Nonlinear SDOF Analytical Model of Mass-Timber Building with PostTensioned Rocking Walls

The dynamic response of post-tensioned rocking walls in a mass timber building can be reduced to a single-degree-of-freedom (SDOF) model. In this model, the rocking wall panel is simplied as a rigid block, while the base rotation represents the degree of freedom of the entire structure. The paper presents an analytical approach to develop and calibrate this nonlinear model using shake table tests of a full-scale two-story building with CLT rocking walls. The experimental data are used to estimate the parameters of the governing equation using least-squares optimization. The correlation between the obtained parameters and the cumulative dissipated energy led to a nonlinear model with degradation behavior captured. After that, the calibrated model was used to assess the fragility functions of the structure under repetitive seismic events.

Under speci c kinematic assumptions, the mechanical model of a frame structure coupled with a rocking wall behaves like an SDOF oscillator. In this section, the authors will demonstrate this observation by deriving the equation of motion of a mechanical model possibly representative of the full-scale two-story mass timber building with post-tensioned rocking CLT walls tested in 2017 at the NHERI@UCSD large outdoor shake table facility. Fig. 1 represents the chosen simpli ed mechanical model of the two-story building. The model consists of a shear-type two-story frame with lumped masses and two rocking walls. The columns and the rocking walls have no mass, and the total mass is lumped in the storeys. The symmetry of both the tested structure and the excitation supported the development of a planar model, neglecting the 3-d effects. The two rocking walls are mutually connected by equivalent shear springs representatives of the UFP steel dissipators. The rocking walls are connected to the frame by sliding constraints. The shear transfer detail devised by [49] releases vertical constrain and allows the sliding between the CLT wall and each story. Each rocking wall possesses a pivot point by one edge of the panel due to the geometrical con guration of the building. The rocking walls are inside the building, and both connected to the storeys. This con guration leads to the identity of displacement between the two rocking walls and the stories. Unbonded prestressed steel cables connected to the foundation stabilize the rocking motion of the CLT walls. The model neglects the stiffness of the sliding constraints at the base of the CLT panels.
Two variables describe the kinematic response of the mechanical system: the rigid rotation of the CLT panels (θ) and the horizontal displacement eld of the CLT panel due to elastic deformation (v(z)), where z spans the CLT panel length. The displacement eld of the rocking walls is the summation of the rigid and deformation contributions, as shown in Eq. 1: u(z) = V(z) + θz; v(x) = θx 1 The following equations express the horizontal constraint between the rocking walls and the frame : u 1 = u h 1 ; u 2 = u(h 1 + h 2 ) 2 where u1 and u2 are the horizontal displacements of the two storeys. The current model represents a possible enhancement of the model proposed by [53]. The main differences between this model and the one by [53] are: This model does not include the stiffening and dissipative contribution of the rocking walls into the equivalent stiffness of a lumpedmass shear-type frame model. The current model represents the rocking walls and the shear-type frame separately, each described by its kinematic variables.
As a consequence of the previous point, this model does not divide the CLT wall into a sequence of segments, each one corresponding to the rocking wall comprised of two storeys. Instead, the CLT wall is modelled as a continuum and has the same displacement and rotation at each storey. Consequently, this model respects the continuity of rotation between segments of the CLT wall, neglected in the frame model, and includes the CLT deformability without segmenting the wall stiffness in the summation of cantilevered-like beam elements.
This model did not explicitly account for the stiffness of the angle brackets connecting the wall to the foundation. The sliding is eliminated by considering a xed pivot point.The model is derived under the assumption of small rotations and displacements. The maximum rotation values of the CLT panel are lower than 0.1. Therefore, the inclusion of the nonlinear terms considered by the Housner nonlinear block model [55] is unnecessary for engineering purposes.
The model neglects the dissipative contribution of impact in the rocking motion [56,57]. Dissipation depends on the UFP dissipators.
The equations of motion are derived from the Theorem of Virtual Works by minimizing the total energy of the mechanical system. There are three energy sources related to the action of elastic, inertial and external forces. The summation of the virtual works associated with the action of the three force typologies is zero: In the following paragraphs, the authors will explicit the virtual work made by the elastic, inertial an external forces, labelled W elastic , W inertial and W external respectively.

Elastic forces
The virtual work done by the elastic forces is the summation of the following addends: ( ) Page  Initially, the authors assumed that the CLT panel behaves like an equivalent Euler-Bernoulli beam. The equivalent beam has null horizontal displacement and bending moment at the base. The pivot point is xed and does not allow sliding motions (V(0) = 0), while the steel tendons are connected to the foundation and do not provide a bending constraint at the panel base (V''(0)). The CLT panel has a free end, associated with null shear forces (V'''(h) = 0) and bending moment (V''(h) = 0). The considered constraint conditions of the CLT panel lead to a null displacement eld and consequently to a null virtual work (W CLT = 0). Straightforward analytical derivation can prove that a mass-less wall does not deform under the considered constraint conditions. This evidence led to the assumption of a rigid-like behaviour of the CLT wall, as anticipated in the introduction. The CLT panel does not deform and behaves like a rigid block, whose motion is described by the θ variable. Accordingly, the virtual work associated with the elastic forces reduces to the summation of the UFP dissipators, the tendons and the columns. Therefore, the following equation holds: where θ is the base rotation, n s the number of UFP dissipators, k s the stiffness of the UFP dissipators, n the number of tendons, N 0 the tendon prestressing force, E s and A i the elastic modulus and cross-section area of the i-th tendons, b i the distance between the pivot point and the i-th tendon, h the height of the CLT panel, k 1 and h 1 the horizontal stiffness and the height of the rst storey, k 2 and h 2 the horizontal stiffness and the height of the second storey. The absence of the CLT panel deformation reduces the degrees of freedom of the mechanical system to the sole base rotation θ. Interestingly, the coupling of a two-degrees of freedom system, a two-story frame, with an internal rocking wall causes the degrees of freedom condensation. This phenomenon does not happen in case of rocking walls coupled with linear or nonlinear devices to the building story's. In that case, the mechanical system possess more degrees of freedom, as demonstrated by [57,58].

Inertial and external forces
The virtual work of the inertial forces is associated with the storeys contributions: W inertial = W 1st − storey + W 2nd − storey = = m 1ü1 +ü g δu 1 + m 2ü2 +ü g δu 2 = = m 1 h 1θ1 +ü g h 1 + m 2 h 2θ +ü g h 2 δθ where * indicates the double time derivative, u 1 and u 2 are the horizontal displacements of the rst and second storey respectively, and ü g is the ground excitation in terms of acceleration. The two displacements are expressed as function of the base rotation.  where I θ is the rotational inertia of the storey masses, I θ is an equivalent rotational spring and S x is the expression of the static moment.
Eq. 8 describes the dynamics of the mechanical system in Fig. 1 in linear elas-ticity, without dissipation. The next paragraphs will discuss the modelling choices of dissipation by including an equivalent viscous term.

Test Description
This paper utilizes the response of a full-scale two-story mass timber building with post-tensioned rocking CLT walls for model calibration.
The authors present a few details of the tested building, signi cant for the current research, before discussing the building dynamics and deriving the equation of motion. Complete details of the building and the tests are in [49]. The building was tested in 2017 at the NHERI@UCSD large outdoor shake table facility. Fig. 2 shows the two-story building and a schematic detail of the post-tensioned rocking walls. The building is symmetric and tested using a uniaxial shake table.
There were two sets of coupled CLT rocking walls installed in the test specimen, as shown in Fig. 2. The walls were coupled using U-shaped exural steel plate (UFP) energy dissipators, analogue to the ones tested by [59]. Each panel had four external post-tensioned steel rods placed symmetrically near the centre of the wall panel (two on each side). A steel saddle detail, installed on the top of the panel, anchored the post-tensioned rods. The rocking-wall lateral system was connected to the diaphragm using constructive details, which allows the uplift of the wall panel and provides out-of-plane bracing of the wall. A dowel-type steel shear key, inserted into a vertically slotted hole, allowed unconstrained uplift of the rocking wall while transferring lateral loads. The main objective of the testing program was to validate the resilient performance of the post-tensioned CLT rocking-wall lateral system at different levels of seismic intensity. The test building was subjected to a total of 14 earthquake excitations selected to represent three hazard levels for a site near Seattle WA.

Discussion Of The Modelling Choices And Estimation Of The System Nonlinearity
The mechanical model in Eq. 8 is designed to be simplistic, reducing a complex 3-d building into an equivalent SDOF system. In this section, the authors discuss the modelling choices by presenting selected results of the test data: The measured base rotation of the CLT panel is compared to the rotation of the storeys with respect to the pivot point (Subsec. Base rotation); The Fast-Fourier-Transform of the measured base rotation (Subsec. Fast-Fourier-Transform); The spectrogram of measured base rotation with indication of the frequency ridge with highest energy content (Subsec. Spectrogram); The measured deformation of the CLT panels measured by vertical strain gauges (Subsec. CLT deformation); The experimental force-displacement plots (Subsec. Hysteresis curve);

Base rotation
The authors prove that the base rotation is the prevalent degree of freedom of the structural system by comparing three measured parameters: . Figure 3 plots the three measured variables, θ, u 1 /θ and u 2 /θ for each of the 14 shake table tests. Despite higher discrepancies in Test No 3,4, and 5, there is a substantial coincidence between the three variables, proving that θ is the prevalent degree of freedom of the two-storey building. Additionally, the plots show that a very low rotation amplitude distinguishes the rocking motion. This fact endorses the adoption of the linear model in Eq. 8 and the consequent neglection of the nonlinear contributions typical of high-amplitude rocking motions. Tab.1 reports the Root Mean Square Error estimated between the rst story rotation (u 1 /h 1 ) and the CLT wall base rotation (θ), and second story rotation (u 2 /h 2 ) and the CLT wall base rotation (θ). The RMSE is minimal and further proves the signi cant agreement between the three variables. The horizontal displacements of the storeys are not independent, but related by the kinematic equations in Eq. 1.

Fast Fourier Transform
A direct proof of the presence of a leading kinematic variable comes from the FFT of the measured θ. Fig. 4 shows the FFT of the measured θ for each of the 14 tests. Table 1 Relative difference between the maximum of the rst (u 1,max )/h 1 ) and second story rotation (u 2,max )/h 2 ) and the maximum of the CLT wall base rotation (θ max ) in % The plots in the logarithmic scale of the y-axis display the presence of a dominant peak in the range 0.3-1 Hz. The observation of the FFT plots reveals a dominant degree of freedom in a pretty wide range. The mechanical system is not elastic and exhibits a signi cant nonstationary or, possibly, nonlinear response due to amplitude-dependent, hysteretic and degradation phenomena. The FFT helps prove the predominance of a single degree of freedom, but it does not provide information about the time-dependency of the response.

Spectrogram
Accordingly, the authors estimated the spectrograms of the θ signals. Spectro-grams are visual representations of the spectrum of frequencies of a signal as it varies with time. Fig. 5 shows the spectrograms of the measured θ for each of the 14 tests using an Hamming window with a 20% overlap. The spectrograms, represented by contour plots in greyscale, exhibits a moving high-amplitude and sharp peak in white, characterized by a signi cantly varying response. The authors evidenced the moving peak by plotting the high-amplitude frequency ridge in red. The inspection of the spectrograms reveals two aspects: (i) the sharpness of the peak further con rms the presence of a dominant degree of freedom; (ii) The signi cant variation of the frequency ridge proves that the system is not elastic and the model in Eq. 8 is not adequate. It must include the effect of dissipation and the time-variability of the structural parameters.
The spectrograms can also reveal the possible nature of the system nonlinearity, helpful in enhancing the elastic model in Eq. 8. Fig. 6(a) superposes the frequency ridges estimated in Fig. 5 for each of the 14 tests. Fig. 6(a) demon-strates the high variability of the peak response, but it does not provide information about the structural properties. Still, there is an evident reduction of the fundamental frequency from the 1st to the 14th test due to possible degradation phenomena (e.g, the shift of the pivot point due to the yielding of the base beam).
Figure 6(b) shows the frequency ridges as a function of the amplitude. The obtained plot is a rough estimation of the structural system's Frequency Response Function (FRF). The estimation of the FRF from seismic response data is an unconventional practice. Accurate FRF estimates should derive from sweep sine tests. However, despite the roughness of the estimation, the dots gather along two main branches, revealing a marked softening response. The fundamental frequency is amplitude-dependent, and the softening effect is related to nonlinear phenomena due to nonlinear terms in the governing equation. The stable and unstable curves of the FRF are quite evident, although the highamplitude dots are limited. A low number of points in correspondence of the high-amplitude response does not allow reliable estimation of the nonlinear structural parameters from the tting of the FRF. However, this qualitative outcome is used in the next section to enhance the mathematical model of the two-story building, inclusive of time-dependent parameters and nonlinear terms.

CLT wall deformation
A direct consequence of the constraints adopted in Fig. 1 is the lack of deformation of the CLT panel, behaving like a rigid block. String pots were installed to measure the exural and shear deformation of the panel.  Figure 7 prove that the deformation of the CLT panel is minimal, lower than the string pots resolution and, possibly, negligible in a practiceoriented structural model.

Hysteresis
Most dissipation in timber buildings depends on the hysteretic phenomena of steel connectors or dissipation devices. In the considered building, the pre-eminent dissipation source is the plasticization of the UFP connectors. Pei et al. [53] developed a ag-type hysteretic model representative of the nonlinear response of the tested building. The ag-type hysteretic model is characteristic of post-tensioned rocking structures. The hysteresis loop is very narrow compared to other hysteresis curves of timber-based systems due to the lack of pinching and manifest degradation phenomena.
The following considerations led the authors to model the rotational stiffness K θ as elastic and reproduce the dissipation phenomena with an equivalent viscous term.
The hysteresis model developed by [49] has three phases. In the considered structural model, the total mass is lumped in the storeys.
Consequently, the CLT panel does not possess any self-weight stabilizing moment, and the panel must rotate to achieve equilibrium with the external loads. There-fore, adopting the hysteretic model by [49], which distinguishes between the rocking and non-rocking phase, is inconsistent with the modelling choices in Fig. 1.
The narrowness and stability of the hysteresis loop may justify the modelling of dissipation with an equivalent viscous term. The shape of elliptic hysteresis curves, associated with viscosity, is not very dissimilar from agtype hysteresis. Therefore, a displacement or energy equivalence can drive the assessment of an equivalent viscous term representative of a ag-type hysteresis curve. Besides, the direct displacement-based design procedure (DDBD) uses equivalent viscous damping and secant stiffness as proxies for the estimation of the nonlinear behaviour of structures. However, the available hysteretic models of timber structures are very elaborate, and the inelastic timedomain simulations using hysteretic models are not easily manageable. Hence, the adoption of equivalent viscous damping (EVD) is a crucial feature of any DDBD method.
The hysteresis curves obtained from the experimental data by plotting the oor acceleration (proportional to the inertial forces) vs the storey displacement are very erratic and do not resemble the shape of any speci c hysteresis curve. Fig. 8 shows the estimated backbone of a sample data set, obtained by selecting the measurement points. The backbone curve is not linear and exhibits two phases, a rst with lower stiffness, a second very irregular. Furthermore, the backbones of all the 14 tests in Fig. 9 con rm that the hysteretic response of the structural system is very unpredictable and do not resemble the backbone of any speci c hysteresis model. Therefore, the high uncertainty associated with estimating the hysteretic parameters sup-ported a more straightforward approach based on equivalent viscosity, which mirrors the global structural dissipation without focusing on the hysteretic response.

Identi cation Of The Structural Parameters
The selected results presented in the previous section showed that the SDOF mathematical model in Eq. 8 could be a reasonable compromise between computational e ciency and prediction accuracy. However, Eq. 8 has no dissipative neither nonlinear terms. Therefore, the authors upgraded Eq. 8 by including a viscous term representative of dissipation phenomena and nonlinear terms representative of the softening response.

First step of system identi cation
The authors estimated the unknown parameters by maximizing the rank correlation [60] between the solution of Eq. (13) and the experimental measurement: where (\cdot) is the inner product, the norm operator, θ e and θ s the experimental and simulated responses, respectively.
Table5 lists the estimated parameters of Eq. 13, while Fig. 10 shows the superposition between the experimental and simulated time histories of the base rotation (\theta) for each of the 14 shake table tests. There is an optimum agreement between the experimental and simulated values of θ. The root mean square error between the two time-histories is minimal and validates the chosen mechanical model in Fig. 1.  Table 3 Estimate of the mechanical parameters in Eq.13.
Table3 reports the estimated mechanical parameters and the values of those assumed for their estimation following Eq. 26-27.

Second step of system identi cation
The variation of the modelling parameters between the 14 tests depends on minor degradation phenomena compared to traditional mass timber structures. Although these phenomena are trivial in a single seismic simulation, their effect can be more consistent under repeated earthquakes. Therefore, following a conventional approach in structural engineering, the authors correlated the parameters in Tab.5 to the cumulative dissipated energy between the 14 tests.

Discussion on the parameters variability
The averaged parameters estimated in the rst four tests refer to the SLE condition, characterized by a lower PGA level up to 0.2g. The averaged param-eters in Tab.6 are used to predict the response under different excitation levels characterized by PGA in the range 0.2-1-2g. The analysis assesses if there is a specif level for the calibration of the parameters minimizing the error of the estimates. Tab.7 lists the error related to the estimate of the displacement in the 14 tests in terms of Root Mean Square Error (RMSE), Maximum error (ME) and Relative Maximum Error (RME) between the experimental ({\theta }_{e}) and simulated ({\theta }_{s}) base rotations. As expected, predicting the displacement response in the rst four tests is entirely accurate, leading to an average error of 1%. The same parameters yield a higher error, 20% on average, in the tests characterized by higher PGA. Still, the prediction can be considered reason-ably accurate for engineering purposes, despite the variability related to the model approximation.  Table 7 Root Mean Square Error (RMSE), Maximum error (ME) and Relative Maximum Error (RME) between the experimental (θe) and simulated (θs) response in terms of base rotation. The use of the parameters in Tab.6 leads, on average, to an underestimation of the structural response at higher displacement levels. This effect mostly depends on the correct choice of the natural frequency related to minor degradation phenomena (reducing the pivot length due to local plasticization, e.g.). However, a close inspection of Tab.7 reveals that the error is both positive and negative even at higher displacement levels. Therefore, the error cannot be removed by simply reducing the natural frequency or adopting other intensity levels to calibrate the parameters. On the other hand, the SLE excitation yields satisfactory results at higher PGAs, proving that the structural system exhibits a mostly linear response related to minor degradation phenomena.

Estimate Of The Limit State Of The Post-tensioned Tendons
The assessment of the fragility functions of the considered structural model with degradation entails the evaluation of the base rotation associated with a given limit state. The authors assume that the limit state of the structure corresponds to the limit state of the posttensioned tendons. The axial deformation of the i-th steel bar, i, can be written as a function of the base rotation θ: {ϵ}_{s}^{i}=\sqrt{{\theta }^{2}+{\left(1+\theta \frac{{b}_{i}}{h}\right)}^{2}}-1 By assuming a conservative collapse condition corresponding to the bar yield-ing, the assumed limit steel deformation, {ϵ}_{s,y}, is equal to 0.0021%. The angle associated with the limit state of the post-tensioned bar is: The failure angle in Eq. 35 is the minimum between the values obtained with the two bi, equal to the distance between the pivot point and the bar direction, see Tab.8 The fragility functions do not express the collapse probability, but the prob-ability of exceeding a speci c limit state, associated with the yielding of the post-tensioning bars, as formulated in Eq. 33. Therefore, the assumption of the considered limit state is in line with the modelling choices, which do not include structural nonlinearities due to failures of the structural components. Besides, the assumption of this limit state does not require a model extrapolation because the tested building experienced the post-tensioning bar yielding during the shake table tests used for the model validation. Following the approach in [13,61], the list of 41 Italian earthquake records with magnitude ranging between 5 and 6.5, given in Tab. 9, represented the base for generating 41 arti cial earthquakes, scaled to the same Peak Ground Acceleration (PGA) and optimized to match the design spectrum. Each earthquake is con-catenated three times, resulting in a sequence of three identical seismic events. The earthquake records are homogenized to the same intensity level and optimized to match the design spectrum expected in L'Aquila (Italy) according to the national seismic code using the algorithm by [62]. A lognormal cumulative distribution function ts the fragility function from data collected from nonlinear dynamic analyses [63]: where P\left(C\right|IM=x) is the probability that a ground motion with IM=x will cause the structure to exceed the considered limit state; {\Phi } is the standard normal cumulative distribution function (CDF); \theta is the median of the fragility function (the IM level with 50% probability of exceeding the considered limit state); and \beta is the standard deviation of ln IM.  Figure 13 shows the response of the structure under the repetition of earth-quake No 41 (see Tab.9). The effect of damage accumulation produces an increment of the maximum θ value attained during the structural response. The plot of the maximum \thetas in Fig. 14(a) as a function of the spectral accel-eration (Sa) demonstrates the effect of stiffness reduction between the three earthquake repetitions, identi ed by three different colours: black the rst repetition, red the second repetition and blue the third repetition. If successive earthquakes occur, the structure reaches the limit state associated with the bar yielding with a lower Sa.
The effect of damage accumulation is evident from the estimated fragility functions, plotted in Fig. 14(b). Earthquake repetitions determine a shift of the fragility function in the left direction, displaying a more fragile performance.

Conclusions
This paper investigates the seismic response of a resilient two-storeys mass timber building from shaking table tests. The authors enhanced the mechanical model proposed by [53], which was used to derive the fragility curves of the considered building under repeated earthquakes. The primary outcomes of this research are: The presence of a rocking post-tensioned Cross-Laminated Timber (CLT) panel, free to uplift, and connected to each storey, determines the substantive reduction of the building degrees of freedom to the sole base rotation. Under reasonable kinematic assumptions based on the geometrical and constructive features of the building, the authors derived the governing equation of the structural response.
The model's hypotheses are validated through selected experimental results, which proved that the base rotation is the prevalent degree of freedom and that the CLT wall behaves substantially like a rigid block.
The experimental data of the 14 shake table tests led to the estimation of the parameters of the governing equation, corrected by rstorder approximations to include nonlinear and degradation effects. Furthermore, the simulated and experimental responses in terms of base rotation prove a satisfactory agreement.
The model is used to estimate the fragility functions of the building under three repeated earthquakes. The structural response is very stable after repeated earthquakes, compared to traditional mass timber buildings, which manifest a dramatic reduction of their initial   Single side Fourier Spectrum of the base rotation (θ).

Figure 5
Spectrogram of the base rotation (θ) with indication of the frequency ridge with highest amplitude.  Longitudinal deformation of the CLT panel.

Figure 8
Estimation of the backbone curve from the measured force-displacement curve of the rst story in Test No 1. The resisting force is proportional to the inertial forces, therefore it is propositional to the measured acceleration, shown in the y-axis.

Figure 9
Superposition of the estimated backbone curves in terms of acceleration and oor displacement for all tests and the rst (a) and second story (b).

Figure 10
Comparison between the experimental and simulated response in terms of base rotation θ.

Figure 11
Dissipated energy of the SDOF in each of the 14 tests.

Figure 12
Correlation between the estimated model parameters and the cumulative maximum dissipated energy.

Figure 13
Response of the considered SDOF oscillator with.

Figure 14
Results of the Truncated Incremental Dynamic Analyses (a) and fragility curves (b) using three concatenations of each of the 41 earthquakes in Tab.9, Rep. stands for the number of earthquake repetition.