Superelevation Basics

When a vehicle travels along a horizontal curve, it experiences a centrifugal force that acts to push it outward, away from the center of the curve. To counter this lateral force and ensure passenger comfort and vehicle stability, the roadway cross-section is tilted or “banked.” This banking is called superelevation ($e$).

For the PE Civil Transportation exam, the physics, fundamental equations, side friction relationships, and minimum radius calculations are governed by the NCEES PE Civil Reference Handbook and AASHTO’s Green Book (GDHS), Chapter 3.

The Physics of Superelevation

The forces acting on a vehicle traversing a horizontal curve include gravity, lateral tire friction, and centrifugal force. Balancing these forces yields the fundamental superelevation equation:

$$e + f = \frac{V^2}{15 R}$$

Where:

  • $e$ = Superelevation rate (expressed as a decimal, e.g., $0.06$ for a $6\%$ cross slope).
  • $f$ = Side friction factor (dimensionless, representing the lateral friction coefficient between the tires and pavement).
  • $V$ = Design speed (mph).
  • $R$ = Radius of the horizontal curve (ft).
  • 15 = Conversion constant that reconciles gravity ($g = 32.2\text{ ft/s}^2$) and units (converting mph to ft/s).

Maximum Superelevation Rates ($e_{max}$)

The maximum rate of superelevation ($e_{max}$) is restricted by environmental and operational factors:

  1. Climate and Terrain: In areas subject to ice and snow, $e_{max}$ is typically limited to 6% to 8% to prevent slow-moving vehicles from sliding down the banked surface toward the inside of the curve. In southern climates, rates up to 10% to 12% are acceptable.
  2. Urban vs. Rural Context:
    • Urban Streets: Limited to 4% to 6% because low speeds, frequent stop-and-go traffic, intersections, and driveways make steep banking hazardous and uncomfortable.
    • Rural Highways and Freeways: Typically designed with $e_{max}$ of 8% to 10%.

Side Friction Factors ($f_{max}$)

The maximum comfortable side friction factor ($f_{max}$) represents the threshold at which passengers feel a distinct outward pull and begin to feel uncomfortable. It is not the physical limit of tire-pavement adhesion.

  • Speed Dependency: $f_{max}$ decreases as design speed increases. This is because drivers are more sensitive to lateral acceleration at higher speeds.
  • Typical $f_{max}$ Values (AASHTO Green Book):
    • $30\text{ mph}$: $f_{max} = 0.16$
    • $40\text{ mph}$: $f_{max} = 0.15$
    • $50\text{ mph}$: $f_{max} = 0.14$
    • $60\text{ mph}$: $f_{max} = 0.12$
    • $70\text{ mph}$: $f_{max} = 0.10$
    • $80\text{ mph}$: $f_{max} = 0.08$

Minimum Radius of Curve ($R_{min}$)

The minimum radius of a curve ($R_{min}$) represents the sharpest curve that can be safely traversed at a given design speed, using the maximum allowable superelevation ($e_{max}$) and the maximum side friction factor ($f_{max}$):

$$R_{min} = \frac{V^2}{15(e_{max} + f_{max})}$$

Worked Example: Minimum Radius and Friction Demand

Problem Statement

A new rural highway segment is being designed. The design parameters are as follows:

  • Design Speed ($V$): $65\text{ mph}$
  • Maximum allowable superelevation ($e_{max}$): $8.0\%$ ($0.08$)
  • Maximum side friction factor ($f_{max}$) at $65\text{ mph}$: $0.11$
  1. Calculate the minimum allowable radius ($R_{min}$) for the horizontal curve.
  2. If the designer uses a radius of $R = 2,200\text{ ft}$ for a curve on this highway, and the design superelevation rate ($e_d$) is set to $5.4\%$ ($0.054$), calculate the actual side friction demand ($f$) felt by a vehicle traveling at $65\text{ mph}$. Verify if it is within comfortable limits.

Solution

  1. Calculate the Minimum Radius ($R_{min}$): Use the minimum radius formula:

    $$R_{min} = \frac{V^2}{15(e_{max} + f_{max})}$$

    $$R_{min} = \frac{65^2}{15(0.08 + 0.11)} = \frac{4225}{15(0.19)}$$

    $$R_{min} = \frac{4225}{2.85} = 1482.46\text{ ft}$$

    Rounding up for design: $1,485\text{ ft}$ (or the exact math value of $1,482.5\text{ ft}$ for exam selection).

  2. Calculate the Side Friction Demand ($f$): Use the fundamental superelevation equation solved for $f$:

    $$f = \frac{V^2}{15 R} - e$$

    Substitute the given values:

    $$f = \frac{65^2}{15(2200)} - 0.054$$

    $$f = \frac{4225}{33000} - 0.054$$

    $$f = 0.12803 - 0.054 = 0.074$$

    Verify with Comfort Limit: The friction demand is $0.074$. The maximum comfortable friction factor at $65\text{ mph}$ is $0.11$. Since $0.074 \le 0.11$, the lateral friction is well within the comfortable range for drivers.

Answer

  1. Minimum curve radius ($R_{min}$): 1,482.5 ft
  2. Actual side friction demand ($f$): 0.074 (satisfactory).

Crucial Exam Tips

  • Decimal vs. Percent Form: Always convert superelevation percentages to decimals before calculating. For example, a superelevation of $6\%$ must be entered as $0.06$, not $6$. Entering $6$ will throw off your answer by a factor of 100.
  • The Comfort vs. Limit Trap: Do not assume that every curve is designed to maximize friction. In practice, designers use AASHTO Tables 3-7 to 3-12 to lookup the design superelevation rate ($e_d$) for a given radius, which distributes the centrifugal force such that the friction demand is much lower than $f_{max}$. Only use $f_{max}$ when calculating the limiting condition ($R_{min}$).
  • Friction Sign Conventions: If a vehicle is traveling below the design speed on a heavily banked curve, the friction force can actually act upward along the slope to prevent the vehicle from sliding down. The equation is self-correcting: if you calculate a negative $f$, it simply means friction is acting to prevent the vehicle from sliding down the slope.