Backbone Curves Research Articles

Stochastic Nonlinear Model Updating in Structural Dynamics Using a Novel Likelihood Function within the Bayesian-MCMC Framework

The study presents a novel approach for stochastic nonlinear model updating in structural dynamics, employing a Bayesian framework integrated with Markov Chain Monte Carlo (MCMC) sampling for parameter estimation by using an approximated likelihood function. The proposed methodology is applied to both numerical and experimental cases. The paper commences by introducing Bayesian inference and its constituents: the likelihood function, prior distribution, and posterior distribution. The resonant decay method is employed to extract backbone curves, which capture the non-linear behaviour of the system. A mathematical model based on a single degree of freedom (SDOF) system is formulated, and backbone curves are obtained from time response data. Subsequently, MCMC sampling is employed to estimate the parameters using both numerical and experimental data. The obtained results demonstrate the convergence of the Markov chain, present parameter trace plots, and provide estimates of posterior distributions of updated parameters along with their uncertainties. Experimental validation is performed on a cantilever beam system equipped with permanent magnets and electromagnets. The proposed methodology demonstrates promising results in estimating parameters of stochastic non-linear dynamical systems. Through the use of the proposed likelihood functions using backbone curves, the probability distributions of both linear and non-linear parameters are simultaneously identified. Based on this view, the necessity to segregate stochastic linear and non-linear model updating is eliminated.

Open-loop swept frequency response of nonlinear structures subjected to weak coupling

AbstractThe present study demonstrates a common behaviour of a forced nonlinear structure with smooth nonlinearity, while coupled dynamics are apparent, originating from the attached electrodynamic shaker. This appears as a variation in the transmitted forcing amplitude and is often subjected to a hysteretic (multi-state) behaviour for up and down open-loop sweeping. This situation differs from the ideal constant amplitude harmonic excitation, on which parameter extraction and engineering comprehension are based on. Untreated or ignored, this can lead to the misinterpretation of the underlying dynamics through the measured nonlinear frequency response curves and their force-normalised version, often called quasi-frequency response function. In this paper, a post-processing solution is introduced for the correct interpretation of frequency response curves at constant forcing amplitudes through the open-loop construction and resectioning of the so-called frequency response surface. The phenomenon and the proposed methodology are demonstrated using a two-degrees-of-freedom model on a shaker-nonlinear beam structure. First, open-loop frequency sweeps are executed on the mechanical system to create the nonlinear frequency response surface, where their actual amplitudes and hysteresis widths are significantly different from the ideal constant forcing amplitude case. The response surface is then sectioned at the assumed constant forcing values by using an appropriate interpolation law. These resectioned curves represent the forced nonlinear standalone structure under ideal constant harmonic excitation. The frequency response surfaces are characterised and resectioned on a nonlinear structure with stiffening and softening cases. Furthermore, an improvement in the operational resonance decay (ORD) method in its filtering and automation is shown to extract the backbone curves (BBCs). The BBC and the resectioned surface provide a complete picture and cross-validation of the underlying dynamics. Finally, the BBC and its distortion are also shown in the response surfaces in relation with the excitation normalization.

Seismic response prediction of composite plate shear walls- concrete filled (C-PSW/CF) using machine learning methods

In the seismic design of composite plate shear walls filled with concrete, referred to as C-PSW/CF, cyclic backbone curves provide invaluable information for effectively withstanding seismic disturbances. The curves evaluate lateral strength, ductility, and energy dissipation capabilities, and guide performance-based design strategies. This study harnesses machine learning (ML) techniques to predict critical parameters of the cyclic backbone curve of C-PSWs/CF. Various types of existing C-PSWs/CF from the literature were categorized for this analysis. ML models, comprising Random Forests (RF), Support Vector Regression (SVR), eXtreme Gradient Boosting (XGB), and Gaussian Process Regression (GPR), were developed to facilitate the predictive tasks. The findings indicate that all models demonstrate robust performance in predicting maximum strength and ultimate strength, yielding R2 scores close to 1.00 for training and testing, and exceeding 0.90 for cross-validation. Additionally, RF, SVR, and XGB exhibit commendable accuracy in predicting yield strength and its corresponding drift ratio.

Sub‐assemblage testing and fragility analysis of connections of continuous‐plasterboard suspended ceiling systems

AbstractPast earthquake reconnaissance reports highlighted extensive seismic damages to suspended ceiling components and connections, including instances of complete ceiling failures. The seismic qualification of these nonstructural elements typically requires comprehensive evaluation through full‐scale shake table testing. However, such experimental evaluation is ordinarily not possible for every change made in various components and connections of ceiling systems due to the cost and effort involved. A feasible alternative is to obtain the behavior of components and connections from sub‐assemblage testing and incorporate them in appropriate numerical ceiling models to derive mechanical responses for developing alternative or new installation schemes. This paper considers critical connections of three continuous‐plasterboard suspended ceiling systems that were evaluated using shake table‐generated motions. The connections were classified according to their attachment to typical floors and walls of building structures, and sub‐assemblage testing was conducted using both monotonic and cyclic displacement loading. The observed failure modes in each connection were detailed, and appropriate damage states were assigned as the basis for constructing connection fragility curves. The results of the sub‐assemblage testing were presented in terms of the hysteresis responses, envelope curves, nonlinear backbone curves, and cumulative energy dissipation curves. Additionally, multilinear models of the connections were derived by approximating nonlinear backbone curves’ initial‐ and post‐yield behaviors. These multilinear models were further idealized to derive three equivalent linearized models for simplified yet reasonably accurate results for linear structural analyses. Finally, fragility curves were derived for all connections, considering their cyclic displacement failure capacities.

The analytical curvature distribution model of columns and mathematical solution for pushover analysis

AbstractThe acquisition of a flexural backbone curve for columns primarily relies on finite element analysis and experiments, of which the complexity and high cost are significant challenges. Hence, this study proposes an analytical curvature distribution model (CDM) along the full height of the column, which is the key point of formula derivation and theoretical solution of the column backbone curve. The proposed CDM is a piecewise function model composed of linear and quadratic functions, with the characteristics of continuity and differentiability at the intersection point. The deformation compatibility equations, equilibrium equations, and material constitutive equations based on the variables of CDM are constructed to form an equation system, the solution of which can be theoretically determined by an iterative algorithm. Furthermore, 155 test specimen columns of different materials, for example, reinforced concrete, high‐strength reinforced concrete, and shape memory alloy are used to validate the proposed CDM and mathematical solutions of pushover analysis through three crucial indicators, that is, goodness of fit of backbone curve, ultimate displacement, and peak force, indicating strong practicability, high accuracy, and wider applicability. This theoretical method can be applied to deal with the key issues of interest during pushover analysis, that is, predicting the flexural backbone curves with different materials, determining the curvature distribution throughout the entire process of pushover analysis, and characterizing the evolution process of the plastic region.

Influence of edge scour on lateral responses of monopiles with precast microbial reinforcement

Microbiologically Induced Calcite Precipitation (MICP) technology offers a promising method for stiffness reinforcement of offshore wind turbines (OWTs). However, edge scour around microbial reinforcement raises concerns about potential stiffness degradation. This study examines the effects of edge scour on the lateral responses of rigid piles reinforced with precast microbial reinforcement using a low-pH one-phase grouting method. Results from static tests, validated by numerical simulations, demonstrated that MICP technology bonded loose sand grains with the pile, forming a bio-reinforced pile with a larger diameter in the shallow soil layer, which significantly enhanced the original pile's bearing capacity and stiffness. However, edge scour reduced the embedment depth of the bio-reinforced pile, leading to a decrease in its bearing capacity and stiffness. Geometrically, protection width was found to have a relatively greater influence on stiffness and capacity compared to protection thickness. Additionally, symmetric cyclic loading tests were conducted to evaluate the effects of edge scour on backbone curves, secant stiffness, and damping ratio. Although MICP-based reinforcement notably enhanced both the secant stiffness and damping ratio of the piles, its effectiveness was completely lost once the scour depth reached the reinforcement thickness of the bio-reinforced soil block.

Image‐based backbone reconstruction for non‐slender soft robots

AbstractOne advantage of soft robots to conventional robots is their continuous deformation, which makes them highly adaptable to their surroundings. To describe the kinematics of soft robots it is not sufficient to only know the position of the end‐effector. Instead, the steady state of a soft robot can be described by its backbone curvature. An image‐based backbone reconstruction method is used and adopted for a non‐slender soft robot to reconstruct the backbone curve using polynomials to describe an arc length‐dependent curvature. A constant curvature and cubic curvature polynomial have been tested to reconstruct the backbone curve. In the case of a pressurized soft robot without externally applied loads, the constant curvature polynomial can be used to reconstruct the backbone of the soft robot using an image‐based approach, yielding good agreement with the captured images. However, some care has to be put into the consideration of the soft robot geometry in this method, to guide the optimization procedure. From our findings, we can conclude that it is not necessary to apply markers to the soft robot to reconstruct the backbone, nor is it necessary to have a detailed geometric model of the soft robot to be used in a differential rendering process. An image‐based iterative‐closest‐point optimization method, with some carefully placed estimate points, to integrate the volumetric structure of the soft robot can yield good results for the reconstructed backbones.

Sequence-Similar Protein Domain Pairs With Structural or Topological Dissimilarity.

For a variety of applications, protein structures are clustered by sequence similarity, and sequence-redundant structures are disregarded. Sequence-similar chains are likely to have similar structures, but significant structural variation, as measured with RMSD, has been documented for sequence-similar chains and found usually to have a functional explanation. Moving two neighboring stretches of backbone through each other may change the chain topology and alter possible folding paths. The size of this motion is compatible to a variation in a flexible loop. We search and find domains with alternate chain topology in CATH4.2 sequence families relatively independent of sequence identity and of structural similarity as measured by RMSD. Structural, topological, and functional representative sets should therefore keep sequence-similar domains not just with structural variation but also with topological variation. We present BCAlign that finds Alignment and superposition of protein Backbone Curves by optimizing a user chosen convex combination of structural derivation and derivation between the structure-based sequence alignment and an input sequence alignment. Steric and topological obstructions from deforming a curve into an aligned curve are then found by a previously developed algorithm. For highly sequence-similar domains, sequence-based structural alignment better represents the chains motion and generally reveals larger structural and topological variation than structure-based does. Fold-switching protein pairs have been reported to be most frequent between X-ray and NMR structures and estimated to be underrepresented in the PDB as the alternate configuration is harder to resolve. Here we similarly find chain topology most frequently altered between X-ray and NMR structures.

Modeling of residual stiffness phenomenon in modified Iwan model of bolted joints and its application

Bolted joints have been widely used in various mechanical structures. Due to the presence of contact interfaces, the joints exhibit complex nonlinear behavior under dynamic loading. Effective prediction of the dynamic response of bolted structures requires the construction of appropriate dynamic models. This paper proposes a modified Iwan model which gives a more comprehensive description of joints than the previous Iwan models, especially for the phenomenon of residual stiffness in macro slip. The equations of the model's backbone curve, hysteresis curve, and energy dissipation are derived. The parameter identification procedure is also provided. Subsequently, connection elements based on the modified Iwan model are integrated into a single bolted joint and a thin-walled cylinder containing multiple bolted joints, the responses under quasi-static unidirectional loading, quasi-static cyclic loading and constant-frequency excitation are investigated. The physical interpretation of the parameters in the model is discussed, thus explaining the relationship between the bolted joint's physical parameters and some important variables. The results indicate that the model can effectively characterize the nonlinear mechanical behavior of the bolted joint for both micro and macro slip regime, with significant improvement in computational efficiency.

Backbone curve orientated parameter identification for systems with coupled nonlinearity

Backbone curve attracts much concern from nonlinear system identification studies because it directly orientates the nonlinearity embedded in a system. Although extensive approaches have been proposed for normal nonlinearities that are symmetric or decoupled in the modal space, it is still challenging to extend the backbone-based identification technique for better applicability. To this end, this paper considers a series of representative 2-degree-of-freedom systems with asymmetric and coupled nonlinearities. First, the expression of the backbone curve, derived by the method of multiple scales (MMS), is augmented from only the driving-point distribution to the transfer-point distribution because of the coupling. Then, a hierarchical decoupling scheme is constructed to decompose the coupling-induced multi-modes in the transfer-point responses, making the backbone extraction technique valid for systems with coupled nonlinearities. Finally, the system parameters are identified by fitting the extended backbone functions to the decomposed samples. Numerical and experimental examples are both provided to demonstrate the detailed operations of the identification algorithm. The results show that by introducing the augmented backbone curves, the system parameters can be identified completely, indicating a successful improvement to the existing backbone-based identification technique.

An Experimentally Validated Numerical Model for Generating the Cyclic Backbone Curve of LYP Links

An Experimentally Validated Numerical Model for Generating the Cyclic Backbone Curve of LYP Links

Qualitative validation on a numerical model for the direct stability assessment of surf-riding and broaching

A previously developed numerical model for surf-riding and broaching is qualitatively validated based on the criteria of direct stability assessment (DSA) proposed in SDC 7. Roll decay tests are conducted to obtain roll backbone curves, which are compared with calculated GZ curve to demonstrate their consistency. Then, roll response curves under different wave frequencies in regular beam wave are obtained. It is verified that roll response curve “fold around” the newly defined “peak backbone curve”. Cases with fixed yaw under different Fn are chosen to demonstrate the numerical model's capability of reproducing surf-riding equilibrium. Turning circle test is conduced to demonstrate correct modelling of maneuvering forces and the capability to reproduce heel due to ship turning. Wave surging forces are calculated under different wave numbers and amplitudes. The correct tendency of phase difference between wave surging force and the incident wave is observed, which reveals the correct modelling of wave forces in the model. Moreover, a preliminary quantitative validation is conducted under various cases. Through comparison with experimental results, it is recommended that the wave diffraction correction of the wave surging forces should be included in the simulation model in order to avoid overestimation of surf-riding and broaching.

Prediction of parameters in the Ibarra–Medina–Krawinkler model for reinforced concrete columns using random forest and active learning

The lumped plasticity model is widely used in the practical modeling of reinforced concrete (RC) columns because of its computational efficiency. However, existing formulations for estimating the backbone curve and cyclic deterioration parameters often fail to accurately predict new data with high variability and necessitate laborious calibrations. To address it, a machine-learning (ML) approach utilizing the random forest (RF) algorithm to predict seven parameters in the Ibarra–Medina–Krawinkler lumped plasticity model for depicting hysteretic response is proposed in this study, where a comprehensive data of 475 RC columns, encompassing diverse material properties, geometric features, and failure modes, is used. In addition, an active-learning framework is integrated to address the limited availability of labeled data in supervised ML tasks, which reduces the exhausted labeling process. The proposed RF surpasses existing research and commonly used ML models regarding coefficient of determination. Additionally, the predicted parameters more accurately simulate the behavior of strength and stiffness degradation in the hysteretic loop of columns than empirical regression formulas. These results will benefit the high-fidelity seismic risk assessments of RC buildings.

Stress field on intermediate length scales during triaxial compression

Defining a continuum stress field in a granular medium involves an approximation, as the underlying contact force network between particles is inhomogeneous and random. The continuum approximation introduces some baseline noise or variability in stress, which becomes more significant as the length scale of interest approaches the typical size of the grains. In this study, the authors evaluate how the stress variability resulting from the homogenisation of intergranular forces changes as the homogenisation scale is increased. This is done by obtaining the probability distribution function of stresses from a large number of repeated discrete-element method-based simulations of the triaxial compression of granular samples. The numerical experiments share sample dimensions, generation protocols and imposed stresses, but involve randomly generated sets of granular packings made of different numbers of particles. The results from this study are used to propose and develop a consistent general framework to rigorously describe mesoscale stress fields and hence fill the gap between existing micro- and macromechanical approaches. A backbone curve is obtained expressing the change in stress variability as the mesoscale is traversed. The results deduced from the backbone curve are in good agreement with those of independent numerical and physical experiments on granular soils.

Nonlinear oscillation of elastic sagged cable using refined mode truncation: A comparative study

A nonlinear sagged cable, due to its initial curvature, leads to various challenges of empirical mode truncation used by routine Galerkin method when constructing reduced-order model. It is recently elucidated that ( Guo and Rega, 2023a ), the key for refined mode truncation (and thus for correct nonlinear dynamics prediction) is to first eliminate low-order nonlinear terms of spatial continuous structures. This paper focuses on refined truncation of nonlinear sagged cable by leveraging the recent low-order elimination perspective, which is realized by a normal form development. Further comparative studies for both primary resonant and two-to-one internally resonant dynamics of the sagged cable, including nonlinear frequency responses, backbone curves, and Poincaré mapping, demonstrate notable differences between the two different types of models built by either routine or refined truncation, which confirms necessity of the refined mode truncation used for geometrically nonlinear structures like sagged cables.

Backbone curve tailoring via Lyapunov subcenter manifold optimization

We present a technique for the direct optimization of conservative backbone curves in nonlinear mechanical systems. The periodic orbits on the conservative backbone are computed analytically using the reduced dynamics of the corresponding Lyapunov subcenter manifold (LSM). In this manner, we avoid expensive full-system simulations and numerical continuation to approximate the nonlinear response. Our method aims at tailoring the shape of the backbone curve using a gradient-based optimization with respect to the system’s parameters. To this end, we formulate the optimization problem by imposing constraints on the frequency-amplitude relation. Sensitivities are computed analytically by differentiating the backbone expression and the corresponding LSM. At each iteration, only the reduced-order model construction and sensitivity computation are performed, making our approach robust and efficient.

Seismic response assessment of ductile reinforced concrete columns affected by corrosion and axial load variations

The present study investigates the effect of reinforcement corrosion and axial compression ratio (ACR) on the seismic response of large-scale ductile reinforced concrete (RC) columns. 3D numerical models of sound and corroded RC columns were developed and validated with detailed experimentation. Various parameters, such as hysteresis and backbone curves, stiffness degradation, ductility, and energy dissipation, were evaluated for all the specimens. The numerical findings were then compared with the experimental findings. Subsequently, a thorough parametric study was executed to analyse the structural behaviour under varying corrosion degrees and ACR level combinations. It was observed that both sound and corroded specimens experienced significant degradation in terms of their strength, deformability, stiffness, and ductility due to elevated ACR levels. Moreover, increased ACR levels in both specimen types led to a decline in pre-peak and post-peak response trajectories, resulting in an early achievement of peak response at lower drift levels. The study’s outcome advocates the significance of incorporating ACR levels into the design philosophy of ductile reinforcements for RC columns.

Dynamic properties of alkali residue-based foamed concrete under cyclic load

Alkali residue-based foamed concrete (A-FC), consisting of cement, alkali residue, blast furnace slag, and foam, has great potential for application in subgrade engineering. A series of dynamic triaxial tests were conducted, and the influences of confining pressure, vibration frequency, and curing age on the dynamic properties of A-FC was revealed. Test results indicated that the backbone curves of A-FC exhibit a hyperbolic shape, effectively captured by the H-D model. After 7 days of curing, A-FC possessed robust dynamic bearing capacity. As the dynamic strain increases, the dynamic elastic modulus of A-FC exhibits an initial rise followed by a gradual decline. Concurrently, the damping ratio demonstrates an initial decrease and subsequent gradual increase. Elevated confining pressure, prolonged curing age, and higher vibration frequency were found to contribute to a progressive increase in the dynamic elastic modulus of A-FC. Throughout the damping ratio growth stage, the increasing curing age and confining pressure led to the decrease of damping ratio, with minimal influence from vibration frequency. A calculation model for the maximum dynamic elastic modulus of A-FC was proposed, considering confining pressure, vibration frequency, and curing age. This research underscores the capacity of A-FC to withstand cyclic loading, thereby offering valuable support for its engineering applications.

Identification of secondary resonances of nonlinear systems using phase-locked loop testing

One unique feature of nonlinear dynamical systems is the existence of superharmonic and subharmonic resonances in addition to primary resonances. In this study, an effective vibration testing methodology is introduced for the experimental identification of these secondary resonances. The proposed method relies on phase-locked loop control combined with adaptive filters for online Fourier decomposition. To this end, the monotonic evolution of the phase lag around secondary resonances is exploited for their identification. The method is demonstrated using two systems featuring cubic nonlinearities, namely a numerical Duffing oscillator and a physical experiment comprising a clamped–clamped thin beam. The obtained results highlight that the control scheme can accurately characterize secondary resonances as well as track their backbone curves. A particularly salient feature of the developed algorithm is that, starting from the rest position, it facilitates an automatic and smooth dynamic state transfer toward one point of a subharmonic isolated branch, hence, inducing branch switching experimentally.

Numerical investigation of nonlinear soil-structure interaction effects on response of irregular RC buildings

Kathmandu, located in a high seismic zone, predominantly features irregular structures among its building stock. These structures are particularly susceptible to severe damage during seismic events, primarily due to torsional effects. Traditional seismic designs rely particularly on fixed-base conditions that underestimate forces and displacement primarily on soft soil conditions leading to irrational design practices. This study aims to quantify the seismic performance of buildings on various base conditions through fully nonlinear Soil-structure Interaction (SSI). The soil nonlinear behaviour was modelled using the Pressure-Independ-Multi-Yield (PIMY) material with an octahedral shear stress-strain backbone curve. Three distinct soil types were considered, and structures with irregular plan configurations were modelled using finite elements in both Opensees and STKO platforms. Structural performance was analyzed through nonlinear dynamic analysis, and outputs were evaluated based on seismic parameters. Comparing nonlinear SSI with linear SSI and fixed-base conditions revealed a significant increase in structural response, expressed in terms of displacement, drift ratio, and base shear. The magnitude of diaphragm rotation was found to be influenced by a combination of building torsional irregularity and SSI effects. It is suggested that the conventional practice of using the torsional irregularity ratio as a measure of torsional irregularity be revised and enhanced to better account for these influences. It has been quantified that torsional irregularity has a relatively lesser impact on displacements, drifts, and base shear compared to SSI. In all cases, fixed-base conditions consistently exhibited the minimum response. The study explored that linear SSI and fixed-base conditions tend to underestimate structural responses, while nonlinear SSI coupled with dynamic analysis provides a more accurate representation of realistic structural behaviour for seismic design particularly in soft soil cases.