PLATE BENDING ELEMENTS
|
|
- Mildred Wood
- 6 years ago
- Views:
Transcription
1 8. PLATE BENING ELEMENTS Plate Bendng s a Smple Etenson of Beam Theor 8. INTROUCTION { XE "Plate Bendng Elements" }Before 960, plates and slabs were modeled usng a grd of beam elements for man cvl engneerng structures. Onl a small number of closed form solutons ested for plates of smple geometr and sotropc materals. Even at the present tme man slab desgns are based on grd models. Ths classcal appromate approach, n general, produces conservatve results because t satsfes statcs and volates compatblt. However, the nternal moment and shear dstrbuton ma be ncorrect. The use of a converged fnte element soluton wll produce a more consstent desgn. The fundamental dfference between a grd of beam elements and a plate-bendng fnte element soluton s that a twstng moment ests n the fnte element model; whereas, the grd model can onl produce one-dmensonal torsonal moments and wll not converge to the theoretcal soluton as the mesh s refned. { XE "Plate Bendng Elements:Thn Plates" }The followng appromatons are used to reduce the three-dmensonal theor of elastct to govern the behavor of thn plates and beams:. { XE "Plate Bendng Elements:Reference Surface" }It s assumed that a lne normal to the reference surface (neutral a of the plate (beam) remans straght n the loaded poston. Ths dsplacement constrant s the same as statng that the n-plane strans are a lnear functon n the thckness drecton. Ths assumpton does not requre that the rotaton of the normal lne to be equal to the rotaton of the reference surface; hence, transverse shear deformatons are possble.. In addton, the normal stress n the thckness drecton, whch s normall ver small compared to the bendng stresses, s assumed to be zero for both beams and plates. Ths s accomplshed b usng plane stress materal propertes n-plane as defned n Chapter. Note
2 8- STATIC AN YNAMIC ANALYSIS that ths appromaton allows Posson s rato strans to est n the thckness drecton. 3. { XE "Krchhoff Appromaton" }If the transverse shearng strans are assumed to be zero, an addtonal dsplacement constrant s ntroduced that states that lnes normal to the reference surface reman normal to the reference surface after loadng. Ths appromaton s attrbuted to Krchhoff and bears hs name. { XE "Plate Bendng Elements:Shearng eformatons" }{ XE "Shearng eformatons" }Classcal thn plate theor s based on all three appromatons and leads to the development of a fourth order partal dfferental equaton n terms of the normal dsplacement of the plate. Ths approach s onl possble for plates of constant thckness. Man books and papers, usng complcated mathematcs, have been wrtten based on ths approach. However, the Krchhoff appromaton s not requred to develop plate bendng fnte elements that are accurate, robust and eas to program. At the present tme, t s possble to nclude transverse shearng deformatons for thck plates wthout a loss of accurac for thn plates. In ths chapter, plate bendng theor s presented as an etenson of beam theor (see Append F) and the equatons of three-dmensonal elastct. Hence, no prevous background n plate theor s requred b the engneer to full understand the appromatons used. Several hundred plate-bendng fnte elements have been proposed durng the past 30 ears. However, onl one element wll be presented here. The element s a three-node trangle or a fournode quadrlateral and s formulated wth and wthout transverse shearng deformatons. The formulaton s restrcted to small dsplacements and elastc materals. Numercal eamples are presented to llustrate the accurac of the element. The theor presented here s an epanded verson of the plate bendng element frst presented n reference [] usng a varatonal formulaton. 8. THE QUARILATERAL ELEMENT { XE "Quadrlateral Element" }Frst, the formulaton for the quadrlateral element wll be consdered. The same approach apples to the trangular element. A quadrlateral of arbtrar geometr, n a local - plane, s shown n Fgure 8.. Note that the parent four-node element, Fgure 8.a, has 6 rotatons at the four node ponts and at the md-pont of each sde. The md-sde rotatons are then rotated to be normal and tangental to each sde. The tangental rotatons are then set to zero, reducng the number of degrees-of-freedom to, Fgure 8.b. The sdes of the element are constraned to be a cubc functon n u and four z
3 PLATE BENING ELEMENTS 8-3 dsplacements are ntroduced at the corner nodes of the element, Fgure 8.c. Fnall, the md-sde rotatons are elmnated b statc condensaton, Fgure 8.d, and a OF element s produced (a) 6 (b) 5 s 3 r 3 θ (d) (c) u z θ Fgure 8. Quadrlateral Plate Bendng Element The basc dsplacement assumpton s that the rotaton of lnes normal to the reference plane of the plate s defned b the followng equatons: θ ( r, θ ( r, 8 N ( r, θ + N ( r, θ N ( r, θ N ( r, θ The eght-node shape functons are gven b: N ( r)( / N ( + r)( / N ( + r)( + )/ N ( r)( + )/ N 3 s 5 ( r )( s 7 ( r )( s s 6 ( + r)( s 8 ( r)( s (8.) )/ N )/ (8.) N + )/ N )/ { XE "Herarchcal Functons" }Note that the frst four shape functons are the natural blnear shape functons for a four-node quadrlateral. The four shape functons for the md-sde nodes are an addton to the blnear functons and are
4 8- STATIC AN YNAMIC ANALYSIS often referred to as herarchcal functons. A tpcal element sde s shown n Fgure 8.. θ j θ j L α m θ θ,,3, j,3,, m 5,6,7,8 θ Fgure 8. Tpcal Element Sde The tangental rotatons are set to zero and onl the normal rotatons est. Therefore, the and components of the normal rotaton are gven b: θ sn α θ cosα θ θ (8.3) Hence, Equaton (8.) can be rewrtten as: θ ( r, θ ( r, 8 N ( r, θ + N ( r, θ M M ( r, θ ( r, θ (8.) { XE "Plate Bendng Elements:Postve splacements" }The number of dsplacement degrees-of-freedom has now been reduced from 6 to, as ndcated n Fgure 8.b. The three-dmensonal dsplacements, as defned n Fgure 8.3 wth respect to the - reference plane, are:
5 PLATE BENING ELEMENTS 8-5,u z,u z h θ θ,u Fgure 8.3 Postve splacements n Plate Bendng Element u ( r, z θ ( r, u ( r, z θ ( r, (8.5) Note that the normal dsplacement of the reference plane u z ( r, has not been defned as a functon of space. Now, t s assumed that the normal dsplacement along each sde s a cubc functon. From Append F, the transverse shear stran along the sde s gven b: γ (uzj - uz) - ( θ + θ j) - θ (8.6) L 3 From Fgure 8., the normal rotatons at nodes and j are epressed n terms of the and rotatons. Or, Equaton (8.6) can be wrtten as: sn α α ( θ + θ j) + cos ( θ + θ j) - (8.7) γ (uzj - uz) - θ L 3 Ths equaton can be wrtten for all four sdes of the element.
6 8-6 STATIC AN YNAMIC ANALYSIS It s now possble to epress the node shears n terms of the sde shears. A tpcal node s shown n Fgure 8.. θ k α k k γ θ j u z θ,,3, j,3,, k,,,3 γ z γ γ k z α θ Fgure 8. Node Pont Transverse Shears The two md-sde shears are related to the shears at node b the followng stran transformaton: γ cosα sn α γ z (8.8) γ k cosα k sn α k γ z Or, n nverse form: γ z sn α k cosα k γ (8.9) γ z det sn α cosα γ k where det cosα sn α cosα sn α. k k The fnal step n determnng the transverse shears s to use the standard four-node blnear functons to evaluate the shears at the ntegraton pont. 8.3 STRAIN-ISPLACEMENT EQUATIONS { XE "Plate Bendng Elements:Stran-splacement Equatons" }{ XE "Stran splacement Equatons:Plate Bendng" }Usng the three-dmensonal strandsplacement equatons, the strans wthn the plate can be epressed n terms of the node rotatons. Or:
7 PLATE BENING ELEMENTS 8-7 u ε z θ ( r,, u ε z θ ( r,, u u γ + z [ θ ( r,, θ ( r, ] (8.0) Therefore, at each ntegraton pont the fve components of stran can be epressed n terms of the 6 dsplacements, shown n Fgure 8.c, b an equaton of the followng form: ε z z θ X ε θ Y γ 0 0 z 0 0 b or d Bu a( z) b( r, u (8.) uz γ z z θ γ Hence, the stran-dsplacement transformaton matr s a product of two matrces n whch one s a functon of z onl. 8. THE QUARILATERAL ELEMENT STIFFNESS { XE "Plate Bendng Elements:Propertes" }From Equaton (8.), the element stffness matr can be wrtten as: where T B EBdV k b b da T (8.) T a Ea dz (8.3) After ntegraton n the z-drecton, the 5 b 5 force-deformaton relatonshp for orthotropc materals s of the followng form: M M M V V z z ψ ψ ψ γ z γ z (8.)
8 8-8 STATIC AN YNAMIC ANALYSIS The moments M and shears resultant V are forces per unt length. As n the case of beam elements, the deformatons assocated wth the moment are the curvature ψ. For sotropc plane stress materals, the non-zero terms are gven b: 3 Eh ( ν ) 3 νeh (8.5) ( ν ) 5Eh 55 ( +ν ) 8.5 SATISFYING THE PATCH TEST { XE "Patch Test" }{ XE "Plate Bendng Elements:Patch Test" }For the element to satsf the patch test, t s necessar that constant curvatures be produced f the node dsplacements assocated wth constant curvature are appled. Equaton (8.) can be wrtten n the followng form: ψ θ ψ b b θ ψ (8.6) b b w γ z θ γ z { XE "Plate Bendng Elements:Constant Moment" }where, for a quadrlateral element, b s a 3 b matr assocated wth the node dsplacements ( θ,θ, w ) and b s a 3 b matr assocated wth the ncompatble normal sde rotatons ( θ ). In order that the element satsfes the constant moment patch test, the followng modfcaton to b must be made: b b b da (8.7) A The development of ths equaton s presented n the chapter on ncompatble elements, Equaton (6.). 8.6 STATIC CONENSATION { XE "Statc Condensaton" }The element 6 b 6 stffness matr for the plate bendng element wth shearng deformatons s obtaned b numercal ntegraton. Or:
9 PLATE BENING ELEMENTS 8-9 K K T K B B da (8.8) K K where K s the b matr assocated wth the ncompatble normal rotatons. The element equlbrum equatons are of the followng form: K K u F (8.9) K K θ 0 where F s the node forces. Because the forces assocated wth θ must be zero, those deformaton degrees-of-freedom can be elmnated, b statc condensaton, before assembl of the global stffness matr. Therefore, the b element stffness matr s not ncreased n sze f shearng deformatons are ncluded. Ths quadrlateral (or trangular) plate bendng element, ncludng shear deformatons, s defned n ths book as the screte Shear Element, or SE. 8.7 TRIANGULAR PLATE BENING ELEMENT { XE "Plate Bendng Elements:Trangular Element" }The same appromatons used to develop the quadrlateral element are appled to the trangular plate bendng element wth three md-sde nodes. The resultng stffness matr s 9 b 9. Appromatel 90 percent of the computer program for the quadrlateral element s the same as for the trangular element. Onl dfferent shape functons are used and the constrant assocated wth the fourth sde s skpped. In general, the trangle s stffer than the quadrlateral. 8.8 OTHER PLATE BENING ELEMENTS The fundamental equaton for the dscrete shear along the sdes of an element s gven b Equaton (8.6). Or: γ (uzj - uz) - ( θ + θ j) - θ (8.0) L 3 { XE "Plate Bendng Elements:PQ" }If θ s set to zero at the md-pont of each sde, shearng deformatons are stll ncluded n the element. However, the nternal moments wthn the element are constraned to a constant value for a thn plate. Ths s the same as the PQ element gven n reference [], whch s based on a second order polnomal appromaton of the normal dsplacement. The dsplacements produced b ths element tend to have a small error; however, the nternal moments for a coarse mesh tend to have a sgnfcant error. Therefore, ths author does not recommend the use of ths element.
10 8-0 STATIC AN YNAMIC ANALYSIS If the shear s set to zero along each sde of the element, the followng equaton s obtaned: 3 3 θ (w j - w ) - ( θ + θ j) (8.) L Hence, t s possble to drectl elmnate the md-sde relatve rotatons drectl wthout usng statc condensaton. Ths appromaton produces the screte Krchhoff Element, KE, n whch transverse shearng deformatons are set to zero. It should be noted that the SE and the KE for thn plates converge at appromatel the same rate for both dsplacements and moments. For man problems, the SE and the KE tend to be more fleble than the eact soluton. 8.9 NUMERICAL EXAMPLES { XE "Plate Bendng Elements:Convergence" }{ XE "Plate Bendng Elements:Eamples" }Several eamples are presented to demonstrate the accurac and convergence propertes of quadrlateral and trangular plate bendng elements wth and wthout transverse shear deformatons. A four-pont numercal ntegraton formula s used for the quadrlateral element. A three-pont ntegraton formula s used for the trangular element One Element Beam To llustrate that the plate element reduces to the same behavor as classcal beam theor, the cantlever beam shown n Fgure 8.5 s modeled as one element that s nches thck. The narrow element s 6 nches b 0. nch n plan. E0,000 ks G3,86 ks.0 k Fgure 8.5 Cantlever Beam Modeled usng One Plate Element 0.
11 PLATE BENING ELEMENTS 8- { XE "Plate Bendng Elements:KE" }{ XE "Plate Bendng Elements:SE" }The end dsplacements and base moments are summarzed n Table 8. for varous theores. Table 8. splacement and Moment for Cantlever Beam THEORY and ELEMENT Tp splacement (nche Mamum Moment (kp-n.) Beam Theor Beam Theor wth Shear eformaton SE Plate Element KE Plate Element PK Plate Element Ref. [] Ths eample clearl ndcates that one plate element can model a onedmensonal beam wthout the loss of accurac. It s worth notng that man plate elements wth shear deformatons, whch are currentl used wthn computer programs, have the same accurac as the PQ element. Hence, the user must verf the theor and accurac of all elements wthn a computer program b checkng the results wth smple eamples Pont Load On Smpl Supported Square Plate { XE "Plate Bendng Elements:Pont Load" }To compare the accurac of the SE and KE as the elements become ver thn, a b mesh, as shown n Fgure 8.6, models one quadrant of a square plate. Note that the normal rotaton along the pnned edge s set to zero. Ths hard boundar condton s requred for the SE. The KE elds the same results for both hard and soft boundar condtons at the pnned edge.
12 8- STATIC AN YNAMIC ANALYSIS θ 0 θ E 0.9 ν 0.3 h., P.0 u z 0 0., at and 0.0, center θ 0 n 0.00, at sdes Fgure 8.6 Pont Load at Center of Smpl Supported Square Plate The mamum dsplacement and moment at the center of the plate are summarzed n Table 8.. For a thn plate wthout shear dsplacements, the dsplacement s proportonal to /h 3. Therefore, to compare results, the dsplacement s normalzed b the factor h 3. The mamum moment s not a functon of thckness for a thn plate. For ths eample, shearng deformatons are onl sgnfcant for a thckness of.0. The eact thn-plate dsplacement for ths problem s.60, whch s ver close to the average of the KE and the SE results. Hence, one can conclude that SE converges to an appromate thn plate soluton as the plate becomes thn. However, SE does not converge for a coarse mesh to the same appromate value as the KE. Table 8. Convergence of Plate Elements b Mesh Pont Load splacement tmes h 3 Mamum Moment Thckness, h KE SE KE SE To demonstrate that the two appromatons converge for a fne mesh, a 6 b 6 mesh s used for one quadrant of the plate. The results obtaned are summarzed n Table 8.3.
13 PLATE BENING ELEMENTS 8-3 Table 8.3 Convergence of Plate Element 6 b 6 Mesh Pont Load splacement tmes h 3 Mamum Moment Thckness h KE SE KE SE One notes that the KE and SE dsplacements converge to the appromatel same value for a pont load at the center of the plate. However, because of stress sngulart, the mamum moments are not equal, whch s to be epected Unform Load On Smpl Supported Square Plate To elmnate the problem assocated wth the pont load, the same plate s subjected to a unform load of.0 per unt area. The results are summarzed n Table 8.. For thn plates, the quadrlateral KE and SE dsplacements and moments agree to three sgnfcant fgures. Table 8. Convergence of Quad Plate Elements 6 b 6 Mesh - Unform Load splacement tmes h 3 Mamum Moment Thckness h KE SE KE SE Evaluaton of Trangular Plate Bendng Elements The accurac of the trangular plate bendng element can be demonstrated b analzng the same square plate subjected to a unform load. The plate s modeled usng 5 trangular elements, whch produces a 6 b 6 mesh, wth each quadrlateral dvded nto two trangles. The results are summarzed n Table 8.5. For thn plates, the quadrlateral KE and SE dsplacements and moments agree to four sgnfcant fgures. The fact that both moments and dsplacements converge to the same value for thn plates ndcates that the trangular elements ma be more accurate than the quadrlateral elements for both thn and thck plates. However, f the trangular mesh s changed b dvdng the quadrlateral on the other dagonal the results are not as mpressve.
14 8- STATIC AN YNAMIC ANALYSIS Table 8.5 Convergence of Trangular Plate Elements - Unform Load Thckness h splacement tmes h 3 Mamum Moment KE SE KE SE * * Quadrlateral dvded on other dagonal It should be noted, however, that f the trangular element s used n shell analss, the membrane behavor of the trangular shell element s ver poor and naccurate results wll be obtaned for man problems Use of Plate Element to Model Torson n Beams { XE "Plate Bendng Elements:Torson" }For one-dmensonal beam elements, the plate element can be used to model the shear and bendng behavor. However, plate elements should not be used to model the torsonal behavor of beams. To llustrate the errors ntroduced b ths appromaton, consder the cantlever beam structure shown n Fgure 8.7 subjected to a unt end torque. FIXE EN τ E0,000,000 ν τ z z γ z 0 0. T.0 Fgure 8.7 Beam Subjected to Torson Modeled b Plate Elements The results for the rotaton at the end of the beam are shown n Table 8.6. Table 8.6 Rotaton at End of Beam Modeled usng Plate Elements Y-ROTATION KE SE
15 PLATE BENING ELEMENTS 8-5 free fed The eact soluton, based on an elastct theor that ncludes warpage of the rectangular cross secton, s 0.03 radans. Note that the shear stress and stran boundar condtons shown n Fgure 8.6 cannot be satsfed eactl b plate elements regardless of the fneness of the mesh. Also, t s not apparent f the - rotaton boundar condton should be free or set to zero For ths eample, the KE element does gve a rotaton that s appromatel 68 percent of the elastct soluton; however, as the mesh s refned, the results are not mproved sgnfcantl. The SE element s ver fleble for the coarse mesh. The results for the fne mesh are stffer. Because nether element s capable of convergng to the eact results, the torson of the beam should not be used as a test problem to verf the accurac of plate bendng elements. Trangular elements produce almost the same results as the quadrlateral elements. 8.0 SUMMARY { XE "FLOOR Program" }{ XE "SAFE Program" }A relatvel new and robust plate bendng element has been summarzed n ths chapter. The element can be used for both thn and thck plates, wth or wthout shearng deformatons. It has been etended to trangular elements and orthotropc materals. The plate bendng theor was presented as an etenson of beam theor and three-dmensonal elastct theor. The KE and SE are currentl used n the SAFE, FLOOR and SAP000 programs. In the net chapter, a membrane element wll be presented wth three OF per node, two translatons and one rotaton normal to the plane. Based on the bendng element presented n ths chapter and membrane element presented n the net chapter, a general thn or thck shell element s presented n the followng chapter. 8. REFERENCES. { XE "Ibrahmbegovc, Adnan" }Ibrahmbegovc, Adnan Quadrlateral Elements for Analss of Thck and Thn Plates, Computer Methods n Appled Mechancs and Engneerng. Vol. 0 (993)
16 8-6 STATIC AN YNAMIC ANALYSIS
MEMBRANE ELEMENT WITH NORMAL ROTATIONS
9. MEMBRANE ELEMENT WITH NORMAL ROTATIONS Rotatons Mst Be Compatble Between Beam, Membrane and Shell Elements 9. INTRODUCTION { XE "Membrane Element" }The comple natre of most bldngs and other cvl engneerng
More informationAPPENDIX F A DISPLACEMENT-BASED BEAM ELEMENT WITH SHEAR DEFORMATIONS. Never use a Cubic Function Approximation for a Non-Prismatic Beam
APPENDIX F A DISPACEMENT-BASED BEAM EEMENT WITH SHEAR DEFORMATIONS Never use a Cubc Functon Approxmaton for a Non-Prsmatc Beam F. INTRODUCTION { XE "Shearng Deformatons" }In ths appendx a unque development
More informationOne Dimensional Axial Deformations
One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the
More informationPlate Theories for Classical and Laminated plates Weak Formulation and Element Calculations
Plate heores for Classcal and Lamnated plates Weak Formulaton and Element Calculatons PM Mohte Department of Aerospace Engneerng Indan Insttute of echnolog Kanpur EQIP School on Computatonal Methods n
More informationInstituto Tecnológico de Aeronáutica FINITE ELEMENTS I. Class notes AE-245
Insttuto Tecnológco de Aeronáutca FIITE ELEMETS I Class notes AE-5 Insttuto Tecnológco de Aeronáutca 5. Isoparametrc Elements AE-5 Insttuto Tecnológco de Aeronáutca ISOPARAMETRIC ELEMETS Introducton What
More informationApplication to Plane (rigid) frame structure
Advanced Computatonal echancs 18 Chapter 4 Applcaton to Plane rgd frame structure 1. Dscusson on degrees of freedom In case of truss structures, t was enough that the element force equaton provdes onl
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationDUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationFINITE DIFFERENCE ANALYSIS OF CURVED DEEP BEAMS ON WINKLER FOUNDATION
VOL. 6, NO. 3, MARCH 0 ISSN 89-6608 006-0 Asan Research Publshng Network (ARPN). All rghts reserved. FINITE DIFFERENCE ANALYSIS OF CURVED DEEP BEAMS ON WINKLER FOUNDATION Adel A. Al-Azzaw and Al S. Shaker
More informationChapter Eight. Review and Summary. Two methods in solid mechanics ---- vectorial methods and energy methods or variational methods
Chapter Eght Energy Method 8. Introducton 8. Stran energy expressons 8.3 Prncpal of statonary potental energy; several degrees of freedom ------ Castglano s frst theorem ---- Examples 8.4 Prncpal of statonary
More informationSTATIC ANALYSIS OF TWO-LAYERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION
STATIC ANALYSIS OF TWO-LERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION Ákos József Lengyel István Ecsed Assstant Lecturer Emertus Professor Insttute of Appled Mechancs Unversty of Mskolc Mskolc-Egyetemváros
More informationOPTIMISATION. Introduction Single Variable Unconstrained Optimisation Multivariable Unconstrained Optimisation Linear Programming
OPTIMIATION Introducton ngle Varable Unconstraned Optmsaton Multvarable Unconstraned Optmsaton Lnear Programmng Chapter Optmsaton /. Introducton In an engneerng analss, sometmes etremtes, ether mnmum or
More informationNumerical Methods. ME Mechanical Lab I. Mechanical Engineering ME Lab I
5 9 Mechancal Engneerng -.30 ME Lab I ME.30 Mechancal Lab I Numercal Methods Volt Sne Seres.5 0.5 SIN(X) 0 3 7 5 9 33 37 4 45 49 53 57 6 65 69 73 77 8 85 89 93 97 0-0.5 Normalzed Squared Functon - 0.07
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationEVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES
EVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES Manuel J. C. Mnhoto Polytechnc Insttute of Bragança, Bragança, Portugal E-mal: mnhoto@pb.pt Paulo A. A. Perera and Jorge
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationUNIVERSITY OF BOLTON RAK ACADEMIC CENTRE BENG(HONS) MECHANICAL ENGINEERING SEMESTER TWO EXAMINATION 2017/2018 FINITE ELEMENT AND DIFFERENCE SOLUTIONS
OCD0 UNIVERSITY OF BOLTON RAK ACADEMIC CENTRE BENG(HONS) MECHANICAL ENGINEERING SEMESTER TWO EXAMINATION 07/08 FINITE ELEMENT AND DIFFERENCE SOLUTIONS MODULE NO. AME6006 Date: Wednesda 0 Ma 08 Tme: 0:00
More informationPART 8. Partial Differential Equations PDEs
he Islamc Unverst of Gaza Facult of Engneerng Cvl Engneerng Department Numercal Analss ECIV 3306 PAR 8 Partal Dfferental Equatons PDEs Chapter 9; Fnte Dfference: Ellptc Equatons Assocate Prof. Mazen Abualtaef
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationProfessor Terje Haukaas University of British Columbia, Vancouver The Q4 Element
Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to
More informationMODELLING OF ELASTO-STATICS OF POWER LINES BY NEW COMPOSITE BEAM FINITE ELEMENT Bratislava
ODING OF ASTO-STATICS OF POW INS BY NW COPOSIT BA FINIT NT urín Justín 1 rabovský Jura 1 Gogola oman 1 utš Vladmír 1 Paulech Jura 1 1 Insttute of Automotve echatroncs FI STU n Bratslava Ilkovčova 3 812
More informationFrame element resists external loads or disturbances by developing internal axial forces, shear forces, and bending moments.
CE7 Structural Analyss II PAAR FRAE EEET y 5 x E, A, I, Each node can translate and rotate n plane. The fnal dsplaced shape has ndependent generalzed dsplacements (.e. translatons and rotatons) noled.
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationSecond Order Analysis
Second Order Analyss In the prevous classes we looked at a method that determnes the load correspondng to a state of bfurcaton equlbrum of a perfect frame by egenvalye analyss The system was assumed to
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationCHAPTER 4. Vector Spaces
man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear
More informationFinite Difference Method
7/0/07 Instructor r. Ramond Rump (9) 747 698 rcrump@utep.edu EE 337 Computatonal Electromagnetcs (CEM) Lecture #0 Fnte erence Method Lecture 0 These notes ma contan coprghted materal obtaned under ar use
More informationTorsion Stiffness of Thin-walled Steel Beams with Web Holes
Torson Stffness of Thn-walled Steel Beams wth Web Holes MARTN HORÁČEK, JNDŘCH MELCHER Department of Metal and Tmber Structures Brno Unversty of Technology, Faculty of Cvl Engneerng Veveří 331/95, 62 Brno
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationA REVIEW OF ERROR ANALYSIS
A REVIEW OF ERROR AALYI EEP Laborator EVE-4860 / MAE-4370 Updated 006 Error Analss In the laborator we measure phscal uanttes. All measurements are subject to some uncertantes. Error analss s the stud
More informationCHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION
CAPTER- INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr
More informationNUMERICAL RESULTS QUALITY IN DEPENDENCE ON ABAQUS PLANE STRESS ELEMENTS TYPE IN BIG DISPLACEMENTS COMPRESSION TEST
Appled Computer Scence, vol. 13, no. 4, pp. 56 64 do: 10.23743/acs-2017-29 Submtted: 2017-10-30 Revsed: 2017-11-15 Accepted: 2017-12-06 Abaqus Fnte Elements, Plane Stress, Orthotropc Materal Bartosz KAWECKI
More informationLecture 8 Modal Analysis
Lecture 8 Modal Analyss 16.0 Release Introducton to ANSYS Mechancal 1 2015 ANSYS, Inc. February 27, 2015 Chapter Overvew In ths chapter free vbraton as well as pre-stressed vbraton analyses n Mechancal
More informationC PLANE ELASTICITY PROBLEM FORMULATIONS
C M.. Tamn, CSMLab, UTM Corse Content: A ITRODUCTIO AD OVERVIEW mercal method and Compter-Aded Engneerng; Phscal problems; Mathematcal models; Fnte element method. B REVIEW OF -D FORMULATIOS Elements and
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationLecture Note 3. Eshelby s Inclusion II
ME340B Elastcty of Mcroscopc Structures Stanford Unversty Wnter 004 Lecture Note 3. Eshelby s Incluson II Chrs Wenberger and We Ca c All rghts reserved January 6, 004 Contents 1 Incluson energy n an nfnte
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationFUZZY FINITE ELEMENT METHOD
FUZZY FINITE ELEMENT METHOD RELIABILITY TRUCTURE ANALYI UING PROBABILITY 3.. Maxmum Normal tress Internal force s the shear force, V has a magntude equal to the load P and bendng moment, M. Bendng moments
More information1 GSW Iterative Techniques for y = Ax
1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informatione i is a random error
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown
More informationPlease review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
ME 270 Sprng 2014 Fnal Exam NME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS
More informationConstitutive Modelling of Superplastic AA-5083
TECHNISCHE MECHANIK, 3, -5, (01, 1-6 submtted: September 19, 011 Consttutve Modellng of Superplastc AA-5083 G. Gulano In ths study a fast procedure for determnng the constants of superplastc 5083 Al alloy
More informationThe Fundamental Theorem of Algebra. Objective To use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions
5-6 The Fundamental Theorem of Algebra Content Standards N.CN.7 Solve quadratc equatons wth real coeffcents that have comple solutons. N.CN.8 Etend polnomal denttes to the comple numbers. Also N.CN.9,
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationModeling of Dynamic Systems
Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationNovember 5, 2002 SE 180: Earthquake Engineering SE 180. Final Project
SE 8 Fnal Project Story Shear Frame u m Gven: u m L L m L L EI ω ω Solve for m Story Bendng Beam u u m L m L Gven: m L L EI ω ω Solve for m 3 3 Story Shear Frame u 3 m 3 Gven: L 3 m m L L L 3 EI ω ω ω
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More information( ) [ ( k) ( k) ( x) ( ) ( ) ( ) [ ] ξ [ ] [ ] [ ] ( )( ) i ( ) ( )( ) 2! ( ) = ( ) 3 Interpolation. Polynomial Approximation.
3 Interpolaton {( y } Gven:,,,,,, [ ] Fnd: y for some Mn, Ma Polynomal Appromaton Theorem (Weerstrass Appromaton Theorem --- estence ε [ ab] f( P( , then there ests a polynomal
More informationGEOSYNTHETICS ENGINEERING: IN THEORY AND PRACTICE
GEOSYNTHETICS ENGINEERING: IN THEORY AND PRACTICE Prof. J. N. Mandal Department of cvl engneerng, IIT Bombay, Powa, Mumba 400076, Inda. Tel.022-25767328 emal: cejnm@cvl.tb.ac.n Module - 9 LECTURE - 48
More informationANALYSIS OF TIMOSHENKO BEAM RESTING ON NONLINEAR COMPRESSIONAL AND FRICTIONAL WINKLER FOUNDATION
VOL. 6, NO., NOVEMBER ISSN 89-668 6- Asan Research Publshng Network (ARPN). All rghts reserved. ANALYSIS OF TIMOSHENKO BEAM RESTING ON NONLINEAR COMPRESSIONAL AND FRICTIONAL WINKLER FOUNDATION Adel A.
More informationTHE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES
THE SMOOTH INDENTATION OF A CYLINDRICAL INDENTOR AND ANGLE-PLY LAMINATES W. C. Lao Department of Cvl Engneerng, Feng Cha Unverst 00 Wen Hwa Rd, Tachung, Tawan SUMMARY: The ndentaton etween clndrcal ndentor
More informationGEO-SLOPE International Ltd, Calgary, Alberta, Canada Vibrating Beam
GEO-SLOPE Internatonal Ltd, Calgary, Alberta, Canada www.geo-slope.com Introducton Vbratng Beam Ths example looks at the dynamc response of a cantlever beam n response to a cyclc force at the free end.
More informationUniformity of Deformation in Element Testing
Woousng Km, Marc Loen, ruce hadbourn and Joseph Labu Unformt of Deformaton n Element Testng Abstract Unform deformaton s a basc assumpton n element testng, where aal stran tpcall s determned from dsplacement
More informationPlease review the following statement: I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.
ME 270 Fall 2013 Fnal Exam NME (Last, Frst): Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS
More informationWeek 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product
The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationImportant Instructions to the Examiners:
Summer 0 Examnaton Subject & Code: asc Maths (70) Model Answer Page No: / Important Instructons to the Examners: ) The Answers should be examned by key words and not as word-to-word as gven n the model
More informationKinematics in 2-Dimensions. Projectile Motion
Knematcs n -Dmensons Projectle Moton A medeval trebuchet b Kolderer, c1507 http://members.net.net.au/~rmne/ht/ht0.html#5 Readng Assgnment: Chapter 4, Sectons -6 Introducton: In medeval das, people had
More informationCorrelation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
More informationONE DIMENSIONAL TRIANGULAR FIN EXPERIMENT. Technical Advisor: Dr. D.C. Look, Jr. Version: 11/03/00
ONE IMENSIONAL TRIANGULAR FIN EXPERIMENT Techncal Advsor: r..c. Look, Jr. Verson: /3/ 7. GENERAL OJECTIVES a) To understand a one-dmensonal epermental appromaton. b) To understand the art of epermental
More informationBuckling analysis of single-layered FG nanoplates on elastic substrate with uneven porosities and various boundary conditions
IOSR Journal of Mechancal and Cvl Engneerng (IOSR-JMCE) e-issn: 78-1684,p-ISSN: 30-334X, Volume 15, Issue 5 Ver. IV (Sep. - Oct. 018), PP 41-46 www.osrjournals.org Bucklng analyss of sngle-layered FG nanoplates
More informationNew Method for Solving Poisson Equation. on Irregular Domains
Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad
More informationA comprehensive study: Boundary conditions for representative volume elements (RVE) of composites
Insttute of Structural Mechancs A comprehensve study: Boundary condtons for representatve volume elements (RVE) of compostes Srhar Kurukur A techncal report on homogenzaton technques A comprehensve study:
More informationModal Identification of the Elastic Properties in Composite Sandwich Structures
Modal Identfcaton of the Elastc Propertes n Composte Sandwch Structures M. Matter Th. Gmür J. Cugnon and A. Schorderet School of Engneerng (STI) Ecole poltechnque fédérale f de Lausanne (EPFL) Swterland
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationEuler-Lagrange Elasticity: Differential Equations for Elasticity without Stress or Strain
Journal of Appled Mathematcs and Physcs,,, 6- Publshed Onlne December (http://wwwscrporg/ournal/amp) http://dxdoorg/46/amp74 Euler-Lagrange Elastcty: Dfferental Equatons for Elastcty wthout Stress or Stran
More informationINDETERMINATE STRUCTURES METHOD OF CONSISTENT DEFORMATIONS (FORCE METHOD)
INETNTE STUTUES ETHO OF ONSISTENT EFOTIONS (FOE ETHO) If all the support reactons and nternal forces (, Q, and N) can not be determned by usng equlbrum equatons only, the structure wll be referred as STTIY
More informationORIGIN 1. PTC_CE_BSD_3.2_us_mp.mcdx. Mathcad Enabled Content 2011 Knovel Corp.
Clck to Vew Mathcad Document 2011 Knovel Corp. Buldng Structural Desgn. homas P. Magner, P.E. 2011 Parametrc echnology Corp. Chapter 3: Renforced Concrete Slabs and Beams 3.2 Renforced Concrete Beams -
More informationTHE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS OF A TELESCOPIC HYDRAULIC CYLINDER SUBJECTED TO EULER S LOAD
Journal of Appled Mathematcs and Computatonal Mechancs 7, 6(3), 7- www.amcm.pcz.pl p-issn 99-9965 DOI:.75/jamcm.7.3. e-issn 353-588 THE EFFECT OF TORSIONAL RIGIDITY BETWEEN ELEMENTS ON FREE VIBRATIONS
More informationLecture 13 APPROXIMATION OF SECOMD ORDER DERIVATIVES
COMPUTATIONAL FLUID DYNAMICS: FDM: Appromaton of Second Order Dervatves Lecture APPROXIMATION OF SECOMD ORDER DERIVATIVES. APPROXIMATION OF SECOND ORDER DERIVATIVES Second order dervatves appear n dffusve
More informationGrid Generation around a Cylinder by Complex Potential Functions
Research Journal of Appled Scences, Engneerng and Technolog 4(): 53-535, 0 ISSN: 040-7467 Mawell Scentfc Organzaton, 0 Submtted: December 0, 0 Accepted: Januar, 0 Publshed: June 0, 0 Grd Generaton around
More informationInternational Mathematical Olympiad. Preliminary Selection Contest 2012 Hong Kong. Outline of Solutions
Internatonal Mathematcal Olympad Prelmnary Selecton ontest Hong Kong Outlne of Solutons nswers: 7 4 7 4 6 5 9 6 99 7 6 6 9 5544 49 5 7 4 6765 5 6 6 7 6 944 9 Solutons: Snce n s a two-dgt number, we have
More informationPreliminary Design of Moment-Resisting Frames
Prelmnary Desgn of Moment-Resstng Frames Preprnt Aamer Haque Abstract A smple method s developed for prelmnary desgn of moment-resstng frames. Preprnt submtted to Elsever August 27, 2017 1. Introducton
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationIncrease Decrease Remain the Same (Circle one) (2 pts)
ME 270 Sample Fnal Eam PROBLEM 1 (25 ponts) Prob. 1 questons are all or nothng. PROBLEM 1A. (5 ponts) FIND: A 2000 N crate (D) s suspended usng ropes AB and AC and s n statc equlbrum. If θ = 53.13, determne
More informationInterconnect Modeling
Interconnect Modelng Modelng of Interconnects Interconnect R, C and computaton Interconnect models umped RC model Dstrbuted crcut models Hgher-order waveform n dstrbuted RC trees Accuracy and fdelty Prepared
More informationThe equation of motion of a dynamical system is given by a set of differential equations. That is (1)
Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence
More informationSCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.
SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.
More informationMeasurement and Uncertainties
Phs L-L Introducton Measurement and Uncertantes An measurement s uncertan to some degree. No measurng nstrument s calbrated to nfnte precson, nor are an two measurements ever performed under eactl the
More informationModeling and Simulation of a Hexapod Machine Tool for the Dynamic Stability Analysis of Milling Processes. C. Henninger, P.
Smpack User Meetng 27 Modelng and Smulaton of a Heapod Machne Tool for the Dynamc Stablty Analyss of Mllng Processes C. Hennnger, P. Eberhard Insttute of Engneerng project funded by the DFG wthn the framework
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More information1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys
More informationFastener Modeling for Joining Composite Parts
AM-VPD09-006 Fastener Modelng for Jonng Composte Parts Alexander Rutman, Assocate Techncal Fellow, Sprt AeroSystems Chrs Boshers, Stress ngneer, Sprt AeroSystems John Parady, Prncpal Applcaton ngneer,
More informationCHAPTER 9 CONCLUSIONS
78 CHAPTER 9 CONCLUSIONS uctlty and structural ntegrty are essentally requred for structures subjected to suddenly appled dynamc loads such as shock loads. Renforced Concrete (RC), the most wdely used
More informationDESIGN OPTIMIZATION OF CFRP RECTANGULAR BOX SUBJECTED TO ARBITRARY LOADINGS
Munch, Germany, 26-30 th June 2016 1 DESIGN OPTIMIZATION OF CFRP RECTANGULAR BOX SUBJECTED TO ARBITRARY LOADINGS Q.T. Guo 1*, Z.Y. L 1, T. Ohor 1 and J. Takahash 1 1 Department of Systems Innovaton, School
More informationFirst Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.
Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act
More informationPhysics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.
Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays
More informationThe Finite Element Method
The Fnte Element Method GENERAL INTRODUCTION Read: Chapters 1 and 2 CONTENTS Engneerng and analyss Smulaton of a physcal process Examples mathematcal model development Approxmate solutons and methods of
More information