Lecture 9. Stability of Elastic Structures. Lecture 10. Advanced Topic in Column Buckling
|
|
- Carol Elliott
- 5 years ago
- Views:
Transcription
1 Lecture 9 Stabiity of Eastic Structures Lecture 1 Advanced Topic in Coumn Bucking robem 9-1: A camped-free coumn is oaded at its tip by a oad. The issue here is to find the itica bucking oad. a) Suggest a simpe form of the bucked of the coumn, satisfying kinematic boundary conditions. b) Use the Rayeigh-Ritz quotient to find the approximate vaue of the bucking oad. c) Come up with another bucking shape which woud give you a ower vaue for the bucking oad. d) Find the exact soution of the probem and show the convergence of the approximate soution to the exact soution. Foow the exampe of a pin-pin coumn, which is presented in the notes of Lecture 9. 1
2 robem 9-1 Soution: a) Kinematic boundary condition, in term of shape function I( x ), for a camped-free coumn is () '() Choose a bucking shape ( x) x '( x ) x ''( x ) b) Use Rayeigh-Ritz Quotient, the itica bucking oad is N c I'' I'' dx II ' 'dx u dx x u xdx N c L
3 c) Choose a bucking shape simiar to a cantiever beam I( x ) x Lx I '( x ) x 6Lx I ''( x ) 6x 6L N c I'' I'' dx II ' 'dx ³ 6x 6L dx x 6Lx dx ³ 1L 4 L 5 5 Compare to the resut in b), N c.5 L N c, this bucking shape givers a ower vaue L d) Choose bucking shape I( x) S 1 cos x L
4 S S I '(x) sin x L L S I ''( x ) cos x L L N c I'' I''dx II ' 'dx ª S S º «cos x» dx «L L ¼» S sin x dx L L ³ S 4L N c.47 L Check for oca equiibrium of the soution ª S 4 S S w IV N c w '' A «cos x cos x «L L 4L L L»¼ This is the exact soution to the camped-free bucking 4
5 robem 9-: Consider a camped-free coumn oaded by a compressive force at the free end. a) Determine the itica senderness ratio E it distinguishing between the eastic and pastic bucking response. What is the bucking stress and strain? b) Cacuate the itica pastic bucking oad for E.5E it and the corresponding stress and strain. c) Cacuate the itica eastic bucking oad for E E it and the corresponding stress and strain. d) Compare a three resuts. robem 9- Soution: a) First, find the bending oad For Camed-Free coumn S L S 4L Second, find the bucking stress and strain Reca that S A 4AL V bucking 5
6 I I r r A A Then Reca that Er E V bucking 4L 4L r L r Er E V 4L 4 V S E 4E H Third, find when V bucking V yied E E it when V y, which is S E V y 4E it S E 4V y E it E V y E it b) E.5E it, the coumn yieds + hits pastic bucking 6
7 S E S t E t p (From Lecture Note, equation 9.7) 1 E it 4 Ei t ¹ p S S n n 1 E it 4 Eit c) E it, the coumn wi bucke easticay E E E V 4 4E 16E it it V S E H 16E it d) Compare the three resuts Yied y y Eastic Bucking E.67.7 y.67 it it astic Bucking E 4 it y 4 it it E To simpy our comparison, assume n., E t.5e (*) and reca (*)In order to compare pastic bucking to eastic+yied, we need to make future assumption about the materia properties. V y E it 7
8 robem 9-: Consider the pin-pin coumn. a) Suggest a poynomia bucking shape function ( x) to improve the approximate soution derived in ecture note. Note that the one used in cass was the paraboic shape. b) Determine the accuracy reative to the exact soution. E I L x robem 9- Soution: S x x a) The exact soution is w sin, use the none-dimensioned vaue F, the Tayor L L series expansion is So we know the shape function must be SF sin SF SF 6 IF ( ) C 1 F C F For x L, the boundary conditions are The first boundary condition gives this doesn t hep. The second boundary condition gives I 1 I' F ¹ I C 1 C 8
9 d x 1 where d F dx dx C C ' 1 1 ' C1 C 4 C 4 C 1 So we have 4 ( ) C1 C1 4 ( ) C1 We can us the Rayeigh-Ritz Quotient N '' dx ' dx '( ) 1 4 x C1 d dx 4 '( ) C d dx 1 ''( x) 8C d dx ''( ) 64C d dx Since we have considered the shape function for d x d L, we must adjust the imits on the integra N ' '' 64C1 d dx dx C d dx dx 1 L dx dx x 1 dx after engthy agebra 4 x x dx 4 9
10 N 1 L b) The resut are compared with the poynomia used in cass and the exact soution Exact Soution araboic Cubic C sin C C 1 C C 1 Coefficient Error N/A 1.5% 1.% Notice how we significanty reduce the error by incuding a higher order term. 1
11 robem 9-4: resent a step-by-step derivation of the bucking soution of the pin-camped coumn from the oca equiibrium equation. E I L x robem 9-4 Soution: Boundary condition for this probem Start with 4 th order ODE ww wl w' w''l w IV w'' IV w w'' We have an eigenvaue probem 4, i 1 4 Define K 4 ik wc C x C sin Kx C cos Kx
12 Use the boundary conditions to sove for constants C, 1 C, C and C 4 w w ' w L C 1 C L w L Substitute C C 1, into the above expression w'' L w C1C4 C C 1 4 w' x C KC cos KxKC sin Kx 4 w' C KC C KC C sin KL C 4 cos KL C KLsin KL C 1cos KL 4 w'' x K C sin KxK C cos Kx w'' L K C sin KLK C cos KL 4 4 det> KL sin KL 1cos KLC K sin KL K cos KL C4 K KL KL KL K KL KL cos sin sin 1cos KL cos KL sin KL sin KL KL tan KL cos KL So the equation to sove in order to find is tan KL KL The smaest roots are KL and KL 4.49, we choose KL
13 L L.7L.7L 1
14 robem 9-5: a) Derive the soution for an imperfect camped-free coumn (ike that considered in probem 9-1, foowing a simiar derivation given in the notes for a pin-pin coumn in the notes. b) Find the ratio of current defection ampitude to the ampitude of the initia imperfection such that the resuting oad is 8% of the theoretica bucking oad of a perfect coumn. robem 9-5 Soution: a) wx : shape of initia imperfection wx : actua bucked shape w o x : ampitude of initia imperfection w o : end ampitude of actua imperfection Moment equiibrium of imperfect coumn w w'' w w o erfect coumn w x Assume that the initia imperfection is in the same shape as the bucking shaper From boundary condition o 1cos w x w Ox w x w o 1 cos Ox wl 14
15 w o O cos OL From moment equiibrium of imperfect coumn erfect coumn L n 1 w wo cosx wo 1 1cosx wo w wo w o S E I O 4L Imperfect coumn w o w o w o w o 1 4L wo wo 1 w o b) When.8 w w o o 1. w w o o 5 15
16 robem 9-6: The pin-pin eastic coumn of ength L (shown beow) is an I section can bucke in either pane. a) Determine the bucking oad in terms of L, b 1,b, t and E. Assume that t<<b. b) What shoud the ratio of b 1 /b be in order for the probabiity of bucking in either of the bucking panes to be the same? Bonus: What coud happen for very arge width to thickness ratio? b b 1 L x robem 9-6 Soution: a) The moment s of inertia for an I shape oss-section is I I zz 1 b b t 1 1 ¹ 1 tb b 1 6b 1 yy tb 1 1 tb 1 1 tb 1 tb
17 If I yy I zz, the coumn wi bucke in x-z pane S yy S E tb b 6b 1 1 If I yy I zz, the coumn wi bucke in x-y pane S S zz E tb 6 1 b) For the probabiity of bucking in either of the panes to be the same, we want I yy I zz The ony physica soution is 1 tb b 6b 1 1 tb b 1 b1 1 b b b b c) If b 1!! t, b!! t, then oca pate bucking my deveop. 17
18 MIT OpenCourseWare / 1.57J Structura Mechanics Fa 1 For information about citing these materias or our Terms of Use, visit:
Lecture 6: Moderately Large Deflection Theory of Beams
Structura Mechanics 2.8 Lecture 6 Semester Yr Lecture 6: Moderatey Large Defection Theory of Beams 6.1 Genera Formuation Compare to the cassica theory of beams with infinitesima deformation, the moderatey
More informationCE601-Structura Anaysis I UNIT-IV SOPE-DEFECTION METHOD 1. What are the assumptions made in sope-defection method? (i) Between each pair of the supports the beam section is constant. (ii) The joint in
More informationModule 22: Simple Harmonic Oscillation and Torque
Modue : Simpe Harmonic Osciation and Torque.1 Introduction We have aready used Newton s Second Law or Conservation of Energy to anayze systems ike the boc-spring system that osciate. We sha now use torque
More information3.10 Implications of Redundancy
118 IB Structures 2008-9 3.10 Impications of Redundancy An important aspect of redundant structures is that it is possibe to have interna forces within the structure, with no externa oading being appied.
More informationVTU-NPTEL-NMEICT Project
MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 VTU-NPTE-NMEICT Project Progress Report The Project on Deveopment of Remaining Three Quadrants to NPTE Phase-I under grant in aid
More informationWork and energy method. Exercise 1 : Beam with a couple. Exercise 1 : Non-linear loaddisplacement. Exercise 2 : Horizontally loaded frame
Work and energy method EI EI T x-axis Exercise 1 : Beam with a coupe Determine the rotation at the right support of the construction dispayed on the right, caused by the coupe T using Castigiano s nd theorem.
More informationNonlinear Analysis of Spatial Trusses
Noninear Anaysis of Spatia Trusses João Barrigó October 14 Abstract The present work addresses the noninear behavior of space trusses A formuation for geometrica noninear anaysis is presented, which incudes
More informationTHE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS
ECCM6-6 TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS, Sevie, Spain, -6 June 04 THE OUT-OF-PLANE BEHAVIOUR OF SPREAD-TOW FABRICS M. Wysocki a,b*, M. Szpieg a, P. Heström a and F. Ohsson c a Swerea SICOMP
More information1D Heat Propagation Problems
Chapter 1 1D Heat Propagation Probems If the ambient space of the heat conduction has ony one dimension, the Fourier equation reduces to the foowing for an homogeneous body cρ T t = T λ 2 + Q, 1.1) x2
More informationCHAPTER 9. Columns and Struts
CHATER 9 Coumns and Struts robem. Compare the ratio of the strength of soid stee coumn to that of the hoow stee coumn of the same cross-sectiona area. The interna diameter of the hoow coumn is /th of the
More informationSeparation of Variables and a Spherical Shell with Surface Charge
Separation of Variabes and a Spherica She with Surface Charge In cass we worked out the eectrostatic potentia due to a spherica she of radius R with a surface charge density σθ = σ cos θ. This cacuation
More informationTorsion and shear stresses due to shear centre eccentricity in SCIA Engineer Delft University of Technology. Marijn Drillenburg
Torsion and shear stresses due to shear centre eccentricity in SCIA Engineer Deft University of Technoogy Marijn Drienburg October 2017 Contents 1 Introduction 2 1.1 Hand Cacuation....................................
More informationFinite element method for structural dynamic and stability analyses
Finite eement method for structura dynamic and stabiity anayses Modue-9 Structura stabiity anaysis Lecture-33 Dynamic anaysis of stabiity and anaysis of time varying systems Prof C S Manohar Department
More information221B Lecture Notes Notes on Spherical Bessel Functions
Definitions B Lecture Notes Notes on Spherica Besse Functions We woud ike to sove the free Schrödinger equation [ h d r R(r) = h k R(r). () m r dr r m R(r) is the radia wave function ψ( x) = R(r)Y m (θ,
More informationUnit 48: Structural Behaviour and Detailing for Construction. Deflection of Beams
Unit 48: Structura Behaviour and Detaiing for Construction 4.1 Introduction Defection of Beams This topic investigates the deformation of beams as the direct effect of that bending tendency, which affects
More informationLecture Notes for Math 251: ODE and PDE. Lecture 34: 10.7 Wave Equation and Vibrations of an Elastic String
ecture Notes for Math 251: ODE and PDE. ecture 3: 1.7 Wave Equation and Vibrations of an Eastic String Shawn D. Ryan Spring 212 ast Time: We studied other Heat Equation probems with various other boundary
More informationTechnical Data for Profiles. Groove position, external dimensions and modular dimensions
Technica Data for Profies Extruded Profie Symbo A Mg Si 0.5 F 25 Materia number.206.72 Status: artificiay aged Mechanica vaues (appy ony in pressing direction) Tensie strength Rm min. 245 N/mm 2 Yied point
More informationUI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE
UI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE Juan Huang, Ronghui Wang and Tao Tang Coege of Traffic and Communications, South China University of Technoogy, Guangzhou, Guangdong 51641,
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
odue 2 naysis of Staticay ndeterminate Structures by the atri Force ethod Version 2 E T, Kharagpur esson 12 The Three-oment Equations- Version 2 E T, Kharagpur nstructiona Objectives fter reading this
More informationCS229 Lecture notes. Andrew Ng
CS229 Lecture notes Andrew Ng Part IX The EM agorithm In the previous set of notes, we taked about the EM agorithm as appied to fitting a mixture of Gaussians. In this set of notes, we give a broader view
More informationSEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l
Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed
More informationInstructional Objectives:
Instructiona Objectives: At te end of tis esson, te students soud be abe to understand: Ways in wic eccentric oads appear in a weded joint. Genera procedure of designing a weded joint for eccentric oading.
More informationGauss Law. 2. Gauss s Law: connects charge and field 3. Applications of Gauss s Law
Gauss Law 1. Review on 1) Couomb s Law (charge and force) 2) Eectric Fied (fied and force) 2. Gauss s Law: connects charge and fied 3. Appications of Gauss s Law Couomb s Law and Eectric Fied Couomb s
More informationApplication of the Finite Fourier Sine Transform Method for the Flexural-Torsional Buckling Analysis of Thin-Walled Columns
IOSR Journa of Mechanica and Civi Engineering (IOSR-JMCE) e-issn: 78-1684,p-ISSN: 3-334X, Voume 14, Issue Ver. I (Mar. - Apr. 17), PP 51-6 www.iosrjournas.org Appication of the Finite Fourier Sine Transform
More informationThe EM Algorithm applied to determining new limit points of Mahler measures
Contro and Cybernetics vo. 39 (2010) No. 4 The EM Agorithm appied to determining new imit points of Maher measures by Souad E Otmani, Georges Rhin and Jean-Marc Sac-Épée Université Pau Veraine-Metz, LMAM,
More informationDavid Eigen. MA112 Final Paper. May 10, 2002
David Eigen MA112 Fina Paper May 1, 22 The Schrodinger equation describes the position of an eectron as a wave. The wave function Ψ(t, x is interpreted as a probabiity density for the position of the eectron.
More informationMath 124B January 31, 2012
Math 124B January 31, 212 Viktor Grigoryan 7 Inhomogeneous boundary vaue probems Having studied the theory of Fourier series, with which we successfuy soved boundary vaue probems for the homogeneous heat
More informationLECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Eectromagnetism II October, 202 Prof. Aan Guth LECTURE NOTES 8 THE TRACELESS SYMMETRIC TENSOR EXPANSION AND STANDARD SPHERICAL HARMONICS
More informationAPPENDIX C FLEXING OF LENGTH BARS
Fexing of ength bars 83 APPENDIX C FLEXING OF LENGTH BARS C.1 FLEXING OF A LENGTH BAR DUE TO ITS OWN WEIGHT Any object ying in a horizonta pane wi sag under its own weight uness it is infinitey stiff or
More information4 1-D Boundary Value Problems Heat Equation
4 -D Boundary Vaue Probems Heat Equation The main purpose of this chapter is to study boundary vaue probems for the heat equation on a finite rod a x b. u t (x, t = ku xx (x, t, a < x < b, t > u(x, = ϕ(x
More informationLecture Notes for Math 251: ODE and PDE. Lecture 32: 10.2 Fourier Series
Lecture Notes for Math 251: ODE and PDE. Lecture 32: 1.2 Fourier Series Shawn D. Ryan Spring 212 Last Time: We studied the heat equation and the method of Separation of Variabes. We then used Separation
More informationMATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES
MATH 172: MOTIVATION FOR FOURIER SERIES: SEPARATION OF VARIABLES Separation of variabes is a method to sove certain PDEs which have a warped product structure. First, on R n, a inear PDE of order m is
More informationMalaysian Journal of Civil Engineering 30(2): (2018)
Maaysian Journa of Ci Engineering 3():331-346 (18) BUBNOV-GALERKIN METHOD FOR THE ELASTIC BUCKLING OF EULER COLUMNS Ofondu I.O. 1, Ikwueze E. U. & Ike C. C. * 1 Dept. of Mechanica and Production Engineering,
More informationFOURIER SERIES ON ANY INTERVAL
FOURIER SERIES ON ANY INTERVAL Overview We have spent considerabe time earning how to compute Fourier series for functions that have a period of 2p on the interva (-p,p). We have aso seen how Fourier series
More informationСРАВНИТЕЛЕН АНАЛИЗ НА МОДЕЛИ НА ГРЕДИ НА ЕЛАСТИЧНА ОСНОВА COMPARATIVE ANALYSIS OF ELASTIC FOUNDATION MODELS FOR BEAMS
СРАВНИТЕЛЕН АНАЛИЗ НА МОДЕЛИ НА ГРЕДИ НА ЕЛАСТИЧНА ОСНОВА Милко Стоянов Милошев 1, Константин Савков Казаков 2 Висше Строително Училище Л. Каравелов - София COMPARATIVE ANALYSIS OF ELASTIC FOUNDATION MODELS
More informationFourier Series. 10 (D3.9) Find the Cesàro sum of the series. 11 (D3.9) Let a and b be real numbers. Under what conditions does a series of the form
Exercises Fourier Anaysis MMG70, Autumn 007 The exercises are taken from: Version: Monday October, 007 DXY Section XY of H F Davis, Fourier Series and orthogona functions EÖ Some exercises from earier
More informationLecture Notes 4: Fourier Series and PDE s
Lecture Notes 4: Fourier Series and PDE s 1. Periodic Functions A function fx defined on R is caed a periodic function if there exists a number T > such that fx + T = fx, x R. 1.1 The smaest number T for
More information1 Equivalent SDOF Approach. Sri Tudjono 1,*, and Patria Kusumaningrum 2
MATEC Web of Conferences 159, 01005 (018) IJCAET & ISAMPE 017 https://doi.org/10.1051/matecconf/01815901005 Dynamic Response of RC Cantiever Beam by Equivaent Singe Degree of Freedom Method on Eastic Anaysis
More informationThe Bending of Rectangular Deep Beams with Fixed at Both Ends under Uniform Load
Engineering,,, 8-9 doi:.6/eng..7 Pubised Onine December (ttp://.scirp.org/journa/eng) Te Bending of Rectanguar Deep Beams it Fied at Bot Ends under Uniform Load Abstract Ying-Jie Cen, Bao-Lian Fu, Gang
More informationSTABILITY OF A PARAMETRICALLY EXCITED DAMPED INVERTED PENDULUM 1. INTRODUCTION
Journa of Sound and Vibration (996) 98(5), 643 65 STABILITY OF A PARAMETRICALLY EXCITED DAMPED INVERTED PENDULUM G. ERDOS AND T. SINGH Department of Mechanica and Aerospace Engineering, SUNY at Buffao,
More informationMore Scattering: the Partial Wave Expansion
More Scattering: the Partia Wave Expansion Michae Fower /7/8 Pane Waves and Partia Waves We are considering the soution to Schrödinger s equation for scattering of an incoming pane wave in the z-direction
More informationAST 418/518 Instrumentation and Statistics
AST 418/518 Instrumentation and Statistics Cass Website: http://ircamera.as.arizona.edu/astr_518 Cass Texts: Practica Statistics for Astronomers, J.V. Wa, and C.R. Jenkins, Second Edition. Measuring the
More informationarxiv: v1 [hep-th] 10 Dec 2018
Casimir energy of an open string with ange-dependent boundary condition A. Jahan 1 and I. Brevik 2 1 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM, Maragha, Iran 2 Department of Energy
More informationLecture 8 February 18, 2010
Sources of Eectromagnetic Fieds Lecture 8 February 18, 2010 We now start to discuss radiation in free space. We wi reorder the materia of Chapter 9, bringing sections 6 7 up front. We wi aso cover some
More informationLaboratory Exercise 1: Pendulum Acceleration Measurement and Prediction Laboratory Handout AME 20213: Fundamentals of Measurements and Data Analysis
Laboratory Exercise 1: Penduum Acceeration Measurement and Prediction Laboratory Handout AME 20213: Fundamentas of Measurements and Data Anaysis Prepared by: Danie Van Ness Date exercises to be performed:
More informationLobontiu: System Dynamics for Engineering Students Website Chapter 3 1. z b z
Chapter W3 Mechanica Systems II Introduction This companion website chapter anayzes the foowing topics in connection to the printed-book Chapter 3: Lumped-parameter inertia fractions of basic compiant
More informationMath 124B January 17, 2012
Math 124B January 17, 212 Viktor Grigoryan 3 Fu Fourier series We saw in previous ectures how the Dirichet and Neumann boundary conditions ead to respectivey sine and cosine Fourier series of the initia
More informationSTRUCTURAL ANALYSIS - I UNIT-I DEFLECTION OF DETERMINATE STRUCTURES
STRUCTURL NLYSIS - I UNIT-I DEFLECTION OF DETERMINTE STRUCTURES 1. Why is it necessary to compute defections in structures? Computation of defection of structures is necessary for the foowing reasons:
More informationStrain Energy in Linear Elastic Solids
Strain Energ in Linear Eastic Soids CEE L. Uncertaint, Design, and Optimiation Department of Civi and Environmenta Engineering Duke Universit Henri P. Gavin Spring, 5 Consider a force, F i, appied gradua
More informationMECHANICAL ENGINEERING
1 SSC-JE SFF SELECION COMMISSION MECHNICL ENGINEERING SUDY MERIL Cassroom Posta Correspondence est-series16 Rights Reserved www.sscje.com C O N E N 1. SIMPLE SRESSES ND SRINS 3-3. PRINCIPL SRESS ND SRIN
More informationMath 220B - Summer 2003 Homework 1 Solutions
Math 0B - Summer 003 Homework Soutions Consider the eigenvaue probem { X = λx 0 < x < X satisfies symmetric BCs x = 0, Suppose f(x)f (x) x=b x=a 0 for a rea-vaued functions f(x) which satisfy the boundary
More informationAdd Math (4044/02) (+) x (+) 2. Find the coordinates of the points of intersection of the curve xy 2 the line 2y 1 x 0. [5]
Add Math (444/) Requirement : Answer a questions Tota mars : 7 Duration : hour 45 minutes. Sove the inequaity 5 and represent the soution set on the number ine. [4] 5 4 From the setch on number ine, we
More information22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 2: The Moment Equations
.615, MHD Theory of Fusion ystems Prof. Freidberg Lecture : The Moment Equations Botzmann-Maxwe Equations 1. Reca that the genera couped Botzmann-Maxwe equations can be written as f q + v + E + v B f =
More informationHYDROGEN ATOM SELECTION RULES TRANSITION RATES
DOING PHYSICS WITH MATLAB QUANTUM PHYSICS Ian Cooper Schoo of Physics, University of Sydney ian.cooper@sydney.edu.au HYDROGEN ATOM SELECTION RULES TRANSITION RATES DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS
More information4 Separation of Variables
4 Separation of Variabes In this chapter we describe a cassica technique for constructing forma soutions to inear boundary vaue probems. The soution of three cassica (paraboic, hyperboic and eiptic) PDE
More informationBending Analysis of Continuous Castellated Beams
Bending Anaysis of Continuous Casteated Beams * Sahar Eaiwi 1), Boksun Kim ) and Long-yuan Li 3) 1), ), 3) Schoo of Engineering, Pymouth University, Drake Circus, Pymouth, UK PL4 8AA 1) sahar.eaiwi@pymouth.ac.uk
More information> 2 CHAPTER 3 SLAB 3.1 INTRODUCTION 3.2 TYPES OF SLAB
CHAPTER 3 SLAB 3. INTRODUCTION Reinforced concrete sabs are one of the most widey used structura eements. In many structures, in addition to providing a versatie and economica method of supporting gravity
More informationTHINKING IN PYRAMIDS
ECS 178 Course Notes THINKING IN PYRAMIDS Kenneth I. Joy Institute for Data Anaysis and Visuaization Department of Computer Science University of Caifornia, Davis Overview It is frequenty usefu to think
More informationRelated Topics Maxwell s equations, electrical eddy field, magnetic field of coils, coil, magnetic flux, induced voltage
Magnetic induction TEP Reated Topics Maxwe s equations, eectrica eddy fied, magnetic fied of cois, coi, magnetic fux, induced votage Principe A magnetic fied of variabe frequency and varying strength is
More informationA PROCEDURE ON STUDYING A CRITICAL FORCE ON FEM AS PLASTIC DEFORMATIONS ARISE WITHIN STRAIGHT TRUSSES
11 th Nationa Congress on Theoretica and Appied Mechanics, -5 Sept. 009, Borovets, Bugaria A OCDU ON STUDYING A CITICAL FOC ON FM AS LASTIC DFOMATIONS AIS WITHIN STAIGHT TUSSS IVAN VAISILOV VSU L. Karaveov,
More information1 Equations of Motion 3: Equivalent System Method
8 Mechanica Vibrations Equations of Motion : Equivaent System Method In systems in which masses are joined by rigid ins, evers, or gears and in some distributed systems, various springs, dampers, and masses
More informationProblem Set 6: Solutions
University of Aabama Department of Physics and Astronomy PH 102 / LeCair Summer II 2010 Probem Set 6: Soutions 1. A conducting rectanguar oop of mass M, resistance R, and dimensions w by fas from rest
More informationC1B Stress Analysis Lecture 1: Minimum Energy Principles in Mechanics Prof Alexander M. Korsunsky Hilary Term (January 08)
CB Stress Anaysis ecture : Minimum Energy Principes in Mechanics Prof Aexander M. Korsunsky Hiary Term (January 8) http://users.ox.ac.uk/~engs6/4me6.htm This course introduces seected chapters in Stress
More informationChemical Kinetics Part 2
Integrated Rate Laws Chemica Kinetics Part 2 The rate aw we have discussed thus far is the differentia rate aw. Let us consider the very simpe reaction: a A à products The differentia rate reates the rate
More informationBuckling of elastic structures under tensile loads
Acta Mech 229, 881 900 (2018) https://doi.org/10.1007/s00707-017-2006-1 ORIGINAL PAPER F. G. Rammerstorfer Bucking of eastic structures under tensie oads This paper is dedicated to the memory of Franz
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING. Question Bank. Sub. Code/Name: CE1303 Structural Analysis-I
KINGS COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING Question Bank Sub. Code/Name: CE1303 Structura Anaysis-I Year: III Sem:V UNIT-I DEFLECTION OF DETERMINATE STRUCTURES 1.Why is it necessary to
More informationAn approximate method for solving the inverse scattering problem with fixed-energy data
J. Inv. I-Posed Probems, Vo. 7, No. 6, pp. 561 571 (1999) c VSP 1999 An approximate method for soving the inverse scattering probem with fixed-energy data A. G. Ramm and W. Scheid Received May 12, 1999
More informationWeek 6 Lectures, Math 6451, Tanveer
Fourier Series Week 6 Lectures, Math 645, Tanveer In the context of separation of variabe to find soutions of PDEs, we encountered or and in other cases f(x = f(x = a 0 + f(x = a 0 + b n sin nπx { a n
More information6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17. Solution 7
6.434J/16.391J Statistics for Engineers and Scientists May 4 MIT, Spring 2006 Handout #17 Soution 7 Probem 1: Generating Random Variabes Each part of this probem requires impementation in MATLAB. For the
More informationVersion 2.2 NE03 - Faraday's Law of Induction
Definition Version. Laboratory Manua Department of Physics he University of Hong Kong Aims o demonstrate various properties of Faraday s Law such as: 1. Verify the aw.. Demonstrate the ighty damped osciation
More information$, (2.1) n="# #. (2.2)
Chapter. Eectrostatic II Notes: Most of the materia presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartoo, Chap... Mathematica Considerations.. The Fourier series and the Fourier
More informationNonperturbative Shell Correction to the Bethe Bloch Formula for the Energy Losses of Fast Charged Particles
ISSN 002-3640, JETP Letters, 20, Vo. 94, No., pp. 5. Peiades Pubishing, Inc., 20. Origina Russian Text V.I. Matveev, D.N. Makarov, 20, pubished in Pis ma v Zhurna Eksperimenta noi i Teoreticheskoi Fiziki,
More informationStructural Analysis III Revised Semester 2 Exam Information. Semester /9
Structura naysis III Structura naysis III Revised Semester Exam Information Semester 008/9 Dr. oin aprani Dr.. aprani Structura naysis III. Exam Format Introduction The exam format is being atered this
More informationSVM: Terminology 1(6) SVM: Terminology 2(6)
Andrew Kusiak Inteigent Systems Laboratory 39 Seamans Center he University of Iowa Iowa City, IA 54-57 SVM he maxima margin cassifier is simiar to the perceptron: It aso assumes that the data points are
More informationPhysics 127c: Statistical Mechanics. Fermi Liquid Theory: Collective Modes. Boltzmann Equation. The quasiparticle energy including interactions
Physics 27c: Statistica Mechanics Fermi Liquid Theory: Coective Modes Botzmann Equation The quasipartice energy incuding interactions ε p,σ = ε p + f(p, p ; σ, σ )δn p,σ, () p,σ with ε p ε F + v F (p p
More informationAutomobile Prices in Market Equilibrium. Berry, Pakes and Levinsohn
Automobie Prices in Market Equiibrium Berry, Pakes and Levinsohn Empirica Anaysis of demand and suppy in a differentiated products market: equiibrium in the U.S. automobie market. Oigopoistic Differentiated
More informationIntroduction. Figure 1 W8LC Line Array, box and horn element. Highlighted section modelled.
imuation of the acoustic fied produced by cavities using the Boundary Eement Rayeigh Integra Method () and its appication to a horn oudspeaer. tephen Kirup East Lancashire Institute, Due treet, Bacburn,
More informationSolution of Wave Equation by the Method of Separation of Variables Using the Foss Tools Maxima
Internationa Journa of Pure and Appied Mathematics Voume 117 No. 14 2017, 167-174 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-ine version) ur: http://www.ijpam.eu Specia Issue ijpam.eu Soution
More informationIn-plane shear stiffness of bare steel deck through shell finite element models. G. Bian, B.W. Schafer. June 2017
In-pane shear stiffness of bare stee deck through she finite eement modes G. Bian, B.W. Schafer June 7 COLD-FORMED STEEL RESEARCH CONSORTIUM REPORT SERIES CFSRC R-7- SDII Stee Diaphragm Innovation Initiative
More informationQuantum Mechanical Models of Vibration and Rotation of Molecules Chapter 18
Quantum Mechanica Modes of Vibration and Rotation of Moecues Chapter 18 Moecuar Energy Transationa Vibrationa Rotationa Eectronic Moecuar Motions Vibrations of Moecues: Mode approximates moecues to atoms
More informationDelhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web: Ph:
Seria : 0 GH1_ME Strength of Materia_1019 Dehi Noida hopa Hyderabad Jaipur Lucknow Indore une hubaneswar Kokata atna Web: E-mai: info@madeeasy.in h: 011-5161 LSS TEST 019-00 MEHNIL ENGINEERING Subject
More informationC. Fourier Sine Series Overview
12 PHILIP D. LOEWEN C. Fourier Sine Series Overview Let some constant > be given. The symboic form of the FSS Eigenvaue probem combines an ordinary differentia equation (ODE) on the interva (, ) with a
More information7. CREST-TO-TROUGH WAVE HEIGHT DISTRIBUTION
7. CREST-TO-TROUGH WAVE HEIGHT DISTRIBUTION 7.1. Introduction In Chater 5, it has been mentioned that, in the wide sectrum case, the assumtion of H η does not hod even in the narrow case (considering that
More informationFirst-Order Corrections to Gutzwiller s Trace Formula for Systems with Discrete Symmetries
c 26 Noninear Phenomena in Compex Systems First-Order Corrections to Gutzwier s Trace Formua for Systems with Discrete Symmetries Hoger Cartarius, Jörg Main, and Günter Wunner Institut für Theoretische
More informationStrauss PDEs 2e: Section Exercise 1 Page 1 of 7
Strauss PDEs 2e: Section 4.3 - Exercise 1 Page 1 of 7 Exercise 1 Find the eigenvaues graphicay for the boundary conditions X(0) = 0, X () + ax() = 0. Assume that a 0. Soution The aim here is to determine
More informationA Brief Introduction to Markov Chains and Hidden Markov Models
A Brief Introduction to Markov Chains and Hidden Markov Modes Aen B MacKenzie Notes for December 1, 3, &8, 2015 Discrete-Time Markov Chains You may reca that when we first introduced random processes,
More informationLegendre Polynomials - Lecture 8
Legendre Poynomias - Lecture 8 Introduction In spherica coordinates the separation of variabes for the function of the poar ange resuts in Legendre s equation when the soution is independent of the azimutha
More information2.1. Cantilever The Hooke's law
.1. Cantiever.1.1 The Hooke's aw The cantiever is the most common sensor of the force interaction in atomic force microscopy. The atomic force microscope acquires any information about a surface because
More informationLecture 17 - The Secrets we have Swept Under the Rug
Lecture 17 - The Secrets we have Swept Under the Rug Today s ectures examines some of the uirky features of eectrostatics that we have negected up unti this point A Puzze... Let s go back to the basics
More informationVibrations of beams with a variable cross-section fixed on rotational rigid disks
1(13) 39 57 Vibrations of beams with a variabe cross-section fixed on rotationa rigid disks Abstract The work is focused on the probem of vibrating beams with a variabe cross-section fixed on a rotationa
More informationChemical Kinetics Part 2. Chapter 16
Chemica Kinetics Part 2 Chapter 16 Integrated Rate Laws The rate aw we have discussed thus far is the differentia rate aw. Let us consider the very simpe reaction: a A à products The differentia rate reates
More informationSupporting Information for Suppressing Klein tunneling in graphene using a one-dimensional array of localized scatterers
Supporting Information for Suppressing Kein tunneing in graphene using a one-dimensiona array of ocaized scatterers Jamie D Was, and Danie Hadad Department of Chemistry, University of Miami, Cora Gabes,
More informationCourse 2BA1, Section 11: Periodic Functions and Fourier Series
Course BA, 8 9 Section : Periodic Functions and Fourier Series David R. Wikins Copyright c David R. Wikins 9 Contents Periodic Functions and Fourier Series 74. Fourier Series of Even and Odd Functions...........
More informationV.B The Cluster Expansion
V.B The Custer Expansion For short range interactions, speciay with a hard core, it is much better to repace the expansion parameter V( q ) by f(q ) = exp ( βv( q )) 1, which is obtained by summing over
More informationModal analysis of a multi-blade system undergoing rotational motion
Journa of Mechanica Science and Technoogy 3 (9) 5~58 Journa of Mechanica Science and Technoogy www.springerin.com/content/738-494x DOI.7/s6-9-43-3 Moda anaysis of a muti-bade system undergoing rotationa
More informationModel Solutions (week 4)
CIV-E16 (17) Engineering Computation and Simuation 1 Home Exercise 6.3 Mode Soutions (week 4) Construct the inear Lagrange basis functions (noda vaues as degrees of freedom) of the ine segment reference
More informationSIMULATION OF TEXTILE COMPOSITE REINFORCEMENT USING ROTATION FREE SHELL FINITE ELEMENT
8 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS SIMULATION OF TEXTILE COMPOSITE REINFORCEMENT USING ROTATION FREE SHELL FINITE ELEMENT P. Wang, N. Hamia *, P. Boisse Universite de Lyon, INSA-Lyon,
More informationMethods for Ordinary Differential Equations. Jacob White
Introduction to Simuation - Lecture 12 for Ordinary Differentia Equations Jacob White Thanks to Deepak Ramaswamy, Jaime Peraire, Micha Rewienski, and Karen Veroy Outine Initia Vaue probem exampes Signa
More informationDYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE
3 th Word Conference on Earthquake Engineering Vancouver, B.C., Canada August -6, 4 Paper No. 38 DYNAMIC RESPONSE OF CIRCULAR FOOTINGS ON SATURATED POROELASTIC HALFSPACE Bo JIN SUMMARY The dynamic responses
More informationCantilever Beam Static and Dynamic Response Comparison with Mid-Point Bending for Thin MDF Composite Panels
Cantiever Beam Static and Dynamic Response Comparison with Mid-Point Bending for Thin MDF Composite Panes John F. Hunt, a, * Houjiang Zhang, b Zhiren Guo, b and Feng Fu c A new cantiever beam apparatus
More information