Introduction to Finite Element Analysis for Structure DesignSlide 2Slide 33D Stress-Strain RelationshipSlide 5FEM Solution for Structural DesignSlide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Finite Element Analysis by Pro- MechanicaSteps in FEA using Pro-MechanicaStep 1: Creation of the partStep 2: Starting Pro-MechanicaStep 3: Choosing the model typeStep 4: Applying the ConstraintsStep 5: Applying the loadsStep 6: Assigning the materialModeled part with constraints and loadsStep 7: Running the AnalysisStep 8: Viewing the resultsPost-processing ResultsIntroduction to Finite Introduction to Finite Element Analysis for Element Analysis for Structure DesignStructure DesignDr. A. Sherif El-GizawyDr. A. Sherif El-Gizawyl ddlAFElasticity PrinciplesElasticity PrinciplesFF : Applied Force & A : Area: Applied Force & A : Area l : Initial Lengthl : Initial LengthStress = Stress =  = F/ A = F/ Adl= displacement (deformation)dl= displacement (deformation)Strain = Strain =  = dl/l = dl/lElastic Deformation ZoneStress-Strain RelationStress-Strain Relation (Hooke’s Law) (Hooke’s Law) Modulus of Elasticity = Modulus of Elasticity = E = E = // = E * = E * xx  in the x-direction in the x-direction = = /E/EPlastic Work (Deformation Energy)Plastic Work (Deformation Energy) Plastic Work/ Unit Volume =Plastic Work/ Unit Volume = dW = F x dl/Volume dW = F x dl/Volume =F x dl/ (A x l) = =F x dl/ (A x l) =  xx dW = dW =  xx 3D Stress-Strain Relationship3D Stress-Strain Relationshipxx = 1/E*((= 1/E*((xx- - ((yy + + zz))))WhereWherex x :: normal strain along normal strain along xx direction direction : Poisson Ration: Poisson RationShear Strain, Shear Strain, xyxy = = xyxy / G/ Gxyxy : Shear Stress: Shear StressG : Shear Modulus of ElasticityG : Shear Modulus of ElasticityEffective Stress (Von-Mises)Effective Stress (Von-Mises) = ((x- y)2 + (y- z)2 + (z- x)2)1/2 when when  reaches a certain value (yield stress), reaches a certain value (yield stress), the applied stress state will cause yieldingthe applied stress state will cause yielding Effective StrainEffective Strain = ( (x2+ y2+ z2))1/22132FEM Solution for Structural DesignFEM Solution for Structural DesignlK ll lAFFFsp = K x Fsp = K x ll = E = E XX  (Hooke’s Law) (Hooke’s Law)F = A F = A XX(E (E XX ) = A ) = A XX E (E (l /l)l /l)F = (A F = (A XX E/l) E/l) XX llThis is an analogy to spring This is an analogy to spring Force withForce withF = KeqF = Keq XX llA x E/l = Element StiffnessA x E/l = Element Stiffness = Keq= KeqF = KeqF = Keq XX llElement Stiffness = Element Stiffness = A x E/l = K A x E/l = K eqeqThe Applied force F is given.The Applied force F is given.Deformation (deflection or displacement)Deformation (deflection or displacement)l = F / Kl = F / Keqeq   = Strain = = Strain = l /l (calculated)l /l (calculated)  = Stress = E x = Stress = E x Introduction to the Finite Element MethodIntroduction to the Finite Element Method•Typically, for the structural stress analysis, it is required to determine the stresses and deformation (strains) throughout the structure which is in equilibrium and is subjected to applied loads.•The finite element method involves modeling of the structure using small units (finite elements).•A displacement function is associated with each finite element. The followings are the steps used in finite element method. This will be followed by illustration of the application of these steps on springs and plane stress cases.The problem to be solved is specified in a) the physical domain and b) the discretized domain used by FEADeveloping a Model for Finite Element AnalysisDeveloping a Model for Finite Element AnalysisLine ElementLine ElementTwo-dimensional ElementsTwo-dimensional ElementsThree-dimensional ElementsThree-dimensional ElementsAxisymmmetric ElementAxisymmmetric ElementStep 1. Discretize and Select Element TypesStep 1. Discretize and Select Element TypesDivide the structure into an equivalent system of finite elements with associated nodes.The simplest line elements, Fig.1.a has two nodes, one at each end. The basic two-dimensional elements, Fig. 1.b are loaded by forces in their own plane (plane stress). They are triangular or quadrilateral elements. The common three dimensional elements, Fig.1.c, are tetrahedral and hexahedral (brick) elements. They are used to perform three dimensional stress analysis in 3-D solid bodies.Developing a Model for Finite Element AnalysisOverall steps in FEA softwareStep 2. Select a Displacement FunctionStep 2. Select a Displacement Function•Choose a displacement function within each element using the nodal values of the element. Linear, quadratic, and polynomials are frequently used functions.Step 3. Define the Stress/Strain Step 3. Define the Stress/Strain RelationshipsRelationships= dl/l= dl/l = E= EStep 4.Step 4. Derive the Element Stiffness Matrix and Equations Derive the Element Stiffness Matrix and Equations•The stiffness matrix and element equations relating nodal forces and displacements are obtained using force equilibrium conditions or the principle of minimum potential energy.Step 5.Step 5.Assemble the Element Equations to obtain the Global Assemble the Element Equations to obtain the Global EquationsEquationsStep 6.Step 6.Solve for the Unknown DisplacementsSolve for the Unknown DisplacementsStep 7.Step 7.Solve for the Element Strains and StressesSolve for the Element Strains and StressesStep 8.Step 8. Interpret the Results Interpret the Results•The final goal is to interpret and analyze the results for use in the design process.Von Mises stress in ¼ model of thin plate under tension using 1st order elementsA disaster waiting to happen using first order elementsApproximation of stress function in a modelA mesh of solid tetrahedral (4 nodes) h-elementsA mesh of tetrahedral p-elements produced by MECHANICATwo common convergence measures using p-elementsFinite Element Analysis Finite Element Analysis by Pro- Mechanicaby Pro- MechanicaSteps in FEA using Pro-MechanicaSteps in FEA using Pro-MechanicaStep 1: Draw part in