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 straight forward 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.
Your first task is to investigate if there is a reasonable degree of correlation between uncertainty and actual stress in the section. If there are reasons to believe that correlation exists between certain factors then a regression analysis needs to be performed. A sample consisting of 22 data items, which is shown in Table 1, is then collected.
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.
Question: Is there sufficient evidence to accept the manufacturer’s claim that the mean yield stress of the material is 130 MPa? What would you recommend? Do you have reason to suspect “noise” from the data set?
You then feel that the vague classification of uncertainty into three arbitrary categories, although this serves the first task adequately well, does not fit very well with a rigorous riskbased modelling framework. The feasibility of this framework is to be illustrated with a “design case”. This proposed design case involves a given load, and the structure is to be designed using a given material.
3 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. risk of 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 .
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:
For the worked example you are to formulate a deterministic and a probabilistic solution procedure for the example problem.
Your first task is to solve the problem deterministically, employing equations (11) – (13), using Excel spreadsheets or MathCAD. The members of staff of the company are familiar with the deterministic solution procedure and this should be used as a starting point for the worked example before introducing the probabilistic solution procedure of simulation.