Advanced Road Profile Analysis-Fundamental Calculus
Transcript of Advanced Road Profile Analysis-Fundamental Calculus
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Advanced Road Profile Analysis-
Fundamental Calculus
PURDUE ROAD SCHOOL
INDIANA, 2014
Christopher R. Byrum, Ph.D., P.E. Soil & Materials Engineers, Inc.
Plymouth, MI Office
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Profile data was “written” onto a paper roll
N.A.S.-N.R.C Publication 1061, “The AASHO Road Test”, 1962
AASHO Profilometer
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AASHO Profilometer
Device measures a continuous SLOPE profile
SLOPE VARIANCE found strongly correlated to expert panel
N.A.S.-N.R.C Publication 1061, “The AASHO Road Test”, 1962
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AASHO data analyzed using Electric-Analog Chart Recorder
N.A.S.-N.R.C Publication 1061, “The AASHO Road Test”, 1962
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Test Site Locations
Over 1,800 profiles analyzed
MEPDG SHRP/LTPP Data
GPS3 and GPS4 Sites
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HIGH-SPEED PROFILER
Height Sensor
Special Vehicle
inside
Accelerometer
Distance Measurement
Computer
For most LTPP Data: *1 Elevation measurement per inch, raw data.
*Elevation reported every 6” (Avg. of surrounding 12, 1” samples)
* 6” Data also has 300-ft Long Wavelength Reduction Filter
50 mph
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PAVEMENT ELEVATION PROFILE:
Uniformly Spaced Point Elevations
LWP
RWP
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IRI Units = L/L Suspension Movement per Mile
Most Common “Roughness Index”
Profile z = f(x)
IRI
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r2 = 0.93
Std Error = 15.7
Error Plot:
0
50
100
150
200
250
300
0 50 100 150 200 250 300
Measured Total Faults
Pred
icte
d
GPS3 Fault Model: No Dowels
-50
0
50
100
150
200
250
300
0 5 10 15 20 25 30 35 40
Pavement Age, years
Fau
lts,
mm
/152m
Measured Faulting
Predicted Faulting
MEPDG IRI and Faulting
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Agg Only Dowels
Range Range INPUT DATA Case 1 Case 2 Case 3 Case 4 Case 5
-0.1 to 0.43 -0.09 to +0.32 Slab Curvature, ft-1
x 1000 0.1 0.15 0.2 0.25 0.3
8 to14.3 6.4 to 13.3 PCC thickness, in. 11 11 11 11 11
2 to 6.6 mill. 3.35 to 5.68 PCC E-modulus, psi 4700000 4700000 4700000 4700000 4700000
460 to1000 466 to 738 PCC Split Tensile, psi 600 600 600 600 600
11 to 20 14 to 30 Joint Spacing, ft 20 20 20 20 20
100 to 680 103 to 376 Subgrade Overburden, psf 220 220 220 220 220
2.5 to 34 4.5 to 27.5 Subgrade w% 12 12 12 12 12
2 to 92 1 to 97 Subgrade P200% 25 25 25 25 25
33 to 181 40 to 170 Days with precip/yr 151 151 151 151 151
0 to 3396 0 to 2280 Freeze Index, degF-days/yr 1211 1211 1211 1211 1211
6 to 58 8 to 57 Ann. Precipitation, in 38 38 38 38 38
1 to 203 2 to 187 Days < 32 degF/yr 164 164 164 164 164
0 to 106 3 to 176 Days > 90 degF/yr 5 5 5 5 5
18 to 1470 18 to 820 Initial KESAL/yr 300 300 300 300 300
-0.05 to 0.2 -0.05 to 0.17 Traffic Growth Rate, %/yr 0.05 0.05 0.05 0.05 0.05
45000 Total Faulting at 20-yr 60 72 87 104 122
Avg. Fault at 20-yr 2.4 2.9 3.5 4.2 4.9
Avg KESAL/yr (15-yr) 425 425 425 425 425
seed value, mm/500 feet 56.54 56.54 56.54 56.54 56.54
delay, yr -9.7 -8.1 -7.7 -7.6 -7.6
growth rate 0.02 0.03 0.03 0.04 0.04
MaxFlt - Cum Trf parameter ######## ######## ######## ######## ########
0
100
200
300
400
0 5 10 15 20 25 30 35 40Pavement Age, yr
To
tal F
au
ltin
g, m
m/5
00
ft Case 1
Case 2
Case 3
Case 4
Case 5
Joints With Dowels
Model Data Range
0
100
200
300
400
0 5 10 15 20 25 30 35 40Pavement Age, yr
To
tal F
au
ltin
g, m
m/5
00
ft Case 1
Case 2
Case 3
Case 4
Case 5
Aggregate Interlock Only-No Dowels
SPREADSHEET
Same Model Form
Different Coefficients
Traffic Model
Data Ranges
Input Table
Trend Graphs
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M EId z
dx
2
2
MEh z
xx
3
2
2
212 1( )
3-D Analyses:
2-D Analyses:
PAVEMENT STRESS ANALYSIS:
Soil-Structure Interaction Engineering
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M EId z
dx
2
2
MEh z
xx
3
2
2
212 1( )
3-D Analyses:
2-D Analyses:
PAVEMENT STRESS ANALYSIS:
Soil-Structure Interaction Engineering
SLAB/BEAM
CURVATURE
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PAVEMENT PROFILE:
A 2-D Slice through a 3-D Solution LWP
RWP
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Profile Elevation, z = f(x)
x
z
Major Lesson #1 Continuity
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Function z = f(x)
x
z Discontinuities
No Fault Fault
Pavements are “Piecewise Continuous Functions”
with Fuzz (texture, measurement error…)
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Function z = f(x)
x
z Points of Inflection
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Function z = f(x)
x
z
Downward Curvature
Upward Curvature
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Function z = f(x)
x
z
SLOPE = “rise/run” =
First Derivative dx
dz
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x
z R, Radius of Curvature 1/ =
CURVATURE of f(x) =
Second Derivative
d z
dx
2
2
1
R
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G P S 3 S i t e 5 5 - 3 0 0 9
- 1 5
- 1 0
- 5
0
5
1 0
1 5
5 0 6 0 7 0 8 0 9 0 1 0 0
D i s t a n c e , m
Ele
va
tio
n,
mm
L W P
R W P
NOTES:
1. Highest IRI of all GPS3 pavements less than 10 years old.
2. High faulting at an early age.
3. Sandy clay subgrade (PI=14.5, w%=13), sand and gravel with 11% P200 base.
4. Built in 1984, 300-350 KESAL/yr, 220 mm PCC, no dowels.
Profile Date: 6/19/1992
ProfileTime: 3:51:44 PM
10± mm
Profile Date: June 19, 1992
Profile Time: 15:51:44
* “Thermally” Curved-Down at this time of day?
* High Level of “Locked-In” Up-Warp and Faulting.
Curved Way Up
R
Example Profile, z = f(x)
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nin
ninn
xx
slopeslopecurvature
nin
ninn
xx
zzslope
d z
dx
2
2 =
dx
dz=
Finite Difference Methods are Ideal
For Constant Interval Profile Data
BREAKING DOWN PROFILE DATA
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Discrete Slope
-0.04
-0.03
-0.02
-0.01
0
0.01
0 15 30 45 60 75Distance, m
Slo
pe,
l/l
Discrete Curvature
-0.3
-0.2
-0.1
0.1
0.2
0.3
0 15 30 45 60 75
Distance, m
Cu
rva
ture
,
1/m
x10
00
Elevation
-15
-10
-5
0
5
10
15
0 15 30 45 60 75Distance, m
Ele
va
tio
n,
mm
LWP
RWP
Upward Translation = “Rocking”
Moment Hinge
55-3009
z = f(x)
dx
dz
d z
dx
2
2
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Discrete Slope
-0.04
-0.03
-0.02
-0.01
0
0.01
0 15 30 45 60 75Distance, m
Slo
pe,
l/l
Discrete Curvature
-0.3
-0.2
-0.1
0.1
0.2
0.3
0 15 30 45 60 75
Distance, m
Cu
rva
ture
,
1/m
x10
00
Elevation
-15
-10
-5
0
5
10
15
0 15 30 45 60 75Distance, m
Ele
va
tio
n,
mm
LWP
RWP
Upward Translation = “Rocking”
Moment Hinge
55-3009
Initial Shape.
Patterns Grow
dx
dz
d z
dx
2
2
z = f(x)
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Consider Profile as a Matrix
X1 L1 R1
X2 L2 R2
X3 L3 R3
. . .
. . .
Xn Ln Rn
Raw Profile
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Split Profile into 2 Sub-Matrices
X1 L1 R1
X2 L2 R2
X3 L3 R3
. . .
. . .
Xn Ln Rn
= +
Raw Profile Discontinuities Continuous
Segments
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= +
Discontinuities Continuous
Segments
Faulting Evaluation
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= +
Discontinuities Continuous
Segments
Slab Curvature, Waves, Tilt, and
Texture Evaluation
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Typical Faulting distress
Analysis of Discontinuities
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Typical Faulting distress
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Fault Size, H
Travel Direction
Structural Discontinuity in
the Road Surface Layer
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How much suspension movement
will occur when the IRI quarter
Car goes over the Fault of Size H?
Fault Size, H
Travel Direction
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Fault Size, H
Travel Direction
Transient Responses with
Exponential/Linear decay,
A function of damping/friction
about 20 m per cycle
about 2 m per cycle
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How much suspension movement
will occur when the IRI quarter
Car goes over the Fault of Size H?
Fault Size, H
Travel Direction
Current MEPDG Design Guide, IRIH = 1.5 H
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How much suspension movement
will occur when the IRI quarter
Car goes over the Fault of Size H?
Fault Size, H
Travel Direction
Current MEPDG Design Guide , IRIH = 1.5 H
Results of My Study, IRIH = 1.75 H
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Site ID: 553008
Profile Date: 7/29/98
ProfileTime: 6:53:01
500-ft Summary Statistics LWP RWP
# Imperfections detected 34 33
Average Size, mm 8.8 11.1
Maximum Size, mm 14.2 16.7
Minimum Size, mm 2.4 4.4
Total Imperfections, mm 298.7 367.8
Total Imperfections, m/km 1.96 2.41
Calculated IRI, m/km 4.42 5.43
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The Primary Cause of IRI- Faults
Faulting vs. Roughness for GPS3 Profiles
IRI(faults) = 1.762(TotalFaults) + 1.104
R2 = 0.83
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Total sum of profile faults, m/km
IRI,
m/k
m
Fault Gain Factor
All GPS3 Data
Y-intercept
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WHAT CAN WE UNDERSTAND
FROM FIRST-DERIVATIVE ANALYSIS
Subgrade- Clay, Silt, Sand, Gravel, Muck, Peat…
Subbase Base
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JOINTED CONCRETE PAVEMENT
UNDER-SLAB EROSION - pumping
Subgrade- Clay, Silt, Sand, Gravel, Muck, Peat…
Subbase Base
Trapped Water
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Traffic Direction
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Traffic Direction
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Traffic Direction
Sediment Transport
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Tilting Data
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553009 19-Jun-92 15:51:44
02040
6080
100120140
160180200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
-0.04
-0.03
-0.02
-0.01
0
0.01
0 50 100 150 200 250 300 350 400 450 500
Distance, feet
Slo
pe
LWP and RWP shown
All-Data
6-in Interval Slope Distribution
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553009 19-Jun-92 15:51:44
02040
6080
100120140
160180200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
-0.04
-0.03
-0.02
-0.01
0
0.01
0 50 100 150 200 250 300 350 400 450 500
Distance, feet
Slo
pe
LWP and RWP shown
All-Data
6-in Interval Slope Distribution
Faults
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553009 19-Jun-92 15:51:44
02040
6080
100120140
160180200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
-0.04
-0.03
-0.02
-0.01
0
0.01
0 50 100 150 200 250 300 350 400 450 500
Distance, feet
Slo
pe
LWP and RWP shown
All-Data
6-in Interval Slope Distribution
Wisconsin 19’-18’-13’-14’ Slab Pattern
Tilted
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553009 19-Jun-92 15:51:44
02040
6080
100120140
160180200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
-0.04
-0.03
-0.02
-0.01
0
0.01
0 50 100 150 200 250 300 350 400 450 500
Distance, feet
Slo
pe
LWP and RWP shown
All-Data
6-in Interval Slope Distribution
Wisconsin 19’-18’-13’-14’ Slab Pattern
Level Tilted
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553009 19-Jun-92 15:51:44
02040
6080
100120140
160180200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
-0.04
-0.03
-0.02
-0.01
0
0.01
0 50 100 150 200 250 300 350 400 450 500
Distance, feet
Slo
pe
All-Data
Slab Slope Data Std. Dev. = f(texture, warp, bumps…)
6-in Interval Slope
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553009 19-Jun-92 15:51:44
02040
6080
100120140
160180200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
-0.04
-0.03
-0.02
-0.01
0
0.01
0 50 100 150 200 250 300 350 400 450 500
Distance, feet
Slo
pe
All-Data
Fault Slope Data
Slab Slope Data
6-in Interval Slope
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0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
Continuous Slabs-Only
All-Data
6-in Interval Slope
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0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
NEW PAVEMENT
Slabs-Only
All-Data
6-in Interval Slope
![Page 54: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/54.jpg)
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
INITIATION OF
FAULTING
NEW
Slabs-Only
All-Data
6-in Interval Slope
![Page 55: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/55.jpg)
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
FAULTING(t)
NEW
Slabs-Only
All-Data
6-in Interval Slope
![Page 56: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/56.jpg)
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
ROCKING(t) FAULTING(t)
NEW
Slabs-Only
SLABS ONLY
Without FAULTS
All-Data
6-in Interval Slope
![Page 57: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/57.jpg)
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
ROCKING FAULTING
NEW
Slabs-Only
SLABS ONLY
Without FAULTS
All-Data
6-in Interval Slope
![Page 58: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/58.jpg)
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
0
50
100
150
200
-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015
Slope Magnitude
# O
bse
rva
tio
ns
Slabs-Only
SLABS ONLY
Without FAULTS
AVERAGE SLAB
SLOPE INDEX
All-Data
6-in Interval Slope
![Page 59: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/59.jpg)
The “mean” slab
Average Fault Spacing, L
L/2
10’
RI H = ½(Rocking Fault Size)
Average Per Slab Volume ½(L/2)(H)(Lane Width)
Per Mile Volume Index = (Per Slab Volume)*5280/L
RI = Slab Rocking Index, reported in inches per 10 feet
Rocking Fault Size = RI(L)/10
Rocking
Deformation
![Page 60: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/60.jpg)
ALL DATA LWP RWP
maximum slope 0.0067 0.0078
minimum slope -0.0250 -0.0335
average slope -0.0001 -0.0001
slope stdev 0.0043 0.0052
median slope 0.0007 0.0010
SLABS ONLY LWP RWP
maximum slope 0.0067 0.0068
minimum slope -0.0057 -0.0045
average slope 0.0011 0.0014
slope stdev 0.0022 0.0022
median slope 0.0013 0.0015
slope trimmed mean (remove outer 40%) 0.0012 0.0015
Slab Rocking Index-inches per 10-ft = 0.142 0.182Average Joint/Crack Spacing, feet = 15.2 15.2
Average Fault Size from Rocking Index, mm = 5.48 7.00
Average Fault Size from Profile Data, mm = 5.28 6.85
Volume Pumped, cubic feet per mile per 12-ft lane = 142 182
Average, cubic feet per mile per 12-ft lane = 162
![Page 61: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/61.jpg)
GPS3 Test Group
0
50
100
150
200
250
300
0 5 10 15 20 25
Pavement Age, yr
Cu
mu
lati
ve
Vo
lum
e P
um
pe
d,
cu
bic
fe
et
pe
r m
ile
0
50
100
150
200
250
0 5 10 15 20 25
Pavement Age, yr
Vo
lum
e p
er
Kilo
-ES
AL
cu
bic
in
ch
es
pe
r m
ile
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
0 5 10 15 20 25
Pavement Age, yr
Cu
mu
lati
ve
KE
SA
L
Generally Constant Rate per ESAL !!!!
![Page 62: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/62.jpg)
For LTPP Site 55-3009
18,900 ESAL per lb-pumped per Slab
![Page 63: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/63.jpg)
WHAT CAN WE UNDERSTAND FROM
SECOND-DERIVATIVE ANALYSIS
Subgrade- Clay, Silt, Sand, Gravel, Muck, Peat…
Subbase
![Page 64: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/64.jpg)
JOINTED CONCRETE PAVEMENT
CURLING AND WARPING
Subgrade- Clay, Silt, Sand, Gravel, Muck, Peat…
Subbase Base
Trapped Water
![Page 65: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/65.jpg)
55-3009
dx
dz
d z
dx
2
2
z = f(x)
![Page 66: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/66.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
What is the Curvature of This?
GPS3 55-3009
“Structurally Continuous”
CURLED & WARPED SLAB
Includes: Macro/Mega Texture, Slab Creep, Warp, Curl,
Abrasion, Wheel-Path Location, Sensor Accuracy, Inertial Plane,
Possible Plastic Hinges (pre-hair-line cracking)……….many..
![Page 67: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/67.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
d 0.0141 feet
2)2
(
)(2
Lstringline
d
STRING-LINE METHOD
•Generally under-estimates
•Wont quantify variability
3-Point Circle/Arc
![Page 68: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/68.jpg)
Curvature = ''2
2
zdx
zd (1)
L
zLz
L
dxxzL
mean
)0(')(')(''0
Mean Value Theorem for “Continuous”
Slab Functions from Calculus
![Page 69: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/69.jpg)
361
361.5
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
Slope = -0.00235
Slope = +0.00372
L
L
zLz
L
dxxzL
mean
)0(')(')(''0
![Page 70: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/70.jpg)
361
361.5
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
Slope = -0.00235
Slope = +0.00372
L
zLz
L
dxxzL
mean
)0(')(')(''0
All Have Same Average Curvature
But Different Statistics: std. deviation,
median, max., min…..
SINE-WAVE PERTUBATION
![Page 71: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/71.jpg)
Average Curvature Over These Spans = Zero!
Slope = Zero Slope = Zero
AVG CONCEPT NOT VALID!
•Free Rotation at Hinges, = undefined, infinity or zero
•Vertical Offsets = Faults, = undefined, not connected
• through Fault equal and opposite to Actual Avg Slab
Discontinuity!!! E
levati
on
![Page 72: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/72.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
6th Order
2nd Order
L
zLz
L
dxxzL
mean
)0(')(')(''0
0
L BEST-FIT POLYNOMIALS
•End slopes vary and can be way off
•Wont quantify variability
z = f(x)
![Page 73: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/73.jpg)
SLOPE at Zero SLOPE at L/2 = 0
L, average slab length
Westergaard’s Infinite Strip Shape Function
![Page 74: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/74.jpg)
z’= dx
dz =
nin
ninn
xx
zzslope
z’’= 2
2
dx
zd
nin
ninn
xx
slopeslopecurvature
BCI- For Discrete Profile Data
![Page 75: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/75.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
BCI Method for Single Slabs
End Slope Samples = Average
Std. Deviation = Variation of
34 Curvature Samples
per Wheel Path
L1
![Page 76: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/76.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
BCI Method for Single Slabs
End Slope Samples
32 Curvature Samples
per Wheel Path
L2
![Page 77: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/77.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
BCI Method for Single Slabs
End Slope Samples
30 Curvature Samples
per Wheel Path
L3
![Page 78: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/78.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
BCI Method for Single Slabs
End Slope Samples
28 Curvature Samples
per Wheel Path
L4
![Page 79: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/79.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
BCI Method for Single Slabs
End Slope Samples
26 Curvature Samples
per Wheel Path
L5
![Page 80: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/80.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
BCI Method for Single Slabs
End Slope Samples
24 Curvature Samples
per Wheel Path
L6
![Page 81: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/81.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
BCI Method for Single Slabs
End Slope Samples
22 Curvature Samples
per Wheel Path
L7
![Page 82: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/82.jpg)
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
BCI Method for Single Slab Segments
End Slope Samples
•0.5 to 4 feet from ends
•16 End Slopes per Slab
•8 Curvature Strings per Slab
• 392 Arc Samples
20 Curvature Samples
per Wheel Path
L8
![Page 83: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/83.jpg)
0
500
1000
1500
2000
2500
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
0.5 Interval Range for Curvature (ft-1
)1000
# A
rcs
Global
1-ft
2-ft
3-ft
4-ft
5-ft
6-ft
7-ft
8-ft
GLOBAL STATISTICS-All Arcs
Standard Deviation = 1.67
#arcs = 8753
Std Error of Mean = 0.018
Avg = 0.498
Median = 0.502
TrimMean = 0.498
4-ft Arc STATISTICS
Standard Deviation = 0.46
#arcs = 889
Std Error of Mean = 0.015
Avg = 0.502
Median = 0.501
TrimMean = 0.511
8-ft Arc STATISTICS
Standard Deviation = 0.20
#arcs = 401
Std Error of Mean = 0.010
Avg = 0.448
Median = 0.458
TrimMean = 0.448
1-ft Arc STATISTICS
Standard Deviation = 2.45
#arcs = 1422
Std Error of Mean = 0.065
Avg = 0.499
Median = 0.656
TrimMean = 0.532
NOTE: 3-5 ft arc length classes are best
ft-1
= 0.3048 m-1
Arc Class
GPS3 Site 55-3009 has 33 Slabs in 500 feet
![Page 84: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/84.jpg)
361
361.5
0
0.005
0.01
0.015
0.02
0.025
340 345 350 355 360 365
Distance, feet
Ele
vati
on
, fe
et
Slope = -0.00235
Slope = +0.00372
down
up down up
up
Lets Compare The Methods
![Page 85: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/85.jpg)
-0.001
-0.0005
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
342 344 346 348 350 352 354 356 358 360 362 364
Distance, feet
Cu
rva
ture
, 1
/ft
2nd Order
3rd Order
4th Order
5th Order
6th Order
4-ft Arcs
3-ft Arcs
5-ft Arcs
Constant
Linear
Parabolic
''
2
2
zdx
zd
![Page 86: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/86.jpg)
0
500
1000
1500
2000
2500
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
0.5 Interval Range for Curvature (ft-1
)1000
# A
rcs
Global
1-ft
2-ft
3-ft
4-ft
5-ft
6-ft
7-ft
8-ft
GLOBAL STATISTICS-All Arcs
Standard Deviation = 1.67
#arcs = 8753
Std Error of Mean = 0.018
Avg = 0.498
Median = 0.502
TrimMean = 0.498
4-ft Arc STATISTICS
Standard Deviation = 0.46
#arcs = 889
Std Error of Mean = 0.015
Avg = 0.502
Median = 0.501
TrimMean = 0.511
8-ft Arc STATISTICS
Standard Deviation = 0.20
#arcs = 401
Std Error of Mean = 0.010
Avg = 0.448
Median = 0.458
TrimMean = 0.448
1-ft Arc STATISTICS
Standard Deviation = 2.45
#arcs = 1422
Std Error of Mean = 0.065
Avg = 0.499
Median = 0.656
TrimMean = 0.532
NOTE: 3-5 ft arc length classes are best
ft-1
= 0.3048 m-1
Arc Class
GPS3 Site 55-3009 has 33 Slabs in 500 feet
![Page 87: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/87.jpg)
Avg 0.5/1000 ft-1
= 1/R, R = 2000 ft
For GPS3 Site
55-3009
R
![Page 88: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/88.jpg)
50% = +/- 0.067
80% = +/- 0.121
90% = +/- 0.166
95% = +/- 0.206
Confidence Intervals For Day-Time Average Slab Curvature
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60
Annual Precipitation, inches
Av
g.
Sla
b C
urv
atu
re (
1/f
t)1
00
0Mean Value
50%
80%
90%
95%
RCI
Mean Trend 3 Deg F per inch Gradient
Warp Variation is Huge, down or up
![Page 89: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/89.jpg)
50% = +/- 0.067
80% = +/- 0.121
90% = +/- 0.166
95% = +/- 0.206
Confidence Intervals For Day-Time Average Slab Curvature
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60
Annual Precipitation, inches
Av
g.
Sla
b C
urv
atu
re (
1/f
t)1
00
0Mean Value
50%
80%
90%
95%
RCI
Individual
Test Sites
Average of Cluster = Warp
Variation of Cluster = Curl
![Page 90: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/90.jpg)
“Moving Arcs & Slopes”
Continuous
Segments
1. “Moving” 1-ft to 8-ft long arcs are
calculated along each continuous
segment at 0.5 ft intervals
2. Avg., Std. Dev., Median, Count, Std
Error…. for each Arc Length Class
“Generalized Analysis of
Variation of Slope and Curvature
for Discrete Data”
![Page 91: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/91.jpg)
Elevation
-0.2
-0.1
0
0.1
0.2
0.3
0.4
350 355 360 365 370 375 380 385 390 395 400Distance, ft.
Ele
va
tio
n,
in.
LWP
RWP
3 Faults
3 Faults
55-3009 Classic Real Faults with Fine Texture
Tilt Faults
Curvature
![Page 92: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/92.jpg)
IRITotal = IRIFaults + IRIWarp + IRITexture + IRIWaves
For PCC and AC Pavements
![Page 93: Advanced Road Profile Analysis-Fundamental Calculus](https://reader034.fdocuments.net/reader034/viewer/2022052313/588f03ec1a28abce3a8b66c9/html5/thumbnails/93.jpg)
The Primary Cause of IRI- Faults
Faulting vs. Roughness for GPS3 Profiles
IRI(faults) = 1.762(TotalFaults) + 1.104
R2 = 0.83
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Total sum of profile faults, m/km
IRI,
m/k
m
Fault Gain Factor
All GPS3 Data
Y-intercept
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Avg. Slab Curvature vs. IRI
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Sensitivity Analysis for the IRI Regression Model
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Faulting Avg. Slab Curvature Stdev Curvature All Parameters
IR
I ,
m/k
m
"Average" LTPP GPS3 Profile
High
Low
Very Curved
Flat
None
All High
All Low
High
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IRITotal = IRIFaults + IRIWarp + IRITexture + IRIWaves
For PCC and AC Pavements
IRITexture, m/km = 0.065(6 Texture Band Size, mm)
2mm (very fine texture) < 6 < 10 mm (very coarse texture)
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0.2 mm Sample Interval
0.5 mm Laser Footprint Diameter
MiDOT Texture Laser Sample Pattern Used
No use having really short spacing with big footprint, low resolution,…..
……………………..
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I-69 Tex F 12:34:09 9-Jul-03
ASTM MPD, mm = 1.486
ASTM ETD, mm = 1.389
ASTM Peak Value, mm = 1.721
ASTM Avg Height, mm = 0.235
Texture Tortuosity, mm/mm = 1.161
Macro Median Value, mm = 0.410
Macro Spline StDev, mm = 0.849
Residue StDev, mm = 0.251
# of Sharp 1.1 mm (+) = 22
Avg Size of 1.1+, mm = 2.24
Total Sum of 1.1+, mm = 49.38
Avg Spacing of 1.1+, mm = 8.70
CVT for 1.1 = 2.80
% too smooth = 0.078
4mm "smooth" stdev, mm = 0.20
Curvature 0.2 stdev = 91.49
0.4 stdev = 30.11
0.6 stdev = 16.84
0.8 stdev = 11.16
1.0 stdev = 8.03
1.2 stdev = 6.27
1.4 stdev = 5.06
1.6 stdev = 4.35
SRI(0.3 without), lb/lb = 0.427
SRI(0.3 with), lb/lb = 0.381
%Contact = 68.4%
Tire Bottom Elev, mm = -0.073
Volume Beneath Tire, in^3 = 0.468
Area Beneath Tire, in^2 = 0.067
# Voids beneath rubber = 15
Avg. Void Width, in = 0.14121
Avg Void Depth, in = 0.03156
0.5" Flow Distance Index, mm = 333.4
Avg Shear Stress, psi = 15.17
Median Shear Stress, psi, = 13.80
Shear Stress StDev, psi = 11.30
Max Shear Stress, psi, = 66.35
MacroTexture + Residue (microtexture + scanner "noise")
-3.000
-2.000
-1.000
0.000
1.000
2.000
3.000
0 25 50 75 100 125 150 175 200
Distance, mm
He
igh
t, m
m
Texture Height Distribution
0
100
200
300
400
500
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3Height, mm
Fre
quency
1 mm Spline
1 mm Median
+/- 50% line
Spline Residue
Texture Elev. Proportion Curve
-3
-2
-1
0
1
2
3
0 10 20 30 40 50 60 70 80 90 100
% greater than
Heig
ht,
mm
Texture Shear Stress
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20 25 30 35 40 45 50Shear Stress, psi
Fre
quency
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M EId z
dx
2
2
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