uncertainty analysis, verification and validation of a
TRANSCRIPT
UNCERTAINTY ANALYSIS, VERIFICATION AND VALIDATION OF STRESS CONCENTRATION IN A CANTILEVER BEAM Soudabeh Kargar, Dr.Bardot University of Alabama in Huntsville
COMSOL Conference 2010 Boston Presented at the
INTRODUCTION
Stress Concentration in a Loaded Cantilever Beam which has a Hole
Structural Mechanics Module in COMSOL Version 4.0a
Simulations are Compared with Theoretical Values and Experimental Results
Uncertainty in Simulation and Experimentation Monte Carlo Method was Used to Estimate the
Total Uncertainty (Random & Systematic) Parametric Sweep Function in COMSOL
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EXPERIMENT SETUP
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High Strength Aluminum Beam
3 Strain Gages next to the Hole Digital Strain Indicator P-3500
Hanger
Weights
4th Strain Gage at Nominal Strain
Position
Bending Fixture
SIMULATION IN COMSOL
CAD Device in 3-D Space was Used Parametric Geometry U
niversity of Alabam
a in Huntsville
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Fixed End
Side View
Top View d w
t
lnom
P
L
GRID CONVERGENCE STUDY
Grid Convergence Study for Optimizing Tetrahedral Mesh Size for the Simulations.
Nine Different Mesh Sizes
1.0
1.2
1.4
1.6
1.8
2.0
0 5000 10000 15000
Kt
No. of Elements
Stress Concentration, Kt, vs
No of Elements Mesh Size Kt No of Elements
Average Element Quality
Solution Time (s)
Extremely Coarse
1.100 84 0.2674 1.765
Extra Coarse
1.292 102 0.3396 1.656
Coarser 1.197 191 0.5082 1.563
Coarse 1.232 259 0.5874 1.593
Normal 1.718 644 0.7070 1.734
Fine 1.842 1340 0.7377 1.984
Finer 1.773 3340 0.7746 2.500
Extra Fine 1.776 6082 0.7950 2.860
Extremely Fine
1.781 16080 0.8213 4.422
5
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untsville
SIMULATION VERIFICATION
Both Simulations Bracket the Theoretical Value of
Deflection.
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Max End Def.
0.138"
0.141"
0.142"
Simulation, Beam without the Hole
Theory
Simulation, Beam with the Hole
Max End Deflection from Theory: EI
PL
3
3
0.006 0.006
0.006 0.006
0.006 0.006
UNCERTAINTY IN EXPERIMENT
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labama in H
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7 0
200
400
-5.4
0-4
.05
-2.7
0-1
.35
0.00
1.35
2.70
4.05
5.40
Random Uncertainty (ε1)
0100200300400
-12.
10-9
.06
-6.0
2-2
.98
0.05
3.09
6.13
9.17
12.2
1Systematic
Uncertainty (ε1)
0200400
-0.0
02-0
.001
5-0
.001
-0.0
005
8.67
3…0.
0005
0.00
10.
0015
0.00
20.05% of Reading
0200400
-12.
00-9
.00
-6.0
0-3
.00
0.00
3.00
6.00
9.00
12.0
0
3µε
0200400600
-2.0
0-1
.50
-1.0
0-0
.50
0.00
0.50
1.00
1.50
2.00
Resolution 1µε
0100200300400
496.
1149
8.61
501.
1150
3.61
506.
1050
8.60
511.
1051
3.60
516.
0951
8.59
521.
09M
ore
Total Uncertainty (ε1)
Repeating Experiment
10 times
Accuracy and Resolution of Equipment
Human Error Equipment Environment
EXPERIMENTAL UNCERTAINTY ANALYSIS
0.014
0.014
0.015
0.015
0.016
0.016
0.017
0.017
1.645
1.646
1.647
1.648
1.649
1.650
1.651
1.652
1.653
0 1000 2000 3000
Number of Samples
Convergence of Stress
concentration, Kt
Average Standard Deviation
0
50
100
150
200
250
300
350
1.5
84
1.5
95
1.6
07
1.6
19
1.6
31
1.6
42
1.6
54
1.6
66
1.6
78
1.6
89
1.7
01
More
Nu
mb
er o
f R
ead
ing
s
Stress Concentration (Kt)
Histogram of Stress
Concentration, Kt
Experiment Gaussian Distribution
Histogram of Kt for the Experiment Convergence of Average and Standard Deviation of Kt for
Experiment
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UNCERTAINTY IN SIMULATION WITH COMSOL
Input Parameters: t, L, w, d, lnom, P, E
Monte Carlo, 3000 Samples
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d w
t
lnom
L
0
200
400
-… -… -… -… -… -… -… -…1.
7347
…0.
0010
…0.
0021
…0.
0032
…0.
0043
…0.
0054
…0.
0065
…0.
0076
…0.
0087
…
Random Uncertainty (t)
0100200300400
-0.0
040
-0.0
033
-0.0
025
-0.0
018
-0.0
010
-0.0
002
0.00
050.
0013
0.00
200.
0028
0.00
35M
ore
Systematic Uncertainty (t)
0200400
0.24
220.
2446
0.24
700.
2494
0.25
180.
2542
0.25
660.
2590
0.26
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Total Uncertainty (t)
P
10 times Repeating
Measurements Accuracy of Equipment
Time Consuming
PARAMETRIC SWEEP
Variables in Parametric Format
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Optimizing Uncertainty Studies
Geometry Material Properties
Boundary Conditions
POST PROCESSING
The Stress or Strain Concentration Factor:
nom
tK
max
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POST PROCESSING
12
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STRAIN DISTRIBUTION
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Strain Distribution at
the Hole
Strain Distribution at Nominal Strain
nom
tK
max
CONVERGENCE OF NUMBER OF SAMPLES IN SIMULATION
Computing Power, a Controlling Factor Importance of Convergence Study Simple versus Complicated Models
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.070
1.79
1.80
1.81
1.82
1.83
1.84
1.85
0 1000 2000 3000
Number of Samples
Convergence for Simulation
Average Standard Deviation
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0
200
400
600
1.6
0
1.6
4
1.6
7
1.7
1
1.7
5
1.7
9
1.8
3
1.8
6
1.9
0
1.9
4
1.9
8
2.0
1
2.0
5
Nu
mb
er o
f R
ead
ing
s
Stress Concentration (Kt)
Histogram of Stress concentration
Kt
Simulation Gaussian Distribution
Histogram of Kt for Uncertainty in COMSOL Simulation in Comparison to
Gaussian Distribution.
Convergence of Average and Standard Deviation of Kt for
COMSOL Simulation.
COMPARISON OF HISTOGRAMS
Histogram of Kt for Experiment and Simulation
0
50
100
150
200
250
300
1.5
84
1.6
11
1.6
38
1.6
66
1.6
93
1.7
21
1.7
48
1.7
76
1.8
03
1.8
31
1.8
58
1.8
86
1.9
13
1.9
40
1.9
68
1.9
95
2.0
23
2.0
50
Nu
mb
er o
f R
ead
ing
s
Stress Concentration (Kt)
Simulation Experiment
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labama in H
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1.6"
1.8"
0.015 0.015
0.05 0.05
SUMMARY AND RESULTS In this Paper, we Looked at Experimentation, Computational
Simulation and the Uncertainty Propagation. The Stress Concentration Factor:
In Experiment :1.646 ± 0.0141 and 1.646 ± 0.0157 In Simulation: 1.80 ± 0.050. Thus we have a Comparison Error6 Implying there is an Un-modeled, Un-
simulated Effect such as the Strain Gauge Sensor Glue; or, Perhaps there is an Experimental or Input Uncertainty that is not Captured.
Parametric Sweep in COMSOL is a Simple Tool to Perform Uncertainty Analysis with Monte Carlo Technique in our Computational Simulations.
The Same Method can be Used to Learn which Parameter has More Effect on the Final Result.
In Order to Prevent Consuming Time and Money on Doing Different Experiments with Different Uncertainty in Parameters, we can use Uncertainty Analysis in Simulation to Find out which Parameter(s) are Controlling Factors.
Defining Cut-line in Parametric format causes Problem.
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REFERENCES 1. Hugh W. Coleman, W Glenn Steele, Experimentation,
Validation, and Uncertainty Analysis for Engineers 3rd Edition, page 64, John Wiley & Sons, Inc., Hoboken, New Jersey (2009).
2. Instrumentation Division Measurements Group, INC, P-3500 Digital Strain Indicator Instruction Manual, page 8 and 14.
3. Introduction to COMSOL Multiphysics Version 4.0a (2010) 4. S. P. Timoshenko, J. N. Goodier, Theory of Elasticity, page 26,
27, 28, 65, 66, 67, 68, 90, 91, 92. McGraw-Hill Book Company, New York (1970).
5. J. A. Gilbert, C. L. Carmen, MAE/CE 370 Mechanics of Materials Laboratory Manual Version 1.0, page numbers. Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville (2000).
6.The American Society of Mechanical Engineers, ASME V&V 20-2009 – Standard for Verification and Validation in Computational Fluids and Heat Transfer”, November 2009.
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STRESS DISTRIBUTION AROUND CIRCULAR HOLES
Strain Distribution:
At the Edge of the Hole where,
R(in) Z1(in) Z2(in) Z3(in)
0.125 0.145 0.185 0.325
42
ii
iZ
RC
Z
RBA
)(44.5)(86.5 3221 C
CB 2.1)(49.3 21
CBA 522.0743.01
1Z
R
CBA max
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R: Radius of Hole Zi: Distance from Strain Gage to the Center of the Hole A,B,C: Constants
EXPERIMENTAL RESULTS The Uncertainty of Kt is Found by Two Methods:
1) Random and Systematic Uncertainties for Kt are Calculated as Standard Deviation and then are Combined with:
2) Value of Kt is Found for All 3000 Samples Including the Random and Systematic Uncertainty and then the Total Uncertainty for Kt is Calculated by Calculating the Standard Deviation
The Average Value of Kt for both Methods: Average = 1.646
The Total Uncertainty for both Methods: Method (1) = 0.0141 Method (2) = 0.0157
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tKtKtK Sbu
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PARAMETRIC SWEEP
Defined Variables for: Random Error Systematic Error
dummy Dummy Variable in Ascending
Order
L Length of the Beam
w Width of the Beam
t Thickness of the Beam
d Diameter of the Hole
lnom Distance, Nominal Stress to Fixed
End
P Load
E Modulus of Elasticity
Parameter Average Random
Error
Systematic
Error
L 11.260 0.009 0.003
w 1.002 0.001 0.001
t 0.252 0.002 0.001
d 0.246 0.002 0.001
lnom 1.021 0.005 0.001
P1 4.010 0.000 0.010
P2 0.064 0.002 0.010
E 1.04E+07 2.5% 0
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3 DIMENSIONAL GRAPH OF THE STRESS DISTRIBUTION AROUND THE HOLE
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Strain Distribution at the Hole