1 Introduction

The use of geogrids to facilitate the construction and to improve the structural performance of low-volume roads over soft subgrades is a common practise in the industry. Geogrid reinforcement is known to increase the composite modulus over 50% in comparison with unreinforced granular base, which implies a substantial saving in pavement thickness or extending the service life of the pavement [1]. The benefits of geogrids in low-volume unpaved roads have been supported by past studies [2,3,4,5,6,7,8,9,10,11].

Over the last 4 decades, numerous studies have been undertaken to evaluate the use of geosynthetics for reinforcing granular base. These studies have provided information on the mechanics of the geosynthetics reinforcement, membrane effect of geosynthetics and confinement effect of the base as summarised in Giroud and Noiray [12] and Perkins and Ismeik [13, 14]. Current design practise uses simplified stress distribution models and one or two reinforcement mechanisms with the unpaved pavement structure for large deformation such as methods proposed in Giroud et al. [15], Milligan and Love [16] and Giroud and Han [17, 18]. These methods are generic and assumed a large deformation of permanent unpaved roads.

The geogrid reinforcement is accounted for in the AASHTO design guide for flexible pavements [19]. The design approach involves quantifying the structural contribution of geogrids by the increase of the bearing capacity mobilisation coefficient of the granular base and/or the reduction in the thickness of the base layer. Perkins et al. [20] used FEM to develop the mechanistic–empirical (M–E) modelling of geosynthetics-reinforced flexible pavements based on the unreinforced M–E model [21]. Perkins et al. [22] used the same model to incorporate the influence of pore-water pressure generation in the subgrade. They found that the geosynthetics-reinforced sections are more stable than unreinforced sections.

In Australia, Austroads [23] provides broad coverage on haul road and temporary unbound granular pavements with a focus on geotextiles. Austroads [24] does not provide guidelines on the enhanced stiffness of the geogrid-reinforced granular base layer, rather the geogrid-reinforced granular material is referred to as selected subgrade material. Geogrid reinforcement cannot be modelled directly as a separate layer of the linear elastic model, CIRCLY [25]. By assuming the unreinforced and geogrid-reinforced granular base as selected subgrade, Christopher and Wardle [26] used CIRCLY to determine the pavement thickness of the granular base layer. They found that Austroads [24] approach fails to recognise the improvement by the addition of geogrid reinforcement. This low-risk approach reflects the lack of local performance studies, although there is evidence in the international studies to support the use of geogrids to reduce thickness of the base layer. For the treatment of soft subgrades with California Bearing Ratio (CBR) of less than 3%, DTMR [27] has provided a design table for the suggested granular base working platform wrapped in geotextile and their confinement benefit consideration. RMS [28] has suggested a minimum of 200 mm bound granular base and assumes a subgrade design CBR of 3%. With the exception of DTMR and RMS, other Australasian Road Agencies do not provide additional guidance for subgrade treatment for CBR of less than 3%.

In the UK, the pavement foundation is designed in advance and separately from the pavement structure [29, 30]. This design approach has no perceived benefit and unnecessarily restricts design options. The French method suggests possible treatments by lowering the water table or replacing 500 mm of the soft subgrade or using in situ lime stabilisation [31]. However, there is no method specified to calculate thicknesses using these treatments.

In M–E design, the structural failure is considered primarily due to surface deformation (i.e. rutting). The pavement design approach is to limit the vertical compressive strain at the top of subgrade to a tolerable level throughout its service life. Traditionally, the M–E procedure incorporated the limiting subgrade strain criterion to determine the pavement thickness required to cater for design traffic. There are numerous alternative design methods and their subgrade failure criteria as discussed in Sect. 2. However, a subgrade failure criterion cannot be compared with other in isolation because they are inseparable part of a particular pavement design method.

In this study, an unpaved road pavement damage model (limiting subgrade strain model) has been proposed to limit rutting to less than 20 mm [32]. Considering the fact that it is a small rut, it is reasonable to assume that the resilient strain induced provides an indication of the plastic strain to predict the permanent deformation. This approach allows geogrid to be incorporated into the pavement design using the M–E approach by modifying the limiting subgrade strain equation for rutting to determine the allowable number of traffic repetitions. In this way, contribution of geogrid reinforcement to the structural strength of the pavement can be accounted for using empirical limiting subgrade strain equation for rutting. However, this approach is not currently in Austroads [33] design guide due to lack of local performance studies.

2 Unpaved Road Pavement Performance

Low-volume roads mainly comprise unreinforced or geogrid-reinforced granular materials laid over subgrade. In such pavements, the unreinforced or geogrid-reinforced granular material is the main structural layer to carry the traffic load. The structural performance for this type of pavement is dictated primarily by rutting or loss of surface shape attributed to the compressive vertical compressive strain at the top of the subgrade. A common damage model (transfer function or failure criteria) used to define the relationship between the limiting vertical compressive strain on the subgrade and the number of standard axle load repetitions to cause failure by excessive rutting has been used extensively in M–E design of flexible pavements [25, 34, 35]. Past researchers [36, 37] used AASHTO road test data to develop their subgrade strain criterion and found the strain instead of stress criterion is independent of the subgrade stiffness. Dorman [38] evaluated the design charts and found that the allowable vertical compressive strain at the top of the subgrade was practically a constant irrespective of the pavement thickness. The allowable vertical compressive stress for design is a function of the subgrade CBR but it does not depend on the number of load repetitions. It is for this reason, the theory of vertical stress criteria proposed by Peattie [39] never caught on.

Past researchers and several agencies [22, 33, 37, 40,41,42,43,44,45,46,47,48,49,50,51,52,53,54] have used a power model as shown in Eq. (1) to relate vertical compressive strain at the top of the subgrade (εv) to the allowable number of traffic load (N) before an unacceptable level of pavement surface deformation develops.

$$N = {\left[\frac{k}{{\upvarepsilon }_{v}}\right]}^{B},$$

where k is the material constant, B is the damage exponent and ɛv is the induced vertical compressive strain (in terms of microstrain) at the top of the subgrade.

Historically, the vertical compressive strain is estimated by layered-elastic analysis [33]. This approach assumes that the permanent strain is proportional to elastic strain [55]. By limiting the elastic strain to some prescribed value also limits the plastic strain. The integration of the permanent strain over the depth of the pavement structure provides an indication of the rut depth. Therefore, by limiting the elastic strain at the top of the subgrade, the magnitude of the rut is controlled. Table 1 summarises different values of coefficients for the subgrade failure in Eq. (1) used by different agencies and research literatures.

Table 1 Rutting model coefficients

The empirical power model in Eq. (1) is used to determine the pavement thickness required to cater for the design traffic. Each failure criterion is an inseparable part of a specific pavement design method, which implies that it cannot be extracted and applied outside the context it was developed. When the limiting subgrade vertical compressive strain is analysed for both unreinforced and geogrid-reinforced granular base for the same pavement structure and for a specified rut depth, the ratio NR (allowable traffic loading for geogrid-reinforced pavement) over NUR (allowable traffic loading for unreinforced pavement) is equal to traffic benefit ratio (TBR). By substituting Eq. (1), the relationship between TBR and subgrade strains can be developed and presented in Eq. (2).

$$\mathrm{TBR }= \frac{{N}_{R}}{{N}_{\mathrm{UR}}}{=\left[\frac{{\upvarepsilon }_{v-R}}{{\upvarepsilon }_{v-\mathrm{UR}}}\right]}^{-B},$$

where the symbols UR and R denote unreinforced and geogrid-reinforced granular base.

3 Problem Statement and Study Objective

Although the basic functions of geogrid-reinforced granular base are reasonably understood by road agencies to extend the service life of pavement, to reduce pavement thickness for equivalent service life, or a combination of the two, there is no guidelines for designers. The literature review revealed that there is no known rutting model for geogrid-reinforced granular base on soft subgrade soils with CBR of 3% or less. The objective of this study is to develop a limiting subgrade strain criterion (damage model) and to produce pavement design chart for low-volume unpaved roads on soft subgrade with geogrid-reinforced granular base. The work performed in this study addresses the use of geogrid reinforcement in conjunction with the design by the M–E procedure.

4 Finite Element Modelling

Though three-dimensional (3D) model can better replicate the actual tyre imprint of the wheel load on the pavement structure than the circular loaded area as restricted in two-dimensional (2D) axisymmetric analysis or the infinite strip load as in 2D plane strain analysis, it requires more computational time and computer memory. It is time-consuming and not practical to run multiple cases for parametric studies. The 2D axisymmetric model is known for its simplicity, greater efficiency regarding computation time, reduction of data and adaptability to pavement models with more cost-effective solutions than 3D modelling. As such, a 2D axisymmetric model is developed in ABAQUS to analyse the two-layer soil system for both with and without geogrid reinforcement of granular base. This approach is consistent with the past studies [9, 63,64,65,66,67].

4.1 Model Geometry

The accuracy of computational results is dependent on the model’s geometry [9]. Several past studies have used varying model geometry and defined the domain size using equivalent radius (r) of the circular loaded area as summarised in Table 2.

Table 2 Studies on model geometry

The dimensions of the axisymmetric model in this study are selected in such a way that the effect of load dissipation is contained to control the computational time. If the maximum induced stress at the outer boundary is less than 0.01% of the applied contact pressure, then it can be considered negligible [72] and acceptable in pavement design practise. In addition, the model dimensions are selected to minimise the undesirable resonance effects of the boundary’s reflection. Kim et al. [74] found that the FEM results are in good agreement with KENLAYER results when adopting 140 times the radius in the vertical and 20 times the radius in the horizontal dimension. The final adopted FEM dimensions are 2.3 m radius cylinder with 0.2 to 0.8 m thick granular base on 12.75 m soft soil to replicate infinite depth as illustrated in Fig. 1. This equates to approximately 25 times and 145 times the radius of circular loaded area for the model radius and height, respectively, to represent the outer model boundaries. As the pavement model is axisymmetric, only half of the whole system is modelled.

Fig. 1
figure 1

a 2D axisymmetric pavement model, b FEM mesh, c boundary conditions and load

4.2 Boundary Conditions

The conventional kinematic boundary conditions are adopted in this study whereby boundaries are assigned to the outer perimeter, the rotation axis and the bottom of the model. However, the top surface of the model is unrestrained and the outer perimeter is restrained in the radial direction but allowed to move in the vertical direction. The bottom of the model is assumed to be encastred (fixed) and restrained from moving both vertically and horizontally.

4.3 Traffic Loading

The FEM model in this study is for simulating single axle dual tyre (SADT) applying a load of 80 kN including the gravity load in the initial loading step of the analysis. The axle load is simulated by applying the contact pressure of 750 kPa on a circular area with a radius of 92.1 mm at the surface [33]. A haversine-shaped load wave of 0.1 s [84,85,86] with the maximum magnitude of 20 kN is adopted to simulate a single wheel load of a standard axle. The cyclic loading is implemented into ABAQUS with the help of subroutine DLOAD. However, as 2D axisymmetric model can only replicate a single circular load, it is not possible in this geometrical formulation to handle dual tyres or multiple axles loading conditions. Considering the compressive strain on top of the subgrade is very small and the surface deformations are recoverable under repeated application of traffic loads, it is reasonable to assume the pavement model performance is within the elastic range. Linearity implies the applicability of the superposition principle. Huang [52] used this technique to develop strain conversion factor charts to determine the strain factor for dual wheels. Australia [77] proposed a simplified superimposing method and successfully used in the finite element study as illustrated in Fig. 2. In this study, Austroads [77] method is adopted for simplicity.

Fig. 2
figure 2

Superposition of strains in axisymmetric model (adapted from Austroads [77])

The modelling of multiple circular loads such as SADT is performed by superimposing the single load prediction responses at various offsets [77]. Standard axle loading consists of a dual tyred single axle, applying a 80 kN load and with circular tyre contact stress of 750 kPa. The four-wheel loads are numbered from 1 to 4 as shown in Fig. 2. The critical response within the pavement is assumed to occur along the vertical axis located symmetrically between a pair of dual wheels, at the top of the subgrade. Therefore, the effect of the four loads at the centre of inner wheel is accounted for by adding all the strains obtained from each load. It should be noted that the superposition of stresses and strains occurs at the end of the numerical analysis.

4.4 Finite Element Type

Eight-node biquadratic axisymmetric quadrilateral solid elements (CAX8R) are used to discretise the granular base and subgrade, whilst two-node linear axisymmetric membrane elements (MAX1) with the thickness of 3 mm are used to discretise the geogrid reinforcement [87]. A finer mesh near the surface wheel load is used to capture the steep stress and strain gradient within this area. By implementing edge biassed structural meshing pattern in ABAQUS, a smooth transition from finer mesh at the loaded area to coarser mesh in the region away from the loading area. The same mesh pattern is used to simulate all layers to preserve the continuity of nodes between consecutive layers. The optimum number of finite elements meshes for the FEM is determined by trial and error with the mesh sensitivity tests until the numerical analysis converges to a unique solution.

4.5 Materials

The modulus of unbound granular materials and subgrade are dependent on the stress level at which they operate and the modulus of the underlying layers. This infers that the modulus of granular material decreases with depth and is influenced by the modulus of the subgrade. Mohr–Coulomb elastoplastic models are selected to represent the nonlinear material properties whereby the shear strength developed is a function of internal friction, cohesion and applied stress. The granular base is assumed to be purely frictional granular soil. But to enhance the stability of numerical analysis, the cohesion for granular base is set as 0.1 kPa [88, 89]. Past researchers have successfully used Mohr–Coulomb elastoplastic model [7, 67, 90].

For simplicity, Austroads [33] sub-layering approach is used to replicate the cross-anisotropic characteristics of subgrade and granular base. Khodakarami and Moghaddam [91] used the sub-layering technique to study the performance of geogrid-reinforced in rehabilitated paved roads by 2D FEM models. This approach assumes that the modulus of unbound granular base layer varies with the underlying subgrade support with CBR ≥ 3%, which better reflects in-service performance to estimate the average resilient modulus. However, soft subgrade materials with CBR < 3% are unlikely to behave elastically when loaded [27]. This is attributed to the fact that soft subgrade does not provide a stable working platform required to obtain good compaction of the granular layer, thereby resulting in lower modulus value. This approach prevents the use of maximum modulus of a granular material to be developed regardless of the thickness of the granular layer or the underlying subgrade strength. Under this condition, the modulus ratio of granular to subgrade is usually taken to be 1 to 5 as recommended by Heukelom and Klomp [40] and a ratio of 2 to 5 are selected in this study. A band of modulus values of granular base corresponds to lower and upper bound subgrade support values, respectively, shown in Table 3. In addition, other presumptive material input parameters used in this study are based on published data [33, 92, 93].

Table 3 Material properties for FEM analysis

Past researchers have found that the influence of the associated flow rule is significant for higher friction angle (ϕ > 30°) up to 15% [90, 94, 95]. Soil dilatancy is accounted for in Mohr–Coulomb model by allowing the yield surface to expand based on plastic straining. In this study, the associated flow rule is replicated by selecting the dilatancy angle ψ equal to internal friction angle less than 30° [89, 93, 96].

4.6 Geogrid-Reinforced Pavement Model

For the condition of a small rut of less than 20 mm, the induced strain in the geogrid is very small and is considered within the elastic range. Hence, a linear elastic model is used to describe the behaviour of geogrid material. In addition, the maximum geogrid tensile strength (axial stiffness) at 2% strain from the material in the machine and cross-machine directions is used. This means that the direction-dependent character of the geogrid is not considered for simplification. Other researchers have successfully used the same modelling assumption [11, 69, 97, 98]. The mechanical stabilising effect of geogrid extends over 100 mm from the geogrid plane and is simulated in the model by assigning stress concentrations of 63 kPa at the geogrid elevation and decreases linearly at a constant rate to nominal 21 kPa at 100 mm above the geogrid plane within the influence zone [9, 99]. The medium stiff geogrid is selected for this study because it is commonly used in engineering practise.

The soil–geogrid interface model is implemented by adopting the ABAQUS contact interaction feature. This feature entails one surface acting as a rigid body. This is defined as a ‘master’ surface and the other deformable body surface provides the ‘slave’ surface. Under this arrangement, the nodes on the slave surface are constrained not to penetrate to the master surface and the nodes of the master surface can, in principle, penetrate into the slave surface. Once the contact pair is defined, the programme automatically generates a family of contact elements. The clearance and relative shear sliding for these elements are measured at each integration point. These kinematic measures are then used, together with the appropriate penalty friction formulation techniques to introduce surface interaction theories.

In the unreinforced case, the interface between granular base and subgrade is assumed to be fully bonded using surface-based tie constraint with node-to-surface instead of surface-to-surface formulation. The surface-to-surface formulation cannot accommodate a mixture of rigid and deformable portions of a surface. This restriction is overcome with the adoption of node-to-surface formulation. For geogrid-reinforced case, the granular base–geogrid–subgrade interface is assumed to be kinematic coupling constraints using node–node formulation with a reference node. This approach will ensure that the geogrid is held in place at the interface when the model is in motion under the applied load.

The geogrid is simplified as continuous membrane embedded between the granular base and subgrade. With the assumption of continuity for geogrid in the finite element model, it is not feasible to directly simulate the interlocking between the geogrid and surrounding pavement materials. This is addressed using Coulomb friction model to simulate the shear resistance interaction at the interface between the geogrid and pavement materials [80, 100,101,102]. This is implemented using the friction coefficient of the interfaces μ and the elastic slip input data into ABAQUS. An elastic slip of 0.001 m is adopted based on Perkins and Cuelho [103] and Perkins [69]. The friction coefficients of geogrid-granular base of 0.84 and geogrid-subgrade of 0.62 have been selected for this study based on BOSTD [104].

5 Verification of Finite Element Model

For model verification, the same granular base thickness and material properties used by Leng and Gabr [7] are adopted. Prior to parametric study, the FEM results are validated with published results of Leng and Gabr [7]. The vertical stress distribution underneath the centre of the uniformly loaded circular area through the granular base and subgrade layers is shown in Fig. 3. The stress distribution is not smooth for the elastoplastic FEM at the interface of granular base–subgrade as compared with linear elastic model as in KENLAYER programme and Boussinesq’s homogeneous elastic solution. This is attributed to shear resistance interfaces and is more pronounced when the geogrid-reinforced granular base becomes stiffer owing to tensile resistance and limit tensile yielding at the bottom of the granular base layer resulting in lower vertical stress in the granular base layer.

Fig. 3
figure 3

Comparison of vertical stress and stain results based on Hbase = 0.15 m, Ebase = 50 MPa, Esubgrade = 10 MPa, Egeogrid = 100 MPa (Data source: Leng and Gabr [7])

The effect of material characteristics can be seen in Fig. 3a. KENLAYER and Boussinesq’s elastic models are stiffer pavement system (constant elastic modulus) with lower vertical stress for depths greater than 0.4 m in the subgrade layer. In all cases, the elastoplastic FEM predicted lower vertical stress in the granular base layer which is more pronounced with geogrid reinforcement.

The vertical strain distribution through the granular base and subgrade layers along the centreline of the loading area is shown in Fig. 3b. The results show that there is a large vertical strain at the bottom of the unreinforced granular base layer resulting from yielding of the granular layer due to low tensile strength causing the outward lateral spreading of the granular base material. It also shows that the vertical strains at the bottom of granular base are significantly reduced when geogrid reinforcement is included. These results illustrate the geogrid reinforcement ability to limit lateral spreading of granular base and is fundamentally linked to the stress response of the pavement structure.

Overall, the vertical stress and strain results from this study show good agreement with elastoplastic FEM analysis conducted by Leng and Gabr [7].

6 Results and Discussion

6.1 Design Equations and Curves

Empirical design chart by Austroads [24] for lightly trafficked roads is used to determine the thickness of unreinforced granular base for five sets of traffic loading (i.e. 1 × 103, 5 × 103, 1 × 104, 5 × 104 and 1 × 105 ESA) and repeated for six sets of subgrades CBR values (2, 3, 4, 5, 6 and 7%). The results show a linear relationship between the thickness (H) of unreinforced granular base and the design traffic (N) for each subgrade CBR value as depicted in Fig. 4.

Fig. 4
figure 4

Granular base versus number of repetitions

When the data points are fitted with a semi-log trend line for each subgrade CBR value, it shows a distinctive semi-log equation form as in Eq. (3).

$$H = A\;{\text{ln}}\;N + B,$$

where A and B are parameters determined from each curve in Fig. 4.

This process was repeated for unreinforced granular base with internal friction angle ϕ = 40°. The granular material is similar to normal crushed rock base quality material. The results from this seem to be more conservative when compared with Austroads [24]. This is attributed to the nonlinear constitutive model being used in FEM.

The parameters A and B determined from each curve for subgrade CBR of 1, 2 and 3% were then plotted against the subgrade CBR and used the polynomial trend line to establish the good fit to the data as shown in Fig. 5.

Fig. 5
figure 5

a A versus subgrade CBR b B versus subgrade CBR

The regression equations for A and B obtained for the curve are used in conjunction with Eq. (3) to derive design curves, and hence to determine thickness of granular base. Similar approach is followed for medium stiff geogrid-reinforced granular base with internal friction angle (ϕ) = 40°. The regression equations for unreinforced granular base are presented in Eqs. (4) and (5).

$$A = { 1}.{\text{1965CBR}}^{{2}} {-}{ 9}.{\text{5895CBR}} + { 45}.{393}$$
$$B = { 8}.{\text{545CBR}}^{{2}} {-}{ 1}0{1}.{\text{65CBR}} + { 353}.0{9}$$

The equations for medium stiff geogrid-reinforced granular base are in Eqs. (6) and (7).

$$A = \, 0.{\text{3255CBR}}^{{2}} {-}{ 2}.{\text{2795CBR}} + { 24}.{972}$$
$$B = {\text{ 8CBR}}^{{2}} {-}{\text{ 91CBR}} + { 3}0{5}$$

where A and B are regression function parameters relating to subgrade CBR values. These equations are valid for low-volume roads with subgrade CBR less than or equal to 3%.

6.2 Comparison with Other Methods

The thickness of pavement with and without a geogrid reinforcement obtained from the proposed design equations is used to compare with other established methods. For unreinforced granular base over subgrade CBR of 3% and 2%, the design curve obtained from this study shows close agreement with mechanistic–empirical method using CIRCLY programme as shown in Figs. 6a and b. When compared with Austroads [24] and AASHTO [19], the design curve is bounded by these two methods with the similar trend. The proposed design curve underestimates the base thickness in comparison with the Giroud and Han (G–H) [17, 18], whereas Holz et al. [105] method is comparable for number of repetitions less than 103 ESA. The commonly used Road Note 31 [48] empirical approach widely used in tropical and sub-tropical countries appears to underestimate the thickness of pavement and is more applicable to traffic exceeding 3 × 106 ESA. In the case of subgrade CBR 3%, South Africa Method [106], Indian Road Congress [107] and ARRB [108] approach appear to have a leaner design. This could be attributed to low reliability level of 50 to 80% being used. HIPAVE [35] method is overly conservative for standard axle loading and is applicable to higher loadings condition used at ports and container terminals [109]. The failure model for HIPAVE [35] is based on USACE CBR method [59] calibrated with aircraft loading. Figure 6a and b clearly shows that the use of HIPAVE method is not applicable to standard axle loading condition. For soft subgrade with CBR less than 3%, the DTMR [27] empirical method appears overly conservative when compared with all other methods as shown in Fig. 6b and c.

Fig. 6
figure 6

Comparison of pavement thickness with other approaches for unreinforced base

For geogrid-reinforced base over subgrade CBR of 3% and 2%, the pavement thickness obtained from this study shows good agreement with Holz et al. [105] and Giroud and Han [17, 18] methods as shown in Fig. 7a and b. In all cases, Perkin [110] empirical method shows similar trend but appears to underestimate when compared with other methods including the proposed design curves. For soft subgrade CBR 1% as shown in Fig. 7c, there is slight deviation in trend with increase in number of repetitions exceeding 103 ESA. This could be attributed to the unstable nature of the models for very weak soil conditions.

Fig. 7
figure 7

Comparison of pavement thickness with other approaches for geogrid-reinforced base

It is evident from this study, that the pavement thickness on soft subgrade soil using the proposed design curve derived from the proposed design equations is comparable with the established methods.

6.3 Limiting Subgrade Strain Curves

A common damage model used to define the relationship between the limiting vertical compressive strain on the subgrade and the number of standard axle repetitions to cause failure by excessive rutting has been extensively used in M–E design of flexible pavements [25, 34, 35]. In this study, results from FEM for 20 mm rut depth for both unreinforced and geogrid-reinforced granular bases over soft subgrade are fitted with power trend lines as shown in Fig. 8.

Fig. 8
figure 8

Maximum allowable subgrade strain criterion

Figure 8 shows that the vertical compressive strain at the top of subgrade decreases with the increase in the number of load repetitions. For a given number of load repetition, the allowable subgrade strain is consistently higher for the geogrid-reinforced granular base case as compared to the unreinforced case similar to a shifted line for reinforced granular base. This implies that for a given subgrade strain value geogrid reinforced granular base case has higher number of repetitions as compared to unreinforced case. Based on the FEM results, this study proposes the following limiting subgrade strain criterion for unreinforced and medium geogrid-reinforced granular base for low-volume roads:

$${N}_{\mathrm{UR}} = {\left(\frac{8350}{\mathrm{\mu \varepsilon }}\right)}^{7.2}$$
$${N}_{R} = {\left(\frac{9388}{\mathrm{\mu \varepsilon }}\right)}^{7.5},$$

where NUR and NR are allowable number of standard axle load repetitions for unreinforced and geogrid-reinforced granular base, respectively.

The validity of these damage models is checked by comparing with the results from established methods [22, 33, 43, 50, 53, 56] and results are shown in Fig. 9.

Fig. 9
figure 9

Comparison of maximum allowable subgrade strain criteria

The proposed subgrade strain line is comparable with Austroads [33], Shell [43], French Design Method (Corte and Goux [50]), Indian Road Congress [53] and Perkins et al. [22]. The lower allowable strain line developed by Brown and Brunton [42] assumes a greater proportion of permanent deformation in the asphalt, which is more appropriate for full-depth asphalt pavements and British conditions whereas, Asphalt Institute [56] line relates to climatic conditions of California. Shell [43, 44] and Brown and Brunton [42] developed their curves from the observed performance of in-service roads subjected to truck loadings with dual tyre up to around 5 tonnes. Shell [43, 44] lines are derived from the AASHTO road test with the strong influence of the freeze–thaw cycle appropriate to Illinois. However, the use of AASHTO data to derive limiting subgrade strain criterion need to be used with caution. It is based exclusively on the silty clay soil at one moisture content and may not be applicable for other materials (gravel, sand, silt, or clay) nor for other moisture conditions. In addition, the use of elastic parameter such as subgrade strain to predict a plastic deformation may not be a rational approach. Furthermore, the estimates of the limiting strain effects were based on relatively thick asphalt surfacing which were subject to freeze–thaw cycles not applicable to bulk of Australian road network conditions.

The study by Perkins et al. [22] is based on Georgia subgrade which is appropriate for American conditions. Based on large field performance data of bituminous pavement across India under various climatic conditions, Indian Road Congress [53] specifies the rutting life relationships for subcontinental conditions. However, all the subgrade criteria assume that the granular base has no influence on pavement performance. It assumes that the pavement as a whole may rut by idealising the subgrade deforming with the granular base deflecting bodily on it without thinning, meaning that the permanent deformation for the granular base layer is not included in the procedure. To the best of authors’ knowledge, there is no suitable model to reliably predict the rutting in granular base layer under traffic [24]. In the case of reinforced granular base, there are no published literatures available for benchmarking as evident in Fig. 9 where all the published data are for unreinforced granular base.

The damage models in Eqs. (8) and (9) can be incorporated into the M–E framework to evaluate the strain (ɛv) at the top of the subgrade for a given pavement structure to estimate the number of traffic passes with and without geogrid-reinforced granular base and the TBR value. Using this approach, the benefit of geogrid reinforcement can be quantified. For the same subgrade strain value, the ratio of NR over NUR is equal to TBR. Based on Eq. (2), the proposed limiting subgrade strain criterion for unreinforced and geogrid-reinforced granular base, the estimated TBR at 20 mm rut depth for medium stiff geogrid reinforcement is approximately 3.1. This result is comparable with TBR results at 25 mm rut ranges from 1.6 to 4.1 obtained by Cuelho et al. [111]. Based on Cuelho et al. [111], it can be inferred that higher allowable rut will lead to higher TBR values. It should be mentioned that the proposed equations are validated by a relatively small database and it is important to further verify using field tests on actual full-scale geogrid-reinforced granular base pavements in the future.

6.4 Limitations and Applicability

The pavement design curves and the limiting subgrade strain curves (and associated equations) developed in this study are valid only for a set of specific conditions. First, the subgrade is assumed to be saturated with low permeability and it behaves in an undrained manner under traffic loading. Second, the design curves are valid for the number of allowable traffic loads not exceeding 105 ESA for a terminal rut depth of less than 20 mm. Third, there is no allowance for the effects of dynamic loading factor to account for cornering, acceleration, braking and surface unevenness. Forth, these design curves only apply to medium stiff geogrids laid on a horizontal plane at the interface of granular base and subgrade. Fifth, the design curves assume that the granular base is a normal standard crushed rock (CBR ≥ 80%) commonly used by road agencies [24]. Sixth, the geogrids should meet the minimum standard specification as per DTMR [112] or AASHTO [113]. Seventh, single axle with dual tyres (SADT) applying a load of 80 kN and uniformly loaded circular areas of equal area (radius 92.1 mm) with tyre contact stress of 750 kPa is used to develop equations and curves. Eighth, the simulated maximum vehicle speed is 50 km/h [114]. Ninth, each layer is homogeneous with a finite thickness, except that the subgrade layer is infinite in thickness. Lastly, the curves should not be extrapolated beyond the data base used to develop them.

7 Conclusion and Recommendations

An M–E pavement design approach for geogrid-reinforced granular base on low-volume unpaved roads has been developed using a 2D axisymmetric finite element model. The objective of this paper is to analyse the critical responses of the pavement layers and to establish limiting subgrade strain models that relates to long-term pavement performance or the reduction of granular base layer thickness for equivalent service life or a combination of both. This approach allows to incorporate geogrid reinforcement in the M–E pavement design method by modifying the limiting subgrade strain equations for rutting and to determine the allowable number of traffic repetitions (service life). The contribution of geogrid reinforcement to the pavement structural strength can also be quantified. The proposed regression equations can be used in lieu of design curves to determine the thickness of the granular base. The limiting subgrade strain equations for unreinforced and geogrid-reinforced granular base can be used to determine the TBR. The FEM results show that the TBR value of 3.1 at 20 mm rut depth can be achieved for thin granular base built over soft subgrade using medium stiff geogrid. However, the proposed regression equations are appropriate to a locality that is dependent on the performance of granular base, subgrade and geogrid that were used to develop the FEM. The developed failure criterion is an inseparable part of the design curves, which means it cannot be directly used to compare with other methods in isolation.