Introduction to Finite Introduction to Finite Element Analysis for Element Analysis for

Structure DesignStructure Design

Dr. A. Sherif El-GizawyDr. A. Sherif El-Gizawy

l

ddl

A

F

Elasticity PrinciplesElasticity Principles

FF : Applied Force & A : Area: Applied Force & A : Area

l : Initial Lengthl : Initial Length

Stress = Stress = = F/ A = F/ A

dl= displacement (deformation)dl= displacement (deformation)

Strain = Strain = = dl/l = dl/l

Elastic 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/E

Plastic 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))))

WhereWherex x :: normal strain along normal strain along xx direction direction : Poisson Ration: Poisson Ration

Shear Strain, Shear Strain, xyxy = = xyxy / G/ G

xyxy : Shear Stress: Shear StressG : Shear Modulus of ElasticityG : Shear Modulus of Elasticity

Effective 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+ y

2+ z2))1/2

2

1

3

2

FEM Solution for Structural DesignFEM Solution for Structural Design

l K

l

l

l

A

F

F

Fsp = 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 ll

This is an analogy to spring This is an analogy to spring Force withForce with

F = KeqF = Keq XX ll

A x E/l = Element StiffnessA x E/l = Element Stiffness = Keq= Keq

F = 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 FEA

Developing a Model for Finite Element AnalysisDeveloping a Model for Finite Element Analysis

Line ElementLine Element

Two-dimensional ElementsTwo-dimensional Elements

Three-dimensional ElementsThree-dimensional Elements

Axisymmmetric ElementAxisymmmetric Element

Step 1. Discretize and Select Element TypesStep 1. Discretize and Select Element Types

Divide 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 Analysis

Overall steps in FEA software

Step 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 EquationsEquations

Step 6.Step 6. Solve for the Unknown DisplacementsSolve for the Unknown Displacements

Step 7.Step 7. Solve for the Element Strains and StressesSolve for the Element Strains and Stresses

Step 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 elements

A disaster waiting to happen using first order elements

Approximation of stress function in a model

A mesh of solid tetrahedral (4 nodes) h-elements

A mesh of tetrahedral p-elements produced by MECHANICA

Two common convergence measures using p-elements

Finite Element Analysis Finite Element Analysis by Pro- Mechanicaby Pro- Mechanica

Steps in FEA using Pro-MechanicaSteps in FEA using Pro-Mechanica Step 1: Draw part in Pro-EngineerStep 1: Draw part in Pro-Engineer Step 2: Start Pro-MechanicaStep 2: Start Pro-Mechanica Step 3: Choose the Model TypeStep 3: Choose the Model Type Step 4: Apply the constraintsStep 4: Apply the constraints Step 5: Apply the loadsStep 5: Apply the loads Step 6: Assign the materialStep 6: Assign the material Step 7: Run the AnalysisStep 7: Run the Analysis Step 8: View the results by post-processingStep 8: View the results by post-processing

Step 1: Creation of the partStep 1: Creation of the part

Use Protrusion by Sweep to create this Use Protrusion by Sweep to create this part (bar.prt)part (bar.prt)

Step 2: Starting Pro-MechanicaStep 2: Starting Pro-Mechanica

In Pro-Engineer window, go to In Pro-Engineer window, go to ApplicationsApplications Mechanica to start Pro- Mechanica to start Pro-Mechanica. Mechanica.

The part (bar.prt) will be loaded in Pro-The part (bar.prt) will be loaded in Pro-Mechanica with a new set of icons for Mechanica with a new set of icons for Structural, thermal AnalysisStructural, thermal Analysis

Step 3: Choosing the model typeStep 3: Choosing the model type

In Mechanica menu, select In Mechanica menu, select Structure Structure Model Model Model Type Model Type Four different models can be created:Four different models can be created:

3D Model3D Model Plane StressPlane Stress Plane StrainPlane Strain 2D Axisymmetric2D Axisymmetric

We will select 3D ModelWe will select 3D Model

Step 4: Applying the ConstraintsStep 4: Applying the Constraints

Create a new constraint by Create a new constraint by ModelModel Constraints Constraints New New SurfaceSurface Give a name for the constraints Give a name for the constraints

(fixed_face) and select the surface to be (fixed_face) and select the surface to be constrainedconstrained

Specify the constraints (in our case will be Specify the constraints (in our case will be fixed for all degrees of freedom)fixed for all degrees of freedom)

Preview and press OkPreview and press Ok

Step 5: Applying the loadsStep 5: Applying the loads

Similar to Constraints, create a new load Similar to Constraints, create a new load by Model by Model Load Load New New Surface Surface

Give a name for the applied load (endload)Give a name for the applied load (endload) Select the surface where the load will be Select the surface where the load will be

appliedapplied Specify the loads (Fx:500, Fy:-250, Fz:0)Specify the loads (Fx:500, Fy:-250, Fz:0) Preview and press OkPreview and press Ok

Step 6: Assigning the materialStep 6: Assigning the material

Model Model Materials Materials A window will pop up with the list of Pro-A window will pop up with the list of Pro-

Mechanica materials. Add the required Mechanica materials. Add the required material and then assign the material to material and then assign the material to the part.the part.

Click on Edit if any change in material Click on Edit if any change in material properties are to be made.properties are to be made.

Press OkPress Ok

Modeled part with constraints and Modeled part with constraints and loadsloads

Step 7: Running the AnalysisStep 7: Running the Analysis In Mechanica menu, select AnalysisIn Mechanica menu, select Analysis Select File Select File New Static in “Analysis and New Static in “Analysis and

Design Studies” dialog box and give a name for Design Studies” dialog box and give a name for the analysis (bar).the analysis (bar).

The constraints and loads are automatically The constraints and loads are automatically loaded. loaded.

In Convergence tab, select Quick Check to In Convergence tab, select Quick Check to check for errors and then select Multi-pass check for errors and then select Multi-pass adaptive for the reliable and accurate results. adaptive for the reliable and accurate results. Change the order of the polynomial and Change the order of the polynomial and percentage of convergence as required.percentage of convergence as required.

Finally, click on Run icon to start the analysis Finally, click on Run icon to start the analysis (click on Display Study Status to view the current (click on Display Study Status to view the current status and completion of the analysis)status and completion of the analysis)

Step 8: Viewing the resultsStep 8: Viewing the results

For post-processing, select Results from Pro-For post-processing, select Results from Pro-Mechanica windowMechanica window

A new window will open, and click on “Insert a A new window will open, and click on “Insert a New Definition” icon. In the dialog box, select the New Definition” icon. In the dialog box, select the folder where the analysis is saved.folder where the analysis is saved.

Select Fringe as Display type, Stress as Select Fringe as Display type, Stress as Quantity and von-mises as the stress Quantity and von-mises as the stress component to displaycomponent to display

Similarly, other quantities can be displayed in Similarly, other quantities can be displayed in one window.one window.

Post-processing ResultsPost-processing Results