\n", "Task 0:\n", "Tasks to complete during the workshop are provided in boxes like this.\n", "
\n", "\n", "Explanations in the solution are provided in boxes like this.\n", "
WeibullMin | 15.1 | 0.48 |\n",
"\n",
"Note that you should define the WeibullMin distribution as WeibullMin(17, 2.2) using OpenTURNS (the inputs are the shape and scale parameters).the water surface elevation assumes a high tide situation, and the normal distribution represents the variability in high tide elevation observed throughout the year.\n",
"\n",
"We know that there are two main hydraulic loads, caused by the wind, which govern overtopping discharge: wind setup and waves. Both of these are also dependent on the water depth, $d=h-ob$.The figure below illustrates the magnitude of each, as a function of the water depth for the 95th percentile wind speed.\n",
"\n",
"![loads](./WS_03_figures/water_depth_loads.JPG \"Effect of water depth on hydraulic loads.\")\n",
"\n",
"We can see that both effects have opposite relationships to water depth. Since both effects are also proportional to the overtopping discharge, it becomes difficult to establish what the \"critical case\" is to use in our dike design. Probabilistic methods can help us (by integrating across all combinations of loads), but in the end we still need to determine one set of design values which we will use to actually build the dike (in this case, freeboard). Finding an answer to this question and understanding the role of uncertainty in the solution is one of the main themes of this workshop."
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"\n", "Task 1:\n", "There are at least two critical cases to consider for this problem. Imagine you are performing a deterministic analysis (but are still acknowledging that a few variables are random). List at least two design cases (qualitatively) and describe how you might select the load values to use in your overtopping calculation. Justify your decision.\n", "\n", "
\n", "\n",
"The two cases are: 1) large waves with smaller setup (deeper water), and 2) large setup with smaller waves (shallower water). The design loads should be selected based on the maximum allowable failure probability. If we need a failure probability of 0.001, we could choose the 90th percentile for each random variable (0.1$^3$=0.001).\n",
"
\n",
" There are two things to note here: 1) there are infinite combinations of design variable exceedance values that could lead to the same joint exceedance probability, and 2) if we can identify the most critical cases, we
\n",
"We call this a 'semi-probabilistic' analysis because we use probability to derive the design values for one or more 'critical cases', but don't consider 'all' of the combinations (i.e., we don't integrate across the failure region!).\n",
"
\n", "Task 2:\n", "Define the three random variables as OpenTURNS distributions, then check whether you can evaluate the limit-state function properly by evaluating the overtopping discharge at the 90th percentile value of each random variable. Based only on this calculation, what do you think the probability of exceedance is for this overtopping discharge?\n", "\n", "
\n", "\n", "Task 3:\n", "Define the limit-state function (incorporating the maximum allowable discharge of $q=1$ [L/m/s]), multivariate probability distribution and set up the OpenTURNS analysis and solve the reliability problem using MCS and FORM.\n", "
\n", "\n", "The solution is in the code cell below.\n", "
\n", "Task 4:\n", "Interpret the reliability analyses. Be sure to consider: pf, beta, the design point in the x and u space, linearity of the limit-state function, importance and sensitivity.\n", "
\n", "\n", "Let's first print and calculate a few of the additional results:\n", "