--- title: "Road earthwork, 80 m length (v)" subject: "Estimating and Costing" chapter: 6 key: "semester_7/estimating_and_costing:road-earthwork-v-0-80" source: https://www.bctengineer.com/q/semester_7/estimating_and_costing/road-earthwork-v-0-80 site: BCT Engineering --- # Road earthwork, 80 m length (v) Estimating and Costing · Chapter 6: Detailed Estimate - Asked in: 2065 Shrawan (old course), 10 marks - Video: [Calculation of Volume of Earthwork from Cross Sections | Road Estimate | Earthwork Calculation](https://www.youtube.com/watch?v=zMl6OVLeKWM) — Civil Engineering Edu - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/estimating_and_costing/road-earthwork-v-0-80 - Whole chapter: https://www.bctengineer.com/md/semester_7/estimating_and_costing/chapter-6 ## Question Estimate the quantity of earthwork in cutting and filling from the following data for a portion of road 80 m length. Formation width = 10 m. Side slopes in banking 2:1. Side slopes in cutting 1:1. | Chainage | Depth of cutting at centre line | Height of banking | Cross slope of ground | |---|---|---|---| | 0 m | 0.60 | — | 10:1 | | 20 m | 0.70 | — | 12:1 | | 40 m | 0.50 | — | 15:1 | | 60 m | | 0.30 | 12:1 | | 80 m | | 0.70 | 10:1 | ## Short answer Earthwork in cutting = 330.59 m³; earthwork in filling = 162.29 m³. ## Answer #### Given data and assumptions - Formation width $B = 10$ m; side slope in banking $2:1$, in cutting $1:1$. - Sections 20 m apart: cutting at 0, 20, 40 m and filling at 60, 80 m. #### Method Ground has a transverse (cross) slope $n:1$ (H:V), so each section is a side-hill section. With centre height $h$ (cutting +, filling −) above/below formation level, formation width $B$ and side slope $s:1$: $$ \begin{aligned} x_u &= \frac{n\,(B/2 + s h)}{n - s},\quad d_u = h + \frac{x_u}{n}\\ x_d &= \frac{n\,(B/2 + s h)}{n + s},\quad d_d = h - \frac{x_d}{n}\\ A &= \tfrac12 B h + \tfrac12\left[\tfrac{B}{2}(d_u + d_d) + h\,(x_u + x_d - B)\right] \end{aligned} $$ Here $x_u, x_d$ are the horizontal distances of the up-hill and down-hill toes from the centre line and $d_u, d_d$ the heights of the ground above formation at the toes (formulas for cutting). For a section wholly in filling the roles swap: the down-hill toe is the far one, $x_d = n(B/2 + s h)/(n - s)$ and $x_u = n(B/2 + s h)/(n + s)$, with $h$ the filling height and $s$ the filling slope. Where the ground line crosses the formation level inside the road width (part cutting, part filling), the cutting and filling areas are found separately from the co-ordinates of the section (toe points from the same intersection equations). Volume between sections = mean area × distance, taken separately for cutting and filling. #### Sectional areas | Chainage | Centre ht. $h$ (m) | Cutting area (m²) | Filling area (m²) | |---|---:|---:|---:| | 0+000 | cut 0.60 | 6.677 | 0.000 | | 0+020 | cut 0.70 | 7.717 | 0.000 | | 0+040 | cut 0.50 | 5.385 | 0.000 | | 0+060 | fill 0.30 | 0.089 | 3.698 | | 0+080 | fill 0.70 | 0.000 | 8.833 | Working for one section: Take chainage 0+020: centre cutting depth $h = 0.70$ m, $n = 12$, $B = 10.0$ m. - Ground height above formation at the uphill edge $= 1.117$ m, at the downhill edge $= 0.283$ m. - Toe of the side slope on the uphill side at $x = 6.218$ m from the centre line (ground above formation by 1.218 m); downhill toe at $x = 5.262$ m (ground above formation by 0.262 m). - Area from the co-ordinates of the polygon: cutting $= 7.717$ m², filling $= 0.000$ m². #### Volumes | Between | Distance (m) | Cutting (m³) | Filling (m³) | |---|---:|---:|---:| | 0+000 – 0+020 | 20.0 | 143.94 | 0.00 | | 0+020 – 0+040 | 20.0 | 131.02 | 0.00 | | 0+040 – 0+060 | 20.0 | 54.74 | 36.98 | | 0+060 – 0+080 | 20.0 | 0.89 | 125.31 | | **Total** | 80.0 | **330.59** | **162.29** | **Answer: Earthwork in cutting = 330.59 m³; earthwork in filling = 162.29 m³.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. At 0+020 (cut 0.70 m, n = 12, B = 10 m) the down-hill edge ground is: A. 0.283 m above formation B. 0.283 m below formation C. 1.117 m above formation D. 0.417 m below formation Answer: A. 0.70 - 5/12 = +0.283 m, so the ground is still above formation and there is no filling. (Key point: Edge height) 2. At 0+060 the centre is in filling (0.30 m) yet there is a cutting area of 0.089 m^2. Why? A. Filling slope is 1:1 B. Up-hill edge ground rises 5/12 = 0.417 m, above the 0.30 m filling C. Wrong cross slope used D. Banking height is negative Answer: B. Up-hill edge height = -0.30 + 0.417 = +0.117 m, so a small part is in cutting. (Key point: Cutting in a filling section) 3. Between 0+040 (cutting 5.385) and 0+060 (cutting 0.089, filling 3.698), the filling volume is: A. About 36.98 m^3 B. About 73.96 m^3 C. About 54.74 m^3 D. About 18.49 m^3 Answer: A. Filling mean area (0 + 3.698)/2 = 1.849 m^2 x 20 m = 36.98 m^3. (Key point: Volume with changeover) 4. Cutting areas 6.677 and 7.717 m^2, 20 m apart, give about: A. 72.0 m^3 B. 287.9 m^3 C. 143.9 m^3 D. 7.2 m^3 Answer: C. Mean 7.197 m^2 x 20 = about 143.94 m^3. (Key point: Cutting volume) ## Close variants set in other exams - [Road earthwork, chainage 0-80 (s)](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Aroad-earthwork-s-table-0-80) — Estimating and Costing, chapter 6, asked 2× ## Related in Estimating and Costing - [Hill road earthwork, cutting 4 m / banking 6 m](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-k-cut4-fill6) — Estimating and Costing, chapter 6 - [Hill road earthwork, chainage 0-150 (50 m)](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-j-formation-10) — Estimating and Costing, chapter 6 - [Hill road earthwork, chainage 12-15](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-q-chain-12-15) — Estimating and Costing, chapter 6 - [Hill road earthwork, chainage 130-280](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-formation-8-table-a) — Estimating and Costing, chapter 6 - [Hill road earthwork, cross slope 1 in 5 (0-60)](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-r-cross-slope-1-5) — Estimating and Costing, chapter 6 - [Hill road earthwork with cross sections](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Ahill-road-u-cross-sections) — Estimating and Costing, chapter 6 ## Related in other subjects - [Earthwork volume, formation 1201.85, 1.65 m depth](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-earthwork-formation-1201-85) — Surveying I, chapter 9 - [Filling volume, 50 m, formation 1081.50](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-filling-50m-gl-1080-50) — Surveying I, chapter 9, asked 4× - [Cut and fill volume, 20 m formation width](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-road-20m-width-cut-fill-1-1-1-2) — Surveying I, chapter 9 - [Cutting volume, 60 m, GL 1181.50, formation 1179.00](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Avolume-cutting-60m-gl-1181-50-formation-1179) — Surveying I, chapter 9, asked 8×