15.2 Swell, Shrinkage & Load Factor Calculations

Swell, Shrinkage & Load Factor Calculations

Mathematical Mechanics of Soil Swell and Swell Percentage

When heavy equipment excavates native earth from its consolidated in-situ cut, the material expands in volume. In earthmoving engineering, this phenomenon is quantified as the Swell Percentage (often denoted as $\% \text{Swell}$). Swell represents the percentage increase in volume that occurs when soil transitions from its undisturbed bank state (BCY) to its disturbed loose state (LCY).

Swell percentage can be calculated either from physical volumetric measurements or from laboratory unit weight (density) testing:

• Volumetric Swell Formula: $\% \text{Swell} = \left(\frac{\text{Loose Volume (LCY)} - \text{Bank Volume (BCY)}}{\text{Bank Volume (BCY)}}\right) \times 100$
• Density-Based Swell Formula: Because total solid mass remains constant while volume expands, density decreases proportionally. Swell is calculated from unit weights as: $\% \text{Swell} = \left(\frac{\text{Bank Density}}{\text{Loose Density}} - 1\right) \times 100$

Once the swell percentage is established, calculating the loose volume produced by an in-place excavation cut is straightforward: $\text{Loose Volume (LCY)} = \text{Bank Volume (BCY)} \times \left(1 + \frac{\% \text{Swell}}{100}\right)$

For example, if an excavator removes 10,000 BCY of dense plastic clay exhibiting a 35% swell factor, the loose volume requiring hauling is: $\text{LCY} = 10{,}000\text{ BCY} \times (1 + 0.35) = 13{,}500\text{ LCY}$

Different soil classifications exhibit vastly different swell characteristics. Uniform cohesionless sands swell only 10% to 15% because rounded sand grains quickly re-settle into loose packing. Highly plastic cohesive clays swell 30% to 45% because sticky clay platelets adhere irregularly, trapping large air pockets. Solid rock blasted from an intact ledge swells between 50% and 75% because jagged, irregular rock fragments cannot pack tightly without mechanical crushing.

The Load Factor: Converting Loose Volume to Bank Measure

While estimators plan cut quantities in Bank Cubic Yards, operators on the grade measure production by counting haul truck loads or weighing haul units on axle scales. Because haul trucks carry loose material, contractors require a direct mathematical coefficient to convert loose measurements back into bank volume for billing and progress tracking. This multiplier is the Load Factor (LF).

The Load Factor is defined as the ratio of loose density to bank density, which is also the exact mathematical reciprocal of the swell multiplier: $LF = \frac{\text{Loose Density}}{\text{Bank Density}} = \frac{1}{1 + \% \text{Swell}/100}$

By multiplying loose volume (LCY) by the Load Factor, the equivalent bank volume (BCY) is immediately determined: $\text{BCY} = \text{LCY} \times LF$

Consider an earthmoving operation where a fleet of articulated dump trucks hauls 450 loads of common earth in a shift. Each truck carries a heaped payload of 20 LCY, yielding a total hauled volume of $450 \times 20 = 9{,}000\text{ LCY}$. Geotechnical testing shows the common earth has a bank density of 2,800 lb/BCY and a loose density of 2,240 lb/LCY.

To determine how many Bank Cubic Yards were excavated from the cut:

1. Calculate the Load Factor: $LF = \frac{2{,}240\text{ lb/LCY}}{2{,}800\text{ lb/BCY}} = 0.80$
2. Convert loose volume to bank volume: $\text{BCY} = 9{,}000\text{ LCY} \times 0.80 = 7{,}200\text{ BCY}$

The Load Factor is always a decimal less than 1.0 (except in rare expansive mineral conditions), reflecting the fact that one loose cubic yard contains less solid mineral mass than one undisturbed bank cubic yard.

Soil Shrinkage and Compaction Factor Calculations

When excavated soil is transported to an embankment, spread in thin horizontal lifts, and compacted by vibratory smooth-drum or padfoot rollers, mechanical energy expels air voids. The soil particles are forced into a tight, dense configuration, resulting in a volume that is generally smaller than the original undisturbed bank cut. This volumetric contraction is known as Shrinkage.

Shrinkage Percentage ($\% \text{Shrinkage}$) quantifies the percentage reduction from the original in-situ bank volume to the final compacted fill volume:

• Volumetric Shrinkage Formula: $\% \text{Shrinkage} = \left(\frac{\text{Bank Volume (BCY)} - \text{Compacted Volume (CCY)}}{\text{Bank Volume (BCY)}}\right) \times 100$
• Density-Based Shrinkage Formula: $\% \text{Shrinkage} = \left(1 - \frac{\text{Bank Density}}{\text{Compacted Density}}\right) \times 100$

The decimal equivalent of shrinkage is the Shrinkage Factor ($SF = \% \text{Shrinkage} / 100$). The reciprocal relationship connecting bank and compacted volumes is expressed through the Compaction Factor ($CF$): $\text{Compaction Factor} = 1 - SF = \frac{\text{Compacted Density (Bank)}}{\text{Compacted Density (Fill)}} = \frac{\text{CCY}}{\text{BCY}}$ $\text{CCY} = \text{BCY} \times (1 - SF)$

Determining Required Borrow Pit Excavation

A critical calculation in site grading and highway construction involves determining how many Bank Cubic Yards must be excavated from an off-site borrow pit to construct an engineered embankment of specified CCY dimensions. Because the soil shrinks during compaction, excavating only the embankment volume would result in a severe material shortfall.

To calculate the required borrow cut volume, divide the required compacted fill volume by the quantity $(1 - SF)$: $\text{Required BCY} = \frac{\text{Required CCY}}{1 - SF} = \frac{\text{Required CCY}}{\text{Compaction Factor}}$

For example, if a highway overpass embankment requires 50,000 CCY of compacted clay fill, and geotechnical testing indicates a 15% shrinkage factor ($SF = 0.15$): $\text{Required BCY} = \frac{50{,}000\text{ CCY}}{1 - 0.15} = \frac{50{,}000}{0.85} = 58{,}824\text{ BCY}$ The contractor must excavate 58,824 BCY from the borrow source to produce 50,000 CCY of completed, compacted embankment.

Technical Comparison: Typical Swell and Shrinkage Percentages by Soil Type

Earthwork engineers and machine operators utilize standard reference ranges to estimate volumetric changes when specific geotechnical laboratory Proctor tests are pending:

Haul Fleet Logistics: Volume-Limited vs. Weight-Limited Truck Payload Calculations

A central operational responsibility in earthmoving logistics is pass matching and truck fleet payload balancing. Every haul truck—whether a highway dump truck, an articulated dump truck (ADT), or a rigid-frame quarry hauler—operates under two distinct engineering limits:

1. Volumetric Capacity: Defined by the physical dimensions of the truck body. Truck bodies carry two volumetric ratings established by SAE/ISO standards:
   • Struck Capacity: The water-level volume of the truck body when struck completely flat along the top edge of the sideboards.
   • Heaped Capacity: The maximum volume of material the body can hold when crowned above the sideboards at a standardized 2:1 slope angle (2 units horizontal to 1 unit vertical).
2. Gravimetric (Weight) Capacity: The maximum allowable net payload weight, determined by the truck's Gross Vehicle Weight Rating (GVWR), structural frame integrity, suspension limits, tire load ratings, and public highway bridge laws.

Determining the Limiting Haul Factor

To prevent dangerous vehicle overloading or inefficient partial hauling, the fleet supervisor must calculate whether the fleet is volume-limited ("volumed out") or weight-limited ("weighed out"): $\text{Potential Payload Weight} = \text{Heaped Body Capacity (LCY)} \times \text{Material Loose Density (lb/LCY)}$

• If $\text{Potential Payload Weight} > \text{Truck Rated Weight Capacity}$, the machine is weight-limited. The truck cannot be filled to its heaped volumetric capacity without exceeding structural or legal weight ratings. Operators must load only to the weight limit.
• If $\text{Potential Payload Weight} \le \text{Truck Rated Weight Capacity}$, the machine is volume-limited. The truck can be safely loaded to full heaped capacity without exceeding gross weight limits.

Total Haul Cycles Formula

Once usable capacity per truck is established, the total number of truck loads required to complete a project is calculated: $\text{Required Truck Loads} = \frac{\text{Total Project Loose Volume (LCY)}}{\text{Usable Capacity per Load (LCY)}}$ Always round fractional loads up to the next whole number.

Worked Engineering Examples: Step-by-Step Earthwork Calculations

Example 1: Calculating Swell and Load Factor from Field Densities

A cut section in a highway project contains native glacial till. Nuclear gauge testing determines that the in-situ bank density is 3,120 lb/BCY. When excavated into a loose stockpile, the material exhibits a loose density of 2,400 lb/LCY.

1. Calculate Swell Percentage: $\% \text{Swell} = \left(\frac{3{,}120\text{ lb/BCY}}{2{,}400\text{ lb/LCY}} - 1\right) \times 100 = (1.30 - 1) \times 100 = 30.0\%$
2. Calculate Load Factor: $LF = \frac{2{,}400\text{ lb/LCY}}{3{,}120\text{ lb/BCY}} = 0.769$

Example 2: Determining Borrow Pit Excavation for a Highway Embankment

A civil contractor must construct a highway bridge approach embankment requiring 72,000 Compacted Cubic Yards (CCY) of compacted cohesive fill. Geotechnical tests indicate a shrinkage factor of 18% ($SF = 0.18$) from the designated borrow pit.

1. Calculate Required Borrow Bank Volume (BCY): $\text{Required BCY} = \frac{\text{Required CCY}}{1 - SF} = \frac{72{,}000\text{ CCY}}{1 - 0.18} = \frac{72{,}000}{0.82} = 87{,}805\text{ BCY}$
2. Calculate Material to Haul (Assuming 28% Swell in Borrow Cut): $\text{Loose Volume (LCY)} = 87{,}805\text{ BCY} \times 1.28 = 112{,}390\text{ LCY}$

Example 3: Truck Fleet Cycle Sizing for a Major Cut-to-Waste Project

An excavation cut requires removing 24,000 BCY of dense clay (bank density: 3,000 lb/BCY; swell: 25%; loose density: 2,400 lb/LCY). The contractor deploys articulated dump trucks with a heaped capacity of 20 LCY and a maximum manufacturer payload rating of 46,000 lb (23 tons).

1. Check Weight Limitation: $\text{Heaped Load Weight} = 20\text{ LCY} \times 2{,}400\text{ lb/LCY} = 48{,}000\text{ lb}$ Because 48,000 lb exceeds the 46,000 lb rated payload, the truck is weight-limited!
2. Determine Maximum Usable Volume per Truck: $\text{Usable Volume} = \frac{46{,}000\text{ lb}}{2{,}400\text{ lb/LCY}} = 19.17\text{ LCY per load}$
3. Calculate Total Loose Volume to Move: $\text{Total LCY} = 24{,}000\text{ BCY} \times 1.25 = 30{,}000\text{ LCY}$
4. Calculate Required Truck Loads: $\text{Loads} = \frac{30{,}000\text{ LCY} \times 2{,}400\text{ lb/LCY}}{46{,}000\text{ lb/load}} = 1{,}565.2 \to 1{,}566\text{ truck loads}$ (Dividing by the rounded 19.17 LCY per load gives 1,564.9, which would wrongly round to 1,565; carry full precision so the final round-up is not missed.)