Geometric Design Standards: Crest Vertical Curves 2014 | 2-4 September, South Africa Geometric...

SARF/IRF 2014 | 2-4 September, South Africa

Geometric Design Standards: Crest Vertical Curves

Prepared by:

Rudi von Fintel

Contents: • Background • Introduction • Stopping sight distance on a vertical curve • The shadow area & the effect of: gradient

deceleration rate eye height speed current AASHTO standards

• Findings • Conclusion

Background: • South African standards based on AASHTO • Developed in the 1950’s and 1960’s

with periodic revisions; big changes in vehicle capabilities;

big changes in traffic composition!

• Designs produced in less time environmental, OHS, procurement processes;

tendering for work, reduced margins; designers with less experience; more use of minimum standards; increased reliance on software!

Introduction: • Parabolic curves

y = ax² + bx + c where: a = A/200L but: K = L/A therefore a = 1/200K

• Rate of curvature (K)

LVC > SSD: LVC < SSD:

Where: K = rate of curvature S = stopping sight distance h1 = eye height h2 = object height A = algebraic difference in gradient

For 100km/h: S = 200m; h1 = 1,05m; h2 = 0,6m;

K = 60

Introduction: Typical table showing minimum K values for vertical curves (TRH17; UTG; SATCC):

Introduction: Table showing minimum K values for vertical crest curves (SANRAL):

Introduction: • Stopping Sight Distance (SSD):

Distance for Perception / Reaction + Braking for V = 100km/h; S = 200m

Note: No gradient taken into account

• SSD on grade:

for V = 100km/h; G = -10%; S = 270m

or

• SSD:

Perception / Reaction + Braking

Introduction:

0

20

40

60

80

100

120

1650 1750 1850 1950

Distance (m)

Spe

ed

(km

/h)

2,5 seconds @ V km/h

(0,694 = 2,5/3,6)

Braking distance:

Using: u²=v²-2as but: v=0 thus: u²=-2as a = deceleration rate => a<0 therefore: u² = 2as or: s = u² / 2a but: u is in m/s => u = V/3,6 thus: s = V²/(3,6² x 2a) adjusting a for grade: aG = 9,81 x ((a/9,81) ± G) thus: s = V²/(3,6² x 2 x 9,81 x ((a / 9,81) ± G)) but: 3,6² x 2 x 9,81 = 254 and: 1/254 = 0,004

therefore:

s = 0,004V²/((a/9,81) ± G)

SSD on a crest curve: El

evat

ion

(m

)

Distance (m)

BV

C

EVC

0

20

40

60

80

100

120

1650 1750 1850 1950

Distance (m)

Spee

d (

km/h

)

s=0

v=0

Calculating braking distance on curves: • Iterative Process…


evat

ion

(m

)

Distance (m)

BV

C

EVC

0

20

40

60

80

100

120

1650 1750 1850 1950

Distance (m)

Spee

d (

km/h

)

s=0

v=0

s=1

u1

G1

Calculating braking distance on curves: • Iteration #1


evat

ion

(m

)

Distance (m)

BV

C

EVC

0

20

40

60

80

100

120

1650 1750 1850 1950

Distance (m)

Spee

d (

km/h

)

s=0

v=0

G2 s=2

u2



evat

ion

(m

)

Distance (m)

BV

C

EVC

0

20

40

60

80

100

120

1650 1750 1850 1950

Distance (m)

Spee

d (

km/h

)

s=0

v=0

G3 s=3

u3



evat

ion

(m

)

Distance (m)

BV

C

EVC

0

20

40

60

80

100

120

1650 1750 1850 1950

Distance (m)

Spee

d (

km/h

)

v=0

s=n

un

Gn s=0

Calculating braking distance on curves: • Iteration #n

where un = V/3,6


evat

ion

(m

)

Distance (m)

BV

C

EVC

0

20

40

60

80

100

120

1650 1750 1850 1950

Distance (m)

Spee

d (

km/h

)

Using:

u² = 2aGs

u = 2aGs

un = 2aGnsn

with iterations:

s = u² / 2aG

re-arranging:

or:


evat

ion

(m

)

Distance (m)

BV

C

EVC

0

20

40

60

80

100

120

1650 1750 1850 1950

Spee

d (

km/h

)

sn = n x 1m

u2 = 2 x 2,019 x 2 = 2,842 m/s

u3 = 2 x 2,022 x 3 = 3,483 m/s

u1 = 2 x 2,019 x 1 = 2,009 m/s

…

Distance (m) un = v179 = 2 x 2,165 x 179 = 27,842 m/s

Using:

un = 2aGnsn ; aiming for vn = 27,778 m/s (100km/h):


evat

ion

(m

)

0

20

40

60

80

100

120

1650 1750 1850 1950

Distance (m)

Spee

d (

km/h

)

• n = 179 (179 iterations for u >= 27,778 m/s)

un = u179 = 2 x 2,165 x 179 = 27,842 m/s

179m

• since n = 179; braking distance = 179m note: braking distance rounded up to the nearest m

• perception / reaction: 2,5s @ 27,778 m/s = 69,444m say 70m 70m

• SSD = 70 + 179 = 249m for an object @ EVC (km 1,900) 249m


evat

ion

(m

) SS

D (

m)

Distance (m)

BV

C

EVC

SSD

249m

200m

The shadow area: • A plot tracing the top of the hidden object at the end of

the stopping sight distance along the length of influence of a crest vertical curve.

The shadow area:

Impact on shadow area: • due to changing gradient

• i.e. change in ‘h2’ due to changes in ‘G’

Changing gradient: El

evat

ion

(m

) H

eigh

t (m

)

Distance (m)

Top of hidden object

BV

C

EVC

Gradeline


evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


0,6m object hidden


evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


0,6m object hidden

1,3m object hidden

Impact on shadow area: • due to changing deceleration rate

• i.e. change in ‘h2’ due to changes in ‘a’ Note: 1) TRH17, SATCC, UTG, G2: a ranges from 3,7 m/s² to 2,8 m/s² (40km/h to 120km/h)

2) SANRAL: a = 3,0 m/s² 3) AASHTO:

a = 3,4 m/s²

Changing deceleration rate: El

evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


1,3m object hidden

0,6m object hidden


evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


0,6m object hidden


evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


Impact on shadow area: • due to changing eye height

• i.e. change in ‘h2’ due to changes in ‘h1’ Note: 1) TRH17, SATCC, UTG, G2, SANRAL: h1 = 1,05 m 2) AASHTO:

h1 = 1,08 m

Changing eye height: El

evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


1,3m object hidden

0,6m object hidden


evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


0,6m object hidden


evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


Impact on shadow area: • due to changing speed

• i.e. change in ‘h2’ due to changes in ‘V’

• Note ‘V²’ and ‘S²’

Changing speed: El

evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


Changing speed: El

evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


0,6m object hidden

Changing speed: El

evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


1,3m object hidden

0,6m object hidden

Changing speed: El

evat

ion

(m

) H

eigh

t (m

)

Distance (m)

Gradeline

BV

C

EVC


1,3m object hidden

0,6m object hidden

Changing speed: El

evat

ion

(m

) H

eigh

t (m

)

Distance (m)

Gradeline

BV

C

EVC


1,3m object hidden

0,6m object hidden

Note: @120km/h, for h2 < 0,6m; a = 6m/s², twice the value specified!

Impact on shadow area: • current AASHTO standards

• i.e. change in ‘h2’ due to changes in: a = 3,4 m/s² h1 = 1,08 m K = 52

Current AASHTO standards: El

evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


1,3m object hidden

0,6m object hidden

Current AASHTO standards: El

evat

ion

(m

) H

eigh

t (m

)

Distance (m)

BV

C

EVC

Gradeline


0,6m object hidden

Findings: • Minimum K values from the manuals apply to

exit gradients of 0%

• Heights and deceleration rates of vehicles have a significant impact on the visibility of object within the SSD

• Speed has a profound impact on the visibility of object within the SSD

References: • Committee of State Road Authorities. TRH17: Geometric Design

of Rural Roads. Department of Transport, Pretoria 1988.

• CSIR. Geometric Design Guidelines. South African National Roads Agency SOC Limited, Pretoria.

• American Association of State Highway and Transportation Officials. A policy on Geometric Design of Highways and Streets, 6th edition. Washington, 2011

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