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Risk-Based Approach to Data Analysis in Engineering Decision Making

Objectives of the Assignment

On successful completion of this assignment, you will:

1. Know how to perform correlation and regression analyses on a set of given data and interpret the results.

2. Perform straightforward statistical inferences.

3. Practice the principle of risk-based approach to data analysis through a mathematically case study with analytical and numerical approaches.

4. Use the probabilistic-based method so derived to support decision making under  uncertainty.

5. Be able to carry out your own literature research prior to solving engineering  decision making problems (in the form of an independent learning project) and present the result with an in depth discussion.

Organise/sort the data to see if patterns can be observed. Perform correlation and regression analysis on this set of data. Explain and interpret the results as clearly as possible.

Your second task is to look into variability in the yield stress of the material used. According to the supplier of the material, the yield stress of the material is 130 MPa. A sample consisting of 10 specimens have been prepared and tested. The results are shown in Table 2. Analyse the data by plotting histogram and/or x-y plot.

On the basis of this analytical framework, set up a spreadsheet to calculate the numerical values of the probability that the excess capacity <= 0 (i.e. riskof structure yielding) over a range of excess capacity (recommended range: 10 kN – 100 kN). A sample spreadsheet is shown in Table 3.

You may also want to plot the sensitivity of the problem (over variation of one or more parameters), the sensitivity of the decision problem over a range of yield stress’s standard deviation is shown in Figure 2.

The risk-based model can then be used to determine an acceptable level of failure probability, and hence the optimum excess capacity (or margin) as contrast to the  safety factor approach. The main consideration is the trade-off between additional material cost to provide a given level of excess capacity and the penalty cost

incurred by failed components. For this exercise, the following data applies:

Annual production rate, N = 100,000 pieces Material cost

Cm

= £9,000 per m2

Penalty cost = £200 per failed component,

CP, plus replacement material cost

ka = 20,000 where ka is a coefficient used to compute additional material cost 1.1

arg arg Add.material cost N area C N (area ka) = ´

Prob Cm Penalty = N ´ failure prob´C + area ´ N ´ failure ´

Performing Correlation and Regression Analyses

is the additional area of a given component to provide a given level of excess capacity.

The total cost is simply the sum of penalty cost and additional material cost. Minimising this total cost therefore yields the optimum level of risk (purely from the consideration of cost). The resultant computation can be summarised as shown . It is also possible to calculate the variation of optimum risk level corresponding to changes in yield stress deviations of 10 and 15 MPa, assuming the former material cost is 10% higher (say) and the later 10% lower.

Your second task is to model the variables associated with the problem.

? Eccentricity, e, can be modelled using a Beta distribution having:

a = 4 ,

b = 2 ,  = 60mm,

B(max) = 90

? Length of the section, l, can be modelled by a normal distribution having a mean of 350 mm and a standard deviation of 20 mm.

? Width of the section, w, can be modelled by a normal distribution having a mean of 250 mm and a standard deviation of 12 mm.

? To make “reasonable adjustments” in view of the Covid-19, you have a choice of modelling the load, P, using only ONE of the following: (a) The load, P, can be modelled by a triangle distribution having a mode value, M, of 1000 kN, a low value, L, of 700 kN, and a high value, H, of 1200  kN, as shown in Figure 7. The triangle distribution is used when one is unable to model the parameter confidently with a precise distribution. In this case, the designer can only give a low, high and mode values of the parameters. (b) Alternatively, P can be modelled as a Beta distribution,

You need only to model load based on (a) or (b). (Note – load P, original assignment – triangle distribution, adjusted current assignment released 2 April – modified, either triangle distribution or Beta distribution).

Perform the Monte Carlo simulation to obtain the distributions of the following output parameters: The axial stress,

s a

The flexural (bending) stress,

s f

The combined compressive stress,

s f +s a

The combined tensile stress,

s f −s a

The distance, a, of axis of zero stress.

The simulation should be performed with no less than 1,000 random sampled data points for all the parameters.

The statistics and distributions of the output parameters are to be presented in your report, together with samples of all intermediate computations.

Your last task is to explore the effect of eccentricity has on the tensile failure, and hence determine an optimum quality (as measured in terms of  variability) eccentricity on the basis of cost. A batch size of 10,000 is to be assumed for the cost calculations.

The low safety margin against tensile failure is a reason for concern as the concrete struts had already been fabricated. However, by using different setup processes the characteristics of eccentricity, e, can be varied. Essentially e can be modelled by Beta distributions between 60 and 90 mm but with different.