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Dr. P. NANJUNDASWAMY Department of Civil Engineering S J College of Engineering Mysore – 570 006 [email protected]
Dr. P. NANJUNDASWAMYDepartment of Civil Engineering
S J College of EngineeringMysore – 570 006
Outline
Introduction
Background on Stress and Strain in Flexible pavements
Approaches for Stress Analysis
Multi-Layer Computer Programs
Introduction
Typical Flexible Pavement Section can be idealized as a multi-layered system
Soil Subgrade
Sub-base course
Base course
Surface course
having different material properties
Introduction
Methods of designing flexible pavements
Empirical with or without a soil test
Limiting shear failure
Mechanistic empirical
Currently, the design is largely empirical
Mechanistic design is becoming more prevalent
Introduction
Mechanistic approach requires the accurate evaluation of
StressesStrainsDeflections
in pavements due to wheel loads
Basics
Stress
Deflection/Deformation
Strain
Stiffness
Poisson’s Ratio
Hooke’s Theory of Elasticity
Principle of Superposition
Approaches
To compute Stresses, Strains & Deflections
Layered elastic methods
Two-dimensional (2D) FE modeling
Three-dimensional (3D) FE modeling
Layered Elastic Approach
Is the most popular and easily understood procedure.
In this method, the system is divided into an arbitrary number of horizontal layers
The thickness of each individual layer and material properties may vary from one layer to the next.
But in any one layer the material is assumed to be homogeneous and linearly elastic.
Layered Elastic Approach
Although the layered elastic method is more easily implemented than finite element methods, it still has severe limitations:
materials must be homogenous andlinearly elastic within each layer
the wheel loads applied on the surface must be axi-symmetric
2D Finite Element Analysis
Plane strain or axis-symmetric conditions
are generally assumed.
It can rigorously handle material anisotropy,
material nonlinearity, and a variety of
boundary conditions – more applicable to
practical situations
Unfortunately, 2D models can not
accurately capture non-uniform tire contact
pressure and multiple wheel loads.
3D Finite Element Analysis
To overcome the limitations inherent in
2D modeling approaches, 3D finite
element models are becoming more
widespread.
With 3D FE analysis, we can study the
response of flexible pavements under
spatially varying tire pavement contact
pressures.
Single Layer Elastic Solutions
P
MaterialE and µ
Vertical Stress
HorizontalRadial Stress
Horizontal Tangential Stress
Shear Stress
Shear Stress
Point LoadHomogeneous Half-Space
Single Layer Elastic Solutions
Cylindrical Coordinates
Boussinesq Theory – Point Load
Stresses
Boussinesq Theory – Point Load
Strains
Boussinesq Theory – Point Load
Deflections
Circular Load – Uniform Vertical Stress
Vertical Stress
MaterialE and µ
Shear Stress
Shear Stress
Homogeneous Half-Space
CircularLoad
Horizontal Tangential Stress
HorizontalRadial Stress
Circular Load – Axis of symmetry
Stresses
At r = 0
Circular Load – Axis of symmetry
Strains
At r = 0
Circular Load – Axis of symmetry
Vertical Deflection
When Poisson’s ratio is 0.5
On the Surface (z = 0)
Circular Load – Uniform Vertical Stress
Foster and Ahlvin Charts (1954)
In the charts
Circular Load – Vertical Stress
After Foster and Ahlvin (1954)
Circular Load – Radial Stress
After Foster and Ahlvin (1954)
Circular Load – Tangential Stress
After Foster and Ahlvin (1954)
Circular Load – Shear Stress
After Foster and Ahlvin (1954)
Circular Load – Vertical Deflection
After Foster and Ahlvin (1954)
Stresses in Layered Systems
Comparison of calculated and measured stressesSource : Herner, HRB, 1955
Two Layer Elastic Solutions
Two Layer Elastic system
p
h E1, µ1
2 a
∞E2, µ2
Two Layer Elastic Solutions
Burmister’s Theory
Vertical stress Distribution (h/a = 1 and μ = 0.5)Source : Burmister, HRB 177, 1958
Two Layer Elastic Solutions
Vertical interface stresses (Source : Huang, 1969)
Surface Deflections - Burmister
Flexible plate
Rigid plate
Surface Deflections - Burmister
E2 / E1
h/a
Vertical surface Deflections (Source : Burmister, 1943)
Vertical Interface Deflections
Vertical Interface Deflections (Source : Huang, 1969)
Vertical Interface Deflections
Vertical Interface Deflections (Source : Huang, 1969)
Vertical Interface Deflections
Vertical Interface Deflections (Source : Huang, 1969)
Vertical Interface Deflections
Vertical Interface Deflections (Source : Huang, 1969)
Equivalent Single Layer Concept
Odemark approximate method
Equivalent Thickness
Utilising Single layer solutions
Equivalent Single Layer Concept
Odemark approximate method
Equivalent Thickness for two layer system
when μ1 = μ2
Utilising Single layer solutions
Three Layer Systems
Three Layer Elastic system
σz2
σ'r2
σr2
σz1
σ'r1
σr1
p
h1E1, µ1
2 a
∞
E2, µ2
E3, µ3
h2
Interface 1
Interface 2
Jones’ Tables
Jones’ Tables
Jones’ Tables
(Source : Jones, 1962)
Jones’ Tables
Stress factors : ZZ1, ZZ2, ZZ1-RR1, ZZ2-RR2
Jones’ Tables
Peattie’s Charts
(Source : Peattie, 1962)
Peattie’s Charts
(Source : Peattie, 1962)
Peattie’s Charts
(Source : Peattie, 1962)
Computer Programs
To calculate stresses, strains and deflections of a layered elastic system
Gradually became more sophisticated in capability to handle
• Linear elastic materials
• Nonlinear elastic granular materials
• Vertical and horizontal loads
• Elastic multilayer systems under multiple wheel loads
• Stress dependent materials
• Finite element linear and nonlinear analysis
Computer Programs
Examples Include :
• BISTRO and BISAR (from Shell)
• ELSYM5 (from Chevron)
• ALIZEIII (LCPC) and CIRCLY (from MINCAD)
• DAMA (from Asphalt Institute)
• SAPIV and ELSYM5 (from University of California)
• ILLI-PAVE (Raad and Figueroa, 1980)
• PDMAP (Finn et al., 1986)
• MICH-PAVE (Harichandran et al., 1989)
• KENLAYER (Huang, 1993)
• Everstress (Washington State DOT, 1995)
Thank you
