A parabolic arch is a common structural shape used in bridges and buildings, known for its efficiency in carrying loads. It is named for its parabolic curve. The shape of a parabolic arch is defined by its key dimensions: the span 'l', which is the horizontal distance between its supports, and the rise 'h', which is the maximum vertical height of the arch from its supports to its crown.

Parabolic Arch Equation Derivation

To find the equation of a parabolic arch with a given span 'l' and rise 'h', we can set up a coordinate system. Let's place the origin of our coordinate system at the left support of the arch. This means the coordinates of the left support are \((0, 0)\).

• The right support will be at a horizontal distance equal to the span 'l' from the origin, so its coordinates are \((l, 0)\).
• The highest point of the arch, known as the crown or vertex, is located exactly midway along the span 'l' and at a height equal to the rise 'h'. Therefore, the coordinates of the crown are \(\left(\frac{l}{2}, h\right)\).

The general equation for a parabola can be written as:

\[y = Ax^2 + Bx + C\]

We use the coordinates of the three known points to find the constants A, B, and C.

Applying Boundary Conditions for the Parabolic Arch

1. At the left support \((0, 0)\):

   Substitute \(x = 0\) and \(y = 0\) into the general equation:

   \[0 = A(0)^2 + B(0) + C\]

   This simplifies to:

   \[C = 0\]

   So, the equation becomes: \(y = Ax^2 + Bx\)

2. At the right support \((l, 0)\):

   Substitute \(x = l\) and \(y = 0\) into the simplified equation \(y = Ax^2 + Bx\):

   \[0 = Al^2 + Bl\]

   Factor out \(l\):

   \[0 = l(Al + B)\]

   Since \(l\) is the span and cannot be zero, we must have:

   \[Al + B = 0\]

   From this, we find \(B = -Al\).

   Now substitute \(B = -Al\) back into the equation \(y = Ax^2 + Bx\):

   \[y = Ax^2 - Alx\]

   Factor out \(Ax\):

   \[y = Ax(x - l)\]

   For a parabolic arch that opens downwards (which is typically the case for an arch supporting a load from above), the coefficient \(A\) must be negative. To make the equation easier to work with, we can rewrite it as \(y = -A'x(x - l)\) or \(y = A'x(l - x)\), where \(A'\) is a positive constant representing the magnitude of \(A\). Let's use the form \(y = Kx(l - x)\) where K is a positive constant.

3. At the crown \(\left(\frac{l}{2}, h\right)\):

   Substitute \(x = \frac{l}{2}\) and \(y = h\) into the equation \(y = Kx(l - x)\):

   \[h = K\left(\frac{l}{2}\right)\left(l - \frac{l}{2}\right)\]

   \[h = K\left(\frac{l}{2}\right)\left(\frac{l}{2}\right)\]

   \[h = K\frac{l^2}{4}\]

   Now, solve for \(K\):

   \[K = \frac{4h}{l^2}\]

Substitute the value of \(K\) back into the equation \(y = Kx(l - x)\):

\[\mathbf{y} = \frac{{\mathbf{4h}}}{{{\mathbf{l}}^2}}{\mathbf{x}}\left( {{\mathbf{l}} - {\mathbf{x}}} \right)\]

This is the required equation for a parabolic arch with span 'l' and rise 'h', with the origin at one of the supports.

Checking Parabolic Arch Equation Validity

We can quickly verify this parabolic arch equation by substituting the coordinates of the known points:

• At \(x = 0\):

  \[y = \frac{4h}{l^2}(0)(l - 0) = 0\]

  This correctly gives \(y = 0\), corresponding to the left support.

• At \(x = l\):

  \[y = \frac{4h}{l^2}(l)(l - l) = \frac{4h}{l^2}(l)(0) = 0\]

  This correctly gives \(y = 0\), corresponding to the right support.

• At \(x = \frac{l}{2}\) (the crown):

  \[y = \frac{4h}{l^2}\left(\frac{l}{2}\right)\left(l - \frac{l}{2}\right)\]

  \[y = \frac{4h}{l^2}\left(\frac{l}{2}\right)\left(\frac{l}{2}\right)\]

  \[y = \frac{4h}{l^2}\left(\frac{l^2}{4}\right)\]

  \[y = h\]

  This correctly gives \(y = h\), which is the rise 'h' at the crown.

The derived equation satisfies all the conditions for a parabolic arch with the specified span 'l' and rise 'h'.