Period Elongation Research Articles

A model/solution‐adaptive explicit‐implicit time‐marching technique for wave propagation analysis

SummaryIn this work, an explicit‐implicit time‐marching procedure with model/ solution‐adaptive time integration parameters is proposed for the analysis of hyperbolic models. The two time integrators of the methodology are locally evaluated, enabling their different spatial and temporal distributions. The first parameter defines the explicit/implicit subdomains of the model, and it is defined in a way that stability is always ensured, as well as period elongation errors are reduced; the second parameter controls the dissipative properties of the methodology, allowing spurious high‐frequency modes to be properly eliminated, rendering reduced amplitude decay errors. In addition, the proposed explicit‐implicit approach allows contracted systems of equations to be obtained, reducing the computational effort of the analysis. The main features of the novel methodology can be summarized as follows: (i) it is simple; (ii) it is locally defined; (iii) it has guaranteed stability; (iv) it is an efficient noniterative single‐step procedure; (v) it provides enhanced accuracy; (vi) it enables advanced controllable algorithmic dissipation in the higher modes; (vii) it considers a link between the temporal and the spatial discretization; (viii) it stands as a single‐solve framework based on reduced systems of equations; (ix) it is truly self‐starting; and (x) it is entirely automatic.

Pseudo-Dynamic Test for Soil-Structure Interaction Analysis using Shake Tables

This paper presents an effective pseudo-dynamic (PSD) test method for analyzing the response of a multi degree of freedom structural system. In addition to the physical and numerical substructuring of the structure, the effect of the soil-structure interaction (SSI) is numerically modeled without a large soil box. The governing equation is separated into those for the superstructure and the base. These are coupled equations that can be solved recursively with the prediction of the superstructure acceleration, for which polynomial extrapolation is employed. The input for the structure is then obtained by solving the coupled equilibrium equations for experimental and analytically modeled systems. The restoring force applied underneath the superstructure is measured and used to update the superstructure acceleration. The stability condition for the proposed method is investigated, revealing a longer range of the convergence criterion than the Newmark explicit method. The implicit feature of the proposed method enables PSD tests irrespective of the modal properties, if the sampling frequency condition is satisfied. Additionally, the amplitude decay and period elongation are investigated for accuracy comparison in the structural responses. A series of PSD tests are performed using a two degree of freedom shear-building model to validate the proposed method. The experiment is performed on one part of the physical structure, while a numerical simulation is performed on the other part of the structure and on the SSI. The test results confirm that the proposed PSD algorithm is reliable for simulating the substructure and SSI effect.

Accounting for spectral shape in simplified fragility analysis of case-study reinforced concrete frames

This paper deals with the selection of optimal intensity measures (IMs) for displacement-based seismic demand assessment and fragility derivation of case-study mid-rise reinforced concrete (RC) frames. The considered frames represent distinct RC vulnerability classes in the Mediterranean region. Optimal IM selection is performed by means of probabilistic seismic demand models considering multiple IMs and various engineering demand parameters (EDPs). Based on findings from previous and concurrent studies, a small subset of potential IMs is investigated here, including conventional peak IMs and two advanced scalar IMs accounting for spectral shape over a range of periods. Probabilistic seismic demand models are built on data obtained from analysis of the case-study frames subjected to over nine hundred ground motions by employing an innovative capacity spectrum method using inelastic response spectra derived from actual earthquake accelerograms to estimate seismic demand and derive fragility curves. This approach has the advantage of simplicity and rapidity over other methods as nonlinear dynamic analysis.This study concludes that advanced IMs, and particularly the ones accounting for period elongation (due to the nonlinear structural behavior) and structure-specific parameters, can effectively satisfy all the selection criteria, including the hazard computability criterion.

Seismic response of RC frames under far-field mainshock and near-fault aftershock sequences

Engineered structures built in seismic-prone areas are affected by aftershocks in addition to mainshocks. Although aftershocks generally are lower in magnitude than that of the mainshocks, some aftershocks may have higher intensities; thus, structures should be able to withstand the effect of strong aftershocks as well. This seismic scenario arises for far-field mainshock along with near-field aftershocks. In this study, four 2D reinforced concrete (RC) frames with different numbers of stories were designed in accordance with the current Iranian seismic design code. As a way to evaluate the seismic response of the case-study RC frames, the inter-story drift ratio (IDR) demand, the residual inter-story drift ratio (RIDR) demand, the Park-Ang damage index, and the period elongation ratio can be useful engineering demand parameters for evaluating their seismic performance under mainshock-aftershock sequences. The frame models were analyzed under a set of far-field mainshock, near-fault aftershocks seismic sequences using nonlinear dynamic time-history analysis to investigate the relationship among IDR, RIDR, Park-Ang damage index and period ratio experienced by the frames. The results indicate that the growth of IDR, RIDR, Park-Ang damage index, and period ratio in high-rise and short structures under near-fault aftershocks were significant. It is evident that engineers should consider the effects of near-fault aftershocks on damaged frames that experience far-field mainshocks as well.

Uni-axial shaking table tests and validations of a seismic isolation system made of spring tube braces

A new seismic isolation system based on spring tube braces was studied. An isolation story, which consisted of pin ended steel columns and spring tube braces, is arranged at the foundation level and/or any intermediate storey of the building. A series of experimental and analytical studies were carried out to evaluate the capability of the proposed seismic isolation system. Three storey ¼ scale 3D steel frames, with the proposed seismic isolation storey, were tested on the shake table using the real earthquake records. The transmitted accelerations from ground to floor levels were largely diminished by means of the isolation system, contrary to the fixed based specimen. Results showed no stability or self-centring problems for the proposed isolation system. The energy balance equations and dynamic force-displacement relations indicated that the equivalent damping arising from the seismic isolation storey was roughly 20%. This relatively high damping property, which can be attributed to the friction on the pinned connections of beam to columns as well as the period elongation, is effective for diminishing seismic response.

A Novel Sub-Stepping Method with Numerical Dissipation Control for Time Integration of Highly Flexible Structures

Sub-stepping time integration methods attempt to march each time step with multiple sub-steps. Generally, for the first sub-step, a single-step method is applied and the following sub-steps are solved using a method that utilizes the data obtained from two or three previous equilibrium points. Despite the robust stability in problems, control of numerical dissipation in sub-stepping schemes is a tough task due to applying different algorithms on a time increment. In order to overcome this insufficiency, a new sub-stepping time integration scheme, which uses two sub-steps in each time increment, is proposed. Newmark and quadratic acceleration methods are applied on the first and second sub-steps, respectively. Both methods utilize constant parameters that enable the control of numerical dissipation in the analysis. For the proposed method, the stability analysis revealed the unconditional stability region for the pertinent parameters. Additionally, the precision investigation disclosed an advantage of the proposed method with the presence of minor period elongation error. Finally, the application of the proposed method is illuminated via several numerical examples. In addition to numerical dissipation control, the proposed method proved to have an outstanding advantage over other methods in solving highly flexible structures more efficiently and more accurately.

The Bathe time integration method with controllable spectral radius: The ρ∞-Bathe method

We consider the Bathe implicit time integration method and focus on the time step splitting ratio and the spectral radius at large time steps to improve and generalize the scheme. The objective is to be able to prescribe the amplitude decay (dissipation) and period elongation (dispersion) for the numerical integration, and to achieve this aim in a direct and optimum manner with the minimum number of parameters. We show that the use of the time step splitting ratio and spectral radius is effective to prescribe in a smooth manner no amplitude decay to very large amplitude decays, with correspondingly small period elongation to very large period elongations while maintaining second-order accuracy. We analyze the effects of the splitting ratio and spectral radius on the stability and accuracy of the scheme and illustrate the use of these parameters in comparison with previously published methods. Furthermore, we show that with a proper setting of these parameters more accurate results may be obtained in some analyses.

Seismic performance enhancement of high-voltage post insulators by a polyurethane spring isolation device

High-voltage (HV) post insulators are vulnerable components of the electrical substations during strong earthquakes. It is rather hard to repair them in service. There are two controversial approaches to increase their seismic safety. They are application of the base stiffening techniques and utilization of the base isolation systems. A new low-cost seismic isolation device that is mounted underneath of HV post insulator is being proposed here to provide period elongation and supplementary damping. The seismic isolator consists of two steel plates, four polyurethane springs and a steel rod. Its cost is insignificant compared with price of HV post insulator. Quasi-static and shake table tests were performed on fixed base and isolated specimens. A set of historical acceleration records selected according to IEEE-693 were utilized in the dynamic tests. The supplementary damping provided by the isolation system was determined between 4 and 8%. The accelerations as well as base shear and base moment of HV post insulator were reduced about 20–70% thanks to the isolation system. The generated numerical models reproduced the experimental results. Consequently, the isolation system is capable to enhance the seismic safety of HV post insulators.

Nonlinear structural dynamic analysis by a stabilized central difference method

In this work, a simple stabilized central difference technique is discussed, to analyse nonlinear models. The proposed technique is unconditionally stable for linear systems and it provides enhanced stability features for nonlinear applications. The proposed method stands as a direct single step procedure, avoiding any iterative computations when solving nonlinear models; thus, it is very efficient. In addition, it is extremely accurate, providing much reduced period elongation and amplitude decay errors, compared to standard methods. The present work also introduces a criterion for updating the nonlinear system matrices, significantly reducing the computational complexity of the simulation and enhancing the efficiency of the technique. Numerical results are presented along the manuscript, illustrating the performance and accuracy of the proposed methodology.

Strongly Stable Multi-time Stepping Method with the Option of Controlling Amplitude Decay in Responses

Recently, multi-time stepping methods have become very popular among scientist due to their high stability in problems with critical conditions. One important shortcoming of these methods backs to their high amount of uncontrolled amplitude decay. This study proposes a new multi-time stepping method in which the time step is split into two sub-steps. The first sub-step is solved using the well-known Newmark method and for the second sub-step an extended version of Newmark method is applied. In fact, similarity in basic formulas of the mentioned methods makes it available to control the amount of amplitude decay in responses obtained by the proposed method; in other words, the amplitude decay in the proposed method is controlled through constant parameters of the two methods applied on each sub-step. The precision assessment of the proposed method is performed using numerical approaches and revealed the minor period elongation error of the proposed method in comparison with other existing methods. In addition to this, the unconditional stability region of constant parameters is also determined through computation of spectral radius of the proposed method. Finally, practical assessment of the proposed method is performed through several numerical examples.

Field evidence of SSI from full-scale structure testing

A full-scale symmetric structure was constructed in the Euroseistest experimental facility in the North of Thessaloniki, Greece, to promote studies of wave propagation and soil-structure interaction in real field conditions. The 3-by-3-by-5 m model structure, called EuroProteas, comprises of a reinforced concrete slab resting on the soil surface, and four steel columns (with or without bracing) supporting one (or two) reinforced concrete slabs acting as the superstructure mass. The reconfigurable stiffness and mass of the structure, in conjunction with the well-known soil profile in Euroseistest, allow for the investigation of soil-structure interaction for different structure-to-soil stiffness ratios. More than 80 accelerometers and seismometers on the structure and on the soil surface, as well as down to a depth of 12 m, were used to record the response of the soil-structure system during a set of field tests under ambient noise, free- and forced-vibration loading. Following a detailed description of the EuroProteas testing facility and the experimental campaign, field evidence on critical SSI issues is reported based on recorded data. Particular emphasis is placed on the period elongation and damping of the SSI system due to the foundation soil compliance by employing well-known data interpretation methods. Softening of the soil-structure system, reflected in the elongation of the SSI period with respect to its fixed-base counterpart, and attenuation of soil response away from the oscillating structure with increasing amplitude of loading, are discussed.

An enhanced explicit technique for the solution of non-Fourier heat transfer problems

In this work, an enhanced explicit technique is proposed to analyze hyperbolic heat conduction models. As usual, the explicit approach allows the solution of the problem to be carried out without dealing with any system of equations, featuring a very efficient methodology. In addition, the proposed technique enables algorithmic dissipation, allowing the influence of spurious high modes to be properly eliminated, without introducing significant period elongation and amplitude decay errors into the analysis. As an explicit approach, the technique is conditionally stable; however, it exhibits high stability limits (its critical time-step is around 1.8 times that of the Central Difference Method), emphasizing its effectiveness. The technique is very accurate, truly self-starting and extremely direct to implement. At the end of the manuscript, numerical results are presented, illustrating the good performance of the discussed technique.

An Explicit Time Integration Scheme Based on B-Spline Interpolation and Its Application in Wave Propagation Analysis

An explicit time integration scheme for hyperbolic equations is proposed using B-spline interpolation and weighted residual method. It has simple formulation and calculation procedure. With one adjustable algorithmic parameter, new scheme has higher accuracy when compared with other excellent explicit schemes. New scheme has controllable and also desirable period elongation which is verified by theoretical analysis and numerical simulations. Especially, a demonstrative dispersion analysis coupled with the corresponding wave propagation demonstrate the desirable numerical dissipation property and the effectiveness of the proposed scheme for wave propagation problems.

Regional subsidence effects on seismic soil-structure interaction in soft clay

Regional subsidence effects on dynamic soil properties and ground layering deformation are often ignored in practice, when dealing with seismic soil-structure interaction analyses. Nevertheless, these effects can substantially change the frequency content and spectral accelerations in both free field and in the soil-structure system. Pore pressure variations over the project economic life are due to both regional subsidence as well as dissipation of excess pore pressure caused by the structure weight. These variations lead to changes in effective stresses, which in turn, modify the dynamic properties such as shear wave velocity distribution and modulus degradation and damping curves, as well as soil layer thickness and shape. These changes can be substantial in highly compressible very soft clay, such as that found in Mexico City valley. This paper presents a numerical study on the seismic response of a conventional five-story building supported by a compensated box foundation built in soft clay, considering these effects. Three-dimensional finite difference models were developed with the software FLAC3D. Initially, the evolution of effective stresses with pore pressure was established based on in-situ piezometer measurements of an instrumented site, and laboratory data. Then, changes in dynamic properties were taken into account based on the results gathered from series of resonant column tests conducted for several effective consolidation stresses, and a PS suspension logging test. The static behavior of the soil-structure system was assessed. For the cases studied herein, the complex interplay between soil nonlinearities, which lead to fundamental period elongation of the soil deposit, Tp, and the overall tendency of ground consolidation to shorten it, controls the variations in the spectral ordinates depending on how close Tp is of the predominant period of the excitation.

A centrifuge-based experimental verification of Soil-Structure Interaction effects

A series of prototype dynamic centrifuge experiments is carried out to investigate the influence of soil properties and structural parameters on the Soil Structure Interaction (SSI) effect. Established analytical models are herein experimentally verified, and are proven accurate in estimating the system's natural frequency characteristics. It is observed that period elongation is strongly correlated to the relative superstructure-foundation stiffness. Although the present study deals exclusively with the small-strain near-linear range, the experimental response indicates occurrence of nonlinearity. The identified damping results remarkably larger than its analytical estimate and proves highly strain-dependent, raising questions on the reliability of existing analytical methods in capturing the actual dissipation mechanisms. An extended experimental dataset is formed under realistic stress and strain soil conditions, and is implemented, for the first time, for verification of existing analytical models offering valuable insight into the theory and serving as a benchmark for engineering practice.

Development of a direct time integration method based on Bezier curve and 5th-order Bernstein basis function

In this study, a robust unconditionally stable method for linear analysis of structures based on Bezier curves and Bernstein polynomials is proposed. The Bezier curve is used as interpolation function and Bernstein basis functions are applied for interpolation. The spectral radius, period elongation and amplitude decay are investigated for stability analysis, numerical dispersion and dissipation of proposed method, and results are compared with other methods that are the best in these properties. It is also shown that the behavior of the proposed method in analysis of finite element system is effective and reliable. To show the robustness and features of proposed method, a challenging problem with a very stiff and flexible response, a Howe truss under impact load, a frame under harmonic loading and a rectangular domain in plane strain condition are considered, and derived results are compared with references solutions and other results reported in the literature.

Structure-dependent improved Wilson-θ method with higher order of accuracy and controllable amplitude decay

• Proposed method yields more accurate responses than Newmark's family of methods. • An important system property is inserted by using α to the acceleration formula. • The method reaches the unconditional stability if θ ≥ 1.38 is adopted. The Wilson-θ method has been proven to have unconditional stability as a numerical direct time integration method in structural dynamics when θ ≥ 1.37 is adopted. Notwithstanding this great advantage of the method, it has also been proven that the method suffers from two shortcomings: high amount of uncontrollable amplitude decay and period elongation. In other words, the unconditional stability allows time step to be stretched, but as the time step grows longer, amplitude decay and period elongation errors grow higher, resulting in a low level of accuracy. The improved version of the Wilson-θ method negates the disadvantages of the classic method by raising the order of acceleration to vary in quadratic scheme over time step domain and by introducing an accelerator coefficient to the acceleration formula in order to control the amount of amplitude decay and lessen the period elongation error. The stability and accuracy of the proposed method has been analyzed, and the results show that unconditional stability is obtained if θ ≥ 1.38 is adopted. A formula is derived for the accelerator coefficient to make it applicable to various types of structural dynamic problems. Numerical examples are presented to provide a practical assessment of the method, along with the classic Wilson-θ and other methods of similar class.

Vector-valued intensity measures for incremental dynamic analysis

In incremental dynamic analysis (IDA), use of an efficient intensity measure (IM) is of vital importance. Efficiency means good explanatory power of the IM in regard to specific engineering demand parameter (EDP) that can help reduce the number of records used to estimate response of structures under given accuracy. According to dimension of the parameters, intensity measures can be classified into scalar-valued IMs and vector-valued IMs. While scalar-valued IMs have been studied systematically, vector-valued IMs have not been investigated comprehensively. Besides, vector-valued IMs considering the effect of higher modes have not been proposed. Hence in order to provide a systematic investigation on vector-valued IMs, this paper proposes five vector-valued IMs with two considering higher mode effect and three incorporating period elongation effect. Then a low-rise and a middle-rise reinforced concrete frames are modeled and analysed by PERFORM-3D and IDA under fifteen suites ground motion records for each structure. To evaluate efficiency of the proposed IMs, residual sum of squares (RSS) and R2 were calculated by logistic regression of IDA data. Results verified the better efficiency of vector-valued IMs than scalar-valued IMs. It is also proved that the relationship between IM and EDP can be expressed as a linear regression of logarithm for both scalar-valued IMs and vector-valued IMs. It turns out that a desirable IM should be selected based on the features of the specific structure. For structures dominated by the first mode, the impact of nonlinearity is of vital importance and should be mainly considered when choosing desirable IMs. Furthermore, for IMs considering nonlinearity effect, efficiency is relevant to the number of spectral acceleration incorporated; for IMs incorporating higher mode effect, the proposed multi-valued IM is more efficient than the two-valued type.

Optimal intensity measure based on spectral acceleration for P-delta vulnerable deteriorating frame structures in the collapse limit state

This study proposes an “optimal” spectral acceleration based intensity measure (IM) to assess the collapse capacity of generic moment frames vulnerable to the P-delta effect. The IM is derived from the geometric mean of the spectral pseudo-acceleration over a certain period interval. The optimized IM includes for first time a flexible lower limit for the period interval, corresponding to the structural period associated with the exceedance of 95% of the total effective mass. This flexible lower limit bound provides an efficient IM, independently of the contribution of higher modes to the total response. The upper bound period is 1.6 times the fundamental period to account for period elongation due to inelastic deformations and gravity loads. In a parametric study on generic frames, structural parameters are varied to quantify the performance of this IM compared to classical benchmark IMs. The “optimal” IM provides minimum, or close to the minimum, dispersion for the entire set of frames with different fundamental periods of vibration, number of stories, and P-delta vulnerability.

Estimating the peak structural response of high-rise structures using spectral value-based intensity measures

Summary With increasing trend towards performance-based design in earthquake engineering, running nonlinear time history analysis is becoming the routing process to quantify the relationship between ground motions intensity measure (IM) and the structural responses. Because a high-rise structure contains many higher modes, a newly proposed spectral value-based IM is presented in this paper to quantify the structural response of high-rise structures. The newly proposed IM uses the modal participation masses to combine higher modes. An actual high-rise structure is taken as an example to demonstrate the efficiency of using the newly proposed IM to quantify the peak structural response of high-rise structures. Five alternative IMs were compared in this study: (a) PGA - peak ground acceleration; (b) S1 - spectra acceleration with only 1 mode; (c) S* - modified S1 with the consideration of period elongation after structure yielded; (d) S12- spectra acceleration with 2 modes; and (e) S123 - spectra acceleration with 3 modes. Linear regression is fitted between the peak structural response and the IM considered. The IM with the highest correlation coefficient to the engineering demand parameter is considered the most efficient IM. The results show that S1 has better correlation to the structural response compared with PGA. S123 has better correlation than S* and S12. It is found that the IM with higher modes can provide better correlation than IM with lower number of structural information. For engineering applications, IM with up to 3 modes (S123) is sufficient to produce an accurate prediction to quantify the structural response of high-rise structures.