Introduction Route surveys involve measuring and computing horizontal and vertical angles, elevations, and horizontal distances. The results of these surveys are used to prepare detailed plan and profile base maps of proposed roadways. roadways. In addition, the elevations determined in the survey serve as the basis for calculation of construction cut and fill quantities, and in determining roadway banking. This section presents a review of basic terminology, concepts, and standard procedures used in highway surveys. The review begins with some basic definitions. Highway curves can be either circular arcs or spirals. simple curve is a circular are connecting two straight lines !tangents". compound curve consists of two or more circular arcs of different radii tangent to each other with their centers on the same side of the common tangent. #ompound curves where two circular ares having centers on the same side are connected by a short tangent are called broken-back curves. A reverse curve curve is two circular arcs tangent to each other but with their centers on opposite sides of the common tangent. curve whose radius decreases uniformly from infinity to that of the curve it meets is called a spiral curve. $piral curves with the proper superelevation !banking" provide safe and smooth riding qualities. #ircular and spiral curves are used for curves in the horizontal plane. Tangents in the vertical plane are %oined by parabolic curves !also referred to simply as vertical curves" route surveying system usually us ually contains four separate but interrelated processes& ' Reconnaissance and planning ' (orks design ' Right of way acquisition ' #onstruction of works
DEFINITION OF TERMS SIMPLE CURVE • •
)ost commonly used for highways and railroads construction. #ircular arc, e*tending from one tangent to the ne*t
Introduction Route surveys involve measuring and computing horizontal and vertical angles, elevations, and horizontal distances. The results of these surveys are used to prepare detailed plan and profile base maps of proposed roadways. roadways. In addition, the elevations determined in the survey serve as the basis for calculation of construction cut and fill quantities, and in determining roadway banking. This section presents a review of basic terminology, concepts, and standard procedures used in highway surveys. The review begins with some basic definitions. Highway curves can be either circular arcs or spirals. simple curve is a circular are connecting two straight lines !tangents". compound curve consists of two or more circular arcs of different radii tangent to each other with their centers on the same side of the common tangent. #ompound curves where two circular ares having centers on the same side are connected by a short tangent are called broken-back curves. A reverse curve curve is two circular arcs tangent to each other but with their centers on opposite sides of the common tangent. curve whose radius decreases uniformly from infinity to that of the curve it meets is called a spiral curve. $piral curves with the proper superelevation !banking" provide safe and smooth riding qualities. #ircular and spiral curves are used for curves in the horizontal plane. Tangents in the vertical plane are %oined by parabolic curves !also referred to simply as vertical curves" route surveying system usually us ually contains four separate but interrelated processes& ' Reconnaissance and planning ' (orks design ' Right of way acquisition ' #onstruction of works
DEFINITION OF TERMS SIMPLE CURVE • •
)ost commonly used for highways and railroads construction. #ircular arc, e*tending from one tangent to the ne*t
PC • •
PT • •
+oint of the curvature The point where the curve leaves the first tangent
+oint of the tangency The point where the curve leaves the second tangent
PC and PT •
Tangent points
VERTEX •
+oint of the intersection of the two tangents
TANGENT DISTANCE (T) •
istance from the verte* to the +# and +T
EXTERNAL DISTANCE (E) •
istance from the verte* to the curve
MIDDLE ORDINATE (M) •
The line %oining the middle of the curve and the mid-point of the chord %oining the +T and +#
DEGREE OF CURVE (D) •
•
enerally used for highway practice !when the radius of the curve is usually small" It is the angle of the center subtended by an arc of /0m !$I" or 1002!3nglish"
A. ARC BASIS
4
1. $I
/. 356I$H
B. CHORD BASIS •
The degree of the curve is the angle subtended by a chord of /0m !$I" or 1002 !3nglish"
ELEMENTS OF A SIMPLE CURVE
7
Tangent ditan!e (T)
Tan I8/ 9 T8R T9R tan I8/ Midd$e %#dinate (M)
cosI8/ 9 R-)8R Rcos I8/ 9 R-) ) 9 R-Rcos I8/ ) 9 R!1-cos I8/"
E"te#na$ ditan!e (E)
cos I8/ 9 R8R:3 !R:3" cosI8/ 9 R R:3 9 Rsec I8/ 39RsecI8/-R 39 R!secI8/-1" Lengt& %' !&%#d (LC)
sin I8/ 9 6#8/ 8R 6# 9 /Rsin I8/
Lengt& %' !#e (LC)
;
6#u8I 9 /08 6# 9 /0I8 EXAMPLES*
1. The tangent distance of a 4< simple curve is = of its radius. etermine& ngle of intersection !I" 6# rea of the fillet of the curve • • •
$olution& 94<, T918/R T9RtanI8/ 18/R9RtanI8/ tanI8/90.; I8/9tan 0.; I9;4.14<
6#9/0I8 6#9/0!;4.14"84< 6#94;7./m
9T!R"- R> o84?0< R9117;.@1?8 R94A1.@B sqm
9!1@0.@@"!4A1.@B"!4A1.@B" !;4.14"84?0< 9;40;.A@sqm
/. The point of intersection of tangents on a simple curve is inaccessible falling within a river. •
+oints C and # on the tangents are connected by measurements on the ground.
Digure&
?
R3EFIR3& • •
istance # and the length of curve rea of the cross-hatched section
$olution&
use sine law I91A0<-7;.7A< I9147.;/<
o91A0<-7B.;<-AB.0/< o97;.7A 1/A.01;8sino9T-;4.;A8sinAB.0/< T9/4/.AAm
B
$ine law& /4/.A?-#8sin7B.;<91/A.01;8sin7;.7A< #9100.;1
6#u9RI, /0I8 T9RtanI8/ /4/.AA9Rtan147.;/<8/ R9@B.?1m
rea
6#9@B.?1!147.;/<"!G81A0" 6#9//@.Bm
sec-triangle R>o84?0<-18/!1A0.0;"!4B.B4" 94A7.B@sqm
6#9/!@B.?1"sin147.;/<8/ 6#91A0.0;m cosI8/9*8R *9 4B.B4m
If station +I9sta1:0;7 Req2d&
sta+#91:0;7-/4/.AA sta+#90:A/1./1
sta+T9sta+#:6# sta+T90:A/1.1/://@.1B sta+T91:0;0./@
1. Two tangents C and C# intersecting in at an angle of /70<. point + is located /1.04 from point C and has a parallel distance of /.B@m from line C.
A
Digure&
sine law& R8sin B0
39R!secI8/" 39R!sec /7<8/-1" 39R!1/<-1" 390.0//41/
sino9/.B@8/1.04 o9B<;B2 9@0<-1/<-B<4B2 9B0
R8sinB0
o91A0<-- o91A0<-10;<4@2-B0
o94<;A2 using sine law& R8sin B0I84?0<
GIVEN*
C9$?;<4023 C#9$/;<4023J 1B0.B;m #9$;7
required& R9K I9K $tation +T if L is at sta /0:170
I911@<;02 1B0.B;m9T1:T/ 1B0.B;m9RtanI18/:RtanI/8/ 1B0.B;m9Rtan70<8/:RtanB@<;028/ 1B0.B;m9R!tan/0<:tan4B
R917/.//m T9RtanI8/ T917/.//tan!11@<;028/" T9/7;.;Bm sta +#9sta L-I sta +#9 /0:170-/@;.;Bm sta +c91@:
[email protected] sta +T9sta +#:6# 6#9RI 81A0< 6#917/.//!11@<;02" 81A0< sta +T91@:
[email protected]:/@B.;B sta +T9/0:1@/.0? GIVEN+FIGURE*
11
tan/0<9?00m-MI8;AB.@?m MI94A?m
I+9 !;AB.@?">: !/17"> I+9 ?/;.?@m cosine law& M+>9?/;.?@>:4A?> M+9/!?/;.?@"!4A?"cos110< M+9A70.0;m o9K sine law& 4A?8sino9A70.0;8sin110< o9/;.;A< 91A0<-110<-/;.;A< 977.7/<
1/
cosine law& ?00>9A70.0;>:!+*"-/!A70.0;" !+*"cos/;.;A< +*9/
[email protected];m
6#9!?00"!4/.A1<"!G81A0<" 6#9474.;@m sta *9sta:6# sta *9;0:000:474.;@m sta *9;0:474.;@
COMPOUND CURVES •
•
PCC •
#omposed of two or more consecutive simple curve having different radii but whose center lie on the same side of the curve. ny two consecutive curves must have a common tangent on their meeting +T.
+oint of compound curvature the +T on the common tangent the through which the two curves %oin.
14
EXAMPLES*
1. The long chord from the +# to the +T of a compound curve is 400m long and the angle that it makes the longer and shorter tangents are 1/< and 1;< respectively. If the common tangent is parallel to the long chord. Required& R 1 R / $tation +T if +# is at sta 10:/07.40 • •
•
17
sine law& 400m8sin1??<40296#18sinB<40296#/8sin?< 6#191?B.B7m 6#/9147.44m 6#9/RsinI8/ 6#19/R 1sinI18/ 1?B.B7m9/!R 1"sin?<8/ R 19A0/.4?m
6#u19R 1I1! G81A0<" 6#u19A0/.4?n!?<"! G81A0<" 6#u191?A.0;m
6#/9/R /sinI/8/ 147.44m9/!R /"sinB<4028/ R /9;17.;Bm
6#u/9R /I/! G81A0<" 6#u/9;17.;Bm!B<402"! G81A0<" 6#u/9147.B1m
sta +T9sta+#:6#u 1:6#u/ sta +T910:/07.40:1?A.0;:147.B1 sta +T910:;0B.0?
1;
I19/?A<402-/7B<;02 I19/0<702 I/9/A/<;02-/?A<402 I/917
T19R 1tanI18/ T19/A?.7B@tan!/0<7028/" T19;/./4m
T/9B?.7/m-T1 T/9B?.7/m-;/.74m T/9/7.1@m
T/9R /tanI/8/ /7.1@9R /tan!17
sta+#9sta-T1:6#u1 sta +#910:010.7?-;/./4:104.44 sta +#910:0?1.;?
6#u19! R 1I1"! G81A0<" 6#u19!/A?.7B@"!/0<702"! G81A0<" 6#u19104.44msta +T9sta+##:6#u/ sta+T910:0?1.;?:7A.14 sta+T910:10@.?@ 6#u/9!R/I/"! G81A0<" 1?
6#u/9!1@/.4A"!17
REVERSE CURVES •
PRC • •
#omposed of two consecutive circular simple curves having a common tangent but lie on the opposite side. +oint of the reverse curvature. The point along the common tangent to which the curve reversed in its direction.
FOUR T,PES OF REVERSE CURVES*
1B
1A
1@
EXAMPLE*
The parallel tangent of a reversal curve are 10m apart the long chord from the +# to the +T is equal to 1/0m determine the following& Radius of the curve 6ength of the common tangent • •
$olution& sinI8/91081/0 I9@<442 sinI9108/T /T9?0./Bm T940.17m
T9RtanI8/ 40.179Rtan!@<4428/" R94?0.A/m
/0
EXAMPLE*
Two converging tangent have azimuth of 400< and @0< respectively common tangent C has an azimuth of 4/0<. The distance from the point of intersection of two converging tangent and that of the verte* of the second curve has a distance of 100m. if the radius of the first curve is /A;.7m between. etermine& R / sta +R# and sta +T if station of L1 is 10:070 •
• •
Isolate triangle C#
Cy sine law& 1008sin/0<9C8sin40< C917?.1@m
/1
C9T1:T/ 17?.1@9R 1tanI18/:R /tanI/ R /9/0;.;@m station +R#9 staL 1-T1:6#1 station +T9 sta+R#:6# / sta +R#910:070-;0.4/:/A;.70!/0"! G81A0<" sta +R#9 10:
[email protected] sta +t910:
[email protected]:!/0;.;@"!;0"! G81A0<" sta +T9 10:/?A.B1 EXAMPLE*
Two tangents /0m apart are to be connected by a reversed curve. The radius of the curve passing thru +# is A00m. if the total length of chord from +# to +T is 400m and stationing of +# is 10:?/0. etermine& I R / $tation of +T • •
•
//
sin I8/9/08400 I9B<4A2 1st way to get the R /&
C91;0-;?9T1:T/ 1;0-;?9R 1tanI18/:R /tanI/8/ R /917;?.A@
/nd way&
4009/R 1sinI8/:/R /sinI8/ R /917;4.7B
4rd way&
cosI9A00-b8A00 b9B.0@ a91/.@1 cosI9R /-1/.@18R / R /917;?.A;
station +T9sta+#:6#1:6#/ 6#19R 1I! G81A0<" 6#/9R /I! G81A0<" sta +T910:@/0.?B E"a-$e*
/4
$olution&
N>9R>:100>OO.eqn1 N>9!R-100">:700>OO.eqn/ R>:100>9R>-/00R:100>:700>
/00R9700> R9A00m
tano91008A00 o9B.14< tan97008/00 9/@.B7<
I9-o I9/@.B7<-B.14< I9//.?1<
VERTICAL PARABOLIC CURVES • •
curve used to connect two intersecting gradelines curve tangent to two intersecting gradelines
T,PES OF VERTICAL PARABOLIC CURVES /. S,MMETRICAL PARABOLIC CURVES
parabolic curve wherein the horizontal length of the curve from the +# to the verte* is equal to the horizontal length from the verte* to the +T.
/7
ELEMENTS OF A S,MMETRICAL PARABOLIC CURVE
1. /. 4. 7. ;.
L3RT3N !+I" +# +T C#P(R T535T DMR(R T535T ?. g1 and g/ !R3$" GUIDING PRICIPLES FOR S,MMETRICAL PARABOLIC CURVES
1. given grade or slope ! in Q" is numerically the rate at which an elevation changes in a horizontal distance. eg ;Q 9 g
/. The vertical offset fro the tangent to the curve is proportional to the squares of the distances from the point of tangency. !$quared +roperty of a +arabola" y1 8 *1 9 H 8 !68/" / 9 y/ 8 !*/"> 1. The curve bisects the distance between the verte* and the midpoint of the long chord. CD 8 !68/"> 9 # 8 !6"> /. If g1 - g/ !:" 9 summitS
g1 - g/ !:" 9 sagS
4. 5o of stations to the left equal to the no of stations to the right.
/;
7. The slope of the parabola varies uniformly along the curve. r 9 g/ - g1 8 n J n 9 /0m stationing LOCATION OF THE HIGHEST OR LO0EST POINT OF THE CURVE
1. DRM) +# $1 9 g16 8 g1 - g/ /. DRM) +T $/ 9 g/6 8 g/ - g1 UNS,MMETRICAL PARABOLIC CURVES •
• •
#onsist of a symmetrical parabolic curve from +# to +T. ,C another symmetrical parabolic curve tangent to that point and +T Fsed in provide a smooth and continues curve transition from +# to +T +oint is the common tangent point
/?
EXAMPLE*
iven& g/9-AQ g19;Q 61970m 6/9?0m Required&
Height of fill needed to cover the outcrop 3levation at station ?:A/0 3levation of the H+
/B
Required& 3levation of the curve of the underpass If elevation curve is //.?A4;m $tationing of the H+ of the curve for question/ • • •
/H8619!g1-g/"6/861:6/ 619/H6/8!g1-g/"6/-/H 1?09/H!1/0"8!0.11"6/-/H H94.BBm y8!?0">9H8!1/0"> y90.@7 elevation of the curve9 elevation L-!?0"!0.07"-y-h elev curve940-/.7-0.07-7.7/ elev curve9//./7m
/A
elev curve9//.?A4;m 6/9K H97.7/m!remains the same" 619/H6/8!g1-g/"6/-/H 1?0m9/H6/8!0.11"6/-/HOO..eq2n1 3lev //.?A4;9elevL-!?0"!0.07"-y-7.7/ //.?A4;940-/.7-y-7.7/ y90.7@A; 0.7@?;8!6/-?0">9H8!6/"> H90.7@?;!6/">8!6/-?0"> H94.10m 1?09/!4.10"6/8!0.11"6/-/!4.10" 6/9100m g1618/ K H !0.0B"!?0"8/ K 4.10 ;.? U 4.10 $/9g/6/8/H !from point +T" $/9!0.07"!100"8/!4.10" $/9?7.;/m station H+9staL:4;.7Am station H+91/:/00:4;.7Am station H+91/:/4;.7A
/@
SPIRAL BASEMENT CURVE (TRANSITION SPIRAL CURVE) •
•
curve of ranging radius introduced at the outer edges of the roadway or track in order to allow the vehicle or train to pass gradually from the tangent to the circular curve. curve provided to smooth the elevation from the super elevation of the tangent to the ma*imum super elevation at the circular curve.
40
PRINCIPLES OF A SPIRAL CURVE •
The super elevation varies directly with the length of the space. e8ec9686c where& e super elevation of the spiral curve at any point e! super elevation at $# L length of the spiral from T$ to any point L! length of the spiral curve
•
The degree of curve varies directly with the length of the spiral 8c9686c where& D degree of the curve of the spiral at any point D! degree of the spiral at $#
•
The spiral angle at any point on the spiral curve $96>8/Rc6c
•
The deflection angle varies directly at the square at the lengths from T$ i8ic9!6">86c> where& i deflection angle at any point i! deflection angle at $#
•
The deflection angle is 184 of the spiral angle i9$84
FORMULAS
41
•
$piral angle, S where& spiral angle of any point along the spiral S! spiral angle at $#
9117;.@1?8R 9P8R
c9117;.@1?8Rc let 1 117;.@1?
8c9686c !P8R"8!Pc8Rc"9686c $olving R& R9Rc6c86 OO.eq1 69ro d69RdsOO..eq/ d69Rc6c86 ds V ds9V 6d68Rc6c $918!Rc6c"V 6d6 $96>8/Rc6c !in radius" t $#& $9$c & 696c $c96c8/Rc !in radius" Recall full arc length is equal to /0m
/0m9cRc Rc9/0m8c 4/
$ubstitution to $c96c8/Rc $c96cc870 !in degree" •
Mffset from tangent to spiral curve !*,Nc" where& " offset from tangent to any point along the spiral curve X! offset at $# or #$
5ote& for small angle $ sin$ almost9 $ sin$9d*8d6 d*9sins d6 d*9$d6 Vd*9V 6>8/Rc6c d6 *918/RdcV 6>d6 *96W8?Rc6c
at $# or #$ & N9Nc, 696c Nc96c>8?Rc
•
istance along the tangent 44
$lope correction formula&
c>9a>:h> c>-a>9h> !c:a"!c-a"9h>
!/c"!c-a"9h> c almost equal to a
Dor spiral curves& a9dy b9d* c9d6 dy9d6-!d*>8/d6" sin$9d*8d6 $9d*8d6 dy9d6-$>!d6">8/d6 dy9d6-!6>8/Rc6c">d6
V dy9 V d6-V6 d68ARc>6c> y9V d6-!18ARc>6c>" V 6yd6 y96-!6 870Rc>6c>"
at $#&
y9yc 696c
yc96c-!6cW870Rc>"
47
•
eflection angle, i where& i deflection angle at any point along the spiral curve
sini9N86 i9N86
N96W8?Rc6c i96>8?Rc6c
$96>8/Rc6c i9$84 •
6ength of throw, P
+9K Ts9K 3s9K 1A0<9I:/ X9@0<-Ic8/-$c
Ts9K sinI8/9Ts-@8Rc:3s 4;
1A0<9I:!@0<-Ic8/-$c" 09I-Ic-/$c Ic9I-/$c
Ts9Rc:3s!sinI8/:@"
+9K cos$c9Rc-!Nc-+"8Rc +9Nc-Rc!1-cos$c" or +9Nc87J +96c>8/7Rc 3s9K cosI8/9Rc:+8Rc:3s 3s9!Rc:+"secI8/-Rc
EARTH0OR1S • •
reas and Lolumes of earthworks istribution nalysis !HE6 and )$$ IR)"
ROUTE SURVE,ING DEFINITION
Route $urveying is a survey which supplies data necessary to determine the alignment, grades, and earthworks quantities necessary for the location and construction of engineering pro%ects. This includes highways, drainage, canal, pipelines, railways, transmission lines, and other civil engineering pro%ects that do not close upon the point of beginning ROUTE LOCATION STAGES OF HIGH0A, SURVE,S
evelopment of the interstate highway system and more general acceptance of the limited access principle for ma%or highways have resulted in a more and more highway pro%ects being to serve local traffic, surveys for highway pro%ects where new location is being considered start with a general study of the entire area between termini, proceed to more specific studies of possible alternative routes, and finally conclude with a detailed survey of the selected route and staking of the final centerline on the ground.
4?
These procedures are generally carried in three stages& • • •
R3#M5I$$5#3 +R36I)I5RY $FRL3Y 6M#TIM5 $FRL3Y
RECONAISSANCE
Includes a general study of the entire area the development of one or more alternative routes or corridors, and the study of each of these corridors in sufficient detail to enable the proper officials to decide which will provide the optimum location.
PRELIMINAR, SURVE,
Is a survey of selected corridors in sufficient detail to permit staking of the final centerline on the ground in some cases, the preliminary survey may be completed and staked in the field without variation in other instances, )inor ad%ustments may be required during the location survey. LOCATION SURVE,
#onsists in staking the final centerline and obtaining all additional information necessary to enable the design engineer to prepare completed plans, specifications, and estimates of earthwork quantities and to prepare deeds and descriptions covering the rights of way to be acquired. EARTH0OR1S
3RTH(MRP$ Z the construction of large open cuttings or e*cavations involving both cutting and filling of material other than rock. 3N#LTIM5 Z is the process of loosening and removing earth or rock from its original position in a cut and transporting it to a fill or to a waste deposit. 3)C5P)35T Z the term embankment describes the fill added above the low points along the roadway to raise the level to the bottom of the pavement structure material for embankment commonly comes from roadway cuts or designated borrow areas. SETTING STA1ES FOR EARTH0OR1
4B
The first step in connection with earthwork is staking out or setting slope stakes as it is commonly called. Two important parts of the work of setting slope stakes& I. II.
$etting the $takes Peeping the 5otes
The data for setting the stakes are& . C. #. .
The ground with center stakes set at every station. record of benchmarks and of elevations and rates of grades established. The base and side slopes of the cross section for each class of material. In practice, notes of alignment, a full profile, and various convenient data are commonly given in addition to the above mentioned data.
$ide $lopes most commonly employed for cuts and fills. )T3RI6 3N#LTIM5 MRI5RY 3RTH #MFR$3 RL36 6MM$3 RM#P $M6I RM#P $MDT #6Y MR $5
$I3 $6M+3 1.;0 & 1.00 1.00 & 1.00 0.;0 & 1.00 0./; & 1.00 / or 4 & 1.00
SETTING THE STA1ES
$etting the stakes work consists of& a" )aking upon the back of the center stakes the cut or fill in feet or meters and tenths, as # / 4 or D 7 B b" $etting side stakes or slope stakes at each side of centerline at the point where the side slope intersects the surface of the ground and marking upon the inner side of the stake, cut or fill at that point.
Digure
4A
Digure
+rocess of determining the height of cut or fill at the center stake or at any other points between the center space and slope stake.
4@
Digure
6et HI 9 elevation of the line of sight or telescope refereed fro known or assumed datum. rade RM 9 difference in elevations between the line of sight !HI" and the grade elevation round RM 9 HI Z rad 3levation #FT 9 rade RM Z round RM Digure
70
(hen the instrument is set up above the grade or subgrade& rade RM 9 !HI" Z rade 3levation DI66 9 round RM Z rade RM 1. (hen the instrument is set up below the grade or subgrade& rade RM C 9 rade 3levation Z !HI" C DI66 9 rade RM C : round RM C SETTING SIDE OF SLOPE (FIELD PROCEDURES)
The cross Z sectioning is done after the grade lines have been determined in the office. The amounts of cut and fill at the center are computed, the distances and their heights above the base, or below it of the slope stakes are determined as follows& 1. n engineer2s level is set up and rod readings are taken at the center and at trial point. ssume that the third trial point is on the slope, compute the distance fro the center using the following formulas& 6 9 C 8 / : $ H6
$ 9 Horizontal 8 Lertical
R 9 C 8 / : $ HR 71
(here& $ 9 $ide $lope C 9 Case pr (idth HR 9 $ide Height Right
H6 9 $ide Height 6eft R 9 istance out right 6 9 istance out left
/. )easure the distance from the center to the trial point, if this distance is less than the calculated distance, the rod is to be moved out for another trial pointJ if greater, the rod is to be moved in, if equal, the point is correctly located. stake is placed here indicating the right of the slope point above or below the base or sub grade.
ILLUSTRATION*
Digure
. If the measured distance is greater than the calculated distance, then the trial point is too far out the center line of the roadway and the direction to the rodman is to move in.
C. If the measured distance is less than the calculated distance, the trial point is too near to the centerline of the roadway and the direction to the rodman is to move out.
7/
Digure
#. If the measured distance is e*actly equal to the calculated distance, the point is correctly located and the slope stake is at on the ground indicating the height of the slope point above or below the ground.
Digure
ROAD CROSS SECTION
74
A. LEVEL SECTION
If the ground level in a direction transverse to the centerline, the only rod reading necessary is that the centerstake, and the distance to the slope stake can be calculated once the center cut or fill has been determined, such a cross-section is called level section.
/. LEVEL SECTION IN CUT
Digure
#enterheight 9 1.A4m Case for #ut 9 A.00m $$ for #ut 9 1&1 R 9 6 9 C 8 / : $# 9 7 : 1 !1.A4" 9 ;.A4 2. LEVEL SECTION IN FILL
Digure
77
#enterheight 9 1.;0m Case for Dill 9 B.00m $$ for Dill 9 -1.;0 & 1.00 R 9 6 9 C 8 / : $# 9 4.;0 : 1.;0 !1.;0" 9 ;.B; A. THREE LEVEL SECTION
(hen Rod readings are taken at each slope stake in addition to readings taken at the center as will normally be done whre the ground is sloping the cross-section is called Three 6evel $ection.
Digure
7;
Case for #ut 9 A.00m $$ for #ut 9 1.00&1.00 6 9 C 8 / : $ H6 9 7.00 : 1!0.?4" 9 7.?4m R 9 C 8 / : $ HR 9 7.00 : 1!7.@?" 9 A.@?m
Digure
Case for Dill 9 B.00m $$ for Dill 9 1.;0&1.00 6 9 C 8 / : $ H6 7?
9 4.;0 : 1.;0!4.1/" 9 A.1Am R 9 C 8 / : $ HR 9 4.;0 : 1.;0!/.?/" 9 B.74 B. FIVE LEVEL SECTION
(hen rod reading is taken at the centerside the slope stake and at points on each side of the center of the distance of half the width of the road bed, the cross section is called a DIL3 63L36 $3#TIM5. Digure
Case for Dill 9 B.00m $$ for Dill 9 1.;0&1.00 6 9 C 8 / : $ H6 9 4.;0 : 1.;0!/.7/" 9 B.14m R 9 C 8 / : $ HR 9 4.;0 : 1.;0!4./A" 9 @./4m SAMPLE PROBLEM (Setting S$%e Sta3e)
In setting slope stakes, the height of cut at the center has been found to be 1.74m, the ground readings at center )S and trial point on the slope are /.44m and 1.7?m, respectively, and the measured distance from the center line of the roadway to the trial point is A./7m. If the base of the roadway is @m and the side slope is 1.;0&1.00, should the trial point be moved in or outK 7B
Digure
$olution& rade Rod Z rade Rod [ ) 9 /.44: 1.74 9 4.B? )easured istance ! )" 9 A./7 #alculated istance ! #" 9 C 8 / : $HR (here& C 8 / 9 7.;m HR 9 4.B? Z 1.7? 9 /.40m 1.;0 8 1.00 9 $HR 8 /.40 $HR 9 /.40 !1.;0" 8 1.00 $HR 9 4.7; # 9 7.; : 4.7; 9 B.@; $ince # \ ) --- )ove In Digure
7A
Case for #ut 9 A.00m $$ for #ut 9 1&1 6 9 C 8 / : $ H6 9 7.00 : 1!/.B;" 9 ?.B;m R 9 C 8 / : $ HR 9 7.00 : 1!4.?0" 9 B.?0m
Digure
Case for Dill 9 B.00m 7@
$$ for Dill 9 1.;0&1.00 6 9 C 8 / : $ H6 9 4.; : 1.;0!/.A7" 9 B.B?m R 9 C 8 / : $ HR 9 4.;0 : 1.;0!/.@/" 9 B.AAm C. IRREGULAR SECTION IN CUT
cross section for which observation is taken to points between center and slope stakes at irregular intervals is called irregular section. Digure
Case for #ut 9 A.00m $$ for #ut 9 1&1 6 9 C 8 / : $ H6 9 7.00 : 1!/.?0" 9 ?.?0m R 9 C 8 / : $ HR 9 7.00 : 1!4.7B" 9 B.7Bm D. SIDE HILL SECTION
;0
(here the cross-section passes through from cut to fill, it is called a $I3 HI66 $3#TIM5 and an additional observation is made to determine the distance from center to the grade point. That is the point where subgrade will intersect the natural ground surface. peg is usually driven to grade at this point and its position is indicated by a guard stake marked radeS. In this case also crosssection is taken additional plus station.
Case for #ut 9 A.00m Case for Dill 9 B.00m 6 9 C 8 / : $ H6 9 4.;0 : 1.;0!4.?0" 9 A.@@m R 9 C 8 / : $ HR 9 7.00 : 1!4.?B" 9 B.7Bm PROBLEMS*
In two ways, find the areas of each of the following cross-section note, given the corresponding bases and side slope if not given they are to be computed . C$3 (ITH A.00m $I3 $6M+3 1.;0&1.00 K 8 1/.A7 : /.A7
K 8 /.A7
C. C$3 (ITH K ;1
$I3 $6M+3 K ;.B@ 8 -1.A? Z 1./B
?.04 8 -/.0/
#. C$3 (ITH K $I3 $6M+3 K B.A; 8 4.0A 7.00 8 4.?; : 4./B . C$3 (ITH A.00m $I3 $6M+3 1.;0&1.00 K 8 -4.;? ?./A 8 -/./A -/.4/ 3. C$3 (ITH A.00m !#ut" $I3 $6M+3 1.00&1.00 ?.@B 8 -4.7B -0.?1 1.07 8 0.00
7.00 8 /.A4
1.00 8 -1.11 K 8 -/.B7
A.0; 8 4./7
B.;0 8 -4.A/
;.00 !Dill" 4.?? 8 -;.@7
4.77 8 /.77
;/
A. LEVEL SECTION IN CUT
DIFR3
B. THREE LEVEL SECTION IN FILL
DIFR3
;4
C. FIVE LEVEL SECTION IN CUT
DIFR3
D. IRREGULAR SECTION IN FILL
DIFR3
;7
E. SIDE HILL SECTION
DIFR3
METHODS OF DETERMINING VOLUMES O EARTHE0OR1S
DIFR3
;;
A. B4 Ae#age End A#ea
L 9 6 8 / !1 : /" (here& L 9 Lolume of $ection of 3arthworks between $ta 1 and /, mW 1 , / 9 #ross Z sectional area of end stations, m> 6 9 +erpendicular istance between the end station, m NOTES*
1. The above volume formula is e*act only when 1 9 / but is appro*imate
1 \U /. /. #onsidering the facts that cross-sections are usually a considerable distance apart and that minor inequalities in the surface of the earth between sections are not considered, the method of end areas is sufficiently precise for ordinary earthwork. 4. Cy where heavy cuts or fills occur on sharp curves. The computed volume of earthwork ay be corrected for curvature out of ordinarily the corrected is not large enough to be considered. A. B4 P#i-%ida$ F%#-$a
L 9 6 8 ? !1 : ) : /" (here& L 9 Lolume of section of earthwork between $ta 1 and / of volume of prismoid, mW 1 , / 9 cross Z sectional area of end sections, m> ) 9 rea of mid section parallel to the end sections and which will be computed as the averages of respective end dimensions, mW 5MT3$& 1. +rismoidal is a solid having for its two ends any dissimilar parallel plane figures of the same no. of sides, and all the sides of the solid plane figures. lso, any prismoid may be resolve into prisms, pyramids and wedges, having a common altitudes the perpendicular distance between the two parallel end plane cross Z section. /. s far as volume of earthworks are concerned, the use of +rismoidal formula is %ustified only if cross-section are taken at short intervals, is a small surface deviations are observed, and if the areas of successive crosssection cliff or widely usually it yields smaller values than those computed from average end areas. ;?
PRISMOIDAL CORRECTION FORMULA
Digure
# 9 6 8 1/ !b 1 Z b/"!h1 Z h/" (here& # 9 +rismoidal #orrection, It is subtracted algebraically from the volume as determined by the average and the areas method to give the more nearly correct volume as determined by the +rismoidal formula, mW 6 9 +erpendicular distance between / parallel and sections, m b1 9 istance between slope stakes at end section C# where the altitude is h 1, m b/ 9 istance between slope stake at end section 3D where the altitude is h /, m h1 9 ltitude of end section C# at $ta 1, m h/ 9 ltitude of end section 3D at $ta /, m PRISMOIDAL CORRECTION FOR IRREGULAR SECTION
In prismoid, there should be equal number of slope in both bases so that on equal number triangles can be found. The +rismoidal correction can then be found. The +rismoidal correction can then be found using either the fundamental formula of correction, # 9 6 8 1/ !b 1 Z b/"!h1 Z h/" or any of the formulas derive from it, where, however one base or any a five level section or three level section and other. five level section !or irregular section" or both bases are irregular sections or, if one base is a five level section and the other irregular section, the formulas cannot be directly applied without making certain assumptions because there are more triangles formed in one section than in ;B
the other. The determination determination of the the correction is at best best only appro*imate. appro*imate. Dor the purpose of determining the +rismoidal +rismoidal correction, the following following may be used& . 5eglect 5eglect the intermedia intermediate te heights heights thereby reducin reducingg the sections sections into three level or level sections this is the most convenient method. C. +lot the irregul irregular ar or five level level sections sections on cross cross sections sections paper. paper. raw on this section two equalizing lines starting from the same point or the center height such that the error added equal equal the areas subtracted appro*imately by estimating the center height as well as the distances in the right right or in the left can then be be scaled. This is more accurate than method but involves more work. #. Reduce Reduce the five level level or irregul irregular ar section section by calcula calculation tion to equivalent level or three level sections as follows& /. T% LE LEVE VEL LS SEC ECTI TION ONS S
a. The area area of a level level section section C# : $# !C is the the base, base, # is the cente centerr point, point, and $ is the side slope." b. 3quate this area forced forced per the irregular or five-level five-level section c. Case $$ being being known, known, a quadrat quadratic ic formula formula in one one unknown unknown is formed formed from from which # is determined. d. $olve $olve for the the corre correspon spondin dingg value value of #. /. T% TH THRE REE E LEVE LEVEL L SECT SECTIO IONS NS
Digure
;A
Total rea of three level section in cut 9 1 : / (here& 1 9 C 8 7 !H 6 : HR " / 9 # 8 7 !C : $" !H6 : HR " Then P 9 C# 8 / : !H6 : HR " !C 8 7 Z #$ 8 /" 5MT3& The unknowns are #, H R and H6. Two these should should be assumed assumed and the third computed. It is simpler to to covert to level level section.
PROBLEM*
1. iven the the following following cross-se cross-section ction notes notes of a roadway roadway with a base of ?m and $$ of 1./;&1.00, between the volume of the prismoid between the two-end sections by the following methods& . 35 35 R3 R3 )3T )3THM HM C. +RI$ +RI$)M )MI I6 6 DMR) DMR)F6 F6 #. 35 R3 R3 )3THM )3THM and and +RI$)MI6 +RI$)MI6 #MRR3#TIM5 #MRR3#TIM5 DMR)F6 DMR)F6
STATION
10 : 000 10 : 0/0
CROSS 5 SECTION NOTES
:?.;; : /.A7 :B.;; : 4.?7
:/.A7 :1.A;
:?.;; : /.A7 :4.?; : 0.;/
$M6FTIM5& #ompute for the area at each station cross-section and at mid-section Digure
;@
#heck for #ut distances R1 9 61 9 C 8 / : $ HR 9 1 8 / !?m" : 1./;!/.A7" 9 ?.;;m rea by method of triangle and rhombus 1 9 C# : $#> 9 /B.1/m> Digure
?0
#heck for the distances R/ 9 C 8 / : $ HR/ 9 1 8 / !?" : 1./;!0.;/" 9 4.?;m 6/ 9 C 8 / : $ H6/ 9 1 8 / !?" : 1./;!4.?7" 9 B.;;m rea by method of triangle / 9 a : 6 : c : d 9 1 8 / !4"!4.?7" : 1 8 / !1.A;"!B.;;" : 1 8 / !1.A;"!4.?;" : 1 8 / !0.;/"!4" / 9 1?.?0m>
#ompute for the dimensions of the mid sections Digure
?1
Rm 9 1 8 / ! R1 : R/" 9 1 8 / !?.;; : 4.?;" Rm 9 ;.10m
HRm 9 1 8/ !H R1 : HR/" 9 1 8 / !/.A7 : 0.;/" HRm 9 1.?Am
6m 9 1 8 / !61 : 6/" 9 1 8 / !?.;; : B.;;" 6m 9 B.0;m
H6m 9 1 8 / !H61 : H6/" 9 1 8 / !/.A7 : 4.?7" 9 4./7m H#m 9 1 8 / !H #1 : H#/" 9 1 8 / !/.A7 : 1.A;" H#m 9 /.47;m
#heck for #ut distances Rm 9 C 8 / $ HRm 9 1 8 / !?" : 1./;!1.?A" Rm 9 ;.10m 6m 9 C 8 / $H6m 9 1 8 / !?" : 1./;!4.//" 6m 9 B.0;m rea by method of triangle m 9 e : f : g : h 9 1 8 / !4"!4./7" : 1 8 / !B.0;"!/.47;" : 1 8 / !;.10"!/.47;" : 1 8 / !4"!1.?A" m 9 /1.?Am COMPUTE FOR THE VOLUME OF EARTH0OR1 VOLUME OF CUT IN BET0EEN THE T0O STATIONS
Digure
?/
/. B4 End A#ea Met&%d
Le 9 6 8 / ! 1 : /"
(here& 6 9 !10 : 0/0" Z !10 : 000" 9 /0m 1 9 /B.1/m> / 9 1?.?0m> Then, Le 9 /0 8 / !/B.1/ : 1?.?0" 9 74B./0m> 2. B4 P#i-%ida$ F%#-$a
L p 9 6 8 ? ! 1 : 7m : /" (here& 6 9 /0m 1 9 /B.1/m> / 9 1?.?0m> m 9 /1.?Bm>
Then, L p 9 /0 8 ? !/B.1/ : 7]/1.?B : 1?.?0" 9 747.14mW /. P#i-%ida$ F%#-$a '%# C%##e!ti%n
# p 9 6 8 / ! 1 : /"!b1 Z b/"
5ote& Resolve the given prismoid into a series of triangular prismoid into a series of triangular prismoid. # p 9 # pa : # pb : # pc : + pd (here& # pa 9 /0 8 1/ !/.A7 Z 4.?7"!4-4" 9 0 # pb 9 /0 8 1/ !/.A7 Z 1.A;"!?.;; Z B.;;" 9 -1.?;mW # pc 9 /0 8 1/ !/.A7 Z 1.A;"!?.;; Z 4.?;" 9 7.BA;mW + pd 9 /0 8 1/ !/.A7 Z 0.;/"!4-4" 9 0 ?4
Then, # p 9 -1.?; : 7.BA; 9 4.14;mW 2. C%##e!ted V%$-e
Lc 9 Le - # p 9 74B./0 Z 4.14; Lc 9 747.0?;mW iven the following cross section notes, determine the volume of the prismoid b end areas method and apply the +rismoidal formula. The roadway base is ?m with side slope of 1./;&1.00 STATIONS
10 : 070 10 : 0;0
:7.0; :0.A7 :B.A0 :4.A7
CROSS6SECTION NOTES
:4.00 : 4.;0 :/.00 :/./7
:/.A; :4./;
:4.00 :/.1/ :7.00 :/.;0
:B.0; :4./7 :;.?; :/.1/
$M6FTIM5& 1. #ompute for the end areas of the end sections Digure
?7
#heck for distances& R1 9 C 8 / : $ HR1 9 4 : 1./;!4./7" R1 9 B.0;
61 9 C 8 / : $ H61 9 4 : 1./;!0.A7" 61 9 7.0;
1 9 1 : / : 4 : 7 9 1 8 / !4.;"!1.0;" : 1 8 / !/.A;" : 4.;!4" : 1 8 / !/.1/ : /.A;"!4" : 1 8 / !/.1/"!7.0;" 1 9 /4.11m> Digure
#heck for distances& R/ 9 C 8 / : $ HR/ 9 4 : 1./;!/.1/" R/ 9 ;.?;m
6/ 9 C 8 / : $ H6/ 9 4 : 1./;!4.A;" 6/ 9 B.A0m
/ 9 a : b : c : d : e : f 9 1 8 / !/.1/ : 4.A7"!;.A0" : 1 8 / !/.7/ : 4./;"!/" : 1 8 / !/.;0 : 4./;"!7" : 1 8 / !/.1/ :/.;0"!1.?;" Z 1 8 / !7.00"!4.A7" Z 1 8 / !/.?;"!/.;/" / 9 /B.11m> CONVERT THE END SECTIONS TO AN E7UIVALENT LEVEL SECTIONS
?;
Digure
1 9 five level section Z 1 !equivalent level section" 1.11 9 14 H#e1 : $ !H#e1"> 4.11 9 ? H#e1 : 1./; !H#e1">J 6et #1 9 H#e1 4.11 9 ? #1 : 1./; #1> Cy quadratic formula #1 9 /.;/m Re1 9 C 8 / : $H#e1J Re1 9 6e/ 9 4 : 1./;!/.;/"
Digure
/ 9 irregular section 9 / !equivalent level section" ??
/B.11 9 CH#e/ : $!H#e/"> /B.11 9 ?H#e/ : 1./;!H#e/"> 6et #e 9 H#e/ Cy quadratic formula
#/ 9 H#e/ re/ 9 C 8 / : $HRe/ 9 4 : 1./;
6e/ 9 Re/
3*. $olve for the following& a" b" c" d" e" f" g"
$tationing of the limits of free haul $tationing of the limits of economical haul Lertical volume 6ength of overhaul #ost of haul #ost of waste #ost of borrow
iven& DH 9 ;0m #ost of borrow 9 + 7.008m4 #ost of e*cavation 9 + 4.;08 m4 #ost of haul 9 + 0./08m station 35 R3 )/ $tation
#FT
1:7?0
DI66 70
1:B?0
0
/:0?0
?0
Calance point
63H 9 #b#h!#" : DH 9 7!/0"0./ : ;0 63H 9 7;0 m a* 9 ?0400 J a 9 ?0*400 - eq. !1" b;0-* 9 70400 J b 9 /000-70*400 ?B
0.;a* 9 0.;b!;0 Z *" - eq. !/" 3quate 1 and 4J / and 4& * 9 //.7B m J ;0 Z * 9 /B.;4 m J
a 9 7.7@ m/ b 9 4.?B m/
a" 6eft limit 9 1:B?0 Z /B.;4 9 1:B4/.7B Right limit 9 1:B?0 : //.7B 9 1:BA/.7B $ 9 ;0 m 9 DH a'y 9 60300
J
a2 9 60y300 - eq. !1"
b'400-y 9 40300
J
b2 9 16,000-40y300 - eq. !/"
0.;!b2 : b : b"!700 - y" 9 0.;!a2 : a : a"!y" !b2 : B.47"!700 - y" 9 !a2 : A.@A"!y" - eq. !4" 3quating !1" and !4"& y 9 1BB.A1m equating !/" and !4"& 700 Z y 9 //0.1@ m a2 9 4;.@? m/ b2 9 /@.4? m/ b" 6imits of 63H& 6eft limit 9 1:B?0 Z /B.;4 Z //0.1@ 9 1:;1/./A Right limit 9 1:B?0 : //.7B :
[email protected] 9 1:@?/./A 63H 9 7;0 c" MH Lol. 9 0.;!a2 : a : a"!y" 9 0.;!4;.@? : A.@A"
[email protected]" 9 7070.44 m4 MH Lol. 9 0.;!b2 : b : b"!700 - y" 9 0.;!/@.4? : B.4@"!//0.1@" 9 7070.7@ m4 MH Lol. 9 0.;!7070.44 : 7070.7@" 9 8989.8/ -:
d) TN6 9 AiXi
7070.7@N6 9 0.;!b2"!700 - y"!/84"!700 - y" : b!700 - y"!0.;"! 700 - y" 9 0.;!/@.4?"!//0.1@"! /84"! //0.1@" : !4.?B"! //0.1@"!0.;"! //0.1@" N6 9
[email protected]; m TNR 9 iNi 7070.44NR 9 0.;!a2"!y"!/84"!y" : !a"!y"0.;!y" 9 0.;!/@.4?"!//0.1@"!/84"!//0.1@" : !4.?B"!//0.1@"!0.;" !//0.1@" NR 9 114.A@ m ?A
