Essential Working Features of Asphalt Airport Pavement Revealed by Structural State-of-Stress Theory

Highlights

What are the main findings?

• Proposes the method for deriving the state variables from the tested strain and deformation data;
• Proposes the method for expressing the state of stress of the asphalt airport pavement using state variables;

What is the implication of the main finding?

• Proposes the method for expressing the state of stress of the asphalt airport pavement using state variables;
• What is the implication of the main finding?

Abstract

The National Airport Pavement Test Facility (NAPTF) in USA obtained the strain and deformation data of the asphalt airport pavement numbered as Track 3 under the wheel load traveling in the north area of Construction Cycle 7 (CC7). But, the classic theories and methods still could not find out the definite and essential working characteristics, such as the starting point of the asphalt pavement’s failure process and the ending point of the normal working process. This study reveals the essential working characteristics of the asphalt airport pavement by modeling the tested strain and deformation data based on structural state-of-stress theory. Firstly, the tested data are modeled as state variables to build the state-of-stress mode and the parameter characterizing the mode. Then, the slope increment criterion detects the mutation points in the evolution curve of the characteristic parameter with a wheel load traveling number increase. Correspondingly, the mutation features are verified by investigating the evolution curves of the state-of-stress modes. The mutation points define the failure starting point and the elastoplastic branch (EPB) point in the working process of the asphalt airport pavements. The strain state-of-stress mode (Δε_t) and characteristic parameters ($E_j$ and ${\Phi }_j$) presented an obvious mutation feature around the EPB point; in addition, the deformation state-of-stress mode (ΔD_t) showed that the total deformation of the pavement changed evidently before and after the failure starting point, and the characteristic parameters ($E_j$ and ${\Phi }_j$) also presented an obvious mutation feature around the failure starting point, so both characteristic points could address the classic issues in the load-bearing capacity of asphalt airport pavements. Furthermore, the EPB point could be directly taken as the design point, and the failure starting point could be taken as the limit-bearing traffic capacity. Hence, this study could open a new way to address the classic issues in the load-bearing capacity of asphalt airport pavements and provide a new reference for their safe estimation and rational design.

1. Introduction

Asphalt pavements met the ever-increasing requirements of airport pavements due to their advantages of flexible performance [1], wear resistance [2], easy construction and durability [3], so they were widely used in the engineering field of airport pavements [4]. But, the asphalt airport pavement in the service period would yield rutting [5], cracking [6] and subsidence [7], which could threaten the service safety and even lose serviceability before the design life. So, asphalt airport pavements had to pay a great cost in maintenance or reconstruction [8], which needed an accurate estimation of the pavement’s working state. However, it is difficult for the analytical theories and methods to accurately judge or calculate the working performance of asphalt airport pavements from the measured response data due to the effects of various complex factors, like compacted degree [9], thickness, temperature [10,11,12] and so on.

Researchers have been trying to establish the relationship between the proposed parameters and the mechanical behavior of asphalt pavements by developing analytical models. Al-Suleiman [13] and Bailey [14] estimated the damage of asphalt pavement under tire–pavement contact loads using the international roughness index. Garber [15] and Yin [16] combined the analytical models involving environmental effects and material properties to evaluate the damage evolution of asphalt pavements. Wen [17] developed a damage-based fatigue model to calculate the damage behavior of pavements. Sharma [18] presented an approach of a probabilistic fatigue curve for asphalt concrete mixtures to predict the fatigue life of asphalt pavement. Brill [19,20] proposed a cumulative damage index to study the failure process of asphalt pavements. Chen [21] predicted the damage behavior of asphalt pavements subjected to the tire contact loading process using the international roughness index. Guo [22] and Wang [23] also developed a fatigue index referring to the viscoelastic parameter to judge the fatigue cracking degree of the pavement. These parametric research studies could show the damage evolution of asphalt pavements to an extent, but they commonly focused on the ultimate states of asphalt pavements to analyze the failure mechanism or to predict damages. Since the pavements’ ultimate state had the attribute of uncertainty, it surely leads to a difficulty in accurately predicting the failure behavior through parametric analysis.

The layers of an asphalt pavement have different thicknesses and material properties (strengths and moduli), together with inherent variation. So, the mechanical responses of different depths inside the asphalt pavement under repeated loads, such as the distributions of stress and strain at the bottoms of layers, are quite variable, particularly in the failure process of the asphalt pavement. Garg [24] observed that the fatigue destruction of the asphalt concrete layer was related to traffic passes and the maximum tensile strain. Tarefder [25] studied the relationship between permanent deformation and base strength based on the stress distribution in the base layer. Zhuang [26] analyzed the structural dynamic response of the asphalt pavements under repeated loads and found that the tensile strain on the bottom of the semi-rigid base layer increased with the gradual compacting of the asphalt layer. However, the tensile strain decreased rapidly when the repeated loads reached a certain number. Huang [27] concluded that the increment in the tensile stress on the bottom of the semi-rigid base would accelerate the development of cracks and reduce the service level of the asphalt pavements. Ren [28] found that both transverse and longitudinal strains were compressive–tensile alternating [29] at the bottom of the structural layer, which led to an accumulation of fatigue damage. Ungureanu [30] studied the mechanical response of a recycled asphalt road using accelerate testing, implying that the strain and deformation measurements were constant at the bottom of the asphalt layer and that the strain started to increase before the first micro-cracks appeared. Ma [31] investigated the distress of the top-down creaking of the asphalt pavement using accelerated pavement testing (APT), indicating that the combination of shear and tensile effects contributed to the appearance of top-down cracking. Liu [32] proposed a method of numerical simulation and guided the rutting prediction and maintenance decision of semi-rigid asphalt pavement. Jiang [33] studied the deformation of an inverted pavement structure using the accelerated pavement test (APT), comparing with the conventional flexible pavement, and concluded that the inverted pavement structure contributed to the rutting performance and reducing the reflective cracks with the pass increasing. It could be seen that the investigation into the pavements above could reflect the exterior working phenomena and present the interior working properties to an extent. But, the exact characteristic points in the working process of the asphalt pavement could not be defined accurately and only estimated empirically and statistically. In addition, the investigation on the experimental data seemingly anticipated some new theory and methods to further mine out the unseen knowledge in the asphalt pavement’s working process.

Generally, the traffic pass number defining the asphalt airport pavement failure was estimated based on its experimental and simulative analysis. For instance, Thompson etc. [34,35,36], Garg [37] and Sarker [38] used the site data of asphalt airport pavements to study the mechanical responses of different layers and associated the damage to the traffic numbers of the pavements; Wang [39] and Peng [40] used numerical simulation and statistical methods to predict the number of the pavement’s failure traffic passes. But, the estimation results were quite inaccurate as the real failure traffic passes were about two to three times as many as the predicted passes. Therefore, it is unknown at present whether a research theory and method exist to enable one to predict or to define the definite number of the pavement’s failure traffic passes or not.

Lately, Zhou [41] established structural state-of-stress theory, based on the recognition that the experimental and simulative response data of structures surely include the working features governed by the natural law from a quantitative change to qualitative change of a system. This resulted in the discovery of two essential and definite characteristic points in structural working processes through modeling the experimental/simulative response data of structures to characterize structural state-of-stress evolutions. So far, many researchers have applied the state-of-stress theory together with the derived modeling methods to various structures under different loading cases, such as concrete airport pavement [42], continuous steel box bridges [43,44], short spiral reinforced concrete columns [45], short stainless steel tubular concrete columns [46], reinforced masonry shear walls [47], parabolic CFST arches [48] and so on. These state-of-stress analyses all verified the existence of the failure starting points and EPB points in structural experimental and simulative strain/displacement data.

In this study, the experimental strain and deformation data of the asphalt airport pavement were modeled to express the state-of-stress mode and the parameter characterizing the mode, called the state-of-stress characteristic pair. The proper state-of-stress characteristic pair could present the mutation points in their evolution curves with a traffic pass increase as the specific embodiment of the natural law from a quantitative change to qualitative change of a system. The mutation points could be detected by the slope increment criterion, which defined the elastoplastic branch (EPB) point in the pavement’s normal working process and the starting point of the pavement’s failure process as well as the progressive failure point in the pavement’s failure process. Hence, this study revealed the essential working features of the asphalt pavement and determined the traffic pass numbers corresponding to the characteristic points, which could lead to the accurate estimation of the load-bearing capacity and safety extent for asphalt pavements.

2. Structural State-of-Stress Theory and Methods

2.1. Brief of Structural State-of-Stress Theory

Structural state-of-stress theory studies the essential and general working behaviors of any structure or its parts under a load case [41]. Structural working behavior is manifested by the structural state-of-stress characteristic pair, the state-of-stress mode [49] and its characteristic parameter [42] composed of the state variables derived by modeling the tested strains and displacements. The evolution curves of the characteristic pair with the load increase will present the mutation features [45,46] (characteristic points) at certain load levels. The mutation points can be detected by the proposed criteria, such as the slope increment criterion [50] introduced below. The mutation points are defined as the EPB point, the failure starting point and the progressive failure point. The EPB point [51,52] implies the end of the structural normal working state, which can be directly taken as the design reference. The failure starting point is the starting point of the structural failure process, which also provides the reference to the structural design and structural estimation of safety. Both characteristic points are the embodiment of the objective law commonly existing in the working processes of various structures under individual loading cases rather than occasional phenomena [41].

2.2. Structural State-of-Stress Analysis Methods

The methods for building the structural state-of-stress characteristic pair are as follows. The structural response data under the jth load value F_j can be modeled as state variables (s) to form the structural state-of-stress characteristic pair (${\mathbf{M}}_j$,${\lambda }_j$). The structural state-of-stress mode ${\mathbf{M}}_j$ can be expressed as a vector or matrix:

$${\mathbf{M}}_j={\left[s_{1j},s_{2j},\dots ,s_{mj}\right]}^{\mathrm{T}}$$

(1)

$${\lambda }_j=\sum _{i=1}^m{s}_{ij}$$

(2)

where $s_{ij}$ denotes the ith state variable, which can be derived from strain and displacement at the jth loading pass; m is the total number of measured points. ${\lambda }_j$ is the parameter characterizing ${\mathbf{M}}_j$, which can be different forms depending on the analytical intentions; for example, ${\lambda }_j$ can be the sum of state variables ($s_{ij}$).

Strains, stresses and displacements have directionality so that they can be complex to form the state-of-stress characteristic pairs. Hence, in structural state-of-stress analysis, the strain and stress (${\epsilon }_{ij}$ and ${\sigma }_{ij}$) at the ith measured point under the jth loading pass are generally transformed into generalized strain energy density (GSED) values as state variables $e_{ij}$:

$$e_{ij}={\int }_0^{{\epsilon }_{ij}}{\sigma }_{ij}\mathrm{d}\epsilon$$

(3)

The state-of-stress mode ${\mathbf{S}}_j$ and its characteristic parameter $E_j$ based on the state variables can be built as

$${\mathbf{S}}_j={[e_{1j},e_{2j},\dots ,e_{mj}]}^{\mathrm{T}},E_j={\sum }_{i=1}^m{e}_{ij}$$

(4)

The method for detecting characteristic points in the state-of-stress evolution curve is as follows. The evolution curves of the structural state-of-stress characteristic pair can present the characteristic points, such as the EPB point and the failure starting point. In this study, the slope increment criterion is used to judge the turn points in the evolution curve of the structural state-of-stress characteristic parameter with the cyclic loading number increase:

$$R_j={\displaystyle \frac{\left(E_{j-}{E}_1\right)\mathrm{m}\mathrm{a}\mathrm{x}\left(E_{ij}\right)}{(j-1)\mathrm{m}\mathrm{a}\mathrm{x}\left(E_{i1}\right)}}\ge \delta$$

(5)

where $R_j$ is the slope increment ratio at the jth traffic pass; δ is the threshold based on the empirical and statistical analysis, such as δ = 1.

3. Experiment of Asphalt Airport Pavement

3.1. Experimental Asphalt Airport Pavement

The tested asphalt airport pavement was a perpetual pavement at the FAA William J. Hughes Technical Centre in Atlantic City International Airport, USA [53]. The pavement was 12,192 mm in the west–east direction and 10,058 mm in the north–south direction. As shown in Figure 1, the pavement consisted of three layers: a P-401 asphalt surface layer with a thickness of 203 mm, a P-154 base layer with a thickness of 1041 mm and a low-strength subgrade with a thickness of 2413 mm. The California bearing ratio (CBR) was 5.5.

3.2. Loading Scheme

A loading device with three axles and six wheels was adopted to apply cyclic mobile loading, as shown in Figure 2a. The wheelbase was 1448 mm and the dual-wheel spacing was 1372 mm, as shown in Figure 2b. For each wheel, the load was 244 kN and the tire pressure was 1758 kPa. The device moved at a speed of 4 km/h.

The loading device moved along 9 tracks in the east–west direction and the arrangement of the tracks is shown in Figure 3. The distance between the center lines of the tracks and the southern edge of the tested pavements ranged from 3532 mm to 5612 mm. The wheels on one side directly moved along the pavement centerline when in track (−3) and track (3).

In each traffic pattern, the loading device moved 66 times in individual tracks. To simulate the standard deviation specified by the Federal Aviation Administration, the number of passes from the track (−4) to track (4) was set as 4, 6, 8, 10, 10, 10, 8, 6, and 4 times, respectively.

3.3. Arrangement of Measured Points

To measure the mechanical responses under loading, dynamic sensors, including six asphalt strain gauges, one pressure cell and six multi-depth deflectometers, were arranged, as shown in Figure 4. Three longitudinal gauges were along the east–west direction (LS-0~LS-2) and three transverse gauges were along the north–south direction (TS-0~TS-2) at the bottom of the asphalt layer. Gauge TS-1 was located at the centerline of the pavement and gauges TS-0 and TS-2 were 686 mm at offset. The longitudinal strain gauges aligned with TS-0 and the interval was 610 mm. The distance between the transverse strain gauges and the west boundary of the pavement was 3658 mm.

The pressure cell PC-4 was located at the center of the tested pavement at the bottom of the base layer. As the loading surface of the sensor was very small compared to the tested pavement, it was possible to neglect the size of the sensor and regard the measured pressure as stress distribution. The multi-depth deflectometers (MDDs) were arranged at six depths at the different layers of the pavement, The shallowest MDD-A was installed at the bottom of the P-401 asphalt layer and MDD-B was located on the top of the P-154 base layer, and the distance between components A and B was controlled within the range of 13 to 25 mm; MDD-D and MDD-E were located at the bottom of the P-154 layer and the top of the base layer, respectively, and the two instrumentations were also close to the critical surface between the base layer and the P-154 asphalt layer; MDD-C was installed in the middle of the base layer; the component MDD-F was the deepest in the subgrade layer.

When the mobile loading moved in track (3) and track (−3), the wheels passed directly above the TS-1 and PC-4 (Figure 3), and the measured data could accurately reflect the synchronous change in the strain at the bottom of the asphalt layer and the stress at the bottom of the base layer inside the pavement. Actually, only 6 passes ran over the location of the strain gauge TS-1 in a pattern with 66 passes. But, when the moving load traveled in two directions on the same track, the tested data were approximately the same. Therefore, the data for 3 passes from west to east in track (3) were selected to conduct the state-of-stress analysis in this study. All of the selected passes were renumbered from 0 for convenience to plot the state-of-stress evolution curves. Once the characteristic points were determined in the working process of the pavement, their definite pass numbers could be obtained as well.

4. The State-of-Stress Analysis of the Tested Pavement

4.1. Procedure of State-of-Stress Analysis

In this study, the state-of-stress modeling and analysis of the tested data can be summarized as the procedure as shown in Figure 5. Firstly, the mechanical response data measured in the test of the pavement are converted into state variables; then, the state variables are used to build the state-of-stress mode and corresponding characteristic parameters; next, the evolution curve of the characteristic parameter is plotted and the criterion is used to detect the mutation points in the curve; further, the evolution curve of the structural state-of-stress mode is investigated to verify the mutation features at the detected points; finally, the mutation points, called characteristic points, are defined as the failure starting point, EPB point and progressive point of the pavement, respectively.

4.2. Evolution Feature of Characteristic Parameter ${\omega }_j$

In the order of loading passes (j), the state variables (${\omega }_j$) are derived using the strains (${\epsilon }_{ij}$) from gauge TS-1 and the stresses (${\sigma }_{ij}$) from pressure cell PC-4 at the same location in the tested pavement:

$$e_{ij}={\int }_{{\epsilon }_{i-1j}}^{{\epsilon }_{ij}}{\sigma }_{ij} d\epsilon ={\displaystyle \frac{1}{2}}({\sigma }_{ij}-{\sigma }_{i-1j})({\epsilon }_{ij}-{\epsilon }_{i-1j})={\displaystyle \frac{1}{2}}∆ {\sigma }_{ij}\times ∆ {\epsilon }_{ij}$$

(6)

where i = 1, 2, …, n is the number of strains or stresses measured in a load pass. Thus, the state-of-stress characteristic parameter ${\omega }_j$ can be set as

$${\omega }_j={\sum }_{i=1}^n{e}_{ij}$$

(7)

Now, the ${\omega }_j-j$ curve is investigated to see the change in ${\omega }_j$ with the increase in loading passes. Figure 6 plots the ${\omega }_j-j$ curve and its probability distribution curve. It can be seen that the ${\omega }_j$ values present the obvious variation but have the normal distribution feature. Meanwhile, the ${\omega }_j$ values significantly fluctuate near the starting point (A) and the ending point (B) of the normal distribution peak domain, whose traffic pattern numbers are j = 249 and j = 366, respectively. Points A and B could be the characteristic points in the pavement’s working process. Such characteristic points are referred to in the existing analysis and design of pavements.

4.3. Evolution of Characteristic Parameter $E_j$

For the state variables derived in Equation (7), ${\omega }_j$ (j = 1, 2, …, n), the state-of-stress characteristic parameter $E_j$ can be set as

$$E_j={\sum }_{s=1}^j{\omega }_s$$

(8)

Figure 7 plots the $E_j-j$ curve, and the slope increment criterion detects the mutation points A, B and C. According to structural state-of-stress theory, point A is the EPB point, the corresponding traffic pattern number j = 249 and the real traffic pass number is 5762; point B is the failure starting point, the corresponding traffic pattern number j = 367 and the real traffic pass number is 10,004; point C is the progressive failure point, the corresponding traffic pattern number j = 422 and the real traffic pass number is 11,548. Compared to the characteristic points A and B presented in Section 4.1, the characteristic points A and B in the structural state-of-stress analysis have obvious physical meanings. In addition, the traffic pattern numbers of points A and B are basically the same in the statistical analysis and the structural state-of-stress analysis. This implies that some results achieved in traditional statistical or empirical analysis are quite close to the definite characteristic points governed by the physical law; but, since the past structural analysis judged and defined structural load-bearing capacities based on structural ultimate/peak states with considerable uncertainty, this leads to difficulties in judging whether the definite characteristic points existed in structural working process.

4.4. Evolution Feature of Characteristic Parameter ${\Phi }_j$

The asphalt strain gauges measured many strain values inside the asphalt pavement in every traffic pass. In one traffic process, the measured strain value at a certain point changed with the location of the wheel load. When the wheel load approached the area where the strain gauge was located, the measured strain values gradually increased, and the maximum strains were measured when the wheel loads were directly above the strain gauge. When the wheel load kept away from the area where the strain gauge was located, the measured strain values gradually decreased to zero. It is obvious that the non-maximum strain value is measured by the strain generated by the interaction between the wheel load and the pavement through the surrounding pavement material to the strain gauge; therefore, the maximum strain values could better reflect the state-of-stress evolution features of the pavement. The maximum strain value measured at gauge TS-1 in the jth cyclic loading process can be used as the state variable (${\epsilon }_{j,\mathrm{m}\mathrm{a}\mathrm{x}}$), and the state-of-stress characteristic parameter can be set as ${\Phi }_j$:

$${\Phi }_j={\sum }_{h=1}^j{\epsilon }_{h,\mathrm{m}\mathrm{a}\mathrm{x}}$$

(9)

Figure 8 plots the ${\Phi }_j-j$ curve, and characteristic points (A and B) are detected by the slope increment criterion. It can be observed that the curve presents a turning feature at point A and then evidently rises. At point B, the ${\Phi }_j-j$ curve also presents the turning feature evidently. So, the curves of characteristic parameters ${\Phi }_j$ and $E_j$ have consistent state-of-stress evolution features.

4.5. Evolution Feature of State-of-Stress Mode Δε around Point A

According to structural state-of-stress theory, the internal or external working behavior characteristics of the structure can be presented by the state-of-stress mode. The mechanical response values, such as strains and displacements, are the most direct expression of the structural stressing state and suitable for constituting the state-of-stress mode and characteristic parameter. Before point EPB, the pavement mainly presents elastic deformation. After point EPB, the plastic deformation starts gradual accumulation. Hence, the state-of-stress evolution of the pavement will present a transition from elastic behavior to plastic behavior around the EPB point according to the natural law from a quantitative change to qualitative change of a system. For the state-of-stress mode Δε built using the state variables derived by the tested strains, three dates are selected within a week before and after the date presenting point A. Here, the state variables are set as the difference between the strains in the individual selected traffic passes (1, 2, 3, k).

Table 1. The state variables derived by the strains from gauge TS-1.

	Date Presenting Point A (13 January 2015)				Three Dates before and after Point A (Before: 30 December 2014 → 5 January 2015; After: 15 January 2015 → 22 January 2015)
No. of pass (1, 2, 3, k)	1	2	3	Last k	1	2	3	Last k
Strains (l = 1, 2, …, n)	εl,1	εl,2	εl,3	εl,k	ε′l,1	ε′l,2	ε′l,3	ε′l,k
State variables (Δεt)					ε′l,1 − εl,1	ε′l,2 − εl,2	ε′l,3 − εl,3	ε′l,k − εl,k

shows the method used for the state variables, and the state-of-stress mode Δε_t is composed as

$$\Delta \mathsf{\epsilon }={\left(\begin{array}{l}{\epsilon }_{1,1}-{\epsilon }_{1,1}^{,} {\epsilon }_{2,1}-{\epsilon }_{2,1}^{,} {\epsilon }_{3,1}-{\epsilon }_{3,1}^{,} \cdots {\epsilon }_{n,1}-{\epsilon }_{n,1}^{,}\\ {\epsilon }_{1,2}-{\epsilon }_{1,2}^{,} {\epsilon }_{2,2}-{\epsilon }_{2,2}^{,} {\epsilon }_{3,2}-{\epsilon }_{3,2}^{,} \cdots {\epsilon }_{n,2}-{\epsilon }_{n,2}^{,}\\ {\epsilon }_{1,3}-{\epsilon }_{1,3}^{,} {\epsilon }_{2,3}-{\epsilon }_{2,3}^{,} {\epsilon }_{3,3}-{\epsilon }_{3,1}^{,} \cdots {\epsilon }_{n,1}-{\epsilon }_{n,3}^{,}\\ {\epsilon }_{1,k}-{\epsilon }_{1,k}^{,} {\epsilon }_{2,k}-{\epsilon }_{2,k}^{,} {\epsilon }_{3,k}-{\epsilon }_{3,k}^{,} \cdots {\epsilon }_{n,k}-{\epsilon }_{n,k}^{,}\end{array}\right)}_t$$

(10)

where t is one of the selected dates (30 December 2014; 5 January 2015; 12 January 2015; 14 January 2015; 15 January 2015; 22 January 2015) before and after the date presenting point A (13 January 2015); n is the number of the strains measured in a selected loading pass.

Figure 9 plots the Δε_t-n curves, which show an evident mutation feature around point A. The Δε_t-n curves before point A do not change basically in form, as shown in Figure 9a. However, the Δε_t-n curves after point A presents a significant mutation feature, as shown in Figure 9b. Obviously, the state-of-stress modes before and after point A are different in form, and the quantitative accumulation of strains inside the pavement has reached the critical value.

The analysis above indicates that the evolution of the state-of-stress mode presents an obvious mutation feature around the EPB point (point A). In other words, the evolutions of the state-of-stress characteristic pair (Δε_t, ${\mathsf{\Phi }}_j$) or (Δε_t, $E_j$) start an essential trend change from point A onward as the specific embodiment of the natural law rather than an occasional phenomenon.

4.6. Evolution Features of State-of-Stress Mode $\Delta {\mathsf{D}}_{\mathit{t}}$ around Points A, B and C

The deformation level is an important criterion for judging the working incapacity of the pavement. So, the state-of-stress mode composed of the deformation could characterize the evolution around the characteristic points A, B and C. Applying the state variables obtained in

Table 2. The state variables derived by the deformations from deflectometers MDD-F and MDD-A.

	The Date Presenting Point A (13 January 2015) and the Date before and after Point A (Before: 6 January 2015; After: 20 January 2015)
No. of selected pass	1	2	…	12
MDD-A (l = 1, 2, …, n)	dAl,1	dAi,2,	…	dAl,12
MDD-F (l = 1, 2, …, n)	dFl,1	dFl,2	…	dFl,12
State variables ($\Delta {\mathbf{D}}_{\mathit{t}}$)	dFl,1 − dA,1l	dFl,2 − dAl,2	…	dFl,12 − dAl,12

MDD-X: multi-depth deflectometer-measured deformations in traffic pass.

, the state-of-stress mode $\mathsf{\Delta }{\mathsf{D}}_t$ is built as

$$\Delta {\mathbf{D}}_t={\left[\begin{array}{c}{d}_{\mathrm{F}\mathrm{1,1}}-d_{\mathrm{A}\mathrm{1,1}} d_{\mathrm{F}\mathrm{2,1}}-d_{\mathrm{A}\mathrm{2,1}} \dots d_{\mathrm{F}n,1}-d_{\mathrm{A}n,1}\\ d_{\mathrm{F}\mathrm{1,2}}-d_{\mathrm{A}\mathrm{1,2}} d_{\mathrm{F}\mathrm{2,2}}-d_{\mathrm{A}\mathrm{2,2}} \dots d_{\mathrm{F}n,1}-d_{\mathrm{A}n,2}\\ .\\ .\\ .\\ d_{\mathrm{F}\mathrm{1,12}}-d_{\mathrm{A}\mathrm{1,12}} d_{\mathrm{F}\mathrm{2,12}}-d_{\mathrm{A}\mathrm{2,12}} \dots d_{\mathrm{F}n,12}-d_{\mathrm{A}n,12}\end{array}\right]}_t$$

(11)

where t is the selected date around point A and $d_{\mathrm{F}l,p}$ and $d_{\mathrm{A}l,p}$ are the lth deformation values recorded by deflectometers MDD-F and MDD-A in the selected traffic pass (p = 1, 2, …, 12) within the date t.

Figure 10 plots the $\mathsf{\Delta }{\mathsf{D}}_t-n$ curves around Point A, i.e., the evolution curves of state-of-stress modes $\mathsf{\Delta }{\mathsf{D}}_t$ on the dates 6, 13 and 20 January 2015. The negative sign represents the downward direction of deformations. It can be seen that the shape of $\mathsf{\Delta }{\mathsf{D}}_t$ does not change obviously, which is different from the Δε_t-n curves. This might imply that the strains could present the state-of-stress evolution features more evidently than the deformation data. In addition, this also provides evidence that point A is the elastoplastic branch point because the pavement is still in the normal working state. After point A, the plastic deformation of the pavement will increase more quickly than that before point A.

Figure 11 plots the $\mathsf{\Delta }{\mathsf{D}}_t-n$ curves around point B (19 February 2015), i.e., the state-of-stress modes $\mathsf{\Delta }{\mathsf{D}}_t$ on the dates 18, 19 and 21 February and 2 March 2015. It can be seen that $\mathsf{\Delta }{\mathsf{D}}_t$ keeps basically the same shape before the failure starting point B. But, after point B, $\mathsf{\Delta }{\mathsf{D}}_t$ changes in amplitude with the maximum deformations. In the working process of the asphalt pavement from point EPB to the failure starting point, the strain level entered a new accumulation stage, i.e., the plastic deformation gradually accumulates and becomes dominant. During this period, three maximum deformations gradually tend to be the same and become smaller when the pavement becomes gradually closer to the failure starting point B. Regarding the aspect of structural design, the load-bearing state of the pavement has entered the abnormal stage because the plastic deformation is accumulated in each time traffic load process.

Figure 12 shows the evolution of $\mathsf{\Delta }{\mathsf{D}}_t$ at the progressive failure point C (2 March 2015). It can be seen that the shapes of $\mathsf{\Delta }{\mathsf{D}}_t$ enter instable failure states from 2 March 2015. Compared with the deformation evolution feature around the failure starting point, three deformation peaks have disappeared around the progressive failure point, and the deformation of the original peak location develops toward the upward direction. Also, the deformation is less than that at the failure starting point, implying that the pavement loses its service capacity as the broken pavement releases its plastic deformation.

5. Discussion

The state-of-stress modes and their characteristic parameters proposed in this study present the EPB point, the starting point and the progressive point of the tested pavement, which were unseen in the previous research of pavements. This indicates that the response data of the pavement at the limited key measured points certainly include the state-of-stress evolution law as long as the proper methods are proposed to model the state of stress using the tested data. The reason could be that characteristic points A, B and C are the reflection of the objective/physical law rather than random phenomena.

The EPB point (point A) could be used as an accurate design reference value for the load-bearing capacity of the tested pavement, as the ending point of the pavement’s normal working state. The EPB design point has two margins of safety: (1) one is from the EPB point to the failure starting point, with the attribute of certainty; (2) the other is from the failure starting point to the ultimate point, with the attribute of certainty. Thus, the load-bearing capacity design of the tested pavement might be accurate as it has a definitely objective law to obey. As a result, the design of pavements could be not only simple and reliable but also have a higher performance–cost ratio than existing design codes of pavements.

Here, it should be underlined that the state-of-stress analysis methods are important for presenting the characteristic points (A, B and C). But, if the state-of-stress characteristic pair derived by some analysis methods does not present the state-of-stress mutation feature obviously, this could just be due to the insufficiency of the method rather than the inexistence of characteristic points. Hence, structural state-of-stress analysis needs to try to propose a proper method that can present the mutation features in the state-of-stress evolution.

6. Conclusions

This study proposes methods for modeling the state variables and state-of-stress characteristic pairs based on the tested strain and deformation data of the asphalt airport pavement. The analysis of the characteristic pairs reveals the essential working features of the asphalt airport pavement from the tested data, which might update the traditional knowledge on the working behavior of the pavement. The conclusions can be drawn as follows:

• The proposed methods can derive the state variables and form the state-of-stress modes and the characteristic parameters that embody the essential features in the working process of the asphalt airport pavement. The slope increment criterion can detect the essential state-of-stress mutation features in the evolution curves of the characteristic parameters. Correspondingly, the investigation into the evolution curves of the state-of-stress mode also presents the mutation features. The mutation points reveal the EPB point at the traffic pattern number 249 and the real traffic pass number 5762, the failure starting point at 367 and 10,004 and the progressive point at 422 and 11,548. The characteristic points of the asphalt airport pavement are the reflection of the natural law of a quantitative change to qualitative change.
• The evolution of the state-of-stress characteristic parameter presents the mutation feature around the EPB point with the strain accumulation inside the pavement. But, the state-of-stress mode $\mathsf{\Delta }{\mathsf{D}}_t$ has no obvious change around the EPB point, implying that elastic deformation is a major part of the total deformation before the EPB point.
• The deformation is obviously reflected by the evolution of the state-of-stress mode $\mathsf{\Delta }{\mathsf{D}}_t$ around the failure starting point. Compared with the state-of-stress mode $\mathsf{\Delta }{\mathsf{D}}_t$ of the EPB point and failure starting point, it is obvious that the plastic deformation gradually accumulates and becomes dominant from point EPB to the failure starting point. When the plastic accumulation develops to the failure starting point, the evolution curves of $\mathsf{\Delta }{\mathsf{D}}_t$ mutate to the other shape, which embodies the plastic deformation around the failure starting point from a quantitative change to qualitative change.
• The deformation around the progressive failure point is smallest even opposite among all the deformations in the working process of the pavement. This indicates that the elastic/plastic deformation has completely disappeared and that the pavement has entered a broken state. The state-of-stress mode and the characteristic parameter show that the evolution features are different from those before the progressive failure point.
• The EPB point is the normal working branch point of the pavement service, so it could be directly taken as the design point of the pavement’s load-bearing capacity. The failure starting point is the starting point of the pavement’s failure process, which provides a reference to the design and accurate estimation of safety for asphalt pavements. Therefore, this study could explore a new way to analyze the working law of asphalt airport pavements.