SURVEYING-II

Horizontal Alignment

Horizontal Alignment Along circular path, vehicle undergoes centripetal acceleration towards center of curvature (lateral acceleration). Balanced by superelevation and weight of vehicle (friction between tire and roadway). Design based on appropriate relationship between design speed and curvature and their relationship with side friction and super elevation.

Vehicle Cornering  Fcp Fcn Fc W Wp Wn Ff 1 ft e Rv

Figure illustrates the forces acting on a vehicle during cornering Figure illustrates the forces acting on a vehicle during cornering. In this figure,  is the angle of inclination, W is the weight of the vehicle in pounds with Wn and Wp being the weight normal and parallel to the roadway surface respectively.

Ff is the side frictional force, Fc is the centrifugal force with Fcp being the centrifugal force acting parallel to the roadway surface, Fcn is the centrifugal force acting normal to the roadway surface, and Rv is the radius defined to the vehicle’s traveled path in ft.

Some basic horizontal curve relationships can be derived by summing forces parallel to the roadway surface. Wp + Ff = Fcp From basic physics this equation can be written as

Where fs is the coefficient of side friction, g is the gravitational constant and V is the vehicle speed (in ft per second). Dividing both the sides of the equation by W cos ;

The term tan is referred to as the super elevation of the curve and is denoted by ‘e’. Super elevation is tilting the roadway to help offset centripetal forces developed as the vehicle goes around a curve. The term ‘fs’ is conservatively set equal to zero for practical applications due to small values that ‘fs’ and ‘’ typically assume.

With e = tan, equation can be rearranged as

In actual design of a horizontal curve, the engineer must select appropriate values of e and fs. Super-elevation value ‘e’ is critical since high rates of super-elevation can cause vehicle steering problems at exits on horizontal curves and in cold climates, ice on road ways can reduce fs and vehicles are forced inwardly off the curve by gravitational forces. Values of ‘e’ and ‘fs’ can be obtained from AASHTO standards.

Horizontal Curve Fundamentals For connecting straight tangent sections of roadway with a curve, several options are available. The most obvious is the simple curve, which is just a standard curve with a single, constant radius. Other options include; compound curve, which consists of two or more simple curves in succession , and spiral curves which are continuously changing radius curves.

Basic Geometry Horizontal Curve Tangent Tangent

Tangent Vs. Horizontal Curve Predicting speeds for tangent and horizontal segments is different May actually be easier to predict speeds on curves than tangents Speeds on curves are restricted to a few well defined variables (e.g. radius, superelevation) Speeds on tangents are not as restricted by design variables (e.g. driver attitude)

Elements of Horizontal Curves    PT M L T PC E PI

Figure shows the basic elements of a simple horizontal curve Figure shows the basic elements of a simple horizontal curve. In this figure R is the radius (measured to center line of the road) PC is the beginning point of horizontal curve T is tangent length PI is tangent intersection  is the central angle of the curve PT is end point of curve M is the middle ordinate E is the external distance L is the length of the curve

Degree of Curve It is the angle subtended by a 100-ft arc along the horizontal curve. Is a measure of the sharpness of curve and is frequently used instead of the radius in the actual construction of horizontal curve. The degree of curve is directly related to the radius of the horizontal curve by

R    PT M L T PC E PI

A geometric and trigonometric analysis of figure, reveals the following relationships

Stopping Sight Distance and Horizontal Curve Design Sight Obstruction s Rv Critical inside lane Highway Centerline Ms SSD

Adequate stopping sight distance must also be provided in the design of horizontal curves. Sight distance restrictions on horizontal curves occur when obstructions are present. Such obstructions are frequently encountered in highway design due to the cost of right of way acquisition and/or cost of moving earthen materials.

When such an obstruction exists, the stopping sight distance is measured along the horizontal curve from the center of the traveled lane. For a specified stopping sight distance, some distance, Ms, must be visually cleared, so that the line of sight is such that sufficient stopping sight distance is available.

Equations for computing SSD relationships for horizontal curves can be derived by first determining the central angle, s, for an arc equal to the required stopping sight distance.

Assuming that the length of the horizontal curve exceeds the required SSD, we have Combining the above equation with following we get;

Rv is the radius to the vehicle’s traveled path, which is also assumed to be the location of the driver’s eye for sight distance, and is again taken as the radius to the middle of the innermost lane, and s is the angle subtended by an arc equal to SSD in length.

By substituting equation for s in equation of middle ordinate, we get the following equation for middle ordinate; Where Ms is the middle ordinate necessary to provide adequate stopping sight distance. Solving further we get;

Max e Controlled by 4 factors: Climate conditions (amount of ice and snow) Terrain (flat, rolling, mountainous) Frequency of slow moving vehicles which influenced by high superelevation rates Highest in common use = 10%, 12% with no ice and snow on low volume gravel-surfaced roads 8% is logical maximum to minimized slipping by stopped vehicles

Radius Calculation (Example) Design radius example: assume a maximum e of 8% and design speed of 60 mph, what is the minimum radius? fmax = 0.12 (from Green Book) Rmin = _____602__________ 15(0.08 + 0.12) Rmin = 1200 feet

Radius Calculation (Example) For emax = 4%? Rmin = _____602_________ 15(0.04 + 0.12) Rmin = 1,500 feet

Sight Distance Example A horizontal curve with R = 800 ft is part of a 2-lane highway with a posted speed limit of 35 mph. What is the minimum distance that a large billboard can be placed from the centerline of the inside lane of the curve without reducing required SSD? Assume p/r =2.5 and a = 11.2 ft/sec2 SSD = 1.47vt + _________v2____ = 246 ft 30( a___  G) 32.2

Horizontal Curve Example Deflection angle of a 4º curve is 55º25’, PC at station 238 + 44.75. Find length of curve,T, and station of PC. D = 4º  = 55º25’ = 55.417º D = _5729.58_ R = _5729.58_ = 1,432.4 ft R 4

Horizontal Curve Example D = 4º  = 55.417º R = 1,432.4 ft L = 2R = 2(1,432.4 ft)(55.417º) = 1385.42ft 360 360

Horizontal Curve Example D = 4º  = 55.417º R = 1,432.4 ft L = 1385.42 ft T = R tan  = 1,432.4 ft tan (55.417) = 752.29 ft 2 2

Stationing Example Stationing goes around horizontal curve. For previous example, what is station of PT? PC = 238 + 44.75 L = 1385.42 ft = 13 + 85.42 Station at PT = (238 + 44.75) + (13 + 85.42) = 252 + 30.17

Suggested Steps on Horizontal Design Select tangents, PIs, and general curves make sure you meet minimum radii Select specific curve radii/spiral and calculate important points (see lab) using formula or table (those needed for design, plans, and lab requirements) Station alignment (as curves are encountered) Determine super and runoff for curves and put in table (see next lecture for def.) Add information to plans

Geometric Design – Horizontal Alignment (1) Horizontal curve Plan view, profile, staking, stationing type of horizontal curves Characteristics of simple circular curve Stopping sight distance on horizontal curves Spiral curve

Plan view and profile plan profile

Surveying and Stationing Staking: route surveyors define the geometry of a highway by “staking” out the horizontal and vertical position of the route and by marking of the cross-section at intervals of 100 ft. Station: Start from an origin by stationing 0, regular stations are established every 100 ft., and numbered 0+00, 12 + 00 (=1200 ft), 20 + 45 (2000 ft + 45) etc.

Horizontal Curve Types

Curve Types Simple curves with spirals Broken Back – two curves same direction (avoid) Compound curves: multiple curves connected directly together (use with caution) go from large radii to smaller radii and have R(large) < 1.5 R(small) Reverse curves – two curves, opposite direction (require separation typically for superelevation attainment)

Straight road sections 1. Simple Curve Circular arc R   Straight road sections

Straight road sections 2. Compound Curve Circular arcs R1 R2 Straight road sections

Straight road sections 3. Broken Back Curve Circular arc Straight road sections

Straight road sections 4. Reverse Curve Circular arcs Straight road sections

5. Spiral R = Rn R =  Straight road section

Angle measurement (a) degree (b) Radian

As the subtended arc is proportional to the radius of the circle, then the radian measure of the angle. Is the ratio of the length of the subtended arc to the radius of the circle

Define horizontal Curve: Circular Horizontal Curve Definitions Radius, usually measured to the centerline of the road, in ft. = Central angle of the curve in degrees PC = point of curve (the beginning point of the horizontal curve) PI = point of tangent intersection PT = Point of tangent (the ending point of the horizontal curve) T = tangent length in ft. M = middle ordinate from middle point of cord to middle point of curve in ft. E = External distance in ft. L = length of curve D = Degree of curvature (the angle subtended by a 100-ft arc* along the horizontal curve) C = chord length from PC to PT

Key measures of the curve Note converts from radians to degrees

Example: A horizontal curve is designed with a 2000-ft radius, the curve has a tangent length of 400 ft and the PI is at station 103 + 00, determine the stationing of the PT. Solution:

Spiral Curve: Spiral curves are curves with a continuously changing radii, they are sometimes used on high-speed roadways with sharp horizontal curves and are sometimes used to gradually introduce the super elevation of an upcoming horizontal curve

Length of Spiral AASHTO also provides recommended spiral length based on driver behavior rather than a specific equation. Superelevation runoff length is set equal to the spiral curve length when spirals are used. Design Note: For construction purposes, round your designs to a reasonable values; e.g. round Ls = 147 feet, to use Ls = 150 feet.

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