D21SR Safety Risk And Reliability

Q1. A steel cable should support a weight with a mean of 15 kN and COV of 0.12. Steel available for the cable has yield stress with a mean of 410 N/mm2 and COV of 0.10. Find the required cross-sectional area of the cable if the target reliability index βT (i.e., theminimum acceptable value of the reliability index) for it is3.4. What is the corresponding central safety factor? Assume that the yield stress of steel and the weight can be modelled by independent normalrandom variables. Neglect any possible variation in the cross-sectional area of the cable

Q2. A solid circular cross-section tie-bar has been designed to carry a mean load of 15 kN with a COV of 0.12. The tie-bar is made of steel with a mean yield stress of 260 N/mm2 and a COV of 0.10. The mean diameter of the bar is 12 mm and its COV is 0.05. Assume that all uncertain parameters can be modelled as independent normal random variables.

(a) Calculate the safety index and the probability of failure of the bar using the FOSM method.

(b) Calculate the safety index and the probability of failure of the bar using the AFOSM method. Use Excel for the calculations.

(c) Compare the results obtained in (a) and (b) and comment on their difference and reasons for that.

Q3. It is required to design an 8-m span simply supported beam made of glued laminated timber, which is loaded by uniformly distributed dead load w and live load q  The ultimate limit state (ULS) is associated with bending strength, for which the target safety index βT = 3.5. In this case the limit state function can be formulated where fb is the bending strength of glulam timber and S=bh2 /6 is the elastic section modulus. The statistical properties of the basic random variable of the problem are given in the table below. The beam span l should be treated as a deterministic parameter.

Variable Mean Coefficient of variation (COV) Distribution type

fb 30 N/mm2

0.15 Lognormal

h ? 0.04 Normal

b 250 mm 0.03 Normal

w 20 kN/m 0.10 Normal

q 15 kN/m 0.18 Gumbel

(a) Find the required mean depth of the beam cross-section, for which β = βT, using the AFOSM method. Neglect in this analysis the information about the distribution types of the basic random variables (i.e. assume that all random variables are normally distributed).

(b) For the found mean depth of the beam estimate the safety index by the FORM taking into account the information about the distribution types of the basic random variables. If the calculated β is less than βT increase the mean depth of the beam until β ≥ βT. Examine the sensitivity factors obtained in the solution and discuss if some of the basic random variables considered in the problem could be treated as deterministic parameters (equal to their mean values) without having significant effect on the estimated safety index.

(c) Check the safety index and the probability of failure obtained in (b) using Monte Carlo simulation. Before carrying out the analysis estimate the required number of simulations if the desired error is 10% with 95% confidence (for this purpose use the probability of failure obtained in (b)). Compare the calculated probability of failure with that obtained in (b) and comment on the results, their difference and possible reasons for that. Use Excel for the calculations. If necessary, formulations of the limit state function different from (but equivalent to) that given above may be used.

Q4. A medium-size crane carries out 2000 lifts per annum. The load in one lift is modelled as a normal random variable with mean of 70 kN and COV of 0.15. The power span of the crane is modelled as a 6-m cantilever with the plastic section modulus Z, which is treated as a normal random variable with mean 2×10-3 m 3 and COV of 0.04. The cantilever is made of steel, whose yield strength, fy, is treated as a lognormal random variable with mean of 380 N/mm2 and COV=0.10. The annual probability of failure of the crane in bending is estimated using Monte Carlo simulation with the limit state function formulated as where W is the load and L=6 m is the length of the power span. The latter should be treated as a deterministic parameter.

(a) Perform one simulation trial manually using the following random numbers: u1=0.197, u2=0.469 and u3=0.834.

(b) Adapt the Visual Basic code for Monte Carlo simulation provided with the course materials and run the program with sufficient number of simulation trials to ensure that the COV of the Pf estimate is less than 0.05. Use Excel for the calculations.

Q5. It is assumed that the distribution of a crack size a in a pipeline can be described by the following lognormal PDF where T is the age of the pipeline in years and λ=1.0 and ξ=0.40 are the distribution parameters. (a) What is the probability based on this PDF that the crack size after 7 years will exceed 10 mm?

(b) After 7 years the pipeline is inspected for cracks using a device whose probability of detection (POD) is given by the exponential distribution with a mean crack detection size of 6 mm. If no crack has been detected by the inspection, what is the probability that there is, in fact, a crack exceeding 10 mm? Use Excel for the calculations.