Uncertainty Analysis of Identified Damping Ratios in Nonlinear Dynamic Systems

Obtaining accurate estimates of damping parameters from experimental measurements of structures has been a challenge in the civil engineering community. In general, the confidence level on the identified damping is much lower than that of the identified natural frequencies and mode shape components. Most system identification methods are based on linear system and control theories, while in reality most structures behave nonlinearly as the amplitude of the input force they are subjected to increases. When these linear system identification methods are applied to nonlinear structural response, dynamic properties of an equivalent linear system are obtained. This equivalent linear system will have similar response time history and input energy. The un-modeled sources of energy dissipation such as hysteretic/yielding energy will be included in the equivalent linear viscous energy. The first part of this study investigates the effects of different sources of uncertainty such as noise level, order of the linear model to be identified, level of response nonlinearity, and type of input excitation on the identified modal damping based on nonlinear response of two single-degree-of-freedom systems with different nonlinear material models, simulated using OpenSees structural analysis software. In the second part of this paper, equivalent modal parameters of a full-scale seven-story reinforced concrete wall building are identified based on its measured response to four historic seismic base excitations reproduced on the UCSD-NEES shake table. It is observed that in general, with increasing level of structural damage and amplitude of excitation, the equivalent viscous damping ratios increase while the equivalent natural frequencies decrease. Comparison of the identified equivalent viscous damping energy to the hysteretic (yielding) energy computed from a well calibrated finite element model (FEM) provides insight into the un-modeled sources of energy dissipation typically modeled as linear viscous damping in the FEM. It should be emphasized that these “inflated” equivalent viscous damping ratios are not to be used to represent the viscous damping component in a nonlinear FE model of a structure that explicitly accounts for the material inelastic behavior (hysteretic energy dissipation). Introduction Obtaining an accurate estimate of damping in structures has been a challenge in the civil engineering community. A realistic energy dissipation model is essential for accurate structural analysis and correct prediction of the structural response. Different mechanisms contribute to energy dissipation in real structures, including internal damping (mainly due to internal friction at grain boundaries of materials), friction in structural connections, opening/closing microcracks in concrete, and friction between structural and non-structural components [1]. Experiments have shown that the internal damping generally increases with response amplitude and is not frequency dependent [2]. Linear viscous damping is the most commonly used damping model in the numerical models of structure. Although an approximation, this type of damping is mathematically convenient. Linear viscous damping, which is directly proportional to velocity, is frequency dependent, but is not amplitude dependent. There have been several studies to investigate the consequence of using linear viscous damping in the analysis of inelastic structural systems [3-4]. In the Ruaumoko user manual [5], Carr states that “during the past two decades, the author and his research colleagues have been increasingly concerned at the effects that poor choices on the damping model have on the inelastic response of structures.” In this paper, the authors investigate the uncertainty/variability of the linear viscous damping identified based on nonlinear response of structures. Most system identification methods are based on linear system and control theories, while in reality most structures behave nonlinearly as the amplitude of the input force they are subjected to increases. When these linear system identification methods are applied to nonlinear structural response, Proceedings of the IMAC-XXVII February 9-12, 2009 Orlando, Florida USA ©2009 Society for Experimental Mechanics Inc. dynamic properties of an equivalent linear system are obtained. This equivalent linear system will have similar response time history and total energy. The un-modeled sources of energy dissipation such as hysteretic/yielding energy will be included in the equivalent linear viscous energy. The first part of this study investigates the effects of different sources of uncertainty/variability on the identified modal damping based on nonlinear response of two single-degree-of-freedom (SDOF) systems simulated using OpenSees structural analysis software. The two SDOF systems considered in this study are based on different material nonlinear behavior, namely (1) elastic-perfectly plastic (elasto-plastic) model, and (2) Takeda model [6]. The variability of the identified damping ratios for each system is quantified through analysis-of-variance (ANOVA) [7] due to variability of the following input factors: (1) type of input excitation (I), (2) excitation amplitude or level of response nonlinearity (A), (3) order of the linear model to be identified (O), and (4) measurement noise level (N). A full factorial design of experiments is considered for these five input factors. The Deterministic-Stochastic Subspace Identification (DSI), an input-output system identification method, was used in this study for parameter identification. In the second part of this paper, equivalent modal parameters of a full-scale seven-story reinforced concrete wall building are identified based on its measured response to four historic seismic base excitations reproduced on the UCSD-NEES shake table. The relative input energies calculated based on measured and identified model simulated response are then compared. Single-Degree-of-Freedom (SDOF) Models The dynamic response of an inelastic SDOF system to ground motions is influenced by the hysteretic forcedeformation behavior assigned to the nonlinear spring. In this study two hysteretic force-deformation relations are considered: the elastic-perfectly plastic model, and the Takeda hysteretic model [6]. These two systems are modeled in OpenSees structural analysis software [8] by defining a corresponding uniaxial material to the crosssection of a nonlinear truss element of unit cross-section area and unit length. In OpenSees, the elasto-plastic material is readily available, while the Takeda model is implemented by adapting an existing general hysteretic material model. An initial fundamental period of Tn = 0.5s is assumed (i.e., the initial natural frequency of both SDOF systems is 2Hz). The Newmark’s average acceleration method is used for time integration with a time step of 0.01s. It is worth noting that the Newmark’s average acceleration method has no numerical damping [1]. For both SDOF models a modal damping ratio of 5% is assumed in the model. Figure 1(a) illustrates the hysteretic force-deformation behavior of the elasto-plastic model. In this model, the loading and unloading stiffness k0 remains unchanged as long as the force does not exceed the yield strength Fy. Once the force reaches the yielding strength, the system stiffness drops to zero. In this study, a yield strength reduction factor of Ry = 4 is considered for the Imperial Valley, 1940 earthquake ground motion recorded at the El Centro station. This implies that the yield strength is one quarter of the minimum strength required for the system to remain elastic during this base excitation. The hysteretic force-deformation behavior of structural systems and components may exhibit stiffness degradation, strength deterioration, and/or pinching, which reduce the area of the hysteresis loops, and therefore, the elasto-plastic model tends to overestimate the amount of energy dissipated in most structural systems. An example of a stiffness degrading hysteretic model is the Takeda model. Figure 1(b) shows the hysteretic forcedeformation behavior of the Takeda model used for the second SDOF system. The Takeda model has been widely used to represent the behavior of reinforced concrete structures. There are three parameters that define the form of the hysteretic loops of this model: the post yielding stiffness ratio (r), the unloading stiffness parameter (), and the reloading stiffness parameter (). The adopted values are r = 0.05,  = 0.5 and  = 0.0 which are representative of reinforced concrete walls or columns [9]. During cyclic loading, the unloading and reloading branches of this hysteretic model are established by Equation (1): y u o m d k k d         (1) where ku is the unloading stiffness of the material, k0 is the initial stiffness, dm is the maximum deformation attained in the direction of the loading during the previous loading cycles, dy is the deformation of the member at yielding, and  is a parameter that determines the degraded unloading stiffness.