Nonlinear Single-degree-of-freedom Research Articles

Progressive collapse analysis of steel frames via a nonlinear SDOF model

AbstractProgressive collapse is a dynamic loading incident triggered by the sudden loss of critical elements. Design codes provide simplified rules to consider the dynamic effects via the use of Dynamic Increased Factors in order to avoid complex and time‐consuming nonlinear time history analyses. In this paper, a nonlinear single degree of freedom (SDOF) system is presented that captures the dynamic response of steel frames subjected to a column removal scenario. The model takes into account inertia, damping effects and assumes an elastoplastic nonlinear resistance for the system. A closed‐form analytical solution for the dynamic equation of motion is derived and then is applied to a frame to simulate the response under a column removal scenario. A nonlinear finite element model developed in OpenSees was used to verify the accuracy of the analytical model. The results showed that the nonlinear response of the frame for the selected hazard event can be well captured with the proposed model.

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.

Illustration of the homotopy perturbation method to the modified nonlinear single degree of freedom system

Nonlinear Single degree of freedom (SDOF) systems are crucial for understanding the behavior of various real-world systems, designing and analyzing mechanical and dynamical systems. In this article we have modified the SDOF model by introducing the nonlinearity as well as damping with/without external force and applied the homotopy perturbation method (HPM) to explain its various phenomena. We have compared the analytical solutions obtained via the HPM to the provided numerical results by the fourth-order Runge-Kutta (RK4) method for verifying the precision and validity of the solutions. The presentation and explanation of the dynamic results can add a new dimension to the research field by influencing the importance of nonlinear SDOF systems.

Numerical study on dynamic response of hollow and cavity type clay brick masonry infill panels subjected to blast loading

Laboratory and prototype study of masonry panel subjected to earthquake, wind and impact is relatively easy however, the experimental study of masonry panel under the action of blast phenomenon is quite a risky and restrictive work area to perform. But computational advancement has made it within our capabilities and is deemed to yield acceptable results. Therefore, this study focuses on investigating the dynamic blast response of novel hollow and cavity type clay brick masonry (HBM and CBM) infill panels enclosed in reinforced concrete (RC) frames using explicit finite element (FE) method. The properties of HBM and CBM required for defining the numerical FE model were derived from the experimental tests. Moreover, due to lack of experimental study on HBM and CBM subjected to the action of blast, the FE peak displacement observed at mid-height of masonry panels were agreeably validated with the peak displacement obtained from an equivalent non-linear single degree of freedom (SDOF) model wherein, a link between positive phase duration of blast and natural time-period of structure is established. Further, the effects of nature of explosion, standoff distance, weight of explosive and height of detonation from ground level were investigated for HBM and CBM walls. Thereafter, HBM panel was strengthened using a low-cost technique, making use of steel bars for enhancing the out-of-plane stiffness of the wall against blast loading. Reinforced hollow brick masonry (RHBM) panel was found to be an excellent alternative for blast load mitigation.

New equivalent damping for the ATC nonlinear static procedure

The Nonlinear Static Procedure, also known in the literature as Pushover Analysis procedure or Capacity Spectrum Method (CSM), is mainly developed in the report of Applied Technology Council entitled “Seismic evaluation and retrofit of concrete buildings” ATC (1996). This is a widely practical engineering method to calculate the seismically induced ductility demand of a System, Structure or Component (SSC) without performing nonlinear transient analyses. Many authors have improved this method by introducing different equivalent linear concepts. However, discrepancy between predictive results by CSM and references obtained by transient nonlinear analyses remains quite important. In this paper, we present a new equivalent linear concept, in which the conventional equivalent frequency corresponding to the secant stiffness is adopted, and new equivalent damping value is calibrated so that the maximum displacement of equivalent linear oscillator is the same as the one of nonlinear oscillator. The new proposed damping values are derived from systematic analysis of the response of a series of nonlinear single degree of freedom (SDOF) oscillators to seismic type input motions. Oscillators have a bilinear elastic–plastic constitutive relationship with a hardening slope equal to 10% or 20% of the initial stiffness. Their natural frequency (f0) is such that f0/fc varies between 0.1 and 2.0, where fc is the central frequency of the input signal and their damping in elastic regime remained at 5%. Input motion is a set of one thousand synthetic filtered white noise signals. The novelty of the proposed approach is that new equivalent damping values exhibit strong frequency dependence on the f0/fc ratio, which has not been discussed so far. In order to validate the proposed equivalent linear concept, predictive results for nonlinear SDOF oscillators undergoing one hundred natural signals are performed by CSM with the new equivalent damping values and compared to reference results obtained from nonlinear transient analyses. It is concluded that the CSM using the new proposed equivalent damping values can determine the seismically induced ductility demand of SDOF oscillators with a better accuracy than methods used at the moment as a routine.

Active Vibration Absorber for Superharmonic Resonance Condition

The present paper, analyses vibration suppression of a nonlinear single degree of freedom (SDOF) spring-mass-damper system under hard harmonic excitation by attaching a piezoelectric based active dynamic vibration absorber (ADVA). The host structure consists of spring with cubic nonlinear stiffness and negligible damping. The acceleration feedback is used to provide the actuating force through ADVA on the vibrating host structure. The hard harmonic excitation on the host structure induces superharmonic resonance condition i.e. when excitation frequency nearly matches one-third of the natural frequency of the host structure. The optimal tuning ratio and damping ratio of the vibration absorber are considered to depend upon the actuating force so that one can actively tune the operating frequency of the ADVA. The method of multiple scales (MMS) is used to obtain the reduced equations to study the frequency response of the system. The analysis is carried out by studying the effect of active force and cubic nonlinear stiffness on the responses of the host structure. The results obtained by MMS are compared numerically showing good agreement.

Recent advances, future application and challenges in nonlinear flutter theory of long span bridges

Polynomial-type mathematical models were presented to describe the nonlinear self-excited forces of nonlinear single degree of freedom (SDOF) torsional flutter of bluff bridge decks and nonlinear coupled flutter of streamline-like flat closed box bridge decks. A full bridge analysis method for nonlinear SDOF torsional flutter of long span bridges with bluff decks was then proposed based on the aeroelastic strip theory, and verified via wind tunnel test of full bridge aeroelastic model. A full bridge analysis method of nonlinear coupled flutter of long span bridges was also established based on the aeroelastic strip theory and an approximate linear complex-modal decomposition (ALCMD), and was applied then to the nonlinear coupled flutter analysis of a 688m main-span cable-stayed bridge. The analysis results show that behaviors of self-limiting and single-frequency oscillation of the nonlinear coupled flutter can be exhibited by using the proposed coupled flutter analysis approach. A new concept of performance-based graded evaluation and fortification criteria against flutter of long span bridges was then put forward for further discussions, which will possibly be an important and prosperous field of future applications of the nonlinear flutter theory. To this end, some challenging issues was discussed for further developing nonlinear flutter theory.

Dynamical systems approach for the evaluation of seismic structural collapse and its integration into PBEE framework

The development of collapse fragility curves is an essential requirement for the assessment of the collapse risk of structures. These fragility curves depend on the structural collapse capacity that is evaluated in terms of either the intensity measure (IM) or the damage measure (DM); such as the engineering demand parameter (EDP). In turn, collapse capacity estimates are sensitive to the method employed for their assessment. Conventionally, the IM and DM rules are employed in conjunction with incremental dynamic analyses (IDA) to quantify the collapse capacity of a structure. However, this approach has been criticised for being subjective in nature, since it depends on the structural response approaching predetermined threshold values. Therefore, it does not relate to the actual dynamic instability. Although the selection of these thresholds stems from the results of experimental investigations and equivalent numerical models and serve as an indirect check for dynamic instability, the present study seeks to provide a mathematical basis for defining collapse criteria. A dynamical system approach is used to formulate a mathematical criterion for defining P-Delta instability induced seismic collapse in single-degree-of-freedom (SDOF) structures. The non-linear SDOF structure is considered as a non-autonomous, non-smooth system, and is studied as an ensemble of different sub-systems. Collapse is defined as the point when the dominant system eigenmode of the structure changes from stable to unstable, and remains unstable as the structure collapses. This approach can be applied to first mode governed multi-degree-of-freedom (MDOF) structures when studied as an equivalent SDOF structure. It is found that the dynamical systems approach results in higher deformations at collapse when compared to the conventional IM/DM rule based approach, suggesting the conservatism involved in the latter. However, it results in lower deformations at collapse when compared to the energy criterion, which relies on the occurrence of large deformations to predict collapse. Furthermore, the derived fragility curves show that the proposed approach yields lower probabilities of collapse when compared to the conventional method. Therefore, the proposed method can be used an alternative method for the performance design of structures.

Period elongation of deteriorating structures under mainshock-aftershock sequences

The elongation of the fundamental period of structure is a typical indicator of structural inelasticity during seismic events and of consequent damage accumulation. This study investigates the lengthening of the fundamental periods of reinforced concrete (RC) structures subjected to mainshock-aftershock sequences. RC structures are schematized herein as nonlinear SDOF (Single Degree Of Freedom) systems with deteriorating properties. Inelastic response spectra are developed considering both ground motion characteristics (e.g. site condition, epicentral distance, PGAs ratio of aftershock to mainshock, and duration) and structural properties, such as ductility, softening, pinching, accumulated damage and unloading stiffness. Furthermore, analytical equations estimating the elongated fundamental period are proposed as a function of the elastic (undamaged) vibration period and the significant structural parameters. The proposed equations can be reliably used for the design of structures against mainshock-aftershock sequences in future generations of seismic design codes.

Deep Convolutional Neural Network for Structural Dynamic Response Estimation and System Identification

This study presents a deep convolutional neural network (CNN)-based approach to estimate the dynamic response of a linear single-degree-of-freedom (SDOF) system, a nonlinear SDOF system, and a full-scale 3-story multidegree of freedom (MDOF) steel frame. In the MDOF system, roof acceleration is estimated through the input ground motion. Various cases of noise-contaminated signals are considered in this study, and the conventional multilayer perceptron (MLP) algorithm serves as a reference for the proposed CNN approach. According to the results from numerical simulations and experimental data, the proposed CNN approach is able to predict the structural responses accurately, and it is more robust against noisy data compared with the MLP algorithm. Moreover, the physical interpretation of CNN model is discussed in the context of structural dynamics. It is demonstrated that in some special cases, the convolution kernel has the capability of approximating the numerical integration operator, and the convolution layers attempt to extract the dominant frequency signature observed in the ideal target signal while eliminating irrelevant information during the training process.

Capacity evaluation of voided driven piles using embedded data collectors

This paper presents an application of a method that will be implemented in the embedded data collector (EDC) system in the near future, to estimate the capacity of driven piles with a combined solid and voided cross section. Data from accelerometers and strain gauges located in the solid sections at both the top and the bottom of a pile are used to independently estimate the pile’s skin friction and tip resistance. Wave propagation along the pile is modeled as a one-dimensional wave equation, with a nonuniform cross section and with nonlinear static skin friction and viscous-damping soil resistances acting along multiple segments of the pile. The static skin friction is extracted by least-squares fitting of computed particle velocities with measured data at both the top and the bottom of the pile. The pile tip is modeled as a nonlinear single degree of freedom with viscous damping. Static tip resistance (nonlinear stiffness–displacement relationship) is extracted by least-squares fitting of the predicted total force and energy with the measured tip data. The new EDC method was applied to four combined solid–voided cross section driven piles with capacities varying from 2800 to 6700 kN. The results of the data evaluated with the new EDC method are consistent with those from the static load tests to within 15%.

Real-time hybrid simulation of structures equipped with viscoelastic-plastic dampers using a user-programmable computational platform

A user-programmable computational/control platform was developed at the University of Toronto that offers real-time hybrid simulation (RTHS) capabilities. The platform was verified previously using several linear physical substructures. The study presented in this paper is focused on further validating the RTHS platform using a nonlinear viscoelastic-plastic damper that has displacement, frequency and temperature-dependent properties. The validation study includes damper component characterization tests, as well as RTHS of a series of single-degree-of-freedom (SDOF) systems equipped with viscoelastic-plastic dampers that represent different structural designs. From the component characterization tests, it was found that for a wide range of excitation frequencies and friction slip loads, the tracking errors are comparable to the errors in RTHS of linear spring systems. The hybrid SDOF results are compared to an independently validated thermalmechanical viscoelastic model to further validate the ability for the platform to test nonlinear systems. After the validation, as an application study, nonlinear SDOF hybrid tests were used to develop performance spectra to predict the response of structures equipped with damping systems that are more challenging to model analytically. The use of the experimental performance spectra is illustrated by comparing the predicted response to the hybrid test response of 2DOF systems equipped with viscoelastic-plastic dampers.

Development of seismic collapse capacity spectra for structures with deteriorating properties

Evaluation on the sidesway seismic collapse capacity of the widely used low- and medium-height structures is meaningful. These structures with such type of collapse are recognized that behave as inelastic deteriorating single-degree-of-freedom (SDOF) systems. To incorporate the deteriorating effects, the hysteretic loop of the nonlinear SDOF structural model is represented by a tri-linear force-displacement relationship. The concept of collapse capacity spectra are adopted, where the incremental dynamic analysis is performed to check the collapse point and a normalized ground motion intensity measure corresponding to the collapse point is used to define the collapse capacity. With a large amount of earthquake ground motions, a systematic parameter study, i.e., the influences of various ground motion parameters (site condition, magnitude, distance to rupture, and near-fault effect) as well as various structural parameters (damping, ductility, degrading stiffness, pinching behavior, accumulated damage, unloading stiffness, and P-delta effect) on the structural collapse capacity has been performed. The analytical formulas for the collapse capacity spectra considering above influences have been presented so as to quickly predict the structural collapse capacities.

Application of Kalman Filtering Methods to Online Real-Time Structural Identification: A Comparison Study

System identification refers to the process of building or improving mathematical models of dynamical systems from the observed experimental input–output data. In the area of civil engineering, the estimation of the integrity of a structure under dynamic loadings and during service condition has become a challenge for the engineering community. Therefore, there has been a great deal of attention paid to online and real-time structural identification, especially when input–output measurement data are contaminated by high-level noise. Among real-time identification methods, one of the most successful and widely used algorithms for estimation of system states and parameters is the Kalman filter and its various nonlinear extensions such as extended Kalman filter (EKF), Iterated EKF (IEKF), the recently developed unscented Kalman filter (UKF) and Iterated UKF (IUKF). In this paper, an investigation has been carried out on the aforementioned techniques for their effectiveness and efficiencies through a highly nonlinear single degree of freedom (SDOF) structure as well as a two-storey linear structure. Although IEKF is an improved version of EKF, results show that IUKF generally produces better results in terms of structural parameters and state estimation than UKF and IEKF. Also IUKF is more robust to noise levels compared to the other approaches.

A nonlinear SDOF model for the simplified evaluation of the displacement demand of low-rise URM buildings

The determination of displacement demands for masonry buildings subjected to seismic action is a key issue in the performance-based assessment and design of such structures. A technique for the definition of single-degree-of-freedom (SDOF) nonlinear systems that approximates the global behaviour of multi-degree-of-freedom (MDOF) 3D structural models has been developed in order to provide useful information on the dependency of displacement demand on different seismic intensity measures. The definition of SDOF system properties is based on the dynamic equivalence of the elastic properties (vibration period and viscous damping) and on the comparability with nonlinear hysteretic behaviour obtained by cyclic pushover analysis on MDOF models. The MDOF systems are based on a nonlinear macroelement model that is able to reproduce the in-plane shear and flexural cyclic behaviour of pier and spandrel elements. For the complete MDOF models an equivalent frame modelling technique was used. The equivalent SDOF system was modelled using a suitable nonlinear spring comprised of two macroelements in parallel. This allows for a simple calibration of the hysteretic response of the SDOF by suitably proportioning the contributions of flexure-dominated and shear-dominated responses. The comparison of results in terms of maximum displacements obtained for the SDOF and MDOF systems demonstrates the feasibility and reliability of the proposed approach. The comparisons between MDOF and equivalent SDOF systems, carried out for several building prototypes, were based on the results of time-history analyses performed with a large database of natural records covering a wide range of magnitude, distance and local soil conditions. The use of unscaled natural accelerograms allowed the displacement demand to be expressed as a function of different ground motion parameters allowing for the study of their relative influence on the displacement demand for masonry structures.

Nonlinear SDOF Model for Dynamic Response of Structures under Progressive Collapse

AbstractThe transient dynamic response of the substructures above a removed column is the most direct and realistic to demonstrate the structural performance against progressive collapse. To efficiently obtain the transient dynamic response, a nonlinear single degree-of-freedom (SDOF) model is proposed with a tri-linear resistance function that is capable of describing various forms of structural resistance, including catenary action and softening resistance and a loading function consisting of a linear ascending part and an ensuing constant force that considers the effect of column removal time. Moreover, the model accounts for nonzero initial conditions, which are very likely to occur under blast loading. The closed-form analytical solutions of the model are derived with Laplace transform techniques and verified with dynamic experimental results of steel and reinforced concrete assemblies subjected to column removal scenarios. The verification indicates the generality and the accuracy of the SDOF model ...

Experimental and numerical investigations of glass curtain walls subjected to low- level blast loads

A series of three full-scale, nearly-conventional, curtain wall specimens were blast tested in the open arena of the Infrastructure Security and Emergency Responder Research and Training (ISERRT) Facility in Gastonia, NC. The specimens were subjected to low-level blast loads produced from the detonation of high explosives. Low-level blast loads, similar to those produced during the tests, are typical of small charge weights (i.e. satchel charges) at short-to-moderate standoffs. A simple finite element (FE) model that effectively represents the nonlinear dynamic response of glass curtain walls subjected to blast loads was developed, and simulation results were compared with the test results. It was shown that, with the judicious choice of modeling parameters, the FE model effectively represents the response of glass curtain walls subjected to blast loads while being computationally economical. The calibrated FE model was used to evaluate the efficacy of a nonlinear single-degree-of-freedom (NSDOF) design expression for analytically approximating the blast resistance of curtain wall systems. The design expression is based on a procedure in which a nonlinear resistance function of the system is input to an energy expression that models the maximum nonlinear dynamic deflection due to an ‘impulsive’ loading. It was shown that the maximum impulse predicted by the design expression, when the expression was used in conjunction with a satisfactory resistance function, compared reasonably well with the FE simulation results. The expression could be used as a starting point for design or to supplement more advanced models of curtain walls subjected to blast loads.

Dynamic behaviour of concrete filled FRP tubes subjected to impact loading

Numerous experimental investigations have demonstrated the advantages of Concrete Filled FRP Tubes (CFFTs) over reinforced concrete members under various loading conditions. None of the studies, however, investigated whether these advantages of CFFTs extended to resisting dynamic impact loads. This presented a fertile field for investigation as it was well known that the addition of the tube confined and protected the concrete core and increased its load carrying capacity, and by extension, its energy absorption. This investigation aimed at understanding the dynamic behaviour of CFFTs under impact loading and to develop a procedure for their analysis and design. These aims were achieved by testing six specimens, three of which were CFFTs and three were reinforced concrete. The specimens were tested in pairs to facilitate comparisons. The first pair were tested monotonically to act as a benchmark and tie into previous research. The remaining two pairs were tested under impact loading. The parameters investigated were the presence of the GFRP tube, the internal steel reinforcement ratio, and the input kinetic energy. The monotonic tests showed that the addition of the tube increased the flexural capacity and maximum displacement by 112%, which translated into a 487.5% increase in energy absorption, when compared to the reinforced concrete counterpart. The addition of the tube increased the impact specimens’ energy absorbing capacity by 467% when the steel reinforcement ratio was 1.2% and 1223% when the steel reinforcement ratio was 2.4%, compared to the reinforced concrete specimens. The GFRP tube also protected the concrete core and prevented its spalling and crushing during the impact tests. These encouraging experimental results led to the development of a nonlinear single degree of freedom (SDOF) model capable of capturing the behaviour of the system under multiple impacts to be used as a sophisticated analysis and design tool. Additionally, a simplified design method based on the conservation of energy was outlined.

Identification of cubic nonlinearity in disbonded aluminum honeycomb panels using single degree-of-freedom models

Prior work on a disbonded aluminum honeycomb panel showed evidence of a quadratic stiffness nonlinearity, as well as the presence of an unknown cubic nonlinearity. Approximations to higher order nonlinear single degree of freedom (SDOF) models were solved using the method of multiple scales. These approximations were then used to fit displacement data from a sinusoidal excitation test and determine the coefficients of the model as a function of damage size. Confirmation of the quadratic stiffness nonlinearity was achieved through examination of force restoration curves excited at one-half the primary resonance in conjunction with coefficient fitting of the test data to the model. The data were fit against the higher order models to determine whether the cubic nonlinearity could be stiffness or damping related. The coefficient fitting shows that the cubic nonlinearity is a stiffness nonlinearity. This confirmed what was seen in the force restoration curves when the system was excited at one-third the primary resonance. The ability to match the vibratory behavior of the damage to a SDOF model shows that the use of single frequency excitation at lower frequencies can isolate the nonlinear behavior of the damaged area and identify what damage mechanisms may be involved.

Blast wave clearing behavior for positive and negative phases

The purpose of the research was to improve prediction of response of buildings to blast waves by including the negative phase and considering clearing of both positive and negative phases. Commonly used structural design practices, which trace their origins to military design manuals, often ignore the negative phase as well as positive phase clearing. For high explosive threats, this approach is conservative in most circumstances. However, negative phase clearing had not previously been studied for blast waves, and the implications for structural response had not been evaluated. This paper presents results of modeling negative phase blast clearing behavior for a typical blast wave and discusses the differences from positive phase clearing. The implications of including positive and negative phase clearing in building blast damage analysis are also investigated through single-degree-of-freedom (SDOF) analyses.Blast waves from explosion sources like a vapor cloud explosion (VCE), pressure vessel burst or high explosive exhibit both positive and negative phases, and the relative magnitude of the positive and negative phases varies among explosion sources and the specific circumstances of each source. A fully reflected blast wave is produced if an incident blast wave were to strike an infinitely tall and wide wall in a normal orientation. Both the positive and negative phases of the blast wave are enhanced by the reflection process. However, when an incident blast wave strikes a wall of finite size in a normal orientation, rarefaction waves are created at the edges of the wall, and the rarefactions sweep down from the roof and inward from sides. The rarefaction waves result in a clearing effect for both the positive and negative phases.Clearing relieves some of the applied blast load on the reflected wall for the positive phase. However, this is not always the case for the negative phase. As shown by the results presented in this paper, clearing may either relieve or enhance the applied negative phase blast load, depending on the duration of the blast wave and the wall dimensions.The impact of negative phase clearing on structural response for generic building components was also investigated. Nonlinear SDOF methods were used to characterize response in terms of peak positive and negative displacements. It was found that the influence of the negative phase is significant and the peak structural response can occur during negative (outward) displacement.