May 2007 Hua Fan University, Taipei An Introductory Talk on Reliability Analysis With contribution from Yung Chia HSU Jeen-Shang Lin University of Pittsburgh

Supply vs. Demand  Failure takes place when demand exceeds supply.  For an engineering system: –Available resistance is the supply, R –Load is the demand, Q –Margin of safety, M=R-Q  The reliability of a system can be defined as the probability that R>Q represented as:

Risk  The probability of failure, or risk  How to find the risk? –If we known the distribution of M; –or, the mean and variance of M; –then we can compute P(M<0) easily.

Normal distribution: the bell curve For a wide variety of conditions, the distribution of the sum of a large number of random variables converge to Normal distribution. (Central Limit Theorem)

IF M=Q-R is normal When Because of symmetry Define reliability index

Example: vertical cut in clay If all variables are normal,

Some basics Negative coefficient

Engineers like Factor of safety  F=R/Q, if F is normal reliability index

Lognormal distribution  The uncertain variable can increase without limits but cannot fall below zero.  The uncertain variable is positively skewed, with most of the values near the lower limit.  The natural logarithm of the uncertain variable follows a normal distribution. F is also often treated as lognormal

In case of lognormal Ln(R) and ln(Q) each is normal

 The MFOSM method assumes that the uncertainty features of a random variable can be represented by its first two moments: mean and variance.  This method is based on the Taylor series expansion of the performance function linearized at the mean values of the random variables. First order second moment method

 Taylor series expansion

Example: vertical cut in clay If all variables are normal, 1-normcdf(1.8896,0,1)1-normcdf(1.8896,0,1) MATLAB

Slope stability 2 (H): 1(V) slope with a height of 5m

Reliability Analysis  The reliability of a system can be defined as the probability that R>Q represented as:

FS contour,, 0.21.

First Order Reliability Method Hasofer-Lind (FORM)  Probability of failure can be found obtained in material space  Approximate as distance to Limit state

Distance to failure criterion  If F=1 or M=0 is a straight line  Reliability becomes the shortest distance

Constraint Optimization:Excel

May get similar results with FOSM FOSM 1-normcdf(1. 796,0,1)1-normcdf(1. 796,0,1)= MATLAB

Monte Carlo Simulation correlation=0 Monte Carlo=0.0495

Monte Carlo Simulation correlation=0.5 FORM=0.0362

FS=1.0 (M=0) UNSAFE Region FS<1 or M<0 The matrix form of the Hasofer-Lind (1974)

The matrix form of the Hasofer-Lind (1974) FOS=1 Soil properties>0 Soil properties

FS=1.0 UNSAFE Region FS<1 or M<0 Correlation=0 Correlation=-.99 Correlation=.99

 The distance

FOSM maybe wrong  FOSM

A projection Method  Check the FOSM  Use the slope, projected to where the failure material is  Use the material to find FS  If FS=1, ok

May 2007 Hua Fan University, Taipei