NONLINEAR ANALYSIS OF PLATE BENDING
|
|
- Sarah Brooks
- 5 years ago
- Views:
Transcription
1 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 Eampls Nonlinar Plat Bnding: 1
2 THE FIRST-ORDER SHEAR DEFORMATION THEORY Displacmnt Fild of th FSDT u (,, z,) t u(,,) t z (,,) t 1 u (,, z,) t v(,,) t z (,,) t u( zt,,,) wt (,,) 3 Nonlinar strains z z w dw d w z dw = d dw = d ij 1 u u i j um um j i i j z Von Karman Nonlinar strains u 1 1 um um u1 1 u1 1 u 1 u Nonlinar Plat Bnding:
3 NONLINEAR STARINS OF THE FSDT Actual Nonlinar strains u 1 1 u 3 u 1 w 0 1 z z u 1 u 3 v 1 w 0 1 z z u u u1 u 3 3 u v w w z 0 1 z u u w u u w z z z z, z z Virtual Nonlinar strains u w w z z v w w z z u v w w w w z z 0 1 w w 0 0 z z, z z
4 PRINCIPLE OF VIRTUAL DISPLACEMENTS 0 = W W W I E z Q M N W z z h I = h Ω ( ) ( ) ( z ) K s z z } 0 K sz z dz dd h WE = ( ) ( ) h nn un zn ns us zs nzw dzds Γ 0 Ω ( q kw) w dd} = N M N M N M Ω Qz Q z qw dd ( ) Q M N M M N N Mnn N ns M ns Nnn un Nns us Mnn n Mns s Qn w ds Γ Q n N nn Plats (Nonlinar): 4
5 THE FIRST-ORDER SHEAR Equations N @ = I = I q = I Q = Q = z z w dw d Strss rsultants w z z dw = d dw = d Z h= µ Q = K s ¾ z dz = K s A 55 Z h= µ Q = K s ¾ z dz = K s A h= M = M = D D M = @ Nonlinar Plat Bnding: 5
6 THE FIRST-ORDER SHEAR DEFORMATION THEORY Wak forms Z 0 N I 0 dd ±un n ds Z 0 N I 0 dd ±vn ns ds 0 = @ I I 0 ±wq dd ±wq n ds Z 0 I Á ±Á Q I dd ±Á M n ds Z 0 I Á ±Á Q I dd ±Á M ns ds Nonlinar Plat Bnding: 6
7 Finit Elmnt Modls of Th First-ordr Plat Thor (FSDT) (Continud) Finit Elmnt Approimation u(; ; t) = Á (; ; t) = mx u j (t)ã j (; ); v(; ; t) = j=1 w(; ; t) = mx v j (t)ã j (; ) j=1 nx w j (t)ã j (; ) j=1 px Sj 1 (t)ã j (; ); Á (; ; t) = j=1 px Sj (t)ã j (; ) j=1 Finit Elmnt Modl 6 4 M M M M M >< 7 5 >: Äu Äv Äw ÄS ÄS 9 >= >; K 11 K 1 K 13 K 14 K u K 1 K K 3 K 4 K 5 >< v >= 6 K 31 K 3 K 33 K 34 K 35 7 w 4 K 41 K 4 K 43 K 44 K 45 5 >: S >; K 51 K 5 K 53 K 54 K 55 S = 8 >< >: F 1 F F 3 F 4 F 5 9 >= >; Nonlinar Plat Bnding: 7
8 Full Dicrtizd Modl and Itrativ Schm Full Discrtizd Finit Elmnt Modl [ ^K] s1 f g s1 = f ^F g s;s1 [ ^K] s1 = [K] s1 a 3 [M] s1 f ^F g s;s1 = ff g s1 [M] s1 (a 3 f g s a 4 f g _ s a 5 f g Ä s ) a 3 = ( t) ; a 4 = t ; a 5 = 1 1 Acclrations and Vlocitis f Ä g s1 = a 3 (f g s1 f g s ) a 4 f _ g a 5 f Ä g s f _ g s1 = f _ g s a f Ä g s a 1 f Ä g s1 whr a 1 = t and a = (1 ) t. Nwton-Raphson Itrativ Schm f g s1 r1 = [ ^K T ] r 1 s1 f ^Rg r ; [ ^K T ] r g # s1 r Nonlinar Plat Bnding: 8
9 Stiffnss Cofficints (tpical) Z Kij 11 = Z Kij 1 = K 13 ij = 1 Z A A i dd dd = Kji j A Z µ = A Kij 3 = 1 Z Kij 31 @' µ 0 A µ 0 @w j # dd @ @ # dd Nonlinar Plat Bnding: 9
10 Tangnt Stiffnss Cofficints (tpical) R T R K F u v w S S 5 n( ) i ij =, i = ik k i, i i, i i, i i, i i, = = = = i = i j = 1 k= 1 K T K F K ( ) 5 n( ) 5 n( ) ik ij = ik k i = k ij j = 1 k= 1 = 1 k= 1 j T = K = ( K ), T = K = ( K ) T T T = K = ( K ), T = K = ( K ) T T K K T K w K 5 n( ) 1 n ik 13 ik 13 ij = k ij = k ij = 1 k= 1 wj k= 1 wj (1) () () 1 i w j w j = A 11 A1 Ω (1) () () i w j w j A66 dd K = K K = K = T ij ij ij ji Plats (Nonlinar): ij
11 Tangnt Stiffnss Cofficints (tpical) 5 n( ) 3 n( ) K ik 33 Kik Kik K ik Tij = Kij k = Kij uk vk wk = 1 k= 1 w j k 1 wj wj w = j () () () () () () () () 33 i j i j u i j i j u = Kij A11 A1 A 66 dd Ω () () () () () () () () i j i j v i j i j v A 1 A A66 dd Ω () () () () () () () () w i j w i j w w i j i j A11 A A66 Ω () () () () () () () () 1 w i j w i j w w i j i j ( A1 A6 6 ) dd Plats (Nonlinar): 11
12 Tangnt Stiffnss Cofficints (tpical) () () u w v 1 w i j Tij = Kij A11 A1 ( 1 66 ) A A dd Ω () () v w u 1 w i j A A1 ( A1 A66 ) dd Ω () () () () u v w w 1 w w A66 i j i j ( A1 A66 ) dd Ω () () () () 33 i j i j () () Tij = A55 A44 ki j dd Ω () () () () () () () () i j i j i j i j N N N N Ω () () () () w w i j i j ( A1 A66 ) () () () () w w i j w w i j A11 A66 A66 A dd 1
13 Shar and Mmbran Locking (Rvisit) Shar Locking Us rducd intgration to valuat all shar stiffnsss (i.., all K ij that contain transvrs shar trms) Mmbran Locking Us rducd intgration to valuat all mmbran stiffnsss (i.., all K ij that contain von Kármán nonlinar trms) Nonlinar Plat Bnding: 13
14 Post-Computation of Strss Componnts Q Q z Q55 0 z Q1 Q 0, z 0 Q 55 z 0 0 Q 66 E E E Q, Q, Q, Q G, Q G, Q G, u 1 w w v 1 w z z, z w u v ww Plats (Nonlinar): 14
15 TYPICAL SIMPLY SUPPORT CONDITIONS for Pur Bnding cas CPT: FSDT: CPT: w w 0 w 0 Smmtr conditions: b b w w 0; FSDT: w 0 w CPT: w 0 FSDT: w 0 Computational domain w a a CPT: w 0 FSDT: w 0 w w CPT: 0 at 0; 0 at 0 FSDT: 0 at 0; 0 at 0 Plat bnding: 15
16 Th ffct of rducd intgration, thicknss, and msh rfinmnt on th linar cntr dflctions and strsss of a simpl supportd, isotropic (ν = 0.5) squar plat undr a uniform transvrs load of intnsit q 0. F full intgration M Mid intgration linar linar linar quadratic Eactz a=h Intg. ¹w ¹¾ ¹w ¹¾ ¹w ¹¾ ¹w ¹¾ ¹w ¹¾ 10 F M F M F M F M F M CPT(N) CPT(C) ¹w = weh 3 10 =q 0 a 4, ¹¾ = ¾ (A; A; h)h =q 0 a, A = 1 4 a (1 1 linar), 1 8 a ( linar), 1 a (4 4 linar), 0:0583a ( 16 quadratic). Plat bnding: 16
17 Gauss Point Locations (basd on rducd Intgration Gauss points) for Strss Computation ( a / ) ( a/ 83, b/ 8) ( 3a/ 83, b/ 8) ( a/ 8, b/ 8) ( 3a/ 8, b/ 8) ( b/ ) ( a / ) ( b/ ) Msh of 4-nod (linar) lmnts Msh of 9-nod (quadratic) lmnts b( 3 1) b( 3 1), = ( a, b) Plat bnding: 17
18 REMARKS Th nin-nod lmnt givs virtuall th sam rsults for full (3 3 Gauss rul) and mid (3 3 and Gauss ruls for bnding and shar trms, rspctivl) intgrations. Howvr, th rsults obtaind using th mid intgration ar closst to th act solution. Full intgration givs lss accurat rsults than mid intgration, and th rror incrass with an incras in sid-to-thicknss ratio (a/h). This implis that mid intgration is ssntial for thin plats, spciall whn modld b lowr-ordr lmnts. Full intgration rsults in smallr rrors for quadratic lmnts and rfind mshs than for linar lmnts and/or coarsr mshs. Plat bnding: 18
19 u = w = = b Nonlinar Analsis of Simpl Supportd Plat (SS-1) 0 SS-1 v = w = = 0 Dflction vrsus load paramtr for simpl supportd (SS1) plat undr uniforml distributd load. 3.0 b.5 a a v = w = = 0 u = w = = 0 v = w = = b b 0 SS- u = w = = 0 Dflction, w/h SS SS1 u = w = = a 0 a v = w = = Load paramtr, P Nonlinar Plat Bnding: 19
20 Clampd Circular Plat undr UDL w 0 /h Dflction, E = 10 6 psi, ν = 0.3 a = 100 in., h = 10 in. u = 0, φ at = 0 = 0 E = 10 6 psi, ν = 0.3 h = 10 in a = 100 in. v = u = w = = = 0 0 φ φ on th clampd dg v = 0, φ = 0 at = Msh of 5-Q9 lmnts Load paramtr, (q 0 a 4 /Eh 4 ) Plats (Nonlinar): 0
21 Simpl Supportd (SS) Orthotropic* Plat, 0 ( ) Eprimntal [8] CLPT FSDT Gomtr and Matrial Proprtis a = b = 1 in, h = in E 1 = psi, E = psi G 1 = G 3 = G 13 = psi ν 1 = ν 3 = ν 13 = Linar Nonlinar [8] Zaghloul, S. A. and Knnd, J. B., ``Nonlinar Bhavior of Smmtricall Laminatd Plats, Journal of Applid Mchanics, 4, 34-36, Prssur, q 0 (psi) Nonlinar Plat Bnding: 1
22 Dflction vs. load paramtr for plats undr uniforml distributd load w Dflction, w qa 0 0 w =, P = h Eh SS-1 (FSDT) SS-1 (CPT) SS-3 (CPT) SS-3 (FSDT) Load paramtr, P 4 4 σ Strsss, a = Eh SS-1 (FSDT) SS-3 (FSDT) SS-1 (CPT) SS-3 (CPT) Mmbran strsss Load paramtr, P Nonlinar Plat Bnding:
23 Cntr Dflction vs. Tim for a Simpl Supportd Isotropic Plat Undr Suddnl Applid Uniforml Distributd Prssur Load Cntr dflction, w0 (cm) Tim, t (s) (ms) a = b = 43.8 cm, h = cm, ρ ρ = N-s /cm 4, E 1 = E = N/cm, ν 1 = 0.5 q 0 = N /cm, t = s = 5ms Figur (SS-) q0 (N/cm ) Dflction, w 0 (cm) q 0 ( t = 5.0 ms) q 0 ( t = 5.0 ms) 5q 0 ( t =.5 ms) 10q 0 ( t =.5 ms)
24 SUMMARY In this lctur w hav covrd th following topics: Govrning Equations of FSDT Finit lmnt modls of FSDT Tangnt stiffnss cofficints Shar and mmbran locking Programming aspcts (including strss computation) Numrical ampls Nonlinar Plat Bnding: 4
Nonlinear 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 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 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 informationOTHER TPOICS OF INTEREST (Convection BC, Axisymmetric problems, 3D FEM)
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
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 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 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 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 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 informationNon-Linear Analysis of Interlaminar Stresses in Composite Beams with Piezoelectric Layers
7TH ITERATIOA OFEREE O OMPOSITE SIEE AD TEHOOGY on-inar Analysis of Intrlaminar Strsss in omosit Bams with Piolctric ayrs MASOUD TAHAI 1, AMIR TOOU DOYAMATI 1 Dartmnt of Mchanical Enginring, Faculty of
More informationBuckling Analysis of Piezolaminated Plates Using Higher Order Shear Deformation Theory
Intrnational Journal of Composit Matrials 03, 3(4): 9-99 DOI: 0.593/j.cmatrials.030304.0 Buckling Analsis of Pizolaminatd Plats Using Highr Ordr Shar Dformation hor Rajan L. Wankhad, Kamal M. Bajoria Dpartmnt
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 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 informationGAS FOIL BEARING ANALYSIS AND THE EFFECT OF BUMP FOIL THICKNESS ON ITS PERFORMANCE CHARACTERISTICS USING A NON-LINEAR MATRIX EQUATION SOLVER
GAS FOIL BEARING ANALYSIS AND THE EFFECT OF BUMP FOIL THICKNESS ON ITS PERFORMANCE CHARACTERISTICS USING A NON-LINEAR MATRIX EQUATION SOLVER T. Moasunp. Jamir 1)*, S. K. Kakoty 1), Karuna Kalita 1) 1)
More informationAn adaptive Strategy for the Multi-scale Analysis of Plate and Shell Structures with Elasto-plastic Material Behaviour
TECHNISCHE MECHANIK, 36, 1-2, (2016), 142 154 submittd: Sptmbr 7, 2015 An adaptiv Stratgy for th Multi-scal Analysis of Plat and Shll Structurs with Elasto-plastic Matrial Bhaviour W Wagnr, F Gruttmann
More informationThermal buckling analysis of skew bre-reinforced composite and sandwich plates using shear deformable nite element models
Composit Structurs 49 () ±85 www.lsir.com/locat/compstruc Thrmal buckling analsis of skw br-rinforcd composit and sandwich plats using shar dformabl nit lmnt modls T. Kant *, C.S. Babu Dpartmnt of Ciil
More informationME311 Machine Design
ME311 Machin Dsign Lctur 4: Strss Concntrations; Static Failur W Dornfld 8Sp017 Fairfild Univrsit School of Enginring Strss Concntration W saw that in a curvd bam, th strss was distortd from th uniform
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 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 informationCHAPTER 9. Interpolation functions for 2D elements. Numerical Integration. Modeling Considerations
HAPTER 9 Intrpolation functions for D lmnts Numrical Intgration Modling onsidrations Pascal s triangl Dgr of Numbr of Elmnt with th complt trms in th nods polnomial polnomial Triangular Elmnts (Figur not
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 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 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 informationDynamic Characteristics Analysis of Blade of Fan Based on Ansys
Powr and Enrgy Enginring Confrnc 1 Dynamic Charactristics Analysis of Blad of Fan Basd on Ansys Junji Zhou, Bo Liu, Dingbiao Wang, Xiaoqian li School of Chmical Enginring Zhngzhou Univrsity Scinc Road
More informationResponse Sensitivity for Nonlinear Beam Column Elements
Rspons Snsitivity for Nonlinar Bam Column Elmnts Michal H. Scott 1 ; Paolo Franchin 2 ; Grgory. Fnvs 3 ; and Filip C. Filippou 4 Abstract: Rspons snsitivity is ndd for simulation applications such as optimization,
More informationFree Vibration of Pre-Tensioned Electromagnetic Nanobeams
IOSR Journal of Mathmatics (IOSR-JM) -ISSN: 78-578, p-issn: 39-765X. Volum 3, Issu Vr. I (Jan. - Fb. 07), PP 47-55 www.iosrjournals.org Fr Vibration of Pr-Tnsiond Elctromagntic Nanobams M. Zaaria& Amira
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 informationAn Investigation on the Effect of the Coupled and Uncoupled Formulation on Transient Seepage by the Finite Element Method
Amrican Journal of Applid Scincs 4 (1): 95-956, 7 ISSN 1546-939 7 Scinc Publications An Invstigation on th Effct of th Coupld and Uncoupld Formulation on Transint Spag by th Finit Elmnt Mthod 1 Ahad Ouria,
More informationKeywords- Active vibration control, cantilever composite beam, Newmark-β method
Pratik K. Gandhi, J. R. Mvada / Intrnational Journal of Enginring Rsarch and Applications (IJERA) ISSN: 8-96 www.ijra.com Vol., Issu, May-Jun, pp.9-95 A Finit Elmnt Modl And Activ Vibration Control Of
More informationRPT nite-element formulation for linear dynamic analysis of orthotropic plates
Scintia Iranica B (01) 5(), 13{3 Sharif Univrsity of Tchnology Scintia Iranica Transactions B: Mchanical Enginring http://scintiairanica.sharif.du RPT nit-lmnt formulation for linar dynamic analysis of
More informationSt. Venants Torsion Constant of Hot Rolled Steel Profiles and Position of the Shear Centre
NSCC2009 St. Vnants Torsion Constant of Hot Rolld Stl Profils and Position of th Shar Cntr M. Kraus 1 & R. Kindmann 1 1 Institut for Stl and Composit Structurs, Univrsity of Bochum, Grmany BSTRCT: Th knowldg
More informationAnalysis of Structural Vibration using the Finite Element Method
Sminar: Vibrations and Structur-Born Sound in Civil Enginring Thor and Applications Analsis of Structural Vibration using th Finit Elmnt thod John.A. Shiwua 5 th April, 006 Abstract Structural vibration
More informationTwist analysis of piezoelectric laminated composite plates
wist analysis of pizolctric laminatd composit plats Mchatronics Enginring Dpartmnt, Faculty of Enginring, Intrnational Islamic Univrsity Malaysia, Malaysia raisuddin@iiu.du.my ABSAC cntly scintists ar
More informationFinite Element Modelling for Static and Free Vibration Response of Functionally Graded Beam
69 Finit Elmnt Modlling for Static and Fr Vibration Rspons of Functionally Gradd Bam Abstract A 1D Finit Elmnt modl for static rspons and fr vibration analysis of functionally gradd matrial (FGM) bam is
More information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More informationA Comparative study of Load Capacity and Pressure Distribution of Infinitely wide Parabolic and Inclined Slider Bearings
Procdings of th World Congrss on Enginring 2 Vol II WCE 2, Jun 3 - July 2, 2, London, U.K. A Comparativ study of Load Capacity and Prssur Distribution of Infinitly wid Parabolic and Inclind Slidr Barings
More informationDynamic Modelling of Hoisting Steel Wire Rope. Da-zhi CAO, Wen-zheng DU, Bao-zhu MA *
17 nd Intrnational Confrnc on Mchanical Control and Automation (ICMCA 17) ISBN: 978-1-6595-46-8 Dynamic Modlling of Hoisting Stl Wir Rop Da-zhi CAO, Wn-zhng DU, Bao-zhu MA * and Su-bing LIU Xi an High
More information3 Finite Element Parametric Geometry
3 Finit Elmnt Paramtric Gomtry 3. Introduction Th intgral of a matrix is th matrix containing th intgral of ach and vry on of its original componnts. Practical finit lmnt analysis rquirs intgrating matrics,
More informationElastic Analysis of Functionally Graded Variable Thickness Rotating Disk by Element Based Material Grading
Journal of Solid Mchanics ol. 9, No. 3 (017) pp. 650-66 Elastic Analysis of Functionally Gradd ariabl hicknss Rotating Disk by Elmnt Basd Matrial Grading A.K. hawait 1,*, L. Sondhi 1, Sh. Sanyal, Sh. Bhowmick
More informationInstantaneous Cutting Force Model in High-Speed Milling Process with Gyroscopic Effect
Advancd Matrials sarch Onlin: -8-6 ISS: 66-8985, Vols. 34-36, pp 389-39 doi:.48/www.scintific.nt/am.34-36.389 rans ch Publications, Switzrland Instantanous Cutting Forc Modl in High-Spd Milling Procss
More informationFree Vibration Analysis of Stiffened Laminated Plates Using a New Stiffened Element
TEHNISHE MEHANIK, Band 8, Hft -4, (8), 7-6 Manusriptingang: 7. Otobr 7 Fr Vibration Analysis of Stiffnd Laminatd Plats Using a Nw Stiffnd Elmnt Tran Ich Thinh, Ngo Nhu Khoa A nw 9-nodd rctangular iffnd
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 informationNONCONFORMING FINITE ELEMENTS FOR REISSNER-MINDLIN PLATES
NONCONFORMING FINITE ELEMENTS FOR REISSNER-MINDLIN PLATES C. CHINOSI Dipartimnto di Scinz Tcnologi Avanzat, Univrsità dl Pimont Orintal, Via Bllini 5/G, 5 Alssandria, Italy E-mail: chinosi@mfn.unipmn.it
More informationJacob Fish and Kamlun Shek 1 Departments of Civil and Mechanical Engineering Rensselaer Polytechnic Institute Troy, NY Abstract.
Finit dformation plasticity basd on th additiv split of th rat of dformation and hyprlasticity Jacob Fish and Kamlun Shk 1 Dpartmnts of Civil and Mchanical Enginring nsslar Polytchnic Institut Troy, NY
More informationUNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES
UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES A Thesis by WOORAM KIM Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationDynamic response of a finite length euler-bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force
Journal of Mchanical Scinc and Tchnology 2 (1) (21) 1957~1961 www.springrlink.com/contnt/1738-9x DOI 1.17/s1226-1-7-x Dynamic rspons of a finit lngth ulr-brnoulli bam on linar and nonlinar viscolastic
More informationBending behaviors of simply supported rectangular plates with an internal line sagged and unsagged supports
Songklanakarin J. Sci. Tchnol. 30 (1) 101-107 Jan. - F. 2008 http://www.sjst.psu.ac.th Original Articl Bnding haviors of simpl supportd rctangular plats with an intrnal lin saggd and unsaggd supports Yos
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 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 informationFinite Element Model of a Ferroelectric
Excrpt from th Procdings of th COMSOL Confrnc 200 Paris Finit Elmnt Modl of a Frrolctric A. Lópz, A. D Andrés and P. Ramos * GRIFO. Dpartamnto d Elctrónica, Univrsidad d Alcalá. Alcalá d Hnars. Madrid,
More informationINC 693, 481 Dynamics System and Modelling: The Language of Bound Graphs Dr.-Ing. Sudchai Boonto Assistant Professor
INC 693, 48 Dynamics Systm and Modlling: Th Languag o Bound Graphs Dr.-Ing. Sudchai Boonto Assistant Prossor Dpartmnt o Control Systm and Instrumntation Enginring King Mongkut s Unnivrsity o Tchnology
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 informationMEEN 617 Handout #12 The FEM in Vibrations A brief introduction to the finite element method for modeling of mechanical structures
MEEN 67 Handout # T FEM in Vibrations A brif introduction to t finit lmnt mtod for modling of mcanical structurs T finit lmnt mtod (FEM) is a picwis application of a variational mtod. Hr I provid you wit
More informationMHD Effects in Laser-Produced Plasmas
MHD Effcts in Lasr-Producd Plasmas OLEG POLOMAROV and RICCARDO BETTI Fusion Scinc Cntr and Laboratory for Lasr Enrgtics Univrsity of Rochstr Abstract Th implmntation of th magnto-hydrodynamic (MHD) modul
More informationMechanical Properties
Mchanical Proprtis Elastic dformation Plastic dformation Fractur Mchanical Proprtis: Th Tnsion Tst s u P L s s y ΔL I II III For matrials proprtis, rplac load-dflction by strss-strain Enginring strss,
More informationA New Approach to the Fatigue Life Prediction for Notched Components Under Multiaxial Cyclic Loading. Zhi-qiang TAO and De-guang SHANG *
2017 2nd Intrnational Conrnc on Applid Mchanics, Elctronics and Mchatronics Enginring (AMEME 2017) ISBN: 978-1-60595-497-4 A Nw Approach to th Fatigu Li Prdiction or Notchd Componnts Undr Multiaxial Cyclic
More informationNumerical Analysis of Transient Responses for Elastic Structures Connected to a Viscoelastic Shock Absorber Using FEM with a Nonlinear Complex Spring
Numrical Analysis of Transint Rsponss for Elastic Structurs Connctd to a Viscolastic Shock Absorbr Using FEM with a Nonlinar Complx Spring Takao Yamaguchi, Yusaku Fujii, Toru Fukushima, Akihiro Takita,
More informationAt the end of this lesson, the students should be able to understand:
Instructional Objctivs: At th nd of this lsson, th studnts should b abl to undrstand: Dsign thod for variabl load Equivalnt strss on shaft Dsign basd on stiffnss and torsional rigidit Critical spd of shaft
More informationFinite Element Analysis of Magneto-Superelastic Behavior of Shape Memory Alloy Composite Actuator
Procdings of th Intrnational MultiConfrnc of Enginrs and Computr cintists 28 Vol II IMEC 28, 19-21 March, 28, Hong Kong Finit Elmnt Analysis of Magnto-uprlastic Bhavior of hap Mmory Alloy Composit Actuator
More informationThe Weak Patch Test for Nonhomogeneous Materials Modeled with Graded Finite Elements
Th Wak Patch Tst for Nonhomognous Matrials Modld with... Glaucio H. Paulino paulino@uiuc.du Univrsit of Illinois at Urbana Champaign Dpt. of Civil and Environmntal Eng. Urbana, IL 680. USA Jong-Ho Kim
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More information2. Background Material
S. Blair Sptmbr 3, 003 4. Background Matrial Th rst of this cours dals with th gnration, modulation, propagation, and ction of optical radiation. As such, bic background in lctromagntics and optics nds
More informationLarge Scale Topology Optimization Using Preconditioned Krylov Subspace Recycling and Continuous Approximation of Material Distribution
Larg Scal Topology Optimization Using Prconditiond Krylov Subspac Rcycling and Continuous Approximation of Matrial Distribution Eric d Sturlr*, Chau L**, Shun Wang***, Glaucio Paulino** * Dpartmnt of Mathmatics,
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 informationDevelopment of solid-shell elements for large deformation simulation and springback prediction
Faculté ds Scincs Appliqués Anné Académiqu 8-9 Dvlopmnt of solid-shll lmnts for larg dformation simulation and springback prdiction ravail présnté par Nhu Huynh NGUYEN Ingéniur mécanicin pour l obtntion
More informationCIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7
CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - In som pplictions, it m mor dsirl to us n lmntl rprsnttion of th domin tht hs four sids, ithr rctngulr or qudriltrl in shp. Considr
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 informationFINITE BEAM ELEMENT WITH PIEZOELECTRIC LAYERS AND FUNCTIONALLY GRADED MATERIAL OF CORE
ECCOMAS Congrss 20 II Europan Congrss on Computational Mthods in Applid Scincs and Enginring M. Papadrakakis,. Papadopoulos, G. Stfanou,. Plvris (ds.) Crt Island, Grc, 5 0 Jun 20 FINITE BEAM ELEMENT WITH
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 informationFINITE ELEMENT ANALYSIS OF SLOSHING IN LIQUID-FILLED CONTAINERS
FINIE ELEMEN ANALYSIS OF SLOSHING IN LIQUID-FILLED CONAINERS Mustafa Arafa Lcturr, Dpartmnt of Mchanical Dsign and Production Enginring, Cairo Univrsity, Cairo, Egypt mharafa@gmail.com, mharafa@yahoo.com
More informationSME 3033 FINITE ELEMENT METHOD. Bending of Prismatic Beams (Initial notes designed by Dr. Nazri Kamsah)
Bnding of Prismatic Bams (Initia nots dsignd by Dr. Nazri Kamsah) St I-bams usd in a roof construction. 5- Gnra Loading Conditions For our anaysis, w wi considr thr typs of oading, as iustratd bow. Not:
More informationPHYS-333: Problem set #2 Solutions
PHYS-333: Problm st #2 Solutions Vrsion of March 5, 2016. 1. Visual binary 15 points): Ovr a priod of 10 yars, two stars sparatd by an angl of 1 arcsc ar obsrvd to mov through a full circl about a point
More informationSelf-Adjointness and Its Relationship to Quantum Mechanics. Ronald I. Frank 2016
Ronald I. Frank 06 Adjoint https://n.wikipdia.org/wiki/adjoint In gnral thr is an oprator and a procss that dfin its adjoint *. It is thn slf-adjoint if *. Innr product spac https://n.wikipdia.org/wiki/innr_product_spac
More informationChapter 3 Lecture 14 Longitudinal stick free static stability and control 3 Topics
Chaptr 3 Lctur 14 Longitudinal stick fr static stability and control 3 Topics 3.4.4 Rquirmnt for propr stick forc variation 3.4.5 Fl of th stability lvl by th pilot Exampl 3.3 3.5 Dtrmination of stick-fr
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 informationSelective Mass Scaling (SMS)
Slctiv Mass Scaling (SMS) Thory and Practic Thomas Borrvall Dynamor Nordic AB Octobr 20 LS DYNA information Contnt Background Is SMS nwsworthy? Thory and Implmntation Diffrnc btwn CMS and SMS Undr th hood
More informationGH. Rahimi & AR. Davoodinik
US ntrnational Journal of nginring Scinc Vol 9 No5-8 Pag 5- HRMAL BHAVOR ANALYSS OF H FUNONALLY GRADD MOSHNKO'S BAM GH Raimi & AR Davoodinik Abstract: intntion of tis stud is t analsis of trmal bavior
More information( ) Abstract. 2 FEDSS method basic relationships. 1 Introduction. 2.1 Tensorial formulation
Displacmnt basd continuous strss rcovry procdur, Mijuca D, Brkoviæ M. and Draškoviæ Z., Advancs in Finit Elmnt Tchnology, ISBN 0 948749 4, Ed. B.H.V.Topping, Civil Comp Prss, 7-34 (996). Abstract In this
More informationThe influence of electron trap on photoelectron decay behavior in silver halide
Th influnc of lctron trap on photolctron dcay bhavior in silvr halid Rongjuan Liu, Xiaowi Li 1, Xiaodong Tian, Shaopng Yang and Guangshng Fu Collg of Physics Scinc and Tchnology, Hbi Univrsity, Baoding,
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More information843. Efficient modeling and simulations of Lamb wave propagation in thin plates by using a new spectral plate element
843. Efficint modling and simulations of Lamb wav propagation in thin plats by using a nw spctral plat lmnt Chunling Xu, Xinwi Wang Stat Ky Laboratory of Mchanics and Control of Mchanical Structurs aning
More informationThe pn junction: 2 Current vs Voltage (IV) characteristics
Th pn junction: Currnt vs Voltag (V) charactristics Considr a pn junction in quilibrium with no applid xtrnal voltag: o th V E F E F V p-typ Dpltion rgion n-typ Elctron movmnt across th junction: 1. n
More informationStatic and Dynamic Analysis of Bistable Piezoelectric- Composite Plates for Energy Harvesting
5rd AIAA/ASME/ASCE/AHS/ASC Structurs, Structural Dnamics and Matrials Confrncth AI - 6 April, Honolulu, Hawaii AIAA -49 Static and Dnamic Analsis of Bistabl Pizolctric- Composit Plats for Enrg Harvsting
More informationCHAPTER 2 LAGRANGIAN AND EULERIAN FINITE ELEMENTS IN ONE DIMENSION
CHAPTER 2 LAGRANGIAN AND EULERIAN FINITE ELEMENTS IN ONE DIMENSION by Td Blytschko Northwstrn Univrsity @ Copyright 1997 2.1 Introduction In this chaptr, th quations for on-dimnsional modls of nonlinar
More informationEffects of Couple Stress Lubricants on Pressure and Load Capacity of Infinitely Wide Exponentially Shaped Slider Bearing
Procdings of t World Congrss on Enginring and Computr Scinc 200 Vol II, Octobr 20-22, 200, San Francisco, USA Effcts of Coupl Strss Lubricants on Prssur and Load Capacity of Infinitly Wid Eponntially Sapd
More informationMassachusetts Institute of Technology Department of Mechanical Engineering
Massachustts Institut of Tchnolog Dpartmnt of Mchanical Enginring. Introduction to Robotics Mid-Trm Eamination Novmbr, 005 :0 pm 4:0 pm Clos-Book. Two shts of nots ar allowd. Show how ou arrivd at our
More informationFerroelectrics 342:73-82, 2006 Computational Modeling of Ferromagnetic Shape Memory Thin Films
Frrolctrics 4:7-8 6 Computational Modling of Frromagntic Shap Mmory Thin Films J. Liakhova M. Luskin and T. Zhang School of Mathmatics 6 Church St. SE Univrsity of Minnsota Minnapolis MN 55455 USA Email:
More information( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition
Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
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 informationConvergence Study for FEM- and FIT-Based Eigenvalue Solvers Applied to a TESLA 1.3 GHz Test Structure
Convrgnc Stud for FEM- and FIT-Basd Eignvalu Solvrs Applid to a TESLA 1.3 GHz Tst Structur W. Ackrmann, H. D Grsm, W. F. O. Müllr Institut Thori Elktromagntischr Fldr, Tchnisch Univrsität Darmstadt Status
More informationChapter 6 2D Elements Plate Elements
Institute of Structural Engineering Page 1 Chapter 6 2D Elements Plate Elements Method of Finite Elements I Institute of Structural Engineering Page 2 Continuum Elements Plane Stress Plane Strain Toda
More informationACCURACY OF DIRECT TREFFTZ FE FORMULATIONS
COMPUTATIONAL MECHANICS Nw Trnds and Applications S. Idlsohn, E. Oñat and E. Dvorkin (Eds.) c CIMNE, Barclona, Spain 1998 ACCURACY OF DIRECT TREFFTZ FE FORMULATIONS Vladimír Kompiš,L ubor Fraštia, Michal
More informationUsing a C 1 triangular finite element based on a refined model for analyzing buckling and post-buckling of multilayered plate/shell structures
Using a C 1 triangular finit lmnt basd on a rfind modl for analyzing buckling and post-buckling of multilayrd plat/sll structurs F. Dau O. Polit M.Touratir LAMEFIP/ENSAM, Esplanad ds arts t métirs, 33405
More informationME469A Numerical Methods for Fluid Mechanics
ME469A Numrical Mthods for Fluid Mchanics Handout #5 Gianluca Iaccarino Finit Volum Mthods Last tim w introducd th FV mthod as a discrtization tchniqu applid to th intgral form of th govrning quations
More information682 CHAPTER 11 Columns. Columns with Other Support Conditions
68 CHTER 11 Columns Columns with Othr Support Conditions Th problms for Sction 11.4 ar to b solvd using th assumptions of idal, slndr, prismatic, linarly lastic columns (Eulr buckling). uckling occurs
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More informationLecture Outline. Skin Depth Power Flow 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3e
8/7/018 Cours Instructor Dr. Raymond C. Rumpf Offic: A 337 Phon: (915) 747 6958 E Mail: rcrumpf@utp.du EE 4347 Applid Elctromagntics Topic 3 Skin Dpth & Powr Flow Skin Dpth Ths & Powr nots Flow may contain
More informationVTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS
Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More informationDifferential Equations
UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs
More information