Chapter 4 Lecture-11,12,13-Transition-Curves.pptx

Surveying and super elevation with transition curve

1. Surveying-II I CT-66453 1 T rans i t i on cur v es Lecture 11,12,13 Engr. Md Al Amin Akhnd

2. Curves Transition Curves A curve of varying radius is called a transition curve. It is also ca l l e d Spiral Cur v e o r Easeme n t curv e . It is used on both highway & railway between tangent and a circular curve in order to have a smooth transition from tangent to the curve and from curve to the tangent. It is also inserted between two branches of compound curve. 2

3. 1.ক্রান্তি বাঁক 2.সুপার এলিভেশন 3.Mathmatical Relation

4. Super Elevation

5. Super Elevation

6. Curves Transition Curves The nee d for T r an s ition Cur v e s Circular curves are limited in road designs due to the forces which act on a vehicle as they travel around a bend. Transition curves are used to built up those forces gradually and u n i f orm l y t h u s e ns u r i ng t he safe t y o f pa s se nge r . Allows for gradual application of Super elevation or Cant. The S uper e l e v a tion i s de s i g n e d s uch t h a t the r oad s u r f a c e is near perpendicular to the resultant force of gravity and c e n t r i fugal iner t ia . 6

7. Calculate Length Of Transition Curve

8. Transition Curve Math

9. Curves Transition Curves The nee d for T r an s ition Cur v e s In order to transition from a flat roadway to a fully super elevated section and still maintain the balance of forces, the degree or sharpness of the curve must begin at zero and increase steadily until maximum super elevation is reached. This is precisely what a Spiral Curve does. 9

10. Curves Transition Curves Ob j e c ts o f providin g a t r an s ition curve To accomplish gradually the transition from tangent to the circular curve or from circular curve to the tangent. To obtain a gradual increase of the curvature from zero at a tangent point to the specified quantity at junction of the t ransition c u r v e wi t h t he ci r c u l a r c u r v e . To Provide a satisfactory mean of obtaining gradual increase of super elevation from zero on the tangent to the specified amount on the main circular curve. 10

11. Curves Transition Curves Condition fulfill by a T r an s ition Cur ve It should meet the original straight (tangent ) tangentially . I t sho u l d m ee t ci r c u l a r c u r v e t ang e nt ia l ly . Its radius at a junction with circular curve should be same as t hat of t he ci r c u l a r c u r v e . The rate of increase of the curvature along the transition curve should be the same as that of the super elevation. Its length should be such that full super elevation is attained at the junction with the circular curve. 11

12. Curves A c l o thoid or s p i r al A c u bi c para b ola 3 ) A l e m nisca t e On l y m a t he m a tical di f f e ren ce are here. I n or d e r t o admi t t he t ransition cur v e , t he main ci r cu l a r cur v e used in railway used in highway Transition Curves Types of the transition curve in common use are re q uired t o be shi f t e d inward. • W h e n t he t ransition c u r v e is ins e r t e d a t eac h e nd o f t he m ain ci r cu l a r cur v e t he res u lt i ng cur v e is ca l l e d com bin e d or com posi t e curve. 12

13. Curves A c l o thoid or s p i r al A c u bi c para b ola 3 ) A l e m nisca t e On l y m a t he m a tical di f f e ren ce are here. I n or d e r t o admi t t he t ransition cur v e , t he main ci r cu l a r cur v e used in railway used in highway Transition Curves Types of the transition curve in common use are re q uired t o be shi f t e d inward. • W h e n t he t ransition c u r v e is ins e r t e d a t eac h e nd o f t he m ain ci r cu l a r cur v e t he res u lt i ng cur v e is ca l l e d com bin e d or com posi t e curve. 13

14. Curves Transition Curves Super Elev ation When vehicle moves from tangent on to the cur v e t h e for c e s ac t i n g on i t a re Weight of the vehicle Ce n t r i fug a l for c e , Bo t h a c t i n g t h ro u gh t h e c e n t e r o f gr a v it y of t h e v e h i c l e . T h e e ff e ct of t h e c e n t r i f u g a l for c e i s t o p u sh t h e v e h i c l e off t h e r a i l or ro a d. T o cou n t e r a ct t h e a c t i on t h e o u t e r r a i l or o u t e r e dge of t h e ro a d i s r a i s e d a b o v e t h e raising of outer edge of rail or road above t h e i nn e r o n e i s c a ll e d Supe r elevatio n or Cant . T h e a m o u n t of Sup e r e l e v a ti on dep e n ds upon Sp e e d of t h e v e h i c l e R a d i us of t h e cur v e 11

15. Curves P = 𝑅 𝑚𝑣 2 = 𝑤𝑣 2 𝑔 𝑅 :: w= mg P = 𝑣 2 𝑤 𝑔𝑅 . Transition Curves Supe r Elev ation Let W = w e i g ht of t he v e hicle P = cent r i fugal Force V = S p ee d of t he v e hicl e , m /s g = A cce l e ra t i o n due t o g r a v i t y , m/s 2 R = R adiu s of t he c u r v e , m h = S u p e r e l e v a tion, m b = W i d t h of t he roa d , m F o r e q u ilibr i um t he res u lta n t R of t he P & W m u s t b e e q u a l & opposi t e t o t he rea c t i on p e rp endicu l a r t o road or r ai l s u r f ace 12

16. Curves ℎ = 𝑑𝑐 = 𝑃 𝑏 𝑎 𝑐 𝑊 tan θ = tan θ = ℎ = 𝑃 = 𝑣 2 𝑏 𝑊 𝑔 𝑅 h = b tan θ 𝑔 𝑅 h = b 𝑣 2 ………………. On highway h = b 𝐺𝑣 2 ………………. On railway 𝑔𝑅 w h e re G = Dis t anc e b/ w t he c e n t re of t he ra i l Super elevation is gradually applied along a transition curve. Full super elevation is attained at junction of the transition c u r v e wi t h t he ci r c u l a r c u r v e . Transition Curves Supe r Elev ation 13

17. Curves L = Length of transition curve, m h = super elevation, m 1 in n = Rate of canting Transition Curves Length o f the T r an s ition curve It may be determine in different ways: As an Arbitrary: V a l ue f r om pa s t e xp e r i m e nt say 5 m . By an Arbitrary Gradient (slope) : Length of the transition curve may be such that super e l e v a t i on i s app l ie d a t t he u n i f orm ra t e of 1 i n n , Where n = 300 to 1200 i.e h feet rise in n feet length 1 feet rise in n feet length Therefore L = n h Where 14

18. Curves t = 𝐿 , sec 𝑣 Super elevation attain in this time: h = 𝑎 t h = 𝑎 𝐿 𝑣 L = ℎ 𝑣 𝑎 L = Length of the transition curve, m h = amount of super elevation, cm V = Speed of the vehicle, m/s 𝑎 = Time rate (cm/sec) Transition Curves Length o f the Tra n s i tion curve 3 ) B y the T ime R ate Transition curve may be of such length that cant ( super e l e v a t i on ) i s ap p l ie d a t a n arbi t rary t im e ra t e of “ a ” c m /s e c, W h e re 𝒂 v ar i e s f r om 2 . 5 c m /s e c t o 5 c m /s e c Time taken by vehicle in passing over the transition curve: 18

19. Curves Centrifugal force = P = 𝑚𝑣 2 𝑅 Radial acceleration = a = 𝑣 2 , m/sec 2 t α = a = 𝑅 Rate of change of radial acceleration divide by the time v 2 R L v = v 3 R L 𝐿 = v 3 α R Out of these methods the 4th method is commonly used in de t e rm i n i n g l e n g t h of t h e t r a n s iti on cur v e . Transition Curves Length o f the tran s ition curve 4) By the Rate of change of Radial Acceleration This rate should be such that the passengers should not experience any sensation of discomfort when the train is travelling over the curve. It is taken as 30 cm/sec 2 , which is maximum that will pass unnoticed. 19

20. Curves 𝑔 𝑔 𝑅 𝑃 = 𝑅 𝑣 2 𝑊 𝑔𝑅 Max v a l u e of C. R a t i o on road i s 1 4 Max v a l u e of C. R at i o on railway i s 1 8 On Road: 𝑣 2 𝑔 𝑅 = 4 = > 𝑣 = 1 𝑔 𝑅 4 On Railway: 𝑣 2 1 𝑔 𝑅 = 8 = > 𝑣 = 𝑔 𝑅 8 𝑣 3 𝛼 𝑅 𝐿 = = 𝑔 𝑅 4 3 𝛼 𝑅 … I 𝑣 3 𝛼 𝑅 𝐿 = = 𝑔 𝑅 4 3 𝛼 𝑅 … II Transition Curves Length o f the tran s ition curve 4) By the Rate of change of Radial Acceleration By centrifugal ratio The ratio of centrifugal force to weight is called centrifugal ratio. 𝑃 = 𝑚𝑣 2 𝑥 𝑔 = 𝑤 𝑣 2 17

21. Curves Transition Curves Ideal T r ans ition cur ve The intrinsic equation of the ideal transition curve (clothoid s p i r a l ) m a y be ded u c e d as: TB = Initial tangent T = Beginning of the transition curve E = Junction point transition curve with circular curve M= Any point of transi t ion curve , 𝑙 m along it from T. ρ =Radius of transition curve at M R = Radius of the circular curve ∅ = Inclination of tangent to the transition curve at M to initial tangent TB. ∅ 1 = angle b/w tangent TB and tangent to the T. Curve at junction E (spiral angle) L = Length of the transition curve. 18

22. Curves ρ 1 = m 𝑙 𝑑𝑙 For all curves, 𝑑∅ = Curvature = 1 ρ d∅ = 𝑑𝑙 = 𝑚 𝑙 𝑑𝑙 ρ I n t e g r a t i ng, ∫ 𝑑∅ = ∫ 𝑚 𝑙 𝑑𝑙 ∅ = 𝑚 𝑙 2 + 𝐶 2 When 𝑙 = then ∅ = then C = ∅ = 𝒎 𝒍 𝟐 𝟐 … ( 𝑎 ) Transition Curves Ide a l T r an s ition curve The fundamental requirement of spiral curve is that its radius of curvature at any point shall vary inversely as a distance 𝑙 from b e g i nning of t he c u r v e . ρ 𝖺 1 or ρ = 𝑚 , m = constant of proportionality 𝑙 𝑙 19

23. Curves :: 1 = 1 = 𝑚 𝐿 𝑜𝑟 𝑚 = 1 𝑅 𝐿 ρ 𝑅 ∅ 1 = 1 x 𝑙 2 𝑅 𝐿 2 𝐿 2 𝑅 ….. (b) ∅ = At point M: ∅ = ∅ , 𝑚 = 1 𝑅 𝐿 1 𝑥 𝑙 2 𝑅 𝐿 2 𝑙 2 … (c) ∅ = ∅ = ∅ = 2 𝑅 𝐿 𝑙 2 𝑙 2 2 𝑅 𝐿 2 𝐾 = , 𝐾 = 𝑅 𝐿 Transition Curves Ide a l T r an s ition curve At point E: 𝑙 = L , ∅ = ∅ 1 (spiral has here max value) 20

24. Curves Transition Curves Ide a l T r an s ition curve If the curve is to be set out by offsets from the tangent at the commencement of the curve (T), it is necessary to calculate the rectangular (Cartesian) co-ordinates, the ‘axes of co- or d ina t e ’ b e in g t he t ang e nt a t T a s t he x - axi s an d a line p e rp endicu l a r t o i t a s t he y - axis. 24

25. Curves Transition Curves Ide a l T r an s ition curve M and N be the two points at a distance 𝜹𝒍 apart on the curve. Let the co-ordinate of M and N be M => (x , y) N => (x + 𝜹 x, y + 𝜹 y ) And respective inclinations of the tangents at M and N to the in i t ia l t ang e nt (TB) a t T , ∅ and ∅ + 𝜹 ∅ 25

26. Curves Transition Curves Ideal T r ans ition cur ve 𝑐𝑜𝑠∅ = 𝑑𝑥 and sin ∅ = 𝑑𝑦 𝑑𝑙 𝑑𝑙 𝑑𝑥 = 𝑑𝑙 𝑐𝑜𝑠∅ and 𝑑𝑦 = 𝑑𝑙 𝑠𝑖𝑛∅ 2 ! 4 ! 𝑐𝑜𝑠∅ = ( 1 − ∅ 2 + ∅ 4 − 𝑒𝑡𝑐) 3 ! 5 ! 𝑠𝑖𝑛∅ = ( ∅ − ∅ 3 + ∅ 5 − 𝑒𝑡𝑐) 𝑑𝑥 = 1 − ∅ 2 + ∅ 4 − … . 𝑑𝑙 2! 4! 𝑑 𝑦 = ∅ − ∅ 3 + ∅ 5 − ⋯ . 𝑑𝑙 3 ! 5 ! 𝑑𝑥 = ( 1 − ( 𝑙 2 2 1 2 𝐾 2 ! ) * + ( 𝑙 2 4 1 2 𝐾 4 ! ) * − …) 𝑑𝑙 I n t e g r a ting 𝑑𝑥 𝑙 5 40 𝑘 2 𝑥 = 𝑙 − + 𝑙 9 3456 𝑘 4 … . 𝒙 = 𝒍 𝟏 − 𝒍 𝟒 𝟒𝟎 𝒌 𝟐 + 𝒍 𝟖 𝟑𝟒𝟓𝟔 𝒌 𝟒 … …(A) 26

27. Curves Transition Curves Ideal T r ans ition cur ve A t ∅ = = 𝑙 2 𝑙 2 2 𝑘 2 𝑅 𝐿 𝟏𝟎 𝒙 = 𝒍 𝟏 − ∅ 𝟐 + ∅ 𝟒 𝟐𝟏𝟔 … ……..(A 1) 𝑙 2 3 1 𝑑𝑦 = ( 𝑙 2 − ( ) * 2 𝐾 2 𝐾 3! 𝑙 2 5 1 2 𝐾 5 ! + ( ) * − …) 𝑑𝑙 I n t e g r a ting 𝑑𝑦 𝒍 𝟑 𝒍 𝟕 𝟔 𝑲 𝟑𝟑𝟔 𝒌 𝟑 − + 𝒍 𝟏𝟏 𝟒𝟐𝟐𝟒𝟎 𝒌 𝟓 𝒚 = 𝒚 = 𝒍 𝟑 ∅ 𝟒 𝟔 𝑲 𝟏 𝟒 𝟒𝟒𝟎 𝟏 − ∅ 𝟐 + … … … (B) … . ( 𝐁𝟏) 1) Rejecting all the terms of equation A and B except 1 st : A t 𝑥 = 𝑙 𝑦 = = 𝑙 3 𝑥 3 6 𝑅 𝐿 6 𝑅 𝐿 … e q n of cub i c p a r a b o l a 27

28. Curves Transition Curves Ideal T r ans ition cur ve 2 ) I f w e t ak e 1 s t t erm only of eq u atio n B 𝑦 = = 𝑙 3 𝑙 3 6 𝐾 6 𝑅 𝐿 … E qn of Cub i c Sp i r a l 𝑙 --- a l o n g cur v e , 𝑦 --- off s e t 3 ) T a k i n g 1 st a n d 2 nd t e rms of e q n s A a n d B 𝑥 = 𝑙 1 − = 𝑙 1 − 𝑙 4 𝑙 4 40 𝑘 2 40 (𝑅𝐿)2 𝑦 = 1 − 𝑙 3 𝑙 4 = 𝑙 3 6 𝐾 56 𝑘 3 6 𝐾 1 − 𝑙 4 56 (𝑅𝐿)3 From w h i ch t h e c o - o r d i n a t e of a n y p o i n t on t h e true or clothiod spiral nay be obtained, the length 𝑙 measured along curve. 28

29. Curves Transition Curves Ideal T r ans ition cur ve Now ta n 𝛼 = 𝑦 , w h e r e 𝛼 = def l e c t i on an g l e i . e t h e a n g l e 𝑥 MTB between the tangent and the line from T to any p o i n t M on t h e cur v e . 1 − ∅ 2 + tan 𝛼 = 6 𝐾 14 440 𝑙 3 ∅ 4 … 𝑙 1 − ∅ 2 + ∅ 4 10 216 … ta n 𝛼 = ∅ ( 1 + ∅ 2 + . . ) , n e g l e c t i n g o t h e r t e rms 3 35 ta n 𝛼 = ∅ 3 Since ∅ is usually small( a small fraction of a radian) 𝛼 = 3 ∅ , B u t ∅ = 𝑙 2 2 𝑅 𝐿 𝛼 = 𝑙 2 2 𝑅 𝐿 = 𝑙 2 3 6 𝑅 𝐿 r a d i a n s 𝜋 𝑅 𝐿 𝛼 = 1800 𝑙 2 minutes 𝛼 = 1800 𝑙 2 60 𝜋 𝑅 𝐿 degrees 26

30. Curves Transition Curves Ch a r a cteri s tics o f a Tr a n s ition cur ve 27

31. Curves Transition Curves Ch a r a cteri s tics o f a Tr a n s ition cur ve L e t T B = or i g i n a l t a n g e n t T = comme n c e me n t of t h e t r a n s iti on cur v e E = e n d of t h e t r a n s iti on cur v e EE 2 = t a n g e n t t o bo t h t h e t r a n s iti on an d c i rcu l a r curve at E Y = EE 1 = offset to junction point E of both curve X = T E 1 = x c o - ord i n a t e of E E E ’ = r e du n da n t c i rcu l a r cur v e T 1 = point of intersection of line (OE`) T to tangent a t t h e C i r. Cur v e a t E ` a n d o r i g i n a l t a n g e n t T B S = E’ T 1 = s h i ft of t h e c i rcu l a r cur v e N = p o i n t i n w h i ch O E ’ cu t s t h e t r a n s iti on cur v e ∅ 1 = spiral angle (EE 2 B) b/w common tangent EE 2 a n d o r i g i n a l t a n g e n t T B R = r a d i us of c i rcu l a r cur v e L = l e n g t h of t r a n s iti on cur v e 28

32. Curves 1 a) E E ’ = R ∅ ; 1 but ∅ = Transition Curves Ch a r a cteri s tics o f a Tr a n s ition cur ve 𝐿 2 𝑅 EE’ = 𝐿 x R = 𝐿 2 𝑅 2 But EN is very nearly equal to EE’ EN = 𝐿 … A 2 That is the shift E`T 1 bisect the transition Curve at N Hence TN = 𝐿 2 b) Draw EG perpendicular to OE’ S = E`T 1 = GT 1 – GE ` =EE1 – GE` S = Y – R(1 – cos ∅ 1 ) Or S = Y – 2 R sin 2 ∅ 1 𝐿 3 But Y = = 2 𝐿 2 6 𝑅 𝐿 6 𝑅 and ∅ = 𝐿 2 𝑅 𝐿 2 6 𝑅 2 S = – 2 R sin ( ∅ 1 ) 2 = 1 𝐿 2 6 𝑅 2 – 2 R ( ∅ 1 ) 2 29

33. Curves Transition Curves Ch a r a cteri s tics o f a Tr a n s ition cur ve Neglecting higher power of ∅ 1 S = 𝐿 2 – R 𝑥 ( 𝐿 2 𝑅 ) 2 S = 6 𝑅 2 𝐿 2 – 𝐿 2 6 𝑅 8 𝑅 𝐿 2 24 𝑅 S = …… (B) 1 Also NT = TN 3 6 𝑅 𝐿 = 𝐿 2 / = 3 2 6 𝑅 𝐿 48 𝑅 ( ) L = S 33 2 1 NT = S 2 i.e Transition curve bisect the shift.

34. Curves Transition Curves Ch a r a cteri s tics o f a Tr a n s ition cur ve Tangent length for Transition Curve: BT = BT 1 +T 1 T BT = (R + S) tan ∆ + 𝐿 2 2 Length of Circular Curve: Length of EE` = 𝜋 R ( ∆ −2∅ 1 ) 180 𝑜 Length of Total Combined Curve: = 180 𝑜 𝜋 R ( ∆ −2∅ 1 ) + 2 L 34

35. Transition Curves 35 Proble m 1 : The fu l l da t a refer t o a c o mposi t e curve. Defl e c t i o n angl e ( ∆ ) = 6 o 30’ M a x spe e d 6 mil e s /hour C e n t r i fugal ra t i o ¼ Max ratio of radial acc. ( 𝛼 ) 1feet/sec Chainage of intersection point at 8565 feet Determine: R ad i us of Ci r cul a r cur v e Leng t h of t ransi t i o n cur v e The chainage of the beginning and end of t r a nsiti o n cur v e an d a t t he j unc t i o ns o f t he t ransiti o n cur v e wi t h t he ci r cul a r cur v e. 1 mile = 5280 feet

36. Transition Curves Solution: Problem 01 1) Radius of Circular curve: V = 60 mph V = 60 𝑥 5280 𝑓𝑡 = 88 𝑓𝑡 60 𝑥 60 𝑠𝑒 𝑐 𝑠𝑒𝑐 C.R = 𝑣 2 = 1 𝑔 𝑅 4 R = 4 𝑣 2 = 4 88 2 = 961.99 𝑓𝑒𝑒𝑡 𝑔 32.2 2) Length of Transition curve: 𝛼 = 𝑣 3 1 = 𝑅 𝐿 (88) 3 961.99 𝐿 L = 798.40 feet 3) Tangent Length: BT = (R + S) tan ∆ + 𝐿 Shift= S = 𝐿 2 24 𝑅 2 2 = 21.74 𝑓𝑒𝑒𝑡 2 BT = (961.99 + 21.74) tan ∆ + 798.40 = 927.89 𝑓𝑒𝑒𝑡 33

37. Transition Curves Solu t ion: Prob l e m 01 4 ) E E ’ = l e ng t h of Cir. C urv e = π 𝑅 (∆ − 2 ∅ ) 180 𝑜 ∅ = 𝐿 2 𝑅 𝜋 𝑥 180 𝑜 = 21 o 5’45.69’’ 37 E E ’ = 3 7 . 3 9 feet ( l eng t h o f C i r . Cur v e ) Chainage of intersection point B = 8 5 65 M i n u s t angen t l eng t h = - 9 2 7 . 11 Chainage of T = 7 6 3 7 . 11 P l us l eng t h of T. C u r v e = + 7 8 . 40 Chainage of E = 8 3 4 5 . 51 Plus leng t h o f Ci r . cur v e = + 3 7 . 39 Chainage of E’ = 8 6 5 2 . 90 P l us l eng t h of T. C u r v e = + 7 8 . 40 Chainage of T’ = 9 3 6 1 . 3 feet

38. Transition Curves 38 Assignment No 2 E x ampl e 1 , 2 , 3 an d 4 page 1 9 3