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Engineering · Civil Engineering
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Problem 9 (for all students) Monte Carlo simulation is often used to analyze probabilities of failure of geotechnical engineering structures. a) For a case, where the limit state function, G, can be described by three normally distributed random variables such that G-Xi-X2x X3, describe a step-by-step methodology (i.e. an algorithm; MatLab code is not needed) that calculates the probability of failure with Monte Carlo simulation. (3 p) I. Il Running the algorithm that you proposed in a), you can visualize the result as a histogram, as shown in the figure (next page). Explain how the probability of failure is visualized in the figure, estimate its value roughly from the figure and state the reasons for your answer. (2 p) In a Monte Carlo simulation, the number of simulations affect the computational time considerably. What is the likely computational result of the probability of failure, if the number of simulations is far too low? Why? (2 p) IlI. b) The probability of failure can also be visualized in a graph with the two random variables on the horizontal axes and probability density in the third direction (toward you). The second figure (next page) illustrates a case where the limit state function, G, is described by a resistance R and a load S. Both random variables are normally distributed. The joint probability density function is indicated through the contour lines (rings), such that the highest probability density is in the smallest ring. Explain in text how the figure illustrates the probability of failure. Indicate also the failure region on the figure (submit the figure with your solution) (2 p) I. II. Are R and S correlated? State the reasons for your answer. (1 p) II Redraw the joint probability density function (the rings) in the figure so that the new probability density function corresponds to a case with negativel correlated random variables. Explain in text the new shape of the joint probability density function. (2 p)x 10 2 1.5 0.5 0 -20 10 10 20 30 40 50 G [kN] Figure to question a-II): histogram visualising the Monte Carlo simulation 20 15 Probability density function of R Joint probability density function of R and S 20 30 Probability density function ofS tion and its two random variables. Figure to question b): illustration of a limit state func Submit this page if you have drawn on it as part of your solution!

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