(Solved): Problem 2 The safety factor, SF, for an axially loaded column is defined as follows: SF=SR where ...

Problem 2 The safety factor, SF, for an axially loaded column is defined as follows: $\mathrm{SF}=\frac{\mathrm{R}}{\mathrm{S}}$ where, $\mathbf{R}=$ the strength (axial load carrying capacity) of the column and $\mathbf{S}=$ the axial load acting on the column. $\mathrm{R}$ is a random variable with a mean value of ${?}_{\mathrm{R}}=140\mathrm{kN}$ and a coefficient of variation of ${?}_R=0.15$. The random variable, $S$ has an average value of ${?}_{\mathrm{S}}=70\mathrm{kN}$ and a coefficient of variation of $?\mathrm{s}=0.20$. R and $\mathrm{S}$ are statistically independent, and both are lognormally distributed. a) Find the distribution parameters ( ${?}_{\mathrm{R}},{?}_{\mathrm{R}}$ and ${?}_{\mathrm{s}},{?}_{\mathrm{s}})$ of the lognormal random variables $\mathrm{R}$ and $\mathrm{S}$. b) Since R and S are random variables, SF will also be a random variable. Accordingly, obtain the probability distribution of SF and state the name of this distribution. At the same time, find the mean value and coefficient of variation of SF. Use the exact solution. c) If the safety factor, SF, is greater than $3.5$, the structure is considered to be "safe". What is the probability that the structure will be rated as safe? d) Obtain the expected value and coefficient of variation of the safety factor, SF, by using the firstorder approximation (i.e. linearization based on the Taylor series expansion at the mean values).