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.† 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 g1 to a straight grade of g2, the elevations of the stations are given byy = g2 − g12L x2 + g1x + E − g1L2.Here y is the elevation of the vertical curve in feet, g1 and g2 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 the figure shown below.) The station x = 0 is the very beginning of the vertical curve, so the station x = 0 lies where the straight section with grade g1 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 g2. Assume that the vertical curve you want to design goes over a slight rise, joining a straight section of grade 1.33% to a straight section of grade −1.72%. Assume that the length of the curve is to be 500 feet (so L = 5) and that the elevation of the intersection is 1010.69 feet.(a) What is the equation for the vertical curve described above? Don't round the coefficients.y = (b) What are the elevations of the stations for the vertical curve? (Round your answers to two decimal places.)first station ftsecond station ftthird station ftfourth station ftfifth station ftlast station ft(c) Where is the highest point of the road on the vertical curve? (Give the distance along the vertical curve and the elevation. Round your answers to two decimal places.)The highest point is feet along the vertical curve and feet elevation.

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Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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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.† 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

g₁

to a straight grade of

g₂,

the elevations of the stations are given by

y =

g₂ − g₁

x² + g₁x + E −

g₁L

Here y is the elevation of the vertical curve in feet,

g₁

and

g₂

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 the figure shown below.) The station

x = 0

is the very beginning of the vertical curve, so the station

x = 0

lies where the straight section with grade

g₁

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

g₂.

Assume that the vertical curve you want to design goes over a slight rise, joining a straight section of grade 1.33% to a straight section of grade

−1.72%.

Assume that the length of the curve is to be 500 feet (so

L = 5)

and that the elevation of the intersection is 1010.69 feet.

(a) What is the equation for the vertical curve described above? Don't round the coefficients.

y =

(b) What are the elevations of the stations for the vertical curve? (Round your answers to two decimal places.)

first station	ft
second station	ft
third station	ft
fourth station	ft
fifth station	ft
last station	ft

(c) Where is the highest point of the road on the vertical curve? (Give the distance along the vertical curve and the elevation. Round your answers to two decimal places.)

The highest point is feet along the vertical curve and feet elevation.

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