Subgrade Models for Rigid Pavements

Subgrade Models for Rigid Pavements

Development of theories for analyzing rigid pavements include the choice of a subgrade model. When the chosen model closely represents the actual situation, the resulting analyses are useful tools for design engineers.

The model for analyzing a concrete pavement consists generallyof an elastic plate that rests on a foundation model. Whereas the equations for plates that are subjected to mechanical load and thermaleffects are well established , the modeling of the supporting base is not.

The simplest and most popular model for the foundation base isthe one proposed by Winkler (1867), consisting of closely spaced independent springs. For static loads, the response of this base modelis the same as the response of a liquid base. An early solution for the infinite plate on a Winkler base subjected to a vertical load was presented by Hertz (1884). Solutions for a semi-infinite plate with a “free” boundary - for interior, edge, and corner loads - were presentedby Westergaard (1926, 1939, 1947). These solutions have been at the

heart of slab design since the 1920’s. According to Iounnides et al.(1985), several investigators have noted repeatedly that althoughthe Westergaard solution agreed fairly well with their observation for the interior condition, it failed to give even a close estimate of the response for edge and corner loads.

It appears that this failure has less to do with the accuracy of theWestergaard solutions than with the inadequacy of the Winkler foundation model, which consists of closely spaced independent springs. From an interior load, the interaction between these springs is provided by the pavement, whereas along the “free” edge, these interactions - which take place in an actual subgrade - are missing. They are included in the Pasternak and Kerr subgrade models.

kw

P

w

k

G

P

(I) Winkler Model (II) Pasternak Model

P

G

kl

ka

(III) Kerr Model

0

1.0

0.6

0.2

1.4

q=3472

lb/in2

Dis

pla

cem

ets,

in

-0.4 -0.2 0 0.2 0.4

x/L

Kerr ModelPasternak ModelFinite Element

Winkler Model

L/2 L/2

q

H(E,v)

π

-0.4 -0.2 0 0.2 0.4

0

1.0

0.6

0.2

1.4

1.8

2.2

Pasternak ModelKerr Model

Winkler Model

Finite Element

lb/in

H(E,v)

L/2 L/2

F F

F

x/L

Dis

pla

cem

ents

, in

π

-0.4 -0.2 0 0.2 0.4

0

200

600

1000

Pasternak Model

Winkler Model

Kerr Model

Finite Element

x/L

Mo

men

ts,

Kip

s in

/in

-5000

-4000

-3000

-2000

-1000

0 -0.4 -0.2 0 0.2 0.4

x/L

Mo

men

ts,

Kip

s in

/in

Pasternak Model

Kerr Model

Finite Element

Winkler Model

P

Most Foundation models are in between these two.

Winkler FoundationP(x,y)=k w(x,y)

w

Figure 1

Elastic Foundation

P

(a)

Figure 2

Semi-infinite continuum (isotropic solid) more difficult mathematically. Does not represent all foundation materials {relative to deflection away from loaded region}.

Improvement of foundation modeling is approached on the basis of assuming some kind of interaction between spring elements.

P

Figure 2 Continued

(b)

Filonenko-Borodich Foundation

• Top end of springs are connected to a stretched elastic membrane of constant- tension T

• Equilibrium in the z direction of a membrane element yields a load displacement relation:

Laplace operator in the x and y direction

Also,Het’enyl Foundation

D= Flexural rigidity of a plate

Spring interaction provided by an embedded plate

wTκwp 2

:

wDκwp 22

T T

stretched membrane, plate in bonding, or shear layer

Spring interaction is a function of T

Figure 3

Assumes existence of shear interaction between spring elements. Connection of springs by incompressible vertical elements which deform by transverse shear.

Pasternak Foundation

δx

δw

w

x x

z

Figure 4

Shear Intaction

dxδx

δww

dxx

xzτ

dyδ

δNN

y

yy

y

y+dy

y

x

xNy

PNx

dxδ

δNN

x

xx

qs

Vertical Equilibrium

Figure 4 Continued

dxx

Pasternak Foundation

Valasov Foundation

Uses a ‘continuum’ approach by means of a variational method. Certain restrictions are imposed on possible deformations of the elastic layer. A load-deflection relation is obtained which can be re-dined to match the Pasternak expression.

Plate Element

Element of Shear Layer

dxδ

δNN

x

xx dy

δ

δNN

y

yy

qs

y

y+dyy

x x+axx

qs

msy

msx

dxdy

Figure 12

Assuming homogenous & isotropic in the x-y plane

Shear force per unit length of shear layer

δy

δwGdzτN yzy10

δx

δwGdzτN xzx10

Equilibrium Equation

0qpδy

δN

δxδN

syx

κwqs

wGκwp 2

Effect of shear interaction of vertical element

GGG yx

δx

δxGγGτ xzxxz

δy

δwGγGτ yzyyz

Reissner Foundation• Based on equations of a continuum.•Assuming in-plane stresses throughout the foundation layer tobe small:

and that horizontal displacements are zero at the upper and lowersurfaces of the foundation layer yields:

where E,G = elastic soil constantsH = soil layer thicknessIf c1= k & c2=G then the pasternak foundation is matched for a constant p

H

Efc

1

3

HGc f

2

0τσσ xyyx

p4c

cpwcwc 2

1

2221

p4κ

GpwGκw 22

0

A consequence of the above assumption is that the shear stresses zx and xy are independent of z and thus are constant throughout the depth of the foundation for a given surface point (x,y); physically a nonrealistic result particularly for relatively thick foundation layers. However, in view of the fact that foundation models are introduced to study the response of the foundation surface to loads and not the stresses caused within the foundation, this deficiency may in general be of no serious consequence.

Deflection profiles for typical highway pavement under a point load

Interior Loading Deflction Profiles

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 20 40 60 80 100 120 140 160

Horizontal offset (in.)

Vert

ical

Def

lecti

on

(in

.)

Elastic Solid

Dense Liquid

Kerr

Two-Parameter

Figure 2a

Edge Loading Deflection Profiles

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 10 20 30 40 50 60 70 80 90 100

Horizontal Offset (in.)

Ve

rtic

al

De

fle

cti

on

(in

.)

Dense Liquid

Elastic Solid

Kerr

Two-Parameter

Figure 2a

Corner Loading Deflection Profiles

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 20 40 60 80 100 120 140 160

Horizontal Offset (in.)

Ve

rtic

al D

efl

ec

tio

n (

in.)

Kerr

Two-Parameter

Elastic Solid

Dense Liquid

Figure 2a

Deflection profiles for typical highway pavement under a uniformly distributed load

Interior Loading Deflection profiles

0

0.002

0.004

0.006

0.008

0.01

0.012

0 20 40 60 80 100 120 140 160

Horizontal Offset (in.)

Ver

tica

l D

efle

ctio

n (

in.)

Elastic Solid

Kerr

Two-Parameter

Dense Liquid

Figure 2b

Edge Loading Deflection Profiles

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 10 20 30 40 50 60 70 80 90

Horizontal Offset (in.)

Ver

tica

l D

efle

ctio

n (

in.)

Elastic Solid

Two-Parameter

Kerr

Dense Liquid

Figure 2b

Corner Loading Deflection Profiles

0

0.02

0.04

0.06

0.08

0.1

0.12

0 20 40 60 80 100 120 140 160

Horizontal Offset (in.)

Ver

tica

l D

efle

ctio

n (

in.)

Elastic Solid

Two-Parameter

Kerr

Dense Liquid

Figure 2b

Shear LayerPlate in Bending

Figure 7p

q q

Reaction Distribution

Bending Moment

Figure 8

Plate R

FoundationRegion

F

tn

Figure 9

D

c

G

k

qPlate in Bending

Shear Layer

Figure 10 (a)

q

p

Figure 10 (b)

q

r

a a

Figure 11