Analysis and Implementation of Resilient Modulus Models for Granular Solids

Constitutive equations based upon stress dependent moduli, like K-θ and Uzan-Witczak, are widely used to characterize the resilient response of granular materials for the analysis and design of pavement systems. These constitutive models are motivated by the observation that the granular layers used in pavement structures shake down to (nonlinear) elastic response under construction loads and will, therefore, respond elastically under service loads typically felt by these systems. Due to their simplicity, their great success in organizing the response data from cyclic triaxial tests, and their success relative to competing material models in predicting the behavior observed in the field, these resilient modulus constitutive models have been implemented in many computer programs used by researchers and design engineers. This paper provides an analysis of the nonlinear solution algorithms that have been used in implementing these models in a conventional nonlinear 3D finite-element framework. The analysis shows that these conventional algorithms are destined to fail at higher load levels. The paper offers two competitive methods for global analysis with these models. A comparative study of eight possible implementations of the algorithms described in the paper is made through two simulation examples.     

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.     

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.     

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.     

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.     

Creep-Effect on Mechanical Behavior of Concrete Confined by FRP under Axial Compression

Despite many successes in concrete creep studies, its effect on the mechanical behavior of concrete members is far from a thorough case-specific understanding. For the members that have been subjected to a long-term load, the classical stress-strain models describing the short-term behavior of either confined or unconfined concrete are unsuitable. In order to investigate this creep-effect, an experiment on eight concrete cylindrical columns confined by fiber-reinforced polymer (FRP) is carried out. Based on the theory of plasticity for concrete, a constitutive model that takes into account the effect of creep on mechanical behavior of concrete confined by FRP is presented. In the model, the creep law inspired in the microprestress-solidification theory is generalized to triaxial stress condition for the calculation of the creep of the concrete columns confined by FRP. The predictions of the model agree well with the experimental results. The present study indicates that the creep increases the elastic modulus, slightly decreases the compressive strength, and degrades the deformation capability of the concrete confined by FRP.     

Variable Bulk Modulus Constitutive Model for Sand

A quasi-linear elastic constitutive model is proposed to describe the behavior of sand well below failure. It is based on isotropic compression tests and captures the increase in stiffness due to confinement—a unique property that makes sand stiffer under certain applied loads. The model traces the increase in stiffness by describing the instantaneous bulk modulus in terms of the effective mean stress and two constitutive parameters: the initial bulk modulus and the ultimate volumetric strain. The validity of the model is examined by checking how closely the proposed mathematical formulation represents high quality isotropic compression tests reported in the literature.     

Neural Network Modeling of Resilient Modulus Using Routine Subgrade Soil Properties

Artificial neural network (ANN) models are developed in this study to correlate resilient modulus with routine properties of subgrade soils and state of stress for pavement design application. A database is developed containing grain size distribution, Atterberg limits, standard Proctor, unconfined compression, and resilient modulus results for 97 soils from 16 different counties in Oklahoma. Of these, 63 soils (development data set) are used in training, and the remaining 34 soils (evaluation data set) from two different counties are used in the evaluation of the developed models. A commercial software, STATISTICA 7.1, is used to develop four different feedforward-type ANN models: linear network, general regression neural network, radial basis function network, and multilayer perceptrons network (MLPN). In each of these models, the input layer consists of seven nodes, one node for each of the independent variables, namely moisture content (w), dry density (γd), plasticity index (PI), percent passing sieve No. 200 (P200), unconfined compressive strength (Uc), deviatoric stress (σd), and bulk stress (θ). The output layer consists of only one node—resilient modulus (MR). After the architecture is set, the development data set is fed into the model for training. The strengths and weaknesses of the developed models are examined by comparing the predicted MR values with the experimental values with respect to the R2 values. Overall, the MLPN model with two hidden layers was found to be the best model for the present development and evaluation data sets. This model as well as the other models could be refined using an enriched database.     

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.     

Pressure-Dependent Elasticity and Energy Conservation in Elastoplastic Models for Soils

This paper presents a study on the consequences of combining energy conservative or non-conservative elasticity within a plasticity framework. Toward this end, a versatile energy potential function is first presented and examined. It is shown to cover a wide range of existing empirical relations for pressure-dependent stiffness of soils. Utilization of these functions within hyperplastic constitutive framework allows for the resulting models to satisfy the Law of Energy Conservation for both elastic and plastic components of soil behavior. Apart from the theoretical rigor, a very important result of this approach is that it automatically implies stress-induced cross-anisotropy of the elastic component of soil behavior and dilatancy term occurs due to shear modulus dependency on pressure. Proper modeling of these phenomena, normally neglected by conventional hypoelastic-plastic models, has been shown to have a significant effect on the accuracy of the model predictions of undrained behavior of overconsolidated clays both in laboratory tests and in tunnel excavation problem.     

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.     

Suction-Controlled Laboratory Test on Resilient Modulus of Unsaturated Compacted Subgrade Soils

Conventionally, the resilient modulus test is conducted in the laboratory under different moisture content in which matric suction is unknown during the test. To investigate the influence of the matric suction on the resilient modulus, this study integrated the suction-controlled testing system and developed a modified testing procedure for the resilient modulus test of unsaturated subgrade soils. Based on the axis-translation technique, two cohesive soils were tested to investigate the effect of matric suction on resilient modulus. In the modified testing procedure, in order to fulfill the equilibrium in matric suction, the number of load cycles at each loading sequence of the resilient modulus test (AASHTO T 292-91) needs to be increased significantly. Experimental data indicate that matric suctions measured in the specimen after consolidation and resilient modulus tests are consistent with the matric suctions deduced from the soil-water characteristic curve corresponding to the same moisture content. In general, the resilient modulus obtained by the suction-controlled resilient modulus test appears to be reasonable. The trends of resilient modulus obtained by the suction-controlled resilient modulus test are consistent with those obtained by the conventional resilient modulus test. However, the suction-controlled resilient modulus test provides better insights that can help in interpreting the test results.     

Energy Evaluation for Ramberg–Osgood Hysteretic Model

Response of structures to static and dynamic loads may be assessed in terms of energy. The energy evaluation depends on hysteretic models used to describe inelastic behavior of structures and their elements. One of the hysteretic models often employed in structural analysis is the Ramberg–Osgood model. In the paper a physically motivated model, which leads to the Ramberg–Osgood force–displacement relationship under monotonic loading and exhibits Masing type of behavior for unloading/reloading, is described. Based on the model formulas to calculate recoverable elastic strain energy and irrecoverable hysteretic energy for systems/elements under monotonic loading are derived. It is also shown how recoverable elastic strain energy and irrecoverable hysteretic energy can be evaluated at any point of unloading/reloading curves for a system/element of Masing type. The application of the derived formulas is illustrated by evaluating the energy of a single-degree-of-freedom system subjected to impulse and seismic loading.     

Axial Compression of Footings in Cohesionless Soils. I: Load-Settlement Behavior

The results of 167 full-scale field load tests were used to examine several issues related to the load-displacement behavior of footings in cohesionless soils under axial compression loading, including (1) method to interpret the “failure load” from the load-settlement curves; (2) correlations among interpreted loads and settlements; and (3) generalized load-settlement behavior. The L1-L2 method was found to be more appropriate than the “tangent intersection” and “10% of the footing width” methods for interpreting the failure load. The interpreted loads and displacements indicate that footing load-settlement behavior is less elastic and more nonlinear than that of drilled foundations. The results show that the footing behavior will be beyond the elastic limit for designs where a traditional factor of safety between 2 and 3 is used. A normalized curve was developed by approximating the load-settlement curve for each load test in the database by hyperbolic fitting, and the uncertainty in this curve was quantified. This normalized curve can be used in footing design that considers capacity and settlement together. Where possible or warranted, the normalized curve can be subdivided as a function of initial soil modulus.     

Three‐Dimensional Finite Element Analysis in Airport Pavement Design

This paper discusses work being performed at the Federal Aviation Administration (FAA) Airport Technology R&D Branch in the development of a three‐dimensional finite element‐based airport pavement design procedure for rigid airport pavements. The structure of the pavement design procedure and the function of the finite element structural model within it are described. A major focus of current FAA research and development efforts is on reducing run time. A simplified, single‐slab mesh runs on a personal computer and returns a maximum edge stress in a fraction of the time required by the full nine‐slab mesh. Results are presented for the simplified mesh for various aircraft types and slab sizes and compared to the larger mesh. Two types of foundation models are considered to represent a subgrade of infinite depth. A subgrade model consisting of discrete springs at nodal points approximates the distributed spring (Winkler) foundation with subgrade modulus k used in Westergaard analysis. An alternative model makes use of infinite elements to represent a linear elastic foundation with elastic modulus E and Poisson’s ratio μ. Stress computations using both models show that the Winkler foundation model is significantly more sensitive to slab size than the infinite element model for dual‐tridem (six‐wheel) aircraft gear loads. In a recent project at the FAA Center of Excellence (COE) for Airport Pavement Research, the open source code software (Nike3D) used in the three‐dimensional finite element computations to include the infinite element formulation. The infinite element was implemented as a new material type applicable to standard eight‐node elements in the Nike3D element library.     

Stability and Collapse of Semirigidly Connected Portal Frames

A numerical procedure for the nonlinear elastic‐plastic instability analysis and collapse of semirigidly connected portal frames, with elastic rotational restraints at the supports, is presented. The procedure is based on nonlinear kinematic relations and linearly elastic material behavior except at the plastic regions (concentrated plasticity). The nonlinear flexible connections are represented by polynomial models. A computational technique for incorporating the stability and strength into the analysis is described in detail. It is found that several important parameters affect the failure modes and consequently the critical loads. These parameters are the slenderness ratio, support restraints, type of connections, and the loading conditions. It is also demonstrated that the connection flexibility has considerable effect on the critical load and the deformation. It is further concluded that for design application the assumption of linear (instead of nolinear, polynomial) connection behavior is adequate for portal frames only if the loading conditions do not produce a significant amount of bending moment at the joints.     

Finite Element Analysis of Concrete Pavements Over Culverts

Accelerated distress of Portland cement concrete pavements (PCCP) over structures such as culverts, pipes, and tunnels beneath roadways is a common occurrence. In this article, finite element analysis is employed to analyze the response of concrete pavements over such structures. The factors that influence the overlying pavement slabs include: (1) cover depth, (2) pavement slab thickness and length, (3) cement concrete elastic modulus, (4) foundation modulus, and (5) backfill soil modulus. The tensile stresses at the bottom and top of the slab induced by wheel loads are predicted. In the traditional pavement design only the tensile stress at the bottom of the slab is considered to be significant. However, this study shows that the tensile stress at the top surface of pavement slabs over culverts may also cause the concrete pavements to fail. A laboratory model was employed to study the mechanical characteristics of Portland cement concrete pavement slabs over culverts and to verify the theoretical analysis.     

Finite Element Studies of Asphalt Concrete Pavement Reinforced with Geogrid

Many geotechnical applications are becoming more sophisticated and solutions derived from simplistic procedure are no longer reasonable or solutions do not exist. This paper describes two-dimensional finite element studies that analyzed the behavior of reinforced asphalt pavement under plane strain conditions and subject to monotonic loading. The asphalt material and soils were expressed using triangular elements of elastoplastic behavior that obeys Mohr–Coulomb criteria with associated and nonassociated flow rules. The geogrid was modeled using a one-dimensional linear elastic bar element. The finite element procedure was validated by comparing the results of analysis with the results obtained from a series of model tests. The load–settlement relationships, settlement profile, and strains in the geogrid were compared. The failure load obtained by assuming subgrade foundation with nonassociated flow rule was smaller than that of associated flow rule. There was only minor difference between the results obtained from the associated and nonassociated plastic models. The finite element procedure was capable of determining most measured quantities satisfactorily except the tensile strain in the geogrid, which was assumed linear elastic. The effects of the stiffness of geogrid reinforcement, thickness of asphalt layer, and strength of subgrade foundation were also investigated. The finite element procedure is a versatile tool for enhanced design of reinforced pavement systems.     

Unified DSC Constitutive Model for Pavement Materials with Numerical Implementation

Need for unified and mechanistic constitutive models for pavement materials for evaluation of various distresses has been recognized; however, such models are not yet available. There have been efforts to develop unified models; however, they have been based usually on ad hoc combinations of models for special properties such as elastic, plastic, creep and fracture, often without appropriate connections to various coupled responses of bound and unbound materials, they may result and in a large number of parameters, often without physical meanings. The disturbed state concept (DSC) provides a modeling approach that includes various responses such as elastic, plastic, creep, microcracking and fracture, softening and healing under mechanical and environmental (thermal, moisture, etc.) within a single unified and coupled framework. A brief review is presented to identify the advantages of the DSC compared to other available models. The DSC has been validated and applied to a wide range of materials: geologic, asphalt, concrete, ceramic, metal alloys, and silicon. It allows for evaluation of various distresses such as permanent deformations (rutting), microcracking and fracture, reflection cracking, thermal cracking, and healing. The DSC is implemented in two- and three-dimensional finite-element (FE) procedures, which allow static, repetitive, and dynamic loads including elastic, plastic, creep, microcracking leading to fracture and failure. A number of examples are solved for various distresses considering flexible (asphalt) pavements; however, the DSC model is applicable to rigid (concrete) pavements also. It is felt that the DSC and the FE computer programs provide unique and novel approaches for pavement engineering. It is desirable to perform further research and applications including validation with respect to simulated and field behavior of pavements.