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

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

INPUTS

• SLAB h• K-VALUE• ESAL• PCC M.R.

STATISTICAL

REGRESSION

MODEL

OUTPUTS• PSI

Mechanistic Approach

INPUTS

• SLAB h• K-VALUE• AXLE LOAD & VOL.• PCC M.R.

SLAB

STRESS

MODEL

σ

FATIGUE

DAMAGE

MODEL

n

N

CALIBRATION

WITH SLAB

CRACKING

OUTPUTS

• CRACKING

Components of PCC Mechanistic Design

Design Inputs

-Material Properties K value Modulus of rupture E Value-Traffic level and distribution-Climatic factors

Design Options

-Subbase-Shoulder-Joint type and spacing-Service level Cracking level Design reliability

-Select Trial Thickness-Axle load curves and stresses-Curling Stress

Structural Model

-Traffic distribution-Allowable application

Fatigue Damage Analysis

-Fatigue Damage vs. performance

Evaluate Design

Final DesignDesign Iterations

DowelBar

Subgrade

Critical Stress forMid Slab Loading

SlabThickness

Slab Length (L)

Single Axle Loading

Agg

Subbase

he

w

s

a

Traffi

c Lan

e

Should

erH

inge

Join

t

Hin

ge Jo

int

Do

3

4 2

Eh=

12 1- k

Finite Element Slab Layout for Single Axle Load.

12”

15”

Wheel Load

Typical Element

Typical Nodal Point

15’

12’

30” 24” 24” 12” 12” 24” 24” 30”

12”

24”

24”

24”

24”

24”

12”

Loading Conditions for Westergaard Equations

2 21e 2 2

a a6Pσ 0.489log 0.091 0.027h

(b)

(a)e 2 2

3P1 ν 1 3P1 ν2σ ln γ2π 2 2πh ha

(c)'6

1c 2

a3Pσ = 1-h

a2a

a 1

2e

σ hs= Dimensional stressP

Medium Thick Plate Equation

Z

X

Y

x

yz

zx zy

y

xz

y

x

xy

z

Notation & SignConvention

Stress Element – Medium Thick Plate

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.

0

&0

0

,,

yz

xyyxxz

z

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):

vv yxz

yxz

yzxzz

)(

&

0

Plane Strain Condition:

Is found in terms of

Poisson’s Ratio

h

y zy

dzdx

xy

Stress Acting on a lamina of thickness dz at a distance of z from themiddle surface

yx

x

x

dy

0

0Z

z xz yz

F

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.

0z

E. The stress resultants are defined in terms of the stresses: (per unit length of mid surface)

In Plane Stress

2

2

2

2

2

2

0

0

0

h

h

xyxy

h

h

yy

h

h

xx

dzN

dzN

dxN

Normal force per unit width - No membrane forces

Transverse Shears:

2

2

2

2

0

0

h

x xzh

h

y yzh

V dz

V dz

Force/Unit Length

NOTE: The transverse shears are determinedby statics. They cannot be determined fromthe stress - strain relations since we have assumed xz = yz = 0. This is the samesituation as exists in beam theory. The transverse shears are necessary for equilibriumeven though the strains associated with themhave been assumed to be zero.

Sign Convention for Stress

Resultants

x

Mx

Nx

Vx

Nxy

Mxy

Nyx

MyxMy

Ny

Vyz

Y

Bending Moments per unit length

2

2

2

2

2

2

h

h

xyxy

h

h

yy

h

h

xx

dzzM

dzzM

dzzM

(Positive when compression occurs on top of slab)

Twisting moment

Equilibrium Equations

0

0

0

02 2

Z

yxx x y y

yx

y

yxxx x yx yx

xx x

F

VVpdxdy V dy V dx dy V dx V dy dx

x y

VVp

x y

M

MMM dy M dx dy M dx M dy dx

x y

V dx dxV dx dy V dy

x

02

Using 0

yxxx

yxxx

x

y xyy

MM Vx dxdxdy dxdy dxdy V dxdy

x y x

MMV

x y

M

M MV

y x

Second order term = 0

(2)

(3)

Substituting equation 2 & 3 into equation 1 gives:

pyx

M

y

M

yx

M

x

M xyyxyx

2

2

22

2

2

Based on statics; no material properties

for beams all

*NOTE: BEAMS

py

M

yx

M

x

M yxyx

2

22

2

2 2

0y

pdx

Md

2

2 dVp

dx

dMV

dx

M may be statically determinate - depending on the boundaryconditions. 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.