AASHTO’s guideline for geometric design of highways and similar guidelines require that roadside areas on the inside of horizontal curves be cleared of high objects to provide stopping sight distance. The guidelines have analytical models for determining the extent of clearance, known as the horizontal sightline offset or clearance offset, for simple curves. Researchers in the past have developed analytical models for clearance offsets for spiraled and reverse curves. Very few researchers developed analytical models for available sight distances for compound curves. Still missing are models for horizontal sightline offsets and locations of the offsets for compound curves. The objective of this paper is to present development of analytical models and charts for determining horizontal sightline offsets and their locations for compound curves. The paper considers curves whose component arcs are individually shorter than stopping sight distance. The resulting models and the charts have been verified with accurate values determined using graphical methods. The models and the charts will find application in geometric design of highway compound curves.
Guidelines for geometric design have analytical models for determination of clearance offsets on the inside of simple or circular horizontal curves. For example, the model for the offsets in the guideline by the American Association of State Highways and Transportation Officials (AASHTO) [
M = R [ 1 − cos ( 28.65 × S R ) ] (1)
where
M is the clearance offset at the middle of a simple curve,
R is the radius of driver’s path in the sharpest lane,
S is the design sight distance, here considered as stopping sight distance.
The model in Equation (1) was derived with an assumption that both a driver and a dangerous object are on curved part of a simple horizontal curve, as stated in the guideline. Therefore, the equation is best suited for curves that are longer than design sight distance. However, there is sufficient literature that has addressed clearance offsets for short and also for long simple curves [
Literature search for studies on horizontal offsets for compound curves found only a few studies, the earliest being that pioneered by Easa [
Hassan et al. [
Two papers, one by Liu and Wang [
The author of this paper is aware that the AASHTO’s guideline recommends use of graphical methods in determining offsets for any type of horizontal curve when there is a technical reason for not applying Equation (1). The graphical method produces accurate offsets for all and compound curves. However, the method is tedious and time consuming. Tedium and time consumption are due to lack of mathematical representation of the method. Lack of mathematical representation makes it even mismatch universities’ learning outcomes that require development of abilities to apply mathematics in formulating and solving complex engineering problems.
The objective of this paper is to present development of analytical models that will act as a rational way of determining clearance offsets for new compound curves as required by codes prior to approval of designs. The models use design sight distances and curve geometry to yield maximum offsets and locations of the offsets. The resulting offsets may therefore be applied as criteria of controlling location of roadside retaining walls, pillars, billboards, and back slopes so as to provide sight distances that are longer than design sight distances. The paper also presents graphical charts for determination of offsets and their locations, their verification, and comparison with values produced from other verified models. The methodology presented in this paper may further be used to extend the guideline for horizontal offsets (18) to cover compound curves.
Engineering intuition may suggest that there are two maximum offsets at a compound curve, one for each circular arc. The intuition is accurate only for compound curves whose each arc is at least long as design sight distance. In that case, Equation (1) is the appropriate model for its offsets and their locations. If one or both arcs are shorter than design sight distance, which is the focus of this paper, preliminary investigation suggests that there is only one maximum offset at a compound curve. That offset may be located within the sharper arc, at the point of common curvature, or within the flatter arc. Before modeling of the magnitude and location of the maximum offset, consider
Using
R1 is the radius of the flatter arc,
L1 is the length of the flatter arc,
R2 is the radius of the sharper arc,
L2 is the length of the sharper arc,
PC is the point of curvature,
PCC is the point of common or compound curvature,
PT is the point of tangency,
PC-S is the origin and the start of influence area of a compound curve,
d is the location of a driver downstream of PC-S,
o is the location of a dangerous object, at distance S downstream of the driver,
α is the angle at which the sightline inclines from the approach tangent.
The methodology in this paper introduces new parameters which simplify complex modeling of many variables to linear equations. The parameters are simply thresholds of L1 and L2 which are used for determining model regimes based on location of objects “o” and offsets. The parameters are defined as:
1) L2cr is the critical value of L2 at which the maximum offset M is at PCC, for a given value of L1. If L2 is longer than L2cr, the offset M is within L2, if L2 equals L2cr the offset M is at PCC, and if L2 is shorter than L2cr the offset M is within L1.
2) L2u is the upper value of L2 at which M is determined with the dangerous object “o” being on the driver’s path at PT, for a given value of L1. Values of L2 that are smaller than L2u correspond to the object being on the departure tangent, else, the object is within L1.
3) L1cr is the critical value of L1 at which L2cr = L2u. The offset M is at PCC and the object is at PT.
The new parameters or thresholds of L1 and L2 were obtained after conducting geometric examination or inspection of a combination of S, L1, L2, R1 and R2. For a simple curve, location of M is simply fixed at the middle of the curve but for compound curves the location of M changes with how values of S, L1, L2, R1 and R2 combine, as stated in the definitions above. Ranges of values of L1 and L2 control modeling of M and its location. The ranges of values of L1 and L2 were observed to naturally form four (4) regimes or cases. Of the 4 cases, 2 are based on ranges of length L1 relative to its critical value L1cr. Each of the 2 cases of L1 has 2 subcases based on ranges of lengths L2 relative to L2cr and L2u values. The cases are:
Case A: Short L1 (i.e. L1 ≤ L1cr). This case also has the property of L2cr ≤ L2u.
a - 1) L2cr ≤ L2 ≤ L2u, the object is on the departure tangent.
b - 2) L2u < L2 ≤ S, the object is within the flatter arc L1.
Case B: Long L1 (i.e. L1 > L1cr). This case also has the property of L2cr > L2u.
a - 1) L2cr ≤ L2 ≤ S, the object is within L1.
b - 2) L2u ≤ L2 < L2cr, the object is within L1.
This case considers short arcs such that both a driver and an object are located on tangents. The conditions L1 < S, L2 < S and L1 + L2 ≤ S hold. The first step in modeling this case is determination of values of the parameters L2cr and L2u. Consider
The critical length of the sharper arc L2cr is obtained when the sightline is parallel to the common tangent at PCC (in
Determination of L2cr is accomplished through programing by finding a positive value of L2 that minimizes M for given values of L1, R1, R2, and S, while maintaining the condition of the sightline being parallel to the common tangent at PCC. The objective function is presented by Equation (2a). The offset M in Equation (2a) is a sum of two parts as seen in
min M = x 2 ⋅ sin ( α ) + T 2 ⋅ sin ( α ) (2a)
Since α = Δ 2 and T 2 = R 2 ⋅ tan ( Δ 2 / 2 ) , α can be replaced with L 2 / R 2 in the equation. Therefore:
min M = x 2 ⋅ sin ( L 2 R 2 ) + R 2 ⋅ tan ( L 2 2 R 2 ) ⋅ sin ( L 2 R 2 ) (2b)
Subject to:
x 2 + L 2 + L 1 + x 1 = S (3)
where;
x 1 is the distance between PT and object location,
x 2 is the distance between PC and driver location.
The solution to Equation (2) is obtained at L2 = L2cr. The driver location d and the angle of inclination α corresponding to L2cr are also at their critical values dcr and αcr, respectively. The Critical driver location dcr and critical angle αcr are calculated using the obtained critical length L2cr as shown in Equation (4) and Equation (5).
α c r = L 2 c r / R 2 = Δ 2 (4)
d c r = S − x 2 = L 2 c r + L 1 + x 1 (5)
The dashed line (in
To determine L2u for a given L1, the term x1 is removed from Equation (3) and the dangerous object “o” is fixed at PT. It is worth to state that when solving for L2u, the sightline is not conditioned to be parallel to the common tangent at PCC. However, when values of L2u and L2cr are equal, the sightline is parallel to the common tangent and the value of L1 is also at its critical value L1cr.
After obtaining the value of L2u, the driver location corresponding to L2u is calculated using Equation (6).
d u = S − x 2 = L 2 u + L 1 (6)
The angle of inclination of a sightline corresponding L2u is presented by Equation (7).
α u = tan − 1 ( y P T x P T − d u ) (7)
where;
x P T is the x-coordinate of PT measured along the approach tangent from PC-S,
y P T is the y-coordinate of PT, positive on the inside of curve.
After the parameters L2cr, L2u, and L1cr are determined, equations for driver location and angle of inclination of a sightline corresponding to M are formed. It was found in this study that the equations are linear functions of lengths L2 (of
sharper arcs). The linear equations are presented by Equation (8) and Equation (9). It can be seen that the fractions are slopes formed by using the range L2cr ≤ L2 ≤ L2u of L2 for case A(a).
d M = d c r + d u − d c r L 2 u − L 2 c r ( L 2 − L 2 c r ) , L 2 c r ≤ L 2 < L 2 u (8)
α M = α c r + α u − α c r L 1 u − L 1 c r ( L 2 − L 2 c r ) , L 2 c r ≤ L 2 < L 2 u (9)
where;
dM is the driver location corresponding to M,
αM is the sightline angle corresponding to M.
These quantities dM and αM in Equation (8) and Equation (9), respectively, are then plugged in Equation (10) for the maximum offset. The value of the offset changes to that of Equation (1) if L2 = S at which dM becomes S and the object is at PCC.
M = ( S − d M ) sin ( α M ) + R 2 ( 1 − cos ( α M ) ) (10)
Location of offset downstream of PC is given by Equation (11):
X M = R 2 α M (11)
where;
XM is the location of M downstream of PC.
The location of M may also be expressed as a linear function of L2 as follows:
X M = X c r + X u − X c r L 2 u − L 2 c r ( L 2 − L 2 c r ) , L 2 c r ≤ L 2 ≤ L 2 u (12)
where:
X c r = R 2 α c r = R 2 Δ 2 c r = L 2 c r ,
X u = R 2 α u .
Note that values of L2 that are shorter than L2cr in Case A are not considered in the modeling since they are shorter than minimum values recommended in the AASHTO’s guideline [
This case considers short flat arcs and long sharp arcs. Since all values of L2 are longer than L2u, dangerous objects are on the short flat arcs. Values of L2u, du, and αu are determined as described in Case A(a) above. Equations for driver location and inclination of sightline corresponding to M are as presented by Equation (13) and Equation (14).
d M = d u + S − d u S − L 2 u ( L 2 − L 2 u ) , L 2 u ≤ L 2 < S (13)
α M = α u + α S − α u S − L 2 u ( L 2 − L 2 u ) , L 2 u ≤ L 2 ≤ S (14)
where:
α S = 0.5 S / R 2 .
These dM and αM values are inputs to Equation (10) above for calculating the offset M. The location of the offset is calculated using Equation (15).
X M = X u + 0.5 S − X u S − L 2 u ( L 2 − L 2 u ) , L 2 u ≤ L 2 ≤ S (15)
This case considers that both flat and sharp arcs are long. Equations for critical driver locations and critical sightline angles are the same as Equations (4) and (5) in Case A(a) except that L1 is long and x1 is not applicable. The critical values of driver location and angle of sightline inclination are calculated with Equation (16) and Equation (17).
α M = α c r + α S − α c r S − L 2 c r ( L 2 − L 2 c r ) , L 2 c r ≤ L 2 ≤ S (16)
d M = d c r + S − d c r S − L 2 c r ( L 2 − L 2 c r ) , L 2 c r ≤ L 2 ≤ S (17)
The maximum offset is calculated with Equation (10) presented above in Case A(a). The location of the offset is given by Equation (18):
X M = X c r + 0.5 S − X c r S − L 2 c r ( L 2 − L 2 c r ) , L 2 c r ≤ L 2 ≤ S (18)
This is the case that involves long flat arcs and short sharper arcs. Equations for critical driver locations and critical sightline angles are the same as Equations (4) and (5) in Case A(a). Equations for driver locations and sightline angles corresponding to the maximum offset M are as follows:
α M = α u + α c r − α u L 2 c r − L 2 u ( L 2 − L 2 u ) , L 2 u ≤ L 2 ≤ L 2 c r (19)
d M = d u + d c r − d u L 2 c r − L 2 u ( L 2 − L 2 u ) , L 2 u ≤ L 2 ≤ L 2 c r (20)
Location of the offset M is downstream of PCC. The offset is calculated with Equation (21). The three components of the equation are presented in the not-to-scale
M = ( S − d M ) sin ( α M ) + ( R 2 − Q ) cos ( θ ) + R 1 [ 1 − cos ( θ ) ] (21)
The term θ is the central angle intercepting the arc between PCC and XM. The central angle is given by Equation (22) and illustrated by
θ = X M − L 2 R 1 = α M − Δ 2 (22)
Length Q in the second term of Equation (21) is given by Equation (23) and shown in
Q = R 2 sin ( 0.5 π + α M − Δ 2 ) sin ( 0.5 π − α M ) (23)
Location of the offset is within the flatter arc [unlike in Cases A and B(a)]. That location is determined with the linear Equation (24):
X M = X u + X c r − X u L 2 c r − L 2 u ( L 2 − L 2 u ) , L 2 u ≤ L 2 ≤ L 2 c r (24)
where
X u = L 2 u + ( α u − Δ 2 u ) R 1 ,
Δ 2 u = L 2 u / R 2 .
Note that in this sub case, values of L2 that are shorter than L2u are not considered in the modeling since they are shorter than minimum values recommended in the AASHTO’s guideline [
Consider a highway with speed of 40mi/h and stopping sight distance of 305 ft. The maximum offset M for a compound curve with R1 = 750 ft, L1 = 180 ft, R2 = 500 ft, and L2 = 150 ft is found by first classifying the compound curve whether it is Case A or B.
R 1 R 2 = 750 500 = 1.5
L 1 S = 180 305 = 0.59 > 0.535 this is Case B in
L 1 S = 0.59 gives a value of L 2 c r S = 0.31 on the vertical axis of
Therefore, L 2 c r = 0.31 × 305 = 94.55 ft .
L 1 S = 0.59 gives a value of L 2 u S = 0.275 on the vertical axis of
L 2 u = 0.275 × 305 = 83.88 ft .
Since L 2 u < L 2 c r < L 2 , it is Case B(a).
The critical value of sightline angle is calculated with Equation (4).
α c r = L 2 c r / R 2 = 94.55 / 500 = 0.1891 rads .
The critical value of driver location is read on
The critical driver location is d c r = 0.84 × 305 = 256.2 ft .
Equations (16) and (17) are then used to calculate driver location and sightline angle corresponding to M as presented in Equation (25) and Equation (26).
α M = 0.1891 + 0.5 × 305 / 500 − 0.1891 305 − 94.55 ( 150 − 94.55 ) = 0.22 rads (25)
d M = 256.2 + 305 − 256.2 305 − 94.45 ( 150 − 94.55 ) = 269.06 ft . (26)
Resulting values in Equation (25) and (26) are plugged in Equation (10) as follows:
M = ( 305 − 269.06 ) sin ( 0.22 ) + 500 ( 1 − cos ( 0.22 ) ) = 19.91 ft . (27)
The offset location downstream of PC is calculated with Equation (18), as Equation (28) below.
X M = 94.55 + 0.5 × 305 − 94.55 305 − 94.55 ( 150 − 94.55 ) = 109.82 ft . (28)
This location is not mid-arc of L2 as it would have been for a simple curve with R2 = 500 ft and L2 = 150 ft.
In
and Case B(a) are spot-on with the dashed line, the offset ratios are the same as those of the dashed line L1cr/S = 0.535. The dotted line is for offsets where sharper arcs are at their critical lengths or critical L2/S ratios i.e. L2cr/S.
If the calculation example presented in the previous section was to use the chart to determine the offset, the procedure would have been brief as follows:
L 1 S = 180 305 = 0.59
L 2 S = 150 305 = 0.49
Drawing a vertical line at the value of 0.49 on the horizontal axis meets the curve for L1/S = 0.59 (spot-on with the dashed line) to give an offset ratio of 0.875 on the vertical axis.
Equation (1) yields an offset of:
M = 500 [ 1 − cos ( 305 2 × 500 ) ] = 23.08 ft (29)
The needed offset is M = 0.875 × 23.08 = 20.2 ft which is comparable to 19.91 ft depending on the precision of reading the charts in both methods.
In
PCC station. Location values of lines on the right of the dotted line imply that maximum offsets M are in sharper arcs while those on the left of the dotted line imply that the offsets are in flatter arcs.
For the example presented in the previous section, values of L2/S and L1/S are 0.49 and 0.59, respectively. The values result in an XM/S ratio of 0.36. The location of the offset is thus:
X M = 0.36 × 305 = 109.8 ft .
The value 109.8 ft is the same as 109.82 ft obtained earlier using equations. The maximum offset is located 109.8 ft downstream of PC or 220.2 ft upstream of PT. If configuration of a compound curve begins with a flatter arc, the location of M will still be measured from PC but with a small modification to its math. Suppose that geometry of the above example begins with the flatter arc. The location of M is determined as follows:
X M = L 1 + L 2 − ( X M / S ratio ) × S
X M = 180 + 150 − 0.36 × 305 = 220.2 ft from PC of flat arc.
Note, since quantities in
This section presents a few examples of maximum offsets and their locations that are verified with offsets that are generated using the graphical procedure. Although the procedure is not an advanced method, offset values generated with it are accurate and are trusted by researchers and practitioners. For example, the AASHTO guideline [
The data in
| Scenario: | 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|---|
| Speed (mi/h) | 50 | 40 | 35 | 30 | |||
| S (ft) | 518.37 | 469.16 | 425 | 305 | 250 | 200 | |
| L1 (ft) | 171.70 | 206.04 | 340 | 305 | 200 | 120 | |
| L2 (ft) | 114.47 | 137.36 | 170 | 244 | 200 | 160 | |
| R2 (ft) | 656.17 | 1312.34 | 700 | 500 | 350 | 250 | |
| Offset M (ft) | 32.64 | 15.59 | 26.50 | 22.64 | 21.64 | 19.34 | |
| Graphical | 32.62 | 15.57 | 26.40 | 22.64 | 21.66 | 19.36 | |
| Easa (19) | 32.81 | - | - | - | - | - | |
| Hassan et al. (20) | - | 15.58 | - | - | - | - | |
| Location from PC (ft) | 110.42 | 130.27 | 141.95 | 136.64 | 112.00 | 89.60 | |
| Graphical | 110.44 | 130.30 | 141.41 | 136.84 | 112.01 | 89.46 | |
| Easa (19) | 114.47 | - | - | - | - | - | |
This paper presents development of analytical models for determination of clearance offsets on the inside of compound curves. The paper introduces three geometric parameters which simplify the problem to a set of linear equations. The paper demonstrates how to apply the equations but also presents design charts. In addition, the paper presents verification of the models and the charts with accurate offsets prepared using the graphical method. The charts use unitless quantities hence they can be used in imperial and metric systems.
In research, the equations and the charts form a foundation on which to base future research on determination of minimum offsets at other than locations of maximum offsets, and also extension to 3D alignments.
In practice, geometric designers are required to provide clearance offsets that ensure provision of design sight distance at curves. Failure to comply with the requirement ends in disapproval of designs by jurisdictions. The equations and charts presented in this paper provide an analytical and rational way of determining the offsets required by design guidelines without having to incur negatives of graphical methods but yield as accurate offsets. The equations and charts will be suited for sites with alignments that are level, are on constant grade, or have flat vertical curvature, or have no vertical curvature. Inclusion of vertical curvature is in the next phase based on what has been developed in this paper.
The author declares no conflicts of interest regarding the publication of this paper.
Mauga, T. (2024) Analytical Determination of Size and Location of Roadside Horizontal Sightline Offsets for Compound Curves. Journal of Tra- nsportation Technologies, 14, 119-135. https://doi.org/10.4236/jtts.2024.141008