2001 in Ballyclare, Northern Ireland, UK. The field trial uses the two different UGMs tested (NI. Good and NI Poor) where the asphalt cover thickness varies from ...
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The application of shakedown approach to granular pavement layers Greg Arnold Nottingham Centre for Pavement Engineering (NCPE), University of Nottingham, UK
Andrew Dawson Nottingham Centre for Pavement Engineering (NCPE), University of Nottingham, UK
David Hughes School of Civil Engineering, Queens University Belfast, UK
Des Robinson School 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 to describing 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 a stable 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 a shakedown 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 cover thicknesses. 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] assume that rutting occurs only in the subgrade. The thickness of the unbound granular sub-base layers is determined from the subgrade condition (California Bearing Ratio and/or resilient modulus) and design traffic (including traffic during construction). The assumption that rutting occurs only within the 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 based and 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 granular materials. 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 UGM’s 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 assess the suitability of these alternative materials for use at various depths within the pavement (e.g. subbase and lower sub-base). Thus, current pavement design methods and material specifications should consider the repeated load deformation performance of the UGM layers. Thus, as an approach 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 Drucker Prager 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 be described by a single equation. Several researchers [Wekmeister et al 1999, Sharp and Booker 1984] who related the magnitude of the accumulated permanent (plastic) strain to shear stress level concluded 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. no further 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 increase linearly. 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 described using 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 four categories of material response under repeated loading: 1. 2.
3.
4.
purely elastic, where the applied repeated stress is sufficiently small all deformations are fully recovered and the response is termed purely elastic. elastic shakedown, where the applied repeated stress is slightly less than that required to produce 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 to have “shaken down” and the maximum stress level at which this condition is achieved is termed the “elastic shakedown limit”. plastic shakedown, where the applied repeated stress is slightly less than that required to produce 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 a purely resilient response has been obtained the material is said, once again, to have “shaken down” and the maximum stress level at which this condition is achieved is termed the “plastic shakedown limit” 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 plastic strains 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 bound shakedown limits can be calculated for boundary value problems using Melan’s static or Koiter’s kinematic 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 observed behaviour 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 stress conditions that cause the various shakedown ranges for use in pavement design. The shakedown ranges are: ?? Range A is the plastic shakedown range and for this to occur the response shows high strain rates 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 response becomes 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 should be 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) is either 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 load cycles 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 may be 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 heading downwards 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 increasing and 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
1
2
3
4
5
6
7
8
9
10
1,000000
0,100000
C 0,010000
0,001000
0,000100
B
0,000010
0,000001
A -3
Permanent vertical strain [10 ] A, B, C
shakedown-range
35 560
140 630
210 700
280
350
420
? D [kPa]
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 Northern Ireland 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 red colour. These aggregates are later referred to as NI Good and 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 of sandstone type deposits. Greywacke is one of the major sources of coarse aggregate in Northern Ireland. 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 tests were 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 materials typically result in straight lines in p – q stress space and may be referred to as Drucker-Prager yield criteria. Yield Stresses Plotted in p-q Stress Space 1000 Best fit yield line (NI Good UGM)
900 800 700
q (kPa)
600 500 Best fit yield line (NI Poor UGM)
400 300 200 100 0 0
100
200
300
400
500
p (kPa)
Figure 2. Yield stresses from tri-axial static shear-failure tests plotted in p – q stress space. It was assumed that RLT permanent strain tests conducted at stress levels close to the yield line plotted 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 this basis stress levels were chosen by keeping p constant and varying q. Figures 2 and 3 detail in p – q stress 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 shakedown ranges. 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 kept constant while q (principle stress difference) was increased for each subsequent test of 50,000 load cycles. 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 new stress level. The affects of stress history will be investigated in the future. Permanent Deformation (%) - Test Stresses - NI Good 1000 900 800 700
q (kPa)
600
Best fit monotonic yield line
500 400 300 200
Stress paths
100 0 0
50
100
150
200
250
300
350
p (kPa)
Figure 3. RLT permanent strain testing stresses for NI Good UGM.
400
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Permanent Deformation (%) - Test Stresses - NI Poor 1000 900 800 700
q (kPa)
600 500 400 Best fit monotonic yield line
300 200 100
Stress paths
0 0
50
100
150
200
250
300
350
400
p (kPa)
Figure 4. RLT permanent strain testing stresses for NI Poor UGM. 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 to determine 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 cumulative permanent strain it was assumed that at the start of each new test stage the deformations were nil (or zero). 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 4.5 4 2f
Permanent Strain (%)
3.5 Test 2a 2b 2c 2d 2e 2f
3 2.5 2
p (kPa) 250 252 253 252 253 252
q (kPa) 402 457 511 555 607 658
Cell (kPa) 116 100 83 67 51 33
2e 1.5 1
2d 2c
0.5
2b
2a 0 0
50000
100000
150000
200000
250000
300000
Load Cycles
Figure 5. Typical RLT permanent strain test result for NI Good UGM. 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 type B. The 2a result hasn’t 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 lie somewhere on a B/C transition. Thus the stress levels associated with the boundaries between A and 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. For comparison 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 below the yield line and nearly parallel. Stresses that cause a shakedown range A response in the UGM where 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 in the UGM pavement layer to avert pre-mature failure in this layer.
ARNOLD ET AL. Permanent Strain vs Permanent Strain Rate - NI Good (Test 2 - p=250 kPa, q varies - 2a lowest to 2f highest) Permanent Strain (10 -3) 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Permanent Strain Rate (10-3/load cycle)
1.E-01
1.E-02 2f
1.E-03 2e
2a
1.E-04
2d 2c 2b 1.E-05
Figure 6. Cumulative permanent strain versus permanent strain rate for NI Good UGM test 2 (Figure 5). Shakedown Range (A, B & C) Stresses - NI Good 1000 900 Best fit yield line 800
peak q (kPa)
700 600 500
Best fit range B-C
400 300
Best fit range A-B
200 100 0 0
50
100
150
200
250
300
350
peak p (kPa)
Figure 7. Shakedown range stress boundaries for NI Good UGM.
400
ARNOLD ET AL. Shakedown Range (A, B & C) Stresses - NI Poor 1000 900 800 700 peak q (kPa)
Best fit yield line 600 500 400
Best fit range B-C
300 200
Best fit range A-B
100 0 0
50
100
150
200
250
300
350
400
peak p (kPa)
Figure 8. Shakedown range stress boundaries for NI Poor UGM.
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 further check as to whether or not shakedown occurs is undertaken. For example, the pavement design can be 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 this design 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 (NI Good and NI Poor) where the asphalt cover thickness varies from 40 to 100mm. Several design checks 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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300 mm
p
Asphalt (E, ? ?
t (40 to 100 mm)
UGM (E, ? , ? , ? 0c, ? , c)
450 mm
Rigid (rock) subgrade Figure 9. Pavement cross-section of proposed field trial. Table 1. Field trial pavement design material parameters. Material Parameters
NI Good
NI Poor
0, 40, 60, 80 & 100
0, 40, 60, 80 & 100
Modulus, E (MPa)
3,000
3,000
Poisons ratio, ?
0.35
0.35
Modulus, E (MPa)
200
200
Poisons ratio, ?
0.35
0.35
54.9
29.2
45
140
Asphalt: Thicknesses, d (mm)
UGM:
1
Yield line angle, ?
1
Yield line y-intercept, d
1
Yield stress for ? 3 =0, ? 0c
84.6
172.3
2
Friction angle, ?
45.9
26.5
2
Cohesion, 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, at this 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 criteria then 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/permanent deformation 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 be predicted. 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 shear strength tests (i.e. yield line, Figure 2) it is proposed to use the shakedown range stress boundary AB 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/stable behaviour occurs below. ?
0c
?
d (1 ? 1 3 tan ? )
(1)
where, d = y-intercept on the yield line (or in this case shakedown limit line) ? ? = angle of the yield line (or in this case shakedown limit line). Table 1 lists the design parameters used for pavement analysis in the ABAQUS finite element model 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 of increasing p and reducing q with the effect of increasing the strength of the UGM. If residual stresses were not included the result is unrealistic asphalt cover thicknesses (> 160 mm) where as the NI Good UGM has been used successfully in Northern Ireland for asphalt cover thicknesses of 50 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 design parameters in Table 1 and a tyre pressure P of 550kPa. Figure 10 illustrates in p-q stress space the effect of including residual stresses in relation to the shakedown limit lines for the NI Good and NI Poor UGMs. Centreline stresses in UGM for range of asphalt cover (AC) thicknesses 500 Shakedown limit line - NI Good 450 400 Shakedown limit line - NI Poor 350 AC=40mm
q (kPa)
300
AC=60mm
250
AC=80mm
200
AC=100mm
150
AC=110mm 100 50 0 0
50
100
150
200
250
300
350
400
450
500
p (kPa)
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 Mohr–Coulomb 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 cohesion were derived via geometry, ? and c by direct transform from the yield line angle (? ) and y-intercept (d) as plotted in p – q stress space (see Figure 2), their values are listed in Table 1. The lower bound shakedown solution uses Melan’s 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 ? ? eij = 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 certain combination of p 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 (larger p) 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. The upper 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 an elastic/perfectly plastic material failing according to Mohr Coulomb’s criterion, characterised by a cohesion 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 rigid blocks 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 lowest shakedown stress was calculated is shown in Figure 12.
P h
X
?? ? '?
Direction of travel
?? ??
Basecourse E b , ? b ? b ? cb
Y
Subgrade E s , ? s ?s ????, cs Z
Velocity continuity gives ? i?? b?? i ’?? s
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 by the 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 not shakedown would occur in the UGM for different asphalt cover thicknesses and aggregate type (NI Good and NI Poor). Material parameters used and layer thicknesses are detailed in Table 1 and Figure 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 for thinner 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 shakedown limit 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 stresses provide additional horizontal confinement to the UGM which increases its strength. Future research will 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 bound shakedown theorems. Shakedown occurs in UGM (Yes or No)? F.E.M
Lower Bound1
Upper Bound
Asphalt thickness (mm): 0
No
Yes
Yes
40
No
Yes
Yes
60
Yes
Yes
Yes
80
Yes
Yes
Yes
100
Yes
Yes
Yes
Asphalt thickness (mm): 0
No
No
No
40
No
Yes
Yes
60
No
Yes
Yes
80
Yes
Yes
Yes
100
Yes
Yes
Yes
UGM = NI Good
UGM = NI Poor
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 cumulative permanent 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 not shakedown 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 not shakedown 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 materials is 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 for the 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 static tri-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 of defining the allowable stress limit to be used in upper and lower bound analyses. The present analytical 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 lower and 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 ARC Project 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 of the 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 Civil Engineering, 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 elastic and 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, ASCE Journal 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 granular materials 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 using discontinuious stress fields. Computer Methods in Applied Mechanics and Engineering, Vol. 167, pp. 209-222.
8. KEYWORDS UGM = RLT =
Unbound Granular Material Repeat Load Tri-axial
