Reference no: EM132281887

Learning Outcomes

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 simple statistical inferences.

3. Understand the principle of risk-based approach to data analysis through a mathematically simple 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.

Tasks

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.

Table 1: Data for Correlation & Regression Analysis

Serial Number	Mean Load (KN)	Load variation factor	Design sd (MPa)	Actual sa (MPa)	sa/sd
1	97	2	130	135	1.038
2	90	3	120	180	1.500
3	83	2	120	117	0.975
4	95	3	120	234	1.950
5	88	2	120	122	1.017
6	101	1	120	96	0.800
7	89	2	120	115	0.958
8	86	3	120	208	1.733
9	85	1	120	89	0.742
10	92	3	120	247	2.058
11	87	1	130	105	0.808
12	102	1	120	80	0.667
13	84	2	120	108	0.900
14	93	3	120	195	1.625
15	90	2	130	104	0.800
16	99	1	120	78	0.650
17	93	3	130	205	1.577
18	97	1	130	100	0.769
19	94	3	120	240	2.000
20	100	2	120	150	1.250
21	97	3	120	180	1.500
22	93	2	120	122	1.017

Task 1 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.

Task 2 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.

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?

Table 2: Yield Test Result from the Sample

S/No	Yield Stress
1	131.6
2	123.8
3	108.0
4	116.7
5	131.5
6	120.6
7	145.0
8	115.4
9	124.3
10	126.3

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 risk- based 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.

Task 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.

Task 4 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.

Task 5 Comment on the appropriateness of current safety factor, using data from Table 1. Recommend revised level of safety factor. Note: you can assume the current safety factor(s) to be constant(s).

Tasks

For the worked example you are to formulate a deterministic and a probabilistic solution procedure for the example problem.

Task 6 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.
You may use the following data:
e = 85 mm
l = 350 mm w = 250 mm c = 175 mm
I = wl³/12

P = 1000 kN

What is the safety factor/margin against tensile failure?

Task 7 Your second task is to model the variables associated with the problem.

• Eccentricity, e, can be modelled using a Beta distribution having: α = 4, β = 2 , A(min)= 60mm, B(max)= 90 mm, as shown in Figure 6..
• 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.
• 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 4. 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.
• The value of c can be taken as L/2.

You are to model the parameters with the given information. Random sampling of the parameters for the simulation process is to be done on the basis of above distributions.

Task 8 Perform the Monte Carlo simulation to obtain the distributions of the following output parameters:

The axial stress, σ_a.
The flexural (bending) stress, σ_f. The combined compressive stress, σ_f

σ_a.

The combined tensile stress, σ_f - σ_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.

Task 9 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 α and β values (i.e. different skewness).

Attachment:- Data_analysis_Assignment_Portfolio.rar