**1. Introduction**

Airports and highways have a huge network of rigid pavement infrastructures. To maintain these infrastructures, their structural assessment is a routine activity by infrastructure owners. The study of the structural condition of the rigid jointed pavement is traditionally achieved by the use of Falling weight deflectometer(FWD). Given that there have been few developments on the efficiency of structural assessment, new technology must be developed or applied to benefit the network owners. This will help the owners to do maintenance cost-effectively by reducing the time required to do such assessments and thus allowing the infrastructure to be more available to its users. In this study, the potential of a novel application of Rolling wheel Deflectometer(RWD) technology for rigid pavement is developed. The aim is to develop the analysis which will help to understand the measurements.

Continuous pavement structural measurement devices operate at the traffic speed and require no traffic disruption. Devices such as the RWD has been developed with the potential to reveal the structural information on a routine basis. For example a continuous, contact-based sensor device such as the rolling dynamic deflectometer(RDD) has been shown to be effective for jointed pavement rehabilitation [1]. A project based research study by using RDD deflection data interpretation and its application on evaluating existing concrete pavement has been carried out in [2]. The RDD is a truck-mounted system that dynamically loads the pavement and simultaneously monitors the pavement response while continuously moving at about 1.6 km/hr. The major components include an electro-hydraulic dynamic loading system, a force measurement system, an array of 4 rolling sensors that are located underneath the RDD, and a distance measurement system as described in [3]. RDD measures at the center of the vehicle at a position 0.6 m far from the tire loads,capturing the influence from both wheels. The positioning of sensors is very far and the magnitude of signals decreases with distance and it is challenging to be capture signals accurately. In a project-level study, measurement speed was limited by pavement contact nature of the sensors. Recent developments indicate a new generation of devices with non-contact continuous measurement technology which has increasingly started to show its capability [4,5]. Recently, Dynatest developed a new RWD device called the Rapid Pavement Tester (Raptor) for rapid evaluation of pavements. The research presented in this study shows its potential for evaluation of rigid pavements, where it has already been demonstrated for flexible pavements [6]. Given that it is a very new technology and there may be developments required on the technology itself, but it is evident that this can be applied to rigid pavements. In contrast to the RDD, this device is faster, the senors are closer to the loads and numbers of sensors are significantly more in RWD.

It is a challenge to understand and draw conclusions from these devices since the measurements themselves are taken in different locations relative to the load. This happens due to the device-specific geometries which are different compared to FWD. In studies with a RDD, the conclusions on joint conditions and load transfer efficiency were formed by setting threshold values for the project and thus they cannot be applied to a different maintenance project [7]. RDD uses rolling geophones with limited sensor capabilities demonstrating the potential of continuous deflection measurements for jointed pavements. However, there have been no significant studies done to investigate the response of jointed rigid pavement with an array of non-contact sensors with a continuously moving load. Therefore, deflections obtained from a continuous deflection measurement device cannot be used to conclude about structural conditions of joints until more investigations are done.

The first step to a back-calculation is a mechanistic model which can accurately and rapidly predict the response under a moving wheel. Historically, FWD measurements use different mechanistic models to do back-calculations, but these same models cannot be applied to the RWD in case of the jointed pavements. Several FEM models of rigid pavements have been able to predict deflections under all types of environmental and design loads [8]. To include and characterize the load transfer mechanisms, advanced modelling strategies e.g., Enrichment of Finite elements to include discontinuities has been shown to work [9,10]. Such advanced modelling strategies are challenging for a practical back-calculation method.

The study aims to develop and test the analysis for the use of RWD technology. A three dimensional analytical model of the rigid pavement joint is developed. This model is then used to analyze the experimental data. The experimental data is gathered by performing a reference beam experiment. This experiment aims to show the potential of the use of non-destructive testing across a rigid pavement discontinuity.

### **2. Methods**

To assess the rigid pavements, different methods exist based on type of assessment and maintenance activity under consideration. In the structural assessment category, the strength of slabs and the subgrade along with joints condition are checked for their performance. Most failures arise from the failure of joints and corners as they fail to perform as a rigid pavement structure, as it ages over its use. From structural assessment point of view, jointed pavements are tested for load transfer capability by measuring deflection ratios at the edges. Where as from modelling point of view, this ability to transfer load can be defined based on definitions based on stresses, strains and deflections induced under the loads. A vertical deflection-based definition of load transfer is classical and most used, as it is easy to calculate.

## *2.1. Mathematical Model*

Various modelling methods exists that can provide deflections. Methods such as Empirical-mechanistic methods and Mechanistic methods have been used in pavement engineering for decades. Such equations were developed from experience and modelling. These models were used for back-calculation. Though accurate but these equations provide deflection values for fewer predefined positions of the load. Intending to predict vertical deflections at positions surrounding the load and the joint, based on a structural formulation as close as possible to a physical problem, a mechanical model simulating the jointed rigid pavement is set up.

From a purely modelling perspective, numerical and analytical methods are two categories where the numerical methods are more popular and handle any geometry. Analytical methods can only handle simple and idealized geometry as for complicated geometries, an analytical solution might not exist. A model of two jointed semi-infinite slabs resting on a Pasternak foundation is a geometrically simplified model to handle a single joint. Therefore, a 3D semi-analytical formulation that already existed in the literature has been implemented here. This formulation has been solved in this study. Thus, a method to predict the deflections all over the slab irrespective of the load's position is developed.

### 2.1.1. Formulation

To predict the response of a jointed concrete pavement, a static 3D semi-analytical solution is developed. This forward model aims to be a sufficiently good approximation to real rigid pavements while being fast to calculate, e.g., in comparison with more numerically intensive approaches like finite element modelling. An efficient forward model is a foundation for the development of efficient back-calculation methods. Figure 1 presents a schematic of the model. The origin of the coordinate system is at the position of the load and *x* is the driving direction, *y* is the transverse direction and *z* is the vertical direction. The formulation is based on two semi-infinite jointed concrete slabs resting on a Pasternak foundation with subgrade reaction *k* and independent spring modulus *G*. The load transfer efficiency *δ* in Equation (1) is the ratio of the vertical deflection on the unloaded (*wUL*) and loaded (*wL*) slab right next to the joint at *x* = *c*.

**Figure 1.** Coordinate system for the problem formulation.

$$
\delta = \frac{w\_{IL}}{w\_L} \tag{1}
$$

This formulation has a vertical load of pressure *p* with a rectangular contact area 2*a* by 2*b* at a distance *c* from the joint. The slab is of thickness *h* with Young's Modulus *E*. The model is derived from the equilibrium equation of the system. The boundary conditions imply zero vertical displacements at infinity in both *x* and *y* directions. The load pressure is assumed uniform and shear loads are not included in the model. The solution method is presented in [11], but the numerically challenging implementation is done in this study.

The model follows linear elasticity and a small strain framework. Static loading is assumed and thermal effects are ignored. The load transfer in the *y*-direction is assumed constant here.

### 2.1.2. Two Semi-Infinite Slabs Resting on a Pasternak Foundation

The equilibrium equation in terms of the vertical deflection *w* can be written as in Equation (2), where *D* is the flexural rigidity of the slab in Equation (3). The relation between the radius of relative stiffness *l* and the flexural rigidity is expressed in Equation (4).

$$
\left(\frac{\partial^2}{\partial \mathbf{x}^2} + \frac{\partial^2}{\partial y^2}\right) \left(\frac{\partial^2 w}{\partial \mathbf{x}^2} + \frac{\partial^2 w}{\partial y^2}\right) - \frac{G}{D} \left(\frac{\partial^2 w}{\partial \mathbf{x}^2} + \frac{\partial^2 w}{\partial y^2}\right) + \frac{kw}{D} = \frac{p}{D} \tag{2}
$$

$$D = \frac{Eh^3}{12\left(1 - v^2\right)}\tag{3}$$

$$\frac{G}{D} = \frac{2g}{l^2} \text{ and } \frac{k}{D} = \frac{1}{l^4} \tag{4}$$

To solve Equation (2), both the load and deflection are expressed as double Fourier integrals, which is shown in Equation (6) for deflection. The ratio between the deflection on both sides of the joint is given by the definition of the load transfer efficiency in Equation (1). On each side of the joint, the solution is expressed as a linear combination of a particular solution *w* to the inhomogeneous equation as in Equation (5).

$$w\_L = \left(w + A(s)w\_a + B(s)w\_b\right); \quad w\_{IL} = w + C(s)w\_c + D(s)w\_d\tag{5}$$

$$w = \frac{p}{\pi k} \frac{1}{\sqrt{1 - g^2}} \int\_0^\infty \frac{\cos(sy/l) \sin(sb/l)}{s(s^4 + 2gs^2 + 1)}$$

$$\left\{ e^{-(x-a)a/l} \left[ \sqrt{1 - g^2} \cos[(x-a)\beta/l] + \left(s^2 + g\right) \sin[(x-a)\beta/l] \right] \right. \tag{6}$$
 $1 - e^{-(x+a)a/l} \left[ \sqrt{1 - g^2} \cos[(x+a)\beta/l] + \left(s^2 + g\right) \sin[(x+a)\beta/l] \right] \right\} ds$ 

And the two solutions to the homogeneous equation *w<sup>a</sup>* and *w<sup>b</sup>* for the loaded slab and *w<sup>c</sup>* and *wd* for unloaded slab are shown in Equation (7).

$$\begin{aligned} w\_{a} &= \frac{p}{\pi k} \frac{1}{\sqrt{1-\mathcal{g}^{2}}} \int\_{0}^{\infty} [A(s)\cos(\beta \mathbf{x}\,/l)] e^{a\mathbf{x}\cdot/l} \frac{\cos(sy/l)\sin(sb/l)}{s} ds \\ w\_{b} &= \frac{p}{\pi k} \frac{1}{\sqrt{1-\mathcal{g}^{2}}} \int\_{0}^{\infty} [B(s)\sin(\beta \mathbf{x}\,/l)] e^{a\mathbf{x}\cdot/l} \frac{\cos(sy/l)\sin(sb/l)}{s} ds \\ w\_{c} &= \frac{p}{\pi k} \frac{1}{\sqrt{1-\mathcal{g}^{2}}} \int\_{0}^{\infty} [\mathbb{C}(s)\cos(\beta \mathbf{x}\,/l)] e^{-a\mathbf{x}\cdot/l} \frac{\cos(sy/l)\sin(sb/l)}{s} ds \\ w\_{d} &= \frac{p}{\pi k} \frac{1}{\sqrt{1-\mathcal{g}^{2}}} \int\_{0}^{\infty} [D(s)\sin(\beta \mathbf{x}\,/l)] e^{-a\mathbf{x}\cdot/l} \frac{\cos(sy/l)\sin(sb/l)}{s} ds \end{aligned} \tag{7}$$

Where two new auxiliary parameters have been introduced in Equation (8).

$$\begin{aligned} a^2 &= \frac{1}{2} \left[ \sqrt{\left(s^2 + g\right)^2 + 1 - g^2} + \left(s^2 + g\right) \right] \\\ \beta^2 &= \frac{1}{2} \left[ \sqrt{\left(s^2 + g\right)^2 + 1 - g^2} - \left(s^2 + g\right) \right] \end{aligned} \tag{8}$$

By the fourth-order partial differential equation, four conditions are required to couple the solution across the discontinuity at the joint. By relating deflections, forces and moments, Equations (9)–(12), can be written at the joint at *x* = *c* in Figure 1. Equation (9) is relating the deflections between unloaded

and loaded slabs. Equations (10) and (11) are the cancellation of the moment at the edge of the loaded slab and unloaded slab respectively. Equation (12) is equality of the shear forces at the the joint.

$$\mathcal{S}\left(w + A(s)w\_a + B(s)w\_b\right) = w + \mathcal{C}(s)w\_c + D(s)w\_d \tag{9}$$

$$\left(\frac{\partial^2}{\partial x^2} + v \frac{\partial^2}{\partial y^2}\right) \left(w + A(s)w\_a + B(s)w\_b\right) = 0\tag{10}$$

$$\left(\frac{\partial^2}{\partial x^2} + v \frac{\partial^2}{\partial y^2}\right)(w + \mathcal{C}(s)w\_c + D(s)w\_d) = 0 \tag{11}$$

$$\begin{aligned} & \left( \frac{\partial^3}{\partial x^3} + (2 - v) \frac{\partial^3}{\partial x \partial y^2} - \frac{2g}{l^2} \right) (w + A(s)w\_d + B(s)w\_b) \\ &= \left( \frac{\partial^3}{\partial x^3} + (2 - v) \frac{\partial^3}{\partial x \partial y^2} - \frac{2g}{l^2} \right) (w + C(s)w\_c + D(s)w\_d) \end{aligned} \tag{12}$$

### *2.2. Model Validation*

(a)

The response from the semi-analytical model is compared to the result of a FEM solution from EverFE. EverFE is a free FEM tool that models the response of jointed slab systems due to various load configurations [12]. EverFE considers load transfer [13] and effects such as aggregate interlock and dowel properties [8]. Figure 2 shows a comparison of the modelled responses from the semi-analytical model and EverFE for two different slab moduli. Note how the semi-analytical model is very close to the FEM solution except immediately under the load (which is due to discretization error in the FEM solution). The deflection in Figure 2 is in the plane passing through the centre of the load. The comparison in Figure 2 validates the 3D semi-analytical model.

**Figure 2.** This comparison shows analytical solution compared to FEM solution for Young's Modulus: (**a**) *E* = 28,000 MPa (**b**) *E* = 30,000 MPa.

With the mechanical model ready, it is aimed that accurately measured deflections with a rolling wheel can be used to back-calculate load transfer efficiency. That will then indicate the condition of joints and predict the state of rigid pavements. With the trust in being able to predict vertical deflections, an experiment needs to be set up to test the viability of this idea. The simplest way to measure vertical deflections by using a rolling wheel is by using a reference beam set up across the joint.

### *2.3. A Reference Experiment*

The whole measurement system comprises of the lasers, odometer(encoder), a reference beam, an ethernet switch, a power supply, a moving load and a recording application (Figure 3).

To measure the edge deflection response of jointed slabs under the influence of a moving wheel load, a beam mounted with distance lasers is setup. This beam is placed across the joints in the centre of two jointed slabs. The beam is mounted with 7 distances lasers such that the middle laser is placed across the joint formed by both the slabs. It is a symmetric setup across the joints. The beam is 6 m in length and mounted on two supports, one at each end. The beam has a rectangular cross-section with channels for mounting brackets and supports.

The load consists of a moving wheel load carrying 5 tons. The loading wheel is a part of a trailer which has an independent suspension in the rear axle. The trailer is attached to a truck and driven at a slow and controlled speed for the experiment. There are other axles and their influence is observable but ignored for this study. Initially, the trailer is moved to a far location from the joint considered for measurement. During the experiment, the load is moved parallel to the beam. The aim is to get as close as possible to the beam in the longitudinal direction to get a good signal.

An accurate odometer device, which measures the moving position of the wheel load at all times is used. This device is known as 'encoder' and is attached to the moving wheel. The signal from this sensor comes from a cable attached to it. On the other end, it is connected to the whole measurement setup via a cable. The output from this device is saved via the recording application and is linked in time to all the lasers.

**Figure 3.** This is a schematic of the reference beam experiment.

### 2.3.1. Experiment at the Vaerlose Airbase

To begin with, the site with joints is selected after visual inspection. After inspection of selected slabs forming the joint, the setup is placed across it symmetrically. The distance of the beam is measured from parallel edges of the slabs to make sure it is in the centre. Then, coloured markings on the slabs at incremental distances parallel to the laser signal are done to help the person driving the truck for visual guidance.

After all the verifications, the measurement starts. The truck is at least 4 slabs distance away from the joint in consideration with its front axle on the edge of the 4th slab. With a slow and constant driving speed of 10 km/h, the truck is carefully driven to avoid vibrations and maintain parallel distance to the laser signal, when it arrives closer to the beam. It is shown in Figure 4. After the loaded wheel axle of the trailer carrying the 5 tons has passed in front of beam and slabs onto the other site, measurement is stopped, and the collected data is saved. This sequence of process is repeated for several joints at the site.

**Figure 4.** Lasers mounted on the beam during the experiment.
