUHN-TUCKER TRIPLES IN OPTIMUM DESIGN PROBLEMS IN THE PRESENCE OF PARAMETRIC SINGULARITIES
COMPUTATION OF KUHN-TUCKER TRIPLES IN OPTIMUM DESIGN PROBLEMS IN THE PRESENCE OF PARAMETRIC SINGULARITIES J. R. Jagannatha Rao* Assistant Professor, Department of Mechanical Engineering The University
More informationA VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010
A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics
More informationEquilibrium of rigid bodies Mehrdad Negahban (1999)
Equilibrium of rigid bodies Mehrdad Negahban (1999) Static equilibrium for a rigid body: A body (or any part of it) which is currently stationary will remain stationary if the resultant force and resultant
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationThe Effect of a Rotational Spring on the Global Stability Aspects of the Classical von Mises Model under Step Loading
Copyright cfl 2001 Tech Science Press CMES, vol.2, no.1, pp.15-26, 2001 The Effect of a Rotational Spring on the Global Stability Aspects of the Classical von Mises Model under Step Loading D. S. Sophianopoulos
More informationMeaning of the Hessian of a function in a critical point
Meaning of the Hessian of a function in a critical point Mircea Petrache February 1, 2012 We consider a function f : R n R and assume for it to be differentiable with continuity at least two times (that
More informationThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems: Non-Linear Dynamics Part I
The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems: Non-Linear Dynamics Part I Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos Agathos Lecture 5/Part A - 23 November,
More information1. Select the unique answer (choice) for each problem. Write only the answer.
MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +
More information11 a 12 a 21 a 11 a 22 a 12 a 21. (C.11) A = The determinant of a product of two matrices is given by AB = A B 1 1 = (C.13) and similarly.
C PROPERTIES OF MATRICES 697 to whether the permutation i 1 i 2 i N is even or odd, respectively Note that I =1 Thus, for a 2 2 matrix, the determinant takes the form A = a 11 a 12 = a a 21 a 11 a 22 a
More informationCE 201 Statics. 2 Physical Sciences. Rigid-Body Deformable-Body Fluid Mechanics Mechanics Mechanics
CE 201 Statics 2 Physical Sciences Branch of physical sciences 16 concerned with the state of Mechanics rest motion of bodies that are subjected to the action of forces Rigid-Body Deformable-Body Fluid
More informationMathematical Properties of Stiffness Matrices
Mathematical Properties of Stiffness Matrices CEE 4L. Matrix Structural Analysis Department of Civil and Environmental Engineering Duke University Henri P. Gavin Fall, 0 These notes describe some of the
More informationMechanics of Earthquakes and Faulting
Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body
More informationStability of Mass-Point Systems
Simulation in Computer Graphics Stability of Mass-Point Systems Matthias Teschner Computer Science Department University of Freiburg Demos surface tension vs. volume preservation distance preservation
More informationAuthor(s) Senda, Kei; Petrovic, Mario; Nakani. Citation Acta Astronautica (2015), 111:
TitleWrinkle generation in shear-enforce Author(s) Senda, Kei; Petrovic, Mario; Nakani Citation Acta Astronautica (2015), 111: 110- Issue Date 2015-06 URL http://hdl.handle.net/2433/196845 2015 IAA. Published
More informationEnergy Problem Solving Techniques.
1 Energy Problem Solving Techniques www.njctl.org 2 Table of Contents Introduction Gravitational Potential Energy Problem Solving GPE, KE and EPE Problem Solving Conservation of Energy Problem Solving
More informationPhysics 170 Week 5, Lecture 2
Physics 170 Week 5, Lecture 2 http://www.phas.ubc.ca/ gordonws/170 Physics 170 Week 5 Lecture 2 1 Textbook Chapter 5:Section 5.5-5.7 Physics 170 Week 5 Lecture 2 2 Learning Goals: Review the condition
More informationAdvanced Vibrations. Elements of Analytical Dynamics. By: H. Ahmadian Lecture One
Advanced Vibrations Lecture One Elements of Analytical Dynamics By: H. Ahmadian ahmadian@iust.ac.ir Elements of Analytical Dynamics Newton's laws were formulated for a single particle Can be extended to
More informationAPPENDIX A. Background Mathematics. A.1 Linear Algebra. Vector algebra. Let x denote the n-dimensional column vector with components x 1 x 2.
APPENDIX A Background Mathematics A. Linear Algebra A.. Vector algebra Let x denote the n-dimensional column vector with components 0 x x 2 B C @. A x n Definition 6 (scalar product). The scalar product
More informationCh 10.1: Two Point Boundary Value Problems
Ch 10.1: Two Point Boundary Value Problems In many important physical problems there are two or more independent variables, so the corresponding mathematical models involve partial differential equations.
More informationChapter #4 EEE8086-EEE8115. Robust and Adaptive Control Systems
Chapter #4 Robust and Adaptive Control Systems Nonlinear Dynamics.... Linear Combination.... Equilibrium points... 3 3. Linearisation... 5 4. Limit cycles... 3 5. Bifurcations... 4 6. Stability... 6 7.
More informationComputation of Equilibrium Paths in Nonlinear Finite Element Models
Computation of Equilibrium Paths in Nonlinear Finite Element Models 1,a 1 Abstract. In the present paper, equilibrium paths are simulated applying the nonlinear finite element model. On the equilibrium
More informationSymmetry and Properties of Crystals (MSE638) Stress and Strain Tensor
Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor Somnath Bhowmick Materials Science and Engineering, IIT Kanpur April 6, 2018 Tensile test and Hooke s Law Upto certain strain (0.75),
More informationNonlinear Theory of Elasticity. Dr.-Ing. Martin Ruess
Nonlinear Theory of Elasticity Dr.-Ing. Martin Ruess geometry description Cartesian global coordinate system with base vectors of the Euclidian space orthonormal basis origin O point P domain of a deformable
More informationMidterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015
Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015 The test lasts 1 hour and 15 minutes. No documents are allowed. The use of a calculator, cell phone or other equivalent electronic
More information