Square Root Of Sum Of Squares Research Articles

MAXIMUM SEISMIC RESPONSE PREDICTION BASED ON CAPACITY SPECTRUM METHOD FOR VIBRATION-CONTROLLED CEILING STRUCTURE EMPLOYED PULLEY MECHANISM

This paper proposes a new procedure for predicting the maximum seismic response based on the capacity spectrum method (CSM) for the passively controlled suspended ceiling structures named “Pulley Damper Ceiling System (PDCS)”. The structural system has a supplemental nonlinear viscous damper and a bilinear hysteretic damper in parallel with displacement amplification mechanisms of block and tackle pulley system. The single degree-of-freedom model of the PDCS is initially discussed. The accuracy of a simplified calculation method for obtaining the structural equivalent damping coefficient, determined by the sum of the inherent damping and the square root of sum of squares (SRSS) of each damping provided by the nonlinear viscous damper and the bilinear hysteretic damper, is verified by comparing with the full-scale shake table test results of the PDCS. Then, the pseudo capacity curve of the structure is generated by continuously plotting the transition of maximum force and displacement values under different amplitudes of sinusoidal waves. Finally, the availability of the response prediction based on proposed CSM procedure considering floor response spectrum (FRS) is examined by the comparison of time history analysis results, and the sufficient accuracy to evaluate the seismic response is confirmed.

Two-stage deep learning for accelerated 3D time-of-flight MRA without matched training data

Time-of-flight magnetic resonance angiography (TOF-MRA) is one of the most widely used non-contrast MR imaging methods to visualize blood vessels, but due to the 3-D volume acquisition highly accelerated acquisition is necessary. Accordingly, high quality reconstruction from undersampled TOF-MRA is an important research topic for deep learning. However, most existing deep learning works require matched reference data for supervised training, which are often difficult to obtain. By extending the recent theoretical understanding of cycleGAN from the optimal transport theory, here we propose a novel two-stage unsupervised deep learning approach, which is composed of the multi-coil reconstruction network along the coronal plane followed by a multi-planar refinement network along the axial plane. Specifically, the first network is trained in the square-root of sum of squares (SSoS) domain to achieve high quality parallel image reconstruction, whereas the second refinement network is designed to efficiently learn the characteristics of highly-activated blood flow using double-headed projection discriminator. Extensive experiments demonstrate that the proposed learning process without matched reference exceeds performance of state-of-the-art compressed sensing (CS)-based method and provides comparable or even better results than supervised learning approaches.

Seismic Performance Evaluation of Steel Buildings with Oil Dampers Using Capacity Spectrum Method

This study proposes a capacity spectrum Method (CSM)-based procedure to estimate the maximum seismic performance of steel buildings passively controlled with bilinear oil dampers. In the proposed CSM, the maximum seismic response of a building was estimated, in the acceleration-displacement response spectrum, as the intersection between the capacity curve and the damping-adjusted demand curves, using the equivalent linearization method. The building equivalent damping ratio was determined by the sum of the inherent damping, and the square root of sum of squares (SRSS) of the hysteretic damping and the viscous damping of the supplemental oil devices. The calculation steps of the proposed CSM are explained in detail based on the equivalent single degree of freedom (ESDOF) system, and its accuracy was examined by comparison with time history analysis (THA) results. Two model steel buildings of 4 and 10 stories, uniformly equipped with oil dampers along the height, were subjected to six selected earthquake ground motions scaled to be compatible with Level-2 earthquakes, as defined in the Japanese Building Standard Law. The seismic performance of the buildings was estimated by the proposed CSM procedure and compared with the results of nonlinear THA in terms of the maximum story displacements and the shear forces. It was observed that the proposed CSM scheme provided a satisfactory accuracy to assess the maximum nonlinear response of steel buildings passively controlled with oil dampers.

Interval Type-2 Fuzzy Hybrid Control of a High-Rise Building Including Soil–Structure Interaction Under Near-Field and Far-Field Ground Motions

The interval type-2 fuzzy logic controller (IT2FLC) has emerged as one of the most effective high-tech control strategies to improve the performance of structures under near-field and far-field ground motions. The present study discusses the application of this strategy in the hybrid control of the behaviour of a real high-rise shear building by considering soil–structure interaction. In this regard, a hybrid control technique, consisting of a tuned mass damper (TMD) and a magneto-rheological (MR) damper is proposed. Using an MR damper on a TMD (MR + TMD) helps to reduce the seismic responses of a high-rise building in soil–structure interaction. The utilization of this hybrid device is very useful because it does not require huge power sources. By applying the observer–teacher–learner–based optimization (OTLBO) method, the fuzzy rules of IT2FLC, the TMD parameters, and the voltage of the MR damper have been optimized for a fifteen-storey shear building under four seismic near-field and far-field excitations. Minimization of the peak displacement of the top floor by using the square root of sum of squares (SRSS) of each response to each earthquake was utilized as the optimization criterion. Then, the optimized type-2 fuzzy rules and optimized TMD parameters were examined under seven new earthquake records in the fifteen-storey building. The results obtained by comparing the proposed control and uncontrolled systems in terms of the structure interaction with soft and dense soils show that the IT2FLC- using OTLBO is an efficient control method. It is also shown that the responses of this building to the hybrid controller have been effectively improved.

Three-Dimensional Numerical Model for Seismic Analysis of Structures

Response Spectrum Analysis (RSA) and modal combination techniques are widely used to estimate the peak response of structures subjected to earthquake vibrations. This paper presents the numerical modeling and seismic analysis of structures. The response spectra based on Eurocode 8 is implemented for a five storey moment-resisting 3D structure to estimate peak response to earthquake-induced vibration. Eigenvector analysis is carried out to determine undamped free vibration mode shapes and frequencies. The lumped mass matrix, stiffness matrix of the full 3D structure is derived and natural periods, modal shapes, modal participation factors, and effective modal mass ratios are calculated. The structural response in different modes is combined to estimate the total dynamic response. The Square Root of Sum of Squares (SRSS), Complete Quadratic Combination (CQC), and Absolute Sum (AbsSum) rules are discussed and compared. The results of the SRSS and CQC rule for the examined building are almost identical as all cross-correlation coefficients are approximately equal to zero, and the responses of the individual modes are completely uncorrelated. CQC rule is appropriate when the natural frequencies are close to each other. AbsSum method shows an upper bound estimate of the total response since it assumes that all modal peaks occur at the same time. The simulation for numerical modeling and response spectrum analysis (RSA) is implemented in MATLAB® software.

Seismic Performance of Gravity Load-Designed RC Frame Buildings in Jordan: a Prelude into the Effect of Masonry Infills

Fifty four infilled reinforced concrete (RC) frame buildings were selected and designed to represent typical Jordanian low and medium rise buildings designed to resist gravity loads only. Using SAP2000 software, nonlinear static analysis was performed on three dimensional models of the representative buildings. The study parameters included building height; horizontal and vertical irregularities; as well as type and layout of walls. The inverted triangular and Square Root of Sum of Squares (SRSS) lateral load patterns were used for irregular buildings. Analysis results indicated that commonly encountered horizontal and vertical irregularities associated with presence of re-entrant corners and soft stories negatively affects the elastic stiffness, energy dissipation and lateral resisting capacities of the investigated buildings. Analysis results confirmed that using RC walls, rather than masonry walls, in the staircase unit greatly enhances the seismic performance of buildings provided the RC walls followed a symmetrical plan arrangement about one of the principal axes. The majority of the investigated buildings exhibited a maximum inter-story drift ratio below 1.5% at the yielding strength and 3.8-4.8% at ultimate strength indicating that severe structural damage may take place under strong earthquake excitations. Plastic hinge formation signified the use of faulty design concepts violating the traditional strong column-weak beam concept of earthquake-resistant design. Compared with the inverted triangular lateral load pattern, using the SRSS pattern for pushover analysis of irregular buildings had a negligible impact on lateral load resistance but significantly affected stiffness and energy dissipation values.

Seismic fragility assessment of a super tall building with hybrid control strategy using IDA method

To investigate the interstory drift control effect under different hybrid control schemes using buckling restrained braces (BRB) and viscous dampers (VD), and to study the effect of square root of sum of squares (SRSS) combined intensity measures (IMs) with engineering demand parameters (EDPs) on fragility assessment, incremental dynamic analysis (IDA) based seismic fragility assessment is performed on three numerical models of a super tall building (729 m) in its both structural principal directions. One is the uncontrolled model, the other two are damped models where the BRBs serve as the diagonal member of outrigger truss (BRB outriggers) at the upper part of structure while the improved viscously damper outriggers (IVDOs) are set at the lower location of the building, or conversely. The results suggest that structure can have smaller maximum interstory drift as well as exceeding probabilities if the IVDOs locate at the lower part of structure meanwhile the BRB outriggers are installed at the upper part of structure. Moreover, IDA demonstrates that the SRSS combined IMs and engineering demand parameters (EDPs) may have higher efficiency compared with results from non-combined ones, and the combined EDP limit obtained by the limit in one direction multiply the square root of sum of 1 and λ2, where λ is the ratio of PGA of seismic records input in the two structural principal directions, is also suitable for fragility analysis. Additionally, studies on the fragility analysis using peak ground velocity (PGV) demonstrate that, for tall buildings, PGV is not only suitable for IDA but also works well in fragility analysis based on its high efficiency and reasonable exceeding probabilities.

Seismic vulnerability of skew bridges under bi-directional ground motions

Past earthquakes have demonstrated that skew bridges are more vulnerable to earthquake induced failure than a normal bridge due to their complex load transfer mechanism. This study deals with the effect of skew angle on seismic vulnerability of these bridges under horizontal two-component (bi-directional) ground motions. For this purpose, representative models of two-span simply supported, typical highway overpasses with varying skew angles have been considered. Finite element models of these bridges, which have been designed using modern seismic provisions, are developed using a widely used software. A suite of bi-directional ground motions with varying strong motion properties and representing different hazard levels is used for nonlinear dynamic analysis and subsequent seismic vulnerability estimation using fragility curves. Deck unseating and damage of columns are considered as the damage measures for the vulnerability analysis. The effect of ground motion directionality on the vulnerability of bridges has also been investigated in this study. Results of this study indicate that as the skew angle increases, there is an increase in the probability of failure for a given ground motion intensity. It is also found that the dispersion of fragility curves for bi-directional ground motions can be minimized by using the square root of sum of squares (SRSS) of peak ground acceleration (PGA) of all components than the PGA of any component.

Prediction of Column Axial Forces in Inverted V-braced Seismic Steel Frames Considering Brace Buckling

Brace buckling in inverted-V-braced frames induces the vertical unbalanced force. The columns in the braced bay should be designed, per the capacity design concept, to remain elastic for gravity load actions and additional column axial forces that result from the brace buckling. However, owing to the difficulty in accumulating the buckling-induced column forces from different stories, empirical and often conservative approaches have been used in design practice. In this paper, three combination rules for a rational estimation of the column axial forces are proposed. The idea central to the three methods is to detect the stories with high buckling potential as precisely as possible by using pushover analysis and/or simple demand-to-capacity analysis. The vertical unbalanced forces in the stories detected as high buckling potential are then summed in a linear manner, whereas those otherwise are combined by following the SRSS (square root of sum of squares) rule. The accuracy and design advantage of the three methods is evaluated on the basis of extensive inelastic dynamic analyses. The mode shape-based method (MSBM), which is both simple and accurate, is recommended as the method of choice for practicing engineers among the three proposed.

A study of a load combination method considering the correlation among dynamic loads (2nd report, Load combination for the seismic response analysis of a liquid storage tank - Part 2)

This report presents a load combination methodology for dynamic response analysis problems, based on a stochastic approach which has been practically introduced in the field of random vibration theory. The methodology proposed in this report has the advantageous point that the appropriate correlation effect among combined load component is taken into consideration and is evaluated in terms of the dynamic correlation factor ρ. The specific combination problem discussed here is the calculation of the maximum base shear coefficient of a large-scaled cylindrical liquid storage tank which can be simulated by a simplified lumped mass model, whereby the sloshing effect of the liquid is neglected. It is concluded that the equations of combination constructed here could be applicable for the case where the fundamental natural frequency of a tank is higher than half of the dominant frequency of input acceleration such as seismic ground excitation ; while an equation proposed in a previous report is effective for other case. The accuracy of this method is statistically examined by use of the random vibration technique ; and its advantage over SRSS (square root of sum of squares) law and ABS (absolute value summation) law is examined with the aid of the numerical computation of the practical examples.

A Study on Load Combination Method Considering the Correlation among Dynamic Loads : 1st Report, Load Combination for seismic Response Analysis of Liquid Storage Tank

This report presents a load combination methodology for dynamic response analysis problems, based on a stochastic approach which has been practically introduced in the field of random vibration theory. The methodology proposed in this report has the advantageous point that the appropriate correlation effect among combined load compoents is take into consideration and is evaluated in terms of the dynamic correlation factor p. The specific combination problem discussed here is the calculation of the maximum base shear coefficient of a large-scale cylindrical liquid storage tank which can be simulated by a simplified lumped mass model, whereby the sloshing effect of the liquid is neglected. It is concluded that the equations of combination constructed here are applicable for the case where the fundamental natural frequency of input acceleration such as seismic ground excitation; while an equation proposed in a previous report is effective for other cases. The accuracy of this method is statistically examined by use of the random vibration technique; and its advantage over SRSS (square root of sum of squares) law and ABS (absolute value summation) law is examined with the aid of the numerical computation of practical examples.

A study on a load combination method considering the correlation among dynamic loads. (1st report, Load combination for seismic response analysis of liquid storage tank)

This report deals with fundamental study concerning various kinds of dynamic load combination problems. Up to the present, the square root of sum of squares (SRSS) law has been widely utilized for various load combination problems. However, considerable error is produced during the combination based upon SRSS law, particularly when correlation among loads to be combined becomes significant. In this study a load combination technique introducing an evaluation of the correlation is proposed. It is applied to the seismic response analysis problem for large-scale liquid storage tank inducing sloshing effect in order to evaluate the base shear coefficient and the overturning moment. It has been made clear that the proposed method is reasonable by comparing with the SRSS-based law and the absolute value summation. And, this merit becomes dominant when combination is carried out not based upon ordinary absolute acceleration response spectrum but relative acceleration response spectrum as input load.

Methods for reliability of structures under multiple time varying loads

Current probabilistic studies on load combination have been concentrating on the modeling and linear combination of loads and load effects. Herein, a methodology for the assessment of life-time reliability of structural systems under multiple time varying loads is proposed. The method is based on considerations of load occurrences (including coincidence) and conditional probability of failure given occurrence whereby current methods of structural reliability and random vibration analysis can be incorporated. It has a wide range of applications. The emphasis is on the time domain load behavior, structural capacity uncertainty and non-linear dynamic load effects. The accuracy and computational advantage of this method as compared with other methods are examined by a numerical example of a simple frame under loads modeled as Poisson renewal pulse processes. Also, the errors implied in the widely used ‘Turkstra’ rule, a recently proposed modification of this rule, the load reduction factor method, and a SRSS (square root of sum of squares) rule for load combinations are quantified for linear combination and the implications in risk evaluation mentioned.