Compound and Reverse Curves

While simple horizontal curves are the most common horizontal alignment elements, complex alignments sometimes require combining multiple curves. Two such configurations are compound curves and reverse curves.

For the PE Civil Transportation exam, you must understand their geometric relationships, coordinate calculations, stationing workflows, and critical design limitations set by AASHTO’s Green Book (GDHS), Chapter 3.

Compound Curves

A compound curve consists of two or more circular curves of different radii curving in the same direction, meeting at a common Point of Compound Curvature (PCC).

       Back Tangent
==================[PC] (Radius R_1)
                     \
                      \    Curve 1 (Angle Delta_1)
                       \
                       [PCC] (Common Tangent Line)
                         \
                          \    Curve 2 (Angle Delta_2)
                           \
                           [PT] (Radius R_2)
                              ================== Forward Tangent

Key Geometry:

  • Total Deflection Angle ($\Delta$): The total angle of deflection between the back and forward tangents is the sum of the individual curve central angles: $$\Delta = \Delta_1 + \Delta_2$$
  • Stationing: Stationing runs continuously along the curve: $$\text{PCC Station} = \text{PC Station} + L_1$$ $$\text{PT Station} = \text{PCC Station} + L_2$$ Where $L_1$ and $L_2$ are the arc lengths of Curve 1 and Curve 2, calculated using their respective radii ($R_1, R_2$) and central angles ($\Delta_1, \Delta_2$).

AASHTO Design Criteria (Radius Ratios)

To prevent drivers from experiencing sudden shifts in lateral acceleration, the transition from a larger radius ($R_1$) to a smaller radius ($R_2$) must be gradual:

  • High-Speed Highways: The ratio of the larger radius to the smaller radius should not exceed 1.5: $$\frac{R_{large}}{R_{small}} \le 1.5$$
  • Low-Speed Intersections and Ramps: The ratio of the larger radius to the smaller radius should not exceed 2.0: $$\frac{R_{large}}{R_{small}} \le 2.0$$

Reverse Curves

A reverse curve consists of two circular curves of equal or different radii curving in opposite directions, meeting at a common Point of Reverse Curvature (PRC).

                    Curve 1 (Left Curve)
                   /
================[PC]
                 \
                  \
                  [PRC] (Point of Zero Curvature)
                     \
                      \
                      [PT] (Right Curve)
                          ====================

The Superelevation Hazard in Reverse Curves

Reverse curves present a significant safety issue on high-speed roads because the roadway must transition from full superelevation in one direction (e.g., $+6\%$) to full superelevation in the opposite direction (e.g., $-6\%$).

  • The Flat Spot: At the PRC, the cross slope must pass through $0\%$. This creates a flat area of pavement with zero cross slope, leading to severe drainage problems and hydroplaning risks.
  • Driver Control: A driver crossing the PRC must quickly rotate the steering wheel in the opposite direction while the vehicle’s lateral acceleration flips, creating instability.

AASHTO Guidelines for Reverse Curves:

  • High-Speed Routes: Reverse curves are strongly discouraged. If they must be used, they must be separated by a tangent section of sufficient length to accommodate the superelevation runoff of both curves, or by spiral transition curves.
  • Low-Speed Routes: Acceptable at low speeds (such as temporary detours or parking lot lanes) where lateral acceleration forces are minimal.

Worked Example: Compound Curve Verification and Stationing

Problem Statement

A compound curve is proposed for an interchange exit ramp. The design speed of the ramp is $35\text{ mph}$ (low-speed facility).

  • Curve 1: Radius ($R_1$) = $600\text{ ft}$, Central Angle ($\Delta_1$) = $24^\circ 00'$
  • Curve 2: Radius ($R_2$) = $320\text{ ft}$, Central Angle ($\Delta_2$) = $38^\circ 00'$
  • PC Station: $200+50.00$
  1. Verify whether the proposed compound curve radii meet the AASHTO design criteria for a low-speed ramp.
  2. Calculate the arc lengths of Curve 1 ($L_1$) and Curve 2 ($L_2$).
  3. Determine the stationing of the Point of Compound Curvature (PCC) and the Point of Tangency (PT).

Solution

  1. Verify AASHTO Radius Ratio Criterion:

    • Larger radius ($R_{large}$) = $600\text{ ft}$
    • Smaller radius ($R_{small}$) = $320\text{ ft}$
    • Calculate the ratio: $$\text{Ratio} = \frac{R_1}{R_2} = \frac{600\text{ ft}}{320\text{ ft}} = 1.875$$
    • For low-speed facilities, the AASHTO limit is $2.0$.
    • Since $1.875 \le 2.0$, the proposed compound curve satisfies the AASHTO criterion.

  2. Calculate Arc Lengths ($L_1$ and $L_2$):

    • Curve 1 Length ($L_1$): $$L_1 = R_1 \Delta_1 \left( \frac{\pi}{180} \right) = 600\text{ ft} \times 24^\circ \times 0.0174533 = 251.33\text{ ft}$$
    • Curve 2 Length ($L_2$): $$L_2 = R_2 \Delta_2 \left( \frac{\pi}{180} \right) = 320\text{ ft} \times 38^\circ \times 0.0174533 = 212.23\text{ ft}$$

  3. Determine Stationing:

    • PC Station: $200+50.00$
    • PCC Station: $$\text{PCC Station} = \text{PC} + L_1 = (200+50.00) + 251.33\text{ ft}$$ $$\text{PCC Station} = 20050.00 + 251.33 = 20301.33 \implies \text{Station } 203+01.33$$
    • PT Station: $$\text{PT Station} = \text{PCC} + L_2 = (203+01.33) + 212.23\text{ ft}$$ $$\text{PT Station} = 20301.33 + 212.23 = 20513.56 \implies \text{Station } 205+13.56$$

Answer

  1. The radius ratio is 1.875, which meets the AASHTO limit ($\le 2.0$).
  2. Curve lengths: $L_1 = \mathbf{251.33\text{ ft}}$ and $L_2 = \mathbf{212.23\text{ ft}}$.
  3. PCC Station: 203+01.33; PT Station: 205+13.56.

Crucial Exam Tips

  • Identifying the Speed Limit: On the exam, pay close attention to whether the roadway is high-speed ($\ge 50\text{ mph}$) or low-speed ($\le 45\text{ mph}$). Applying the wrong radius ratio limit (e.g., using $2.0$ instead of $1.5$ on a highway) is a common trap.
  • No Short Cuts through the PI: In compound curves, you cannot calculate the PT station by working through a single PI tangent length like you would for a simple curve. You must calculate the lengths of the individual curve arcs ($L_1, L_2$) and add them sequentially starting from the PC.
  • PRC Stationing in Reverse Curves: If a question asks for the stationing of a reverse curve, remember that: $$\text{PRC Station} = \text{PC Station} + L_1$$ $$\text{PT Station} = \text{PRC Station} + L_2$$ Just like compound curves, the stationing runs strictly along the centerline arc of the roadway.