DOC PREVIEW
The Direct Stiffness Method Part I

This preview shows page 1-2-22-23 out of 23 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 23 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 23 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 23 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 23 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 23 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

The DirectStiffness MethodPart IIntroduction to FEMIFEM Ch 2 – Slide 1The Direct Stiffness Method (DSM)A democratic method, works the same no matter what the element:Obvious decision: use the truss to teach the DSMImportance: DSM is used by all major commercial FEM codesBar (truss member) element, 2 nodes, 4 DOFsTricubic brick element, 64 nodes, 192 DOFsQuadratic thin-plate element, 6 nodes, 12 DOFsIntroduction to FEMIFEM Ch 2 – Slide 2Model Based Simulation(a simplification of diagrams of Chapter 1)Physical systemModeling + discretization + solution errorDiscretization + solution errorSolution errorDiscrete modelDiscretesolutionMathematical modelIDEALIZATIONDISCRETIZATIONSOLUTIONFEMVERIFICATION & VALIDATIONIntroduction to FEMIFEM Ch 2 – Slide 3Introduction to FEMIdealization Process(a) Physical System(b) Idealized Sytem: FEM-Discretized Mathematical ModelIDEALIZATION;;;;;;jointsupportmemberIFEM Ch 2 – Slide 4;;;;;;;;FEM model:Remove loads & supports:Disassemble:Localize: longeronsbattensdiagonalslongeronsGeneric element:Introduction to FEMDSM: Breakdown StepsIFEM Ch 2 – Slide 5;;;;;;;;Solve for jointdisplacements:Merge:Apply loadsand supports:Formelements: longeronsbattensdiagonalslongeronsGlobalize:Introduction to FEMDSM: Assembly & Solution StepsIFEM Ch 2 – Slide 6The Direct Stiffness Method (DSM) StepsDisconnectionLocalizationMember (Element) FormationGlobalizationMergeApplication of BCsSolutionRecovery of Derived QuantitiesBreakdown(Chapter 2)Assembly & Solution(Chapter 3)Introduction to FEMStarting with: IdealizationIFEM Ch 2 – Slide 7A Physical Plane TrussIntroduction to FEMjointsupportmemberToo complicated to do by hand. We will use a simpler one to illustrate DSM stepsTypical of those used for building roofs and short span bridges.IFEM Ch 2 – Slide 8The Example Truss: Physical andPin-Jointed IdealizationIntroduction to FEM;;;;123Idealization as Pin-Jointed Trussand FEM DiscretizationIFEM Ch 2 – Slide 9The Example Truss - FEM Model:Nodes, Elements and DOFsIntroduction to FEM45o45o1(0,0)2(10,0)3(10,10)xyx3x3f , uy2y2f , uy1y1f , ux2x2f , ux1x1f , uy3y3f , u(1)(2)(3)(3)(3)(3)(3)E = 100, A = 2 2, L = 10 2, ρ = 3/20 (1)(1)(1)(1)E = 50, A = 2, L = 10, ρ = 1/5 (2)(2)(2)(2)E = 50, A = 1, L = 10, ρ = 1/5 IFEM Ch 2 – Slide 10The Example Truss - FEM Model BCs:Applied Loads and Supports (Saved for Last)Introduction to FEM;;;;;;;;f = 2x3f = 1y3123xy(1)(2)(3)IFEM Ch 2 – Slide 11Master (Global) Stiffness Equationsf =fx1fy1fx2fy2fx3fy3u =ux1uy1ux2uy2ux3uy3fx1fy1fx2fy2fx3fy3=Kx1x1Kx1y1Kx1x2Kx1y2Kx1x3Kx1y3Ky1x1Ky1y1Ky1x2Ky1y2Ky1x3Ky1y3Kx2x1Kx2y1Kx2x2Kx2y2Kx2x3Kx2y3Ky2x1Ky2y1Ky2x2Ky2y2Ky2x3Ky2y3Kx3x1Kx3y1Kx3x2Kx3y2Kx3x3Kx3y3Ky3x1Ky3y1Ky3x2Ky3y2Ky3x3Ky3y3ux1uy1ux2uy2ux3uy3f = KuNodal forcesMaster stiffness matrix NodaldisplacementsLinear structure:orIntroduction to FEMIFEM Ch 2 – Slide 12Member (Element) Stiffness Equations¯f = K¯u¯fxi¯fyi¯fxj¯fyj=¯Kxixi¯Kxiyi¯Kxixj¯Kxiyj¯Kyixi¯Kyiyi¯Kyixj¯Kyiyj¯Kxjxi¯Kxjyi¯Kxjxj¯Kxjyj¯Kyjxi¯Kyjyi¯Kyjxj¯Kyjyj¯uxi¯uyi¯uxj¯uyjIntroduction to FEM_IFEM Ch 2 – Slide 13First Two Breakdown Steps:Disconnection and Localization Introduction to FEMThese steps are conceptual (not actually programmed as part of the DSM)123(3)(1)(2)y_(1)x_(1)y_(2)x_(2)y_(3)x_(3);;;;123Remove loads and supports,and disconnect pinsxyIFEM Ch 2 – Slide 14The 2-Node Truss (Bar) Element Introduction to FEMiijjdLxEquivalent spring stiffness−FFf , u xixi__f , u xjxj__f , u yjyj__f , u yiyi__y__sk = EA / LIFEM Ch 2 – Slide 15Truss (Bar) Element Formulation by Mechanics of Materials (MoM)K =EAL10−1000 00−10 1000 00F= ksd =EALdF =¯fxj=−¯fxi,,d =¯uxj−¯uxi¯fxi¯fyi¯fxj¯fyj=EAL10−1000 00−10 1000 00¯uxi¯uyi¯uxj¯uyjExercise 2.3Element stiffnessmatrix in local coordinatesElement stiffnessequations in local coordinatesIntroduction to FEMfrom whichIFEM Ch 2 – Slide 16Where We Are So Far in the DSMDisconnectionLocalizationMember (Element) FormationGlobalizationMergeApplication of BCsSolutionRecovery of Derived QuantitiesBreakdown(Chapter 2)Assembly & Solution(Chapter 3)Introduction to FEMwe are done with this ...we finish Chapter 2 withIFEM Ch 2 – Slide 17¯uxi= uxic + uyis, ¯uyi=−uxis + uyicγ¯uxj= uxjc + uyjs, ¯uyj=−uxjs + uyjcγNode displacements transform asixyc = cosϕs = sinϕin whichGlobalization: Displacement TransformationIntroduction to FEMxyjϕuxiuyiuxjuyjuyi___uxi_uxj_uyj_IFEM Ch 2 – Slide 18Displacement Transformation (cont'd)In matrix form oreeeNote:global on RHS,local on LHSIntroduction to FEMu = T uuuuu=c−s00sc 0000c−s00scuxixiuyiyiuxjxjuyjyj_____IFEM Ch 2 – Slide 19Globalization: Force TransformationNode forces transform asorxyijϕfxifyifxjfyjfxifyifxjfyj=c−s00sc0000c−s00scNote:global on LHS,local on RHSIntroduction to FEMfxifyifxjfyj____fyi_fxi_fxj_fyj_f = (T ) feeTe_IFEM Ch 2 – Slide 20ue==Teueuefe= TeTfeKe= TeT¯¯¯fe¯KeT¯KeeKe=EeAeLec2sc −c2−scsc s2−sc −s2−c2−sc c2sc−sc −s2sc s2Globalization: Congruential Transformationof Element Stiffness MatricesExercise 2.8Introduction to FEM(())IFEM Ch 2 – Slide 21The Example Truss - FEM Model(Recalled for Convenience)Insert the geometric &physical properties ofthis model intothe globalized memberstiffness equationsIntroduction to FEM45o45o1(0,0)2(10,0)3(10,10)xyx3x3f , uy2y2f , uy1y1f , ux2x2f , ux1x1f , uy3y3f , u(1)(2)(3)(3)(3)(3)(3)E = 100, A = 2 2, L = 10 2, ρ = 3/20 (1)(1)(1)(1)E = 50, A = 2, L = 10, ρ = 1/5 (2)(2)(2)(2)E = 50, A = 1, L = 10, ρ = 1/5 IFEM Ch 2 – Slide 22We Obtain the Globalized Element Stiffness Equations of the Example Trussfx1fy1fx2fy2= 1010−100000−10100000ux1uy1ux2uy2fx2fy2fx3fy3= 50000010−100000 −10 1ux2uy2ux3uy3fx1fy1fx3fy3=


The Direct Stiffness Method Part I

Download The Direct Stiffness Method Part I
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view The Direct Stiffness Method Part I and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view The Direct Stiffness Method Part I 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?