Q8. Which of the following methods estimates the best volume of earthwork of an irregular embankment?
📚 Detailed Explanation: Simpson’s Method Gives the Most Accurate Earthwork Volume
When an embankment or cutting has an irregular profile, the cross-sectional area varies in a non-linear way between sections. The choice of volume estimation method determines how well this variation is captured. Simpson’s method (prismoidal rule) is the most accurate because it fits a parabolic curve through three consecutive cross-sections instead of a straight line.
It approximates the solid between sections as a parabolic prismoid rather than a straight-sided trapezoid. The parabola tracks surface undulations far better than a linear assumption, making it the preferred method for irregular earthwork.
Comparison of All Four Methods
| Method | Assumption | Accuracy for Irregular Sections | Typical Error |
|---|---|---|---|
| Simpson’s (Prismoidal) | Parabolic variation between sections | Highest | Least — captures curvature |
| Trapezoidal | Linear variation between sections | Moderate | Overestimates for tapered solids |
| Mid-ordinate | Constant value at midpoint represents interval | Moderate | Depends on section regularity |
| Average ordinate | Mean of all ordinates applied uniformly | Lower | Least accurate; ignores end-area weighting |
Why the Other Options Are Wrong
A (Average ordinate method): This method simply averages all offset or ordinate values and multiplies by the total base length. It assigns equal weight to every ordinate including the end ones, which misrepresents the geometry and gives the least accurate result.
B (Mid-ordinate method): The mid-ordinate method takes the ordinate at the midpoint of each interval and multiplies by the interval length. It is more accurate than the average ordinate method but still assumes a constant value within each interval, not a parabolic curve.
D (Trapezoidal method): The trapezoidal method assumes linear variation between consecutive cross-sections. While it is widely used for its simplicity, it systematically overestimates the volume of embankments that taper (where area decreases progressively). A prismoidal correction must be applied for accurate results.
Key Concepts for Students
- Prismoidal correction: When using the trapezoidal method, the prismoidal correction Cᵣ = (D/6)(c₁ − c₂)(d₁ − d₂) can be subtracted from the trapezoidal volume to obtain the more accurate prismoidal volume. This shows that the trapezoidal method always overestimates compared to Simpson’s for tapered solids.
- Odd-section requirement: Simpson’s method requires an odd number of cross-sections (even number of intervals). If the number of sections is even, the last interval must be treated with the trapezoidal rule separately.
- Exam shortcut: If a question asks which method is “most accurate” or “best estimates” earthwork volume for irregular sections, the answer is always Simpson’s / prismoidal method. This is a frequently repeated question pattern.