### Exercise 2 - Politecnico di Milano

```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)
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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:
M q  Rq  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 KMN viMN  0
 The buckling analysis includes the effects of preloads (force, moment,
pressure)
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Linear perturbation – modal analysis
4
 Starting from general dynamic equation:
M q  Rq  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

 iR MN  K MN  MN  0
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Exercise 1 - buckling
F
5
F
T
Part: 2D beam planar
Material: E=210 GPa, ν=0.3
Boundary: bottom U1=U2=0; top U1=0
Problem:
1. Perform buckling analysis with 1 step
perform buckling analysis with 2 steps
3. Compare the results btw the analysis
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Exercise 1 – results 1st configuration
1st freq: 1449 Hz
2nd freq: 4852 Hz
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3rd freq: 8504 Hz
Exercise 1 – results 2nd configuration
1st freq: 14.5 Hz
2nd freq: 48.5 Hz
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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
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
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Exercise 2 – definition of frequency and
Natural Frequencies:
Dynamic Response:
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Exercise 2 – definition of harmonic step
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
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```