Evaluating Mechanical Properties of Earth Material During Intelligent Compaction (2020)

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27 Findings from Numerical Model Introduction This chapter presents the development of a 3D FE model simulating the roller compaction of one- and two-layer geosystems. Different levels of complexity were considered in the model, including the use of linear and nonlinear geomaterial approaches and simulations of the roller operation ranging from a static load to a stationary vibratory load to moving vibratory loads. The responses of the model to these various geomaterial properties were numerically assessed, and correlations among the responses were established to study (a) whether the simplified model can account adequately for the behavior of the geomaterials under compaction and (b) whether these relationships can be used to simplify the modeling. Development and Limitations of Numerical Simulation of IC The multi-purpose FE program LS-DYNA, which makes use of explicit and implicit time integration techniques, was used for simulating roller compaction. A 3D mesh was assembled to simulate a roller’s drum in the process of proof mapping geomaterials at a given loading amplitude and vibrating frequency (see Figure 3-1). The drum of the roller was simulated as a rigid body containing shell elements with the commercially available regular dimensions of an IC roller (i.e., 2 m [80 in.] wide and 1.5 m [60 in.] diameter). A section of the geomaterial layer, 4 m (160 in.) wide, 4 m (160 in.) long, and 2.5 m (100 in.) deep, was modeled with nonreflective boundaries. A mesh consisting of brick elements was used to represent the geosystem. The mesh was composed of approximately 64,000 elements. Smaller elements with 50 × 50 × 50 mm (2 × 2 × 2 in.) dimensions were used near the roller up to 0.5 m (20 in.) in depth, 0.6 m (24 in.) longitudinally, and 1.2 m (48 in.) transversally from the center of the drum, beyond which larger elements were used. The interaction between the drum and the geosystem was simulated using an automatic single-surface contact type that allows the decoupling of the drum from the soil surface as it occurs in the field. To that end, about 75,000 shell elements were used to simulate the drum. The centrifugal force caused by the rotation of the eccentric masses inside the drum induces an excitation force, Fe, defined as F t m e te cos , (3-1)0 0 2( ) ( )= Ω Ω where t = time, W = the rotational frequency, and e0 = the eccentricity of the rotating mass, m0. C H A P T E R 3

28 Evaluating Mechanical Properties of Earth Material During Intelligent Compaction Typical values used for the simulated drum are shown in Table 3-1. The vibratory motion of the roller was maintained for 200 ms, equivalent to six load cycles. The stress, strain, and displacement time histories were calculated for every time interval of 1 ms underneath the center of the roller. Rayleigh constants were defined as α = 25 and µ = 0.0002, as recommended by Mooney and Facas (2013) to minimize the dilatational and shear wave reflections. Ooi et al. (2004) proposed a resilient modulus material model in which the resilient modulus, MR, is defined as MR k P P P a a k oct a k 1 1 , (3-2)1 2 3 = ′ θ +   τ +   ′ ′ where q = bulk stress, toct = octahedral shear stress, Pa = atmospheric pressure, and k ′1,2,3 = regression constants. (a) IC-equipped roller (b) FE model of soil and drum (c) Soil-drum interface Drum Drum Sliding permitted Node penetration check (d) Contact interaction Soil-Drum Contact Interface Soil Figure 3-1. Schematic representation of drum-soil system. Operating Parameter Symbol Value Width of drum (compaction width) L 2.0 m (80 in.) Diameter of drum d 1.5 m (60 in.) Mass of drum md 6,000 kg (34.3 lb. × s2/in.) Weight of drum md g 58,840 N (13,200 lb.) Mass-eccentricity m 0e0 5.36 kg·m (1.20 lb. × s2) Centrifugal force (Vertical excitation force) Fev 170 kN (38 kips) Frequency f 28 Hz (1,680 vpm) Frequency Ω 176 rad/s Operating speed v 0.9 m/s (3.24 km/h, 2.0 mph) Table 3-1. Specifications for simulated drum.

Findings from Numerical Model 29 This model was incorporated as a user defined material subroutine into LS-DYNA to account for the nonlinear behaviors of geomaterials under loading conditions. Mazari et al. (2014) found that this model yielded more representative responses of modulus-based devices as compared to the standard material model incorporated in the MEPDG. Considering the practical problems the use of this model may cause for highway agencies that utilize the MEPDG material model, the authors also provided simple relationships for convert- ing the more common MEPDG model parameters into the nonlinear parameters shown in Equation 3-2. Development of Comprehensive Database of Pavement Sections A comprehensive database of linear and nonlinear 3D dynamic cases with different input parameters was assembled for single-layer and two-layer geosystems. The information stored in the database was used to evaluate the sensitivity of the geosystem responses to the various input parameters. That database was also used to develop an optimized model that simulated the response of different geomaterials subjected to a vibratory roller with a variety of levels of sophistication. The database contained the following types of data: • Roller operating parameters, including drum dimensions, mass of drum, frequency, vertical excitation force, and operating speed. • Geosystem structure and geomaterial properties, including layer thickness, nonlinear k ′ parameters of layers, and the representative resilient modulus per layer. • Level of sophistication of the FE model, including the type of analysis (static, quasi-static, or dynamic); geomaterial constitutive model (linear elastic or nonlinear); and contact type (roller load applied directly to the geosystem or by means of a contact model). • Geosystem responses obtained after simulation of roller compaction, including maximum surface vertical displacement, maximum stress observed under the load, and depth of influence. Three groups of geosystems were simulated, consisting of one single-layer (subgrade-only) system and two-layer systems with top layer (base) thicknesses of 150 mm (6 in.) and 300 mm (12 in.) on top of the subgrade. Feasible ranges of k nonlinear parameters (proposed by Velasquez et al. [2009]) were used for the coarse- and fine-grained geomaterials, as shown in Table 3-2. For each geosystem, 200 randomly generated cases were initially selected considering a uniform distribution within the feasible range of values shown in Table 3-2. This prototype database contained information about the distributions of stress, strain, displacement, and modulus (when applicable), and was used to study the feasibility of different concepts. As soon as a concept was deemed feasible, a more expanded strategic database relevant to that concept was developed. The representative resilient modulus of the base was constrained to a range between 70–700 MPa (10–100 ksi), whereas the representative resilient moduli of the subgrade were constrained to a range between 35–350 MPa (5–50 ksi). Material Type Nonlinear Parameters k1 k2 k3 Coarse-grained 400–3,000 0.2–1.0 -0.9 – -0.1 Fine-grained 1,000–4,000 0.01–0.5 -6.0 – -1.5 Source: Velasquez et al. (2009) Table 3-2. Feasible range of layer properties.

30 Evaluating Mechanical Properties of Earth Material During Intelligent Compaction The research team considered six levels of sophistication of the FE model, as described in Table 3-3. In terms of the surface displacements and stresses at critical points, the geo- system responses were obtained directly under the drum of single- and two-layer geosystems simulating the drum as static and vibratory. The main levels of sophistication consisted of the following items: • Linear vs. Nonlinear Behavior of Geomaterials: The use of the nonlinear material models requires iterative procedures to update the state of stress during the simulation, which leads to longer execution times. For this reason, the linear elastic material models are commonly used. The responses of the linear models were compared with their comparable nonlinear models to explore the possibility of establishing relationships that could estimate the non- linear response knowing the linear response and the geomaterial k ′ nonlinear parameters. • Static vs. Vibratory Drum: For static loading conditions, a quasi-static analysis was imple- mented in which the load was applied in 1 ms as a ramp load until the peak excitation force was reached, and then the load was maintained at a constant magnitude for the following 19 ms. In that manner, the impact of inertia was reduced, allowing the contact elements to accommodate the drum. The simulation of a vibratory load consisted of a sinusoidal load with peak vertical force of 170 kN (38 kips) and a frequency of 28 Hz, in addition to the weight of the drum. At that frequency, six load cycles were produced in 200 ms of simulation time. • Stationary vs. Moving/Rolling Drum: A prescribed motion to the drum was considered in vibratory moving cases, where velocity, angular velocity, and direction of movement were specified. These assumptions lend to slower executions if the nonlinear behavior of the geomaterials is considered, due to the iterative process required to update the state of the stress. Stationary drums were simulated at a unique position, yet the vibrating load applied to the drum was still incorporated. Example comparisons of the displacements obtained with the nonlinear (SSN/VSN) and linear (SSL/VSL) models are shown in Figures 3-2 and 3-3 for single- and two-layer geo- systems, respectively. These two parameters are correlated with some uncertainty, as judged by the number of cases lying outside the ±20% uncertainty bounds. The stiffer top (base) layer of the two-layer systems reduced the effect of the nonlinear behavior of the subgrade as the stresses attenuated more. Table 3-4 provides a summary of the slopes, coefficients of determination (R2 values) and normalized standard errors of estimate (SEEs) of the regression lines from single-layer and two-layer 150-mm base on top-of-subgrade scenarios. NSEE is calculated from Normalized standard error of estimate SEE Y Y Y i ii n ii n , (3-3) 2 1 2 1 ∑ ∑ ( ) ( ) ( ) = ′ − ′ = = FE Model Characteristics Label Load Type Constitutive Model Roller Velocity Static Stationary Linear SSL Static Linear Elastic -- Static Stationary Nonlinear SSN Static Modified MEPDG -- Vibratory Stationary Linear VSL Dynamic Linear Elastic -- Vibratory Stationary Nonlinear VSN Dynamic Modified MEPDG -- Vibratory Moving Linear VML Dynamic Linear Elastic 0.9 m/s (2 mph) Vibratory Moving Nonlinear VMN Dynamic Modified MEPDG 0.9 m/s (2 mph) Table 3-3. Characteristics of different levels of sophistication of FE model for parametric study.

Findings from Numerical Model 31 where Y ′i = the estimated displacement obtained from the linear equation of the fitted trend, Yi = the displacement from the FE simulation, and n = the total number of points. In general, displacement pairs correlated with R2 values greater than 0.85 and typically greater than 0.90, while normalized errors of estimate were typically less than 0.20. These descriptive statistics suggest that the surface deflections as obtained from the more sophis- ticated FE models may be estimated using relationships that adjust the responses of the less- sophisticated (i.e., less computationally intense) FE models. The level of sophistication impact y = 1.10x R² = 0.85 SEE = 0.52 mm 0 2 4 6 8 0 2 4 6 8 SSL FE Model Surface Displacement (mm) ±20% Error Line Line of Equality (a) SS N F E M od el Su rf ac e D is pl ac em en t ( m m ) y = 0.92x R² = 0.94 SEE = 0.27 mm 0 2 4 6 8 0 2 4 6 8 SSN FE Model Surface Displacement (mm) ±20% Error Line Line of Equality (b) V SN F E M od el Su rf ac e D is pl ac em en t ( m m ) Figure 3-2. Relationship of surface displacement under roller between (a) linear (SSL) to nonlinear (SSN) static stationary and (b) vibratory (VSN) to static (SSN) stationary nonlinear FE models for a single-layer (subgrade) geosystem. y = 1.04x R² = 0.94 SEE = 0.21 mm 0 1 2 3 4 5 0 1 2 3 4 5 SSL FE Model Surface Displacement (mm) ±20% Error Line Line of Equality (a) SS N F E M od el Su rf ac e D isp la ce m en t ( m m ) y = 0.96x R² = 0.96 SEE = 0.16 mm 0 1 2 3 4 5 0 1 2 3 4 5 SSN FE Model Surface Displacement (mm) ±20% Error Line Line of Equality (b) V SN F E M od el Su rf ac e D isp la ce m en t ( m m ) Figure 3-3. Relationship of surface displacement under roller between (a) linear (SSL) to nonlinear (SSN) static stationary and (b) vibratory (VSN) to static (SSN) stationary nonlinear FE models for two-layer geosystem with 150 mm (6 in.) base layer on top of subgrade.

32 Evaluating Mechanical Properties of Earth Material During Intelligent Compaction on the analysis time and the relationships between all the models are described in detail in Appendix D, which is available as part of the downloadable “Appendices.pdf” file on the NCHRP Research Report 933 webpage). Establishing Depth of Influence of IC The depth of influence was defined as the depth at which the geomaterial’s response dimin- ishes to 10% of the peak response. Different depths can be obtained based on the response criterion used being either the displacements, stresses, or strains underneath the drum. Table 3-5 presents a summary of the normalized depth of influence with respect to the contact width (z/B) calculated using the displacement criterion for all models with different levels of sophistication. The average influence depth slightly increases for the vibratory moving drums as compared to the stationary vibratory or static conditions. Nevertheless, the differences in the mean values among the six different cases is less than 11%. Based on these case studies, for practical purposes one can approximate the depth of influence to about six times the effective contact width (i.e., about 1.8 m [70 in.] in depth). When material nonlinearity is introduced, the depth of influence increases as k′2 increases (i.e., as the geomaterial becomes more granular) and decreases as the absolute value of k ′3 increases (i.e., the geomaterial becomes less cohesive). As shown in Figure 3-4, the effect of the Model Level of Sophistication of FE Model Single-Layer System Two-Layer System (150 mm base on top of subgrade) Slope of Fitted Linear Relationship Slope of Fitted Linear Relationship SSL SSN VSL VSN VML VMN SSL SSN VSL VSN VML VMN SSL 1 1.10 0.98 1.04 1.00 1.13 1 1.04 0.97 1.01 1.01 1.15 SSN 1 0.85 0.92 0.87 1.00 1 0.93 0.96 0.96 1.10 VSL 1 1.05 1.02 1.14 1 1.03 1.04 1.18 VSN 1 0.95 1.09 1 0.99 1.14 VML 1 1.13 1 1.14 VMN 1 1 Model Coefficient of Determination (R2) Coefficient of Determination (R2) SSL SSN VSL VSN VML VMN SSL SSN VSL VSN VML VMN SSL 1 0.85 1.00 0.92 0.99 0.93 1 0.94 0.99 0.99 0.99 0.93 SSN 1 0.83 0.95 0.79 0.90 1 0.93 0.96 0.92 0.90 VSL 1 0.91 0.99 0.82 1 0.96 0.92 0.91 VSN 1 0.90 0.93 1 0.98 0.93 VML 1 0.85 1 0.94 VMN 1 1 Model Normalized Standard Error of Estimate (SEE) Normalized Standard Error of Estimate (SEE) SSL SSN VSL VSN VML VMN SSL SSN VSL VSN VML VMN SSL -- 0.22 0.03 0.15 0.04 0.19 -- 0.11 0.04 0.04 0.03 0.11 SSN -- 0.23 0.12 0.23 0.15 -- 012 0.09 0.13 0.14 VSL -- 0.16 0.05 0.21 -- 0.06 0.05 0.13 VSN -- 0.16 0.12 -- 0.06 0.11 VML -- 0.19 -- 0.10 VMN -- -- Table 3-4. Descriptive statistics for various levels of sophistication of FE model.

Findings from Numerical Model 33 material nonlinearity on the depth of influence was more significant in single-layer geosystems than in two-layer geosystems. This was also reflected in the standard deviation values of the depths of influence shown in Table 3-5, where higher standard deviations occurred in single- layer systems as well. Detailed analyses for the stress criterion are provided in Appendix D. The depth of influence decreased to four times the drum contact width (i.e., about 1.2 m [48 in.]). Impact of Geomaterial Properties on ICMVs The effect of the nonlinear k′ parameters on the roller responses were also quantified. The influence of the nonlinear nature of the geomaterials on the pavement responses was studied using Spearman’s correlation (McDonald 2014). Different levels of sophistication of the FE Normalized Depth of Influence (z/B) Levels of Sophistication of FE Model One- Layer System Two-Layer System One-Layer System Two-Layer System 150 mm Base 300 mm Base 150 mm Base 300 mm Base Static Stationary Linear Geomaterial (SSL) Nonlinear Geomaterial (SSN) Mean 5.89 6.12 6.29 5.82 6.01 6.18 Median 5.90 6.13 6.30 5.91 6.03 6.03 Standard Deviation 0.03 0.17 0.26 0.58 0.37 0.37 Vibratory Stationary Linear Geomaterial (VSL) Nonlinear Geomaterial (VSN) Mean 6.09 6.31 6.46 5.94 6.11 6.24 Median 6.11 6.30 6.46 5.99 6.13 6.30 Standard Deviation 0.06 0.19 0.28 0.40 0.30 0.28 Vibratory Moving Linear Geomaterial (VML) Nonlinear Geomaterial (VMN) Mean 6.12 6.33 6.49 5.91 6.08 6.24 Median 6.13 6.34 6.51 5.99 6.09 6.25 Standard Deviation 0.05 0.17 0.27 0.43 0.32 0.28 Table 3-5. Descriptive statistics of normalized depths of influence with respect to displacement for various levels of sophistication of FE model. 2 4 6 8 10 0 1 2 3 k'2, subgrade (a) kꞌ1 = 100–1,000 kꞌ1 = 1,001–2,000 kꞌ1 = 2,001–3,000 In flu en ce D ep th (z /B ) k'3, subgrade 2 4 6 8 10 -4-3-2-10 (b) kꞌ1 = 100–1,000 kꞌ1 = 1,001–2,000 kꞌ1 = 2,001–3,000 In flu en ce D ep th (z /B ) Figure 3-4. Variation in influence depth with nonlinear k’ parameters of subgrade for single-layer systems based on displacement criterion.

34 Evaluating Mechanical Properties of Earth Material During Intelligent Compaction models were also taken into consideration. Table 3-6 shows that subgrade nonlinear parameters k ′1 (related to stiffness) and k′2 (granularity causing stress hardening) impact surface displace- ment the most, even for two-layer systems. In addition, material granularity (i.e., k′2) seemed to have a more significant impact on the surface stress directly under the drum than did other parameters. This greater impact is shown in Table 3-7 for single-layer and two-layer geo- systems, respectively. The nonlinear parameters of the subgrade tend to influence the surface stresses less significantly, especially as the base thickness increases. Impact of Roller Operating Features on Geomaterials Responses Roller parameters significantly affect both the roller measurements and the geomaterials’ responses during the mapping process. Aside from the pavement structure and mechanical properties of geomaterials (e.g., modulus and nonlinear k′ parameters), the impact of the Level of Sophistication of FE Model Spearman’s Correlation Coefficients Base Parameters Subgrade Parameters k′1 k′2 k′3 k′1 k′2 k′3 Single-Layer (Subgrade Only) SSN -- -- -- -0.37 -0.42 -0.18 VSN -- -- -- -0.42 -0.35 -0.15 VMN -- -- -- -0.32 -0.37 -0.30 150 mm (6 in.) Base Thickness SSN -0.08 -0.14 -0.19 -0.45 -0.36 -0.15 VSN -0.12 -0.10 -0.18 -0.56 -0.24 -0.06 VMN -0.17 0.07 -0.29 -0.49 -0.30 -0.13 300 mm (12 in.) Base Thickness SSN -0.12 -0.19 -0.28 -0.49 -0.20 -0.08 VSN -0.21 -0.06 -0.23 -0.55 -0.22 -0.03 VMN -0.24 0.03 -0.31 -0.54 -0.23 -0.05 Table 3-6. Impact of nonlinear material parameters on surface displacement. Level of Sophistication of FE Model Spearman’s Correlation Coefficients Base Parameters Subgrade Parameters k′1 k′2 k′3 k′1 k′2 k′3 Single Layer (Subgrade Only) SSN -- -- -- 0.09 -0.14 -0.12 VSN -- -- -- -0.29 0.31 0.26 VMN -- -- -- 0.52 0.35 -0.02 150 mm (6 in.) Base Thickness SSN -0.07 0.16 0.07 0.20 0.37 0.16 VSN -0.14 -0.08 -0.05 -0.14 0.33 0.18 VMN 0.22 0.30 0.08 0.15 0.31 0.15 300 mm (12 in.) Base Thickness SSN -0.09 0.09 -0.02 -0.06 0.14 0.08 VSN -0.17 0.01 -0.17 -0.14 0.38 0.18 VMN 0.10 -0.05 0.13 0.05 0.26 -0.09 Table 3-7. Impact of nonlinear material parameters on surface stress for two-layer system.

Findings from Numerical Model 35 roller’s operating features (e.g., operating weight and dimensions of the drum) on the pave- ment responses should be taken into consideration. To evaluate the effect of the roller operating features on the geomaterials responses, 13 rollers with different operating features were simu- lated (see Table 3-8). The rollers were identified using a code that summarizes the imparted force plus drum weight, length, and diameter. The soil responses determined under a static load exerted by a stationary drum were evaluated on a set of 200 nonlinear geomaterials systems that were made up of single-layer and two-layer pavement systems, comprising a total of 600 SSN FE models per simulated drum. The conclusions from this study are summarized next. The reader is referred to Appendix D for further details on the analysis presented briefly in this section. Impact of Weight Three rollers with different drum dimensions were considered (Cases 1, 3, and 9, as listed in Table 3-8). As shown in Figure 3-5, increasing the load imposed on the soil by a factor of two Case Model Code* Drum Weight (kN) Centrifugal Force (kN) Length (m) Diameter (m) No. of SSN Cases 1 22.6W_1.00L_0.60D 7.45 15.12 1.00 0.60 600 2 45.1W_1.00L_0.60D 14.90 30.24 1.00 0.60 600 3 38.5W_1.20L_0.70D 23.93 14.60 1.20 0.70 600 4 77.1W_1.20L_0.70D 47.86 29.20 1.20 0.70 600 5 118.7W_1.50L_1.10D 88.55 30.20 1.50 1.10 600 6 118.7W_1.50L_0.55D 88.55 30.20 1.50 0.55 600 7 166.8W_1.50L_1.10D 88.55 78.30 1.50 1.10 600 8 166.8W_1.50L_0.55D 88.55 78.30 1.50 0.55 600 9 113.9W_2.00L_1.50D 29.42 84.50 2.00 1.50 600 10 227.8W_2.00L_1.50D 58.84 169.00 2.00 1.50 600 11 227.8W_2.00L_0.75D 58.84 169.00 2.00 0.75 600 12 227.8W_1.00L_1.50D 58.84 169.00 1.00 1.50 600 Total: 7,200 * W = operating weight + eccentric force, L = length of drum, D = diameter of drum. Table 3-8. Simulated rollers with different operating features. y = 2.17x R² = 0.95 SEE = 0.10 mm 0.0 2.0 4.0 6.0 8.0 0.0 2.0 4.0 6.0 8.0 Surface Displacement (mm) (a) Single-Layer Geosystems Line of Equality +/- 20% Error Line Su rf ac e D is pl ac em en t w it h Im po se d W ei gh t D ou bl ed (m m ) y = 2.00x R² = 0.96 SEE = 0.06 mm 0.0 2.0 4.0 6.0 8.0 0.0 2.0 4.0 6.0 8.0 Su rf ac e D is pl ac em en t w it h Im po se d W ei gh t D ou bl ed (m m ) Surface Displacement (mm) (b) Two-Layer Geosystems Line of Equality +/- 20% Error Line Figure 3-5. Evaluation of weight impact on surface displacement.

36 Evaluating Mechanical Properties of Earth Material During Intelligent Compaction led to an increase in the surface displacement, with a factor of 2.17 and 2.00 for single-layer and two-layer geosystems, respectively. Like the surface displacements, the surface vertical stresses directly under the drum increased by about a factor of two when the magnitude of the imposed weight increased by a factor of two. More variability was observed in the surface stresses as compared to the surface displacements, however, which may be attributed to the effects that the nonlinear parameters of the top-layer geomaterial have on the contact area. Impact of Drum Length Case 10 had a drum weight plus peak centrifugal force of 228 kN, a drum length of 2.0 m, and a drum diameter of 1.50 m. Case 10 was compared to Case 12, which involved another roller with identical features except that it had half the drum length (1.0 m). As shown in Figure 3-6(a), with the drum length halved, the surface displacement essentially doubled, increasing nearly 2.1 times for single-layer systems and nearly 1.8 times for two-layer systems. Likewise, as shown in Figure 3-6(b), the surface stress increased by about 100% as the drum length was shortened by 50% (increasing by a factor of 2 for single-layer systems and slightly more for two-layer systems). The increase in the contact area with an increase in the drum length results in a reduction in the surface vertical stress. Impact of Drum Diameter Three rollers (Cases 5, 7, and 10) were selected to assess the effect of the drum diameter on soil responses. Their drum diameters were halved while the imposed weight and drum length were kept constant, resulting in the roller conditions described in Cases 6, 8, and 11. No significant change occurred in the surface displacements for both the single-layer and two-layer systems (see Figure 3-6). The surface stress increased when the diameter was halved, ranging from 6% for the smaller and lighter roller (Case 5) to 34% for the heavier and larger roller (Case 11). 0.0 0.5 1.0 1.5 2.0 2.5 Weight Doubled Drum Length Halved Drum Diameter Halved N or m al iz ed S ur fa ce D is pl ac em en t (a) Impact on Surface Displacement 0.0 0.5 1.0 1.5 2.0 2.5 Weight Doubled Drum Length Halved Drum Diameter Halved N or m al iz ed Su rf ac e St re ss (b) Impact on Surface Vertical Stress Single-Layer Systems Two-Layer Systems Figure 3-6. Evaluation of impact of weight, length, and diameter of drum on surface displacement and stress.

Findings from Numerical Model 37 Evaluation of Approaches for Developing Forward Models The traditional methods for modeling and optimizing complex drum-soil compaction systems require huge amounts of computational resources. For this reason, a simplified model is necessary to predict the pavement responses with minimal computational effort and reason- able accuracy. Artificial intelligence (AI), with its predictive analytics and machine-learning components, provides powerful predictive capabilities commensurate to traditional methods in modeling the complex behavior of materials without incurring high levels of computing time and effort. Two models were developed using symbolic regression, one using a genetic programming (GP) approach and the other using an artificial neural network (ANN) approach. A database was generated that consisted of 7,200 cases of stationary static nonlinear (SSN) FE models with different operating features as listed in Table 3-8, and geosystems with different properties using the feasible ranges of nonlinear k’ parameters shown in Table 3-2. Displacement For both single-layer and two-layer systems, the general form of the mathematical model proposed using the GP approach for predicting surface displacement underneath the center of the drum, dSSN, consisted of a function, defined as follows: , , , , , , , , , , (3-4)1 2 3 1 2 3d f k k k k k k h L D WSSN b b b s s s( )= ′ ′ ′ ′ ′ ′ where W = the weight (in kN), which includes the total force due to the drum weight, FD, and the peak eccentric vertical force, Fev; L = the drum length (in m); D = the drum diameter (in m); h = the thickness of the base (in mm); and k ′i b and k ′is = nonlinear parameters of the base and subgrade, respectively. To predict the maximum surface displacement under ordinary static drums with differing operating features, the dSSN calculation could be adapted by incorporating variables to represent the various operating features along with an operating index, y, The operating index, y, was defined as follows: , (3-5) L D Wψ = × where L = the drum length (in m), and D = the drum diameter (in m), and the variables C1 through C8 were used to capture the operating features of the drums. Equation 3-6 shows the new equation as constructed to predict displacement for two-layer geosystems. To predict displacement for single-layer geosystems, the last term of Equation 3-6 is excluded. d C C W C k C W C W k k k C k W k C W L D C h k k k SSN s s s s s s b b b cos cos 223 , (3-6) • • • • 1 2 3 1 2 5 1 2 3 6 1 1 7 8 1 2 3 ( ) ( )( )= + + ′ + + ′ + ′ + ′ + ψ ′ + ′ +     + ψ ′ ′ ′

38 Evaluating Mechanical Properties of Earth Material During Intelligent Compaction where C1 = 0.00425, C2 = 0.0139, C3 = 205, C4 = 0.075, C5 = 5.58 × 10–6, C6 = 2.98 × 10–10, C7 = 0.0004, C8 = 4.65 × 10–5, and y = the operating index (as defined in Equation 3-5). Figure 3-7 compares the GP-predicted surface displacements under the drum to the cor- responding surface displacements as determined by FE modeling. The GP approach estimated the peak surface displacement under rollers with different operating features fairly well, as most of the cases fell within the ±20% uncertainty bounds, with an R2 value of 0.73 and standard error of the estimate of 0.39 mm. To further improve the estimation of the surface displacements directly under the drum, an ANN-based method was developed. As shown in Figure 3-8, the ANN-based method predicted the surface displacements more accurately than the GP method. With the ANN method, the error of the estimate was less than 15% in 85% of the cases. The ANN method predicted surface displacements with an R2 of 0.99 and a SEE of 0.10 mm (see Figure 3-9). y = 0.97x R² = 0.72 SEE = 0.39 mm 0 2 4 6 8 10 0 2 4 6 8 10 FE-Determined Surface Displacement (mm) Line of Equality +/- 20% Error Line G P- Pr ed ic te d Su rf ac e D is pl ac em en t (m m ) Figure 3-7. Comparison of GP-predicted surface displacement to FE-determined surface displacement obtained from stationary drums with different operating features. 0% 20% 40% 60% 80% 100% Error (%) 0% 20% 40% 60% 80% 100% More Predicted Displacement Using ANN Model Predicted Displacement Using GA Model C um ul at iv e Fr eq ue nc y (% ) Figure 3-8. Cumulative distribution of estimation error for the predicted surface displacement using GP- and ANN-based methods.

Findings from Numerical Model 39 Stiffness Stiffness is defined as the resistance to deformation of a material under an applied load. As such, stiffness is not a unique material property; rather, it is the response of the pavement system to the load. Roller-measured soil stiffness can normally be derived from the vertical force equilibrium of the vibrating drum. In this study, the soil stiffness during pre-mapping as well as during the mapping process can be determined as the ratio of force over surface displacement (k = W/dSSN) for the SSN FE models. The general form of the proposed model for prediction of geomaterials stiffness for single-layer (ks-SUBG) and two-layer (ks-COMP) geosystems is expressed using Equation 3-7. , , , , , , , , , , (3-7)1 2 3 1 2 3d f k k k k k k h L D WSSN b b b s s s( )= ′ ′ ′ ′ ′ ′ where ks = the stiffness of geomaterials (in MN/m); W = weight (in kN), including the total force due to the drum weight, FD, and the peak eccentric vertical force, Fev; L = the drum length (in m); D = the drum diameter (in m); h = the thickness of the base (in mm); and k ′i b and k ′i s = nonlinear parameters of the base and subgrade, respectively. The composite stiffness of compacted geomaterials in two-layer system, ks, can be predicted using the following GP equation: , (3-8)1 1 2 2 1 3 4 3 5 2 6 2 1 2 2k C k C hk Lk e C C k C hk e C h k k s s b s k s b k b b s s( )= ′ + ′ + ′ + ′ + ′ + ψ ′ ′ ′ ′ where C1 = 0.0252, C2 = 0.135, C3 = 0.0339, C4 = 0.00616, C5 = –0.0143, C6 = –0.0399, and y = an operating index, as defined in Equation 3-5. 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 A N N -P re di ct ed D is pl ac em en t ( m m ) FE-Determined Displacement (mm) Line of Equality +/- 10% Error Line R² = 0.99 SEE = 0.10 mm Figure 3-9. Comparison of predicted surface displacement from ANN method versus FE model.

40 Evaluating Mechanical Properties of Earth Material During Intelligent Compaction For the prediction of stiffness for single-layer geosystems, Equation 3.8 reduces to: , (3-9)1 1 1 3 4 32k C k Lk e C C ks s s k ss ( )= ′ + ′ + ′′ where coefficients C1, C3, and C4 are defined as shown in Equation 3-8. Figure 3-10 compares the GP-predicted composite stiffness values to the stiffness values as determined from the FE responses. Both single-layer and two-layer systems are evaluated and included in the figure. The figure shows that the proposed GP method can predict the stiffness favorably, as judged by the number of cases falling inside the 20% error lines. The best regres- sion line passing through the results shows an R2 value of 0.87 and a SEE value of 34.7 MN/m. As an alternative option, an ANN-based method also was developed with the purpose of improving the prediction of stiffness. Figure 3-11 compares the ANN-predicted composite stiffness for both single-layer and two-layer systems with the stiffness determined from the SSN FE models. The ANN method yielded a more favorable prediction of stiffness than the GP method, as shown by the higher R2 value (0.99) and the lower SEE value (11.8 MN/m). Thus, the ANN method proves to be an efficient tool that can reproduce the results provided by FE models in an expedited manner without conceding accuracy, making it suitable for implementation in field operations. R² = 0.99 SEE = 11.8 MN/m 0 200 400 600 800 0 200 400 600 800 A N N -P re di ct ed S tif fn es s (M N /m ) FE-Determined Stiffness (MN/m) Line of Equality +/- 20% Error Line Figure 3-11. Comparison of ANN-predicted stiffness of layered geomaterials to FE-determined stiffness obtained from stationary drums with different operating features. y = 0.98x R² = 0.87 SEE = 34.7 MN/m 0 200 400 600 800 0 200 400 600 800 G P- Pr ed ic te d St iff ne ss (M N /m ) FE-Determined Stiffness (MN/m) Line of Equality +/- 20% Error Line Figure 3-10. Comparison of GP-predicted stiffness of layered geomaterials to FE-determined stiffness obtained from stationary drums with different operating features.