8_Lesson-4.3-Horizontal-Alignment.pdf

Appearance standards vary from agency to agency. Current California standards for instance, require a minimum vertical curve length of 60 m where grade breaks are less than 2% or design speeds are less than 60 km/h, the minimum vertical curve length is given by L = 2V, where L in the vertical curve...

Appearance standards vary from agency to agency. Current California standards for instance, require a minimum vertical curve length of 60 m where grade breaks are less than 2% or design speeds are less than 60 km/h, the minimum vertical curve length is given by L = 2V, where L in the vertical curve length in metres and V is the design speed in Km/h. Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University Determine the minimum length of a crest vertical curve between a +0.5% grade and a -1.0% grade for a road with a 100- km/h design speed. The vertical curve must provide 190-m stopping sight distance and meet the California appearance criteria. Round up to the nearest 20 m interval. Given: 𝑔1 = +0.5% 𝑔2 = -1.0% S = 190 California appearance criteria: 𝐹𝐴𝐿𝑆𝐸 1. Design speed less than 60km/h Required: minimum length (𝐿𝑚𝑖𝑛 ) 100 km/h > 60 km/h Therefore, check second criteria. Solution: Assume S ≤ L 2. Grade breaks less than 2% 𝑇𝑅𝑈𝐸 𝐴𝑆 2 (0.5 − (−1.0))(1902 ) 𝐿= = = 134.0 𝑚 Grade break: 0.5% + 1% = 1.5% 200( ℎ1 + ℎ2 )2 200( 1.070 + 0.150)2 1.5% < 2% Therefore, L = 60m 134.0 m < 190 m, S > L assumption is wrong (use other equation) 2 200 ℎ1 + ℎ2 200 1.070+ 0.150 2 𝐿 = 2𝑆 − = 2(190) − = 380 − 269.5 = 110.5 𝑚 (0.5− −1.0 ) 𝐴 Therefore, use 120 m vertical curve.  Sight Distances Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University CE4 - HIGHWAY AND RAILROAD ENGINEERING  Horizontal Alignment Horizontal tangents are described in terms of their lengths (as expressed in the stationing of the job) and their directions. Directions may be either expressed as bearings or as azimuths and are always defined in the direction of increasing station. Azimuths are expressed as angles turned clockwise from due north; bearings are expressed as angles turned either clockwise or counterclockwise from either north or south. For instance, the azimuth 280is equivalent to the bearing north 80west (or N80W). Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 2  Circular Curves Horizontal curves are normally circular. Horizontal curves are described by radius (R), central angle (Δ) (which is equal to the deflection angle between the tangents), length (L), semi tangent distance (T), middle ordinate (M), external distance (E), and chord (C). The curve begins at the tangent-to-curve point (TC) and ends at the curve-to-tangent point (CT). Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 4  Circular Curves In the past, severity of curvature was sometimes expressed in degree of curvature. Although obsolete in the metric system, degree of curvature may still be encountered in some situations. Degree of curvature may be defined in two ways. The arc definition is the angle subtended by a 100 ft arc. The chord definition is the angle subtended by a 100 ft chord. Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 6 POINTS PC – Point of Curve PI/V – Point of Intersection / Vertex PT – Point of Tangent ANGLE ∆ – Angle of Intersection LINES R – Radius T – Tangent Distance C – Long Chord M – Middle Ordinate E – External Distance  Circular Curves 36000 5729.58 ∆ ∆ 𝐷= = 𝑇 = 𝑅𝑡𝑎𝑛 𝑀 = 𝑅 − 𝑅𝑐𝑜𝑠( ) 2𝜋𝑅 𝑅 2 2 R = radius of curvature (feet) 𝑅 ∆ 𝐸= −𝑅 𝐶 = 2𝑅 sin( ) ∆ 2 2𝜋𝑅∆ cos 2 𝐿= = 𝑅∆𝑟𝑎𝑑 360° ∆ = central angle of the curve ∆𝑟𝑎𝑑 = measured in radians Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 8  ∆ 𝑇 = 𝑅𝑡𝑎𝑛 2 ∆ 𝐶 = 2𝑅 sin( ) 2 ∆ 𝑀 = 𝑅 − 𝑅𝑐𝑜𝑠( ) 2 𝑅 𝐸= −𝑅 ∆ cos 2  Horizontal Alignment Circular curves are usually laid out in the field by occupying the tangent-to-curve point TC with a transit and then establishing successive points by turning deflection angles and measuring chords, as shown in the figure below. Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 10  Horizontal Alignment The deflection angle in radians 𝑑𝑥 to a point on the curve at a distance x from the TC is given by 𝑥 𝑑𝑥 = ( )𝑟𝑎𝑑 2𝑅 The chord 𝑐𝑥 to this point is given by 𝑐𝑥 = 2𝑅 sin 𝑑𝑥 Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 11  Horizontal Alignment Table 4.3 gives deflection angles and chords at 20 m intervals for a 500 m radius curve with a deflection angle of 15° and a TC at station 17+25. Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 12  Horizontal Alignment Design standards for horizontal curves establish their minimum radii and, in some cases, their minimum lengths. Minimum radius of horizontal curve is most commonly established by the relationship between design speed, maximum rate of superelevation, and curve radius. In other cases, minimum radii or curve lengths for highways may be established by the need to provide stopping sight distance or by appearance standards. Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 13  Horizontal Alignment Figure 4.15 illustrates the relationship between curve radius, stopping sight distance, and the setback distance to obstructions to vision. Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 14  Horizontal Alignment The relationship between the radius of curvature R, the setback distance m, and the sight distance s is given by 28.65𝑠 𝑅 −1 𝑅−𝑚 𝑚 = 𝑅[1 − cos( )] 𝑠= [𝑐𝑜𝑠 ( )] 𝑅 28.65 𝑅 where the angles in the formula are measured in degrees. Since these formula are hard to solve for R, design charts or tables are normally used to find the minimum radius of curvature that will provide stopping sight distance. Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 15  Superelevation Curved section roadways are usually super elevated. Provisions for gradual change from one to the other have to be considered. The centerline of each individual roadway at profile grade is maintained while raising the outer edge and lowering the inner edge to attain the desired elevation. A vehicle travelling on a horizontal curve exerts an outward force called centrifugal force. To resist this force and maintain the desired design speed, highway curves need to be superelevated. Superelevation may be defined as the rotation of the roadway cross section in such a manner as to overcome the centrifugal force that acts on a motor vehicle traversing a curve. Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 16  Superelevation On the superelevated highway, the centrifugal force can be resisted by: 1. The weight component of the vehicle parallel to the superelevated surface 2. The side friction between the tires and the pavement 3. Introduction of transition curves 4. Pavement widening. It is impossible to balance centrifugal force by superelevation alone, because for any given curve radius, a certain superelevation rate is exactly correct for only one operating speed around the curve. At all other speeds, there will be a side thrust outward or inward relative to the curve centre which must be offset by side friction Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 17  𝑤𝑣 2 𝑣2 𝐶𝐹 = 𝐶𝑅 = 𝑔𝑟 𝑔𝑟 𝑤 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑣𝑒ℎ𝑖𝑐𝑙𝑒 𝑉 = 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑣𝑒ℎ𝑖𝑐𝑙𝑒 𝑘𝑚/ℎ m CF 𝑣 = 𝑠𝑝𝑒𝑒𝑑 𝑜𝑓 𝑣𝑒ℎ𝑖𝑐𝑙𝑒 𝑎𝑡 𝑐𝑢𝑟𝑣𝑒 in s = 0.25V 𝑅 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑐𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒 μ = 𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑟𝑜𝑎𝑑 A 𝑎𝑛𝑑 𝑡𝑖𝑟𝑒𝑠 𝐶𝐹 = 𝑐𝑒𝑛𝑡𝑟𝑖𝑓𝑢𝑔𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝐶𝑅 = 𝑐𝑒𝑛𝑡𝑟𝑖𝑓𝑢𝑔𝑎𝑙 𝑟𝑎𝑡𝑖𝑜 e = superelevation of road, raising of outer edge at a rate of 1 horizontal to e vertical 𝑒 tan α = 1 e 𝑣2 𝑒+μ= α 𝑔𝑟  Widening Curve The provision for a wider roadway is necessary on sharp curve for two lane pavement under the following reasons: 1. To force the drivers to shy away from the pavement edge. 2. To increase the effective transverse vehicle width for non-tracking of front and rear wheels. 3. To give additional width due to the slanted position of the front wheel to the roadway centerline. 4. For a 7.20 meter wide roadway, an additional width of 0.30 is necessary on an open curve highway. Engr. Reina Mae Chong | Highway and Railroad Engineering| Catanduanes State University | 20