Residual Force Equations
|
|
- Allyson Lucas
- 6 years ago
- Views:
Transcription
1 3 Residual Force Equations NFEM Ch 3 Slide 1
2 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 Indicial form r (u, Λ ) = 0 i j k NFEM Ch 3 Slide 2
3 Black Box Interpretation Control: Λ State u Residual evaluation Residual r NFEM Ch 3 Slide 3
4 Example Nonlinear FEM r 1 = 4 u 1 u 2 + u 2 u = 0 r 2 = 6 u 2 u 1 + u 1 u = 0 r 3 = 4 u 3 + u 1 u = 0 Vector form: [ ] r1 (u, Λ) [ ] 4 u1 u 2 + u 2 u r = r(u, Λ) = r 2 (u, Λ) r 3 (u, Λ) = 6 u 2 u 1 + u 1 u u 3 + u 1 u = 0 with u = [ u1 u 2 u 3 ] Λ = [ 1 2 ] NFEM Ch 3 Slide 4
5 Correlation with Linear FEM Master stiffness equations: K u = f Transforming to force residual: r = K u f = 0 Control parameters are not needed in linear FEM. NFEM Ch 3 Slide 5
6 Conservative System: Derivation From Energy total potential energy Π r = = 0 u Physical meaning: force equilibrium is associated with total potential energy Π being stationary (its gradient of r w.r.t. state vector u vanishes) NFEM Ch 3 Slide 6
7 Balanced Force Residual Form Split total residual as r = p f = 0 Move f to RHS p(u) = f(u,λ) internal force vector external force vector NFEM Ch 3 Slide 7
8 Balanced Force Residual Form: Black Box Control: Λ State u Ext force evaluation Int force evaluation f p Residual r NFEM Ch 3 Slide 8
9 Previous Example Nonlinear FEM r 1 = 4 u 1 u 2 + u 2 u = 0 r 2 = 6 u 2 u 1 + u 1 u = 0 r 3 = 4 u 3 + u 1 u = 0 [ ] r1 (u, Λ) [ ] 4 u1 u 2 + u 2 u r = r(u, Λ) = r 2 (u, Λ) r 3 (u, Λ) = 6 u 2 u 1 + u 1 u u 3 + u 1 u = 0 [ ] p1 (u) [ ] 4 u1 u 2 + u 2 u 3 [ ] f1 (Λ) [ ] 2 1 p = p 2 (u) p 3 (u) = 6 u 2 u 1 + u 1 u 3 4 u 3 + u 1 u 2, f = f 2 (Λ) f 3 (Λ) = NFEM Ch 3 Slide 9
10 Conservative System: Derivation from Energy total potential energy Π = U W internal energy external work potential p = U u, f = W u, NFEM Ch 3 Slide 10
11 (Tangent) Stiffness and Control Matrices K = r u with entries K ij = r i u j Q = r Λ with entries Q ij = r i j An example next NFEM Ch 3 Slide 11
12 Residual Rate Forms Nonlinear FEM Define state and control in terms of pseudo time t : u = u(t) Λ = Λ(t) The first two derivatives of r with respect to t are, using indicial form ṙ i = r i u j u j + r i j j r i = r [ i 2 r i ü j + u j + u k + u j u k [ 2 r i u k + j u k 2 r ] i k u j + r i j u j k j 2 r i j k k ] j in which a superposed dot denotes derivative wrt t, as in real dynamics NFEM Ch 3 Slide 12
13 Residual Rate ODEs in Matrix Form Nonlinear FEM Recall that K = r u Q = r Λ Using these definitions the previous ODEs in indicial form can be rewritten in matrix form as ṙ = K u Q Λ first order rate form r = Kü + K u Q Λ Q Λ second order rate form Examples in next slide NFEM Ch 3 Slide 13
14 Residual Computation Example Nonlinear FEM A E,A constant B A C' P = λ EA B θ θ u C k = β EA L k L 2L L Only one DOF: vertical motion of midspan joint C Only one control parameter denoted by λ instead of Λ 1 NFEM Ch 3 Slide 14
15 Residual Computation Example (cnt'd) A C' P = λ EA B θ θ u k C' P = λ EA State parameter µ = u/l = tan θ Control parameter λ = P/EA F AC F s F BC NFEM Ch 3 Slide 15
16 Residual Computation Example (cnt'd) Nonlinear FEM r(µ, λ) = µ ( 2 + β ) 2 λ = µ 2 K = r µ = β + 2(1 + µ2 ) 3/2 1 (1 + µ 2 ) 3/2 NFEM Ch 3 Slide 16
17 Residual Computation Example (cnt'd) Load factor λ = P/EA Response using engineering strain measure β = 1 β = 1/10 β = 1/100, 1/1000 (indistinguishable at plot scale) Dimensionless displacement µ = u/l NFEM Ch 3 Slide 17
18 Residual Computation Example (cnt'd) Stiffness coefficient K = dr/dµ Stiffness using engineering strain measure Dimensionless displacement µ = u/l NFEM Ch 3 Slide 18
19 Staging Complicated nonlinear systems, such as structures, are analyzed in stages because the superposition principle no longer applies Staging reduces multiple control parameters to only one over each loading stage. This single parameter is called the staging parameter and will be usually denoted by λ NFEM Ch 3 Slide 19
20 Staging (cont'd) Nonlinear FEM Suppose that we have solved the residual equation for the control parameters Λ A corresponding to point A in the control space. We next want to advance the solution to Λ corresponding to point B Interpolate the control parameters linearly B Λ = (1 λ)λ A + λλ B where λ is the staging parameter that varies from 0 at A to 1 at B The residual equation to ber solve over the stage is with the I.C. u = u at λ = 0. A r (u,λ) = 0 This equation has only one control parameter. It is studied in Chapter 4. NFEM Ch 3 Slide 20
21 To Illustrate Staging We Will Look at a Suspension Bridge NFEM Ch 3 Slide 21
One-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 informationNonlinear FEM. Critical Points. NFEM Ch 5 Slide 1
5 Critical Points NFEM Ch 5 Slide 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
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 informationThe Plane Stress Problem
14 The Plane Stress Problem IFEM Ch 14 Slide 1 Plate in Plane Stress Thickness dimension or transverse dimension z Top surface Inplane dimensions: in, plane IFEM Ch 14 Slide 2 Mathematical Idealization
More informationChapter 3 Variational Formulation & the Galerkin Method
Institute of Structural Engineering Page 1 Chapter 3 Variational Formulation & the Galerkin Method Institute of Structural Engineering Page 2 Today s Lecture Contents: Introduction Differential formulation
More 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 informationMulti-Point Constraints
Multi-Point Constraints Multi-Point Constraints Multi-Point Constraints Single point constraint examples Multi-Point constraint examples linear, homogeneous linear, non-homogeneous linear, homogeneous
More 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 informationStructural Analysis of Truss Structures using Stiffness Matrix. Dr. Nasrellah Hassan Ahmed
Structural Analysis of Truss Structures using Stiffness Matrix Dr. Nasrellah Hassan Ahmed FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS In general, there are three types of relationships: Equilibrium
More 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 informationMechanics of Structures (CE130N) Lab 6. 2 Principle of stationary potential energy
UNIVERSITY OF CALIFORNIA BERKELEY Department of Civil Engineering Spring 0 Structural Engineering, Mechanics and Materials Professor: S. Govindjee Mechanics of Structures (CE30N) Lab 6 Objective The objective
More informationAdvanced Dynamics. - Lecture 4 Lagrange Equations. Paolo Tiso Spring Semester 2017 ETH Zürich
Advanced Dynamics - Lecture 4 Lagrange Equations Paolo Tiso Spring Semester 2017 ETH Zürich LECTURE OBJECTIVES 1. Derive the Lagrange equations of a system of particles; 2. Show that the equation of motion
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 informationChapter 2 Finite Element Formulations
Chapter 2 Finite Element Formulations The governing equations for problems solved by the finite element method are typically formulated by partial differential equations in their original form. These are
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 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 informationInverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros
Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros Computational Design Forward design: direct manipulation of design parameters Level of abstraction Exploration
More informationBAR ELEMENT WITH VARIATION OF CROSS-SECTION FOR GEOMETRIC NON-LINEAR ANALYSIS
Journal of Computational and Applied Mechanics, Vol.., No. 1., (2005), pp. 83 94 BAR ELEMENT WITH VARIATION OF CROSS-SECTION FOR GEOMETRIC NON-LINEAR ANALYSIS Vladimír Kutiš and Justín Murín Department
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 informationKirchhoff Plates: Field Equations
20 Kirchhoff Plates: Field Equations AFEM Ch 20 Slide 1 Plate Structures A plate is a three dimensional bod characterized b Thinness: one of the plate dimensions, the thickness, is much smaller than the
More informationAssumptions: beam is initially straight, is elastically deformed by the loads, such that the slope and deflection of the elastic curve are
*12.4 SLOPE & DISPLACEMENT BY THE MOMENT-AREA METHOD Assumptions: beam is initially straight, is elastically deformed by the loads, such that the slope and deflection of the elastic curve are very small,
More informationMITOCW MITRES2_002S10nonlinear_lec05_300k-mp4
MITOCW MITRES2_002S10nonlinear_lec05_300k-mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources
More informationThermomechanical Effects
3 hermomechanical Effects 3 Chapter 3: HERMOMECHANICAL EFFECS ABLE OF CONENS Page 3. Introduction..................... 3 3 3.2 hermomechanical Behavior............... 3 3 3.2. hermomechanical Stiffness
More informationFREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS
Lecture Notes: STRUCTURAL DYNAMICS / FALL 2011 / Page: 1 FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS : : 0, 0 As demonstrated previously, the above Equation of Motion (free-vibration equation) has a solution
More informationCh 5.7: Series Solutions Near a Regular Singular Point, Part II
Ch 5.7: Series Solutions Near a Regular Singular Point, Part II! Recall from Section 5.6 (Part I): The point x 0 = 0 is a regular singular point of with and corresponding Euler Equation! We assume solutions
More informationNon-linear and time-dependent material models in Mentat & MARC. Tutorial with Background and Exercises
Non-linear and time-dependent material models in Mentat & MARC Tutorial with Background and Exercises Eindhoven University of Technology Department of Mechanical Engineering Piet Schreurs July 7, 2009
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 11 Last class, what we did is, we looked at a method called superposition
More informationMMJ1153 COMPUTATIONAL METHOD IN SOLID MECHANICS PRELIMINARIES TO FEM
B Course Content: A INTRODUCTION AND OVERVIEW Numerical method and Computer-Aided Engineering; Phsical problems; Mathematical models; Finite element method;. B Elements and nodes, natural coordinates,
More informationThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems. Prof. Dr. Eleni Chatzi Lecture 1-20 September, 2017
The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Lecture 1-20 September, 2017 Institute of Structural Engineering Method of Finite Elements II 1 Course
More informationLinearization problem. The simplest example
Linear Systems Lecture 3 1 problem Consider a non-linear time-invariant system of the form ( ẋ(t f x(t u(t y(t g ( x(t u(t (1 such that x R n u R m y R p and Slide 1 A: f(xu f(xu g(xu and g(xu exist and
More informationOptimization Methods for Force and Shape Design of Tensegrity Structures. J.Y. Zhang and M. Ohsaki Kyoto University, Japan
Optimization Methods for Force and Shape Design of Tensegrity Structures J.Y. Zhang and M. Ohsaki Kyoto University, Japan Contents Purpose Optimization techniques are effective for shape and force design
More informationStress analysis of a stepped bar
Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.
More 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 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 informationLecture 27 Introduction to finite elements methods
Fall, 2017 ME 323 Mechanics of Materials Lecture 27 Introduction to finite elements methods Reading assignment: News: Instructor: Prof. Marcial Gonzalez Last modified: 10/24/17 7:02:00 PM Finite element
More informationA NOTE ON RELATIONSHIP BETWEEN FIXED-POLE AND MOVING-POLE APPROACHES IN STATIC AND DYNAMIC ANALYSIS OF NON-LINEAR SPATIAL BEAM STRUCTURES
European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 212) J. Eberhardsteiner et.al. (eds.) Vienna, Austria, September 1-14, 212 A NOTE ON RELATIONSHIP BETWEEN FIXED-POLE
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 informationMECh300H Introduction to Finite Element Methods. Finite Element Analysis (F.E.A.) of 1-D Problems
MECh300H Introduction to Finite Element Methods Finite Element Analysis (F.E.A.) of -D Problems Historical Background Hrenikoff, 94 frame work method Courant, 943 piecewise polynomial interpolation Turner,
More informationEffects Of Temperature, Pre-strain & Support Displacement
Lecture 14: TEMPERTURE, PRESTRIN & SUPPORT Effects Of Temperature, Pre-strain & Support Displacement In the previous sections we have only considered loads acting on the structure. We would also like to
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 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 informationDesign optimization of multi-point constraints in structures
11 th World Congress on Structural and Multidisciplinary Optimization 7 th - 12 th, June 2015, Sydney Australia Design optimization of multi-point constraints in structures Daniel N. Wilke 1, Schalk Kok
More information16.21 Techniques of Structural Analysis and Design Spring 2005 Unit #8 Principle of Virtual Displacements
6. Techniques of Structural Analysis and Design Spring 005 Unit #8 rinciple of irtual Displacements Raúl Radovitzky March 3, 005 rinciple of irtual Displacements Consider a body in equilibrium. We know
More informationElectro-Thermal-Mechanical MEMS Devices Multi-Physics Problems
Electro-Thermal-Mechanical MEMS Devices Multi-Physics Problems Presentation for ASEN 5519 Joseph Pajot Department of Aerospace Engineering CU Boulder November 15, 2004 Typeset by FoilTEX 1 The ETM Problems
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 new closed-form model for isotropic elastic sphere including new solutions for the free vibrations problem
A new closed-form model for isotropic elastic sphere including new solutions for the free vibrations problem E Hanukah Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Haifa
More informationNon-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, Politecnico di Milano, February 17, 2017, Lesson 5
Non-Linear Finite Element Methods in Solid Mechanics Attilio Frangi, attilio.frangi@polimi.it Politecnico di Milano, February 17, 2017, Lesson 5 1 Politecnico di Milano, February 17, 2017, Lesson 5 2 Outline
More informationIntroduction to gradient descent
6-1: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction to gradient descent Derivation and intuitions Hessian 6-2: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction Our
More informationChapter 3 Numerical Methods
Chapter 3 Numerical Methods Part 3 3.4 Differential Algebraic Systems 3.5 Integration of Differential Equations 1 Outline 3.4 Differential Algebraic Systems 3.4.1 Constrained Dynamics 3.4.2 First and Second
More informationContents. Prologue Introduction. Classical Approximation... 19
Contents Prologue........................................................................ 15 1 Introduction. Classical Approximation.................................. 19 1.1 Introduction................................................................
More informationAn Energy Dissipative Constitutive Model for Multi-Surface Interfaces at Weld Defect Sites in Ultrasonic Consolidation
An Energy Dissipative Constitutive Model for Multi-Surface Interfaces at Weld Defect Sites in Ultrasonic Consolidation Nachiket Patil, Deepankar Pal and Brent E. Stucker Industrial Engineering, University
More informationSensitivity and Reliability Analysis of Nonlinear Frame Structures
Sensitivity and Reliability Analysis of Nonlinear Frame Structures Michael H. Scott Associate Professor School of Civil and Construction Engineering Applied Mathematics and Computation Seminar April 8,
More informationStructural Dynamics A Graduate Course in Aerospace Engineering
Structural Dynamics A Graduate Course in Aerospace Engineering By: H. Ahmadian ahmadian@iust.ac.ir The Science and Art of Structural Dynamics What do all the followings have in common? > A sport-utility
More informationLecture 4 Orthonormal vectors and QR factorization
Orthonormal vectors and QR factorization 4 1 Lecture 4 Orthonormal vectors and QR factorization EE263 Autumn 2004 orthonormal vectors Gram-Schmidt procedure, QR factorization orthogonal decomposition induced
More informationLecture 11: The Stiffness Method. Introduction
Introduction Although the mathematical formulation of the flexibility and stiffness methods are similar, the physical concepts involved are different. We found that in the flexibility method, the unknowns
More informationNONLINEAR STRUCTURAL DYNAMICS USING FE METHODS
NONLINEAR STRUCTURAL DYNAMICS USING FE METHODS Nonlinear Structural Dynamics Using FE Methods emphasizes fundamental mechanics principles and outlines a modern approach to understanding structural dynamics.
More information8 A pseudo-spectral solution to the Stokes Problem
8 A pseudo-spectral solution to the Stokes Problem 8.1 The Method 8.1.1 Generalities We are interested in setting up a pseudo-spectral method for the following Stokes Problem u σu p = f in Ω u = 0 in Ω,
More informationUniversity of Groningen
University of Groningen Nature-inspired microfluidic propulsion using magnetic actuation Khaderi, S. N.; Baltussen, M. G. H. M.; Anderson, P. D.; Ioan, D.; den Toonder, J.M.J.; Onck, Patrick Published
More informationFormulation of the displacement-based Finite Element Method and General Convergence Results
Formulation of the displacement-based Finite Element Method and General Convergence Results z Basics of Elasticity Theory strain e: measure of relative distortions u r r' y for small displacements : x
More informationENGN2340 Final Project: Implementation of a Euler-Bernuolli Beam Element Michael Monn
ENGN234 Final Project: Implementation of a Euler-Bernuolli Beam Element Michael Monn 12/11/13 Problem Definition and Shape Functions Although there exist many analytical solutions to the Euler-Bernuolli
More informationSteps in the Finite Element Method. Chung Hua University Department of Mechanical Engineering Dr. Ching I Chen
Steps in the Finite Element Method Chung Hua University Department of Mechanical Engineering Dr. Ching I Chen General Idea Engineers are interested in evaluating effects such as deformations, stresses,
More informationME 563 HOMEWORK # 5 SOLUTIONS Fall 2010
ME 563 HOMEWORK # 5 SOLUTIONS Fall 2010 PROBLEM 1: You are given the lumped parameter dynamic differential equations of motion for a two degree-offreedom model of an automobile suspension system for small
More informationAnalytical Mechanics: Elastic Deformation
Analytical Mechanics: Elastic Deformation Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation / 59 Agenda Agenda
More informationPseudo-Force Incremental Methods
. 19 Pseudo-Force Incremental Methods 19 1 Chapter 19: PSEUDO-FORCE INCREMENTAL METHODS 19 2 TABLE OF CONTENTS Page 19.1. Pseudo Force Formulation 19 3 19.2. Computing the Reference Stiffness and Internal
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 informationOn the adaptive finite element analysis of the Kohn-Sham equations
On the adaptive finite element analysis of the Kohn-Sham equations Denis Davydov, Toby Young, Paul Steinmann Denis Davydov, LTM, Erlangen, Germany August 2015 Denis Davydov, LTM, Erlangen, Germany College
More informationMoment Distribution Method
Moment Distribution Method Lesson Objectives: 1) Identify the formulation and sign conventions associated with the Moment Distribution Method. 2) Derive the Moment Distribution Method equations using mechanics
More informationAnalytical Mechanics: Elastic Deformation
Analytical Mechanics: Elastic Deformation Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 1 / 60 Agenda Agenda
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 informationVerification of assumptions in dynamics of lattice structures
Verification of assumptions in dynamics of lattice structures B.Błachowski and W.Gutkowski Warsaw, Poland 37th SOLD MECHANCS CONFERENCE, Warsaw, Poland September 6 1, 21 Outline of presentation 1. Motivation
More informationThe CR Formulation: BE Plane Beam
6 The CR Formulation: BE Plane Beam 6 Chapter 6: THE CR FORMUATION: BE PANE BEAM TABE OF CONTENTS Page 6. Introduction..................... 6 4 6.2 CR Beam Kinematics................. 6 4 6.2. Coordinate
More informationCode No: RT41033 R13 Set No. 1 IV B.Tech I Semester Regular Examinations, November - 2016 FINITE ELEMENT METHODS (Common to Mechanical Engineering, Aeronautical Engineering and Automobile Engineering)
More informationCh 3.7: Mechanical & Electrical Vibrations
Ch 3.7: Mechanical & Electrical Vibrations Two important areas of application for second order linear equations with constant coefficients are in modeling mechanical and electrical oscillations. We will
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture - 06
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture - 06 In the last lecture, we have seen a boundary value problem, using the formal
More informationThe Direct Stiffness Method II
The Direct Stiffness Method II Chapter : THE DIRECT STIFFNESS METHOD II TABLE OF CONTENTS Page. The Remaining DSM Steps.................2 Assembly: Merge....................2. Governing Rules.................2.2
More informationME751 Advanced Computational Multibody Dynamics
ME751 Advanced Computational Multibody Dynamics Review: Elements of Linear Algebra & Calculus September 9, 2016 Dan Negrut University of Wisconsin-Madison Quote of the day If you can't convince them, confuse
More informationPILE SOIL INTERACTION MOMENT AREA METHOD
Pile IGC Soil 2009, Interaction Moment Guntur, INDIA Area Method PILE SOIL INTERACTION MOMENT AREA METHOD D.M. Dewaikar Professor, Department of Civil Engineering, IIT Bombay, Mumbai 400 076, India. E-mail:
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67
1/67 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 6 Mathematical Representation of Physical Systems II State Variable Models for Dynamic Systems u 1 u 2 u ṙ. Internal Variables x 1, x 2 x n y 1 y 2. y m Figure
More informationOptimization of nonlinear trusses using a displacement-based approach
Struct Multidisc Optim 23, 214 221 Springer-Verlag 2002 Digital Object Identifier (DOI) 10.1007/s00158-002-0179-1 Optimization of nonlinear trusses using a displacement-based approach S. Missoum, Z. Gürdal
More informationThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems. Prof. Dr. Eleni Chatzi Lecture 6-5 November, 2015
The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Lecture 6-5 November, 015 Institute of Structural Engineering Method of Finite Elements II 1 Introduction
More informationLecture 7. Pile Analysis
Lecture 7 14.5 Release Pile Analysis 2012 ANSYS, Inc. February 9, 2013 1 Release 14.5 Pile definition in Mechanical - There are a number of methods that can be used to analyze piled foundations in ANSYS
More informationA Hybrid Method for the Wave Equation. beilina
A Hybrid Method for the Wave Equation http://www.math.unibas.ch/ beilina 1 The mathematical model The model problem is the wave equation 2 u t 2 = (a 2 u) + f, x Ω R 3, t > 0, (1) u(x, 0) = 0, x Ω, (2)
More informationGeometric Misfitting in Structures An Interval-Based Approach
Geometric Misfitting in Structures An Interval-Based Approach M. V. Rama Rao Vasavi College of Engineering, Hyderabad - 500 031 INDIA Rafi Muhanna School of Civil and Environmental Engineering Georgia
More informationContinuum Mechanics and the Finite Element Method
Continuum Mechanics and the Finite Element Method 1 Assignment 2 Due on March 2 nd @ midnight 2 Suppose you want to simulate this The familiar mass-spring system l 0 l y i X y i x Spring length before/after
More informationReduction of Random Variables in Structural Reliability Analysis
Reduction of Random Variables in Structural Reliability Analysis S. ADHIKARI AND R. S. LANGLEY Cambridge University Engineering Department Cambridge, U.K. Random Variable Reduction in Reliability Analysis
More informationFLEXIBILITY METHOD FOR INDETERMINATE FRAMES
UNIT - I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 1. What is meant by indeterminate structures? Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These
More informationIntroduction ODEs and Linear Systems
BENG 221 Mathematical Methods in Bioengineering ODEs and Linear Systems Gert Cauwenberghs Department of Bioengineering UC San Diego 1.1 Course Objectives 1. Acquire methods for quantitative analysis and
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 informationOptimal thickness of a cylindrical shell under dynamical loading
Optimal thickness of a cylindrical shell under dynamical loading Paul Ziemann Institute of Mathematics and Computer Science, E.-M.-A. University Greifswald, Germany e-mail paul.ziemann@uni-greifswald.de
More informationModule 3. Analysis of Statically Indeterminate Structures by the Displacement Method
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 Finite Element Method for the Analysis of Non-Linear and Dynamic Systems
The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos Agathos Lecture 1-21 September, 2017 Institute of Structural Engineering
More informationCS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages:
CS100: DISCRETE STRUCTURES Lecture 3 Matrices Ch 3 Pages: 246-262 Matrices 2 Introduction DEFINITION 1: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n
More informationCOORDINATE TRANSFORMATIONS
COORDINAE RANSFORMAIONS Members of a structural system are typically oriented in differing directions, e.g., Fig. 17.1. In order to perform an analysis, the element stiffness equations need to be expressed
More informationAA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS 1 / 50 AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of n-dof Systems These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations:
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 informationFinite Element Nonlinear Analysis for Catenary Structure Considering Elastic Deformation
Copyright 21 Tech Science Press CMES, vol.63, no.1, pp.29-45, 21 Finite Element Nonlinear Analysis for Catenary Structure Considering Elastic Deformation B.W. Kim 1, H.G. Sung 1, S.Y. Hong 1 and H.J. Jung
More informationMODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Class Meeting #5: Integration of Constitutive Equations
MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #5: Integration of Constitutive Equations Structural Equilibrium: Incremental Tangent Stiffness
More informationLecture 15: Revisiting bars and beams.
3.10 Potential Energy Approach to Derive Bar Element Equations. We place assumptions on the stresses developed inside the bar. The spatial stress and strain matrices become very sparse. We add (ad-hoc)
More informationMarch 24, Chapter 4. Deflection and Stiffness. Dr. Mohammad Suliman Abuhaiba, PE
Chapter 4 Deflection and Stiffness 1 2 Chapter Outline Spring Rates Tension, Compression, and Torsion Deflection Due to Bending Beam Deflection Methods Beam Deflections by Superposition Strain Energy Castigliano
More informationBase force element method of complementary energy principle for large rotation problems
Acta Mech Sin (2009) 25:507 515 DOI 10.1007/s10409-009-0234-x RESEARH PAPER Base force element method of complementary energy principle for large rotation problems Yijiang Peng Yinghua Liu Received: 26
More information