Inverse calculation of elastic moduli in cross-anisotropic and layered pavements by system identification method

Figures & data

Figure 1. Schematics of the FWD for measuring the surface deflections of a layered pavement at sensors s_i with distances d_i from the loading centre.

Figure 2. Schematics of a p-layered pavement half space under a uniform vertical loading q within the circle r = R on the surface.

Figure 3. Schematics of the SID method in moduli inversion of layered pavements using the FWD-measured surface deflections as inputs.

Table 1. Actual material properties and geometry of the three-layer pavement structures with different cross-anisotropic layers and surface deflections “measured” at nine sensors s_i located at distance d_i from the loading centre.

Model 1: Pavement with only cross-anisotropic base layer
Layer			AC						Base						Subgrade
									Horizontal			Vertical
Moduli (MPa)			1500						300			500			30
Poisson’s ratio			0.35						0.4			0.4			0.45
Thickness (mm)			150						240						Infinity
di (mm)	0		203	305			457		610		914			1219	1524		1829
Deflection (μm)	1002.64		867.74	795.73			704.98		624.85		491.79			389.27	312.43		255.54
Model 2: Pavement with both cross-anisotropic asphalt concrete and base layers
Layer			AC						Base						Subgrade
			Horizontal			Vertical			Horizontal			Vertical
Moduli (MPa)			1500			3000			300			500			30
Poisson’s ratio			0.35			0.35			0.4			0.4			0.45
Thickness (mm)			150						240						Infinity
di (mm)		0	203		305			457	610	914			1219		1524	1829
Deflection (μm)		977.79	867.11		798.28			707.63	627.06	493.18			390.05		312.80	255.67

Table 2. Actual material properties of the three-layer pavement with cross-anisotropic base layer only (Model 1), along with the inverted elastic moduli. In the inversion calculation, the Poisson’s ratios and vertical shear modulus G_v are fixed and three types of seed moduli are selected to study the sensitivity of the inverse algorithm on the seed moduli.

Pavement Model 1	Actual modulus (MPa)		Seed modulus (MPa)		Inverted modulus (MPa)	Relative error (%)
AC	Eac	1500	Type 1	30,000	1500.019	0.001
Base	Ebh	300		3000	299.999	0.000
	Ebv	500		3000	499.982	−0.004
Subgrade	Esg	30		300	30.000	0.000
AC	Eac	1500	TypE 2	10,000	1500.016	0.001
Base	Ebh	300		10,000	300.00	0.000
	Ebv	500		10,000	499.989	−0.002
Subgrade	Esg	30		10,000	30.00	0.000
AC	Eac	1500	Type 3	300	1500.022	0.001
Base	Ebh	300		3000	300.000	0.000
	Ebv	500		3000	499.985	−0.003
Subgrade	Esg	30		3000	30.000	0.000

Note for modulus symbols: E_ac = modulus of asphalt concrete, E_bh and E_bv = horizontal and vertical moduli of granular base, and E_sg = modulus of subgrade.

Table 3. Actual material properties of the three-layer pavement with cross-anisotropic base layer only (Model 1), along with the inverted elastic moduli and vertical shear moduli. In the inversion calculation, only the Poisson’s ratios are fixed and three types of seed moduli are selected to study the sensitivity of the inverse algorithm on the seed moduli.

Pavement model 1	Actual modulus (MPa)		Seed modulus (MPa)		Inverted modulus (MPa)	Relative error (%)
AC	Eac	1500	Type 1	30,000	1500.117	0.008
Base	Ebh	300		3000	299.862	−0.046
	Ebv	500		3000	500.127	0.025
	Gv	178.6		1500	178.592	0.012
Subgrade	Esg	30		300	30	0.000
AC	Eac	1500	Type 2	10,000	1500.414	0.028
Base	Ebh	300		10,000	299.802	−0.066
	Ebv	500		10,000	500.308	0.062
	Gv	178.6		5000	178.58	0.006
Subgrade	Esg	30		10,000	30.002	0.007
AC	Eac	1500	Type 3	300	1499.972	−0.002
Base	Ebh	300		3000	299.77	−0.077
	Ebv	500		3000	500.28	0.056
	Gv	178.6		1500	178.61	0.022
Subgrade	Esg	30		30,000	30	0.000

Note for modulus symbols: E_ac = moduli of asphalt concrete, E_bh and E_bv = horizontal and vertical moduli of granular base, G_v = vertical shear modulus of granular base and E_sg = modulus of subgrade.

Table 4. Actual material properties of the three-layer pavement with both cross-anisotropic asphalt concrete and base layers (Model 2), along with the inverted elastic moduli. In the inversion calculation, the Poisson’s ratios and vertical shear modulus G_v are fixed and three types of seed moduli are selected to study the sensitivity of the inverse algorithm on the seed moduli.

Pavement model 2	Actual modulus (MPa)		Seed modulus (MPa)		Inverted modulus (MPa)	Relative error (%)
AC	Each	1500	Type 1	30,000	1502.553	0.170
	Eacv	3000		30,000	2978.868	−0.704
Base	Ebh	300		3000	299.529	−0.157
	Ebv	500		3000	500.093	0.019
Subgrade	Esg	30		300	29.999	−0.003
AC	Each	1500	Type 2	10,000	1502.551	0.170
	Eacv	3000		10,000	2978.861	−0.705
Base	Ebh	300		10,000	299.529	−0.157
	Ebv	500		10,000	500.092	0.018
Subgrade	Esg	30		10,000	29.999	−0.003
AC	Each	1500	Type 3	300	1502.574	0.172
	Eacv	3000		300	2978.925	−0.702
Base	Ebh	300		3000	299.527	−0.158
	Ebv	500		3000	500.09	0.018
Subgrade	Esg	30		30,000	29.999	−0.003

Note for modulus symbols: E_ach and E_acv = horizontal and vertical moduli of asphalt concrete, E_bh and E_bv = horizontal and vertical moduli of granular base, and E_sg = modulus of subgrade.

Table 5. Actual material properties and geometry of the four-layer pavement structures with different cross-anisotropic layers and surface deflections “measured” at nine sensors s_i located at distance d_i from the loading centre.

Model 3: Pavement with both cross-anisotropic base and subbase layers
Layer	AC						Base										Subbase								Subgrade
							Horizontal						Vertical				Horizontal				Vertical
Moduli (MPa)	1500						500						800				300				500				30
Poisson’s ratio	0.35						0.40						0.40				0.40				0.40				0.45
Thickness (mm)	150						240										180								Infinity
di (mm)	0	203			305				457		610				914				1219				1524				1829
Deflection (μm)	722.28	619.67			576.88				530.06		489.77				418.51				356.69				304.24				260.46
Model 4: Pavement with both cross-anisotropic asphalt concrete and base layers
Layer	AC							Base										Subbase								Subgrade
	Horizontal			Vertical				Horizontal						Vertical
Moduli (MPa)	1500			3000				500						800				300								30
Poisson’s ratio	0.35			0.35				0.40						0.40				0.40								0.45
Thickness (mm)	150							240										180								Infinity
di (mm)	0		203			305				457		610				914				1219				1524				1829
Deflection (μm)	717.84		636.14			593.80				543.55		500.14				424.67				360.24				306.12				261.27
Model 5: Pavement with cross-anisotropic asphalt concrete, base and subbase layers
Layer	AC							Base										Subbase								Subgrade
	Horizontal			Vertical				Horizontal						Vertical				Horizontal				Vertical
Moduli (MPa)	1500			3000				500						800				300				500				30
Poisson’s ratio	0.35			0.35				0.40						0.40				0.40				0.40				0.45
Thickness (mm)	150							240										180								Infinity
di (mm)	0		203			305				457		610				914				1219				1524				1829
Deflection (μm)	702.49		622.22			581.37				533.60		492.43				420.14				357.67				304.81				260.75

Table 6. Actual material properties of the four-layer pavement with both cross-anisotropic base and subbase layers (Model 3), along with the inverted elastic moduli. In the inversion calculation, the Poisson’s ratios and vertical shear modulus G_v are fixed and three types of seed moduli are selected to study the sensitivity of the inverse algorithm on the seed moduli.

Pavement Model 3	Actual modulus (MPa)		Seed modulus (MPa)		Inverted modulus (MPa)	Relative error (%)
AC	Eac	1500	Type 1	30,000	1499.965	−0.002
Base	Ebh	500		3000	503.579	0.716
	Ebv	800		3000	799.392	−0.076
Subbase	Esbh	300		3000	299.508	−0.164
	Esbv	500		3000	500.659	0.132
Subgrade	Esg	30		300	30	0.000
AC	Eac	1500	Type 2	10,000	1499.999	0.000
Base	Ebh	500		10,000	500.343	0.069
	Ebv	800		10,000	799.941	−0.007
Subbase	Esbh	300		10,000	299.882	−0.039
	Esbv	500		10,000	500.267	0.053
Subgrade	Esg	30		10,000	30	0.000
AC	Eac	1500	Type 3	300	1499.994	0.000
Base	Ebh	500		3000	500.409	0.082
	Ebv	800		3000	799.916	−0.010
Subbase	Esbh	300		3000	299.923	−0.026
	Esbv	500		3000	500.112	0.022
Subgrade	Esg	30		30,000	30	0.000

Note for modulus symbols: E_ac = modulus of asphalt concrete, E_bh and E_bv = horizontal and vertical moduli of granular base, E_sbh and E_sbv = horizontal and vertical moduli of subbase, and E_sg = modulus of subgrade.

Table 7. Actual material properties of the four-layer pavement with both cross-anisotropic asphalt concrete and base layers (Model 4), along with inverted elastic moduli and vertical shear moduli. In the inversion calculation, only the Poisson’s ratios are fixed and three types of seed moduli are selected to study the sensitivity of the inverse algorithm on the seed moduli.

Pavement Model 4	Actual modulus (MPa)		Seed modulus (MPa)		Inverted modulus (MPa)	Relative error (%)
AC	Each	1500	Type 1	30,000	1500.766	0.051
	Eacv	3000		30,000	3000.726	0.024
	Gacv	1111.11		15,000	1110.515	−0.054
Base	Ebh	500		3000	499.707	−0.059
	Ebv	800		3000	799.498	−0.063
	Gbv	285.7		1500	285.796	0.020
Subbase	Esb	300		3000	299.929	−0.024
Subgrade	Esg	30		300	30	0.000
AC	Each	1500	Type 2	10,000	1500.77	0.051
	Eacv	2000		10,000	3000.712	0.024
	Gacv	740.7		5000	1110.515	−0.054
Base	Ebh	500		10,000	499.707	−0.059
	Ebv	800		10,000	799.495	−0.063
	Gbv	285.7		5000	285.794	0.019
Subbase	Esb	300		10,000	299.929	−0.024
Subgrade	Esg	30		10,000	30	0.000
AC	Each	1500	Type 3	300	1500.778	0.052
	Eacv	2000		300	3000.714	0.024
	Gacv	740.7		150	1110.51	−0.054
Base	Ebh	500		3000	499.697	−0.061
	Ebv	800		3000	799.491	−0.064
	Gbv	285.7		1500	285.796	0.020
Subbase	Esb	300		3000	299.929	−0.024
Subgrade	Esg	30		30,000	30	0.000

Note for modulus symbols: E_ach and E_acv = horizontal and vertical moduli of asphalt concrete, G_acv = vertical shear modulus of asphalt concrete, E_bh and E_bv = horizontal and vertical moduli of granular base, G_bv = vertical shear modulus of granular base, E_sb = modulus of subbase, and E_sg = modulus of subgrade.

Table 8. Actual material properties of the four-layer pavement with cross-anisotropic asphalt concrete, base and subbase layers (Model 5), along with the inverted elastic moduli. In the inversion calculation, the Poisson’s ratios and vertical shear modulus G_v are fixed and three types of seed moduli are selected to study the sensitivity of the inverse algorithm on the seed moduli.

	Actual modulus (MPa)		Seed modulus (MPa)		Inverted modulus (MPa)	Relative error (%)
AC	Each	1500	Type 1	30,000	1498.544	−0.097
	Eacv	2000		30,000	3000.496	0.017
Base	Ebh	500		3000	508.286	1.657
	Ebv	800		3000	798.844	−0.144
Subbase	Esbh	300		3000	298.941	−0.353
	Esbv	500		3000	501.585	0.317
Subgrade	Esg	30		300	30	0.000
AC	Each	1500	Type 2	10,000	1499.652	−0.023
	Eacv	2000		10,000	3000.132	0.004
Base	Ebh	500		10,000	501.633	0.327
	Ebv	800		10,000	799.782	−0.027
Subbase	Esbh	300		10,000	299.786	−0.071
	Esbv	500		10,000	500.283	0.057
Subgrade	Esg	30		10,000	30	0.000
AC	Each	1500	Type 3	300	1499.364	−0.042
	Eacv	2000		300	3000.216	0.007
Base	Ebh	500		3000	503.457	0.691
	Ebv	800		3000	799.526	−0.059
Subbase	Esbh	300		3000	299.565	−0.145
	Esbv	500		3000	500.632	0.126
Subgrade	Esg	30		30,000	30	0.000

Note for modulus symbols: E_ach and E_acv = horizontal and vertical moduli of asphalt concrete, E_bh and E_bv = horizontal and vertical moduli of granular base, E_sbh and E_sbv = horizontal and vertical moduli of subbase, and E_sg = modulus of subgrade.