Backbone Curves Research Articles

Instantaneous phase difference of nonlinear coupled bubbles from nonstationary forcing

The attraction and repulsion regimes of acoustically forced bubbles follow from their phase coupling. The direct evaluation of phase difference has received less attention than the secondary Bjerknes force, though it has important implications for nonlinear, nonstationary forcing regimes. We numerically investigate the response of two nonlinearly coupled bubbles undergoing radial oscillations and translation from chirp forcing. The corresponding scattered acoustic responses are analyzed using the empirical mode decomposition (EMD) and Hilbert transform in terms of respective instantaneous amplitude, frequency, and phase. This allows for determination of the phase differences from the coupling between the two bubbles. The phase coupling results agree with previous analytical theory in the linear acoustic limit. Attraction and repulsion regimes for nonstationary forcing can be classified with respect to the instantaneous phase differences unlike the time averaged secondary Bjerknes force. In the nonlinear regime, the chirp direction (forward, backward) results in significantly different radial and translational responses. This result can be explained with respect to influence of the softening jump discontinuity (backbone curve) on the phase coupling. Notably a backward chirp results in overall greater amplitude response and thus more translation during attraction. Implications for optimal waveforms are highlighted for potential applications.

Curvature response of bridge pier subjected to earthquake action in a saline soil environment

The curvature of bridge piers can be described in terms of their syntagmatic deformation in the cross-sectional tensile and compressive zones. In the present study, 12 reinforced concrete bridge piers were designed and cast. Two of them were control specimens without corrosion, while the other 10 were corroded by immersion in a saline solution through the application of an electrical current; pseudo-static tests were conducted on the bridge piers after corrosion. The failure modes, corrosion ratios of the rebars, and load–curvature backbone curves were obtained and discussed. The evolution in curvature, in relation to different axial forces and corrosion ratios, was compared and analysed. It was found that the effective diameter of the rebars was reduced by corrosion, resulting in considerable reductions in the expected deformability (curvature) of the bridge pier cross-section. A robust load–curvature model was derived based on classical lamination theory and by dividing the bridge pier cross-section into concrete and steel reinforcement bar layers. The modelled curvature values match the results of the pseudo-static tests well. The experimental data and model can help lay out improved guidelines for bridge design, particularly for the seismic design of bridge piers in saline soil environments.

Seismic experimental assessment of beam-through beam-column connections for modular prefabricated steel moment frames

This study proposes a beam-through moment-resisting joint for H-section (wide-flange) beams and columns that can be prefabricated and easily assembled onsite for low-rise and mid-rise steel structures. Three groups (six in total) of beam-through beam-column connections are designed and tested under quasi-static loading protocols. The output parameters include joints' deformation, failure mechanism, hysteretic behavior, backbone curves, bearing capacity, ductility, energy dissipation capacity, joint rotational stiffness. The results demonstrate that the combination of through-beam continuity plate stiffeners, panel zone doubler plates, and reduced beam sections provide optimal seismic performance. A simplified analytical method is also proposed to calculate the initial rotational stiffness of the through-beam column joint.

Effect of strain rate on triaxial extension behavior of silty clay

Traditional methods of suction caisson foundation design only consider the soil is under either purely drained or purely undrained conditions and ignore the effects of the excess pore water pressure (EPWP). A series of triaxial extension stress path tests were conducted in this article to evaluate the effects of the loading rate on the development of EPWP and soil strength. The soil specimens were consolidated under pressures of 60, 80, and 100 kPa and then unloaded at seven different loading rates. It is found that the EPWP and soil strength are heavily influenced by the loading rate. The EPWP tends to be negative at small axial strain and then increases to be positive with the increase of axial strain. The normalized deviatoric stress versus loading rate curves are in the backbone curve. The minimum deviatoric strength happens when the normalized loading rate V ref=5 and linearly increases with the increase of consolidation pressure with the slope of −0.715. The magnitude of the slope of the critical state line in p′ – q space is heavily influenced by the loading rate.

Periodic Motion and Frequency Energy Plots of Dynamical Systems Coupled With Piecewise Nonlinear Energy Sink

Abstract The frequency-energy plots (FEPs) of two-degree-of-freedom linear structures attached to a piecewise nonlinear energy sink (PNES) are generated here and thoroughly investigated. This study provides the FEP analysis of such systems for further understanding to nonlinear targeted energy transfer (TET) by the PNES. The attached PNES to the considered linear dynamical systems incorporates a symmetrical clearance zone of zero stiffness content where the boundaries of the zone are coupled with the linear structure by linear stiffness elements. In addition, linear viscous damping is selected to be continuous during the PNES mass oscillation. The underlying nonlinear dynamical behavior of the considered structure-PNES systems is investigated by generating the fundamental backbone curves of the FEP and the bifurcated subharmonic resonance branches using numerical continuation methods. Accordingly, interesting dynamical behavior of the nonlinear normal modes (NNMs) of the structure-PNES system on different backbones and subharmonic resonance branches has been observed. In addition, the imposed wavelet transform frequency spectra on the FEPs have revealed that the TET takes place in multiple resonance captures where it is dominated by the nonlinear action of the PNES.

Battered minipile response to low-frequency cyclic lateral loading in very dense sand

This experimental study investigates the response of vertical and battered minipiles to two-way symmetrical low-frequency (0.1 Hz) cyclic lateral loading. Laboratory (1-g) tests were performed on scaled-down minipiles in very dense cohesionless soil, for batter angles of 0°, 25° and 45°. The cyclic loading is classified into two categories: multi-amplitude and long-term single amplitude, where force-controlled load was applied at a constant frequency. The minipiles were instrumented with optic fibres, and strain profiles were obtained at each loading stage, in both compression and tension stroke. The results are presented in terms of hysteresis loops, variation of normalised stiffness, minipile strain and bending moments under cyclic loading. In the multi-amplitude loading category, backbone curves show a stiffer force–displacement response in tension stroke than in compression stroke. For the single-amplitude category, the area of the hysteresis loop is largest for 45° battered minipiles with the lowest accumulated deformation. The normalised stiffness at the end of 50 cycles is highest for 25° minipiles with a value slightly greater than one. The strain profiles along the minipiles show stabilisation of measured strain before the number of cycle reaches 50, for all three battered conditions. A multi-surface hardening constitutive model is used to explain the effect of shearing and cyclic loading, with increasing loading amplitude on 25° battered minipiles. These test results are indicative of better performance capability of 25° battered minipiles, in terms of secant stiffness, compared to the vertical and 45° battered cases.

A novel GPR-based prediction model for cyclic backbone curves of reinforced concrete shear walls

Backbone curves are used to characterize nonlinear responses of structural elements by simplifying the cyclic force–deformation relationships. Accurate modeling of cyclic behavior can be achieved with a reliable backbone curve model. In this paper, a novel machine learning-based model is proposed to predict the backbone curve of reinforced concrete shear (structural) walls based on key wall design properties. Reported experimental responses of a detailed test database consisting of 384 reinforced concrete shear walls under cyclic loading were utilized to predict seven critical points to define the backbone curves, namely: shear at cracking point (Vcr); shear and displacement at yielding point (Vy and δy); and peak shear force and corresponding displacement (Vmax and δmax); and ultimate displacement and corresponding shear (Vu and δu). The predictive models were developed based on the Gaussian Process Regression method (GPR), which adopts a non-parametric Bayesian approach. The ability of the proposed GPR-based model to make accurate and robust estimations for the backbone curves was validated based on unseen data using a hundred random sampling procedure. The prediction accuracies (i.e., ratio of predicted/actual values) are close to 1.0, whereas the coefficient of determination (R2) values range between 0.90–0.97 for all backbone points. The proposed GPR-based backbone models are shown to reflect cyclic behavior more accurately than the traditional methods, therefore, they would serve the earthquake engineering community for better evaluation of the seismic performance of existing buildings.

A consistency analysis of phase-locked-loop testing and control-based continuation for a geometrically nonlinear frictional system

Two of the most popular vibration testing methods for nonlinear structures are control-based continuation and phase-locked-loop testing. In this paper, they are directly compared on the same benchmark system, for the first time, to demonstrate their general capabilities and to discuss practical implementation aspects. The considered system, which is specifically designed for this study, is a slightly arched beam clamped at both ends via bolted joints. It exhibits a pronounced softening–hardening behavior as well as an increasing damping characteristic due to the frictional clamping. Both methods are implemented to identify periodic responses at steady-state constituting the phase-resonant backbone curve and nonlinear frequency response curves. To ensure coherent results, the repetition variability is thoroughly assessed via an uncertainty analysis. It is concluded that the methods are in excellent agreement, taking into account the inherent repetition variability of the system.

Cyclic Experiments on Steel Sheet Connections for Standard CFS Framed Steel Sheet Sheathed Shear Walls

The primary objective of this work is to provide connection-level force-deformation response appropriate for standard cold-formed steel (CFS) framed steel sheet sheathed shear walls under cyclic loads. Common CFS framing designs increasingly are exploring thicker framing options so that walls can meet gravity demands, overturning demands, and seismic overstrength requirements. For the seismic performance of self-drilling screw-fastened steel sheet sheathed shear walls, the cyclic nonlinear response of the screw-fastened connection is particularly important and should incorporate the impact of shear buckling of the steel sheet on the strength and ductility of the connection. Minimal cyclic connection-level shear test data exist, especially for combinations of screw-fastened thin steel sheet and thick framing steel. A unique lap shear test following current test standards was proposed to elucidate and characterize the cyclic screw-fastened connection behavior. An asymmetric cyclic loading protocol was selected with a small displacement applied in the direction that buckles the thin steel sheet, followed by progressively larger displacements in the opposite direction. A total of 93 tests were conducted, and characterization of the observed cyclic connection response with a multilinear backbone curve appropriate for use in models is provided. Connection strength is sensitive to whether the thin steel sheet ply is buckling away from or toward the fastener head in some test series. Performance of the screw shear strength as per the standard’s provisions is evaluated. The work is intended to provide critical missing information for CFS framed steel sheet sheathed shear walls for use in both simulation and design.

High-strength steel deep H-shaped and box columns under proposed near-fault and post-earthquake loadings

This paper presents the seismic evaluation of high-strength steel first-story columns in a seven-story dual system, composed of a special moment-resisting frame (SMRF) and buckling-restrained braced frame (BRBF), in near-fault and post near-fault earthquakes. Nonlinear time history analyses were conducted on the frame to develop loading protocols that represent seismic demands on the first-story columns in the dual system. The proposed near-fault displacement protocol has a large displacement pulse from -2 to +4% drift, followed by small displacement cycles oscillated at a residual drift of 2.5%. The proposed far-field displacement protocol is an increasingly cyclic displacement protocol with one cycle after a drift of 2%. Two built-up H-shaped column specimens (HC-20-B and HC-20-C) and two built-up box column specimens (HBC-40-B and HBC-40-C), representing the first-story columns in the dual system, are tested under constant axial force (0.2Py or 0.4Py) and lateral drifts based on the proposed loading protocols. These specimens that satisfy the compactness requirement of highly ductile members in AISC 341-16 (2016) show acceptable seismic performance under the near-fault earthquake loading. However, the flexural strength of HC specimens decreases with increasing drift, and that of HBC specimens increases without strength degradation. The HBC specimens can perform well in the subsequent near-fault or far-fielding earthquake loading, which is not the case for the HC specimens due to significant local buckling. The NIST (2017) backbone curve reasonably predicts the test envelope of the HC specimens, but significantly underestimates the test envelope of the HBC specimens.

Bifurcation Analysis on the Periodic Response of a Comb Drive MEMS Resonator.

In this paper, we investigate the bifurcation characteristics of a comb drive MEMS resonator. The method of averaging and the residue theorem are used to get a more accurate analytical solution for the periodic response. Then, the singularity theory is employed to give the transition sets on the DC-AC voltage plane and the lateral separation-quality factor plane, which divide the planes into 9 persist regions. The corresponding bifurcation diagrams are present to discuss the jump phenomena of the periodic response, and the influences of the parameters on the amplitude-frequency response are studied. We also attempt to analyze the feasibility for the resonators working in the nonlinear regions and give the available frequency range and the available maximum amplitude of the nonlinear response. With the increase of the DC voltage, the amplitude-frequency curves change from hardening to softening, and the lateral separation has the opposite effect. The amplitude-frequency curves increase along the backbone curves with the AC voltage and quality factor. The response curves of softening or hardening characteristics have enough available frequency range and large available amplitudes, which may be more appropriate for the operation of the resonator than those of the mixture characteristics.

Deformation capacity of FRP retrofitted reinforced concrete columns with corroded reinforcing bars

FRP retrofitting is an effective way to promote the degraded seismic capacity of reinforced concrete (RC) columns with corroded reinforcing bars. It is important to evaluate whether the performance of a corroded RC column meets the requirement of safety after FRP retrofitting. However, there is little available guidance for evaluating the deformation capacity of FRP retrofitted corroded columns. In this study, the test data of 63 FRP retrofitted corroded columns and 17 FRP retrofitted uncorroded columns under repeated cyclic loads are collected to establish a database. For each specimen, the rotation angles at yielding and ultimate states are determined from the backbone curve. The empirical prediction models are then developed for the yield and ultimate rotation angles of FRP retrofitted corroded columns, which include two parts: 1) the predicted rotation angle for the FRP retrofitted uncorroded RC column, and 2) a Corrosion Influence Factor (CIF) defined as the ratio between the rotation angles of the FRP retrofitted corroded columns to that of the FRP retrofitted uncorroded ones. The former part is represented using the available empirical prediction models, whereas the latter part is represented using the multi-linear relationships through stepwise regression with a backward elimination procedure. Through comparing with the experimental data, the developed empirical prediction models for the yield and ultimate rotation angles of FRP retrofitted corroded columns provide sufficient accuracy. Finally, the parameter “a” is determined in corresponding to different target probability of exceedance, which is required by the ASCE 41-compliant force versus displacement relation for FRP retrofitted corroded RC columns.

Experimental and Numerical Investigation of a Dissipative Connection for the Seismic Retrofit of Precast RC Industrial Sheds

Past earthquakes have highlighted the seismic vulnerability of prefabricated industrial sheds typical of past Italian building practices. Such buildings typically exhibited rigid collapse mechanisms due to the absence of rigid links between columns, beams, and roof elements. This study aims at presenting the experimental and numerical assessment of a novel dissipative connection system (DCS) designed to improve the seismic performance of prefabricated sheds. The device, which is placed on the top of columns, exploits the movement of a rigid slider on a sloped surface to dissipate seismic energy and control the lateral displacement of the beam, and to provide a recentering effect at the end of the earthquake. The backbone curve of the DCS, and the effect of vertical load, sliding velocity, and number of cycles were assessed in experimental tests conducted on a scaled prototype, according to a test protocol designed accounting for similarity requirements. In the second part of the study, non-linear dynamic analyses were performed on a finite element model of a portal frame implementing, at beam-column joints, either the DCS or a pure friction connection. The results highlighted the effectiveness of the DCS in controlling beam-to-column displacements, reducing shear forces on the top of columns, and limiting residual displacements that can accrue during ground motion sequences.

Machine learning-based adaptive degradation model for RC beams

Reliable seismic damage assessment of structural systems requires analytical models that are able to capture the cyclic deterioration of components. In this paper, an adaptive-updated degradation model (AUDM) for reinforced concrete (RC) beams is proposed based on the machine learning approach. The proposed model is capable of updating its constitutive parameters from the embedded artificial neural networks, based on the analysis results derived from the previous step. It offers the possibility for generating deterioration without constructing empirical equations, updating the hysteretic shape as the accumulation of damage, and accounting for the effect of variable shear-flexural interaction in nonlinear analysis. The details of the proposed model, including the backbone curve, the hysteretic behavior and the cyclic deterioration rule, were first described. An experimental database consisting of 100 cantilever rectangular RC beams under cyclic loading was collected from the existing literature to identify the constitutive parameters in the proposed model. Artificial neural networks for predicting the constitutive parameters in the proposed model were developed and trained based on the identified data, which was then embedded into the open-source computational platform OpenSees to implement AUDM. The accuracy of the proposed model over the existing commonly-used degradation model is revealed in OpenSees based on the test results of RC beams under cyclic loading. It was found that the proposed AUDM was shown to be superior to the existing degradation models in terms of the lateral bearing capacity, the dissipated energy, the cyclic deterioration and the hysteretic shapes under different failure modes, with the relative error of energy dissipation capacity in the range between −0.19 and 0.19 and the maximum root square error for strength deterioration less than 0.259.

Cyclic performance assessment of damaged unreinforced masonry walls repaired with steel mesh reinforced shotcrete

Unreinforced masonry (URM) buildings are rather popular around the globe due to low construction costs, even though they can be prone to substantial damage caused by even moderate earthquakes. Numerous URM buildings that have experienced damages from past earthquakes require to be upgraded or at least return to their undamaged state in order to be able to withstand future earthquakes. In many cases, reconstruction is not the best choice because of financial and time restrictions. As such, repair/retrofit is the best choice, assuring the post-earthquake serviceability. Furthermore, seismic repair/retrofit can be a cost-efficient method to avoid reconstruction complexities and expenses. In this study, steel-mesh reinforce shotcrete (SRS) is proposed as a cheap and simply conducted yet effective repair/retrofit method. To prove the enhanced capability of the proposed system, a series of experimental tests were carried out on typical 1:1.5 scale URM walls while considering influential parameters i.e., vertical load, presence of opening in the walls, and the effect of prior damage in the URM. The walls were first tested without any intervention to improve their response; then, the damaged walls were repaired using shotcrete on one side and tested again. The repaired/retrofitted walls performed even better than the undamaged URM walls. The results show notable improvement in lateral behavior resulting in an average increase of 288% in lateral strength capacity. More importantly, the repaired walls performed as well as the corresponding retrofitted undamaged wall. To quantify the response of the walls, response modification factors were calculated based on the backbone curves derived from the hysteresis curves.

Quantitative evaluation of nonlinearities in frequency response characteristics of two DOF Duffing systems

Nonlinear characteristics of two DOF Duffing oscillator with fifth order term are investigated. The system used in this study has mass and stiffness proportional dampings simultaneously, and two combinations of stiffnesses that have same natural frequencies. In our previous reports for the same system, nonlinear characteristics were qualitatively studied using frequency response curves of modal coordinates. However, the quantitative criterion of the nonlinear characteristics is desirable. For one DOF Duffing oscillator, single backbone curve symbolically describes many frequency response curves for the various amplitude of the response. In this study, the coefficients of the backbone curves are evaluated and compared with the parameters of the system. The calculated examples show that the coefficients were successfully identified for the modal amplitudes with softening characteristics. However, for the case that the modal amplitudes had the softening and hardening characteristics, the backbone curves went out of the frequency response curves. The parameters of backbone curves were independent from the ratio between mass and stiffness proportional dampings.

Motion-Planning Algorithm for a Hyper-Redundant Manipulator in Narrow Spaces

In this study, a hyper-redundant manipulator was designed for detection and searching in narrow spaces for aerospace and earthquake rescue applications. A forward kinematics equation for the hyper-redundant manipulator was derived using the homogeneous coordinate transformation method. Based on the modal function backbone curve method and the known path, an improved modal method for the backbone curves was proposed. First, the configuration of the backbone curve for the hyper-redundant manipulator was divided into two parts: a mode function curve segment of the mode function and a known path segment. By changing the discrete points along the known path, the backbone curve for the manipulator when it reached a specified path point was dynamically obtained, and then the joint positions of the manipulator were fitted to the main curve by dichotomy. Combined with engineering examples, simulation experiments were performed using the new algorithm to extract mathematical models for external narrow space environments. The experimental results showed that when using the new algorithm, the hyper-redundant manipulator could complete the tasks of passing through curved pipes and moving into narrow workspaces. The effectiveness of the algorithm was also proven by these experiments.

Analiza nelinearnih slobodnih vibracija neujednačene aksijalno stepenovane grede na promenljivoj elastičnoj osnovi

The present work investigates the nonlinear free vibration of an axially functionally graded (AFG) beam supported on the variable foundation. The beam geometry is non-uniform, with a linear cross-section variation along the length. The beam material is graded along the axial direction following the power-law relation. A Winkler type of variable elastic foundation is taken, in which variation of stiffness is considered along the length of the foundation. Geometrical nonlinearity produced by the beam's large-amplitude deflection is also considered. To attain the desired objectives, the problem is divided into two parts. The static problem is solved first, then a subsequent free vibration analysis is executed on the statically deformed beam configuration. The governing differential equations of the system are derived using suitable energy methods. A numerical technique of direct substitution with relaxation is utilized to obtain the solution of the derived nonlinear differential equations. A suitable validation study is presented to ensure the appropriateness of the present methodology. Benchmark results are also presented by means of natural frequency, backbone curve, and mode shape plot to investigate the influences of elastic foundation, material gradation, and non-uniform geometry on nonlinear vibration.

Detection of different dynamics of two coupled oscillators including a time-dependent cubic nonlinearity

Vibratory energy channelling between a linear and a nonlinear oscillator is studied at different time scales. The nonlinear system possesses a time-dependent periodic restoring forcing function. Detection of fast and slow system dynamics leads to revealing different dynamical characteristics, namely slow invariant manifold, equilibrium and singular points. We show that the time-dependent nonlinearity produces a phase-dependent slow invariant manifold, frequency responses, and modifications concerning stability borders of its slow invariant manifold and singularities zones. The backbone curves of the system and also isola are detected; the latter should be taken into account carefully if the aim is system control.

Normal form transformations for structural dynamics: An introduction for linear and nonlinear systems.

The aim of this paper is to provide an introduction to using normal form transformations for linear and nonlinear structural dynamics examples. Starting with linear single-degree-of-freedom systems, a series of examples are presented that eventually lead to the analysis of a system of two coupled nonlinear oscillators. A key part of normal form transformations are the associated coordinate transformations.This review includes topics such as Jordan normal form and modal transformations for linear systems, while for nonlinear systems, near-identity transformations are discussed in detail. For nonlinear oscillators, the classical methods of Poincaré and Birkhoff are covered, alongside more recent approaches to normal form transformations. Other important topics such as nonlinear resonance, bifurcations, frequency detuning and the inclusion of damping are demonstrated using examples. Furthermore, the connection between normal form transformations and Lie series is described for both first and second-order differential equations. The use of normal form transformations to compute backbone curves is described along with an explanation of the relationship to nonlinear normal modes. Lastly, conclusions and possible future directions for research are given.