1.1 Quantity and Cost Estimating

1.1 Quantity and Cost Estimating

In transportation engineering, accurate quantity and cost estimation is essential for budgeting, bidding, and resource allocation. Projects transition through several design stages, each requiring a different level of estimating detail and precision.

Cost Estimating Methods

Depending on the project stage, estimators employ three primary methods:

1. Parametric (Conceptual) Estimating: Used in the planning and feasibility phase (typically 1% to 15% design definition). It relies on historical statistical relationships between project parameters and costs (e.g., cost per lane-mile of highway, cost per square foot of bridge deck). It is fast but has an accuracy range of -30% to +50%.
2. Assemblies (Systems) Estimating: Used during preliminary design (15% to 45% design definition). Estimators group individual tasks into functional assemblies (e.g., a complete concrete retaining wall system, which includes excavation, formwork, reinforcing steel, concrete placement, and backfill). The accuracy range is typically -20% to +30%.
3. Unit Price (Detailed) Estimating: Used during final design (90% to 100% design definition). Estimators conduct a detailed takeoff from the construction drawings to establish a Bill of Quantities (BOQ). Each quantity is multiplied by a highly specific unit price (labor, material, equipment, overhead). The accuracy range is narrow, typically -5% to +15%.

Earthwork Takeoff Procedures

Earthwork calculations involve determining the volume of material to be excavated (cut) or deposited (fill) along the roadway alignment.

Average End Area Method

For relatively uniform terrain, the volume ($V_y$) in cubic yards between two parallel cross-sections separated by a distance $L$ (in feet) is calculated as: $V_y = \frac{L}{27} \times \left( \frac{A_1 + A_2}{2} \right)$ Where $A_1$ and $A_2$ are the cross-sectional areas in square feet. Note that dividing by 27 converts cubic feet to cubic yards. If the cross-sections transition from cut to fill, a zero-cut/zero-fill point must be located to compute volumes separately.

Prismoidal Formula

When the area of the cross-sections changes non-linearly or there is a large difference between $A_1$ and $A_2$, the Average End Area method overestimates the volume. The Prismoidal Formula provides a more accurate estimate: $V_y = \frac{L}{6 \times 27} \times (A_1 + 4A_m + A_2)$ Where $A_m$ is the cross-sectional area at the midpoint (computed using midpoint dimensions, not by averaging $A_1$ and $A_2$).

Prismoidal Correction

An alternative to recalculating volumes with the prismoidal formula is to apply a correction ($C_p$) to the volume obtained from the Average End Area method. The corrected volume is: $V_{\text{corrected}} = V_{\text{end area}} - C_p$ For two cross-sections with maximum depths of cut (or fill) $c_1$ and $c_2$, and top widths $w_1$ and $w_2$, the prismoidal correction in cubic yards is: $C_p = \frac{L}{12 \times 27} \times (c_1 - c_2)(w_1 - w_2)$ This correction is subtractive when $(c_1 - c_2)(w_1 - w_2) > 0$. Since the end area method generally overestimates volume, the correction is almost always subtracted.

Soil State and Volume Changes

Soil volume changes significantly during excavation, transport, and compaction. Estimators must track volumes in three distinct states:

1. Bank Measure (Bank Cubic Yards, BCY): Soil in its natural, undisturbed state.
2. Loose Measure (Loose Cubic Yards, LCY): Soil after excavation, which swells due to the introduction of air voids.
3. Compacted Measure (Compacted Cubic Yards, CCY): Soil after mechanical compaction, which shrinks compared to the bank state.

The relationships are defined by the Swell ($Sw$) and Shrinkage ($Sh$) percentages: $\text{Swell (\%)} = \left( \frac{\text{Bank Density}}{\text{Loose Density}} - 1 \right) \times 100$ $\text{Shrinkage (\%)} = \left( 1 - \frac{\text{Bank Density}}{\text{Compacted Density}} \right) \times 100$

Converting between states requires using the Load Factor ($LF$) and Shrinkage Factor ($SF$): $LF = \frac{\text{Bank Volume}}{\text{Loose Volume}} = \frac{1}{1 + Sw}$ $SF = \frac{\text{Compacted Volume}}{\text{Bank Volume}} = 1 - Sh$

Mass Diagram Analysis

A Mass Diagram is a continuous graphical plot of the cumulative net earthwork volume (cut is positive, fill is negative) along the project stationing. It is used to plan haul operations and select earthmoving equipment:

• Rising Curve: Indicates a net cut (excess material).
• Falling Curve: Indicates a net fill (deficit of material).
• Peaks: Transition points from cut to fill.
• Valleys: Transition points from fill to cut.
• Balance Line: Any horizontal line intersecting the mass curve. The volume of cut equals the volume of fill between the intersection points.
• Free Haul Distance (FHD): The maximum distance over which material is hauled at no additional cost beyond the excavation price.
• Overhaul: Excavated material hauled beyond the FHD. Measured in station-yards ($yd^3 \times \text{stations}$, where 1 station = 100 feet).

Pavement Materials Takeoff

Transportation estimators must calculate quantities for pavement courses (asphalt and aggregate base).

Hot-Mix Asphalt (HMA) Takeoff

HMA is typically bid and purchased by weight (tons). The basic equation is: $\text{Weight (Tons)} = \frac{\text{Length (ft)} \times \text{Width (ft)} \times \text{Thickness (ft)} \times \text{Density (lb/ft}^3\text{)}}{2,000 \text{ lb/ton}}$ For estimating purposes, HMA compacted density is typically assumed to be $145 \text{ lb/ft}^3$ (or $110 \text{ lb/yd}^2\text{-in}$ of thickness). $\text{Tons} = \text{Area (sq yd)} \times \text{Thickness (in)} \times \frac{110 \text{ lb/yd}^2\text{-in}}{2,000 \text{ lb/ton}}$

Aggregate Base Takeoff

Aggregate base course is typically measured and bid in either cubic yards compacted ($CCY$) or tons. If bid in tons, the wet density of the compacted aggregate (typically $130$ to $140 \text{ lb/ft}^3$) is used: $\text{Tons} = \text{Volume (CCY)} \times 27 \text{ ft}^3/\text{yd}^3 \times \text{Compacted Density (lb/ft}^3\text{)} \div 2,000 \text{ lb/ton}$

Contingency and Escalation

Because construction occurs in the future and carries uncertainty, estimates must incorporate contingency and escalation:

• Contingency: An allocation of funds for "known-unknowns" (e.g., minor design changes, field conditions, weather delays). Project contingency is a percentage of the base construction cost, typically ranging from 30% for conceptual phases down to 5% or 10% at the final bid.
• Escalation: Adjusts the estimated cost from the time of the estimate to the mid-point of construction to account for price inflation. The formula is: $\text{Escalated Cost} = \text{Estimated Cost} \times (1 + r)^n$ Where $r$ is the annual escalation rate, and $n$ is the number of years between the estimate and construction.