Modeling Permanent Deformation of Unbound Granular Materials under Repeated Loads

Finite-element analysis on a pavement structure under traffic loads has been a viable option for researchers and designers in highway pavement design and analysis. Most of the constitutive drivers used were nonlinear elastic models defined by empirical resilient modulus equations. Few isotropic/kinematic hardening elastoplastic models were used but applying thousands of repeated load cycles became computationally expensive. In this paper, a cyclic plasticity model based on fuzzy plasticity theory is presented to model the long-term behavior of unbound granular materials under repeated loads. The discussion focuses on the model parameters that control long-term behavior such as elastic shakedown. The performance of the constitutive model is presented by comparing modeled and measured permanent strain at various numbers of load cycles. Calculated resilient modulus from the complete stress-strain curve is also discussed.     

Simple Nonlinear Model for Elastic Response of Cohesionless Granular Materials

A constitutive model based on hyperelasticity is proposed to capture the resilient (elastic) behavior of granular materials. Resilient behavior is a widely accepted idealization of the response of unbound granular layers of pavements, following shakedown. The coupling property of the proposed model accounts for shear dilatancy and pressure-dependent behavior of the granular materials. The model is calibrated using triaxial resilient test data obtained from the literature. A statistical comparison is made between the predictions of the proposed model and a few of the prominent models of resilient response. The proposed coupled hyperelastic model yields a significantly better fit to the experimental data. It also offers a computational efficiency when implemented in a classical nonlinear finite elemental framework.     

Mechanistic Corrections for Determining the Resilient Modulus of Base Course Materials Based on Elastic Wave Measurements

The mechanical performance of pavement systems depends on the stiffness of subsurface soil and aggregate materials. The moduli of base course, subbase, and subgrade soils included in pavement systems need to be characterized for their use in the new empirical-mechanistic design procedure (NCHRP 1-37A). Typically, the resilient modulus test is used in the design of base and subbase layers under repetitive loads. Unfortunately, resilient modulus tests are expensive and cannot be applied to materials that contain particles larger than 25 mm (for 125-mm diameter specimens) without scalping the large grains. This paper examines a new methodology for estimating resilient modulus based on the propagation of elastic waves. The method is based on using a mechanistic approach that relates the P-wave velocity-based modulus to the resilient modulus through corrections for stress, void ratio, strain, and Poisson’s ratio effects. Results of this study indicate that resilient moduli are approximately 30% of Young’s moduli based on seismic measurements. The technique is then applied to specimens with large-grain particles. Results show that the methodology can be applied to large-grained materials and their resilient modulus can be estimated with reasonable accuracy based on seismic techniques. An approach is proposed to apply the technique to field determinations of modulus.     

Experimental and Modeling Studies of Permanent Strains of Subgrade Soils

Mechanistic-empirical pavement design guide for flexible pavements as per the AASHTO design guide requires characterization of subgrade soils using the resilient modulus (MR) property. This property, however, does not fully account for the plastic or permanent strain or rutting of subgrade soils, which often distress the overlying pavements. Soils such as silts exhibit moderate to high resilient moduli properties but they still undergo large permanent deformations under repeated loading. This explains the fallacy in the current pavement material characterization practice. A comprehensive research study was performed to measure permanent deformation properties of subgrade soils by subjecting various soils under repeated cycles of deviatoric loads. This paper describes test procedure followed and results obtained on three soils including clay, silt, and sandy soils. The influence of compaction moisture content, confining pressure, and deviatoric stresses applied on the measured permanent deformations of all three soils are addressed. A four-parameter permanent strain model formulation as a function of stress states in soils and the number of loading cycles was used to model and analyze the present test results. The model constants of all three soils were first determined and these results were used to explain the effects of various soil properties on permanent deformations of soils. Validation studies were performed to address the adequacy of the formulated model to predict rutting or permanent strains in soils.     

Finite‐Element Calculations on Alliant FX/80

The finite‐element method has proven to be an invaluable tool for analysis and design of complex, high‐performance systems, such as those typically encountered in the aerospace or automotive industries. However, as the size of the finite‐element models of such systems increases, analysis computation time using conventional computers can become prohibitively high. Parallel processing computers provide the means to overcome these computation‐time limits, provided the algorithms used in the analysis can take advantage of multiple processors. The writers have examined several algorithms for linear and nonlinear static analysis, as well as dynamic finite‐element analysis. The performance of these algorithms on an Alliant FX/80 parallel supercomputer has been investigated. For single load‐case linear static analysis, the optimal solution algorithm is strongly problem dependent. For multiple load cases or nonlinear static analysis through a modified Newton‐Raphson method, decomposition algorithms are shown to have a decided advantage over element‐by‐element preconditioned conjugate gradient algorithms. For eigenvalue/eigenvector analysis, the subspace iteration algorithm with a parallel decomposition is shown to achieve a relatively high parallel efficiency.     

Implementation of a Critical State Two-Surface Model to Evaluate the Response of Geosynthetic Reinforced Pavements

A finite-element model was developed using ABAQUS software package to investigate the effect of placing geosynthetic reinforcement within the base course layer on the response of a flexible pavement structure. A critical state two-surface constitutive model was first modified to represent the behavior of base course materials under the unsaturated field conditions. The modified model was then implemented into ABAQUS through a user defined subroutine, UMAT. The implemented model was validated using the results of laboratory triaxial tests. Finite-element analyses were then conducted on different unreinforced and geosynthetic reinforced flexible pavement sections. The results of this study demonstrated the ability of the modified critical state two-surface constitutive model to predict, with good accuracy, the response of the considered base course material at its optimum field conditions when subjected to cyclic as well as static loads. The results of the finite-element analyses showed that the geosynthetic reinforcement reduced the lateral strains within the base course and subgrade layers. Furthermore, the inclusion of the geosynthetic layer resulted in a significant reduction in the vertical and shear strains at the top of the subgrade layer. The improvement of the geosynthetic layer was found to be more pronounced in the development of the plastic strains rather than the resilient strains. The reinforcement benefits were enhanced as its elastic modulus increased.     

Effect of Stress‐Dependent Base Layer on the Superposition of Flexible Pavement Solutions

Flexible pavement structural analysis for design usage must consider (as a minimum) multiple wheel/axle loading configurations, seasonal variations of material layer properties, and the nonlinear behavior of unbound materials. Although these requirements are all easily within the capabilities of three‐dimensional finite element analysis, the required computation times may be impracticably long for routine design. Compromises between analytical rigor (e.g., three‐dimensionality) and analysis features (e.g., multiple wheels, seasonal property variations, material nonlinearity) must be made. One compromise is to retain seasonal property variations and material nonlinearity within an axisymmetric single wheel finite element model and to approximate multiple wheel effects via superposition. Although this superposition of nonlinear solutions is undeniably invalid from a rigorous theoretical viewpoint, the errors may be well within acceptable magnitudes for practical design. The paper investigates this issue by comparing superimposed nonlinear solutions against computationally rigorous three‐dimensional nonlinear solutions and evaluating the discrepancies in key pavement response quantities. The results suggest that the errors from superimposing nonlinear solutions are acceptably small for key pavement response quantities. Moreover, these errors are substantially smaller than those resulting from neglect of nonlinear unbound material behavior, a modeling compromise that is common in pavement structural analysis today.     

Modeling Nonlinear Anisotropic Elastic Properties of Unbound Granular Bases Using Microstructure Distribution Tensors

The resilient properties of unbound aggregate bases are important parameters in the design of asphalt pavements. Previous studies have shown that these resilient properties exhibit nonlinear and transverse anisotropic characteristics. The paper in hand presents a micromechanics-based approach to model the nonlinear and anisotropic properties of unbound aggregate bases. The anisotropic behavior is captured using two microstructure parameters representing the preferred orientation of aggregate particles, and the ratio of the normal contact stiffness to shear contact stiffness among particles. The nonlinear response is modeled using a relationship that relates the shear modulus to particle packing, material properties, particle size, and confining pressure. The micromechanics model is used to represent the resilient properties for a total of 18 different combinations of material conditions with different aggregate types, moisture contents, and gradation characteristics. Anisotropic and nonlinear resilient properties were measured at ten different stress states for each of the material conditions. The results presented in this paper show that the micromechanics model is capable of successfully representing the experimental measurements.     

Normalizing Behavior of Unsaturated Granular Pavement Materials

One of the important components of a flexible pavement structure is granular material layers. Unsaturated granular pavement materials (UGPMs) in these layers influence stresses and strains throughout the pavement structure, and have a large effect on asphalt concrete fatigue and pavement rutting (two of the primary failure mechanisms for flexible pavements). The behavior of UGPMs is dependent on water content, but this effect has been traditionally difficult to quantify using either empirical or mechanistic methods. This paper presents a practical mechanistic framework for determining the behavior of UGPMs within the range of water contents, densities, and stress states likely to be encountered under field conditions. Both soil suction and generated pore pressures are determined and compared to confinement under typical field loading conditions. The framework utilizes a simple soil suction model that has three density-independent parameters, and can be determined using conventional triaxial equipment that is available in many pavement engineering laboratories.     

Shakedown Approaches to Rut Depth Prediction in Low-Volume Roads

Rutting, due to permanent deformations of unbound materials, is one of the principal damage modes of low traffic pavements. Flexible pavement design methods remain empirical; they do not take into account the inelastic behavior of pavement materials and do not predict the rutting under cyclic loading. A finite-element program, based on the concept of the shakedown theory developed by Zarka for metallic structures under cyclic loadings, has been used to estimate the permanent deformations of unbound granular materials subjected to traffic loading. Based on repeated load triaxial tests, a general procedure has been developed for the determination of the material parameters of the constitutive model. Finally, the results of a finite-element modeling of the long-term behavior of a flexible pavement with the simplified method are presented and compared to the results of a full-scale flexible pavement experiment performed by Laboratoire Central des Ponts et Chaussées. Finally, the calculation of the rut depth evolution with time is carried out.     

Rate-Type Model for Bituminous Mixtures and Its Application to Sand Asphalt

A variety of hot mix asphalt mixtures are used in highway and runway pavement construction. Each mixture caters to specific needs. Mixtures differ from one another in the type and percentage of aggregates and asphalt used, and their response can be markedly different, and thus there is a need to develop constitutive models that can differentiate between the different kinds of mixtures. In this paper, we outline a general procedure for the constitutive modeling of bituminous mixtures. We illustrate the efficacy of this approach by means of an application to sand asphalt. The governing equations for this special problem reduce to a stiff nonlinear ordinary differential equation and this is solved numerically using Gear’s method. We compare the results of the predictions of the model that we have developed with the compressive creep experiments carried out by Wood and Goetz on a typical sand asphalt mixture and find them to be in good agreement.     

“Underlying” Causes for Settlement of Bridge Approach Pavement Systems

A comprehensive field study of 74 bridges in Iowa was conducted to characterize problems leading to poor performance of bridge approach pavement systems. Subsurface void development caused by water infiltration through unsealed expansion joints, collapse and erosion of the granular backfill, and poor construction practices were found to be the main contributing factors. To characterize the problem, International Roughness Index and profile measurements from several sites were used to show that approach pavement roughness is several times higher than the average roadway condition and are most severe at the abutment-to-approach pavement intersection and transverse expansion joints due to large (5–10?cm) joint widths. Further, a settlement time history was documented at one bridge site by measuring the approach slab pavement elevations periodically after completion of bridge construction, revealing a progressive settlement problem under the approach pavement. To better understand the void development under the approach pavement, laboratory compaction tests were performed on granular backfill materials from various bridge sites to quantify their saturated collapse potential in the postconstruction phase. These tests revealed collapse potential of backfill materials in the range of 5–18% (based on volume) with the high values for poorly graded sandy backfill materials, indicating significant settlement problems. Based on the research findings, some relatively simple design and construction modifications are suggested which could be used to alleviate field problems for similar bridge approach pavement systems.     

Unbonded Posttensioned Concrete Bridge Piers. II: Seismic Analyses

The seismic response characteristics of a proposed unbonded posttensioned concrete bridge-pier system are evaluated. Time-history analyses are carried out on prototype designs of single-column piers and two-column bents using detailed nonlinear finite-element (FE) models and equivalent single-degree-of-freedom (SDOF) systems embedded with phenomenological constitutive models. The phenomenological models are based on the hysteretic behavior of the prototype designs from cyclic analyses using nonlinear FE models, which have been calibrated and verified against experiments. The two modeling techniques are compared and evaluated for simulating the response of unbonded posttensioned bridge piers. Extensive time-history analyses are carried out on the SDOF models to study the influence of unbonded posttensioning on seismic response. To assess the adequacy of the proposed bridge-pier system, the seismic demands on the prototype designs are compared to their capacities as established in a companion paper. The applicability of current bridge design specifications to designing the proposed bridge-pier system is discussed.     

Sensitivity of Seismic Applications to Different Shape Memory Alloy Models

Shape memory alloys (SMAs) are known for their superelastic properties, which have been exploited in numerous applications in the biomedical, aerospace, and commercial fields. More recently, these materials have been evaluated for applications in the area of earthquake engineering. One key question that arises when using these materials is the appropriate constitutive material model to use to capture the highly nonlinear behavior of SMAs. This paper explores the effect of using different SMA constitutive models on the resulting response of systems using SMAs. A sensitivity analysis is conducted by using three SMA models with various levels of complexity. The models are implemented in a single-degree-of-freedom system and subjected to three groups of earthquake records with various characteristics. Considering a more accurate trend in modeling incomplete cycles in SMAs has little impact on the structural response. The strength degradation and residual deformation seem to be of more importance than the sublooping behavior. The response is more sensitive to the cyclic effects in the case of records with long durations or large intensities.     

Numerical Study of Reinforced Soil Segmental Walls Using Three Different Constitutive Soil Models

A numerical finite-difference method (FLAC) model was used to investigate the influence of constitutive soil model on predicted response of two full-scale reinforced soil walls during construction and surcharge loading. One wall was reinforced with a relatively extensible polymeric geogrid and the other with a relatively stiff welded wire mesh. The backfill sand was modeled using three different constitutive soil models varying as follows with respect to increasing complexity: linear elastic-plastic Mohr-Coulomb, modified Duncan-Chang hyperbolic model, and Lade’s single hardening model. Calculated results were compared against toe footing loads, foundation pressures, facing displacements, connection loads, and reinforcement strains. In general, predictions were within measurement accuracy for the end-of-construction and surcharge load levels corresponding to working stress conditions. However, the modified Duncan-Chang model which explicitly considers plane strain boundary conditions is a good compromise between prediction accuracy and availability of parameters from conventional triaxial compression testing. The results of this investigation give confidence that numerical FLAC models using this simple soil constitutive model are adequate to predict the performance of reinforced soil walls under typical operational conditions provided that the soil reinforcement, interfaces, boundaries, construction sequence, and soil compaction are modeled correctly. Further improvement of predictions using more sophisticated soil models is not guaranteed.     

Geomechanical Analysis of Unbound Pavements Based on Shakedown Theory

A procedure for analyzing the mechanical response of an unbound pavement to the repeated loading of traffic is presented. The pavement is modeled as a layered elastic/plastic structure, and its response is described by the concepts of shakedown theory. A critical shakedown load is identified as the key design parameter. Pavements operating at higher loads will eventually fail, and those operating at loads less than critical may initially exhibit some distress but will eventually shakedown to a steady state. Estimates of this critical load, for different types of pavement, are found by studying various types of failure mechanisms, such as rut formation and subsurface slip. Optimization procedures are then used to determine the most likely form of failure for a particular pavement. The effects of self-weight, dual loads, moisture content, relative strengths of the various layers, and nonassociated plastic flow are studied. Some preliminary implications for pavement design are discussed.     

Modeling the Nonlinear Behavior of Concrete Masonry Walls Retrofitted with Steel Studs under Blast Loading

This paper presents the results of an analytical investigation of one-way unreinforced masonry (URM) walls retrofitted with externally anchored steel studs and subjected to blast loads. Using the wall geometrical and material properties, deflected shape, and crack pattern as input, a nonlinear model is developed to predict the inward force-displacement relationship of the retrofitted walls. In addition, using a rigid body analysis, a simple bilinear force-displacement relationship is developed to model the outward force-displacement relationship of the walls. Utilizing these two force-displacement relationships (resistance functions), a generalized single-degree-of-freedom (SDOF) model is developed to capture the nonlinear out-of-plane dynamic response of the retrofitted walls under blast loads. The SDOF model captured the experimentally observed displacement responses of the tested walls with reasonable accuracy. The model was also used to investigate the influence of block thickness, wall slenderness ratio, blast load intensity, and blast pulse shape on the out-of-plane dynamic response of retrofitted walls. The results demonstrated that anchored steel-stud systems could significantly enhance the out-of-plane capacity of the retrofitted walls by increasing their out-of-plane capacity and reducing their displacement.     

Degradation of a Granular Base under a Flexible Pavement: DEM Simulation

Flexible pavements are composed by an asphalt concrete layer, granular base and subbase layers, and a natural subgrade. The granular materials forming part of the granular layers are subjected to static and dynamic loads during their engineering life. As a result of these loads particle crushing may occur depending on the strength of the particles forming the granular layers. Particle crushing is important since it is associated with several detrimental effects such as settlements and a reduction in hydraulic conductivity. A computer simulation using the discrete element method (DEM) is presented in order to understand and visualize how crushing initiates and develops inside a simulated pavement structure.     

Numerical Implementation of Polymer Viscoplastic Equations for High Strain-Rate Composite Models

For polymer matrix composites subjected to large strain rates, it is important to correctly characterize the nonlinear and strain-rate dependent response of polymers. Viscoplastic constitutive material models have been developed to account for the effects of hydrostatic effects and inelastic strains in polymer materials. The effective implementation of such viscoplastic models is important for development of composite models geared toward practical applications. Goldberg’s polymer model numerical implementation into a commercial finite-element code constitutes the main objective of this paper. Special attention is given to the use of effective algorithms for solving the model nonlinear rate dependent viscoplastic equations. Existent experimental data are used to verify the accuracy and robustness of the computational polymer model. A phenomenological fiber model and a simplified iso-strain mixture theory used to obtain the resultant stresses in the composite by averaging the stresses of the individual constituents are also defined. The validation of the simplified mixture theory for the composite model will be presented later on.     

Lagrangian Approach to Structural Collapse Simulation

Computer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach is developed, which is a mixed method, where besides displacements, the stress resultants and other variables of state are primary unknowns. The method can potentially be used for the analysis of collapse of structures subjected to severe vibrations resulting from shocks or dynamic loads. The evolution of the structural state in time is provided a weak formulation using Hamilton’s principle. It is shown that a certain class of structures, known as reciprocal structures, has a mixed Lagrangian formulation in terms of displacements and internal forces. The form of the Lagrangian is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and it is solved using an augmented Lagrangian algorithm. In contrast to the traditional displacement-based method, the Lagrangian method provides a generalized formulation which clearly separates the modeling of components from the numerical solution. Phenomenological models of components, essential to simulate collapse, can be incorporated without having to implement model-specific incremental state determination algorithms. The state variables are determined at the global level by the optimization method.