Vertical curves provide gradual changes between tangents of different grades. The simple parabola shown in is used in the highway profile design of vertical curves.

For vertical curve discussion purposes, the following parameters are defined:

$K=\frac{L}{A}$

$E=\frac{AL}{800}$

$y=\frac{4D^{2}E}{L^2}d$
or $y=\frac{D^{2}A}{200L}$

Where:

G

1

and

G

2

=

tangent grades, percent

K =

length of vertical curve per percent change in A (also known as the design control)

A =

algebraic difference in grades (G

1

– G

2

), percent

L =

length of vertical curve, ft

E =

vertical offset at the VPI, ft

y =

ordinate from tangent to curve, ft

D =

distance from nearest VPC or VPT to any point on curve, ft

The minimum lengths of crest and sag vertical curves for different values of A to provide the stopping sight distances for each design speed are shown in and , respectively. The solid lines give the minimum vertical curve lengths on the basis of rounded values of K. These lengths represent minimum K-values based on design speed.

A dashed curve crossing the solid lines indicates where sight distance (S) equals length of vertical curve (L) (i.e., S = L). Note that to the right of the S = L line, the value of K is a simple and convenient expression of the design control. For each design speed, this single value is a positive number that is indicative of the rate of vertical curvature. The design control in terms of K covers all combinations of

A and L for any one design speed; thus, A and L need not be indicated separately in a tabulation of the design values.

The selection of design curves is facilitated because the length of curve is equal to K times the algebraic difference in grades in percent,

L = KA.

Conversely,

the checking of curve design is simplified by comparing all curves with the design value for K.

Where S is greater than L, the values plot as a curve (as shown by the dashed curve extension for 45 mph). For small values of A, the vertical curve lengths are zero because the sight line passes over the apex. Since this relationship does not represent desirable design practice except in limited conditions (see discussion in ), a minimum length of vertical curve is shown. Attention should be given where there are successive vertical curves.

T

his minimum length is not considered a design control

(i.e., a design exception, design waiver would not be required for these minimum length values if the minimum stopping sight distance for the relevant design speed is met).

For sag vertical curves, at least four different criteria for establishing the lengths of sag vertical curves are recognized to some extent. These are:

Generally, a sag vertical curve should be long enough that the light beam distance is nearly the same as the stopping sight distance, especially at intersections located within the vicinity of the sag curve. Accordingly, it is appropriate to use stopping sight distances for different design speeds to establish sag vertical curve lengths. The resulting sag vertical curves for the recommended stopping sight distances for each design speed are shown in with the solid lines representing the rounded K-values.

There is a level point on a vertical curve which can affect drainage, particularly on curbed facilities. This level point will only occur on Type I and Type III vertical curves shown in . Typically, there is no difficulty with drainage on highways if the curve is sharp enough so that a minimum grade of 0.30 percent is reached at a point about 50-ft from the crest or sag. This corresponds to a K-value of 167 which is plotted in and as the drainage threshold. All combinations above or to the left of this line satisfy the drainage criterion. The combinations below and to the right of this line involve flatter vertical curves.

Special attention is needed in these cases to ensure proper pavement drainage.

It is not intended that these values be considered a design maximum, but merely a value beyond which drainage should be more carefully designed.

provides minimum vertical curve lengths and associated K as a function of A.

Longer curves should be used wherever practical.

Because cost and energy conservation considerations are factors in operating continuous lighting systems, headlight sight distance should be generally used in the design of sag vertical curves.

“Comfort control criteria”

is approximately 50 percent of the sag vertical curve lengths required by headlight distance and should be reserved for special use. Instances where the comfort control criteria may be appropriately used include ramp profiles where safety lighting is provided and for economic reasons in cases where an existing element, such as a structure not ready for replacement, controls the vertical profile. Comfort control criteria should be used sparingly on continuously lighted facilities since local, outside agencies often maintain and operate these systems and operations could be curtailed in the event of energy shortages.

See for more information about sag vertical curve design to ensure that overhead sight obstructions such as structures for overpassing roadways, overhead sign bridges, and tree crowns do not reduce stopping sight distance below the appropriate minimum value.

When vertical curves are located within the limits of a bridge, it is preferred to locate the VPI, VPC, VPT’s in the center of the bridge abutment(s) or bent(s) to simplify construction.

Roadway profiles should be designed using vertical curves at vertical points of intersection and intersecting streets where practical. Designing a sag or crest vertical point of intersection without a vertical curve is generally acceptable where the grade difference (A) is:

When a grade change without vertical curve is specified, the construction process typically results in a short vertical curve being built (i.e., the actual point of intersection is “smoothed” in the field). It is recommended to always use vertical curves to address grade changes for the following conditions.

The minor approach (the approach with the lower ADT) of an intersection with yield or stop control may be constructed without vertical curves, provided that a grade change does not adversely affect local vehicle operations. A review of local traffic characteristics should be performed to identify a target design vehicle for each intersection for determining an appropriate grade change without a vertical curve. Typically, a grade change of 3 percent or less accommodates most vehicles. However, in highly constrained conditions, a grade change of up to 8 percent may operate effectively depending on local site conditions.

It is desirable that the intersection of highway and railroad be as level as possible from the standpoint of sight distance, rideability, and braking and acceleration distances. If a railroad crossing is located at the peak of a vertical curve on the highway, the curve should be adequately flat to avoid hanging-up of vehicles and of sufficient length to ensure an adequate view of the crossing consistent with the highway design or operating speed. Refer to TxDOT’s (Chapter 7, Section 1) and for additional information.

Adequate

intersection sight distance

should be provided at intersecting roads.

At a minimum, stopping sight distance (measured from the edge of travel way of the intersecting facility) must be provided.

The grades of intersecting roads should be as level as practical.

Approaching an intersection, drivers might misjudge the effect of steep grades on stopping or accelerating distances.

At intersections where marked or unmarked crosswalks are present, the roadway profile grade within the crosswalks should be limited so that the crosswalk is accessible and usable by all individuals, including those with disabilities. The roadway profile grade across a crosswalk must not exceed:

Accessibility accommodations should be considered when determining appropriate profile grades for all future crosswalk locations. See for additional guidance and when TDLR variances are required.

It is generally preferable to provide a smooth junction at intersecting streets. To accomplish this, the profile, cross slopes, and crowns of both roads are warped to create a common plane at the intersection. This technique is commonly referred to as

“table-topping.”

Design features including but not limited to proper drainage and pedestrian use should be carefully considered when table-topping an intersection.

When table-topping of an intersection is not feasible and minor roadway traffic is expected to stop or travel slowly through the intersection, grades should generally favor the

major roadway

(the roadway with the higher ADT) to provide a smooth profile. This is accomplished by warping the grades on the minor roadway to fit the cross-section of the major roadway. Similarly, for intersections contained within a horizontal curve with superelevation, an attempt should be made to match the superelevated section through the intersection.

Additional profiling of intersection features (e.g., curb, median islands, and curb ramps) may be required to ensure proper drainage and a smooth intersection surface.

Contours at appropriate intervals (e.g., 1-ft major and 0.25-ft minor contours) and key curb elevations should be shown in the plans for proper construction of such intersections.

Low points near intersections must be designed to prevent water ponding within pedestrian access routes.

Minimum vertical clearance for structures should be calculated from the point on the roadway with least clearance including the outside edge of one shoulder to the other. This includes both sides of the structure being passed under to verify all vehicles can clear the structure at any point along the structure including travel lanes and shoulders.

Required vertical clearances also allow for potential future overlays.

shows the minimum vertical clearances required. provides additional guidance for roadways on the THFN.

During construction, refer to TxDOT’s for requirements on maximum time between erection of girders and placement of deck for spans overactive lanes of traffic to avoid over-height impacts.

A clearance of 15-ft is preferable to provide a sufficient buffer between allowed and actual heights to avoid over-height impacts.