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204
book/exercises/example-quiz.json
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@ -1,204 +0,0 @@
[
{
"question": "Choose all of the following that can be included in Jupyter notebooks?",
"type": "many_choice",
"answers": [
{
"answer": "Text and graphics output from Python",
"correct": true,
"feedback": "Correct."
},
{
"answer": "Typeset mathematics",
"correct": true,
"feedback": "Correct."
},
{
"answer": "Python executable code",
"correct": true,
"feedback": "Correct."
},
{
"answer": "Formatted text",
"correct": true,
"feedback": "Correct."
},
{
"answer": "Live snakes via Python",
"correct": false,
"feedback": "I hope not."
}
]
},
{
"question": "Which of these are used to create formatted text in Jupyter notebooks?",
"type": "multiple_choice",
"answers": [
{
"answer": "Wiki markup",
"correct": false,
"feedback": "False."
},
{
"answer": "SVG",
"correct": false,
"feedback": "False."
},
{
"answer": "Markdown",
"correct": true,
"feedback": "Correct."
},
{
"answer": "Rich Text",
"correct": false,
"feedback": "False."
}
]
},
{
"question": "Enter the value of $\\pi$ to 2 decimal places.",
"type": "numeric",
"answers": [
{
"type": "value",
"value": 3.14,
"correct": true,
"feedback": "Correct."
},
{
"type": "range",
"range": [
3.142857,
3.142858
],
"correct": true,
"feedback": "True to 2 decimal places, but you know $\\pi$ is not really 22/7, right?"
},
{
"type": "range",
"range": [
-100000000,
0
],
"correct": false,
"feedback": "$\\pi$ is the AREA of a circle of radius 1. Try again."
},
{
"type": "default",
"feedback": "$\\pi$ is the area of a circle of radius 1. Try again."
}
]
},
{
"question": "Enter the value of $\\pi$ to 2 decimal places.",
"type": "numeric",
"precision": 2,
"answers": [
{
"type": "value",
"value": 3.14,
"correct": true,
"feedback": "Correct."
},
{
"type": "range",
"range": [
3.142857,
3.142858
],
"correct": true,
"feedback": "True to 2 decimal places, but you know $\\pi$ is not really 22/7, right?"
},
{
"type": "range",
"range": [
-100000000,
0
],
"correct": false,
"feedback": "$\\pi$ is the AREA of a circle of radius 1. Try again."
},
{
"type": "default",
"feedback": "$\\pi$ is the area of a circle of radius 1. Try again."
}
]
},
{
"question": "Determine the output of the following Python code:",
"code": "a=\"1\"\nb=\"2\"\nprint(a+b)",
"type": "multiple_choice",
"answers": [
{
"answer": "1",
"correct": false,
"feedback": "No. When strings are operated on by +, they are concatenated."
},
{
"answer": "2",
"correct": false,
"feedback": "No. When strings are operated on by +, they are concatenated."
},
{
"answer": "3",
"correct": false,
"feedback": "No. When strings are operated on by +, they are concatenated."
},
{
"answer": "12",
"correct": true,
"feedback": "Yes. The + operator will concatenate the strings \"1\" and \"2\"."
},
{
"answer": "error",
"correct": false,
"feedback": "No. The + operator for strings performs string concatenation."
}
]
},
{
"question": "The variable mylist is a Python list. Choose which code snippet will append the item 3 to mylist.",
"type": "multiple_choice",
"answers": [
{
"code": "mylist+=3",
"correct": false
},
{
"code": "mylist+=[3]",
"correct": true
},
{
"code": "mylist+={3}",
"correct": false
}
]
},
{
"question": "Which of these is the ratio of a circle's circumference to its diameter?",
"type": "multiple_choice",
"answers": [
{
"answer": "$\\pi$",
"correct": true,
"feedback": "Correct."
},
{
"answer": "$\\frac{22}{7}$",
"correct": false,
"feedback": "$\\frac{22}{7}$ is only an approximation to the true value."
},
{
"answer": "3",
"correct": false,
"feedback": "This is a crude approximation to the true value."
},
{
"answer": "$\\tau$",
"correct": false,
"feedback": "True for the ratio of the circle's circumference to its radius, not diameter."
}
]
}
]

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book/exercises/exercise-dam.md
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@ -1,154 +0,0 @@
# Dam and River
You are tasked to analyse the safety of a dam and the downstream river system. Downstream of the dam there is a dike ring protected by two main dike sections which must be able to contain water released from the reservoir. The dike sections are connected to each other and form a continuous boundary along the same side of the river. Let $P(F_1)=0.01$ and $P(F_2)=0.01$ denote failure probability of dike section 1 and section 2 respectively. There appears to be a correlation between failures of both sections, $\rho_{1,2}=0.9$, and the figure below shows the effect of the correlation coefficient on the joint failure of the two sections.
%commented paragraphs were used for an exam question with a fault tree
%You are tasked to analyse the safety of a dam. The dam itself can fail due to instability, and another failure mechanism is failure of the spillway due to uncontrolled erosion that could reach the body of the dam. If water levels in the reservoir become high, water is released from the reservoir into the spillway due to overflow, which happens approximately once every 10 years.
%When water is released through spillway, there are two situations that could lead to a weak spillway prone to erosion and failure: 1) hidden damage to the spillway structure that cannot be detected in yearly inspections, and 2) a crack in the concrete of the spillway which has been missed during the yearly inspection.
%Downstream of the dam there is an area protected by two main dike sections which must be able to contain water released from the reservoir. A dike section fails if the river discharge exceeds a critical discharge $Q_c$ with a probability of exceedance of 1/100 per year, which leads to flooding of the area. There appears to be a correlation between failures of both sections, $\rho_{1,2}=0.9$, and the figure below shows the effect of the correlation coefficient on the joint failure of the two sections.
```{figure} ../figures/system-corr-coeff.PNG
---
width: 400px
name: corr-coeff-or
---
Effect of correlation coefficient joint failure, $P(A \cap B)$ (assume that $P(A \cap B)=0.003$ at $\rho=+0.9$, where the arrow is located).
```
**Question 1:** Estimate the probability of flooding of the area protected by the two dike sections. Assume that the failure probability of each mechanism is 0.01, and they are _not_ independent; however, you can use the figure above to quantify this.
```{admonition} Answer
:class: tip, dropdown
Flooding of the protected area occurs when at least one of the two dike section fails. This is a series system. Thus, probability of flooding of the protected area can be determined by:
$$
P(\textrm{flooding}) &= P(\textrm{dike sections 1 fails}\:\cup \textrm{dike section 2 fails})\\
P(F_1 \cup F_2) &= P(F_1) + P(F_2) - P(F_1 \cap F_2)
$$
Based on the figure and the correlation $\rho_{1,2} = 0.9$, the joint probability of failure $P(F_1 \cap F_2)=0.003$ and the series failure probability is
$$
P(\textrm{flooding}) = 0.01 + 0.01 - 0.003 = 0.017 \: \mathrm{/year}
$$
```
## Dam repair: probabilistic planning
The dam has to be repaired. The responsible minister wants to make sure the whole project is finished within 12 months, and only accepts a probability of 10% that actual construction works exceed this duration. The dam reinforcement consists of two activities. The second activity can only start when the first one has been finished. The durations of both activities are uncertain and normally distributed. The first activity has a mean duration of $\mu_1=4$ months and a standard deviation of $\sigma_1 = 1$ month; for the second activity $\sigma_2 = 2$ months.
**Question 2:** Compute the mean duration of the activity 2, so that the project duration would fall within the criterion given by the minister.
```{admonition} Answer
:class: tip, dropdown
Let $T_1$ and $T_2$ denote the duration of activities 1 and 2; $T$ denotes the total duration; $T_1$ and $T_2$ are normally distributed:
$$
T_1 \sim \mathcal{N}(
\mu_1=4,\sigma_1=1) \\
T_2 \sim \mathcal{N}(\mu_2=?,\sigma_2=2)
$$
$T = T1 + T2$ is also normally distributed, where
$$
\mu_T &= \mu_1 + \mu_2 \\
\sigma_T &= \sqrt{\sigma_1^2 + \sigma_2^2} = \sqrt{5} = 2.326
$$
The probability of project duration greater than 12 months can be found with the complementary standard normal CDF, and should be a maximum of 0.1:
$$
1-\Phi\left[\frac{12\:\mathrm{months}-\mu_T}{\sigma_T}\right]=0.1
$$
Applying the inverse CDF allows us to solve for $\mu_1$:
$$
\frac{12-\mu_T}{\sigma_T} &=\Phi^{-1}[0.9] \\
\frac{12-(\mu_1 + \mu_2)}{\sigma_T} &=1.2816
$$
$$
\mu_1 &= 12 - 1.2816 \cdot \sigma_T - \mu_2 \\
\mu_1 &= 12 - 1.282(2.326) - 4 = 5.018\:\mathrm{months}
$$
<!-- Limit state equation: total duration of constuction will not exceed 12 months, thus:
$$
Z = 12 - (T_1 + T_2)
$$
The event $Z<0$ reflects that the actual construction works exceed the duration the minister want (12 months). It is acceptable that the actual construction works exceed this duration with a probability of 10%. Thus: $P(Z<0) = 0.1$. This leads to $\beta_z = 1.28155$
Application of Level II reliability calculation:
$$
\mu_z = 12 - (\mu_1 + \mu_2) = 8 - \mu_2 \\
\sigma_z = \sqrt{\sigma_1^2 + \sigma_2^2} = \sqrt{5} = 2.326 \\
8 - \mu_2 = \beta_z \cdot \sigma_z = 2.86 \\
\mu_2 = 5.13 \: \mathrm{months}
$$ -->
```
## Acceptable risk
The government wants to study the acceptable risk and the acceptable probability of dam failure. Two criteria are used for acceptable risk:
- Societal risk (SR): an FN limit line: $P_f \leq 10^{-2}/n^2$.
- Individual risk (IR): $IR \leq 10^{-4}$ per year.
We know that mortality in case of dam failure is $F_{d|f} = 0.1$ and that the population living downstream of the dam is 1000 people.
**Question 3:** Plot the FN limit line and indicate the acceptable probability of failure according to both criteria (individual risk, societal risk).
````{admonition} Answer
:class: tip, dropdown
```{figure} ../figures/exercise-dam-fn-limit-line.png
---
width: 500px
---
```
````
**Question 4:** Provide an advice on the acceptable probability of failure for the dam. Give the proposed numerical value and a short motivation.
```{admonition} Answer
:class: tip, dropdown
Based on individual risk criterion, we have:
$$
IR = P_f P_{d|f} \leq IR_{acc}
$$
where $P_{d|f}=0.1$. Hence, the acceptable probability of failure is:
$$
P_f \leq \frac{IR_{acc}}{P_{d|f}} = \frac{10^{-4}}{0.1} = 10^{-3} \: \mathrm{/year}
$$
Based on societal risk criterion (FN-Curve):
Acceptable failure probability can be determined by the FN-Curve:
$$
P_f = 1 - F_N(n) \leq \frac{C}{n^\alpha} = \frac{10^{-2}}{n^2}
$$
Based on given information above, expected number of casualties per dam failure event is $n = F_{d|f}\cdot 1000 = 100$.
$$
P_f &\leq \frac{10^{-2}}{100^2} \\
P_f &= 10^{-6} \: \mathrm{/year}
$$
The most stringent of the two criteria applies, so the proposed safety standard would be $10^{-6}$ per year for the given inputs.
```

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book/exercises/exercise-fn-curve.md
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(ex_fn_curve)=
# FN Curve
A small town is located in a seismic region threatened by two faults. Fault A has a 10% probability of occurring each year, which would cause 10 fatalities, and Fault B has a 1% chance of occurring, which would cause 100 fatalities. Construct an FN curve for the town. Assume that the town is risk averse and that the tolerable risk limit is governed using a constant of C=1.
**Question 1:** Compute the expected annual fatalities for the town.
```{admonition} Answer
:class: tip, dropdown
$$
E(N) = P_1 \cdot N_1 + P_2 \cdot N_2 = 0.1 \cdot 10 + 0.01 \cdot 100 = 2
$$
```
**Question 2:** If the town improved the safety of its buildings, what would be the maximum allowable number of fatalities that would be considered acceptable due to an earthquake on fault A?
```{admonition} Answer
:class: tip, dropdown
Using the limit curve
$$
1 - F_n(n) = C / n^\alpha
$$
$$
0.1 = 1 / n^2
$$
$$
n = \sqrt{10} \approx 3
$$
```
**Question 3:** What is the maximum value for the probability of an earthquake on Fault B that would be considered acceptable by the town?
```{admonition} Answer
:class: tip, dropdown
Using the limit curve
$$
1 - F_n(n) = C / n^\alpha
$$
$$
1 - F_n(100) = C / 100^2 = 0.0001 = 10^{-4}
$$
```
**Question 4:** Create an FN curve for this situation, including the limit line. Be sure to clearly label the axes and the values of key points on your diagram.
```{note}
If you are not sure what the limit line is, wait for this question until you read the Safety Standards Section (Risk Evaluation chapter).
```
````{admonition} Answer
:class: tip, dropdown
```{figure} ../figures/exercise-FN-curve.png
---
width: 600px
name: FN-curve
---
FN-curve for exercise 4
```
````

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book/exercises/exercise-paint.md
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# Paint System
In this assignment we will consider a paint system that prevents rust and corrosion on a steel structure. A high quality paint system costs €40 per square meter and has a failure probability of 0.002, which assumes that the old paint system is completely stripped off and the steel surface is cleaned prior to installation, which has an additional cost of €20 per square meter.
Common cost cutting measures are to use a cheaper paint that costs half as much, but is five times more likely to fail, or to simply sand the old paint rather than stripping it off prior to applying a new paint, which increases the failure probability by 10, but only costs €5 per square meter.
During operation of the structure, if a rust spot is found it must be repaired immediately by removing the damaged paint and replacing it. The repair has an additional cost of €500 per square meter.
Construct a decision tree to evaluate the different options for steel preparation and paint, then use it to answer the following questions. Assume that the owner wants to minimize total costs, that the paint system only fails once in any reference period and that the probability of failure in each year is independent. All costs and probabilities are calculated on a square meter basis.
%**Question 6:** Upload a photo/scan of your decision tree. Make sure the values and parameters for each branch are clear.
````{admonition} Decision Tree
:class: tip, dropdown
This problem can be visualized using a decision tree as shown below. However, it results in a different interpretation of the problem than that shown in the solution. For example, consider the top branch at the far right, where 'No rust spot' is assigned a probability of 0.998. This is consistent with the formulation of a decision tree, but not consistent with the problem statement, which implies that the €60 cost associated with cleaning and painting is always done when replacing the paint system (thus a probability of 1.0). This is why the values in the decision tree are slightly different than those in Question 3, for example.
Although the decision tree is not part of the exam, this illustrates an important lesson: it is very easy to accidentally make probability calculations that are inconsistent with the situation you are trying to evaluate!
```{figure} ../figures/exercise-decision-tree.png
---
width: 600px
name: decision-tree
---
Decision tree.
```
````
**Question 1:** What is the probability of each paint system failure per square meter over one year?
```{admonition} Answer
:class: tip, dropdown
| Paint | Cleaning | $p_f$ [1/yr] |
|------- |--- | --- |
| High-Quality | Stripping | 0.002 |
| High-Quality | Sanding | 0.02 |
| Low-Quality | Stripping | 0.01 |
| Low-Quality | Sanding | 0.10 |
```
**Question 2:** What is the probability of each paint system failure per square meter over a 10 year period?
```{admonition} Answer
:class: tip, dropdown
$$P_{f,10y} = 1 - (1 - p_{f,1y})^{10}$$
| Paint | Cleaning | $p_f$ [1/10-yr] |
|------- |--- | --- |
| High-Quality | Stripping | 0.02 |
| High-Quality | Sanding | 0.18 |
| Low-Quality | Stripping | 0.10 |
| Low-Quality | Sanding | 0.65 |
```
**Question 3:** What is the expected cost per square meter over a one year period if the owner uses the high quality cleaning and high quality paint?
```{admonition} Answer
:class: tip, dropdown
$$
40 + 20 + 0.002 \cdot (500 + 40 + 20) = \euro{} 61.12 /\mathrm{m^2}
$$
```
**Question 4:** What is the optimal paint system for a one year period?
```{admonition} Answer
:class: tip, dropdown
$$C_{paint} + C_{cleaning} + P_{f,1y} \cdot (C_{paint} + C_{cleaning} + C_{repair})$$
| Paint | Cleaning | Expected cost [€/yr] |
|------- |--- | --- |
| High-Quality | Stripping | 61.12 |
| High-Quality | Sanding | 55.90 |
| Low-Quality | Stripping | **45.40** |
| Low-Quality | Sanding | 77.50 |
Low Quality Paint + High Quality Cleaning
```
**Question 5:** What is the optimal paint system for a 10 year period?
```{admonition} Answer
:class: tip, dropdown
$$C_{paint} + C_{cleaning} + P_{f,10y} \cdot (C_{paint} + C_{cleaning} + C_{repair})$$
| Paint | Cleaning | Expected cost [€/10 yr] |
|------- |--- | --- |
| High-Quality | Stripping | **71.10** |
| High-Quality | Sanding | 144.70 |
| Low-Quality | Stripping | 91.63 |
| Low-Quality | Sanding | 366.94 |
High Quality Paint + High Quality Cleaning
**NOTE:** If you're getting slightly different results, don't panic! Most likely you have approximated the probabilities computed above in question 2.
```
Assume the owner has decided to use the cheap paint system, and is simply going to sand it off (i.e. the cheap cleaning method) and re-apply the same system every year (i.e., for these problems consider a 1-year reference period). You have been asked to assess whether it is worthwhile to use a quick sonic test system to prevent failures during the year, and if it is, the number of tests that should be performed per year. Each test costs about €3 per m2, and if a weak spot is found, a new layer of paint can be easily applied, lowering the failure probability by a factor of 0.50.
**Question 6:** What is the annual expected cost due to paint failure if no sonic inspections are done?
```{admonition} Answer
:class: tip, dropdown
$$
E(D) = P_f \cdot D = 0.1 \cdot 525 = 52.50
$$
```
**Question 7:** What is the annual expected total cost for the paint system if no sonic inspections are done?
```{admonition} Answer
:class: tip, dropdown
Using the decision tree from above: €77.50/m2
```
**Question 8:** What is the change in risk for 3 sonic inspections?
```{admonition} Answer
:class: tip, dropdown
$$
\Delta E(D) = (P_{f,0} - P_{f,new}) \cdot D = 45.94
$$
```
**Question 9:** What is the annual expected cost due to paint failure if 3 sonic inspections are done?
```{admonition} Answer
:class: tip, dropdown
The expected damage is:
$$
E(D)=P_f \cdot D={0.5}^3 \cdot 0.1 \cdot 525=6.56
$$
But the 3 inspections cost €9, thus the total expected cost is €15.56. It is incorrect to multiply the inspection cost by the failure probability. NB: this calculation implies the inspections occur regardles of when the failure happens.
```
**Question 10:** What is the maximum number of inspections per year that can be performed that are cost-effective? Note: Round your answer towards the smallest integer.
```{admonition} Answer
:class: tip, dropdown
For the investments to be cost effective:
$$
\frac{\Delta E(D)}{I}>1
$$
Which results in a maximum of 17 inspections per year.
```
**Question 11:** What is the optimum number of sonic inspections to perform each year? Plot a graph, with the Risk, Investment and Total cost curves and clearly indicate the optimum.
````{admonition} Answer
:class: tip, dropdown
As shown in the figure, the optimum is 3 per year.
```{figure} ../figures/exercise-optimization.png
---
width: 600px
name: optimization-curve
---
Optimization curve for exercise 12
```
````

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book/exercises/exercise-sample-exam.md
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@ -1,248 +0,0 @@
# Sample Exam Questions
<!-- ```{admonition} Sample Exam Answers
:class: tip
Answers to the sample exam will be made available on this page after Monday of Week 2.8.
``` -->
## General Questions
**Question 1:** Indicate which of the following statements is true (multiple answers possible):
1. A parallel system gets stronger if there is a strong positive correlation between failure of the elements.
2. Dependence between elements is only determined by dependence in loads.
3. In case of a full dependence between two elements $A$ and $B$, it holds that $P(B|A)=1$.
4. A flood defence system that is a single line of defence composed of multiple dike sections would best be characterized as a series sytem, not as a parallel system.
5. A prison with multiple walls and fences would best be characterized as a series system, not as a parallel system.
```{admonition} Answer
:class: tip, dropdown
Options 3 and 4 are true.
```
## Flood Protection for a Bay
A flood protection system for an area at a bay consists of a first line of defence (a dam on the coast) and a second line (a dike around the bay) to protect the city against a storm surge that can cause a high water level in the bay. The design team considers two strategies: 1) a single dike around the bay, or 2) a dike around the bay plus a dam (see figure).
```{figure} ../figures/exercise-sample-exam-bay.png
---
width: 400px
name: flood-protection
---
```
**Question 1:** from the perspective of flooding in the city, the function of the dam and dike together is best described as a (choose one):
a. series system
b. parallel system
c. a single component
d. none of the above
```{admonition} Answer
:class: tip, dropdown
Option b is true.
```
**Question 2:** how will dependence between the dam and dike change the system failure probability? (choose one)
a. increase
b. decrease
c. no change
```{admonition} Answer
:class: tip, dropdown
Option a is true.
```
**Question 3:** describe whether or not there is dependence between the dam and dike, and what the source could be.
```{admonition} Answer
:class: tip, dropdown
It is likely that both structures are constructed of the same material (e.g., local soil), which would respond in a similar way to the storm surge and waves. The structures are also likely to be operated and maintained by the same government agency, which would result in dependence due to human error and maintenance problems.
```
---
The damage in case of flood protection system failure is 1 billion $10^9$ Euros. We compare two different investments in the system. In strategy 1 the system failure probability becomes $10^{-3}$ per year at a cost of 10 million ($10 \cdot 10^{6}$). In strategy 2 the system failure probability becomes $3 \cdot 10^{-4}$ per year at a cost of 50 million.
**Question 4:** for which value of the interest rate $r$ is strategy 2 the most interesting? (You may consider an infinite lifetime of investments).
```{admonition} Answer
:class: tip, dropdown
Find $r$ such that the difference in expected damage between strategy 1 and 2, $\Delta p_{f,sys}D/r$, is greater than the difference between the investments, $I_2-I_1$, where the net present value over an infinite lifetime is $1/r$.
$$
r < \frac{\Delta p_{f,sys} D}{I_2-I_1} = \frac{(0.0010-0.0003)\cdot 1000 \text{m€}}{50\: \text{m€}-10\: \text{m€}}=0.0175
$$
Can also set up as solving for $r$ when comparing total costs:
$$
I_1+\frac{p_1 D}{r}>I_2+\frac{p_2 D}{r}
$$
```
---
A full scale risk assessment is made for the system with probabilities $P(N>10)=10^{-3}$ per year and $P(N>100)=5 \cdot 10^{-5}$ per year. A so-called limit line is given with values with $C = 10^{-1}$; and $\alpha = 2$.
**Question 5:** determine whether this situation is acceptable. You may answer using a few sentences, calculations or by submitting a sketch, but either way, you must support your answer quantitatively. If you use a sketch, indicate the scale and relevant points on the plot.
````{admonition} Answer
:class: tip, dropdown
First create an FN curve for the risk assessment
**NOTE:** _the figure below is slightly incorrect; the horizontal lines should be to the right of the "dots;" in other words, the 1E-05 value should be above N=100. This makes it more obvious why the system does not meet the safety criteria._
```{figure} ../figures/exercise-sample-exam-limit-line.png
---
width: 400px
name: limit-line
---
```
Next, check that each point on the line is below the limit line:
$$
1-F_N(n)<\frac{C}{N^α}
$$
The probability of exceeding 100 fatalities is too high since $5E-5>0.1/100^2=1E-5$. The same is true for 10 fatalities.
````
---
The flooding probability is 1/50 per year and damage is 150 million € ($10^6$). It is questioned whether additional protection is needed and what would be the best solution between the following:
- Strategy A: raise the dike
- Strategy B: raise the dike and build a mangrove forest in front which reduces waves
The table below indicates the costs of the strategies and how much they reduce the initial failure probability. Assume the interest rate is 4%.
```{list-table} Mitigation Strategy A and B.
:header-rows: 1
* - Strategy
- Failure probability factor
- Costs, million € ($10^6$)
* - Do Nothing
- 1.0
- 0
* - A
- 0.5
- 10
* - B
- 0.2
- 18
```
**Question 6:** what is the most cost-effective or optimal strategy? Explain your answer.
1. Strategy A
2. Strategy B
3. neither A nor B
````{admonition} Answer
:class: tip, dropdown
Strategy B is best.
Various costs (NPV value) are calculated in the table below, and Strategy B is the lowest total cost we well as highest risk reduction.
```{figure} ../figures/exercise_sample_opt.png
---
width: 500px
---
```
````
---
Consider the individual risk in two areas of the city by the bay, above. The acceptable individual risk is $10^{-5}$ per year. The conditional probability of death due to flooding is dependent on the water depth and indicated next to the graph. The individual risk level in area A is $10^{-5}$ per year and the water depth is 3 m. The individual risk level in area B is $10^{-5}$ per year and the water depth is 2 m.
**Question 7:** what should be the maximum allowable failure probability of the dike on the bay, $P_{\text{flooding}}$, to meet the standard for individual risk, $IR$?
```{figure} ../figures/exercise-sample-exam-mortality.png
---
width: 400px
---
```
````{admonition} Answer
:class: tip, dropdown
Individual risk is:
$$
IR = P_{\text{flooding}} \cdot P_{\text{death}|\text{flooding}}
$$
where the values for the conditional term can be found using the mortality figure, and solve for $P_{\text{flooding}}$.
```{figure} ../figures/exercise_sample_IR.png
---
width: 500px
---
```
The norm is governed by the most strict safety requirement, thus, $P_{\text{flooding}}$ should be 0.0002 per year.
````
## Component Reliability
The shaded region, $\Omega$, in the figure below is the failure domain of an unknown object (the shaded zone extends further towards positive infinity for $x$ and $y$). $p_f$ is the probability of failure of that object, such that:
$$
p_f = \iint_\Omega f(x,y)dxdy
$$
Where $x$ and $y$ are random variables and $f(x,y)$ is the joint PDF.
```{figure} ../figures/exercise-sample-exam-failure-domain.png
---
width: 400px
name: failure-domain
---
```
**Question 1:** identify a specific real-life object of your choosing (it can be anything!) that can be described by this diagram and failure probability. Describe the object and provide a definition of failure using words only, no equations. Mention whether $X$ and $Y$ each acts as a load or a resistance.
```{admonition} Answer
:class: tip, dropdown
It is clear that $X$ and $Y$ are both loads, since high values of both variables lead to failure. There are many possible ‘objects’ that are governed by two loads:
- Dike: x and y are water level and wave load, failure is excessive overtopping
- Steel rod: 2 loads exceed strength
- Beam: self-weight and length are loads (but this is non-linear)
- Island: x and y are sea level rise and settlement, capacity is elevation; failure is flooding
- 2 things on object with limiting capacity. E.g.: people on a couch or bench, birds on a tree branch
- Literally any two loads would be acceptable, as long as the capacity of the object was a sum of both of them (if not a sum, it’s non-linear)
%Honorable mentions: exam grading, stoof pot, student knowledge on exam material, ear damage at a music festival, dog and cat in box, exam desk stability, fingers squeezing a pen, 2 heaters in a room
<!-- see below for a list of examples.
*The following text is included in the exaplanation of this sample exam question to give extra insight; you are not expected to reproduce this on the MUDE exam:
The limit state is a line, which means the variables $X$ and $Y$ should have a linear relationship and can be represented as
$$Z=y_0-(y_0/x_0)\cdot x-y$$
or if $y_0=x_0$ is assumed, simply
$$Z=y_0-x-y$$
We can also describe the random variables x and y; specify possible values for the mean ($\mu_x$, $\mu_y$) and standard deviation ($\sigma_x$, $\sigma_y$)If the object is generally stable the mean values should be below the line; specifically any value such that $mu_x<x_0$, $mu_y<y_0$ and $mu_y<y_0-mu_x$.
Since we want to find the design point we should use FORM, not Monte Carlo. If an object with a non-linear LSF is chosen, or non-normal random varialbes, this will be an approximation of pf but the design point is reliable. If x and y are described by a normal distribution, pf can be solved for exactly using the std normal dist for z with mu_z=y0-mu_x-mu_y and sigma_z=sqrt(sigma_x^2+sigma_y^2). Beta is the distance between point {mu_x, mu_y} and and the LSF such that the line is perpendicular to the LSF. The design point is the point on the LSF where the perpendicular line crosses which is the maximum probability density.
Common mistakes on this question:
- Not describing what x, y, x0 or y0 were
- Using Z=R-S – this was a big error! Note: ok to write Z=R-S only if you clarify that S was a function of X and Y, and that x0 and y0 determine R (not necessarily as random variables)
- Some tried to assign Y as a resistance with no explanation
- Accidentally flipping the signs on R or S
- Correct description of X and Y as loads, but not writing an equation that showed it
- Writing Z = R - (X + Y) without an explanation of what R is
- Combining x and y in the limit-state function in a way that makes it non-linear
- FORM gives you the design point; Monte Carlo does not
Comments about answers that were not quite right (points were not deduced for these):
- Be careful with what you assign as a random variable: technically the length of a beam under load acts as a load variable (longer beam is more fragile), but you wouldn’t always include it as a random variable if the variance in the load is much bigger.
- For large structures that are built once (e.g., a bridge in a valley) you can measure the width and construct the bridge, so it’s not really practical to consider these as the main loads or resistances, even though mathematically speaking (alpha value) it is true.
- Combining load variables that had different units
- Making time a random variable
- Creating an object and/or limit state function that was non-linear (e.g., X*Y instead of X+Y)
Common ‘objects’: -->

7
book/exercises/overview.md
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@ -1,7 +0,0 @@
# Exercises
A number of exercises are included in this chapter that illustrate the previously discussed concepts of risk and reliability in a probabilistic design context. At the end a set of **sample exam** questions are included.
---
_Exercises are largely drawn from decades of old course material from many contributors; this chapter can be referred to using standard Editor attribution._