25 VERTICAL CURVES
Asterisks indicate problems that have partial answers given in Appendix G.
25.1 Why are vertical curves needed on the grade lines for highways and railroads?
25.2 Identify the type of vertical curve (sag or crest) by the following rates of change in grade.
25.3 What factors must be taken into account when designing a grade line on any highway or
railroad?
25.4* A −2.50% grade meets a +1.75% grade at station 44+25 and elevation 3386.96 ft, 400-ft
curve, stakeout at half stations.
(*)
© 2018 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all
25.5 A +4.5% grade meets a +1.50% grade at station 4 + 200 and elevation 605.568 m, 180-
m curve, stakeout at 30-m increments.
25.6 A 455-ft curve, grades of g1 = +2.00% and g2 = −1.50%, VPI at station 78 + 60, and
elevation 7855.35 ft, stakeout at full stations.
25.7 A 500-ft curve, grades of g1 = −3.00% and g2 = −1.25%, VPI at station 38 + 00, and
elevation 560.00 ft, stakeout at full stations.
BVC Station = 35+50.000
BVC Elevation = 567.5
Station
x (Sta)
g1*x
r/2*x²
Elevation
1st Diff
2nd Diff
40+50.00
5.00
−15.00
4.375
556.875
40+00.00
4.50
−13.50
3.544
557.544
−1.60
0.35
39+00.00
3.50
−10.50
2.144
559.144
−1.95
0.35
38+00.00
2.50
−7.50
1.094
561.094
−2.30
0.35
37+00.00
1.50
−4.50
0.394
563.394
−2.65
36+00.00
0.50
−1.50
0.044
566.044
35+50.00
0.00
0.00
0
567.500
© 2018 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all
copyright laws as they currently exist. No portion of this material may be reproduced, in any form
or by any means, without permission in writing from the publisher.
25.8 A 600-ft curve, g1 = −1.50%, g2 = +4.50%, VPI station = 46 + 00, VPI elevation = 1395.00
ft, stakeout at full stations.
BVC Station = 43+00.00
BVC Elevation = 1399.50
Station
x (Sta)
g1*x
r/2*x*x
Elevation
1st Diff
2nd Diff
49+00.00
6.00
−9.00
18.00
1,408.50
48+00.00
5.00
−7.50
12.50
1,404.50
4.00
1.00
47+00.00
4.00
−6.00
8.00
1,401.50
3.00
1.00
46+00.00
3.00
−4.50
4.50
1,399.50
2.00
1.00
45+00.00
2.00
−3.00
2.00
1,398.50
1.00
1.00
44+00.00
1.00
−1.50
0.50
1,398.50
0.00
43+00.00
0.00
0.00
0.00
1,399.50
Minimum elevation = 1,398.375 @ station 44+50.00
25.9 A 60-m curve, g1 = −1.500%, g2 = 2.600%, VPI station = 12 + 280, VPI elevation =
155.600 m, stakeout at 10-m increments.
BVC Station = 12+250.000
BVC Elevation = 156.050
Station
x (Sta)
g1*x
r/2*x*x
Elevation
12+310.000
0.600
−0.900
1.230
156.380
12+300.000
0.500
−0.750
0.854
156.154
12+290.000
0.400
−0.600
0.547
155.997
12+280.000
0.300
−.450
0.308
155.907
12+270.000
0.200
−0.300
0.137
155.887
12+260.000
0.100
−0.150
0.034
155.934
12+250.000
0.000
0.000
0.000
156.050
Minimum elevation = 155.885 @ Sta 12+271.951
Field conditions require a highway curve to pass through a fixed point. Compute a suitable
equal-tangent vertical curve and full-station elevations for Problems 25.10 through 25.12.
25.10 Grades of g1 = −2.50% and g2 = +1.00%, VPI elevation 750.00 ft at station 30 + 00. Fixed
elevation 753.00 ft at station 30 + 00.
25.11 Grades of g1 = −1.50% and g2 = +1.75%, VPI elevation 1430.00 ft at station 15 + 00
Fixed elevation 1436.50 ft at station 14 + 00.
BVC Station = 6+97.07
BVC Elevation = 1442.044
Station
x (Sta)
g1*x
r/2*x*x
Elevation
1st Diff
2nd Diff
23+02.930
16.059
-24.088
26.095
1444.051
23+00.000
16.029
-24.044
26.000
1444.000
22+00.000
15.029
-22.544
22.857
1442.357
1.643
21+00.000
14.029
-21.044
19.917
1440.917
1.440
0.203
20+00.000
13.029
-19.544
17.179
1439.679
1.238
0.202
19+00.000
12.029
-18.044
14.643
1438.643
1.036
0.202
18+00.000
11.029
-16.544
12.310
1437.810
0.833
0.203
17+00.000
10.03
-15.044
10.179
1437.179
0.631
0.202
16+00.000
9.03
-13.544
8.250
1436.750
0.429
0.202
15+00.000
8.03
-12.044
6.524
1436.524
0.226
0.203
14+00.000
7.03
-10.544
5.000
1436.500
0.024
0.202
13+00.000
6.03
-9.044
3.679
1436.679
-0.179
0.203
12+00.000
5.029
-7.544
2.560
1437.060
-0.381
0.202
11+00.000
4.029
-6.044
1.643
1437.643
-0.583
0.202
10+00.000
3.029
-4.544
0.929
1438.429
-0.786
0.203
9+00.000
2.029
-3.044
0.417
1439.417
-0.988
0.202
8+00.000
1.029
-1.544
0.107
1440.607
-1.190
0.202
7+00.000
0.029
-0.044
0.000
1442.000
-1.393
0.203
6+97.070
0
0
0.000
1442.044
Minimum elevation = 1436.485 @ 14+38.236
25.12 Grades of g1 = +4.500% and g2 = +2.000% VPI station 6+300 and elevation 205.930 m.
Fixed elevation 205.620 m at station 6+400. (Use 100-m stationing)
6+500.000
7.515
33.816
-6.400
208.530
6+400.000
6.515
29.316
-4.810
205.620
6+300.000
5.515
24.816
-3.447
202.483
6+200.000
4.515
20.316
-2.310
199.120
6+100.000
3.515
15.816
-1.400
195.530
6+00.000
2.515
11.316
-0.717
191.713
5+900.000
1.515
6.816
-0.260
187.670
5+800.000
0.515
2.316
-0.030
183.400
5+748.533
0.000
0.000
0.000
181.114
25.13 A −1.10% grade meets a +0.60% grade at station 36 + 00 and elevation 800.00 ft. The
+0.60% grade then joins a +2.40% grade at station 39 + 00. Compute and tabulate the
notes for an equal-tangent vertical curve, at half-stations, that passes through the midpoint
of the 0.60% grade.
37+00.000 2.460 -2.706 1.763 800.660
25.14 When is it advantageous to use an unequal-tangent vertical curve instead of an equal-
tangent one?
25.15 A +2.50% grade meets a −1.25% grade at station 60+00 and elevation 3310.00 ft. Length
of first curve 600 ft, second curve 400 ft.
BVC Station = 54+00.00
BVC Elevation = 3295.00
Station
x (Sta)
g1*x
r/2*x²
Elevation
64+00.00
4.00
4.00
−4.50
3305.000
63+00.00
3.00
3.00
−2.531
3305.969
62+00.00
2.00
2.00
−1.125
3306.375
61+00.00
1.00
1.00
−0.281
3306.219
60+00.00
0.00
0.00
0.000
3305.500
CVC
60+00.00
6.00
15.00
−4.500
3305.500
59+00.00
5.00
12.50
−3.125
3304.375
58+00.00
4.00
10.00
−2.000
3303.000
57+00.00
3.00
7.50
−1.125
3301.375
56+00.00
2.00
5.00
−0.500
3299.500
54+00.00
0.00
0.00
0.000
3295.000
25.16 Grade g1 = +1.25%, g1 = +3.75%, VPI at station 62+00 and elevation 1053.95 ft, L1 =
500 ft and L2 = 600 ft.
BVC Station = 57+00.00
BVC Elevation = 1047.70
Station
x (Sta)
g1*x
r/2*x²
Elevation
1st Diff
2nd
Diff
© 2018 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all
copyright laws as they currently exist. No portion of this material may be reproduced, in any form
or by any means, without permission in writing from the publisher.
68+00.00
6.00
15.68
3.41
1076.450
67+00.00
5.00
13.07
2.367
1072.795
3.655
0.189
66+00.00
4.00
10.46
1.515
1069.329
3.466
0.189
65+00.00
3.00
7.84
0.852
1066.052
3.277
0.190
64+00.00
2.00
5.23
0.379
1062.965
3.087
0.189
63+00.00
1.00
2.61
0.095
1060.067
2.898
0.190
62+00.00
0.00
0.00
0.000
1057.359
2.708
CVC
62+00.00
5.00
6.25
3.409
1057.359
61+00.00
4.00
5.00
2.182
1054.882
2.477
0.272
60+00.00
3.00
3.75
1.227
1052.677
2.205
0.273
59+00.00
2.00
2.50
0.545
1050.745
1.932
57+00.00
0.00
0
0.000
1047.700
3.045
25.17 Grades g1 of +3.00% and g2 of −1.00% meet at the VPI at station 4+300 and elevation
268.473 m. Lengths of curves are 200 m and 350 m. (Use 40-m stationing.)
BVC Station = 4+100.000
BVC Elevation = 262.473
Station
x (Sta)
g1*x
r/2*x²
Elevation
4+650.000
2.750
1.591
-2.55
264.973
4+640.000
3.400
1.545
-2.402
265.071
4+600.000
3.000
1.364
-1.870
265.421
4+560.000
2.600
1.182
-1.405
265.705
4+520.000
2.200
1.000
-1.006
265.922
4+480.000
1.800
0.818
-0.673
266.072
4+440.000
1.400
0.636
-0.407
266.157
4+400.000
1.000
0.455
-0.208
266.174
4+360.000
0.600
0.273
-0.075
266.125
4+320.000
0.200
0.091
-0.008
266.010
CVC
4+300.000
2.000
6.000
-2.545
265.928
4+240.000
1.400
4.200
-1.247
265.426
4+200.000
1.000
3.000
-0.636
264.837
4+160.000
0.600
1.800
-0.229
264.044
4+100.000
0.000
0.000
0.000
262.473
25.18 A −1.40% grade meets a +2.75% grade at station 89 + 00 and elevation 6320.64 ft. Length
BVC Elevation = 6324.84
Station
x (Sta)
g1*x
r/2*x²
Elevation
1st Diff
2nd Diff
93+00.00
4.00
3.89
3.56
6,331.64
92+00.00
3.00
2.91
2.00
6,329.11
2.528
0.445
91+00.00
2.00
1.943
0.89
6327.029
2.083
0.445
90+00.00
1.00
0.971
0.222
6325.391
1.638
0.444
89+00.00
0.00
0.000
0.000
6324.197
1.194
CVC
89+00.00
3.00
-4.200
3.557
6324.197
88+00.00
2.00
-2.800
1.581
6323.621
0.576
0.790
87+00.00
1.00
-1.400
0.395
6323.835
-0.214
0.791
86+00.00
0.00
0.000
0.000
6324.840
-1.005
25.19* A manhole is 12 ft from the centerline of a 30-ft
wide street that has a 6-in. parabolic crown. The
street center at the station of the manhole is at
elevation 612.58 ft. What is the elevation of the
manhole cover?
15
25.20 A 40-ft wide street has an average parabolic crown from the center to each edge of 1/4
in./ft. How much does the surface drop from the street center to a point 4 ft from the edge?
0.33 ft y = (20 − 4)(1/4 in.) = 4.00 in. = 0.333 ft
25.21 Determine the station and elevation at the low point of the curve in Problem 25.4.
25.22* Calculate the station and elevation at the high point of the curve in Problem 25.6.
25.23 Compute the station and elevation at the low point of the curve of Problem 25.9.
1398.375 ft @ station 44+50.00
25.24 What are the station and elevation of the low point of the curve of Problem 25.13?
25.25 What are the requirements for sight distances on a vertical curve?
© 2018 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all
25.26* Compute the passing sight distance available in Problem 25.6. (Assume h1 = 3.50 ft and
h2 = =4.25 ft.)
25.27 Similar to Problem 25.26, except for a stopping sight distance with an h2 = 2.00 ft.
25.28 What is the minimum required length of curve in Problem 25.7 assuming a design speed
of 70 mph.
25.29 In determining sight distances on vertical curves, how does the designer determine
whether the cars or objects are on the curve or tangent?
What is the minimum length of a vertical curve to provide a required sight distance for the
conditions given in Problems 25.30 through 25.32?
25.30* Grades of +3.00% and −2.00%, sight distance 600 ft, h1 = 3.50 ft and h2 = 2.00 ft.
25.31 A crest curve with grades of +2.50% and −1.00% sight distance 500 ft, h1 = 3.50 ft and
h2 = 4.25 ft.
© 2018 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all
25.32* A backsight of 4.86 ft is taken on a benchmark whose elevation is 33.86 ft. What rod
reading is needed at that HI to set a blue top at grade elevation of 34.80 ft?
25.33 A backsight of 4.52 ft is taken on a benchmark whose elevation is 658.28 ft. A foresight
of 5.04 ft and a backsight of 6.04 ft are then taken in turn on TP1 to establish a HI. What
rod reading will be necessary to set a blue top at a grade elevation of 657.96 ft?
© 2018 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all
© 2018 Pearson Education, Inc., Hoboken, NJ. All rights reserved. This material is protected under all