Post on 17-Sep-2015
description
1Structural Reliability
Structural limit states are often defined as a difference between Strength (R) and Load (S):
Probability of failure can be defined as
Or, using reliabilityf R
(r),
fS(s
)
S, R
Te
xt
PD
F o
f F
ailu
re
R - S
Text
Text
Text
Text
Text
Text
T e x tT e x tT e x tT e x tT e x tT e x tT e x t
FS(s)FR(r)
fS(s*)FR(s*)
Te
xt
2Reliability Analysis
Uncertainty propagate through physical system
Physical System
Output Uncertainty
Inp
ut
Sta
tist
ica
l U
nce
rta
inty
Model Uncertainty
X S(X)
Finite element,Mathematical modeling,Etc.
S(X)+C(X)S(X)+K(X)S(X)=F(X)
3Deterministic Optimization vs RBDO
Initial Design
Feasibility Check
Design Optimization
Critical G?Engineering
Analysis(e.g.,, FEA)
Converged?
Termination
Design Update
Deterministic Design
Optimization
Preliminary Reliability Analysis
Design Optimization
Critical Gp?Refined
Reliability Analysis
Engineering Analysis
(e.g.,, FEA)
Converged?
Termination
Design Update
4Reliability-Based Design Optimization (RBDO)
Problem formulation
Limit state:
Design variable: d Random variable: X
Target probability of failure:
Target reliability index:
Constraint is given in terms of probability of failure
Need to evaluate PF at every design iteration
5RBDO
X2
0
Fatigue Failure
G1(X)=0
Infeasible Region G i(X)>0
X1Feasible Region
Gi(X)0
Initial Design
Wear Failure
G2(X)=0
Deterministic Optimum
Design 50% Reliability!
Joint PDF
fX(x) Contour
FailSafe
Reliable Optimum
Design
FailSafe
6Monte Carlo Simulation
Probability of failure
0G
0G
7Most Probable Point (MPP)-based Method
Transform the limit state to the U-space
Need to find MPP point
Calculate reliability index
X2
0
Failure Surfaceg(X)=0
Failure Regiong(X)0
Mean Value Design Point
Joint PDFfX(x) Contour
U2
U1
Pf
Failure Regiong(U)0
0
Reliability
Index
Joint PDF fU(u)
Joint PDFfU(u) Contour
MPP u*
Mapping T
8MPP Search Method
Reliability appears as a constraint in optimization
Reliability index approach vs Performance measure approach
9Reliability Index Approach (RIA)
Find S,FORM using FORM in U-space
HL-RF Method
Good for reliability analysis
Expensive with MCS and MPP-based method when reliability is high
MPP-based method can be unstable when reliability if high or the limit state is highly nonlinear
10
RIA-RBDO
RIA-RBDO:
Feasible set
11
Performance Measure Approach (PMA)
Inverse reliability analysis
Advanced mean value method
Not suitable for assessing reliability (t is fixed)
Efficient and stable for design optimization
Optimum: *( )pG u
12
PMA-RBDO
PMA-RBDO
13
Reliability-based Sensitivity Analysis
During optimization, gradient (sensitivity) needs to be calculated at each iteration
For RIA, gradient of reliability index w.r.t. design variable (DV)
For PMA, gradient of performance function w.r.t. DV
RIA:
From U = T(X; d)
14
Reliability-based Sensitivity Analysis
Example Normal distribution
Design variables are d = [m, s] of Normal distribution
Thus,
Example Log-Normal distribution
15
Reliability-based sensitivity analysis
PMA:
Regular sensitivity analysis in optimization can be used
The sensitivity needs to be evaluated at MPP point.
*
,FORM ( )
t
p
i i
G G
d d U u
U
*
,FORM ( ( ; ))
t
p
i i
G G
d d X x
T X d