1. Examples and Solutions

The following offsets were taken from a chain line to an irregular boundary line at an interval of 10 m: 0, 2.50, 3.50, 5.00, 4.60, 3.20, and 0 m. Compute the area between the chain line, the irregular boundary line and the end of offsets by: (a) the Trapezoidal rule, and (b) Simpson’s rule.

Solution: By Trapezoidal rule: Required area=10 / 2{0 + 0 + 2(2.50 + 3.50 + 5.00 + 4.60 + 3.20)} = 537.60 =188 m² By Simpson’s rule: Required area=10/ 3{0 + 0 + 4(2.50 + 5.00 + 3.20) + 2(3.50 + 4.60)} = 10/3{42.80 +16.20}=10/359.00 = 196.66 m²

In Example 1.19, we learn to compute areas using two different numerical methods: the Trapezoidal rule and Simpson's rule. The trapezoidal rule calculates area by approximating the shape under the curve with trapezoids, while Simpson's rule uses parabolic segments for better accuracy.

1. Trapezoidal Rule: We sum the areas of trapezoids formed between the chain line and the irregular boundary. The formula is simplified by summing offset values.

2. Simpson’s Rule: This rule is more accurate as it fits parabolas to the data points. It requires both odd and even indexed offsets and uses a similar approach of summation but with different weights.

The final calculation results show how different methods can reach slightly varied results, providing insight into numerical analysis for area calculations.

Imagine a farmer wanting to find the area of an irregular field. Using the trapezoidal method is like putting down small rectangular plots to cover the entire field and calculating their areas. Simpson's method is like using curved, flexible tape to outline the grass, which allows for a deeper and more accurate assessment of the area because it adheres to the edges better.

An embankment of width 10 m and side slopes 1½:1 is required to be made on a ground which is level in a direction transverse to the centre line. The central heights at 40 m intervals are as: 0.90, 1.25, 2.15, 2.50, 1.85, 1.35, and 0.85 m, calculate the volume of earth work according to (i) Trapezoidal formula, and (ii) Prismoidal formula.

Solution: The cross-sections areas are calculated by ∆= (b + sh)*h

• ∆1= (10 + 1.5*0.90)*0.90 = 10.22 m²
• ∆2= (10 + 1.5*1.25)*0.90 = 14.84 m²
• ∆3= (10 +1.5*1.25)*2.15 = 28.43 m²
• ∆4= (10 +1.5*2.50)*2.50 = 34.38 m²
• ∆5= (10 +1.5*1.85)*1.85 = 23.63 m²
• ∆6= (10 +1.5*1.35)*1.35 = 16.23 m²
• ∆7= (10 +1.5*0.85)*0.85= 9.58 m²

(a) Volume according to trapezoidal formula V= 40 / 2{10.22 + 9.58 + 2(14.84 + 28.43 + 34.38 + 23.63 + 16.23)} = 5096.4 m³ (b) Volume calculated in prismoidal formula: V = 40 / 3 {10.22 + 9.58 + 4(14.84 + 34.38 +16.23) + 2(28.43 + 23.63)} = 5142.9 m³

In Example 1.20, we calculate the volume of earthwork for constructing an embankment using two different methods: Trapezoidal and Prismoidal formulas.

1. Cross-section Area Calculation: First, we calculate the cross-sectional areas based on the widths and heights at specified intervals, considering the side slopes of the embankment.

2. Trapezoidal Volume Calculation: The trapezoidal method merely uses the average of these cross-sectional areas multiplied by the length.

3. Prismoidal Volume Calculation: This more accurate method gives additional weight to the more centrally located cross-section, enhancing the volume calculation precision.'

Think of filling a sandbox with sand where the heights of sand differ at different sections. The trapezoidal method is like taking the average height of different sections to estimate how much sand you'll need, while the prismoidal method adjusts for the central height more accurately, ensuring none of the unevenness is overlooked.

The areas enclosed by the contours in the lake are as follows:

Contour (m) Area (m²) 270 2050 275 8400 280 16300 285 24600 290 31500 Calculate the volume of water between the contours 270 m and 290 m by Trapezoidal formula.

Solution: Volume according to trapezoidal formula: = 5 / 2{2050 + 31500 + 2(8400 + 16300 + 24600)} = 330,250 m³

In Example 1.21, we calculate the volume of water between two contour levels using the trapezoidal formula. Each contour represents a different elevation level, and we know their respective enclosed areas.

1. Areas of Contours: Here, we take areas corresponding to different height levels of the lake and recognize that the volume beneath these contour levels is not uniform.

2. Volume Calculation: Applying the trapezoidal formula allows us to approximate the volume of water between the specified contours effectively, providing critical information for maintaining water levels or planning drainage.

Imagine a multi-layered cake where each layer represents a contour level of a lake. While calculating the volume of cake, we would combine the areas of each layer, just like using the trapezoidal formula to estimate the water volume between the layers accurately.

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