A two-step unconditionally stable explicit method with controllable numerical dissipations

A family of unconditionally stable direct integration algorithm with controllable numerical dissipations is proposed. The numerical properties of the new algorithms are controlled by three parameters α, β and γ. By the consistent and stability analysis, the proposed algorithms achieve the second-order accuracy and are unconditionally stable under the condition that α≥-0.5, β≤ 0.5 and γ≥-(1+α)/2. Compared with other unconditionally stable algorithms, such as Chang's algorithms and CR algorithm, the proposed algorithms are found to be superior in terms of the controllable numerical damping ratios. The unconditional stability and numerical damping ratios of the proposed algorithms are examined by three numerical examples. The results demonstrate that the proposed algorithms have a superior performance and can be used expediently in solving linear elastic dynamics problems.     

Nonlinear evaluations of unconditionally stable explicit algorithms

Two explicit integration algorithms with unconditional stability for linear elastic systems have been successfully developed for pseudodynamic testing.Their numerical properties in the solution of a linear elastic system have been well explored and their applications to the pseudodynamic testing of a nonlinear system have been shown to be feasible. However,their numerical properties in the solution of a nonlinear system are not apparent.Therefore,the performance of both algorithms for use in the solution...     

Response to Maxam and Tamma's discussion (EQE-18-0306) to Kolay and Ricles's paper, “Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation”

This paper presents the authors' response to the discussion by Dean J. Maxam and Kumar K. Tamma of the paper titled “Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation.”     

Implementation and application of the unconditionally stable explicit parametrically dissipative KR‐α method for real‐time hybrid simulation

In real‐time hybrid simulations (RTHS) that utilize explicit integration algorithms, the inherent damping in the analytical substructure is generally defined using mass and initial stiffness proportional damping. This type of damping model is known to produce inaccurate results when the structure undergoes significant inelastic deformations. To alleviate the problem, a form of a nonproportional damping model often used in numerical simulations involving implicit integration algorithms can be considered. This type of damping model, however, when used with explicit integration algorithms can require a small time step to achieve the desired accuracy in an RTHS involving a structure with a large number of degrees of freedom. Restrictions on the minimum time step exist in an RTHS that are associated with the computational demand. Integrating the equations of motion for an RTHS with too large of a time step can result in spurious high‐frequency oscillations in the member forces for elements of the structural model that undergo inelastic deformations. The problem is circumvented by introducing the parametrically controllable numerical energy dissipation available in the recently developed unconditionally stable explicit KR‐α method. This paper reviews the formulation of the KR‐α method and presents an efficient implementation for RTHS. Using the method, RTHS of a three‐story 0.6‐scale prototype steel building with nonlinear elastomeric dampers are conducted with a ground motion scaled to the design basis and maximum considered earthquake hazard levels. The results show that controllable numerical energy dissipation can significantly eliminate spurious participation of higher modes and produce exceptional RTHS results. Copyright © 2014 John Wiley & Sons, Ltd.     

Real‐time hybrid testing using the unconditionally stable explicit CR integration algorithm

Real‐time hybrid testing combines experimental testing and numerical simulation, and provides a viable alternative for the dynamic testing of structural systems. An integration algorithm is used in real‐time hybrid testing to compute the structural response based on feedback restoring forces from experimental and analytical substructures. Explicit integration algorithms are usually preferred over implicit algorithms as they do not require iteration and are therefore computationally efficient. The time step size for explicit integration algorithms, which are typically conditionally stable, can be extremely small in order to avoid numerical stability when the number of degree‐of‐freedom of the structure becomes large. This paper presents the implementation and application of a newly developed unconditionally stable explicit integration algorithm for real‐time hybrid testing. The development of the integration algorithm is briefly reviewed. An extrapolation procedure is introduced in the implementation of the algorithm for real‐time testing to ensure the continuous movement of the servo‐hydraulic actuator. The stability of the implemented integration algorithm is investigated using control theory. Real‐time hybrid test results of single‐degree‐of‐freedom and multi‐degree‐of‐freedom structures with a passive elastomeric damper subjected to earthquake ground motion are presented. The explicit integration algorithm is shown to enable the exceptional real‐time hybrid test results to be achieved. Copyright © 2008 John Wiley & Sons, Ltd.     

Discussion of paper ‘Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation’ by Chinmoy Kolay and James M. Ricles,Earthquake Engineering and Structural Dynamics 2014; 43:1361–1380

It seems that the explicit KR‐α method (KRM), which was developed by Kolay and Ricles, is promising for the step‐by‐step integration because it simultaneously integrates unconditional stability, explicit formulation, and numerical dissipation together. It was shown that KRM can inherit the numerical dispersion and energy dissipation properties of the generalized‐α method [1] for a linear elastic system, and it reduces to CR method (CRM), which was developed by Chen and Ricles [2] if ρ_∞ = 1 is adopted, where ρ_∞ is the spectral radius of the amplification matrix of KRM as the product of the natural frequency and the step size tends to infinity. However, two unusual properties were found for KRM and CRM, and they might limit their application to solve either linear elastic or nonlinear systems. One is the lack of capability to capture the structural nonlinearity, and the other is that it is unable to realistically reflect the dynamic loading. Copyright © 2014 John Wiley & Sons, Ltd.     

Retracted: Discussion of paper ‘Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation’ by Chinmoy Kolay and James M. Ricles,Earthquake Engineering and Structural Dynamics 2014; 43:1361–1380

It seems that the explicit KR‐α method (KRM) is promising for the step‐by‐step integration because it simultaneously integrates unconditional stability, explicit formulation, and numerical dissipation together. It was shown that KRM can inherit the numerical dispersion and energy dissipation properties of the generalized‐α method (GM) for a linear elastic system, and it reduces to CR method (CRM) if ρ_∞ = 1is adopted, where ρ_∞ is the spectral radius of the amplification matrix of KRM as the product of the natural frequency and the step size tends to infinity. However, two unusual properties were found for KRM and CRM, and they might limit their application to solve either linear elastic or nonlinear systems. One is the lack of capability to capture the structural nonlinearity, and the other is that it is unable to realistically reflect the dynamic loading. Copyright © 2014 John Wiley & Sons, Ltd.     

A family of explicit algorithms for general pseudodynamic testing

A new family of explicit pseudodynamic algorithms is proposed for general pseudodynamic testing. One particular subfamily seems very promising for use in general pseudodynamic testing since the stability problem for a structure does not need to be considered. This is because this subfamily is unconditionally stable for any instantaneous stiffness softening system, linear elastic system and instantaneous stiffness hardening system that might occur in the pseudodynamic testing of a real structure. In addition, it also offers good accuracy when compared to a general second-order accurate method for both linear elastic and nonlinear systems.     

A new family of generalized‐α time integration algorithms without overshoot for structural dynamics

A new family of generalized‐α (G‐α) algorithm without overshoot is presented by introducing seven free parameters into the single‐step three‐stage formulation for solution of structural dynamic problems. It is proved through finite difference analysis that these algorithms are unconditionally stable, second‐order accurate and numerical dissipation controllable. The comparison of the new G‐α algorithms with the commonly used G‐α algorithms shows that the newly developed algorithms have the advantage of eliminating the overshooting characteristics exhibited by the commonly used algorithms while their excellent property of dissipation is preserved. The numerical simulation results obtained using a single‐degree‐of‐freedom system and a two‐degree‐of‐freedom system to represent the character of typical large systems coincide well with the results of theoretical analyses. Copyright © 2008 John Wiley & Sons, Ltd.     

摩擦消能支撑钢框架结构的弹塑性地震反应时程分析

本文分析了摩擦消能支撑及框架主体结构弹塑性本构关系，并给出了动力时程分析的计算方法。同时，对六层钢框架模型做了各种工况下的地震反应时程分析。结果表明，摩擦消能支撑钢框架（FEDBF）比抗弯钢框架（MRF）的地震作用明显降低，尤其在强震作用下效果更加明显。     

动力反应分析的显式积分方法及其稳定性

Newm ark-更新精细积分法是动力方程求解的隐式的时域逐步积分法,其稳定性条件非常容易满足。与隐式方法相比较,显式积分方法不需要求解耦联的方程组,可以有效地减少内存占用和机时耗费。因此,根据显式积分方法的特点和优点,基于Newm ark-更新精细积分法的基本思想,提出其显式积分格式。对显式积分方法的精度与稳定性进行了初步的分析,指出该显式积分方法具有极好的稳定性,其精度比隐式积分方法的精度稍低。随着时间步长的增加,其精度优于传统的方法。     

Stability analysis of SDOF real‐time hybrid testing systems with explicit integration algorithms and actuator delay

Real‐time hybrid testing is a method that combines experimental substructure(s) representing component(s) of a structure with a numerical model of the remaining part of the structure. These substructures are combined with the integration algorithm for the test and the servo‐hydraulic actuator to form the real‐time hybrid testing system. The inherent dynamics of the servo‐hydraulic actuator used in real‐time hybrid testing will give rise to a time delay, which may result in a degradation of accuracy of the test, and possibly render the system to become unstable. To acquire a better understanding of the stability of a real‐time hybrid test with actuator delay, a stability analysis procedure for single‐degree‐of‐freedom structures is presented that includes both the actuator delay and an explicit integration algorithm. The actuator delay is modeled by a discrete transfer function and combined with a discrete transfer function representing the integration algorithm to form a closed‐loop transfer function for the real‐time hybrid testing system. The stability of the system is investigated by examining the poles of the closed‐loop transfer function. The effect of actuator delay on the stability of a real‐time hybrid test is shown to be dependent on the structural parameters as well as the form of the integration algorithm. The stability analysis results can have a significant difference compared with the solution from the delay differential equation, thereby illustrating the need to include the integration algorithm in the stability analysis of a real‐time hybrid testing system. Copyright © 2007 John Wiley & Sons, Ltd.     

Development of a novel sacrificial-energy dissipation outrigger system for tall buildings

The frame-core tube-outrigger structural system is widely used in tall buildings, in which outriggers coordinate the deformation between the core tube and the moment frame, leading to a larger structural lateral stiffness. Existing studies indicate that outriggers can be designed as “fuses” of tall buildings through dissipating seismic energy after yielding, to protect the main structure. To date, both conventional and buckling-restrained brace (BRB) outriggers have been applied in practice. Subjected to the maximum considered earthquake (MCE), the hardening effect of BRB outriggers increases the damage of other structural components. Meanwhile, conventional outriggers are difficult to repair, owing to the local buckling-induced severe deterioration and damage. To overcome these problems, this study proposes a novel sacrificial-energy dissipation outrigger (SEDO) to improve the seismic resilience of tall buildings. The chords of SEDO are made of high-strength steel and remain elastic. The inclined braces of the SEDO are composed of a sacrificial part and an energy-dissipating part. Therefore, the SEDO remains elastic under design-based earthquakes (DBEs) and dissipates inelastic energy under MCEs. Moreover, the detailing of this novel SEDO is proposed on the basis of experimental studies. The optimal strength ratio between the sacrificial part and the energy-dissipating part is determined in the range of 6:4 to 4:6 on the basis of nonlinear time history analyses (THAs) and parametric studies. Afterwards, the SEDOs are used in an actual tall building to verify their seismic performances through nonlinear THAs. The results indicate the proposed SEDO is able to protect other structural components and effectively improve the seismic resilience of tall buildings.     

Newmark精细积分格式及其在地震反应分析中的应用

在Newmark精细直接积分法的基础上,应用高斯积分与精细指数运算,提出该方法的两种逐步积分格式。文中对两种积分格式的稳定性和精度进行了分析。经过分析比较,第1种逐步积分格式计算精度较高,其稳定性明显地满足算法稳定性分析的条件;而第2种积分格式计算精度相对较差,且是不稳定的。因此本文将第1种积分格式应用于结构的地震反应分析中。算例表明,该逐步积分格式对地震作用有很好的适应性。     

Dynamic analysis method of a combined energy dissipation system and its experimental verification

A combined energy dissipation system is developed in this paper. In this system lead rubber dampers and their parallel connection with oil dampers are used in the braces of a structural frame. A dynamic analysis method of the system, including the modelling of the lead rubber damper and the oil damper, is proposed. In the analysis method, the restoring force characterestics of the lead rubber damper is simulated by the Bouc–Wen hysteretic model, and the behaviour of the oil damper is simulated by a velocity and displacement‐related model in which the contributions of the oil damper to the damping force and stiffness of the system are considered. A series of shaking table tests of a three‐storey steel frame with the combined energy dissipation system are carried out to evaluate the performance of the system and to verify the analysis method. The test and analysis show that the performance of the combined energy dissipation system is quite satisfactory and there is a good agreement between the analysis and test results, which indicates that the analysis method proposed in this paper is valid and suitable for the dynamic analysis of the combined energy dissipation system. Copyright © 2002 John Wiley & Sons, Ltd.     

消能减震结构设计的阻尼比研究

阻尼比是消能减震结构设计的重要参数,对消能减震结构设计具有决定性影响。消能减震结构设汁巾阻尼是时变参数,消能减震结构是非经典阻尼结构。采用复模态设计方法、强解耦振型分解法、基于变形能的等效阻尼法三种方法分别对不同情况的消能减震结构阻尼比进行研究,得到选择消能减震设计疗法和不同方法计算精度的规律,对消能减震结构设计具有参考价值。     

基于位移的RC框架结构与附加消能减震装置一体化设计

消能减震结构中主体结构和附加消能减震装置通常是单独设计的,由于主体结构和附加消能减震装置之间的耦合特性,为实现期望的性能目标需要进行迭代计算。本文提出一种基于位移的主体结构与附加消能减震装置一体化设计方法,首先,基于所选地震动记录和预期性能目标,得出结构阻尼比需求曲面;其次,推导了阻尼器-支撑部件参数计算公式,在已知主体结构基本信息及预期性能目标时,根据阻尼比需求曲面及所提出公式即可获得所需附加消能减震装置的支撑刚度及粘滞系数;再次,引入无量纲成本指数,以成本指数最小为目标函数,在可实现目标性能的附加消能减震装置参数中识别最优设计参数;最后,基于所提出的设计方法,分别以等效SDOF体系、MDOF体系及SAP2000设计的6层RC框架结构为例,对所提出的设计方法的有效性进行了验证,结果表明:经一体化设计的结构体系均可实现性能目标。