Methods for Measuring and Assessing Irregularities of Stone Pavements—Part I

Abstract

Stone pavements are the historical, architectural, and cultural heritage of lots of cities in Italy and the world. Road managers should be able to make decisions on the global conditions to define the most suitable strategies and maintenance interventions for every type of pavement. There are no standard monitoring methods or criteria for evaluating stone pavement performance. These pavements have more uneven surfaces than traditional pavements, but this characteristic could be accepted if type of vehicles and relative travel conditions are considered. Therefore, it is useful to define criteria for assessing roughness considering the comfort experienced by users in different vehicles. In this research, both traditional and innovative methodologies for assessing irregularities have been investigated using true stone surface profiles. In this regard, traditional performance indicators such as the International Roughness Index (IRI) defined by the ASTM E1926, the ISO 8608 classification, and the frequency-weighted vertical acceleration (a_wz) provided by ISO 2631-1 for comfort assessment have been considered. In the case of comfort assessment, three dynamic vehicle models (bike, automobile, and bus) have been adopted. Finally, this two-part paper also proposes an innovative straightedge analysis for stone pavements (SASP) to evaluate the effect on traffic of both pavement profile roughness and localized irregularities. In this way, the authors aim to provide an effective tool to monitor stone pavements.

1. Introduction

Historic stone pavements are of undisputed aesthetic value and are the architectural and cultural heritage of several cities around the world, but often they lead to management issues in urban road networks [1]. These pavements were not designed to accommodate modern traffic categories. They consist of juxtaposed modular elements of non-flat shapes with interposed joints, and their final configuration is an irregular surface in all wavelengths of the worldwide accepted definition [2], even when the pavement has just been built. In particular, the irregularity of a new stone pavement depends on the pavers’ size, the construction technique (flat, concave, bush-hammered surface, etc.), the joints’ width and type (open, sealed with mortar, etc.) [3,4], and the bedding layer [5]. Factors negatively affecting the functional performances of these pavements are climate change [6] and motor vehicle traffic. It has to be noted that stone pavements often have more irregularities than traditional asphalt concrete (AC) or cement concrete (CC) ones, and that they may not guarantee acceptable conditions of safety and comfort for vehicles and users [7]. Concerning the safety of vehicles moving on uneven surfaces, it is generally observed that the risk of rollover and skidding increases on uneven pavements [8]. Natural stone pavers can cause unsafe conditions for motorcyclists by reducing the amount of friction (skid resistance) or creating an uneven rolling surface [9]. Indeed, pavement roughness negatively affects the contact between the wheel and the surface [10,11], as confirmed by [12,13].

The functional issues of stone pavements spark debates amongst both expert critics and common public opinion. It is not always possible to find a compromise, and the most frequent interventions are as follows:

• The removal of stone pavements from roads with high traffic volumes and reuse of the stones in low- and light-traffic areas [14];
• The implementation of speed reduction to allow circulation in urban roads of all traffic categories and ensure riding comfort [15].

Whatever the admitted road users, the network manager should guarantee a minimum level of pavement regularity to ensure riding comfort and minimize the impact on the surrounding environment in terms of rolling noise and vibrations compared to traditional pavements [16,17]. Experimental data allow assessing the pavement state of health by adopting one or more indices about structural and/or functional conditions [18]. The comparison between the measured values and consolidated threshold values enables us to assess the current pavement condition and schedule the maintenance program. This decision-making process is part of the Pavement Management System (PMS), which allows for identifying priorities and planning interventions based on collected data and future projections, feasibility analysis, and budget optimization [19].

The identification of the monitoring and evaluation criteria for modular stone pavements in the scientific literature [20] has been marginally addressed. Garilli et al. [21] proposed a method to evaluate the functional and safety characteristics of stone pavements used by pedestrians with wheeled trolleys.

A proper method to assess roughness conditions cannot overlook the operating conditions (e.g., speed) and the users (e.g., pedestrians, two-wheeled vehicles, vehicles, buses). Road pavement roughness is a pivotal functional requirement adopted by road managers as a key performance indicator (KPI) to plan maintenance [22]. The International Roughness Index (IRI) is nowadays accepted for both AC and CC carriageways [23]. Accurate IRI decay curves allow road managers to plan network maintenance that optimizes the available resources [24]. Several studies have been proposed to define appropriate IRI thresholds concerning different operational and functional requirements such as the length of the road section [23,25,26], the effects on dynamic loads [27], comfort [28], or the contribution of the joints [4]. Other researchers have investigated the relationship between IRI and the comfort experienced on-board different types of vehicles [29,30,31] whilst also considering the operating speeds of the road [32]. All these studies have demonstrated the versatility of this index in evaluating the condition of paved surfaces. However, there is a limitation concerning the operating speed and the types of assessed distress. In regard to speed, different thresholds have been proposed as a function of the operating speed [33,34,35]. These studies have shown that while the threshold values provided by road agencies refer to travel speeds above 80 km/h, higher IRI values can be generally accepted on roads travelled at lower speeds. Moreover, while the evaluation through the IRI is ideal for widespread defects, it performs poorly in interpreting isolated and localized defects [36]. In this regard, an alternative approach has been proposed to evaluate the longitudinal roughness through users’ perception [37].

Another interesting approach to evaluating pavement roughness is the Power Spectral Density (PSD) of the profiles according to ISO 8608 [38]. Several studies have adopted this procedure, which simulates the response of the vehicle suspension system to generate artificial profiles and simulates the behavior of vehicles on uneven pavements [39]. In the field of vehicle design, surface profile classification according to [9] has been adopted to generate artificial road profiles that simulate different conditions to optimize the mechanical components or to verify the suspension parameters of vehicles [40,41,42]. Other scholars have realized artificial road profiles with given IRI values [43,44,45].

Smartphone apps or low-cost Inertial Measurement Unit (IMU) systems are modern alternative systems for evaluating pavement conditions in urban areas where driving speed is generally low, localized distresses are widespread, and different types of vehicles move at different speeds. Several applications for smartphones [46] and validated IMU systems [47] allow the measuring of accelerations on-board a vehicle moving at a fixed speed using as a reference the comfort evaluation parameters provided by the international standard ISO 2631-1 [48]. According to the reference standard, the time history of vertical accelerations calculated with a mechanical model of a vehicle moving on a pavement profile allows the calculation of the frequency-weighted vertical acceleration (a_wz).

Finally, the authors proposed a Straightedge Analysis for Stone Pavements (SASP). This derives from the Procedures for Measuring Airfield Pavement Roughness defined by the Federal Aviation Administration (FAA) [49] to obtain the Boeing Bump Index (BBI).

Whatever the adopted method, it is necessary to give straightforward and widely accepted definitions of both surface defects and threshold values. Based on true stone surface profiles, this paper takes into account existing methodologies and proposes a new methodology to evaluate the irregularity conditions of stone pavements. Firstly, the IRI [50] and ISO 8608 standards [38] have been adopted to characterize the pavement roughness. Then, the whole vertical body vibration using three mechanical models (i.e., a bicycle, an automobile, and a bus) at different running speeds has been considered. Therefore, according to [50], it was possible to classify the surveyed pavements in terms of performance classes. Finally, SASP classified the irregularities of the stone pavements and overcame the limitations of the traditional methods.

2. Materials and Methods

2.1. International Roughness Index

IRI derives from studies in the 1980s [51] involving controlled measurements of road surface roughness for 320 m-long road sections under a variety of conditions and by a variety of instruments and methods [52]. It is based on a mathematical model called quarter-car and was developed to assess not only the ride quality on road pavements but also other detrimental effects (e.g., dynamic load increment on both vehicle and pavement) due to the presence of irregularities on the road surfaces. Equation (1) allows IRI calculation according to [50]:

$$IRI=\frac{1}{l}{{\displaystyle \int }}_0^{l/v}\left|{\dot{z}}_s-{\dot{z}}_u\right|dt$$

(1)

where l is the length of the profile in km, v is the simulated speed equal to 80 km/h, ${\dot{z}}_s$ is the time derivative of vertical displacement of the sprung mass in m, and ${\dot{z}}_u$ is the time derivative of vertical displacement of the unsprung mass in m. IRI is expressed in slope units (e.g., m/km or mm/m).

Over the world, different threshold values for roughness are prescribed in terms of IRI values for pavement quality control, monitoring, maintenance, and repair planning. According to [23], there is a high heterogeneity of adopted IRI thresholds that depend on road surface type (i.e., asphalt or cement concrete pavements), road functional category, average annual daily traffic (AADT), legal speed limit, segment length considered for its calculation, as well as other variables. Among these, the most important are the maximum driving velocity allowed on the road and the type of vehicle adopted to assess riding comfort. In recent years, some authors have proposed that speed-related IRI thresholds be used for the evaluation of ride quality. In particular, Cantisani and Loprencipe [33] considered an assorted sample of 320 m-long 124 profiles of AC pavements (average IRI = 2.11 m/km; maximum IRI = 5.98 m/km, minimum IRI = 0.55 m/km, and standard deviation of IRI = 1.18 m/km) and defined four ride quality levels (

Table 1. IRI thresholds at different speeds [33].

Ride Quality Level	IRI Thresholds at Different Measured Speeds (m/km)
	30 km/h	40 km/h	50 km/h	60 km/h	70 km/h	80 km/h	90 km/h
Very Good	<4.17	<3.41	<2.98	<1.87	<1.60	<1.42	<1.15
Good/Fair	4.17–8.34	3.41–6.83	2.98–5.95	1.87–3.73	1.60–3.20	1.42–2.84	1.15–2.31
Mediocre	8.34–11.92	6.83–9.75	5.95–8.51	3.73–5.33	3.20–4.58	2.84–4.06	2.31–3.30
Poor	>11.92	>9.75	>8.51	>5.33	>4.58	>4.06	>3.30

). For each level, the IRI thresholds depended on the a_wz calculated for measured speed values ranging between 30 and 90 km/h.

The proposed thresholds are valid for the investigated IRI values and AC pavements. An in-depth study is necessary to apply them to stone pavements, whose IRI values and distribution differ from those investigated by [33]. Indeed, higher IRI values and greater IRI variability are to be expected due to the used dynamic model. Figure 1 shows the response of the IRI quarter car filter to different wavelengths varying the speed vehicle. The red curve represents the transfer function between road excitation and the suspension relative velocity, H_Vrel, for the reference quarter-car model with parameters defined for IRI estimation (velocity = 80 km/h). In the same graph, the transfer function H_Vrel for velocity = 40 km/h is represented [46] to compare the effect of speed on pavement roughness evaluation. Both curves in Figure 1 have the same diagram shape, but the wavelengths influencing the IRI shorten, decreasing the simulation speed.

It cannot be overlooked that the profile length to assess IRI is 320 m. In the urban environment, it is very difficult to collect such a profile length, and shorter sections must be considered. The IRI thresholds on different pavements should refer to the same profile lengths.

For stone pavements, it is difficult to define irregularities threshold values as these pavements have higher values than traditional pavements and an unambiguous definition of surface defects. An attempt to define some threshold values was developed by Zoccali et al. [20] for the so-called Sampietrini, a particular type of stone paving used throughout central Italy and made of stone elements of average dimensions 10 × 10 × 17 cm laid on a 5–10 cm-thick sand bed. For such pavements, Zoccali et al. [20] demonstrated that IRI values were comparable to those obtained for rough unpaved roads (Figure 2). Therefore, IRI values can describe the perceived riding comfort for different speed values, and IRI values more than 8 m/km (dotted red line in Figure 2) imply less than 70 km/h riding speed.

2.2. Road Surface Profile Classification According to ISO 8608 Standard

Road surface profile classification according to ISO 8608 standard is based on the PSD of the profile. To determine the PSD, it is necessary to measure the surface profile concerning a reference plane [53]. In particular, the Power Spectral Density is calculated using the Fast Fourier Transform (FFT), windowing each profile signal with the Hanning window [54] and representing the smoothed power spectral density. Finally, Equation (2) allows the calculation of the fitted PSD:

$$G_d\left(n\right)=G_d\left(n_0\right){\left(\frac{n}{n_0}\right)}^{-w}$$

(2)

where G_d(n) is the fitted PSD [m³], n is the spatial frequency in cycles/m, n₀ = 0.1 cycles/m is the reference spatial frequency, and w is the exponent of the fitted PSD, also known as waviness.

As specified by Múčka and Granlund [55], the two parameters defining Gd(n) (i.e., G_d(n₀) and w) are independent; in particular, the second one provides information about wavelength distribution in the investigated spatial frequency range. Values of waviness greater than two mean that long wavelengths are prevalent, while if w is lower than two, the short ones are predominant.

Based on the G_d(n₀) values, a pavement profile is catalogued according to the classes (from A to H; i.e., from best to worst) defined by [38] (

Table 2. ISO 8608 thresholds limits for the road profiles classes.

ISO 8608 Class	Gd(n0) (10−6 m3)
A	<32
B	32–128
C	128–512
D	512–2048
E	2048–8192
F	8192–32,768
G	32,768–131,072
H	>131,072

). Usually, road profiles of AC pavements do not belong to classes worse than D because road agencies set intervention thresholds to restore optimal conditions.

Destroyed roads belong to classes above D. However, artificial class profiles higher than D have been generated in the literature [56]. They imply severe distress conditions expected for unpaved roads. Therefore, the classification proposed by [38] provides a synthetic description of the geometric features of a road pavement surface. However, it overlooks the vehicle type and the driving velocity, and does not properly consider the different vibration levels affecting road users. At any rate, it seems appropriate to use this evaluation method for stone pavements that are characterized by a high level of irregularities.

2.3. Whole-Body Vibration Using Vehicle Mechanical Model Simulation

The international standard ISO 2631-1 allows comfort assessment in public transport [48]. It is based on the analysis of the time-histories of vertical accelerations obtained from direct measurements inside the riding vehicle or simulated by a vehicle mechanical calibrated model. The seat-acceleration time-histories ($a_s\left(t\right)$) are processed to determine the vertical Root Mean Square accelerations ($a_{iz}^{RMS}$) for the ith one-third-octave bands that represent the frequency range of interest for human response to vibrations (i.e., 0.5–80 Hz) according to [48]. Given $a_{iz}^{RMS}$, Equation (3) allows calculation of the vertical weighted Root Mean Square acceleration (a_wz):

$$a_{wz}=\sqrt{{\displaystyle \sum }_{i=1}^{23}{\left(W_{k,i} · a_{iz}^{RMS}\right)}^2}$$

(3)

where W_k,i are the frequency weightings in one-third-octave bands for seated positions provided by [48]; i = 1, …, 23.

The a_wz values can be compared to the threshold values proposed by [48] (

Table 3. a_wz threshold values for public transport [47].

awz Values (m/s2)	Comfort Level
<0.315	Not uncomfortable
0.315–0.63	Little uncomfortable
0.5–1.0	Fairly uncomfortable
0.8–1.6	Uncomfortable
1.25–2.5	Very uncomfortable
>2.0	Extremely uncomfortable

) to identify the comfort level perceived by road users. The current standard provides several comfort levels and introduces overlapping zones between two adjacent levels because several factors (e.g., user age, acoustic noise, temperature, etc.) contribute to the level of perceived discomfort.

Innovative Vehicle Mechanical Model Simulation

In this paper, three different dynamic vehicle models have been considered to quantify users’ comfort level according to [48]:

• A 5-degree-of-freedom (5-dof) bike model developed in this research and calibrated using parameters proposed by [57,58,59,60,61,62,63] to represent the vibration perception of a common biker;
• An 8-dof full-car model developed and calibrated by Cantisani and Loprencipe [33] to represent the vibration perception of a common passenger car (automobile);
• An 8-dof full-car model developed by the authors and calibrated using parameters proposed by [28,64,65] to represent the vibration perception of a common passenger bus;

For each model, the main assumptions are as follows:

• Vehicle body parts are rigidly connected;
• The vehicles move in a straight line, and the longitudinal and transversal variations are assigned by the measured profiles;
• Passenger mass and unsprung mass are considered constant in magnitude during the simulation;
• The input road profiles are responsible for vibration transfer in the vehicle models.

By using Newton’s law, Equation (4) is valid for the three vehicle mathematical models:

$$\left[M\right]\{\ddot{Z}\}+\left[\mathsf{\Gamma }\right]\{\dot{Z}\}+\left[K\right]\left\{Z\right\}=\left[F\right]$$

(4)

where $\left[M\right],$ $\left[\mathsf{\Gamma }\right],$ $\left[K\right]$, and $\left[F\right]$ are mass, damping coefficient, spring stiffness, and force matrices, respectively. For example, for the 5-dof bike model in Figure 3, $\left[M\right],$ $\left[\mathsf{\Gamma }\right],$ $\left[K\right]$, and $\left[F\right]$ are Equations (5)–(8), respectively.

Table 4. Constants and variables adopted in the bike 5-dof mathematical model.

Symbol	Value	Unit	Description
ms	80	kg	Driver body mass
M	20	kg	Bike frame mass without wheels
m1	2	kg	Mass of a front unsprung mass (wheel)
m2	2	kg	Mass of a rear unsprung mass (wheel)
Jθ	13	kg m2	Body (bike) sprung mass pitch moment of inertia
R	0.0	m	Distance of CG from the seat driver
a1	0.6	m	Distance of CG from the front axle
a2	0.4	m	Distance of CG from the rear axle
k1	30,000	N/m	Spring constant suspension front
k2	25,000	N/m	Spring constant suspension rear
kt1	25,000	N/m	Spring constant tire front
kt2	25,000	N/m	Spring constant tire rear
ks	50,000	N/m	Spring constant seat driver
c1	9000	N s/m	Damping constant suspension front
c2	9000	N s/m	Damping constant suspension rear
cs	1000	N s/m	Damping constant suspension seat driver
Z	var	m	Body vertical motion coordinate (1-dof)
z1	var	m	Front-wheel vertical motion coordinate (2-dof)
z2	var	m	Rear wheel vertical motion coordinate (3-dof)
zs	var	m	Driver vertical motion coordinate (4-dof)
θ	var	rad	Body pitch motion coordinate (5-dof)
zv,1	var	m	Road profile elevation–front
zv,2	var	m	Road profile elevation–rear

lists the meaning and the values of the constants and variables.

$$\left[M\right]=\left[\begin{array}{ccccc}{m}_s& 0& 0& 0& 0\\ 0& m& 0& 0& 0\\ 0& 0& J_{\theta }& 0& 0\\ 0& 0& 0& m_1& 0\\ 0& 0& 0& 0& m_2\end{array}\right]$$

(5)

$$\left[\mathsf{\Gamma }\right]=\left[\begin{array}{ccccc}{c}_s& -c_s& r · c_s& 0& 0\\ -c_s& c_s+c_1+c_2& a_2 · c_2-a_1 · c_1-r · c_s& -c_1& -c_2\\ r · c_s& a_2 · c_2-a_1 · c_1-r · c_s& c_1 · a_1^2+c_2 · a_2^2+c_s · r^2& a_1 · c_1& -a_2 · c_2\\ 0& -c_1& a_1 · c_1& c_1& 0\\ 0& -c_2& -a_2 · c_2& 0& c_2\end{array}\right]$$

(6)

$$\left[K\right]=\left[\begin{array}{ccccc}{k}_s& -k_s& r · k_s& 0& 0\\ -k_s& k_s+k_1+k_2& a_2 · k_2-a_1 · k_1-r · k_s& -k_1& -k_2\\ r · k_s& a_2 · k_2-a_1 · k_1-r · k_s& k_1 · a_1^2+k_2 · a_2^2+k_s · r^2& a_1 · k_1& -a_2 · k_2\\ 0& -k_1& a_1 · k_1& k_1+k_{t1}& 0\\ 0& -k_2& -a_2 · k_2& 0& k_2+k_{t2}\end{array}\right].$$

(7)

$$\left[F\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ k_{t1}& 0\\ 0& k_{t2}\end{array}\right]$$

(8)

The differential equations have been implemented in MATLAB^® using the State Space representation that describes the set of all possible states of the system [66]. Each coordinate is a state variable whose values completely describe the state of the system [67].

In continuous-time, a state-space model has the following form Equations (9) and (10):

$$\begin{array}{c}\dot{z}=Az+Bu\\ f=Cz+Du\end{array}$$

(9)

where

$$\left[A\right]=\left[\begin{array}{cc}\left[Z_{5x5}\right]& \left[I_{5x5}\right]\\ \left[-\frac{M}{K}\right]& \left[-\frac{M}{C}\right]\end{array}\right] \hspace{1em} \left[B\right]=\left[\begin{array}{c}\left[Z_{5x2}\right]\\ \left[M/F\right]\end{array}\right] \hspace{1em} \left[C\right]=\left[\left[I_{10x10}\right]\right] \hspace{1em} \left[D\right]=\left[\left[Z_{10x2}\right]\right]$$

(10)

The matrix $\left[Z\left(t\right)\right]$ Equation (11) contains the discrete time-history of all the parameters to calculate the system dynamic of model:

$$\left[Z\left(t\right)\right]=\left[\begin{array}{c}{z}_s\left(t\right)\\ z\left(t\right)\\ \theta \left(t\right)\\ z_1\left(t\right)\\ z_2\left(t\right)\\ \dot{z_s}\left(t\right)\\ \dot{z}\left(t\right)\\ \dot{\theta }\left(t\right)\\ \dot{z_1}\left(t\right)\\ \dot{z_2}\left(t\right)\end{array}\right]$$

(11)

where z_s(t) is the seat vertical motion coordinate in m, z(t) is the body vertical motion coordinate in m, θ(t) is the body pitch motion coordinate in rad, z₁(t) is the front wheel vertical motion coordinate in m, z₂(t) is the rear wheel vertical motion coordinate in m, $\dot{z_s}\left(t\right)$ is the derivate seat vertical motion in m/s, $\dot{z}\left(t\right)$ is the derivate body vertical motion in m/s, $\dot{\theta }\left(t\right)$ is the derivate body pitch motion in rad/s, $\dot{z_1}\left(t\right)$ is the derivate front wheel vertical motion in m/s, and $\dot{z_2}\left(t\right)$ is the derivate rear wheel vertical motion in m/s.

Finally, Equation (12) gives the acceleration time-history of the bike driver:

$$a_s\left(t\right)=\frac{k_s·\left[z\left(t\right)-z_s\left(t\right)\right]+c_s·\left[\dot{z}\left(t\right)-\dot{z_s}\left(t\right)\right]+k_s·r·\theta \left(t\right)+c_s·r·\dot{\theta }\left(t\right)}{m_s}$$

(12)

Similarly, the MATLAB^® State Space formulation was used to simulate the dynamics of the automobile and the bus using the adopted full-car models. Appendix A lists the matrices and all the geometric and mechanical parameters of the adopted models to assess a_wz.

The flow chart in Figure 4 shows the MATLAB^® code that carried out the simulations. For each pavement profile, six a_wz values were obtained (two driving speed values for each of the three modelled vehicles). Data relating to driving speed and the parameters of the model can be modified, but only those adopted in this study are herein reported.

In the case of the comfort assessment, the modelled vehicle and the driving speed played a pivotal role in calculating the vertical accelerations perceived by users because they affected the transfer function of the pavement irregularities. In this respect, the contribution of the punctual defects of the pavement cannot be overlooked in a proper evaluation of the examined road section.

2.4. Straightedge Analysis for Stone Pavements

All the methods described in the previous sections allow for a global evaluation of pavement branches for which one or more profiles are known, but when they are used to evaluate the stone paving they have some limits that should be known. In the discussed methods, the results can be influenced by the length of the pavement profile, and it is not possible to identify the contribution of localized geometric defects unless sections of pavement of limited length are cut (however, it is not advisable to go below 20 m). Furthermore, IRI values depend on the position where the localized irregularity occurs (i.e., potholes or bumps, if at the beginning or the end of the profile).

In the case of the comfort assessment, the vehicle used to calculate the vertical acceleration bearing by the user plays a fundamental role for its physical-geometric characteristics and the driving speed as these parameters determine the transfer function of the pavement irregularities. This method also gives a global evaluation of the examined section, and, with difficulty, it is possible to identify the contribution of the punctual defects of the pavement.

For stone pavements, it is necessary to overcome the discussed issues with a method capable of evaluating the effects of localized irregularities. Therefore, this study proposes an innovative approach derived from [49] to identify the position of the most severe irregularities that must be restored. The proposed method is based on the geometric analysis of the true profile of the road collected by profilometric instruments in contact or by 3D surface surveys [64] that allow the extraction of profiles with adequate resolution. The global evaluation of stone pavements consists of the measurement of Bump Heights ($BH$s) using different theoretical lengths of straightedge (SL_i) that connect two generic points of the L-long profile. The measurement of the $BH$ is carried out for all the possible straightedges, whose lengths range between the minimum value SL_min ($S{L}_{min}=2·\Delta x$) and the maximum one ($S{L}_{max}$), with step Δx. Equation (13) gives the total number of SL_i (N_SL):

$$N_{SL}=\frac{S{L}_{max}-S{L}_{min}}{\Delta x}+1$$

(13)

For example, if $\Delta x$ = 0.25 m and $S{L}_{max}=$ 3 m a total of 11 N_SL should be considered. Therefore, a true profile can be reduced to $\frac{L}{\Delta x}+1$ points ($X_j,Y_j)$ where j = 1, 2, …, $\frac{L}{\Delta x}+1$. For each profile point the straightedge should be positioned to link two points (e.g., $X_s,Y_s$ and $X_t,Y_t$ such that $t>s$).Therefore, for each $X_j$ with $s\le j$ $\le t$, the Bump Height is the difference between $Y_j$ and the corresponding straightedge elevation in $X_j$. Each $BH$ value can be associated to a Bump Length ($BL$) that is the distance between the $X_j$ point where the $BH$ occurs and the closest of the straightedge extremes. It has to be noted that $BL$ ranges between the minimum value ($B{L}_{min}$) equal to Δx and the maximum value ($B{L}_{max}$) equal to $S{L}_{max}/2$. This analysis should be implemented for all the defined $S{L}_i$. For example, Figure 5 represents a true road profile (black line) investigated with Δx = 0.25 m, and different straightedges (dotted red lines) whose lengths are d = 1.5 m; m = 2 m; n = 0.75 m, o =1 m; p = 0.5 m; q = 0.5 m; r = 0.5 m.

For each point ($X_j,Y_j$), pairs of $BL$ and $BH$ values are identified; values equal to 0 refer to non-measurable points (Figure 6). For each $B{L}_k$ (with k = 1, …, $N_{SL}$ + 1/2), the maximum and average $BH$ values are calculated as $B{H}_{max}$ and $B{H}_{avg}$, respectively.

Finally, the pairs $B{L}_k$, $B{H}_{max,k}$ as well as $B{L}_k$, $B{H}_{avg,k}$ are represented in a scatterplot to identify the irregularities of the profile. For example, the Figure 7 shows the curves based on $S{L}_{max}$ equal to 12 m and referring to a true measured profile.

In order to evaluate the effect of irregularities on wavelengths comparable to the dimensions of the vehicle, the proper choice of $S{L}_{max}$ is in accordance with the expected traffic. In the event of roads travelled by vehicles with very different pitches, it is better to draw $BH$ curves considering $S{L}_{max}$ according to the vehicle pitches.

Therefore, in this paper, the authors provide a reference length $S{L}_{max}$ for three typical vehicles in the urban environment (Figure 8):

• Bike $S{L}_{max}$ = 1 m ($B{L}_{max}$ = 0.5 m);
• Automobile $S{L}_{max}$ = 3 m ($B{L}_{max}$ = 1.5 m);
• Bus $S{L}_{max}$ = 6 m ($B{L}_{max}$ = 3 m).

In regards to less than 0.5 m $BL$ values, it was appropriate to refer to the value of $BH$ determined with $S{L}_{max}$ = 1 m because they were representative of the real travel conditions for the bike category. On the other hand, $BL$ values between 0.5 and 1.5 m were properly obtained considering $S{L}_{max}$ = 3 m in order to identify the wavelengths that affected the riding conditions of automobiles. For buses, $BL$ values between 1.5 and 3.0 m were considered. The envelope of all the $BH$ curves enabled us to synthesize the geometric analysis of the examined pavement profile. The final $BH$ curves were composed of branches from the $BH$ curves obtained for each modelled vehicle ($E_{max}$ and $E_{avg}$ from $B{H}_{max}$ and $B{H}_{avg}$ curves, respectively), having regard to the discussed BL ranges. Figure 9 represents $E_{avg}$ (green solid curve) derived from bike $B{H}_{avg}$ (dotted blue curve), automobile $B{H}_{avg}$ (dotted black curve) and bus $B{H}_{avg}$ (dotted red curve), and $E_{max}$ (yellow solid curve) derived from bike $B{H}_{max}$ (dashed blue curve), automobile $B{H}_{max}$ (dashed black curve) and bus $B{H}_{max}$ (dashed red curve).

According to Figure 9, the envelope curves are not continuous for $BL$ values equal to 0.5 m and 1.5 m corresponding to the vehicle category change that occurs (bike/automobile and automobile/bus).

3. Conclusions

In recent years, the interest in stone pavements has grown because of their historic heritage and physical properties, as well as environmental issues. Due to their resistance, stone pavers generally require limited and timely interventions to restore defects that cause unsafe or uncomfortable journeys. However, no standard monitoring methods or criteria for evaluating stone pavements are available in the scientific literature to assess their functional performance and implement a pavement management system. The current lack of an effective assessing method for this type of pavement has consequences in terms of road safety and cost management.

Several methods allow for measuring and assessing irregularities of asphalt and concrete pavements; these methods provide different approaches based on the geometric profiles detected. In particular, in this paper, four methods have been investigated to describe pavement unevenness: the International Roughness Index, the surface profile classification according to ISO 8608, the vehicle mechanical model to evaluate the comfort index according to ISO 2631-1 (a_wz), and SASP, which is able to evaluate the effect of localized irregularities. This paper provides a critical analysis of both traditional and innovative methods of monitoring stone pavements by measuring and assessing their irregularities in order to then implement a pavement management system. IRI is defined for continuous pavements and does not provide thresholds valid for modular pavements; the classification according to ISO 8608 allow comparison between pavements but it does not support the maintenance decisions of road managers; the analysis according to ISO 2631-1 uses vehicle mechanical models (in this study, bikes, automobiles, and buses) to investigate riding conditions, but it does not allow identification of single defects to be restored.

This study focuses on the definition of SASP to overcome these aforementioned limitations. SASP allows for the identification of the most severe irregularities (for example with respect to the running conditions of the vehicles) and also their position. It will be implemented in the second part of the paper, and threshold curves will be proposed for different vehicles and operating conditions. In particular, several urban profiles have been investigated to evaluate the effects of localized irregularities taking into account different urban users (bike, automobile, and bus). The results, presented in part II, allowed the authors to propose four classes to describe geometric and comfort stone pavement conditions that consider also their effects on vulnerable road users.