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An Introductory Talk on Reliability Analysis

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## An Introductory Talk on Reliability Analysis

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**An Introductory Talk on Reliability Analysis**Jeen-Shang Lin University of Pittsburgh With contribution from Yung Chia HSU Hua Fan University, Taipei**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**First order second moment method**• 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) 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**May get similar results with FOSM**FOSM 1-normcdf(1. 796,0,1)=0.0362 MATLAB**Monte Carlo Simulationcorrelation=0**Monte Carlo=0.0495**Monte Carlo Simulationcorrelation=0.5**FORM=0.0362**The matrix form of the**Hasofer-Lind (1974) FS=1.0 (M=0) UNSAFE Region FS<1 or M<0**Soil properties**FOS=1 Soil properties>0 The matrix form of the Hasofer-Lind (1974)**FS=1.0**Correlation=.99 UNSAFE Region FS<1 or M<0 Correlation=-.99 Correlation=0**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