MDOF Model Research Articles

Seismic assessment of Isolator-Damper hybrid structural system based on analysis of MDOF and modal 2DOF models

A hybrid structural system is introduced and assessed; by combining two widely used passive devices, isolator and damper, along with the use of the partial plan isolation technique in the conventional buildings. Four partial isolation cases in comparison with three limit cases including a full-base fixing, full-base isolating and ideal story damping (Skyhook) are used on a finite element model of a 12-story building. The mentioned four cases have been selected by changing the number of fixed and isolated columns which caused isolated mass ratio between 28% and 71%. Also, a simplified manifest of the seismic performance of this proposed hybrid system is presented in the form of a 2DOF model. Time history analyzes under far- and near-faults earthquakes, frequency analysis and stationary random analysis under white noise are performed. Various structural actions are investigated, including the root mean squares (rms) and maximum of drift, displacement, shear, acceleration, maximum total and plastic deformation and energies of different parts of this system. It can be inferred from the results that desired efficiency extent of this system occurs in the range of 30–70% for the mass isolation ratio. In addition, the novel approach has prevented the full isolation of the base columns (even up to 70% reduction in isolation capacity) and by removing operational limitations such as providing a large performance gap in dense urban areas, compared to a full-isolated structure, it reduces the cost of construction.

Extending substitute frame model as a rapid supersede for moment frames and X-braced structures in seismic response estimations

Substitute Frame (SF) model is a recently introduced equivalent simplified MDOF model by the authors which is applicable to moment-resisting frames, facilitating the modeling efforts and output management besides reducing the analysis time. The use of this model is deliberated only in the code-based OpenSEES software generally used by researchers and is not suitable for the practical works of engineers. This paper aims to develop the SF model to be usable in the practical engineering platforms that are widely used by engineers, and authenticate its accuracy and applicability in such platforms. Moreover, another objective is to develop this model to be applicable to concentric X-braced steel frames with equal or non-equal bay lengths. These objectives will be fulfilled with different examples, and the accuracy of the simplified models will be comprehensively evaluated by means of modal, nonlinear static, and nonlinear dynamic analyzes. The nonlinear time history analyses (NTHA) are performed with 20 real ground motion records. It has been shown that the proposed simplified model can estimate the responses of moment frames and X-braced systems with acceptable accuracy in a considered practical engineering platform. The proposed model could estimate the natural periods of all the original models with less than 1% error. In addition, the proposed SF model could predict the overall behavior of the original models in the pushover curve and estimate the maximum IDRs, PFAs, and story shears of all stories with acceptable accuracy in nonlinear dynamic analysis. Moreover, the use of the SF model can considerably reduce the computational time (by about 85%) for performing the analyses, and also facilitate the modeling efforts and output management. The supplementary example files for the present study would be shared through the link “https://web.nit.ac.ir/∼h.hamidi/software_data.htm”.

Optimum shapes and dimensions of rubber bumpers in order to reduce structural pounding during seismic excitations

Large displacement of structures observed during seismic excitations may lead to collisions between two adjacent, insufficiently-separated buildings and may result in major damages of both of them. In many building codes, appropriate equations or approximately recommended distances between structures in order to avoid pounding hazard have been introduced. Unfortunately, further, more detailed considerations show that safety situation or economic aspects are not always satisfied due to the collisions between buildings and the cost of land, respectively. Hence, researchers have studied other approaches of reducing the negative pounding effects. Such methods include the use of tuned mass or liquid dampers. Moreover the increase in stiffness of building or reduction of mass of the structure are still considered. In this paper, another approach is considered by the application of rubber bumpers placed between buildings. The bumpers are attached at each story to absorb energy during impact. Several different shapes and dimensions of bumper elements were numerically investigated so as to find the most effective ones most effective in reducing structural pounding negative effects. For this purpose, two MDOF models of 3-story and 4-story buildings were firstly considered. Such parameters as lateral displacement, damage index, dissipated energy and impact forces were calculated and depicted as the results of numerical study. Then, different shapes and dimensions of bumpers were parametrically investigated.

Vibration reduction of monopile-supported offshore wind turbines based on finite element structural analysis and active control

A finite element (FE) structural analysis based on a commercial software is combined with an observer-based active control to study the real-time performance of an Active Tuned Mass Damper (ATMD) in reducing the fore-aft vibration of a monopile-supported offshore wind turbine (OWT). Firstly, a reduced order MDOF model was established and used to design a Linear Quadratic Regulator. Secondly, the controller was combined with an observer to minimize the required number of sensors. Thirdly, co-simulations between the high-fidelity FE software and the optimized controller were performed. The aim was to test the robustness of the MDOF model-based controller in terms of OWT vibration reduction when the ATMD is used within a structural FE commercial software. The optimized controller applied to the FE mechanical model of the OWT proved to be efficient, the RMS reduction ratio of the tower top displacement being around 60%. Given the increasing size and capacity of OWTs, the proposed co-simulation technique combining a large deformation structural analysis based on a FE commercial software with an optimized observer-based active control can serve as a tool for the final design stage of the next-generation dynamically-sensitive OWT structures to accurately capture their dynamic responses in the presence of ATMDs.

Dynamics analysis on the MDOF model of ball screw feed system considering the assembly error of guide rails

The dynamics of the ball screw feed system is very important in high-precision machining of CNC machine tools. However, the existing literatures rarely study the dynamic behaviors of the coupled feed system and didn’t take the influence of the assembly error of the guide rails into consideration. Because the stiffness of the feed system is mainly affected by the ball screw, linear guide and ball bearings, this paper presents the fourteen degree-of-freedom dynamic model of the feed system by deriving the nonlinear restoring force functions of the screw nut, linear guide and ball bearings and the screw shaft’s stiffness. An algorithm of combining the Newmark Method and the Monte Carlo simulation is proposed to numerically solve for the vibration responses of the feed system, and the proposed dynamic model is experimentally verified. Based on the established dynamic model, the effects of the excitation amplitude, system parameters and assembly error on the dynamic characteristics of the feed system are discussed by means of the amplitude-frequency curve and 3-D frequency spectrum, which demonstrate that the vibration responses of the feed system for the fourteen degrees of freedom influence each other, and the assembly error affects the nonlinear dynamic characteristics of the feed system.

Closed-form critical response of undamped bilinear hysteretic MDOF system under pseudo-double impulse for estimating resonant response under one-cycle sine wave

Approximate closed-form maximum interstory drifts under the critical pseudo-double impulse (PDI) are derived for undamped multi-story shear building models with bilinear hysteresis by using the proposed updated mode-controlled energy-based approach (UMEA). PDI was proposed in the previous paper to simulate the critical responses of elastic-plastic MDOF models under near-fault fling-step motions. Although a double impulse (DI) is treated as a ground motion, PDI is treated as a set of impulsive lateral forces. PDI excites only the fundamental-mode responses of elastic proportionally-damped MDOF models because the fundamental participation vector is applied to the influence coefficient vector of PDI. UMEA is a kind of displacement control analyses, and the displacement increment, which is proportional to the fundamental mode evaluated by the tangent story stiffnesses, is given to the model. It is demonstrated that UMEA effectively captures the main characteristics of the maximum interstory drifts under the critical PDI and the critical one-cycle sine wave because most of the kinetic energy, which the fundamental mode has just before yielding, is given to the plastic fundamental mode just after yielding except for the case that the post-yielding stiffness is almost zero. The use of UMEA leads to much more efficient estimation of the maximum interstory drifts than the time-history response analysis because UMEA requires only 2N-time evaluations of the elastic and plastic fundamental modes at most for N-story shear mass models with bilinear hysteresis. It is shown through numerical examples that the maximum interstory drifts under the critical PDI and the corresponding one-cycle sine wave correspond well to those evaluated by the proposed method. It is also shown that the correspondence of the responses of elastic-plastic MDOF models under recorded near-fault motions and the estimated responses by the proposed method is fairly good.

Bounding of earthquake response via critical double impulse for efficient optimal design of viscous dampers for elastic‐plastic moment frames

AbstractAn input velocity adjustment method of the critical double impulse (DI) is presented for efficient design of viscous dampers for elastic‐plastic moment frames. The input velocity is adjusted so that the input energy to the lowest eigenmode under the critical DI is equal to those under the selected recorded ground motions. This adjustment makes the critical DI work as the active earthquake. The response bounding property of the critical DI is supported by, (1) the multi‐modal property of the critical DI, (2) the usual excitation of the lowest mode response under recorded ground motions, (3) the large proportion of the instantaneous input energy to the total input energy in the critical DI. The proposed method can treat not only pulse‐like near‐fault ground motions but also ground motions of random nature. To validate the proposed method, the input energies and the maximum interstory drifts under the recorded ground motions and the critical DI with the adjusted input velocity are investigated for elastic single‐degree‐of‐freedom (SDOF) models, elastic proportionally damped MDOF models, and elastic‐plastic proportionally damped MDOF models. The optimization method presented in the previous paper is extended so that the critical DI can be treated for the design of viscous dampers for elastic‐plastic moment frames.

Optimal sliding friction coefficients for isolated viaducts and bridges: A comparison study

The aim of this work is to evaluate the influence of the pier–abutment–deck interaction on the seismic response of bridges isolated by single concave sliding pendulum isolators (friction pendulum system [FPS]) through a comparison with the results of the seismic response of isolated bridges without considering the presence of the rigid abutment (i.e., isolated viaducts). Two different multidegree-of-freedom (mdof) models are properly defined to carry out this comparison. In the both mdof models, five vibrational modes are considered to describe the elastic behavior of the reinforced concrete pier, and an additional degree of freedom is adopted to analyze the response of the infinitely rigid deck isolated by the seismic devices. The FPS isolator behavior is described through a widespread velocity-dependent model. By means of a nondimensional formulation of the motion equations with respect to the seismic intensity, a parametric analysis for several structural properties is performed in order to investigate the differences between the two mdof models in relation to the relevant response parameters. The uncertainty in the seismic input is taken into account by means of a set of natural records with different characteristics. Finally, multivariate nonlinear regression relationships are provided to estimate the optimum values of the sliding friction coefficient able to minimize the pier displacements relative to the ground as a function of the structural properties considering or neglecting the presence of the abutment.

General Substitute Frame Model (GSF) for efficient estimation of seismic demands of steel and RC moment frames

This paper proposes an approach to simplify 2D moment-resisting frames (with equal or non-equal bay lengths) to an equivalent single-bay framed MDOF model. This model, named the General Substitute Frame (GSF), is a promising development of the recently introduced Substitute Frame (SF) model by the authors that was restricted to equal-bay moment frames. The proposed GSF model which resolves the limitation of the SF model is applicable for both the steel and RC moment frames from low-rise to high-rise buildings. In order to evaluate the accuracy of the GSF model rigorously, a combination of 24 full-frame 2D moment frames including 4-, 12-, and 20-story buildings was designed and the results of nonlinear dynamic analyses were compared with their corresponding GSF models in terms of engineering demand parameters. In addition, modal and pushover analyses were also performed. Earthquake ground motions employed in this study include mild far-fault, moderate far-fault, and severe near-fault records selected by the Conditional Spectrum method. To accurately perform the required analyses, OpenSEES software was used, and 3600 runs of nonlinear time history analysis have been carried out. Based on the results of the analyses, it was demonstrated that the GSF model could predict the seismic response of moment frames well enough into both the linear and nonlinear ranges with high accuracy and can be a reliably practical substitution for moment-resisting frames. Moreover, the GSF model can be easily extracted from the primary full-frame based on the proposed equations.

A design method for multi-degree-of-freedom aeroelastic model of super tall buildings

Wind tunnel test models for super tall buildings mainly include synchronized pressure models, high-frequency force balance models, forced vibration models and aeroelastic models. Aeroelastic models, especially MDOF aeroelastic models, are relatively accurate, and designing MDOF model is an important step in aero-model wind tunnel tests. In this paper, the authors propose a simple and accurate design method for MDOF model. The purpose of this paper is to make it easier to design MDOF models without unnecessary experimentation, which is of great significance for the use of the aero-model for tall buildings.

Α mass‐reduction design concept for seismic hazard mitigation

SummaryA mass reduction concept for seismic hazard mitigation is investigated herein. The proposed method is implemented through floating slabs, ie, slabs that have been seismically isolated from the skeleton of the structure. The investigation is based on time history analyses of MDOF models under scaled strong‐motion seismic records complying with an EC8 spectrum. The purpose of these slabs is twofold; for selected short isolation periods, they act as a mass‐damping system for the overall response of the structure, employing significantly more mass than traditional TMDs, while for longer isolation periods they provide seismic protection on their contents while effectively reducing the seismic mass of the structure. In the latter case, it is found that the response of the skeleton can be evaluated accurately from a corresponding reduced‐mass model. The proposed design method does not necessarily aim at replacing existing seismic design approaches; it rather provides design versatility in the hands of the practicing engineers.

Correlation of earthquake intensity measures and spectral displacement demands in building type structures

Seismic damage and its correlation with the earthquake intensity measures (IMs) are important for both record selection and probabilistic seismic risk assessment studies. A lot of studies considering the various structural properties were performed to investigate the correlations of mainly PGA and PGV under different strength reduction (R) and ductility factors (μ). In this study, correlation of many scalar IMs named as PGA, PGV, arms, vrms, drms, Ia, CAV, ASI, SI are studied. Building represented by single degree of freedom (SDOF) models were classified according to lateral strength ratios (Vt/W), ratio of strength capacity at yield (Vt) to seismic weight (W), varying from 10% to 50%. Effects of post yield stiffness capacities, hysteretic behavior type and the soil properties on deformation demand of SDOF models are also taken into account. Post yield stiffness ratios of 0%, 5% and 10% considered and assigned to building models. Elastic-perfectly plastic and modified Clough hysteretic models were used in the non-linear dynamic analyses. Strong motion records obtained from firm and soft soil sites were classified according to Vs30 and inelastic deformation demands obtained from both soil conditions are calculated and compared. Variation of correlation coefficients of different IMs indicates that SI is the best earthquake parameter especially along the period range of 0.5–2 s which can be thought to represent building type structures. Furthermore, it was observed that sensitivity of SI parameter to post yield stiffness, hysteretic behavior type and the soil properties is very limited with respect to considered IMs. Correlation coefficients obtained from SDOF models are compared to that of multi-story buildings and the compatibility of both SDOF and MDOF models are presented.

Strength or force reduction factors for steel buildings: MDOF vs SDOF systems

The nonlinear seismic responses of steel buildings with perimeter moment resisting frames (MRF), modeled as complex 3D MDOF systems, are calculated and the ductility demands, ductility reduction factors and the force reduction factors (R), are studied. Equivalent 3D models with spatial MRF, two-dimensional models, and equivalent SDOF systems (SD), are also considered. Results indicate that the global and local force reduction factors significantly vary from one structural representation to another and that they are much larger for the SD models. One of the reasons for this is that, although there is equivalence between the SD and MDOF models, the dissipated energy and the number of incursions in the inelastic range are significantly larger for the SD models. In addition, while for the SD models total plasticization occurs, for the SAC and EQ models, even for significant yielding, plastic hinges are developed only in a relatively small number of structural members; therefore, using the R factors of the SD models may be conservative. According to the results obtained in this research for the more realistic representation of the steel building (3D), the value of 8 suggested in many codes for the R factor for ductile MRF cannot be justified. It is only justified for the SD systems. The implication of this is that non-conservative designs may be obtained if so large value is used. More transparence is needed in the codes regarding the magnitude and the components involved in the R factor.

Tuned mass damper system of high-rise intake towers optimized by improved harmony search algorithm

The seismic safety and proper functioning of intake towers in a major earthquake are very crucial to the whole hydraulic project, since the controlled release of the reservoir could help to prevent catastrophic failure of a dam after an earthquake by reducing the water pressure. In this study, the tuned mass damper (TMD) system was introduced into the seismic design of high-rise intake towers. The installation of TMD can effectively dissipate the seismic energy acting on the intake tower and thus overcome the whiplash effect of the hoist room. The 3D finite element model of the intake tower was simplified into a 2D MDOF model. Improved harmony search (IHS) algorithm was used to determine the optimal parameters for the TMD systems using Matlab, and the robustness of the TMD systems was investigated. The simulation elements of the TMD system were incorporated into ADINA to simulate and evaluate the effect of the TMD system on the dynamic displacement and stress responses of various key points of the intake tower to different seismic excitations.

Prediction of Critical Distance Between Two MDOF Systems Subjected to Seismic Excitation in Terms of Artificial Neural Networks

This study focuses on preventing collisions between structuresduring seismic excitation based on gap size. Several approximatedequations in order to estimate separation distancebetween buildings are collected and evaluated to measure gapsize in order to avoid impact between them when large lateraldisplacements occurred due to earthquake. Artificial neuralnetworks are utilized to estimate the required distance betweenstructures. The majority of building codes suggest separationdistances based on maximum lateral displacements of eachbuilding or height of buildings in order to provide safety gapsize between them. Subsequently, researchers have proposedseveral equations to predict the critical distance. In currentstudy, some MDOF models are equivalently modelled and optimumgap size between buildings is approximately estimatedand finally a new equation for separation distance is suggestedand the accuracy of formula is numerically investigated.

主系と副系から成る建物を対象とした動的縮約モデルによる層間変位分布の復元

A two degree-of-freedom (2-DOF) model is employed for mid-story isolation buildings or a tuned mass damper mounted on buildings in practical design. These systems are classified into a primary-secondary structural system. A seismic response analysis using this model is an empirical evaluation method in design practice. However, the accuracy of the peak response distribution computed by the 2-DOF model has not been clarified. The objective of this study is to investigate the error in the inter-story drift calculated by the 2-DOF model. The employed 2-DOF model is derived from its original multi-degree of freedom system (MDOF) based on dynamic reduction technique. Parameters of interest are a tuning ratio and mass ratio. The tuning ratio is a frequency ratio to secondary to the primary system. The mass ratio is a ratio of the secondary mass to the effective mass of the primary system. The mass ratio is set within the parameter of 0.02 to 0.5. The inter-story drifts in primary system are recovered from generalized displacements (modal displacements) of the 2-DOF model and classical modal vectors with respect to the primary system. A stochastic vibration analysis is conducted in both original MDOF model and corresponding the 2-DOF model. RMS responses are obtained by solving a Lyapunov equation governing the stochastic process. The displacement responses of the MDOF model are decomposed into coupled and decoupled components. The decoupled components are the higher modal responses. The coupled component is the remaining response and corresponds to a controlled response. The controlled mode is limited to the fundamental mode in this study. The response of the controlled mode is compared to the one from the 2-DOF model. The analysis reveals that the 2-DOF model estimates accurate floor displacements. The relative errors are ten percent at most. However, the 2-DOF model underestimates inter-story drifts in the upper story of the primary system. As the mass ratio becomes larger, the error becomes large. A convenient modal superposition technique is proposed to improve the accuracy of inter-story drift obtained by the 2-DOF model. The method is as follows. The CQC modal combination is applied with modal damping factors evaluated by classical modal vectors. Then, amplitude of the response is slightly corrected to be consistent with the one obtained from the MDOF model. It is confirmed that reasonable accuracy is obtained by the proposed method.

Performance-based seismic design of flexible-base multi-storey buildings considering soil–structure interaction

A comprehensive parametric study has been carried out to investigate the seismic performance of multi-storey shear buildings considering soil–structure interaction (SSI). More than 40,000 SDOF and MDOF models are designed based on different lateral seismic load patterns and target ductility demands to represent a wide range of building structures constructed on shallow foundations. The cone model is adopted to simulate the dynamic behaviour of an elastic homogeneous soil half-space. 1, 5, 10, 15 and 20-storey SSI systems are subjected to three sets of synthetic spectrum-compatible earthquakes corresponding to different soil classes, and the effects of soil stiffness, design lateral load pattern, fundamental period, number of storeys, structure slenderness ratio and site condition are investigated. The results indicate that, in general, SSI can reduce (up to 60%) the strength and ductility demands of multi-storey buildings, especially those with small slenderness ratio and low ductility demands. It is shown that code-specified design lateral load patterns are more suitable for long period flexible-base structures; whereas a trapezoidal design lateral-load pattern can provide the best solution for short period flexible-base structures. Based on the results of this study, a new design factor RF is introduced which is able to capture the reduction of strength of single-degree-of-freedom structures due to the combination of SSI and structural yielding. To take into account multi-degree-of-freedom effects in SSI systems, a new site and interaction-dependent modification factor RM is also proposed. The RF and RM factors are integrated into a novel performance-based design method for site and interaction-dependent seismic design of flexible-base structures. The adequacy of the proposed method is demonstrated through several practical design examples.

Using experimental data to reduce the single-building sigma of fragility curves: case study of the BRD tower in Bucharest, Romania

The lack of knowledge concerning modelling existing buildings leads to significant variability in fragility curves for single or grouped existing buildings. This paper aims to investigate the uncertainties of fragility curves, with special consideration of the single-building sigma. Experimental data and simplified models are applied to the BRD tower in Bucharest (Romania), a RC building with permanent instrumentation. A 3-step methodology is applied, (1) adjustment of a linear MDOF model for experimental modal analysis using a Timoshenko beam model and based on Anderson’s criteria, (2) computation of the structure’s response to a large set of accelerograms simulated by SIMQKE software, considering twelve ground motion parameters as intensity measurements (IM), and (3) construction of the fragility curves by comparing numerical inter-storey drift with the threshold criteria provided by the Hazus methodology for the slight damage state. By introducing experimental data into the model, uncertainty is reduced to 0.02 considering Sd(f1) as seismic intensity IM and uncertainty related to the model is assessed at 0.03. These values must be compared with the total uncertainty value of around 0.7 provided by the Hazus methodology.

A method for earthquake response analysis of tall flexible structure

The earthquake responses are studied for the tall flexible structures such as TV towers when the vertical eccentricities between the discrete nodes and the corresponding centroids of investigated lumps are considered. In practical analyses, the tall flexible structures can be made into a spatial-discrete system of some certain length of beam elements with different lengths and cross-sectional areas. These elements are used to construct the investigated lumps in this paper. The different cross-sectional areas and the different lengths of two adjacent elements lead to the appearance of vertical eccentricity between the discrete node and the centroid of investigated lump within the same investigated lump. Firstly, the governing equations are established for a typical investigated lump. Secondly, the calculating formulae of the forces and moments acting on the investigated lump are derived and provided. Finally the new dynamic equilibrium equations with modified mass matrix and assemblage of stiffness matrix have been derived for the stick MDOF model based on beam theory when the existing vertical eccentricities are considered. Numerical results demonstrate that these vertical eccentricities should be considered in order to obtain the accurate earthquake responses for the tall flexible structures.

Simplified models for assessment and optimal redesign of irregular planar frames

This paper presents a simplified methodology for predicting the response of buildings irregular in elevation, based on the non-linear dynamic multi-modal spectral seismic response of an equivalent simplified MDOF linear system. Making use of the MDOF model, it is also proposed a methodology for the optimum strengthening design of buildings, based on convex approximation methods. Three structural upgrading optimization problems (SUOPs) are implemented. The proposed models are validated against the results of a series of PsD tests carried out on a full-scale four-storey RC structure, representative of the construction practice in southern Mediterranean countries in Europe until the late 1960s.