Mechanistic Models

7/27/2019 Mechanistic Models

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LAYERED ELASTIC MODEL

A layered elastic model can compute stresses, strains and deflections at any point in a

pavement structure resulting from the application of a surface load. Layered elastic models

assume that each pavement structural layer is homogeneous, isotropic, and linearly elastic. In

other words, it is the same everywhere and will rebound to its original form once the load is

removed. The origin of layered elastic theory is credited to Joseph Valentine Boussinesq

who published his classic work in 1885. Today, Boussinesq influence charts are still widely

used in soil mechanics and foundation design. This section covers the basic assumptions,

inputs and outputs from a typical layered elastic model.

ASSUMPTIONS

The layered elastic approach works with relatively simple mathematical models and thus,

requires some basic assumptions. These assumptions are:

Pavement layers extend infinitely in the horizontal direction. The bottom layer (usually the subgrade) extends infinitely downward. Materials are not stressed beyond their elastic ranges.

INPUT

A layered elastic model requires a minimum number of inputs to adequately characterize a

pavement structure and its response to loading. These inputs are:

Material properties of each layero Modulus of elasticityo Poissons ratio

Pavement layer thicknesses Loading conditions

o Magnitude: The total force (P) applied to the pavement surfaceo Geometry: Usually specified as being a circle of a given radius (r or a), or the

radius computed knowing the contact pressure of the load (p) and the

magnitude of the load (P). Although most actual loads more closely represent

an ellipse, the effect of the differences in geometry becomes negligible at a

very shallow depth in the pavement.

o Repetitions: Multiple loads on a pavement surface can be accommodated bysumming the effects of individual loads. This can be done because we are

assuming that the materials are not being stressed beyond their elastic ranges.

Figure 1 shows how these inputs relate to a layered elastic model of a pavement system.


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Fig. 1 layered elastic inputs

The use of a layered elastic analysis computer program will allow one to calculate the

theoretical stresses, strains, and deflections anywhere in a pavement structure. However,

there are a few critical locations that are often used in pavement analysis as given in Table 1

and shown in Figure 2.

Table 1. Critical Analysis Locations in a Pavement Structure

Location Response Reason for Use

Pavement Surface Deflection

Used in imposing load

restrictions during spring thaw

and overlay design (for

example)

Bottom of bituminous mix

layerHorizontal Tensile Strain

Used to predict fatigue failure

in the bituminous mix

Top of Intermediate Layer

(Base or Subbase)Vertical Compressive Strain

Used to predict rutting failure in

the base or subbase

Top of Subgrade Vertical Compressive StrainUsed to predict rutting failure in

the subgrade


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Fig. 2 Critical Analysis Locations in a Pavement StructureOUTPUT

The outputs of a layered elastic model are the stresses, strains, and deflections in the

pavement:

Stress. The intensity of internally distributed forces experienced within the pavementstructure at various points. Stress has units of force per unit area (N/m2, Pa or psi).

Strain. The unit displacement due to stress, usually expressed as a ratio of the changein dimension to the original dimension (mm/mm or in/in). Since the strains in

pavements are very small, they are normally expressed in terms of micro strain (10-6).

Deflection. The linear change in a dimension. Deflection is expressed in units oflength (mm or m or inches or mils).

MODULUS OF ELASTICITY

Elastic modulus is sometimes called Youngs modulus after Thomas Young who published

the concept back in 1807. An elastic modulus (E) can be determined for any solid material

and represents a constant ratio of stress and strain (stiffness):


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A material is elastic if it is able to return to its original shape or size immediately after being

stretched or squeezed. Almost all materials are elastic to some degree as long as the appliedload does not cause it to deform permanently. Thus, the flexibility of any object or

structure depends on its elastic modulus and geometric shape.

The modulus of elasticity for a material is basically the slope of its stress-strain plot within

the elastic range (as shown in Figure 1). Figure 2 shows a stress versus strain curve for steel.

The initial straight-line portion of the curve is the elastic range for the steel. If the material is

loaded to any value of stress in this part of the curve, it will return to its original shape. Thus,

the modulus of elasticity is the slope of this part of the curve and is equal to about 207,000MPa (30,000,000 psi) for steel. It is important to remember that a measure of a materials

modulus of elasticity is not a measure of strength. Strength is the stress needed to break or

rupture a material (as illustrated in Figure 1), whereas elasticity is a measure of how well a

material returns to its original shape and size.

Fig. 1 Stress-strain plot showing the elastic range


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Fig. 2 Example stress-strain plot for steel

Stress Sensitivity Of Moduli

Changes in stress can have a large impact on resilient modulus. Typical relationships are

shown in Figures 3 and 4.

Fig. 3 Resilient modulus vs. bulk stress for unstabilized coarse grained materials
http://www.pavementinteractive.org/wp-content/uploads/2007/08/Mr_vs_bulk_stress.gif


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Fig. 4 Resilient modulus vs. deviator stress for unstabilized fine grained materials

E BS = base course resilient modulus ESB = subbase course resilient modulus

MR(or ESG) = roadbed soil (subgrade) resilient modulus (used interchangeably)

EAC = asphalt concrete elastic modulus Source: AASHTO Gui de (1993)

TYPICAL VALUES

Table 2 shows typical values of modulus of elasticity for various materials.

Table 2. Typical Modulus of Elasticity Values for Various Materials

MaterialElastic Modulus

MPa psi

Diamond 1,200,000 170,000,000

Steel 200,000 30,000,000

Aluminum 70,000 10,000,000

Wood 7,000-14,000 1,000,000-2,000,000

Crushed Stone 150-300 20,000-40,000

Silty Soils 35-150 5,000-20,000

Clay Soils 35-100 5,000-15,000

Rubber 7 1,000
http://www.pavementinteractive.org/wp-content/uploads/2007/08/Deviator_stress.gif


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POISSONS RATIO

An important material property used in elastic analysis of pavement systems is Poissons

ratio. Poissons ratio is defined as the ratio of transverse to longitudinal strains of a loaded

specimen. This concept is illustrated in Figure 1.

Fig. 1 Poissons Ratio Example

In realistic terms, Poissons ratio can vary from initially 0 to about 0.5 (assuming no

specimen volume change after loading). Generally, stiffer materials will have lower

Poissons ratios than softer materials (see Table 1). You might see Poissons ratios larger

than 0.5 reported in the literature; however, this implies that the material was stressed to

cracking, experimental error, etc.

Table 1: Typical Values of Poissons Ratio

Material Poissons RatioSteel 0.250.30

Aluminum 0.33

PCC 0.150.20*

Flexible Pavement

Bituminous Concrete 0.35 ()

Crushed Stone 0.40 ()

Soils (fine-grained) 0.45 ()

*Dynamic determination of could approach 0.25 for PCC (Neville, 1975)

= - D / L

Where

= Poissons ratio

D= D/D = strain along thediametrical (horizontal) axis

L= L/L = strain along the

longitudinal (vertical) axis
http://www.pavementinteractive.org/wp-content/uploads/2007/08/Poisson.jpg

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