CVEN2401_Week10(1)_Vertical Alignment_Part 1 (6).pdf

CVEN2401 Sustainable Transport and Highway Engineering Week 10: Vertical Alignment (Part 1) Kasun P. Wijayaratna

Review Assignment Progress Check…

•

Introduction - What is Geometric Design? / Design Standards (L7(1))

•

Road and Design Characteristics (L7(1))

•

Basic Kinematics (L7(1))

If you answer “No” to any of these questions, you need to catch up and get organised!!!

•

Speed Parameters (L7(2))

-

•

Sight Distance (L7(2))

•

Do you know what design group you are in? Have you spoken to others in your design group?

Horizontal Alignment: Part 1 (Basics and Sight Distance) (L8(1))

•

-

Horizontal Alignment: Superelevation and Cross Sections (L9(1))

•

Horizontal Alignment: Transition Curves (L9(1),(2))

Have you or members in your group downloaded and installed Infraworks360?

•

Satisfying the Guidelines (L9(2))

-

Has your group developed a base model of the study area?

-

Has your group completed an analysis of the existing road conditions using Infraworks360?

-

Has your group finalised your horizontal alignment?

-

Has your group thought about how to present the assignment, reports, technical drawings?

–

Completed a full worked example according to AGRD

CVEN 2401: Sustainable Transport and Highway Engineering

2

Overview •

Vertical Alignment: Why do we need it?

•

Vertical Alignment: Controls

•

Vertical Alignment: Design Procedure – Grading – Vertical Curves

•

Vertical Curves: Key Definitions

•

Vertical Curve Fundamental Calculations

•

Vertical Curve Design Calculations – Stopping Sight Distance Revisited – Crest Curves

– Sag Curves •

Minimum Length of Vertical Curves

Historic Columbia River Highway Scenic Byway , Oregon, USA http://industry.traveloregon.com/industry-resources/productdevelopment/oregon-scenic-byways/

Reference: Austroads Guide to Road Design (AGRD) Part 3: Geometric Design

CVEN 2401: Sustainable Transport and Highway Engineering

3

Vertical Alignment: Why do we need it? Remember, why didn’t we think this was an appropriate design?

Not very efficient in terms of earth works…

Earthworks are dependent on the longitudinal profile

Vertical alignment is the longitudinal profile along the centre line of a road. It is made up of a series of grades and vertical curves. CVEN 2401: Sustainable Transport and Highway Engineering

4

Vertical Alignment: Why do we need it? Longitudinal Profile of the natural terrain along the straight line road from the previous slide Can a vehicle travel along this terrain in the existing state?

Maybe… but it will most likely be uncomfortable or unsafe.

CVEN 2401: Sustainable Transport and Highway Engineering

5

Vertical Alignment: Why do we need it? Longitudinal Profile of the natural terrain along the straight line road from the previous slide Select a grading – for the road design Rough RoughEstimate… Estimate… Even with this design, we still have the pictured issue.

CVEN 2401: Sustainable Transport and Highway Engineering

Cut>>>Fill Cut>>>Fill

6

Vertical Alignment: Why do we need it? •

Vertical alignment is the longitudinal profile along the centre line of a road. It is made up of a series of grades and vertical curves. – The natural terrain must be graded to achieve consistent slopes along the road. – In order to achieve a smooth transition between grades we need to design vertical curves.

Natural Terrain

CVEN 2401: Sustainable Transport and Highway Engineering

Graded - Design Terrain

7

Vertical Alignment: Controls •

The level of a road at any point along its route (vertical alignment) is controlled by the features that exist along the route. (Section 8.2 AGRD) – Topography – Geotechnical conditions – Existing intersections – Property entrances – Overpasses and underpasses – Pedestrian access – Service utility assets – Median openings

Source: http://kboi2.com/news/local/walmart-truck-getsstuck-under-overpass-in-oregon-11-21-2015

CVEN 2401: Sustainable Transport and Highway Engineering

8

Vertical Alignment: Design Procedure •

Grading (Section 8.3 AGRD3) – Step 1: Identify all major controls on the alignment and categorise as mandatory or discretionary (refer to controls discussed in previous slide and Section 8.2) – Step 2: Prepare a horizontal alignment in accordance with Section 7 of AGRD3 (Week 9 of the course) – Step 3: Select appropriate grading points (selecting the centreline for a 2-way, 2lane road) – Step 4: Prepare a longitudinal section with an appropriate vertical exaggeration (commonly 10:1) showing natural levels relative to the grading point (centreline) – Step 5: Prepare a trial gradeline, taking into account the vertical control including culverts and coordination of horizontal and vertical alignments. – Step 6: Calculate earthworks quantities (Week 11-12) – Step 7: Adjust the vertical alignment so that: o all mandatory controls are met o discretionary controls are met as far as possible o ensuring minimum sight distance and critical cross-fall controls are met o earthworks are minimised.

CVEN 2401: Sustainable Transport and Highway Engineering

9

Vertical Alignment: Design Procedure •

Grading: Grades (Section 8.5 AGRD3) – “Grades should generally be as flat as possible, consistent with economy and longitudinal drainage requirements (where kerbing is to be incorporated. Flat grades permit all vehicles to operate at the same speed. Steeper grades introduce variation in speeds between vehicles with varying power to weight rations in the uphill and downhill direction.” o The speed variation can result in higher rear end vehicle crash rates. o Increases the likelihood of queuing and makes it difficult to overtake.

Baldwin Street, NZ, “Worlds steepest street Source: https://au.pinterest.com/pin/299911656409359649/

CVEN 2401: Sustainable Transport and Highway Engineering

10

Vertical Alignment: Design Procedure •

Grading: Grades (Section 8.5 AGRD3), Effect of grade on vehicle type – Grades are generally expressed as a percentage of the vertical component divided by the horizontal component.

CVEN 2401: Sustainable Transport and Highway Engineering

11

Vertical Alignment: Design Procedure •

Grading: Grades (Section 8.5 AGRD3), Maximum Grades

CVEN 2401: Sustainable Transport and Highway Engineering

12

Vertical Alignment: Design Procedure •

Vertical Curves (Section 8.6 AGRD3) – “The vertical alignment of a road consists of a series of straight grades joined by vertical curves.” – Vertical curves are parabolic in shape and selected based on 3 controlling factors. o Sight distance: safety requirement o Riding comfort o Appearance: pertinent in flat topography where it is used to alleviate boredom (safety) and enhance aesthetics.

CVEN 2401: Sustainable Transport and Highway Engineering

13

Vertical Curves: Key Definitions •

Classification of Vertical Curves: – Crest Curves: convex vertical curves that meet at a summit (high point) – Sag Curves: concave vertical curves that meet at a trough (low point) PVC 𝑦

PVI

•

Properties of Vertical Curves –

𝑮𝟏 : initial roadway grade in percent or m/m (also referred to as initial tangent grade)

–

𝑮𝟐 : final road way grade in percent or m/m (also referred to as final tangent road way grade)

–

𝑨: absolute value of the difference in grades in percent or m/m (|𝐺2 − 𝐺1 |)

–

𝑳: length of curve in stations or metres measure in a constant-elevation horizontal plane.

–

𝑷𝑽𝑪: point of vertical curve (initial point of the curve, also referred to as “beginning of vertical curve” (𝐵𝑉𝐶) or as tangent point 1 (𝑇𝑃1 ))

–

𝑷𝑽𝑰: point of vertical intersection (intersection of grades)

–

𝑷𝑽𝑻: point of vertical tangent (final point of the curve, also referred to as “end of vertical curve” (𝐸𝑉𝐶) or as tangent point 2 (𝑇𝑃2 ))

PVT

𝑳 𝑥

𝑦 𝑥

𝑥

CVEN 2401: Sustainable Transport and Highway Engineering

14

Vertical Curves: Key Definitions •

Properties of Vertical Curves – In general (and for this course), curves will be equal-tangent vertical curves, 𝐿 where the 𝑃𝑉𝐼 point is distance from the 𝑃𝑉𝐶 point and to the 𝑃𝑉𝑇 point. 2

𝑦

𝐿

𝑌

𝑥

𝑌𝑀

𝑥 𝑌𝑓

𝐿 2

PVC

𝐿 2

PVI

– 𝒀: offset at any distance 𝑥 from the PVC (in metres) – 𝒀𝒎 : mid-curve offset (in metres) – 𝑌𝑓 : offset at the end of the vertical curve (PVT) (in metres)

PVT

CVEN 2401: Sustainable Transport and Highway Engineering

15

Vertical Curve Fundamental Calculations •

General parabolic equation

𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄

Where; 𝒚: roadway elevation at distance 𝑥 from the beginning of the vertical curve (𝑃𝑉𝐶) in stations/metres 𝒙: distance from the beginning of the vertical curve in stations/metres 𝒂, 𝒃, 𝒄: coefficients of the parabola. PVC

PVI

PVT

𝒚 𝒙

𝒚 𝒙

CVEN 2401: Sustainable Transport and Highway Engineering

16

Vertical Curve Fundamental Calculations • •

General parabolic equation 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 Defining 𝒂, 𝒃 and 𝒄 – Slope at the origin (𝑥 = 0) is equal to the slop of the initial tangent, 𝐺1 : 𝑑𝑦 = 2𝑎𝑥 + 𝑏 𝑑𝑥 𝑑𝑦 = 2𝑎 0 + 𝑏 = 𝑏 = 𝐺1 𝑑𝑥 𝑥=0 ∴ 𝒃 = 𝑮𝟏 – The average rate of change of slope is the difference between 𝐺 −𝐺

the gradients divided by the length of the curve, 2 1 : 𝐿 2 𝑑 𝑦 𝐺2 − 𝐺1 = 2𝑎 = 𝑑𝑥 2 𝐿 𝑮𝟐 − 𝑮𝟏 ∴𝒂= 𝟐𝑳 – The 𝑦-intercept of the parabola = 𝑐, which corresponds to the elevation of the 𝑃𝑉𝐶 at 𝑥 = 0: ∴ 𝒄 = 𝑷𝑽𝑪𝒆𝒍𝒆𝒗𝒂𝒕𝒊𝒐𝒏 CVEN 2401: Sustainable Transport and Highway Engineering

17

Vertical Curve Fundamental Calculations • •

General parabolic equation 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 Defining 𝒂, 𝒃 and 𝒄 – Slope at the origin (𝑥 = 0) is equal to the slop of the initial tangent, 𝐺1 : 𝑑𝑦 = 2𝑎𝑥 + 𝑏 𝑑𝑥 𝑑𝑦 = 2𝑎 0 + 𝑏 = 𝑏 = 𝐺1 𝑑𝑥 𝑥=0 ∴ 𝒃 = 𝑮𝟏 – The average rate of change of slope is the difference between 𝐺 −𝐺

the gradients divided by the length of the curve, 2 1 : 𝐿 2 𝑑 𝑦 𝐺2 − 𝐺1 = 2𝑎 = 𝑑𝑥 2 𝐿 𝑮𝟐 − 𝑮𝟏 ∴𝒂= 𝟐𝑳 – The 𝑦-intercept of the parabola = 𝑐, which corresponds to the elevation of the 𝑃𝑉𝐶 at 𝑥 = 0: ∴ 𝒄 = 𝑷𝑽𝑪𝒆𝒍𝒆𝒗𝒂𝒕𝒊𝒐𝒏 CVEN 2401: Sustainable Transport and Highway Engineering

18

Note: These are not formulae and should be derived for each question. They may need to be adjusted depending on the context of a question and the set-up of axes.

Vertical Curve Fundamental Calculations • Example A 200 metre equal-tangent sag vertical curve has the PVC at station 3+700.000 and elevation at 321m. The initial grade is -3.5% and the final grade is +0.5%. Determine the stationing and elevation of the PVI, the PVT, and the lowest point on the curve.

𝑦

PVC (3+700.000) Elevation: 321m

𝐿 = 200

𝑥

PVI?

PVT?

CVEN 2401: Sustainable Transport and Highway Engineering

19

Vertical Curve Fundamental Calculations • Example A 200 metre equal-tangent sag vertical curve has the PVC at station 3+700.000 and elevation at 321m. The initial grade is -3.5% and the final grade is +0.5%. Determine the stationing and elevation of the PVI, the PVT, and the lowest point on the curve. – Measuring along the horizontal plane and as 𝐿 = 200: 𝐿

o PVI will be 100m from PVC: 𝑃𝑉𝐼 = 𝑃𝑉𝐶 + = 3 + 700.000 + 0 + 100.000 = 2 𝟑 + 𝟖𝟎𝟎. 𝟎𝟎𝟎 o PVT will be 200m from PVC: 𝑃𝑉𝑇 = 𝑃𝑉𝐶 + 𝐿 = 3 + 700.000 + 0 + 200.000 = 𝟑 + 𝟗𝟎𝟎. 𝟎𝟎𝟎 – Elevation at PVI and PVT: o 𝑃𝑉𝐼𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 = 𝑃𝑉𝐶𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 − 𝐺1

𝐿 2

𝑷𝑽𝑰𝒆𝒍𝒆𝒗𝒂𝒕𝒊𝒐𝒏 = 321 − 0.035 × 100 = 𝟑𝟏𝟕. 𝟓𝐦 o 𝑃𝑉𝑇𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 = 𝑃𝑉𝐼𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 + 𝐺2

𝐿 2

𝑷𝑽𝑰𝒆𝒍𝒆𝒗𝒂𝒕𝒊𝒐𝒏 = 317.5 + 0.005 × 100 = 𝟑𝟏𝟖. 𝟎𝐦

CVEN 2401: Sustainable Transport and Highway Engineering

20

Vertical Curve Fundamental Calculations • Example A 200 metre equal-tangent sag vertical curve has the PVC at station 3+700.000 and elevation at 321m. The initial grade is -3.5% and the final grade is +0.5%. Determine the stationing and elevation of the PVI, the PVT, and the lowest point on the curve. – Low point is equivalent to the minimum of the parabolic function (given that the low point is definitely between PVC and PVT) – For 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, let –

𝑑𝑦 𝑑𝑥

𝑑𝑦 𝑑𝑥

= 0 to determine location of low point:

= 2𝑎𝑥 + 𝑏 = 0

– When 𝑥 = 0, 𝑏 = 𝐺1 = −0.035 – Consider the average rate of change of slope: o 𝑎=

𝐺2 −𝐺1 2𝐿

=

0.005− −0.035 2 200

= 0.0001

∴ 2 0.0001 𝑥 + −0.035 = 0 𝑥 = 175m – Stationing of low point: 3 + 700.000 + 0 + 175.000 = 𝟑 + 𝟖𝟕𝟓. 𝟎𝟎𝟎 – Elevation of low point: 𝑦 = 0.0001(175)2 + −0.035 175 + 321 = 𝟑𝟏𝟕. 𝟗𝟒𝟎𝐦

CVEN 2401: Sustainable Transport and Highway Engineering

21

Vertical Curve Fundamental Calculations •

Further derivation:

𝐺2 − 𝐺1 2 𝑥 + 𝐺1 𝑥 + 𝑃𝑉𝐶𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 2𝐿 – Offset distance, 𝑌, is the distance between the initial tangent and curve 𝐺2 − 𝐺1 2 ∴ 𝑌 = 𝑦𝑡𝑎𝑛𝑔𝑒𝑛𝑡 − 𝑦𝑐𝑢𝑟𝑣𝑒 = 𝐺1 𝑥 + 𝑃𝑉𝐶𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 − 𝑥 + 𝐺1 𝑥 + 𝑃𝑉𝐶𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛 2𝐿 |𝐺2 − 𝐺1 | 2 𝐴 = 𝑥 = 𝑥2 2𝐿 200𝐿 𝑨 𝒀= 𝒙𝟐 𝟐𝟎𝟎𝑳 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 → 𝑦 =

Where, 𝐴 = absolute value of the difference in grades expressed as a percentage

– Similarly the mid-curve offset (𝑌𝑚 ) and end of curve offset (𝑌𝑓 ) can be derived as follows: 𝐿 2

At 𝑥 = → 𝑌𝑚 and 𝑥 = 𝐿 → 𝑌𝑓 2

𝐴𝐿2 𝑨𝑳 = = 200𝐿 × 4 𝟖𝟎𝟎 𝐴 𝑨𝑳 ∴ 𝒀𝒇 = 𝐿 2= 200𝐿 𝟐𝟎𝟎

𝐴 𝐿 ∴ 𝒀𝒎 = 200𝐿 2

CVEN 2401: Sustainable Transport and Highway Engineering

22

Vertical Curve Fundamental Calculations •

Further derivation (considering the case presented):

– 𝒀: offset at any distance 𝑥 from the PVC (in metres) 𝑨 𝒀= 𝒙𝟐 𝟐𝟎𝟎𝑳 – 𝒀𝒎 : mid-curve offset (in metres) 𝑨𝑳 𝒀𝒎 = 𝟖𝟎𝟎 – 𝑌𝑓 : offset at the end of the vertical curve (PVT) (in metres) 𝑨𝑳 𝒀𝒇 = 𝟐𝟎𝟎

𝑦

𝐿

𝑌

𝑥

𝑌𝑀

𝑥 𝑌𝑓

𝐿 2

PVC

CVEN 2401: Sustainable Transport and Highway Engineering

𝐿 2

PVI

PVT

23

Vertical Curve Fundamental Calculations •

Constancy of the rate of change of slope (second derivative) – Vertical curves are parabolas, considering 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐: 𝑑2 𝑦 ∴ 2 = 2𝑎 → constant 𝑑𝑥 – Using this property, the horizontal distance, 𝐾 necessary to change the slope of the vertical curve by 1% can be estimated as follows: 𝐾=

Curve Length Absolute change in grade over the curve length 𝑳 𝑲= 𝑨

Where,

𝐾: horizontal distance in metres required to affect a 1% change in the slope of the vertical curve 𝐿: length of the curve in metres 𝐴: absolute value of the difference in grades |𝐺2 − 𝐺1 |

CVEN 2401: Sustainable Transport and Highway Engineering

24

Vertical Curve Fundamental Calculations •

Computation of high and low points on a vertical curve – The K-value can also be used to compute the high and low point locations of crest and sag vertical curves, respectively (provided the high or low point does not occur at the PVC or PVT) 𝑑𝑦

– For a high/low point to occur = 0 and this is located 𝑥𝐻𝐿 distance from 𝑃𝑉𝐶: 𝑑𝑥 𝑑𝑦 = 2𝑎𝑥 + 𝑏 = 0 𝑑𝑥 𝑏 ∴ 𝑥𝐻𝐿 = − 2𝑎 Substitute 2𝑎 =

𝐺2 −𝐺1 𝐿

and 𝑏 = 𝐺1

𝐺1 𝐺2 − 𝐺1 𝐿 is a distance, take the absolute value: 𝐺1 |𝐺1 | |𝐺1 | |𝐺1 | 𝑥𝐻𝐿 = − = = = = 𝐾 × 𝐺1 𝐴 1 |𝐺2 − 𝐺1 | 𝐺2 − 𝐺1 𝐿 𝐾 𝐿 𝐿 𝑥𝐻𝐿 = −

As 𝑥𝐻𝐿

∴ 𝒙𝑯𝑳 = 𝑲 × 𝑮𝟏 CVEN 2401: Sustainable Transport and Highway Engineering

25

Vertical Curve Design Calculations •

Stopping Sight Distance Revisited – The sight of a driver is affected by the presence of a crest. – There must be adequate sight distance to a distant object so that the driver can stop the vehicle prior to striking the object whilst traversing the vertical curve – For a given set of grades and operating speed, longer curves provide more SSD but are longer to construct – so estimate the minimum curve length and provide a length that is equal to or greater than the minimum.

CVEN 2401: Sustainable Transport and Highway Engineering

26

Vertical Curve Design Calculations •

Stopping Sight Distance Revisited – Sight Distance (S) may be greater than or smaller than the vertical curve designed. – Thus, these instances govern the estimation of the curve length. – Crest Curve Example 𝐿

𝑆

𝑆 𝐿

𝑆

CVEN 2401: Sustainable Transport and Highway Engineering

27

Vertical Curve Design Calculations •

Stopping Sight Distance Revisited – Sight Distance (S) may be greater than or smaller than the vertical curve designed. – Thus, these instances govern the estimation of the curve length. – Sag Curve Example 𝐿

𝑆

𝑆 𝐿

𝑆

CVEN 2401: Sustainable Transport and Highway Engineering

28

Vertical Curve Design Calculations •

Crest Vertical Curve Design (AGRD3) – Vertical Curve Length (L) 𝑳 = 𝑲𝑨 – Sight Distance Criteria 𝑺𝟐 𝑲= 𝟐𝟎𝟎 𝒉𝟏 + 𝒉𝟐 𝟐𝑺 𝟐𝟎𝟎 𝑲= − 𝑨

𝟐

𝒉𝟏 + 𝒉𝟐 𝑨𝟐

when 𝑺 < 𝑳

𝟐

when 𝑺 > 𝑳

Where: 𝐿 = length of vertical curve (m) 𝐾 = length of vertical curve in metres for 1% change in grade (m) 𝐴 = algebraic grade change |𝐺2 − 𝐺1 | (%) 𝑆 = sight distance (m) ℎ1 = driver eye height, as used to establish sight distance (m) (Table 5.1) ℎ2 = object height, as used to establish sight distance (m) (Table 5.1)

– To determine minimum vertical curve length, set 𝑺 = 𝑺𝑺𝑫.

CVEN 2401: Sustainable Transport and Highway Engineering

29

Vertical Curve Design Calculations •

Crest Vertical Curve Design: A note on overtaking sight distance (OSD) – If overtaking is allowed (2-lane, 2-way road sections) the vertical curve length for a curve should be compared with the OSD. – Overtaking sight distance is only a factor in crest curves. Sag curves offer adequate visibility to conduct passing manoeuvres. – Vertical Curve Length (L): 𝑳 = 𝑲𝑨 – Overtaking Sight Distance Criteria 𝑺𝟐 𝑲= when 𝑺 < 𝑳 𝟐 𝟐𝟎𝟎 𝒉𝟏 + 𝒉𝟐 𝟐𝑺 𝟐𝟎𝟎 𝑲= − 𝑨

𝒉𝟏 + 𝒉𝟐 𝑨𝟐

𝟐

when 𝑺 > 𝑳

Where: 𝐿 = length of vertical curve (m) 𝐾 = length of vertical curve in metres for 1% change in grade (m) 𝐴 = algebraic grade change |𝐺2 − 𝐺1 | (%) 𝑆 = sight distance (m) 𝒉𝟏 = 𝒉𝟐 = driver eye height, as used to establish sight distance (m) (Table 5.1)

– To determine minimum vertical curve length, set 𝑺 = 𝑶𝑺𝑫. CVEN 2401: Sustainable Transport and Highway Engineering

30

Vertical Curve Design Calculations •

Crest Curve Vertical Design (AGRD3) – Minimum crest curve lengths (𝑆 < 𝐿)

CVEN 2401: Sustainable Transport and Highway Engineering

31

Vertical Curve Design Calculations •

Crest Vertical Curve Design (AGRD3) – Appearance criteria: “ At very small changes of grade, a vertical curve has little influence other than appearance of the profile and may be omitted.”

CVEN 2401: Sustainable Transport and Highway Engineering

32

Vertical Curve Design Calculations •

Sag Vertical Curve Design (AGRD3) – Vertical Curve Length (L) 𝑳 = 𝑲𝑨 – Sight Distance Criteria 𝑺𝟐 𝑲= when 𝑺 < 𝑳 𝟐𝟎𝟎 𝒉 + 𝑺𝒕𝒂𝒏(𝒒) 𝑲=

𝟐𝑺 𝟐𝟎𝟎 𝒉 + 𝑺𝒕𝒂𝒏(𝒒) − when 𝑺 > 𝑳 𝑨 𝑨𝟐

Where: 𝐿 = length of vertical curve (m) 𝐾 = length of vertical curve in metres for 1% change in grade (m) 𝐴 = algebraic grade change (𝑔2 − 𝑔1 ) (%) 𝑆 = sight distance (m) ℎ = mounting height of headlights (m) (taken as 0.65m) 𝑞 = elevation angle of beam 1 degree (+upwards) (𝑞 = 1°) – To determine minimum vertical curve length, set 𝑺 = 𝑺𝑺𝑫.

CVEN 2401: Sustainable Transport and Highway Engineering

33

Vertical Curve Design Calculations •

Sag Vertical Curve Design (AGRD3) – Vertical Curve Length (L) 𝑳 = 𝑲𝑨 – Appearance and Comfort Criteria – “A person subjected to rapid changes in vertical acceleration feels discomfort. To minimise such discomfort when passing from one grade to another, it is usual to limit the vertical acceleration generated on the vertical curve to a value less than 0.05 times the acceleration due to gravity”. 𝑽𝟐 𝑲= 𝟏𝟐𝟗𝟔𝒂 Where: 𝐿 = length of vertical curve (m) 𝐾 = length of vertical curve in metres for 1% change in grade (m) 𝑎 = vertical acceleration (m/sec2) = 0.05𝑔 𝑉 = speed of the vehicle (km/h) 𝑔 = acceleration due to gravity = 9.81 m/sec2

CVEN 2401: Sustainable Transport and Highway Engineering

34

Vertical Curve Design Calculations •

Sag Vertical Curve Design (AGRD3) – Vertical Curve Length (L) 𝑳 = 𝑲𝑨 – Overhead obstruction criteria (if overhead obstructions are present) 𝑲=

𝑺𝟐 𝟐𝟎𝟎

𝑯 − 𝒉𝟏 + 𝑯 − 𝒉𝟐

𝟐

when 𝑺 > 𝑳

Where: 𝐿 = length of vertical curve (m) 𝐾 = length of vertical curve in metres for 1% change in grade (m) 𝐻 = height of overhead obstruction 𝑆 = sight distance (m) ℎ1 = truck driver eye height (2.4m) ℎ2 = object height (0.6m) – To determine minimum vertical curve length, set 𝑺 = 𝑺𝑺𝑫.

CVEN 2401: Sustainable Transport and Highway Engineering

𝑯

35

Vertical Curve Design Calculations •

Sag Vertical Curve Design (AGRD3) – K-values graph.

CVEN 2401: Sustainable Transport and Highway Engineering

36

Minimum Length of Vertical Curves •

Short Calculated Lengths – “When there are changes of grade less than 1%, the calculated curve lengths can be too short for practical construction” o Define the minimum length of vertical curve as the either 𝑳𝐦𝐢𝐧 = 𝟎. 𝟔𝑽 where 𝑽 is the speed of the vehicle in km/hr or use Table 8.10 (for new construction) or Table 8.11 (for reconstruction. o Design road without a vertical curve (See Table 8.12)

CVEN 2401: Sustainable Transport and Highway Engineering

37

Minimum Length of Vertical Curves •

Short Calculated Lengths – “When there are changes of grade less than 1%, the calculated curve lengths can be too short for practical construction” o Define the minimum length of vertical curve as the either 𝑳𝐦𝐢𝐧 = 𝟎. 𝟔𝑽 where 𝑽 is the speed of the vehicle in km/hr or use Table 8.10 (for new construction) or Table 8.11 (for reconstruction. o Design road without a vertical curve (See Table 8.12)

CVEN 2401: Sustainable Transport and Highway Engineering

38

Minimum Length of Vertical Curves •

Short Calculated Lengths – “When there are changes of grade less than 1%, the calculated curve lengths can be too short for practical construction” o Define the minimum length of vertical curve as the either 𝐿min = 0.6𝑉 where 𝑉 is the speed of the vehicle in km/hr or use Table 8.10 (for new construction) or Table 8.11 (for reconstruction. o Design road without a vertical curve (See Table 8.12)

CVEN 2401: Sustainable Transport and Highway Engineering

39