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Comparison of FEM and Meshfree Method for Non-linear Beam Analyses

Comparison of FEM and Meshfree Method for Non-linear Beam Analyses. Speaker: Tsu-Han Chang Adviser: Dr. Pai-Chen Guan. Outline. Basic Formulation Non-Linear Beam Theory Method of Virtual Work (or Galerkin Weak From) Finite Element Method (FEM) Direct iteration model Numerical Examples.

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Comparison of FEM and Meshfree Method for Non-linear Beam Analyses

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  1. Comparison of FEM and Meshfree Method for Non-linear Beam Analyses Speaker:Tsu-Han Chang Adviser: Dr. Pai-Chen Guan

  2. Outline • Basic Formulation • Non-Linear Beam Theory • Method of Virtual Work (or Galerkin Weak From) • Finite Element Method (FEM) • Direct iteration model • Numerical Examples

  3. Basic Formulation • Non-Linear Beam Theory • Displacement Field : Euler Beam Theory • Non-linear Strains • Define Degree of Freedom: Nodal displacements and Nodal forces

  4. Displacement Field • Euler beam : displacement for x-axis : displacement for y-axis : displacement for z-axis

  5. Non-linear Strains Non-linear strain Substitute the displacement into

  6. Define Degree of Freedom: Nodal displacements and Nodal forces Nodal displacements Nodal forces

  7. Method of Virtual Work : Distributed force for z-axis : Distributed force for x-axis : Nodal forces : Nodal displacements Internal energy External energy

  8. Internal energy Strains Finally

  9. Method of Virtual Work : extensional stiffness : extensional-bending stiffness : bending stiffness • When the x-axis is taken along the geometric centroidal axis

  10. Method of Virtual Work Energy balance Substitute theapproximation function and into

  11. K Matrix

  12. Finite Element Method (FEM) • Approximatematerial behavior • Use shape function to build K matrix • General one-dimensional shape function Projection N1 N2 Projection

  13. Hermit Shape Function • Rotational degree of freedom • Third-order polynomial Hermit Shape Function First differential Quadratic differential

  14. Direct iteration model Displacement Force f1 f2

  15. Direct iteration model Initial displacement Force step Iterative equation No Tolerance error Yes

  16. Numerical Examples • L = 100 • W= 10 • Young's modulus = 30*10^6 • Node = 9 • Tolerance Error = 10^-3 W L

  17. FEM Linear

  18. FEM Non-Linear

  19. References • Chen, J. S., C. H. Pan, et al. (1996). "Reproducing kernel particle methods for large deformation analysis of non-linear structures." Computer Methods in Applied Mechanics and Engineering139(1-4): 195-227. • Chen, J. S., C. T. Wu, et al. (2001). "A stabilized conforming nodal integration for Galerkin mesh-free methods." International Journal for Numerical Methods in Engineering50(2): 435-466. • Liu, W. K., S. Jun, et al. (1995). "REPRODUCING KERNEL PARTICLE METHODS." International Journal for Numerical Methods in Fluids20(8-9): 1081-1106.

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