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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 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
Basic Formulation • Non-Linear Beam Theory • Displacement Field : Euler Beam Theory • Non-linear Strains • Define Degree of Freedom: Nodal displacements and Nodal forces
Displacement Field • Euler beam : displacement for x-axis : displacement for y-axis : displacement for z-axis
Non-linear Strains Non-linear strain Substitute the displacement into
Define Degree of Freedom: Nodal displacements and Nodal forces Nodal displacements Nodal forces
Method of Virtual Work : Distributed force for z-axis : Distributed force for x-axis : Nodal forces : Nodal displacements Internal energy External energy
Internal energy Strains Finally
Method of Virtual Work : extensional stiffness : extensional-bending stiffness : bending stiffness • When the x-axis is taken along the geometric centroidal axis
Method of Virtual Work Energy balance Substitute theapproximation function and into
Finite Element Method (FEM) • Approximatematerial behavior • Use shape function to build K matrix • General one-dimensional shape function Projection N1 N2 Projection
Hermit Shape Function • Rotational degree of freedom • Third-order polynomial Hermit Shape Function First differential Quadratic differential
Direct iteration model Displacement Force f1 f2
Direct iteration model Initial displacement Force step Iterative equation No Tolerance error Yes
Numerical Examples • L = 100 • W= 10 • Young's modulus = 30*10^6 • Node = 9 • Tolerance Error = 10^-3 W L
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.