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# Monte Carlo Simulation

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In this lab, we are trying to predict the possibility of failure of a beam with a given cross-sectional area of 78.55m^2 using Matlab. Monte carlo simulation is a technique will be using in this lab to predict that. The material of the beam used in this simulation was 304 stainless steel. We suppose to build a matlab code and plot the results on a graph in matlab.. The simulation was created in matlab based on given dimensions of the beam and material properties. The ultimate tensile strength of the stainless steel 304 material is 505MPa.

Using the relationship between force and a cross-sectional area of material. The magnitude of this stresses can be found by;

F= force (in N)

𝞼= Ultimate tensile stress (505MPa for stainless steel 304)

A= area (m^2)

d= diameter of the beam

r=d/2

For AISI 304 Stainless Steel with a cross-sectional area of , the given value of ultimate tensile stress is 505 MPa. Since the best way to measure the diameter is by using Vernier calipers, then this what we did for this simulation exercise too. Since the accuracy of a pair of Vernier calipers is given to the nearest ±0.01cm, or 0.1mm. For a diameter of 10 mm, the uncertainty in the calculated area is, using the ultimate tensile stress with the nominal cross-sectional area values given above, the theoretical failure load.

The Monte Carlo simulation was running two hundreds thousand times (N=200000) in a for loop function. A for loop function in this case was applied to help determining the magnitude of failure stress over the number of runs (N=200000). After creating a histogram of force vs number of runs (N=200000), it was realized that a normal (Gaussian) distribution of stress failures based on changing cross-section and material properties was obtained as shown above.

From the matlab plot, it is clear that most of the samples are high at Force=0 position which represents a histogram at the 40th position (since there are 80 histograms). This point represents the mean point of our data. However, the probability of failure increases in 1.5*10^8 from this point and decreases in -1.5*10^8 from this position. As the forces increase, the magnitude of induced stresses increases and the rate of material failure increases. Whenever we move away from the mean position of the force, the data move away from the expected sets. This means that the data is becoming more not accurate.

From the plot, it is clear that it follows a perfect normal distribution trend .This shows that all parameters were correctly entered in the matlab code and giving correct results.

By using matlab it was possible to apply Monte Carlo simulation, and the random variables were been used to determine the “truth value” of the normal stress failures for AISI 304 Stainless Steel with a cross-sectional area of 78.55 mm^2 . After running the simulation it was realized that it created a normal (Gaussian) distribution graph after running the program. The exercise was successfully worked on matlab and it helped me to learn a new technique of predicting the possibility of failure of a beam with given parameters.

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