Volume of DirtRobert Lotze George H. Moody Middle School Henrico County Schools
Developed with funding from the American Council of Engineering Companies of Virginia and the Math Science Innovation Center |
Question(s) |
How do engineers calculate the amount of fill dirt needed to build ramps for bridges in highway overpass projects? |
Grade/Subject |
Grade 8 Math, Pre-Algebra
Virginia Standards of Learning: 2009 Math 8 (8.3, 8.4, 8.7) |
21^{st} Century Curriculum |
Engineering: Nature of STEM (1.23); Civil Engineering (4.44) |
Background |
Moving earth is a costly endeavor for construction companies. When a highway construction company can reduce the amount of fill hauled in from remote locations, the cost of construction is reduced, often substantially. This can be done a number of ways. Typically, the first option is to use earth that has to be removed from places on-site for fill where needed. This way the earth when removed can be placed immediately or stored on-site to be filled later. Another is to reduce the height of ramps, thus reducing the amount of fill dirt needed.
The Pocahontas Parkway Airport Connector Project that connects I-895 to Richmond International Airport involves three bridges, two of them over a road and another over rail tracks. In addition, there are a number of approaches and exits to and from the existing raised level I-895 and for the bridges. Each of these require earth fill to construct these ramps.
This lesson is a direct application of calculations used on-site in the Pocahontas Parkway Airport Connector Project by Dewberry, the engineering consulting firm, for determining the volume of earth needed to construct these ramps.
The method used is by averaging the area of cross section areas based on the shape and height of the ramps. By using the area of the cross section, which, in general, is the shape of a trapezoid, at a given point in the ramp, and then doing the same for a point a given distance along the ramp, the volume of fill needed can be calculated by averaging the two and multiplying by the distance between the two cross sections. The formula for finding the area of a trapezoid is:
If the ramp is not tapered out but uses retaining walls, such as in the case of ramps where there is not room for side tapers, the shape is a rectangle. Thus, the area of a rectangle is: or depending on your perspective, in this case the first one is applicable. |
Materials |
· Embankment building worksheet (cut and paste below onto a sheet for student use) · Calculator · Pencils |
Safety |
Not applicable for this lesson, as it is all done on paper. |
Procedure |
Students will use the following diagram of a typical embankment. This is called the Average End Area Method to find the embankment necessary between given ends.
The volume of earth needed to fill the embankment will be found by the equation:
is the area of the cross section of the embankment at the smaller end and is the area of the cross section of the embankment at the larger end. These areas are found by using the equation for the area of a trapezoid (given above).
is the length of the embankment. Using the above information, and given that the distance is 120 meters and the cross section trapezoid dimensions are as shown on the diagram, find the volume of earth needed to fill the given embankment. |
Data Analysis / Results |
See below for the calculations.
The volume of earth needed for the given embankment would be 33,750 cubic meters of earth.
Extending this thinking, most roads are not built on graded embankments that are straight and of regular height. Trapezoids are not always the exact shape of the cross section area due to curved roads and surfaces that need banking curves. Add to this the median strip and shoulders that do not need as thick a blacktop, thus more fill. Thus, fill can take on irregular shapes, providing opportunity for lessons that expect thinking about cutting the cross sections into smaller sections. This leads to thinking in terms of pre-calculus problems of finding the area under a line or curve for a given limits in the value of x. This lesson can be revised to go beyond Pre-algebra into Algebra1 and on into Calculus. |
Conclusion / Questions |
Have students answer the following questions:
1.
Figure the amount of earth needed to extend the given embankment
down to nothing, i.e. where the road began to rise from normal ground height,
at the same rate of incline as given? 2.
Without going through specific calculations, but by looking at the
drawing, how would the volume of earth change if the shape is more like a
rectangle for the cross section area due to the road needing to have vertical
retaining walls, due to space limitations on the size of the extended
embankment. (It would cut the amount in half.) 3.
If the shape changed so that there is no longer a trapezoid with
top and bottom parallel sides, how would you go about figuring the area? What
about if there were a median strip? What if the road had to be banked? What
if the road has to curve? (Cut up the cross section into smaller parts for
greater accuracy. This could be accomplished in a lesson by making 3 groups
and giving them different gradations of detail into which the cross section
could be cut, and then tabulating the data into one form, and comparing, thus
proving that the smaller the size of the cuts, the greater the accuracy of
the calculations.) Review with the students the ideas of how a road is built, and given real life situations, what other variables would have to be taken in to consideration in making such an embankment? |
References |
TeachEngineering.org
Dewberry Engineers – Sample calculations provided by Kyle LaClair
MathScience Innovation Center |