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4 Inelastic structural models cannot represent all the energy dissipation mechanisms involved in the measured damping. It is mentioned in [2] section that most of the structural damping [should] be modeled directly in the analysis through hysteretic response of the structural components. And then, depending on the type and characteristics of the nonlinear model, additional viscous damping may be used to simulate the portion of energy dissipation arising from both structural and nonstructural components (e.g. cladding, partitions) that is not otherwise incorporated in the model. Concerning now the choice of the critical damping ratios to be considered for adding viscous damping, the general principle to be appealed for is expressed in the following terms in [2] section 6.4.4: When used, viscous damping should be consistent with the inherent damping in the structure that is not already captured by the nonlinear hysteretic response that is directly simulated in the model. It is stated in [4] that many of the currently available guidelines on damping are intended for use with elastic dynamic analysis. Nevertheless, the value of 5% of critical damping is presented as common practice for nonlinear studies of low to midrise buildings. In [5], the dynamic behavior of steel-concrete composite structures is modeled and the authors conclude that the two critical damping ratios of 1% and 5% can be viewed as reasonable lower and upper bounds when energy dissipation due to material hysteretic behavior is already modeled explicitly. In works developed with the finite element computer program Vector [6], it is advocated not to add damping and thus to use the inelastic structural model as the sole source of seismic energy dissipation. 2.3 Adding viscous damping The inelastic earthquake response of RC structures is computed according to the following discrete equations of motion: MÜ(t)+C(t) U(t)+R(t) = F sta M Üg(t) (1) where M, C(t) are the mass and damping matrices and R(t) is the nonlinear restoring forces vector; F sta is the applied static forces, and M Üg(t) represents the applied seismic forces with the matrix indicating the active dynamic degrees of freedom for directional mass and Ü g (t) the applied ground acceleration. There are several methods for introducing added viscous damping (C(t) U(t)) in numerical simulations: the so-called Rayleigh s or Caughey s methods or modal damping. We only present here Rayleigh s methods that will be used for the numerical simulations presented in the next section: C [i] (t) = (a M M+b K K 0 ) [1], (a M M+b K K tan (t)) [2] or (a M (t)m+b K (t)k tan (t)) [3] (2) wherek 0 andk tan (t) are the initial and tangent stiffness matrices. The coefficientsa M andb K determine the critical damping ratio for two chosen structural frequencies. Although all of these methods have clear significance and computational benefits in elastic simulations, they might be inappropriate in inelastic analyses and thus have to be handled with care. Several researchers presented seismic analyses of nonlinear structural systems using different damping models from Eqs. (2) [7, 8, 9, 10]. It was concluded that C [3] (t) was the best model to maintain a constant value of added viscous damping throughout the analysis and avoid spurious internal damping forces. However, the update of parameters a M (t) and b K (t) with the progress of the solution is not practical from a computational aspect and damping models C [1] or C [2] (t) are generally preferred. The coefficients a M and b K are most often computed from 4

5 the initial dynamic properties of the structures, but pushover analysis (or other approximations) could also be used to predict the secant stiffness at the target ductility to estimate an elongated natural period of vibration in the computation ofa M andb K. The purpose of the following section is to investigate how far such commonly used added viscous Rayleigh s damping models can be considered as consistent with the hysteretic damping coming from the inelastic structural model. Then, to a posteriori quantify this consistency, an indicator based on energy quantities is suggested. 3 ENERGY AS AN INDICATOR OF CONSISTENCY FOR ADDED DAMPING In this section, numerical inelastic seismic analyses of a RC moment-resisting frame tested on a shaking table are compared. Different computer programs available to practitioners and researchers are used to combine different RC inelastic and added viscous damping models, and to compare their relative amount of energy dissipated during the seismic time-history. 3.1 Description of the RC frame structure and ground motions Figure 1: RC frame tested on a shaking table. Fig. 1 shows the geometry and reinforcing steel of the half scale RC moment-resisting frame that was tested on a shaking table some years ago by Filiatrault et al. [3] at École Polytechnique of Montreal. The structure was designed according to the Canadian National Building Codes CNBC 1995 for a global displacement ductility demand R = 2. A detailed description of the frame is given in [3]. The two bays, two stories frame is 5 m wide, 3 m high with rectangular cross-sections of 15x16 cm and 14x15 cm for the 1 st and 2 nd floor beams, 17x13 cm and 18x13 cm for the external and internal columns. Resulting properties of RC are as follows: concrete Young s modulus 25,200 MPa, compressive strength 31 MPa and Poisson s ratio 0.17; the longitudinal steel Young s modulus 224,600 MPa with a yield strength 438 MPa and the 5

6 ultimate strength 601 MPa. Four inverted U shape concrete blocks attached in each span of the beams were used to simulate concentrated gravity loads from framing joints. The centers of gravity of the added masses were computed such that they coincide with the center of gravity of the beams. Service cracks were induced by the added masses. The total weight of the frame was 95 kn. The fundamental period was measured at 0.36 s in a free-vibration test with a first modal damping ratio of 3.3%. Mode 1 is preponderant. The ground motion record that was selected for the test program corresponds to the N04W component of the accelerogram recorded in Olympia, Washington (April 13, 1949). Fig. 2 presents the feedback record measured on the shaking table during the test initially scaled to a peak ground acceleration PGA = 0.21 g, as well as the corresponding response spectrum. Figure 2: Seismic input motion: (a) shaking table acceleration, (b) response spectrum (5% critical damping ratio). The frame structure was designed according to the CNBC The structural response observed during the shaking table test, expressed in terms of displacements, shear forces and plastic hinges positions, is in accordance with the building code. 3.2 Structural models It is well known that all the inelastic analyses do not provide identical results and it is not realistic to aim at modeling all the energy dissipative phenomena that can occur during seismic time-history. However, as mentioned in [11], there are phenomena that affect the behavior at or near collapse and that cannot be detected and evaluated by means of conventional elastic analysis techniques ; among others, the author focuses on i) structure P-delta effects, ii) deterioration in strength and stiffness and, for moment-resisting frame structures, iii) the capacity design strong column weak girder concept for which excessive plastic hinging in columns should be avoided. The inelastic structural models used thus have to be at least capable of reproducing these key issues. The model developed with the computer program Ruaumoko [12] (Fig. 3a) is based on beamcolumn elements with plastic hinges lumped at the member ends. The Q-HYST degrading rule version of the modified Takeda model (Fig. 3b) was used to represent the inelastic momentrotation behavior in plastic hinges. The backbone of the hysteretic curve was obtained from a plane section analysis program. At each node the connection was modeled using rigid end beam and column offsets (infinitely stiff zones whose length is equal to the depth of the adjacent 6

7 beam or column) and plastic hinges whose lengths are equal to half the height inside the stirrups. Ruaumoko allows using the damping modelsc [1] orc [2] (t) defined in Eqs. 2. Figure 3: Ruaumoko lumped plasticity model: (a) geometry; (b) modified Takeda hysteretic model. We also used the computer program Perform3D [13] allowing for the discretisation of the various frame cross-sections into fibers. A division of the sections into 6 concrete and steel layers for the columns and 8 layers for the beams was used. A 1D behavior law was associated to each material; Fig. 4a shows the stress-strain behavior of the concrete material models. The mesh is the same as in Fig. 3a. For the structural model referred to as Perform3D 1, unconfined concrete material was assigned to each concrete fiber and 5 numerical integration points (NIPs) per element were considered. For the structural model Perform3D 2, the concrete material was defined as confined concrete inside the stirrup and as unconfined outside (Fig. 4b), and only 2 NIPs per element were considered. Connections at each node were modeled by using rigid end beam and column offsets, that is, for Perform3D program, zones whose length is equal to the depth of the adjacent beam or column and whose stiffness is 10 times those of the element. In Perform3D, Rayleigh s damping models are not exactly defined as in Eqs. 2. We consequently define bothc [F1] as a Rayleigh s damping model computed according to the initial stiffness, as for model C [1], and C [F2] as a Rayleigh s damping model computed according to the reduced stiffness associated to a global displacement ductilityr = 2, in the spirit of modelc [2]. To this, the methodology is described in [13]. Figure 4: Perform3D fiber element model: (a) concrete model, (b) confined and unconfined fibers. Stiffness degradation is modeled and there is no strength in tension. Degradations, and thus energy dissipation phenomena, were observed in the beam-column joints during the test. For more detailed structural models, rigid end zones should thus be replaced by inelastic connections. 7

8 3.3 Results analysis The total weight of the structurep and the fundamental elastic periodt ela are first indicated in Tab. 1. A posteriori, with the experimental results at hand, the values obtained fort ela can be validated as follows. Because of the dead load, there is significant initial damage in the structure and the experimental fundamental period that is initially measured (T ini,exp F V = 0.36 s) is longer than the elastic one. The elastic dynamic properties of the mock-up are thus not known. For Perform3D, we evaluate the initial fundamental period by weakly exciting the structure after the dead load has been applied, letting it return to rest in free vibrations and measuringtfv ini (mode 1 is predominant). Then, the maximum top-displacement d max is indicated. The quantities E T, E D and E H correspond to the total seismic energy dissipated by the numerical model, the energy dissipated by the added viscous damping model and by the inelastic structural model. Finally, the fundamental period T fin FV is computed when the structure returns to rest at the end of the seismic motion. Table 1: Earthquake response analyses of a RC moment-resisting frame using different inelastic structural and added viscous damping models (excerpt of the results). Our purpose here is to show that it is somehow difficult to a priori assess that an added damping model is consistent with the inelastic structural model. Indeed, even for Rayleigh s damping models defined with small critical damping ratios, E D can become very larger than E H. This sounds not acceptable for most of the performance-based analyses. One can hope that this non-realistic behavior would be avoided with a good inelastic structural model. However, there is a priori no reason not to check whether added damping is consistently modeled or not and the computation of the amounts of energy dissipated all along the seismic time-history seems to provide a pertinent indicator for this. 8

9 The following tendencies can be observed from Tab. 1: The E H/D /E T ratio is affected by the choice of a structural model. 1) For the lumped plasticity models (Ruaumoko), hysteretic energy is preponderant, which is no more the case for the Perform3D fiber element models; this is illustrated in Fig. 5. 2) For Perform3D models, the assumptions made on the concrete behavior law also significantly affect the energy ratios. For a given computer program, the larger the E H /E T ratio is, the larger and closer to the experimental response is the maximum top-displacement. This observation supports some researchers opinion, who advocate not to introduce added damping in inelastic simulations [6]. For both Ruaumoko and Perform3D models, the energy repartition is not significantly affected by the choice of a Rayleigh s damping model of the C [(F)1] or C [(F)2] kind. Defining added viscous damping according to a reduced stiffness matrix with respect to the initial one nevertheless leads to a slight reduction of thee D /E T ratio. For both Ruaumoko and Perform3D models, the energy ratios are sensitive to ξ 1,2, the critical damping ratio attributed to modes 1 and 2. For all Ruaumoko models, the final fundamental structural frequency is roughly T fin FV = 0.45 s. This indicates that the global structural stiffness degradation is independent from the E H /E T ratio. On the contrary, for Perform3D models, there is a clear correlation between the amount of energy dissipated by the model and the degradation of the global stiffness. Considering Perform3D models only, the best approximation is obtained when there is almost no added damping. Figure 5: Energy dissipation sources: (a) RuaumokoC [1] -3.3%, (b) Perform3D 2C [F2] -1.5%. Besides, computing energy quantities allows to compare the relative importance of the inelastic mechanisms accounted for in the structural model. For instance, for detailed material constitutive models developed in the theoretical frameworks of continuum mechanics and thermodynamics with interval variables, it is often difficult, in the context of seismic inelastic timehistory analysis, to assess whether enriching the model with new local mechanism is meaningful or not. However, for this kind of models, it is often straightforward to compute the energy dissipated by each local phenomenon such as plasticity or damage [14], and it is thus possible to quantify the relative importance of each local phenomena. 9

10 4 CONCLUSIONS In the particular context of RC moment-resisting frame structure in seismic loading, and without considering any interaction between the structure and its surrounding environment: Facts. The seismic energy imparted to a structure is dissipated consequently to inelastic mechanisms, that are irreversible modifications, in the structure. Issue. Performance-based methods ideally require that the irreversible modifications in the structure are predicted. One of the main issues related to performance-based engineering thus is: how to model the energy dissipation mechanisms in the structure? Methodological aspects. The need for modeling internal energy dissipation provides methodological indications for the development of inelastic analyses. Energy dissipation in a structural numerical model should be mostly hysteretic to predict the internal modifications. However, it is not always satisfied for inelastic numerical simulations, even when added viscous damping is associated to a small critical damping ratio. Therefore, developments should be oriented towards both increasing the capacity of inelastic structural models to reproduce the physical mechanisms that dissipate the imparted energy and reducing the portion of added damping. Results interpretation. Results of inelastic time-history analyses can be better interpreted according to energetic quantities, which indicate whether the energy dissipation is, as expected, mostly hysteretic or not. In the case most of the energy is dissipated by added damping, it can be inferred that the numerical simulation failed to predict the internal modifications in the structure (strength and stiffness loss, drift, etc.). Elastic simulations are an extreme case: all the seismic energy is dissipated by added damping and, consequently, the structure after the earthquake is the same as before. Computing the portion of the total energy dissipated by a particular nonlinear mechanism during the seismic time-history would also allow to decide whether it is worth modeling this mechanism or not. For these reasons, expressed in the context of RC frame structures in seismic loading, and without considering any interaction between the structure and its surrounding environment, modeling energy dissipation is a paradigm [15] for performance-based engineering. REFERENCES [1] M.J.N. Priestley, G.M. Calvi, M.J. Kowalsky, Displacement-based seismic design of structures. IUSS Press, Pavia, Italy, [2] FEMA P695, Quantification of Building Seismic Performance Factors. FEMA, Washington DC, June [3] A. Filiatrault, E. Lachapelle, P. Lamontagne, Seismic performance of ductile and normally ductile concrete moment resisting frames - Part I: Experimental study. Canadian Journal of Civil Engineering, 25, , [4] PEER-2010/111 or PEER/ATC-72-1, Modeling and Acceptance Criteria for Seismic Design and Analysis of Tall Buildings. PEER, Richmond (CA), October

11 [5] A. Zona, M. Barbato, J.P. Conte, Nonlinear Seismic Response Analysis of Steel Concrete Composite Frames. ASCE Journal of Structural Engineering, 134(6), , [6] S. Saatci, Behavior and modeling of reinforced concrete structures subjected to impact loads. PhD Thesis, University of Toronto, Department of Civil Engineering, Canada, http : // [7] J. Wang, Intrinsic damping: Modeling techniques for engineering systems. ASCE Journal of Structural Engineering, 135(3), , [8] F.A. Charney, Unintended consequences of modeling damping in structures. ASCE Journal of Structural Engineering, 134(4), , [9] J.F. Hall, Problems encountered from the use (or misuse) of Rayleigh damping. Earthquake Engineering and Structural Dynamics, 35, , [10] P. Léger, S. Dussault, Seismic-Energy Dissipation in MDOF Structures. ASCE Journal of Structural Engineering, 118(6), , [11] H. Krawinkler, Importance of good nonlinear analysis. The Structural Design of Tall and Special Buildings, 15, , [12] A.J. Carr, Ruaumoko manual, theory and user s guide to associated programs, University of Cantabury, Christhurch, New Zealand, [13] CSI, Perform3D user s manual, California, [14] P. Jehel P, A. Ibrahimbegovic, P. Léger, L. Davenne, Towards robust viscoelastic-plasticdamage material model with different hardenings / softenings capable of representing salient phenomena in seismic loading applications. Computers and Concrete, 7(4), , [15] T.S. Kuhn, The Structure of Scientific Revolutions, 2 nd edition. In International Encyclopedia of Unified Science. The University of Chicago Press, Chicago,

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