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Horizontal Curves.Pdf Geospatial Science RMIT ENGINEERING SURVEYING 1 HORIZONTAL CURVES CIRCULAR CURVES, COMPOUND CIRCULAR CURVES, REVERSE CIRCULAR CURVES TRANSITION CURVES AND COMPOUND CURVES R.E.Deakin, August 2005 1. TYPES OF HORIZONTAL CURVES The types of horizontal curves usually encountered in engineering surveying application may be broadly categorised as (i) Circular curves: curves of constant radius joining two intersecting straights. C θ nt ge rc tan a A B chord A' B' s iu d a R r θ O Figure 1.1 In Figure 1.1, a circular curve of constant radius R, centred at O, joins two straights A'A and BB' which intersect at C. A and B are tangent points to the circular arc. OA and OB are radials, which meet the straights at right angles, and the angle at O is equal to the intersection angle at C. Horizontal Curves.doc 1 Geospatial Science RMIT (ii) Compound circular curves: two or more consecutive circular curves of different radii. C θ = θ 12 + θ D A B R A' 1 2 R B' θ1 O 1 R 2 11 RR − 2 R θ2 O2 Figure 1.2 In Figure 1.2, a compound circular curve ADB joins two straights A'A and BB' which intersect at C. A and B are tangent points to circular arcs of radii R1 and R2 respectively. D is a common tangent point. Horizontal Curves.doc 2 Geospatial Science RMIT (iii) Reverse circular curves: two or more consecutive circular curves, of the same or different radii whose centres lie on different sides of a common tangent point. O 2 A' θ A 2 C1 R 2 θ1 1 R 2 D R R 1 θ1 O1 θ B' C2 2 B Figure 1.3 In Figure 1.3, a reverse circular curve ADB joins two straights A'A and BB'. A and B are tangent points to circular arcs of radii R1 and R2 respectively. D is a common tangent point. C1 and C2 are intersection points and the line CC12 is perpendicular to the line between the centres O1 and O2 . (iv) Transition curves: curves with gradually changing radius, often referred to as spirals. A' A Straight Tran sition c urve D C ir cu la r R c u r v e O Figure 1.4 In Figure 1.4, a transition curve AD joins the straight A'A and the circular curve of radius R whose centre is O. The transition curve has an infinite radius at A, decreasing gradually to a radius of R at D. Horizontal Curves.doc 3 Geospatial Science RMIT (v) Combined curves: consisting of consecutive transition and circular curves. Combined curves are used in road and railway surveying. A' A C Straight Tran sition c D urve C ir cu 1 la R r c u r v e E T r O a 1 n s i t O i 2 o R2 n e F v r u c r a l u c ir C e G rv cu n tio si an Tr ht B ig ra St B' Figure 1.5 In Figure 1.5, a combined curve ADEFGB joins the straights A'A and BB' which intersect at C. Horizontal Curves.doc 4 Geospatial Science RMIT 2. GEOMETRY OF CIRCULAR CURVES C θ D M E P α −θ 4 θ −θ − A 4 X 2 B R A' 2α B' θ −θ − 2 2 O Figure 2.1 Figure 2.1 shows a circular curve APMB of radius R, centre O, joining two straights A'A and B'B which intersect at C. The angle of intersection is θ . A and B are tangent points and the radials OA and OB intersect the straights at right angles. M is the mid-point of the circular arc AB and the mid-point of the line DE. DE and the chord AB are parallel and X is the mid-point of the chord AB. The chord AB is perpendicular to the straight line OXMC. In the quadrilateral OACB, the angles A and B both equal 90° and C = 180D −θ , therefore O = θ . OACB is known as a cyclic quadrilateral, (a quadrilateral inscribed within a circle whose opposite angles add to 180°). Due to symmetry AOC== BOC θ /2 and ACO==− OAB 90D θ / 2 . Hence, in the right-angle triangles AXC and BXC, CAB== CBA θ /2. Therefore, the angle between the tangent AC and the chord AB is half the angle subtended at the centre of the circle by the chord AB. This is a general property of chords and tangents to circles. The following formulae may be deduced from Figure 2.1. θ Tangent length AC TR= tan (2.1) 2 Arc length AB A = Rθ (2.2) θ Chord length AB CR= 2sin (2.3) 2 Horizontal Curves.doc 5 Geospatial Science RMIT ⎛θ ⎞ Mid ordinate distance XM MR=−⎜1cos ⎟ (2.4) ⎝⎠2 ⎛θ ⎞ Secant distance MC SR= ⎜sec− 1⎟ (2.5) ⎝⎠2 3. GEOMETRY OF COMPOUND CIRCULAR CURVES C θ = θ 12 + θ T 1 B = C AC = T C t D t 2 1 1 2 C2 t 1 t 2 A a B A' R 2 1 R θ B' 1 b O1 R 2 11 RR − 2 c R θ2 O2 Figure 3.1 In Figure 3.1, a compound circular curve ADB joins two straights A'A and BB' which intersect at C. A and B are tangent points to circular arcs of radii R1 and R2 respectively, whose centres are O1 and O2 . D is a common tangent point and the line CC12 is tangential to both circular curves and perpendicular to the line DO12 O . TAC1 = , TBC2 = are tangent lengths and A1 = arc AD , A2 = arc DB are arc lengths of the circular curves. There are nine elements of a compound circular curve, θ,,θθ12, R1 , R2 , T1 , T2 , A1 and A2 and the following formulae linking these elements may be deduced from Figure 3.1. θ = θθ1+ 2 (3.1) A11= Rθ1 (3.2) A22= R θ2 (3.3) Horizontal Curves.doc 6 Geospatial Science RMIT In the polygon OOACBO21 2 the algebraic sum of the projections of the sides onto any one side must be zero. In Figure 3.1, considering the projections of the sides onto the radius OB2 we may write Ca=−− BO22 O c O 1 b or TRRRR1sinθθθ=− 2( 21 −)cos 21 − cos =−RR22cosθθθ 21 + R cos 21 − R cos =−RR2212()()1 cosθθθ + cos − cos =−RR221()1cosθ +−−−() 1cosθθ () 1cos2 which simplifies to TR11sinθ =−( 1 cosθ) +−−( RR 21)(1 cosθ2) (3.4) Similarly, projecting onto the radius OA1 gives TR22sinθ =−( 1 cosθ) −−−( RR 21)(1 cosθ1) (3.5) Expressions for the tangent distances T1 and T2 can be obtained by considering the tangent distances t1 and t2 θ tR= tan 1 (3.6) 11 2 θ tR= tan 2 (3.7) 22 2 and using the sine rule in triangle CCC12 sinθ CC=−=+ T t() t t 2 11112sinθ giving sinθ Tt=+() tt + 2 (3.8) 11 12sinθ and similarly sinθ Tt=+() tt + 1 (3.9) 22 12sinθ Horizontal Curves.doc 7 Geospatial Science RMIT In some compound curve computations, the equations above are not convenient for solving unknowns. In such circumstances an "equivalent" circle of radius R, which is tangential to all three lines AA', BB' and CC12 may be introduced and equations developed. C θ = θ 12 + θ T 1 B = C AC = T C D M 2 1 C2 A P Q B R 1 A' R B' θ1 R O 1 R 2 θ1 θ2 O θ2 O2 Figure 3.2 In Figure 3.2, the circular curve (dotted) PMQ of radius R, centred at O, is tangential to the two straights AA' and BB' and the line CC12. The tangent points are P, M and Q. Using the formula for tangent length θ θθ PA=−= PC AC Rtan11 − R tan =−() R R tan 1 11 221 12 Similarly θ θθ QB=−= BC QC Rtan22 − R tan =−() R R tan 2 112222 2 Now, since PA= DM and QB = DM then PA= QB hence θ θ DM=−() R Rtan 1 =() R − R tan 2 (3.10) 1222 Re-arranging the equation gives the radius of the equivalent circular curve θ12θ RR12tan+ tan R = 22 (3.11) θθ tan11+ tan 22 Horizontal Curves.doc 8 Geospatial Science RMIT Also ACCPDM= − (3.12) BC= CQ+ DM (3.13) θ where CP== CQ R tan (3.14) 2 D D D Example: Given: θ = 75 , θ1 = 30 , θ2 = 45 , AC = 180.000 m and BC = 215.000 m. Compute: R1 and R2 . Using equations (3.12), (3.13) and (3.14) 180 = CP− DM 215 =+CP DM From which we obtain 23(CP) = 95 thus CP = 197.500 m and DM = 17.500 m. Since CP is now known and θ = 75D , then from (3.14) R = 257.387 m. Since DM is now known, then from (3.10) R1 = 192.076 m and R2 = 299.636 m 4. GEOMETRY OF REVERSE CIRCULAR CURVES O 2 A' θ A 2 C1 R 2 θ θ1 1 R 2 D R R 1 θ1 b c O 1 θ a θ C B' C2 2 B Figure 4.1 In Figure 4.1, a reverse circular curve ADB joins two straights A'A and BB'. A and B are tangent points to circular arcs of radii R1 and R2 respectively. D is a common tangent point. C1 and C2 are intersection points and the line CC12 is perpendicular to the line between the centres ODO12.
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