91Ó°ÊÓ

When a road is being built, it usually has straight sections, all with the same grade, that must be linked to each other by curves. (By this we mean curves up and down rather than side to side, which would be another matter.) It's important that as the road changes from one grade to another, the rate of change of grade between the two be constant. \({ }^{56}\) The curve linking one grade to another grade is called a vertical curve. Surveyors mark distances by means of stations that are 100 feet apart. To link a straight grade of \(g_{1}\) to a straight grade of \(g_{2}\), the elevations of the stations are given by $$ y=\frac{g_{2}-g_{1}}{2 L} x^{2}+g_{1} x+E-\frac{g_{1} L}{2} . $$ Here \(y\) is the elevation of the vertical curve in feet, \(g_{1}\) and \(g_{2}\) are percents, \(L\) is the length of the vertical curve in hundreds of feet, \(x\) is the number of the station, and \(E\) is the elevation in feet of the intersection where the two grades would meet. (See Figure 5.105.) The station \(x=0\) is the very beginning of the vertical curve, so the station \(x=0\) lies where the straight section with grade \(g_{1}\) meets the vertical curve. The last station of the vertical curve is \(x=L\), which lies where the vertical curve meets the straight section with grade \(g_{2}\). Assume that the vertical curve you want to design goes over a slight rise, joining a straight section of grade \(1.35 \%\) to a straight section of grade \(-1.75 \%\). Assume that the length of the curve is to be 500 feet (so \(L=5\) ) and that the elevation of the intersection is \(1040.63\) feet. a. What phrase in the first paragraph of this exercise assures you that a quadratic model is appropriate? b. What is the equation for the vertical curve described above? Don't round the coefficients. c. What are the elevations of the stations for the vertical curve? d. Where is the highest point of the road on the vertical curve? (Give the distance along the vertical curve and the elevation.)

Step by step solution

01

Determine Reason for Quadratic Model

The phrase that indicates a quadratic model is appropriate is "the rate of change of grade between the two be constant." This implies a uniform change, which suggests parabolic, or quadratic behavior.

02

Substitute Given Values into the Equation

We have the grades: \( g_1 = 1.35\% \) or 0.0135, \( g_2 = -1.75\% \) or -0.0175, the curve length \( L = 5 \) (in hundreds of feet), and intersection elevation \( E = 1040.63 \) feet. Substitute these values into the equation:\[y = \frac{-0.0175 - 0.0135}{2 \times 5} x^2 + 0.0135 x + 1040.63 - \frac{0.0135 \times 5}{2}.\]

03

Simplify the Vertical Curve Equation

Perform the calculations for each term:1.\( \frac{-0.031}{10} = -0.0031 \),2.\( -\frac{0.0675}{2} = -0.03375. \)So, the equation becomes:\[ y = -0.0031 x^2 + 0.0135 x + 1040.63 - 0.03375. \]Simplify further:\[ y = -0.0031 x^2 + 0.0135 x + 1040.59625. \]

04

Calculate Elevations of Stations

For each station \( x \) from 0 to 5, substitute \( x \) into the equation \( y = -0.0031 x^2 + 0.0135 x + 1040.59625 \), and find:- Station 0: \( y = 1040.59625 \).- Station 1: \( y = 1040.60665 \).- Station 2: \( y = 1040.61705 \).- Station 3: \( y = 1040.62745 \).- Station 4: \( y = 1040.63785 \).- Station 5: \( y = 1040.64825 \).

05

Calculate the Vertex for the Highest Elevation

To find the highest point on the vertical curve, determine the vertex of the parabola \( y = ax^2 + bx + c \). The vertex \( x \) occurs at \( x = -\frac{b}{2a} \). Calculate using \( a = -0.0031 \) and \( b = 0.0135 \):\[ x = -\frac{0.0135}{2 \times -0.0031} = 2.177. \]Substitute \( x = 2.177 \) back into the curve equation to find elevation \( y \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Model

When it comes to the design of roads, especially in engineering, understanding why a quadratic model is often used is crucial. One main reason behind the application of a quadratic model in road design, especially for vertical curves, is to ensure a **constant rate of change** of the grade. This requirement means that the surface must transition smoothly from one grade to another. Whenever you hear the phrase "the rate of change of grade between the two be constant," it indicates that a quadratic equation is the most appropriate model.

Mathematically, quadratic equations can describe a parabolic shape. In the context of roads, this is valuable for creating smooth vertical transitions between different slope (grade) sections. A quadratic model with a constant term, linear term, and quadratic term captures the slightly rising or falling nature of these transitions, ensuring the ride is smooth and predictable.

Elevation Calculation

The elevation calculation along a vertical curve is central to designing roads that change grades sensibly. To compute the elevation at any station on a vertical curve, you use a quadratic equation like: \[ y = \frac{g_{2} - g_{1}}{2L} x^2 + g_{1} x + E - \frac{g_{1}L}{2} \] **Breaking it down into parts:**:

• The term \( \frac{g_{2} - g_{1}}{2L} x^2 \) is derived from the need for a constant changing slope. It accounts for how the grade shifts across the length of the curve.
• The linear term \( g_{1} x \) adjusts for the initial incline or decline, extending the impact of the initial grade throughout the curve.
• The constant terms \( E - \frac{g_{1}L}{2} \) provide the baseline elevation at the specific intersection point. They adjust the vertical shift of the entire curve to match the starting and transitional prerequisites of the design.

This calculation becomes particularly useful to ensure each station, marked by their distance, has an exact elevation reading that maintains a smooth and safe gradient throughout the transition.

Parabola Vertex

In the context of vertical curves, the highest or lowest point on the curve is known as the **vertex** of the parabola. Finding the vertex is important because it determines the maximum or minimum elevation on the curve.

For a quadratic equation in the standard form of \( y = ax^2 + bx + c \), the vertex can be calculated using the formula:
\[ x = -\frac{b}{2a} \]

This x-value represents the station number where the peak or trough of the elevation occurs. By substituting this x-value back into the original quadratic equation, the corresponding elevation (y-value) at this point can be found. This calculation ensures that the transition design accounts for natural highs and lows appropriately, which is crucial for road safety and efficiency.

Rate of Change of Grade

The **rate of change of grade** in road design refers to how quickly the elevation of the road changes over a distance. For the vertical curve, it's important that this rate is not only constant but also appropriately gentle for safe vehicular travel.

This consistent rate of change is why the quadratic model is ideal; it represents a gradual transition from one grade (slope) to another, using a parabolic curve. In mathematical terms, the rate of change along a quadratic curve, unlike a linear one, isn't constant across its entirety but instead varies smoothly, ensuring more control over road design.

Properly predicting and applying the rate of change is critical in determining where to place vertical curves and how to connect them seamlessly with straight grades, directly influencing travel comfort and safety. The primary goal is to enhance smoothness and safety by mitigating abrupt elevation changes that could be hazardous.