OTHER TPOICS OF INTEREST (Convection BC, Axisymmetric problems, 3D FEM)
|
|
- Elijah Cooper
- 5 years ago
- Views:
Transcription
1 OTHER TPOICS OF INTEREST (Convction BC, Axisymmtric problms, 3D FEM) CONTENTS 2-D Problms with convction BC Typs of Axisymmtric Problms Axisymmtric Problms (2-D) 3-D Hat Transfr 3-D Elasticity Typical 3-D Finit Elmnts 2-D Problms: 1
2 CONVECTION HEAT TRANSFER u u a a f x x y y x y n u u a n a n u u q x y ( ) i h i h h h i x y i h i h i n w u w u a a wf dxdy x x y y u u w a n a n ds x y w u w u a a wf dxdy w q x x y y ( ) ( ) h i h i h i h i n u u ds w u w u a a wf dxdy w u ds w q u ds x x y y 2-D Problms: 2
3 CONVECTION HEAT TRANSFER n 0 Ku ij j fi Qi Ku ij j Fi or Ku F j1 j1 K a a dxdy ds i j i j ij i j x x y y n i i i n i i i F f dxdy ( q u ) ds f Q P H ij Convctiv hat transfr contributions ( H ) ij and Pi ar only from lmnt boundary Th contributions nd to b calculatd only for lmnts with convctiv boundary 2-D Problms: 3
4 CONVECTION HEAT TRANSFER (continud) y y _ y y _ b 1 a 2 x _ x b 1 5 a 2 x _ x H i j ds i j ds ds i j ds i j ds ij i j D Problms: 4
5 Conditions for Solution Symmtry Th solution of a problm may b symmtric about a lin or plan, allowing on to modl only a part of th domain and thrby rducing th computational ffort. Th solution is symmtric about a lin or plan, if and only if (a) gomtry is symmtric, (b) matrial proprtis ar symmtric, (c) loads ar symmtrically applid, and (d) boundary conditions ar symmtric about th lin or plan. Us of th solution symmtry allows us to idntify a subdomain whos analysis yilds th solution in th ntir domain. Th subdomain ncssarily will hav boundaris that coincid with th lins or plans of symmtry, and on must idntify th boundary conditions along ths lins or on th plans of symmtry Axisymm. & 3-D Problms 5
6 Rduction of Problm Siz from 3-D using solution symmtry z 3-D Modls Gnral loading, boundary conditions, and matrial proprtis, all of which may chang along th lngth and around th circumfrnc (i.., with z and θ) 2-D Modls z ( r, θ, z) θ r θ ( r, θ ) r rz, ) Th loading, boundary r conditions, and matrial proprtis do not chang along th lngth Th loading, boundary conditions, and matrial proprtis do not chang around th circumfrnc Axisymm. & 3-D Problms 6
7 Rduction of Problm Siz from 3-D 1-D Modls Th loading, boundary conditions, and matrial proprtis do not chang around th circumfrnc as wll as th lngth r r Axisymm. & 3-D Problms 7
8 AXISYMMETRIC PROBLEMS (2-D) Govrning Equation 1 u u ra a f ( r, z) r r 11 r z 22 z L R 0 z Typical axisymmtric plan r Wak Form 1 u u 0 w i ra a f ( r, z) rdrdz r r 11 r z 22 z wi u wi u a a wif rdrdz qnwirds r r z z dv rd dz dr rdr d dz u u q qnˆ a11 n a22 n r z n r z Axisymm. & 3-D Problms 8
9 AXISYMMETRIC PROBLEMS (cont.) Finit Elmnt Modl u u ( rz, ) n h j j j1 0 n i j i j u j a a rdrdz j r r z z 1 n KufQ j1 f rdrdz q w rds ij j i i i n i i j i j Kij a11 a22 rdrdz f f rdrdz, Q q w rds i i i n i r r z z Axisymm. & 3-D Problms
10 SINGLE-VARIABLE PROBLEMS IN 3-D Govrning Equation u u u a a a f( xyz,, ) x 11 x y 22 y z 33 z Boundary Conditions Spcify: u or q q q u u u q qnˆ a11 n a22 n a33 n x y z q ( uu ) cnd cnv n cnd x y z cnv Axisymm. & 3-D Problms 10
11 3-D HEAT TRANSFER (continud) ds n z nˆ n y Parts of th boundary z y n x Domain Ω Surfac Γ Ω Γ x A six-fac 3-D finit lmnt Wak Form wi uh wi uh wi u h 0 a11 a22 a33 dxdydz wiuhds x x y y z z f w dxdydz q u w ds i n i Axisymm. & 3-D Problms 11
12 3-D HEAT TRANSFER (continud) Finit lmnt approximation Finit lmnt modl n uh ujj( xyz,, ) n j1 0 KufQ j1 ij j i i K a a a dxdydz ds i j i j i j ij i j x x y y z z f f dxdydz, Q q u ds i i i n i Axisymm. & 3-D Problms 12
13 Equations of Motion 3-D + f x = f y = f z = 2 Strain-Displacmnt Rlations " xx ; 2" xy ; 2" yz " yy ; " zz 2" xz 3-D Elasticity 13
14 3-D ELASTICITY (continud) Constitutiv Rlations 8 >< >: ¾ xx ¾ yy ¾ zz ¾ xz ¾ yz ¾ xy >= >; = 2 c 11 c 12 c c 12 c 22 c c 13 c 23 c c c c >< 7 5 >: " xx " yy " zz 2" xz 2" yz 2" xy Th matrial axs ar assumd coincid with th global axs and th matrial is orthotropic with rspct to th global axs. Boundary Conditions t x ¾ xx n x + ¾ xy n y + ¾ xz n z = ^t x t y ¾ xy n x + ¾ yy n y + ¾ yz n z = ^t y t z ¾ xz n x + ¾ yz n y + ¾ zz n z = ^t z = >= >; ; on ¾ or u = ^u on u 3-D Elasticity 14
15 3-D ELASTICITY (continud) MATRIX FORM OF THE GOVERNING EQUATIONS Notation ¾ = D T = 8 >< >: @=@y >< >= < f x = < u x = " = ; f = f : y ; ; u = u : y ; f z u z >: >; ¾ xx ¾ yy ¾ zz ¾ xy ¾ xz ¾ yz Govrning quations D T ¾ + f = ½Äu ¾ = C" " = Du; 3 " xx " yy " zz 2" xz 2" yz 2" xy >= ; >; 3-D Elasticity 15
16 3-D ELASTICITY (continud) Principl of virtual displacmnts (in matrix form) 0 = Z (D±u) T C (Du) + ½u T Äu Z I dx (±u) T f dx (±u) T t ds Finit lmnt approximation (in matrix form) u = ª = 8 < : u x u y u z = ; = ª ; w = ±u = 8 < : ±u x ±u y ±u z = ; = ª± à à : : : à n à à 2 0 : : : à n à à 2 0 : : : 0 à n = f u 1 x u 1 y u 1 z u 2 x u 2 y u 2 z : : : u n x u n y u n z g T D Elasticity 16
17 3-D ELASTICITY (continud) Finit Elmnt Modl whr K = F = M Ä + K = F + Q Z Z h B T CB dx; M = ½h ª T ª dx Z I ª T f dx; Q = ª T t ds At ach nod ( uvw,, ) D Elasticity 17
18 TYPICAL 3-D FINITE ELEENTS nods Axisymm. & 3-D Problms 18
19 TYPICAL 3-D FINITE ELEMENTS Linar ttrahdral lmnt L 3 = fª g = 2 8 >< >: L 1 L 2 L 3 L 4 L 1 = 0 3 L 4 = 0 >= >; Quadratic ttrahdral lmnt fª g = Axisymm. & 3-D Problms 1 8 >< >: 3 L 1 (2L 1 1) L 2 (2L 2 1) L 3 (2L 3 1) L 4 (2L 4 1) 4L 1 L 2 4L 2 L 3 4L 3 L 1 4L 1 L 4 4L 2 L 4 4L 3 L 4 >= >;
20 TYPICAL 3-D FINITE ELEMENTS Linar prism lmnt ζ L 3 = 0 η ξ L 2 = 0 4 ζ = L 1 = ζ = 1 2 u (,, ) h c c c c c c fª g = >< >: L 1 (1 ³) L 2 (1 ³) L 3 (1 ³) L 1 (1 + ³) L 2 (1 + ³) L 3 (1 + ³) Axisymm. & 3-D Problms 20 >= >;
21 TYPICAL 3-D FINITE ELEMENTS (cont ) Quadratic prism lmnt fª g = >< >: L 1 [(2L 1 1)(1 ³) (1 ³ 2 )] L 2 [(2L 2 1)(1 ³) (1 ³ 2 )] L 3 [(2L 3 1)(1 ³) (1 ³ 2 )] L 1 [(2L 1 1)(1 + ³) (1 ³ 2 )] L 2 [(2L 2 1)(1 + ³) (1 ³ 2 )] L 3 [(2L 3 1)(1 + ³) (1 ³ 2 )] 4L 1 L 2 (1 ³) 4L 2 L 3 (1 ³) 4L 3 L 1 (1 ³) 2L 1 (1 ³ 2 ) 2L 2 (1 ³ 2 ) 2L 3 (1 ³ 2 ) 4L 1 L 2 (1 + ³) 4L 2 L 3 (1 + ³) 4L 3 L 1 (1 + ³) >= >; Axisymm. & 3-D Problms 21
22 TYPICAL 3-D FINITE ELEMENTS (cont ) Linar brick lmnt ξ 6 u ( h,, ) c0 c1 c2 c3 c4 c5 c6 c7 8 (1»)(1 )(1 ³) (1 +»)(1 )(1 ³) fª g = 1 > (1 +»)(1 + )(1 ³) < >= (1»)(1 + )(1 ³) 8 (1»)(1 )(1 + ³) (1 +»)(1 )(1 + ³) > : (1 +»)(1 + )(1 + ³) > ; (1»)(1 + )(1 + ³) ζ η Axisymm. & 3-D Problms 22
23 TYPICAL 3-D FINITE ELEMENTS (cont ) Quadratic Brick Elmnt ζ ξ η fª g = >< >: (1»)(1 )(1 ³)(» ³ 2) (1 +»)(1 )(1 ³)(» ³ 2) (1 +»)(1 + )(1 ³)(» + ³ 2) (1»)(1 + )(1 ³)(» + ³ 2) (1»)(1 )(1 + ³)(» + ³ 2) (1 +»)(1 )(1 + ³)(» + ³ 2) (1 +»)(1 + )(1 + ³)(» + + ³ 2) (1»)(1 + )(1 + ³)(» + + ³ 2) 2(1» 2 )(1 )(1 ³) 2(1 +»)(1 2)(1 ³) 2(1» 2 )(1 + )(1 ³) 2(1»)(1 2)(1 ³) 2(1»)(1 )(1 ³ 2 ) 2(1 +»)(1 )(1 ³ 2 ) 2(1 +»)(1 + )(1 ³ 2 ) 2(1»)(1 + )(1 ³ 2 ) 2(1» 2 )(1 )(1 + ³) 2(1 +»)(1 2)(1 + ³) 2(1» 2 )(1 + )(1 + ³) 2(1»)(1 2)(1 + ³) >= >;
24 SUMMARY In this lctur, th following topics wr covrd: 2-D problm with convction Conditions for solution symmtry Typs of axisymmtric problms FEM of axisymmtric problms (2-D) 3-D hat transfr 3-D lasticity Typical 3-D finit lmnts Axisymm. & 3-D Problms 24
The Finite Element Method
The Finite Element Method 3D Problems Heat Transfer and Elasticity Read: Chapter 14 CONTENTS Finite element models of 3-D Heat Transfer Finite element model of 3-D Elasticity Typical 3-D Finite Elements
More informationAS 5850 Finite Element Analysis
AS 5850 Finit Elmnt Analysis Two-Dimnsional Linar Elasticity Instructor Prof. IIT Madras Equations of Plan Elasticity - 1 displacmnt fild strain- displacmnt rlations (infinitsimal strain) in matrix form
More informationFinite Element Models for Steady Flows of Viscous Incompressible Fluids
Finit Elmnt Modls for Stad Flows of Viscous Incomprssibl Fluids Rad: Chaptr 10 JN Rdd CONTENTS Govrning Equations of Flows of Incomprssibl Fluids Mid (Vlocit-Prssur) Finit Elmnt Modl Pnalt Function Mthod
More informationFinite element discretization of Laplace and Poisson equations
Finit lmnt discrtization of Laplac and Poisson quations Yashwanth Tummala Tutor: Prof S.Mittal 1 Outlin Finit Elmnt Mthod for 1D Introduction to Poisson s and Laplac s Equations Finit Elmnt Mthod for 2D-Discrtization
More informationNonlinear Bending of Strait Beams
Nonlinar Bnding of Strait Bams CONTENTS Th Eulr-Brnoulli bam thory Th Timoshnko bam thory Govrning Equations Wak Forms Finit lmnt modls Computr Implmntation: calculation of lmnt matrics Numrical ampls
More informationNONLINEAR ANALYSIS OF PLATE BENDING
NONLINEAR ANALYSIS OF PLATE BENDING CONTENTS Govrning Equations of th First-Ordr Shar Dformation thor (FSDT) Finit lmnt modls of FSDT Shar and mmbran locking Computr implmntation Strss calculation Numrical
More informationMiddle East Technical University Department of Mechanical Engineering ME 413 Introduction to Finite Element Analysis
Middl East Tchnical Univrsity Dpartmnt of Mchanical Enginring ME Introduction to Finit Elmnt Analysis Chaptr 5 Two-Dimnsional Formulation Ths nots ar prpard by Dr. Cünyt Srt http://www.m.mtu.du.tr/popl/cunyt
More information1 Isoparametric Concept
UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric
More informationMiddle East Technical University Department of Mechanical Engineering ME 413 Introduction to Finite Element Analysis
Middl East Tchnical Univrsity Dpartmnt of Mchanical Enginring ME 43 Introduction to Finit Elmnt Analysis Chaptr 3 Computr Implmntation of D FEM Ths nots ar prpard by Dr. Cünyt Srt http://www.m.mtu.du.tr/popl/cunyt
More informationDerivation of Eigenvalue Matrix Equations
Drivation of Eignvalu Matrix Equations h scalar wav quations ar φ φ η + ( k + 0ξ η β ) φ 0 x y x pq ε r r whr for E mod E, 1, y pq φ φ x 1 1 ε r nr (4 36) for E mod H,, 1 x η η ξ ξ n [ N ] { } i i i 1
More informationMiddle East Technical University Department of Mechanical Engineering ME 413 Introduction to Finite Element Analysis
Middl East Tchnical Univrsity Dpartmnt of Mchanical Enginring ME 4 Introduction to Finit Elmnt Analysis Chaptr 4 Trusss, Bams and Frams Ths nots ar prpard by Dr. Cünyt Srt http://www.m.mtu.du.tr/popl/cunyt
More informationCOMPUTATIONAL NUCLEAR THERMAL HYDRAULICS
COMPUTTIONL NUCLER THERML HYDRULICS Cho, Hyoung Kyu Dpartmnt of Nuclar Enginring Soul National Univrsity CHPTER4. THE FINITE VOLUME METHOD FOR DIFFUSION PROBLEMS 2 Tabl of Contnts Chaptr 1 Chaptr 2 Chaptr
More informationChapter 5. Introduction. Introduction. Introduction. Finite Element Modelling. Finite Element Modelling
Chaptr 5 wo-dimnsional problms using Constant Strain riangls (CS) Lctur Nots Dr Mohd Andi Univrsiti Malasia Prlis EN7 Finit Elmnt Analsis Introction wo-dimnsional init lmnt ormulation ollows th stps usd
More informationElectromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology
Elctromagntic scattring Graduat Cours Elctrical Enginring (Communications) 1 st Smstr, 1388-1389 Sharif Univrsity of Tchnology Contnts of lctur 8 Contnts of lctur 8: Scattring from small dilctric objcts
More informationVSMN30 FINITA ELEMENTMETODEN - DUGGA
VSMN3 FINITA ELEMENTMETODEN - DUGGA 1-11-6 kl. 8.-1. Maximum points: 4, Rquird points to pass: Assistanc: CALFEM manual and calculator Problm 1 ( 8p ) 8 7 6 5 y 4 1. m x 1 3 1. m Th isotropic two-dimnsional
More informationDirect Approach for Discrete Systems One-Dimensional Elements
CONTINUUM & FINITE ELEMENT METHOD Dirct Approach or Discrt Systms On-Dimnsional Elmnts Pro. Song Jin Par Mchanical Enginring, POSTECH Dirct Approach or Discrt Systms Dirct approach has th ollowing aturs:
More informationFEM FOR HEAT TRANSFER PROBLEMS دانشگاه صنعتي اصفهان- دانشكده مكانيك
FEM FOR HE RNSFER PROBLEMS 1 Fild problms Gnral orm o systm quations o D linar stady stat ild problms: For 1D problms: D D g Q y y (Hlmholtz quation) d D g Q d Fild problms Hat transr in D in h h ( D D
More informationHIGHER-ORDER THEORIES
HIGHER-ORDER THEORIES THIRD-ORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1 Third-Order Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) +
More information16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity
6.20 HANDOUT #2 Fall, 2002 Review of General Elasticity NOTATION REVIEW (e.g., for strain) Engineering Contracted Engineering Tensor Tensor ε x = ε = ε xx = ε ε y = ε 2 = ε yy = ε 22 ε z = ε 3 = ε zz =
More informationTwo Posts to Fill On School Board
Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationHigher-Order Discrete Calculus Methods
Highr-Ordr Discrt Calculus Mthods J. Blair Prot V. Subramanian Ralistic Practical, Cost-ctiv, Physically Accurat Paralll, Moving Msh, Complx Gomtry, Slid 1 Contxt Discrt Calculus Mthods Finit Dirnc Mimtic
More informationDivision of Mechanics Lund University MULTIBODY DYNAMICS. Examination Name (write in block letters):.
Division of Mchanics Lund Univrsity MULTIBODY DYNMICS Examination 7033 Nam (writ in block lttrs):. Id.-numbr: Writtn xamination with fiv tasks. Plas chck that all tasks ar includd. clan copy of th solutions
More informationAE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.
AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
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 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 information12. Stresses and Strains
12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)
More informationMANY BILLS OF CONCERN TO PUBLIC
- 6 8 9-6 8 9 6 9 XXX 4 > -? - 8 9 x 4 z ) - -! x - x - - X - - - - - x 00 - - - - - x z - - - x x - x - - - - - ) x - - - - - - 0 > - 000-90 - - 4 0 x 00 - -? z 8 & x - - 8? > 9 - - - - 64 49 9 x - -
More informationJim Lambers MAT 280 Summer Semester Practice Final Exam Solution. dy + xz dz = x(t)y(t) dt. t 3 (4t 3 ) + e t2 (2t) + t 7 (3t 2 ) dt
Jim Lambers MAT 28 ummer emester 212-1 Practice Final Exam olution 1. Evaluate the line integral xy dx + e y dy + xz dz, where is given by r(t) t 4, t 2, t, t 1. olution From r (t) 4t, 2t, t 2, we obtain
More informationNumerical methods for PDEs FEM implementation: element stiffness matrix, isoparametric mapping, assembling global stiffness matrix
Platzhaltr für Bild, Bild auf Titlfoli hintr das Logo instzn Numrical mthods for PDEs FEM implmntation: lmnt stiffnss matrix, isoparamtric mapping, assmbling global stiffnss matrix Dr. Nomi Fridman Contnts
More informationMultiple Integrals and Vector Calculus: Synopsis
Multiple Integrals and Vector Calculus: Synopsis Hilary Term 28: 14 lectures. Steve Rawlings. 1. Vectors - recap of basic principles. Things which are (and are not) vectors. Differentiation and integration
More informationA DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any
Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»
More informationMULTIVARIABLE INTEGRATION
MULTIVARIABLE INTEGRATION (PLANE & CYLINDRICAL POLAR COORDINATES) PLANE POLAR COORDINATES Question 1 The finite region on the x-y plane satisfies 1 x + y 4, y 0. Find, in terms of π, the value of I. I
More informationINC 693, 481 Dynamics System and Modelling: Linear Graph Modeling II Dr.-Ing. Sudchai Boonto Assistant Professor
INC 69, 48 Dynamics Systm and Modlling: Linar Graph Modling II Dr.-Ing. Sudchai Boonto Assistant Profssor Dpartmnt of Control Systm and Instrumntation Enginring King Mongkut s Unnivrsity of Tchnology Thonuri
More informationWave and Elasticity Equations
1 Wave and lasticity quations Now let us consider the vibrating string problem which is modeled by the one-dimensional wave equation. Suppose that a taut string is suspended by its extremes at the points
More informationMECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso
MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium
More informationOWELL WEEKLY JOURNAL
Y \»< - } Y Y Y & #»»» q ] q»»»>) & - - - } ) x ( - { Y» & ( x - (» & )< - Y X - & Q Q» 3 - x Q Y 6 \Y > Y Y X 3 3-9 33 x - - / - -»- --
More informationNONLINEAR CONTINUUM FORMULATIONS CONTENTS
NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell
More informationUnit IV State of stress in Three Dimensions
Unit IV State of stress in Three Dimensions State of stress in Three Dimensions References Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationMath 304, Differential Forms, Lecture 1 May 26, 2011
Math 304, Differential Forms, Lecture 1 May 26, 2011 Differential forms were developed and applied to vector fields in a decisive way by the French Mathematician Élie artan (1869 1951) in the first quarter
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 informationHIGHER-ORDER THEORIES
HIGHER-ORDER THEORIES Third-order Shear Deformation Plate Theory Displacement and strain fields Equations of motion Navier s solution for bending Layerwise Laminate Theory Interlaminar stress and strain
More informationVariational principles in mechanics
CHAPTER Variational principles in mechanics.1 Linear Elasticity n D Figure.1: A domain and its boundary = D [. Consider a domain Ω R 3 with its boundary = D [ of normal n (see Figure.1). The problem of
More informationSection 4.5. Integration and Expectation
4.5. Integration and Expectation 1 Section 4.5. Integration and Expectation Note. In this section we consider integrals of scalar valued functions of a vector and matrix valued functions of a scalar. We
More informationLecture 8. Stress Strain in Multi-dimension
Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]
More informationME FINITE ELEMENT ANALYSIS FORMULAS
ME 2353 - FINITE ELEMENT ANALYSIS FORMULAS UNIT I FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS 01. Global Equation for Force Vector, {F} = [K] {u} {F} = Global Force Vector [K] = Global Stiffness
More informationFigure 25:Differentials of surface.
2.5. Change of variables and Jacobians In the previous example we saw that, once we have identified the type of coordinates which is best to use for solving a particular problem, the next step is to do
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 informationELASTICITY (MDM 10203)
LASTICITY (MDM 10203) Lecture Module 5: 3D Constitutive Relations Dr. Waluyo Adi Siswanto University Tun Hussein Onn Malaysia Generalised Hooke's Law In one dimensional system: = (basic Hooke's law) Considering
More informationEAcos θ, where θ is the angle between the electric field and
8.4. Modl: Th lctric flux flows out of a closd surfac around a rgion of spac containing a nt positiv charg and into a closd surfac surrounding a nt ngativ charg. Visualiz: Plas rfr to Figur EX8.4. Lt A
More informationArtificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.-Ing. John Nassour J.
Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik Robot Dynamics Dr.-Ing. John Nassour 25.1.218 J.Nassour 1 Introduction Dynamics concerns the motion of bodies Includes Kinematics
More informationUnderstand basic stress-strain response of engineering materials.
Module 3 Constitutive quations Learning Objectives Understand basic stress-strain response of engineering materials. Quantify the linear elastic stress-strain response in terms of tensorial quantities
More informationCONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS
Chapter 9 CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS Figure 9.1: Hooke memorial window, St. Helen s, Bishopsgate, City of London 211 212 CHAPTR 9. CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS 9.1 Mechanical
More informationLOWELL WEEKLY JOURNAL
Y -» $ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - $ { Q» / X x»»- 3 q $ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q
More informationDifference -Analytical Method of The One-Dimensional Convection-Diffusion Equation
Diffrnc -Analytical Mthod of Th On-Dimnsional Convction-Diffusion Equation Dalabav Umurdin Dpartmnt mathmatic modlling, Univrsity of orld Economy and Diplomacy, Uzbistan Abstract. An analytical diffrncing
More information3D and Planar Constitutive Relations
3D and Planar Constitutive Relations A School on Mechanics of Fibre Reinforced Polymer Composites Knowledge Incubation for TEQIP Indian Institute of Technology Kanpur PM Mohite Department of Aerospace
More informationLie Groups HW7. Wang Shuai. November 2015
Li roups HW7 Wang Shuai Novmbr 015 1 Lt (π, V b a complx rprsntation of a compact group, show that V has an invariant non-dgnratd Hrmitian form. For any givn Hrmitian form on V, (for xampl (u, v = i u
More informationGetting started: CFD notation
PDE of p-th order Getting started: CFD notation f ( u,x, t, u x 1,..., u x n, u, 2 u x 1 x 2,..., p u p ) = 0 scalar unknowns u = u(x, t), x R n, t R, n = 1,2,3 vector unknowns v = v(x, t), v R m, m =
More informationMAE4700/5700 Finite Element Analysis for Mechanical and Aerospace Design
MAE4700/5700 Finit Elmnt Analysis for Mchanical and Arospac Dsign Cornll Univrsity, Fall 2009 Nicholas Zabaras Matrials Procss Dsign and Control Laboratory Sibly School of Mchanical and Arospac Enginring
More informationCHAPTER 1. Introductory Concepts Elements of Vector Analysis Newton s Laws Units The basis of Newtonian Mechanics D Alembert s Principle
CHPTER 1 Introductory Concpts Elmnts of Vctor nalysis Nwton s Laws Units Th basis of Nwtonian Mchanics D lmbrt s Principl 1 Scinc of Mchanics: It is concrnd with th motion of matrial bodis. odis hav diffrnt
More informationModule 2: Governing Equations and Hypersonic Relations
Module 2: Governing Equations and Hypersonic Relations Lecture -2: Mass Conservation Equation 2.1 The Differential Equation for mass conservation: Let consider an infinitely small elemental control volume
More informationPotential energy of a structure. Vforce. joints j
Potntial nrgy of a strctr P y P x y x P y P x ij ij ij V K ( ) Vforc P V ij ij j j K ( ) P [ K] r Elastic Potntial nrgy of mmbr ij Potntial nrgy of applid forc at joint i i i i mmbrs joints j Find th st
More informationThe Finite Element Method
Th Finit Elmnt Mthod Eulr-Brnoulli and Timoshnko Bams Rad: Chaptr 5 CONTENTS Eulr-Brnoulli bam thory Govrning Equations Finit lmnt modl Numrical ampls Timoshnko bam thory Govrning Equations Finit lmnt
More informationCIE4145 : STRESS STRAIN RELATION LECTURE TOPICS
CI445 : STRSS STRAIN RLATION LCTUR TOPICS Stress tensor Stress definition Special stress situations Strain tensor Relative displacements Strain definition Strain tensor 3 Tensor properties Introduction
More informationHeat/Di usion Equation. 2 = 0 k constant w(x; 0) = '(x) initial condition. ( w2 2 ) t (kww x ) x + k(w x ) 2 dx. (w x ) 2 dx 0.
Hat/Di usion Equation @w @t k @ w @x k constant w(x; ) '(x) initial condition w(; t) w(l; t) boundary conditions Enrgy stimat: So w(w t kw xx ) ( w ) t (kww x ) x + k(w x ) or and thrfor E(t) R l Z l Z
More information[2] (a) Develop and describe the piecewise linear Galerkin finite element approximation of,
269 C, Vese Practice problems [1] Write the differential equation u + u = f(x, y), (x, y) Ω u = 1 (x, y) Ω 1 n + u = x (x, y) Ω 2, Ω = {(x, y) x 2 + y 2 < 1}, Ω 1 = {(x, y) x 2 + y 2 = 1, x 0}, Ω 2 = {(x,
More informationReview of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis
uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods
More informationBasic Equations of Elasticity
A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ
More informationThe Transmission Line Wave Equation
1//5 Th Transmission Lin Wav Equation.doc 1/6 Th Transmission Lin Wav Equation Q: So, what functions I (z) and V (z) do satisfy both tlgraphr s quations?? A: To mak this asir, w will combin th tlgraphr
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationLecture M1 Slender (one dimensional) Structures Reading: Crandall, Dahl and Lardner 3.1, 7.2
Lecture M1 Slender (one dimensional) Structures Reading: Crandall, Dahl and Lardner 3.1, 7.2 This semester we are going to utilize the principles we learnt last semester (i.e the 3 great principles and
More informationFinite Strain Elastic-Viscoplastic Model
Finit Strain Elastic-Viscoplastic Modl Pinksh Malhotra Mchanics of Solids,Brown Univrsity Introduction Th main goal of th projct is to modl finit strain rat-dpndnt plasticity using a modl compatibl for
More informationFinite Element Method (FEM)
Finite Element Method (FEM) The finite element method (FEM) is the oldest numerical technique applied to engineering problems. FEM itself is not rigorous, but when combined with integral equation techniques
More informationDr. D. Dinev, Department of Structural Mechanics, UACEG
Lecture 6 Energy principles Energy methods and variational principles Print version Lecture on Theory of Elasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACEG 6.1 Contents
More informationChapter 7: Exponents
Chapter : Exponents Algebra Chapter Notes Name: Algebra Homework: Chapter (Homework is listed by date assigned; homework is due the following class period) HW# Date In-Class Homework M / Review of Sections.-.
More informationModelling ice shelf basal melt with Glimmer-CISM coupled to a meltwater plume model
Modelling ice shelf basal melt with Glimmer-CISM coupled to a meltwater plume model Carl Gladish NYU CIMS February 17, 2010 Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 1 / 24 Acknowledgements
More informationPlane and axisymmetric models in Mentat & MARC. Tutorial with some Background
Plane and axisymmetric models in Mentat & MARC Tutorial with some Background Eindhoven University of Technology Department of Mechanical Engineering Piet J.G. Schreurs Lambèrt C.A. van Breemen March 6,
More informationPartial derivatives, linear approximation and optimization
ams 11b Study Guide 4 econ 11b Partial derivatives, linear approximation and optimization 1. Find the indicated partial derivatives of the functions below. a. z = 3x 2 + 4xy 5y 2 4x + 7y 2, z x = 6x +
More informationReview for the Final Test
Math 7 Review for the Final Test () Decide if the limit exists and if it exists, evaluate it. lim (x,y,z) (0,0,0) xz. x +y +z () Use implicit differentiation to find z if x + y z = 9 () Find the unit tangent
More informationPart 8: Rigid Body Dynamics
Document that contains homework problems. Comment out the solutions when printing off for students. Part 8: Rigid Body Dynamics Problem 1. Inertia review Find the moment of inertia for a thin uniform rod
More information3D Elasticity Theory
3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.
More informationUNIT I PARTIAL DIFFERENTIAL EQUATIONS PART B. 3) Form the partial differential equation by eliminating the arbitrary functions
UNIT I PARTIAL DIFFERENTIAL EQUATIONS PART B 1) Form th artial diffrntial quation b liminating th arbitrar functions f and g in z f ( x ) g( x ) ) Form th artial diffrntial quation b liminating th arbitrar
More information4.2 Design of Sections for Flexure
4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt
More information( ) Chapter 6 ( ) ( ) ( ) ( ) Exercise Set The greatest common factor is x + 3.
Chapter 6 Exercise Set 6.1 1. A prime number is an integer greater than 1 that has exactly two factors, itself and 1. 3. To factor an expression means to write the expression as the product of factors.
More informationChapter 7: Exponents
Chapter 7: Exponents Algebra 1 Chapter 7 Notes Name: Algebra Homework: Chapter 7 (Homework is listed by date assigned; homework is due the following class period) HW# Date In-Class Homework Section 7.:
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More information11.1 Three-Dimensional Coordinate System
11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into
More informationProblem Solving 1: Line Integrals and Surface Integrals
A. Line Integrals MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics Problem Solving 1: Line Integrals and Surface Integrals The line integral of a scalar function f ( xyz),, along a path C is
More informationLOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort
- 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [
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 informationis an appropriate single phase forced convection heat transfer coefficient (e.g. Weisman), and h
For t BWR oprating paramtrs givn blow, comput and plot: a) T clad surfac tmpratur assuming t Jns-Lotts Corrlation b) T clad surfac tmpratur assuming t Tom Corrlation c) T clad surfac tmpratur assuming
More informationFinite Element Analysis
Finit Elmnt Analysis L4 D Shap Functions, an Gauss Quaratur FEA Formulation Dr. Wiong Wu EGR 54 Finit Elmnt Analysis Roamap for Dvlopmnt of FE Strong form: govrning DE an BCs EGR 54 Finit Elmnt Analysis
More informationPhys 402: Nonlinear Spectroscopy: SHG and Raman Scattering
Rquirmnts: Polariation of Elctromagntic Wavs Phys : Nonlinar Spctroscopy: SHG and Scattring Gnral considration of polariation How Polarirs work Rprsntation of Polariation: Jons Formalism Polariation of
More informationMultiple Integrals and Vector Calculus (Oxford Physics) Synopsis and Problem Sets; Hilary 2015
Multiple Integrals and Vector Calculus (Oxford Physics) Ramin Golestanian Synopsis and Problem Sets; Hilary 215 The outline of the material, which will be covered in 14 lectures, is as follows: 1. Introduction
More informationThe Rotating Inhomogeneous Elastic Cylinders of. Variable-Thickness and Density
Applied Mathematics & Information Sciences 23 2008, 237 257 An International Journal c 2008 Dixie W Publishing Corporation, U. S. A. The Rotating Inhomogeneous Elastic Cylinders of Variable-Thickness and
More information16. Electromagnetics and vector elements (draft, under construction)
16. Elctromagntics (draft)... 1 16.1 Introduction... 1 16.2 Paramtric coordinats... 2 16.3 Edg Basd (Vctor) Finit Elmnts... 4 16.4 Whitny vctor lmnts... 5 16.5 Wak Form... 8 16.6 Vctor lmnt matrics...
More informationLOWELL WEEKLY JOURNAL
Y G y G Y 87 y Y 8 Y - $ X ; ; y y q 8 y $8 $ $ $ G 8 q < 8 6 4 y 8 7 4 8 8 < < y 6 $ q - - y G y G - Y y y 8 y y y Y Y 7-7- G - y y y ) y - y y y y - - y - y 87 7-7- G G < G y G y y 6 X y G y y y 87 G
More information