Several researchers have modelled permanent deformation. A number of these have related the permanent deformation to the number of load repetitions (Barksdale, 1972, Khedr, 1985, PAUTE, 1988, Sweere, 1990, Wolff et al., 1994), others linked it to the applied stresses (Duncan and Chang, 1970, Lashine, 1971, Lentz and Baladi, 1981, Pappin, 1979); and others (Lekarp et al., 2000a, Li and Selig, 1996, Puppala et al., 1999, Puppala et al., 2009) modified
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these models through introducing different conditions and parameters, such as moisture change and strength properties. A large amount of interest has been paid to the stress level and the number of load repetitions in the modelling of permanent deformation. However, there are many other effects that have influence on progressing permanent deformation in unbound granular materials and subgrade soils, such as the soil type, particle size distribution, fines content and moisture changes. Table 2.3 summarises the permanent deformation models.
Table 2. 3 Permanent deformation models
Model Reference Equation Parameters
Models that Relate Permanent Deformation to the Number of Load Repetitions
π1,π= π + π. πππ (π) Barksdale (1972) 2.23 πstrain 1,π=accumulated permanent
N = number of load repetitions a, b, m, A and B= materialsβ constants
π1,π= π ππ Sweere (1990) 2.24
π1,π= (π. π + π). (1 β πβππ ) Wolff & Visser (1994) 2.25 π1,π
π = π΄. π
βπ Khedr (1985) 2.26
π1,π= π΄. [1 β (100π)βπ΅] Paute et al. (1996) 2.27 Models that Relate Permanent Deformation to the Stress state
Ι1,π=
π πΎ.π3π
1β[(π π .π)/2(πΆ.πππ π+π3.π πππ)(1βπ πππ) ]
Barksdale (1972) 2.28 ππ ,π = accumulated permanent shear strain; πππ = shape factor; L = stress path length;
πΒ° = modified deviator stress =β23 π;
πΒ°= modified mean normal stress β3. π
Ι1,p(ππππ)=accumulated
permanent axial strain at a given number of cycles Nref;
Nref >100; L = length of stress
path
q= deviatoric stress p= mean normal stress
(q/p)max= maximum stress ratio
Po= reference stress ππ ,π= (πππ). πΏ. (πΒ°πΒ°)πππ₯2.8 Pappin (1979) 2.29 Ι1,π(ππππ) (π0πΏ) = π. ( π π)πππ₯ π Lekarp and Dawson (1998) 2.30
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Continued
Models that Relate the Resilient Strain to Plastic Strain
π1,π= π. ππ. ππ Veverka (1979) 2.31 πΞ΄π = resilient strain
a = permanent deformation for the layer/sublayer (in)
π = number of traffic repetitions ΙΒ°, π½ and π = material properties Ιπ = resilient strain imposed in laboratory test to obtain the above listed material properties, ΙΒ°, π½ and π (in/in)
Ιπ£ = average vertical resilient strain in the layer/sublayer as obtained from the primary response model (in/in)
β = thickness of the layer/sublayer (in) πΏπ(π) = π½ππΊ(ΙΙΒ° π) π β(ππ)π½Ι π£β
Tseng and Lytton
(1989) 2.32
πππβ°π
β°π = ππ+ ππβ ππππ Uzan (2004) 2.33
Models include the number of loads and stress state
ππ= π΄ππΌ[πππ§]π½ Ullidtz (1993) 2.34 ππ = permanent deformation PD=permanent deformation (mm) Οd = deviator stress Οs = static strength ππππ‘ = π1+π32+π3 πππ‘π = atmosphere pressure ππππ‘=(Ο1+2Ο3)/3, ππππ‘= (β23) (π1β π3); ππ§ is vertical effective stress Οv is vertical stress on top of
pavement foundation (kPa); e is the base of the natural logarithm
s = the slope;
c= the intersect of the straight line relationship between A and N on a log-log scale
πΌ1, πΌ2, πΌ3, πΌ4, a, b, m, A, Ξ± and Ξ² are regression parameters π1,π= π. (πππ π ) πππ Li and Selig (1996) 2.35 ππ= π΄π[πππππ‘ ππ‘π] α΅ Puppala (1999) 2.36 ππ= πΌ1ππΌ2(πππππ‘ ππ‘π) πΌ3(ππππ‘ πππ‘π) πΌ4 Puppala (2009) 2.37 ππ· = π΄ (ππ΅ππ£β 1) πππ΄ = π . ππ(π) + π or π΄ = ππππ ππ· = ππ ππ (ππ΅ππ£β 1) Theyse (1997) 2.38a 2.38b 2.38c
Using Deflection as a Parameter in a Permanent Deformation Model
Log (RR) = -7.424 + 1.151 Log (d) + 0.486 Log (N18) + 1.26 Log (Οc)
Chen and Lin
(1999) 2.39
RR= the rate of rutting d= surface deflection in mm N18= load repetitions/(105) in 80KN ESALs
Οc= compressive stress on top of base in kPa.
PD= permanent deformation N=the number of load repetitions a, b and m are regression coefficients
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The number of load repetitions given by Equations 2.23 to 2.27 determines the permanent deformation without direct consideration of the stress level. Monismith (1988) concluded that the parameter (b) in power model (π1,π= π ππ) depends only on soil type and parameter (a)
plays the main role in the development of permanent deformation. A number of researchers have found the power model acceptable for practice especially for fine-grained soils (Elliott et al., 1998). They concluded that the confining pressure is not a significant factor for permanent deformation development.
To account for the influence of the stress level on the permanent deformation, a number of authors have proposed relationships between the permanent deformation, the deviatoric stress and confining pressure (e.g. Lekarp and Dawson, 1998). Two such models proposed by Barksdale (1972) and Pappin (1979), which do not take into account the number of load cycles. These two models neglect the effect of the number of load repetitions.
As mentioned previously (section 2.2.3) an important property of the unbound granular materials in pavement design is the resilient modulus. Resilient properties of the subgrade materials and their resistance to the permanent deformation are necessary for an appropriate pavement design. Veverka (1979), Equation 2.31, correlated resilient and plastic properties. According to Lekarp et al. (2000a), Sweere (1990) could not confirm this equation for base course materials and sands. However, MEPDG (2004) used Tseng and Lyttonβs equation (Tseng and Lytton, 1989), Equation 2.32, for determining the rutting of the subgrade layer in a pavement system. Also Uzan (2004) proposed a model for clayey subgrades to predict the ratio of permanent deformation to resilient strain from the number of load repetitions as presented by Equation 2.33; these two models include resilient strain in the model. In Chapter Five an equation has been modified to consider deviatoric stress and resilient modulus for subgrade soils investigated in this research.
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Lekarp and Dawson (1998) conducted a series of repeated load triaxial tests on five different aggregates and compared the data with the models (Equations 2.23 to 2.31). Their work showed that the Paute et al. (1996) model has the least error in most cases. Then a model developed by Lekarp et al. (1998), Equation (2.30), that includes the maximum shear stress ratio (q/p) max and the length of the stress path (L). The increase of the maximum shear stress ratio results in an increase in accumulation of permanent strain.
Li and Selig (1996) modified the power model which relates the rate of cumulative plastic strain to the number of repeated load applications. They believed that introducing some properties of the soil such as moisture content and density directly into the model is neither common nor convenient. Therefore they incorporated the static strength of the soil to represent moisture content and density in the general power model, see Equation 2.35. Li and Selig (1996) constructed a railroad track to check the validity of the modified model and found a good correlation between the subgrade plastic deformations predicted from the model and the experimental test results.
Puppala et al. (1999) modified the model developed by (Ullidtz, 1993), Equation 2.33, which included the vertical stress effects. They introduced the octahedral stress (Οoct) into the model described by Equation 2.36 to account for confining pressure which they believed it had influence on permanent deformation of soils. Puppala (1999) carried out repeated load triaxial tests on three soil types to find the parameters of the model and he found a coefficient of determination ranging from 0.79 to 0.93. Field evaluations also verified the predicted plastic deformation.
Puppala et al. (2009) reported the drawback of models that combine the effect of deviatoric stress and confining pressure on plastic strains. To account for this they developed a four-
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parameter permanent strain model which separated the influence of deviatoric stress and confining pressure, see Equation 2.37.
Theyse (1997) used the data collected from a Heavy Vehicle Simulator (HVS) test to develop a model of permanent deformation that related the permanent deformation to the number of repeated loads and the vertical stress on the top of the pavement foundation. The data obtained from a Multi-Depth Deflectometer (MDD) was used to develop the model further. βThe MDD system is basically a stack of linear variable displacement transducers (LVDTs) referred to as MDD modules, installed at predetermined depths in the pavement structure with a reference point at the anchor, normally at 3 m depthβ (De beer et al., 1989). In the model developed by Theyse (1997), the permanent deformation, PD, is determined as presented in Equation 2.38a. From the relationship of the number of load repetitions and coefficient (A) Equation 2.38b is obtained. Substituting A in Equation 2.38a yields Equation 2.38c.
Chen and Lin (1999) used data from the Texas Mobile Load Simulator (MLS) to develop a model predicting permanent deformation. The model relates the rutting to surface deflection, load repetitions and compressive stress at the top of the base, see Equation 2.40.
R = -5.841+ 25.419 d + 0.29 N18 β 0.015 Οc (2.40) Where R is rutting in mm; d is surface deflection in mm; N18 is load repetitions/(105) in 80KN ESALs; Οc is compressive stress on top of base in kPa.
Then a ten-based logarithm was applied to Equation 2.38 to obtain Equation 2.41:
Log (R) = -3.17 + 0.141 Log (d) + 0.564 Log (N18) + 1.505 Log (Οc) (2.41) Then Equation 2.39 has been transformed to model the rate of rutting (i.e. the future condition) as presented by Equation 2.39 in Table 2.3.
As demonstrated in this section the progress of the permanent deformation were modelled using different parameters such as stress state, resilient strain, surface deflection, material
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properties and the number of load repetitions. In this research a performance model for predicting the permanent deformation was attempted for unstabilised and modified subgrade soils in which the deviatoric stress, resilient modulus and the number of load repetitions is considered.
Summary
In this section a detailed literature was reviewed on permanent deformation modelling. The development of permanent deformation in these models is related to the number of load repetitions, stress state, resilient strain, deflections and material properties. However, some researchers developed their model taking into consideration one or two of these parameters. The performance model comprises a major part of analytical pavement design therefore a performance model was developed in this research to take into consideration a higher number of parameters, especially when the deterioration of the modified subgrade soils is included in the design procedure. Chapter Five in addition to presenting the development of the performance model for prediction of the permanent deformation demonstrates in detail the factors affecting the development and progress of permanent deformation for stabilised and unstabilised subgrade soils.