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 Properties

Based 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

R

SDRM

65.28cos1

R

G

aV

VtSSD 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 curves

Without 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

RC

VL

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 70mph

Level 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 Curves

Given – 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: L

GGr 12

G1%

G2%

A

L

Axx

GEE 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