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