Consider a simple bolted connection between structural members. In the structural system at hand, all bolts always carry equal loads. If there are n bolts in the connection, and the total load being carried by the connection is L [kN], then each bolt carries L/n.
For the critical failure mode, the load capacity of individual bolts can be modeled as a Normal random variable with mean μ = 100 [kN] and standard deviation σ = 2 [kN].
For a two-bolt connection, if both bolts were exactly average, the load capacity would be 2μ = 200 [kN]. However, the connection fails if either of the two bolts fail. Thus, if one of the bolts was weaker than average with an individual load capacity less than μ, say 95 [kN], then the load capacity of the connection would be 2 · 95 = 190 [kN], which is less than 2μ.
The load capacity of an n-bolt connection is n times the strength of the weakest bolt. Thus, the strength of the weakest bolt determines the load capacity of the connection.
- What is the average and standard deviation of the load capacity for one-bolt connections? (Yes, this is the obvious answer.)
- What is the average and standard deviation of the load capacity for two-bolt connec- tions? (No, there is no simple analytic formula for this question.)
- Estimate and plot and/or tabulate the averages and standard deviations of the load capacity of n-bolt connections, for n = 1, 2, . . . , 10. (Yes, you will want to use the computer on this problem.)
- Plot and/or tabulate the number of bolts in a connection "n" (on the abscissa: hor- izontal axis), versus the probability that the load capacity of a connection is greater than or equal to nμ (on the ordinate: vertical axis). The plot and/or table should include the range n = 1, 2, . . . 10. Hint: this is not a direct extension of the previous parts. This question can be answered using a relatively simple analytic formula.