3.3 2B2B VERTICAL ALIGNMENT
EXAMPLE 3.4 VERTICAL CURVE DESIGN USING OFFSETS
A vertical curve crosses a 4-ft diameter pipe at right angles. The pipe is located at station 110 + 85 and its centerline is at elevation 1091.60 ft. The PVI of the vertical curve is at station 110 + 00 and elevation 1098.4 ft. The vertical curve is equal tangent, 600 ft long, and connects an initial grade of +1.20% and a final grade of 1.08%. Using offsets, determine the depth, below the surface of the curve, of the top of the pipe and determine the station of the highest point on the curve.
SOLUTION
The PVC is at station 107 + 00 (110 + 00 minus 3 + 00, which is half of the curve length), so the pipe is 385 ft (110 + 85 minus 107 + 00) from the beginning of the curve (PVC). The elevation of the PVC will be the elevation of the PVI minus the drop in grade over one-half the curve length,
1098.4 í (3 stations u 1.2 ft/station) = 1094.8 ft Using this, the elevation of the initial tangent above the pipe is
1094.8 + (3.85 stations u 1.2 ft/station) = 1099.42 ft
Using Eq. 3.7 to determine the offset above the pipe at x = 385 ft (the distance of the pipe from the PVC), we have
2
200 Y A x
L
2
1.2 1.08
385 2.82 ft 200 600
Y
Thus the elevation of the curve above the pipe is 1096.6 ft (1099.42 2.82). The elevation of the top of the pipe is 1093.60 ft (elevation of the centerline plus one-half of the pipe’s diameter), so the pipe is 3.0 ft below the surface of the curve (1096.6 1093.6).
To determine the location of the highest point on the curve, we find K from Eq. 3.10 as
600 263.16
1.2 1.08
K
and the distance from the PVC to the highest point is (from Eq. 3.11)
1 263.16 1.2 315.79 ft xhl uK G u
This gives the station of the highest point at 110 + 15.79 (107 + 00 plus 3 + 15.79). Note that this example could also be solved by applying Eq. 3.1, setting Eq. 3.2 equal to zero (for determining the location of the highest point on the curve), and following the procedure used in Example 3.1.
3.3.2 6B6BStopping Sight Distance
Construction of a vertical curve is generally a costly operation requiring the movement of significant amounts of earthen material. Thus one of the primary challenges facing highway designers is to minimize construction costs (usually by making the vertical curve as short as possible) while still providing an adequate level of safety. An appropriate level of safety is usually defined as that level of safety that gives drivers sufficient sight distance to allow them to safely stop their vehicles to avoid collisions with objects obstructing their forward motion. The provision of adequate roadway drainage is sometimes an important concern as well, but is not discussed in terms of vertical curves in this book (see [AASHTO 2011]). Referring back to the vehicle braking performance concepts discussed in Chapter 2, we can compute this necessary stopping sight distance (SSD) as the summation of vehicle practical stopping distance (Eq. 2.47) and the distance traveled during driver perception/reaction time (Eq. 2.49). That is,
12
SSD = + 1
2
r
V V t
g a G
g
§§ · r · u
¨¨ ¸ ¸
©© ¹ ¹
(3.12)
where
SSD = stopping sight distance in ft, V1 = initial vehicle speed in ft/s,
g = gravitational constant, 32.2 ft/s2, a = deceleration rate in ft/s2,
G = roadway grade (+ for uphill and for downhill) in percent/100, and tr = perception/reaction time in s.
Recall from Sections 2.9.5 and 2.9.6 that a value of 11.2 ft/s2 for a and a value of 2.5 s for tr were recommended for roadway design purposes. The design speed of the highway is defined as the maximum safe speed at which a highway can be negotiated assuming near-worst-case conditions (wet-weather conditions). The application of Eq. 3.12 (assuming G = 0) produces the stopping sight distances presented in Table 3.1.
3.3 Vertical Alignment
59
distance Note: Brake reaction distance is based on a time of 2.5 s; a deceleration rate of 11.2 ft/s2 is used to determine calculated stopping sight distance.
Source: American Association of State Highway and Transportation Officials, A Policy on Geometric Design of Highways and Streets, 6th Edition, Washington, DC, 2011.
3.3.3 7B7BStopping Sight Distance and Crest Vertical Curve Design
The length of curve (L in Fig. 3.3) is the critical element in providing sufficient SSD on a vertical curve. Longer curve lengths provide more SSD, all else being equal, but are more costly to construct. Shorter curve lengths are less expensive to construct but may not provide adequate SSD due to more rapid changes in slope. What is needed, then, is an expression for minimum curve length given a required SSD. In developing such an expression, crest and sag vertical curves are considered separately.
The case of designing a crest vertical curve for adequate stopping sight distance is illustrated in Fig. 3.6. To determine the minimum length of curve for a required sight distance, the properties of a parabola for an equal tangent curve can be used to show that
Lm = minimum length of vertical curve in ft,
A = absolute value of the difference in grades (|G1G2|), expressed as a percentage, and Other terms are as defined in Fig. 3.6.
For the sight distance required to provide adequate SSD, current AASHTO design guidelines [2011] use a driver eye height, H1, of 3.5 ft and a roadway object height, H2, of 2.0 ft (the height of an object to be avoided by stopping before a collision). In applying Eqs. 3.13 and 3.14 to determine the minimum length of curve required to provide adequate SSD, we set the sight distance, S, equal to the stopping sight distance, SSD (note that the relatively small distance from the driver’s eye position to the front of the vehicle is ignored). Substituting AASHTO guidelines for H1 and H2 and letting S = SSD in Eqs. 3.13 and 3.14 gives
For SSD < L
SSD2 m A2158
L u (3.15)
For SSD > L
2 SSD 2158 Lm
u A (3.16)
Figure 3.6 Stopping sight distance considerations for crest vertical curves.
S = sight distance in ft, PVC = point of vertical curve (the initial point of the curve), H1 = height of driver’s eye above roadway
surface in ft, PVI = point of vertical intersection (intersection of initial and final grades), and
H2 = height of object above roadway surface
in ft, PVT = point of vertical tangent, which is the final point of the vertical curve (the point where the curve returns to the final grade or, equivalently, the final tangent).
L = length of the curve in ft,
3.3 Vertical Alignment