Bispec_Paper_Structures04 - BISPEC: Interactive Software...

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BISPEC: Interactive Software for the Computation of Unidirectional and Bidirectional Nonlinear Earthquake Spectra Mahmoud M. Hachem 1 , PhD, PE 1 Wiss, Janney, Elstner Associates, Inc., 2200 Powell Street, Suite 925, Emeryville, CA 94608; PH (510) 428-2907; FAX (510) 428-0456; email: Introduction The concept of an earthquake design spectrum is an integral part of design codes in the United States and around the world. Such design spectra are idealizations of actual earthquake spectra. The understanding of earthquake spectra and their relationship to the response of a single degree of freedom system is essential to the understanding of the design and performance of structures under earthquakes of any magnitude. As opposed to linear spectra, nonlinear spectra represent the response of structures that undergo yielding and damage during a seismic event, and are hence more representative of actual structural performance under a moderate or significant earthquake event. The computation of nonlinear spectra is complex and involves a significant amount of computation and synthesis of results. Bispec is an interactive computer program that facilitates the process of computing various types of linear and nonlinear spectra. By applying common dynamic analysis algorithms for the computation of the response of one or two degree of freedom systems, and providing access to complex results through a simple user interface, Bispec can be used to analyze simple systems, and to gain a deeper understanding of earthquake ground motions and their effect on structures of various shapes and sizes. Dynamics of a Simple Pendulum System Consider a simple pendulum system as shown in Figure 1, which is commonly referred to as a single degree of freedom system (sdof). The general differential equation representing the response of a sdof to ground acceleration is the following: ) ( )) ( ( ) ( ) ( t u m t u f t u c t u m g s & & & & & = + + (Equation-1) where: m : mass of the sdof system m k u u : relative deformation of sdof with respect to the ground u & : relative velocity with respect to the ground u & & : relative acceleration with respect to the ground c : coefficient of viscous damping f s : resisting force, which is a function of u g u & & : ground acceleration Figure 1:sdof system 2004 Structures Congress
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For a linear system, f s is equal to ku where k is the stiffness of the sdof. Additionally, the undamped natural frequency of such a system can be computed as: T m k π ω 2 = = This allows the reduction of Equation 1 to the following: (Equation-2) g u u u u & & & & & = + + 2 2 ζ where ζ is the ratio of the damping coefficient c to the critical damping coefficient km c cr 2 = . Linear Response. A structure that responds linearly possesses a constant stiffness regardless of the level of force it is subjected to. Such a structure can be thought of as infinitely strong. For linear systems, magnifying the excitation results in a response that is magnified by the same amount i.e. that scales linearly with the excitation. Despite the fact that many structures are
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This note was uploaded on 05/09/2008 for the course CE 227 taught by Professor Mahin during the Spring '06 term at University of California, Berkeley.

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Bispec_Paper_Structures04 - BISPEC: Interactive Software...

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