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Uncertainty Quantification in the Comparison of Structural Criterions of Failure Yaşar Yanık
Advisor: Samuel da Silva Co-Advisor: Americo Barbosa da Cunha Junior Universidade Estadual Paulista - UNESP Faculdade de Engenharia de Ilha Solteira Departamento de Engenharia Mecânica
Ilha Solteira, SP
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1
Introduction
2
Objectives
3
Maximum Entropy Principle
4
The Stochastic Explanation for the Simple Deflection Problem
5
Probabilistic design results
6
Conclusion and Future Work
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Introduction
The aim behind failure theory is the investigation of predicting the circumstances under which solid materials under the processing of external loads A model can have uncertainties on its forecasts, because of conceivable wrong presumptions made during its originations
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Objectives
A critical comparison between the Tresca and Von Mises failure criterions Uncertainty quantification Monte Carlo Simulation Parametric probabilistic approach Probabilistic design in ANSYS APDL
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Stochastic Explanation for Failure Criterions
The stochastic version of the stress tensor "
σ=
σx τxy τxy σy
#
Where the parameters σx and σy are the normal stresses and τxy is shear stress respectively. In this part of the work, the value of σy is given as 2 Mpa
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Stochastic Explanation for Failure Criterions
Afterward, random values are used by applying uncertainty quantification method considering τxy and σx . These values were examined in the the non-standard experimental distribution graph considering Maximum Entropy Principle and the σ values were obtained. |[σ] − λI|υ = 0 According to this equation, λ1 and λ2 eigenvalues were obtained.
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Stochastic Explanation for Failure Criterions In this equation that represents how to obtain stochastic version of the equation of safety factor for Von Mises and Tresca failure criterion. σvm =
q
(σ2 − σ1 )2 + σ21 + σ22
Fs(V onM ises) =
Fs(T resca) =
Yaşar Yanık (UNESP)
σe σvm
σe σ2 − σ1
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Results and Discussion Considering Two Uncertainties 2.5 VonMises - Histogram VonMises - PDF Tresca - Histogram Tresca - PDF
Probability Density Function
2
1.5
1
0.5
0 1.5
2
2.5
3
3.5
4
4.5
5
F s (Safety Factor)
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Results and Discussion Considering Two Uncertainties
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Maximum Entropy Principle
px (x) is probability density function of the Maximum Entropy Principle. According to stochastic equivalent of the approximation, it is also assumed that support, µ (mean) and σ (variance) of the random variables are known px (x) = eλ0 e−λ1 x−λ2 x
Yaşar Yanık (UNESP)
2
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Maximum Entropy Principle
According to Maximum Entropy Principle, the stochastic equivalent of the approximation shown below Z +∞ −∞
px (x)dx − 1 = 0
Z +∞ −∞
Z +∞ −∞
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xpx (x)dx = µ = 0
x2 px (x)dx − µ2 − σ 2 = 0
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Maximum Entropy Principle
0.025
Pdf (probability density function)
Generated Theoretical
0.02
0.015
0.01
0.005
0 120
140
160
180
200
220
240
260
280
300
Normal stress (kpa)
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Examination of Simple Deflection Problem Considering Uncertainty Quantification
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The Stochastic Explanation for the Simple Deflection Problem
The stochastic version of equation of the elastic curve and deflection and slope at A were determined as shown below M( x) d2 v = 2 dx EI
M(x) = F (L − x)
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The Stochastic Explanation for the Simple Deflection Problem If the equation multiplying both members by the constant EI and integrate considering variable x, the equation will be obtained as shown below EI
1 dv = FLx − Fx2 + C1 dx 2
We now observe that at the fixed end B we have v(0) = 0 and θ = dv/dx = 0. Substituting these values into this equation and solving for C1 , then the slop function is determined as shown below EI
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dv 1 = FLx − Fx2 dx 2
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The Stochastic Explanation for the Simple Deflection Problem if the equation integrating both members of the slop function considering at B we have x = L , y = 0. Then the stochastic version of equation is determined as shown below 1 1 EIv = FLx2 − Fx3 + C2 2 6 if the value of C2 carrying back into the deflection function, then the stochastic version equation of the deflection function is determined as shown below 1 3 1 1 2 FLx − Fx v(x) = EI 2 6
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The Stochastic Explanation for the Simple Deflection Problem
The Hooke-Lamé’s Law in Cartesian Coordinates is shown below
The Stochastic Explanation for the Simple Deflection Problem The eliminated equations of Hooke-Lamé are determined as shown below h
σx
i
h
τxy
=
i
h
λ + 2G
=
h
G
ih
ih
ξxx
γxy
i
i
Afterwards, the slope function at x = L will be equal to displacement εx . dv 1 = ξx = dx EI
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1 2 FL 2
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The Stochastic Explanation for the Simple Deflection Problem The shear (G) and lambda (λ) modulus are determined as shown below λ=
νE (1 + ν)(1 − 2ν)
G=
E 2(1 + ν)
Also, moment of inertia of the cantilever beam is determined as shown below Z Z +h/2 bh3 I = y 2 dA = y 2 dy = 12 −h/2
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The Stochastic Explanation for the Simple Deflection Problem
Table: Parameters used in Hooke-Lamé’s equation in a deterministic way.
E[P a] 210 ∗ 109
Yaşar Yanık (UNESP)
ν 0.3
h[m] 0.025
b[m] 0.05
L[m] 1
σe [P a] 5 ∗ 109
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The Stochastic Explanation for the Simple Deflection Problem 0.025
Pdf (probability density function)
Generated Theoretical
0.02
0.015
0.01
0.005
0 150
200
250
300
350
400
Force (Newton)
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Results and Discussion for Simple Deflection Problem
4 VonMises - Histogram VonMises - PDF Tresca - Histogram Tresca - PDF
Probability Density Function
3.5 3 2.5 2 1.5 1 0.5 0 1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
F s (Safety Factor)
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Results and Discussion for Simple Deflection Problem
4.5
Convergence of F s (Tresca)
4
3.5
3
2.5
2 0
1000
2000
3000
4000
5000
6000
7000
8000
Number of Samples
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Probabilistic design results
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Probabilistic design results
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Probabilistic design results
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The comparison between results of Ansys and Matlab
4
5 VonMises - Histogram VonMises - PDF Tresca - Histogram Tresca - PDF
4.5
Probability Density Function
Probability Density Function
4
VonMises - Histogram VonMises - PDF Tresca - Histogram Tresca - PDF
3.5
3.5 3 2.5 2 1.5
3 2.5 2 1.5 1
1 0.5
0.5 0
0 1
1.2
1.4
1.6
1.8
2
2.2
F s (Safety Factor)
Yaşar Yanık (UNESP)
2.4
2.6
2.8
3
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
F s (Safety Factor)
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Conclusion
The critical comparison between Von Mises and Tresca failure criterions The stochastic model of the problem The probability density function and a convergence graphic about Von Mises and Tresca failure criterions considering two uncertainty The description of simple deflection problem considering uncertainty quantification The results of simple deflection problem considering uncertainty quantification in ANSYS APDL
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Future Work
For future work it is going to be studied about the comparison of failure criterions with examination of the frame of formula car in ANSYS APDL Presentation of experimental examples and comparison between deterministic and experimental results considering Von Mises and Tresca failure criterions
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Acknowledgments
Thank you very much for your attention! Yaşar Yanık (UNESP)
