University of Illinois at Chicago
Civil and Materials Engineering Department
CME 402
"GEOMETRIC DESIGN OF HIGHWAY FACILITIES"
Lecture 8 & 9: Cross Section
Fall 2019
1
Cross Section Elements
2
Traffic Volume
▪ Annual Average Daily Traffic (AADT)
• it is the total volume of vehicle traffic of a highway or road
for a year divided by 365 days.
▪ Design Hourly Volume (DHV):
• Officially: the 30th highest hourly volume in a year
(i.e. the hourly volume that is exceeded for 29 hrs/year)
• Practically: a percentage of the Annual Average Daily Traffic
o
For urban highways: DHV ≈ 0.08 to 0.12 AADT
o
For rural highways: DHV ≈ 0.12 to 0.18 AADT
3
Travel lanes
▪ Lane width depends on
▪ Design speed
▪ Design Hourly Volume (DHV)
▪ Lane width affects safety and LOS (Level of Service).
▪ Safety increases as lane width increases up to a width of
about 12 ft.
• Wider lane widths might result in the lanes to be
shared by more than one vehicle, thus reducing safety.
▪ Tables are usually available in manuals for lane widths.
4
Shoulders
▪ Function:
Shoulder
• to provide on area along the
highway for vehicles to stop.
Rumble strip
• to provide lateral support to the pavement structure.
• to be used by emergency vehicles and police
• “Bus on shoulders” during peak hours, only if paved & strong
▪ The usable width of shoulder is the actual width that can be used
and is measured from the edge of the traveled lane.
5
Shoulders
▪ Undivided rural roadways
• Shoulder widths are provided in manual tables.
▪ Divided rural roadways:
• Right shoulder width for all designations and design
speeds: 10 ft
• Left shoulder width for all designations and design
speeds for:
• 4-lane divided roadways: 5 ft
• more than 4 lanes: 8 ft
6
Medians
Function:
• to separate opposing traffic
• to provide refuge for pedestrians
• to provide storage area for left-
and U-turn vehicles
• to provide a recovery area for
out-of-control vehicles
• to reduce headlight glare
• to provide temporary lanes during maintenance
• to provide space for potential additional lanes
7
Medians
▪ The median width is the lateral distance
between the inner (left) edges of the
opposite travel lanes.
▪ Median widths could be between
1-100 ft, that includes the left shoulder.
▪ For median widths above 10 ft, the
alignments of each direction is done independent of the other
▪ Median widths for rural typical sections are normally larger
than those for urban typical sections.
8
Cross-slopes
▪ Straight sections on two-lane highways
are sloped from the middle downward
(crown-section), to provide for drainage.
▪ On multi-lane highways, several arrangements can be used
▪ Sloped section can be straight or curved.
▪ Recommended rate for cross-slope on good pavements is 2%,
which can be increased if the pavement quality is worse.
▪ Shoulder slope is usually higher than the lane slope (4-5%).
9
Cross-slopes
▪ Cross slope is expressed moving outwards from Center Line (CL).
▪ If upwards, then slope is +ve, and if downwards, slope is –ve.
▪ Maximum cross slope:
o
If slope is > 6%, stopped vehicles might slide sideways on ice.
o
If slope is > 8%, moving vehicles might slide sideways on ice
• Therefore, slope is ≤ 8% for rural and ≤ 6% for urban roads.
• If no snow, maximum slope depends on degree of comfort
when the car is stopped, and also on the possibility of
tipping of trucks.
10
Other elements
▪ Side Slopes
• Provide stability for earth work
• In cut sections, a down slope and a drainage ditch may be
added
▪ Right-of-way:
• The total area required for the construction of the highway
• Should accommodate all cross sectional elements
• Should include the area needed for future road expansion
▪ Other:
• Barriers, curbs and gutters, sidewalks, guardrails, bridges,
utility placement, special purpose lanes (passing lanes,
climbing lanes, right turn lanes, left turn lanes, parking
lanes, transit lanes, bike lanes, ramps, etc.)
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Cross Section Elements
12
Earthwork
▪ The construction of a highway in a terrain (existing ground)
involves cutting some earth along some parts of the highway and
filling some earth along other parts.
▪ Its cost constitutes a large portion of the construction cost of the
highway.
▪ In designing and constructing highways we aim at minimizing the
quantities of earthwork (cut/excavation & fill/embankment)
▪ Balancing the cut and fill quantities is required to avoid borrowing
or wasting any material and/or damaging the natural environment
13
Earthwork
Illustration of a typical earthwork cross-section
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Earthwork Cross Section Notations
▪ Cut and fill slopes are usually expressed as dimensionless ratios,
e.g. 2:1 means 2 horizontal and 1 vertical
▪ The points of intersection between the cut and fill slopes and the
existing ground are known as catch points
▪ Normal surveying practice is such that one slope stake is placed at
each catch point, with the horizontal and vertical distances from
the centerline and profile grade are noted on the stake.
15
Earthwork Cross Section Notations
▪ The existing ground elevations used in drawing earthwork cross
sections are usually determined from topographic maps or by a
surveyor in the field.
▪ We assume straight lines between points. Therefore, the surveyor
should choose sufficient points such that this assumption is valid.
▪ At a minimum, the elevations of the catch points and the
intersection of the highway centerline with ground should be
determined.
16
Earthwork Cross Section Notations
▪ The following notation is usually used in
practice for recording Cross section notes:
FYl
FYc
CYr
Xl
0
Xr
where:
• X is the horizontal distance of the point from the centerline.
• Y is the vertical distance of the point from the grade profile.
• The subscript ‘r’ is used for points to the right of the
centerline, while the subscript ‘l’ is used for points to the left
of the centerline, and the subscript ‘c’ is for the point at the
roadway centerline.
• C denotes Cut and F denotes Fill, or simply, treat C as “above
profile grade” and F as “below profile grade”.
17
Calculation of Cross Section Area
▪ Divide the cross section area into triangles and trapezoids.
▪ In the case where the elevations of only three points (usually the
two catch points and centerline point) are available, we can
calculate the cross section area by a simple formula:
18
Calculating cross sectional area using coordinates
▪ In this method, choose any point (preferably
point O (i.e. the top surface of the road at the
Centerline) as the origin, and beside any other
point write down its coordinates with their
algebraic signs with respect to that origin.
▪ Then start from any point (preferably point O), go around clockwise,
end with the start point, and construct a table similar to the one
below. Ys are on top row and Xs are on the bottom row:
Ys
Y1
Y2
Y3
Y4
Y5
Y1
Xs
X1
X2
X3
X4
X5
X1
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Calculating cross sectional area using coordinates
Ys
Y1
Y2
Y3
Y4
Y5
Y1
Xs
X1
X2
X3
X4
X5
X1
▪ Then use the following formula to calculate the area:
Area = 1/2 [(sum of products of the two solid arrows numbers)
-(sum of products of the two dashed arrows numbers)]
▪ In the case of the above example:
Area = 1/2[(Y1X2+Y2X3+Y3X4+Y4X5+Y5X1)-(Y2X1+Y3X2+Y4X3+Y5X4+Y1X5)]
20
Earthworks Transition
21
Earthwork Calculations
In calculating the earthwork between two points 1 and 2 that are
separated by a distance of L:
Let’s call the cross section area at point 1 as A1 and at point 2 as A2:
a) If both cross sections are only cut (c) or only fill (f), the earthwork
volume is calculated using the following formula:
(𝐴1 + 𝐴2 )
𝑉=
𝐿
2
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2)
Earthwork Calculations
b) If both cross sections have both cut and fill parts, such that cut
parts of the two cross sections are on the same side, and both of
fill parts are on the other side of the two cross sections, then the
cut and fill volumes will be calculated as in part a), i.e.:
𝑉𝑐 =
𝐿
𝑉𝑓 =
(𝐴𝑐1 +𝐴𝑐2 )
2
(𝐴𝑓1 +𝐴𝑓2 )
2
𝐿
𝑉𝑓 =
(𝐴𝑓1 +𝐴𝑓2 )
2
𝐿
𝐿
23
Earthwork Calculations
c) If the cross sections are completely
opposite, i.e. one is only cut (c) while
the other one is only fill (f), or vice
versa, one should first locate the
zero point where the cut and fill switch,
and then calculate the earthwork:
𝐿𝑐 =
𝐿𝑓 =
𝐴𝑐
(𝐴𝑐 +𝐴𝑓 )
𝐴𝑓
(𝐴𝑐 +𝐴𝑓 )
𝐿
𝑉𝑐 =
𝐿
𝑉𝑓 =
𝐴𝑐
2
𝐴𝑓
2
𝐿𝑐
𝐿𝑓
24
Earthwork Calculations
d) If the cut parts of the two cross sections are on the opposite (left
and right) sides of the two cross sections, and the fill parts are also
on the other opposite sides of the two cross sections, then the cut
and fill volumes will be calculated for each side separately using
the procedure in part c), i.e. by first finding the zero point
between the stations:
25
Earthwork Calculations
e) If one of the cut or fill parts in one of the cross sections is zero,
then use a combination of the formulas learned in previous parts,
including finding the zero point for the side on which the cut and
fill parts exist. For example, for the case shown below:
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Earthwork Volume
▪ To calculate the volume of cut (or fill) between 2 stations:
• For each station, calculate the area of cut (or fill), A1 and
A2, at their cross sections,
• Compute the approximate volume of cut (or fill) as:
V = (A1+A2)L/2, where:
A1 and A2 = the cut (or fill) areas at the two cross sections
L = the distance between the two stations
Note: This method is not always 100% accurate.
It is accurate only when A1 = A2!
27
Mass Diagram
▪ Mass Diagram (also called Bruckner Curve) is a graph of cumulative
volume of earthwork versus distance from the start point.
▪ Cut volumes have positive sign and fill volumes have negative sign.
▪ We use a Mass Diagram to determine the optimum haul strategies
(where to get/dump the earth) and the optimum earthwork cost.
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Mass Diagram
▪ Mass diagrams are plotted from tables known as Summary Sheets.
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Mass Diagram or Bruckner Curve
In order to construct a Mass Diagram, first provide a table with the
following information, and then use it to draw the Bruckner curve:
a) Write down the chainage of all Stations (as the X values),
b) Calculate the Cut and Fill cross sectional areas at each station,
c) Find the locations of zero points (if any) between successive
stations and add them to the table as new points between the
corresponding stations,
d) Calculate X (the distance of each point from the first station),
e) Calculate the distances between each two successive points,
f) Calculate the Cut volumes between successive points (as a
positive number),
30
Mass Diagram or Bruckner Curve
g) Calculate Fill volumes between successive points (as a negative
number),
h) Calculate the Adjusted Fill volumes between successive points (as
a negative number) using the soil compaction factor,
i) Calculate the net Cut/Fill volumes (algebraic sum of the Cut and
Adjusted Fill volumes) between successive points,
j) Calculate Y (cumulative sum of the net Cut/Fill from the first point
to each point),
k) Having the X (chainage) and Y (cumulative earthwork) values,
draw the Mass Diagram (Bruckner Curve).
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Mass Diagram or Bruckner Curve
32
Mass Diagram
▪
▪
▪
▪
▪
Any horizontal line spanning a loop is called a Balance Line.
The amount of cut and fill within the loop balance each other.
Within the loop, an element dV would be hauled a distance x.
The product of the two, xdV, is referred to as ‘haul’.
Therefore, the total amount of haul is represented in the diagram
by the areas under the curve between given limits
33
Optimum Haul Strategies
▪ The Optimum Haul strategies are based on:
• The properties of the mass diagram,
• Prices of various earthwork items for which contractors are
paid.
▪ Common pay items include:
1. Borrow
It includes the cost of material obtained off the job site to
make a fill.
34
Optimum Haul Strategies
2. Excavation:
In addition to the cost of removing the material from its
existing state, it also includes hauling the material and
placing it in a properly compacted fill.
At the beginning of the haul, there is a “Free Haul
Distance”. Therefore, the cost will be he cost of hauling
material beyond the free haul distance.
3. Waste
It includes the cost of disposing of excess cut material.
35
Optimum Haul Strategies
▪ In order to determine an optimum haul strategy, we need to
determine the most economical location for balance lines on
the mass diagram.
▪ Logically, there is a maximum haul distance beyond which
borrowing material would be cheaper than overhauling it.
▪ This distance is called the Limit of Economic Haul (LEH)
▪ At Limit of Economic Haul (LEH):
Borrow cost = Haul cost
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Optimum Haul Strategies
▪ CostB = CostRateOH*(LEH – F)
where:
CostB = the borrow price $/ft3),
CostRateOH = the overhaul price per mile ($/ft3/Station mile)
F = the free haul distance (mile)
▪ Or:
Limit of Economic Haul: LEH = F + CostB/CostRateOH
The LEH is the longest distance material would ever be hauled,
hence the longest balance line that would ever be used.
37
Optimum Haul Strategies
▪ Locating balance lines for minimum cost is not simple, but 3
rules cover most cases:
1. In all big loops, put a balance line with its length = LEH
• A “big” loop is one that has its LEH distance not
overlapping with any LEH distance of an adjacent loop
38
Optimum Haul Strategies
2. In groups with an even number of loops, in order to
achieve optimum haul strategy, sum of balance lines in
peaks should be equal to sum of balance lines in dip, i.e:
a+c=b+d
• Each balance line is ≤ LEH
39
Optimum Haul Strategies
3. In groups with an odd number of loops, in order to achieve
optimum haul strategy, difference in sum of balance lines
in peaks and dips should be equal to LEH, i.e.:
|(a + c) – b| = LEH
• Each balance line is ≤ LEH.
40
Mass Diagram or Bruckner Curve
41
Mass Diagram or Bruckner Curve
Example: In either of the Mass Diagrams below, what are the total
earthwork to be moved?
42
Correction for Curvature
X is the centroid of the cross section
𝑉𝑝 = 𝑉𝑒 − 𝐶𝑒
𝑉𝑝 = 𝑉𝑒 ± 𝐶𝑒
Vp = Corrected volume
Ve = Uncorrected volume
Ce = Correction coefficient for curvature
𝑉𝑝 = 𝑉𝑒 + 𝐶𝑒
𝐿
𝐶𝑒 =
𝐴 𝑒 ± 𝐴2 𝑒2
2𝑅 1 1
Ce = Correction coefficient for curvature
L = Length
R = Radius
A = Area of section
43
e = ex-centric
Correction for Curvature
44