Simple Method for Pushover Curves of Asymmetric Structure with Displacement-Dependent Passive Energy Dissipation Devices

This paper presents a simple method to calculate pushover curves for asymmetric structure with displacement-dependent passive energy dissipation devices (DDPEDDs). The method analyzes the deformation of a symmetric structure in translation and in torsion. These results are then combined in order to calculate the pushover curve for an asymmetric structure with DDPEDDs. The numerical results obtained by using the simple analysis method are then compared to the results obtained from the analysis of the models using the software SAP2000. The results show that the simple analysis method can be an effective tool for engineering analysis. Introduction The technique of passive energy dissipation has been rapidly growing and it is now widely used in many parts of the world. This technique reduces the structure response subjected to wind and earthquake through mounting energy dissipation devices into the buildings (Soong 1997). Contrary to semi-active and active systems, there is no need for an external supply of power. In recent years, variable kinds of dampers have been studied by several researchers and serious efforts have been undertaken to develop the concept of energy dissipation or supplemental damping into a workable technology (Li 2003). A number of these devices, such as passive metallic yielding, visco-elastic, and viscous energy dissipation devices have been installed in structures throughout the world (Soong 1997). The primary objective of adding energy dissipation systems to building frames has been to focus the energy dissipation during an earthquake into disposable elements specifically designed for this purpose, and to substantially reduce energy dissipation in the gravity-load-resisting frame. Moreover, since energy dissipation devices are not part of the gravity-load-resisting frame they can easily be replaced after an earthquake without compromising the structural integrity of the frame. Two different types of passive energy dissipation systems include the displacement dependent damper and the velocity dependent damper (Code 2001). Due to the material presented here, this paper will focus on displacement dependent dampers, i.e. metallic and friction dampers. The pushover analysis method, as a foundation for performance-based design has attracted the interest of many researchers. Regarding the pushover analysis procedure of regular structures, there are two widely used methods, i.e. FEMA273 (Federal 1996) and ATC-40 (ATC 1996). These methods are used by most engineers as a standard tool for estimating seismic demands for buildings, because they analyze the progress of the pushover method numerically. The non-linear static pushover analysis is a simple option for estimation the strength capacity in the post-elastic range. This procedure involves applying a predefined lateral load pattern that is distributed along the building height. These methods are considered to be better than other methods such as using linear analysis and ductility-modified response spectra, because they are based on a more accurate estimate of the distributed yielding within a structure, rather than an assumed, uniform ductility. The generation of the pushover curve also provides the engineer with a good feel for the non-linear behavior of the structure under lateral load. However, it is important to remember that pushover methods have no rigorous theoretical basis, and may be inaccurate if the assumed load distribution is incorrect. For example, the use of a load pattern based on the fundamental mode shape may be inaccurate if higher modes are significant, and the use of any fixed load pattern may be unrealistic if yielding is not uniformly distributed, so that the stiffness profile changes as the structure yields. In order to extend the pushover analysis into more areas of application, various researchers have studied the pushover method for asymmetric structures. Kilar and Fajfar (1997) present a simple method for the non-linear static analysis of complex buildings subjected to monotonically increasing horizontal loading. Their method is designed to be a part of new methodologies for the seismic design and evaluation of structures. It is based on the extension of a pseudo-three-dimensional mathematical model of a building structure into the non-linear range. The structure consists of planar macro elements. For each planar macro element, a simple bilinear or multi linear base shear-top displacement relationship is assumed. Furthermore, Chopra and Goel (2004) developed a modal pushover analysis based on the structural dynamics theory and extended this method to asymmetric–plan buildings. In this method, the seismic demand due to individual terms in the modal expansion of the effective earthquake forces is determined by non-linear static analysis using the inertia force distribution for each mode, which for asymmetric buildings includes two lateral forces and torque at each floor level. At last, by the CQC rule to obtain an estimate of the total seismic demand for structure Barros and Almeida (2005) simplify the process for the static non-linear dynamic response of a structure, utilizing a load pattern proportional to the shape of the fundamental mode of vibration of the structure. In this paper, a simple method to calculate pushover curves for asymmetric structure with DDPEDDs is presented. Basic Theory Asymmetric structures are characterized for having their center of stiffness and center of mass located in different positions. Due to this characteristic, it is rather complex to analyze the effect of lateral forces on such structures because the structure will react both in torsion and translation to such forces. In this paper, a simple method for calculating the effect of lateral loads on an asymmetric structure with DDPEDDs is presented. Such technique relies on decomposing the deformation of an asymmetric structure into translation and torsion on symmetric structures and then combining the results. Related basic theory is introduced as follows. Structural Deformation In order to analyze the deformation of an asymmetric structure (3-D) under lateral loading, two reactions, translation and rotation for a symmetric structure (2-D), need to be taken into consideration. Torsion effects have a considerable impact on the response of a structure reacting to lateral loading as shown in Fig 1. Furthermore, this shows the importance of taking rotational effects on asymmetric structures into consideration for engineering analysis. Fig 1 shows the difference on the pushover curve after taking torsion effects into consideration The deformation imposed on an asymmetric structure due to lateral loading can be calculated by the following method. First, calculate the translational effect of the lateral loading on a planar symmetric structure. Second, calculate rotational effects on the planar symmetric structure by applying torques on the structure. Finally, combine both effects as shown in Fig 2 to obtain the effect that the loading would have on an asymmetric structure. (a) Combined effect of asymmetric (b) Translation of symmetric (c) Torsion of symmetric Fig 2 simple method for estimating the effect of an earthquake on an asymmetric structure Dynamic Analysis of Asymmetric Structure Equations of asymmetric structure with DDPEDDs: The equation of motion for the multiple degrees of freedom, MDOF, structures with passive energy dissipative devices can be written as [ ]{ } [ ]{ } [ ]{ } { } [ ][ ]{ } g U R M F U K U C U M & & & & & − = + + + (1) where 、 and represent the mass, damping, and stiffness matrices respectively, while { symbolizes a matrix of force offered by DDPEDDs. Furthermore, the vectors ] [M ] [C ] [K } F { } U& & ,{ } U& and { } U represent the acceleration, velocity and displacement respectively. Moreover, these parameters are defined as vector quantities i.e. { } { } θ , ,v u U = ,where , , u v θ represent the magnitude of the displacement in the different directions. In addition ( ) g U& & represents the acceleration contributed by the earthquake, and represents a matrix of units. Finally, ] [R { } { } g g g g v u U θ& & & & & & & & , , = represents the acceleration contributed by the earthquake. The matrices for the different parameters are shown below