How to calculate the offsets for a curve from a long chord? /Finding ordinates from a long chord to set out a curve. ~ PARAM VISIONS

Eg:

Calculate the offsets from a long chord at 8m. interval to set out a curve. The length of the long chord is 80m. & the mid-ordinate measures 5m.

Given data:

Mid-ordinate = Oo = 5m.

Length of long chord = L = 80m.

Ordinate interval over long chord = x = 8m.

To find:

Length of ordinates at 8m. intervals to set out a curve.

Calculation:

Ordinate at given intervals is calculated by the formula

Ox = [√ (R² - x²) - ( R - Oo) ]

Where x 👉 Distance of ordinates over the long chord. = 8m.

R 👉 Radius of the curve =?

Oo 👉 Length of mid-ordinate = 5m.

Here, the value of R is unknown. So, we have to find the value of R by using the following formula.

Oo = [ R - √ {R² - ( L/2)²} ]

5m. = [ R - √ { R² - ( 80/2)²} ]

5 = R - √ { R² - 40²}

√ { R² - 40²} = (R -5)

By squaring the above equation

[√ { R² - 40²}]² = (R -5)²

R² - 40² = [R² - (2 x R x 5) + 5²]

Note: Solved as per algebra equation (a -b)² = [a² - (2 x a x b) + b²]

R² - 1600 = [R² - 10R + 25]

Canceling R² on both sides,

- 1600 = -10R + 25

- 1600 -25 = - 10R

R = 1625 / 10

R = 162.50 m.

Now, putting the value of R in the above formula,

1. 1st ordinate at 8m. interval

O8 = [√ ( R² - x²) - ( R - Oo) ]

= [√ (162.5² - 8²) - ( 162.5 - 5)]

= [√ 26,342.25 - (157.5)]

= [162.30 - 157.50]

O8 = 4.80m.

Similarly,

2. 2nd ordinate at 16m. interval

O16 = [√ (162.5² - 16²) - ( 162.5 - 5)]

= [√ 26,150.25 - (157.5)]

= [161.71 - 157.50]

O16 = 4.21m.

3. 3rd ordinate at 24m. interval
O24 = [√ (162.5² - 24²) - ( 162.5 - 5)]
= [√ 25,830.25 - (157.5)]
= [160.717 - 157.50]
  O24 = 3.218m.

4. 4th ordinate at 32m. interval
O32 = [√ (162.5² - 32²) - ( 162.5 - 5)]
= [√ 25,382.25 - (157.5)]
= [159.318 - 157.50]
  O32 = 1.818m.

Note:
As the length of the long chord L = 80m., half of the long chord L/2 = 40m. So the length of the 5th ordinate at 40m. distance should be equal to zero.

Check:
6. 5th ordinate at 40m. interval
O40 = [√ (162.5² - 40²) - ( 162.5 - 5)]
= [√ 24,806.25 - (157.5)]
= [157.50 - 157.50]
  O40 = 0m. ✔