Pavement Engineering
Stress and Strain in flexible Pavements
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Flexible pavements • Flexible pavements were classified by a pavement structure having a relatively thin asphalt wearing course with layers of granular base and subbase being used to protect the subgrade from being overstressed. • This type of pavement design was primarily based upon empiricism or experience, with theory playing only a subordinate role in the procedure. 2
Flexible pavements… • However, the recent design and construction changes brought about primarily by heavier wheel loads, higher traffic levels and the recognition of various independent distress modes contributing to pavement failure (such as rutting, shoving and cracking) have led to the introduction and increased use of stabilized base and subbase material.
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HOMOGENEOUS MASS • The simplest way to characterize the behavior of a flexible pavement under wheel loads is to consider it as a homogeneous half-space . • A half-space has an infinitely large area and an infinite depth with a top plane on which the loads are applied . • The original Boussinesq (1885) theory was based on a concentrated load applied on an elastic half space. The stresses, strains, and deflections due to a concentrated load can be integrated to obtain those due to a circular loaded area . 4
HOMOGENEOUS MASS • The theory can be used to determine the stresses , strains, and deflections in the subgrade if the modulus ratio between the pavement and the subgrade is close to unity, as exemplified by a thin asphalt surface and a thin granular base.
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Stress • Force per unit area
Load P s = = Area A • Units: MPa, psi, ksi • Types: bearing, shearing , axial 6
Strain • Ratio of deformation caused by load to the original length of material
e =
Change in Length Original Length
=
DL L
• Units: Dimensionless
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Deflection (D) • Change in length. • Deformation. • Units: mm or linear
D
Figure shows a homogeneous half-space subjected to a circular load with a radius a and a uniform pressure q. The half-space has an elastic modulus E and a Poisson ratio v . A small cylindrical element with center at a distance z below the surface and r from the axis of symmetry is shown . Because of axisymmetry, there are only three normal stresses, and σz, σr ,σt , and one shear stress, rz , which is equal to zr. These stresses are functions of q, r/a, and z/a .
Components of stress under axisymetric loading
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(
Stresses at a point • From theory it can be shown that at a given point within any layer, 9 stresses exist. • These stresses are comprised of 3 normal stresses (σz, σr ,σt) acting perpendicular to the element face and • 6 shearing stresses acting parallel to the face. Static equilibrium conditions on the element show that the shear stresses acting on intersecting faces are equal. Thus 10
Foster and Ahlvin (1954) resented charts for determining vertical stress σz, radial stress σr , tangential stress σt , shear stress rz , and vertical eflection w, as shown in Figures. The load is applied over a circular area with a radius a and an intensity q . Because the Poisson ratio has little effect on stresses and deflection Foster and Ahlvin assumed the half space to be imcompressible with a Poisson ratio of 0.5
Solution by charts
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Stresses at a point After the stresses are obtained from the charts, the strains can be obtained from If the contact area consists of two circles, the stresses and strains can be computed by superposition .
In applying Boussinesq's solutions, it is usually assumed that the pavement above the subgrade has no deformation, so the deflection on the pavement surface is equal to that on the top of the subgrade . 16
Solutions at Axis of Symmetry When the load is applied over a single circular loaded area, the most critical stress , strain, and deflection occur under the center of the circular area on the axis of symmetry, where rz = 0 and σr = σt , so σz = σr are the principal stresses .
Flexible plate The load applied from tire to pavement is similar to a flexible plate with a radius a and a uniform pressure q. The stresses beneath the center of the plate can be determined from
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Solutions at Axis of Symmetry Note that σz , is independent of E and v, and σz, is independent of E. From following Eq we can get
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The vertical deflection w can be determined from
When v = 0 .5, above Eq. can be simplified to
On the surface of the half-space, z = 0 ; from Eq .
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Rigid Plate All the above analyses are based on the assumption that the load is applied on a flexible plate, such as a rubber tire . If the load is applied on a rigid plate, such as that used in a plate loading test, the deflection is the same at all points on the plate , but the pressure distribution under the plate is not uniform.
20 Differences between flexible and rigid plates.
The pressure distribution under a rigid plate can be expressed as (Ullidtz,1987 )
in which r is the distance from center to the point where pressure is to be determine d and q is the average pressure, which is equal to the total load divided by the area . The smallest pressure is at the center and equal to one-half of the average pressure . The pressure at the edge of the plate is infinity. By integrating the point load over the area , it can be shown that the deflection of the plate is A comparison of with above Eq. indicates that the surface deflection under a rigid plate is only 79% of that under the center of a uniformly distributed load . This is reasonable because the pressure under the rigid plate is smaller near the center of the loaded area but greater near the edge 21
Nonlinear mass Boussinesq's solutions are based on the assumption that the material that constitutes the half-space is linear elastic . It is well known that subgrade soils are not elastic and undergo permanent deformation under stationary loads . However, under the repeated application of moving traffic loads, most of the deformations are recoverable and can be considered elastic. Iterative Method To show the effect of nonlinearity of granular materials on vertical stresses and deflections, Huang (1968a) divided the half-space into seven layers, as shown in Figure, 22
Nonlinear mass
After the stresses are obtained, the elastic modulus of each layer is determined from
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in which is the stress invariant, or the sum of three normal stresses ; E is the elastic modulus under the given stress invariant; E0 is the initial elastic modulus, or the modulus when the stress invariant is zero ; and is a soil constant indicating the increas in elastic modulus per unit increase in stress invariant . Note that the stress invariant should include both the effects of the applied load and the geostatic stresses ; it can be expressed as 24
in which σz, σr and σt , 0 r, and at are the vertical, radial, and tangential stresses due to loading ; y is the unit weight of soil ; z is the distance below ground surface at which the stress invariant is computed; and Ko is the coefficient of earth pressure at rest. 25
Approximate Method One approximate method to analyze a nonlinear half-space is to divide it into a number of layers and determine the stresses at the mid height of each layer by Boussinesq's equations based on linear theory.
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• Example • Figure shows a flexible pavement surface subjected to two circular loads, each 10 in. in diameter and spaced at 20 in. on centers. The pressure on the circular area is 50psi. Determine the vertical stress, strain and deflection at point A, which is located 10in. below the center of one circle. (E=10,000 psi and a Poisson ratio =0.5).
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Layered system concept
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Layred system Generally the analytical solution to the state of stress or strain has several assumptions. They are: The material properties of each layer are homogeneous, that is the property at point Ai is the same at point Bi Each layer has finite thickness except for the lower layer, and all are infinite in the lateral directions 29
Layred system… Each layer is isotropic, that is, the property at a specific point such as Ai is the same in every direction or orientation Full friction is developed between layers at each interface Surface shearing forces are not present at the surface The stress solutions are characterized by two material properties for each layer. They are Poisson’s ratio µ and the elastic modulus E. 30
Two – Layer system
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Two – Layer system
• Typical flexible pavements are composed of layers so that the moduli of elasticity decrease with depth. • In the solution of the two – layer problem, certain essential assumptions are made: – The materials in the layers are assumed to be homogeneous, isotropic and elastic. – The surface layer is assumed to be infinite in extent in the lateral direction but of finite depth, whereas the underlying layer is infinite in both the horizontal and vertical direction. – the layers are in continuous contact and that the surface layer is free of shearing and normal stresses outside the loaded area. 32
Two – Layer system… • Stress and deflection values as obtained by Burmister are dependent upon the strength ratio of the layers, E1/E2, • where E1 and E2 are moduli of the reinforcing and subgrade layers respectively
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The vertical stress values under the center of a circular plate for the two-layer system
Vertical stress distribution in a two - layer system . (After Burmister (1958) .)
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Vertical interface stresses for two-layer systems . (After Huang (1969b) . ) 35
Vertical surface deflections for two-layer systems. (After Burmister (1943) )
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Vertical surface deflection Vertical Surface Deflection Vertical surface deflections have been used as a criterion of pavement design. Above Figure can be used to determine the surface deflections for two-layer systems. The deflection is expressed in terms of the deflection factor F2 by
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Vertical surface deflection The deflection factor is a function of E1/E2 and h1/a. For a homogeneous half-space with h1la = 0, F2 = 1, so above Eq. is identical to Eq . when v = 0 .5 . If the load is applied by a rigid plate, then, from Eq ,
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Vertical interface deflection The vertical interface deflection has also been used as a design criterion. Following Figure can be used to determine the vertical interface deflection in a two-layer system (Huang, 1969c) . The deflection is expressed in terms of the deflection factor F by
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Vertical interface deflections for two-layer systems . (After Huang (1969c) .)
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Interface deflection factor Chart
Numbers on curves indicate offset distances in radii.41
Interface deflection factor Chart
Numbers on curves indicate offset distances in radii.42
Interface deflection factor Chart
Numbers on curves indicate offset distances in radii.
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Total Surface Deflection
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Critical tensile strain The tensile strains at the bottom of asphalt layer have been used as a design criterion to prevent fatigue cracking . Two types of principal strains could be considered. One is the overall principal strain based on all six components of normal and shear stresses . The other, which is more popular, is the horizontal principal strain based on the horizontal normal and shear stresses only. The overall principal strain is slightly greater than the horizontal principal strain, so the use of overall principal strain is on the safe side . 45
• Huang (1973a) developed charts for determining the critical tensile strain at the bottom of layer 1 for a two-layer system . The critical tensile strain is the overall strain and can be determined from
in which e is the critical tensile strain and Fe is the strain factor, which can be determined from the charts .
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Single wheel Following Figure presents the strain factor for a twolayer system under a circular loaded area . In most cases, the critical tensile strain occurs under the center of the loaded area where the shear stress is zero. 47
However, when both h1/a and E1/E2 are small, the critical tensile strain occurs at some distance from the center, as the predominant effect of the shear stress. Under such situations, the principal tensile strains at the radial distances 0, 0.5a, a, and 1 .5a from the center were computed, and the critical value was obtained and plotted in above Figure.
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Dual wheels • Because the strain factor for dual wheels with a contact radius a and a dual spacing Sd depends on Sd/a in addition to E1/E2 and hi /a, the most direct method is to present charts similar to above Figure, one for each value of Sd/a .
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Three – layer System • Burmister’s work provided analytical expression for stresses and displacements in three-layer systems.
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• The representative three-layer pavement structure along with the stresses that can be solved by stress factor values provided is shown in figure:
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Three – layer pavement system
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Solutions for the five stresses
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• Vertical stress solutions have been obtained by Peattie and are shown in graphical form:
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Three-Layer Vertical Stress Factor (ZZ1)
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Three-Layer Vertical Stress Factor (ZZ2)
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• The horizontal stress solutions have been obtained from Jones and are shown in tabular form:
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Three-Layer Horizontal Stress Factor
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• The stress values are all along the axis of symmetry of a single load. • It should be noted that the figures and the tables have been developed for =0.5 for all layers. • The sign convention is positive for compression. • While interpolation of the stress factor is necessary for many problem solutions, no extrapolation is allowed. 59
• Both the vertical stress (graphical solutions) and the tabular solutions for the horizontal stresses use the following parameters. • k1 or K1 =E1/E2 k2 or K2= E2/E1 • a1 or A =a/h2 H=h1/h2
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Vertical Stresses • The vertical stresses can be obtained by the diagrams above. From these diagrams, a stress factor value (ZZ1 or ZZ2) is obtained for the particular K1, K2, A and H values of the pavement system. • The stresses are then:
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Horizontal stresses
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