Journal of Engineering Mechanics

In this two-paper contribution, a general formulation for the nonuniform warping analysis of composite beams of arbitrary simply or multiply connected cross sections, under arbitrary external loading and general boundary conditions, is presented. Part I is devoted to the theoretical development and numerical implementation of the solution method, whereas Part II discusses the examined numerical applications illustrating the efficiency, accuracy, and range of applications of the proposed method as well as the effects arising because of nonuniform warping. The nonuniform warping distributions are taken into account by using four independent warping parameters multiplying a shear warping function in each direction and two torsional warping functions, respectively, which are obtained by solving the corresponding boundary value problems, formulated exploiting the longitudinal local equilibrium equation. A shear stress correction is also performed to improve the stress field arising from the kinematical considerations used. Ten boundary value problems are formulated with respect to the displacement and rotation components as well as to the independent warping parameters and solved using the analog equation method, a boundary element method (BEM)-based technique. The warping functions and the geometric constants including the additional ones due to warping are evaluated with a pure BEM approach.

In this two-paper contribution, a general formulation for the nonuniform warping analysis of composite beams of arbitrary simply or multiply connected cross sections, under arbitrary external loading and general boundary conditions, is presented. In Part I, the governing differential equations have been derived, leading to 10 boundary value problems with respect to the kinematical components and independent warping parameters as well as to seven two-dimensional boundary value problems for the involved shear and torsional warping functions. The aforementioned boundary value problems for the beam are numerically solved using the analog equation method (AEM), a boundary element method (BEM)-based technique. The warping functions and the geometric constants, including the additional ones due to warping, are evaluated with a pure BEM approach. In this Part II, numerical applications of great practical interest are worked out to demonstrate the efficiency and accuracy of the developed method compared with FEM results and results available in the literature. In these examples, the effects arising owing to nonuniform warping in homogeneous and composite beams are illustrated.

A novel approximate analytical technique is developed for determining the nonstationary response probability density function (PDF) of randomly excited nonlinear multidegree-of-freedom (MDOF) systems. Specifically, the concept of the Wiener path integral (WPI) is used in conjunction with a variational formulation to derive an approximate closed-form solution for the system response PDF. Notably, determining the nonstationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by existing alternative numerical path integral solution schemes, which rely on a discrete version of the Chapman-Kolmogorov (C-K) equation. In this manner, the analytical WPI-based technique developed by the authors is extended and generalized herein to account for hysteretic nonlinearities and MDOF systems. This enhancement of the technique affords circumventing approximations associated with the stochastic averaging treatment of the previously developed technique. Hopefully the technique can be used as a convenient tool for assessing the accuracy of alternative, more computationally intensive stochastic dynamics solution methods. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.

A multilevel adaptive FEM is proposed for multiscale mechanical analysis of heterogeneous materials. The algorithm is developed based on the extended multiscale FEM (EMSFEM), which has been proven to be an efficient method for multiscale mechanical analysis. A residual force-based error estimate is introduced, which is able to detect both global and local errors economically. The concept of microscopic numerical base function (NBF) is introduced, which can be used to refine the local results at the regions with a high-stress gradient level by level. Two efficient methods for updating the microscopic NBFs were developed, i.e., the hierarchical NBFs scheme and the residual force NBFs scheme. A hierarchical linear solver is designed to solve the multilevel equation system in each iteration step. In the algorithm, the calculations at the current iteration step take full advantage of the results of previous iteration steps, which is helpful to save computational resources. The numerical examples demonstrate the stability, flexibility, and convergence of the method.

A Bernoulli noncontact gripper can generate suction force to achieve noncontact handling by use of axisymmetric radial airflow. An experimental and theoretical study of the factors that affect the suction force and its variation is conducted. First, a theoretical model of the pressure distribution is developed, taking into consideration the inertial effect of airflow deceleration, viscous effect, total pressure of the central air-supply hole, and gap height between the gripper and the workpiece. Based on the model, the suction force is calculated. Then, the pressure distribution and suction force are measured experimentally. The theoretical calculation and experimental results reveal that the inertial effect causes a negative pressure distribution and resulting suction force, while the viscous effect and the total pressure of the central air supply give rise to positive distributions and a resulting repulsive force. Because the repulsive force is inversely proportional to the gap height to a higher power (the third power) than the suction force, the repulsive force is absolutely dominant when the gap height is very small and is gradually prevailed over by the suction force as the gap height is increased. As a result, the curve of the resultant force and gap height (the $F\text{-}h$ curve) has a part with a positive slope (referred to as the stable levitation region), where the levitation of the workpiece is stable. It also has another part with a negative slope (referred to as the unstable levitation region), where the levitation of the workpiece is unstable. Finally, the effect of the separation phenomenon that occurs behind the entrance to the gap is investigated, and it is found that it can retard the decline of the suction force in the unstable levitation region, which results in an expansion of the levitation region.

The time evolution of the parameter sensitivity of a nonlinear dynamical system is essential to its successful design, control, and identification. For an important class of systems, the governing equations of motion can be expressed as a linear system of ordinary differential equations augmented with a low-rank parameter-dependent (generally nonlinear) vector function. This paper builds on previous efforts by the authors to devise an approach for computing not only the responses of such systems but also the sensitivities of their responses to changes in design parameters in a computationally efficient manner. The proposed solution procedure first segregates the degrees of freedom appearing in the augmented functions from those that involve only linear deterministic operations. Second, the responses and sensitivities of the augmented terms are determined by solving the Volterra integral equations of reduced dimension. Finally, the responses and sensitivities of all remaining desired outputs are computed through a convolution approach using the fast Fourier transform to further increase the computational efficiency. The proposed method is applied to two examples, one simple linear model where exact solutions can be computed, and a second, more realistic, example of a wind-excited 20-story building augmented with nonlinear tuned mass dampers on its roof. The proposed method is shown to possess accuracy levels similar to those obtained through the time integration of the corresponding analytical sensitivity equations; however, it is several orders of magnitude faster than conventional sensitivity analysis approaches.

The deterministic frequency response analysis and the stationary random vibration of large, linear structural models are often computationally intensive tasks. The computational expense is only exacerbated when stiffness and damping variabilities are present in the system’s description. This paper proposes an approach for the highly computationally efficient and accurate computation of the sensitivities, to changes in design parameters that are spatially localized within some small portion of the computational model, of the transfer functions (TFs), power spectral densities, and mean-square responses of a structural system. The proposed method is derived, first for system TFs and then their sensitivities; the corresponding spectral densities are then derived using the TF approach. Finally, the proposed method is applied to two examples, one simple model where exact solutions can be computed, and a second, more realistic, building example with a high-order model that causes significant difficulties for standard state-space (SS) approaches in MATLAB as a result of numerical inaccuracies in the computations for very high-order SS models. The proposed method is shown to be highly accurate, with relative errors on the order of ${10}^{-10}$, but is 4–6 orders of magnitude faster than conventional approaches.

This paper deals with nonlinear random vibrations of a beam comprising a fractional derivative element; the nonlinear term arises from the assumption of moderately large beam displacements. It is shown that the beam response can be determined reliably via an optimal statistical linearization procedure. Specifically, the solution is obtained by utilizing an appropriate iterative representation of the stochastic response spectrum, which involves the linear modes of vibration of the beam. Such a representation allows retaining the nonlinearity in the time-dependent part of the response, which, in turn, is linearized in a stochastic mean square sense. The reliability of the proposed approximate solution is assessed in relation to the results of relevant Monte Carlo simulations. In this regard, a boundary integral method (BIM)–based algorithm is employed, in conjunction with a Newmark integration scheme, for estimating the beam response from spectrum-compatible realizations of the excitation, while accounting for the memory effect related to the fractional derivative element. The presented analytical and Monte Carlo results pertain to simply supported beams excited by a random uniform load of a given power spectral density function (colored white noise). Nevertheless, problems involving other boundary conditions can be solved readily by the method presented herein.

This paper considers a receding contact problem for two elastic layers (with different elastic constants and heights) supported by two elastic quarter planes. The lower layer is supported by two elastic quarter planes, and the upper elastic layer is subjected to a symmetrical distributed load whose length is $2a$ on its top surface. It is assumed that contact between all surfaces is frictionless, and the effect of gravity force is neglected. First, the problem is formulated and solved using the theory of elasticity and integral transform technique. Using the integral transform technique and boundary conditions of the problem, the problem is reduced to a system of singular integral equations in which contact pressures and contact areas are unknown. The system of singular integral equations is solved numerically by using the Gauss-Jacobi integration formulation. Second, the receding contact problem has been developed based on the FEM ANSYS software. Two-dimensional analysis of the problem is carried out. The ANSYS and analytical results for the contact pressures, contact areas, and normal stresses (${\sigma }_x$ and ${\sigma }_y$) along the axis of symmetry are given for various dimensionless quantities. The ANSYS results are verified by comparison with analytical results.

Stiffeners are used to reduce the probability of local failures in structures. Local stiffening is of interest in civil, mechanical, and aerospace engineering. In this paper, a stiffener is modeled by a massless spring. The spring introduces a fall (inverse jump) in a function of displacement. For example, a torsion spring produces a fall in the tangent to the lateral displacement of a flexural member. The effect of stiffeners on the free-vibration analysis of structural frames is considered, and governing equations are derived. In the case of simple members, the closed-form solution, and in the case of stiffened structural frames, the finite-element formulation, are proposed. The formulation is implemented in a computer program. Through analyses of three examples, the accuracy, efficiency, and robustness of the work are verified.

This paper studies fractal patterns forming at elastic-plastic transitions in soil- and rock-like materials. Taking either friction or cohesion as nonfractal vector random fields with weak noise-to-signal ratios, it is found that the evolving set of plastic grains (i.e., a shear-band system) is always a monotonically growing fractal under increasing macroscopic load in plane strain. Statistical analysis is used to assess the anisotropy of those shear bands. All the macroscopic responses display smooth transitions, but as the randomness vanishes, they turn into a sharp response of an idealized homogeneous material. Parametric study shows that increasing hardening or friction makes the transition more rapid. In addition, randomness in cohesion has a stronger effect than randomness in friction, whereas dilatation has practically no influence. Adapting the concept of scaling functions, the authors find the elastic-plastic transitions in random Mohr-Coulomb media to be similar to phase transitions in condensed-matter physics: the fully plastic state is a critical point, and with three order parameters (reduced Mohr-Coulomb stress, reduced plastic volume fraction, and reduced fractal dimension), three scaling functions are introduced to unify the responses of different materials. The critical exponents are demonstrated to be universal regardless of the randomness in various constitutive properties and their random noise levels.

The basic problem of a uniformly loaded rectangular plate with rotationally restrained edges is studied. An efficient, accurate method of eigenfunction superposition is used to determine important design parameters such as maximum deflection, maximum moment, maximum shear, and magnitude of the corner force. As the rotational spring stiffness decreases, oscillatory behavior of the edge shear gradually becomes monotonic. Corner forces exist whenever at least one adjacent edge is not clamped and become maximal when both edges are simply supported.

A simple but effective hybrid grid method is proposed to solve a two-dimensional, multiscale, moving-boundary flow problem. Two unsteady flows of different length scales and degrees of boundary movability are involved in the numerical simulation: (1) the generation and propagation of cnoidal waves, and (2) the induced oscillatory boundary-layer flow over a ripple bottom. For wave generation by a given motion pattern of waveplate, the fully nonlinear, free-surface conditions are applied to get the flow solution in a moving boundary-fitted grid. The generated cnoidal wave is developed to its permanent form after five periods without introduction of a wave theory. On the other hand, under nonlinear wave motion, the thin oscillatory boundary layer with intermittent separation around the wavy solid boundary is resolved by a very fine and stationary grid. The unified hybrid grid enables the authors to carry out the detailed flow analysis and track particle trajectories in this study.