National Academies Press: OpenBook

Selecting Ramp Design Speeds, Volume 1: Guide (2021)

Chapter: Section 2. Ramp Elements Related to Ramp Design Speed

« Previous: Section 1. Introduction
Page 23
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 23
Page 24
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 24
Page 25
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 25
Page 26
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 26
Page 27
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 27
Page 28
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 28
Page 29
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 29
Page 30
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 30
Page 31
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 31
Page 32
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 32
Page 33
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 33
Page 34
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 34
Page 35
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 35
Page 36
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 36
Page 37
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 37
Page 38
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 38
Page 39
Suggested Citation:"Section 2. Ramp Elements Related to Ramp Design Speed." National Academies of Sciences, Engineering, and Medicine. 2021. Selecting Ramp Design Speeds, Volume 1: Guide. Washington, DC: The National Academies Press. doi: 10.17226/26415.
×
Page 39

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

23 Section 2. Ramp Elements Related to Ramp Design Speed The definition of ramp design speed indicates a combination of contextual issues such as the type of intersecting highways, area type, ramp configuration, and site constraints that should be considered when selecting a ramp design speed. This section of the design guidelines discusses in greater detail the most relevant elements that affect selecting an appropriate ramp design speed. For completeness, additional elements are discussed that do not impact the selection of an appropriate ramp design speed, but are key elements of ramp design. The elements addressed in this section include: • System and service interchanges. • Ramp configuration. • Entrance and exit ramps. • Freeway mainline ramp terminal configuration (i.e., taper vs. parallel). • Type of crossroad ramp terminal (i.e., stop control, signal control, free-flow). • Ramp grade. • Type of horizontal curvature (i.e., simple, spiral, compound, and reverse curves). • Number of lanes (i.e., single vs. multilane ramps). • Crossover crown line. • Superelevation and superelevation transition. • Ramp metering. • Collector-distributor (C-D) and frontage roads. • Physical site constraints, environmental and social impacts, and right-of-way costs. For some of the ramp elements, guide values for ramp design speeds are provided. These should be considered general guide values for ramp design speeds related to the individual ramp elements. However, no single element is the primary determinant in selecting an appropriate ramp design speed. Rather, a combination of contextual considerations should be made in selecting an appropriate ramp design speed. The elements presented in this section, and associated recommended ramp design speeds, should be considered collectively to select an appropriate ramp design speed for a given ramp. In Section 2.14, a table of guide values for ramp design speeds for system and service interchanges is provided. 2.1 System and Service Interchanges A system interchange, which connects two or more freeways, and a service interchange, which connects a freeway to lesser facilities (i.e., non-controlled access facilities), differ from one another in how they relate to ramp design speed. With a system interchange, vehicles exit a high- speed facility and enter another high-speed facility, so it is desirable to provide a ramp between the two facilities that enables vehicles to maintain a speed as close as possible to freeway speeds, minimizing the need for deceleration or acceleration along the ramp. In rural or unconstrained areas, drivers expect minimal acceleration or deceleration along a ramp of a system interchange; while in urban and constrained areas, drivers expect moderate acceleration or deceleration along the ramp of a system interchange.

24 For situations where the design speeds of both intersecting freeways of a ramp are equivalent, the highest possible ramp design speed for the connecting ramp would be the design speed of the intersecting freeways. However, due to physical constraints and related right-of-way costs, in most cases it is not practical for the ramp design speed to be equivalent to the design speed of the two intersecting freeways. Therefore, the ramp design speed may be less than the design speed of the two intersecting freeways, but as close as practical. For the situation where two intersecting freeways do not have equivalent design speeds, the ramp design speed may be a speed between the two highway design speeds of the intersecting freeways; but again, for practical purposes this may not be feasible so the ramp design speed may be less than the lower design speed of the two intersecting freeways. With a service interchange, vehicles either accelerate from a near-stop condition at the crossroad ramp terminal to a higher speed on the freeway (e.g., in the case of entrance ramps) or decelerate from a higher speed on the freeway to a stop or near-stop condition at the crossroad ramp terminal (e.g., in the case of exit ramps). In either case, the ramp needs to be designed to accommodate the necessary acceleration or deceleration. In rural or unconstrained areas where the intersecting crossroad is a primary highway or major street, minimal to moderate acceleration or deceleration along the ramp is expected by the driver; while if the intersecting crossroad is a minor arterial or collector, moderate acceleration or deceleration is expected by the driver along the ramp. In urban or constrained areas where the intersecting facility is a primary highway or major street, moderate to significant acceleration or deceleration along the ramp is expected by the driver; while if the intersecting road is a minor arterial or collector, significant acceleration or deceleration is expected by the driver along the ramp. Table 2 summarizes the anticipated acceleration/deceleration adjustment (i.e., either speed reduction or increase) by drivers in rural/unconstrained and urban/constrained areas by intersecting facility type and interchange type and corresponding ramp design speed. Table 2. Anticipated Driver Behavior and Ramp Design Speeds by System and Service Interchanges (adapted from Leisch, 2005) Intersecting Facility Rural / Unconstrained Urban / Constrained Se rv ic e In te rc ha ng e Primary highway or major arterial Minimal to moderate acceleration/deceleration adjustment Moderate to substantial acceleration/deceleration adjustment High to low ramp design speed Intermediate to low ramp design speed Minor arterial or collector Moderate acceleration/deceleration adjustment Substantial acceleration/deceleration adjustment Intermediate to low ramp design speed Low ramp design speed Sy st em In te rc ha ng e Freeway Minimal acceleration/deceleration adjustment Moderate acceleration/deceleration adjustment High ramp design speed Intermediate ramp design speed

25 As a guide, high ramp design speeds generally correspond to values of 50 mph and above; intermediate ramp design speeds are in the range of 35 to 45 mph; and low ramp design speeds correspond to values of 30 mph or less. 2.2 Ramp Configuration When selecting the appropriate ramp design speed for a particular ramp configuration, the primary considerations are the intended purpose of the ramp and the controlling feature of the ramp. For example, is the ramp intended to facilitate movement between two high-speed facilities in the case of a system interchange or is the ramp intended to facilitate larger speed transitions between high- and low-speed facilities? Some ramp configurations, such as direct and semidirect ramps, are primarily used at system interchanges, while diagonal ramps are primarily used at service interchanges. Loop ramps are commonly used at both system and service interchanges. Another consideration is the controlling feature of the ramp. Is the horizontal alignment the primary feature of the ramp that will control speeds, or will speed primarily be controlled by the operations of the terminals? Several considerations are discussed below regarding the selection of an appropriate ramp design speed for the following ramp configurations and are relatively consistent with guidance provided in the Green Book (AASHTO, 2018): • Direct connection ramps • Semidirect connection ramps • Loop ramps • Diagonal ramps Direct connection ramps (see Figure 8) do not deviate greatly from the intended direction of travel. They can be designed as right exit/right entrance ramps or as left exit/left entrance ramps that utilize the shortest and most direct path with curvilinear, diagonal, or reverse curve alignment. Direct connection ramps are commonly used at system interchanges. As such, they are intended to encourage high-speed, free-flow merging with freeway traffic and may be designed with ramp design speeds ranging from 35 to 80 mph. Direct connection ramps are also used at service interchanges, and in these situations the ramp design speeds may range from 30 to 65 mph. For both system and service interchanges, the horizontal alignment is commonly the controlling feature of direct connection ramps. For direct connection ramps, the sharpest curve on the ramp proper that significantly affects vehicle speed is considered the controlling curve. Semidirect connection ramps (see Figure 7) provide a less direct path than direct connection ramps and typically include smaller curve radii than direct connection ramps, and thus may have associated lower ramp design speeds compared to direct connection ramps. Semidirect connection ramps are commonly used at system interchanges and are intended to encourage high-speed, free-flow merging with freeway traffic. As such, high ramp design speeds are preferred but may not be practical due to the curvilinear alignment. For semidirect connection ramps, ramp design speeds may range from 30 to 65 mph. For a semidirect connection ramp, the horizontal alignment is commonly the controlling feature of the ramp. For semidirect connection ramps, the sharpest curve on the ramp proper that significantly affects vehicle speed is considered the controlling curve.

26 Loop ramps (see Figure 7) may be used in both system and service interchanges. The design speed of a loop ramp is limited by the radius of the controlling curve. The radii of controlling curves on loop ramps generally range from 150 to 250 ft, with corresponding design speeds of about 25 to 30 mph. Loop ramps with design speeds above 30 mph involve large land areas rarely available in urban areas. Loop ramps with ramp design speeds above 30 mph are costly and require drivers to travel a considerable extra distance, but if less restrictive conditions exist, a ramp design speed above 30 mph may be used and the corresponding radius may be increased. For both system and service interchanges, the horizontal alignment is the controlling feature of loop ramps. Diagonal ramps are used for service interchanges and typically have both a left- and right-turn movement at the crossroad ramp terminal. Diagonal ramps consist of a high-speed freeway mainline ramp terminal, a tangent or curved alignment on the ramp proper, and stop or yield conditions at the crossroad ramp terminal. Depending on the configuration of the crossroad, a free-merge right-turn maneuver may be provided. The “shape” of a diagonal ramp largely depends on the pattern of turning traffic and the right-of- way available. Diagonal ramps may be designed with (1) a tangent throughout most, or all, of the ramp; (2) a continuous curve to the right, with a spur to the left for left turns; (3) or a combination of tangent and reverse curvature. For diagonal ramps where the horizontal alignment is curvilinear, the horizontal alignment will be the controlling feature of the ramp. For diagonal ramps where the horizontal alignment is relatively flat (i.e., where the horizontal alignment has little or no impact on the speeds of vehicles), the ramp design speed is based on the operational characteristics of the freeway mainline ramp terminal for an entrance ramp and the operational characteristics of the crossroad ramp terminal for an exit ramp. For an entrance ramp, the design speed of the tangent section of the ramp proper (i.e., the ramp design speed) should be consistent with the speed entering the freeway mainline ramp terminal. For an exit ramp, the design speed of the tangent section of the ramp proper (i.e., the ramp design speed) should be consistent with the operating conditions at the crossroad ramp terminal. Because diagonal ramps are used for service interchanges and the controlling feature of a diagonal ramp may be either the horizontal alignment or the operational characteristics of one of the terminals, diagonal ramps may be designed with ramp design speeds as low 15 to 20 mph and potentially ranging as high as 45 to 70 mph. Diagonal ramps are unique compared to other ramp configurations in that the controlling feature of a diagonal ramp may be either the horizontal alignment or the operational characteristics of one of the terminals. With the other ramp configurations, the horizontal alignment of the ramp will likely be the controlling feature. 2.3 Entrance and Exit Ramps Entrance and exit ramps are more associated with service interchanges than with system interchanges. The primary components of entrance and exit ramps include the freeway mainline ramp terminal, the ramp proper, and the crossroad ramp terminal. With system interchanges, ramps function more as a freeway mainline ramp terminal, a ramp proper, and another freeway mainline ramp terminal. There is no crossroad ramp terminal component to a system interchange ramp, which makes it difficult to classify ramps at system interchanges as entrance or exit ramps. Therefore, the discussion below applies primarily to ramps at service interchanges.

27 Entrance Ramps On entrance ramps, vehicles travel from the crossroad ramp terminal to the ramp proper to the freeway mainline ramp terminal. At the crossroad ramp terminal vehicles speeds are low and increase along the ramp proper. On entrance ramps, the controlling feature is either a curve on the ramp proper immediately upstream of the freeway mainline ramp terminal or vehicle operations near the freeway mainline ramp terminal. Where the horizontal alignment of the ramp is curvilinear, the last curve encountered along the ramp proper immediately upstream of the freeway mainline ramp terminal that significantly affects vehicle speed is considered the controlling curve (i.e., it will control the speeds of vehicles transitioning from the ramp proper to the freeway mainline ramp terminal). For design purposes, it may be assumed that horizontal curves with radii less than 1,000 ft will significantly control vehicle speeds. For entrance ramps where the horizontal alignment is relatively flat (i.e., diagonal ramps), the ramp design speed is based on the operational characteristics of the freeway mainline ramp terminal so the design speed of the tangent section of the ramp proper (i.e., the ramp design speed) should be consistent with the speed entering the freeway mainline ramp terminal. The boundary between the ramp proper and the freeway mainline ramp terminal is dependent upon the line of sight of the driver and the horizontal alignment of the ramp. Typically, the gore point serves as the boundary between the ramp proper and freeway mainline ramp terminal. Research shows that drivers tend to begin the merge, acceleration process only after gaining a clear view of the freeway right-traffic lane. Thus, if the driver’s view from the ramp is obstructed, the acceleration length will not begin until near the gore point where the view of the freeway becomes unobstructed (Hunter and Machemehl, 1997). However, if the driver’s view from the ramp is unobstructed, the acceleration length may extend upstream from the gore point where the horizontal curvature has a radius less than 1,000 ft, thus extending the freeway mainline ramp terminal upstream of the gore point. Entrance ramps can be designed with taper- or parallel-type freeway mainline ramp terminal configurations. A taper-type freeway mainline ramp terminal is a general form of a speed-change lane that provides direct entry at a flat angle (see Figure 10). Vehicles merge onto the freeway from a long, uniform taper between the outer edge of the acceleration lane and the edge of the through-traffic lane. A parallel-type freeway mainline ramp terminal is a general form of a speed-change lane that has an added lane for acceleration adjacent to the freeway mainline (see Figure 11). The type of freeway mainline ramp terminal configuration, taper or parallel, has no direct impact on the selection of an appropriate ramp design speed for an entrance ramp. Entrance ramps with either type of freeway mainline ramp terminal configuration are designed in the same manner for a given ramp design speed. For the same assumed entrance speed at the beginning of the freeway mainline ramp terminal onto the acceleration lane and the same assumed merge speed at the end of the freeway mainline ramp terminal, the minimum acceleration length would be exactly the same for a taper-type entrance ramp and a parallel-type entrance ramp. In a recent study (Torbic et al., 2012), it was found that vehicles are more likely to use the full length of a tapered speed- change lane to accelerate to near-freeway speeds before merging; in contrast to a parallel speed- change lane, where vehicles may merge earlier along the speed-change lane and at lower speeds. However, for design purposes, vehicles are assumed to accelerate at the same rates on both taper- and parallel-type entrance ramps and use the full acceleration length prior to merging. Therefore,

28 acceleration lengths are equivalent for both taper- and parallel-type entrance ramps given the same initial speed (VAcc Length (i)) and speed reached (VAcc Length (f)), so the type of freeway mainline ramp terminal configuration (i.e., taper or parallel) does not affect the selection of the ramp design speed. Differences in the general alignment of ramps between taper- and parallel-type entrances, however, could impact whether the selection of the ramp design speed is based on the last curve that significantly influences vehicle speeds on the ramp proper prior to the freeway mainline ramp terminal or the operational characteristics of the freeway mainline ramp terminal. With a taper-type entrance providing direct entry at a flat angle, there is a greater likelihood that the controlling feature of the ramp will be the operational characteristics of the freeway mainline ramp terminal. With a parallel-type entrance, there is a greater likelihood that the horizontal alignment immediately upstream of the acceleration length on the ramp proper will be curvilinear, affect the speed of vehicles, and be the controlling feature of the ramp. For both taper- and parallel-type entry ramps, acceleration lengths should include only the portions of the ramp proper from which drivers can clearly view the freeway right-lane traffic. Because the controlling feature of an entrance ramp will always be associated with speeds or operational characteristics towards the end of the ramp near the freeway mainline ramp terminal, ramp design speeds are generally at least 35 mph or greater for entrance ramps. Exit Ramps Exit ramps should be designed to facilitate deceleration from the mainline freeway speed to a speed slightly below the ramp design speed, and then potentially to a stop condition (at service interchanges). Exit ramps may be designed as taper-type exits that provide direct exits at flat angles and fit the preferred path of most exiting drivers and allow vehicles to leave the through lane of the freeway at relatively high speeds (see Figure 10). Deceleration can take place off of the freeway mainline, as the exiting vehicle moves along the taper onto the ramp proper. Exit ramps may also be designed with parallel-type exits that have an added lane adjacent to the freeway for deceleration (see Figure 11). Similar to entrance ramps, the type of freeway mainline ramp terminal configuration (i.e., taper or parallel) of an exit ramp has no direct impact on the selection of an appropriate ramp design speed. For the same assumed diverge speed from the mainline freeway onto the deceleration lane at the beginning of the deceleration length and the same assumed speed at the end of the deceleration length, the minimum deceleration length would be exactly the same for a taper-type exit ramp and a parallel-type exit ramp. The key point, though, is that the assumed speed at the end of the freeway mainline ramp terminal and the beginning of the first section of the ramp proper would be the same whether an exit ramp is designed as a taper-type or parallel-type. Therefore, the type of freeway mainline ramp terminal configuration (i.e., taper or parallel) has no direct impact on the selection of the appropriate ramp design speed for an exit ramp. Differences in the general alignment of ramps between taper- and parallel-type exits, however, may affect whether the horizontal alignment or the operations near the crossroad ramp terminal is the controlling feature of the ramp and will impact the selection of the ramp design speed. With a taper-type exit providing direct exit at a flat angle, there is a greater likelihood that the controlling feature of the ramp will be vehicle operations near the crossroad ramp terminal. With

29 a parallel-type exit, there is a greater likelihood that the horizontal alignment immediately downstream of the deceleration lane on the ramp proper will be curvilinear, affect the speed of vehicles, and be the controlling feature of the ramp. If the controlling feature of the ramp is the horizontal alignment immediately downstream of the deceleration lane on the ramp proper, the ramp design speed may be higher than if the operational characteristics of the crossroad ramp terminal are the controlling feature and ramp design speeds will be lower. In general, the controlling feature of an exit ramp may be the horizontal alignment or operational characteristics of the crossroad ramp terminal. A wide range of ramp design speeds are applicable to exit ramps ranging from 15 mph to 65 mph or higher. In particular, where the horizontal alignment of the ramp is curvilinear and the first curve encountered on the exit ramp is the controlling feature, ramp design speeds will likely be higher. If the horizontal alignment of the ramp is relatively straight and the operational characteristics of the crossroad ramp terminal are the controlling feature, the ramp design speed may be as low as 15 or 20 mph. 2.4 Freeway Mainline Ramp Terminal Configuration As discussed in Section 2.3, the type of freeway mainline ramp terminal configuration (i.e., taper or parallel), has no direct impact on the selection of an appropriate ramp design speed. For design purposes, vehicles are assumed to accelerate at the same rates on both taper- and parallel- type entrance ramps and use the full acceleration length prior to merging. Similarly, for exit ramps, vehicles are assumed to decelerate at the same rates on both taper- and parallel-type exit ramps. As a result, the manner in which speed transitions occur between adjoining sections of a ramp is the same whether the freeway mainline ramp terminal is designed with a taper- or parallel-type configuration. Thus, the type of freeway mainline ramp terminal configuration does not impact the ramp design speed of a given ramp. 2.5 Type of Crossroad Ramp Terminal A crossroad ramp terminal, associated with a service interchange, is that portion of the ramp that connects the ramp proper to the crossroad. Crossroad ramp terminals can be classified in several ways based upon distinguishing factors such as number of ramp legs, number of left-turn movements, location of crossroad left-turn storage, and type of traffic control. In many cases, the controlling feature of an exit ramp may be the horizontal alignment immediately downstream of the freeway mainline ramp terminal. However, where the horizontal alignment of an exit ramp is relatively straight and has little impact on vehicle speeds, which is possible at diagonal ramps, then the operational characteristics of the crossroad ramp terminal will be the controlling feature of the ramp, and the type of traffic control at the crossroad ramp terminal (stop-controlled, signal controlled, or free-flow) will directly affect the selection of the ramp design speed. The Highway Safety Manual (HSM) (AASHTO, 2010; AASHTO, 2014) indicates that the average speed of vehicles at the point where the ramp connects to the crossroad is typically around 15 mph where the crossroad ramp terminal is stop-, yield-, or signal-controlled and around 30 mph for all other ramps at service interchanges, which implies ramps with free-flow operations. However, for stop-, yield-, and signal-controlled terminals, which would be the case for all diagonal exit ramps, the ramps need to be designed to accommodate queued vehicles

30 stopped for the traffic control. Thus, rather than designing to accommodate vehicles traveling 15 mph at the intersection of the ramp and crossroad, the ramp design speed needs to be selected based on vehicles stopped at the end of the queue storage leading to the crossroad. Thus, for all diagonal exit ramps where the horizontal alignment is relatively straight and has little impact on vehicle speeds and the crossroad ramp terminal is stop-, yield-, or signal-controlled, for practical purposes the ramp design speed should be 15 to 20 mph, which will still require vehicles to decelerate to a complete stop upon entering the functional area of the crossroad ramp terminal. As described in Section 2.3.1, the controlling feature of an entrance ramp will always be based upon speeds or operational characteristics towards the end of the ramp near the freeway mainline ramp terminal. The speeds of vehicles entering a ramp will be affected by the type of traffic control at the crossroad ramp terminal; but as the length of the ramp increases, the speeds of vehicles towards the end of the ramp near the freeway mainline ramp terminal become less of a function of the vehicle speeds entering the ramp. Therefore, the type of crossroad ramp terminal of an entrance ramp indirectly impacts the selection of an appropriate ramp design speed. As ramp length increases, the type of traffic control at the crossroad ramp terminal of an entrance ramp has less impact on the selection of an appropriate ramp design speed. 2.6 Ramp Grade The vertical profile of a ramp usually assumes the shape of the letter “S” with a sag curve at the lower end, a central portion on an appreciable grade, followed by a crest vertical curve at the upper end. Additional vertical curvature may be needed for ramps that cross other roadways. Most passenger vehicles can readily negotiate grades as steep as 4 to 6 percent without any appreciable loss in speed, but the effect of grades on truck speeds is much more pronounced. Little is known about the relationship between roadway grades and design speeds for conventional roadways (i.e., arterials, collectors, etc.), and the same is true for the relationship between ramp grades and ramp design speeds (AASHTO, 2018). Therefore, ramp grades are not directly related to selecting appropriate ramp design speeds; however, ramps with higher ramp design speeds should generally have flatter gradients than ramps with lower ramp design speeds. The Green Book provides the following general criteria for upgrades on ramps: • Upgrades on ramps with ramp design speeds of 45 to 50 mph should be limited to 3 to 5 percent. • Upgrades on ramps with ramp design speeds of 35 to 40 mph should be limited to 4 to 6 percent. • Upgrades on ramps with ramp design speeds of 25 to 30 mph should be limited to 5 to 7 percent. • Upgrades on ramps with ramp design speeds of 15 to 20 mph should be limited to 6 to 8 percent. The Institute of Transportation Engineers (ITE) Freeway and Interchange Handbook (Leisch, 2005) provides the following guidance for maximum gradients on ramps: • Normal conditions: 4 to 6 percent. • Heavy truck traffic: 3 to 4 percent.

31 • Rugged terrain: 6 to 8 percent. The lower values for each range of gradient are applicable to the higher ramp design speeds, particularly for major interchanges. While the selection of ramp grade is important, the AASHTO Green Book states that “adequate sight distance is more important than a specific gradient control and should be favored in design,” but acknowledges that these two controls are usually compatible (AASHTO, 2018). 2.7 Type of Horizontal Curvature The horizontal alignment of an interchange ramp consists of a combination of tangents and curves. When pieced together, the individual tangents and curves form the shape of the ramp. Four basic types of horizontal curves are used in the design of interchange ramps to achieve the desired shape and alignment: • Simple curves with constant or uniform radii. • Spiral curves with varying radii. • Compound curves which consist of two or more simple curves with different radii, joined together, curving in the same direction. • Reverse curves which consist of two simple curves joined together, curving in the opposite direction. The specific shape of a ramp may be influenced by factors such as the traffic pattern, traffic volume, ramp design speed, topography, culture, intersection angle, and type of ramp terminal (AASHTO, 2018). In particular, compound and spiral curves may be used to obtain the desired alignment and fit the natural path of vehicles; but from the perspective of selecting an appropriate ramp design speed, there is no difference in the type of horizontal curves used in the design of a ramp. In other words, for a given ramp design speed and maximum superelevation rate, a simple curve, a spiral curve, and the individual simple curve of a compound or reverse curve are designed with the same curve radius and would be expected to result in the same operational conditions. With respect to the horizontal curvature and selection of the appropriate ramp design speed, the key is designing the ramp to provide appropriate speed transitions from one adjoining section of the ramp to the next. Each individual horizontal curve and tangent has its own design speed, and the design speeds of adjoining sections should incrementally increase or decrease by no more than 10 to 15 mph to accommodate sequential increasing or decreasing speeds along the ramp. 2.8 Number of Lanes As traffic volumes continue to increase and right-of-way for infrastructure becomes more limited, a potential treatment to add capacity at interchanges is to construct multilane ramps or convert single-lane ramps to multilane ramps. With higher traffic volumes and increased capacity, multilane ramps may naturally be considered or viewed as higher functional classification facilities compared to single-lane ramps; and it is common for higher functional class facilities to have

32 higher design speeds than lower functional class facilities. For example, multilane loop ramps have higher capacities and typically consist of: • More significant movements. • Larger curve radii (ranging from 180 to 200 ft). • Higher ramp design speeds. Single-lane loop ramps, on the other hand, have lower capacities and typically consist of: • Less significant movements. • Smaller curve radii (ranging from 100 to 170 ft). • Lower ramp design speeds. However, the higher ramp design speeds found on multilane loop ramps are directly related to accommodating the more significant movements, and not to the number of lanes on the ramp. Acceleration and deceleration properties of vehicle speeds and capabilities and driver behaviors are expected to be the same whether the ramp is single-lane or multilane ramp. For example, minimum acceleration lengths for entrance terminals and minimum deceleration lengths for exit terminals are assumed to be the same for design purposes of both single-lane and multilane ramps. Furthermore, research on single-lane and multilane loop ramps indicates that vehicle speeds and driver behaviors are comparable in either lane of multilane ramps (Torbic et al., 2016), which supports using the same ramp design speed for single-lane and multilane ramps. 2.9 Crossover Crown Line At the boundary between the right lane of the freeway and the adjacent auxiliary lane of a freeway mainline ramp terminal, the cross slopes of the right lane of the freeway and auxiliary lane may differ. This boundary location is referred to as the crossover crown line and should not to be confused with the crown line normally provided at the centerline of the roadway or the cross-slope break between the traveled way and the shoulder. The maximum allowable difference in cross slopes at the crossover crown line, measured as is the algebraic difference in cross slopes of the right lane of the freeway and adjacent auxiliary lane, is an important control in developing superelevation along the freeway mainline ramp terminal and should not be designed such that it could cause driver discomfort or impact control of the vehicle. Chapter 10 of the Green Book states that the maximum algebraic differences in the cross-slope rates at the crossover crown line should be based on the values in Green Book Table 9-18 which are intended for use with turning roadway terminals at intersections (AASHTO, 2018). Green Book Table 9-18, presented here as Table 3, provides suggested maximum algebraic differences in cross-slope rates at the crossover crown line for design speeds of exit and entrance curves for three speed categories: 20 mph and under, 25 and 30 mph, and 35 mph and over. The suggested maximum algebraic differences decrease as the speed categories increase.

33 Table 3. Maximum Algebraic Difference in Cross Slope at Turning Roadway Terminals (AASHTO, 2018) Design Speed of Exit or Entrance Curve (mph) Maximum Algebraic Difference in Cross Slope at Crossover Crown Line (%) 20 and under 5.0 to 8.0 25 and 30 5.0 to 6.0 35 and over 4.0 to 5.0 From a design perspective, it seems logical to limit the maximum algebraic difference in cross- slope rates at the crossover crown line based on speed. At higher speeds of 35 mph and over, the maximum algebraic difference in cross slope at the crossover crown line is 4.0 to 5.0 percent. As speeds decrease, the maximum algebraic difference in cross slope at the crossover crown line increases to as high as 8.0 percent, and recent research (Torbic et al., 2016) shows that under normal scenarios vehicles will be able to recover when encountering a cross slope break of 8 percent or less. It is only under the most extreme departure scenarios that some types of heavy trucks may not recover (i.e., rollover) when encountering a cross slope break of 8 percent or less. At the same time, there is no logical reason for the maximum allowable algebraic difference in cross slope at the crossover crown line to directly affect the selection of an appropriate ramp design speed for a given ramp. After the ramp design speed is selected, the ramp should be designed consistent with the current design criteria for the cross slope break at the crossover crown line. 2.10 Superelevation and Superelevation Transition Superelevation is the rotation of the pavement surface on the approach to and through a horizontal curve. As necessary, horizontal curves are designed with superelevation to offset a portion of the lateral acceleration of the vehicle. The practical limits of superelevation on horizontal curves are controlled by several factors including climate conditions, terrain conditions, adjacent land use, and frequency of slow-moving vehicles (AASHTO, 2018). There is no universally applicable maximum superelevation rate for every region. Transportation agencies typically establish their own policies concerning the maximum superelevation rate that will be used on horizontal curves within their jurisdiction. Most transportation agencies use maximum superelevation rates of either 6 or 8 percent; but in regions where snow and ice are not a factor, a superelevation rate of 12 percent represents a practical maximum limit. One maximum superelevation rate should be used within a region to promote design consistency. The maximum superelevation rate of a curve does not impact the selection of an appropriate ramp design speed for a given ramp. It will impact the minimum radius that can be designed for a given ramp design speed, but the maximum superelevation rate used by an agency does not factor into the selection of an appropriate ramp design speed for a given ramp. The critical issue related to superelevation and ramp design speed is that adjoining sections of the ramp need to be of sufficient length to allow for appropriate speed transitions between sections and superelevation transition of the pavement between sections. For longer ramps, providing sufficient lengths to accommodate superelevation transitions should not be an issue; however, in constrained areas where individual components of ramps are designed near their

34 minimum length requirements, superelevation transition of the pavement could become the controlling factor in the design of the adjoining sections of ramps. 2.11 Ramp Metering Ramp metering regulates the flow of vehicles at entrance ramps to reduce turbulence in the merge area to improve the efficiency of freeway operations and reduce crash frequency. Ramp metering consists of traffic signals installed on the ramp proper near the freeway mainline ramp terminal to control the number of vehicles entering the freeway from the ramp. Ramp meters generally only operate during peak hours. Ramp metering does not directly impact the selection of an appropriate ramp design speed because ramps should be designed for the higher ramp operating speeds that would be present when ramp meters are not operational. Other design considerations such as the number of lanes at the ramp meter stop bar, ramp traffic volume, ramp grade, and freeway design speed factor into the design of the length and geometry of the ramp proper and longer acceleration lengths necessary during metering operations. With the exception of the freeway design speed, these other design considerations do not affect the selection of the ramp design speed. 2.12 Collector-Distributor and Frontage Roads The purpose of a collector-distributor (C-D) road is to eliminate weaving on the mainline freeway and reduce the number of entrances and exits on the through roadways while satisfying the demand for access to and from the freeway. The design speeds of C-D roads are usually equal to or less than the design speeds of the mainline freeway roadways. Frontage roads serve numerous functions depending on the facility they serve and the character of the surroundings. For example, frontage roads may control access to an arterial, function as a street facility serving adjoining properties, and maintain circulation of traffic on either side of the arterial. Frontage roads separate local traffic from higher speed through traffic and provide access to residences and commercial establishments along the highway. C-D and frontage roads function as one of the intersecting highways connected by a ramp. A C- D road can serve as the equal or lower speed intersecting facility where connected to a freeway so the ramp design speed for a ramp connecting a freeway to a C-D road would be similar to that of a ramp for a system interchange. A C-D road can also serve as the higher speed facility where the ramp connects the C-D road to a surface street or crossroad. In this situation, the ramp design speed for a ramp connecting a C-D road to a surface street or crossroad would be similar to that of ramp for a service interchange. Similarly, the ramp design speed for a ramp connecting a freeway to a frontage road would be similar to that of a ramp for a service interchange. Otherwise, C-D and frontage roads do not affect the selection of the ramp design speed. 2.13 Physical Site Constraints, Environmental and Social Impacts, and Right-of-Way Costs A combination of contextual considerations factor into the process for determining interchange type and location, and the length and orientation of each interchange ramp, including:

35 • Physical site constraints such as topography, bodies of water (e.g., streams, lakes, and rivers), existing infrastructure, buildings, and right-of-way limits. • Environmental conditions such as wetlands and endangered species. • Social impacts related to displacement, barrier effects, accessibility, visual quality, land use, and noise. • Accommodation of pedestrians and bicyclists. • Economic factors such as construction, maintenance, vehicle operating, and right-of-way costs. In general, any combination of these contextual considerations may affect the selection of an appropriate ramp design speed for a given ramp, and any one aspect of these contextual considerations does not necessarily carry more weight in the decision-making process. In other words, physical site constraints may be just as relevant and carry equal weight as environmental and social impacts and economic factors in the decision-making processes for determining interchange type and location and the length and orientation of each interchange ramp. Ultimately, these contextual considerations impact the overall design of each ramp, and since ramp design speed is the selected speed used to determine the various geometric design features of a given ramp, these contextual considerations directly impact the selection of the ramp design speed. Conceptually, where these contextual considerations factor into the selection process for determining an appropriate ramp design speed for a given ramp, these contextual considerations can be viewed from a qualitative perspective and classified as either creating constrained or unconstrained conditions. For example, where the physical site characteristics for a given ramp may be considered typical or less restrictive than normal for a given rural or urban area, the site should be classified as being unconstrained. On the other hand, where physical site characteristics exist beyond what should be considered as typical or normal for a given rural or urban area, the site should be classified as constrained. Similarly, sites should be classified as unconstrained or constrained as they pertain to environmental and social impacts and economic factors. As a general rule of thumb, ramp design speeds are higher for unconstrained ramps and lower for constrained ramps, consistent with driver behavior and expectations. 2.14 Ramp Design Speed Guide Values Sections 2.1 through 2.13 address key elements related to interchange ramp design and discuss how the various elements directly or indirectly impact the selection of an appropriate ramp design speed or do not factor into the selection of ramp design speed. In particular, selection of an appropriate ramp design speed is most influenced by the type of interchange (i.e., system or service) and the type of ramp configuration. However, none of these elements (i.e., type of interchange, ramp configuration, entrance vs. exit ramp, freeway mainline ramp terminal configuration, type of crossroad ramp terminal, ramp grade, type of horizontal curvature, number of lanes, crossover crown line, superelevation, ramp metering, and C-D and frontage roads) is the single primary determinant in selecting an appropriate ramp design speed. In addition to the key ramp elements that directly or indirectly impact the selection of an appropriate ramp design, a combination of contextual considerations such as physical site constraints, environmental and social impacts, and right-of-way costs factor into the processes for determining interchange type

36 and location and the length and orientation of each ramp and directly impact the selection of the ramp design speed as part of the geometric design process. Collectively, these key ramp elements and contextual considerations factor into the selection of an appropriate ramp design speed for a given ramp. Based on the key elements and contextual considerations that factor into the geometric design process for ramps, Table 4 provides a range of guide values for ramp design speeds for system and service interchanges. In general, higher ramp design speeds are associated with ramps: • For system interchanges compared to service interchanges. • For ramp configurations that have mild or gentle horizontal curvature compared to sharp curvature. • In rural areas compared to urban areas. • At service interchanges that intersect higher functional class roadways (i.e., primary/major arterials) compared to lower functional class roadways (i.e., minor arterials/collectors). • In unconstrained conditions compared to constrained conditions. Table 4 provides a range of guide values given the context, ramp configuration, and type of interchange. For the specified design condition, it is recommended that the ramp design speed for a given ramp be within the range of guide values. The lower guide values are generally associated with constrained conditions, and the higher guide values are associated with unconstrained conditions. Typically, the range in guide values for a given context and ramp configuration is between 5 and 15 mph. The maximum range in guide values for a given context and ramp configuration is 30 mph. Where no guide value(s) is provided in the table, it is implied that the type of context and ramp configuration is not applicable. For example, it is uncommon for freeways in urban areas to designed with design speeds of 80 to 85 mph. Similarly, system interchanges always connect two or more freeways. Thus, at a minimum the design speed of the intersecting freeway will be at least 50 mph. Therefore, no guide value or values for ramp design speed are provided for these uncommon design conditions. Where the horizontal alignment of the ramp is curvilinear, the ramp design speed applies to the controlling curve on the ramp proper. For an entrance ramp, the last curve encountered along the ramp proper that significantly affects vehicle speed is considered the controlling curve (see Figure 13). For an exit ramp, the controlling curve is the first curve encountered along the ramp proper that significantly affects vehicle speed. For design purposes, it can be assumed that horizontal curves with radii less than 1,000 ft will significantly affect vehicle speeds. For direct, semidirect, and outer connection ramps, the sharpest curve on the ramp proper is considered the controlling curve (see Figure 14). As noted at the bottom of Table 4, where the horizontal alignment of the ramp is relatively straight and has little or no influence on vehicle speeds, the ramp design speed applies to the primary tangential section of the ramp proper. For entrance ramps that are relatively straight, it is recommended that the ramp design speed be within 15 to 20 mph of the highway design speed. For exit ramps that are relatively straight, it is recommended that the ramp design speed be around 15 to 20 mph due to operational characteristics of the crossroad ramp terminals.

37 Table 4. Range of Guide Values for Ramp Design Speed as Related to Highway Design Speed, Interchange Type, Ramp Configuration, and Contextual Considerations Context Ramp configuration Highway design speed (mph) 30 35 40 45 50 55 60 65 70 75 80 85 Ramp design speed (mph) System Interchanges Rural Direct connections - - - - 40-45 40-50 40-55 45-60 45-65 50-70 50-75 50-80 Semidirect connections - - - - 30-40 30-45 35-45 35-50 40-55 40-60 40-65 45-65 Loop ramps - - - - 25-30 25-30 25-35 25-35 30-35 30-40 35-45 35-45 Urban Direct connections - - - - 35-40 35-40 35-50 40-50 40-55 40-60 - - Semidirect connections - - - - 30-35 30-40 30-45 30-45 35-50 35-55 - - Loop ramps - - - - 20-25 20-30 25-30 25-30 25-35 25-40 - - Service Interchanges Rural: Primary or major arterial Direct connections - - - 35-40 35-45 40-45 40-45 40-50 45-50 45-55 50-60 50-65 Diagonal ramps1 20-25 20-25 25-30 25-35 30-40 30-45 30-50 30-55 35-55 40-60 45-65 45-70 Loop ramps 15-20 20-25 20-25 20-25 20-25 25-30 25-30 25-30 30-35 30-40 35-45 35-45 Minor arterial or collector Diagonal ramps1 20 20-25 25 25-30 25-35 30-40 30-45 30-50 35-50 40-55 45-60 45-65 Loop ramps 15-20 15-20 15-20 15-25 20-25 20-30 20-30 25-30 25-35 30-40 30-45 30-45 Urban: Primary or major arterial Direct connections - - - 30-35 35-40 35-40 35-40 40-45 40-50 40-55 - - Diagonal ramps1 15-20 20-25 25-30 25-30 30-35 30-40 30-45 30-45 35-50 40-55 - - Loop ramps 15-20 15-20 15-20 20-25 20-25 20-30 25-30 25-30 25-35 25-40 - - Minor arterial or collector Diagonal ramps1 15 20 20-25 20-30 25-35 25-40 25-45 30-45 30-50 35-55 - - Loop ramps 15-20 15-20 15-20 15-25 15-25 20-30 20-30 20-30 20-35 25-40 - - 1 Where the horizontal alignment of a diagonal entrance ramp is relatively straight and has little impact on vehicle speeds, the ramp design speed is dependent on the operational characteristics of the freeway mainline ramp terminal. For this situation, it is recommended that the ramp design speed be within 15 to 20 mph of the highway design speed. Where the horizontal alignment of a diagonal exit ramp is relatively straight and has little impact on vehicle speeds, the ramp design speed is dependent on the operational characteristics of the crossroad ramp terminal. The recommended ramp design speed for this situation is 15 or 20 mph, as it is expected that the crossroad ramp terminal at all diagonal ramps will either be stop-, yield-, or signal-controlled.

38 Figure 13. Primary Components and Controlling Curves at Entrance and Exit Ramps at Service Interchanges Figure 14. Primary Components and Controlling Curve for Outer Connection Ramp at a System Interchange

39 By providing a range of guide values for ramp design speeds, Table 4 provides designers with flexibility in determining the contextual and design considerations that apply for a given ramp (e.g., in determining whether a ramp is located in an unconstrained or constrained area). The range of guide values in Table 4 were developed to be relatively consistent with guidance presented in the Green Book (AASHTO, 2018). The Green Book states that for direct connections ramp design speeds between the middle and upper ranges shown in Green Book Table 10-1 should be used, and 40 mph is preferably the minimum ramp design speed for a direct connection. As presented in Table 4, for a majority of the conditions, the minimum ramp design speed is 40 mph or greater for direct connections, and the range of values includes values in the middle and upper ranges of Green Book Table 10-1. For semidirect connections, again the Green Book states ramps design speeds between the middle and upper ranges shown in Green Book Table 10-1 should be used, and ramp design speeds for semidirect connections are typically 30 to 40 mph. The Green Book continues indicating that ramp design speeds less than 30 mph should not be used for semidirect connections; and for a short single-lane ramp, a ramp design speed of 50 mph is generally not practical. As presented in Table 4, for all of the conditions, the minimum ramp design speed is 30 mph or greater; and for all but one of the conditions, values of 30, 35, and/or 40 mph are included in the range of guide values. It is only for the rural system interchange with a highway design speed of 80 mph where values of 30, 35, and/or 40 mph are not included in the range of guide values. For this condition (i.e., rural system interchange with a highway design speed of 80 mph), the minimum ramp design speed is 45 mph. For loop ramps, the Green Book states that upper-range values of ramp design speeds shown in Green Book Table 10-1 are generally not attainable. Ramp design speeds of 30 mph for loops encompass large land areas rarely available in urban areas. Long loop ramps needed for higher design speeds are costly and require drivers to travel a considerable extra distance. In addition, minimum values usually control; but for highway design speeds of 55 mph and greater, the ramp design speed should be no less than 20 mph. Where less restrictive conditions exist, the ramp design speed and consequently the radius of the loop ramp may be increased. As presented in Table 4, for a majority of the conditions for service interchanges, the minimum ramp design speed for a loop ramp is either 15, 20, or 25 mph; and for a majority of the conditions for system interchanges, the range of values includes values in the lower range of Green Book Table 10-1. In some cases, the range of guide values in Table 4 extend slightly beyond the range of values suggested in the Green Book. This was done intentionally. A table of guide values should not be too prescriptive that it limits flexibility and inhibits designers from exercising engineering judgement. In other cases, extending beyond the range of values suggested in the Green Book was necessary to address the fact that many states are posting freeway speed limits as high as 75, 80, and 85 mph, and assistance for selecting ramp design speeds to address such conditions is necessary. Selecting the appropriate ramp design speed is only the first step in designing a ramp that meets driver expectations. Another critical step in the design process for designing a ramp that meets driver expectations is designing the other components and sections of the ramp consistent with the overall ramp design speed such that appropriate speed transitions between adjoining components and sections can take place. For this to occur, in most situations, the change in design speed between adjoining sections should be limited to no more than 10 to 15 mph.

Next: Section 3. Guidelines for Designing Ramps in a Consistent Manner Based on the Selected Ramp Design Speed »
Selecting Ramp Design Speeds, Volume 1: Guide Get This Book
×
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

Selection of a design speed should be based upon the anticipated operating speed, topography, adjacent land use, modal mix, and functional classification of the roadway.

The TRB National Cooperative Highway Research Program's NCHRP Web-Only Document 313: Selecting Ramp Design Speeds, Volume 1: Guide provides further detail for selecting an appropriate ramp design speed than presented in the 2018 Green Book, to address several overarching challenges that may lead to confusion or inconsistent interpretation of existing AASHTO guidance for selecting an appropriate ramp design speed.

Supplemental to the document are NCHRP Web-Only Document 313: Selecting Ramp Design Speeds,Volume 2: Conduct of Research Report and Ramp Speed Profile Model worksheets.

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  6. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  7. ×

    View our suggested citation for this chapter.

    « Back Next »
  8. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!