SURVEYING AND TRANSPORTATION ENGINEERING CE ELECTIVE 3

CORRECTIONS IN MEASURING DISTANCES PROBABLE VALUE AND ERROR

 Probable Error (P.E.)
where:
CD – correct distance
RD – recorded distance/actual reading
C – all applicable corrections
 Standard Error (S.E.)
 Correction due to Temperature (CT)

where:
k – coefficient of linear expansion  Standard Deviation (S.D.)
= 0.0000116/˚C
= 0.00000645/˚F
Tm – temperature during measurement
Ts – standard temperature

 Correction due to Pull (CP)  Precision of Measurement

where:  Weight
Pm – pull during measurement Probable Value
Ps – standard pull
A – cross sectional area of tape
E – modulus of elasticity of tape

 Correction due to Sag (CSAG)

Correct Value
where:
ω – weight of tape per linear meter
= wtape/Ltape

 Normal Tension (PN)

where:
M – no. of measurement
TAPE CORRECTION D – distance
P.E. – probable error

+ too long
EARTH’S CURVATURE AND ATMOSPHERIC REFRACTION (hCR)
- too short

HCR – distance in m
PACE FACTOR
k – distance in km

1|P ag e Engr. Kristalyn P. Cabardo, CE

 SURVEYING AND TRANSPORTATION ENGINEERING CE ELECTIVE 3

THREE HILLS PROBLEM AREA COMPUTATION

 Trapezoidal Rule

 Simpson’s 1/3 Rule

Note: Applicable only to odd number of offsets

SIMPLE CURVE
Elements:
PC = Point of curvature. It is the beginning of curve.

STADIA FORMULA PT = Point of tangency. It is the end of curve.

 Horizontal Sight PI = Point of intersection of the tangents. Also called vertex

T = Length of tangent from PC to PI and from PI to PT. It is
known as subtangent.
R = Radius of simple curve, or simply radius.
 Inclined Sight L = Length of chord from PC to PT. Point Q as shown below is
the midpoint of L.
Lc = Length of curve from PC to PT. Point M in the the figure is
the midpoint of Lc.
E = External distance, the nearest distance from PI to the
curve.
where:
m = Middle ordinate, the distance from midpoint of curve to
f/i – stadia interval factor midpoint of chord.
f + c – stadia constant I = Deflection angle (also called angle of intersection and
s – stadia intercept central angle). It is the angle of intersection of the tangents.
The angle subtended by PC and PT at O is also equal to I,
i – distance between stadia hair where O is the center of the circular curve from the above
note: figure.
Internal focusing telescope x = offset distance from tangent to the curve. Note: x is
perpendicular to T.
f/i = 100
θ = offset angle subtended at PC between PI and any point in
f+c=0
the curve
D = Degree of curve. It is the central angle subtended by a
LATITUDE AND DEPARTURE length of curve equal to one station. In English system, one
station is equal to 100 ft and in SI, one station is equal to 20
m.
Sub chord = chord distance between two adjacent full
stations.

MISSING SIDES
 Case I – Unknown bearing and distance of a line
 Case II – Two unknown distances of the two lines
 Case III – not case I or case II
Step 1 – Compute the resultant of the unknown lines
Step 2 – Draw the resultant line of the known lines with the
two unknown lines then analyze to compute the unknown
data.

2|P ag e Engr. Kristalyn P. Cabardo, CE

 SURVEYING AND TRANSPORTATION ENGINEERING CE ELECTIVE 3

Formulas: SPIRAL CURVE

PARABOLIC CURVE

TS = Tangent to spiral
SC = Spiral to curve
CS = Curve to spiral
ST = Spiral to tangent
LT = Long tangent
ST = Short tangent
R = Radius of simple curve
Ts = Spiral tangent distance
Tc = Circular curve tangent
L = Length of spiral from TS to any point along the spiral
Ls = Length of spiral
PI = Point of intersection
For symmetrical parabolic curve: I = Angle of intersection
Ic = Angle of intersection of the simple curve
p = Length of throw or the distance from tangent that the
- location of the highest point 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
Y = Distance along tangent to any point on the spiral
Yc = Distance along tangent from TS to point at right angle to
For unsymmetrical parabolic curve: SC
Es = External distance of the simple curve
θ = Spiral angle from tangent to any point on the spiral
- location of the highest point
θs = Spiral angle from tangent to SC
i = Deflection angle from TS to any point on the spiral, it is
If proportional to the square of its distance
is = Deflection angle from TS to SC
D = Degree of spiral curve at any point
Dc = Degree of simple curve
If

3|P ag e Engr. Kristalyn P. Cabardo, CE

 SURVEYING AND TRANSPORTATION ENGINEERING CE ELECTIVE 3

Formulas: SIGHT DISTANCE

At L = Ls, Y = Yc, thus: v – running speed in m/s

t – perception time + brake reaction time
f – coefficient of friction between tires and pavement
G – grade/slope of the road
Deceleration:
At L = Ls, X = Xc, thus:

Braking time:

PARABOLIC SUMMIT CURVE

At L = Ls, θ = θs, thus: L>S

L<S

If h1 and h2 are not given:

Passing sight distance:
h1 = h2 = 1.143 m or 3.75’
If given velocity, k in kph Stopping sight distance:
h1 = 1.143 m or 3.75’
h2 = 0.15 m or 0.50’

PARABOLIC SAG CURVE

L>S

COMPUTATION OF VOLUME: EARTHWORKS

Where:
 End Area Formula (Ve)

L<S

 Prismoidal Formula (Vp)

If h1 and h2 are not given:
Replace 800h by 122 + 3.5S

Prismoidal correction
ALLOWABLE SPEED IN PARABOLIC SAG CURVE

Note:
1 m/s = 3.6 kph
1 mph = 1.609 kph

4|P ag e Engr. Kristalyn P. Cabardo, CE

 SURVEYING AND TRANSPORTATION ENGINEERING CE ELECTIVE 3

SIGHT DISTANCE IN SIMPLE OR HORIZONTAL CURVE

L>S  Rate of Flow/Volume of Traffic

q – rate of flow in vehicles/hour

L<S k – density in vehicles/km or density of traffic

us – space mean speed in kph

 Minimum time headway (hrs)

HIGHWAY TRAFFIC ACCIDENT ANALYSIS

 Accident rate for 100 million vehicles per miles of travel in
a segment of a highway:
 Spacing of vehicles (km)

A – number of accidents during period of analysis

ADT – average daily traffic  Peak Hour Factor (PHF)
N – time period in years
L – length of segment in miles

 Accident rates per million entering vehicles in an

intersection:
DESIGN OF HORIZONTAL CURVE
 Minimum radius of curvature

A – number of accidents during period of analysis

ADT – average daily traffic entering
R – minimum radius of curvature
N – time period in years
e – superelevation in m/m
f – coefficient of side friction or skid resistance
 Severity Ratio (S.R.)
v – design speed in m/s
g = 9.81 m/s2

 Centrifugal ratio or impact factor

 Space mean speed of vehicle (us)

v – design speed of car in m/s

g = 9.81 m/s2
R – radius of curvature in meters

POWER TO MOVE A VEHICLE

 Time mean speed (ut)

P – power needed to move the vehicle in watts
V – velocity of vehicle in m/s
R – sum of various resistance in Newtons

where:
Ʃt – sum of the time traveled by all vehicles
Ʃd – sum of the distance traveled by all vehicles
Ʃ(1/ui) – sum of reciprocal of spot speed
Ʃui – sum of all spot speed
n – no of vehicles

5|P ag e Engr. Kristalyn P. Cabardo, CE

 SURVEYING AND TRANSPORTATION ENGINEERING CE ELECTIVE 3

DESIGN OF PAVEMENT EXAMPLES:

 Rigid Pavement without dowels 1.) A 100 m steel tape is used to measure the distance of the
line and found to be 1539.28 m long. During measurement,
the tap is supported at the ends under a pull of 10 kg and the
observed mean temperature is 31˚C. The tape is of standard
length at 20˚C and tension of 12 kg. The cross-sectional area
of tape is 0.035 cm2. The coefficient of linear expansion is
 Rigid Pavement with dowels 0.0000116/˚C and modulus of elasticity of steel is 2.0x10
6
2
kg/cm . Determine the following:
a.) Total correction for temperature
b.) Total correction due to pull
c.) If the tape weighs 1.5 kg, what is the total sag correction?
d.) What is the correct length of the line?
e.) Compute the normal tension
t – thickness of pavement
W – wheel load
2.) The distance of line AB was measured four times and
f – allow tensile stress of concrete recorded as follows:
Trial Distance
 Flexible Pavement 1 102.1 m
2 102.3 m
3 102.2 m
4 102.4 m
f1 – allow bearing pressure of subgrade a.) Find the probable value of the distance
r – radius of circular area of contact between wheel load and b.) Determine the probable error
pavement
c.) Determine the precision of measurement

 Thickness of pavement in terms of expansion pressure

3.) Three hills A, B and C have elevations 135 m, 146 m and
154 m respectively above sea level. Distance AB is 3.90 km
while distance BC is 3.10 km.
a.) Compute the effect of earth’s curvature and atmospheric
refraction between A and B.
 Stiffness factor of pavement
b.) Compute the effect of earth’s curvature and atmospheric
refraction between B and C.
c.) What would be the height of tower to be constructed so
that the line of sight will clear hill B by 2.50 m?

Es – modulus of elasticity of subgrade

Ep – modulus of elasticity of pavement 4.) The lot has the following data:
LINE BEARING DISTANCE
AB N15˚24’E 31.20 m
BC S82˚32’E 28.5 m
CD S33˚09’E -
DE - 33.0 m
EA N65˚21’W 24.23 m

5.) Two tangents adjacent to each other having bearings

N65˚30’E and S85˚10’E meet at station 11+157.98. If the
radius of the simple curve connecting these two tangents is
249.17 m determine the following:
a.) Tangent distance
b.) Length of curve
c.) Stationing of PT

6|P ag e Engr. Kristalyn P. Cabardo, CE

 SURVEYING AND TRANSPORTATION ENGINEERING CE ELECTIVE 3

6.) A compound curve has the following elements: 12.) In a certain underpass, the length of the parabolic sag
I1 = 30˚ I2 = 24˚ curve is 240 m. The height of object at the instant of
D1 = 4˚ D2 = 5˚ perception is 1.0 m and the driver’s eye is 1.6 m above the
If the stationing of the vertex is 4+620, road. The approach grade is -4% and the forward tangent is
a.) Determine the stationing of PC +2.4%. If the sight distance is 320 m,
b.) Find the stationing of PCC
a.) Determine the clearance at the center of the curve.
c.) Determine the stationing of PT
b.) Find the location of from the PC where the catch basin
7.) The perpendicular distance between two parallel tangents should be installed.
of the reverse curve is 35 m. The azimuth of the back tangent
is 270˚ while the common tangent is 300˚. The first radius of
curve is 160 m and the stationing of PRC is 2+578. Determine 13.) Five vehicles were traversing a 2-km highway and the
the following: following data were taken:
a.) Radius of the second curve VEHICLE TIME(min)
b.) Stationing of PT
1 1.8
c.) Stationing of PC
2 1.4
8.) A spiral curve having a length of 100 m is to be laid out in a
3 1.6
certain portion of road. The degree of the central curve is 6˚.
a.) Find the offset distance at the first quarter point of spiral. 4 1.5
b.) Determine the spiral angle at the third quarter point of 5 1.3
spiral.
c.) Compute the maximum speed of the car that could pass a.) Find the density of traffic in vehicles/km
thru the spiral without skidding. b.) Find the space mean in kph

9.) An unsymmetrical parabolic curve connects a +4.2% grade c.) Compute the time mean speed in kph
and a -3.4% grade. The length of curve on the left side of the
vertex is 80 m and 110 m on the other side. If the stationing
14.) A vehicle weighing 15 kN passes through a highway curve
of the point of intersection is 4+460 and its elevation is
having a radius of 120 m and an angle of embankment equal
145.20 m. Determine the following:
to 9˚. The location of the center of gravity of vehicle is 0.8 m
a.) Location of the summit from PT above the roadway and the distance between the front
b.) Stationing of the summit
wheels is 1.2 m. If friction is great enough to prevent skidding,
c.) Elevation of the summit
at what speed would overturning impend?
10.) The following data are cross-section notes of the ground
which will be excavated for a roadway. 15.) Determine the thickness of pavement from the following
conditions:
Station 4+120
a.) The pavement is rigid and to carry a maximum wheel load
of 60 kN. Neglect effects of dowels. f’c = 20 MPa and use an
allowable tensile stress of concrete pavement equal to 0.06f’c.
b.) The concrete pavement has an expansion pressure of 0.15
Station 4+160 kg/cm2 and a pavement density of 0.0025 kg/cm3.

The base of road is 10 m and the side slopes are 1.5:1.

a.) Find the volume of excavation by end area method.
b.) Compute the volume by prismoidal formula.
c.) Determine the prismoidal correction.

11.) A driver travelling at 50 mph is 80 m from a wall ahead. If

the driver applies the brake immediately at a brake reaction
2
time of 2 seconds and begins slowing the vehicle at 10 m/s .
a.) Find the distance from the stopping point to the wall.
b.) Determine the braking time or time during deceleration.
c.) Assuming that the brake efficiency of the vehicle is 70%,
find the average skid resistance of the pavement.

7|P ag e Engr. Kristalyn P. Cabardo, CE