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LOCAL
BUCKLING STRESS FOR A FLANGED COMPONENT UNDER LONGITUDINAL LOAD
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Description
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Limit
State Function
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Random
Variables
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Analysis
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Reference
This is an example of local buckling stress for a flanged component under
longitudinal compressive load. Local instability or local buckling of a section
is an elastic type of failure, which may occur at low stresses. The critical
local buckling stress, sCT,
can be computed using the formula provided here. The stress response, s,
can be computed from Finite Element Analysis or manual calculation (in this
example, s is assumed to be a random
variable with the specified statistical data). The geometry of the flange
component is shown in Figure
1. The local
buckling stress coefficient, KL, for the compression flange under
longitudinal compressive load may be obtained from Figure
1.
where:
h
= Plastic correction
E
= Young’s modulus
where bL, bf are defined in Figure
1.
n
= Poisson’s ratio
s
= Stress
PlastC=
h:
Weibull (m=2.5, s=0.75, XL=0)
Young
= E: Gamma (m=3*107, s=6*106)
bl
= bL: Lognormal (m=4.75*10-1, s=4.75*10-2)
bf
= bf: Lognormal (m=1.45, s=1.45*10-2)
tl
= tL: Lognormal (m=5*10-2, s=5*10-4)
poisson= n: Gamma (m=2.5*10-1, s=7.5*10-2)
stress
= s:
Gamma (m=1*105, s=2.5*104)
Figure 1 Local
Buckling Stress Coefficients for Lips Under Longitudinal Compression
Anal1:Calculate the failure probability Pf = Prob(g17 £0)
defined in Eq. 1
using different methods. Compare and discuss the results.
CASE
1. Second-Order Reliability Method (SORM): Use different MPP identification
methods ( e.g., U-space X-linearized MPPL Method) and Point Fitting/Curvature
Fitting Second-Order Probability Method.
CASE
2. Simulation Method (SM): Use 10000 as the maximum number of g-function
calculations/sampling points and 0.05 as COV.
CASE
3. Importance Sampling Method (ISM): Use different MPP identification methods
and Directional Simulation Method with 10000 as the maximum number of g-function
calculations/sampling points and 0.05 as COV.
CASE 4.
Importance Sampling Method (ISM): Use different MPP identification methods and
Sphere Based Importance Sampling Method with 1000 as the maximum number of
g-function calculations, 10000 as the maximum number of sampling points, the
first-order reliability index as the radius of sphere, and 0.05 as COV.
Anal2:
Identify key design variables in the Anal1 by the result from sensitivity
analysis.
Anal3:
CDF/PDF Analysis: Develop the cumulative density function of the critical
buckling stress, sCT (i.e., g18
in Eq. 2) from 1.0E-3 to 0.999 by using SORM/Curvature
Fitting Method.
Anal4:
Plot g as a function of b in Anal
3.
Anal5: Use the Response Surface method (RSM) to calculate the failure
probability Pf = Prob(g17£0).
CASE
1. Expend the limit-state function around the mean point with ±one
standard deviation ranges and use SORM to calculate probability of the
approximated limit state function.
CASE
2. Expend the limit-state function around the MPP obtained from Anal1 and use
SORM to calculate probability. If the RSM fails to approximate the limit-state
function with the second-order polynomial equation, try a smaller step length in
the Box Behnken Matrix.
Anal6:
Use the Mean Value-Based Method to calculate the failure probability Pf
= Prob(g17£0)
CASE 1. Mean Value First-Order Second Moment Method.
CASE 2. Mean Value (U-space) Method
CASE 3. Mean Value (X-space) Method
CASE 4. Advanced Mean Value (U-space) Method
CASE 5. Advanced Mean Value (X-space) Method
Anal7: Use the Latin Hyper Cube Simulation method with 10000 sample points to
calculate the failure probability Pf = Prob(g17£0).
In Latin Hypercube method the sampling space is divided
into subsets of equal probability in the standard normal space. The number of
samples is reduced by representing every subset of each independent random
variable with only one value. The result is listed in the table in Case 2 of
Anal1.
Anal8: Use Inverse Probability Analysis to calculate the level of g18
corresponding to the failure probability of 0.001 by
CASE
1. FORM with U-space MPP/U-Linearized MPPL Method.
CASE
2. FORM with U-space MPP/X-Linearized MPPL Method.
CASE
3. FORM with Non Gradient Based MPP Searching Method.
CASE
4. SM with 1000000 as the maximum number of g-function calculation/sampling
points and 0.05 as COV.
CASE
5. ISM with 1000000 as the maximum number of g-function calculation/sampling
points and 0.05 as COV.
CASE
6. RSM with Expending the limit-state function around the mean point with ±s
ranges and FORM for probability calculation.
CASE
7. MVBM with U-space MV method.
CASE
8. MVBM with X-space AMV method.
Anal
9: Verify the results of Anal8
by the Probability Analysis with same method, parameters, and pre-defined value
of limit-state function obtained in each case.
Lin
and Khalessi, “Identification of the MPP in Original Space - Applications to
Structural Reliability”, Proceedings of the 34th Structures, Structural
Dynamics, and Material Conference, AIAA/ASME/ASCE/AHS/ASC, April 19-22, 1993,
pp.2791-2800.
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