h CIVIL ENGINEERING FLUID MECHANICS section. ± G = percent grade divided by 100 (uphill grade "+")

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1 FLUID MECHANICS section. TRANSPORTATION U.S. Customary Units a = deceleration rate (ft/sec ) A = absolute value of algebraic difference in grades (%) e = superelevation (%) f = side friction factor ± G = percent grade divided by 100 (uphill grade "+") h 1 = height of driver s eyes above the roadway surface (ft) h = height of object above the roadway surface (ft) L = length of curve (ft) L s = spiral transition length (ft) R = radius of curve (ft) S = stopping sight distance (ft) t = driver reaction time (sec) V = design speed (mph) v = vehicle approach speed (fps) W = width of intersection, curb-to-curb (ft) l = length of vehicle (ft) y = length of yellow interval to nearest 0.1 sec (sec) r = length of red clearance interval to nearest 0.1 sec (sec) Vehicle Signal Change Interval y = t + v a! 64. 4G r = W l v + Stopping Sight Distance S = 147. Vt + V 30 cb a l! Gm CIVIL ENGINEERING

2 Transportation Models See INDUSTRIAL ENGINEERING for optimization models and methods, including queueing theory. SPEED v (mph) VOLUME q (veh/hr) CAPACITY SPEED v (mph) CAPACITY DENSITY k (veh/mi) DENSITY k (veh/mi) VOLUME q (veh/hr) Vertical Curves: Sight Distance Related to Curve Length S L S > L Crest Vertical Curve General equation: L = AS 100( h + h ) 1 ( ) 1 00 h + h L = S A Standard Criteria: h 1 = 3.50 ft and h =.0 ft: Sag Vertical Curve (based on standard headlight criteria) Sag Vertical Curve (based on riding comfort) Sag Vertical Curve (based on adequate sight distance under an overhead structure to see an object beyond a sag vertical curve) L = L = AS,158 L = 800 AS S AS h1+ h ( C ) L = L = S,158 A AV 46.5 L = S ( ) S A ( ) 800 h1+ h L = S C A C = vertical clearance for overhead structure (overpass) located within 00 feet of the midpoint of the curve Horizontal Curves Side friction factor (based on superelevation) Spiral Transition Length Sight Distance (to see around obstruction) V 0.01e+ f = 15R V L s = RC C = rate of increase of lateral acceleration [use 1 ft/sec 3 unless otherwise stated] 8.65S HSO = R [ 1 cos( ) R ] HSO = Horizontal sight line offset CIVIL ENGINEERING 163

3 Horizontal Curve Formulas D = Degree of Curve, Arc PC = Point of Curve (also called BC) PT = Point of Tangent (also called EC) PI = Point of Intersection I = Intersection Angle (also called Δ) Angle Between Two Tangents L = Length of Curve, from PC to PT T = Tangent Distance E = External Distance R = Radius LC = Length of Long Chord M = Length of Middle Ordinate c = Length of Sub-Chord d = Angle of Sub-Chord l = Curve Length for Sub-Chord R = D R = LC sin_ I i T = Rtan_ I i = LC cos_ I i L = RI r 180 = D I 100 M = R81 - cos_ I ib R cos I E + R = _ i R - M R = cos_ I i c = Rsin_ d i LATITUDES AND DEPARTURES + Latitude Departure + Departure Latitude l = Rdb r l 180 E = R = 1-1 cos_ I i G D 164 CIVIL ENGINEERING

4 Vertical Curve Formulas BACK TANGENT PVC TANGENT OFFSET x y L PVI E PVT FORWARD TANGENT g g 1 Y PVC DATUM VERTICAL CURVE FORMULAS NOT TO SCALE L = Length of Curve (horizontal) g = Grade of Forward Tangent PVC = Point of Vertical Curvature a = Parabola Constant PVI = Point of Vertical Intersection y = Tangent Offset PVT = Point of Vertical Tangency E = Tangent Offset at PVI g 1 = Grade of Back Tangent r = Rate of Change of Grade x = Horizontal Distance from PVC to Point on Curve g gl 1 x m = Horizontal Distance to Min/Max Elevation on Curve = - a = g - g Tangent Elevation = Y PVC + g 1 x and = Y PVI + g (x L/) Curve Elevation = Y PVC + g 1 x + ax = Y PVC + g 1 x + [(g g 1 )/(L)]x g g1 y ax a E a L L g g = = - r = b l = - L EARTHWORK FORMULAS Average End Area Formula, V = L(A 1 + A )/ Prismoidal Formula, V = L (A 1 + 4A m + A )/6, where A m = area of mid-section, and L = distance between A 1 and A Pyramid or Cone, V = h (Area of Base)/ AREA FORMULAS Area by Coordinates: Area = [X A (Y B Y N ) + X B (Y C Y A ) + X C (Y D Y B ) X N (Y A Y N 1 )] / Trapezoidal Rule: Area = w c h 1 + h + h + h3 + h4 + f + h - 1m w = common interval n n n - n - 1 Simpson s 1/3 Rule: Area = = wh > + e! h o + 4e! h o + hh 3 n must be odd number of measurements 1 k k k = 35,, f k = 4,, f n w = common interval CIVIL ENGINEERING 165

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9 Dilemma Zone: X = V ( Y + AR) D = X ( w + l) Intersection Operation: ( G +Y AR) - l EffectiveG reen= + LOS: FFS = 75.4 f lw f lc 3.TRD 0.84 V N = MSF i x PHF x f V PHF = V 15 4 f HV = 1+ P T 1 ( - 1) E T HV x f P ( ) G+ Y + AR l lane capacity = s C s = 3600 h s Capacity = MSF D = v p / S E x N x f HV x f p xphf Signal Timing: s = s 0 1+ P T C min = X c - L X c CLV s n i=1 ( G + Y + AR) i = CLV s i CLV s i i ( C L) + l i

10 PCE by Type of Terrain Vehicle Level Rolling Mountainous Trucks and buses, E T RVs, E R Right-Side Lateral Clearance (ft) Lanes in One Direction *values in table above are for f LC in mi/hr Average Lane Width (ft) Reduction in FFS, f LW (mi/h)

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