LEC17 Vertical Curves

Vertical Curves

Introduction to Vertical Curves

• Vertical curves are designed where two tangents with different grades meet.
• Surveyors and engineers traditionally use the term "vertical curve" to describe the use of parabolic curves in design work.
• Applications include highways, railways, dam spillways, landscape design, and roller coasters.
• Vertical curves provide a smooth transition between constant grades.
• Without smoothing, the rate of change, $r$, would be too abrupt.
• Vertical curves are unnecessary when the algebraic difference of the intersection grade is less than 0.5%.

Types of Vertical Curves

• Two main types:
  • Crest curve (summit curve)
  • Sag curve

Factors to Consider for Designing Tangent Grades and Curves

1. Providing a good fit with the existing ground profile to minimize cuts and fills.
2. Balancing the volume of cut material against fill.
3. Maintaining adequate drainage.
4. Not exceeding maximum specified grades.
5. Meeting fixed elevations such as intersections with other roads.

• Curves must:
  • Fit the grade lines they connect.
  • Have lengths sufficient to meet specifications for a maximum rate of change of grade (affecting vehicle occupant comfort).
  • Provide sufficient sight distance for safe vehicle operation.

Vertical Curve Elements

• PVC (Point of Vertical Curvature) or BVC (Beginning of Vertical Curve): Where the curve begins.
• PVI (Point of Vertical Intersection) or V (Vertex): Where the grade tangents intersect.
• PVT (Point of Vertical Tangency) or EVC (End of Vertical Curve): Where the curve ends.
• POVC (Point on Vertical Curve): Any point on the curve.
• POVT (Point on Vertical Tangent): Any point on either tangent.
• $g_1$: Grade of the tangent on which the PVC is located (back tangent), measured in percent of slope.
• $g_2$: Grade of the tangent on which the PVT is located (forward tangent), measured in percent of slope.
• $L$: Horizontal length of the curve (in stations), measured from PVC to PVT.

Staking Out Vertical Curves

• Elevations at selected points (full or half stations in the English system, or 20, 30, or 40 m in the metric system) along vertical parabolic curves are usually computed by the tangent-offset method.
• Offsets from the tangent to a vertical curve (tangent offsets) are proportional to the squares of the distances from the point of tangency.
• The point of tangency may be the PVC or PVT.

General Equation of a Vertical Parabolic Curve

• General mathematical expression of a parabola:
  • $Y<em>p = a + bx</em>p + cx_p^2$
    • $Y<em>p$: Ordinate at any point $p$ located at distance $x</em>p$ from the origin of the curve.
    • $a$, $b$, $c$: Constants.
    • $a$: Ordinate at the beginning of the curve, at $X = 0$.
    • $b$: Slope of a tangent to the curve, at $X = 0$.
    • $bx<em>p$: Change in the ordinate along the tangent over distance $x</em>p$.
    • $cx<em>p^2$: Parabola’s departure from the tangent (tangent offset) in distance $x</em>p$.
  • For a crest curve, $b$ has a positive algebraic sign, and $c$ is negative.

Equation of an Equal Tangent Vertical Parabolic Curve

• In an equal tangent curve, the horizontal distance from BVC to V and V to EVC are equal, i.e., $L/2$.
• In the XY axis, X values are measured from the BVC, and Y values are measured from the vertical datum of reference.
• Substituting surveying terminology into the parabola equation:

Equation of an Equal Tangent Vertical Parabolic Curve (Tangent Offset Equation)

• The rate of change of grade, $r$, for an equal-tangent parabolic curve equals the total grade change from BVC to EVC divided by length $L$ (in stations for the English system or 1/10th stations for metric units) over which the change occurs.
• $r = (g<em>2 - g</em>1) / L$
• The value $r$ (negative for a crest curve and positive for a sag curve) controls the rate of curvature and rider comfort.
• Y = Y{BVC} + g1X +
  arc{X^2}{2}
• $Y$ is the elevation at distance $X$ from BVC
• $L$ and $X$ are in stations, $r$, $g<em>1$ and $g</em>2$ are in percentages.

High or Low Point on a Vertical Curve

• High or low point location is important for:
  • Drainage conditions
  • Clearance beneath overhead structures
  • Cover over pipes
  • Sight distance
• At the high or low point:
  • The tangent of the curve will be horizontal.
  • Its slope equals zero.
• $X = -g_1 / r$ (from the PVC)

Example 1

• A grade $g<em>1$ of +3.00% intersects grade $g</em>2$ of –2.40% at a vertex whose station and elevation are 46+70 and 853.48 ft, respectively. An equal-tangent parabolic curve 600 ft long has been selected to join the two tangents. Compute and tabulate the curve for stakeout at full stations.

Example 2

• A grade $g<em>1$ of +3.00% intersects grade $g</em>2$ of –2.40% at a vertex whose station and elevation are 46+70 and 853.48 ft, respectively. An equal-tangent parabolic curve 600 ft long has been selected to join the two tangents. Compute and tabulate the curve for stakeout at full stations. Compute the station and elevation of the curve's high point.

Curve Computation by Proportion

• Basic property of a parabola: Offsets from a tangent to a parabola are proportional to the squares of the distances from the point of tangency.
• Example 3: What's the offset from the tangent to the curve at station 47+00 for the previous example?

Staking a Vertical Parabolic Curve

• Prior to construction, the planned centerline (or an offset one) will normally be staked at full or half stations, as well as other critical horizontal alignment points such as PCs and PTs.
• Slope stakes will be set out perpendicular to the centerline at or near the slope intercepts to guide rough grading.
• Excavation and embankment construction then proceed and continue until the grade is near plan elevation.
• Blue tops:
  • Centerline stations are then staked again using sharpened 2-in. square wooden hubs, usually about 10 in. long.
  • Their tops are driven to grade elevation and colored blue.
  • Contractors request blue tops when excavated areas are still slightly high and embankments somewhat low.
  • After blue tops are set to mark the precise grade, final grading is completed.

Setting Blue Tops at Grade

• A circuit of differential levels is run from a nearby benchmark to establish the HI (Height of Instrument) of a leveling instrument in the project area.
• The difference between the HI and any station’s grade is the required rod reading on that stake.
• Example:
  • HI = 856.20 ft
  • Station 45+00 is to be set.
  • Curve elevation at station 45+00 = 847.62 ft.
  • Required rod reading = 856.20 - 847.62 = 8.58 ft.
  • If the initial rod reading is 8.25 ft, the stake must be driven down an additional 8.58 - 8.25 = 0.33 ft.
  • This process is repeated until the required reading of 8.58 is achieved, whereupon the stake is colored blue.

Additional Considerations for Staking

• This process is continued until all stakes are set.
• The required rod reading at station 46+00, for an HI of 856.20, is (856.20 - 849.0 = 7.20).
• If a rock is encountered and the stake cannot be driven to grade, a vertical offset of, say, 1.00 ft above grade can be marked and noted on the stake.
• When the level is too far away from the station being set, a turning point is established, and the instrument is brought forward to establish a new HI.
• Whenever possible, level circuit checks should be made by closing on nearby benchmarks as blue top work on the project progresses.
• Also, when quitting for the day or when the job is finished, the level circuit must always be closed to verify that no mistakes were made.

Designing a Curve to Pass through a Fixed Point

• Frequently encountered in practice.
• Occurs where a new grade line must meet an existing railroad or highway crossing.
• When a minimum vertical distance must be maintained between the grade line and underground utilities or drainage structures.
• How to solve the problem: Given the station and elevation of the VPI, and grades $g<em>1$ and $g</em>2$ of the back and forward tangents, respectively, the problem consists of calculating the curve length, $L$, required to meet the fixed condition.

Example: Designing a Curve to Pass through a Fixed Point

• Grades $g<em>1$ = -4.00% and $g</em>2$ = +3.80% meet at VPI station 52+00 and elevation 1261.50. Design a parabolic curve to meet a railroad crossing, which exists at station 53+50 and elevation 1271.20.

Sources of Error and Mistakes in Laying Out Vertical Curves

• Sources of Errors:
  1. Making errors in measuring distances and angles when staking the centerline.
  2. Not holding the level rod plumb when setting blue tops.
  3. Using a leveling instrument that is out of adjustment.
• Mistakes:
  1. Arithmetic mistakes.
  2. Failure to properly account for the algebraic signs of $g<em>1$ and $g</em>2$.
  3. Subtracting offsets from tangents for a sag curve or adding them for a crest curve.
  4. Failure to make the second-difference check.
  5. Not completing the level circuit back to a benchmark after setting blue tops.

References

• Charles D. Ghilani and Paul R. Wolf, Elementary Surveying: An Introduction to Geomatics, 16th edition, Pearson, 2015.
• Image sources:
  • Mmola, 2019, Application of Particle Swarm Optimization Algorithm to Optimize Stope Layout for Underground Mines
  • Hyunh, 2010, Design of vertical curves
  • US Navy