Finite Element Primer for Engineers: Part 2

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• Finite Element Primer for Engineers Part 2
• Mike Barton S. D. Rajan

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Contents

• Introduction to the Finite Element Method (FEM)
• Steps in Using the FEM (an Example from Solid
  Mechanics)
• Examples
• Commercial FEM Software
• Competing Technologies
• Future Trends
• Internet Resources
• References

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FEM Applied to Solid Mechanics Problems

• A FEM model in solid mechanics can be thought of
  as a system of assembled springs. When a load is
  applied, all elements deform until all forces
  balance.
• F Kd
• K is dependant upon Youngs modulus and Poissons
  ratio, as well as the geometry.
• Equations from discrete elements are assembled
  together to form the global stiffness matrix.
• Deflections are obtained by solving the assembled
  set of linear equations.
• Stresses and strains are calculated from the
  deflections.

Create elements of the beam
Nodal displacement and forces
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Classification of Solid-Mechanics Problems
Analysis of solids
Dynamics
Static
Advanced
Elementary
Stress Stiffening
Behavior of Solids
Large Displacement
Geometric
Instability
Linear
Nonlinear
Fracture
Plasticity
Material
Viscoplasticity
Geometric Classification of solids
Skeletal Systems 1D Elements
Plates and Shells 2D Elements
Solid Blocks 3D Elements
Trusses Cables Pipes
Plane Stress Plane Strain Axisymmetric Plate
Bending Shells with flat elements Shells with
curved elements
Brick Elements Tetrahedral Elements General
Elements
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Governing Equation for Solid Mechanics Problems

• Basic equation for a static analysis is as
  follows

• K u Fapp Fth Fpr Fma Fpl
  Fcr Fsw Fld
• K total stiffness matrix
• u nodal displacement
• Fapp applied nodal force load vector
• Fth applied element thermal load vector
• Fpr applied element pressure load vector
• Fma applied element body force vector
• Fpl element plastic strain load vector
• Fcr element creep strain load vector
• Fsw element swelling strain load vector
• Fld element large deflection load vector

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Six Steps in the Finite Element Method

• Step 1 - Discretization The problem domain is
  discretized into a collection of simple shapes,
  or elements.
• Step 2 - Develop Element Equations Developed
  using the physics of the problem, and typically
  Galerkins Method or variational principles.
• Step 3 - Assembly The element equations for each
  element in the FEM mesh are assembled into a set
  of global equations that model the properties of
  the entire system.
• Step 4 - Application of Boundary Conditions
  Solution cannot be obtained unless boundary
  conditions are applied. They reflect the known
  values for certain primary unknowns. Imposing
  the boundary conditions modifies the global
  equations.
• Step 5 - Solve for Primary Unknowns The modified
  global equations are solved for the primary
  unknowns at the nodes.
• Step 6 - Calculate Derived Variables Calculated
  using the nodal values of the primary variables.

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Process Flow in a Typical FEM Analysis
Problem Definition
Analysis and design decisions
Stop
Start

• Pre-processor
• Reads or generates nodes and elements (ex ANSYS)
• Reads or generates material property data.
• Reads or generates boundary conditions (loads and
  constraints.)

• Processor
• Generates element shape functions
• Calculates master element equations
• Calculates transformation matrices
• Maps element equations into global system
• Assembles element equations
• Introduces boundary conditions
• Performs solution procedures

• Post-processor
• Prints or plots contours of stress components.
• Prints or plots contours of displacements.
• Evaluates and prints error bounds.

Step 6
Step 1, Step 4
Steps 2, 3, 5
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Step 1 Discretization - Mesh Generation
surface model
airfoil geometry (from CAD program)
mesh generator
ET,1,SOLID45 N, 1, 183.894081 ,
-.770218637 , 5.30522740 N, 2, 183.893935
, -.838009645 , 5.29452965 . . TYPE, 1 E,
1, 2, 80, 79, 4, 5, 83, 82 E,
2, 3, 81, 80, 5, 6, 84,
83 . . .
meshed model
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Step 4 Boundary Conditions for a Solid Mechanics
Problem

• Displacements ??DOF constraints usually specified
  at model boundaries to define rigid supports.
• Forces and Moments ??Concentrated loads on nodes
  usually specified on the model exterior.
• Pressures ??Surface loads usually specified on
  the model exterior.
• Temperatures ??Input at nodes to study the effect
  of thermal expansion or contraction.
• Inertia Loads ??Loads that affect the entire
  structure (ex acceleration, rotation).

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Step 4 Applying Boundary Conditions (Thermal
Loads)
Nodes from FE Modeler
bf, 1,temp, 149.77 bf, 2,temp,
149.78 . . . bf, 1637,temp, 303.64 bf,
1638,temp, 303.63
Temp mapper
Thermal Soln Files
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Step 4 Applying Boundary Conditions (Other Loads)

• Speed, temperature and hub fixity applied to
  sample problem.
• FE Modeler used to apply speed and hub constraint.

antype,static omega,104003.1416/30 d,1,all,0,0,57
,1