fw6_ce121.docx

Laying of a symmetrical parabolic CURVE USING TRANSIT AND TAPE

Table of contents Introduction --------------------------------------------------------------2

Objectives and instruments ---------------------------------------------- 3

Procedure and Computation --------------------------------------------4

Preliminary data sheet ----------------------------------------------------5

Sample Computations -----------------------------------------------------6

Sketch -------------------------------------------------------------------------7 Final data sheet -------------------------------------------------------------8

Pictures -----------------------------------------------------------------------9

Research and discussions ------------------------------------------------10

Conclusion -------------------------------------------------------------------12

Introduction

1


A parabola is a conic section. It's the curve that can be obtained by a plane intersecting a right circular cone parallel to the side elements of the cone. Surveyors and engineers have traditionally used the term vertical curve to describe the use of parabolic curves in design work. They are used for transitions from one straight grade to another in designing crowns for pavements, routes such as highways and railways, as well as dam spillways, landscape design and of course roller coasters.

Vertical curves are used to provide gradual change between two adjacent vertical grade lines. The curve used to connect the two adjacent grades is parabola. Parabola offers smooth transition because its second derivative is constant. Most often vertical curves are used to improve the comfort, safety and appearance of routes. They are just as crucial to good work as horizontal curves.

Objectives:

2


1. To familiarize oneself with the elements of horizontal and vertical parabolic curve.2. To lay a horizontal parabolic curve by using tangent offset method.3. To master the skill in leveling, orienting and using the transit/ theodolite effectively.4. To work harmoniously with one’s group mates and efficiently perform the required

task.

Instruments:

Tape

Used to measure horizontal distances as well as slopes. Usually in 30m, 50m or

100m in length.

A theodolite is a precision instrument used for measuring

angles both horizontally and vertically. Theodolites can rotate

along their horizontal axis as well as their vertical axis.

Marking pins

2 range poles

Straight round stalks, 3 to 4 cm thickand about 2 m long. They are madeof wood or metal. They are used tomark areas and to set out straight lineson the field. They are also used to markpoints which must be seen from adistance, in which case a flag may be attached to improve the visibility.

Procedures:1. The professor gives the following data:

3


Length of the Parabolic Curve = g1 =Full Station = g2 =Sta of PVI (Point of Vertical Intersection) = Preferably not in full station

Elev PVI =

2. Compute the elements of the parabolic curve given the data above. Note that the nomenclature of the parabolic curve adopts that of a simple curve with the additional V for vertical on the middle of the abbreviations or on a subscript.

Solve for:

Set the reference point at PVC, which must have a (0,0,0) coordinate.

Elev PVC=Elev PVI ±g1( L2 )The distances of each intermediate point must be based on PVC using the figure I. if you are using the second figure the distance of each intermediate point is based on either PVC or PVT. Just indicate the different % of grade you are using.

y

x2= H

( L2 )2

3. Set and level the transit at PVC (Point of Vertical Curvature). Level the telescope and set the horizontal and vertical vernier to zero.

4. Assign a datum line. Sight the location of vertex PVI following the given value of g1 and mark the location on the ground assuming PVC is at (0,0,0) origin. The horizontal angle on the transit is computed as:

θ=tan−1 g1100

5. Position the PVI at half the length of the parabolic curve.6. Locate the intermediate points at full stations on the datum line (coplanar with

PVC). Note: in most cases, there will be equal numbers of intermediate stations on both sides of PVI.

7. Mark using marking pins/chalk on the ground, the individual tangent offset of each intermediate points from the back and forward tangents.

8. (OPTIONAL- TO SEE HOW A VERTICAL PARABOLIC CURVE MAY LOOK LIKE) After laying the parabolic curve on the ground, use nylon ropes with lengths equal to tangent offsets and trace each points by tying each marking pins or any stone on the assigned tangent line to the back and forward tangents.

9. Drop the entire assembly on a vertical wall (e.g. Intramuros Wall) holding only the point on the tangent line coinciding with the PVI. Tie both ends of the tangent line (PVC and PVT) at two points having different elevations (as assigned) to verify the accuracy of the parabolic curve that has been traced and transformed from a horizontal parabolic curve into a vertical parabolic curve.

Preliminary Data Sheet:

4


Sample Computations:

5


Sketch:

6


Final Data Sheet

7


Field Work 6 Laying of a Parabolic Curve using Transit and Tape

Date: Nov. 10, 2015 Group No.: 4

Time: 12pm-4:30pm Location: Rizal Park

Weather: Sunny Professor: Engr. Ira Balmoris

Data Supplied:

Length of the Parabolic Curve = 22m g1 = 1Full Station = 3m g2 = -2Sta of PVI (Point of Vertical Intersection) = 0+013 Preferably not in full station

Elev PVI = 100m

STATIONSTATION

IN METERS

DISTANCE FROM PC

PERCENTAGE (% GRADE)

TANGENT ELEVATION

TANGENT OFFSET

ELEVATION

PVC 0+002 0 1 89 0 891 0+003 1 1 90 0.068 89.9322 0+005 4 1 93 1.09 91.913 0+008 7 1 96 3.34 92.664 0+009 10 1 99 6.82 92.18

MID 0+012 11 1 100 8.25 91.755 0+013 13 -2 102 11.52 90.486 0+015 16 -2 105 17.45 87.557 0+018 19 -2 108 24.61 83.39

PVT 0+024 22 -2 111 33 78

Sample Computation:H=1

8L (g1−g2 )=1

8(22 ) ¿

Elev of PC=100m−L2=100−22

2=89m

θ=tan−1(1)=45 ° Tangent Offsets:

y

x2= H

( L2 )2

8


y112

=8.25112

y1=0.068m

y242

=8.25112

y2=1.09m

y372

=8.25112

y3=3.34m

Elevations:

Elevation=Tangent Elevation−Tangent OffsetElev1=90−0.068=89.932mElev2=93−1.09=91.91mElev3=96−3.34=92.66m

Pictures:

Setting the instrument. Establishing PVI Measuring L and stations.

Measuring the vertical height of each stations. The parabolic curve.

9


Research and Discussions:Symmetrical parabolic curve does not necessarily mean the curve is symmetrical at L/2, it simply means that the curve is made up of single vertical parabolic curve. Using two or more parabolic curves placed adjacent to each other is called unsymmetrical parabolic curve. The figure shown below is a vertical summit curve. Note that the same elements holds true for vertical sag curve.

Elements of Vertical Curve

PC = point of curvature, also known as BVC (beginning of vertical curve) PT = point of tangency, also known as EVC (end of vertical curve) PI = point of intersection of the tangents, also called PVI (point of vertical

intersection) L = length of parabolic curve, it is

the projection of the curve onto a horizontal surface which corresponds to the plan distance.

S1 = horizontal distance from PC to the highest (lowest) point of the summit (sag) curve

S2 = horizontal distance from PT to the highest (lowest) point of the summit (sag) curve

h1 = vertical distance between PC and the highest (lowest) point of the summit (sag) curve

h2 = vertical distance between PT and the highest (lowest) point of the summit (sag) curve

g1 = grade (in percent) of back tangent (tangent through PC) g2 = grade (in percent) of forward tangent (tangent through PT) A = change in grade from PC to PT a = vertical distance between PC and PI b = vertical distance between PT and PI H = vertical distance between PI and the curve

Formulas for Symmetrical Parabolic Curve

Properties of Parabolic Curve and its Grade Diagram

1. The length of parabolic curve L is the horizontal distance between PI and PT.2. PI is midway between PC and PT.3. The curve is midway between PI and the midpoint of the chord from PC to PT.

10


4. The vertical distance between any two points on the curve is equal to area under the grade diagram. The vertical distance c = Area.

5. The grade of the curve at a specific point is equal to the offset distance in the grade diagram under that point. The grade at point Q is equal to gQ.

Note that the principles and formulas can be applied to both summit and sag curves.

rise = run × slope

a=12g₁L

b=12g₂

Neglecting the sign of g and g₁ ₂

S₁g₁

= Lg₁+g₂

S₁=g₁Lg₁+g₂

S₂g₂

= Lg₁+g₂

S₂= g₂Lg₁+g₂

vertical distance = area under the grade diagram

h₁=12g₁S₁

h₂=12g₂S₂

Other formulas

H=18L(g₁+g₂)

x2

y=

( 12 L)H

2

These equations are often used to check the design speed of an existing vertical curve. K values are preferred to be used when design a new vertical curve because it provides a better safety distance.

11


Conclusion:

12

fw6_ce121.docx

Documents