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Chapter 3: Problem 103

The article "Reliability-Based Service-Life Assessment of Aging Concrete Structures" \((J\). Struct. Engrg., 1993: 1600-1621) suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads (which can be shown to be \(=1 / \alpha\) ) is \(.5\) year. a. How many loads can be expected to occur during a 2-year period? b. What is the probability that more than five loads occur during a 2 -year period? c. How long must a time period be so that the probability of no loads occurring during that period is at most .1?

Short Answer

Expert verified
a. 4 loads; b. 0.215; c. At least 1.15 years.

Step by step solution

01

Determine the rate 位 of the Poisson process

The mean time between occurrences of loads is given as 0.5 years. This implies that \( \alpha = \frac{1}{0.5} = 2 \) loads per year. Therefore, in a 2-year period, the rate is \( \lambda = 2 \times 2 = 4 \).
02

Calculate expected number of loads in 2 years

The expected number of loads in a 2-year period is equal to the rate \( \lambda \). Hence, the expected number is 4 loads.
03

Calculate the probability of more than five loads in 2 years

To find the probability that more than five loads occur, we sum the probabilities of obtaining six or more loads and subtract from 1:\[P(X > 5) = 1 - P(X \leq 5) = 1 - \sum_{k=0}^{5} \frac{\lambda^k e^{-\lambda}}{k!}\]Substituting \(\lambda = 4\), we calculate:\( P(X \leq 5) = \sum_{k=0}^{5} \frac{4^k e^{-4}}{k!} \approx 0.785 \P(X > 5) = 1 - 0.785 = 0.215 \).
04

Determine the time period for probability of no loads to be at most 0.1

We want \( P(X = 0) = e^{-\lambda_t} \leq 0.1 \) where \( \lambda_t \) is the rate for some time period \( t \). Solving for \( t \):\[e^{-2t} \leq 0.1 \ -2t \leq \ln(0.1) \t \geq \frac{\ln(0.1)}{-2} \approx 1.15 \text{ years} \]Therefore, for the probability of no loads to be at most 0.1, \( t \geq 1.15 \text{ years} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Structural Loads
Structural loads refer to various forces or stresses acting on a structure like a building or bridge. These forces can come from natural events like wind, earthquakes, snow, or consistent factors like the weight of the building materials themselves.

In engineering and construction, it's crucial to anticipate these loads to ensure the safety and durability of structures. Ensuring that a construction can withstand both the predictable and occasional extreme loads is essential for its reliability throughout its service life. Utilizing models like the Poisson process helps in predicting and understanding these occurrences over time. This allows engineers to prepare for the average frequency of loads and design structures that can handle them effectively.

The Poisson process, as applied here, enables professionals to calculate occurrences of these structural loads, making it a fundamental tool in the engineering field.
Basics of Probability
Probability is a measure of the likelihood of an event occurring. In the case of structural loads, it helps determine how often specific forces can impact a structure within a given time frame.

For example, when calculating the probability that more than five loads will occur within two years, we can utilize the Poisson process. This involves summing up probabilities of events (loads) happening up to a certain number and then subtracting from one to find the probability of the event happening more frequently than expected.

Understanding probability allows engineers to make more informed decisions about the safety standards required for a structure, predicting potential risks, and planning accordingly. It forms the cornerstone for modeling and assessing situations where certainty isn't always possible.
Mean Time Between Occurrences: A Key Indicator
The Mean Time Between Occurrences (MTBO) is a crucial metric in predicting how often an event, like a structural load, is expected to occur. It's especially important in contexts where safety is a concern, such as in civil engineering.

Here, MTBO is expressed as the reciprocal of the rate parameter (1/\(\alpha\)), giving insight into the average time span between consecutive load events. For example, if on average there is an occurrence every 0.5 years, then the MTBO is 0.5 years. This means that, statistically speaking, we can expect a load to impact the structure once every half year.

By understanding this metric, professionals can enhance their schedules and maintenance protocols, ensuring that structures are regularly checked and maintained to handle expected loads effectively.
Expected Number of Events
The expected number of events is a predictive measure that estimates how many occurrences, such as structural loads, are likely to happen within a specific period. This measure is crucial in scenarios involving unpredictable or random events.

Using the Poisson process, the expected number can be calculated by multiplying the rate of occurrence by the time period of interest. For instance, with a rate of two occurrences per year, over two years, the expected number of loads would be 4.

This calculation helps in resource planning, budgeting, and designing structures that need to accommodate certain anticipated loads. It provides a practical means for engineers to prepare for future occurrences, ensuring safety and stability in their work.

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Most popular questions from this chapter

Each time a component is tested, the trial is a success \((S)\) or failure \((F)\). Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let \(Y\) denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of \(Y\), and state which \(Y\) value is associated with each one.

A manufacturer of flashlight batteries wishes to control the quality of its product by rejecting any lot in which the proportion of batteries having unacceptable voltage appears to be too high. To this end, out of each large lot ( 10,000 batteries), 25 will be selected and tested. If at least 5 of these generate an unacceptable voltage, the entire lot will be rejected. What is the probability that a lot will be rejected if a. Five percent of the batteries in the lot have unacceptable voltages? b. Ten percent of the batteries in the lot have unacceptable voltages? c. Twenty percent of the batteries in the lot have unacceptable voltages? d. What would happen to the probabilities in parts (a)-(c) if the critical rejection number were increased from 5 to 6 ?

In proof testing of circuit boards, the probability that any particular diode will fail is .01. Suppose a circuit board contains 200 diodes. a. How many diodes would you expect to fail, and what is the standard deviation of the number that are expected to fail? b. What is the (approximate) probability that at least four diodes will fail on a randomly selected board? c. If five boards are shipped to a particular customer, how likely is it that at least four of them will work properly? (A board works properly only if all its diodes work.)

a. Use derivatives of the moment generating function to obtain the mean and variance for the Poisson distribution. b. As discussed in Section 3.4, obtain the Poisson mean and variance from \(R_{X}(t)=\ln\) \(\left[M_{X}(t)\right]\). In terms of effort, how does this method compare with the one in part (a)?

Many manufacturers have quality control programs that include inspection of incoming materials for defects. Suppose a computer manufacturer receives computer boards in lots of five. Two boards are selected from each lot for inspection. We can represent possible outcomes of the selection process by pairs. For example, the pair \((1,2)\) represents the selection of boards 1 and 2 for inspection. a. List the ten different possible outcomes. b. Suppose that boards 1 and 2 are the only defective boards in a lot of five. Two boards are to be chosen at random. Define \(X\) to be the number of defective boards observed among those inspected. Find the probability distribution of \(X\). c. Let \(F(x)\) denote the cdf of \(X\). First determine \(F(0)=P(X \leq 0), F(1)\), and \(F(2)\), and then obtain \(F(x)\) for all other \(x\).
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