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Modelling permanent deformation behaviour of unbound granular materials
Construction and Building Materials, Vol. 12, No. 1, pp. 9]18, 1998 Q 1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0950]0618r98 $19.00 q 0.00
PII:S0950–0618(97)00078–0
Modelling permanent deformation behaviour of unbound granular materials F. LekarpU † and A. Dawson‡ U
Division of Highway Engineering, Royal Institute of Technology, S-100 44 Stockholm, Sweden ‡Department of Civil Engineering, University of Nottingham, Nottingham NG7 2RD, UK
Received 15 April 1997; revised 13 November 1997; accepted 19 November 1997 A state of the art is presented on modelling permanent deformation behaviour of unbound granular materials. The existing numerical models are verified by a series of laboratory repeated load tests. A new model is introduced expressing the accumulated permanent axial strain at any given number of cycles as a function of applied stresses, taking into account the maximum shear stress ratio and the length of the stress path in p-q space. The results show similarities with the concept of the shakedown theory. At low levels of stress ratios the accumulation of permanent strain ceases, resulting in an equilibrium state. However, there are indications of the existence of a threshold stress ratio, known as the shakedown limit, beyond which the material experiences incremental collapse or gradual failure. Q 1998 Elsevier Science Ltd. All rights reserved. Keywords: permanent strain; granular materials; plastic behaviour
Introduction
these small plastic deformation increments could lead to an eventual failure of the pavement due to excessive rutting. For performance prediction, it is of great importance to know whether a given pavement will experience progressive accumulation of permanent deformation, or whether the increase of permanent deformation will cease, resulting in a stable and fully resilient response. The objective of this paper is to increase the understanding of the permanent deformation behaviour of unbound granular materials. The results of previous research from various sources have been collected through a literature survey, which is presented in the first part of this paper. The second part discusses the findings from a series of repeated load laboratory tests conducted at the University of Nottingham to study the behaviour of a number of unbound aggregates. The laboratory tests have been used to study the effect of number of load repetitions and of stress level on plastic strains.
A mechanistic pavement design procedure requires that the constituent materials are efficiently used in regard to their strength and stiffness characteristics. The current knowledge concerning the granular materials employed in pavement structures is limited. To date, no overall framework has been established to explain satisfactorily the behaviour of unbound granular materials under the complex repeated loading which they experience. During its service life, a pavement experiences a large number of stress pulses each consisting of vertical, horizontal and shear stress components. These stresses are transient and change with time as shown in Figure 1. The shear stress is reversed as the load passes and there is thus a rotation of the principal stress axes. Granular materials exhibit two types of deformation when subjected to repeated loading: resilient deformation which could lead to fatigue cracking of the overlying bound Že.g. asphalt. surface, and permanent Žplastic. deformation. Although the permanent deformation during one cycle of loading is normally just a fraction of the total deformation produced by each load repetition, the gradual accumulation of a large number of
Literature review In modelling the long-term behaviour of pavements, it is essential to take into account the gradual accumula-
†Corresponding author. Tel.: q46 8 7908707; fax: q46 8 108124; e-mail: [email protected].
9
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
10
Figure 1 Stresses beneath a rolling wheel load
tion of permanent strain with number of load applications and the important role played by stress condition. Perhaps the main objective of research into long-term behaviour should be to establish a constitutive relationship which allows an accurate prediction of the amount of permanent strain at any number of cycles at any stress level. This section aims to present a brief state of the art on modelling the permanent deformation behaviour based on the literature available to date. Correlation between static and cyclic loading tests Several research workers have attempted to correlate repetitive loading and simple static loading test results 1 ] 5. This approach has met mixed reaction since the behaviour of granular materials is generally regarded to be very complex and cyclic and static loadings do not necessarily induce the same structural response. Lentz and Baladi 3 used the static stress] strain results of samples of sand to predict the cumulative permanent strain of identical samples tested under cyclic loading condition. They suggested that, provided the samples used in the static and cyclic tests are identical in every aspect, the permanent strain under cyclic loading can be expressed by a constitutive expression of the form
« 1, p s « 0.95S ? ln Ž 1 y qrS . q
½
y0 .15
n ? Ž qrS . ? ln Ž N . w 1 y m ? Ž qrS .x
5
Ž1.
where
« 1, p
s accumulated permanent strain after N load repetitions
« 0.95S s static strain at 95% of static strength q s deviator stress s s 1 y s 3 Ž s 1 and s 3 are principal stresses . S s static strength N s number of load cycles n, m s regression parameters, which vary with confining pressure. Lentz and Baladi obtained a good correlation between measured and calculated values in their investigation. However, they pointed out that the equation was based on the results from a single sand subgrade material and additional research was needed. The validity of Equation Ž1. was investigated by Sweere 6 , for both sands and granular base course materials, and rejected. Gerrard et al.5 suggested a new approach by relating the applied stress level to the static shear strength given by the Mohr]Coulomb static failure envelope. In this case the static failure envelope is plotted on a normal Mohr]Coulomb diagram as shown in Figure 2 Ž a.. The Mohr circles of the stresses applied in each repetitive loading test are then drawn and the permanent strains marked on the circles at the point nearest to the static failure envelope. The applied stress level during the cyclic triaxial test is referred to by the ratio of the cyclic shear stress to the corresponding static shear strength, i.e. the ratio CArCB in Figure 2 Ž a.. The permanent strain contours are then drawn after a certain number of load cycles as shown in Figure 2 Ž b .. Whilst this approach allows the relationship of permanent strain to stress to be estimated, it makes no prediction of development with number of load appli-
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
11
Figure 2 Mohr]Coulomb representation of permanent strains 5
cations. Barrett and Smith7 used this approach to express the permanent strain response of a crushed dolerite with a clay binder and reported good correlation with observations. No other verification was found in the literature.
q N
s deviator stress s number of load repetitions.
The experimental results were then used to express the permanent deformation moduli as functions of number of cycles by
Correlation between resilient and plastic beha¨iour Veverka8 studied both resilient and plastic behaviour of granular materials and found a correlation between the two. He proposed a simple relationship between permanent and resilient strains given by
« 1, p s a? « r ? N b
Ž2.
in which « r is resilient strain, and a and b are material parameters. This equation is very simplistic, requiring considerable testing as the stress dependency is not explicit, and has not been confirmed by other researchers. Permanent deformation moduli Most of the work on modelling the permanent deformation behaviour deals with axial and horizontal stresses and strains. Jouve et al.9 introduced a new approach by decomposing both stress and strain fields into volumetric and shear components. In a similar manner to the elasticity theories, plastic deformation ‘moduli’ were defined by p q K pŽ N . s , Gp Ž N . s Ž . «¨ , p N 3 « s, p Ž N .
Ž3.
where K p Ž N . s bulk modulus with respect to permanent deformation Gp Ž N . s shear modulus with respect to permanent deformation « ¨ , p Ž N . s permanent volumetric strain for N ) 100 « s, p Ž N . s permanent shear strain for N ) 100 p s mean normal stress s Ž s 1 q s 2 q s 3 .r3 Ž s 1 , s 2 , and s 3 are principal stresses .
Gp s
A 2'N
'N q D 2
,
Gp A 3'N s Kp 'N q D
Ž4. 3
in which the parameters A 2 , A 3 , D 2 and D 3 are functions of the stress ratio qrp. The parameter A 2 is dependent on the stress ratio by a relationship of the type arx b while A 3 , D 2 and D 3 are connected linearly with this ratio. Permanent strain and number of cycles Barksdale1 performed a comprehensive study of the behaviour of several different base course materials using cyclic load triaxial tests with 10 5 load applications. He found that the accumulated permanent axial strain was proportional to the logarithm of the number of load cycles and expressed the results by log]normal expression of the form
« 1, p s aq b? log Ž N .
Ž5.
in which a and b are constants for a given level of s 1 ] s 3 and s 3 . Once again, the definition of stress dependency is less than ideal. Sweere 6 applied 10 6 load applications in a series of repeated load triaxial tests and observed that the lognormal approach did not fit his test results. He then suggested that for a large number of load repetitions a log-log approach should be employed, and expressed the results Žalso without an explicit stress dependency definition. by
« 1, p s a? N b in which a and b are regression parameters.
Ž6.
12
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
The applicability of the log-log model presented by Sweere was later challenged by Wolff and Visser 10 who performed a full-scale Heavy Vehicle Simulator ŽHVS. testing with several million load applications. They described the permanent deformation behaviour as being composed of two phases. In the HVS testing results, an initial phase, up to 1.2 million load repetitions, was observed with a rapid development of permanent deformation but a constantly diminishing rate of increase. During the second phase, permanent strain development seemed much slower and the rate of increase approached a constant value. The log]log model failed to give reliable estimates of permanent strain at large numbers of load cycles. Wolff and Visser, therefore, suggested an improved stress]strain model given by
« 1, p s Ž m ? N q a. ? Ž 1 y eyb N .
Ž7.
where a, b and m are regression parameters. Equations Ž5. ] Ž7. all imply that accumulation of permanent strain with number of load cycles proceeds indefinitely. Yet several other researchers 11 ] 21 have reported that at least at certain levels of applied stresses the induced permanent strain eventually levels off, resulting in an equilibrium condition. Khedr 13 used repeated load triaxial tests to study the permanent strain behaviour of a crushed limestone and came to the conclusion that the rate of permanent strain accumulation decreases logarithmically with the number of load repetitions according to an expression of the form
«p s A ? Nym N
Ž8.
in which m is a material parameter and A a material and stress-strain parameter given as a function of shear stress ratio and resilient modulus. While Khedr reported correlation values quite close to unity for all tested samples, no other verification of this model was found in the literature. Paute et al.19 suggested that permanent strain increases gradually towards an asymptotic value. Using a similar approach to that of Equation Ž4., they expressed the relationship between permanent axial strain and number of cycles by U « 1, ps
A ? 'N 'N q D
Ž9.
in which « 1,U p is additional permanent axial strain after the first 100 load cycles, and A and D are regression parameters. In a separate, recent study, Paute et al.20 proposed a new approach to express the influence of number of load applications on development of permanent deformation in granular materials. In this case, the accumulated permanent strain after an initial period of 100
cycles is given by
« 1,) p s A ? 1 y
N 100
yB
ž ž / /
Ž 10 .
in which A and B are regression parameters. According to this equation, « 1,U p goes towards a limit value Žequal to A. as N increases towards infinity. The parameter A is, therefore, considered as the limit value for total permanent axial strain. Permanent strains and their relationship to stress Pervious research has shown that stress level has a significant influence on the development of permanent deformation in pavement structures. Several researchers who carried out repeated load triaxial tests on unbound granular materials have found that permanent deformation behaviour is principally governed by some form of stress ratio. Lashine et al.22 conducted repeated load triaxial tests on a crushed stone and found that the measured permanent axial strain settled down to a constant value related to the ratio qmaxrs 3max , where qmax and s 3max are maximum deviator stress and maximum confining pressure, respectively. Similar results have been reported by Brown and Hyde 23 who studied the response of crushed stone under cyclic triaxial condition with constant confining pressure. Brown and Hyde further stated that the same results are obtained from tests with variable confining pressure if the mean value of the applied confining stress is used in the analysis. Barksdale1 used repeated load triaxial tests to relate the permanent axial strain to the ratio of repeated deviator stress and constant confining pressure. He employed the general hyperbolic expression given by Duncan and Chang 24 for static triaxial tests, and found a close fit to plastic stress-strain curves obtained from repeated load test results. Barksdale suggested that for a given number of load applications the variation of permanent axial strain with stresses can be expressed by
« 1, p s
qrK? s 3n Ž R f ? q . r2 Ž C ? cos f q s 3 ? sin f . 1y Ž 1 y sin f .
Ž 11 .
where K? s 3n s relationship defining the initial tangent modulus as a function of confining pressure, s 3 Ž K and n are constants . C s apparent cohesion f s angle of internal friction Rf s a constant relating compressive strength to an asymptotic stress difference. Pappin12 performed triaxial tests with variable confining pressure on specimens of a well-graded crushed limestone. He reported that permanent shear strain
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson rate can be expressed as a function of the length of the stress path in p]q space and the applied shear stress ratio. He also calculated a shape factor for the variation of permanent strain with number of cycles, and expressed the total permanent shear strain by
« s, p s Ž fnN . ? L ?
q8 p8
ž /
Ž 12 . max
accumulated permanent shear strain shape factor stress path length modified deviator stress s 2r3 ? q modified mean normal stress s '3 ? p.
'
Pappin stated that unless the material was stressed close to the static failure limit, large permanent strains did not occur. However, the mathematical expression suggested by Pappin is not asymptotic to failure and predicts finite permanent strain even at or beyond the static failure stress. Other researchers have reported that the amount of permanent strain is determined by how close the applied stresses are to the static failure stress. Barret and Smith7 and Raymond and Williams 25 used the stress ratio qmaxrqfailure to characterise the results of permanent deformation tests. Thom26 , on the other hand, suggested that the permanent shear strain is better related to the stress ratio Ž qfailure y qmax .rqmax . In a recent study, Paute et al.20 defined a limit value for the maximum permanent axial strain, A-value as described in Equation Ž10., and suggested that it varies with maximum shear stress ratio, qmaxrŽ pmax q pU ., according to a hyperbolic expression given by Equation Ž13.. This hyperbolic relationship indicates that A increases when the maximum shear stress ratio increases, and that there is a limit value to the maximum shear stress ratio Žequal to m. for which A becomes infinite. This approach suggests that the static failure line, defined by parameters m and pU , could be estimated using the results from cyclic triaxial tests. On the other hand, if the failure parameters are known, a single triaxial test is needed to define the expression. Lekarp et al.21 discussed this approach and showed that the model generally results in quite unreasonable values of failure parameters if these are estimated, or very low correlation in cases where the failure line is determined by static failure tests. qmax Ž pmax q pU . As qmax b? m y Ž pmax q pU .
ž
pmaxs maximum mean normal stress pU s stress parameter defined by intersection of the static failure line and the p-axis in p]q space m s slope of the static failure line b s regression parameter.
2.8
where
« s, p s fnN s L s q8 s p8 s
13
Shakedown theory The literature review revealed that several researchers1,14,17,18,27 ] 30 , having related the magnitude of permanent strain to shear stress level, concluded that at low levels of stress ratio the resulting permanent strain would eventually reach an equilibrium condition. At high levels of stress ratio, however, they stated that permanent strains are likely to increase rapidly resulting in an eventual failure. This has raised the possibility of the existence of a critical stress level separating the stable and failure condition. A certain theory of plasticity known as the ‘shakedown’ theory has been used for the response of structures subjected to repeated cyclic loading. This theory which was first introduced by Melan 31 has been widely applied to structures such as trusses, frames and plates. Sharp and Booker 30 and Sharp 32 suggested that shakedown principles can also be employed for pavement design. Applied in this manner, the shakedown theory predicts that a pavement is liable to show progressive accumulation of plastic strains under repeated loading if the magnitude of the applied loads exceeds a limiting value called the shakedown load. The pavement is then said to exhibit an incremental collapse or incremental failure. On the other hand, if the applied loads are less than this shakedown load, the growth of plastic strains will eventually level off and the pavement is said to have attained a state of shakedown by means of adaptation to the applied loads. The pavement’s response will then be elastic under additional load applications. It is, therefore, postulated that the life of a pavement under traffic is directly related to its resistance to incremental failure, and a satisfactory pavement is one that shakes down17 . Although the possible application of the shakedown theory as a convenient pavement design tool has been recognised by several researchers, very limited effort has been devoted to developing simple numerical models based on the theory. The few proposed numerical methods found in the literature 15,17,33 deal with the response of the whole pavement as a single, unified structure. No work was found on adapting the shakedown theory to explain the behaviour of granular materials.
Ž 13.
/
where A s limit value for maximum permanent axial strain qmaxs maximum deviator stress
Experimental A series of repeated load laboratory tests was performed at the University of Nottingham to study the permanent deformation behaviour of unbound granular materials. Five different aggregates, mainly used as subbase material in pavement structures, were tested.
14
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
Figure 3 Particle size distributions for the tested materials
The materials consisted of: crushed dolomitic limestone ŽLS., crushed granidiorite ŽGr., crushed slate waste ŽSW., sand and gravel ŽS& G. and Leighton Buzzard sand ŽS.. The particle size distribution curves are given in Figure 3. In this investigation the Repeated Load Triaxial Apparatus ŽRLTA. and Hollow Cylinder Apparatus ŽHCA. available at the University of Nottingham were employed. Detailed description of the testing equipment and the specimen preparation are given elsewhere 34 . A test programme was planned with the objective of studying the development of cumulative permanent strain with number of load applications and its variation with stresses. The test programme was chosen with great consideration to the newly proposed European standard 35 regarding repeated load triaxial testing. Details of the testing programme are given in Table 1. The sand ŽS. was tested in the HCA using constant confining pressure since the existing equipment does not allow application of variable confining pressure. In the case of the other materials, however, the RLTA was employed and both deviatoric and confining stresses were cycled. All the tests were performed in a drained condition to avoid development of excess pore water pressures.
Results and discussion Accumulation of permanent axial strain with number of cycles Development of permanent axial strain with number of load applications is shown for each material in Figure 4 Žsee abbreviations at top of paper.. The data were compared with some of the existing models discussed previously ŽEquations Ž5. ] Ž10... The Paute model of Equation Ž10. showed the least error in most cases and, subjectively, seemed to fit the data best 34 . The predicted results by the Paute model are also given in
Figure 4. Table 2 shows the actual stresses applied in each test and the results of the analysis for the Paute model. In most cases a very high correlation was found with the Paute model, and the rate of increase of permanent strain reduced gradually to such an extent that towards the end of the test the specimen seemed to be in an equilibrium condition. However, in a few cases ŽP1 for Gr, P2 and P4 for LS, and P2 and P3 for S& G. the model did not seem to fit the data well and no stabilisa-
Table 1 Details of the intended test programme Material
Stress path
s3
N
q
Min
Max
Min
Max
Gr DDs 2.09 MCs 1.9
P1 P2 P3 P4 P5
80 000 80 000 80 000 80 000 80 000
0 0 0 60 45
20 50 120 165 150
0 0 0 120 20
300 600 600 700 420
LS DDs 2.26 MCs 5.0
P1 P2 P3 P4 P5 P6
80 000 80 000 80 000 80 000 80 000 80 000
0 0 0 0 60 45
20 50 75 120 165 150
0 0 0 0 120 20
300 600 300 600 700 420
SW DDs 2.16 MCs 4.6
P1 P2 P3 P4
80 000 80 000 80 000 80 000
0 0 0 0
20 100 200 75
0 0 0 0
300 600 600 300
S& G DDs 2.05 MCs 4.0
P1 P2 P3
80 000 80 000 80 000
100 100 100
135 285 220
0 0 0
200 500 400
S DDs 1.52 MCs 0.0
P1 P2 P3
40 000 40 000 40 000
70 70 70
70 70 70
0 0 0
80 90 105
DD, dry density Žgrcm3 .; MC, moisture content Žgrcm3 .; N, number of load applications; s 3 , confining pressure ŽkPa.; q, deviator stress ŽkPa.. In the case of Gr and SW the test P3 had to be stopped at 20 000 cycles due to puncturing of the membrane.
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
15
Table 2 Results of permanent deformation tests Stress path
qmin ŽkPa.
qmax ŽkPa.
pmin ŽkPa.
pmax ŽkPa.
Ž qrp.max
« 1, p Ž100. Ž10y4 .
A Ž10y4 .
B Ž10y4 .
R2 Ž10y4 .
Gr
P1 P2 P3 P4 P5
0.0 0.0 0.0 122.6 23.6
292.7 595.4 585.6 719.5 441.2
0.0 0.0 0.0 103.8 57.4
112.1 245.5 310.2 417.5 311.5
2.61 2.42 1.89 1.72 1.42
9.37 21.96 17.64 19.40 9.57
342.49 25.99 6.25 25.20 3.66
0.004 0.089 0.416 0.040 0.155
0.990 0.991 0.993 0.995 0.997
LS
P1 P2 P3 P4 P5 P6
0.0 0.0 0.0 0.0 131.7 23.1
296.1 593.0 298.1 594.4 726.2 440.6
0.0 0.0 0.0 0.0 107.2 55.8
117.5 245.9 170.8 312.6 419.4 310.6
2.52 2.41 1.74 1.90 1.73 1.42
7.10 20.57 4.72 16.96 9.77 7.42
14.41 608.68 4.16 1520.32 6.97 1.36
0.049 0.018 0.104 0.002 0.165 0.102
0.993 0.845 0.992 0.981 0.996 0.991
SW
P1 P2 P3 P4
0.0 0.0 0.0 0.0
295.4 589.4 591.5 295.4
0.0 0.0 0.0 0.0
117.6 292.5 388.1 170.0
2.51 2.01 1.52 1.74
25.89 211.66 71.60 94.78
1092.59 1226.00 4994.08 276.48
0.004 0.037 0.005 0.054
0.967 0.992 0.997 0.985
S&G
P1 P2 P3
0.0 0.0 0.0
198.0 498.1 395.7
100.0 100.0 100.0
199.3 429.0 346.5
0.99 1.16 1.14
3.73 29.26 14.45
177.58 6093.86 1218.27
0.004 0.002 0.003
0.816 0.980 0.864
S
P1 P2 P3
0.0 0.0 0.0
78.8 91.0 103.4
69.9 68.6 66.6
96.2 98.9 101.1
0.82 0.92 1.02
39.52 30.72 121.01
45.47 666.41 178.10
0.205 0.015 0.197
0.999 0.995 0.998
Material
tion was achieved. A possible explanation for how well each test fits the model is given in the following section where the effect of stresses is discussed. Effect of stresses on permanent deformation beha¨iour None of the models found in the literature seemed to give a satisfactory explanation of the influence of stresses on accumulation of permanent strain. However, further analysis of the test results indicated a possible relationship between accumulated permanent axial strain, the maximum shear stress ratio and the length of the stress path. It was clearly observed that accumulated permanent strain increased with maxi-
mum shear stress ratio. It was then argued that at a constant shear stress ratio the amount of permanent strain should increase with the length of the applied stress path to reach this maximum point, as a result of the greater energy dissipated in hysteresis during the load-unload cycle. A consistent pattern was observed when the ratio of permanent strain at a given number of cycles and stress path length was plotted against the maximum shear stress ratio. The relationship can be given by a simple expression of the form
« 1, p Ž Nr e f . q s a? L p
Figure 4 Cumulative permanent axial strain versus number of load repetitions
ž /
b
Ž 14 . max
16
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
where
« 1, p Ž Nr e f . s accumulated permanent axial strain at a given number of cycles Nr e f , Nr e f ) 100 L s length of stress path q s deviator stress p s mean normal stress Ž qrp. max s maximum stress ratio a, b s regression parameters
w}x wkPax wkPax wkPax w}x w}x
The equation above is not dimensionally correct. This can be avoided by dividing the stress path length by a reference stress of, for instance, 1 kPa. By introducing p 0 as the reference stress Equation Ž14. would become
« 1, p Ž Nr e f . q s a? p Ž Lrp 0 .
ž /
b
Ž 15. max
It can be shown that Equation Ž15. is applicable for any given number of load cycles greater than 100. Figure 5 shows the results for Nr e f s 20 000 cycles. It should be mentioned that for LS and SW the stress path P1 Žwhich has a high stress ratio. shows a very distinct deviation from the general pattern. These extreme cases were regarded as outliers and excluded from the statistical analysis. For the type of analysis explained here, it is very important that a wide range of shear stress ratios are applied. However, the limited test programme used in this investigation does not provide a sufficient range of shear stress ratios for all the materials tested. The three stress paths used for S& G did not provide enough information for the analysis and no correlation was found.
Figure 5 Variation of permanent axial strain with stresses for N s 20 000
Although the Paute model of Equation Ž10. was shown above to fit the strain-number of cycles data in most cases, in some cases the correlation was not quite satisfactory and the observed values indicated signs of progressive accumulation of permanent strain. The analysis of the effect of stresses shows that at low levels of shear stress ratio a very high correlation is found with the Paute model, the growth of permanent strain levels off and the material behaviour approaches an equilibrium state. At high shear stress ratios, however, the correlation with the Paute model is relatively poor and the material exhibits a form of gradual collapse rather than stabilisation. In the case of Gr, for instance, the least correlation with the Paute model is found for the stress path P1 followed by P2, both showing signs of gradual collapse. Figure 5 shows that for Gr the applied shear stress ratio is highest for the stress path P1 followed by P2. For the remaining stress paths, with lower shear stress ratios, the high correlation with the Paute model indicates that the permanent strain increases asymptotically and the material stabilises. This behaviour is even more clear for LS, for which incremental collapse is apparent for the stress paths P2 and P4, with the least correlation obtained for P2. According to Figure 5 the applied stress ratio for LS is highest for the stress path P2 followed by P4. The same argument can be used for the other materials tested. If low levels of stress ratio ultimately result in an equilibrium state and high stress ratios cause gradual collapse of the material, there must be a certain level of stress ratio at which this change of behaviour occurs. This stress ratio can be considered as the boundary which separates the final stabilised behaviour from the incremental failure of the material. This hypothesis shows close similarities with the concept of the shake-
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson down theory in pavement structures, with the existing boundary being the shakedown limit. However, no attempt is made here to determine this shakedown limit, as the number of tests performed and the range of stress ratios applied do not seem to be sufficient and further research is needed in this regard. Previously, several research workers have attempted to relate gradual accumulation of permanent strain under cyclic loading to the ultimate shear strength of the material under static loading. The use by Paute et al.20 of an asymptotic strain value based on failure stress conditions ŽEquation Ž13.. is an example of this. The results of this investigation do not support the existence of such a relationship and it is argued that the failure in granular layers in a pavement under repeated loading is perhaps a gradual process rather than a sudden collapse as in static failure tests. On this basis, ultimate shear strength and stress levels that cause sudden failure are of no great interest for the analysis of the material behaviour when the increase of permanent strain is incremental. The new model of Equation Ž15. explains the material behaviour while the accumulated permanent strain is finite and the applied stresses do not cause sudden failure.
Conclusions The main conclusions from this study can be summarised as follows. 1. Unbound granular materials show a complex behaviour under cyclic loads, with a gradual accumulation of permanent strain with each load application. 2. A new model has been introduced which describes the relationship between accumulated permanent axial strain and stresses at any given number of cycles greater than 100. The model takes into account the maximum shear stress ratio and the length of the stress path in p-q space. The results of the laboratory repeated load tests show very good correlation between the proposed model and the observed values. 3. At low levels of shear stress ratios, the growth of permanent strain seems to level off indicating that the material will eventually reach an equilibrium condition with almost total resilient behaviour. For these cases the Paute model of Equation Ž10. was found to be quite successful in predicting the relationship between permanent axial strain and number of load applications. 4. At high shear stress ratios, the accumulation of permanent strain is more progressive with signs of gradual deterioration of the material. The Paute model, which is based on the assumption of eventual stabilisation of the behaviour, clearly loses its accuracy at high stress ratios. 5. If low shear stress ratios result in an ultimate equilibrium condition and high shear stress ratios cause gradual failure, then a threshold stress ratio must
17
exist at which this change of behaviour occurs. This threshold stress ratio could be a form of the, socalled, shakedown limit. 6. More research is needed to determine the shakedown limit, and to investigate the possibility of using a modified shakedown theory to study the behaviour of granular materials under cyclic loading.
Acknowledgements This study was made possible by a 1-year joint research programme between the University of Nottingham, Nottingham, England and the Royal Institute of Technology ŽKTH., Stockholm, Sweden. The Swedish part of the work was financially supported through the ‘Ballast Research Programme’ at the KTH, by SBUF ŽDevelopment Fund of the Swedish Construction Industry., GMF ŽNational Swedish Sand, Gravel and Crushed Stone Association., KFB ŽSwedish Transport and Communications Research Board., and NUTEK ŽSwedish National Board for Industrial and Technical Development.. Thanks are due to Mr. T. Gleitz, formerly of the Technishe Universitat ¨ Dresden, some of whose triaxial test data was made available to the authors, and to Mr. I.R. Richardson, in the Civil Engineering Department of the University of Nottingham, who carried out the hollow cylinder tests on sands.
References 1 Barksdale, R. D., Laboratory evaluation of rutting in base course materials. In Proceedings of the Third International Conference on Structural Design of Asphalt Pa¨ements, London, 1972, pp. 161]174 2 Lentz, R. W., Permanent deformation of cohesionless subgrade material under cyclic loading. Ph.D. Thesis, Dept. of Civil Eng., Michigan State Univ., East Lansing, 1979 3 Lentz, R. W. and Baladi, G. Y., Constitutive equation for permanent strain of sand subjected to cyclic loading. Transportation Research Record, 1981, 810, 50]54 4 Morgan, J. R., Laboratory tests on fine crushed rock. In Proceedings of the 6th Australian Road Research Board Conference, vol. 6, Part 5, 1972, pp. 179]194 5 Gerrard, C. M., Morgan, J. R. and Richards, B. G., An approach to the design of flexible pavements for Australian conditions. Australian Road Research Report, 1975, 5Ž8. 6 Sweere, G. T. H., Unbound granular bases for roads. Ph.D. Thesis, University of Delft, 1990 7 Barret, J. R. and Smith, D. M., Stress history effects in base course materials. In Proceedings of the 8th Australian Road Research Board Conference, vol. 8, 1976, pp. 30]39 8 Veverka, V., Raming Van de Spoordiepte Bij Wegen met een Bitumineuze Verharding. De Wegentechniek, Vol. XXIV, 1979, 3, 25]45 9 Jouve, P., Martinez, J., Paute, J. L. and Ragneau, E., Rational model for the flexible pavements deformations. In Proceedings of the 6th International Conference on Structural Design of Asphalt Pa¨ements, vol. 1, Ann Arbor, 1987, pp. 50]64 10 Wolff, H. and Visser, A. T., Incorporating elasto-plasticity in granular layer pavement design. In Proceedings of the Institution of Ci¨il Engineers Transport, 1994, 105, pp. 259]272 11 Boyce, J. R., The behaviour of a granular material under repeated loading. Ph.D. Thesis, Department of Civil Engineering, University of Nottingham, 1976 12 Pappin, J. W., Characteristics of a granular material for pave-
18
13 14 15 16 17 18 19
20
21 22
23
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson ment analysis. Ph.D. Thesis, Department of Civil Engineering, University of Nottingham, 1979 Khedr, S., Deformation characteristics of granular base course in flexible pavement. Transportation Research Record, 1985, 1043, 131]138 Chan, F. W. K., Permanent deformation resistance of granular layers in pavements. Ph.D. Thesis, Department of Civil Engineering, University of Nottingham, 1990 Raad, L., Weichert, D. and Haidar, A., Shakedown and fatigue of pavements with granular bases. Transportation Research Record, 1989, 1227, 159]172 Larew, H. G. and Leonards, G. A., A strength criterion for repeated loads. In Proceedings of the 41st Annual Meeting, Highway Research Board, Washington, DC, 1962, pp. 529]556 Sharp, R. W., Pavement design based on shakedown analysis. Transportation Research Record, 1985, 1022, 99]107 Morgan, J. R., The response of granular materials to repeated loading. In Proceedings of the 3rd Australian Road Research Board Conference, Sydney, 1966, pp. 1178]1192 Paute, J. L., Jouve, P., Martinez, J. and Ragneau, E., Modele ` de calcul pour le dimensionnement des chaussees ´ souples. Bulletin de Liaison des Laboratoires des Ponts et Chaussees, ´ 1988, 156, 21]36 Paute, J. L., Hornych, P. and Benaben, J. P., Repeated load triaxial testing of granular materials in the French network of laboratories des ponts et chaussees. ´ In Flexible Pa¨ements, ed. A. G. Correia. Lisbon, 1996, pp. 53]64 Lekarp, F., Richardson, I. R. and Dawson, A., Influences on permanent deformation behavior of unbound granular materials. Transportation Research Record, 1996, 1547, 68]75 Lashine, A. K., Brown, S. F. and Pell, P. S., Dynamic properties of soils. Report No. 2 submitted to KoninklijkerShell Laboratorium, Department of Civil Engineering, University of Nottingham, 1971 Brown, S. F. and Hyde, A. F. L., Significance of cyclic confining stress in repeated load triaxial testing of granular material. Transportation Research Record, 1985, 537, 49]58
24 Duncan, J. M. and Chang, C. Y., Nonlinear analysis of stress and strain in soils. In Proceedings of the American Society of Ci¨il Engineers, vol. 96, 1970, No. SM5, pp. 1629]1653 25 Raymond, G. P. and Williams, D. R., Repeated load triaxial tests on a dolomite ballast. In Proceedings of the American Society of Ci¨il Engineers, vol. 104, 1978, No. GT7, pp. 1013]1029 26 Thom, N. H., Design of road foundations. Ph.D. Thesis, Department of Civil Engineering, University of Nottingham, 1988 27 Dunlap, W. A., Deformation characteristics of granular materials subjected to rapid repetitive loading. Ph.D. Thesis, Texas A& M University, 1966 28 Holubec, I., Cyclic creep of granular materials. Department of Highways, Ontario, Report No. RR147, 1969 29 Knutson, R. M., Thompson, M. R., Mullin, T. and Tajabji, S. D., Materials Evaluation Study. Federal Railroad Administration, Ballast and Foundation Research Program, Report Nos. FRAOR and D-77-02, Washington DC, 1977 30 Sharp, R. W. and Booker, J. R., Shakedown of pavements under moving surface loads. Journal of Transportation Engineering, ¨ol. 110, 1984, 1, 1]14 31 Melan, E., Theorie Statisch Unbestimmer Systeme Aus IdealPlastischen Baustoff. Sitzungsberichte der Akademie der Wissenschaften in Wien, ¨ol. Iia, 1936, pp. 145]195 32 Sharp, R. W., Shakedown analysis and the design of pavements. Ph.D. Thesis, University of Sydney, 1983 33 Collins, I. F., Wang, A. P. and Saunders, L. R., Shakedown theory and the design of unbound pavements. ARRB, Road and Transport Research, ¨ol. 2, 1993, 4, 28]37 34 Lekarp, F., Permanent deformation behaviour of unbound granular materials. Licentiate Thesis, Department of Infrastructure and Planning, Royal Institution of Technology, Stockholm, 1997 35 Unbound and hydraulic bound mixtures for roads } test methods } cyclic load triaxial test. Draft of European Standard Submitted to CEN Members, No. prEN 00227413, Brussels, 1995
PII:S0950–0618(97)00078–0
Modelling permanent deformation behaviour of unbound granular materials F. LekarpU † and A. Dawson‡ U
Division of Highway Engineering, Royal Institute of Technology, S-100 44 Stockholm, Sweden ‡Department of Civil Engineering, University of Nottingham, Nottingham NG7 2RD, UK
Received 15 April 1997; revised 13 November 1997; accepted 19 November 1997 A state of the art is presented on modelling permanent deformation behaviour of unbound granular materials. The existing numerical models are verified by a series of laboratory repeated load tests. A new model is introduced expressing the accumulated permanent axial strain at any given number of cycles as a function of applied stresses, taking into account the maximum shear stress ratio and the length of the stress path in p-q space. The results show similarities with the concept of the shakedown theory. At low levels of stress ratios the accumulation of permanent strain ceases, resulting in an equilibrium state. However, there are indications of the existence of a threshold stress ratio, known as the shakedown limit, beyond which the material experiences incremental collapse or gradual failure. Q 1998 Elsevier Science Ltd. All rights reserved. Keywords: permanent strain; granular materials; plastic behaviour
Introduction
these small plastic deformation increments could lead to an eventual failure of the pavement due to excessive rutting. For performance prediction, it is of great importance to know whether a given pavement will experience progressive accumulation of permanent deformation, or whether the increase of permanent deformation will cease, resulting in a stable and fully resilient response. The objective of this paper is to increase the understanding of the permanent deformation behaviour of unbound granular materials. The results of previous research from various sources have been collected through a literature survey, which is presented in the first part of this paper. The second part discusses the findings from a series of repeated load laboratory tests conducted at the University of Nottingham to study the behaviour of a number of unbound aggregates. The laboratory tests have been used to study the effect of number of load repetitions and of stress level on plastic strains.
A mechanistic pavement design procedure requires that the constituent materials are efficiently used in regard to their strength and stiffness characteristics. The current knowledge concerning the granular materials employed in pavement structures is limited. To date, no overall framework has been established to explain satisfactorily the behaviour of unbound granular materials under the complex repeated loading which they experience. During its service life, a pavement experiences a large number of stress pulses each consisting of vertical, horizontal and shear stress components. These stresses are transient and change with time as shown in Figure 1. The shear stress is reversed as the load passes and there is thus a rotation of the principal stress axes. Granular materials exhibit two types of deformation when subjected to repeated loading: resilient deformation which could lead to fatigue cracking of the overlying bound Že.g. asphalt. surface, and permanent Žplastic. deformation. Although the permanent deformation during one cycle of loading is normally just a fraction of the total deformation produced by each load repetition, the gradual accumulation of a large number of
Literature review In modelling the long-term behaviour of pavements, it is essential to take into account the gradual accumula-
†Corresponding author. Tel.: q46 8 7908707; fax: q46 8 108124; e-mail: [email protected].
9
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
10
Figure 1 Stresses beneath a rolling wheel load
tion of permanent strain with number of load applications and the important role played by stress condition. Perhaps the main objective of research into long-term behaviour should be to establish a constitutive relationship which allows an accurate prediction of the amount of permanent strain at any number of cycles at any stress level. This section aims to present a brief state of the art on modelling the permanent deformation behaviour based on the literature available to date. Correlation between static and cyclic loading tests Several research workers have attempted to correlate repetitive loading and simple static loading test results 1 ] 5. This approach has met mixed reaction since the behaviour of granular materials is generally regarded to be very complex and cyclic and static loadings do not necessarily induce the same structural response. Lentz and Baladi 3 used the static stress] strain results of samples of sand to predict the cumulative permanent strain of identical samples tested under cyclic loading condition. They suggested that, provided the samples used in the static and cyclic tests are identical in every aspect, the permanent strain under cyclic loading can be expressed by a constitutive expression of the form
« 1, p s « 0.95S ? ln Ž 1 y qrS . q
½
y0 .15
n ? Ž qrS . ? ln Ž N . w 1 y m ? Ž qrS .x
5
Ž1.
where
« 1, p
s accumulated permanent strain after N load repetitions
« 0.95S s static strain at 95% of static strength q s deviator stress s s 1 y s 3 Ž s 1 and s 3 are principal stresses . S s static strength N s number of load cycles n, m s regression parameters, which vary with confining pressure. Lentz and Baladi obtained a good correlation between measured and calculated values in their investigation. However, they pointed out that the equation was based on the results from a single sand subgrade material and additional research was needed. The validity of Equation Ž1. was investigated by Sweere 6 , for both sands and granular base course materials, and rejected. Gerrard et al.5 suggested a new approach by relating the applied stress level to the static shear strength given by the Mohr]Coulomb static failure envelope. In this case the static failure envelope is plotted on a normal Mohr]Coulomb diagram as shown in Figure 2 Ž a.. The Mohr circles of the stresses applied in each repetitive loading test are then drawn and the permanent strains marked on the circles at the point nearest to the static failure envelope. The applied stress level during the cyclic triaxial test is referred to by the ratio of the cyclic shear stress to the corresponding static shear strength, i.e. the ratio CArCB in Figure 2 Ž a.. The permanent strain contours are then drawn after a certain number of load cycles as shown in Figure 2 Ž b .. Whilst this approach allows the relationship of permanent strain to stress to be estimated, it makes no prediction of development with number of load appli-
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
11
Figure 2 Mohr]Coulomb representation of permanent strains 5
cations. Barrett and Smith7 used this approach to express the permanent strain response of a crushed dolerite with a clay binder and reported good correlation with observations. No other verification was found in the literature.
q N
s deviator stress s number of load repetitions.
The experimental results were then used to express the permanent deformation moduli as functions of number of cycles by
Correlation between resilient and plastic beha¨iour Veverka8 studied both resilient and plastic behaviour of granular materials and found a correlation between the two. He proposed a simple relationship between permanent and resilient strains given by
« 1, p s a? « r ? N b
Ž2.
in which « r is resilient strain, and a and b are material parameters. This equation is very simplistic, requiring considerable testing as the stress dependency is not explicit, and has not been confirmed by other researchers. Permanent deformation moduli Most of the work on modelling the permanent deformation behaviour deals with axial and horizontal stresses and strains. Jouve et al.9 introduced a new approach by decomposing both stress and strain fields into volumetric and shear components. In a similar manner to the elasticity theories, plastic deformation ‘moduli’ were defined by p q K pŽ N . s , Gp Ž N . s Ž . «¨ , p N 3 « s, p Ž N .
Ž3.
where K p Ž N . s bulk modulus with respect to permanent deformation Gp Ž N . s shear modulus with respect to permanent deformation « ¨ , p Ž N . s permanent volumetric strain for N ) 100 « s, p Ž N . s permanent shear strain for N ) 100 p s mean normal stress s Ž s 1 q s 2 q s 3 .r3 Ž s 1 , s 2 , and s 3 are principal stresses .
Gp s
A 2'N
'N q D 2
,
Gp A 3'N s Kp 'N q D
Ž4. 3
in which the parameters A 2 , A 3 , D 2 and D 3 are functions of the stress ratio qrp. The parameter A 2 is dependent on the stress ratio by a relationship of the type arx b while A 3 , D 2 and D 3 are connected linearly with this ratio. Permanent strain and number of cycles Barksdale1 performed a comprehensive study of the behaviour of several different base course materials using cyclic load triaxial tests with 10 5 load applications. He found that the accumulated permanent axial strain was proportional to the logarithm of the number of load cycles and expressed the results by log]normal expression of the form
« 1, p s aq b? log Ž N .
Ž5.
in which a and b are constants for a given level of s 1 ] s 3 and s 3 . Once again, the definition of stress dependency is less than ideal. Sweere 6 applied 10 6 load applications in a series of repeated load triaxial tests and observed that the lognormal approach did not fit his test results. He then suggested that for a large number of load repetitions a log-log approach should be employed, and expressed the results Žalso without an explicit stress dependency definition. by
« 1, p s a? N b in which a and b are regression parameters.
Ž6.
12
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
The applicability of the log-log model presented by Sweere was later challenged by Wolff and Visser 10 who performed a full-scale Heavy Vehicle Simulator ŽHVS. testing with several million load applications. They described the permanent deformation behaviour as being composed of two phases. In the HVS testing results, an initial phase, up to 1.2 million load repetitions, was observed with a rapid development of permanent deformation but a constantly diminishing rate of increase. During the second phase, permanent strain development seemed much slower and the rate of increase approached a constant value. The log]log model failed to give reliable estimates of permanent strain at large numbers of load cycles. Wolff and Visser, therefore, suggested an improved stress]strain model given by
« 1, p s Ž m ? N q a. ? Ž 1 y eyb N .
Ž7.
where a, b and m are regression parameters. Equations Ž5. ] Ž7. all imply that accumulation of permanent strain with number of load cycles proceeds indefinitely. Yet several other researchers 11 ] 21 have reported that at least at certain levels of applied stresses the induced permanent strain eventually levels off, resulting in an equilibrium condition. Khedr 13 used repeated load triaxial tests to study the permanent strain behaviour of a crushed limestone and came to the conclusion that the rate of permanent strain accumulation decreases logarithmically with the number of load repetitions according to an expression of the form
«p s A ? Nym N
Ž8.
in which m is a material parameter and A a material and stress-strain parameter given as a function of shear stress ratio and resilient modulus. While Khedr reported correlation values quite close to unity for all tested samples, no other verification of this model was found in the literature. Paute et al.19 suggested that permanent strain increases gradually towards an asymptotic value. Using a similar approach to that of Equation Ž4., they expressed the relationship between permanent axial strain and number of cycles by U « 1, ps
A ? 'N 'N q D
Ž9.
in which « 1,U p is additional permanent axial strain after the first 100 load cycles, and A and D are regression parameters. In a separate, recent study, Paute et al.20 proposed a new approach to express the influence of number of load applications on development of permanent deformation in granular materials. In this case, the accumulated permanent strain after an initial period of 100
cycles is given by
« 1,) p s A ? 1 y
N 100
yB
ž ž / /
Ž 10 .
in which A and B are regression parameters. According to this equation, « 1,U p goes towards a limit value Žequal to A. as N increases towards infinity. The parameter A is, therefore, considered as the limit value for total permanent axial strain. Permanent strains and their relationship to stress Pervious research has shown that stress level has a significant influence on the development of permanent deformation in pavement structures. Several researchers who carried out repeated load triaxial tests on unbound granular materials have found that permanent deformation behaviour is principally governed by some form of stress ratio. Lashine et al.22 conducted repeated load triaxial tests on a crushed stone and found that the measured permanent axial strain settled down to a constant value related to the ratio qmaxrs 3max , where qmax and s 3max are maximum deviator stress and maximum confining pressure, respectively. Similar results have been reported by Brown and Hyde 23 who studied the response of crushed stone under cyclic triaxial condition with constant confining pressure. Brown and Hyde further stated that the same results are obtained from tests with variable confining pressure if the mean value of the applied confining stress is used in the analysis. Barksdale1 used repeated load triaxial tests to relate the permanent axial strain to the ratio of repeated deviator stress and constant confining pressure. He employed the general hyperbolic expression given by Duncan and Chang 24 for static triaxial tests, and found a close fit to plastic stress-strain curves obtained from repeated load test results. Barksdale suggested that for a given number of load applications the variation of permanent axial strain with stresses can be expressed by
« 1, p s
qrK? s 3n Ž R f ? q . r2 Ž C ? cos f q s 3 ? sin f . 1y Ž 1 y sin f .
Ž 11 .
where K? s 3n s relationship defining the initial tangent modulus as a function of confining pressure, s 3 Ž K and n are constants . C s apparent cohesion f s angle of internal friction Rf s a constant relating compressive strength to an asymptotic stress difference. Pappin12 performed triaxial tests with variable confining pressure on specimens of a well-graded crushed limestone. He reported that permanent shear strain
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson rate can be expressed as a function of the length of the stress path in p]q space and the applied shear stress ratio. He also calculated a shape factor for the variation of permanent strain with number of cycles, and expressed the total permanent shear strain by
« s, p s Ž fnN . ? L ?
q8 p8
ž /
Ž 12 . max
accumulated permanent shear strain shape factor stress path length modified deviator stress s 2r3 ? q modified mean normal stress s '3 ? p.
'
Pappin stated that unless the material was stressed close to the static failure limit, large permanent strains did not occur. However, the mathematical expression suggested by Pappin is not asymptotic to failure and predicts finite permanent strain even at or beyond the static failure stress. Other researchers have reported that the amount of permanent strain is determined by how close the applied stresses are to the static failure stress. Barret and Smith7 and Raymond and Williams 25 used the stress ratio qmaxrqfailure to characterise the results of permanent deformation tests. Thom26 , on the other hand, suggested that the permanent shear strain is better related to the stress ratio Ž qfailure y qmax .rqmax . In a recent study, Paute et al.20 defined a limit value for the maximum permanent axial strain, A-value as described in Equation Ž10., and suggested that it varies with maximum shear stress ratio, qmaxrŽ pmax q pU ., according to a hyperbolic expression given by Equation Ž13.. This hyperbolic relationship indicates that A increases when the maximum shear stress ratio increases, and that there is a limit value to the maximum shear stress ratio Žequal to m. for which A becomes infinite. This approach suggests that the static failure line, defined by parameters m and pU , could be estimated using the results from cyclic triaxial tests. On the other hand, if the failure parameters are known, a single triaxial test is needed to define the expression. Lekarp et al.21 discussed this approach and showed that the model generally results in quite unreasonable values of failure parameters if these are estimated, or very low correlation in cases where the failure line is determined by static failure tests. qmax Ž pmax q pU . As qmax b? m y Ž pmax q pU .
ž
pmaxs maximum mean normal stress pU s stress parameter defined by intersection of the static failure line and the p-axis in p]q space m s slope of the static failure line b s regression parameter.
2.8
where
« s, p s fnN s L s q8 s p8 s
13
Shakedown theory The literature review revealed that several researchers1,14,17,18,27 ] 30 , having related the magnitude of permanent strain to shear stress level, concluded that at low levels of stress ratio the resulting permanent strain would eventually reach an equilibrium condition. At high levels of stress ratio, however, they stated that permanent strains are likely to increase rapidly resulting in an eventual failure. This has raised the possibility of the existence of a critical stress level separating the stable and failure condition. A certain theory of plasticity known as the ‘shakedown’ theory has been used for the response of structures subjected to repeated cyclic loading. This theory which was first introduced by Melan 31 has been widely applied to structures such as trusses, frames and plates. Sharp and Booker 30 and Sharp 32 suggested that shakedown principles can also be employed for pavement design. Applied in this manner, the shakedown theory predicts that a pavement is liable to show progressive accumulation of plastic strains under repeated loading if the magnitude of the applied loads exceeds a limiting value called the shakedown load. The pavement is then said to exhibit an incremental collapse or incremental failure. On the other hand, if the applied loads are less than this shakedown load, the growth of plastic strains will eventually level off and the pavement is said to have attained a state of shakedown by means of adaptation to the applied loads. The pavement’s response will then be elastic under additional load applications. It is, therefore, postulated that the life of a pavement under traffic is directly related to its resistance to incremental failure, and a satisfactory pavement is one that shakes down17 . Although the possible application of the shakedown theory as a convenient pavement design tool has been recognised by several researchers, very limited effort has been devoted to developing simple numerical models based on the theory. The few proposed numerical methods found in the literature 15,17,33 deal with the response of the whole pavement as a single, unified structure. No work was found on adapting the shakedown theory to explain the behaviour of granular materials.
Ž 13.
/
where A s limit value for maximum permanent axial strain qmaxs maximum deviator stress
Experimental A series of repeated load laboratory tests was performed at the University of Nottingham to study the permanent deformation behaviour of unbound granular materials. Five different aggregates, mainly used as subbase material in pavement structures, were tested.
14
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
Figure 3 Particle size distributions for the tested materials
The materials consisted of: crushed dolomitic limestone ŽLS., crushed granidiorite ŽGr., crushed slate waste ŽSW., sand and gravel ŽS& G. and Leighton Buzzard sand ŽS.. The particle size distribution curves are given in Figure 3. In this investigation the Repeated Load Triaxial Apparatus ŽRLTA. and Hollow Cylinder Apparatus ŽHCA. available at the University of Nottingham were employed. Detailed description of the testing equipment and the specimen preparation are given elsewhere 34 . A test programme was planned with the objective of studying the development of cumulative permanent strain with number of load applications and its variation with stresses. The test programme was chosen with great consideration to the newly proposed European standard 35 regarding repeated load triaxial testing. Details of the testing programme are given in Table 1. The sand ŽS. was tested in the HCA using constant confining pressure since the existing equipment does not allow application of variable confining pressure. In the case of the other materials, however, the RLTA was employed and both deviatoric and confining stresses were cycled. All the tests were performed in a drained condition to avoid development of excess pore water pressures.
Results and discussion Accumulation of permanent axial strain with number of cycles Development of permanent axial strain with number of load applications is shown for each material in Figure 4 Žsee abbreviations at top of paper.. The data were compared with some of the existing models discussed previously ŽEquations Ž5. ] Ž10... The Paute model of Equation Ž10. showed the least error in most cases and, subjectively, seemed to fit the data best 34 . The predicted results by the Paute model are also given in
Figure 4. Table 2 shows the actual stresses applied in each test and the results of the analysis for the Paute model. In most cases a very high correlation was found with the Paute model, and the rate of increase of permanent strain reduced gradually to such an extent that towards the end of the test the specimen seemed to be in an equilibrium condition. However, in a few cases ŽP1 for Gr, P2 and P4 for LS, and P2 and P3 for S& G. the model did not seem to fit the data well and no stabilisa-
Table 1 Details of the intended test programme Material
Stress path
s3
N
q
Min
Max
Min
Max
Gr DDs 2.09 MCs 1.9
P1 P2 P3 P4 P5
80 000 80 000 80 000 80 000 80 000
0 0 0 60 45
20 50 120 165 150
0 0 0 120 20
300 600 600 700 420
LS DDs 2.26 MCs 5.0
P1 P2 P3 P4 P5 P6
80 000 80 000 80 000 80 000 80 000 80 000
0 0 0 0 60 45
20 50 75 120 165 150
0 0 0 0 120 20
300 600 300 600 700 420
SW DDs 2.16 MCs 4.6
P1 P2 P3 P4
80 000 80 000 80 000 80 000
0 0 0 0
20 100 200 75
0 0 0 0
300 600 600 300
S& G DDs 2.05 MCs 4.0
P1 P2 P3
80 000 80 000 80 000
100 100 100
135 285 220
0 0 0
200 500 400
S DDs 1.52 MCs 0.0
P1 P2 P3
40 000 40 000 40 000
70 70 70
70 70 70
0 0 0
80 90 105
DD, dry density Žgrcm3 .; MC, moisture content Žgrcm3 .; N, number of load applications; s 3 , confining pressure ŽkPa.; q, deviator stress ŽkPa.. In the case of Gr and SW the test P3 had to be stopped at 20 000 cycles due to puncturing of the membrane.
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
15
Table 2 Results of permanent deformation tests Stress path
qmin ŽkPa.
qmax ŽkPa.
pmin ŽkPa.
pmax ŽkPa.
Ž qrp.max
« 1, p Ž100. Ž10y4 .
A Ž10y4 .
B Ž10y4 .
R2 Ž10y4 .
Gr
P1 P2 P3 P4 P5
0.0 0.0 0.0 122.6 23.6
292.7 595.4 585.6 719.5 441.2
0.0 0.0 0.0 103.8 57.4
112.1 245.5 310.2 417.5 311.5
2.61 2.42 1.89 1.72 1.42
9.37 21.96 17.64 19.40 9.57
342.49 25.99 6.25 25.20 3.66
0.004 0.089 0.416 0.040 0.155
0.990 0.991 0.993 0.995 0.997
LS
P1 P2 P3 P4 P5 P6
0.0 0.0 0.0 0.0 131.7 23.1
296.1 593.0 298.1 594.4 726.2 440.6
0.0 0.0 0.0 0.0 107.2 55.8
117.5 245.9 170.8 312.6 419.4 310.6
2.52 2.41 1.74 1.90 1.73 1.42
7.10 20.57 4.72 16.96 9.77 7.42
14.41 608.68 4.16 1520.32 6.97 1.36
0.049 0.018 0.104 0.002 0.165 0.102
0.993 0.845 0.992 0.981 0.996 0.991
SW
P1 P2 P3 P4
0.0 0.0 0.0 0.0
295.4 589.4 591.5 295.4
0.0 0.0 0.0 0.0
117.6 292.5 388.1 170.0
2.51 2.01 1.52 1.74
25.89 211.66 71.60 94.78
1092.59 1226.00 4994.08 276.48
0.004 0.037 0.005 0.054
0.967 0.992 0.997 0.985
S&G
P1 P2 P3
0.0 0.0 0.0
198.0 498.1 395.7
100.0 100.0 100.0
199.3 429.0 346.5
0.99 1.16 1.14
3.73 29.26 14.45
177.58 6093.86 1218.27
0.004 0.002 0.003
0.816 0.980 0.864
S
P1 P2 P3
0.0 0.0 0.0
78.8 91.0 103.4
69.9 68.6 66.6
96.2 98.9 101.1
0.82 0.92 1.02
39.52 30.72 121.01
45.47 666.41 178.10
0.205 0.015 0.197
0.999 0.995 0.998
Material
tion was achieved. A possible explanation for how well each test fits the model is given in the following section where the effect of stresses is discussed. Effect of stresses on permanent deformation beha¨iour None of the models found in the literature seemed to give a satisfactory explanation of the influence of stresses on accumulation of permanent strain. However, further analysis of the test results indicated a possible relationship between accumulated permanent axial strain, the maximum shear stress ratio and the length of the stress path. It was clearly observed that accumulated permanent strain increased with maxi-
mum shear stress ratio. It was then argued that at a constant shear stress ratio the amount of permanent strain should increase with the length of the applied stress path to reach this maximum point, as a result of the greater energy dissipated in hysteresis during the load-unload cycle. A consistent pattern was observed when the ratio of permanent strain at a given number of cycles and stress path length was plotted against the maximum shear stress ratio. The relationship can be given by a simple expression of the form
« 1, p Ž Nr e f . q s a? L p
Figure 4 Cumulative permanent axial strain versus number of load repetitions
ž /
b
Ž 14 . max
16
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson
where
« 1, p Ž Nr e f . s accumulated permanent axial strain at a given number of cycles Nr e f , Nr e f ) 100 L s length of stress path q s deviator stress p s mean normal stress Ž qrp. max s maximum stress ratio a, b s regression parameters
w}x wkPax wkPax wkPax w}x w}x
The equation above is not dimensionally correct. This can be avoided by dividing the stress path length by a reference stress of, for instance, 1 kPa. By introducing p 0 as the reference stress Equation Ž14. would become
« 1, p Ž Nr e f . q s a? p Ž Lrp 0 .
ž /
b
Ž 15. max
It can be shown that Equation Ž15. is applicable for any given number of load cycles greater than 100. Figure 5 shows the results for Nr e f s 20 000 cycles. It should be mentioned that for LS and SW the stress path P1 Žwhich has a high stress ratio. shows a very distinct deviation from the general pattern. These extreme cases were regarded as outliers and excluded from the statistical analysis. For the type of analysis explained here, it is very important that a wide range of shear stress ratios are applied. However, the limited test programme used in this investigation does not provide a sufficient range of shear stress ratios for all the materials tested. The three stress paths used for S& G did not provide enough information for the analysis and no correlation was found.
Figure 5 Variation of permanent axial strain with stresses for N s 20 000
Although the Paute model of Equation Ž10. was shown above to fit the strain-number of cycles data in most cases, in some cases the correlation was not quite satisfactory and the observed values indicated signs of progressive accumulation of permanent strain. The analysis of the effect of stresses shows that at low levels of shear stress ratio a very high correlation is found with the Paute model, the growth of permanent strain levels off and the material behaviour approaches an equilibrium state. At high shear stress ratios, however, the correlation with the Paute model is relatively poor and the material exhibits a form of gradual collapse rather than stabilisation. In the case of Gr, for instance, the least correlation with the Paute model is found for the stress path P1 followed by P2, both showing signs of gradual collapse. Figure 5 shows that for Gr the applied shear stress ratio is highest for the stress path P1 followed by P2. For the remaining stress paths, with lower shear stress ratios, the high correlation with the Paute model indicates that the permanent strain increases asymptotically and the material stabilises. This behaviour is even more clear for LS, for which incremental collapse is apparent for the stress paths P2 and P4, with the least correlation obtained for P2. According to Figure 5 the applied stress ratio for LS is highest for the stress path P2 followed by P4. The same argument can be used for the other materials tested. If low levels of stress ratio ultimately result in an equilibrium state and high stress ratios cause gradual collapse of the material, there must be a certain level of stress ratio at which this change of behaviour occurs. This stress ratio can be considered as the boundary which separates the final stabilised behaviour from the incremental failure of the material. This hypothesis shows close similarities with the concept of the shake-
Permanent deformation behaviour of unbound granular materials: F. Lekarp and A. Dawson down theory in pavement structures, with the existing boundary being the shakedown limit. However, no attempt is made here to determine this shakedown limit, as the number of tests performed and the range of stress ratios applied do not seem to be sufficient and further research is needed in this regard. Previously, several research workers have attempted to relate gradual accumulation of permanent strain under cyclic loading to the ultimate shear strength of the material under static loading. The use by Paute et al.20 of an asymptotic strain value based on failure stress conditions ŽEquation Ž13.. is an example of this. The results of this investigation do not support the existence of such a relationship and it is argued that the failure in granular layers in a pavement under repeated loading is perhaps a gradual process rather than a sudden collapse as in static failure tests. On this basis, ultimate shear strength and stress levels that cause sudden failure are of no great interest for the analysis of the material behaviour when the increase of permanent strain is incremental. The new model of Equation Ž15. explains the material behaviour while the accumulated permanent strain is finite and the applied stresses do not cause sudden failure.
Conclusions The main conclusions from this study can be summarised as follows. 1. Unbound granular materials show a complex behaviour under cyclic loads, with a gradual accumulation of permanent strain with each load application. 2. A new model has been introduced which describes the relationship between accumulated permanent axial strain and stresses at any given number of cycles greater than 100. The model takes into account the maximum shear stress ratio and the length of the stress path in p-q space. The results of the laboratory repeated load tests show very good correlation between the proposed model and the observed values. 3. At low levels of shear stress ratios, the growth of permanent strain seems to level off indicating that the material will eventually reach an equilibrium condition with almost total resilient behaviour. For these cases the Paute model of Equation Ž10. was found to be quite successful in predicting the relationship between permanent axial strain and number of load applications. 4. At high shear stress ratios, the accumulation of permanent strain is more progressive with signs of gradual deterioration of the material. The Paute model, which is based on the assumption of eventual stabilisation of the behaviour, clearly loses its accuracy at high stress ratios. 5. If low shear stress ratios result in an ultimate equilibrium condition and high shear stress ratios cause gradual failure, then a threshold stress ratio must
17
exist at which this change of behaviour occurs. This threshold stress ratio could be a form of the, socalled, shakedown limit. 6. More research is needed to determine the shakedown limit, and to investigate the possibility of using a modified shakedown theory to study the behaviour of granular materials under cyclic loading.
Acknowledgements This study was made possible by a 1-year joint research programme between the University of Nottingham, Nottingham, England and the Royal Institute of Technology ŽKTH., Stockholm, Sweden. The Swedish part of the work was financially supported through the ‘Ballast Research Programme’ at the KTH, by SBUF ŽDevelopment Fund of the Swedish Construction Industry., GMF ŽNational Swedish Sand, Gravel and Crushed Stone Association., KFB ŽSwedish Transport and Communications Research Board., and NUTEK ŽSwedish National Board for Industrial and Technical Development.. Thanks are due to Mr. T. Gleitz, formerly of the Technishe Universitat ¨ Dresden, some of whose triaxial test data was made available to the authors, and to Mr. I.R. Richardson, in the Civil Engineering Department of the University of Nottingham, who carried out the hollow cylinder tests on sands.
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