Earthworks and Volumetric Calculations in Surveying

Fundamentals of Surveying II: Introduction to Earthworks

Earthworks involve determining the quantities of soil or earth material to be moved, specifically focusing on volumes and areas. In highway engineering, trapezoidal sections are primarily used for road construction because they provide simple estimates and ensure the stability of the road structure through side slopes.

Fundamental Formulas for Earthwork Volume

There are three primary formulas used to determine the volume of earthworks between two cross-sections:

• Average End Area Formula ($V_e$):     A simple arithmetic average used for estimation.     $V_e = \frac{L}{2} \times (A_1 + A_2)$     Where:

  • $L$ = Distance between the two stations (length).

  • $A_1, A_2$ = Cross-sectional areas of the two stations.

• Prismoidal Formula ($V_p$):     This formula provides a more accurate estimation of volume but is restrictive because it requires the area of the midpoint ($A_m$).     $V_p = \frac{L}{6} \times (A_1 + 4A_m + A_2)$     Note: Midpoint area ($A_m$) is derived from the average of the dimensions (height and width) of the two end sections, not just the average of the areas.

• Prismoidal Correction Formula ($V_c$):     Used to correct the volume calculated by the average end area method for more precision, particularly in three-level sections.     $V = V_e - V_{cp}$     The subsequent formula for the correction volume ($V_{cp}$) is:     $V_{cp} = \frac{L}{12} \times (c_1 - c_2) \times (d_1 - d_2)$     Where:

  • $c_1, c_2$ = Centerline heights of the first and second trapezoids.

  • $d_1, d_2$ = Total width (distance between side slope limits) of the first and second trapezoids.

Types of Level Sections in Highway Engineering

Cross-sections are categorized based on their complexity and the number of points surveyed:

• Three-level sections.

• Five-level sections.

• Side hill level sections.

• Irregular sections (used for complex cuts and fills).

Haul and Mass Diagrams

Haul and mass diagrams are graphical representations used to manage earthwork quantities across multiple stations, especially in large-scale topography-based highway projects where elevations and road requirements vary.

Key Terms and Definitions

• Haul: The product of the volume of excavated earth ($V$) and the distance it must be transported ($D$) to an embankment or waste disposal site.

• Freehaul Distance (FHD): A fixed, pre-determined distance within which the hauling of excavated material is performed at no extra cost (the fee is typically part of the initial bid).

• Overhaul: The hauling of material beyond the Freehaul Distance. It is calculated as the product of the volume in excess of the freehold mass and the length of the overhaul.

• Length of Overhaul: The distance between the center of gravity of the excavation (cut) and the center of gravity of the embankment (fill), minus the FHD.

• Limit of Economical Haul (LEH): The distance where the cost of hauling material equals the cost of wasting the material and borrowing new material from a pit nearby.

  • $LEH = \frac{C_b}{C_h} \times (\text{full station unit}) + FHD$

  • Where $C_b$ is the cost of borrowing/fill and $C_h$ is the cost of hauling per cubic meter.

  • Often, a full station is represented as $20\,m$.

Practical Implications

When excavation becomes too costly due to long transport distances (exceeding the LEH), contractors may choose to "waste" the soil (dispose of it locally) and "borrow" soil from a pita closer to the fill site to save on manpower, fuel, and time.

Sample Problem 1: Cross-Sectional Analysis

Problem Statement: Following are cross-sections of a proposed highway project with equal side slopes and base widths. Determine the side slope, the elevation height ($x$) for a specific area, and the volume using the prismoidal correction.

Data for Station $10+100$:

• Left: $\frac{+2.3}{6.45}$

• Center: $\frac{+1.5}{0}$

• Right: $\frac{+1.0}{4.5}$

Analysis:

• Side Slope ($s$):     By setting up equations for the horizontal distance based on side slope $s$ and width $b$:     $6.45 = 2.3s + \frac{b}{2}$     $4.5 = 1.0s + \frac{b}{2}$     Subtracting the equations: $1.95 = 1.3s \rightarrow s = 1.5$.     Side slope is $1.5:1$.

• Base Width ($b$):     Plugging $s$ into the second equation: $4.5 = 1.0(1.5) + \frac{b}{2} \rightarrow 3 = \frac{b}{2} \rightarrow b = 6\,m$.

• Station $10+200$ Evaluation:     Given Area $A_2 = 14.64\,m^2$ at centerline height $x$.     Using triangle division for a three-level section:     $A = \frac{1}{2} \times [6x + (3 \times 2) + 6.9x + (3 \times 2.6)] = 14.64$     Solving for $x$ results in $x = 1.2\,m$.

• Volume Calculation:

  • $L = 10,200 - 10,100 = 100\,m$

  • $A_1$ (at $10+100$) = $13.163\,m^2$

  • $A_2$ (at $10+200$) = $14.64\,m^2$

  • $V_e = \frac{100}{2}(13.163 + 14.64) = 1390.15\,m^3$

  • $V_{cp} = \frac{100}{12}(1.5 - 1.2)(10.95 - 12.9) = -4.875\,m^3$

  • Total Volume $V = V_e - V_{cp} = 1390.15 - (-4.875) = 1395.03\,m^3$.

Sample Problem 2: Haul and Mass Logic

Problem Constraints:

• Stationing: $10+1460$, $10+1760$, $10+2060$.

• $FHD = 50\,m$.

• Cost of Borrowing = $4.00\,Pesos/m^3$.

• Cost of Excavation = $3.50\,Pesos/m^3$.

• Cost of Haul = $0.20\,Pesos/m\cdot\text{station}$.

Key Calculations:

• Limit of Economical Haul (LEH):     Using the standard formula: $LEH = \frac{4.00}{0.20} \times 20 + 50 = 450\,m$.

• Freehaul Limits:     By similar triangles and volume equality ($V_{cut} = V_{fill}$), the transition points are found.     $x = 22.47\,m$ (offset into the cut).     Stationing FHD Left = $1732.47$; Stationing FHD Right = $1782.47$.

• Overhaul Volume:     Average end area of the mass between FHD and LEH limits.     $V_{OH} = 4040.41\,m^3$.

• Cost of Haul:     $Cost = V_{OH} \times (LOH - FHD) \times \text{Rate} = 10,235.97\,Pesos$.

Earthworks and Parabolic Curves (Past Exam Problem)

Scenario: A road surface follows a symmetrical parabolic curve with $L = 120\,m$. Grades are $g_1 = -3\%$ and $g_2 = +2\%$.

Elevation of Curve Points

To find the cross-sectional area, the elevation of the road center must be compared to the Natural Ground Line (NGL).

• Maximum Offset ($h$):     $h = \frac{L}{8} \times |g_1 - g_2| = \frac{120}{8} \times |-0.03 - 0.02| = 0.75\,m$

• Vertical Offset at any point ($y$):     Using the squared property: $y = (\frac{x}{L/2})^2 \times h$.

• Height of Cut/Fill ($H$):     $H = NGL - \text{Elevation of Curve}$.     Positive results indicate a Cut; negative results indicate a Fill.

Area and Volume Evaluation

For three-level sections with base width $b = 10$ and side slope $s = 1$:

• Formula for area: $A = bH + sH^2$.

• For $H = 1.0\,m$, $A = 10(1) + 1(1)^2 = 11.0\,m^2$.

• For $H = 0.75\,m$, $A = 10(0.75) + 1(0.75)^2 = 8.0625\,m^2$.

• For $H = 1.5\,m$, $A = 10(1.5) + 1(1.5)^2 = 17.25\,m^2$.

Total Volume via Prismoidal Formula:

• Interval volumes are calculated per segment (e.g., $30\,m$ per segment).

• Total Excavation Volume = $94.41\,m^3$ (based on cut sections).

• Total Embankment (Borrow) Volume = $675\,m^3$ (based on fill sections).

Questions & Discussion

Q: How do we determine if a station is in cut or fill?A: It is determined by comparing the Natural Grade Line (NGL) with the design elevation of the road (the curve). If NGL is higher, you cut (positive); if it is lower, you fill (negative).

Q: Why use the squared property of parabola in these problems?A: The road profile often follows a parabolic curve to provide a smooth transition between different grades. The squared property ($y \times x^2$) allows for precise elevation calculation at any horizontal distance from the PC or PT.

Q: Is the mass ordinate always zero at the end of the diagram?A: Only if the total volume of cut equals the total volume of fill. If there is a residual value, it represents either waste material to be disposed of or borrow material required from an external source.