1. Subgrade Characterization For Concrete Pavement Design

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

3. Concrete Pavements: Slab-on-grade-type Structures
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.

4. Most Common Foundation Models
Winkler Foundation P
P(x,y)=k w(x,y) w
Figure 1
Elastic Foundation P
(a)
Figure 2

5. Winkler Elastic Solid
These theories:
1. Provide significantly different responses of the pavement system Represent different assumptions to characterize soil support 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.
Plate
Plate k Eo

6. Winkler Elastic Solid
Deflection (Concentrated load; infinitely large plate)
Subgrade Stress
Bending Stress

7. Part II - The modulus of Subgrade Reaction, k
Design Tip
Part II - The modulus of Subgrade Reaction, k
In 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.

8. What exactly is the purpose of a foundation system
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).

9. 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).

10. For equal deflections:

11. From the AASHTO Design Guide Generally, Eo =1500 CBR
From the AASHTO Design Guide Generally, Eo =1500 CBR =1500 (5)=7500 psi
k = 385 psi/in from PCA (OJ Porter chart), CBR=5 yields k=140 psi
In these designs, NO correlation between Eo and k is apparent

12. Es vs k Esubg (x 10-3) psi k (psi/in) 50 40 30 20 10 0 100 200 300 400
Solid lines are for Equal Slab deflection used on slab Theory 42 50 40 30 20 10
1. Westergaard for Dense Liquid 2. Hogg for Elastic Solid
h=14”
h=12”
h=10”
h=8”
Esubg (x 10-3) psi
Slab Thickness
AASHTO Resilient Modulus=19.4xk
k (psi/in)

13. From Boussinesq for a circular loaded area (on subgrade)P a
Eo  ho= z r
(no plate stiffness involved)
Eo constant
For a=30”/2 & k= Eo=2250= psi
E,k
k constant a
Therefore Eo and k can be correlated only if the size of the loaded area is taken into account.

14. For Ec=4,000,000 psi
c= & 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:

15. Equal Deflections
0.05
0.10
0.15
0.192
0.125 (Hertz)

16. Equal Subgrade Stress
0.05
0.10
0.15
0.192
Note: the rel. between l & lo is diff depending on the respondnce !
0.125 (Hertz)

17. Equal Bending Stress Mr
x=xo
0.05
0.10
0.15 Mt
=0.3

18. 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’

19. Different load area/slab stiffness relationships result from equating
max deflection
max subgrade stress
max Bending stress

20. Pavement Thickness (in.)
Soil Modulus (ksi) 18 16 14 12 10 8 6 4 2
Conditions
Same Max Defl
Same Max B S *
Same Max S S
k=100
Pavement Thickness (in.)
Stiffness Effect on Es vs. K

21. Maximum Subgrade Stress E(S) vs. h14 13 12 11 10 9 8 7 6 5 4 3 2 1
ELASTIC MODULUS OF SOIL (psi) (Thousands)
Thickness (in)
k=50
k=100
k=200
k=400

22. Maximum Bending Stress E(S) vs. h24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2
ELASTIC MODULUS OF SOIL (psi) (Thousands)
Thickness (in)
k=50
k=100
k=200
k=400

23. Maximum Deflection E(S) vs. h45 40 35 30 25 20 15 10 5
ELASTIC MODULUS OF SOIL (psi) (Thousands)
Thickness (in)
k=50
k=100
k=200
k=400

26. Deflection-Time plots for Typical Plate Load Test on Cohesive Soils
0.10
0.075
0.050
0.025
W=0.001 inches in one min
Deflection, inches
Alternate Time Deflection Curves for Plate
10 Sec
Time, Min
Deflection-Time plots for Typical Plate Load Test on Cohesive Soils