12. Rigid Pavement Analysis.pdf

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


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


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


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 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




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)

Documents

12. Rigid Pavement Analysis.pdf