1

Structural 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

2

Reliability 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)

3

Deterministic 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

4

Reliability-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

5

RBDO

X2

0

Fatigue Failure

G1(X)=0

Infeasible Region G i(X)>0

X1

Feasible 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

6

Monte Carlo Simulation

• Probability of failure

0G

0G

7

Most 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

X1

Safe Regiong(X)>0

Mean Value Design Point

Joint PDFfX(x) Contour

U2

U1

Pf

Failure Regiong(U)<0

Most Probable Region to Fail

FORM

Failure Surfaceg(U)=0

SORM

Safe Regiong(U)>0

0

Reliability

Index

Joint PDF fU(u)

Joint PDFfU(u) Contour

MPP u*

Mapping T

8

MPP Search Method

• Reliability appears as a constraint in optimization

• Reliability index approach vs Performance measure approach

9

Reliability 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