Potential Energy Approach: Applied to Truss Elements

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Potential Energy ApproachApplied to Truss
Elements

• Recall

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Potential Energy of a Spring
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Potential Energy of a Truss Element
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Truss Element Strain Energy
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Strain Energy Axial Loading
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Truss Element Loading
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Potential Energy of a Truss Element
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Using Finite Element NotationStrain Energy Term
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Using Finite Element NotationApplied Load Terms

• Concentrated Forces
• Surface Traction

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Applied Load Terms (cont.)
Body Forces Work equivalent concentrated
forces
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Potential Energy Matrix Form
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Linearly Varying Load
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Mesh Refinement
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Mesh Refinement (cont.)
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Mesh Refinement (cont.)
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Mesh Refinement (cont.)
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Mesh Refinement (cont.)
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Chapter 4. Development of Beam Equations

• Recall
• Truss (or bar) elements are subjected to axial
  tensile or compressive forces only (no bending)
  and deform by change in length
• Beam elements (Chapter 4) - deform by bending
• Frame elements (Chapter 5) combined axial,
  bending, and torsional deformation

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Typical problem
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Review Beam Theory
M moment distribution E Youngs modulus
I Moment of Inertia of cross-section v
transverse displacement V shear load w
distributed load
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Beam Theory (cont.)

• If EI constant and only concentrated loads and
  moments are applied, i.e. w(x)0
• Solution (exact)

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Beam Element - Definitions
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Sign Conventions FEA formulatoin vs. Beam theory
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Steps in the Finite Element Method

• Discretize the region and select element type
• Select a displacement function
• Define the strain/displacement and stress/strain
  relations
• Derive the element equations
• Direct Stiffness Method
• Energy Methods
• Method of Weighted Residuals (Galerkins method)
• Assemble global equations and impose boundary
  conditions
• Solve for unknown nodal displacements
• Solve for element strains and stresses
• Interpret results