Single Degree Of Freedom System Research Articles

Direct loss‐based seismic design of reinforced concrete frame and wall structures

AbstractThis paper presents a procedure to design reinforced concrete (RC) buildings to achieve an acceptable target level of earthquake‐induced loss (e.g., deaths, dollars, downtime) under a site‐specific hazard profile. The procedure is called “direct” since the target loss level is specified at the first step of the process, and virtually no iteration is required. The procedure is based on a simplified loss assessment involving a surrogate model for the seismic demand (i.e., probability distribution of peak horizontal deformation given ground‐motion intensity) and simplified loss models for direct and indirect losses. For an arbitrarily‐selected target loss level and structural geometry, the procedure provides the force‐displacement curve of the corresponding equivalent single degree of freedom system. The principles of displacement‐based design are adopted to provide member detailings (beams, columns, walls) consistent with such force‐displacement curve. The procedure is applied to 16 realistic RC case studies with a lateral resisting system composed of frames in one direction and cantilever walls in the perpendicular one. They show different geometries, hazard profiles, and target values of direct economic expected annual loss. A benchmark loss estimation is obtained using cloud‐based non‐linear time‐history analyses of multi‐degree of freedom models. The procedure is conservative since the benchmark loss levels are always smaller than the targets. Such discrepancy is within 10% for 12 out of 32 case studies, between 10% and 20% for 13, between 20% and 31% for the remaining six. Therefore, the proposed procedure is deemed dependable for preliminary design.

A polynomial extrapolation-based wavelet-Galerkin method for dynamic response reconstruction

In this work we present a single scale wavelet-Galerkin method (WGM) for obtaining the linear dynamic response of single degree of freedom (SDOF) and multi-degree of freedom (MDOF) systems, the equation of motion is discretized using a polynomial extrapolation-based time stepping scheme. The aim of the study is to investigate the accuracy of the dynamic response resulting from the application of the proposed method and to explore the Daubechies wavelet types that can be used for unconditional stability. In our work we solve only for the scaling function coefficients of the displacement response, the scaling function coefficients of the velocity and the acceleration responses are obtained from the solved scaling function coefficients of the displacement function and the pertaining connection coefficients. The classical central differences method (CCDM) is used to generate consistent values of the velocity and the acceleration response to replace the distorted ones at the edges of the time interval. Stability analysis shows that the method using D6 wavelets is unconditionally stable for the practical range of the product of the time step and the frequency of vibration. An application of a SDOF system with initial conditions shows that for D6 wavelets the calculated responses at an adequate approximation level correspond well with those obtained using Newmark’s average acceleration method (AAM). The method is also applied to a MDOF system without performing a modal analysis, the resulting dynamic response is found to be accurate enough to be compared to that of the mode superposition method used by the STAAD pro program.

PI-LSTM: Physics-informed long short-term memory network for structural response modeling

Deep learning models have achieved remarkable accuracy for structural response modeling. However, these models heavily depend on having a sufficient amount of training data, which can be challenging and time-consuming to collect. Moreover, data-driven models sometimes struggle to adhere to physics constraints. Therefore, in this study, a physics-informed long short-term memory (PI-LSTM) network was applied to structural response modeling by incorporating physics constraints into deep learning. The physics constraints were modified to accommodate the characteristics of both linear and nonlinear cases. The PI-LSTM network, inspired by and compared with existing physics-informed deep learning models (PhyCNN and PhyLSTM), was validated using the numerical simulation results of the single-degree-of-freedom (SDOF) system and the experimental results of the six-story building. Additionally, the PI-LSTM network underwent thorough investigation and validation across the four cases of the SDOF system and numerical simulation results of the six-story building with the comparison of the regular LSTM. The results indicate that the PI-LSTM network outperformed the regular LSTM models in terms of accuracy. Furthermore, the PI-LSTM network exhibited a more concentrated and higher accuracy range when analyzing the results of both the SDOF system and the six-story building. These findings demonstrate that the PI-LSTM network presents a reliable and efficient approach for structural response modeling.

Significance of post-shaking response of SDOF systems

The present study reveals for the first time that some real earthquake ground motions would induce the absolute maximum response (AMR) of structural systems posterior to the end of the excitation. Under these ground motions, illustrated using single-degree-of-freedom (SDOF) systems, the peak responses calculated within the during-shaking phase might be significantly lower than the actual AMR. Extending the dynamic analysis to the post-shaking phase by half of the structural period is recommended when conducting response history analyses or calculating the response spectra of these ground motions.

Evaluation of Pedestrian Comfort for a Footbridge with Hinged Piers

To study the standardized evaluation method of vibration comfort of pedestrian bridges, the effect of the transient response term is ignored based on the generalized single degree of freedom (SDOF) method with reference to the American Institution of Steel Construction inc (AISC 11) specification. The resonance conditions between the crowd and the bridge structure are also considered, and the pedestrian bridge structure is simplified to an SDOF system. Then, the peak acceleration of the structure under crowd excitation is directly calculated using the generalized crowd load, and the comfort evaluation is carried out according to the German EN03 code and the Chinese code. For a new pedestrian bridge with hinged piers, human-induced vibration tests were carried out to obtain the structural acceleration time response data under various crowd load excitation conditions, and the acceleration values calculated with the numerical simulation method, the frequency response function method, the generalized SDOF method, and the proposed method were compared with field measurements. The results show that for the normal excitation condition with a crowd density less than or equal to 0.2 person/m2, the acceleration maximum error for the proposed method is between 4.22% and 13.28%. The error is only 6.35% compared with the finite element simulation results. The method can derive peak acceleration by performing only modal tests, eliminating errors caused by different testers and errors due to deflection measurements. It saves considerable time cost and economic cost, and improves the speed of pedestrian bridge comfort evaluation. Therefore, the method proposed for the first time in this study can accurately assess the human-caused vibration comfort of pedestrian bridges with articulated piers. Although the hinged-pier and column steel pedestrian bridge meets the code requirement of a vertical first-order frequency greater than 3 Hz, it is recommended to add TMD vibration damping measures to the bridge in order to improve pedestrian comfort.

Development of adjustable variable stiffness restrainer for bridge subjected to seismic excitation.

This paper presents a numerical and experimental assessment of a developed adjustable variable stiffness restrainer (AVSR) utilized for short span bridges. This restrainer has the ability to demonstrate multi stiffness capacity in different stages of bridge's superstructure movement to mitigate the severe damage of bridge due to an earthquake. The multi-level stiffness behavior of developed AVSR is achieved by using multiple mechanical springs with different lengths and placed in parallel in proposed design. A small prototype of developed AVSR has been fabricated and tested under incremental and cyclic loading in order to assess the restrainer performance and the behavior has been validated using finite element analysis. Thereafter, the constitutive model of AVSR was derived for the proposed restrainer in order to implement it in numerical simulations. Furthermore, a parametric study has been conducted numerically to evaluate the effectiveness of different parameters on the restrainer capacity. Moreover, the efficiency of AVSR application in a single degree of freedom system has been assessed by performing seismic analysis on a frame equipped with AVSR subjected to different seismic excitations using Newmark's method. The experimental and finite element results proved the efficiency of developed variable stiffness device to exhibit adjustable action against imposed loads in three designed stages. Furthermore, the parametric study results revealed that increasing the section area of the spring wire leads to increase the restrainer capacity. In contrast, the restrainer resistance is declined by an increase in the mean spring diameter and number of coils for each spring of AVSR. The time history analysis results also indicated that the frame response in terms of displacement, velocity and acceleration is improved by implementing the AVSR in the considered system.

A ground motion model for orientation-independent inelastic spectral displacements from shallow crustal earthquakes

The peak inelastic displacement of a single-degree-of-freedom (SDOF) system is an efficient ground motion (GM) intensity measure (IM) for estimating seismic demands in structures and infrastructures. Most ground motion models (GMMs) that predict this IM are developed to estimate the arbitrary or geometric mean of the horizontal GM components obtained from the two as-recorded orientations. In this study, the median and maximum directional inelastic spectral displacements, denoted as Sdi,RotD50and Sdi,RotD100, respectively, were considered as IMs to develop GMMs. These orientation-independent measures of horizontal motion were calculated by rotating the GMs through all non-redundant rotation angles and have the advantage of removing the sensor orientation as a contributor to overall uncertainty. To do this, several SDOF systems were considered using a subset of the NGA-West2 database to fit GMMs for these IMs. A set of functional forms was developed using a mixed-effects regression approach. The predictor variables were the elastic period, T, the strength ratio, R, and a set of seismological GM causal parameters (i.e. magnitude, distance, soil condition, and so on). Comparison with existing GMMs that consider either inelastic spectral displacement or directionality is provided to highlight key differences and developments. In addition, the response directionality, defined as the ratio Sdi,RotD100/ Sdi,RotD50, was observed to be notably impacted by the strength ratio and elastic period, which has not been discussed in the literature to date. Overall, this GMM has the advantage of quantifying ground motion intensity via an efficient IM in Sdi, which can better highlight some fundamental issues surrounding the directionality of GMs and allow for more informative risk estimates.

1/3 SIMPSON’S RULE FOR ANALYSIS OF STRUCTURE DYNAMIC RESPONSE DUE TO EARTHQUAKE LOAD

In structural analysis, many calculations are encountered which are very complex, making it difficult to do with exact mathematical calculations. For easier analysis, numerical methods are needed to simplify the calculations. Complex building structures such as towers, multi-storey structures and other buildings, are idealized for simplification into a single degree of freedom system (SDOF), assuming that the dynamic response of structures due to earthquake loads is horizontal. The analysis of this model is correlated with numerical analysis, so it can be completed quickly. The numerical method used in this study is the 1/3 Simpson Integral Method, because this method is suitable for calculating dynamic structural responses such as structural displacement responses. The analysis procedure begins by entering the external forces on the structural system and calculating the resulting response value. The analysis can be repeated for a variety of different parameters, such as the mass of the structure, the dumping ratio and the stiffness of the structure. Structural response is calculated by sinusoidal dynamic load type for damped and undamped systems. The results of this study conclude that the relationship between the mass of the structure, the damping of the structure and the stiffness of the structure with the displacement of the structure has an inverse relationship, where with high mass, high damping and high stiffness, it can reduce the structure displacement.

Simulation of the 2022 Mw 6.6 Luding, China, earthquake by a stochastic finite-fault model with a nonstationary phase

We adopted stochastic finite-fault simulations considering the nonstationary frequency characteristics for the 2022 Mw 6.6 Luding, China, earthquake at 16 strong-motion stations using the stochastic finite-fault model based on a dynamic corner frequency (EXSIM) code. The average simulated response spectra of earthquake ground motions were obtained and then compared with response spectra of 5% damped single degree of freedom system under recorded ground motion excitation to verify whether the finite-fault model that considers the nonstationary frequency characteristics can reproduce the critical characteristics of ground motions and improve the simulation accuracy. The high-frequency spectral decay is calculated by the real recordings and the stress drop is determined by the trial-and-error method. In addition, the local site amplifications are estimated from the horizontal-vertical spectral ratio (H/V). An improved EXSIM that considers the slip-related corner frequency is also used in this study. The results indicate that the nonstationary phase can significantly improve the simulation accuracy of EXSIM in the short period and has little effect on the long-period simulation results. However, the effect of the nonstationary phase on the simulation results obtained from the improved EXSIM is not significant. Regarding the peak ground acceleration (PGA), the nonstationary phase can produce waveform characteristics that are more similar to the real waveform, and the synthetic PGA is closer to the observed ones.

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.

Particle swarm optimization of friction tuned mass dampers subjected to ground motion records

Friction tuned mass dampers (FTMDs) are widely used to control the displacement of structures located in seismically active areas. Typically, the frequency and friction ratios of FTMDs are tuned up during design, but this task is complex if real ground motion records are considered. This article proposes a novel and accurate approach to calculate optimum parameters of FTMDs for controlling the displacements of both single degree of freedom (SDOF) systems and multi-story structural frames subjected to real ground motion records. In this study, the SDOF displacement and two FTMD parameters (frequency ratio and friction ratio) are first optimized simultaneously using a Particle Swarm Optimization (PSO) algorithm. A series of sensitivity analyses are then carried out to examine the effect of different structural features (damper movement, variations of optimized parameters and damping) on the optimized SDOF displacements and FTMD parameters given by the PSO. It is shown that, compared to a more established method available in the literature, the PSO algorithm reduces the SDOF displacements by an additional 21% on average. The PSO is then used to obtain optimum parameters of FTMDs and TMDs connected to four moment-resisting frames, and the results from the frames are compared to those from equivalent SDOF systems. This article contributes towards providing more suitable optimization tools for structures fitted with FTMDs, which in turn can lead to more efficient design methods for dampers.

Pressure-impulse diagrams using coupled single degree of freedom systems subjected to blast load

Pressure-impulse diagrams using coupled single degree of freedom systems subjected to blast load

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.

Elasto-plastic analysis of foundations during emergency shutdown due to blast loading

Behaviour of machine foundations is investigated in this study under harmonic loading with varying frequency and pulse loading. Owing to excessive vibrations from the pulse, the machine encounters an emergency shutdown where the machine’s exciting frequency reduces to zero exponentially. The analysis is performed considering an elastic plastic single degree of freedom system with hardening and softening behaviour. Two cases are analysed depending on the time of application of pulse loading to the foundation system. For both the cases, the governing equations of motion are acquired and solved numerically using the fourth order Runge-Kutta method. Using these solutions, displacement-time graphs are obtained for a variety of parameters such as time constant, hardening/softening index, mass of machine and the foundation block, stiffness, pulse load magnitude, damping ratio and decay coefficient to study their influence on the behaviour of the machine-foundation system. The study helps in predicting the dynamic response of foundations under the emergency shutdown conditions due to pulse loading. Further, these also help in taking the decision if there exists a need for vibration barriers to have the displacements well within the permissible limits.

Closed-form solutions to investigate the nonlinear response of foundations supporting operating machines under blast loads

Machine foundations are subjected to significant dynamic impacts. These impacts could spread to the surrounding regions, affecting workers, sensitive equipment in the same institution, or nearby areas. This study analyzes the response of machine-supporting foundations to harmonic and explosive loads under operational conditions and provides closed-form solutions for predicting responses in terms of displacement, velocity, and acceleration time-histories to two common types of blast loads: a more accurate typical profile and simplified triangular profile. The soil-machine foundation system is regarded as a single-degree-of-freedom (SDOF) system that exhibits elastic–perfectly flexible resistance behavior. For the analysis of the SDOF system, two cases are considered: one assumes that the supporting soil keeps elastic during the explosion, and the peak displacement is less than the elastic one, while the other assumes that the blast occurs in an elastic state, and the peak displacement occurs in a plastic state. By using the closed-form analytical solutions, a detailed parametric analysis is carried out to evaluate the impacts of significant soil-foundation system characteristics such as mass, stiffness, and damping ratio on the response-time history of machine foundations. The findings are compared to those reported in the literature, and relevant conclusions are derived. Obtained results demonstrated that, despite its simplicity and usage of only positive phase to simulate blast loads, the simplified model’s response behavior differs significantly from the typical one. Furthermore, the derived solutions are utilized to design the foundations supporting vibrating machines for both harmonic and blast loads in a variety of conceivable scenarios depending on the blast magnitude.

A partitioned combined computational method for multi‐scale dynamic systems

AbstractA partitioned/combined computational method based on Newmark scheme is proposed to analyze dynamic systems with different temporal scales and element scales. To effectively filter spurious high‐frequency vibration content and retain the second‐order accuracy simultaneously, Generalized‐α schemes are investigated and incorporated into the proposed method. The proposed method can decompose a complete domain into several independent computational subdomains (≥3), and several independent substructures can be combined into a complete computational structure. The accuracy and stability of responses in different subdomains can be ensured and adjusted by using their own integration parameters and time‐step sizes. The energy conservation property is preserved in the proposed method. Only one calculation is performed at each time step for all subdomains, and computational information exchange between subdomains is only conducted at the system time step, therefore, the computational efficiency is improved significantly compared with the existing multi‐time‐step methods. The derivation process and theoretical demonstration of the proposed method are given in detail. Two representative examples, namely, a single degree of freedom system split into four subdomains and a sandwich beam subjected to high‐frequency impact loads, are studied to systematically demonstrate the proposed method's accuracy, energy properties, and efficiency compared with the existing multi‐time‐step methods.

AN INVESTIGATION OF DAMPING MODIFICATION FACTORS CORRESPONDING TO DIFFERENT DAMPING RATIOS FOR SDOF SYSTEMS

The damping ratio is an important parameter in dynamic analyses and plays a key role in the design of building structures. Elastic response spectra are widely used in this design to describe the earthquake action in specific site classes. 5% damped response spectra are generally used for most of the conventional structures. However, other types of structures may not have a damping ratio of 5% and can have much lower or much larger damping ratios. The damping ratio is recommended at about 5% for concrete structures whereas it is estimated at 2% for steel structures. Tall slender buildings may have much lower damping ratios and low-rise buildings may be designed using much larger damping ratios than 5%. For these types of buildings, elastic response spectra are modified to account for different levels of damping ratios. This study proposes the damping modification factors (DMFs) which are computed using 5% response spectra as the benchmark. A strong ground motion set has been selected for the stiff site class and the displacement, pseudo-velocity, and pseudo-acceleration response spectra have been computed for single-degree-of-freedom (SDOF) systems. DMFs have been obtained in terms of these spectra and 3%, 10%, 20%, and 30% damping ratios have been considered. Variations of DMFs with different damping ratios have been obtained graphically. It can be seen from the results that DMFs are sensitive to the vibration period of the SDOF system.

Lumped parameter model for vertical vibrations of surface circular foundations on nonhomogeneous soil

PurposeA newly developed frequency-independent lumped parameter model (LPM) is the purpose of the present paper. This new model’s direct outcome ensures high efficiency and a straightforward calculation of foundations’ vertical vibrations. A rigid circular foundation shape resting on a nonhomogeneous half-space subjected to a vertical time-harmonic excitation is considered.Design/methodology/approachA simple model representing the soil–foundation system consists of a single degree of freedom (SDOF) system incorporating a lumped mass linked to a frequency-independent spring and dashpot. Besides that, an additional fictitious mass is incorporated into the SDOF system. Several numerical methods and mathematical techniques are used to identify each SDOF’s parameter: (1) the vertical component of the static and dynamic foundation impedance function is calculated. This dynamic interaction problem is solved by using a formulation combining the boundary element method and the thin layer method, which allows the simulation of any complex nonhomogeneous half-space configuration. After, one determines the static stiffness’s expression of the circular foundation resting on a nonhomogeneous half-space. (2) The system’s parameters (dashpot coefficient and fictitious mass) are calculated at the resonance frequency; and (3) using a curve fitting technique, the general formulas of the frequency-independent dashpot coefficients and additional fictitious mass are established.FindingsComparisons with other results from a rigorous formulation were made to verify the developed model’s accuracy; these are exceptional cases of the more general problems that can be addressed (problems like shallow or embedded foundations of arbitrary shape, other vibration modes, etc.).Originality/valueIn this new LPM, the impedance functions will no longer be needed. The engineer only needs a limited number of input parameters (geometrical and mechanical characteristics of the foundation and the soil). Moreover, a simple calculator is required (i.e. we do not need any sophisticated software).

The role of soil in structure response of a building damaged by the 26 December 2018 earthquake in Italy

Local soil conditions can significantly modify the seismic motion expected on the soil surface. In most cases, the indications concerning the influence of the underlying soil provided by the in-force European and Italian Building Codes underestimate the real seismic amplification effects. For this reason, numerical analyses of the local seismic response (LSR) have been encouraged to estimate the soil filtering effects. These analyses are generally performed in free-field conditions, ignoring the presence of superstructures and, therefore, the effects of dynamic soil-structure interaction (DSSI). Moreover, many studies on DSSI are characterised by a sophisticated modelling of the structure and an approximate modelling of the soil (using springs and dashpots at the foundation level); while others are characterised by a sophisticated modelling of the soil and an approximate modelling of the structure (considered as a simple linear elastic structure or a single degree of freedom system). This paper presents a set of finite element method (FEM) analyses on a fully-coupled soil-structure system for a reinforced concrete building located in Fleri (Catania, Italy). The building, designed for gravity loads only, was severely damaged during the 26 December 2018 earthquake. The soil was modelled considering an equivalent visco-elastic behaviour, while the structure was modelled assuming both the visco-elastic and visco-inelastic behaviours. The comparison made between the results of the FEM analyses and the observed damage is valuable. • Soil filtering phenomena are a key aspect for structure seismic response. • FEM modelling of the soil cannot ignore the dynamic properties of the bedrock. • Modelling of the structures should be performed in the inelastic range. • Earthquake damage is not necessarily a "physical" process for vulnerable buildings. • FEM analyses of DSSI must be encouraged for all buildings.

Parametrically Excited Nonlinear Pneumatic Artificial Muscle Under Hard Excitation: A Theoretical and Experimental Investigation

In this work, a single degree of freedom system consisting of a mass and a Pneumatic Artificial Muscle subjected to time-varying pressure inside the muscle is considered. The system is subjected to hard excitation and the governing equation of motion is found to be that of a nonlinear forced and parametrically excited system under super- and sub-harmonic resonance conditions. The solution of the nonlinear governing equation of motion is obtained using the method of multiple scales. The time and frequency response, phase portraits, and basin of attraction are plotted to study the system response along with the stability and bifurcations. Further, the different muscle parameters are evaluated by performing experiments which are further used for numerically evaluating the system response using the theoretically obtained closed form equations. The responses obtained from the experiments are found to be in good agreement with those obtained from the method of multiple scales. With the help of examples, the procedure to obtain the safe operating range of different system parameters is illustrated.