Formula Sheet: Highway Design

Horizontal Alignment

Degree of Curve

Arc Definition (used in highway design): \[D = \frac{5729.58}{R}\] Chord Definition (used in railroad design): \[D = \frac{5729.58}{R} \times \sin\left(\frac{D}{2}\right)\]

• D = degree of curve (degrees)
• R = radius of curve (ft)
• Arc definition: angle subtended by 100 ft arc
• Chord definition: angle subtended by 100 ft chord

Circular Curve Elements

Radius of Curve: \[R = \frac{5729.58}{D}\] Tangent Length: \[T = R \tan\left(\frac{Δ}{2}\right)\] Length of Curve: \[L = \frac{100Δ}{D} = \frac{πRΔ}{180}\] External Distance: \[E = R\left[\frac{1}{\cos\left(\frac{Δ}{2}\right)} - 1\right] = R\left[\sec\left(\frac{Δ}{2}\right) - 1\right]\] Middle Ordinate: \[M = R\left[1 - \cos\left(\frac{Δ}{2}\right)\right]\] Long Chord: \[LC = 2R\sin\left(\frac{Δ}{2}\right)\]

• Δ = total central angle (deflection angle) (degrees)
• R = radius of curve (ft)
• T = tangent distance from PC or PT to PI (ft)
• L = length of curve along arc (ft)
• E = external distance from PI to curve midpoint (ft)
• M = middle ordinate, perpendicular distance from chord midpoint to curve midpoint (ft)
• LC = long chord from PC to PT (ft)

Superelevation and Side Friction

Fundamental Relationship: \[e + f = \frac{V^2}{15R}\]

• e = superelevation rate (ft/ft, decimal)
• f = side friction factor (dimensionless)
• V = design speed (mph)
• R = radius of curve (ft)

Minimum Radius: \[R_{min} = \frac{V^2}{15(e_{max} + f_{max})}\]

• e_max = maximum superelevation rate (typically 0.04 to 0.12)
• f_max = maximum side friction factor (speed-dependent)

Superelevation Runoff Length: \[L_r = \frac{EDW}{Δ_{rel}}\] Alternative form: \[L_r = \frac{we_dBW}{Δ_{rel}}\]

• L_r = superelevation runoff length (ft)
• ED = edge of traveled way elevation difference (ft)
• W = width from centerline to edge of traveled way (ft)
• Δ_rel = relative gradient (typically 0.005 to 0.008)
• w = number of lanes rotated
• e_d = design superelevation rate (ft/ft)
• BW = bandwidth being rotated (ft)

Spiral Transitions

Spiral Length: \[L_s = \frac{3.15V^3}{RC}\]

• L_s = minimum length of spiral (ft)
• V = design speed (mph)
• R = radius of circular curve (ft)
• C = rate of increase of lateral acceleration (ft/s³), typically 1-3

Spiral Angle: \[Δ_s = \frac{L_s}{2R} \text{ (radians)}\] \[Δ_s = \frac{28.65L_s}{R} \text{ (degrees)}\] Tangent to Spiral (TS): \[T_s = \left(R + p\right)\tan\left(\frac{Δ}{2}\right) + k\] Offset Distance: \[p = \frac{L_s^2}{24R}\] Throw Distance: \[k = \frac{L_s}{2} - \frac{L_s^3}{240R^2}\]

• Δ_s = spiral angle (radians or degrees)
• p = offset distance (ft)
• k = throw or abscissa of spiral (ft)

Vertical Alignment

Vertical Curves - General

Rate of Grade Change: \[K = \frac{L}{A}\] Length of Vertical Curve: \[L = KA\]

• K = rate of vertical curvature (ft per % grade change)
• L = length of vertical curve (ft)
• A = absolute algebraic difference in grades, |G₁ - G₂| (%)
• G₁ = initial grade (%)
• G₂ = final grade (%)

Elevation on Vertical Curve: \[Y = Y_{PVC} + G_1\left(\frac{x}{100}\right) + \frac{A}{200L}x^2\]

• Y = elevation at distance x from PVC (ft)
• Y_PVC = elevation at PVC (ft)
• x = horizontal distance from PVC (ft)
• G₁ = initial grade entering curve (%)
• For sag curves, use -A if G₂ > G₁; for crest curves, use -A if G₂ <>

High/Low Point Location: \[x = -\frac{G_1L}{A}\]

• x = distance from PVC to high or low point (ft)
• Valid only when high/low point is within curve (0 < x=""><>
• For sag curve: low point where G₁ is negative
• For crest curve: high point where G₁ is positive

Crest Vertical Curves

Sight Distance - Case 1 (S <> \[L = \frac{AS^2}{100\left(2h_1^{0.5} + 2h_2^{0.5}\right)^2}\] Using K-value: \[K = \frac{S^2}{100\left(2h_1^{0.5} + 2h_2^{0.5}\right)^2}\] Sight Distance - Case 2 (S > L): \[L = 2S - \frac{100\left(2h_1^{0.5} + 2h_2^{0.5}\right)^2}{A}\]

• S = sight distance (ft)
• L = length of vertical curve (ft)
• h₁ = height of driver's eye above roadway (ft), typically 3.5 ft
• h₂ = height of object above roadway (ft), typically 2.0 ft for stopping sight distance
• A = algebraic difference in grades (%)

Stopping Sight Distance (simplified): \[K = \frac{S^2}{2158} \text{ when } h_1 = 3.5 \text{ ft, } h_2 = 2.0 \text{ ft}\] Passing Sight Distance (simplified): \[K = \frac{S^2}{2800} \text{ when } h_1 = 3.5 \text{ ft, } h_2 = 4.25 \text{ ft}\]

Sag Vertical Curves

Headlight Sight Distance - Case 1 (S <> \[L = \frac{AS^2}{200(H + S\tan β)}\] Using K-value: \[K = \frac{S^2}{200(H + S\tan β)}\] Headlight Sight Distance - Case 2 (S > L): \[L = 2S - \frac{200(H + S\tan β)}{A}\]

• H = headlight height (ft), typically 2.0 ft
• β = upward angle of headlight beam (degrees), typically 1°
• tan(1°) ≈ 0.01746

Stopping Sight Distance (simplified, headlight criterion): \[K = \frac{S^2}{400 + 3.5S} \text{ when } H = 2.0 \text{ ft, } β = 1°\] Comfort Criterion: \[K = \frac{V^2}{46.5}\]

• V = design speed (mph)
• Based on vertical acceleration comfort limit of 1 ft/s²

Appearance Criterion: \[K ≥ 50 \text{ (minimum for acceptable appearance)}\]

Stopping Sight Distance

Stopping Sight Distance Components

Total Stopping Sight Distance: \[SSD = d_1 + d_2\] Brake Reaction Distance: \[d_1 = 1.47Vt\] Braking Distance: \[d_2 = \frac{V^2}{30\left(\frac{a}{g} ± G\right)}\] Combined Formula: \[SSD = 1.47Vt + \frac{V^2}{30\left(\frac{a}{g} ± G\right)}\]

• SSD = stopping sight distance (ft)
• d₁ = brake reaction distance (ft)
• d₂ = braking distance (ft)
• V = design speed (mph)
• t = brake reaction time (s), typically 2.5 s
• a = deceleration rate (ft/s²), typically 11.2 ft/s²
• g = gravitational acceleration, 32.2 ft/s²
• G = grade (decimal, positive for uphill, negative for downhill)
• Use + for upgrade, - for downgrade

Simplified Form (a/g = 11.2/32.2 ≈ 0.348): \[SSD = 1.47Vt + \frac{V^2}{30(0.348 ± G)}\]

Decision Sight Distance

Decision Sight Distance (DSD):

• Significantly longer than SSD
• Allows time for detection, recognition, decision, and maneuver
• Values are tabulated based on design speed and maneuver type
• Typical reaction times: 3.0 to 14.5 seconds depending on complexity

Passing Sight Distance

Passing Sight Distance (PSD): \[PSD = d_1 + d_2 + d_3 + d_4\]

• d₁ = distance traveled during perception and reaction time
• d₂ = distance traveled during acceleration and passing maneuver
• d₃ = clearance distance between passing vehicle and oncoming vehicle
• d₄ = distance traveled by oncoming vehicle during passing maneuver
• Typically tabulated based on design speed
• d₄ is typically equal to 2/3 of d₂

Cross Section Elements

Lane Widening on Curves

Widening Required: \[W = n\left(R - \sqrt{R^2 - L^2}\right) + \frac{V}{10\sqrt{R}}\] Simplified for highways: \[W = \frac{V}{10\sqrt{R}}\]

• W = total widening (ft)
• n = number of lanes
• R = radius of curve (ft)
• L = wheelbase of design vehicle (ft)
• V = design speed (mph)
• First term accounts for vehicle tracking
• Second term accounts for driver comfort

Sight Distance on Horizontal Curves

Middle Ordinate for Sight Distance: \[M = R\left(1 - \cos\left(\frac{28.65S}{R}\right)\right)\] Approximate formula (for small angles): \[M ≈ \frac{S^2}{8R}\]

• M = middle ordinate, minimum clearance from centerline (ft)
• S = sight distance (ft)
• R = radius of curve (ft)
• Used to determine lateral clearance requirements on inside of curves

Traveled Way and Shoulder Width

• Lane width: typically 10, 11, or 12 ft
• Shoulder width: varies by functional class, typically 2-12 ft
• Usable shoulder width: excludes gutter pan and slopes steeper than 1V:4H

Earthwork and Grading

Cross-Sectional Area

Level Section: \[A = d \cdot W\] Side Slope Area (one side): \[A_{ss} = \frac{d^2}{2} \cdot S\] Total Cross-Sectional Area: \[A = W \cdot d + d^2(S_L + S_R)\]

• A = cross-sectional area (ft²)
• d = depth of cut or fill at centerline (ft)
• W = width of roadway at grade (ft)
• S = side slope ratio (horizontal:vertical), e.g., 2:1 = 2
• S_L = left side slope ratio
• S_R = right side slope ratio

Volume Calculations

Average End Area Method: \[V = \frac{A_1 + A_2}{2} \times L\]

• V = volume (ft³)
• A₁, A₂ = end areas of adjacent sections (ft²)
• L = distance between sections (ft)
• Divide by 27 to convert to cubic yards

Prismoidal Formula: \[V = \frac{L}{6}(A_1 + 4A_m + A_2)\]

• A_m = area at midpoint between sections (ft²)
• More accurate than average end area method

Prismoidal Correction: \[C_p = \frac{L}{12}(c_1 - c_2)(w_1 - w_2)\]

• C_p = prismoidal correction (ft³)
• c₁, c₂ = center heights at end sections (ft)
• w₁, w₂ = widths at end sections (ft)
• Subtract from average end area volume to get prismoidal volume

Mass Diagram

Cumulative Volume: \[M_i = M_{i-1} + V_i\]

• M_i = cumulative algebraic sum of earthwork volumes (yd³)
• V_i = volume at section i (yd³, cut positive, fill negative)
• Balance line: horizontal line connecting equal mass diagram ordinates
• Free haul distance: maximum economical haul distance
• Overhaul: volume × distance beyond free haul limit

Intersection Design

Sight Distance at Intersections

Intersection Sight Distance: \[ISD = 1.47 V_m t\]

• ISD = intersection sight distance (ft)
• V_m = design speed of major road (mph)
• t = time for minor road vehicle to enter intersection (s)
• Typical time gap: 6.5 to 9.5 seconds

Distance Along Minor Road: \[d = 1.47 V_a t_g\]

• d = distance from intersection to decision point (ft)
• V_a = approach speed on minor road (mph)
• t_g = perception-reaction time (s), typically 2.5 s

Curb Return Radius

Minimum Curb Radius (based on design vehicle): \[R_{curb} = R_v - \frac{W_v}{2}\]

• R_curb = curb return radius (ft)
• R_v = minimum turning radius of design vehicle (ft)
• W_v = width of design vehicle (ft)
• Values typically tabulated by vehicle type

Roadside Design

Clear Zone Width

• Clear zone: unobstructed, traversable area beyond edge of traveled way
• Width depends on:
  • Design speed
  • Traffic volume (ADT)
  • Roadside slope
• Typical values: 10 to 30 ft from edge of traveled way
• Flatter slopes and higher speeds require wider clear zones

Roadside Slopes

Recoverable Slopes:

• 4:1 or flatter - generally traversable and recoverable

Non-recoverable Slopes:

• 3:1 to 4:1 - traversable but may not be recoverable

Critical Slopes:

• Steeper than 3:1 - non-traversable, requires barrier or flattening

Traffic Control and Capacity

Level of Service (LOS)

Density (for freeways): \[D = \frac{V}{S}\]

• D = density (veh/mi/ln)
• V = flow rate (veh/hr/ln)
• S = average speed (mph)

Service Flow Rate: \[SF_i = c_i \times (v/c)_i \times N \times f_{HV} \times f_p\]

• SF_i = service flow rate for LOS i (veh/hr)
• c_i = capacity per lane (veh/hr/ln)
• (v/c)_i = volume to capacity ratio for LOS i
• N = number of lanes
• f_HV = heavy vehicle adjustment factor
• f_p = driver population factor

Peak Hour Factor

Peak Hour Factor (PHF): \[PHF = \frac{V_h}{4 \times V_{15}}\]

• PHF = peak hour factor (dimensionless, ≤ 1.0)
• V_h = hourly volume (veh/hr)
• V₁₅ = volume during peak 15 minutes (veh/15 min)
• Typical values: 0.80 to 0.98

Pavement Design - Geometric Considerations

Cross Slope

Normal Crown:

• Typical cross slope: 1.5% to 2.5% for drainage
• High type pavement: 1.5% to 2.0%
• Low type pavement: 2.0% to 3.0%

Superelevation Transition:

• From normal crown to superelevation occurs over tangent runout and superelevation runoff

Drainage

Time of Concentration (Kirpich equation): \[t_c = 0.0078 L^{0.77} S^{-0.385}\]

• t_c = time of concentration (min)
• L = flow length (ft)
• S = slope (ft/ft)

Rational Method (runoff): \[Q = CiA\]

• Q = peak runoff rate (ft³/s)
• C = runoff coefficient (dimensionless)
• i = rainfall intensity (in/hr)
• A = drainage area (acres)

Acceleration and Deceleration Lanes

Acceleration Lane Length

Length of Acceleration Lane: \[L_a = 1.47 \bar{V} t + \frac{V_f^2 - V_i^2}{30a}\]

• L_a = acceleration lane length (ft)
• V̄ = average speed during acceleration (mph)
• t = perception-reaction time (s)
• V_f = final speed (mph)
• V_i = initial speed (mph)
• a = acceleration rate (ft/s²)
• Typically designed using tabulated values based on design speed and type of facility

Deceleration Lane Length

Length of Deceleration Lane: \[L_d = 1.47 V_i t + \frac{V_i^2 - V_f^2}{30d}\]

• L_d = deceleration lane length (ft)
• d = deceleration rate (ft/s²), typically 10-11 ft/s²
• Other variables same as acceleration lane

Interchange Design

Ramp Design Speed

Ramp Design Speed Relationship: \[V_R = V_H \times k\]

• V_R = ramp design speed (mph)
• V_H = highway design speed (mph)
• k = factor, typically 0.4 to 0.8 depending on ramp type
• Loop ramps: k = 0.4 to 0.5
• Diagonal/direct ramps: k = 0.6 to 0.7

Ramp Spacing

• Minimum spacing based on weaving section requirements
• Adequate length for lane changes and decision making
• Varies by interchange type and traffic volume

Traveled Way Transitions

Lane Width Transitions

Transition Taper Length: \[L_t = WS\]

• L_t = taper length (ft)
• W = width of offset (ft)
• S = taper rate
• High-speed roads: S = 50:1 to 100:1
• Low-speed roads: S = 15:1 to 25:1

Alternative (AASHTO): \[L = WV\]

• L = taper length (ft)
• W = width of offset (ft)
• V = design or average speed (mph)