Bridges
are one of the most important architectural infrastructures, especially in
today’s world. They help a person to commute from one place to another without
having difficulties to get passed a busy road, dangerous railway line, or water
body. In order to build a bridge, the contractor needs to first measure its
length, from its starting point, till its endpoint in order to build an
effective bridge that connects two lands. The length of the bridge is also
essential to determine whether the bridge is high enough over the water body to
prevent it from getting damaged by natural disasters. The type of bridge constructed
depends on the distance that must be covered and the amount of load it would
have to hold. Other than that, many other factors must be taken into
consideration in order to build a well-made bridge; therefore, engineers use
many mathematical concepts, in particular trigonometry and calculus when
constructing a bridge. But it doesn’t just stop at the construction of bridges.
The pretty designs of the bridges are also influenced by many mathematical
concepts, as a sturdy bridge is less about the materials used and more about
the design incorporated. Social and ethical concepts also influence the
building of bridges. Hence, the strength and stability of a bridge are
immensely important to every person who will make use of this architectural
structure as many of the times, their lives are dependent on it.

Figure 1:
Use of Triangles in Bridges

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Figure 2:
Labeled Suspension Bridge

This topic is very interesting to
me because I’ve always been intrigued by the different designs used in
architecture and architectural engineering itself. My rationale for this
investigation began when the new suspension bridge was built on our island. A suspension
bridge is a bridge in which the weight of the deck is supported by the main
cable and suspender cables.1
They are very useful because they can be very long, yet work effectively. They
have a beautiful design and look fragile from a distance. Its geometric design
and the use of each component made me wonder why each of those was used.
Additionally, the architecture of bridges is very important in today’s
fast-paced world. Due to the increasing population and therefore extreme levels
of traffic, engineers are looking for shorter ways to commute. The inquirer in
me decided to further investigate this and therefore, my aim is to investigate
the math behind building an ideal suspension bridge, taking today’s
infrastructure into consideration.

While
researching I came across that in order to build an effective suspension
bridge, the angle of elevation between the main cable and deck has a range from
30 degrees to 45 degrees. Any angle outside this range will cause a large force
to be exerted on the cables, hence damaging the cables or affecting its
stability. Moreover, when looking at pictures of the most popular bridges, I
noticed that only right, isosceles triangles were used and it brought me to
wonder why no other shapes are used. The bridge is also very symmetric in
nature, considering all its suspender components are equally spaced and each
side of the tower mirrors the other.

Theory:

Figure 3: Stable structure of a
triangle

Figure 4: 45°
diagonal in the square

45°

45°

According to the webpage hunker, the
strongest shape is considered to be a triangle, since all the points are fixed
and connected to another point. So this brings me to wonder, why other polygons
aren’t used. They too connect to other points and all points are fixed in
place. However, while researching, I learnt that triangles are constitutionally
rigid. Therefore, applying a force at any point will evenly spread through all
three sides evenly, as illustrated in figure 3. As for another polygon, the
shape may change or simply collapse. Nevertheless, when looking at bridges,
each has a different design implementing triangles, therefore, with additional
research, I learnt that triangles are used to strengthen
other shapes, especially squares. This is because adding a diagonal to the
square at a 45°
angle, makes the square of opposing triangles, therefore, it’s much stronger. However,
this gets one wondering, why can’t a 60°, 30°
right-angled triangle be used, as that too will provide in the square. A 60°, 30°
right-angled triangle won’t divide the square equally, thus one side will have
more pressure than the other, which may cause the bridge to collapse, or more
internal triangles will be necessary, which requires more materials, thus
increasing the cost of building the bridge. Nonetheless, other angles can be
used in other polygons, for instance, a 60°, 30° right angle triangle would fit
perfectly in a square. A general property about triangles, whether it’s
isosceles, scalene, or a right-angled triangle, is that it’s stable. Therefore,
the use of triangles in polygons, makes the shape stronger, which in return
produces a sturdy bridge.

Adding
on to the geometric aspect of a bridge, symmetry is very important when
designing a bridge. This concept means that one side of the bridge mirrors the
other side. When looking at a suspension bridge, this line of symmetry can be
drawn in the center of the second arch, the one between both towers, as seen in
figure 5. The importance of symmetry is that the length of the bridge must be
able to bear the weight. An asymmetrical bridge does not have the right ratio,
hence will collapse. Each arch on a suspension bridge must be symmetrical, thus
resulting in both sides having a ratio of 1:1. The vertical suspension cables
must also be equally spaced and symmetrical in order to build a well-made
bridge.    

Graph
1: Forces Exerted on the Arch

In order to build the ideal suspension bridge, proper dimensions must
be noted. These consist of the height of the tower from the deck of the bridge,
the length of the deck from the lowest point on the arch to the tower, and the
length of the main cable. Additionally, forces are exerted and these must be
taken into consideration. When looking at the arch of a bridge, it can be seen
that a hyperbola is formed, more specifically a vertical hyperbola. Due to the
graph being concave upward, vertex (point T) will have a horizontal tangent,
therefore it has slope equal to zero. Another force, the weight of the main
cable (point W) acts on the bridge. However, the sum of these two vector
components is zero.TALK ABOUT F
VECTOR. SUM OF ALL THREE IS 0.These vector components are required to find the
right material in order for the bridge to be balanced.

The general equation for
the length of a curve is:

 

Graph 2: y=mx2

Mathematic
Formulation:

In order to find the
length of a line, a parabolic arc must first be considered. The equation for
this curve is

. The general equation to
find the length of a line is:

Graph
3: Line Segment between two points

 From graph 3, it can be seen that the change it can be seen
that the change in the x values is ?x. The rate of change can also be found by
doing the derivative of that equation, hence the change in the y values can be
found by finding its slope (derivative) and multiplying it by ?x, since a slope
can be calculated by calculated by dividing the change in the y values by the
change in the x values. This gives the equation:

This equation can be
simplified. When looking at graph 3, one can see that the change in the x
values is 1, which gives you the tangent at point x (k-1). Hence,
this equation can be integrated in order to find the length of the curve.

This equation calculated
the length of the half the arch on the bridge, which is one-fourth of the
bridge, since a suspension bridge is made up of half an arch, an entire arch,
and finally another half arch. Once the length of the arch has been found,
integrating the equation will let you find the equation of the curve.

In order to simplify
this equation, the second derivative can be found, which removes the integral.

Using substitution,
u(x)= f'(x), u’= f”(x), which in turn gives the equation:

Using the common integral

The integral of dx is
x, therefore the integration of the entire equation is:

Using logs, this
equation can be written as,

Squaring both sides
gives:

As mentioned earlier,
the arch takes a hyperbolic shape and this shown from this equation:

 = cosh (x), which is the equation of a
hyperbolic cosine. This equation doesn’t only help to find the length of the arch,
but when multiplied by the linear density of the material that will be for the
cables, and integrating that as well, will help find the mass of the entire
cable. This helps engineers determine the material of the towers, as they too
require being strong enough in order to hold the cables. In addition, it
determines how many vertical suspension cables are needed, its material, and
the spacing between them to hold the main cable in plain.

Graph:
y=cosh(x)

            Essentially,
the purpose of a bridge is to connect two roads for easier transportation.