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

2. Es02 Es-06 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)

3. 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:  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. Es02 Es-06 Linear perturbation – modal analysis 4  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. Es02 Es-06 Exercise 1 - buckling 5 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 T F F

6. Es02 Es-06 Exercise 1 – results 1st configuration 6 1 st freq: 1449 Hz2 nd freq: 4852 Hz 3 rd freq: 8504 Hz

7. Es02 Es-06 Exercise 1 – results 2nd configuration 7 1 st freq: 14.5 Hz2 nd freq: 48.5 Hz 3 rd freq: 85.04 Hz

8. Es02 Es-06 Exercise 2 – Modal analysis 8 Part: 2D beam, L=1000 mm Section : circular, R=10 mm Material: E=210 GPa, ν=0.3, ρ=7800 kg/m 3 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 T

9. Es02 Es-06 Exercise 2 – definition of frequency and steady-state steps 9 Natural Frequencies: Dynamic Response:

10. Es02 Es-06 Exercise 2 – definition of harmonic step 10 Harmonic Response:

11. Es02 Es-06 Exercise 2 – results (1) 11

12. Es02 Es-06 Exercise 2 – results (2) 12

13. Es02 Es-06 Exercise 2 – results (3) 13 1Hz 100Hz 1000Hz