Inelastic Structures Research Articles

Seismic Structure‐Soil‐Structure Interaction between inelastic structures

AbstractFrequently, buildings in urban areas are designed by considering their stand‐alone response, that is, as single structures with no neighboring buildings. Nevertheless, the existence of a high density of buildings in large metropolitan areas inevitably results in the likelihood of an important seismic interaction between adjacent buildings through the underlying soil. This paper explores the effects of Structure‐Soil‐Structure Interaction (SSSI) on the seismic response of two yielding structures embedded in a linear elastic soil. A simple two‐dimensional nonlinear reduced‐order parametric model is proposed, where different building parameters are considered. A nonlinear phenomenological Bouc–Wen model is assumed for the buildings. A database of 15 strong ground motion records and an additional spectrally matched seismic ground motion are considered. An extensive parametric study comprising over two million nonlinear cases is conducted. The results show important differences between nonlinear SSSI and nonlinear SSI for particular parameter configurations. Nevertheless, due to energy dissipation and increases in damping in the nonlinear case, the effects of SSSI are less relevant compared with the linear case.

Reliability of inelastic wind excited structures by dynamic shakedown and adaptive fast nonlinear analysis (AFNA)

The ever-growing interest in performance-based wind engineering has created a need for assessment frameworks that can efficiently deal with inelasticity. The computationally efficient strain-driven dynamic shakedown approach has provided a solution that is not only capable of identifying failure mechanisms that are potentially critical during extreme winds, e.g., low cycle fatigue and ratcheting, but also allows direct estimation of inelastic deformations. This approach, however, can only solve problems at dynamic shakedown, i.e., with limited nonlinearity, and is not capable of providing response time histories. To address these limitations, this paper presents an efficient framework for reliability assessment of inelastic structures at dynamic shakedown and beyond. To this end, a novel step-by-step integration algorithm is developed for rapid response time history analysis within the setting of dynamic shakedown. The method is based on advancing fast nonlinear analysis through introducing schemes for enabling at each time step the adaptive selection of the step size, number of modes to be included, and number of potentially nonlinear elements. Inelasticity is modeled as distributed at the level of the stress resultants through a return mapping scheme based on the Haar–Kàrmàn principle, therefore enabling the integrated estimation of the state of dynamic shakedown. The scheme is seen to preserve the efficiency of recently developed strain-driven dynamic shakedown algorithms while providing a full range of response time histories at and beyond the state of dynamic shakedown. To enable reliability analysis, the scheme is embedded in a general uncertainty propagation framework.

Simplified Estimation Method of Plastic Energy Dissipation for MDOF Systems Using Force Analogy Method

Plastic energy dissipation is a key factor in the response of inelastic structures subjected to seismic input, and is often regarded as a primary source of structural damage due to the inelastic deformation of structural components. Accurately predicting a structure’s plastic energy dissipation is essential for efficient energy-based design and seismic assessment. While a single-degree-of-freedom (SDOF) system provides a simple and effective method for estimating plastic energy dissipation, few studies have explored the use of nonlinear analysis methods or equivalent SDOF systems for this purpose. Based on the principle-of-force-analogy method, firstly, the formulas of plastic energy dissipation of a multi-degree-of-freedom (MDOF) system and its equivalent SDOF system were established. Secondly, two estimation methods of plastic energy dissipation of MDOF systems were proposed. Finally, numerical simulations were performed on several multi-story and high-rise structures with varying heights and spans to compare the plastic energy dissipation of MDOF systems and the equivalent SDOF systems of different modes. The simulation results demonstrate that the proposed methods and formulas accurately estimate plastic energy dissipation in multi-story and high-rise structures, while also requiring fewer calculations and less storage.

Short-horizon acceleration-predictive control for reducing lateral seismic inertia forces of inelastic frame structures using semi-active fluid viscous dampers

Semi-active control systems effectively improve the resilience of civil structures when subject to dynamic vibrations due to their ability to change their damping properties. In the case of semi-active fluid viscous dampers (SAFVD), the damping coefficient quantity varies according to the tradeoff between the need to increase the energy dissipation capabilities (i.e., increasing the damping coefficient) and the need to decrease the applied shear forces (i.e., reducing the damping coefficient). The operation of a SAFVD device is similar to that of a passive fluid viscous damper except that, based on the status of the control valve, it can deliver damping at two distinct levels (two-stage) or over a wide range between an upper and lower bound (continuously adjustable). This paper employs a continuously adjustable SAFVD assembly and develops its optimal command voltage change at each time step, which regulates the valve that determines the fluid passing through the external path. The control strategy uses total acceleration measurement as the system's closed-loop feedback and utilizes the Kalman filter model to predict the sequential acceleration reading to cater to time delays. The solution derives from a short‐horizon problem whose objective is reducing the difference between the lateral seismic inertia forces and the SAFVD forces, thus, mitigating the columns’ resisting forces. The case study presented in this paper showcases the effectiveness of the SAFVDs in improving the inelastic earthquake response of a 10-story frame structure.

Seismic demand estimation of isolated buildings with nonlinear bearings subjected to linear response spectra

Response spectrum analysis is a commonly used method for the seismic design of linear structures with fixed bases since the seismic input is normally provided in terms of the response spectrum in many codes. This study proposes a more generalized response spectrum method for the seismic design of three-dimensional isolated structures. Firstly, the theory basis of this method is derived in the framework of the random vibration theory. The equivalent parameters of each bearing are expressed in terms of the statistics of the displacement amplitude. Then, the complex mode theory is introduced into this method to consider the effect of the nonclassical damping resulting from the isolation layer. Finally, an iterative analysis procedure is presented, which combines the equivalent linearization with the complex mode theory. Moreover, the convergence and applicability of this method are discussed in detail. The parameter analyses show that the relative error of the base shear of the superstructure estimated by using this method is within 10% compared with that of the nonlinear time history analysis. In addition, a benchmark base-isolated structure is used as a numerical example to examine the effectiveness of the proposed method. The results show that the iteration is convergent and computationally efficient. The results also show that the maximum relative errors of the predicted isolation layer displacement, the inter-story drift of the bottom floor and the absolute acceleration of the top floor are 9.5%, 3.9% and 18.9%, respectively, by comparing them with the nonlinear time history analysis, while these responses are underestimated by 9.8%, 25.9% and 40.2%, respectively, when using the classical complete quadratic combination (CQC) method that neglects the nonclassical damping effect. Although in this study the proposed approach mainly focuses on isolated structures, in essence, this approach can also be applied to inelastic structures or structures with nonlinear dampers.

Optimization of inelastic multistory structures under seismic vibrations using shape-memory-alloy material

This paper develops a novel optimization methodology for designing Shape-memory-alloy resisting devices (SMARDs) and optimally allocating them to inelastic multistory structures. The solution algorithm is a control gains optimization procedure that refers to a formal optimization problem with an objective function subject to the state-space equation and design limitations. The objective function integrates the squared state components in time, and the state-space equation consists of a newly introduced state vector form that reflects the system's inelasticity. The control gains are the number of total Shape-memory-alloy (SMA) wires attached to the devices in each story, and the design limitations dictate the minimum/maximum number of wires. The solution algorithm consists of five iterative steps that employ the defined Hamiltonian gradients in state and gains and cater to the necessary optimality conditions. The numerical example deals with upgrading an eight-story shear-type frame system. It studies the algorithm efficiency and elaborates on the effect of the optimal weighting matrix by investigating three different configurations. In all cases, the algorithm improves the system's inelastic seismic response—showcasing the reliability of the developed design methodology and the utilization of SMA material.

Matrix equations models for nonlinear dynamic analysis of two-dimensional and three-dimensional RC structures with lateral load resisting cantilever elements

In control engineering and structural dynamics, mathematical models such as the state-space representation, equation of motion, and the phase plane are matrix equations describing the system equilibrium. This paper develops novel matrix equations models for linear/nonlinear dynamic analysis of reinforced concrete (RC) buildings with cantilever elements lateral load resisting systems (e.g., RC shear wall, RC core). The models offer a new approach for introducing two-dimensional and three-dimensional cantilever structures to control the theory’s state-space representation and structural dynamics’ equation of motion. The development primarily addresses the stiffness and mass matrices. The proposed displacement-related stiffness matrix of cantilever elements satisfies the necessary conditions of symmetricity and elemental boundary conditions. The nonlinear matrix structural analysis employs a smooth hysteretic model for deteriorating inelastic structures, referring to the relation between the bending moment and the bending curvature through the bending stiffness. The parameters controlling the cyclic behavior regard a composite RC cross section subject to gravitational load and bending simultaneously. The paper includes four examples that exemplify the practical utilization of the matrix equations models in analyzing two-dimensional and three-dimensional structures of linearly elastic and inelastic properties. The four examples demonstrated the idealized applicability of the matrix equations models for modal analysis, pushover analysis, and inelastic earthquake response analysis.

A robust method for load-carrying capacity assessment of semirigid steel frames considering fuzzy parameters

The non-probabilistic approach based on fuzzy logic has been applied in analyzing and assessing steel structures. However, there is still a dearth of studies on the behavior of nonlinear inelastic semirigid frame structures in the presence of fuzzy uncertainty. Moreover, accurate fuzzy analysis of structures often involves high computation cost. In this paper, a robust method for the prediction of the load-carrying capacity of semirigid steel frame structures defined with fuzzy parameters is proposed. For the analysis purpose, the nonlinear inelastic analysis taking account of both material and geometrical nonlinearities is adopted to estimate the ultimate load factor (ULF11ULF is the ultimate load factor of the semi-rigid steel frame structure.) of the structure. Columns and beams are modeled by the beam–column element with a refined plastic hinge at two ends, and the beam-to-column connections are defined by zero-length semirigid joints. Innovatively, a fuzzy estimation procedure named the Moving Extrema Method (MEM) is developed to efficiently capture the variation of ULF due to fuzziness in the structural properties and applied loads. Applications to three semirigid steel frames having a relatively large number of fuzzy parameters are conducted. Numerical results show that the proposed MEM can predict the fuzzy ULF with relatively high accuracy and reasonable computation volume. More insights on the effect of the fuzzy structural parameters on the ULF are also presented.

Effectiveness and robustness of braced‐damper systems with adaptive negative stiffness devices in yielding structures

AbstractThis study applied adaptive negative stiffness devices in the application of braced‐damper systems to propose an adaptive negative stiffness amplifying damper (NSAD), and investigated its effectiveness and robustness for controlling the inelastic seismic responses of yielding structures. The adaptive stiffness behavior of the proposed NSAD perform negative stiffness and damping magnification effect within a certain displacement threshold, thus controlling both structural acceleration and drift at elastic stage; when subjected to strong earthquakes, the proposed NSAD would adaptively develop positive stiffness, which limits the inelastic deformation of yielding structures. A set of nonlinear seismic spectra are established for responses including acceleration, ductility, energy dissipation, and residual deformation. Numerical results validated that the proposed adaptive NSAD is effective for both elastic and inelastic structures. In addition, influences of key parameters like flexible support, negative stiffness ratio, and displacement threshold ratio (ratio of displacement threshold to yielding displacement), are carefully studied. From the point of controlling both acceleration and ductility demands of inelastic structures, the displacement threshold ratio is recommended to have a value close to a unit; typically, a reasonable upper limit of displacement threshold ratio is recommended to be lower than 1.5 to avoid amplifying residual deformation.

Fast and robust RSDM shakedown solutions of structures under cyclic variation of loads and imposed displacements

The Residual Stress Decomposition Method (RSDM) is an iterative numerical procedure developed to directly estimate the kind of asymptotic stress states under cyclic loading of inelastic structures. The method has been modified (RSDM-S) to establish safety margins for elastic shakedown under mechanical and/or thermal loads. In the present work, the formulation of the shakedown problem to account for the coexistence of loads and imposed displacements is presented. An updated RSDM-S procedure in terms of robustness and computational efficiency is also proposed. It is proved that a monotonically decreasing sequence of the iterative steps is created that converges to the shakedown loading factor, thus making the procedure absolutely robust. Based on the monotonicity, a numerical scheme with proven super-linear convergence is suggested that renders the approach very fast. The procedure is then applied to structures under cyclic loadings of loads combined with applied displacements. Shakedown domains, not frequently met in the literature under these two actions, are constructed. Additionally, shakedown values of displacements, simulating quasi-static earthquake loading on structural components, are estimated.

Building structure with elastoplastic bilinear model under multi-dimensional earthquake forces

Abstract Developed herein is an analysis procedure based on closed-form solutions to elastoplastic bilinear model of building structures accounted for different stiffnesses and yielding forces in different directions and rotated yield ellipses in different floor levels due to the layout of buildings and the complexity of structural members. The seismic design often considers earthquake forces on multiple floor levels but usually only in a single direction. However, in reality, the direction of the earthquake is not limited to one particular direction. Therefore, studying the influence of a two-way, furthermore multi-dimensional, earthquake on buildings is of great value. To estimate the total seismic demand on inelastic building structures subjected to multi-dimensional loading, this paper aims to find closed-form solution responses to an input rectilinear force path for the elastoplastic bilinear model of Hong and Liu (1999) which already has available closed-form solution responses to an input rectilinear displacement path. In this paper the elastoplastic bilinear model of building structures and Minkowski spacetime are adapted to accommodate such situations as different stiffnesses and yielding forces in different directions and rotated yield ellipses in different floor levels.

A robust unsupervised neural network framework for geometrically nonlinear analysis of inelastic truss structures

In this study, a robust and simple unsupervised neural network (NN) framework is proposed to perform the geometrically nonlinear analysis of inelastic truss structures. The core idea is to employ the NN to directly estimate nonlinear structural responses without utilizing any time-consuming incremental-iterative algorithms as those done in standard finite element method (FEM). To achieve such an objective, the loss function built via the total potential energy principle under boundary conditions (BCs) is minimized in the suggested NN model whose weights and biases are considered as design variables. In our computational framework, spatial coordinates of truss nodes are treated as input data, whilst corresponding displacement degrees of freedom are taken account of output. At the beginning of each training step, feedforward is performed to get the predicted displacement field, and it is used to derive the loss function based on the physical law. Then, back-propagation is applied to update the parameters of the network. This adjustment, which is the so-called learning process, is repeated until the potential energy is minimized. Once the network is properly trained, the mechanical responses of inelastic structures can be easily obtained. The suggested methodology is also extremely simple to implement, while the unlabeled data is available, small in size, independent of sampling techniques, and without finite element analyses (FEAs). Several benchmark examples regarding geometrical and material nonlinear analysis of truss structures are tested to show the effectiveness and reliability of the proposed paradigm. Obtained outcomes indicate that the developed NN framework is robust and can be extended to apply for other structures.

Factor mapping method for grouped input variables and its application to seismic damage analysis

The study considers the characterization of global seismic damage indices of randomly parametered inelastic structures with a view to estimate the relative importance of underlying random variables with respect to variability in these damage indices. We build on an existing simulation-based approach, namely, the factor mapping method, to tackle the problem. Here, an ensemble of input variables is classified into two categories, depending upon whether a sample vector produces a global response damage measure in a pre-defined interval or not. We propose that a measure of distance between samples in these two categories be established in terms of an estimate of the Bhattacharyya distance. This distance is taken to measure the relative importance of individual/groups of input variables with respect to the damage index of interest. Illustrations include results of incremental dynamic analysis of a randomly parametered four-storied inelastic frame under a suite of scaled recorded ground motions.

Asymmetric Reflection of Shocks in Baltic Dry Index to Istanbul Freight Index

Since the maritime freight markets have inelastic supply structures in the short run, freight is considered an indicator of trade volume. Freight rates can rise rapidly when fleet utilization is high, as supply, which does not increase in the short run due to time of ship building, cannot respond to increases in demand. In this sense, the relationships between the relevant freight indices can be examined in order to determine the regional reflections of global commercial developments. The aim of this study is to examine the effects of the shocks in Baltic Dry Index (BDI), which is an indicator of dry bulk trade in the global sense and considered as a leading indicator of the world economy by many researchers, on Istanbul Freight Index (ISTFIX), which is an indicator of trade in the Mediterranean and Aegean in the regional sense. The dataset covers the period between 31.12.2007 and 19.02.2018, and consists of 524 weekly observations. Asymmetric causality test is used in order to reveal relationship between variables. According to the findings, a significant causality relationship was determined only from negative shocks in BDI to negative shocks in ISTFIX. This situation shows that the contraction in global trade is immediately reflected in the ISTFIX region, while the expansion in trade is not immediately reflected.

Impact of Ground Densification on the Response of Urban Liquefiable Sites and Structures

Ground densification is a common countermeasure against soil liquefaction. The state-of-practice for designing ground densification is typically based on empirical estimates of liquefaction triggering and ground settlement in the free field, ignoring seismic soil-structure interaction (SSI) near building structures. The existing guidelines for the geometry of ground densification also do not take into account constraints introduced by the presence of a neighboring foundation, or the impact of densification on seismic structure-soil-structure interaction (SSSI) and performance of adjacent buildings in urban settings. In this paper, three-dimensional (3D), fully-coupled, nonlinear finite-element analyses, validated with centrifuge experimental results, are used to evaluate the influence of ground densification on the seismic performance of isolated and adjacent, similar and dissimilar, inelastic structures on liquefiable soils. The response is evaluated for treatment of varying dimensions (depth [dDS] and width [WDS]), location (e.g., below one or both neighboring buildings), and symmetry configurations. Ground densification is shown to effectively reduce the permanent settlement of isolated structures when covering the full depth of the critical layer, while potentially amplifying column strains and structural deflections. Depending on the properties of the structure and symmetry of treatment, densification could also adversely impact a foundation’s permanent tilt. The influence of densification on SSSI is shown to depend strongly on the geometry of densification, dynamic properties of the neighboring structures, and building spacing. For the spacings (S) considered, ground densification effectively reduced the permanent settlement of two adjacent mitigated structure(s). However, the combination of SSSI and ground densification typically notably amplified asymmetrical deformations below the foundations (hence, permanent tilt) as well as column strains, particularly for an unmitigated neighbor. This effect was strongest when closely spaced and when WDS=S. The results indicate that ground densification location and geometry must be designed with extreme care in urban settings, particularly when near a taller and weaker neighboring structure.

Parallel projection—An improved return mapping algorithm for finite element modeling of shape memory alloys

We present a novel finite element analysis of inelastic structures containing Shape Memory Alloys (SMAs). Phenomenological constitutive models for SMAs lead to material nonlinearities, that require substantial computational effort to resolve. Finite element analysis methods, which rely on Gauss quadrature integration schemes, must solve two sets of coupled differential equations: one at the global level and the other at the local, i.e. Gauss point level. In contrast to the conventional return mapping algorithm, which solves these two sets of coupled differential equations separately using a nested Newton procedure, we propose a scheme to solve the local and global differential equations simultaneously. In the process we also derive closed-form expressions used to update the internal/constitutive state variables, and unify the popular closest-point and cutting plane methods with our formulas. Numerical testing indicates that our method allows for larger thermomechanical loading steps and provides increased computational efficiency, over the standard return mapping algorithm.

A framework for the efficient reliability assessment of inelastic wind excited structures at dynamic shakedown

The introduction of performance-based wind engineering has opened the door to the potential of designing wind excited structural systems with controlled inelasticity. The possibility of adopting the state of dynamic shakedown as a target system-level limit state is attracting growing interest. However, there is a lack of frameworks that enable the efficient assessment of the reliability of structural systems at dynamic shakedown. To overcome this limitation, a stochastic simulation-based reliability assessment framework is presented in this paper, in which the dynamic shakedown problem is formulated within the setting of distributed stress-resultant plasticity. In addition, the framework allows the estimation of traditional component-level first yield limit states therefore enabling direct comparison between reliabilities associated with system-level inelastic limit states and those aligned with current design practices. A full scale archetype is studied, from which it is seen that the limit state of dynamic shakedown occurs at load levels far beyond those inducing first yield, resulting in significant increases in reliability at dynamic shakedown as compared to component and system first yield. The straightforward estimation of the reliabilities through the proposed framework illustrated the potential of adopting dynamic shakedown as a target system-level limit state for design with controlled inelasticity.

Lumbosacral Nevus Simplex in a Newborn Girl with an Asymmetrical Y-Shaped Gluteal Cleft

Lumbosacral Nevus Simplex in a Newborn Girl with an Asymmetrical Y-Shaped Gluteal Cleft

Effect of Viscous Damping Models on Displacement Ductility Demands for SDOF Systems

The viscous damping is a highly idealized but mathematically convenient way of representing the energy dissipation mechanisms not related to the hysteretic response. In this study, the tangent-stiffness proportional Rayleigh damping (TSPRD) appropriate for inelastic hysteretic structures outfitted with the supplemental damping devices is utilized in inelastic dynamic analysis of simple single-degree-of-freedom (SDOF) structures. A numerical procedure of dynamic equilibrium equation for SDOF systems with TSPRD model is derived for development of the general numerical solution scheme. By specifying various combinations of tangent-stiffness and mass proportional damping terms in TSPRD model, the influence of viscous damping models on displacement ductility demands is investigated. The results indicate that the difference in ductility demands for various damping models is considerable, depending on the periods of vibration, relative lateral strength, hysteretic models, stiffness deterioration and initial damping ratios. The constant-strength displacement ductility spectra, by considering both tangent-stiffness proportional and mass proportional damping, are developed in terms of site conditions, hysteretic rules and initial damping ratios.

The effect of surface free energy on the fracture behaviors of nanoparticle-reinforced composites

By considering the surface free energy effect, an analytical investigation on the fracture behaviors of nanoparticle-reinforced composites was performed in this study. The plastic zones ahead of the crack tips based on the Irwin model were introduced in our analysis, which has a significant influence on the fracture toughness of inelastic structures, especially for polymer and metal matrix composites. Take copper and aluminum nanoparticles as examples, comprehensive parametric studies were conducted to investigate the interaction between a Griffith crack and the nearby nanoparticle. It was found that for nanoparticles with the radius of less than 10 nm, apart from the mechanical property of the nanoparticle, the surface free energy effect plays a dominant role in determining the fracture behaviors of the nanoparticle-reinforced composites. Moreover, taking the surface free energy into account, a decrease in the plastic zone size is obtained, particularly for the crack tip nearer the nanoparticle.