Design of Highway Horizontal Alignment Chapter 16 Dr. TALEB M. AL-ROUSAN.

Design of Highway Horizontal Alignment

Chapter 16

Dr. TALEB M. AL-ROUSAN

Horizontal Alignment Consists of straight sections of the

road (tangents) connected by horizontal curves.

Curves are segments of circles with radii to provide smooth flow of traffic along the curve.

Min radius of horizontal curve depends on design speed, super elevation, and side friction.

R = u2/ (15 (e + f))

Horizontal Alignment Cont. Objects located near the inside edge

of the road may interfere with the view of the driver, which result in reducing the driver sight distance.

It is necessary to design a horizontal curve such that the available sight distance is at least equal to the safe stopping sight distance.

See Figure 16.18 for range of lower values for SSD on horizontal curves.

Horizontal Alignment Cont.Arc : (S/2 R) = (2 θ /360)

S = (2R θ ) / 180 R= radius of horizontal curve S= sight distance = length of arc = distance from the driver to

the object. 2θ = angle subtended at the center of the circle by arc (S)

θ = (28.65)S/R

(R –m)/ R = cos θ = cos ((28.65)S/R)

m = R[1 – cos((28.65)S/R)]

See Figure 16.19 for horizontal curve layout.

Horizontal Curves The design entails the determination of:

Min. radius Length of curve Computation of horizontal offsets from the

tangents to the curve (setting out). Types of horizontal curves:

Simple Compound Reversed Spiral (transition)

Simple Horizontal Curves See Figure 16.19 for simple horizontal

curve layout. The curve is a segment of a circle with radius

R. PC: point of curve (point at which curve begins). PT: point of tangent (point at which curve ends). PI: point of intersection (point at which tangents

intersect) known also as vertex (v). The simple circular curve is described by its:

Radius (e.g. 200-ft radius). Degree of curve: has two definitions (the Arc &

the Chord).

•From Triangle (PI-PC-O) : tan ( /2) =T / R From triangle (PC-O-B) (C/2)/R = sin ( /2)

Simple Horizontal Curves/ Degree of Curve

The Arc: defines the curve in terms of the angle subtended at the center by a circular arc 100 ft in length.

See Figure 16. 20 (a). It means that for a (2o)curve, for example, an arc of 100 ft will

be subtended by an angle of (2o)at the center. If (θ) is the angle in radian subtended at the center by an arc of

the circle, the length of the arc [L = R θ]

If (Dao) is the angle in degrees subtended at the center by

an arc of length L, then θ = ( Da

o) / 180 … (rad)

L = [R ( Dao) / 180] = 100

R = (180 * 100) / ( Dao)

R = 5729.6/ (Dao)

Note that the radius of the curve can be determined if the degree of curve is known.

Simple Horizontal Curves/ Degree of Curve

The chord: defines the curve in terms of the angle subtended at the center by a chord of 100 ft in length.

See Figure 16. 20 (b).

R = 50/ sin(Dao/2)

The arc definition is commonly used for highway work.

The chord definition is commonly used for railway work.

Formulas of Simple Circular Curves Referring to Fig 16.19 and using the

properties of the circle, the two tangent lengths AV and BV are equal = T.

The angle () formed by the two tangents is known as the deflection angle.

Tangent length= T = R tan ( /2) Long Chord = C = 2 R Sin ( /2) External Distance = E = R [(1/cos ( /2) ) -1] Middle ordinate = M = R [1- (cos ( /2))] Length of curve= L = [(R o)/ 180]

Setting Out Simple Horizontal Curve

Usually set out in the field by staking out points on the curve using deflection angles measured from the tangent at (PC) and the length of the chord joining consecutive whole stations.

See Figure 16.21. the first deflection angle (VAp = (1)/2) determined for the

first whole station on the curve, which is usually less than a station a way from the PC.

Next deflection angle VAq = ((1)/2) + (D/2)

The next angle : VAv =((1)/2) + (D/2) + (D/2)= =((1)/2) + D

The next angle : VAs =((1)/2) + 3D/2

The last angle VAB =((1)/2) + 3D/2 +((2)/2) = /2

Setting Out Simple Horizontal Curve Cont.

To find (1 & 2) use the relation: (L1/ 1) = (L/ )= (L2/2)

Where: L1 = length of first arc = [( R )/ 180] R = [(L1*180)/ ( R = [(180* L)/ ( Which results in: (L1/ 1) = (L/ )= (L2/2)

In setting out a simple horizontal curve in the field:1. Locate PC & PT2. Compute deflection angles from PC to each whole station.3. Compute and measure chord distance from preceding station

Note that L1 & L2 are measured along the curve, and the corresponding chord lengths should be calculated, particularly when curves are sharp

Setting Out Simple Horizontal Curve Cont.

Note that L1 & L2 are measured along the curve, and the corresponding chord lengths should be calculated, particularly when curves are sharp.

Chord lengths can be calculated by: First chord = C1 = 2 R sin (1/2) Intermediate chords = C = 2 R sin (D/2) last chord =C2 = 2 R sin (2/2) See Example 16.7 for Design of Horizontal

Curve

Compound Curves Consist of two or more curves in

succession, turning in the same direction, with any two successive curves having a common tangent point.

See Figure 16.23 for typical layout of compound curve.

These curves are used mainly in obtaining desirable shapes of the horizontal alignment in difficult topography conditions.

radii of any two consecutive simple curves forming the compound curve should not be widely different to avoid abrupt changes in alignment.

Compound Curves AASHTO recommends that ratio of flatter

radius to sharper radius should not be greater than 2:1. this is needed to adjust for sudden changes in curvature and speed.

To provide smooth transition from flat to sharp curve, and to facilitate a reasonable deceleration rate, the length of each curve should not be too short.

See Table 16.10 for min. lengths recommended by AASHTO.

Compound Curves Cont. See Figure 16.23 Several solutions can be developed for the

compound curve, but the vertex triangle method is presented here.

= 1 + 2t1 = R1 tan (/2)t2 = R2 tan (/2)

(VG/ sin (2)) = (VH/ sin (1))= (t1 + t2)/ sin (180 – ) = (t1 + t2)/ sin ()

T1 = VG + t1T2 = VH + t2

Compound Curves Cont.

To lay out the curve, find 1 & 2 from plans.

Find required parameters (1, 2, t1, t2. VG, VH, T1, T2).

Deflection angles can then be determined for each simple curve in turn.

See Example 16.8

Reverse Curves Usually consist of two simple curves with equal radii

turning in opposite directions with a common tangent. Used to change the alignment of the highway. See Figure 16.25 for reverse curve with parallel

tangents. Seldom recommended because sudden changes to

the alignment may result in difficulties for drivers to keep their lanes.

When it is necessary to reverse alignment, it is preferable to design two simple curves separated by sufficient length of tangent between them to achieve super elevation. Or

The simple curves may be separated by an equivalent length of spiral.

Reverse Curves Cont. If D and d are known, it is necessary to

determine 1 & 2 to set out the curve. = 1 = 2

angle OWX = 1/2 = 2/2angle OYZ = 1/2 = 2/2

tan ( /2)= d/D d= (R – R cos R – R cos )

= 2R (1 – cos ( ))R = d / 2(1 – cos ( ))

If d and R are known, thenCos = 1- (d/2R)

D = d cot (

Transition (Spiral) Curves Transition curves are placed between

tangents and circular curves or between two adjacent circular curves having different radii.

They provide a vehicle path that gradually increase or decrease the radial force as vehicle inter or leave the circular curve.

Degree of transition = 0o at tangent end, to degree of circular curve at the curve end.

When placed between two circular curves, the Degree of transition curve = varies from that of first curve to that of the second circular curve.

Transition (Spiral) Curves Cont. Min. length of transition curve is given by:

L = (3.15 u3)/ RC L = min length of curve (ft) U = speed (mi/h) R = radius of curve (ft) C= rate of increase of radial acceleration (ft/sec2/sec)

(1 – 3) C: is an empirical factor that indicate the level of

comfort and safety involved (usually used values in highway engineering vary from 1 to 3.

See Table 16.12 for AASHTO recommended values for length of spiral curves.

Superelevation Runoff Superelevation runoff: the length of highway required

to achieve a full superelevated section from a section with adverse crown removed, or vice versa.

Its length depends on: Design speed Rate of superelevation Pavement widthAASHTO recommends that when spiral curves are used

in transition design, the superelevation runoff should be achieved over the length of the spiral curve.

It is recommended : length of spiral curve = length of superelevation runoff.

Table 16.13 shows recommended lengths for superelevation runoff.

Attainment of Superelevation Its is essential that when changing from a crowned

cross section to a superelevated one be achieved without causing any discomfort or creating unsafe conditions.

To achieve this change the following methods can be used on undivided highways: A crowned pavement is rotated about the profile of

the center line. A crowned pavement is rotated about the profile of

the inside edge. A crowned pavement is rotated about the profile of

the outside edge. A straight cross-slope pavement is rotated about the

profile of the outside edge. ٍٍLL LLLSee Figure 16.26.

Most commonly used method (less distortion than other methods).

Superelevation on Divided Highways

Superelevation is achieved on divided highways using three methods:

1. Superelevating the whole cross section ,including the median. mostly rotated about center line Used with narrow medians Moderate superelevations

2. rotating each pavement seperately around the median edges, while keeping the median horizontal. Used for medians with 30 ft or less Can be used for any median, because by keeping

the median in horizontal plane, the difference in elevation between the extreme pavement edges doesn’t exceed the pavement super elevation.

Superelevation on Divided Highways

3. The two pavements are treated separately, resulting in variable elevation differences between the median edges.

used in pavements with medians width of 40 ft or greater.