Backbone Curves Research Articles

Adopting a fiber macroelement for accounting the SSI effect in nonlinear seismic analysis

The main purpose of this paper is to propose a practical approach for considering the soil - shallow foundation - superstructure interaction effect in nonlinear seismic analysis. Soil structure interaction is a complicated task dealing with various engineering problems. Taking into account the soil structure interaction extends beyond the numerical analysis, geotechnical investigation, or structural design fields. The artistry of considering the soil structure interaction effect lies in the balance between incorporating soil conditions and the superstructure into one collective model. Following this methodology alongside classical soil mechanics principles provides additional confidence to the structural engineering society in adopting a complete practical concept. The herein presented study introduces a fiber macroelement that represents the soil beneath shallow foundations. This allows for the direct utilization of the fiber macroelement in nonlinear pushover and dynamic analysis in commercial software, as well as its simple implementation in open software systems. The introduction of a procedure for forming the stiffness matrix of the element and a hysteresis soil model is also a key component. Special attention is devoted to obtaining the material backbone curve through experimental results and Eurocode 8?s procedures. Furthermore, this paper demonstrates the implementation of a macroelement for considering the soil-structure interaction effect in nonlinear seismic analysis within commercial software. The primary aim is to provide structural designers with a streamlined approach for incorporating soil conditions into superstructure analysis. Numerical analysis results from an example building with a reinforced concrete structure in commercial software highlight the significance of not underestimating soil conditions. This work emphasizes that relying solely on fixed models in everyday structural design could lead to dangerous and unexpected consequences.

Simplified Nonlinear Analysis of RC Columns Exposed to Lateral Loads

In this study, the monotonic and hysteretic behavior of poorly detailed reinforced concrete (RC) columns under lateral loads is simply modeled by SAP2000. It has been shown in literature that the flexural, reinforcement slip and shear deformations is contributed to lateral deformation. The monotonic and hysteretic responses of RC columns due to each of these deformations, were determined. Under pushover loading, deformations due to the flexural, reinforcement slip and shear deformations were summed up to create monotonic behavior of the column. The load-deformation curve from the monotonic model were used to create backbone curve of hysteretic pivot model. The monotonic and hysteretic lateral load- displacement curves obtained in this study were compared with the results obtained from experimental studies. It has been shown that SAP2000 and experimental results were in good agreement and the behavior of poorly detailed RC columns are simplified by SAP2000 instead of using the complex analysis methods.

Frequency-Energy Analysis of Coupled Linear Oscillator With Unsymmetrical Nonlinear Energy Sink

Abstract The underlying nonlinear dynamical behavior of a linear oscillator (LO) attached with nonlinear energy sink (NES) is usually revealed by studying the LO-NES system on the frequency-energy plot (FEP). Accordingly, different kinds of periodic motions on the backbone curves of the nonlinear normal modes (NNMs) and their associated subharmonic branches are analyzed. Here, the considered unsymmetrical nonlinear energy sink (UNES) incorporates a cubic stiffness element in one direction of the motion and a weak linear restoring coupling stiffness in both directions of the motion. This UNES is attached with the LO in the LO-UNES system which is studied here on the FEP. Therefore, the obtained FEP of the LO-UNES system is generated and analyzed. Accordingly, several unsymmetrical backbone curves of periodic oscillations between the UNES mass and the LO mass at 1:1 resonance have been obtained for broad range of nonlinear frequency levels.

Modelling the Cyclic Response and Development of the Backbone Curve of Corroded RC Bridge Piers

Modelling the cyclic response for the evaluation of the seismic capacity of existing corroded RC bridges is important as it is well recognised in literature. Procedures exist in the literature for the inclusion of corrosion effects through variation in different parameters like cross-sectional area of reinforcement, stress–strain behaviour, yield strength, ultimate strength and elastic modulus. In the present study, change in the inelastic buckling behaviour of reinforcement and variation in the core concrete damage characteristic under different levels of corrosion are also modelled in detail with OpenSees computational software. Modelling procedures for pristine and corroded bridge piers are validated by comparison of the cyclic response results and backbone curves obtained from analytical modelling with the experimental results reported in the literature. In this paper, the procedure for the cyclic response evaluation of piers for different mass loss percentages for corrosion has been demonstrated. From cyclic response results, the backbone curves for corroded bridge pier specimens are developed for different mass loss percentages of corrosion. The modelling strategy proposed in this paper for the cyclic response evaluation of bridge piers can be adopted for quantifying the effect of corrosion on the cyclic response of similar important structures.

Quantifying Uncertainties in Nonlinear Dynamics of a Modular Assembly Using the Resonance Decay Method

Modular assembling is a promising approach to constructing large spacecraft beyond the size limitations posed by launch vehicles. However, the uncertainties and nonlinearities of the dynamics associated with the assembled structure are deeply concerned with the design stage of such a spacecraft. Conventionally, this concern can be relieved by performing Ground Vibration Testing (GVT) of the structure. Nevertheless, it is challenging for a modular assembly, in which a very low-frequency behaviour and a lack of dynamic testing procedure that can incorporate nonlinearities are two major obstacles. In this regard, the present paper first introduces a demonstrator of Large Structure Assembly (LSA demonstrator), which includes a soft-bungee suspension system, a 6 m long modular assembly, a vibration control system, and a noncontact measurement system. Secondly, a new quantification procedure for the modular assembly, which utilises the resonance decay method, was proposed in this paper. Detailed test steps were illustrated through the demonstrator, in which the backbone curves were treated as key measurement targets in quantifying its nonlinear dynamics. The uncertainties in nonlinear dynamics were also evaluated by assembling and disassembling the structure multiple times. Results have shown that the proposed procedure can efficiently and accurately quantify the dynamics of a highly flexible, large-scale modular assembly.

Backbone model for reinforced concrete block shear wall components and systems using controlled multigene genetic programming

Reinforced concrete block shear walls (RCBSWs)have been used as an effective seismic force resisting system in low- and medium-rise buildings for many decades. However, attributed to their complex nonlinear behavior and the composite nature of their constituent materials, accurate prediction of their seismic performance, relying solely on mechanics, has been challenging. This study adopts multi-gene genetic programming (MGGP)— a class of bio-inspired artificial intelligence, to uncover the complexity of RCBSW behaviors and develop simplified procedures for predicting the full backbone curve of flexure-dominated RCBSWs under cyclic loading. A piecewise linear backbone curve was developed using five secant stiffness expressions associated with: cracking; yielding; 80% ultimate; ultimate; and 20% strength degradation (i.e., post-peak stage) derived through mechanics-controlled MGGP. Based on the experimental results of large-scale cyclically loaded fully-grouted RCBSWs, compiled from previously reported studies, a variable selection procedure was performed to identify the most influential variable subset governing wall behaviors. Subsequently, the MGGP stiffness expressions were trained and tested, and their accuracy was compared to that of existing models employing various statistical measures. In addition, the predictability of the developed backbone model was assessed at the system-level against experimental results of two two-story buildings available in the literature. This study demonstrates the power of the MGGP approach in addressing the complexity of the cyclic behavior of RCBSWs at both component- and system-level—offering an efficient prediction tool that can be adopted by relevant seismic design standards pertaining to RCBSW buildings.

Utjecaj duljine kritičnog područja na nosivost armiranobetonskih greda

This study investigated seismic analysis of the behavior of reinforced concrete beams with various shear spans, stirrup shapes, and tightening zone lengths using the Seismo-struct 2020 program. The load–displacement relations of 24 reinforced concrete beams with different conditions were modeled analytically. The analytical model used in the study was validated by comparing it with the experimental data from the literature. It was found that increasing the length of the tightening zone, particularly in beams with a high shear span, is essential for earthquake safety. In addition, the backbone curve of each beam was extracted, and comparison graphs were analyzed. The results obtained demonstrated that the length of the tightening zone limits the effect of stirrup shape and diameter on beam behavior.

Cyclic axial stress-strain model for circular CFST columns under compression loading

This study presents an experimental investigation of twenty-seven numbers of concrete-filled steel tubular (CFST) columns under cyclic axial compression loading to investigate the influence of grade of concrete, length-to-depth (L/D) ratio, and age of concrete on the cyclic axial performance of CFST columns. The L/D ratios were varied in the range of 3–10, while diameter-to-thickness (D/t) ratios of the tube were selected as 13 and 26. The main parameters studied were the peak axial strengths, displacement ductility, post-peak stiffness, and post-peak strengths of CFST columns. The backbone curves of CFST columns obtained from the cyclic tests were found to be similar to the monotonic load-deformation behavior. Test results were also compared with various confinement models available in the literature. A modified cyclic confinement model has been proposed in this study to accurately predict the cyclic behavior of CFST columns. The predicted stress-strain response using the developed confinement model matched well with the test results.

Experimental Study on the Seismic Performance of Shear Walls with Different Coal Gangue Replacement Rates

To replace conventional concrete with coal gangue concrete in the construction industry, lateral cyclic loading tests were applied to three shear walls with different coal gangue replacement rates in this study, in which the replacement rate of coal gangue was 0%, 50%, and 100%. The load-displacement hysteretic curves and backbone curves of the shear walls obtained from tests were analyzed to compare the failure process and seismic performance of each shear wall. The results indicate that the stress performance and failure morphology of coal gangue concrete shear walls and conventional concrete shear walls are extremely similar, and the characteristics of the hysteretic and backbone curves are approximately the same. With the increase in the coal gangue replacement rate, the bearing capacity and ductility of the three shear walls gradually decrease, the strength degradation gradually becomes significant, and the energy dissipation capacity becomes worse, but the difference is not obvious, and all of them can meet the requirements of seismic performance. In addition, with the increase in the coal gangue replacement rate, the stiffness degradation gradually slows, so it is feasible to construct a shear wall using coal gangue concrete instead of conventional concrete.

Self-Centering Prestressed Concrete Pier Considering the Effect of Vertical Earthquake Motions with External Aluminum Dissipators

A prototype self-centering prestressedconcrete (SCPC) bridge pier andconventional reinforced concrete (RC) pier with similar backbone curves are designed and modeled. The objective of this study is to investigate the seismic performance of the SCPC bridge pier considering the effect of vertical earthquake ground motions. Under a severe vertical earthquake motion, the RC pier damage is much greater than the SCPC pier. The SCPC bridge pier shows a great capability in reducing residual drift at the top of the structure, therefore, the probability of bridge survival is increased. In this study, the external aluminum bars were used to reduce the seismic energy impacts on the bridge pier structure. In addition, the averages of the maximum and residual drifts of bridge piers under a set of 20 earthquake ground motion records impact with different vertical-to-horizontal peak acceleration ratios on the bridge seismic response are presented. The results are compared with the case of horizontal-only excitations, to clarify the effect of the vertical earthquake on the SCPC and RC bridge pier. Hence, the designers can find good solutions for structures in earthquakes resistance.

Nonlinear modeling for composite plate shear walls-concrete filled structures

This paper proposes modeling recommendations, including constitutive model parameters for concrete and steel for nonlinear and macroscopic simulation of Composite Plate Shear Walls-Concrete Filled (C-PSW/CF) structures. Predictive equations are developed to determine the constitutive models of confined and unconfined concrete and steel faceplates, considering the buckling and fracture effects. A comprehensive database consisting of the test results of 37 C-PSW/CF specimens with and without endplates and boundary elements is collected and used to assess the required constitutive model parameters. Based on the validated macro model, predictive equations for backbone curves of unconfined and confined concrete and steel faceplates are proposed, all having high accuracy (R2 values larger than 0.9). The proposed modeling recommendations and constitutive models are further validated using a wide range of finite element analyses performed on prototype composite shear walls under monotonic loading. The proposed equations and modeling recommendations efficiently simulate the seismic behavior of C-PSW/CF structures with aspect ratios larger than or equal to 1.

Deep HystereticNet to predict hysteretic performance of RC columns against cyclic loading

This article attempts to reproduce hysteretic performance of HRB600 bar reinforced concrete columns under cyclic loading by adopting multivariate deep learning methods. Bidirectional long short-term memory (BiLSTM) and bidirectional gated recurrent units network (BiGRU) have been modified for temporal sequence related strength prediction with the consideration of multivariate parameters. To this end, a database with a 79,244 (row) × 12 (column) matrix is prepared for training, validation, and testing purposes. The inputs to the model include time intervals, loading displacement, concrete strength, axial compression ratio, longitudinal reinforcement tensile grade, longitudinal reinforcement diameter, longitudinal reinforcement ratio, stirrup tensile strength, stirrup diameter, spacing, and stirrup reinforcement ratio while the target is the cyclic load. By comparing the energy dissipation, accumulated energy dissipation, stiffness degradation and backbone curve of the hysteretic performance, the capacity of the adopted methods is compared with that of the finite element analysis. The comparison shows that the adopted method can recovery the hysteretic curves very well and the GRU method exhibits the most accurate prediction performance because of the highest R2 and the lowest MAE, MSE, and RMSE. The originality and innovation are mainly reflected in the use of time series data-driven deep learning model to replace the existing empirical formula and finite element method to reconstruct the nonlinear hysteretic behavior, so as to predict the seismic performance of the structure.

Numerical investigation on seismic resilient steel beam-to-column connections with replaceable buckling-restrained fuses

The detailed finite element model (FEM) of a new replaceable fuse steel beam-to-column joint proposed previously, where bolted steel plates located between the beam and C-shaped shear tab flanges in the beam splicing region behave as structural fuses to dissipate energy with their buckling controlled by the surrounding flanges, was developed in this paper and subjected to cyclic loading at the beam end. The simulated results matched the test results well in terms of the hysteresis curves (and the corresponding backbone curves) and failure modes. Based on the validated FEM, subsequent parametric studies were carried out to investigate the influences of the fuse plate type and slenderness ratio, and the slip coefficient between the steel plates, on seismic performance of this joint. The parametric analysis indicated that, the joint could accommodate an ultimate story drift angle of 6% without damaging the beam or shear tab flanges by using flat-type (FP) fuse plates, which are recommended over slotted-type (SP) and double-slotted-type (DSP) fuse plates, as the latter both types developed earlier buckling than the former FP-type. Eccentric-type (EP) fuse plates are also suggested, with which the joint equipped sustained an ultimate story drift angle of at least 8%. In addition, the slenderness ratio of FP-type fuse plates is recommended to be controlled within 44 since the fuse plates with higher slenderness ratios introduced damage through buckling to the shear tab flanges at an earlier stage, while the slip coefficient presented no significant impact on the joint seismic performance.

Combined internal resonances at crossover of slacked micromachined resonators

The dynamics of micro-/nanoelectromechanical systems (M/NEMS) curved beams have been thoroughly investigated in the literature, commonly for curved arch beams actuated with electrodes facing their concave surface. Except for few works on slacked carbon nanotubes, the literature lacks a deep understanding of the dynamics of slacked curved resonators, where the electrode is placed in front of the convex beam surface. This paper investigates the dynamics of slacked curved resonators as experiencing combined internal resonances. The curved slacked resonator is excited using an antisymmetric partial electrode while the electrostatic voltage load is driven to elevated excitations, which breaks the symmetry of the system and affects natural frequencies and corresponding mode shapes. The axial load is tuned to monitor the ratios between the natural frequencies of different vibration modes, which induces simultaneous 1:1 and 2:1 internal resonances between the first and second mode with the third. We observe the interaction of hardening and softening bending of the fundamental backbone curves triggering various patterns of the response scenario and the appearance of coexisting regions of irregular dynamics.

Application of a coupled hydro‐mechanical interface model in simulating uplifting problems

AbstractThis paper presents the detailed formulation of a coupled hydro‐mechanical structure‐soil interface and demonstrates its application in simulating uplifting problems. This interface features real‐time prediction of the pore pressure generation and structure‐soil separation, and thus rate dependency and ‘breakaway’ can be modeled without user intervention. Constitutive relations of this interface were derived by considering the coupling between soil skeleton and fluid along the interface. A complete finite element formulation and numerical implementation of the interface is provided based on an eight‐node element. The performance of this interface is demonstrated by simulating lifting a surface footing at varying rates (spanning across undrained, partially drained and drained conditions), compared with existing theoretical solutions, numerical results and experimental data. The good agreement achieved indicates that this interface is capable of modelling uplift at varying rates, which is an extremely challenging topic in offshore engineering. Sensitivity studies were conducted to investigate the parameters affecting uplifting behaviour. A unified backbone curve was established correspondingly, which is shown to be different from existing studies in compression, due to the difference in the mechanism between the two cases.

Exact augmented perpetual manifolds as a tool for the determination of similar rigid body modes: Corollary for determining backbone curves of autonomous systems

Perpetual points in mathematics have been defined recently, and their significance in describing the dynamics of systems is ongoing research. In mechanical engineering, the perpetual points of linear unforced mechanical systems are associated with rigid body motions, and they form the perpetual manifolds. The systems that admit perpetual manifolds of rigid body motions are called perpetual mechanical systems. The concept of perpetual manifolds has been extended to the exact augmented perpetual manifolds, whereas all the accelerations are equal but not necessarily zero. A theorem defines the conditions of external forces that lead to rigid body motion of externally forced mechanical systems. Herein, based on this theorem, a corollary is proven that, through a very simple framework, defines specifications to mechanical systems resulting in the existence of rigid body similar modes and can serve as a tool in the nonlinear normal modes (NNMs) theory for the determination of their backbone curves of dissipative mechanical systems. Further, on these, backbone curves can be used for passive vibration control design of N-degree-of-freedom (dof) mechanical systems. The verification of the theory has been done with the design of a 5-dof nonlinear perpetual mechanical system with three types of boundaries, linear, nonlinear smooth, and nonsmooth boundaries. The numerical simulations results are in excellent agreement with the theory. Although 5-dof mechanical systems have been examined, the corollary is valid for N-degree-of freedom systems. The corollary provides a simplified framework for the design of mechanical systems to admit rigid body modes for systems with a general configuration that is not obvious using traditional techniques. This work is rather significant in mechanical engineering because it provides a new tool firstly for determining similar NNMs of dissipative mechanical systems and secondly for the design of passive vibration control devices.

Uniaxial hysteretic spring models for static and dynamic analyses of self-centering rocking shallow foundations

Self-centering rocking shallow foundations have been proved to be a potential alternative to conventional foundations that provide superstructures with “fixed-base” constraints. Through appropriate design of foundations to undergo controlled rocking motions, they serve as “seismic fuses” (associated with a rocking isolation mechanism) which enable their supporting structures to remain even undamaged during intense shaking events without compromising the overall resilience of the soil–structure interaction (SSI) system. Recent progress in rocking shallow foundations has heightened the need for simple and efficient models in seismic design and evaluation of structures with rocking isolation. To this end, the paper proposes uniaxial hysteretic rocking spring models to enrich modeling of rocking-dominated foundation response in engineering practice. Two types of models are developed, including a bilinear hysteretic model (BHM) and a smooth hysteretic model (SHM). These models are defined by dimensionless backbone curves and recentering ratio based hysteretic rules obtained by performing monotonic and cyclic push-over analyses using a nonlinear hysteretic Winkler foundation (NHWF). The current implementation is based on square or strip footings on saturated clay foundations. Using the developed models, seismic performance of practical SSI systems consisting of rigid shear walls or deformable bridge piers on rocking shallow foundations are examined.

Experimental investigation and empirical model on damping properties of rock under multistage cyclic loading

The dynamic shear modulus and damping ratio of rock are the main indicators that reflect the dynamic characteristics of rock masses under seismic and dynamic action. The multistage cyclic triaxial compression test is performed on granite and red sandstone under different confining pressures with two stress paths. To reasonably describe the dynamic characteristics of rock, a modified Hardin hyperbolic model is proposed to describe the backbone curve. The relationship between the dynamic shear modulus, damping ratio and cyclic shear strain is analysed under different confining pressure conditions with variable and same amplitude. Furthermore, an empirical formula is established for estimating the dynamic shear modulus and damping ratio. The obtained results show that with increasing shear strain, the damping ratio of red sandstone and granite decreases and then increases under the condition of variable amplitude. When the shear strain is constant, the damping ratio of red sandstone and granite is less affected by the confining pressure. Under the same amplitude conditions, as the shear strain increases, the dynamic shear modulus under different confining pressures shows an increasing trend. The two established empirical formulas can be used to predict the dynamic shear modulus and damping ratio under multistage cyclic loading. This has important reference significance for studying the dynamic characteristics of rock.

Dynamic Characteristics of Rubber Reinforced Expansive Soil (ESR) at Positive and Negative Ambient Temperatures.

Using tire waste rubber reinforced expansive soil (ESR) can modify its poor engineering characteristics. The damping properties of ESR at different temperatures may vary dramatically. Two kinds of rubber Ra (large particle size) and Rb (small particle size) are mixed with expansive soil according to gradient ratio. The backbone curves, dynamic shear modulus, and damping ratio of expansive soil in varying temperature fields of 20 °C, −5 °C, and −15 °C are investigated. The Hardin-Drnevich model can well fit the backbone curves of ESR specimens in various temperature fields. Dynamic triaxial results show that 5–10% Ra rubber can withstand higher shear stress in all temperature fields; Rb rubber can increase the dynamic shear modulus of expansive soil and reach the peak value with 10% rubber content. The damping ratio can be significantly improved by using 10% Ra rubber at room temperature, while the ESR damping ratio in a temperature field of −5 °C does not change significantly with increasing shear strain or even decreases; Ra increases the damping ratio of expansive soils in the temperature field of 15 °C while small particle size Rb decreases the damping ratio of expansive soils. The experimental results validate the effectiveness of ESR in the frozen soil area. In an engineering sense, local temperature needs to be considered to use an appropriate ESR, which can provide effective seismic isolation and damping.

Shear strength model of large-scale reinforced concrete rectangular columns with light transverse reinforcement

Columns with low transverse reinforcement due to poor detail, small rebar size, and wide spacing were commonly used in old buildings. These types of columns often fail in shear with the brittle mode that could lead to a sudden collapse of the entire building. Thus, the accurate determination of the shear strength is significant for estimating the response of reinforced concrete (RC) columns under earthquake events. Numerous experiments were conducted to investigate the shear strength, however, there is a lack of tests on large-scale lightly RC columns under high axial compression load. Therefore, this study tested four large-scale RC columns with dimensions of 800 mm × 800 mm × 3200 mm. The paper highlights the main factors that influence the shear strength of the RC columns and based on the test results, an equation with high predictive power is established. In addition to the experimental results, the analytical/empirical models’ predictions from design codes or other researchers were evaluated. Furthermore, the interaction diagram between bending moment (M)—shear force (V)—axial compression force (N) was conducted to estimate the strength ability of RC columns. It was found that the simple processes to build the backbone curves and the hysteresis loops that were recommended had good agreement with the tested results and experimental database.