In earthwork estimation using the mid-section (mean-section) method, which of the following statements correctly describe how areas and volumes are determined between two consecutive sections?
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AThe mean depth is taken as the arithmetic average of the depths at the two consecutive sections.
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BThe area of the mid-section is calculated using the mean depth (with the section shape assumed constant).
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CThe earthwork volume between the sections is obtained by multiplying the mid-section area by the spacing between the two sections.
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DThe earthwork volume is obtained by multiplying the mid-section area by the distance between some other “original” reference sections (not the current pair).
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E(a), (b) and (c) of the above.
Answer
Correct Answer: (a), (b) and (c) of the above.
Explanation
Introduction / Context:For routine roadway and canal projects, earthwork volumes between stations can be computed by simple approximate methods when the cross-section shape is consistent. The mid-section (mean-section) method is one such technique, lying conceptually between the average-end-area and prismoidal approaches.
Given Data / Assumptions:
- Two consecutive sections at a known spacing L.
- Sectional geometry (e.g., formation width and side slopes) remains the same, while depths vary.
- Mean depth is representative for computing a mid-section area.
Concept / Approach:The method first computes a representative depth as the simple average of depths at the two stations. Using that mean depth and the section geometry, one evaluates the “mid-section area.” The earthwork volume over the interval is then approximated as this mid-section area multiplied by the spacing L between the same two sections.
Step-by-Step Solution:Compute mean depth: d_mean = (d1 + d2) / 2.Compute mid-section area A_m using d_mean in the standard area formula (e.g., for a trapezoidal cutting).Compute volume: V ≈ A_m * L, where L is the distance between the two consecutive sections under consideration.Recognize that the “original sections” phrase in option (d) is a distractor; the spacing used is the same pair’s spacing.
Verification / Alternative check:For small differences between d1 and d2, the mid-section result approaches the prismoidal value; for larger differences, the prismoidal formula gives improved accuracy over this approximation.
Why Other Options Are Wrong:
- Option (d) is incorrect because volume must refer to the distance between the same two sections whose depths were averaged.
Common Pitfalls:Using inconsistent geometry when forming A_m; confusing section spacing; applying the method where section shape changes significantly.
Final Answer:(a), (b) and (c) of the above.