Worksheets/Civil Engineering Road Planning

Read This Before You Begin

This worksheet will utilize your experience last worksheet implementing Simpson's Rule in a spreadsheet. You will be applying this spreadsheet to estimate the volume of concrete and asphalt needed to build a road.

Notes

• Civil engineering applications of Simpson's Rule
• Building a highway going through a town
• Cross-sectional design of highway - arc length, volume of materials
• Design of highway: top X rectangle is asphalt, bottom X rectangle plus cut-out half-circle is concrete, coord system has split down middle
• Curve design is a piecewise curve. Given a set of formulas.
• Use Simpson's Rule to compute the arc length of the road, and report the total length in ft.
• Highway standards require a minimum lane width of 15 feet, 10 feet for side shoulder, 5 feet for inner shoulder, 3 lanes both directions. Total width of X.
• Use Simpson's Rule to find the total volume of asphalt, volume of concrete required (use Pappus' Theorem to obtain the volume via V = A * length traveled)

Back Story

You are working as a civil engineer at a construction firm. You have been tasked with helping the company build a brand new highway between Town A and Town B. The road can't go directly between the two towns because of Lake Stinky. You will be working on the construction of the road curves.

Your engineering coworker Sally has designed the layout of the road and will provide you with information about it. Your engineering coworker Naveen has designed the cross-sectional layout of the road bed, asphalt, and shoulder. It is your job to estimate the amount of construction materials that will be needed to build the highway. You will use the information provided by Sally to estimate the length of the road, and use the information provided by Naveen to estimate the total volume of concrete and asphalt needed.

Background Information (Highway Curves)

Some information about the design of highway curves: point of curve (PC), point of tangency (PT), circular curve connecting the two points. Many factors affecting decision - most important is speed of curve, which determines radius of curve.

During the initial survey work, evenly-spaced points are staked out along the proposed path of the highway. If two straight segments must be connected by a curve, a circle is drawn whose tangent line at the start of the curve, point PC, is the segment of road leading up to the curve. The road follows the arc of this circular curve until the tangent line to the circle is parallel to the segment of road leading away from the curve. This point is the point of tangency (PT) and is where the curve transitions back to a straight section.

In addition to the problem of designing the correct route for the road, the road must be designed to be safe and last a long time. For this reason, highway curves are typically sloped: both to offset the centripetal forces created by a vehicle going around a curve, and to prevent buildup of rainwater or debris. This means the road bed design will be different for a curved section than for a straight section.

The Problem

Route Layout

Your coworker Sally provides you with a map of curves going through the town, along with a coordinate system and piecewise mathematical functions that represent the curves. This information is provided below.

...

(Explanation of drawing)

Information provided:

• Town A coordinates
• Town B coordinates
• X miles from Town A, bearing X degrees NE
• X miles from Town B, bearing X degrees NW
• Curvature of circle

How to specify the circle?

Road Design

Naveen provides you with the cross-sectional design of the road. The cross-sectional design of the road incorporates elements like sloped surfaces to prevent water buildup, and the ability to expand or contract during heating and cooling.

...

(Cross-section of road bed on straight sections)

...

(Cross-section of road bed on curved sections)

...

(Explanation of drawing)

Your Task

Your job in the construction process is to estimate the total volume of concrete and asphalt you will need to order to construct the highway. You can use the mathematical description of the road, provided by Sally, to construct an arc length integral and determine the distance that the road travels. You can then use the cross-sectional information provided by Naveen to compute the cross-sectional area. Using Pappus' Centroid Theorem to estimate the total volume of the concrete and asphalt material on the highway.

Estimating the Volume

To estimate the amount of concrete you need, you can use Pappus' Centroid Theorem, which is a way of calculating the volume formed when a shape of area A travels some distance L, forming a volume. (This is also useful in simplifying problems with volumes by rotation.)

(oops.)

Pappus' Centroid Theorem states that the volume V of a solid generated by the revolution of an formed when a shape of area A travels over some distance is equal to the area of the shape times the total distance traveled by the centroid of the shape.

Similarly, the second theorem of Pappus states that the volume V of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area A of the lamina and the distance d_2 traveled by the lamina's geometric centroid x,

states that if the centroid of a cross-sectional shape travels a distance L to form a volume V, then the volume of concrete or asphalt is related to this distance and the cross-sectional area via:

$ V = A * L $

where A is the cross-sectional area of the material (concrete or asphalt):

$ A = \int_{a}^{b} f(x) - g(x) dx $

and is the area between two curves f(x) and g(x), and L is the length traveled by the centroid,

$ L = \int_{a}^{b} \sqrt{ 1 + \left( \frac{dy}{dx} \right)^2 } dx $

and is the arc length (or total distance traveled by the road).

Calc II Questions

Use Simpson's Rule, and the spreadsheet template you used for the prior worksheet, to complete the following exercises. Submit a report that answers each of the following questions with complete sentences.

1) Compute the total length of the road, in feet, using Simpson's Rule and the arc length formula. Justify your choices of N. Why is the centerline of the road used to find the length?

2) Determine the total volume of asphalt and concrete that will be used using Simpson's Rule together with Pappus' Theorem (Volume = Area * Distance Traveled By Centroid)

3) On the morning of the construction project, Sally, your engineering co-worker who gave you the original map of the site runs up to you in a panic and apologizes, explaining that her map LEFT OUT CRITICAL INFORMATION.

Sally informs you that the road is actually sloped at a 4% grade the entire length of the road. That means, for every 100 feet the road travels, it rises 4 feet. This will affect the actual length of the road, and the amount of material you need.

You don't have your spreadsheets or calculations in front of you - all you have is a pencil, and the back of an envelope. If you want more material in time for the construction job, you'll have to order it before the dump trucks leave for the construction site in 15 minutes - the pressure is on. How much more concrete and asphalt do you need to order?

4) What is the approximate increase in the cost of concrete and asphalt for the construction job that resulted from Sally's mistake?

Resources

Federal Highway Administration: http://safety.fhwa.dot.gov/geometric/pubs/mitigationstrategies/chapter3/3_lanewidth.cfm

Wikipedia Lane: https://en.wikipedia.org/wiki/Lane

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Link to all worksheets idea list: Worksheets

Calc II:

• Archimedes: Don't Disturb my Circles Worksheets/Archimdes_Dont_Disturb_My_Circles

• Simpson's Rule: Worksheets/Simpsons_Rule

• Civil Engineering Road Planning: Worksheets/Civil_Engineering_Road_Planning

• Euler's Method and Circuits: Worksheets/Eulers_Method_Circuits

Calc III:

• Infinite Series: Worksheets/Infinite_Series_Convergence

• Partial derivatives: Worksheets/Van Der Waal Equation Critical Point