4.2 Leveling, Earthworks, and Mass Haul Diagrams

4.2 Leveling, Earthworks, and Mass Haul Diagrams

Vertical control and volumetric computations constitute the core of highway geometric design, railway grading, site development, and drainage infrastructure. This section reviews differential leveling protocols, curvature and refraction corrections, three-level cross-section area formulas, prismoidal volume adjustments, borrow pit grading, and the economic optimization of mass haul diagrams.

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1. Principles of Differential Leveling

Differential leveling determines elevation differences between points by measuring vertical distances on graduated leveling rods relative to a horizontal line of sight established by an automatic level, digital level, or tilting level.

Essential Definitions

• Bench Mark (BM): A permanent or semi-permanent physical marker of precisely known elevation above a reference datum (e.g., Mean Sea Level).
• Turning Point (TP): An intermediate, stable ground point used to advance the level line forward, serving successively as a foresight and a backsight.
• Backsight (BS or Plus Sight): A rod reading taken on a point of known elevation to determine the Height of Instrument (HI).
• Height of Instrument (HI): The elevation of the horizontal line of sight passing through the optical center of the telescope: $\text{HI} = \text{Elevation} + \text{BS}$
• Foresight (FS or Minus Sight): A rod reading taken on a turning point or benchmark of unknown elevation to transfer the vertical datum: $\text{Elevation}_{\text{new}} = \text{HI} - \text{FS}$
• Intermediate Foresight (IFS): Rod readings taken along cross-sections, building corners, or topography from a single instrument setup where no subsequent backsight is made.

Leveling Field Book Structure & Arithmetic Check

In professional surveying and board exam evaluations, leveling notes must satisfy a strict mathematical identity. The algebraic sum of all backsights minus the algebraic sum of all foresights must equal the final elevation minus the initial elevation: $\sum \text{BS} - \sum \text{FS} = \text{Elevation}_{\text{Final}} - \text{Elevation}_{\text{Initial}}$

Check Verification: $99.392 - 100.000 = -0.608\text{ m}$. The notes are mathematically closed.

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2. Earth Curvature and Atmospheric Refraction

Over sight distances exceeding $100\text{ m}$, the earth's curvature and atmospheric refraction introduce systematic optical errors:

1. Earth Curvature ($h_c$): The earth's surface curves away from the tangent line of sight. The rod reading is displaced upward by: $h_c = \frac{K^2}{2R} \approx 0.0785 K^2 \quad (\text{m, with } K \text{ in km})$
2. Atmospheric Refraction ($h_r$): Density gradients in the atmosphere bend optical rays downward toward denser air, causing the sightline to hit the rod lower than the geometric tangent line by approximately one-seventh of curvature: $h_r \approx 0.0110 K^2 \quad (\text{m, with } K \text{ in km})$
3. Combined Correction ($h_{cr}$): Combining both effects yields the standard net correction: $h_{cr} = h_c - h_r = 0.0785 K^2 - 0.0110 K^2 = 0.0675 K^2 \quad (\text{or } 0.067 K^2)$ Where $K$ is the horizontal sight distance in kilometers ($K = D / 1000$).

Application to Rod Readings

Because curvature dominates refraction, the optical line of sight strikes the rod too high. Therefore, the corrected rod reading is: $\text{Corrected Reading} = \text{Observed Reading} - h_{cr}$ $\text{True Elevation} = \text{HI} - (\text{FS} - h_{cr}) = \text{HI} - \text{FS} + h_{cr}$

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3. Three-Wire Leveling & Cross-Sectional Geometry

Three-Wire Leveling

Precision leveling employs reticles equipped with upper, middle, and lower stadia hairs:

• Stadia Interval ($s$): $s = \text{Upper Reading} - \text{Lower Reading}$.
• Sight Distance ($D$): $D = 100 \times s$ (using standard internal focusing instrument stadia constant $K_s = 100$).
• Rod Reading Check: To detect misreadings, ensure that $\text{Middle} = \frac{\text{Upper} + \text{Lower}}{2}$. The tolerance is typically $\pm 1\text{ mm}$ to $2\text{ mm}$.

Three-Level Road Cross-Sections

A standard three-level highway section consists of a roadbed of width $b$, a centerline cut or fill $c$, and slope stakes set where the side slopes intersect original ground:

• Left slope stake: height $h_L$, horizontal offset $w_L = \frac{b}{2} + s_s h_L$
• Right slope stake: height $h_R$, horizontal offset $w_R = \frac{b}{2} + s_s h_R$

Dividing the cross-section into four triangles (two with base $c$ and heights $w_L, w_R$; two with base $b/2$ and heights $h_L, h_R$), the exact area formula is: $A = \frac{1}{2} \left[ c(w_L + w_R) + \frac{b}{2}(h_L + h_R) \right]$

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4. Earthwork Volume Computations

Between consecutive cross-sections separated by horizontal distance $L$, earthwork volume is calculated via prismoidal geometry.

Average End-Area Method ($V_e$)

The average end-area formula assumes the volume varies linearly between end sections: $V_e = \left( \frac{A_1 + A_2}{2} \right) L$ While simple and accepted in standard DPWH contracts, this formula systematically overestimates volume whenever $A_1 \neq A_2$.

Prismoidal Formula ($V_p$)

The mathematically rigorous prismoidal formula accounts for parabolic warping: $V_p = \frac{L}{6} \left( A_1 + 4 A_m + A_2 \right)$ Where $A_m$ is the middle area computed from linear dimensions averaged between Station 1 and Station 2: $c_m = \frac{c_1 + c_2}{2}, \quad w_{L, m} = \frac{w_{L1} + w_{L2}}{2}, \quad w_{R, m} = \frac{w_{R1} + w_{R2}}{2}, \quad h_{L, m} = \frac{h_{L1} + h_{L2}}{2}, \quad h_{R, m} = \frac{h_{R1} + h_{R2}}{2}$

Prismoidal Correction ($C_p$)

For three-level sections, the difference between the average end-area volume and the exact prismoidal volume is the Prismoidal Correction ($C_p$): $C_p = V_e - V_p = \frac{L}{12} (c_1 - c_2)(w_1 - w_2)$ Where $w_1 = (w_{L1} + w_{R1})$ and $w_2 = (w_{L2} + w_{R2})$. Then: $V_p = V_e - C_p$ Because $(c_1 - c_2)$ and $(w_1 - w_2)$ almost always share the same sign, $C_p$ is positive, confirming that $V_e > V_p$.

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5. Borrow Pit Excavation (Unit Base / Grid Method)

For large foundation excavations, basements, and site leveling, the site is staked in a uniform grid of square or rectangular prisms of horizontal area $A_{\text{unit}}$. Corner elevations are surveyed before and after excavation to determine net vertical cuts ($h$):

$V = \frac{A_{\text{unit}}}{4} \left( \sum h_1 + 2 \sum h_2 + 3 \sum h_3 + 4 \sum h_4 \right)$ Where:

• $h_1$ = corner cuts common to exactly one grid unit
• $h_2$ = corner cuts common to two adjoining grid units
• $h_3$ = corner cuts common to three adjoining grid units
• $h_4$ = corner cuts common to four adjoining grid units

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6. Mass Haul Diagrams & Economic Analysis

A Mass Haul Diagram (MHD) is a continuous graphical plot of stationing along the horizontal axis versus cumulative net earthwork volume along the vertical axis.

Fundamental Properties of the Mass Curve

• Cut (Excavation): Slopes upward from left to right (cumulative volume increases).
• Fill (Embankment): Slopes downward from left to right (cumulative volume decreases).
• Peak (Summit): Represents the transition point from cut to fill (maximum net cut).
• Trough (Valley): Represents the transition point from fill to cut (maximum net fill).
• Zero Slope: Indicates zero earthwork cross-section (grade point).

Haul Terminology and Definitions

• Haul: The work performed in transporting material: $\text{Haul} = \text{Volume } (\text{m}^3) \times \text{Distance } (\text{m or stations})$.
• Balance Line: A horizontal line drawn across the mass curve. Where the balance line intersects the curve, cut equals fill ($V_{\text{cut}} = V_{\text{fill}}$).
• Free Haul Distance (FHD): The fixed distance (typically $50\text{ m}$ to $100\text{ m}$, or $4\text{ to }5\text{ stations}$) within which the contractor moves earth without additional transportation payment beyond the unit excavation price.
• Overhaul (OH): The transport of excavated material beyond the Free Haul Distance. Measured in $\text{m}^3\text{-stations}$ or $\text{m}^3\text{-meters}$: $\text{Overhaul Distance (OHD)} = \text{Distance between Cut/Fill Centroids} - \text{FHD}$
• Borrow: Material imported from external borrow pits when local excavation is insufficient to satisfy embankment requirements.
• Waste: Surplus excavation dumped off-site when cut exceeds nearby fill and haul costs exceed disposal costs.

Limit of Economic Haul (LEH)

The Limit of Economic Haul (LEH) represents the maximum haul distance at which it is economically preferable to haul on-site cut rather than wasting the cut and purchasing borrow: $\text{Cost of Hauling} = \text{Cost of Borrowing}$ $C_{\text{overhaul}} \times (\text{LEH} - \text{FHD}) = C_{\text{borrow}}$ $\text{LEH} = \text{FHD} + \frac{C_{\text{borrow}}}{C_{\text{overhaul}}}$ Where:

• $C_{\text{borrow}}$ = unit cost of borrow (₱/$\text{m}^3$)
• $C_{\text{overhaul}}$ = unit cost of overhaul (₱/$\text{m}^3\text{-station}$ or ₱/$\text{m}^3\text{-m}$)

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7. Comprehensive Worked Example

Problem: Three-Level Earthwork & Prismoidal Correction

Given: Two consecutive highway stations separated by $L = 60.00\text{ m}$ have base width $b = 10.00\text{ m}$, side slopes $1.5:1$ (H:V). The cross-section notes are:

• Station 1+000: Center cut $c_1 = 2.00\text{ m}$, Left: $\frac{h_{L1}}{w_{L1}} = \frac{1.60\text{ m}}{7.40\text{ m}}$, Right: $\frac{h_{R1}}{w_{R1}} = \frac{2.40\text{ m}}{8.60\text{ m}}$
• Station 1+060: Center cut $c_2 = 3.60\text{ m}$, Left: $\frac{h_{L2}}{w_{L2}} = \frac{2.80\text{ m}}{9.20\text{ m}}$, Right: $\frac{h_{R2}}{w_{R2}} = \frac{4.20\text{ m}}{11.30\text{ m}}$

Calculate:

1. Cross-sectional areas $A_1$ and $A_2$.
2. Volume by Average End-Area ($V_e$).
3. Prismoidal correction $C_p$ and exact prismoidal volume $V_p$.

Solution:

1. Area Computations: $A_1 = \frac{1}{2} \left[ c_1(w_{L1} + w_{R1}) + \frac{b}{2}(h_{L1} + h_{R1}) \right]$ $w_1 = 7.40 + 8.60 = 16.00\text{ m}, \quad h_{L1} + h_{R1} = 1.60 + 2.40 = 4.00\text{ m}$ $A_1 = \frac{1}{2} [ 2.00(16.00) + 5.00(4.00) ] = \frac{1}{2} [ 32.00 + 20.00 ] = 26.00\text{ m}^2$

   $A_2 = \frac{1}{2} \left[ c_2(w_{L2} + w_{R2}) + \frac{b}{2}(h_{L2} + h_{R2}) \right]$ $w_2 = 9.20 + 11.30 = 20.50\text{ m}, \quad h_{L2} + h_{R2} = 2.80 + 4.20 = 7.00\text{ m}$ $A_2 = \frac{1}{2} [ 3.60(20.50) + 5.00(7.00) ] = \frac{1}{2} [ 73.80 + 35.00 ] = 54.40\text{ m}^2$

2. Average End-Area Volume ($V_e$): $V_e = \left( \frac{A_1 + A_2}{2} \right) L = \left( \frac{26.00 + 54.40}{2} \right) \times 60.00 = 40.20 \times 60.00 = 2,412.00\text{ m}^3$

3. Prismoidal Correction and Volume ($V_p$): $C_p = \frac{L}{12}(c_1 - c_2)(w_1 - w_2)$ $c_1 - c_2 = 2.00 - 3.60 = -1.60\text{ m}$ $w_1 - w_2 = 16.00 - 20.50 = -4.50\text{ m}$ $C_p = \frac{60.00}{12} (-1.60)(-4.50) = 5.00 \times 7.20 = 36.00\text{ m}^3$ $V_p = V_e - C_p = 2,412.00 - 36.00 = 2,376.00\text{ m}^3$