## 2017 Mathematics aaDone by: goh chin ray, alya rasyiqah, joshua lee, lim hong wei, premkumar roshan

### Introduction

According to the Singapore Civil Defense Force (SCDF), in 2005, the SCDF responded to a total 5039 fires in Singapore. (SCDF, 2005) Among the many numbers of fire cases in Singapore, the majority of them happened in residential premises. The number of cases is at an astonishing number of 3056, 60% of the total amount of fire cases in that year happened at residential housing. (SCDF, 2005) Of this there were 2889 fire cases which happened in HDB flats, that is 57% of all the fire cases which were reported in that year. (SCDF, 2005)

With the huge number of fire cases, injury and death cases are bound to happen too. There were 62 fire injuries in 2016. This was a decrease of 44.1% from 2015. (SCDF, 2016) Despite the decrease of the numbers, there is still a plenty amount of deaths and injuries caused by fires every year. How do we prevent it?

Today, there are a vast range of HDB types in Singapore. From the newly-built 40-storey flat, to the older 10-storey flat. However, a wide range of building types usually brings about countless types of different configurations and designs of interiors of the buildings. This not only affects the aesthetic design of the building, but more importantly, it affects the fire evacuation route and the efficiency of it. This would also affect the reliability of results since different kinds of buildings would come about a very different timings. Even though HDB has implemented rules to stick by, such as the rule that the walkway has to be at least 1.5m in length so that the handicap can walk through, we plan to dig deeper.

Fire usually sparks a mix of emotions, such as fear and nervousness. Shoving and pushing as people make their down is a common sight as people attempt to be swift and quick to escape the building. The evacuation time (t)not only depends on the building structure, but also on the feelings and reactions of the residents to the fire. In this experiment, we plan to dig deeper on the variables which affect a fire evacuation and from there, come out with a plan to make evacuation as effective as possible for everyone.

### 1. Variables

There are many factors that we have to put into consideration when it comes to a fire evacuation. Below are some of the possible conditions that we have put into consideration, and the assumptions that we made about each variable to the evacuation time (t):

Height of the building, number of storeys (Chosen)- Different people living in different levels of a particular building would usually have a huge difference in their evacuation time (t). For example, someone who is living in a 40th storey would take a longer time to make their way down due to the fact that they would have to walk a longer distance down to the 1st floor compared to someone who is living in a lower level than them. Therefore, the height of building would affect the evacuation time (t) being achieved.

Width of stairs and area of stairs (Chosen) - The width and area of stairs would greatly affect the evacuation time (t) recordings. This is because the size would determine how many people would be able to go down the stairs at once. With a greater number of people being able to go down the stairs at a time, it is evident that the timing achieve would be much shorter than a stairs which is only able to hold a lesser amount of people.

Reaction time of residents (Chosen) - People’s reaction time varies. Some people would quickly leave their house once they heard there is a fire, while others would spend time collecting their valuables before leaving. Overall, these difference in reaction time, affects the evacuation time (t). Take note, in this experiment, our interpretation of reaction time is that from the time they notice the fire to the when they first step out of the house.

Demographic of residents in the HDB - People living in HDB comes in all shapes and sizes. Some may be young, old, disabled, sick, etc. This would affect evacuation time (t), since different people would react differently to different situations and their performance in running down the stairs varies. This would be harder to represent in the form of data, as they are a lot of exceptions of the types of people.

Distance between houses to the nearest staircase (Chosen)- This is a simple yet impactful variable that affects the evacuation time (t) being achieved. Some houses may be located near the staircase while others might be located at the end of the building. With a longer distance to travel, the evacuation time (t) would be affected.

#### b. What is the relationship between each variable and t?

Height of the building, number of storeys - Linear. This is because as the building goes higher, the evacuation time (t) would increase together with it too. This makes the height of the building / the number of storeys directly proportional to the evacuation timing.

Width of stairs and area of stairs - Inverse. This is because as the width/ area of stairs increases, most people can fit into the stairs, and travel down together at once. This makes evacuation quicker, and timing recorded would be lesser. This also makes the width/area of stairs to the evacuation time (t) inversely proportional.

Reaction timing of residents - Linear. This is because as the reaction timing of the residents is longer, the evacuation time (t) would be even longer too. This is makes the reaction timing of residents directly proportional to the evacuation time (t) .

Distance between houses to the nearest staircase - Linear. This is because as the distance between houses to the nearest staircase increases, people would need to take a longer time to reach the stairs, affecting the evacuation time (t) , making it longer. This makes the distance between houses to the nearest staircase directly proportional to the evacuation time (t).

### 2. Tables and graphs

Around the March holidays, we went about collecting data regarding on our 4 variables. We went to a 25-storey HDB flat at Clementi Ave 2 to collect our data. The data we collected for our variables, and their respective graphs are of the following:

#### 1. Reaction time

As stated before, our interpretation of reaction time is from when the person notices the fire to when he first stepped out of the door. We recorded our reaction time when taking intervals of 5 seconds before heading out of the door and running down the stairs (1 storey). We then gathered the data and plotted the following graph:

This table shows the data that we collected. This data would be then used to plot our graph.

We calculated the rough timing for the person to go down a stairs, as such:

Time to go down the stairs = Total evacuation time (t) - Reaction time

since this experiment is recorded on a 1 storey basis, and that there will be people living at different levels, in which we will be exploring more on that in the next section.

The relationship of the reaction time and the total time is directly proportional. As the reaction time goes up with an increase of 5 seconds, the total time taken to travel 1 storey would be roughly increase by 5 seconds too.

The line of best fit equation would be : y=x+8 or

Evacuation time (t)=Reaction time +8

Subbing in the equation to when t = 10min/600s, you would get:

Evacuation time (t) = Reaction time +8

Reaction time = 600-8

Reaction time = 592 seconds

This maximum timing of 592 seconds is provided as we solely considered reaction time as the only variable and that the resident is living on the second floor. This is because, the timings recorded were based only on a 1 storey basis. However, this is of course impossible as people live in different sections and storeys of the building. In which, we would find out more in the next few sections. **This is if we take 8 seconds as the time to go down 1 storey. This is also if we didn’t add in the height as a variable yet and that the building is only 1 storey. More information is will be mentioned in the later sections**

#### 2. Height of building

Our interpretation of this is the difference in height between each storeys, and from that we can infer to the height of the building. We firstly, recorded the difference in height between each stairs, in which it is 15cm. We then times the number of steps between each storey, and from that we got the height between each storey. Which is:

15cm (Height of stairs) x 16 (Number of stairs) = 240cm/2.4m (Difference in height between each level)

240cm/2.4m =The difference of height between each level

240cm/2.4m =The height of 1 storey

We then did a few test where we time ourselves going from different amount of stories at once. The data we gathered for the time taken to go down 1 to 3 storeys are the following:

This table shows the data that we collected. This data would be then used to plot our graph.

As you can see, the relationship of the number of storeys and the total time is directly proportional. As the height increase by 2.4m, which is 1 storey, the timing would be increase by roughly 8 seconds. This shows that the time for one person to get down 1 storey is about roughly every 8 seconds.

The line of best fit equation would be : y = 8x

or Evacuation time (t) = 8 (Number of stories)

Subbing in the equation to when t = 10min/600s, you would get:

600 = 8(Number of storeys)

8(Number of storeys)=600

Number of storeys=600/8

Number of storeys=75

However, this maximum number of storeys is provided we solely consider number of storeys as the only variable, which is clearly impossible. This is because, for variables such as reaction time, it can never be 0 seconds.

#### 3. Distance between housing and nearest stairs

Our interpretation of this is the distance between housing and the nearest stairs is the measurement of the length between the edge of the stairs to the front slab of the unit. The recording of data regarding this variable is very simple, we simply used a measuring tape and measured the distance between the edge of the stairs to the front slab of the unit before recording down the data. From there, we then recorded the time taken for one person to run from the different slabs to 1 storey below them, recording their timing each time. The data we collected are as follows:

This table shows the data that we collected. This data would be then used to plot our graph.

The relationship of the distance between housing and stairs and the total time is directly proportional. This is because as the distance between the stairs and houses increases, the time the people would take the travel from their house to their stairs would increase too. As seen in the labels of the graph:

The line of best fit equation would be : y=0.51x+0.08

or Evacuation time (t) = 0.51 (Distance between housing and stairs) +0.08

Subbing in the equation to when t=10min/600s, you would get:

Evacuation time (t) = 0.51 (Distance between housing and stairs) +0.08

0.51 (Distance between housing and stairs) = 600-0.08

0.51 (Distance between housing and stairs) = 599.92

Distance between housing and stairs 1,170m (3s.f.)

However, this maximum distance between housing and stairs of 1,170m is provided we solely consider the distance between housing and stairs as the only variable, which is clearly impossible. This is because, for variables such as reaction time, it can never be 0 seconds.

#### 4. Width of stairs

Our interpretation of this is the measurement of the longer side of the stair slab. The recording of data regarding this variable is very simple, we simply used a measuring tape and measured the distance between one end of the stair to the other end before recording down the data. From there, we then recorded the time taken for four person to run from the different amount of width by assigning a specific spot where there can evacuate down only.

The relationship of the width of stairs to the total time is inversely proportional. This is because when the width increases ,more people would be able to evacuate at a time and therefore the time taken would decrease. As seen in the labels of the graph,

x = Number of people able to fit the width at once

y = Total time (seconds)

c=Fixed timing

This is such that to solve issues when the width of stairs is = 1

The best fit equation would be : y=kx+c therefore

y=0.61/x+6.45 or Evacuation time (t)=0.61/Width of stairs +6.45

This total time is used because we are solely considering the width of staircase to the timing of going down 1 level as the only variable. We will be combing all of the variables in the next sections.

### 3. FORMULATING A FORMULA

Here is the list of different variables and their relation with the total evacuation timing (t):

Reaction time

y=x+8, where:

y=Total evacuation timing (t)

x=Reaction time (R)

8=Time taken to travel down 1 storey (T)

It can also be written as: t=R+T

Number of storeys

y=8x, where:

y= Total evacuation time (t)

x=Number of storeys (S)

8=Time taken to travel down 1 storey (T)

It can also be written as: t=T(S)

If we combine what we have so far we will get: t=R+T(S)

* * This is provided we assume that the time taken to travel down 1 storey is at a constant of 8 seconds. We rounded off the results for the first 2 variable so that it is more easy to combine them later. This time to taken to travel down 1 storey is simply an assumption. The value of this time to taken to travel down 1 storey will be answered in the “width of stairs” section. **

Distance between housing and nearest stairs

y=0.51x+0.08, where:

y=Time taken from housing to stairs (ths)

x=Distance between housing and nearest stairs (D)

It can also be written as: ths=0.51(D)+0.08

If we combine what we have so far we will get: t=R+T(S)+0.51(D)+0.08

Width of stairs

y=0.61/x+6.45, where:

y=Time to go down 1 storey (T)

x=Width of stairs (W)

It can also be written as: T=0.61/W+6.45

With all of the variables collected, we will get:

t=R+T(S)+0.51(D)+0.08

Since T=0.61/W+6.45, we will get:

Final formula:

t=R+S(0.61/W+6.45)+0.51(D)+0.08)

To break it down in terms of time, you will get:

R= Reaction time

S(0.61/W+6.45)= Time to travel a given number of stories

(0.51(D)+0.08)= Time to travel from housing to stairs

Technically, in simpler forms, the formula can be broken into,

Total time=Reaction time+Time to travel a given number of stories+Time travel from housing to stairs

This diagram shows a the formula in a pictorial form

We, through this simple idea of the different timing adding up together to give us the total timing (t) we then came out this formula. This is because we have to various formulas of the different relationship have with the time, through this, we can simply combine them.

### 4. Explanation of mathematical models

As stated above, the effect of each variable of the evacuation time (t) is the following:

#### 1. Reaction time

As stated before, the relationship of the reaction time to the evacuation time (t) is directly proportional. Therefore, as the reaction time goes up, the evacuation time (t) would be affected to and it will increase too. This is because with a longer reaction time, the person would carry out their actions later.

#### 2. Number of stories

As stated before, the relationship of the number of storeys and the evacuation time (t) is directly proportional. Therefore, as the number of stories increases, the evacuation time (t) would increase too. This makes sense since the higher up you live, you would take a longer time to go down.

#### 3. Distance between houses to nearest staircase

As stated before, the relationship of the distance between housing and stairs and the evacuation time (t) is directly proportional. Therefore, the distance between the stairs and houses increases, the evacuation time (t) would increase too. This is because with a longer distance, the residents have to spend a longer time running.

#### 4. Width of stairs

As stated before , the relationship of the width of stairs to the evacuation time (t) is inversely proportional. This is because when the width increases ,more people would be able to fit the stairs and evacuate at a time and therefore the time taken would decrease too.

### 5. Analysis and conclusion

From our mathematical model, we have came up to some conclusion regarding on how to make the evacuation timing of the residents more efficient. This are the MAIN requirements:

#### Building should be no taller than 75 storeys (180m) ✓

Through calculations, we have come to a conclusion that HDB buildings should be no taller than 75 storeys/180m. This is because given that during a fire, people would crowd all over the staircase, squeezing together, taking them longer to travel down level by level. By right, this maximum number should be lower since this calculation is if we solely consider it as the only factor. HDB has been fulfilling this requirements, with the highest of their flats being 40 storeys/96m.

#### The width of stairs should not be any lower than 97.5cm ✓

Through calculations, we have come conclusion that in order of a swift evacuation, the width of stairs should be no shorter than 97.5cm, this is so that we can at least 3 people can evacuate at a time. HDB has been fulfilling this requirement, with most of their flats having a width of stairs of at least 1.3m.

Here are some of the methods which we felt that would help improve the evacuation time even further. We then consider if it was feasible or not:

#### Adding fire alarm on every level. ✓ (Feasible)

By adding fire alarm on every level, this would decrease the reaction time for residents, thus this would eventually lead to the allocation of more time for evacuation.

#### By adding more sets of staircase. ✓ (Feasible)

This method is especially effective when it comes to wide buildings where the distance between the house to the stair is a distance away. By adding another set of staircase, this would reduce the time needed to travel from the house to the staircase since the distance is being reduced, it also gives the residents an alternative route in case one of the stairway is inaccessible.

#### Maximum number of residents in each unit. ✓✗ (Possible)

This would be counted as a rule that no more 10 people should reside in the same unit, especially if these people are not related to each other, unless it has passed through inspection by the HDB. This will make sure that all the people in the building will be able to evacuate and there would not be a human congestion at the staircases.

#### Performing regular checks for the equipments and obstruction. ✓(Feasible)

This method is especially effective since it helps prevent cases such as obstruction caused by items such as potted plants during a fire. Furthermore, this helps ensure that all the walkways meets the requirements.

In conclusion, HDB has met the MAIN requirements needed for a swift evacuation. However, there might be some ways to improve it such as adding fire alarms on every levels or adding more set of staircase.

### 6. other references

Singapore Civil Defense Force (2016). Fire Incident Statistics 2016. Retrieved 28 March 2017, from www.scdf.gov.sg%2Fsites%2Fwww.scdf.gov.sg%2Ffiles%2FFire%2520Stats%25202016.pdf

Singapore Civil Defense Force (2005). Fire Incident Statistics 2005. Retrieved 28 March 2017, from www.scdf.gov.sg%2Fcontent%2Fscdf_internet%2Fen%2Fgeneral%2Fnews%2Fstatistics%2F_jcr_content%2Fpar%2Fdownload_12%2Ffile.res%2Ffire_stats_Jan_Dec2005.pdf

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