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
Due to sag:
(subtract only); unsupported length
C=
w 2 L3
24P 2
Due to slope:
(subtract only); measured length
E = 0.6745√
CD = MD (1 +
CD = MD (1 −
∑(x − x̅)
n−1
Probable Error (mean):
∑(x − x̅)
Em =
= 0.6745√
n(n − 1)
√n
E
Proportionalities of weight, w:
𝑤∝
Normal Tension:
0.204W√AE
1
𝐸2
𝑤∝
1
𝑑
𝑤∝𝑛
Area of Closed Traverse
√PN − P
Symmetrical:
L
H = (g1 + g 2 )
8
L 2
x 2 ( 2)
=
L
y
H 1
Error of Closure:
Error of Closure
Perimeter
from South
D2
(h − h2 ) − 0.067D1 D2
D1 + D2 1
Stadia Measurement
Leveling
Horizontal:
Elev𝐵 = Elev𝐴 + 𝐵𝑆 − 𝐹𝑆
D = d + (f + c)
𝑓
D = ( )s +C
𝑖
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 ∙ no. of setups
Double Meridian Distance Method DMD
DMD𝑓𝑖𝑟𝑠𝑡 = Dep𝑓𝑖𝑟𝑠𝑡
DMD𝑛 = DMD𝑛−1 + Dep𝑛−1 + Dep𝑛
DMD𝑙𝑎𝑠𝑡 = −Dep𝑙𝑎𝑠𝑡
2A = Σ(DMD ∙ Lat)
d
[h + hn + 2Σh]
2 1
Double Parallel Distance Method DPD
d
A = [h1 + hn + 2Σh𝑜𝑑𝑑 + 4Σh𝑒𝑣𝑒𝑛 ]
3
Relative Error/Precision:
1 acre =
4047 m2
h = h2 +
Simpson’s 1/3 Rule:
= √ΣL2 + ΣD2
Azimuth
hcr = 0.067K 2
Trapezoidal Rule:
Lat = L cos α
Dep = L sin α
=
e
)
TL
Effect of Curvature & Refraction
Area of Irregular Boundaries
A=
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 =
too long
too short
(add/subtract); measured length
Due to pull:
lay-out
subtract
add
Note: n must be odd
Simple, Compound & Reverse Curves
DPD𝑓𝑖𝑟𝑠𝑡 = Lat𝑓𝑖𝑟𝑠𝑡
DPD𝑛 = DPD𝑛−1 + Lat 𝑛−1 + Lat 𝑛
DPD𝑙𝑎𝑠𝑡 = −Lat 𝑙𝑎𝑠𝑡
2A = Σ(DMD ∙ 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
𝑑𝐿 0 𝑑𝑅
±𝑓𝐿 ±𝑓 ±𝑓𝑅
A=
f
w
(d + dR ) + (fL + fR )
2 L
4
T = R tan
i=
θ
Ls 2
; p=
3
24R
x=
L3
6RLs
I
I
E = R [sec − 1]
L
Ve = (A1 + A2 )
2
I
2
m = R [1 − cos ]
2
Volume (Prismoidal):
L = 2R sin
L
VP = (A1 + 4Am + A2 )
6
L
(c − c2 )(d1 − d2 )
12 1
VP = Ve − Cp
Y=L−
2
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
VT = ABase ∙ Have = A ( )
n
A
VT = (Σh1 + 2Σh2 + 3Σh3 + 4Σh4 )
n
e=
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
L=
L5
I
π
Lc = RI ∙
180°
20 2πR
=
D
360°
1145.916
R=
D
Prismoidal Correction:
v → speed in m/s
t → perception-reaction time
f → coefficient of friction
G → grade/slope of road
L2 180°
∙
2RLs π
2
Volume (End Area):
CP =
θ=
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