Identification methods of material‐based damping for cracked reinforced concrete beam models

Seismic performance evaluation in structural design requires the use of sophisticated numerical models. In particular, to accurately represent the non‐linear behaviour of reinforced concrete (RC) structures when subjected to dynamic loadings, the energy dissipation mechanisms must be accurately represented. However, the classical viscous damping models, which are still widely used, are not based on physical considerations at the material level and the choice of damping parameters is often arbitrary. This paper, thus, proposes a time‐domain damping identification method based on equivalent single‐degree‐of‐freedom (SDOF) systems. The methodology is developed using either an updated linear model or a non‐linear energy‐dissipating constitutive model. Energy dissipative phenomena are cracking, friction and unilateral effects upon crack closure. Both models allow the identification of different damping transient variations: (i) With the updated linear model, intrinsic damping ratios and frequencies are identified to define a simple generic damping model, and (ii) with the non‐linear constitutive model, the identified viscous damping ratios represent the dissipative phenomena not described by the material model. The aim is to propose simple models that can be used by anyone to complement their own models. Applying the method to experimental data allows evaluating effective damping ratio transient variations as functions of variables representative of non‐linear behaviour. It is shown that it is possible to accurately model the energy dissipation that is missing in the non‐linear dynamic constitutive models through effective viscous damping models based on dissipative phenomena internal variables.


Context
The consideration of seismic performance evaluation in structural design leads to the requirement of sophisticated numerical models. In particular, the energy dissipation mechanisms must be accurately represented to adequately capture non-linear structural behaviour. The energy dissipation mechanisms are generally modelled through equivalent viscous damping. However, damping remains misunderstood and is still actively studied in the literature. 1,2 The most classically used equivalent viscous damping models are of the Rayleigh-type (Equation 1) because they can be easily implemented in finite element (FE) codes.
where ℂ, and are the damping, mass and stiffness matrices, respectively, and 0 and 1 are proportionality coefficients. Because the mass and stiffness matrices can be computed from the structural geometry and material properties, the evaluation of the damping matrix is simple on the condition that the coefficients 0 and 1 are known. To evaluate those parameters, two damping ratios ( and ) must be selected for two frequencies ( and ). The values 0 and 1 are prescribed according to frequencies and damping identified for a linear behaviour of the structure. However, in the non-linear behaviour of the structure, the evolution of the energy dissipations is quite arbitrary. In a previous paper, 3 different damping models used in seismic engineering were compared. The analyses were performed on dynamic responses compared to experimental data and energy dissipation mechanisms. Mainly, the tangent stiffness use is to improve the numerical results. However, the major part of energy stays dissipated by the viscous damping, characterizing the unknown dissipative phenomena. Another major conclusion is that the selection of an accurate damping model is strongly dependent on the input signal, the severity of non-linearities or the response quantity of interest. That is why research focuses on proposing identification methods to evaluate those damping ratios based on experimental data 4 or numerical analyses, 5 depending on (i) the type of structure, (ii) the material properties and (iii) the input signal. In addition, many papers also propose damping models as functions of different material or structural modelling parameters: ductility, 6 crack opening 7 or damage indices. 8 Another domain in which damping identification is practical is damage detection and structural health monitoring. The modal properties of a structure evolve with respect to the structural damage state (stiffness reduction). Many studies exist regarding the identification of frequencies or mode shapes. However, the proper identification of damping is still a matter of debate. 9-13

Equivalent viscous damping models
Daneshjoo and Gharighoran 14 explained that the notion of damping is challenging because the associated energy dissipation mechanisms are still not well understood. That is why no generalised mathematical formulae can be developed. To describe the whole dissipation and its evolution with respect to the state of the structure, the concept of equivalent viscous damping has been introduced in Jacobsen. 15 This damping represents the effects of elastic and hysteretic damping 16 and can be decomposed into two parts: where is the damping that develops in the linear range of the studied structure and is the damping that is introduced to model the additional dissipation that appears during the structural non-linear response. First, depends mainly on material and geometrical characteristics as well as boundary conditions: (e.g., Salzmann 17 ). Then, depends mainly on internal state variables: . To evaluate , Jacobsen 15 defined the notions of dissipated and stored energies from where is the dissipated energy, is the stored energy and ℎ − is the dissipated energy for one half-cycle. and represent the maximal force and displacement of the studied half-cycle, respectively. Nevertheless, one challenge here is defining and identifying the state variables associated with dissipation evolutions. For plasticity, one may use the ductility = ( is the maximum displacement and is the displacement in tensile rebar at first yield) or the permanent displacement, , at zero force. For damage or cracking, one may use the degradation (or damage) variable or the crack opening, . Finally, the amplitude of the response, , may be considered. A general formulation for the equivalent viscous damping can then be written as ( , , , , , ) = ( ) + ( , , , , , ) (4) with additional variables, with respect to those introduced, that can be measured or obtained by postprocessing and/or calibration. For example, Rodrigues et al. 6 synthesized numerous equivalent viscous damping models based on ductility 18-23 by studying uniaxial and biaxial load-displacement responses in columns. Then, Salzmann 17 focused on the residual deflection, Δ , and the reinforcement properties. Expressions of the undamaged logarithmic decrement, , for reinforced concrete (RC) and prestressed beams were proposed depending on the ratios of tensile and compressive reinforcement bars and on the amount of prestressed force and its eccentricity. The residual deflection was also considered in the 14 model. The crack opening was also of interest in Chowdhury et al., 7 who proposed evaluating the logarithmic decrement of RC beams and for partially pre-stressed beams using the residual crack width , a function of the instantaneous average crack width. Finally, damage indices were defined in Heitz et al., 8 where a damping model depending on two engineering parameters (the mid-span curvature and a damage index) was proposed, as discussed later in Section 6.1. To develop the damping model, a numerical analysis was performed with an RC non-linear constitutive model identified on experimental data. Another numerical strategy was proposed in Crambuer 24 to evaluate the best updated damping model to dissipate the right amount of energy without requiring a complex constitutive model. Three models were studied: depending only on the damage (crack opening), (ii) ℎ = .Γ depending on the damage (crack opening) and crack surface erosion and (iii) ℎ = .Γ , which is accurate in the case of RC columns and depends on the damage and loading intensity.
In earthquake engineering, equivalent viscous damping is strongly used in the 'capacity spectrum method' (CSM). 25 In such analyses, the aim is to determine the performance point of the studied structure corresponding to the intersection between the structure demand and capacity curves. 25 The equivalent viscous damping contributes to the evaluation of the capacity spectrum, leading to a reduced spectrum that takes into consideration the hysteretic behaviour of the structure. The CSM analyses are recommended in design codes because they are easier to perform than 'non-linear time history analyses' (NLTHA).

Paper objective
The purpose of this paper is to improve the understanding and modelling of damping in NLTHA. Such analyses should give more precise structural responses than CSM. However, Martinelli and Faella 26 indicated three difficulties with NLTHA related to the better accuracy of the method: (i) the use of sophisticated non-linear material models, (ii) the choice of representative accelerograms and (iii) the strong variability in the behaviour of different structural elements. Here, to limit the sources of uncertainties, the analyses will be performed on an RC beam tested experimentally and submitted to dynamic excitations. Therefore, only a typical structural element (RC beam) is studied, and the performed experiments impose the choice of the accelerograms. In addition, an equivalent single-degree-of-freedom (SDOF) system of the RC beam is defined and used for the analyses. The advantage of the SDOF system is that the understanding and accounting of sophisticated material dissipative phenomena is easier. The drawback is the loss of information. However, nearly 90% of the effective modal mass is activated in the first mode. Therefore, an acceptable approximation of a multi-degrees-offreedom (MDOF) behaviour is obtained with the projection using only the first mode. Based on the proposition of a novel time-domain identification method with the SDOF system, this paper presents time-or state-dependent damping models using an advanced non-linear cyclic concrete material constitutive model. Two approaches are developed: (i) identification based on a piecewise linear model given intrinsic damping ratios and frequencies and (ii) identification based on a non-linear constitutive model to evaluate a model-dependent damping ratio. The first strategy corresponds to the discussed equivalent viscous damping , often used in engineering. This model aims to be very generic to allow people to complement their models by using it. However, this is only a first step before going further to define a 'gap damping', , corresponding to the energy not dissipated through the explicitly modelled dissipative phenomena. The combination of this damping with the material dissipation, , in Equation (5) defines F I G U R E 1 Dynamic experimental setup. 28 the physical damping, ℎ : The objective is, thus, to study the evolution of depending on dissipative phenomena internal variables. The results then could be used to calibrate viscous ( ) and hysteretic ( ) damping models and improve the non-linear dynamic analyses of more complex RC structures with different material models.
The work is performed on RC beam equivalent systems. The experimental data used were obtained from previously carried out tests keeping the steel reinforcement in the elastic range. Therefore, the developed work is restricted to RC material non-linearities in the case of low amplitude dynamic excitations. After presenting the RC beam used and the experimental data in Section 2, the SDOF equivalent model of the RC beam is given in Section 3. Next, the proposed identification algorithms are derived and validated in Section 4. The methods are then applied on a damaging test in Section 5, leading to the proposition and validation of damping models in Section 6. Finally, conclusions are discussed in Section 7.

STUDIED EXPERIMENTAL BEAM AND DATA
The experimental data used in this paper come from a campaign carried out in Heitz et al. 27 on RC beams at the French Alternative Energies and Atomic Energy Commission (CEA) on the AZALÉE shaking table. Quasi-static and dynamic tests were performed to evaluate the seismic energy dissipation of RC beams depending on their structural (geometry, steel ratio), material properties ( ′ ) and applied loading.

Reinforced concrete beam
The tested RC beams are 6-m long with a section of 0.2 × 0.4 m 2 , as presented in Heitz et al. 27 This paper considers only the beams referred to as HA16-C1A-1 and HA16-C1A-2. The chosen experiments were the richest and the most welldocumented. The concrete formulation is classified C25/30 in Eurocode 2, and the reinforcements are composed of eight high-adherence (HA) rebar of diameter 16 mm. The material tests performed on concrete cylinders and steel rebar gave the following properties for concrete and steel, respectively. 28

Experimental data
Different tests were carried out: (i) quasi-static tests, (ii) dynamic tests ( Figure 1) and (iii) hammer shock tests to follow the decrease in beam dynamic properties between each forced vibration test. The quasi-static tests were performed on a strong floor with two actuators. When the actuators were acting in phase, the four-point bending test excited the first mode of the beam. It consisted of imposed displacement cycles with increasing amplitudes. Three cycles were performed for each amplitude to stabilise the energy dissipation. The obtained experimental data from an HA16-C1A-1 beam are used to calibrate the non-linear beam model in Section 3.3. Then, two dynamic tests ( Figure 2) successively performed on the HA16-C1A-2 beam are considered to identify damping models. The first white-noise signal, 'WN1' (Figure 2A), did not damage the beam and was used to evaluate the undamaged modal properties of the beam. Then, Figure 2B shows a decreasing sine sweep signal (called 'DSS2') with larger amplitudes cracking the beam. The grey blocks are defined by frequency-imposed signals with decreasing frequency increments of 0.5 Hz (from 8 to 5 Hz). Numerous sensors were used during the tests. For the displacements, linear variable differential transformers (LVDT) associated with the actuators (quasi-static tests) and wire sensors (dynamic tests) were used. Nine accelerometers were placed above the beam to measure accelerations in one or three directions. Two 6-axis load cells were positioned at the beam ends, and strain gauges were placed to validate the elastic behaviour of the support steel blades. Mainly, the load measurements at the supports are of interest in the model calibration and validation of the elastic behaviour of the blades. Then, the displacements and accelerations at the beam mid-span are used in the identification and dynamic analyses. Indeed, the beam mid-span is the point with the largest motions, so the experimental data are less noisy.

DEFINITION OF THE BEAM EQUIVALENT SDOF SYSTEM
In Chambreuil et al., 3 a multi-fibre model of the experimental RC beam was proposed. However, performing 3D or multifibre dynamic analyses is computationally demanding. That is why an equivalent SDOF system (3.2) is proposed to facilitate computation analyses and the understanding of dissipative phenomena. The equivalence is obtained using the continuous field system projection on the beam modal basis (3.1). For the non-linear constitutive model, the hysteretic model proposed in Heitz et al. 8 is used (3.3).

Evaluation of the studied beam modal basis
The first three modes of the experimental beam can be determined with the hammer shock test data on the undamaged beam. For example, Figure 3 presents the imaginary part of the acceleration FFT responses for the beam HA16-C1A-1, in the initial stage and with shocks applied at the mid-span of the beam. The plot corresponds to the mean response of four shocks. A similar figure can be plotted with the shocks at the beam quarter to characterise the second mode. The frequencies obtained for the first three modes are 7.11, 23.1 and 58.2 Hz. If no experimental data are available concerning the undamaged structures, a numerical or analytical modal basis should be evaluated. The studied beam is a complex system composed of a simply supported beam with elastic supports and additional masses placed at one-quarter and three-quarters of the beam. No analytical modal basis of the system can be defined. That is why a strategy is proposed here to approximate the basis. First, the system is decomposed into sub-systems with analytical responses proposed in Karnovsky and Lebed. 29 Then, for the combination of the frequencies, based on the propositions of Dunkerley 30 for the first frequency evaluation and of Low 31 for a system with additional F I G U R E 3 Imaginary part of the acceleration FFT response-beam HA16-C1A-1-shocks at mid-span.

TA B L E 1
Comparison of the first three eigenfrequencies. masses, Equation (6) is proposed for the evaluation of the approximated eigenfrequencies,

Frequencies [Hz
where , , and , are the eigenfrequencies associated with mode of the complete system, the th sub-system and the simply supported beam, respectively. Table 1 compares the experimental and approximated analytical first eigenfrequencies of the beam. The first two frequencies match very well, while a more significant error is obtained for the third. However, the confidence associated with the third mode experimental data is too low to conclude on that value. Therefore, the proposed strategy is accurate for the frequency range of interest.
Then, for the combination of the mode shapes, Rayleigh's quotient, associated with the th mode ( ) and defined in Equation (7), is used for the continuous model of a simply supported beam with (i) translational (stiffness ) and rotational (stiffness ) elastic supports and (ii) two additional masses at one-quarter and three quarters of the beam, where , and are the beam length, section and inertia modulus, respectively; and are the concrete Young's modulus and density, respectively; are the additional masses and and are the translational and rotational stiffness values of the beam supports, respectively. Each mode of the complete system is defined as a linear combination of the subsystem modes , (Equation 8). The proportionality parameters are deduced using the minimisation of Rayleigh's quotient with all parameters.
To improve the evaluated eigenbasis, Gram-Schmidt's algorithm with a mass scalar product is applied to define an orthonormal basis. Figure 4 presents the equivalence between the experimental and approximated analytical modes. Again, the two first modes are well-characterised, and more significant errors are observed with the third one, as discussed for the frequencies.

Lagrangian formulation of the equilibrium
In this section, the Lagrangian formulation associated with the studied beam is developed to express the dynamic equilibrium to solve for the beam equivalent SDOF system.

Kinematics of the beam:
The Euler-Bernoulli beam kinematics are considered in this study. The continuous displacement ( , ) of the beam along the bending direction is written in the modal basis where are the coordinates of the displacement field in the modal basis and are the mode shapes for all ∈ ℕ.

Derivation of energies:
These coordinates are used to express the Lagrangian  of the system as well as the viscous damping energy and the external work  . First, the kinetic energy can be expressed as with being the modal masses for all ∈ ℕ. Then, in the context of moderate non-linearities, the hypothesis of uncoupling of the Helmholtz free energy in the modal basis is considered. The Helmholtz free energy Ψ( ) is written as with being the bending moment. For small damping values, the Basil hypothesis is considered with a decoupling of the viscous damping energy in the modal basis. The damping energy (̇) can be written as where are the damping parameters for all ∈ ℕ. Finally, considering a homogeneous ground acceleration̈, the external work  ( ) is written as Lagrange equation: The equation of equilibrium over the modal basis is obtained with Equation (14): By considering the expressions of the energies (Equations 10-13), the equations of equilibrium (Equation 14) can be written, in the modal basis,̈+̇+

SDOF model:
It was shown in Chambreuil et al. 3 that the beam movement is mostly characterised by its first mode, and the first mode represents 89.6% of the total modal mass. Furthermore, the imposed loading has a frequency content in the range of the beam first eigenfrequency, and a homogeneous distribution of the loading activates only the odd modes (i.e., symmetric mode shapes). According to these elements, the response of the beam can be approximated by considering only the first mode: ( , ) ≈ 1 ( ) ⋅ 1 ( ) with 1 deduced from Section 3.1. Therefore, the problem in Equation (15) reduces to where 1 is a coefficient taking into consideration the input acceleration direction.

Evolution of the system and the associated dissipation
From this section, the choice is made to normalise the mode shape 1 with respect to the mass (i.e., 1 = 1). Two free energy models (Ψ 1 ) will be studied in this paper: piecewise linear (3.3.1) and non-linear (3.3.2) models. By comparing those models, the influence of dissipative phenomena on damping could be of interest, and links between those two quantities could be sought. The signal is decomposed into windows of length ( ∈ [ [1, ]]) and which overlap over a length , corresponding to the time interval

Piecewise linear model
In this first model, the system is characterised by a frequency 1 and a viscous damping ratio 1 (Equation 17) that are free to evolve from one window to another. The hypothesis is that the structure behaviour is linear along the window . So, the two parameters 1, and 1, , that need to be identified, are constant on window , what can be written as The Helmholtz free energy and the viscous damping energy become on window with 0 1 the fundamental frequency of the initial structure. Therefore, the equation of motion deduced from the total equilibrium on window (Equation 16) becomes in the case of mass normalization.

A non-linear constitutive model -'IDEFIX'
In [ 8 ] a hysteretic model called "IDEFIX" is proposed to describe the RC beam behaviour. In this model, the damping ratio (Equation 20) is free to evolve to describe the dissipation not taken into account. However, the hypothesis is still that the damping ratio, the only parameter to be identified, is constant on window .
The equation of motion deduced from the total equilibrium on window becomes, with the IDEFIX model and in the case of mass normalisation, with 0 1 the fundamental frequency of the initial structure and Ψ 1, defined from the thermodynamic framework developed in Heitz et al. 8

on window .
Thermodynamic framework: Based on the internal and observable global variables and with ( , ) ≈ 1 ( ) ⋅ 1 ( ), Helmholtz's free energy Ψ 1 is used to define the state potential. The considered phenomena are (i) damage due to concrete cracking (stiffness reduction) with the unilateral effect for crack reclosure, (ii) friction and (iii) the pinching effect. In the following, the index ± means that the internal material variables take different values with respect to the displacement direction.
± is the damage variable (two variables for two crack families on each side of the beam), is the friction displacement variable, is the variable associated with friction work hardening and is the crack closure variable evolving from 0 (crack completely closed) to 1 (crack opened) to take into account the pinching effect.
where ( ) is the altered stiffness, meaning that the pinching effect appears progressively with the damage evolution.
where is a parameter for the model of friction work hardening.

Energy descriptions:
The dissipated energy is defined in Equation (25)  Model calibration: The IDEFIX model has been implemented in the MATLAB R computer program and is calibrated to reproduce the beam quasi-static response. It is composed of eight parameters. The calibrations were performed using an error minimisation method compared to the experimental data. The parameter calibrations were decomposed between the first parameters influencing the envelope response and then those influencing the hysteresis cycles. Figure 5 presents the quasi-static response of the IDEFIX model. The force-displacement response ( Figure 5A) shows a good correlation between the experimental and numerical responses. Figure 5B presents the evolution of the hysteretic and work-hardening-associated energetic responses as defined in Equations (25)- (27). The IDEFIX model, thus, mainly dissipates energy through friction.
As soon as cracks are initiated, the cyclic motions induce friction between crack surfaces. That is why the evolution of ℎ, shows that the hysteretic friction energy is continuously increasing. In contrast, the damage energy (tensile softening) evolves only when new cracks open or propagate, explaining the step behaviour in ℎ, (damage hysteretic energy).

ALGORITHM AND VALIDATION OF THE DAMPING IDENTIFICATION METHOD
The damping identification method proposed here is inspired by Demarie and Sabia, 32 which aimed at identifying non-linear damping and frequencies of damaged RC elements with a linear material model. The idea was taken up in Heitz et al. 27 and adapted to identify equivalent viscous damping ratios in low amplitude white-noise signals (small non-linearities) based on experimental displacement measurements, and to compare with values from the half-power bandwidth method. Furthermore, dynamic data will be analysed using a physical constitutive material model. Therefore, this paper aims to apply the method to more intense non-linear dynamic tests with two non-linear models: (i) a piecewise linear model presented in Section 3.3.1 (used in Heitz et al. 27 ) and (ii) the IDEFIX model presented in Section 3.3.2 (not used in Heitz et al. 27 ). The related algorithm is presented in Section 4.1. Then, it is validated using the elastic test called WN1 in Section 4.2. Figure 6 presents the method algorithm. First, the input signal of the studied test and the beam experimental dynamic responses are considered. All data are projected on the first mode of the modal basis, as discussed in Section 3.1 and F I G U R E 6 Algorithm of the damping identification method.

Method algorithm
are decomposed into windows. On each window, a minimisation function is used to identify the constant parameters ( 1, , 1, ). The minimisation is based on an error computation between the function to identify  , deduced from the experimental data and the computed function  , obtained with the resolution of dynamic equation (29) or (30) depending on the behaviour model considered. Finally, the parameter state-dependent evolutions ( 1 , 1 ) are deduced after convergence of the minimisation function on all windows. These steps will be further developed for the studied system in the following paragraphs.
Definition of the function to identify  : The displacement ( , ) and acceleration̈( , ) responses at the beam centre are recovered from the experimental campaign presented in Section 2.2. The velocitẏ( , ) response is then computed by integrating the acceleration response. The experimental data can be filtered to eliminate the influence of too low or too high frequency signal contents disturbing the analyses. The three responses are projected on the first mode of the beam to evaluate 1 ( ),̇1 ( ) and The function to identify could be only one of the three dynamic responses. However, it was decided to take all three into account to consider as many responses as possible. Therefore, the function to identify  can be defined as

Problems to solve and parameters to identify
The two studied models are presented in Section 3.3. Equation (19) for the piecewise linear model and Equation (21) for IDEFIX constitutive models are resumed in Equations (29) and (30), respectively. where 1 is the parameter to identify = ( 1 ). In both cases, the input ground acceleration̈is the dynamic signal applied experimentally on the beam (signals in Figures 2A and 2B). By solving the dynamic equations (Equation 29 or 30), displacement 1 , velocitẏ1 and acceleration 1 responses are deduced and defined together with the computed function  (Equation 31) that can be compared to the function to identify  (Equation 28).

Signal decomposition over time windows:
As previously indicated, to obtain a state-dependent evolution of the parameters, the data are split into windows of length . The windows have to be long enough to give a sufficient amount of data to identify, but short enough to consider that the parameters to identify are constant on each window. Indeed, the idea of the method is based on this last hypothesis. To smooth the responses at the window interfaces, they overlap over a length . Therefore, we introduce the index (⋅) , to define the period of study, ∀ ∈ [( − 1)( − ) , − ( − 1) ].
If the parameters are identified on the window − 1 ( , −1 ), is known. Therefore, the initial conditions to solve Equation (29) or (30) on the window are defined as follows: In addition, to identify , , the minimisation function requires initial values of the parameters , . Therefore, the parameters converged in the previous window are considered: , = , −1 . Finally, the parameter evolutions characterise the change in the beam properties. The latter changes cannot physically be too abrupt, which is why the parameter variations from one window to another are limited: , ∈ [(1 − ) × , −1 , (1 + ) × , −1 ] with ∈ [0, 1].

Definition of the error computations:
To evaluate , on the window , the performed minimisation function is based on the gap between the function to identify  , and the computed function  , . The value , = ( , ,  , ) is, thus, defined as an error function between the two functions  , and  , . Metrics are, particularly, proposed in the literature as the Russel and Sprague and Geers ones. 33 Both decompose the errors between the amplitude and the phase errors (expressions in Table 2). Therefore, if a slight phase shift occurs, for example, it does not influence the amplitude response.
The performances of those metrics for the studied signal are interesting because they introduce a normalisation allowing the management of different types of data. That is why the adopted error function is the Russel amplitude metric leading to Equation (32) with the studied problem data: where is the number of terms in  , and  , , , ( ) 2 , and , is the number of time increments in the studied window and with a time step constant between two time increments. Finally, for the studied case,

Minimisation method to solve the problem and numerical scheme:
Using the error defined in Equation (32), parameters are identified on each window considering an error minimisation function. The function fmincon implemented in MATLAB R is used to minimise the error function. This function is based on the 'interior point' method, 34 allowing minimisations to be carried out under constraints. It should be noted that, because negative values can be obtained with Russel's amplitude error, absolute values are considered in the analyses.
Finally, to solve Equations (19) and (21), Newmark's algorithm is used with = 1∕2 and = 1∕4, which is unconditionally stable and induces no numerical dissipation. In addition, for the non-linear parts, a quasi-Newton algorithm is considered based on the approximated tangent modulus = Δ ∕Δ .
TA B L E 3 Identification method characteristics for the computations -WN1 test.

Method validation on an elastic test
A large parametric study was performed to calibrate the parameters of the method. The results of the identification method can be studied with the two non-linear models and the properties described in Table 3. First, the identified accelerations are plotted in Figure 7. A very good correlation is obtained between the experimental data and the identified responses. The global identification error (Equation 32) is 13% for the piecewise model and 12% with the IDEFIX model. The errors are on the same order of magnitude with the two models, yet strong differences are obtained between the dynamic response components, as presented in Table 4. With the piecewise linear model, the displacement is best identified, while with the IDEFIX model, it is the acceleration that is best identified. This demonstrates that both models are not influenced by the same components and that the use of the accelerations, velocities and displacements in the identification is necessary to homogenise the process.
Then, Figure 8 shows the identified parameters. The coloured surfaces around the central line represent the variation of 10% admissible from one window to another. The idea is to check whether this limit does (or does not) influence parameters ( 1, , 1, ) -comparison of the models on the WN1 test. the results. No parameter saturation at the surface limits appears in any of the figures. Therefore, the selected limit variation is admissible. The eigenfrequency (piecewise model) ( Figure 8B) is relatively stable, with a mean value of 6.85 Hz. It represents a variation of 3.7% with the first beam eigenfrequency deduced from the hammer shock tests on the undamaged beam (7.11 Hz). Because this test kept the beam in the linear range, it can accurately find a similar eigenfrequency. However, the slight decrease in the identified eigenfrequency could suggest a beginning of damage at the end of this test. The conclusions on the eigenfrequency validate the performance of the developed method. Then, the most interesting feature is the identified damping ratio ( ) for the RC beam. Figure 8A also shows nearly stable values because the standard deviation is low (0.40%). The mean value of the identified damping ratio is = 4.26% with the piecewise model, and the values oscillate globally between 4 and 5%, which are expected values for such RC undamaged structures. However, a steady increase of equivalent damping ratio from 3 to 5% is observed within the fortyfirst seconds. This phenomenon is due to the choice of the initial value , = 3% and the available variation from one window to another (10% × , −1 ). Therefore, a transition time is necessary to reach the stable damping ratio values representative of the studied system. With the IDEFIX model, Figure 8C shows the identified damping ratio . A damping ratio of approximately = 3.42% can be determined, which is on the same order of magnitude as with the piecewise linear model. This leads to a hysteretic damping ≃ 0.84%, which is very low because of the test linearity.

IDENTIFICATION OF DAMPING DURING A NON-LINEAR TEST
The performance of the damping identification method has been previously validated. Due to the development of nonlinearities in the RC beam, it can now be applied to damaging tests to discuss the evolution of the damping ratio. The properties of the identification method differ from Table 3 for , and , which are now equal to 1.2 s, 0.6 s and 30%, respectively. The damageable test (input signals in Figure 2B) discussed in Section 2.2 is considered. In addition, both models, the piecewise linear and IDEFIX, are studied to characterise the influence of the energy dissipation at the material level.

Dynamic responses
First, the identification errors indicated in Table 5 show a similar error magnitude for each quantity of motion compared with the WN1 test. This can be explained by the stronger amplitude considered here that tends to reduce the signal-to-noise ratio. Then, Figure 9 presents the identified accelerations compared to the experimental data for both models, as well as the identification errors on each window. With the piecewise linear model ( Figure 9A), it is possible to obtain a better representation of the experimental responses because of the adjustments allowed by the frequency identification in addition to the damping ratio ( Figures 10A and 10B). However, the peaks in errors at the block ends show that the beam behaviour in free vibrations is badly represented. Indeed, the frequency evolution in Figure 10B shows an increase in frequency, which is not physical because when the beam is damaged, its frequency decreases. The damping ratio evolution ( Figure 10A) appears complementary to the frequency evolution ( Figure 10B), so a link between both parameters can be defined in Section 6. With IDEFIX ( Figure 9B), the acceleration amplitude decreases are not as important as experimentally measured. However, at the block ends, the acceleration response in free vibrations is best modelled, so the damage state of the beam is more physically characterised. This is linked with the non-linearities developed along with computation without parameter updates. The damage variable evolves in steps, which is physically representative of the RC behaviour because cracks cannot disappear after opening. Finally, for the evolution of the damping ratio ( Figure 10C), stronger variations are obtained than with the piecewise linear model, indicating the strong link between non-linearity development and damping ratio evolution. 1

Dissipation analysis
The integration of Equation (15) leads to the relative energy balance This can be rewritten as where , is the relative kinetic energy, is the viscous damping energy, is the absorbed energy and , is the relative imparted energy.

WN1 test:
This test keeps the beam in its linear range. Therefore, the only energy dissipation is the damping energy , in the cases of the piecewise linear and IDEFIX models. With the IDEFIX model, the absorbed energy is still recoverable because the material energy dissipation is null. 1 It should be noted that the strong damping ratio values are due to acceleration amplitudes close to zero in the transitory phase of the identification method.

DSS2 test:
The development of the non-linearities discussed in Section 5.1 explains the differences between the energy balances obtained with the two models ( Figure 11). With the piecewise model ( Figure 11A), the only dissipated energy is due to the viscous damping energy because the absorbed energy in Figure 11C is recoverable. In contrast, Figure 11C shows that the absorbed energy is not null at the computation end with the IDEFIX model. Therefore, a part of the energy dissipation is due to the material constitutive model ( Figure 11B). The last model is more representative of physical behaviour, even if here the energy dissipation in the non-linear constitutive model is negligible in comparison with the damping energy.
In addition, the difference in final energy between the two models is due to the difference in identified displacements. Indeed, the energy integration is performed from the displacement response. With IDEFIX, the two preponderant phenomena dissipating energy are damage and friction. Their evolutions are plotted in Figure 12 by dividing the energies by their final values. The damage energy evolves in steps when the maximal displacement increases. In contrast, friction occurs after the cracks open and then it continuously dissipates energy due to the cyclic response. In the end, the friction takes more than 95% of the total material model dissipations, so it is clearly the phenomenon dissipating the largest quantity of energy as already observed in Chambreuil et al. 3

DAMPING MODELS
The dynamic and energy responses demonstrated the good performance of the identification method: the dynamic responses accurately represent the experimental behaviour, and the energy balances are satisfied. Therefore, the identified damping values can now be studied to analyse in detail the modelled and missing dissipation that occurs during damaging dynamic loading on the studied RC beam.

Evaluation of the 'gap damping'
From the piecewise linear model: With the piecewise linear model, two parameters (damping ratio and frequency) are identified. Therefore, a damage index can be defined in each window with respect to the eigenfrequency dropoff Then, the identified damping ratio can be plotted depending on this index, as presented in Figure 13 with the data of the DSS2 test. An evolution function is then defined by the ratio of two polynomials , (Equation 36). The function parameters are determined by minimising the errors between the data and the curves.
with = 4.3%, the elastic damping ratio deduced from Figure 8A. This function particularly well represents the substantial increase in for extensive damage. In addition, it gives an almost constant value for damage indices smaller than 0.6, but the values may be slightly too large because they are between 5 and 6%, while a value between 3 and 4% was observed for the undamaged beam. Therefore, a constant value of approximately 4% could be an accurate approximation for damage indices smaller than = 0.6. This last conclusion validates the engineering practices that always consider constant damping ratios. However, Figure 13 shows that this practice is no longer reliable for extensive damage.

From the IDEFIX model:
To better understand the dissipative phenomena that occur during dynamic excitations, the IDEFIX model is considered. Dissipations occur at the material level, so the identified damping ratio corresponds to the missing energy dissipation . Two phenomena particularly dissipate energy: crack opening and friction, associated with variables and , respectively. In each window, we consider the maximal value of those variables ( and ) to evaluate ( ) and ( ). First, Figure 14A presents ( ). However, because this variable increases in steps, it is not a quantity of interest for the proposal of a damping model. Particularly, information about the cyclic dissipations cannot be characterised. However, if the friction is considered, Figure 14B is obtained. The damping ratio ( ) initially seems to increase before decreasing with an inverse function. The damping increases for small friction levels, perhaps because some unknown dissipative phenomena become non-negligible at low levels of non-linearities. However, after a certain amount of damage, the friction can dissipate enough energy to limit the requirement for additional damping, , at the structural level. The evolution function is proposed in Equation (37). Finally, in Heitz et al., 8 a numerical analysis was performed to evaluate the evolution of the equivalent viscous damping ratio as a function of two engineering demand parameters: (i) the mid-span curvature , restricted to values smaller than the first yielding curvature , and (ii) a degradation index Γ corresponding to the maximum historic curvature at mid-span over . Therefore, the idea was to consider the response amplitude to define . This strategy is shown in Figure 15, with defined as the maximal curvature on each window. Depending on Γ, an evolution similar to ( Figure 14A) is obtained, with a decrease in the maximal damping ratio as Γ increases. Depending on , the damping ratio evolution is similar to that with the friction variable ( Figure 14B) for the same reasons as previously discussed. For each Γ, an evolution function could be defined as a function of . Therefore, the missing energy dissipation with the IDEFIX non-linear constitutive model appears to be modelled through damping ratio evolutions (Γ, ) depending on two engineering demand parameters, as observed in Heitz et al. 8 from quasi-static analyses.

6.2
Performance of the proposed models Different damping models have been proposed in the previous section leading to or and . Their calibrations were performed on the experimental data from RC beam HA16-C1A-A. Other beams were tested during the experimental However, for the energy representation (grey data), the constant damping model always gives the poorest results. Therefore, the strategy used in engineering to consider constant damping seems to be performative, provided that the objective is to obtain dynamic quantities as the acceleration. However, it is shown here that it does not allow for accurately characterising the energy dissipation. Furthermore, the evolution of energy is particularly interesting with the damping model based on the friction variable because (i) the final energies are larger than with the two other models and (ii) more dissipation occurs at the material level (green curve corresponding to the absorbed energy). Therefore, the damping model seems well able to complete the non-linear constitutive model inducing .

Summary
Based on experimental data, this study proposes a damping identification method for dynamic analyses with two constitutive models: (i) a piecewise linear model (identification of an intrinsic damping ratio) and (ii) the IDEFIX model, a non-linear hysteretic model considering damage, friction and unilateral effect dissipative phenomena (identification of model-dependent damping equivalent to the lack of dissipative phenomena representation). The aim of identifying damping is then to propose damping models depending on different internal non-linear variables to represent the energy dissipation that is not considered with the non-linear material constitutive model. The SDOF non-linear model is first calibrated based on the quasi-static experimental responses and then used in the identification method. The method is based on an SDOF system equivalent to the studied experimental RC beam. The equivalence is obtained by projecting the experimental data onto the first mode of the beam modal basis. Indeed, it is demonstrated that using only one mode for the projection is sufficient, even if this is only an approximate mode shape of the beam first mode. This last conclusion allows performing such analyses even with complex structures. The method algorithm is developed assuming that the identified parameters ( for all analyses and for the piecewise linear model) are constant in each time window. Then, the method parameters are studied based on the WN1 experimental test, which experimentally kept the beam in its linear range. The method parameters are evaluated, and the identification is finally performed with the two non-linear models. With the piecewise linear model, a viscous damping ratio and a frequency are identified for each window. A damage index can, thus, be deduced from the evolution of frequency. Then, the identification method is applied to the experimental DSS2 test damaging the beam. The damping ratio evolution, as a function of internal variables, is deduced from the damping identification. Finally, the proposed damping models are applied to different experimental data. Two beams similar to the one used for the model identifications are considered, and dynamic non-linear computations are performed using the updated damping models.

Conclusions
The main conclusions obtained in the paper are summarised here: • Because the RC beam remained linear during the WN1 test, we obtained a frequency almost constant during the computation and a mean value of the viscous damping ratio equal to , ≈ 4.3%, a value in accordance with the values used in engineering for RC structures. Then, with the IDEFIX model, a value slightly smaller ( , ≈ 3.4%) was obtained but still on the order of magnitude of the expected values.
• From the damaging DSS2 test and with the piecewise linear model, a damping evolution function is proposed for depending on the damage index ( , ). An important conclusion is that below a damage index of 0.6, considering a constant value of the damping ratio , ≈ 4% is an accurate approximation. This hypothesis of constant damping is considered most of the time in dynamic analyses. This assumption is no longer accurate for significant damage levels of RC structures. Damping updates are required to improve the representativeness of experimental behaviours.
• From the damaging test and with the IDEFIX non-linear model, different non-linear variables were studied. The first is the damage variable, , but it does not exhibit a strong correlation with the evolution. However, the friction variable is a better indicator to evaluate the equivalent viscous damping ratio, and an evolution function of was proposed. For small levels of friction development, the damping ratio increases because insufficient dissipation is adequately captured by the material model. Nevertheless, after a certain amount of damage, enough energy is dissipated through friction, so a small amount of additional damping, , is needed. Finally, the last studied damage indices were two engineering parameters: the mid-span curvature and a degradation index corresponding to the maximum historic curvature at mid-span over the first yielding curvature Γ. A damping evolution similar to the friction variable was obtained with the mid-span curvature, and no influence was observed depending on the second degradation index.
• From the application of the identified damping models, the acceleration responses are similar to the proposed damping models compared to constant damping. However, a significant difference is observed in the energy analyses. The structural energies appear better characterised with the updated damping models. Indeed, using one or another non-linear variable induces a different development of energy dissipation in the beam. Therefore, by choosing the correct model, it is possible to be more representative of the damage evolution of RC structures submitted to dynamic loadings. Now, all computations were performed on equivalent SDOF systems, so the results must eventually be confirmed in the case of MDOF dynamic non-linear analyses. • Between the two updated damping models studied, it was observed that for simple models considering only a damage degradation of the frequency, the model ( ) provides a good approximation of the response. For more advanced models, the damping model based on the friction variable ( ) should be used to characterise the dissipation not modelled by the non-linear constitutive model to study more refined phenomena because it quantifies what was called the 'gap damping' in Equation (5). • The internal state variables depend on the choice of material models used to simulate concrete stress-strain relationships and energy dissipation mechanisms. However, the dissipative phenomena considered the internal variables related to (i) the damage and (ii) the friction. These phenomena are concrete's most important dissipative mechanisms and are generally considered in concrete stress-strain relationships. So, the internal variable associated with those phenomena, in any model, could be used.
The identified damping models are summarised in Table 6 for the two studied material models and for linear or non-linear dynamic tests. Remark. The proposed models have been developed based on experiments during which the steel rebar remained in its elastic range. Therefore, it must be considered that these models only characterise the energy dissipation related to the cyclic non-linear response of concrete.

A C K N O W L E D G E M E N T S
This work was performed using HPC resources from the Mésocentre' computing centre of CentraleSupélec and École Normale Supérieure Paris-Saclay supported by CNRS and Région Île-de-France (http://mesocentre.centralesupelec.fr/). A part of this work has been performed at École Polytechnique de Montréal, thanks to funding from FRQnet.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.