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Buckling and harmonic analysis with FEM E. Tarallo, G. Mastinu

Buckling and harmonic analysis with FEM E. Tarallo, G. Mastinu POLITECNICO DI MILANO, Dipartimento di Meccanica. Summary. Subjects covered in this tutorial An introduction to linear perturbation analysis An introduction to buckling analysis

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Buckling and harmonic analysis with FEM E. Tarallo, G. Mastinu

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  1. Buckling and harmonic analysis with FEM E. Tarallo, G. Mastinu POLITECNICO DI MILANO, Dipartimento di Meccanica

  2. Summary • Subjects covered in this tutorial • An introduction to linear perturbation analysis • An introduction to buckling analysis • An introduction to modal analysis (frequency and complex) • A guided example to evaluate the harmonic response of a simple structure • Other few exercises (to include in exercises-book)

  3. Linear perturbation - buckling • Linear perturbation means impose a δq around the equilibrium position • A general dynamic system is described fully by the basic equation: • In a general static problem, Abaqus solves the following equation: • The buckling solver is generally used to estimate the critical (bifurcation) load of “stiff” structures; Abaqus solves the following equation: • The buckling analysis includes the effects of preloads (force, moment, pressure)

  4. Linear perturbation – modal analysis • Starting from general dynamic equation: • in the “frequency” analysis, Abaqus solves the following equation: • The “frequency” analysis doesn’t include the effects of loads and damping • Following the “frequency” analysis is possible to perform a “complex” analysis where the damping (structural and contact effects) is taken into account.

  5. Exercise 1 - buckling F F Part: 2D beam planar Material: E=210 GPa, ν=0.3 Section: circular radius 10 mm Load F: 1 kN Boundary: bottom U1=U2=0; top U1=0 Problem: Perform buckling analysis with 1 step Add 1 static step with Load T=100 kN and perform buckling analysis with 2 steps Compare the results btw the analysis T

  6. Exercise 1 – results 1st configuration 1st freq: 1449 Hz 2nd freq: 4852 Hz 3rd freq: 8504 Hz

  7. Exercise 1 – results 2nd configuration 1st freq: 14.5 Hz 2nd freq: 48.5 Hz 3rd freq: 85.04 Hz

  8. Exercise 2 – Modal analysis Part: 2D beam, L=1000 mm Section: circular, R=10 mm Material: E=210 GPa, ν=0.3, ρ=7800 kg/m3 Boundary: encastre Analysis: Frequency, Steady-state dynamic, Dynamic-Implicit T 1) Frequency analysis: find first 5 natural frequency 2) Steady-state dynamic: T=-1 kN; frequency range=[1,800] Hz 3) Harmonic response: T=-1000sin(ft) where f=1,100,1000 Hz

  9. Exercise 2 – definition of frequency and steady-state steps Natural Frequencies: Dynamic Response:

  10. Exercise 2 – definition of harmonic step Harmonic Response:

  11. Exercise 2 – results (1)

  12. Exercise 2 – results (2)

  13. Exercise 2 – results (3) 1Hz 100Hz 1000Hz

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