measure too long add too short subtract Measurement Corrections Due to temperature: Probable Errors C = αL(T2 − T1 ) Probable Error (single): (add/subtract); measured length (P2 − P1 )L C= EA (subtract only); unsupported length w 2 L3 24P 2 CD = MD (1 − ∑(x − xΜ ) n−1 ∑(x − xΜ ) Em = = 0.6745√ n(n − 1) √n E Proportionalities of weight, w: Due to slope: (subtract only); measured length π€∝ Normal Tension: 0.204W√AE 1 πΈ2 π€∝ 1 π π€∝π Area of Closed Traverse √PN − P Error of Closure: L H = (g1 + g 2 ) 8 L 2 x 2 ( 2) = L y H 1 Error of Closure Perimeter 1 acre = 4047 m2 from South D2 (h − h2 ) − 0.067D1 D2 D1 + D2 1 Stadia Measurement Leveling Horizontal: Elevπ΅ = Elevπ΄ + π΅π − πΉπ D = d + (f + c) π D = ( )s +C π D = Ks + C Inclined Upward: Inclined: Total Error: Reduction to Sea Level CD MD = R R+h error/setup = −eBS + eFS Subtense Bar Inclined Downward: error/setup = +eBS − eFS D = cot θ 2 eT = error/setup β no. of setups Double Meridian Distance Method DMD DMDππππ π‘ = Depππππ π‘ DMDπ = DMDπ−1 + Depπ−1 + Depπ DMDπππ π‘ = −Depπππ π‘ 2A = Σ(DMD β Lat) d [h + hn + 2Σh] 2 1 Double Parallel Distance Method DPD d A = [h1 + hn + 2Σhπππ + 4Σhππ£ππ ] 3 Relative Error/Precision: = h = h2 + Simpson’s 1/3 Rule: = √ΣL2 + ΣD2 Azimuth hcr = 0.067K 2 Trapezoidal Rule: A= Symmetrical: e ) TL Effect of Curvature & Refraction Area of Irregular Boundaries Lat = L cos α Dep = L sin α Parabolic Curves e ) TL D = Ks cos θ + C H = D cos θ V = D sin θ E=error; d=distance; n=no. of trials C 2 = S 2 − h2 PN = CD = MD (1 + Probable Error (mean): Due to sag: C= E = 0.6745√ too long too short (add/subtract); measured length Due to pull: lay-out subtract add Note: n must be odd Simple, Compound & Reverse Curves DPDππππ π‘ = Latππππ π‘ DPDπ = DPDπ−1 + Lat π−1 + Lat π DPDπππ π‘ = −Lat πππ π‘ 2A = Σ(DMD β Dep) Spiral Curve Unsymmetrical: H= L1 L2 (g + g 2 ) 2(L1 +L2 ) 1 g 3 (L1 +L2 ) = g1 L1 + g 2 L2 Note: Consider signs. Earthworks ππΏ 0 ππ ±ππΏ ±π ±ππ A= f w (d + dR ) + (fL + fR ) 2 L 4 T = R tan I m = R [1 − cos ] L = 2R sin L3 6RLs L (c − c2 )(d1 − d2 ) 12 1 VP = Ve − Cp L5 I Y=L− 2 π Lc = RI β 180° 20 2πR = D 360° 1145.916 R= D Prismoidal Correction: 40R2 Ls 2 Ls I + (R + p) tan 2 2 I Es = (R + p) sec − R 2 Ts = Ls = Volume (Truncated): 0.036k 3 R 0.0079k 2 R D L = DC Ls Σh = A( ) n e= A (Σh1 + 2Σh2 + 3Σh3 + 4Σh4 ) n Stopping Sight Distance Parabolic Summit Curve v2 S = vt + 2g(f ± G) a = g(f ± G) (deceleration) v (breaking time) tb = g(f ± G) f Eff = (100) fave L>S v → speed in m/s t → perception-reaction time f → coefficient of friction G → grade/slope of road x= 2 L VP = (A1 + 4Am + A2 ) 6 VT = θ Ls 2 ; p= 3 24R 2 Volume (Prismoidal): VT = ABase β Have i= I E = R [sec − 1] L Ve = (A1 + A2 ) 2 CP = L2 180° β 2RLs π 2 I Volume (End Area): θ= L= A(S)2 200(√h1 + √h2 ) 2 L<S 200(√h1 + √h2 ) L = 2(S) − A L → length of summit curve S → sight distance h1 → height of driver’s eye h1 = 1.143 m or 3.75 ft h2 → height of object h2 = 0.15 m or 0.50 ft 2 LT → long tangent ST → short tangent R → radius of simple curve L → length of spiral from TS to any point along the spiral Ls → length of spiral I → angle of intersection I c → angle of intersection of the simple curve p → length of throw or the distance from tangent that the circular curve has been offset x → offset distance (right angle distance) from tangent to any point on the spiral xc → offset distance (right angle distance) from tangent to SC Ec → external distance of the simple curve θ → spiral angle from tangent to any point on the spiral θS → spiral angle from tangent to SC i → deflection angle from TS to any point on the spiral is → deflection angle from TS to SC y → distance from TS along the tangent to any point on the spiral Parabolic Sag Curve Underpass Sight Distance Horizontal Curve L>S L>S L>S A(S)2 L= 122 + 3.5S A(S)2 L= 800H L<S L<S 122 + 3.5S L = 2(S) − A 800H L = 2(S) − A R= A → algebraic difference of grades, in percent L → length of sag curve S → sight distance A → algebraic difference of grades, in percent L → length of sag curve L<S L= A(K)2 395 H= C− h1 + h2 2 For passengers comfort, where K is speed in KPH R= S2 8M L(2S − L) 8M L → length of horizontal curve S → sight distance R → radius of the curve M → clearance from the centerline of the road