Geometric Design (II)

Learning Objectives• To calculate minimum radius of horizontal

curve• To understand design concepts for

transition curves and compute min length• To understand the role of SSD in horizontal

and vertical design• To define and apply grade considerations• To develop vertical curves

(Chapter 6.1 ~ 6.4)

• Minimum Curve Radius – Curve requiring the

most centripetal force for the given speed

– Given emax, umax, Vdesign

Horizontal Curve

ue

VR mph

ft

15min

2

)()(

R

Horizontal Curve PropertiesBased on circular

curve• R: radius of curve• D: degree of curve : central angle• T: length of tangent• L: length of curve• LC: long chord• M: middle ordinate dist• E: external dist

Point of Curvature

Point of Tangency

Horizontal Design Iterations

• Design baseline – Curve radius above the minimum– Superelevation and side-friction factor

not exceeding the maximum values• Design is revised to consider:

cost, environmental impacts, sight distances, aesthetic consequences, etc. 

Horizontal Curve Sight Distance

• Sight line is a chord of the circular curve

• Sight Distance is curve length measured along centerline of inside lane

RSDRM 65.28cos1

R

Ga

VVtSSD r

2.3230

47.12

Recall

Horizontal Curve Sight Distance

Figure 6-10

With Transition Curves

Transition Curves• Gradually changing the curvature from

tangents to circular curvesWithout Transition Curves

Transition Curves• Gradually changing the curvature from

tangents to circular curves– Use a spiral curve

L: min length of spiral (ft)V: speed (mph)R: curve radius (ft)C: rate of increase of centrifugal accel (ft/sec3),

1~3

RCVL

315.3

Transitional Curves• Gradually changing the cross-section of

the roadway from normal to superelevated (Figure 6-9)

Keep water drainage in mind while considering all of the available cross-section options

Vertical Alignment

Reduced Speed

Increased Speed

Vertical Alignment• Grade

– measure of inclination or slope, rise over the run

– Cars: negotiate 4-5% grades without significant speed reduction

– Trucks: significant speed changes• 5% increase on short descending grades• 7% decrease on short ascending grades

Grade Considerations• Maximum grade – depends on terrain type,

road functional class, and design speed

Terrain 60mph 70mphLevel 3% 3%Rolling 4% 4%Mountainous 6% 5%

Rural Arterials

Grade Considerations• Critical length of

grade – Maximum length which

a loaded truck can travel without unreasonable speed reduction

– Based on accident involvement rates with 10mph speed reduction as threshold

Grade Considerations

General Design Speed Reduction

Vertical Curves• To provide transition between two

grades• Consider

– Drainage (rainfall)– Driver safety (SSD)– Driver comfort

• Use parabolic curves• Crest vs Sag curves

Vertical Curves

Vertical CurvesGiven

– G1, G2: initial & final grades in percent– L: curve length (horizontal distance)

Develop the actual shape of the vertical curve

PVI

point of vertical curvature

point of vertical intersection

point of verticaltangency

G2%

G1%

Vertical Curves• Define curve so that PVI is at a horizontal distance

of L/2 from PVC and PVT• Provides constant rate of change of grade:

LGGr 12

G1%

G2%

A

LAxxGEE PVCP 200100

21

Example

• G1 = 2%• G2 = -4%• Design speed = 70 mph• Is this a crest or sag curve?• What is A?

Vertical Curves• Major control for safe operation is

sight distance• MSSD should be provided in all

cases (use larger sight distance where economically and physically feasible)

• For sag curves, also concerned with driver comfort (large accelerations due to both gravitational and centrifugal forces)

Crest Vertical Curves• Critical length of curve, L, is where

sight line is tangent to the crest• Assume driver eye height (H1) = 3.5

ft and object height (H2) = 2.0 ft and S=MSSD

Sag VC - Design Criteria• Headlight sight distance• Rider comfort• Drainage control• Appearance