What is a horizontal circular curve?

A horizontal circular curve is an arc between two straight lines known as tangents.

Law of Sines

asin A = bsin B = csin C

Use the Law of Sines when you have a non-right triangle and know either two (2) angles and one (1) side (AAS or ASA), or two (2) sides and an angle opposite one (1) of them (SSA). It's also possible to use it when all three (3) sides are known to find the angles. The key is knowing at least one (1) side-angle pair, where the angle is opposite the given side. 

Law of Cosines

AB² = BC² + AC² - 2(BC)(AC)cos(C) 

Use the Law of Cosines to solve for a missing side or angle in any triangle when you know two (2) sides and the included angle (SAS), or when you know all three (3) sides (SSS). This is in contrast to the law of sines, which is used when you have an angle-side pair (ASA, AAS, or SSA). 

What is the equation for the degree of curvature by arc definition (used for highways / roadways)?

D_a = 5279.578' / R

D_a = degree of curvature by arc definition (°)

R = radius (ft)

What is the equation for the degree of curvature by chord definition (used for railroads)?

D_c = 2sin^-1(50/R)

D_c = degree of curvature by chord definition (°)

R = radius (ft)

Determine angle ‘α’ for the following circular curve: 

A) 44° 40’     

B) 44° 25”

C) 88° 50’

D) 44° 25’ 

[image] 

α = I / 2

I = 180° - 45° 20' - 45° 50’ = 88° 50’ 

α = 88° 50’ / 2 = 44° 25’ 

[image]

Minimum Radius Equation

(U.S. Customary)

R_min = V² / 15(0.01e_max + f_max)

R_min = minimum radius

V = velocity

e_max = maximum rate of superelevation

f_max = maximum side friction factor

Reference: AASHTO GDHS-7 (Eq. 3-8)

What is the minimum radius of curvature based on?

The minimum radius of curvature is based on a threshold of driver comfort that is sufficient to provide a margin of safety against skidding and vehicle rollover. The minimum radius of curvature is also an important control value for determining superelevation rates for flatter curves.

Reference: AASHTO GDHS-7 (Page 3-33)

Where is the Horizontal Control Line located?

Often along the centerline of the alignment.

Reference: AASHTO GDHS-7 (Page 3-33)

What does the variable T_R represent?

Tangent Runout

What does SRR stand for?

Superelevation Runoff Rate

Tangent Runout Distance Equation

T_r = wp / SRR

T_r = tangent runout distance

w = lane width

p = rate of cross slope

SRR = rate of superelevation runoff

Superelevation Runoff Distance Equation

L = we / SRR

L = runoff distance

w = lane width

e = superelevation (rate)

 SRR = rate of superelevation runoff

What is the definition of the distance represented by the radius in the horizontal curve equations?

The horizontal curve equations use a radius measured to a vehicle's center of gravity, which is approximately the center of the innermost travel lane. Furthermore, the equations do not consider the width of the roadway or the horizontal control line. 

Reference: AASHTO GDHS-7 (Page 3-33)

What are the four (4) factors controlling the maximum rates of superelevation used on highways?

1. Climate conditions (frequency and amount of snow and ice)

2. Terrain conditions (i.e., flat, rolling, and mountainous)

3. Area type (i.e., rural or urban)

4. Frequency of very-slow moving vehicles whose operation might be affected by high superelevation rates

True or False

Superelevation may be omitted on low-speed streets in urban areas where severe constraints are present.

True

What is the equation used to calculate sight distance on horizontal curves (U.S. Customary)?

HSO = R[1 - cos(28.65S / R)]

HSO = horizontal sight line offset (ft)

R = radius of curve (ft)

S = sight distance (ft)

Reference: AASHTO GDHS-7 (Eq. 3-37)

Which design code is popularly known as the Green Book?

A Policy on Geometric Design of Highways and Streets

SSD 

Stopping Sight Distance

PSD

Passing Sight Distance

DSD

Decision Sight Distance

ISD

Intersection Sight Distance

What is the definition of sight distance?

Sight distance is the length of roadway ahead that is visible to the driver. 

What is the definition of stopping sight distance?

Stopping Sight Distance is the sum of two distances: (1) the distance traversed by the vehicle from the instant the driver sights an object necessitating a stop to the instant the brakes are applied; and (2) the distance needed to stop the vehicle from the instant the brake application begins. These are referred to as brake reaction distance and braking distance, respectively. 

What is the coefficient f known as?

The coefficient f has been called lateral ratio, cornering ratio, unbalanced centrifugal ratio, friction factor, and side friction factor.

How is stopping sight distance calculated if the side friction factor, f, is unknown?

If the deceleration rate, a, is known, the side friction factor, f, can be replaced with a / g, where g is the acceleration of gravity (9.81 m/s² or 32.2 ft/s²).

What is the equation used to calculate the equilibrium elevation, E, for a railroad? 

E = G_effv² / gR

E = equilibrium elevation

G_eff = center-to-center rail spacing

v = velocity

g = acceleration of gravity (9.81 m/s² or 32.2 ft/s²)

R = radius 

What does the variable G_eff represent?

The effective gauge, G_eff, is the center-to-center rail spacing.

What is the definition of the term Transition Length?

Transition Length is the distance required in transitioning the road from normal to full superelevation. It consists of Runout Length and Runoff Length.

Transition Length = Runout Length + Runout Length

What is the definition of the term Runout Length?

The length of roadway needed to change the cross slope of the road from a "normal crown" (a sloping surface for drainage) to a flat, level surface (zero superelevation).

What is the definition of the term Runoff Length?

The length of roadway needed to transition from the flat, level surface to the full superelevation rate required for the curve.

What is the purpose of the Runoff Length?

To introduce the desired bank, or superelevation, that helps vehicles safely navigate the curve.

What is the purpose of the Runout Length?

To flatten the adverse cross slope of the outer lane before beginning the superelevation. An adverse slope would drain toward the inside of the curve, which is undesirable.

What is the position of the Runout Length?

Occurs immediately before the runoff section when entering a curve. The reverse is true when exiting a curve.

What is the position of the Runoff Length?

Occurs after the runout section and continues into the curve itself.

A certain problem requires the calculation of the Tangent Runout (T_R) and / or Superelevation Runoff (L). However, the Superelevation Runoff Rate (SRR) is missing, what can be done to solve the problem?

If the design speed is given, refer to Table 3-17 in the AASHTO Green Book (7th Edition). The table shows L values based on the design speed. 

What is a Spiral Curve?

A curve of continuously changing radius. It varies from 0° of curvature (infinite radius) at its Tangent to Spiral (TS) to the specific Degree of Curvature (DC), of the curve it connects to at the Spiral to Curve (SC).

What is the advantage of using a Compound Curve?

A Compound Curve is advantageous in effecting desirable shapes of turning roadways for at-grade intersections and interchanges ramps.

What is a Compound Curve?

A Compound Curve is a horizontal alignment composed of two (2) or more circular arcs of different radii that bend in the same direction and are joined at a common point. These curves are often used in situations with space constraints, such as highway interchanges, where fitting a simple curve is not feasible.

When designing a Compound Circular Curve for an open highway, what is the generally accepted ratio of the flatter radius to the sharper radius?

The ratio should not exceed (1.5:1).

Reference: AASHTO GDHS-7 (Pg. 3-90)

For a Compound Circular Curve at an intersection where drivers accept more rapid changes in direction and speed, what is the design ratio of the radius of the flatter arc to the radius of the sharper arc?

The radius of the flatter arc can be as much as 100% greater than the radius of the sharper arc, a ratio of (2:1).

Reference: AASHTO GDHS-7 (Pg. 3-90)

The horizontal alignment of an interchange exit ramp consists of a series of three (3) consecutive and progressively sharper circular curves that form a single compound circular curve. Proceeding in the direction of traffic, if the first curve has a radius of 2,000 ft, what is the minimum radius of the third curve? 

Radius of first curve = 2,000 ft

Minimum radius of the second curve (sharper than the first curve) = Half of the radius of the first curve = 1,000 ft

Minimum radius of the third curve (sharper than the second curve) = Half of the radius of the second curve = 500 ft 

If given a radius and EITHER a superelevation OR a side friction factor, how is the most appropriate advisory speed determined?

Use Table  3-7 in the AASHTO Green Book (Pg. 3-34). Note that the posted speed limit may be given as arbitrary information.

The GDHS determines safe design speed on highway curves. Which of the following criteria normally apply to designing a safe curve?

I.         curve radius

II.       passenger comfort factor

III.      sight distance

IV.       shoulder width

V.        speed limit posting

VI.      superelevation rate

VII.     volume of traffic

VIII.    weather conditions

(A) I, II, III, and IV only

(B) I, III, VI, and VIII only

(C) I, II, III, VI and VIII

(D) I, II, III, IV, V and VIII

(C) I, II, III, VI and VIII

The numbers for airport runways are assigned

(A) according to the landing heading, or azimuth, rounded to the nearest 10° and dropping the last zero.

(B) according the take-off heading, rounded to the nearest 10° and dropping the last zero.

(C) based on the airport design firm's internal design procedure.

(D) based on the heading bearing, rounded to the nearest 10°.

A magnetic compass can show the heading under no power and poor visibility conditions. A universal default numbering system is a fail-safe feature of aviation, so that pilots can land planes with the greatest degree of safety. Without a unified numbering system, confusion would result, especially among international flights.

(A) according to the landing heading, or azimuth, rounded to the nearest 10° and dropping the last zero.

What is the equation used to calculate the total intersection angle (I_T or Δ_T) for a three-centered compound curve?

I_T = I₁ + I₂ + I₃

_OR

Δ_T = Δ₁ + Δ₂ + Δ₃

What is the equation used to calculate the distance (X) from Point of Curvature (PC) to Point of Tangency (PT) for a three-centered compound curve?

What is the equation used to calculate the perpendicular distance (Y) from the Backward Tangent (T_L) to Point of Tangency (PT) for a three-centered compound curve?

What is the equation used to calculate the Short Tangent (T_s) for a three-centered compound curve?

T_s = Y / SIN(I_T)

OR

Y = R₁ - [R₃COS(I_T)] - [(R₁ - R₂)COS(I₁)] - [(R₂ - R₃)COS(I₁ + I₂)]

What is the equation used to calculate the Long Tangent (T_L) for a three-centered compound curve?

What is the equation used to calculate the length of a spiral curve (L_s)?

L_s = 3.15V³ / RC

L_s = length of spiral curve (ft)

V = velocity (mph)

R = radius (ft)

C = rate of change in lateral acceleration (ft/s³)

Reference: AASHTO GDHS-7 (Eq. 3-28)

What is the equation used to calculate the rate of change in degree of curvature for a spiral curve?

rate of change in curvature = D_C / L_s

D_C = degree of curvature (°)

L_s = length of spiral curve (ft or stations)

What is the Turning Point of a vertical curve?

The Turning Point is where the slope of the curve becomes zero. In other words, it is the point of maximum / minimum elevation.

What is the equation used to calculate the turning point of a vertical curve?

x_m = -g₁ / 2a = g₁L / g₁ - g₂

x_m = horizontal distance to min / max elevation on curve

g₁ = slope of back tangent

g₂ = slope of forward tangent

L = length of curve

Reference: NCEES Civil Handbook 2.0.2

What is the equation used to calculate the length of a vertical curve that passes through a fixed point above / below an obstruction?

L = 2w(z + 1) / (z - 1)

w = distance between PVI and fixed point (station)

z = √(ElevA - ElevC / ElevA - ElevB)

The variable z is known as the curve constant and is unitless. Note the variable s is also used to represent the curve constant.

What is the equation used to calculate the length of a vertical curve that passes through the Turning Point?

L = 2(g₂ - g₁)(elev_PVI - elev_TP) / g₁g₂

g₁ = slope of back tangent

g₂ = slope of forward tangent

elev_PVI = elevation of PVI

elev_TP = elevation of Turning Point

What is the equation used to calculate the length of a vertical curve that passes through a fixed point without an obstruction?

What is the equation used to calculate the length of a vertical curve based on sight distance?

L = KA

L = length of vertical curve (ft)

K = rate of vertical curvature (tabulated factor)

A = |g₂ - g₁|

When asked to calculate the length of a CREST vertical curve based on sight distance, which tables are used to find the value of K, the rate of vertical curvature?

Refer to Table 3-35 in AASHTO GDHS-7 for K-values associated with stopping sight distance (SSD).

Refer to Table 3-36 in AASHTO GDHS-7 for K-values associated with passing sight distance (PSD).

When asked to calculate the length of a SAG vertical curve based on sight distance, which table is used to find the value of K, the rate of vertical curvature?

Refer to Table 3-37 in AASHTO GDHS-7 for K-values associated with stopping sight distance (SSD).