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