Approaches to Design

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

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

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

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

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

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

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”

Loading Conditions for Westergaard Equations (b) a1 a a2

Medium Thick Plate Equation

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

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:

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

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

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

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).

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.

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

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.

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

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

Equilibrium Equations

Second order term = 0 (2) (3)

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

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.