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Beam Design for Geometric Nonlinearities

Beam Design for Geometric Nonlinearities. Jordan Radas Kantaphat Sirison Wendy Zhao. Premise. Large deflection Linear assumptions no longer apply Is necessary form many real life applications. Linear. Design Overview. Nonlinear. Geometric Nonlinearity Assumptions. Large deformation

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Beam Design for Geometric Nonlinearities

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  1. Beam Design for Geometric Nonlinearities Jordan Radas KantaphatSirison Wendy Zhao

  2. Premise Large deflection Linear assumptions no longer apply Is necessary form many real life applications

  3. Linear Design Overview • Nonlinear

  4. Geometric Nonlinearity Assumptions • Large deformation • Plane cross section remains plane • Linear elastic material • Constant cross section

  5. Kinematics Location of particle at deformed configuration relative to displacement and original configuration

  6. Kinematics Green Lagrange Strain Tensor Characterize axial strain, shear strain and curvature in terms of the derivatives of the displacement

  7. Strain Displacement Matrix The components of the strain displacement matrix can be determined explicitly by differentiation. [B] where

  8. Tangent Stiffness Matrix R [K][d] Through discretization and linearization of the weak form

  9. Newton-Raphson Method Load Displacement

  10. Newton-Raphson Method

  11. Restoring load Corresponds to element internal loads of current stress state. Definition of deformation gradient From right polar decomposition theorem Spatial Decomposition

  12. Incremental Approximation With From With With Evaluated at midpoint geometry

  13. Nonlinear solution levels Load steps: Adjusting the number of load steps account for: abrupt changes in loading on a structure specific point in time of response desired Substeps: Application of load in incremental substeps to obtain a solution within each load step Equilibrium Iterations: Set maximum number of iterations desired

  14. Substeps Equilibrium iterations performed until convergence Opportunity cost of accuracy versus time Automatic time stepping feature Chooses the size and number of substeps to optimize Bisections method Activates to restart solution from last converged step if a solution does not converge within a substep

  15. Modified Newton-Raphson Incremental Newton-Raphson Initial-Stiffness Newton-Raphson

  16. Displacement iteration As opposed to residual iteration

  17. Ansys Features Predictor Line Search Option

  18. Ansys Features Adaptive Descent

  19. Design challenge: Olympic diving board • L = 96in • b = 19.625in • h = 1.625in • P = -2500lbs • Al 2024 – T6 (aircraft alloy) • E = 10500ksi • v = .33 • Yield Strength = 50ksi

  20. Solid Beam: Linear/nonlinear

  21. Optimization problem • ANSYS Goal Driven Optimization is used to create a geometry where hole diameter is the design variable. • Goals include minimizing volume and satisfying yield strength criterion.

  22. Optimization and element technology • Optimization samples points in the user specified design space. • The number of sampling points is minimized using statistical methods and an FEA calculation is made for each sample. • Samples are chosen based on goals set for output variables, such as volume and safety factor.

  23. Optimization results

  24. Conclusion • Analysis serves as a proof of concept that real-world situations involving large structural displacements benefit from nonlinear modeling considerations • Extra computing power and time is worth it • Recommendations/suggestions

  25. Questions?

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