Estimating Trench
Excavation
By Steven J. Peterson, PE,
MBA
Introduction
Trench excavation is often per-formed by hydraulic backhoe tors. This is because hydraulic excava-tors are not only good at excavating below grade, but they may be used to lift and place utility pipe in the trench, eliminating the need for a second piece of equipment.
The purpose of this article is to give the reader an understanding of how to accurately calculate the quantity of earth that needs to be excavated for trenches. This article discusses deter-mining the volume of excavation using the average-end method and also pres-ents an alternative method that pro-duces more accurate quantities. Factors affecting the production of hydraulic excavators are also dis-cussed.
CSI Divisions and Subdivisions
Trench excavation is part of Division 2 Site Construction Subdivision 02300 Earthwork.
Methods of Measurement
Trench excavation is measured by the cubic yard (cubic meters) of exca-vation or by the lineal foot (meter) of trench. Measuring trench excavation by the lineal foot (meter) of trench is only appropriate when the depth of excavation is constant. For many trenches, particularly sewer trenches, the depth changes over the length of the trench, requiring the trench exca-vation to be measured in cubic yards (cubic meters) of excavation. This
article will limit its discussion to the measuring excavation in cubic yards (cubic meters) of excavation. Feet may be converted to meters by multi-plying the feet by 0.3048, and cubic yards may be converted to cubic meters by multiplying the cubic yards by 0.7646.
Estimating Quantities
A typical trench is shown in Figure 1. In this figure, the sidewalls of the trench consist of both a vertical com-ponent with a constant height of dv and a sloped component with a slope of s feet horizontal per vertical foot. The height of the sloped component of the sides of the trench varies as the depth of the trench varies. The slop-ing of the sides of the trench is com-monly used to prevent the side slopes from caving in. The height of dv and the slope of the sidewalls (S) are set by safety regulations and the soil condi-tions.
Figure 1
One method of determining the volume of excavation is the average-end method. The average-average-end method calculates the volume of exca-vation by taking the average of the areas representing both ends of the trench and multiplying them by the length of the trench. The equation for the average-end method may be writ-ten as follows:
V = L(A1 + A2)/2 Eq. 1 Where:
L = Length of the Trench
A1 = Area of One End of the Trench
A2 = Area of the Other End of the Trench
In the case of the trench in Figure 1, the areas of the ends may be deter-mined as follows:
A = Wd + S(d – dv)2 Eq. 2 Where:
W = Width of the Trench at the Bottom
d1 = Depth at One End of the Trench
d2 = Depth at the Other End of the Trench
dv = Depth of the Vertical Trench Wall
S = Slope of the Trench Walls The slope of the trench walls is cal-culated using the following equation:
S = Horizontal Run/Vertical Rise Eq. 3
The use of the average-end method is demonstrated in the following example:
Example 1:
A utility contractor is to install the sewer line whose plan and profile are shown in Fig. 2. The trench is to have the cross-section shown in Fig. 3. Using the average-end method, deter-mine the volume of excavation in cubic yards needed to install the sewer line.
Figure 2
Figure 3
The depth of excavation at either manhole equals the elevation of the rim less the flow line (FL) of the sewer line plus the distance between the flow line and the bottom of the trench. The depth of excavation (d1) at Station 14+30.20 and the depth of excavation (d2) at Station 15+80.20 are calculated as follows:
d1 = 1148.20 ft – 1140.88 ft + 0.50 ft = 7.82 ft
d2 = 1146.71 ft – 1143.88 ft + 0.50 ft = 3.33 ft
The area of the end of the trench (A1) at Station 14+30.20 and the area of the end of the trench (A2) at Station 15+80.20 are calculated as fol-lows:
A1 = 2 ft(7.82 ft) + 0.75(7.82 ft – 3 ft)2 = 33.1 ft2
A2 = 2 ft(3.33 ft) + 0.75(3.33 ft – 3 ft)2 = 6.7 ft2
The volume of excavation required for the trench is calculated using Eq. 1 as follows:
V = 150 ft (33.1 ft2 + 6.7 ft2)/2 = 2,985 ft3 (1 yd3/27 ft3) = 111 yd3
To simplify these calculations, Eq. 1 and 2 may be set up in a spreadsheet as shown in Fig. 4:
Figure 4
When the length, width, and depths are measured in feet, the slope is measured in feet per foot, and the input variables are set up as shown in Figure 4. The volume in cubic yards is calculated by entering the following equation in Cell B8 of the spreadsheet: = B 1 * ( B 2 * B 3 + B 6 * ( B 3 -B5)^2+B2*B4+B6*(B4-B5)^2)/(2*27) Care must be used when setting up spreadsheet formulas because it is easy to make a mistake that results in inac-curate quantities. After setting up a formula, the estimator must carefully test the formula under a wide variety of conditions. This should be done by changing each one of the variables individually and verifying that the spreadsheet produces the correct answer.
Alternately, the volume of
excava-tion for the trench in Figure 1 may be calculated using the following equa-tion:
V = L[W(d1 + d2)/2 + S(d12 + d1d2 + d22 + 3dv2 – 3d1dv –3d2dv)/3]Eq. 4
Where:
L = Length of the Trench
W = Width of the Trench at the Bottom
d1 = Depth at One End of the Trench
d2 = Depth at the Other End of the Trench
dv = Depth of the Vertical Trench Wall
S = Slope of the Trench Walls Equation 4 is based upon four assumptions. First, the width at the bottom of the trench is constant. Second, the height of the vertical side-wall of the trench is constant. Third, the slope of the sloped sidewalls is con-stant; and as a result, as the trench gets deeper the width of the trench at the surface increases. Fourth, the change in depth of the trench is uniform over the length of the trench; therefore, the depth may be expressed as a linear function of the distance from one end of the trench. Provided these condi-tions are met, Eq. 4 produces an exact quantity for the excavation of the trench. The use of Eq. 4 is shown in the following example:
Example 2:
Solve Example 1 using Eq. 4. From Example 1 the depth of exca-vation at the manholes is 7.82 feet and 3.33 feet. The volume of excavation required for the trench is calculated using Eq. 1 as follows:
V = 150 ft{2 ft(7.82 ft + 3.33 ft)/2 + 0.75[(7.82 ft)2 + (7.82 ft)(3.33 ft) + (3.33 ft)2 + 3(3 ft)2 – 3(7.82 ft)(3 ft) –3(3.33 ft)(3 ft)]/3} V = 2,607 ft3 = 2,607 ft3(1 yd3/27 ft3) = 97 yd3
To simplify these calculations, Eq. 4 may be set up in a spreadsheet as shown in Fig. 5:
Figure 5
When the length, width, and depths are measured in feet, the slope is measured in feet per foot, and the input variables are set up as shown in Fig. 5. The volume in cubic yards is calculated by entering the following equation in Cell B8 of the spread-sheet:
=B1*(B2*(B3+B4)/2+B6*(B3^2+B 3 * B 4 + B 4 ^ 2 + 3 * B 5 ^ 2 3 * B 3 * B 5 -3*B4*B5)/3)/27
Switching From Average-End Method to Eq. 4
Care must be exercised when switching from the average-end method to Eq. 4 because the average-end method can overstate volumes by as much as 50 percent. In Examples 1 and 2, we saw that the average-end method overstated the excavation quantity by 14 percent. This overstate-ment occurs when the sides of the trench are sloped and the depth of the trench is not constant. When a company’s unit costs are based upon overstated quantities calculated by the average-end method, the true unit costs will be understated. If a compa-ny switches to Eq. 4—which produces accurate quantities—and use under-stated costs, they will underbid a job and risk losing money. When convert-ing to Eq. 4 estimators need to verify their unit prices by using Eq. 4 to cal-culate the volumes for completed jobs and determine the unit price based upon Eq. 4.
Factors Affecting Excavation Production
The productivity of a hydraulic backhoe excavator is dependent upon the cycle time, amount of material excavated during each cycle (load), and the operational efficiency (the time that the equipment is being used for excavation). Production may be calculated using the following Equation:
Production = Load(Operational Efficiency)/(Cycle Time)
Eq. 5
The cycle time is the time it takes the hydraulic excavator to load its bucket, swing to the dump location, dump its load and return to the point where it will load its bucket. The more cycles a hydraulic backhoe exca-vator can perform in one hour, the greater its production. The cycle time is dependent on the following factors:
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material being excavated, bucket size, swing angle, depth of excavation, placement of excavated material and obstructions that increase the difficul-ty of excavation.
The material being excavated determines the speed at which an excavation cycle may be completed. It takes longer to load a bucket full of hard-to-dig materials (such as rocky soil, rock and frost-laden soils) than it does to load a bucket full of easy-to-dig materials (such as sand and clay).
The cycle time is also a function of the bucket size: the larger the bucket, the longer the cycle time. This is because large loads require more of
the equipment’s power to excavate and swing the material to the dump location.
The cycle time is also a function of swing angle: the smaller the swing angle, the shorter the cycle time. Swing angles range from 30 degrees when the excavated material is placed in spoil piles next to the trench, to 90 degrees when the excavated material is placed in trucks to the side of the excavator, to 180 degrees when the excavated material is placed in trucks behind the excavator. The placement of the material is a function of the space available for the excavation operation. When a very narrow space
is available for excavation, the trucks must be located behind the excavator. The cycle time is also a function of the depth of excavation. The most optimal depth for excavation is 30 per-cent to 60 perper-cent of the excavator’s maximum digging depth (Schaufelberger, p 145). Excavation depths deeper or shallower than the optimal depth of excavation will increase the cycle time.
Finally, the cycle time is a function of obstacles that the excavator has to work around. The operator must exercise extra care and slow down the excavating process when digging around utilities and other obstacles or
when working around overhead power lines and other above ground obstacles.
In addition to cycle time the amount of material (load) excavated during each cycle affects production. The amount of material is a function of the bucket size—which is a function of the excavator size and bucket width—and the bucket fill factor. When excavating trenches, the width of the trench will limit the maximum width of the bucket. It makes little sense to increase the bucket width to increase production if the extra pro-duction is used to excavate material that does not need to be excavated. The increase in production due to an increase in bucket size more than off-sets the extra cycle time it takes for the larger bucket.
The type of material also deter-mines how much material can be loaded into the bucket. When exca-vating poorly blasted rock, we may only get material equal to 40 percent to 50 percent of the bucket’s rated capacity; whereas, when excavating cohesive materials (such as clay) we may get material equal to 100 percent to 110 percent of the bucket’s rated capacity (Schaufelberger, p 146).
The operational efficiency is expressed as the number of minutes during an hour that the excavator is excavating. Nonproductive activities include bathroom and coffee breaks, waiting for trucks, and performing other tasks (such as placing pipe). A typical operational efficiency for an excavator is 45 to 50 minutes per hour and includes time spent performing other tasks.
Special Risk Considerations
Digging into an existing utility line is not only a hazard for the operator and personnel in the area, but it can be very expensive to repair. Cutting a fiber-optic line can cost the contractor thousands of dollars per minute. Before excavation, the contractor should be familiar with the laws gov-erning the location of existing utilities and should have the existing utilities located. There are often laws regulat-ing how often the utilities must be located, how accurately they are locat-ed and how much time you have to
give the utility company to locate them.
Ratios and Analysis
The cost per cubic yard for one project may be compared to the cost per cubic yard from other projects provided the projects have similar characteristics. This includes types of materials being excavated, swing angle, depth of excavation, placement of excavated material (stockpile versus truck), obstructions, bucket size, buck-et fill factor, and operational efficien-cy. These factors must be similar because they all affect the production of the excavation process.
Conclusion
Trench excavation is most often measured in cubic yards. The volume may be calculated by the average-end method or Eq. 4, with Eq. 4 producing more accurate quantities. When bid-ding trench excavation using unit prices it is important that the unit prices are based upon the same method of determining the volume of excavation as is being used to calcu-late the volume of excavation for the bid. The cost of excavation is a func-tion of producfunc-tion. Producfunc-tion is a function of material being excavated, swing angle, depth of excavation, placement of excavated material, obstructions, bucket size, bucket fill factor, and operational efficiency (the time that the equipment is being used for excavation). Finally, care must be taken when performing trench exca-vation to locate existing utilities to avoid damaging the utilities and to ensure the safety of the workers.
Glossary
Cycle Time – The time it takes a hydraulic excavator to excavate mate-rial, swing to the dump location, dump and return to the excavation location ready to excavate the next load of material.
Hydraulic Backhoe Excavator – A hydraulic-operated excavator with a bucket at the end of a boom, which is drawn towards the excavator during the excavation portion of the cycle. Often referred to as a backhoe or trackhoe.
Rated Capacity – The rated volume of a bucket. The rated capacity is based upon the bucket being heaped full with a 1:1 angle of repose.
Swing Angle – The horizontal angel between the point of excavation and dumping point for a hydraulic backhoe excavator.
References
Schaufelberger, John E., Construction Equipment Management, Prentice Hall, 1999, p. 142–148.
Steven Peterson is an associate professor in the Parson Construction Management Technology program at Weber State University. He teaches courses in estimat-ing, schedulestimat-ing, and construction finance. Prior to teaching at Weber State, Steve worked as a project manager and estimator in the construction industry. He received a MBA and a BS in engineering from the University of Utah. Steven is the author of Construction Accounting and Financial Management and is working on a book entitled Construction Estimating with Excel.
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