Executive Summary
 
 

Investigation of the stress at the extreme fibers of a steel beam was performed. Random numbers were generated to represent a 1,000 number sample size for each of the four variables involved, M, c, I, and f_y. When the stress, s, exceeded the yield stress, f_y, the beam failed. The probability of failure was analyzed and compared to the sample size to establish when the failure probability converged. Hypothesis testing was an integral part of the s analysis. Both chi-square tests for goodness of fit and hypothesis testing on the mean were performed. The distribution types were verified for all the variables as well as for s. In order to determine which variable had the most influence on s, a parametric analysis was performed. The results of that analysis indicated that M had the greatest impact on the stress outcome, which is to be expected. Confidence intervals were also calculated for the probability of failure and the tests showed that as the sample size increased, the confidence intervals converged. Overall, there was some chance for error since the statistics for each variable varied greatly, but the results of the investigation were within reason.
 
 

Table of Contents
 

1. Problem Description
2. Objectives
3. Methodology
4. Probabilistic Characteristics of s

IV-A. Histograms and Cumulative Distribution Functions for M, c, I, and f_y
IV-B. Statistical Analyses for M, c, I, and f_y
IV-C. Discussion of Histogram, CDF, and Statistical Analysis for s
IV-D. Chi-Square Test for Goodness of Fit for s
5. Yield Stress Exceedence Probability and Error Analysis
6. Parametric Analysis
7. Confidence Intervals
8. Additional Items

VIII-A. Test for the Mean with Known Population Variance
VIII-B. Goodness of Fit for Probability Distributions for c, M, I, and f_y
9. Conclusions
10. References
11. Appendices

Tables

Table 1. Probabilistic Characteristics of c, M, I, and f_y
Table 2-1. Random Variable c – Normal Distribution
Table 3-1. Random Variable M- Lognormal Distribution
Table 4-1. Random Variable I – Normal Distribution
Table 5-1. Random Variable f_y – Lognormal Distribution
Table 6. Summary Statistics of Random Variables
Table 7-1. Chi-Square Test for s
Table 8-1. Random Variable s - Identify Failure
Table 8-2. Probability of Failure and Confidence Intervals
Table 9. Hypothesis Testing for means of c, M, I, and f_y
Table 10-1. Best Fit Test for c
Table 10-2. Best Fit Test for M
Table 10-3. Best Fit Test for I
Table 10-4. Best Fit Test for f_y
 
 
 
 

Figures

Figure 1. Histogram and Cumulative Distribution Function of Random Variable c
Figure 2. Histogram and Cumulative Distribution Function of Random Variable M
Figure 3. Histogram and Cumulative Distribution Function of Random Variable I
Figure 4. Histogram and Cumulative Distribution Function of Random Variable f_y
Figure 5. Histogram and Cumulative Distribution Function of s
Figure 6. P_f vs. N
Figure 7. COV (P_f) vs. N
Figure 8. Pf vs. COV (M)
Figure 9. Pf vs. COV (I)
Figure 10. Pf vs. COV ( c)
Figure 11. Confidence Intervals