Journal of Engineering Mechanics

Free-standing bodies, which rock in response to dynamic base excitations, can be found in a wide range of applications. There are many instances, for example a museum artifact seated on a support pedestal, where both the object and its support are free-standing, resulting in a dynamic system where both bodies can rock with respect to their contact interfaces. This paper analyzes the rich dynamics of systems of two stacked rocking bodies. It focuses on the location of the resultant impulses between bodies during impact, considering their effect on the overall stability of the system. The widely-used assumption that the impulses occur at the future rocking corners is shown to be the least conservative scenario and often leads to underestimating the likelihood of failure. The optimum configuration of two stacked blocks is found to be highly sensitive to the characteristics of the input motion and to the location of the impulses.

This paper explores the basic mechanism of a metamaterial I-girder. The metamaterial I-girder is designed by integrating a mass-spring-damper subsystem with an I-girder to act as a multifrequency vibration absorber. The local vibration absorber creates shear force to absorb the vibration energy under the resonant excited state. The efficiency of the metamaterial I-girder is verified via numerical testing on a steel beam and a cable-stayed bridge. The results show that the resonance of the vibration absorber creates stopbands of frequencies that are influenced by the absorber’s mass ratio. Each stopband’s width can be increased by increasing the absorbers’ damping. Moreover, the metamaterial I-girder can be used for the absorption of elastic waves caused by traffic loads and earthquakes on composite cable-stayed bridges.

An upper-bound formulation for performing finite-element limit analysis by using semidefinite programming (SDP) for an axisymmetric stability problem involving the Mohr-Coulomb yield criterion has been presented. The SDP technique has an advantage in that it can deal with the yield criterion directly in its native form in terms of principal stresses and strains without any smoothing of the parent yield surface. The associated flow rule and plastic power dissipation are expressed entirely in terms of principal plastic strain rates. Nodal velocities and element plastic strain rates were framed as basic governing variables without involving stresses. The solution was obtained by using the SDP solver MOSEK in MATLAB. The problem of finding the bearing capacity of a circular footing was dealt with by choosing a planar domain that was discretized with (1) three-noded constant strain finite elements and (2) six-noded linear strain finite elements. The results were obtained with and without the provision of the velocity discontinuities, and were compared with those reported in literature. It was learned that quite accurate solutions can be obtained with the application of six-noded linear strain triangular elements and using velocity discontinuities along all the elements’ interfaces.

This paper proposes an accurate and efficient numerical model by combining the method of characteristics (MOC) and the meshless localized radial basis function collocation method (LRBFCM) to simulate the shallow water flow problems. The shallow water equations (SWEs) are classified into a hyperbolic-type partial differential equations (PDEs) system that easily creates numerically unstable results for the case with discontinuous field values or shock waves. To solve this problem, the SWEs are derived into conservative eigensystem form, and then the MOC is applied to capture the change of conservative variables along the characteristic lines. Specifically, the meshless LRBFCM is used to obtain the field values from the conservative variables; it can ease the complexity of the interpolation procedure on characteristic Lagrangian points and preserve the accuracy in transient problems. For the boundary disposal, a fractional time step skill with the characteristic velocity is considered to determine the boundary requirements. The computational nodes can be generated by the uniform or nonuniform distribution, which reduces the difficulty of node generation to obtain efficient and accurate numerical analysis. Six continuous and discontinuous SWEs benchmark examples are simulated and discussed to verify the proposed model. The excellent agreements with the analytical, experimental, and numerical solutions demonstrate the accuracy and efficiency of the algorithm.

At large hypocentral distances, it is convenient to approximate curved transient seismic wavefronts as planar to estimate rotational ground motions from the single-station recordings of translational ground motions. This paper investigates whether and when this approximation, referred to as the plane-wave approximation, can be considered to be adequate close to the source. A simplistic source model comprising a two-dimensional (2D) kinematic shear dislocation P-SV line source buried in a homogenous, elastic half-space is considered to model the finite fault. The plane-wave rotational motion is then synthesized from the exact translational motion corresponding to the assumed model and compared with the exact rotational motion. The comparison of the two sets of rotational amplitudes in the frequency domain suggests that the plane-wave approximation may lead to significant underestimation of the Fourier amplitudes when the incidence angles of the seismic waves are small or when the wavelengths are appreciably longer than the hypocentral distance. For short wavelengths and other incidence angles, the adequacy of the plane-wave approximation depends critically on the source dip angle.

Taking account of the spatial variation of material properties is of paramount importance to the safety and reliability evaluation of engineering structures. This necessitates the highly efficient and accurate representation of random fields of material properties. Although theoretically it is only an extension of stochastic process representation, in practice this is nontrivial because the dimension of space increases. To reduce the number of basic random variables and improve the accuracy, a stochastic harmonic function (SHF) representation method for homogenous random fields is proposed. Compared to the classical spectral representation method, besides the phase angles, the wave numbers of each harmonic component are also random variables, of which the supports could be specified by the Voronoi partition of the bounded wave-number domain. It is rigorously proved that the proposed SHF representation could reproduce the target wave-number spectral density exactly rather than approximately with a finite number of random variables. In practice, for the same number of random variables and samples, it is numerically proved that the SHF is accurate to the target wave-number spectral density, while spectral representation is only accurate on specific points. Further, it is demonstrated that the SHF fields are homogeneous and asymptotically Gaussian, with the convergence rate being higher than the spectral representation method. In contrast to the Karhunen-Loève expansion, on the other hand, the solution of the integral equation is avoided. The response analysis of a shear wall with random-field material parameters is taken as an example to illustrate the application of the proposed method. It is shown that the scale of fluctuation will affect the variation of dissipated energy greatly, and thereby will affect the reliability. Problems to be further studied are also discussed.

The analysis of the internal forces of long tunnels subjected to seismic loading is an important problem in underground engineering. This study is focused on the seismic responses of a circular tunnel in a layered half-space under a longitudinally asynchronous incident shear-horizontal (SH) wave propagation. The tunnel is assumed to be deep in the ground such that the influence of the ground surface is negligible. A one-dimensional numerical procedure for the free-field responses of the layered half-space due to an obliquely incident seismic wave propagation is presented first. Then the soil–tunnel interaction is considered both in the cross-sectional plane and along the longitudinal direction. The internal forces of the tunnel lining are analytically derived considering not only the effect of the cross-sectional shear strain ${\gamma }_{xy}$ but also the effect of the propagating transverse motion $u_x$ along the longitudinal direction. The final semianalytical solution is obtained from the summation of the two responses. The proposed semianalytical procedure is validated by a comparison with the results from dynamic three-dimensional finite-element analyses.

This paper proposes a novel three-dimensional (3D) discrete solid-element method (DSEM) to calculate the extremely large deformation and high material nonlinearity of continuity. In DSEM, the material is discretized into rigid spherical elements. The two spherical elements on the edge and the diagonal line of the cube model are linked together through springs, which consist of one normal spring and two shear springs. The mechanical behavior of the structure is calculated using the discrete grid system composed of spherical elements and springs. To accurately reflect the mechanical behavior of the material, the principle of energy conservation is used to strictly deduce the spring stiffness, and the relationship between spring stiffness and elastic constants is established. In accordance with the plastic mechanics and the fourth strength theory of the material, the yield equation and elastoplastic force-displacement equations in DSEM are deduced based on the orthogonal flow rule and the consistency condition. The numerical analysis shows that DSEM can effectively address the problems of extremely large deformation and high material nonlinearity.

As an exact method, the Woodbury formula is used to solve local material nonlinearity problems in structural analysis, in which the calculation and factorization of the global stiffness matrix are avoidable compared to the direct method used in the conventional finite-element method (FEM). This study uses the time-complexity theory to evaluate the efficiency of the Woodbury formula. The results show that the time complexity increases as the number of the inelastic degrees of freedom (IDOF) increases, and the Woodbury formula is more efficient than the ${LDL}^T$ factorization method only when nonlinearity appears within local, small regions. For example, for a structure with a total of 10,000 degrees of freedom (DOF), the efficiency of the Woodbury formula is higher than the ${LDL}^T$ method when the number of IDOF is less than 10% of the total DOF (1,000). Although the nonlinearity only occurring in small partial domains is common for most engineering structures, this low efficiency threshold still limits the Woodbury formula application for some structures with large part of nonlinear regions. To extend this efficiency threshold value, a Woodbury approximation method (WAM) is proposed that incorporates the idea of a combined approximations (CA) approach into the framework of the Woodbury formula, in which the reduced-basis method and binomial series expansion are used to solve the system of linear equations whose scale depends on the number of IDOF. The accuracy considerations for the approximate solution are discussed and the convergence criterion for error evaluation is presented. Moreover, the time-complexity analysis indicates that the limit on the efficiency is enhanced greatly by the proposed WAM under the comparable computational accuracy to the Woodbury formula, such as expanding above percentage of 10 to 70%. Finally, a numerical example is presented to prove that the WAM can provide accurate results with high efficiency, and thus, has greater potential for solving nonlinear problems.

This study addresses analysis of the crosswind response of tall buildings under nonstationary wind excitations. The wind load under nonstationary wind excitation was quantified using the force parameters under stationary wind, but the effect of the time-varying mean wind speed was considered in a quasi-stationary manner. For buildings without consideration of nonlinear aerodynamic damping, the evolutionary spectral analysis approach and statistical moment equation approach are presented, which provide analytical solutions of time-varying response standard derivation (STD), extreme response, and fatigue damage. For buildings with consideration of nonlinear aerodynamic damping, the governing equations of statistical moments of the building motion including time-varying STD, or variance/covariance, and kurtosis of response were established and solved using the non-Gaussian moment closure technique. The higher-order moments involved were estimated from a kurtosis-based translation process model. Narrowband response characteristics were applied to simplify the non-Gaussian closure approach. The nonstationary extreme response and fatigue damage were further estimated from time-varying STD and kurtosis of response. The effectiveness of the analysis framework was examined through comparison with a response time history simulation. The characteristics of the crosswind response under nonstationary excitations are also discussed.

Traditional vibration measurement typically requires physically attached sensors, such as accelerometers and strain gauges. However, these discrete-point sensors provide only low spatial resolution vibration measurements, potentially foregoing valuable structural information such as localized damage. Noncontact optical measurement methods such as laser vibrometers can achieve high spatial resolution vibration measurements but only through time-consuming sequential measurements and are not cost-effective. As an alternative to traditional vibration measurement methods, digital video cameras are relatively low-cost, agile, and offer noncontact, simultaneous high spatial resolution measurements where every pixel on the structure becomes a measurement point. However, regular digital video cameras are frame-based where each pixel simultaneously performs temporally uniform (synchronous) measurements containing large amounts of redundant (background) data, which consumes considerable resources for video data measurement, management, and processing. To alleviate such a challenge, this work explores the use of event-based neuromorphic imagers, specifically silicon retinas, an efficient alternative to traditional frame-based video cameras, to perform full-field vibration measurements and operational modal analysis. By imitating biological vision, each silicon retina pixel independently and asynchronously records only intensity change events that contain structural motion information while excluding redundant (background) information. Such an asynchronous event-based data measurement mechanism allows for structural motion to be captured on the microsecond scale in an extremely data-efficient manner, which could benefit real-time vibration measurement and control applications. This study takes the first step toward these applications by formulating an existing video frame-based full-field operational modal analysis technique in the event-based, asynchronous silicon retina measurement framework. Specifically, local phase-based motion extraction and blind source separation are used to automatically and efficiently extract full-field vibration and dynamics parameters from silicon retina measurements. The developed method is validated by laboratory experiments on a bench-scale cantilever beam.

Self-healing materials have recently become more popular due to their capability of self-repairing cracks and rehabilitation of structures. Recent research has revealed that self-healing presents a crucial solution for the strengthening of the materials. This new concept has been termed superhealing. Once the stiffness of the material is recovered, further healing can result in a strengthening of the material. This work presents a refined theory of the superhealing model within the framework of continuum damage mechanics. The proposed refined theory is extended in this paper from linear to nonlinear superhealing theory. The general framework of continuum damage-healing mechanics is first reviewed. Following that, the concepts of healing and superhealing of materials are introduced along with both their isotropic and anisotropic presentation. The proposed linear refined theory of superhealing materials and its anisotropic definition are presented using sound mathematical and mechanical principles. Afterwards, the nonlinear formulation of the refined superhealing theory and its anisotropic presentation are introduced. In addition, the link of the proposed theory with the theory of undamageable materials is outlined. Examples for the case of plane stress and one-dimensional element are demonstrated in which the refined theory of superhealing is applied. The aim of the present work is to present the proposed refined theory and provide guidance regarding the concept of the new theory that can be applied in manufacturing technology in the future. It is hoped that the new theory will open an area of new research in materials science and pave the way for new technologies that can be exploited for rehabilitation of structures.

This paper analytically investigates the buckling and postbuckling of functionally graded composite plates reinforced by single-walled carbon nanotubes (SWCNTs), supported by elastic foundations and loaded by edge-compressive and thermomechanical loadings. The effective properties of a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate were assumed to be graded in the thickness direction and were determined by the extended mixture rule. The basic equations for a geometrically imperfect plate were established according to the classical plate theory (CPT) incorporating nonlinear terms, a two-parameter elastic foundation, and elastic constraints of the in-plane boundary condition. The deflection and stress functions were assumed to satisfy the simply supported condition of the boundary edges, and the Galerkin procedure was adopted to obtain the explicit form results of the critical buckling loads and nonlinear relations of load and deflection. The important effects played by the tangential edge restraints on the nonlinear stability response of the FG-CNTRC plates were highlighted. In addition, the separate and combined influences of the volume percentage and the distribution patterns of the carbon nanotubes, aspect ratios, stiffness of foundations, thermal environments, and imperfection size on the buckling resistance capacity and postbuckling strength of FG-CNTRC plates were analyzed through a variety of numerical examples.