Q$3.$ 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 $q($see Figure Q$3$ below$)$

The ultimate limit state $($ULS$)$ is associated with bending strength, for which the target

safety index ${}_T=3\mathrm{.}5.$ In this case, the limit state function can be formulated as

$G(x)=f_{b}S-\frac{(w+q)l^2}{8}$

where $f_b$ is the bending strength of glulam timber and $S=b\frac{h^2}{6}$ 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.

$($a$)$ Using the FORM estimate the beam reliability and check that the beam satisfies the

reliability requirement for the ULS. If not, increase the mean depth of the beam until the

calculated ${}_T.$ Examine the sensitivity factors obtained in the solution and based on

their values identify those of the basic random variables, which can be treated as

deterministic parameters $($equal to their mean values$)$ without changing the calculated

value of $$ by more than $3\%.$

$($b$)$ Check the probability of failure and the safety index obtained in $($a$)$ using Monte Carlo

simulation. Before carrying out the analysis estimate the required number of simulation

trials if the desired error is $5\%$ with $95\%$ confidence; for this purpose use the probability

of failure obtained in $($a$).$

$($c$)$ Compare the probability of failure calculated by Monte Carlo simulation with that

obtained in $($a$)$ and comment on the results, their difference and possible reasons for that.