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Flexible Pavement Stress Analysis
Dr. Antonis Michael
Frederick University
Notes Courtesy of Dr. Christos Drakos, University of Florida
Topic 3 – Flexible Pavement Stress Analysis
Need to predict & understand stress/strain distribution within the pavement structure as they (σ & ε) relate to failure (cracking & rutting)
Numerical Models• Need model to compute deflections (δ) and strains (ε)• Numerous models available with different:
– Capabilities–––
IDEAL MODEL
Predicts Input Parameters
•• Strains
••••
Topic 3 – Flexible Pavement Stress Analysis
1. Available Models
Most widely used• Reasonable Results• Properties Relatively Simple to Obtain
• Multilayer Elastic Theory• Finite Element Methods• Viscoelastic Theory ( • Dynamic Analysis (inertial effects)• Thermal Models (temperature change)
Falling Weight Deflectometer
• Small trailer• Dropping Weight• Geophones• Deflection Basin
Uses elastic theory to predict the deflection basin for the given load. Then
Topic 3 – Flexible Pavement Stress Analysis
Topic 3 – Flexible Pavement Stress Analysis
2. Multilayer Elastic Theory
E1, ν1
E2, ν2
E3, ν3
∞
z1
z2
z3
a = radius
q = pressure
Point APoint B
Assumptions (p. 60):• Each Layer
– Continuous––– Linearly – Material is – Finite thickness (except last layer)
Topic 3 – Flexible Pavement Stress Analysis
2. Multilayer Elastic Theory (cont.)
Assumptions (cont.):• Surface stresses
– Circular– Vertical– Uniformly distributed
• Full • Each layer continuously supported
Point APoint B
E1, ν1
E2, ν2
E3, ν3
∞
z1
z2
z3
a = radius
q = pressure
Why do we want full friction between layers?
2in
lbspsi =
610−×==in
innmicrostraiµε
Units Guidelines
• Stress:– Reported in psi:
• Strain:– Reported in µε:
• Deflections:– Reported in mils:
1000
inmils =
For homework, exams, and projects, you are expected to convert all of your answers to these units.
Topic 3 – Flexible Pavement Stress Analysis
Topic 3 – Flexible Pavement Stress Analysis
3. One-layer System
3.1 Based on Boussinesq (1885)Half-space:
Z
σz
σz
X
P
r
zσz
2
2
52
zz
P
z
r1
1
2π
3σ
+
=
Point load on an elastic half-space• Examine σ distribution along Z & X
Where:– σz = – r = – z = – P =
Topic 3 – Flexible Pavement Stress Analysis
3.2 One-layer Solutions (Foster & Ahlvin)
Figures 2.2 – 2.6
Developed charts to determine σz, σt, σr, τrz & w (ν=0.5)
• Axisymmetric loading:
– σz =
– σr =
– σt =
– τrz =
– w =
•
2a
q
z
r
σz
σr σt
τrz
a
q
02a 1a3a
0
2a
1a
Offset
Depth
Topic 3 – Flexible Pavement Stress Analysis
3.2 One-layer Solutions (Foster & Ahlvin)
Charts follow similar outline
Depth (z) and offset (r) are
Topic 3 – Flexible Pavement Stress Analysis
3.2.1 Vertical Stress
Given:– Load, P = 9000 lbs– Pressure, q = 80 psi
a
q
r=6”
z=6”σzFind:– Vertical Stress, σz @ z=6” & r=6”
First, we need to
Topic 3 – Flexible Pavement Stress Analysis
3.2.1 Vertical Stress (cont)
z/a = 6/6 =1r/a = 6/6 =1
Topic 3 – Flexible Pavement Stress Analysis
Deflection Profile
3.2.2 Deflection
Flexible Plate Rigid Plate
q qRubber Steel
Ground Reaction
Topic 3 – Flexible Pavement Stress Analysis
3.2.2 Deflection (cont.)
a = 6”
q = 80 psi
∞
h1= 4”
h2= 8”
h3= 12”
Pavement Structure
How can we use one-layer theory to estimate the deflection of the system?
We can assume
Basically:
A
For this case (assuming
Topic 3 – Flexible Pavement Stress Analysis
3.2.2 Deflection (cont.)
Given:z/a=24/6=4r/a=0
Topic 3 – Flexible Pavement Stress Analysis
3.2.2 Deflection (cont.)
a = 6”
q = 80 psi
∞
h1= 4”
h2= 8”
h3= 12”
A
• Examine two cases:
Clay Dense Sand
E=2,500 E=25,000
Subgrade quality
Topic 3 – Flexible Pavement Stress Analysis
• Purpose of the pavement structure:– Protect the subgrade; reduce
4.1 Vertical Stress
• Vertical stress on top of subgrade; important in pvt design as it accounts
• Allowable σz depends on
4. Stresses & Strains for Design
– To combine
– Effect of
Vertical compressive
εc
a
q
∞
h1
h2
E1
E2
E3
Topic 3 – Flexible Pavement Stress Analysis
4.2 Tensile Strain
Horizontal ‘principal’ strain (εt) used as a design criterion.
a
q
∞
h1
h2
E1
E2
E3
ε
• Tensile strain at the bottom of AC layer; used in pvt design as the
• Two types of strain:– Overall minor – Horizontal ‘principal’ strain, εt (not
Topic 3 – Flexible Pavement Stress Analysis
4.2.1 Overall Principal Strains
• Based on all 6 components of normal and shear stresses – σx, σy, σz, τxy, τxz, τyz− Solve cubic equation to get σ1, σ2, & σ3
− Then calculate principal strains ( )( )2133 σσνσE
1ε +−=
Minor principal strain (ε3) considered to be
a
q
AC Minor principal strain (ε3)
Topic 3 – Flexible Pavement Stress Analysis
4.2.1 Horizontal ‘Principal’ Strain
• Based on the horizontal normal and shear stresses only – σx, σy, τxy
• Horizontal ‘principal’ strain (εt) is
–• Maximum • Always acts on the horizontal plane• Used by the program KENLAYER to predict fatigue failure
a
q
AC
Topic 3 – Flexible Pavement Stress Analysis
Developed solutions for:• Vertical deflections (flexible & rigid)• Vertical stresses (limited # of cases)
− σ & δ highly dependent on
5. Two-layer Theory (Burmister)
Topic 3 – Flexible Pavement Stress Analysis
5.1 Two-Layer Deflections
• In one-layer theory we assumed that all layers could be represented as one– δsurface = δtop of the subgrade
• For two-layer theory we have:– Vertical – Vertical
a
q
∞
h1 E1
E2
• Flexible
• Rigid
5.1.1 Surface Deflections
Topic 3 – Flexible Pavement Stress Analysis
5.1.2 Surface Deflections Example
a=6”
q=80 psi
∞
6”E1=50,000 psi
E2=10,000 psi
Given:h1/a=6/6=1E1/E2=5
Topic 3 – Flexible Pavement Stress Analysis
5.1.3 Interface Deflections ExampleF
h1/a
Offset
a=6”
q=80 psi
∞
6”E1=50,000 psi
E2=10,000 psi
Given:h1/a=6/6=1 ;r/a=0E1/E2=5
• For the same example as above
Topic 3 – Flexible Pavement Stress Analysis
5.1.4 Surface Vs Interface Deflections
Compare the results from the example:• Surface deflection = 43 mils• Interface deflection = 40 mils
Topic 3 – Flexible Pavement Stress Analysis
5.2 Two-Layer Vertical Stress
a=6”
q=80 psi
∞
h1E1=500,000 psi
E2=5,000 psi
Maximum allowable σc for clay = 8 psi
Given:σc/q=0.1E1/E2=100
Topic 3 – Flexible Pavement Stress Analysis
5.2 Critical Tensile Strain
a=6”
q=80 psi
∞
6”E1=200,000 psi
E2=10,000 psi
e = εt= critical tensile strainGiven:E1/E2=20h1/a=1
εt
Str
ain
Fact
or,
Fe
Topic 3 – Flexible Pavement Stress Analysis
6. Failure Criteria
6.1 Fatigue Cracking Model
• Based on Miner’s cumulative damage concept– Amount of damage expressed as a damage ratio predicted/allowable
load repetitions
( ) ( )2 3f f
f 1 t 1N f E− −= ε
( ) 5f
d 4 cN f−= ε
f1 =
f2 & f3 =
6.2 Rutting Model
• Allowable number of load repetitions related to εc on top of the subgrade– Does not account for failure in other layers
f4 & f5=
( ) 4.4779
d cN 1.365 10−−= × ε
( ) ( )3.291 0.854
f t 1N 0.0796 E− −= ε
Topic 3 – Flexible Pavement Stress Analysis
7. Sensitivity Analysis
• Sensitivity analyses illustrate the effect of various parameterson pavement responses
• Variables to be considered:– Layer – Layer
Topic 3 – Flexible Pavement Stress Analysis
7.1 Effect of HMA Thickness
Tensile Strain (εt)••
Compressive Strain (εc)•
Topic 3 – Flexible Pavement Stress Analysis
7.2 Effect of Base Thickness
Tensile Strain (εt)•
Compressive Strain (εc)•
Topic 3 – Flexible Pavement Stress Analysis
7.3 Effect of Base Modulus
Tensile Strain (εt)•
•
Compressive Strain (εc)•
Topic 3 – Flexible Pavement Stress Analysis
7.4 Effect of Subgrade Modulus
Tensile Strain (εt)•
Compressive Strain (εc)•