1. Approaches to Design

2. Past Design Practice Fatigue Based – (Equations)
Serviceability (roughness) Based
Systems Approach

3. Design Components Slab length, thickness, and width Concrete strength
Base/subbase materials
Joint type
Subdrainage
Shoulder type
Use of reinforcement

4. Empirical Approach STATISTICAL REGRESSION MODEL INPUTS SLAB h OUTPUTS
K-VALUE
ESAL
PCC M.R.
OUTPUTS
PSI

5. Mechanistic Approach FATIGUE SLAB DAMAGE STRESS MODEL MODEL INPUTS
SLAB h
K-VALUE
AXLE LOAD & VOL.
PCC M.R.
CALIBRATION
WITH SLAB
CRACKING
OUTPUTS
CRACKING

6. Components of PCC Mechanistic Design
Design Options
Design Inputs
-Subbase -Shoulder -Joint type and spacing -Service level Cracking level Design reliability
-Material Properties K value Modulus of rupture E Value -Traffic level and distribution -Climatic factors
Structural Model
-Select Trial Thickness -Axle load curves and stresses -Curling Stress
Design Iterations
Final Design
Evaluate Design
Fatigue Damage Analysis
-Fatigue Damage vs. performance
-Traffic distribution -Allowable application

7. Critical Stress for Mid Slab Loading
Slab Length (L)
Single Axle Loading
Dowel Bar s w
Hinge Joint
Traffic Lane
Hinge Joint a Do
Agg
Shoulder
Slab Thickness he
Subbase
Subgrade
Critical Stress for Mid Slab Loading

8. Finite Element Slab Layout for Single Axle Load.
15’
12”
24”
15”
24”
12”
12’
24”
Typical Nodal Point
Wheel Load
24”
Typical Element
24”
12”
30”
24”
24”
12”
12”
24”
24”
30”

9. Loading Conditions for Westergaard Equations
(b) a1 a a2

10. Medium Thick Plate Equation

11. Stress Element – Medium Thick PlateZ z zx zy X x xy xz
Notation & Sign
Convention yx y yz Y

12. Elastic, Homogenous, “Medium - Thick Plate”
A. Thickness = 1/20 to 1/100 of L
B. Plate can carry transverse loads by flexure rather than in-plane
force (thin membrane) but not so thick that transverse shear
deformation becomes important.
C. Deformations are small - such that in-plane forces produced
by stretching of the middle surface are negligible.
D. To reduce the three-dimensional stress analysis problem to
two-dimensions, two basic assumptions:

13. 1. Applied stress on the boundary faces of the plate are small
compared to other stresses in the plate. Direct stresses in the
thickness direction is negligible.
Plane Stress Condition:
Specifies the
state of stress

14. 2. A line normal to the middle surface before deformation
remains normal and unstretched after deformation. (Similar to
Bernoulli assumption in engineering beam theory):
Plane Strain Condition:
Is found in terms of
Poisson’s Ratio

15. Stress Acting on a lamina of thickness dz at a distance of z from the
middle surface x xy x dy dz dx yx h z y y

16. Assumptions: classical “medium-thick plate” theory
1. All forces on the surface of the plate are perpendicular to the
surface (i.e. no shear or frictional forces).
2. The slab is of uniform cross-section (i.e. constant thickness).
3. In-plane forces do not exist (i.e. no membrane forces).
4. X-Y plane (neutral axis) is located mid-depth within the slab
(i.e. stresses and strains are zero at mid-depth).

17. 5. Deformation within an elemental volume which is normal to the
slab surfaces can be ignored (i.e. plane strain ).
6. Shear deformations are small and are ignored; shear forces are not
ignored.
7. Slab dimensions are infinite. However empirical guidelines have been
developed for the least slab dimension L, required to achieve the
infinite slab condition.
8. Slab on a Winkler foundation-subgrade is represented as discrete
springs beneath the slab.

18. E. The stress resultants are defined in terms of the stresses:
(per unit length of mid surface)
In Plane Stress
Normal force per unit width -
No membrane forces

19. Transverse Shears:
Force/Unit Length
NOTE: The transverse shears are determined
by statics. They cannot be determined from
the stress - strain relations since we have
assumed xz = yz = 0. This is the same
situation as exists in beam theory. The
transverse shears are necessary for equilibrium
even though the strains associated with them
have been assumed to be zero.

20. Sign Convention for Stressx Mx
Mxy Nx My
Myx
Nxy
Nyx Vx Y Ny Vy
Sign Convention for Stress
Resultants z

21. Bending Moments per unit length
(Positive when compression occurs on top
of slab)
Twisting moment

23. Equilibrium Equations

24. Second order term = 0
(2)
(3)

25. Substituting equation 2 & 3 into equation 1 gives:
Based on statics; no material properties
for beams all
*NOTE: BEAMS

26. M may be statically determinate - depending on the boundary
conditions. The plate is intrinsically indeterminate since there are 3 moment Mx, My, and Mxy and 1 equation of statics; to proceed further, consideration of deformations is required.