Journal of Engineering Mechanics

The transportation sector is responsible for 27% of greenhouse gas emissions in the United States, implying that it plays an important role in achieving sustainability goals and slowing down the impact of climate change. To understand the influence of road roughness on vehicle fuel consumption, researchers have recently proposed stochastic pavement models combined with a mechanics-based vehicle model to study the energy dissipated by the vehicle suspension. Solutions were obtained in the frequency domain, where the computational effort increases rapidly as more complicated vehicle types and surface profiles are considered. This paper proposes a new formulation and solution procedure for estimating the energy dissipation in the tire and the suspension of vehicles traversing rough nondeformable pavements. Herein, a filtered white noise model of the road roughness is combined with a quarter-car model of the vehicle to yield an augmented state space representation. The Lyapunov equation governing the covariance of the responses is then solved to obtain the energy dissipation of the vehicle. Subsequently, examples are provided to illustrate the efficacy and flexibility of the proposed approach.

In this paper, a recently developed type of lower-order mixed finite elements is extended to model porous materials based on the microdilatation theory. These mixed finite elements are based on assuming independent linear generalized strain fields and collocating them with the generalized strains derived from primal variables (mechanical displacements and change in matrix volume fraction) at some cleverly chosen points within each element. This mixed formulation is very effective in alleviating the shear locking problem that regular lower-order finite elements suffer from. Hence the accuracy of the predicted mechanical fields (such as displacements and stresses), as well as the fields coupled with them (such as change in matrix volume fraction, which is also called microdilatation), is improved over regular finite-element formulation. The mixed-collocation formulation is also superior over other types of previously published hybrid-mixed finite-element formulations in that it avoids the Ladyzenskaja–Babuška–Brezzi (LBB) stability conditions completely because it does not include any Lagrange multipliers. The paper also presents some numerical examples that help in providing more insight on the effect of porosity-related parameters used in microdilatation theory on the behavior of porous materials. Finally, the paper defines two limits on the coupling number; the first considers the positive definiteness of the stored energy density, whereas the second sets the limit between auxetic (having negative Poisson’s ratio) and nonauxetic material behavior.

Retaining walls are widely used for shoring excavations and stabilizing earth slopes. They often support naturally formed soils above the groundwater table and compacted soils, which are in an unsaturated state. To perform reliable designs, it is important to be able to model how retaining walls interact with unsaturated soils and, in particular, be able to estimate the earth pressures that act on the walls. In this paper, soil responses observed in three retaining wall model tests involving unsaturated soil samples are compared with computations made using slip line theory. Each test consisted of a rigid retaining wall rotating about its toe into a silty sand sample. The soil in the theory was treated as rigid perfectly plastic continuum, with strength controlled by the Mohr-Coulomb criterion. The effective stress was used to incorporate the effects of suction. The suctions inside the unsaturated soil samples made them stiff and brittle. Extensive rupturing occurred in the soil samples adjacent to the wall as rotations occurred. The rupturing caused the soil behavior to divert from that of a continuum, thus the slip line theory could not provide an accurate match to the experimental measurements. Most notably, the measured earth pressure profiles exhibited localized spikes that could not be modeled by the slip line theory. Theoretically computed vertical soil displacements also diverted from those measured experimentally, although the theoretical and measured horizontal displacements were in reasonable agreement. Although these results are negative, they will guide researchers and practitioners on what level of accuracy can be expected when using a rigid perfectly plastic continuum assumption to model highly brittle soils interacting with structures, where improvements are needed, and whether continuum-based theories should be overlooked in favor of others.

A semianalytical method hybridizing hierarchical theory and finite-strip method (FSM) is developed to analyze buckling behavior of composite laminate shells in this paper. The Legendre orthogonal polynomial series are chosen as the hierarchical shape functions for the shell strips. The critical buckling loads and modes of cylinder shells with various radiuses subjected to axial loads are calculated by the proposed hierarchical finite-strip method (HFSM). The efficiency of HFSM is noted as the method combines the advantages of both hierarchical theory and FSM. The convergence and precision of HFSM are also demonstrated by several numerical examples.

The response analysis of sliding objects and structures subjected to ground excitation is most often performed using Newmark’s rigid sliding block with Coulomb friction. This simplified sliding model has been shown to sometimes result in poor predictions of sliding displacement. In an effort to improve sliding response predictions, this paper introduces a model that relaxes the rigidity constraint of the Newmark model and takes into account velocity-dependence of the friction force using a Stribeck friction model. It is shown that the flexibility of the support system in conjunction with velocity-dependent friction at low velocities can result in intermittent stick-slip phases during sliding. A dimensional analysis approach is taken to expose the underlying physical similarities in the sliding problem and to compare the results of the stick-slip model with conventional sliding models, including Newmark’s rigid-block model and coupled sliding models that are not capable of capturing the stick-slip phenomenon. It is demonstrated that implementation of simplified sliding models may result in underestimation of the sliding response.

Understanding the mechanical behavior of fresh cut timber, in which the natural moisture content has been retained, has application in predicting the behavior of living tree elements such as branch/stem joints, potentially inspiring structural designs through biomimicry. This project develops a process of strain imaging using particle image velocimetry to analyze behavior of timber samples freshly cut from oak tree joints and tested in tension. The timber surface was not coated so the fiber response could be visualized directly. Load was applied in steps and different methods for comparing images were trialled, including a sequential method in which strain is accumulated progressively from image to image and a first-to-last comparison. Sample flexibility caused significant deformation in the camera field of view, and so the analysis methodology was modified so that each image was compared sequentially with the first with the target area in the image for strain calculation expanded as the test proceeded. This resulted in strain contour plots able to show the impending failure of the component as the tension is applied.

Structural monitoring systems installed on cable-supported bridges have the potential to generate large data repositories from which a deeper understanding of bridge behavior can be obtained. The core focus of this study is the use of response and environmental data collected by a permanent wireless monitoring system operating on the New Carquinez Bridge (Vallejo, California) since 2010. Given the large amount of data available, the study proposes an automated stochastic subspace identification approach for the extraction of bridge modal properties. The study fits logistic distributions to the extracted modal frequencies and modal damping ratios to provide statistical models for these important bridge properties. Low levels of modal damping well below 0.8% are reported for the majority of structural modes. Bridge modal properties exhibit sensitivity to the environmental and operational conditions of the bridge. Ridge regression and Gaussian process regression (GPR) are used to model the dependency of modal frequency on bridge environmental and operational conditions. The GPR models are shown capable of accurately modeling the relationship between modal frequency and bridge environmental and operational conditions.

Linear elastic fracture mechanics-based discrete crack propagation simulations are presented as the pullout of an anchor group from a concrete matrix. The group was modeled as a periodic arrangement of anchors and load-carrying capacities and crack paths were determined as functions of relative depth and spacing. The results suggest that American Concrete Institute design formulas for predicting the capacity of group anchors are highly conservative, and that three-dimensional fracture mechanics-based simulations offer great promise for improving design formulas associated with specific anchor geometries and loadings.

This study modeled structural performance of bituminous pavements with inelastic deformation and fatigue damage. Two major distresses, namely rutting and fatigue damage, of typical bituminous pavements were modeled by extending the pavement analysis using nonlinear damage approach (PANDA), a recently-developed finite-element approach to model damage-associated pavement performance, by incorporating main features in the PANDA with a fracture mechanics approach so as to improve the capability in predicting rate-dependent localized behavior of bituminous mixtures in pavement structures. More specifically, the Schapery’s nonlinear viscoelasticity and Perzyna-type viscoplasticity featured in the PANDA were used to characterize recoverable and irrecoverable deformation of bituminous paving mixtures, and damage of bituminous mixtures due to multiple microscale and macroscale cracks was characterized using the cohesive zone fracture law, which was incorporated with a relevant fracture test such as a semicircular bend test. To apply the modeling approach to real structures, laboratory tests with two typical bituminous mixtures were conducted to characterize the material properties necessary for the model. Then, model simulation results presenting the two primary structural damage modes, rutting and fatigue cracking, were compared to actual monitored field performance. Simulation results demonstrated that the model can successfully account for the effect of key design variables such as paving materials, structural layer configuration, loading condition, and environmental conditions (such as temperature) on the pavement performance with damage. This implies the potential power and efficacy of this mechanistic modeling approach for the analysis and design of bituminous materials and pavement structures.

In the past two decades, meshfree methods have emerged into a new class of computational methods with considerable success. In addition, a significant amount of progress has been made in addressing the major shortcomings that were present in these methods at the early stages of their development. For instance, essential boundary conditions are almost trivial to enforce by employing the techniques now available, and the need for high order quadrature has been circumvented with the development of advanced techniques, essentially eliminating the previously existing bottleneck of computational expense in meshfree methods. Given the proper treatment, nodal integration can be made accurate and free of spatial instability, making it possible to eliminate the need for a mesh entirely. Meshfree collocation methods have also undergone significant development, which also offer a truly meshfree solution. This paper gives an overview of many classes of meshfree methods and their applications, and several advances are described in detail.

Granular materials are the most recurrent form of solid-state matter on Earth. They challenge researchers and engineers in various fields not only because they occur with a broad variety of grain sizes, shapes and interactions in nature and industry, but also because they show a rich panoply of mechanical states. Despite this polymorphism, all these different types of soils, powders, granules, ores, pharmaceutical products, etc., are instances of the granular matter with the same least common denominator of being sandlike (psammoid in Greek), i.e., solid grains interacting via frictional contacts. This review describes milestone contributions to the field of granular materials since the early elastic-plastic models developed for soils in the 1950s. The research on granular materials has grown into a vast multidisciplinary field in the 1980s with increasing focus on the microstructure and owing to new experimental tools and discrete simulation methods. It turns out that the granular texture, particle-scale kinematics, and force transmission are far more complex than presumed in early micromechanical models of granular materials. Hence, constitutive relations cannot easily be derived from the particle-scale behavior although advanced continuum models have been developed to account for anisotropy, intermediate stress, and complex loading paths. The subtle elastic properties and origins of bulk friction will be discussed, as well as the effects of particle shape and size distributions. The review covers also recent developments in macroscopic modeling such as the thermomechanical approach, anisotropic critical state theory, nonlocal modeling approach, inertial flows, and material instabilities. Finally, a brief account is given of open issues and some new frontiers and challenges in the field.

This work presents a derivation of equivalent loads, coming from the integration of hydrostatic pressure fields on the internal and external walls of a curved pipe, which is modeled as a Euler-Bernoulli beam. To achieve that, the divergence theorem is applied to an infinitesimal-length pipe element. The Frenet coordinate system is used, leading to convenient simplifications. Finally, an expression is obtained for the equivalent distributed load along the pipe, due to pressure fields. Such load has a follower behavior and is curvature-dependent. Discussions are made, revisiting the effective tension and effective moment classical concepts, such as relating the proposed derivation with those concepts. An example of application is made in the context of analytical beam models, which is compared to a solid finite element model, experiencing a hydrostatic pressure field. Lastly, the obtained formulas are applied to an example of nonlinear buckling analysis of an initially straight pipeline, under constant internal pressurization. The postcritical configuration is obtained by using a geometric nonlinear finite element model, composed of beams, which are loaded using the expressions derived in this paper.

In practical engineering, a tuned mass damper (TMD) inherently exhibits nonlinear behavior owing to its large displacement or the application of a limiting stopper. This paper studies the control performance of a TMD when nonlinear behavior caused by nonlinear spring stiffness is taken into account for practical application. A new analysis and design method of a nonlinear TMD is proposed. Based on the analysis of the nonlinear TMD, a modified design of a nonlinear TMD is presented according to the jump frequency. The analytic solutions of the jump frequency that can be used to design the nonlinear TMD are obtained. The results of the simulation prove that the proposed design method considering the nonlinear behavior of the TMD will improve the control performance of the nonlinear TMD compared to the linear-based design method.

In seismic base isolation applications, fiber reinforcement was initially proposed as a potential cost-saving alternative to conventional steel reinforcement in laminated bearings. Steel reinforcement is often assumed to be rigid, but the extensibility of the reinforcement serves as an additional design parameter that must be considered. Similar to the compressibility of the elastomer, the extensibility of the reinforcement has a pronounced effect on important design parameters such as the compression modulus, bending modulus, and shear strains that develop because of compression or rotation. Analytical solutions for the bending modulus developed based on the pressure solution are available for most common pad geometries and can be used to derive the maximum shear strain due to rotation. These solutions are often complex and unsuitable for design purposes. In this study, the analytical solutions for an infinite strip, circular, square, rectangular, and annular pad geometries are derived and simplified to form geometry-specific approximations for the maximum shear strain due to rotation. The simplified approximations account for the reinforcement extensibility and the compressibility of the elastomer. The derived approximations are evaluated based on the analytical solutions and provide accurate values over a wide range of shape factors and values of bulk compressibility and reinforcement extensibility.

This study formulates a frequency-domain computational scheme for simulating nonlinear wave propagation in a homogeneous medium governed by the Westervelt equation. The need for such numerical treatment arises in both engineering and medical imaging applications, where finite-amplitude pressure waves trigger nonlinear effects that may critically affect the sensory data. The primary advantage of the proposed approach over commonly used approximations, which account for nonlinear effects via the Burgers’ equation, lies in its ability to handle nonlinearities due to arbitrarily inclined incident waves, which becomes especially important for focused sound beams with large apertures, i.e., wide ranges of inclination angles. The proposed direction-independent algorithm has a direct mathematical connection with the Westervelt equation, as opposed to the Burger’s equation (that relies on the plane-wave hypothesis), and has computational efficiency that is comparable to that of the traditional approach. The developments are illustrated by numerical examples that verify the method against an analytical solution and highlight the significance of accurately modeling nonlinear waves.

The importance of internal structure on the stress-strain behavior of granular materials has been widely recognized. How to define the fabric tensor and use it in constitutive modeling, however, remains an open question. The definition of fabric tensor requires (1) identifying the key aspects of structure information, and (2) quantifying their impact on material strength and deformation. This paper addresses these issues by applying the homogenization theory to interpret the multiscale data obtained from the discrete element simulations. Numerical experiments have been carried out to test granular materials with different particle friction coefficients. More frictional particles tend to form fewer but larger void cells, leading to a larger sample void ratio. Upon shearing, they form more significant structure anisotropy and support higher force anisotropy, resulting in higher friction angle. Material strength and deformation have been explored on the local scale with the particle packing described by the void cell system. Three groups of fabric tensor are discussed herein. The first group is based on the contact vectors, which are the geometrical links between contact forces and material stress. Their relationship with material strength has been quantified by the Stress-Force-Fabric relationship. The second group is based on the statistics of individual void cell characteristics. Material dilatancy has been interpreted by tracing the void cell statistics during shearing. The last group is based on the void vectors, for their direct presence in the microstructural strain definition, including those based on the void vector probability density and mean void vector. Correlations among various fabric quantifications have been explored. The mean void vector length and the mean void cell area are parameters quantifying the internal structure size and strongly correlated with each other. Anisotropy indices defined based on contact normal density, void vector density, void vector length, and void cell orientation are found to be effective in characterizing loading-induced anisotropy. They are also closely correlated. An in-depth investigation on structural topology may help establish the correlation among different fabric descriptors and unify the fabric-tensor definition. Deformation bands have been observed to continuously form, develop, and disappear over a length scale of several tens of particle diameters. Its relation to and impact on material deformation is an area of future investigation.

In the second-order reliability method (SORM), the failure probability is generally estimated based on parabolic approximation of the performance function. In the present paper, the first four moments (i.e., the mean, standard deviation, skewness, and kurtosis) of the second-order approximation of performance functions are obtained using the definition of the probability moment. Based on the recently developed fourth-moment standardization function, an explicit second-order fourth-moment reliability index is proposed for the estimation of failure probability corresponding to both the simple and general parabolic approximations. The simplicity and accuracy of the second-order fourth-moment reliability index is demonstrated using numerical examples. It can be concluded that the proposed method is applicable to the second-order approximation of performance functions with strong nonnormality and is accurate enough to improve the existing SORM in structural reliability analysis with minimal additional computational effort.

Previous work has shown that the acoustic-impedance model does not accurately predict blast wave transmission through plates of 10 different materials in air. In this work, results are presented for experimental tests of predictions of the acoustic-impedance model for blast wave transmission through the same 10 materials under water. Underwater blasts were created in a laboratory, and peak blast pressures were measured at symmetric locations: one where the blast wave travelled through the material and another where the blast wave reached the high-speed pressure sensor directly through the water. The acoustic-impedance model did not accurately predict underwater blast wave transmission, with a root-mean-square error (RMSE) of 22% in predicted transmission ratios and a correlation of $r=0.69$ between predicted and measured transmissions. Measured transmission ratios were better described by a linear model based on material density, with an RMSE under 5% and a correlation of $r=0.97$.