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

Buckling and harmonic analysis with FEM

E. Tarallo, G. Mastinu

POLITECNICO DI MILANO, Dipartimento di Meccanica

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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)

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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)

QqKqRqM

QqK

00 MNi

MNi

MN vKK

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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.

QqKqRqM

0 qKqM 02 MNMNMN KM

02 MNMNMNMN KRiM

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Exercise 1 - buckling 5

Part: 2D beam planarMaterial: E=210 GPa, ν=0.3Section: circular radius 10 mmLoad F: 1 kNBoundary: bottom U1=U2=0; top U1=0Problem:1. Perform buckling analysis with 1 step2. Add 1 static step with Load T=100 kN and

perform buckling analysis with 2 steps3. Compare the results btw the analysis

T

F F

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Exercise 1 – results 1st configuration 6

1st freq: 1449 Hz 2nd freq: 4852 Hz 3rd freq: 8504 Hz

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Exercise 1 – results 2nd configuration 7

1st freq: 14.5 Hz 2nd freq: 48.5 Hz 3rd freq: 85.04 Hz

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Exercise 2 – Modal analysis 8

Part: 2D beam, L=1000 mmSection: circular, R=10 mmMaterial: E=210 GPa, ν=0.3, ρ=7800 kg/m3

Boundary: encastreAnalysis: 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

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Exercise 2 – definition of frequency and steady-state steps

9

Natural Frequencies:

Dynamic Response:

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Exercise 2 – definition of harmonic step 10

Harmonic Response:

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Exercise 2 – results (1) 11

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Exercise 2 – results (2) 12

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Exercise 2 – results (3) 13

1Hz

100Hz

1000Hz