Study Notes on Cross-Sections and Volume Calculations

Cross-Sections and Volume Calculations

1a. Calculation of Volume of Cut Using Cross-Section Areas

Cross-Section Data at 50 m Intervals

• The cross-section areas at various chainages are given as follows:
  • At 00 m: Area = 50 m²
  • At 50 m: Area = 75 m²
  • At 100 m: Area = 62 m²
  • At 150 m: Area = 58 m²
  • At 200 m: Area = 83 m²
  • At 250 m: Area = 22 m²
  • At 300 m: Area = 51 m²
  • At 350 m: Area = 26 m²
  • At 400 m: Area = 18 m²

i. Volume Calculation Using the Trapezoidal Rule

• The trapezoidal rule is used to approximate the integral of a function based on its values at specific points. The formula is:
  $V = \frac{b - a}{2n} imes (y<em>0 + 2(y</em>1 + y<em>2 + … + y</em>{n-1}) + y_n)$ where:

  • b = final x value (chainage at 400 m)
  • a = initial x value (chainage at 0 m)
  • n = number of intervals (9 intervals for this data)

• For this case:

  • Total width of intervals = 400 m - 0 m = 400 m
  • Number of intervals (n) = 8
  • Area for each interval (y values) = [50, 75, 62, 58, 83, 22, 51, 26, 18]
  • Applying the trapezoidal rule:
  • Volume Calculation:
  1. Sum the areas:
     • $ext{Sum} = 50 + 75 + 62 + 58 + 83 + 22 + 51 + 26 + 18$
     • $ext{Sum} = 405 \, m^2$
  2. Compute the volume:
     • $V = \frac{400}{2 imes 8} imes (50 + 2(75 + 62 + 58 + 83 + 22 + 51 + 26) + 18)$
     • $V = \frac{400}{16} imes (50 + 2 imes 405 - 18)$
     • Calculation leads to final volume.

ii. Volume Calculation Using Simpson's Rule

• Simpson's rule can provide a better approximation for the integral of a function than the trapezoidal rule, particularly when using an even number of intervals. The formula is:
  $V = \frac{b - a}{3n} imes (y<em>0 + 4y</em>1 + 2y<em>2 + 4y</em>3 + … + 4y<em>{n-1} + y</em>n)$ where:

  • b = final x value (chainage at 400 m)
  • a = initial x value (chainage at 0 m)
  • n = number of intervals (must be even)

• For this data:

  • Total width = 400 m
  • Number of intervals = 8 (even)
  • Given y values again as previous with:
  • $ext{Sum} = 50 + 75 + 62 + 58 + 83 + 22 + 51 + 26 + 18$

• Applying Simpson's Rule:

  • $( V = \frac{400}{3 imes 8} imes (50 + 4(75) + 2(62) + 4(58) + 2(83) + 4(22) + 2(51) + 4(26) + 18))$
  • Calculate the individual components and sum them accordingly to find the volume.

1b. Area Calculation of Irregular Boundaries

Land Boundary Measurements

• A tract of land comprises boundaries AB, BD, and an irregular boundary DA.
• The straight boundaries:
  • AB: 135 m
  • BD: 255 m

Offsets from Boundary DA to Irregular Boundary

• Measured offsets at regular intervals of 30 m from D are:
  • At Distance 0.0 m: Offset = 0.0 m
  • At 30 m: Offset = 3.7 m
  • At 60 m: Offset = 4.9 m
  • At 90 m: Offset = 4.2 m
  • At 120 m: Offset = 2.8 m
  • At 150 m: Offset = 3.6 m
  • At 180 m: Offset = 0.0 m

Area Calculation Method

• To calculate the area of the irregular section DA, we can employ the trapezoidal rule or a suitable numerical integration technique, much like the previous volume calculations. The area can be computed by:
  • Determining the total width (length of DA) and applying suitable summation techniques to the individual offsets at each distance to find the land area.
• The resulting area will provide the necessary metric for land use, planning, or purchase assessments.