Subgrade CharacterizationFor Concrete

Pavement Design

Fig. 2. Approximate interrelationships of soil classification and bearing values

Slab is treated as an elastic plate where the response of the supporting soil medium can have considerable effect on the analysis.

The inherent complexity of real soil characterizations has led to idealized models which provide certain aspects of pavement response under specific loading and boundary conditions.

Concrete Pavements: Slab-on-grade-type Structures

P

Most Common Foundation ModelsWinkler Foundation

P(x,y)=k w(x,y)

w

Figure 1Elastic Foundation

P

(a)

Figure 2

Winkler Elastic Solid

These theories:

1. Provide significantly different responses of the pavement system.2. Represent different assumptions to characterize soil support.3. Use parameters (E and k) that have different units.

E: F/L2

k: F/L2/L

Considerable effort has been expended in attempting to establish correlation between k and Eo of the soil.

Platek

PlateEo

Winkler Elastic Solid

4k

Dk

Deflection (Concentrated load; infinitely large plate)

8DPw

2k

Subgrade Stress

2k

z 8Pσ

Bending Stress

2r r

kr

σ 6m /h

2P 1m 1 Log γ 14π γ 2

3

r 2 3o

P Ehσ 0.366 1 Log 0.266h k b

s

o 2s

Ek2 1

2eP

w 3 3D

3

e

o2o

2D;C

EC1

2e

2e

z 33P

93Pσ

Design Tip

Part II - The modulus of Subgrade Reaction, kIn our previous discussion of the AASHTO pavement design method we discussed

the influence of reliability as defined by the 1986 AASHTO Guide for Design of Pavement Structures on pavement design. The variable used in the Rigid pavement thickness determination that accounts for the amount of support supplied by the roadbed and subbase is called the Modulus of Subgrade Reaction or “k”. For this discussion the variables names and definitions we will be using are as follows:

The Roadbed Resilient Modulus, MR (psi)The Subbase Elastic Modulus, Esb (psi)The Modulus of Subgrade Reaction, k (pci)The Composite k value, kc (pci) k value on Rigid Foundations k’ (pci)Loss of Support LS (no units)

Each variable is used to describe the quality of the foundation or a factor contributing to the quality of the foundation beneath the pavement slab.

What exactly is the purpose of a foundation system? According to the AASHTO guide, a foundation system should provide a uniform, stable, and permanent support system for the pavement. The support system minimizes the damaging effects of frost action, prevents pumping of fine-grained soils at joints, cracks, and edges of the pavement slab, and provides a working platform for construction equipment. Subbase layers are frequently used to increase the total strength of the foundation system.

The determination of k of a soil according to ASTM requires the placement of a 30 inch diameter rigid steel plate on the soil and the application of a static load. Knowing the amount of weight with which the plate is loaded and the measured deflection of the plate into the soil, k can be calculated as follows:

Where:

The Pressure on the soil is equal to the loading of the rigid steel plate divided by its area.

From this calculation you can see why k is expressed in units of pounds per cubic inch, (pci). It would probably make more sense if k was expressed as pounds per square inch deflection, (psi/in2/in).

Pressure on the soilk=Deflection of the soil

The history associated with the letter notation of k goes back to basic physics. You may remember, k is the variable name of the spring constant; and when Dr. Westergaard modeled the interaction of a rigid slab resting on a soil, he treated the soil as a bed of springs with a stiffness of “k.”

With that brief history, you are probably wondering why the roadbed resilient modulus is listed? After all, it is also a property indicating the soil strength. The reason for this is that the new AASHTO Guide depends heavily on MR for the design of bitummous pavements. Many of the new mechanistic design methods use MR to characterize the strength of the soils rather than k.

Since the resilient modulus can be calculated from a laboratory test and correlated to k, the new AASHTO Design Guide uses MR. This value replaces the former Soil Support Value used to describe soil strength in the previous editions of the guide. The analytical relationship between a soils resilient modulus and it Modulus, of Subgrade Reaction is a follows:

k=MR/19.4

The actual k used in the design of concrete pavements is modified to reflect increased support resulting from frozen subgrade, high-quality subbases and loss of support to develop a “composite k” (kc).

For equal deflections:8DPw

2k

2eP

w 3 3D

3/41/3

1/3

4/3

4/32/34/3

2k

2e

k5.977DC

DC0.1673k

C28DD33k

338

From the AASHTO Design Guide

Generally, Eo =1500 CBR =1500 (5)=7500 psi

k = 385 psi/infrom PCA (OJ Porter chart), CBR=5 yields k=140 psi

In these designs, NO correlation between Eo and k is apparent

19.4Ek o

Es vs kSolid lines are for Equal Slab deflection used on slab Theory 42

50

40

30

20

10

0 0 100 200 300 400

1. Westergaard for Dense Liquid2. Hogg for Elastic Solid

h=14”

h=12”

h=10”

h=8”

AASHTO Resilient Modulus=19.4xk

Slab Thickness

k (psi/in)

E sub g

(x 1

0-3) p

si

From Boussinesq for a circular loaded area (on subgrade)

Therefore Eo and k can be correlated only if the size of the loaded area is taken into account.

Pa

rzEo

ho=

1.5akwP1.5aE

0.5for E

1.5paw

o

o

(no plate stiffness involved)

a

E,kEo constant

k constantFor a=30”/2 & k=100 Eo=2250= psi

2o o

3e

E C 1

2D C

For Ec=4,000,000 psi

c=0.15 & o=.5

C=13,896

Eo=0.75(13,896)=10,422 psi

Difference is in the slab stiffness and the effect on the loaded area.

For a concrete slab on subgrade, the size of the loaded area is a function of slab stiffness Equating max Deflection:

D33P

8DP 2

e2k

0 1 2 3 4 5 6 1 2 30

0.05

0.10

0.15

0.192

Solid) (Elast.PlDw 2

o

(Hertz)PlDw 2

o1.241ll

oo xr/l xr/l

0.125 (Hertz)

Equal Deflections

Equal Subgrade Stress

0 1 2 3 4 5 60

0.05

0.10

0.15

0.192

1 2 3id)(Elast.sol

Plp2o

(Hertz)Plp2

o0.806ll

0.125 (Hertz)

oo xr/l xr/l

Note: the rel. between l & lo is diff depending on the respondnce !

0 1 2 34 5 6

x=xo

Mt

=0.3

0

0.05

0.10

0.15

PM

Mr

Equal Bending Stress

Wester

RLE W, T, or Q: Elastic

IF we=wk : w, same max Defl

IF r= b : T, same max bending

IF z= sg : Q, Same max subgrade stress\

ESW______ : Elastic Eo

DeFES_____ : Elastic Surf. Defl

SSES______ : Elastic Subg Stress

BSES______ : Elastic layer radial E

EQRE______ : Equivalent ‘k’ (P/w)

EQRL______ : Equivalent ‘kl’

Different load area/slab stiffness relationships result from equating

• max deflection

• max subgrade stress

•max Bending stress

ConditionsSame Max Defl

Same Max B SSame Max S S

Pavement Thickness (in.)Stiffness Effect on Es vs. K

18

16

14

12

10

8

6

4

2

0 6 7 8 9 10 11 12 13 14

k=100

Soil Modulus (ksi)

*

Maximum Subgrade StressE(S) vs. h

k=50 k=100 k=200 k=400

ELA

STIC

MO

DU

LUS

OF

SOIL

(psi

)(T

hous

ands

)

6 8 10 12

1413121110

98

76

543

21

Thickness (in)

k=50 k=100 k=200 k=400

Thickness (in)

Maximum Bending StressE(S) vs. h

24232221201918171615141312111098765432 6 8 10 12

ELA

STIC

MO

DU

LUS

OF

SOIL

(psi

)(T

hous

ands

)

Maximum DeflectionE(S) vs. h

k=50 k=100 k=200 k=400Thickness (in)

6 8 10 12

45

40

35

30

25

20

15

10

5

ELA

STIC

MO

DU

LUS

OF

SOIL

(psi

)(T

hous

ands

)

Same Maximum Deflection

0

50

100

150

200

250

k= 100, 8" K=400, 8" k= 100, 12" K=400, 12"

Foundation Modulus (pci)

Pave

men

t Stre

ss (p

si)

WestergaardElastic SolidElastic Layer

Same Maximum Deflection

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

k= 100, 8" K=400, 8" k= 100, 12" K=400, 12"

Foundation Modulus (pci)

Subg

rade

Str

ess

(psi

)

WestergaardElastic SolidElastic Layer

Deflection-Time plots for Typical Plate

Load Test on Cohesive Soils

10 Sec 1 2 3 4 5

0.10

0.075

0.050

0.025

==DP 10K =250 psi/inW 0.040

Alternate Time Deflection Curves for Plate

P 10K= = =133 psi/inW 0.075W=0.001 inches in one min

Time, Min

Def

lect

ion,

inch

es