12. rigid pavement analysis.pdf
TRANSCRIPT
Rigid Pavement Analysis
2
Critical Rigid Pavement Responses
Pavement responses that have a direct
bearing on individual distress modes
Critical responses occur at specific locations
within the pavement structure
Slab edge
Slab corner
3
Key Rigid Pavement Distresses
Fatigue cracking (bottom-up)
Fatigue cracking (top-down)
Joint faulting (undoweled and doweled JPCP)
Punchouts (CRCP)
4
Fatigue Cracking (Bottom-Up)
5
Fatigue Cracking (Bottom-Up)
Critical response is the tensile stress at the bottom of the PCC slab
Location of critical stress is usually at mid-slab location at the bottom of the slab
Traffic and climatic forces contribute to critical stresses
Critical Stress Location
Traffic
Critical Stress Location
6
Fatigue Cracking (Top-Down)
7
Fatigue Cracking (Top-Down)
Critical response is
tensile stress at the top
of the PCC slab
Critical location varies
with axle configuration
Traffic and climatic
forces contribute to this
critical responseCritical stress location
Shoulder
Traffic
8
Joint Faulting
9
Joint Faulting (Undoweled)
Critical responses are
deflections of loaded
and unloaded slab
Critical locations are at
slab corners
Traffic, foundation
erosion, and moisture
contribute to this critical
response
Critical Stress
Critical locations
Foundation: Base and Subgrade
Traffic
10
Joint Faulting (Doweled)
Critical responses and
responses locations
same as for undoweled
slabs
Dowel-Concrete
bearing stresses are
used by some
researchers
Foundation: Base and Subgrade
Traffic
P
Critical Response Location
11
Punchouts (CRCP)
12
Punchouts (CRCP)
Critical slab structural
response is tensile stress
Critical location is at the
top of the slab between
two adjacent cracks
Crack spacing, material
properties, subgrade
friction, and external
loads affect this response
Pavement
EdgeCritical Stress
Location
Traffic
Transverse
Crack
Punchout
13
Sources of Slab Stresses
Traffic Loads
Thermal Curling
Moisture Warping
Shrinkage from Curing
Contraction and Expansion from
Temperature Changes
14
Traffic-Induced Stresses and Deflections
Major source of stresses in pavements
Traffic load creates a bending stress (tensile
stress at the bottom of the slab)
Repeated applications can result in fatigue
cracking
Critical location for traffic loading is generally
along outside slab edge
15
Temperature-Induced Curling Stresses
Differential temperatures at the top and
bottom of the PCC slab induce curl stresses
Positive (daytime) temperature gradients curl
the slab down at the corners
Negative (nighttime) temperature gradients
curl the slab up at the corners
16
Diurnal Temperature Changes
Positive gradient
Negative gradient
Warmer
Cooler
Cooler
Warmer
17
Slab Curling
18
Temperature-Induced Stresses and Deflections
Positive gradients produce tensile stresses
at the bottom of the pavement slab
Critical when wheel load at slab edge
Negative gradients produce tensile stresses
at the top of the pavement slab
Critical when wheel load at slab corner
Magnitude depends on slab properties,
support conditions, and thermal gradient
19
Temperature Gradients
Temperature
differentials are usually
expressed linear
temperature gradients
Field studies have
shown that temperature
gradients are non-linear Dep
th, in
52 56 60 64 68 72
Temperature, oF
Top of PCC Slab
0
6
3
9
6 AM 11 AM7 PM
3 PM
Linear idealization
of 3 PM gradient
20
Built-in Temperature Gradient
Temperature gradient in the slab just prior to
final set will show up as built-in temperature
gradient of the opposite sign
For daytime construction, the residual
gradient is negative
Positive built-in gradients offset diurnal
daytime gradients and add to nighttime
gradients
21
Warping Stresses
Caused by differences in moisture content
between the top and bottom of the slab
Greater moisture at top of slab results in
downward warping, and vice versa
Moisture contents through slabs in:
Wet climates - fairly constant
Dry climates - top is drier than the bottom
Difficult to measure strains due to moisture
22
Moisture Warping
Slab top wetter than slab
bottom
Slab bottom wetter than slab top
23
Variations in Deflection Responses Due to Moisture
MEASUREMENT DATE
1
4
2
3.5 m
3.5 m
0.23 m2
4
1
1987 1988 1989
1
4
2
3.5 m
3.5 m
0.23 m2
4
1
MONTHLY RAINFALL (mm)
19871987 19881988 19891989
24
Drying Shrinkage Stresses
Loss of moisture as concrete cures leads to
shrinkage of slab
Shrinkage resisted by friction of the base,
which induces the stress development
Introduction of joints in slab reduces
magnitude of shrinkage stresses
25
Temperature Shrinkage Stresses
Daily and seasonal temperature changes
cause PCC slab to expand/contract
Frictional force between slab and base
creates stresses in slab
Magnitude of stress estimated by subgrade
drag formula.
26
Effect of Volume Change on Concrete
Frictional stress
CL L
h
1 (unit width)
Where
fa=coefficient of friction, h: thickness(in), L: slab length(ft)
rc: density of concrete(lb/ft3)
2
a cc
f hLh
Tensile force
27
Combined Load and Curling Stresses
Stresses result from traffic loading and
climatic forces
Combined stress state determined by
superimposing environmentally related
stresses on load-associated stresses
Load and thermal stresses are usually
considered
Calculating Responses In PCC
Pavements
29
Structural Analysis of Rigid Pavements
Analyzing rigid pavement systems is a
complex problem involving aspects of
geotechnical and structural engineering
Structural engineering problems – complex
geometry simple support conditions
Pavement problems – simple geometry complex
support conditions
30
Requirements for Structural Modeling of Rigid Pavements
Accurate representation of pavement layers and foundation (subgrade)
Ability to model slab curling
Ability to model cracks and joints in the pavement
Ability to model multi-wheel loading
Ability to model multiple slabs
Ability to model multiple layers
31
Methods for Structural Analysis
Closed-form equations Westergaard’s slab on Winkler foundation
Slab on elastic solid foundation
Finite Element Methods (FEMs)
FEM-based analytical expressions Zero-Maintenance equations for edge stress
NCHRP 1-26 equations for load and curl
RPPR equation for edge stress
NAPCOM equation for corner deflection
32
Westergaard’s Solutions
Stress and deflection equations for three loading conditions Interior
Edge
Corner
Solutions were also available for curl stresses at edge and interior locations
Solutions based on medium-thick plate resting on a Winkler foundation
33
Westergaard’s Assumptions
Slab is homogeneous, isotropic elastic solid
Fully characterized by E and m
Shear forces ignored
Infinite slab dimensions
No load transfer
Winkler foundation
Circular contact area for interior and corner;
semicircular or circular contact area for edge
34
Westergaard’s Loading Conditions
Corner loadingEdge loading
Interior loading
35
Important Concepts—Winkler Idealization
Foundation type
originally proposed in
1867
Subgade is represented
using a series of
independent springs
Modulus of subgrade
reaction or k value is
used to represent
subgrade
PCC Slab
Subbase
Subgrade
PCC Slab
Subbase
Subgrade
PCC Slab
Subbase
Subgrade
PCC SlabPCC SlabPCC Slab
36
Important Concepts—Radius of Relative Stiffness, l
Radius of relative stiffness was introduced to
measure the stiffness of the slab relative to
the subgrade
42
3
112 μ)(
Ehl
where,
E = PCC modulus of elasticity
m = PCC Poisson’s ratio
37
0.6
2
3 2[1 ( ) ]c
P a
h
Stress Deflection
Corner Loading by Westergaard (1939)
a: radius of contact area
l : radius of relative stiffness
k: modulus of subgrade reaction
2
2[1.1 0.88( )]c
P a
k
38
Modified Westergaard eq. by FEM
a: radius of contact area c=1.772al : radius of relative stiffnessk: modulus of subgrade reaction
0.72
2
3[1 ( ) ]c
P c
h
Corner Loading by Ioannides (1985)
39
Closed Form Solutions
도로포장공학, 구미서관(2004), 남영국저
40
2
3(1 )(ln 0.6159)
2i
v P
h b
hhab 675.06.1 22
b=a
Stress
Else (a<1.724h),
Deflection
Interior Loading by Westergaard (1939)
102
0.3164log 1.069i
P
h b
2
2
1{1 [ln( ) 0.673]( ) }
8 2 2i
P a a
k
when v=0.15
b: radius of equivalent distribution of pressure (in)
a: radius of contact area, v: Poisson's ratio
l : radius of relative stiffness, k: modulus of subgrade reaction
If a>1.724h,
41
Interior Loading by Losberg (1960)
Stress
Deflection
42
Stress
Edge Loading by Westergaard (1939)
b: radius of equivalent distribution of pressure (in)
v: Poisson's ratio l : radius of relative stiffness
k: modulus of subgrade reaction
Deflection
10 102
0.8034log 0.666log 0.034e
P a
h a
3
0.76 0.42 1.21e
v avP
Eh k
Westergaard eq.
43
Edge Loading by Ioannides et al. (1985)
l
a
ka
Eh
he
2
4
2
3
2)21(50.0
3
484.3
100ln
)3(
)1(3m
m
m
m
Equation for edge stress due to semi-circular load
Equation for edge deflection due to semi-circular load
a 0.17 + 0.323 - 1
)k h (E
) 1.2 + (2 P = 2
0.5 3
0.5
e
mm
44
Curling Stress in Slab Edgeby Westergaard (1939)
2
T E C = T
e
Equation for curl stress at slab edge
) + ( 2 + 2
2 - 1C
tanhtan
sinhsin
coshcos
where,
T = PCC coefficient of thermal expansion
T = Temperature differential between slab top and bottom
45
Curling Chart by Bradbury (1938)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 164 8 10 146 122
Ratio L/l
Cu
rlin
g s
tres
s co
effi
cien
t, C
46
Edge Stress due to Traffic and Thermal Loads by Ioannides and Salsilli-Murua (1989)
./.max Westt
.maxt
.West
= maximum combined tensile stress under curling and load
= maximum tensile stress predicted by Westergaard (ΔT=0)
TC
TB
TA
laClaBA
8713.0
6215.1
9152.00.1
/log/ 10
Traffic and Thermal Loading
47
Solutions to Westergaard Equations
Manual Methods
Influence Charts (Pickett and Ray)
Computer Programs (WESTY, WESTER)
PROBLEMS
1) Edge stress calculations—Westergaard
2) Edge stress sensitivity
49
Problem 1– Definition
Given
Slab thickness = 10 in.
PCC modulus of elasticity = 4,000,000 lbf/in2
Poisson's ratio of PCC = 0.15
Wheel load = 9,000 lb
Tire pressure = 80 lbf/in2
Temperature differential = 25 oF
Modulus of subgrade reaction = 100 lbf/in2/in
Slab length =15 ft
Slab width = 12 ft
PCC thermal coefficient of expansion = 5.5 x 10-6/oF
50
Problem 1– Definition
Determine
• Stress and deflection responses for the edge,
interior, and corner loading using Westergaard's
equations
• Curl stress at the slab interior and slab edge
using Westergaard's equations
51
Problem 1– Solution
Use the Westergaard.xls MS Excel® Spreadsheet to
solve the equations
52
Problem 1– Solution
Slab Position
Stress, lbf/in2
Deflection, in.
Curl Stress, lbf/in2
Interior 128.0 0.0060 149.7
Edge (semi-circle) 254.9 0.0196
Edge (circle) 251.7 0.0186
135.5
Corner 168.2 0.0452
53
Problem 2– Definition
Given
Using the inputs in Problem 1, discuss the sensitivity of
the edge stress solution to the following variables:
Slab thickness = 8, 10, and 12 inches
PCC modulus of elasticity = 2, 4, and 6 million psi
PCC thermal coefficient of expansion = 4.5 x 10-6/ oF, 5.5 x 10-6/ oF,
6.5 x 10-6/ oF
Modulus of subgrade reaction, k-value = 50, 100, 200 psi/
Slab length = 10, 15, 20 ft
Temperature differential = 10, 25, 40 oF
54
Stress Versus Thickness
0
100
200
300
400
500
600
6 7 8 9 10 11 12 13
Slab Thickness, in
Ed
ge
Str
ess,
psi
Load Curl Combined
55
Stress Versus PCC Modulus
0
100
200
300
400
500
600
2000000 3000000 4000000 5000000 6000000 7000000
PCC Elastic Modulus, psi
Ed
ge
Str
ess,
psi
Load Curl Combined
56
Stress Versus Temp. Diff.
0
100
200
300
400
500
600
0 10 20 30 40 50
Temperature Differential, oF
Ed
ge
Str
ess,
psi
Load Curl Combined
57
Stress Versus k-Value
0
100
200
300
400
500
600
0 50 100 150 200 250
Modulus of Subg. Reaction (k), psi/in
Ed
ge
Str
ess,
psi
Load Curl Combined
58
Stress Versus Slab Length
0
100
200
300
400
500
600
0 5 10 15 20 25
Slab Length, ft
Ed
ge
Str
ess,
psi
Load Curl Combined
59
Stress Versus PCC Coefficient of Thermal Expansion
0
100
200
300
400
500
600
4.00E-06 5.00E-06 6.00E-06 7.00E-06 8.00E-06
PCC Coef. Therm. Exp., in/in/oF
Ed
ge
Str
ess,
psi
Load Curl Combined
Problem 2–Sensitivity
-40
-30
-20
-10
0
10
20
30
40
-50 -30 -10 10 30 50
Percent Change of Independent Value
Perc
en
t C
ha
ng
e o
f C
om
bin
ed
Ed
ge S
tress
PCC Thickness PCC ModulusTemp. Diff. k-valueSlab Length PCC Coeff. of Therm. Exp
61
Limitations of Westergaard Theory
Only interior, edge, and corner stresses and
deformations can be calculated
Shear and frictional forces on slab surface may not
be negligible
Winkler foundation extends only to slab edge
Assumes slab is fully supported
Does not allow for multiple wheel loads
Load transfer between joints and cracks is not
considered
62
Other Forms of Subgrade Characterization
In the Winkler idealization, shear interaction in the subgrade is ignored
Other theories to more accurately model subgrade have been proposed Slab on elastic foundation (Hogg and Holl)
Two-parameter ―in-between‖ approach (Pasternek, Kerr, Vlasov)
These are difficult to implement in closed-form solutions
Rigid Pavement Analysis Programs
64
Finite Element Methods (FEMs)
The complexity of modeling a slab-joint-
foundation system has contributed greatly to
the popularity of numerical techniques to
analyze rigid pavements
The use of numerical methods has increased
with the advent of modern computers
FEMs are the method of choice today
65
Finite Element Methods (FEMs)
Finite element techniques have been used to
solve problems in civil, mechanical, and
electrical engineering where closed-form
solutions are not readily available
Can handle geometric and load-related
complexities common to rigid pavement
systems
66
General FEM Concepts
Finite element method discretizes a structure as an assemblage of interconnected small parts (elements)
Each element is of a simple geometry and is much easier to analyze than the actual structure
A complicated solution is thus approximated by a model that consists of piecewise-continuous simple solutions
67
Mesh Discretization
Shoulder
Joint
Transverse
Joint
Wheel
Loads
Nodes Finite
elements
68
FEM Options
General purpose programs
Very powerful
Usage requires advanced knowledge of solid
mechanics and mechanics of materials
Careful attention to element selection, mesh
discretization, and geometry and load definition
required
Examples: ABAQUS, LS-DYNA, ANSYS
69
FEM Options
―Pavement-specific‖ programs Usage is simpler
Modeling options function of the program used
Three-dimensional examples: EVERFE
Two-dimensional examples: ILLI-SLAB, JSLAB, KENSLAB
Two-dimensional FE programs are popular today due to their expediency, accuracy, and ease of use
70
ILLI-SLAB
First developed at the University of Illinois
Continuously improved over the last two decades by several researchers
Features Can model up to 10 slabs in x- and y-directions
Several subgrade characterization options
Several load transfer options
Temperature and load analysis
Partial slab-foundation contact modeling
71
J-SLAB
Developed by Construction Technology
Laboratories
Features
Winkler foundation
Temperature and load analysis
Dowel and aggregate load transfer options
Variable support conditions
72
KENSLAB
Developed at the University of Kentucky
Features
Two-layer system can be modeled
Dense-liquid or layered foundation
Load and temperature loading handled
Partial contact modeling
Variable slab thickness
Damage analysis
73
74
KENSLABS Numbering Method
75
ISLAB2000
76
ISLAB2000
Proprietary software of ERES Consultants
Retains all the positive features of ILLI-SLAB
but is more computationally efficient and
user-friendly
Has capabilities that are not available in the
other 2D finite element codes
77
ISLAB2000
Improved input and output formats
Automatic mesh generator
Nonlinear temperature and moisture analysis
―Unlimited‖ number of nodes and layers
―Mismatched joints
Corrected void analysis
Variable bond analysis
More efficient solver
78
Steps for Executing ISLAB2000
Pre-processing
Define pavement geometry, material properties,
joints, loading, and voids
Define finite element mesh
Generate input file
Run ISLAB2000
Post-process results
De f l e c t i o n s
Flat Slab Condition, Tridem Axle
Loading
Flat Slab Condition, Tridem Axle Loading
81
KENSLABS Program
Based on the finite element method, in which
the slab is divided rectangular finite elements
Can be applied to a maximum of 6 slabs, 7
joints, and 420 nodes
Damage analysis up to 12 periods
S t r e s s e s i n Y- d i r e c t i o n
Flat Slab Condition, Tridem Axle Loading
De f l e c t i o n s
Day Time Curling, Tridem Axle Loading
S t r e s s e s i n Y- d i r e c t i o n
Day Time Curling, Tridem Axle Loading
85
Applications of FEMs
Verification of Westergaard equations
Extension of Westergaard equations
Edge stress equation—FHWA Zero-Maintenance
study, NCHRP 1-26 study, FHWA RPPR study
Corner deflection equation—FHWA PRS study
Neural Network based rapid solutions
86
Rapid Solution Methodologies
Several reasons to adopt FE-based rapid
solution methods to obtain critical responses
To facilitate wider adaptation of finite element
solutions in pavement design
To facilitate numerous response calculations
performed in incremental damage analysis based
M-E design
87
Rapid Solution Methodologies
Rapid solution schemes
Regression algorithms—used by many
researchers in the past
Neural networks—becoming more popular
88
Long-Term Dowel Bearing Stresses
Effectiveness of dowel bars depends on
magnitude of dowel-concrete bearing stress
Excessive bearing stresses can lead to
fracture of the concrete, dowel "socketing,"
poor load transfer, and excessive faulting
Bearing stresses can be calculated using
pioneering work done by Friberg or by using
finite element techniques
89
Dowel Bearing Stress
Load Diagram
P
90
EverFE
http://www.civil.umaine.edu/everfe/tutorial_1.htm
http://www.civil.umaine.edu/everfe/tutorial_2.htm
91
FEAFAA
92
FEAFAA Screen Shots
Airplane
Selection
Pavement
Structure
Joint
Modeling3D Mesh
Generation
93
FEAFAA Plots
3D-FEM Mesh Showing 6-Wheel
Gear Occupying 4 Slabs
Contours of Stress in x-Direction
(Top of Slab)