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Buckling and harmonic analysis with FEM E. Tarallo, G. Mastinu POLITECNICO DI MILANO, Dipartimento di Meccanica Summary 2 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) Es02 Es-06 Linear perturbation - buckling 3 Linear perturbation means impose a δq around the equilibrium position A general dynamic system is described fully by the basic equation: M q Rq K q Q In a general static problem, Abaqus solves the following equation: K q Q The buckling solver is generally used to estimate the critical (bifurcation) load of “stiff” structures; Abaqus solves the following equation: K MN 0 i KMN viMN 0 The buckling analysis includes the effects of preloads (force, moment, pressure) Es02 Es-06 Linear perturbation – modal analysis 4 Starting from general dynamic equation: M q Rq K q Q in the “frequency” analysis, Abaqus solves the following equation: M M q K q 0 2 MN K MN MN 0 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. M 2 MN iR MN K MN MN 0 Es02 Es-06 Exercise 1 - buckling F 5 F T 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: 1. Perform buckling analysis with 1 step 2. Add 1 static step with Load T=100 kN and perform buckling analysis with 2 steps 3. Compare the results btw the analysis Es02 Es-06 Exercise 1 – results 1st configuration 1st freq: 1449 Hz 2nd freq: 4852 Hz Es02 Es-06 6 3rd freq: 8504 Hz Exercise 1 – results 2nd configuration 1st freq: 14.5 Hz 2nd freq: 48.5 Hz Es02 Es-06 7 3rd freq: 85.04 Hz Exercise 2 – Modal analysis T 8 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 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 Es02 Es-06 Exercise 2 – definition of frequency and steady-state steps Natural Frequencies: Dynamic Response: Es02 Es-06 9 Exercise 2 – definition of harmonic step Harmonic Response: Es02 Es-06 10 Exercise 2 – results (1) 11 Es02 Es-06 Exercise 2 – results (2) 12 Es02 Es-06 Exercise 2 – results (3) 13 1Hz 100Hz 1000Hz Es02 Es-06