4.1.1. Dry pavement skid resistance based on theories of contact mechanics

Modeling of tire-pavement friction based on theories of contact mechanics became a topic of research interest in the last century after World War II, following the dramatic increase of road vehicle volume as a result of global automobile ownership boom that started in the 1950s. However, it was not until the beginning of the 21st century that theoretical procedures for quantitative estimation of tire-pavement friction were developed. Heinrich (1997) and his fellow researchers (Heinrich et al., 2000; Kluppel and Heinrich, 2000) published a theoretical concept that relates the friction force of rubber sliding on a rigid self-affine road surface to the hysteretic energy losses arising from rubber deformations caused by road surface asperities. The basic assumption was that the dynamic damping behavior of the rubber was dependent on the texture properties of the road surface profile represented by its power spectral density. By applying the concept of fractal geometry to characterize a multiple-scaled rough road surface, they analyzed the hysteretic frictional dynamics of rubber under stochastic excitations over the rough surface, and derived a general expression of the coefficient of hysteretic friction as a function of surface texture parameters and sliding velocity.${\mu }_{\text{h}}=\frac{F}{F_{\text{n}}}=\frac{1}{2{\left(2\text{π}\right)}^2}\frac{<z_{\text{p}}>}{{\sigma }_{0}V}{\int }_{{\omega }_{\min }}^{{\omega }_{\max }}\omega E^{″}\left(\omega \right)S\left(\omega \right)\text{d}\omega$where F is the frictional force, Fn is the normal force, <zp> is the mean penetration depth of rubber into the road surface texture, σ0 is the normal stress equal to Fn divided by the nominal contact area A0, V is the sliding velocity, S(ω) is the power spectral density function of the road surface in frequency scale ω, and $E^{″}\left(\omega \right)$ is the imaginary part of energy dissipation calculated from the local stresses and strains which are determined by the heights of the rough surface. The lower and upper cut-off frequency, ωmin and ωmax, correspond to the upper and lower cut-off wavelengths λmax and λmin respectively, to be determined from the road surface roughness.

In about the same period, adopting essentially a similar approach in calculating hysteretic energy dissipation of rubber, but instead of applying the notion of penetration depth, Persson (2001) proposed a concept of nominal and “real” contact area between the rubber and the substrate surface. Although the proposed contact mechanism was developed for random road surface roughness, explicit solutions were presented only for self-affine fractal surface profiles. The coefficient of hysteresis friction at sliding velocity v is given by the following equation.${\mu }_{\text{h}}\approx \frac{1}{2}{\int }_{q_L}^{q_1}{q}^{3}C\left(q\right)P\left(q\right)\text{d}q{\int }_0^{2\text{π}}\cos \left(\varphi \right)\text{Im}\left(\frac{E\left(qV\cos \left(\varphi \right)\right)}{\left(1-{\nu }^2\right){\sigma }_0}\right)\text{d}\varphi$where wavenumber q ​= ​2π/λ, λ is wavelength, C(q) is spectral density of rough surface, P(q) is A(q)/A(qL) ​= ​A(λ)/A(L) ​= ​A(λ)/A0, L is length of nominal contact area A0 between rubber and the road surface, ϕ is the angle between the wave propagation direction and the rubber sliding direction, ν is Poisson ratio of rubber material, V is sliding velocity, the imaginary part of the complex modulus of rubber E(ω) ​= ​E(qVcos(ϕ)). The lower and upper cut-off of integration are q0 ​= ​2π/L and q1 ​= ​2π/λ1, respectively, where λ1 is the shortest wavelength of the road surface roughness.

The afore-mentioned theories of hysteretic friction does not include the contribution from cohesion. Lorenz et al. (2011) suggested that the cohesion friction component could be derived from experimental measurements. Cohesion induced friction is generally considered to be insignificant at normal vehicle speed and on wet road surfaces (Heinrich et al., 2000; Persson, 2001).

It is noted that in applying either the Heinrich or the Persson approach, the determination of the actual contact area between the sliding rubber and the rough surface is dependent on the calculation of either the average penetration depth <zp> or the contact area fraction function P(q). Both <zp> and P(q) vary nonlinearly with sliding velocity, normal load, viscoelastic modulus of rubber, and road roughness parameters. The magnitudes of both are affected by the choice of the upper and lower cut-off road surface wavelengths. Studies by pavement engineering researchers indicated that to arrive at the correct calculated coefficient of friction of a road surface, a proper selection of the upper and lower cut-off wavelength is critical (Alhasan et al., 2018; Deng et al., 2021; Kanafi and Tuononen, 2017; Pratico et al., 2021).

It is recognized that under typical wheel loads of passenger cars and trucks, tire-pavement contact would occur mostly along the upper portions of the top asperities (Almqvist et al., 2011; Deng et al., 2021; Heinrich, 1997; Kanafi and Tuononen, 2017; Kluppel and Heinrich, 2000; Lorenz et al., 2011; Persson et al., 2005b). In the case of asphalt and concrete pavements, the depth of penetration would vary with the mix design of the surface course, including the type, size and gradation of the aggregates used. Since road surface roughness would deteriorate under traffic abrasive actions, the depth of rubber penetration would also vary with the degree of wear and polishing suffered by a road surface. Unfortunately, even with the knowledge of all the influencing factors (i.e., sliding velocity, normal load, viscoelastic modulus of rubber, and road roughness parameters), the current theories of contact mechanics are unable to determine the depth of penetration and actual area of contact. Studies on common road pavement materials have found that the depth of penetration ratio could vary from 10% to 50% (Deng et al., 2021; Kanafi and Tuononen, 2017; Pratico et al., 2021). Hence, to determine the sliding coefficient of hysteresis friction of rubber on a road surface, a calibration procedure involving experimental determination of the coefficient of friction is necessary to estimate the depth of penetration and area of contact for a given set of rubber and road roughness conditions.

Heinrich and Klüppel (2008) and Persson (2011) had both extended the rubber contact theory to study the sliding friction of vehicle tires on road pavements. Persson proposed a two-dimensional mass-spring tire model together with his rubber friction model to calculate the speed-dependent coefficient of friction for different slip ratios during braking and/or cornering. Heinrich and Kluppel applied a brush model for sliding tires to analyze the load dependency of sliding friction coefficient under different slip ratios during cornering and braking.

The dependency of rubber friction on temperature and sliding velocity has an important effect on the magnitude of tire-pavement friction generated. Heat is generated in a sliding rubber locally at the contact asperities. When rubber begins to slide on a pavement at a low speed, friction increases due to rubber hysteresis caused by sliding movement on unevenness of pavement surface texture. However, as the sliding speed increases, there exits a heating effect caused by the phenomenon of rubber glass transition, and it tends to reduce rubber hysteresis (Heinrich, 1997; Kluppel and Heinrich, 2000; Persson, 2006). On road pavements, the normal operating speeds of vehicles is more than one order of magnitude higher than the sliding speed at which the peak rubber friction occurs. Hence, within the range of vehicle speeds relevant for the study of skid resistance of road pavements, the effect of rubber glass transition must be considered. Overall, as depicted in Fig. 2 which shows the variation trends of measured skid resistance with speed, dry pavement skid resistance may or may not decrease with speed. Whether or not skid resistance would decrease with speed depends on the net magnitude of hysteresis friction generated.

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Fig. 2

4.1.2. Dry pavement skid resistance by test measurements

Skid resistance testing of dry road pavement is rarely performed in practice. This is because the dry skid resistance of a typical pavement would far exceed the maximum skid resistance required for safe driving. The skid resistance requirements of a pavement is governed by its skid resistance performance under wet-weather conditions. All standard skid resistance tests are conducted on wet pavements with a layer of applied water of 0.5 or 1.0 ​mm thick (ASTM, 2015c; BSI, 2000, 2006; FAA, 2016). If for any reason the dry skid resistance of a pavement is to be tested, it can be measured using one of the standard test methods without applying water. Standard wet skid resistance test methods are described in Section 4.3.3 “Measurements of wet pavement skid resistance”.

4.2.1. Wetted pavement skid resistance based on theories of contact mechanics

The techniques of calculating the coefficient of hysteresis friction of a pavement from its surface texture characteristics based on rubber-pavement contact mechanics, as presented in Section 4.1.1, can also be applied to predict friction of wetted pavements. According to Persson et al. (2005a), Hartikanen et al. (2014) and Kienle et al. (2020), the presence of trapped water would affect the depth of penetration into the rubber by texture asperities, and the actual tire-pavement contact area. By appropriately selecting the cut-off depth in the calculation of the actual contact area, it was found that a good prediction of the coefficient of hysteresis friction could be achieved for a wetted pavement. As in the case of dry pavement, a calibration analysis against actual measured tire-pavement friction is necessary for the determination of an appropriate cut-off depth. The value of appropriate cut-off depth depends on the characteristics of pavement surface texture, properties of the rubber, and the degree of wetness.

It is appropriate to mention at this juncture that although the wetted skid resistance calculated using the contact theories would decrease as the sliding speed increases, the rate of skid resistance reduction is essentially caused by hysteresis losses of the tire rubber. This rate of skid resistance loss cannot be applied to the case of a wet or flooded pavement (Section 4.3). The presence of a layer of free water on a wet or flooded pavement would generate hydrodynamic effects due to a dynamic tire-water-pavement interaction when a tire moves on the pavement surface. The dynamic tire-water-pavement interaction mechanism is totally different from the tire-pavement contact mechanisms presented in this section.

4.2.2. Measurements of wetted pavement skid resistance

Skid resistance measurement is as a rule not performed on a wetted pavement, just as it is not normally conducted on a dry pavement. Standard skid resistance tests are performed on a wet pavement covered with a thin film of water. A wetted pavement with only some water trapped within its surface voids, though having reduced skid resistance compared with its skid resistance in the dry state, would still over-estimate the skid resistance of a wet pavement. In addition, there are two further reasons that make skid resistance measurement of wetted pavement impractical: (i) rubber-pavement friction fell considerably as the degree of wetness increases (Qian et al., 2016), where wetness refers to the degree of saturation of water in the surface voids; (ii) there is no easy practical way of measuring the degree of wetness of a wetted pavement.

4.3.1. Wet pavement skid resistance based on theories of mechanics

The presence of a layer of water on a wet pavement means that the rubber-pavement contact theories, described earlier for friction determination on dry and wetted pavements, are no longer applicable in tire-pavement skid resistance determination. In the present case, the pavement surface is saturated and submerged in a layer of water, the dynamic interaction of tire, flowing water and pavement surface texture presents a highly complex contact situation. This complex tire-fluid-pavement interaction could not be explained by the contact theories presented in Sections 4.1 Determination of skid resistance of dry pavements, 4.2 Determination of skid resistance of wetted pavements. Typically, during an acceleration phase of a road vehicle, the tire-pavement contact area continues to decrease as the vehicle speed increases, as illustrated in Fig. 3. The reduction in tire-pavement contact area is due to hydrodynamic uplift pressure build-up in the water trapped between the tire and pavement surface, which causes the tire rubber to deform and separate from the pavement surface. The uplift pressure increases in magnitude as the vehicle speed picks up, thereby resulting in decreasing tire-pavement contact area in the process.

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Fig. 3

A different set of theories of mechanics, including theories of hydrodynamics and solid mechanics, is required to solve the problem of wet pavement skid resistance. The first systematic attempt by pavement engineering researchers to solve numerically for a theoretical solution of the wet-weather pavement skid resistance and/or hydroplaning problem, taking into consideration the pavement surface characteristics, was made by the research group at the National University of Singapore (NUS) (Fwa and Ong, 2008; Ong et al., 2005; Ong and Fwa, 2007a, b). The NUS research group applied commercial finite-element software ADINA (2005) to solve the tire-fluid-pavement interaction problem numerically. The computer analysis produced predicted results in good agreement with field measured skid resistance and hydroplaning speeds. Other commercial finite-element software, such as ABAQUS (2010) and ANSYS (2009), have been subsequently employed by other researchers successfully in solving the wet pavement skid resistance problem (Anupam et al., 2013; Liu et al., 2019; Peng et al., 2021b; Srirangam et al., 2014; Zhang et al., 2013a, b; Zheng et al., 2018).

The key to calculating the available skid resistance to a vehicle traveling on a wet pavement is to correctly and accurately solve the tire-fluid-pavement interaction problem involving a moving tire rolling and/or sliding on a pavement covered with a thin film of water. It has become feasible today for researchers in general with the widely available efficient computational fluid dynamics software and powerful computing facilities. Computer based 3-dimensional computational fluid dynamics (CFD) analysis employing the full Navier–Stokes equations has become a relative common tool for solving the turbulent flow field involved in the wet pavement skid resistance and hydroplaning problem. For example, Ong and Fwa (2007a, b) applied a 3-D finite volume model based on the Navier–Stokes equations, and the turbulence flow model in terms of the turbulent kinetic energy (k) and rate of dissipation of turbulent kinetic energy (ε) using the standard k-ε method, integrated with a fluid–solid interaction (FSI) approach, to simulate tire-fluid-pavement interaction and calculate hydroplaning speed and skid resistance at different vehicle speeds.

There are three sub-models in finite element simulation modeling of the wet pavement skid resistance problem, namely the tire, fluid and the pavement sub-models. Fig. 4 presents input and output variables of the three sub-models, and the flow of processed information among them. Table 1 presents a summary of the input variables required for the simulation analysis. The interaction among the three elements causes the pneumatic tire wall to deform, which in turn leads to changes in the tire-pavement contact area (i.e., the shape and area of the tire footprint) as shown in Fig. 3. Fig. 5 shows the finite element mesh of a skid resistance simulation model. As accurate determination of tire deformation is critical to the calculation of skid resistance, the complete structural elements of a tire is fully represented in the tire sub-model. As shown in Fig. 5, the structural elements include the ply, belt, double bead, rigid metal rim, and rubber components of tire tread, sidewall, inner liner, filler, and chafer. Surface membrane elements with embedded rebar layers are employed to model radial ply and layered steel belts. In view of the complex structure composition of a vehicle tire, a carefully conducted full-scale calibration of the tire's load–deformation characteristics is essential to the correct simulation of tire-pavement skid resistance.

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Fig. 4

Table 1. Input parameters for mechanistic tire-pavement-fluid interaction analysis.

Pavement parameter	Environmental and vehicle operating condition
	Vehicle parameter	Environmental parameter
• Surface texture properties (e.g., microtexture and macrotexture; or effects of microtexture and drainage)	• Tire dimensions	• Water film thickness
	• Tire structural properties	• Water temperature and viscosity
	• Tire tread pattern
	• Tire inflation pressure
	• Wheel load
• Longitudinal gradient	• Tire slip ratio
• Cross slope	• Tire slip angle
• Permeability coefficient (porous pavement)	• Tire temperature
	• Vehicle speed

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Fig. 5

Due to the large difference between the elastic moduli of typical pavements and vehicle tires, pavements are without exception modeled as a rigid surface in skid resistance studies. The interaction between pavement and surface water, and that between tire and pavement, are straight forward. However, the dynamic interaction between a rotating tire and flowing water is computationally complex, requiring a “two-way coupling” solution procedure which is an iterative process to couple the responses of the fluid and tire sub-models. To save computer simulation time, instead of modeling a tire rolling on a pavement, the rolling tire is taken as the frame of reference. That is, an angular velocity is applied to the tire, and a layer of water and the pavement are given a linear speed moving toward the tire. In the computer simulation, the method of Coupled Eulerian–Lagrangian (CEL) is employed to model the interaction between the Eulerian fluid and the rotating Lagrangian tire.

The pavement sub-model is often represented by a solid plane surface in the skid resistance simulation models today. This is unsatisfactory as it does not realistically represent the actual surface texture of a pavement. An important outcome of the computer skid resistance simulation analysis is the frictional force generated at areas where the tire is in contact with the pavement. The magnitude of the contact frictional force is a function of the values of coefficient of friction, the actual contact area, and the actual normal contact force, all of which are dependent on the vehicle speed, water film thickness, water pressure, wheel load, and the pavement microtexture and macrotexture. According to Fig. 4 and Table 1, the determination of wet pavement skid resistance requires input information on pavement properties in the form of pavement surface microtexture and macrotexture, and the coefficient of kinetic friction between tire and pavement under a submerged condition. Pavement surface microtexture and macrotexture would affect drainage of surface water under a rolling tire, as well as the magnitude of the coefficient of tire-pavement friction. Drainage and friction coefficients are two key pavement properties that affect the wet skid resistance performance of a pavement. Unfortunately, the large range of geometric scale ranging from the order of μm of pavement microtexture to more than 1 ​m in the diameter of a truck tire presents a critical modeling problem. It is not only impractical, but also theoretically infeasible currently to model pavement microtexture using the finite element method. This explains why pavement is represented by a solid plane surface in finite element simulation of pavement skid resistance. Hence, other measures are required to model the effects of pavement microtexture and macrotexture in the finite element simulation analysis in order to correctly solve the wet pavement skid resistance problem. Some of the measures that have been adopted by researchers are described in the next section.

4.3.2. Prediction of wet pavement skid resistance performance

It has been noted in the preceding section that finite element skid resistance simulation models are unable to include pavement microtexture and macrotexture in the pavement sub-model. That is, it is unable to directly model mechanistically the effects of pavement texture on water drainage, and the generation of tire-pavement kinetic friction. The drainage effect of pavement texture is customarily represented through surface roughness indices such as Manning roughness coefficient (Reed and Kibler, 1983), and roughness height and roughness constant (NCHRP, 2021) in fluid flow analysis. This has been an accepted practice in pavement engineering, and can be conveniently incorporated into a skid resistance simulation analysis, provided that values of appropriately calibrated roughness indices are available for the common range of pavement texture profiles encountered.

Compared with the representation of drainage effect of pavements, the determination of the input value of coefficient of wet kinetic friction is much more complex. Two main approaches have been adopted by researchers for this purpose. The first approach directly provides as an input an estimated value of friction coefficient, often as a function of vehicle speed. Such a function can be obtained experimentally by performing dynamic friction tester (DFT) tests at different speeds on a test surface (Hu et al., 2016; Wasilewska et al., 2016). This method does not consider the influence of water thickness and hydrodynamic effect. Several studies applied Persson's contact theory of rubber friction (Sections 4.1.1 Dry pavement skid resistance based on theories of contact mechanics, 4.2.1 Wetted pavement skid resistance based on theories of contact mechanics) to calculate the coefficient of hysteresis friction as the input value to skid resistance simulation analysis (Liu et al., 2019; Zhu et al., 2017, 2021). According to Persson et al. (2005a), this is applicable at low vehicle speed lower than approximately 30 ​km/h when there is sufficient time for surface water to be squeezed out from the tire-pavement contact area. However, at higher vehicle speeds, the friction coefficient would be over-estimated due to the presence of hydrodynamic pressure and nonlinear deformations of tire walls. Furthermore, the tire-pavement contact area may be fragmented and not continuous in the case of ribbed tires. Besides physical measurements of pavement microtexture and macrotexture, this method also requires experimental calibration of problem parameters, such as the ratio of actual contact area, that are dependent on the characteristics of pavement texture.

The second approach is the back-calculation method first proposed by the research group at the National University of Singapore (Fwa and Ong, 2008). This NUS back-calculation method has been validated using actual field measured skid resistance data of various modes of vehicle and aircraft operations (Fwa and Chu, 2021; Ju et al., 2013; Ong and Fwa, 2010a, b; Pasindu et al., 2011; Peng et al., 2021a). The concept of the approach is based on the understanding that the skid resistance of a wet or flooded pavement is dependent on its microtexture and macrotexture. According to the Penn State model proposed by researchers at the Pennsylvania State University (Henry, 1986; Leu and Henry, 1983), the kinetic friction of a wet pavement can be expressed as a negative function of vehicle speed with two parameters of pavement texture characteristics, as given by the following equation.${\mu }_V={\mu }_0{\text{e}}^{-\beta V}$where ${\mu }_V$ is the coefficient of kinetic friction at speed V, ${\mu }_0$ is the initial coefficient of friction when the relative sliding begins, which is given by the intercept at V ​= ​0, and β is a regression constant. Both ${\mu }_0$ and β are to be estimated from experimental data. Research studies have found the exponential decay function of Eq. (4) to be a representative deterioration trend of the dynamic friction coefficient between rubber and a large class of materials (Kuwajima et al., 2006; Oden and Martins, 1985).

Experimental studies have shown that on a given pavement covered with a known thickness of water film, field measured skid numbers fitted well with Eq. (4); and that the terms μ0 and β were highly correlated with pavement texture related indices (Hall et al., 2009; Henry, 2000). μ0 was found to be highly correlated with pavement microtexture; while β which is related to the rate of change of coefficient of friction with speed V, was found to be closely governed by pavement macrotexture. According to ASTM standard E274 (ASTM, 2011), skid number SNV at speed V is defined as 100 times the measured friction coefficient ${\mu }_V$ defined as follow.${\text{SN}}_V=100{\mu }_V=100F/W$where F is the horizontal tractive force applied to the test tire, and W is the vertical load on the test tire.

Other statistical regression relationships similar to Eq. (4) have been proposed by researchers. Such relationships can also be used for the purpose of the back-calculation analysis. For example, adopting the exponential decay function, the deterioration trend of rubber-wet pavement contact friction from an initial ${\mu }_0$ to a terminal ${\mu }_{\text{k}}$, can be represented by the following expression (Kuwajima et al., 2006; Oden and Martins, 1985).${\mu }_V={\mu }_{\text{k}}+\left({\mu }_0-{\mu }_{\text{k}}\right){\text{e}}^{-\beta V}$

The value of ${\mu }_0$ can be taken as (SN0/100) on the understanding that at very low test speed, the hydrodynamic effect of surface water is negligible and there is sufficient time for water to be squeezed from the tire-pavement contact. The terminal ${\mu }_{\text{k}}$ has the value of zero theoretically at the hydroplaning speed of the wet pavement concerned. Hydroplaning refers to the state of tire-pavement contact when the hydrodynamic uplift acting on the tire increases and becomes equal to the wheel load. Under this condition, the tire loses contact with the pavement and the driver would lose steering (i.e., directional) and braking control. In actual field measurements, the practical hydroplaning phenomenon would take place before the theoretical value of zero, due to the presence of fluid drag force. A small friction coefficient ${\mu }_{\text{k}}$ of the order of 0.05 may be chosen (Agrawal, 1983; Pasindu et al., 2016; Zhu et al., 2021), as shown schematically in Fig. 6.

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Fig. 6

The flow diagram in Fig. 7 shows the steps involved in the application of the back-calculation approach in the finite element skid resistance simulation analysis. As there are two unknowns in either Eqs. (4) or (6), at least two skid resistance data measured at different test speeds are required for the back-calculation analysis. Although the analysis involves a trial-and-error process, the problem is well behaved and with an appropriate initial choice of ${\mu }_0$, a good match between the predicted and measured skid resistance could be achieved in less than four or five trials. A good initial choice of ${\mu }_0$ can be taken as (SN0/100) where a good estimate of SN0 can be made from the two skid resistance data, which are typically measured at the test speeds of 48 and 64 ​km/h. Alternatively, though not practically feasible in most field operations, SN0 can be obtained from a low-speed skid resistance test, such as the British pendulum test as explained in Section 4.3.3.

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Fig. 7

The back-calculated input parameter μ0 represents largely the effect of pavement microtexture on tire-pavement contact friction, although the presence of pavement macrotexture effect could not be ruled out, especially on rough-textured or grooved pavement surfaces (Fwa et al., 2003; Han et al., 2020; Liu et al., 2004). The other back-calculated input parameter β represents the deterioration rate of ${\mu }_V$ with vehicle speed V. This speed dependent deterioration of contact friction, together with the hydrodynamic effect of surface water related to pavement macrotexture, contributes to the overall deterioration of skid resistance SNV with vehicle speed V.

After the input parameters related to pavement texture have been calibrated by means of the back-calculation method, the finite element skid resistance simulation model can be applied to predict the wet skid resistance of the pavement concerned under any speed, water film thickness, slip ratio, and slip angle. Fig. 8 illustrates schematically the family of skid resistance curves that can be obtained from the calibrated simulation model of a given pavement.

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Fig. 8

This back-calculation based approach of computer simulation of wet pavement skid resistance has been validated by research studies for different vehicle and aircraft types operating under different wet pavement conditions, based on a large number of full-scale field test data reported in past experimental studies. Fig. 9 shows graphically examples of validation on measured skid resistance data obtained by past research studies. These studies indicate that the finite element simulation modeling together with the back-calculation calibration method presents a practically reliable method for the determination and prediction of wet skid resistance of a pavement.

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Fig. 9

4.3.3. Measurements of wet pavement skid resistance