dawson and arnold isap_2002_paper

8/3/2019 Dawson and Arnold ISAP_2002_Paper

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ARNOLD ET AL.

The application of shakedown approach to granular pavement layers

Greg ArnoldNottingham Centre for Pavement Engineering (NCPE), University of Nottingham, UK

Andrew DawsonNottingham Centre for Pavement Engineering (NCPE), University of Nottingham, UK

David HughesSchool of Civil Engineering, Queens University Belfast, UK

Des RobinsonSchool of Civil Engineering, Queens University Belfast, UK

ABSTRACT: The concept of shakedown was developed for describing the material and structural

response to the repeated application of a cyclic load. In this paper the concept is applied todescribing the behaviour of unbound granular materials in the repeated load tri-axial permanent

strain test. Behaviour was categorised into 3 possible shakedown ranges A, B or C, where A is astable shakedown response and C is incremental collapse while B is intermediate. From this data

test stresses near or at the boundary of shakedown range A and B were determined to define ashakedown limit line stress boundary in p (mean normal stress) q (principle stress difference)stress space. The shakedown limit line was applied as a yield criteria in the finite element model of

the field trial to predict whether or not shakedown occurs in the UGM for a range of asphalt coverthicknesses. For comparison the lower and upper bound shakedown theorems were also applied to

the field trial cross-sections.

1. INTRODUCTION

Most current pavement thickness design guides [HMSO 1994, TRL 1993, Austroads 1992] assumethat rutting occurs only in the subgrade. The thickness of the unbound granular sub-base layers isdetermined from the subgrade condition (California Bearing Ratio and/or resilient modulus) and

design traffic (including traffic during construction). The assumption that rutting occurs only withinthe sub-grade is assumed to be assured through the requirement of the unbound granular materials

(UGMs) to comply with material specifications. These specifications for UGMs are recipe basedand typically include criteria for aggregate strength, durability, cleanliness, grading and angularity,none of which is a direct measure of resistance to rutting caused by repeated loading.

The repeated load tri-axial (RLT), hollow cylinder [Chan 1990] and k-mould [Semmelink et al,1997] apparatuses can in various degrees simulate pavement loading on soils and granularmaterials. Permanent strain tests commonly show a wide range of performances for granular

materials even though all comply with the same specification [Thom and Brown 1989].Accelerated pavement tests show the same results and also report that 30% to 70% of the surface

rutting is attributed to the UGMs layers [Little 1993 and Pidwerbesky 1996].Furthermore, recycled aggregates and other materials considered suitable for use as unbound

sub-base pavement layers can often fail the highway agency material specifications and thus restrict

their use. There is potential of the permanent strain test in the RLT (or similar) apparatus to assessthe suitability of these alternative materials for use at various depths within the pavement (e.g. sub-

base and lower sub-base). Thus, current pavement design methods and material specificationsshould consider the repeated load deformation performance of the UGM layers. Thus, as anapproach to overcome the limitations of current practice, the shakedown concept was used to

determine appropriate pavement design parameters from permanent strain tests of UGMs. These


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design parameters were then used in a ABAQUS Finite Element Model to define the linear DruckerPrager yield criteria for the UGM .

2. SHAKEDOWN CONCEPT

The performance of UGMs in permanent strain RLT tests is highly non-linear with respect to stress.

There are a range of permanent strain responses to stress level and load cycles that cannot bedescribed by a single equation. Several researchers [Wekmeister et al 1999, Sharp and Booker1984] who related the magnitude of the accumulated permanent (plastic) strain to shear stress levelconcluded that the resulting permanent strains at low levels of additional stress ratio, ? ? 1/? 3,

eventually reach an equilibrium state after the process of post-compaction stabilisation (i.e. nofurther increase in permanent strain with increasing number of loads). At slightly higher levels of

additional stress ratio, however, permanent deformation does not stabilise and appears to increaselinearly. For even higher levels of additional stress ratio, however, permanent deformation

increases rapidly and results in failure of the specimen. These range of behaviours can be describedusing the shakedown concept.

The shakedown concept has been used to describe the behaviour of conventional engineering

structures under repeated cyclic loading. In summary, the concept maintains that there are fourcategories of material response under repeated loading:

1. purely elastic, where the applied repeated stress is sufficiently small all deformations arefully recovered and the response is termed purely elastic.

2. elastic shakedown, where the applied repeated stress is slightly less than that required toproduce plastic shakedown. The material response is plastic for a finite number of

stress/strain excursions. However the ultimate response is elastic. The material is said tohave shaken down and the maximum stress level at which this condition is achieved istermed the elastic shakedown limit.

3. plastic shakedown, where the applied repeated stress is slightly less than that required toproduce a rapid incremental collapse. The material achieves a long-term steady state

response, i.e. no accumulation of plastic strain and each response is hysteretic. Once apurely resilient response has been obtained the material is said, once again, to have shakendown and the maximum stress level at which this condition is achieved is termed the

plastic shakedown limit4. incremental collapse or ratcheting [Johnson, 1986], where the applied repeated stress is

relatively large. A significant zone of material is in a yielding condition and the plasticstrains accumulate rapidly with failure occurring in the relatively short term.

For design purposes, the shakedown limit can be used to define a critical surface (or wheel)

stress level between stable and unstable conditions in a pavement. Lower and upper boundshakedown limits can be calculated for boundary value problems using Melans static or Koiterskinematic theorems respectively [Lubliner, 1990 cited in Boulbibane, 1999].

2.1 Shakedown Ranges for UGMs

Dawson and Wellner [1999] have applied the shakedown concept to describe the observedbehaviour of UGMs in the RLT permanent strain test. The results of the RLT permanent strain

tests are reported as either shakedown range A, B or C. This allows the determination of stressconditions that cause the various shakedown ranges for use in pavement design. The shakedownranges are:

?? Range A is the plastic shakedown range and for this to occur the response shows high strainrates per load cycle for a finite number of load applications during the initial compaction period.

After the compaction period the permanent strain rate per load cycle decreases until the responsebecomes entirely resilient and no further permanent strain occurs. This range occurs at low


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stress levels and Werkmeister et al [2001] suggest that the cover to UGMs in pavements shouldbe designed to ensure stress levels in the UGM will result in a Range A response to loading.

?? Range B is the plastic creep shakedown range and initially behaviour is like Range A during the

compaction period. After this time the permanent strain rate (permanent strain per load cycle) iseither decreasing, constant or slightly increasing. Also for the duration of the RLT test the

permanent strain is acceptable and the response does not become entirely resilient. However, it

is possible that if the RLT test number of load cycles were increased to perhaps 2 million loadcycles the result could either be Range A or Range C (incremental collapse).

?? Range C is the incremental collapse shakedown range where initially a compaction period maybe observed and after this time the permanent strain rate increases or remains constant with

increasing load cycles.Cumulative permanent strain versus permanent strain rate plots can be used to aid in determining

the shakedown range [Werkmeister et al 2001]. An example plot is shown in Figure 1 [Figure 3,Werkmeister et al 2001]. Range A response is shown on this plot as a vertical line headingdownwards towards very low strain rates and virtually no change in cumulative permanent strain.

The horizontal lines show a Range C response where the strain rate is not changing or is increasingand cumulative permanent strain increases. A Range B response is between a Range A and Range

C response, typically exhibiting a constant, but very small, plastic strain rate per cycle.

0,000001

0,000010

0,000100

0,001000

0,010000

0,100000

1,000000

0 1 2 3 4 5 6 7 8 9 10

Permanent vertical strain [10-3]

35 140 210 280 350 420

560 630 700

A, B, C shakedown-range

?D[kPa]

A

C

B

Figure 1. Cumulative permanent strain versus strain rate plot showing shakedown ranges

[Werkmeister et al 2001].

3. RLT PERMANENT STRAIN TESTS

A series of Repeated Load Tri-axial (RLT) permanent strain tests were conducted on two NorthernIreland unbound granular materials (UGMs). The two aggregates chosen for testing were located at

the same quarry. They are both greywackes, where one is of lower quality and has a slight redcolour. These aggregates are later referred to as NI Goodand NI Poor. The greywacke, according


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to BS 812 Part 1: 1975 is part of the gritstone group of aggregates that embraces a large number ofsandstone type deposits. Greywacke is one of the major sources of coarse aggregate in NorthernIreland. The aim was to determine the range of stress conditions that cause the various shakedown

range responses either A, B or C. In the RLT permanent strain tests the cell pressure (confinement)was held constant while the vertical load was cycled.

3.1 Tests

To assist in setting the RLT permanent strain test stress conditions, tri-axial static shear-failure testswere undertaken at different cell pressures. The maximum yield stress results were plotted in p

(mean normal) q (deviatoric) stress space (Figure 2). Plots of this type for granular materialstypically result in straight lines in p q stress space and may be referred to as Drucker-Prageryield

criteria.

Yield Stresses Plotted in p-qStress Space

0

100

200

300

400

500

600

700

800

900

1000

0 100 200 300 400 500

p (kPa)

q(kPa)

Best fit yield line

(NI Good UGM)

Best fit yield line

(NI Poor UGM)

Figure 2. Yield stresses from tri-axial static shear-failure tests plotted inp q stress space.

It was assumed that RLT permanent strain tests conducted at stress levels close to the yield lineplotted in p q stress space would result in the highest deformations. Further, permanent strain

results were needed over the full range of stress conditions that occur within a pavement. On thisbasis stress levels were chosen by keeping p constant and varying q. Figures 2 and 3 detail in p qstress space the stress levels and paths used in the permanent strain tests, where q (principle stress

difference) is also the cyclic vertical deviator stress.The initial testing reported in this paper were screening tests designed to determine the best test

conditions for future research in determining the stress conditions that cause the various shakedownranges. These screening tests aimed to cover the full range of stress conditions by using the same


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sample for tests at different stress levels. For the same sample p (mean normal stress) was keptconstant while q (principle stress difference) was increased for each subsequent test of 50,000 loadcycles. This method of testing allows the full spectra of stresses to be tested while only using 3

samples. As stress history is likely to affect results ideally a new sample is required for each newstress level. The affects of stress history will be investigated in the future.

Permanent Deformation (%) - Test Stresses - NI Good

0

100

200

300

400

500

600

700

800

900

1000

0 50 100 150 200 250 300 350 400

p (kPa)

q(kPa)

Stress paths

Best fit monotonic

yield line

Figure 3. RLT permanent strain testing stresses forNI GoodUGM.


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Permanent Deformation (%) - Test Stresses - NI Poor

0

100

200

300

400

500

600

700

800

900

1000

0 50 100 150 200 250 300 350 400

p (kPa)

q(kPa)

Stress paths

Best fit monotonic

yield line

Figure 4. RLT permanent strain testing stresses forNI PoorUGM.

3.2 Results

For each material three multi-stage RLT permanent strain tests were conducted. The results were

analysed to determine cumulative deformation as shown for one test in Figure 5. As discussed todetermine which stress level caused the various shakedown ranges (A, B or C) cumulative

permanent strain versus permanent strain rate plots were produced. For calculation of cumulativepermanent strain it was assumed that at the start of each new test stage the deformations were nil (orzero). Figure 6 shows the cumulative permanent strain plot versus permanent strain rate for the test

results shown in Figure 5.


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Permanent Deformation (%) - Test 2

(p=250kPa, q varies on same specimen) - NI Good

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50000 100000 150000 200000 250000 300000

Load Cycles

PermanentStrain(%)

2a 2b

2c

2d

2e

2f

Test p (kPa) q (kPa) Cell (kPa)

2a 250 402 116

2b 252 457 100

2c 253 511 83

2d 252 555 67

2e 253 607 51

2f 252 658 33

Figure 5. Typical RLT permanent strain test result forNI GoodUGM.

At each test stress level the permanent strain response in terms of shakedown range (A, B or C)

was estimated. For example, the data of Figure 6 indicates that the 2b, 2c and 2d results are of typeB. The 2a result hasnt had enough cycles of loading to fully stabilise as illustrated in Figure 1, but

is rapidly doing so. Thus it is of Type A response. The 2f result is Type C while 2e appears to liesomewhere on a B/C transition. Thus the stress levels associated with the boundaries between Aand B and between B and C response and between B and C response can be defined. The stress

levels at the boundary between the shakedown ranges were plotted in p q stress space. Forcomparison the yield line was also plotted and the results are shown in Figures 5 and 6. From the

results it can be seen that the boundary between shakedown ranges A and B is significantly belowthe yield line and nearly parallel. Stresses that cause a shakedown range A response in the UGMwhere stable behaviour results are ideal from a design perspective. The stress boundary between

shakedown responses B and C are close to the yield line and stresses this high should be avoided inthe UGM pavement layer to avert pre-mature failure in this layer.


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Permanent Strain vs Permanent Strain Rate - NI Good

(Test 2 - p=250 kPa, q varies - 2a lowest to 2f highest)

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Permanent Strain (10-3)

PermanentStrainRate(10-3/loadcycle)

2f

2e

2d

2c

2a

2b

Figure 6. Cumulative permanent strain versus permanent strain rate forNI GoodUGM test 2

(Figure 5).

Shakedown Range (A, B & C) Stresses - NI Good

0

100

200

300

400

500

600

700

800

900

1000

0 50 100 150 200 250 300 350 400

peak p (kPa)

peakq(

kPa

)

Best fit yield line

Best fit range B-C

Best fit range A-B

Figure 7. Shakedown range stress boundaries forNI GoodUGM.


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Shakedown Range (A, B & C) Stresses - NI Poor

0

100

200

300

400

500

600

700

800

900

1000

0 50 100 150 200 250 300 350 400

peak p (kPa)

peak

q

(kPa)

Best fit yield line

Best fit range B-C

Best fit range A-B

Figure 8. Shakedown range stress boundaries forNI PoorUGM.

4. PAVEMENT DESIGN OF UGM LAYER

The design process proposed is simply a check as to whether or not shakedown will occur in the

UGM pavement layer. Should shakedown not occur then the pavement is re-designed and a furthercheck as to whether or not shakedown occurs is undertaken. For example, the pavement design canbe altered by either increasing the asphalt cover, demoting the UGM layer to a lower sub-base or

capping layer, or replacing the UGM with a better quality UGM. For this research project thisdesign process has been applied to the pavement design of a field trial to be constructed in October

2001 in Ballyclare, Northern Ireland, UK. The field trial uses the two different UGMs tested (NIGoodand NI Poor) where the asphalt cover thickness varies from 40 to 100mm. Several designchecks as to whether or not shakedown occurs were undertaken for each UGM and asphalt cover

thicknesses of 0, 40, 60, 80 and 100mm. Figure 9 shows the field trial cross-section and material

parameters (E, ?, ?, ? 0c, ? , c) used for design are detailed in Table 1.


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Asphalt (E, ? ?

UGM

(E, ? , ?, ?0c,? , c)

t (40 to 100 mm)

450 mm

Rigid (rock) subgrade

p

300 mm

Asphalt (E, ? ?

UGM

(E, ? , ?, ?0c,? , c)

t (40 to 100 mm)

450 mm

Rigid (rock) subgrade

p

300 mm

Figure 9. Pavement cross-section of proposed field trial.

Table 1. Field trial pavement design material parameters.

Material Parameters NI Good NI Poor

Asphalt:

Thicknesses, d(mm) 0, 40, 60, 80 & 100 0, 40, 60, 80 & 100

Modulus,E(MPa) 3,000 3,000

Poisons ratio, ? 0.35 0.35

UGM:

Modulus,E(MPa) 200 200

Poisons ratio, ? 0.35 0.35

1Yield line angle, ? 54.9 29.2

1Yield line y-intercept, d 45 140

1Yield stress for ? 3 =0, ? 0c 84.6 172.32Friction angle, ? 45.9 26.5

2Cohesion, c (kPa) 73.4 94.5

1 These parameters are derived from shakedown limit boundary A-B (Figures 7 and 8);

2 These parameters are derived from the yield line at static shear failure (Figure 2).

4.1 Finite Element Modelling

The finite element model developed for the pavement (Figure 8) in ABAQUS is not complicated, atthis stage, as it simply computes principle stresses from linear elastic theory spatially within the

pavement and checks for yield. Should the stresses at any point meet or exceed the yield criteriathen perfectly plastic behaviour is assured to result. If yielding occurs the ABAQUS finite element

model of the pavement either fails to compute a solution and aborts or plastic/permanentdeformation results. Therefore, if the response to a static wheel load is purely elastic (i.e. yielding


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has not occurred) then the pavement design is acceptable as shakedown/stable behaviour would bepredicted.

A yield line in p-q stress space is defined in ABAQUS using the yield line angle (?) and yield

stress value where the cell pressure or ? 3 (principle stress in horizontal direction) is zero (? 0c ,Equation 1). Although, the correct method for defining the yield line is from static tri-axial shearstrength tests (i.e. yield line, Figure 2) it is proposed to use the shakedown range stress boundary A-

B for pavement design purposes (see Figures 7 and 8). It is thus more appropriate to refer to the yield line as a shakedown limit line that defines a stress boundary where shakedown/stablebehaviour occurs below.

)tan311(0

??

??

dc (1)

where,d= y-intercept on theyield line (or in this case shakedown limit line)

? ? = angle of theyield line (or in this case shakedown limit line).

Table 1 lists the design parameters used for pavement analysis in the ABAQUS finite elementmodel of the pavement. A residual stress due to compaction, active earth pressure and yield

condition of the UGM pavement layer of 30kPa is estimated. This residual stress has the effect ofincreasing p and reducing q with the effect of increasing the strength of the UGM. If residualstresses were not included the result is unrealistic asphalt cover thicknesses (> 160 mm) where as

the NI GoodUGM has been used successfully in Northern Ireland for asphalt cover thicknesses of50 mm. Principle stresses were computed under the wheel load centre for a range of asphalt cover

thicknesses to illustrate the effect including a horizontal residual stress of 30 kPa using the designparameters in Table 1 and a tyre pressure P of 550kPa. Figure 10 illustrates in p-q stress space theeffect of including residual stresses in relation to the shakedown limit lines for the NI Goodand NI

PoorUGMs.

Centreline stresses in UGM for range of asphalt cover (AC) thicknesses

0

50

100

150

200

250

300

350

400

450

500

0 50 100 150 200 250 300 350 400 450 500p (kPa)

q(kPa)

AC=40mm

AC=60mm

AC=80mm

AC=100mm

AC=110mm

Shakedown limit line - NI Poor

Shakedown limit line - NI Good

Figure 11. Centreline stresses in UGM caused by a contact tyre stress of 550kPa for a range of

asphalt cover thicknesses, where horizontal residual stresses are assumed = 30 kPa.


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4.2 Lower and Upper Bound Shakedown Theorems

Static shear failure MohrCoulomb parameters ? (friction angle) and c (cohesion) are needed for

the lower and upper bound shakedown theorems developed for pavement analysis by researchers[Yu and Hossain 1998; Boulbibane and Collins 1999 respectively]. The friction angle and cohesionwere derived via geometry, ? and c by direct transform from the yield line angle (?) and y-intercept

(d) as plotted inp q stress space (see Figure 2), their values are listed in Table 1.The lower bound shakedown solution uses Melans static shakedown theorem that states, if the

combination of a time-independent, self equilibrated residual stress field and the elastic stresses

can be found which does not violate the yield condition (i.e. as defined by c and? ) anywhere in the

region, then the material will shakedown. The total stresses, ? ij, are therefore:

? ij = ??eij + ?

rij ( 2)

where,? ?

rij = residual stresses

? ?= the shakedown factor

? ?e

ij= the elastic stresses resulting from the application of a unit load distribution.The determination of the best lower bound is usually considered as a problem of linear

programming: the maximisation of the load multiplier ?? subject to the constraint that the yield

condition is not violated. The equilibrium conditions, yield criteria and boundary conditions are

then used to form the finite element formulation. Bearing in mind that yield is defined by a certaincombination ofp and q, it is found that the highest possible critical shakedown stress (i.e. the largest

?? is associated with the maximum possible residual compressive stress as this increases

confinement (largerp) and thus increases the strength (higher q allowable).Collins and Cliffe [1987, cited in Boulbibane 1999] showed the more realistic three-dimensional

problem could be solved relatively easily using of the dual kinematic upper bound approach. Theupper bound approach requires the optimisation and calculation of loads for a range of competing

pavement failure mechanisms. The material in the various pavement layers is modelled as anelastic/perfectly plastic material failing according to Mohr Coulombs criterion, characterised by acohesion c and angle of internal friction ? . The approach widely used in limit analysis [Chen 1975,

cited in Boulbibane 1999] of considering failure mechanisms consisting of sliding or rotating rigidblocks is used to compute the upper bound shakedown load. Six possible failure mechanisms were

investigated by Boulbibane [1999] and the optimum failure mechanism where the lowestshakedown stress was calculated is shown in Figure 12.

P

Basecourse Eb , ? b

Subgrade E s , ? s

Direction of travel

Y

X

Z

?s????, cs

??

???

?

? '?h

Velocity continuity gives ? i?? b?? i?? s

?b? cb

Figure 12. Optimum failure mechanism used in calculation critical shakedown stress using upper

bound method by Boulbibane [1999].

Both the lower and upper bound solutions for the field trial pavement analysis were computed bythe researchers Yu and Boulbibane whom refined the methods for pavements. At this stage of the

research the results were used for comparison with the finite element method developed.


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4.3 Results

The results of the pavement analysis of the field trial were a prediction as to whether or notshakedown would occur in the UGM for different asphalt cover thicknesses and aggregate type (NI

Goodand NI Poor). Material parameters used and layer thicknesses are detailed in Table 1 andFigure 9. Shakedown predictions from the finite element modelling (assuming a horizontal residual

stress of 30 kPa) and lower and upper bound shakedown theorems are listed in Table 2.The results show that both the lower and upper bound methods predict shakedown to occur forthinner asphalt cover than the finite element method (FEM). Thus the differences in Table 2 are

greater for the NI Poor material where the difference between the c and ? yield line and the

shakedown limit line is greatest. This is predictable as the lower and upper bound methods use the

static shear failure line that allows higher stresses before yield occurs than using the shakedownlimit line as used in the FEM. Further, the lower and upper bound methods compute the residual

stresses in the UGM that build up after compaction and repeated loading. These residual stressesprovide additional horizontal confinement to the UGM which increases its strength. Future researchwill compare these predictions of shakedown to those that actually occurred in the field trial. The

field trial results will also be used to investigate the use of the shakedown limit line for the yield

criteria in the lower and upper bound shakedown theorems.

Table 2. Prediction of shakedown using three different methods, FEM, lower and upper boundshakedown theorems.

Shakedown occurs in UGM (Yes or No)?

F.E.M Lower Bound1 Upper Bound

UGM =NI Good

Asphalt thickness (mm): 0 No Yes Yes

40 No Yes Yes

60 Yes Yes Yes

80 Yes Yes Yes

100 Yes Yes Yes

UGM =NI PoorAsphalt thickness (mm): 0 No No No

40 No Yes Yes

60 No Yes Yes

80 Yes Yes Yes

100 Yes Yes Yes

1 A shear ratio (shear/normal load) of 0.5 was assumed.

5. CONCLUSIONS

RLT permanent strain test results for two Northern Ireland unbound granular materials UGM (NI

Good and NI Poor) showed a range of responses. These responses were categorised into three


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possible shakedown ranges A, B and C. Shakedown range A is where the rate of cumulativepermanent deformation decreases with increasing load cycles until the response is purely elastic.Range C is incremental collapse or failure and range B is between A and C responses. The RLT

test stresses at the boundary between shakedown range A and B were plotted in p (mean normal) q (deviatoric) stress space. A best fit line was then derived to define the stress boundary or

shakedown limit line between stable (acceptable) behaviour and unstable behaviour. This

shakedown limit line was then used as the yield criteria for the UGM to predict whether or notshakedown occurs for the field trial pavement for a range of asphalt cover thicknesses. For

comparison, lower and upper bound shakedown theorems were also used to predict whether or notshakedown occurs for the field trial pavements. However, the lower and upper bound shakedown

theorems used the static shear failure Mohr-Coulomb ? ??friction angle) and c (cohesion) to define

the UGM yield criteria. In summary it was shown that:

?? A practical method for determining design criteria from RLT permanent tests on UGM materialsis possible and that this can be applied as a yield criteria in a finite element model for pavement

design;

?? The predictions of asphalt cover needed to ensure stable/shakedown behaviour in the UGM forthe field trial appeared reasonable using the finite element method with the shakedown limit line

yield criteria;?? The lower and upper bound theorems predicted shakedown to occur in thinner asphalt cover than

predicted using the FEM although this was expected as the yield criteria was derived from statictri-axial failure tests and not repeated load tests.

It is anticipated that a limit line based on RLT test results will provide an improved means ofdefining the allowable stress limit to be used in upper and lower bound analyses. The presentanalytical study using this limit approach, but with a simple elastic/perfectly-plastic UGM

model would then be superceded by the bounding analyses.

6. ACKNOWLEDGEMENTS

The authors acknowledge Professor Hai-Sui Yu (University of Nottingham), Dr M Boulbibane(University of Leicester), and Professor Ian Collins (University of Auckland) for using their lowerand upper bound shakedown methods for predicting whether or not shakedown occurs for the field

trial pavements.

7. REFERENCES

AUSTROADS, 1992, Pavement Design - A Guide to the Structural Design of Road Pavement,Austroads, Sydney, Australia.

Boulbibane, M. and Collins, I.F, 1999,A geomechanical analysis of unbound pavements based

upon shakedown theory. University of Auckland, 1999.

Chan, FWK, 1990, Permanent deformation resistance of granular layers in pavements, Ph.D.Thesis, University of Nottingham, Nottingham.

Chen, W.F. (1975). Limit analysis and soil plasticity. Elsevier, Amsterdam.

Collins, I.F. and Cliffe, P.F, 1987, Shakedown in frictional materials under moving surface loads.

Int. J. Num. Anal. Methods. Geomechanics, 11, pp. 409-420.

Dawson A R and Wellner F. 1999. Plastic behavior of granular materials, Final Report ARCProject 933, University of Nottingham Reference PRG99014, April 1999.

HMSO, 1994,Design manual for roads and bridges, Vol 7, HD 25/94, Part 2, Foundations.

Johnson K L, 1986, Plastic flow, residual stresses and shakedown in rolling contact. Proceedings ofthe 2nd International Conference on Contact Mechanics and Wear of Rail/Wheel Systems,University of Rhode Island, Waterloo Ontario 1986.


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Little, Peter H., (1993) The design of unsurfaced roads using geosynthetics, Dept. of CivilEngineering, University of Nottingham.

Lubliner, J, 1990, Plasticity theory. Macmillan, New York.

Pidwerbesky, B. Fundamental Behaviour of Unbound Granular Pavements Subjected to Various

Loading Conditions and Accelerated Trafficking. PhD Thesis, University of Canterbury,Christchurch, New Zealand, 1996.

Semmelink, CJ, Jooste, FJ & de Beer, M, 1997, Use of the K-mould in determination of the elasticand shear properties of road materials for flexible pavements, 8th Int. Conf. on Asphalt

Pavements, August, Seattle, Washington, USA.

Sharp R and Booker J, Shakedown of pavements under moving surface loads, pp. 1-14, ASCEJournal of Transportation Engineering, No 1, 1984.

Thom, N and Brown, S, 1989, The mechanical properties of unbound aggregates from various

sources. Proceedings of the Third International Symposium on Unbound Aggregates in Roads,UNBAR 3, Nottingham, United Kingdom, 11-13 April 1989.

TRL, 1993,A guide to the structural design of bitumen-surfaced roads in tropical and sub-tropical

countries, RN31, Draft 4th Edition.

Werkmeister, S. Dawson, A. Wellner, F, 2001, Permanent deformation behaviour of granularmaterials and the shakedown concept. Transportation Research Board, 80th Annual Meeting,Washington D.C. January 7-11, 2001.

Yu, H.S. and Hossain, M.Z. (1998). Lower bound shakedown analysis of layered pavements usingdiscontinuious stress fields. Computer Methods in Applied Mechanics and Engineering, Vol.167, pp. 209-222.

8. KEYWORDS

UGM = Unbound Granular Material

RLT = Repeat Load Tri-axial

dawson and arnold isap_2002_paper

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