Journal of Engineering Mechanics

This paper presents an information-theoretic approach for identifiability assessment of model parameters in nonlinear finite-element (FE) model updating problems. Rooted in the Bayesian inference method, the proposed approach uses the Shannon information entropy as a measure of uncertainty in the model parameters. The difference in the entropy of a priori and a posteriori probability distribution functions of model parameters, which is referred to as the entropy gain, is used as a measure of information contained in each measurement channel about the model parameters. The entropy gain approach can be used for selection of estimation parameters, optimal sensor placement, and design of experiment. In this study, an approximate expression for the entropy gain is derived, and a three-step process is suggested for the identifiability assessment. The application of the proposed approach is demonstrated for a nonlinear structural system identification problem. Although the focus of this study is on nonlinear structural FE model identifiability, the provided approach can be used for identifiability assessment of other types of linear/nonlinear dynamic models.

Quasi-brittle materials exhibit strain softening. Their modeling requires regularized constitutive formulations to avoid instabilities on the material level. A commonly used model is the implicit gradient-enhanced damage model. For complex geometries, it still shows structural instabilities when integrated with classical backward Euler schemes. An alternative is the implicit–explicit (IMPL-EX) integration scheme. It consists of the extrapolation of internal variables followed by an implicit calculation of the solution fields. The solution procedure for the nonlinear gradient-enhanced damage model is thus transformed into a sequence of problems that are algorithmically linear in every time step. Therefore, they require one single Newton–Raphson iteration per time step to converge. This provides both additional robustness and computational acceleration. The introduced extrapolation error is controlled by adaptive time-stepping schemes. This paper introduced and assessed two novel classes of error control schemes that provide further performance improvements. In a three-dimensional compression test for a mesoscale model of concrete, the presented scheme was about 40 times faster than an adaptive backward Euler time integration.

An implicit numerical integration algorithm is presented for the unified yield criterion, which could encompass most other yield criteria. The modification matrix, which is used to convert the continuum tangent modular matrix into the consistent tangent modular matrix, is derived for the return to planes, lines, and the apex of the unified yield criterion with multisurface plasticity with discontinuities. The stress update and consistent tangent modular matrix are first derived in closed form in the principal stress space, and then they are transformed back into the general stress space by coordinate transformation. Three numerical examples are used to demonstrate the effectiveness of the presented algorithm. The correctness of the developed algorithm is validated by the analytical solution and ABAQUS solution with the built-in Mohr-Coulomb model. The developed algorithm is also demonstrated to be least twice more efficient than the ABAQUS built-in algorithm. The presented algorithm for the unified yield criterion can facilitate the understanding of the effect the intermediate principal stress. With the increase in $b$, the force versus deflection curve at the midspan increases for the beam under three-point bending, and the critical radius of the elastoplastic interface decreases (i.e., the plastic zone becomes small) for the circular tunnel under hydrostatic pressure.

Laminated glass consists of at least two monolithic glass lites bonded together by an elastomeric interlayer. Existing mathematical models using the finite-difference method or the nine-node quadrilateral finite-element method were developed to numerically characterize the nonlinear behavior of laminated glass lites under bending and were benchmarked against available test data. The finite-difference solution was predicated on the well-known von Kármán equations, which are generally limited to the case of thin plates, while the nine-node quadrilateral finite element was predicated on the nonlinear Reissner-Mindlin plate formulation applicable to thick and thin plates but could result in a system of nonlinear equations that are computationally inefficient to solve. Therefore, a nonlinear four-node quadrilateral finite-element model for laminated glass based on the Reissner-Mindlin formulation is advanced. The assumed transverse shear strain fields method is employed to prevent shear locking and all the required stiffness terms are fully integrated. Hourglassing effects due to the reduced integration technique commonly used to prevent shear locking are mitigated and the stability of the numerical solution is preserved. The numerical solution obtained from the four-node element is in good agreement with available test data as well as the finite-difference solution.

This paper investigates the effects of rock foundation roughness on the shear strength of dam–rock interfaces and dam sliding stability. For this purpose, bathymetric and light detection and ranging (LiDAR) surveys of existing rock foundation surfaces were carried out close to existing dam sites and processed to obtain realistic dam–rock interface geometries differing by their roughness. The generated rock profiles are implemented into nonlinear finite-element models to conduct stability analyses of two gravity dams differing by size. A detailed analysis of the nonlinear response of dam–rock interfaces is presented in terms of limit friction angles, sliding safety factors, dilation angles, apertures, dam displacements, and shear stresses. It is shown that global roughness along dam–rock interfaces can substantially increase their shear strength. Natural local shear keys at a dam–rock interface may greatly improve the sliding stability of gravity dams; however, their effect is found to be sensitive to dam size and rock mechanical properties. Roughness effects on shear strength are found to generally decrease for larger dams. The results also reveal that the influence of rock strength parameters is more significant when the rock foundation surface includes prominent natural shear keys.

Based on a literature review of the assumed kinematics in the so-called higher-order displacement-based shear-deformation theories, a generalization of this kinematics is first proposed and used to formulate variationally consistent field equations and boundary conditions for the bending and vibration of a flat plate of a rectangular platform. Second, attention is focused on displacement-based polynomial shear-deformation plate theories. It is shown that all the kinematics of polynomial third-order theories ({3,0}-order polynomial) proposed in the open literature are special cases of the present theory. Furthermore, it is concluded that the {3,0}-order polynomial kinematics of all the theories is the same when the maximum transverse shear strain is used as generalized displacement coordinates. A deep analysis of the static and dynamic behavior of simply supported rectangular plates in cylindrical bending is performed in order to substantiate the general conclusion that all the {3,0}-order polynomial displacement-based shear-deformation theories give the same numerical results, i.e., they are kinematically equivalent, although not all are statically equivalent.

This paper provides a new approach to incorporating the stochastic nature of damage constitutive relations in the finite-element analysis of concrete structures. Within the framework of stochastic damage mechanics, the spatial variability of concrete was modeled as a two-scale stationary random field. At the microlevel, the damage evolution law of concrete was mapped to a random field corresponding to the microscopic fracture strain. At the macrolevel, the strength distribution of any concrete component forms a lognormally distributed random field. The connection between the two-scale random fields was established by a covariance constraint such that the scale of fluctuation of the random material property was satisfied in both scales. Taking advantage of the stochastic finite-element method, both the microscopic random damage evolution of concrete and the fluctuation of macroscopic structural responses can be numerically represented. Stochastic structural modeling and damage analyses for a conventional cantilever beam and a plane frame were carried out to illustrate the proposed method.

Laboratory experiments were conducted to directly measure the drag force induced by vegetation patches under unidirectional flow conditions. The drag force was measured with a highly responsive load cell for six patch models, of which the vegetation stems were simulated using rigid cylindrical rods. The measured drag coefficient varies with the Reynolds number and the number of rods included in the patch model. It generally reduces with increasing Reynolds number, and the reduction appears more significant for a patch model with fewer rods. To take into account effects of the patch porosity, this study proposed to redefine the drag coefficient and Reynolds number based on new length and velocity scales, that is, the vegetation hydraulic radius and the average velocity for the flow through the patch. This change produces a monotonic relationship between the redefined drag coefficient and redefined Reynolds number, which is generally applicable for all patch models.

Expansion of concrete caused by external sulfate attack (ESA) is a multiscale chemomechanical response, mainly including internal stress evolution at the dormant period and significant cracking expansion for a detrimental duration in hardened cement paste (HCP). The internal stress generated in the dormant period of ESA can result in the microcracking of HCP and further significant volume expansion of concrete, which can reduce the durability of concrete structures. Based on the ESA-induced expansion mechanism, a multiscale model is developed to describe the expansion response of HCP in the dormant period of ESA. In this model, a saturated and homogeneous HCP and a representative volume element (RVE), which is composed of a pore and nearby cement paste, are regarded as macroscopic and microscopic objects, respectively. By using Fick’s law and reaction kinetics, a diffusion-reaction model is proposed to simulate the macroscopic sulfate diffusion in HCP and microscopic chemical reaction to produce ettringite in RVE. According to the crystallization pressure theory and continuum mechanics, some mechanical models are constructed to investigate the microscopic mechanical response caused by the supersaturation of ettringite in RVE and its induced macroscopic mechanical response in HCP. Taking a HCP cylinder under different sulfate exposures as an example, a numerical simulation is performed to investigate its expansion response in the dormant period of ESA.

In this study, the full analytical solution of one-dimensional space-fractional advection–diffusion equation (fADE) is derived using the homotopy analysis method (HAM). Apart from the several applications of the fADE, it has been applied in sediment-laden turbulent flows to find the distribution of suspended sediment particles when the effect of nonlocal transport is present. In such flows, due to the presence of turbulent-bursting particles ejected from the bed level up to the free surface performing nonlocal vertical jumps, which cannot be captured through traditional integer-order concentration gradient, the fractional derivative of concentration, therefore, comes into promise. The fractional derivative considered here contains a convolution of suspension concentration and concentration gradient which captures the effect over all points throughout the domain. The HAM is a powerful method to solve the nonlinear fractional differential equations and has been applied here to obtain the solutions. The advantage of this proposed HAM solution over previous analytical solutions of fADE, in terms of the Mittag-Leffler function, is that it allows to choose the region of convergence of the series solution with greater freedom, and thus broadens the applicability of the model under many difficult situations. The HAM solutions are obtained for two different choices of sediment diffusivity and they are also compared with analytical solution and numerical solutions. The solutions are also validated with the experimental data. The comparison results are satisfactory and it shows the advantage of using the homotopy analysis method. Finally, the variations of the model with the parameters are also presented and the reasons are justified physically.

This paper simulates fracture in notched mortar beams under three-point bending using an extended finite-element method (XFEM) and peridynamics. A three-phase microstructure (i.e., cement paste, aggregates, and paste–aggregate interface) is used for the constitutive modeling of the mortar in order to obtain the elastic properties for simulation. In the XFEM approach, the simulated homogenized elastic modulus is used along with the total fracture energy of the cement mortar in a damage model to predict the fracture response of the mortar, including crack propagation and fracture parameters [Mode I stress intensity factor, $K_{IC}$, and critical crack tip opening displacement (${\mathrm{CTOD}}_C$)]. The damage model incorporates a maximum principal stress-based damage initiation criterion and a traction-separation law for damage evolution. In the peridynamics approach, a bond-based model involving a prototype microelastic brittle (PMB) material model is used and implemented in LS-DYNA. The elastic properties and fracture energy release rates are used as inputs in the PMB model, along with the choice of peridynamic horizon size. Comparisons with experimental fracture properties ($K_{IC}$, ${\mathrm{CTOD}}_C$) and crack propagation paths from digital image correlation show that both approaches yield satisfactory results, particularly for $K_{IC}$ and crack extension. Thus, both methods can be adopted for fracture simulation of cement-based materials.

To estimate life-cycle cost and future seismic capacity, the life-cycle analysis in seismic design of bridges is formulated as an interactive multiobjective decision-making and optimization problem. Specifically, preference information based on engineering judgement and theoretical analysis is incorporated in the optimization procedure. The posteriori evolutionary multiobjective optimization (EMO) algorithm (nonpreference) and the preference-based interactive EMO algorithm are both applied to the seismic design of an reinforced concrete (RC) pier with two, three, and four objectives, namely, flexural strength coefficient, shear strength coefficient, reliability index of drift, and life-cycle cost coefficient. In terms of a rational displacement ductility, the safety preference information is applied to establish a value function after every few optimization runs, progressively directing the search of the EMO algorithm to more preferred solutions. By comparing the results of both algorithms, it is found that when there are more than three objectives, it is difficult for the nonpreference optimization to converge with the global Pareto frontier, resulting in a local optimal solution as the final design. The combination of preference structure and multiobjective optimization effectively avoids the challenges of global search and purposefully converges with the most preferred design.

For system identification of structures using responses comprised of contributions from multiple modes, one usually decomposes responses into individual modes prior to identifying parameters such as natural frequency, damping, and mode shapes. Conventional mode decomposition techniques such as the frequency domain decomposition (FDD) and the blind source separation (BSS) have been successfully applied to estimate modal matrix (mode shapes) from output-only data. Most of these techniques are formulated in physical coordinates, so the classical damping assumption is implicitly invoked. However, structures with auxiliary damping devices, which have been increasingly incorporated into modern structures over the last several decades, represent a nonclassically damped system. This arrangement may result in modal responses that are not completely decomposed by conventional techniques. In addition, these techniques may suffer from failing to decompose closely spaced modes. To overcome these limitations, a new state space–based mode decomposition (SSBMD) framework is proposed here that involves an optimization procedure to estimate the modal matrix from the eigen problem. Furthermore, the framework can be expanded to identify mode shapes not only in the state-space coordinates but also in physical coordinates, depending on the established performance index. To validate the proposed framework, this study examined a heavily damped three-degree-of-freedom (DOF) nonclassically damped system by way of numerical simulation and the wind-induced full-scale acceleration response of a 40-story steel-framed building with closely spaced modes.

This paper makes use of the Duffing oscillator equation to represent the dynamic nonlinearity of rectangular plates. The response of the nonlinear oscillation of a beam leads to parameters that are determined and compared to experiments. The resulting technique used for their evaluation are then extended to plates considering both simple and clamped boundary conditions. The programs MATLAB and ABAQUS are combined for these analyses.