Reliability analysis in engineering applications
Exercises for block 4 “Resistance modeling”
Excercise 4.1
Consider a structural member with volume V built up of n elementary volumes Vo, i.e. V=nVo. The strength fo of volumes Vo is assumed to have a two parameter Weibull distribution, so that
where u and k are the parameters of the distribution.
Assume that
- the strengths of elementary volumes Vo are independent
- the strength f of the volume V is governed by the weakest of the elementary volumes Vo
Determine the cumulative distribution function for f and calculate the strength ratio f/fo for the mean level and the characteristic value defined as the 5th percentile.
Excercise 4.2
Consider a post-tensioning tendon unit built up by N individual and parallel strands. The tension capacity R of the tendon unit is given by
where Riis the the capacity of strand number i. Assume that the various Riare identically distributed normal variables, N (186, 9) with units of kN. Calculate mean, variance and 5th percentile for R as a function of N under the assumption that
a) The strength of the various strands are independent
b) The correlation between the various Ri is jk = = 0.8, for all jk.
Exercise 4.3
To estimate the variability of strength for reinforcing steel to be used for a nuclear reactor containment structure, tests were made on 12 specimens. The average yield strength was then found to be 527 Mpa with a standard deviation of 10 Mpa.
However, it was found that all the 12 specimens from the producer had been taken from the same batch of steel. The large amount of material to be used during construction was expected to come from the same supplier, but not from the same batch. Estimate the standard deviation for the strength underthese circumstances. Hint: Look in the chapter for reinforcing steel in in the JCSS “Probabilistic model code”, which is available on
What can be said about the mean value of strength for the reinforcement to be used in the construction of the containment.
Exercise 4.4
The compressive strength fc of concrete for a bridge was tested on standard cylinder specimens produced in the concrete plant during construction. The average strength was determined to 52.4 Mpa with a standard deviation of 3,6 Mpa. The in-situ strength fc,is (the strength in the structure) is generally lower than that obtained from standard specimens manufactured from the fresh concrete and can be written as
fc,is = fc
where is areduction factor with mean value = 0.85 and coefficient of variation 6 %. Assume that both fcand are lognormal and determine mean and standarddeviation for the in-situ strength fc,is.
Exercises for block 5 “Code calibration”
Exercise 5.1
Consider a structure with load bearing capacity R loaded with permanent load G and variable load Q, defined as the annual maximum value of Q(t). The following limit state equation is assumed to be valid
where C is a random variable describing model uncertainty. The statistical properties for the basic variables are given in the table below
Variable / Distribution / Mean / Coefficient of variation % / Characteristic value, fractileC / Lognormal / 1.0 / 5 / 0.5
R / Lognormal / R / 10* / 0.05
G / Normal / G / 5 / 0.5
Q / Gumbel / Q / 40 / 0.98
* includes dimensional uncertainties
The deterministic design format for the above described problem is given by
where index k denotes characteristic value for the property in question, and i are partial safety coefficients. The partial safety coefficients defined in the Swedish code BKR 03 as well as in Eurocode are given in the table below.
Partial coefficient / Swedish code / Eurocodemn (reinforcement steel) / 1.38 / 1.15
G / 1.0 / 1.35
Q / 1.3 / 1.50
a) By use of COMREL, calculate the safety indices S and E for the Swedish code and Eurocode respectively, as a function of the ratio between permanent and variable load defined by
Consider the interval 0.2 1.0.
b) Perform the same analysis for a simple safety factor format defined by
where s =1.5 is a single safety factor. This latter format is close to the safety principles used in Sweden before the limit state design principles were introduced about 20 years ago. Compare the results with todays format.
Exercises for block 6 “Reliability of structural systems”
Exercise 6.1
Consider spot welds in cold formed channels. The shear capacity of the spotwelds determines the capacity of the cantilever beam shown in Figure a. The spotwelds can be considered as a parallel system since they are ductile so that all of them must fail before the system fails, see Figure b. The shear force per weld is denoted Viand is a constant. The weld capacity for each weld is Ri. Two extreme cases are considered:
a) Perfectly correlated welds
b)The welds are statistically independent
Determine the coefficient of variation for the two extreme cases, and discuss how they would relate to the coefficient of variation for a general case of partially correlated welds.
a)b)
Exercise 6.2
Consider the three-bar structural system shown in Figure a. The system model used for reliability analysis is shown in Figure b. Let Ri be the strength of the ith element. Assume that R1 and R2 are normally distributed with mean values of 10 kN and standard deviations of 2 kN. Assume that R3 is lognormally distributed with a mean value of 20 kN and a standard deviation of 3,5 kN. The strengths are all independent.
a) What is the probability of failure of the system when it is loaded with a deterministic force Q=15 kN?
b) Repeat step a for a load of Q = 20kN.
Exercise 6.3
Repeat problem 6.2, part a, assuming that all strengths are perfectly correlated.
Exercise 6.4
Repeat problem 6.2, part a, assuming that R1 and R2 are perfectly correlated, but R3is independent from both R1 and R2.
övn-block 4-6