Design Guidelines for Horizontal Sightline Offsets (2019)

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59 A P P E N D I X A Computation of HSOs This appendix addresses the computation of horizontal sightline offsets. A.1 Presentation of HSOs in the AASHTO Green Book The Green Book (AASHTO 2011) presents the fundamental equation for the distance from the driver’s eye to a roadside object that should be provided so that the roadside object does not become a horizontal sight obstruction. This distance is known to surveyors as the middle ordinate of a horizontal curve, typically designated as m or mmax. This distance was designated as m in the Green Book through 2001, but since the 2004 edition of the Green Book it has been designated as the HSO. The HSO is illustrated in Figure A-1 (based on Green Book Figure 3-23) and its value is determined in Green Book Equation (3-36) as: (A-1) where HSO = horizontal sightline offset (ft); S = design stopping sight distance (ft); and R = radius of horizontal curve measured along the centerline of the inside lane (ft). The Green Book states that Equation (A-1) is exact only when both the vehicle and the object are located within the limits of a simple horizontal curve. The Green Book states that when the vehicle or the sight obstruction is situated beyond the limits of the simple curve (or if both are within the limits of a spiral curve or compound curve), the value obtained from Equation (A-1) is only approximate. In fact, the value provided by Equation (A-1) represents the maximum value of horizontal sightline offset that can occur at any point along a simple curve, and this maximum value applies only in the middle portion of curves whose length exceeds the design stopping sight distance (L > S). The HSO will be smaller than HSO in Equation (A-1) toward the ends of a simple curve. Furthermore, the Green Book does not state, but should, that where the length of a horizontal curve is less than the design stopping sight distance (L < S), the HSO that is needed will be less than the maximum value indicated by HSO in Equation (A-1) throughout the entire curve. The Green Book suggests two approaches to estimating actual HSO values less than the maximum indicated in Equation (A-1). These are use of graphical procedures and computational methods. Both of these approaches are discussed here.

60 Design Guidelines for Horizontal Sightline Offsets Figure A-1. Diagram illustrating components for determining horizontal sight distance (AASHTO 2011) Determination of the actual horizontal sightline offset needed at a specific point on a horizontal curve, rather than the maximum needed anywhere within a curve, is potentially valuable to highway agencies because such knowledge can avoid unnecessary construction or right-of-way costs that might be incurred in providing the maximum HSO along the entire curve. In some cases, such as in rock cuts or along retaining walls, determination of the actual HSO needed could substantially reduce costs and might indicate situations where no design exception is needed because the planned or existing design already provides the design stopping sight distance along the entire curve. A computational method would be preferable to an approximate graphical method, because a computational method could be incorporated in CADD systems so that the actual area on the inside of a horizontal curve that should be free of sight obstructions can be displayed to the designer. A.2 Early Studies Raymond (1972) used an unspecified computational method for determining offset ratios for specified points along a horizontal curve. The offset ratio (OR) is a dimensionless quantity defined as: (A-2) Thus, the offset ratio is the horizontal sightline offset actually needed at any given point on a horizontal curve divided by the maximum horizontal sightline offset determined with Equation (A-1). Figure A-2 presents the chart developed by Raymond. In this chart, the horizontal and vertical axes and the multiple curves are labeled with three ratios: • The distance from the spiral-to-curve (SC) point or the PC to the sight obstruction measured along the curve divided by the design stopping sight distance; • The length of the spiral curve divided by the design sight distance, or Ls/S; and

Computation of HSOs 61 • The offset to the horizontal sight obstruction that is needed to provide the design stopping sight distance divided by the maximum offset determined with Equation (A-2), or m/M. This chart has been reviewed by Mauga (2015), who found it to be suitable for sites with spiraled horizontal curves and for simple curves whose length exceeds the design stopping sight distance (L S). Raymond’s work does not address curves whose length is shorter than the design stopping sight distance (L < S). Raymond’s work also developed a correction factor (CF) that should be added to the computed value of OR computed with Equation (A-2) for small values of the ratio of curve radius to stopping sight distance, R/S. The correction factor is determined as shown in Figure A-3. The correction factor is determined with two ratios: the Z/S ratio used in Figure A-2 and the curve radius divided by the design stopping sight distance, R/S. Figure A-2. Offset ratios for determining HSO (Raymond 1972) Figure A-3. Correction factor used in determining HSO (Raymond 1972)

62 Design Guidelines for Horizontal Sightline Offsets Glennon (1987) used a graphical method to develop offset ratios similar to those developed by Raymond. The review by Mauga (2015b) indicates that Glennon’s results are also suitable only for long curves (L S). Other studies of the mathematics of horizontal sightline offsets have been performed by Waissi and Cleveland (1987), and Easa (1991). A.3 Graphical Method Mauga (2015b) indicates that a graphical method for estimating minimum offsets for horizontal sightlines can be performed drawing chords to the horizontal curve from a series of points on the road representing the driver’s eye to a corresponding point at a distance equal to design stopping sight distance (S) downstream of the driver’s eye, which represents the sightline along which the driver needs to be able to see. A curve is then drawn so that the curve is tangent to all of the chords. This curve defines the “clearance envelope,” or the area on the inside of the horizontal curve that should be clear of sight obstructions. Figure A-4 illustrates this graphical method. Figure A-4. Graphical method for determining minimum HSOs (Mauga 2015) A.4 Computational Method for Horizontal Sightline Offsets Developed by Olson et al. Olson et al. (1984) developed mathematical derivations of the horizontal sightline offsets on the inside of horizontal curves for any placement of the observer (i.e., the driver), a roadside obstacle which serves as a sight obstruction, and the object in the roadway to be seen. The observer and the object to be seen are, as defined in the AASHTO Green Book (2011), always along the centerline of the inside lane of the roadway. The roadside obstacle is on the inside of the horizontal curve at a specified distance, m, from the center of the inside lane, where m is often less than the value of HSO in Equation (A-1). Olson et al. considered both the long curve (L S) and short curve (L < S) cases. Mauga (2014, 2015b) found the derivation by Olson et al. to be only approximately correct. Furthermore, Olson et al. did not present a closed-form method for computing the value of m actually needed at any point on a curve. Therefore, Mauga’s method for computing horizontal sightline offsets is preferred to Olson’s method.

Computation of HSOs 63 A.5 Computational Method for HSOs Developed by Mauga Two related papers by Mauga (2014, 2015b) present an alternative computational method for determining HSO for horizontal curves. Six cases are addressed: • Case 1(a) – Long curve (L > S), driver on the approach tangent and object on the curve; • Case 1(b) – Long curve (L > S), both driver and object on the curve; • Case 1(c) – Long curve (L > S), driver on the curve tangent and object on the departing tangent; • Case 2(a) – Short curve (L < S), driver on the approach tangent and object on the curve; • Case 2(b) – Short curve (L < S), driver on the approach tangent and object on the departing tangent; and • Case 2(c) – Short curve (L < S), driver on the curve tangent and object on the departing tangent. These cases together define the area that should be clear of sight obstructions, which Mauga refers to as the clearance envelope, like that shown in Figure A-4. The area that should be clear of sight obstructions begins at a distance equivalent to stopping sight distance in advance of the PC of the curve (i.e., PC – S) and ends at a distance equivalent to stopping sight distance beyond the PT of the curve (PT + S). Mauga designates the horizontal sightline offset needed at any location between PC – S and PT + S as m. Mauga’s computational method treats the plan view of a roadway (see Figure A-5) as an x-y coordinate system in which the x-axis lies along the approach tangent to the horizontal curve (with positive values in the direction from the PC to the PT) and the y-axis is perpendicular to the x-axis (with positive values toward the inside of the curve). Table A-1 defines the notation based on Mauga (2015b).

64 Design Guidelines for Horizontal Sightline Offsets C1…C6 = constant values (defined by equations in text) Figure A-5. Plan view showing approach tangent, horizontal curves, and clearance envelope showing definition of x-y coordinate system (Mauga 2015) Table A-1. Summary of notation used in the Mauga Derivation of HSO equations for horizontal curves (adapted from Mauga 2015b) R = radius of the curve (ft) L = length of the curve (ft) from PC to PT ∆ = central angle or deflection angle of horizontal curve (degrees) S = design stopping sight distance (ft) X = station corresponding to driver position at any point along the curve or its tangents m = horizontal sightline offset to the area that needs of be clear of sight obstructions at any point along the curve represented by Station X + S xd, yd = coordinates of the driver’s position on the horizontal curve or its tangents (see accompanying text for explanation of this coordinate system) x, y = coordinates of the point on the edge of the clearance envelope when the drive’s position is at xd, yd (see accompanying text for explanation of this coordinate system) = angle between the horizontal sightline and the approach tangent (degrees) TH = length of the sightline from the driver to the point at which the sightline touches the clearance envelope tangentially (ft) TM = maximum length of TH (ft); occurs at the point where the horizontal sightline offset is maximum dchord = length of the chord of the circular roadway curve from the PC to the position of driver’s eye when the driver is on the curve (ft) = angle between the curve radius at the PC of the horizontal curve and the line from the center of curvature to object location when the object is on the departing tangent (degrees) (see Figure A-6)

Computation of HSOs 65 A.5.1 Case 1(a) – Long Curve (L > S), Driver on the Approach Tangent and Object on the Curve Mauga develops equations to determine the horizontal sightline offset for a driver at any station along the road, X. For Case 1(a), the driver location is in the range PC – S X PC. The object location (which, by definition, is at X + S) is, therefore, in the range PC X + S PC + S. The horizontal sightline offset is determined through the following sequence of equations: (A-3) (A-4) (A-5) (A-6) (A-7) (A-8) (A-9) A.5.2 Case 1(b) – Long Curve (L > S), Driver and Object Both on the Curve For Case 1(b), the driver location is in the range PC X PT - S. The object location (which, by definition, is at X + S) is, therefore, in the range PC + S X + S PT. The horizontal sightline offset is determined through the following equation which is equivalent to Equation (A-1): (A-10) A.5.3 Case 1(c) – Long Curve (L > S), Driver on the Curve and Object on the Departing Tangent For Case 1(c), the driver location is in the range PT - S X PT. The object location (which, by definition, is at X + S) is, therefore, in the range PT X + S PT + S. The horizontal sightline offset is determined through the following sequence of equations: (A-11) (A-12) (A-13) (A-14) (A-15)

66 Design Guidelines for Horizontal Sightline Offsets (A-16) (A-17) (A-18) (A-19) (A-20) (A-21) (A-22) (A-23) (A-24) Figure A-6 illustrates the definitions of the angles Δ and γ for this case and for Case 2(c). Figure A-6. Plan View Showing Angles Considered for an Object on the Departing Tangent Beyond the PT of a Horizontal Curve (Mauga 2015)

Computation of HSOs 67 A.5.4 Case 2(a) – Short Curve (L < S), Driver on the Approach Tangent and Object on the Curve For Case 2(a), the driver location is in the range PC – S X PC – S + L. The object location (which, by definition, is at X + S) is, therefore, in the range PC X + S PT. The horizontal sightline offset is determined through the following sequence of equations: (A-25) (A-26) (A-27) (A-28) (A-29) (A-30) (A-31) A.5.5 Case 2(b) – Short Curve (L < S), Driver on the Approach Tangent and Object on the Departing Tangent For Case 2(b), the driver location is in the range PC – S + L X PC. The object location (which, by definition, is at X + S) is, therefore, in the range PT X + S PT + S - L. The horizontal sightline offset is determined through the following equation: (A-32) (A-33) (A-34) (A-35) (A-36) (A-37) (A-38) (A-39) (A-40)

68 Design Guidelines for Horizontal Sightline Offsets (A-41) (A-42) (A-43) (A-44) A.5.6 Case 2(c) – Short Curve (L < S), Driver on the Curve and Object on the Departing Tangent For Case 2(c), the driver location is in the range PC X PT. The object location (which, by definition, is at X + S) is, therefore, in the range PT + S - L X + S PT + S. The horizontal sightline offset is determined through the following sequence of equations: (A-45) (A-46) (A-47) (A-48) (A-49) (A-50) (A-51) (A-52) (A-53) (A-54) (A-55) (A-56) (A-57)

Computation of HSOs 69 A.6 Comparison of Methods Figure A-7 shows a comparison by Mauga (2014) of his analytical model with the graphical procedure and the approaches used by Raymond (1972), Glennon (1987), and Olson et al. (1984). The comparison shows good agreement in clearance offsets (i.e., horizontal sightline offsets) except with the Olson et al. model for offsets less than 15 ft. There is a minor difference with the Glennon model for clearance offsets less than 5 ft, but that is of no practical importance since it would fall within the inside lane (see below). Figure A-7. Data plots comparing Mauga’s analytical model to the graphical procedure and other computational models (Mauga 2014) (1972) (1987) (1984)

70 Design Guidelines for Horizontal Sightline Offsets A.7 Computation of Roadside Clear Areas Since sight distance is measured from the centerline of the inside lane and the travel lanes and shoulder are, by definition, clear of sight obstructions (with the possible exception of stopped vehicles on the shoulder), the value of horizontal sightline offset, m, is only of practical importance to design where: (A-58) where m = horizontal sightline offset on the inside of the curve (ft); LW = lane width for the inside lane on the curve (ft); and SW = shoulder width (ft). The width of the roadside area (outside of the shoulder) that should be clear of sight obstructions at any location along the curve is: (A-59) where mroadside is portion of horizontal sightline offset that is on the roadside area outside of the shoulder (ft). If mroadside is less than or equal to zero, there is no portion of the roadside that needs to be clear of sight obstructions. A.8 CADD Implementation The output of the Mauga (2014, 2015b) method is the horizontal sightline offset (m) needed to provide design stopping sight distance at any point near a horizontal curve from Station PC-S to Station PT+S. The Mauga method may be too complex for manual application, but is suitable for implementation of CADD systems. A limitation of the Mauga method is that it is applicable only in two dimensions (i.e., to curves in a horizontal plane). The Mauga method should generally be very accurate for horizontal curves with grades that do not exceed three percent.