1 Introduction

The subgrade layer is the natural or prepared soil layer beneath the pavement structures, and it plays a crucial role in providing the support and stability required for the overlying pavement. However, the level of support as characterized by the subgrade strength or modulus can vary spatially due to a combination of factors such as natural deposition processes, variation in construction quality and environmental change. Spatial variability in the subgrade layer can have significant impacts on pavement performance. Non-uniform settlement and deformation of the pavement may arise due to this variability [19], resulting in undesirable and excessive pavement distress such as cracking, rutting, and roughness [37]. Furthermore, localized bearing capacity failure may also occur in regions where the subgrade is very weak or soft, leading to the ingress of moisture and the contamination of the aggregate base material with underlying soft subgrade soil. All these consequences can exacerbate the risk of premature failure of the pavement system.

Spatial variability is a significant contributor to material uncertainty in geotechnical engineering [32]. It can be effectively characterized using random field theory, primarily through two key parameters: the coefficient of variation (COV) and the scale of fluctuation (SOF). The COV describes the extent of dispersion in material properties. The SOF, which characterizes the pattern of property dispersion, is the distance at which the correlation between two points in the field decays to a negligible level. The random field finite element analysis (RFFEA), or the random field finite difference analysis (RFFDA), both of which integrate numerical simulation with random field theory and Monte Carlo Simulation, are probabilistic frameworks widely employed to characterize the spatial variability of materials and assess its statistical impact on the structural response [11, 19, 20, 22, 23, 27, 41].

Numerous studies have investigated the influence of subgrade spatial variability on the pavement response with the two probabilistic frameworks. Lua and Sues [27] performed a comprehensive probabilistic analysis that accounted for various uncertainties, including spatial variability arising from the asphalt layer and foundation layers (i.e., the base and subgrade layers). They concluded that ignoring spatial variability can lead to inaccurate prediction of pavement life. Ali et al. [4] examined the impact of subgrade spatial variability on pavement responses and revealed that the rutting life is more sensitive to the subgrade spatial variability as opposed to fatigue life. Vaillancourt et al. [37] investigated the effect SOF of subgrade stiffness on the roughness of flexible pavements, as quantified by the International Roughness Index (IRI). They found that SOF had a direct impact on the wavelength of the resulting deformations and the IRI value. Alhasan et al. [3] employed geospatial models to define the spatial variability and systematic measurement errors of base and subgrade layers and consequently established a mechanistic-empirical (ME) pavement performance model based on reliability analysis. They concluded that: (1) when performing risk and reliability analysis, it is essential to distinguish the inherent spatial variability from other uncertainties and (2) the location of critical responses can be different from the deterministic situation described in the ME pavement design guides such as AASHTO [1]. Choudhuri and Chakraborty [13] investigated how the horizontal and vertical SOF of the subgrade soil affect the pavement bearing capacity using an anisotropic random field.

The above studies focus on the influence of spatial variability in subgrade layers on pavement performance, consistently revealing the adverse impact of such variability on the structural response of pavements. However, there are limited studies effectively delving into the statistical and mechanical mechanisms behind these effects. Additionally, it remains uncertain whether there exists a specific “worst” SOF that leads to the most unfavorable statistical consequences for pavement response.

Moreover, the impacts of spatial variability in subgrade layers have received scant attention in various practical scenarios commonly encountered in practice. For example, the use of geogrid reinforcement is a widely adopted practice in pavement engineering due to its potential to improve pavement performance, such as reducing critical strains, mitigating rut depth and crack development [15, 16, 21]. However, the impact of subgrade spatial variability on the efficacy of geogrid reinforcement remains largely unexplored. The effectiveness of geogrid reinforcement relies on various design variables, such as the type and properties of geosynthetic [10, 33], the placement position of geosynthetic [29, 43], the thickness of design section [15], the loading magnitude and conditions [10, 33], and the subgrade strength [9, 16, 39]. Correia et al. [16] found that the reduction in the asphalt surface vertical displacements resulting from the geogrid-reinforced asphalt layers was more significant for pavement on comparatively weak subgrades. Notably, contradictory findings have been reported regarding the impact of subgrade strength on the effectiveness of geogrid reinforcement used in the base course layer. Cancelli et al. [9] observed a reduction in the reinforcement benefit as subgrade strength increases, while Webster [39] found that a higher California Bearing Ratio (CBR) in the subgrade resulted in a greater reinforcement benefit. Given the limited number of studies available, it remains inconclusive whether or not the geogrid reinforcement in the base course layer is more beneficial on a weaker subgrade. Additionally, the impacts of weak subgrade on the geogrid reinforcement placed at other positions within the pavement system have not been adequately explored. The situation becomes more complex when considering a subgrade with spatial variability, as it introduces different strain and stress distributions within the multilayer pavement system compared to deterministic conditions. These variations can significantly influence the effectiveness of geogrid reinforcement. Nevertheless, it remains unclear whether the subgrade spatial variability enhances or hampers the geogrid reinforcement. This knowledge gap highlights the necessity for further research to address these issues.

This study investigates the effects of subgrade spatial variability on critical pavement responses and the effectiveness of geogrid reinforcement using the random field finite difference analysis (RFFDA). The critical pavement response variables analyzed are the maximum horizontal tensile strain at the bottom of the asphalt layer (\({\varepsilon }_{h}^{{\text{ac}}}\)) and the maximum vertical compressive strain on top of the subgrade layer (\({\varepsilon }_{v}^{{\text{sub}}})\). These strains are normally used to estimate the pavement fatigue life (using \({\varepsilon }_{h}^{{\text{ac}}}\)) and rutting life (using \({\varepsilon }_{v}^{sub}\)), respectively, i.e., the respective number of load repetitions for fatigue and rutting failures [1, 7]. To avoid the complex coupling effect of different sources of uncertainties, only the subgrade modulus is considered as spatially variable, assuming all other parameters to be homogeneous and deterministic. Parametric studies were conducted to examine the effects of two key influencers, namely the coefficient of variation (\({{\text{COV}}}_{E}\)) and the scale of fluctuation (SOF) of subgrade modulus. Further investigation was conducted to uncover the statistical and mechanical mechanisms underlying the impact of subgrade spatial variability by analyzing the critical strain distributions and their correlations with both the overall modulus and the local spatial variability of the key influence zone. Finally, the influence of subgrade spatial variability on the effectiveness of geogrid reinforcement in mitigating critical strains is also explored, considering various locations and moduli of the reinforcements.

2 Random field finite difference analysis (RFFDA)

2.1 Finite difference model of the pavement structure

A typical three-layer pavement structure subjected to a load of 53 kN from a single-tired single-axle was chosen for analysis in this study. The load is represented by a uniform contact stress of 800 kPa over two areas with a width of D = 204.8 mm spaced by 2130 mm [7]. Though the deformation of pavements under wheel loading is a 3D problem, 2D pavement models are often employed in numerical simulations for simplification [6, 16, 21, 28]. To enhance the computational efficiency of RFFDA, the pavement structure is simulated with a plane strain model by constraining it to a 2D plane in FLAC3D software. As shown in Fig. 1, the mesh sizes of the pavement model are 0.05 m horizontally and vary vertically depending on their distance from the loading and the thickness of the layer. The vertical boundaries are fixed in horizontal directions and the bottom is fixed in all directions. The base and subgrade layers are simulated with the Mohr–Coulomb constitutive model, and the asphalt layer with a linear elastic material [5, 16, 31]. The material properties are listed in Table 1.

Fig. 1
figure 1

Finite difference model of the three-layered pavement

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Table 1 Material properties of the pavement model [7]
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In deterministic scenarios, the critical strains (\({\varepsilon }_{h}^{{\text{ac}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\)) would occur along the vertical axis directly beneath the tire pressure [7]. However, in probabilistic analysis incorporating spatial variability, the occurrence of critical strains may not necessarily align with the locations observed in the deterministic scenarios. This issue will be thoroughly investigated and discussed in the following sections.

To investigate the impact of subgrade spatial variability on geogrid reinforcement, this study incorporates six types of reinforcement arrangements. These arrangements consider two levels of geogrid stiffness (G1 and G2) and three placement positions (L1, L1-2, L2). Specifically, L1 represents the location at the interface between the asphalt and base course, L1-2 the middle of the base course, and L2 the interface between the base course and subgrade. The secant stiffnesses (J) at 2% of strain are 2500 kN/m and 5000 kN/m for G1 and G2, respectively. Linear shear behavior is assumed for the interfaces between the geogrid and the surrounding materials.

Table 2 lists the detailed simulation scheme and the interface properties of the six geogrid reinforcements. These geogrid parameters were determined based on existing field and laboratory tests [2, 10, 14, 30]. The shear moduli of the soil/aggregate–geogrid interface are influenced by multiple factors [10, 34], including geogrid type, interface type, vertical pressure on the geogrids, and even environmental temperature. These factors are not examined in this study but will be explored in future research to explore their possible impact on geogrid’s strain alleviation capacity under subgrade spatial variability. The geogrids are simulated with geogridSEL in FLAC3D.

Table 2 Simulation programs of the six reinforcement arrangements and interface properties of the geogrids [2, 10, 14, 30]
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2.2 Validation of finite difference modeling

Correia et al. [16] performed 2D finite element analyses on a large-scale paved model test, and the results showed the efficiency of using a 2D model to simulate the 3D problem when proper models and material properties are adopted. The 3D large-scale paved model test by Correia et al. [16] was employed to check the validity of the constitutive models used in this study. Figure 2 shows the finite difference model of the test under plane strain assumption in FLAC3D software. The material properties reported by Correia et al. [16] are used in the validation simulation.

Fig. 2
figure 2

Finite difference model of the large-scale paved model test by Correia et al. [16]

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The maximum asphalt surface displacement reported in the test was 1.8 mm [16]. The numerical prediction shows a close result of 1.9 mm, representing a 5% deviation. Figure 3 compares the experimental and predicted vertical stress distribution across pavement layers and the geogrid strain distribution. Figure 3a demonstrates that the finite difference model accurately predicted the vertical stress observed in the experiment. A good match has been observed at four critical locations: the interface between asphalt layers, the bottom of the lower asphalt layer, the middle of the base course, and 10 cm below the top of the subgrade layer. In Fig. 3b, the strain distribution of the numerical model aligns well with the experimental results. The geogrid strain peaks right beneath the wheel load center and decreases with distance from the wheel load. However, discrepancies between numerical predictions and experimental findings are noticeable near the wheel load and in regions extending 400 mm to 600 mm from it. These inconsistencies might stem from the limitations of linear elastic models in simulating volumetric changes in AC material and the nonlinear behavior of geogrids [26]. Despite this, the numerical model well predicts both the magnitude and location of the peak geogrid strain. Additionally, it shows no geogrid mobilization beyond 300 mm from the wheel load, aligning with the experimental findings.

Fig. 3
figure 3

Comparison of experimental and predicted results: a vertical stress distribution; b geogrid strain distribution

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Thus, the 2D finite difference model accurately predicted the key performance of the 3D large-scale paved model test. This validation confirms the efficacy of the finite difference modeling procedure, including the plane strain assumption and constitutive models, to represent the actual 3D geogrid-reinforced pavement model.

2.3 Modeling and discretization of random field

The spatial variability of subgrade modulus is simulated with a continuous and stationary 2D random field. This field is characterized by a vector of random variables \(E({{\varvec{X}}}_{{\varvec{i}}})\) following the lognormal distribution with its mean \({\mu }_{E}\) = 60 MPa and different coefficients of variation (\({{\text{COV}}}_{E})\). The lognormal distribution is chosen to avoid the generation of negative material properties [14]. The exponential autocorrelation function adopted by Huang et al. [24] is used to define the correlation of the random variables \(E({{\varvec{X}}}_{{\varvec{i}}})\) at two locations:

$$\rho \left( {x,y} \right) = \exp \left( {{{ - 2\left| x \right|} \mathord{\left/ {\vphantom {{ - 2\left| x \right|} {\delta_{x} }}} \right. \kern-0pt} {\delta_{x} }}} \right)\exp \left( {{{ - 2\left| y \right|} \mathord{\left/ {\vphantom {{ - 2\left| y \right|} {\delta_{y} }}} \right. \kern-0pt} {\delta_{y} }}} \right)$$
(1)

where \({\delta }_{x}\) represents the horizontal scale of fluctuation (SOF), while \({\delta }_{y}\) represents the vertical SOF. The variables x and y denote the relative horizontal and vertical coordinates of any two points, respectively. This study focuses only on the 2D isotropic random field of the subgrade layer. The spatial variation along the alignment direction is not considered. Consequently, the horizontal and vertical scale of fluctuation (\({\delta }_{x}\) and \({\delta }_{y}\)) are assumed to be equal and will be collectively referred to as SOF henceforth.

The random field \(E({{\varvec{X}}}_{{\varvec{i}}})\) is obtained through two steps:

  1. (a)

    Generating a group of normally distributed random field \(G({{\varvec{X}}}_{{\varvec{i}}})\) using zero mean, unit variance, an autocorrelation function \(\rho \left(x,y\right)\), and an appropriate Monte Carlo simulation number. The Karhunen–Loève expansion method is chosen to generate the random field for its excellent approximation to the original random field and computational efficiency in high-dimensional problems [25, 42].

  2. (b)

    Deriving the lognormally distributed random field \(E({{\varvec{X}}}_{{\varvec{i}}})\) from \(G({{\varvec{X}}}_{{\varvec{i}}})\) using the following transformation equations:

    $${\zeta }_{{\text{ln}}E}=\sqrt{{\text{ln}}(1+{{\text{COV}}}_{E}^{2})}$$
    (2)
    $${\lambda }_{{\text{ln}}E}={\text{ln}}{\mu }_{E}-\frac{1}{2}{\zeta }_{{\text{ln}}E}^{2}$$
    (3)
    $$E\left({{\varvec{X}}}_{{\varvec{i}}}\right)=exp\left[{\lambda }_{{\text{ln}}E}+{\zeta }_{{\text{ln}}E}G({{\varvec{X}}}_{{\varvec{i}}})\right]$$
    (4)

where \({\zeta }_{{\text{ln}}E}\) and \({\lambda }_{{\text{ln}}E}\) are the mean and standard deviation of the normal distribution of \({\text{ln}}(E)\), respectively.

Generally, the required number of Monte Carlo simulations depends on precision requirements and available computational resources. A convergence analysis was conducted by progressively increasing the simulation number from 20 to 1400 in RFFDA with \({{\text{COV}}}_{E}\) =0.5 and SOF = 0.25 m. It was observed that, when the number of simulations exceeds 200, the mean and COV of both critical subgrade strain (\({\varepsilon }_{v}^{{\text{sub}}}\)) and critical asphalt strain (\({\varepsilon }_{h}^{{\text{ac}}}\)) stabilize with an error percentage below 1.5%. Hence, 200 Monte Carlo simulations are utilized for subsequent RFFDA.

2.4 Key influence zone (KIZ)

In probabilistic analysis, the magnitudes of critical strains, particularly \({\varepsilon }_{v}^{{\text{sub}}}\), exhibit variations among different realizations due to the spatially variable modulus in the subgrade. The magnitude of \({\varepsilon }_{v}^{{\text{sub}}}\) is primarily governed by the magnitude and spatial distribution of subgrade modulus in the vicinity of its critical location, known as the key influence zone (KIZ) for \({\varepsilon }_{v}^{{\text{sub}}}\). Outside the KIZ, the distribution of subgrade modulus makes an insignificant contribution to the difference of pavement responses between spatially variable and deterministic cases, as it is statistically equivalent to the deterministic case and far away from the critical location.

To locate the KIZ, Fig. 4 presents the contour of vertical compressive stress in the subgrade within 1 m of the base’s underside. At a horizontal distance of 0.15 m (equivalent to 0.75D) from the loading centerline at the subgrade top, the stress reduces to less than 60% of the maximum value. This indicates that the critical subgrade strain and stress are predominantly within a width region of 0.3 m (i.e., 1.5D) beneath the loading area. Therefore, the region depicted in Fig. 4, with a width of 1.5D and a thickness of 35 mm, is identified as the KIZ. The random distribution of Young’s modulus of the subgrade may slightly affect the local mean values in the KIZ, but may not greatly change the ratio between the moduli of the base course and the subgrade within the KIZ. The size of the KIZ is mainly affected by the ratio between the asphalt, base course and subgrade moduli, and the thicknesses of the asphalt and the base course layers. As long as the ratios and the thicknesses do not vary too much, the size of KIZ will not vary significantly. For simplification, in the analyses, the size of the KIZ was kept constant in all the analyses. However, if the thickness and moduli of the pavement layers significantly differ from those in Table 2, the dimensions of KIZ may change accordingly.

Fig. 4
figure 4

Key influence zone (KIZ) for the critical subgrade strain

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To quantify the overall modulus of the KIZ, a local modulus, \({E}_{{\text{Local}}}\), is defined as the arithmetic mean modulus of meshes in the KIZ. The statistics of critical strains in spatially variable cases are significantly influenced by \({E}_{{\text{Loca}}l}\) and the internal spatial variability within the KIZ. Further elaboration on this aspect will be provided in subsequent sections.

2.5 Parametric study

A parametric study was conducted to explore how changes in the subgrade spatial variability affect the critical pavement responses (\({\varepsilon }_{h}^{{\text{ac}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\)). Two key input parameters, namely the coefficient of variation (\({{\text{COV}}}_{E}\)) and the scale of fluctuation (SOF) of the subgrade modulus, are systematically varied within predefined ranges. The coefficient of variation is site-specific and can vary depending on factors such as the soil type, testing methods employed, and the inherent variability in the subgrade material [18, 32, 36, 40]. Timm et al. [36] reported a \({{\text{COV}}}_{E}\) ranging from 5 to 50%. White et al. [40] found the \({{\text{COV}}}_{E}\) to be within 38–97% based on the light weight deflectometer (LWD) tests and dynamic cone penetrometer (DCP) tests. Phoon and Kulhawy [32] revealed \({{\text{COV}}}_{E}\) ranging from 9 to 92% for various types of soil. Considering the typical range, this study assumes \({{\text{COV}}}_{E}\) values of 0.3 and 0.5 for the parametric analysis. In comparison, information on SOF is relatively limited.

Typical values of SOF can range from 0.1 to 60 m for a variety of geotechnical parameters [8, 32]. The strength and variability of subgrade soil within 1 m below the base course is usually assessed to determine the subgrade design parameters (such as modulus and CBR), and the subgrade within 1 m of the underside of the base often shows vertical stratification [7]. To capture the stratifications within this 1 m depth, the SOF of the subgrade layer adopted in the simulations need to be less than 1 m. When SOF is set larger than 1 m, it cannot capture the spatial variability within this crucial 1 m depth subgrade region. A prior study conducted by Lua and Sues [27] adopted SOFs ranging from 0.1 to 1 m for investigating the impact of subgrade spatial variability on pavement performance. Therefore, the SOF is selected between 0.1 and 1 m in this study.

Table 3 presents the parameter combinations for 24 spatial variable cases with SOF ranging from 0.1–1.0 m and \({{\text{COV}}}_{E}\) values of 0.3 or 0.5. A length ratio \({R}_{L}={\text{SOF}}/D\) is defined to assess the relative length of SOF over the loading areas. Specifically, the SOF of Case 1–3 and Case 2–3 equals D, the width of the loading area; and the SOF in Case 1–5 and Case 2–5 equals 1.5D, the widths of KIZ. Figure 5 shows 6 realizations of random fields with \({{\text{COV}}}_{E}=0.5\) at different SOF levels. All SOFs considered in this study exceed the maximum subgrade mesh size within 1 m below the base layer, ensuring that the discretization of the random field does not introduce substantial inconsistency with the continuous solution in this region [12].

Table 3 Parameter combinations of the 24 spatial variable cases
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Fig. 5
figure 5

Random fields of Young's modulus in the subgrade layer of different SOFs (\({{\text{COV}}}_{E}\) = 0.5)

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3 Result and analysis

3.1 Effect of spatial variability on the statistics of critical strains

Table 4 lists the statistics of critical strains for the 24 spatial variable cases in Table 3, including the mean values, COVs, and extremes. The corresponding deterministic values are 1.283 millistrain for \({\varepsilon }_{h}^{{\text{ac}}}\) and 3.709 millistrain for \({\varepsilon }_{v}^{{\text{sub}}}\). All the strains in Table 4 are within reasonable range of pavements [7].

Table 4 Statistics of critical strains of the 24 spatial variable cases
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Subsequent sections will assess the influence of SOF and \({{\text{COV}}}_{E}\) on these statistics and the locations of critical strains. The analysis aims to elucidate the statistical and mechanical mechanisms underlying the impacts of subgrade spatial variability.

3.1.1 Critical locations

This section aims to investigate and assess the influence of subgrade spatial variability on the critical locations of the maximum tensile strain in the asphalt layer (\({\varepsilon }_{h}^{{\text{ac}}}\)) and the maximum vertical compressive strain in the subgrade layer (\({\varepsilon }_{v}^{{\text{sub}}}\)). Figure 6 shows the potential critical positions, where P0 and Q0 represent the critical positions for \({\varepsilon }_{h}^{{\text{ac}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\) in deterministic scenarios, respectively. Note that P0 is not located at the centerline of the loading area, as the centerline of the loading area differs from the centerline of the pavement model.

Fig. 6
figure 6

Possible critical locations for \({\varepsilon }_{h}^{{\text{ac}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\) in spatial variable cases

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Figure 7 presents the distribution of critical subgrade strain at different locations for different values of SOF and \({{\text{COV}}}_{E}\). In the majority of cases, more than 50% of realizations show a critical subgrade strain occurring away from Q0, suggesting that assuming Q0 as the critical location may result in an undervalued \({\varepsilon }_{v}^{{\text{sub}}}\) with a likelihood of over 50%. Additionally, when SOF is 0.3 m, i.e., the width of the KIZ, the number of realizations with \({\varepsilon }_{v}^{{\text{sub}}}\) at Q0 reaches the minimum regardless of \({{\text{COV}}}_{E}\). When SOF increases beyond 0.3 m, the number of realizations with a Q0 critical strain tends to increase. This can be attributed to the homogenization of the KIZ, as SOF transitions from a value smaller than 1.5D to a significantly larger value. On the other hand, the comparison of Fig. 7a and b indicates that, at a larger \({{\text{COV}}}_{E}\), the likelihood of the critical subgrade strain occurring at Q0 reduces.

Fig. 7
figure 7

Distribution of \({\varepsilon }_{v}^{{\text{sub}}}\) at different locations for different SOF under a \({{\text{COV}}}_{E}\)= 0.3 and b \({{\text{COV}}}_{E}\)= 0.5

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It is anticipated that the critical locations of \({\varepsilon }_{h}^{{\text{ac}}}\) would be relatively unaffected by the subgrade spatial variability. This is attributed to the spatial averaging effect, where larger areas or longer distances tend to smooth out local variations, resulting in a more homogeneous and deterministic response [38]. This expectation is confirmed by Fig. 8. For \({{\text{COV}}}_{E}\) values of 0.3 and 0.5, P0 is determined as the critical location in 99% and 97% of the realizations, respectively. Noticeable deviations from P0 are observed only when both SOF and \({{\text{COV}}}_{E}\) are of considerably high values, as shown in Fig. 8b.

Fig. 8
figure 8

Distribution of \({\varepsilon }_{h}^{ac}\) at different locations for different SOF under a \({{\text{COV}}}_{E}\)= 0.3 and b \({{\text{COV}}}_{E}\)= 0.5

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Overall, subgrade spatial variability has a much greater impact on the critical locations of \({\varepsilon }_{v}^{{\text{sub}}}\) compared to \({\varepsilon }_{h}^{{\text{ac}}}\). This effect highlights the potential underestimation of \({\varepsilon }_{v}^{{\text{sub}}}\) when assuming a fixed location for maximum subgrade strain in RFFDA. The upcoming section will evaluate the extent of underestimation resulting from this assumption.

3.1.2 Mean values of critical strains

Since the magnitudes of critical strains vary among different realizations due to the spatially variable modulus in the subgrade, their mean values could be different from their deterministic counterparts. To assess the impact of spatial variability on the mean critical strains, a relative strain amplification ratio \({R}_{A}\) is defined as:

$$R_{A} = \left( {{{\overline{\varepsilon }} \mathord{\left/ {\vphantom {{\overline{\varepsilon }} {\varepsilon_{{{\text{det}}}} }}} \right. \kern-0pt} {\varepsilon_{{{\text{det}}}} }} - 1} \right) \times 100\%$$
(5)

where \(\overline{\varepsilon }\) denotes the mean value of the critical strains in a spatially variable case, and \({\varepsilon }_{{\text{det}}}\) represents the corresponding critical strains in the deterministic case.

Figure 9a shows how the variation of SOF affects \({R}_{A}\) for \({\varepsilon }_{v}^{{\text{sub}}}\) under different values of \({{\text{COV}}}_{E}\). In the figure, two kinds of critical subgrade strain are presented for each case: the actual critical strain (\({\varepsilon }_{v}^{{\text{sub}}}\)) and the strain at Q0 (\({\varepsilon }_{v, Q0}^{{\text{sub}}}\)). It shows that the impact of SOF on \({R}_{A}\) is more pronounced for \({\varepsilon }_{v}^{{\text{sub}}}\) compared to \({\varepsilon }_{v, Q0}^{{\text{sub}}}\). For \({\varepsilon }_{v}^{{\text{sub}}}\), the value of \({R}_{A}\) increases as SOF increases from 0.05 to approximately 0.3 m and then decreases as SOF increases from 0.3 to around 0.7 m. Subsequently, \({R}_{A}\) remains relatively stable. On the other hand, \({R}_{A}\) for \({\varepsilon }_{v, Q0}^{{\text{sub}}}\) increases as SOF increases from 0.05 m to approximately 0.3 m and then stabilizes. Notably, the maximum \({R}_{A}\) values for \({\varepsilon }_{v}^{{\text{sub}}}\) and \({\varepsilon }_{v, Q0}^{{\text{sub}}}\), as well as the greatest discrepancy between \({R}_{A}\) for \({\varepsilon }_{v}^{{\text{sub}}}\) and \({\varepsilon }_{v, Q0}^{{\text{sub}}}\), are observed when SOF is approximately 0.3 m. Specifically, when SOF = 0.3 m and \({{\text{COV}}}_{E}=0.5\), the mean values of \({\varepsilon }_{v}^{{\text{sub}}}\) and \({\varepsilon }_{v, Q0}^{{\text{sub}}}\) are increased by 22% and 13%, respectively, relative to the deterministic strain. This indicates that the amplifying effect of subgrade spatial variability on \({\varepsilon }_{v, Q0}^{{\text{sub}}}\) can be as low as 60% of that on \({\varepsilon }_{v}^{{\text{sub}}}\). Hence, the subgrade strains extracted at Q0 might not represent the actual critical subgrade strains and can result in a significant underestimation of the amplifying impact resulting from subgrade spatial variability. These findings highlight the need for careful determination of the critical strain location for subgrade strain when considering the subgrade spatial variability, particularly in situations where SOF is approximately 1.5D.

Fig. 9
figure 9

Variation of \({R}_{A}\) for a \({\varepsilon }_{v}^{{\text{sub}}}\) and b \({\varepsilon }_{h}^{{\text{ac}}}\) with increasing SOF under different values of \({{\text{COV}}}_{E}\)

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Figure 9b depicts the variation of \({R}_{A}\) for \({\varepsilon }_{h}^{ac}\) with SOF. The value of \({R}_{A}\) initially increases with increasing SOF and maximizes at SOF = 0.4 m (i.e., 2D), after which it remains relatively stable. Compared with \({\varepsilon }_{v}^{{\text{sub}}}\), the impact of SOF on mean \({\varepsilon }_{h}^{ac}\) is minimal, with \({R}_{A}\) values ranging from 1.8% to 2.7% for \({{\text{COV}}}_{E}\) = 0.3 and from 5.0% to 7.6% for \({{\text{COV}}}_{E}\) = 0.5. Furthermore, there is a negligible discrepancy between \({\varepsilon }_{h}^{{\text{ac}}}\) and the tensile strain at P0 (\({\varepsilon }_{h, P0}^{ac}\)), which confirms that the critical location of \({\varepsilon }_{h}^{ac}\) is relatively unaffected by the subgrade spatial variability.

The underestimation of critical strains, particularly for \({\varepsilon }_{v}^{{\text{sub}}}\), is substantial when assuming the same critical locations in spatial variable cases as in deterministic case. Therefore, the actual critical strains, rather than solely those at the P0 or Q0 positions, will be considered for the analysis in the subsequent sections.

3.1.3 Coefficient of variation (COV) of the critical strains

Figure 10 shows the variations in the coefficient of variation (COV) for the critical strains as SOF increases under different values of \({{\text{COV}}}_{E}\). The figure reveals that increasing spatial variability, as characterized by the higher \({{\text{COV}}}_{E}\), leads to an elevated COV for the critical strains, particularly for \({\varepsilon }_{v}^{{\text{sub}}}\). This observation aligns with Ali et al. [4] who found a proportional relationship between the variability of pavement response and that of the subgrade modulus. Moreover, the COV of \({\varepsilon }_{v}^{{\text{sub}}}\) exhibits much higher values and displays a distinct trend with SOF compared to that of \({\varepsilon }_{h}^{{\text{ac}}}\). Specifically, the COV of \({\varepsilon }_{v}^{{\text{sub}}}\) initially rises with increasing SOF and peaks at SOF = 0.3 m, after which it stabilizes. On the other hand, the COV of \({\varepsilon }_{h}^{{\text{ac}}}\) generally increases with SOF, but the impact of subgrade spatial variability is negligible for small SOF values due to the spatial averaging effect. Notably, the COVs for both \({\varepsilon }_{h}^{{\text{ac}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\) are considerably smaller compared to \({{\text{COV}}}_{E}\). For example, for \({{\text{COV}}}_{E}=0.5\), COV for \({\varepsilon }_{h}^{ac}\) is below 7% and COV for \({\varepsilon }_{v}^{{\text{sub}}}\) is below 30%. The findings suggest a reduction of uncertainty during the uncertainty propagation from material properties to structural performance. So, in a probabilistic analysis of pavements that incorporates spatial variability, it is inappropriate to directly equate the COV of material properties with the COV of critical structural responses.

Fig. 10
figure 10

Variation of COV for critical strains with increasing SOF under different values of \({{\text{COV}}}_{E}\)

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3.1.4 The maximum and minimum critical strains

The extreme values, i.e., the maximum and minimum critical strains, depict the range of distribution and provide insight into the dispersion of critical strains in probabilistic analysis. To assess the relative deviation of these extreme values from the deterministic strains, two deviation ratios, \({R}_{D,{\text{max}}}\) and \({R}_{D,{\text{min}}}\), are introduced. These ratios are calculated as follows:

$$R_{{D,{\text{max}}}} = {{\varepsilon_{{{\text{max}}}} } \mathord{\left/ {\vphantom {{\varepsilon_{{{\text{max}}}} } {\varepsilon_{{{\text{det}}}} }}} \right. \kern-0pt} {\varepsilon_{{{\text{det}}}} }} - 1$$
(6)
$$R_{{D,{\text{min}}}} = 1 - {{\varepsilon_{{{\text{min}}}} } \mathord{\left/ {\vphantom {{\varepsilon_{{{\text{min}}}} } {\varepsilon_{{{\text{det}}}} }}} \right. \kern-0pt} {\varepsilon_{{{\text{det}}}} }}$$
(7)

where \({\varepsilon }_{{\text{max}}}\) and \({\varepsilon }_{{\text{min}}}\) correspond to the maximum and minimum critical strains observed in a specific case.

Figure 11 depicts the variations of \({R}_{D,{\text{max}}}\) and \({R}_{D,{\text{min}}}\) for the critical strains as SOF increases at different values of \({{\text{COV}}}_{E}\). The figure shows that \({R}_{D,{\text{max}}}\) consistently exceeds \({R}_{D,{\text{min}}}\) for both \({\varepsilon }_{v}^{{\text{sub}}}\) and \({\varepsilon }_{h}^{{\text{ac}}}\). For instance, when \({{\text{COV}}}_{E}\)=0.3, \({R}_{D,{\text{max}}}\) for \({\varepsilon }_{v}^{{\text{sub}}}\) ranges from 24.3 to 86.1%, whereas \({R}_{min}\) ranges from only 13.8% to 44.5%. Moreover, as the degree of spatial variability intensifies with higher values of \({{\text{COV}}}_{E}\), the disparity between \({R}_{D,{\text{max}}}\) and \({R}_{D,{\text{min}}}\) widens even further. These findings indicate that the maximum strains deviate more significantly from the deterministic strains than the minimum ones, suggesting left-skewed distributions for both \({\varepsilon }_{h}^{{\text{ac}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\). Notably, a smaller \({\varepsilon }_{v}^{{\text{sub}}}\) is associated with a stiffer key influence zone (KIZ), while a higher \({\varepsilon }_{v}^{{\text{sub}}}\) corresponds to a softer KIZ. Hence, the left-skewed distribution of \({\varepsilon }_{v}^{{\text{sub}}}\) implies that the softer KIZ has a greater impact on critical strain compared to the stiffer KIZ, a phenomenon known as the low modulus dominating effect [19]. A detailed explanation of this effect will be provided later.

Fig. 11
figure 11

Variation of deviation ratios for a \({\varepsilon }_{v}^{{\text{sub}}}\) and b \({\varepsilon }_{h}^{{\text{ac}}}\) with increasing SOF under different values of \({{\text{COV}}}_{E}\)

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Figure 11a and b also reveals notable distinctions between \({\varepsilon }_{h}^{{\text{ac}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\) in terms of the deviation ratios. In Fig. 11a, \({R}_{D,{\text{max}}}\) and \({R}_{D,{\text{min}}}\) show non-monotonic patterns as SOF increases, with the highest deviation ratios observed at intermediate levels of SOF within the predefined ranges. Conversely, in Fig. 11b, the deviation ratios of \({\varepsilon }_{h}^{{\text{ac}}}\) show increasing trends with SOF. Additionally, the deviation ratios of \({\varepsilon }_{h}^{{\text{ac}}}\) are significantly lower than those of \({\varepsilon }_{v}^{{\text{sub}}}\) in each case. This confirms that the impact of subgrade spatial variability on critical subgrade strain significantly surpasses its effect on critical asphalt strain.

3.2 Statistical and mechanical explanation for amplifying effect and worst SOF for \({{\varvec{\varepsilon}}}_{{\varvec{v}}}^{\mathbf{s}\mathbf{u}\mathbf{b}}\)

As found in Sect. 3.1, subgrade spatial variability significantly amplifies the pavement response, particularly the critical subgrade strain, \({\varepsilon }_{v}^{{\text{sub}}}\). Additionally, both the highest amplification ratio (\({R}_{{\text{mean}}}\)) and COV for \({\varepsilon }_{v}^{{\text{sub}}}\) are observed when SOF is approximately 0.3 m (i.e., 1.5D). This section aims to uncover the statistical and mechanical mechanisms of these observations by examining the correlations among \({\varepsilon }_{v}^{{\text{sub}}}\), \({E}_{{\text{Local}}}\), and the local spatial variability within the key influence zone (KIZ). Figure 12 illustrates the correlation between \({E}_{{\text{Local}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\) for Case 2–5 (SOF = 0.25 m and \({{\text{COV}}}_{E}\)= 0.5). The data points are fitted using a power regression model and divided into four quadrants based on the deterministic subgrade strain (i.e., 3.71 millistrain, denoted as \({\varepsilon }_{v,{\text{det}}}^{{\text{sub}}}\)) and deterministic subgrade modulus (i.e., 60 MPa, denoted as \({E}_{{\text{det}}}\)). The number of data points in each quadrant is shown in the figure. Additionally, the results from 200 deterministic analyses are included, encompassing subgrade moduli ranging from 20 to 150 MPa.

Fig. 12
figure 12

Correlation between \({E}_{{\text{Local}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\) for Case 2–5 (SOF = 0.25 m, and \({{\text{COV}}}_{E}\) = 0.5)

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In Fig. 12, the critical subgrade strain \({\varepsilon }_{v}^{{\text{sub}}}\) demonstrates an increasing trend as \({E}_{{\text{Local}}}\) decreases in spatial variable situations, which is consistent with the deterministic situations. However, despite the statistical equivalence of the subgrade modulus between spatially variable and deterministic cases, the number of realizations with \({\varepsilon }_{v}^{{\text{sub}}}\)above \({\varepsilon }_{v,{\text{det}}}^{{\text{sub}}}\) far exceeds the number below \({\varepsilon }_{v,{\text{det}}}^{{\text{sub}}}\). Among 200 realizations, \({\varepsilon }_{v}^{{\text{sub}}}\)exceeds \({\varepsilon }_{v,{\text{det}}}^{{\text{sub}}}\) in 151 realizations. Specifically, for realizations with \({E}_{{\text{Local}}}\) \(\le\) \({E}_{{\text{det}}}\), 98% of \({\varepsilon }_{v}^{{\text{sub}}}\) is greater than \({\varepsilon }_{v,{\text{det}}}^{{\text{sub}}}\), while for realizations with \({E}_{Local} > {E}_{det}\), 48% of them still exhibit \({\varepsilon }_{v}^{{\text{sub}}}\) higher than \({\varepsilon }_{v,{\text{det}}}^{{\text{sub}}}\). Consequently, the number of realizations with \({\varepsilon }_{v}^{{\text{sub}}}\) above \({\varepsilon }_{v,{\text{det}}}^{{\text{sub}}}\) (i.e., 151) significantly surpasses that with \({E}_{{\text{Local}}}\) below \({E}_{{\text{det}}}\)(i.e., 110). This discrepancy is due to the combined impact of the low modulus dominating effect and the local spatial variability within the KIZ. As discussed by Fenton and Griffiths [19], the low modulus dominating effect indicates that the softer regions exert a greater influence on critical strain compared to the stiffer regions. This claim is also supported by the power regression trend observed in Fig. 12, where \({\varepsilon }_{v}^{{\text{sub}}}\) increases at an accelerating rate with decreasing \({E}_{{\text{Local}}}\). It is worth noting that even when \({E}_{{\text{Local}}}\) is very close in certain realizations, the corresponding \({\varepsilon }_{v}^{{\text{sub}}}\) can exhibit substantial variations. This can be attributed to the local spatial variability within the KIZ. Generally, a substantial degree of local spatial variability intensifies the low modulus dominating effect, consequently exacerbating the impairment of the deformation resistance of the KIZ.

To further explore the mechanical and statistical reasons behind the varying impact of subgrade spatial variability for different SOF, the coefficient of variation of \({E}_{{\text{Local}}}\) (denoted as \({{\text{COV}}}_{E,1}\)) and the average coefficient of variation for the moduli (denoted as \({{\text{COV}}}_{E,2}\)) in the six meshes of the KIZ (Fig. 4) are plotted against SOF in Fig. 13. Note that \({{\text{COV}}}_{E,2}\) quantifies the extent of local spatial variability within the KIZ. Figure 13 shows that \({{\text{COV}}}_{E,1}\) rapidly increases with SOF when SOF < 0.25 m and reaches its maximum at SOF = 0.3 m. Subsequently, \({{\text{COV}}}_{E,1}\) gradually approaches the coefficient of variation in the subgrade modulus (i.e., \({{\text{COV}}}_{E}\)=0.5). As for \({{\text{COV}}}_{E,2}\), it exhibits a swift increase with SOF when SOF < 0.15m and reaches its maximum at SOF = 0.15 m. Afterward, \({{\text{COV}}}_{E,2}\) gradually declines with increasing SOF. Although \({{\text{COV}}}_{E,1}\) and \({COV}_{E,2}\) cannot reach their respective maximum at the same SOF due to their different trends, their combined value reaches its maximum at SOF = 0.3 m. Therefore, the SOF of 0.3 m signifies a situation where both the variation of \({E}_{{\text{Local}}}\) and the local spatial variability within the KIZ are notably significant. Consequently, this explains why an SOF value of 0.3 m induces the most unfavorable effect on \({\varepsilon }_{v}^{{\text{sub}}}\), such as the maximum amplification ratio and the highest variability.

Fig. 13
figure 13

Effects of SOFs on \({{\text{COV}}}_{E,1}\) and \({{\text{COV}}}_{E,2}\)

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Notably, all the statistics of \({{\varvec{\varepsilon}}}_{{\varvec{h}}}^{{\varvec{a}}{\varvec{c}}}\), especially the COVs and the extreme values (as shown in Fig. 10 and Fig. 11), display a similar trend to that of \({{\text{COV}}}_{E,1}\) as SOF increases. This similarity can be attributed to the spatial averaging effect, which smooths out the local variations in the KIZ. Consequently, the effect of subgrade spatial variability on \({{\varvec{\varepsilon}}}_{{\varvec{h}}}^{{\varvec{a}}{\varvec{c}}}\) is primarily governed by \({{\text{COV}}}_{E,1}\) rather than \({{\text{COV}}}_{E,2}\).

3.3 Effect of subgrade spatial variability on the effectiveness of geogrid reinforcement

3.3.1 Difference of the effectiveness of geogrid between deterministic and spatially variable scenarios

Previous studies have shown that geogrid reinforcement in pavements can alleviate critical pavement strains, as evidenced by laboratory tests and deterministic numerical simulations [8, 12, 23]. It is reasonable to expect that these benefits can also apply to situations involving subgrade spatial variability This section compares the effectiveness of geogrid reinforcement between deterministic and spatially variable scenarios. The objective is to investigate whether subgrade spatial variability enhances or reduces the efficacy of geogrid reinforcement in reducing critical strains. To achieve this, further analysis was conducted based on Case 2–2 (SOF = 0.25 m, \({{\text{COV}}}_{E}\)=0.5) from Table 3. Six types of geogrid reinforcement (G1_L1, G2_L1, G1_L1-2, G2_L1-2, G1_L2, G2_L2) will be evaluated. For brevity, G_L1 refers to geogrid reinforcement positioned at the interface between the asphalt and base layer, and similar nomenclature applies to the other configurations. To evaluate the effectiveness of geogrid in reducing critical strains, a strain alleviating ratio (SAR) is defined as the percentage of critical strain reduction achieved by the geogrid reinforcement. A higher value of SAR indicates a greater ability of the geogrid reinforcement to alleviate critical strains.

Figure 14 shows the variation of mean SAR under different geogrid conditions in both deterministic and spatially variable scenarios. The results indicate that, regardless of subgrade spatial variability, G_L1 performs better in reducing \({\varepsilon }_{h}^{{\text{ac}}}\) compared to \({\varepsilon }_{v}^{{\text{sub}}}\). Conversely, G_L1-2 exhibits good performance in reducing both \({\varepsilon }_{h}^{{\text{ac}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\). In comparison, G_L2 is more effective in reducing \({\varepsilon }_{v}^{{\text{sub}}}\) than \({\varepsilon }_{h}^{{\text{ac}}}\). Comparing the results of deterministic and spatially variable scenarios reveals that the strain-alleviating ability of the geogrids is greatly affected by subgrade spatial variability, and the nature and level of the influence depend on the type of critical strain and the placement position of the geogrid. Figure 14a shows that due to the subgrade spatial variability, the effectiveness of G_L1 in reducing \({\varepsilon }_{h}^{{\text{ac}}}\) is slightly promoted while G_L1-2 and G_L2 are slightly undermined. Figure 14b indicates, due to the subgrade spatial variability, that the effectiveness of G_L1 in reducing \({\varepsilon }_{v}^{{\text{sub}}}\) is slightly enhanced, while geogrids placed at the other two positions, particularly G_L2, are significantly impaired. Specifically, in the deterministic case, G1_L2 reinforcement reduces \({\varepsilon }_{v}^{{\text{sub}}}\) by 20.6%, whereas in spatially variable cases, the reduction is only 10.9%. Furthermore, the stiffer for G_L1-2 and G_L2, the greater the difference in their ability to reduce \({\varepsilon }_{v}^{{\text{sub}}}\) between the deterministic and spatially variable scenarios.

Fig. 14
figure 14

Mean SAR of a \({\varepsilon }_{h}^{{\text{ac}}}\) and b \({\varepsilon }_{v}^{{\text{sub}}}\) under different geogrid reinforcement schemes

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3.3.2 Statistical and mechanical explanation for the detrimental impact of subgrade spatial variability on the effectiveness of G_L2

Figure 15 depicts the relationship between \({E}_{{\text{Local}}}\) and the strain alleviating ratio (SAR) of G1_L2 reinforcement for \({\varepsilon }_{v}^{{\text{sub}}}\). It reveals the SAR values of most realizations (187 out of 200) in the spatially variable scenario are lower than the deterministic value (i.e., 20.6%, denoted as \({\varepsilon }_{v,{\text{det}}}^{{\text{sub}}}\)). Moreover, among the 200 realizations, 110 of them have \({E}_{{\text{Local}}}\) lower than \({E}_{{\text{det}}}\), and 2.7% of those (3/110 = 2.7%) have a SAR value higher than SARdet. On the other hand, in the remaining 90 realizations with \({E}_{{\text{Local}}}\) greater than \({{\text{E}}}_{{\text{det}}}\), 11.1% of them (10/90 = 11.1%) have a SAR value higher than SARdet. These findings suggest that as the key influence zone (KIZ) becomes stiffer, there is an increased probability of the SAR value being higher than the deterministic value, which is also supported by the linear regression trend in Fig. 15. This implies that a higher subgrade modulus, especially within the KIZ, enhances the capacity of G_L2 to reduce critical subgrade strains.

Fig. 15
figure 15

Relationship of SAR for \({\varepsilon }_{v}^{{\text{sub}}}\) versus \({E}_{{\text{Local}}}\) under G1_L2 reinforcement

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In general, it has been widely reported [16, 43] that large deformations facilitate the mobilization of geogrid reinforcement and improve the performance of reinforced pavement. However, the amplified critical subgrade strains observed in this study do not result in greater strain alleviating effects of G_L2 compared to the deterministic case. One possible explanation is that the locally weak KIZ not only induces greater subgrade strains but also alters the magnitude of stresses transmitted to the KIZ or the base course–subgrade interface, which might negatively affect the geogrid performance. To verify this hypothesis, the average stress transmitted to the top of KIZ (\({\sigma }_{v}^{{\text{sub}}}\)) was computed, and its relationship with \({E}_{{\text{Local}}}\) and the strain alleviating ratio (SAR) are demonstrated in Fig. 16. Figure 16a shows that, in most realizations (154 out of 200), \({\sigma }_{v}^{{\text{sub}}}\) is lower than the deterministic value (i.e., \({\sigma }_{{\text{v}},{\text{det}}}^{{\text{sub}}}\)). Specifically, for realizations with \({E}_{{\text{Local}}}\) \(\le\) \({E}_{{\text{det}}}\), 100 out of 110 have \({\sigma }_{v}^{{\text{sub}}}\) lower than \({\sigma }_{{\text{v}},{\text{det}}}^{{\text{sub}}}\), whereas for the other 90 realizations with \({E}_{{\text{Local}}}> {E}_{{\text{det}}}\), 54 of them have \({\sigma }_{v}^{{\text{sub}}}\) higher than \({\sigma }_{{\text{v}},{\text{det}}}^{{\text{sub}}}\). Consequently, the number of realizations with \({\sigma }_{v}^{{\text{sub}}}\) below \({\sigma }_{v,det}^{sub}\) (154 out of 200) is significantly higher than the number of realizations with \({E}_{{\text{Local}}}\) below \({E}_{{\text{det}}}\)(110 out of 200). This indicates that a softer and spatial variable KIZ reduces the vertical pressure transmitted to the top of KIZ. Mechanically, this can be attributed to the fact that a softer KIZ induces higher vertical subgrade strain, causing more vertical stress to be transmitted to the relatively stiffer area outside the KIZ, and eventually resulting in reduced \({\sigma }_{v}^{{\text{sub}}}\). Note that this reduction in \({\sigma }_{v}^{{\text{sub}}}\) in turn mitigate the large subgrade strain, but not to the extent of fully reversing the strain amplifying effect of subgrade spatial variability.

Fig. 16
figure 16

Relationship of \({\sigma }_{v}^{{\text{sub}}}\) with a \({E}_{{\text{Local}}}\) and b SAR for \({\varepsilon }_{v}^{{\text{sub}}}\) under G1_L2 reinforcement

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Figure 16b indicates an increasing trend of SAR with increasing \({\sigma }_{v}^{{\text{sub}}}\)For the 154 realizations with \({\sigma }_{v}^{{\text{sub}}}\) lower than the deterministic value (\({\sigma }_{v,det}^{sub}\)), only 3.9% have a SAR higher than SARdet (i.e., 20.6%), while for the other 46 realizations with \({\sigma }_{v}^{{\text{sub}}}\) \(> {\sigma }_{v,det}^{sub}\), 15% of them have a SAR higher than SARdet. Figures 15 and 16b collectively indicate that high levels of strain and stress in the proximity of base course–subgrade improve the strain-alleviating ability of G1_L2, which is in accordance with previous studies [16, 43]. However, in the context of subgrade spatial variability, the adverse impact of reduced vertical stress at the base course–subgrade interface outweighs the positive effect of the amplified subgrade strain. Consequently, the strain-alleviating ability of G1_L2 is generally compromised. Furthermore, out of the 46 realizations with \({\sigma }_{v}^{{\text{sub}}}\) \(>{\sigma }_{v,{\text{det}}}^{{\text{sub}}}\), only 7 realizations have a SAR higher than SARdet. This discrepancy can be attributed to the local spatial variability within KIZ. When the subgrade modulus changes abruptly over a short distance, considerable differential deformation or fluctuation of the subgrade surface occurs. This will adversely affect the generation of the membrane effects in the reinforcements, ultimately undermining the strain-alleviating ability of G1_L2.

Overall, when geogrids are placed within or at the bottom of base course, their strain-alleviating ability is negatively affected by the subgrade spatial variability. This finding has practical implications, particularly when the subgrade layer exhibits significant spatial variation. To counteract or compensate for the negative effects caused by the subgrade spatial variability on geogrid effectiveness, measures such as using stronger geogrids or implementing treatments like surface subgrade material replacement and stabilization should be considered.

4 Conclusions and discussions

The subgrade modulus is one of the most crucial factors in determining pavement thickness, composition, and performance. However, the subgrade modulus can vary spatially, leading to detrimental effects on pavement performance. This study employs random field finite difference analysis (RFFDA) to examine the impact of subgrade modulus spatial variability on two critical pavement strains: the critical asphalt strain (\({\varepsilon }_{h}^{{\text{ac}}}\)) and critical subgrade strain (\({\sigma }_{v}^{{\text{sub}}}\)). A parametric study was conducted on two key influencers: the coefficient of variation (\({{\text{COV}}}_{E}\)) and the scale of fluctuation (SOF) of the subgrade modulus. To uncover the statistical and mechanical mechanisms underlying the impact of subgrade spatial variability, the correlations between \({\varepsilon }_{v}^{sub}\), the overall modulus (\({E}_{{\text{Local}}}\)) of the key influence zone (KIZ), and the local spatial variability within the KIZ were analyzed. The study also explores the influence of subgrade modulus spatial variability on the effectiveness of geogrid in reducing critical strains, considering two kinds of geogrid stiffness and three placement positions: asphalt–base course interface (L1), mid-depth of base course (L1-2) and base course–subgrade interface (L2). The geogrid’s efficacy in reducing critical strains was measured by a strain alleviating ratio (SAR), which describes the reduction of the critical strains induced by the geogrid reinforcement. The key findings are as follows:

  1. 1.

    The spatial variability of the subgrade modulus can have a substantial amplifying effect on critical pavement strains. Specifically, compared to the homogeneous cases, at \({{\text{COV}}}_{E}=0.5\), the critical subgrade strains can increase by 10–22% depending on the SOF. This finding suggests that assuming subgrade homogeneity can lead to an underestimation of the critical subgrade strain and, consequently, an overestimation of pavement life.

  2. 2.

    Subgrade spatial variability introduces considerable uncertainties in critical strains, posing a significant challenge for predicting pavement life. Furthermore, the coefficients of variation (COV) of the critical strains are lower than \({{\text{COV}}}_{E}\), which implies a reduction of uncertainty during the uncertainty transmission from material properties to structural performance. Thus, when performing probabilistic analysis that incorporates spatial variability, the COV of material properties cannot be directly used as the COV of structural response.

  3. 3.

    Subgrade spatial variability also induces notable variation in the location of maximum pavement strains. The extent of this variability is influenced by the magnitudes of SOF and \({{\text{COV}}}_{E}\), and the specific type of critical strains. Therefore, assuming fixed locations for maximum strains can underestimate critical strains, as it fails to account for the possibility of maximum strains occurring at different locations than those observed in deterministic scenarios.

  4. 4.

    The impact of subgrade spatial variability on \({\varepsilon }_{v}^{sub}\) is more direct and pronounced compared to its effect on \({\varepsilon }_{h}^{{\text{ac}}}\) due to the spatial averaging effect. Subgrade spatial variability leads to a greater amplifying effect, higher values of COV, and increased variation of the critical position for \({\varepsilon }_{v}^{{\text{sub}}}\) compared to \({\varepsilon }_{h}^{{\text{ac}}}\). Additionally, for \({\varepsilon }_{v}^{{\text{sub}}}\), there exists a worst value of SOF (i.e., 1.5D) that results in the most unfavorable statistical effect. In contrast, for \({\varepsilon }_{h}^{{\text{ac}}}\), the maximum mean value is observed when SOF is approximately 2D, while its COV and extreme values show an increasing trend as SOF increases. These findings also imply that subgrade spatial variability has a more significant impact on the pavement’s rutting life compared to its fatigue life.

  5. 5.

    The strain-alleviating ability of the geogrids is impacted by subgrade spatial variability, with the impact varying based on the type of critical strain and geogrid location. When positioned at L1, the strain-alleviating ability of geogrid is enhanced, while geogrids placed at the other two positions (G_L1-2 or G_L2), are apparently impaired. Particularly, the mean values of SAR of G_L2 for \({\varepsilon }_{v}^{{\text{sub}}}\) in spatially variable situations are only half of those observed in deterministic scenarios. Furthermore, as geogrid stiffness increases, the disparity in the ability of G_L1-2 and G_L2 in reducing \({\varepsilon }_{v}^{{\text{sub}}}\) between deterministic and spatially variable scenarios becomes more pronounced. These findings imply that when the geogrid is positioned within or at the bottom of a base course, measures such as employing stronger geogrids or implementing specialized soil treatments should be taken to compensate for the adverse effects arising from subgrade spatial variability.

  6. 6.

    The influence of subgrade spatial variability on the statistics of critical strains can be attributed to the low modulus dominating effect, which depends on two factors: \({E}_{{\text{Local}}}\) and the local spatial variability within the KIZ. A KIZ with lower \({E}_{{\text{Local}}}\) and higher local spatial variability results in reduced deformation resistance, leading to higher \({\varepsilon }_{v}^{{\text{sub}}}\) but lower vertical stress transmitted to the KIZ. Higher level of strain or stress tends to improve the strain-alleviating ability of geogrid. However, for G_L2, the negative effect of the reduced vertical stress outweighs the positive effect of the amplified subgrade strain. Consequently, the ability of G_L2 in reducing \({\varepsilon }_{v}^{{\text{sub}}}\) is compromised.

Pavement design methods employ the reliability concept to compensate for variations and uncertainties in material properties and loading conditions. This study quantified the influence of subgrade spatial variability on the two kinds of critical strains: \({\varepsilon }_{h}^{{\text{ac}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\). Since \({\varepsilon }_{h}^{{\text{ac}}}\) and \({\varepsilon }_{v}^{{\text{sub}}}\) are closely linked to the pavement fatigue life and rutting life, respectively, the influence of subgrade spatial variability on both fatigue and rutting life can also be assessed. The impact of asphalt spatial variability on critical strain and pavement life has recently been explored by the authors [41]. The effect of spatial variability in the base course remains under-researched and will be addressed in future. All these studies necessitate further study on the coupling effects of spatial variability across multiple pavement layers and facilitate a more advanced and refined reliability-based design to better capture the spatial variability-induced uncertainties. Of course, to systematically integrate spatial variability with established reliability principles, further research is required to ensure accurate probabilistic characterization of material spatial variability for natural soil and man-made asphalt layer.