Dynamic Analysis of Resilient Rocking Wall Structures: A Numerical Study on Performance Demands

Abstract

Dynamic time history analysis has long been regarded as an acceptable and reliable method for the seismic design of structures. The methodology for conducting such analyses, particularly for modern structures with advanced seismic resisting systems, is generally not covered by codal guidelines and is often categorized as “alternative” analysis. Resilient rocking wall systems with low-damage hold-downs fall within the “alternative” design category for most international standards, and designs must include dynamic time history analysis. However, the analysis results are influenced by factors such as ground motion selection, scaling methodologies, modeling considerations employed, and the assumptions embedded within the numerical model. This study takes a practical approach and assesses their impact on the structural response and seismic demand determination of a selected mass timber archetype featuring a rocking wall system with friction connections. The investigation into modeling considerations explores various damping models, time history analysis methods, and the associated variables within these models. It is demonstrated that varied seismic demands can result from different selections and modeling assumptions. However, with careful and rational engineering judgment and selection during the analysis process, reasonably close and acceptable seismic demands can be achieved. Furthermore, the authors provide recommendations and insights to enhance the analysis and design demand determination process.

1. Introduction

Shear walls and rocking walls have long been recognized for their ease of construction and reliability, demonstrating outstanding performance in seismic events over recent decades. Only a small proportion of such structures have collapsed or sustained catastrophic damage. There has been a great deal of advancement in the field of rocking wall systems in recent years, both for concrete and mass timber construction. In part, this is due to the fact that a rocking wall is more efficient and capable of dissipating energy through controlled rocking, regardless of whether it is coupled or decoupled from the gravity frame. Generally, the rocking wall, as a result of its own weight or further gravitational load, is expected to return to its original position and re-center. Furthermore, it can be combined with additional seismic performance-enhancing systems, such as post-tensioning [1,2] or low-damage friction hold-downs [3,4], in order to enhance seismic performance and increase energy dissipation capacity. Numerous numerical studies and experimental studies have been conducted to evaluate and investigate the performance of such innovative and resilient systems; many have shown great potential, and some have been implemented in real-life projects and structures, including the 10-story Hive project in Vancouver, BC [5,6,7,8,9,10]. Despite this, such innovative systems are limited in that current seismic codes, such as NZS1170, ASCE7, and NBCC2020, do not address or provide provisions for the design of such systems [11]. For instance, ASCE has not yet assigned an R factor to any of these systems. As a result, such systems are presently categorized under the “alternative” system category. For the design of an “alternative” system, dynamic time history analysis (DTHA) is the prescribed and acceptable method. Rocking wall systems exhibit highly dynamic and nonlinear responses, and the higher mode effects become more prominent with the increasing height. DTHA allows for the simulation of these complex behaviors under different ground motions, which are difficult to capture accurately with simpler, more conventional analysis methods. There are, however, factors that affect the analysis result, including the selection of the ground motions, scaling method, and assumptions used in the numerical model. In addition to codes providing guidance on selecting and scaling records, there are research studies that propose new or improved methods for selecting and scaling records. This paper takes a practical approach and focuses on the investigation of the impact of the selection and scaling of ground motions on the structural response of selected archetype rocking wall systems with low-damage friction hold-downs, which have gained momentum in recent years, especially in the field of mass timber buildings. The investigation into modeling considerations examines different damping models, time history analysis methods, and the variables inherent in these models that can affect the structural response and analysis results. This paper provides valuable insights for those interested in adopting friction connections for low-damage design of resilient rocking wall structures.

2. Numerical Modeling Considerations

2.1. Ground Motion Scaling

Ground motion recordings from large and rare events are scarce. Hence, designers often utilize records from smaller events, which are more abundant by nature. Each record is individually scaled to the intensity of the larger “design earthquake” event. This is illustrated in Figure 1.

Figure 1a depicts two records that have been amplitude scaled [12]. Equation (1) shows the scale factor ${SF}_{S_a}$ required to produce the best fit between the record’s response spectrum $S_a^{unscaled}$ and the target response spectrum $S_a^{target}$. This value is found by minimizing the sum of squared errors ${SSE}_{S_a}$ in Equation (2).

(1)${SF}_{S_a}={\displaystyle \frac{\sum _{i=1}^N\left[S_a^{unscaled}\left(T_i\right)·S_a^{target}\left(T_i\right)\right]}{\sum _{i=1}^N{\left[S_a^{unscaled}\left(T_i\right)\right]}^2}}$

(2)${SSE}_{S_a}=\sum _{i=1}^N{\left[{SF}_{S_a}·S_a^{unscaled}\left(T_i\right)-S_a^{target}\left(T_i\right)\right]}^2$

Alternatively, the scale factor ${SF}_{\ln S_a}$ shown in Equation (3) applies when scaling in logarithmic ordinates to minimize the error ${SSE}_{\ln S_a}$ shown in Equation (4). This is the preferred method in some design codes like NZS1170.5 and NBCC2020 because ground motion responses assume a lognormal distribution [13]. However, for practical purposes other design codes employ scaling in linear coordinates, which emphasize a better fit across the shorter periods.

(3)${SF}_{\mathit{ln}{S}_a}=\mathit{exp}\left({\displaystyle \frac{1}{N}}{\sum }_{i=1}^N\left[\mathit{ln}{S}_a^{target}\left(T_i\right)-\mathit{ln}{S}_a^{unscaled}\left(T_i\right)\right]\right)={\left[{\prod }_{i=1}^N{\displaystyle \frac{S_a^{target}\left(T_i\right)}{S_a^{unscaled}\left(T_i\right)}}\right]}^{{\displaystyle \frac{1}{N}}}$

(4)${SSE}_{\mathit{ln}{S}_a}={\sum }_{i=1}^N{\left[\mathit{ln}\left({SF}_{\mathit{ln}{S}_a}·S_a^{unscaled}\left(T_i\right)\right)-\mathit{ln}\left(S_a^{target}\left(T_i\right)\right)\right]}^2$

Designers are often interested only in the mean or median response across several ground motions. As shown in Figure 1c, spectral matching achieves this purpose by suppressing the variance, and this enables a quicker convergence on the mean or median with fewer analyses [14]. Essentially, spectral matching modifies the ground motion record by inserting a series of $n$ wavelets at the time of peak responses corresponding to the $n$ oscillator periods. Thus, the peak responses at $n$ periods can be adjusted. The magnitude of each wavelet is determined such that their combined effect provides the necessary adjustment at each of the $n$ periods (i.e., the difference between the scaled and target response spectra). More formally, the spectral matching procedure is described by Equations (5) and (6) [15]. The element ${\mathit{c}}_{\mathit{i}\mathit{j}}$ in the matrix $\mathbf{C}$ represents the peak response of an oscillator with period $i$, when subjected to a wavelet of unit amplitude and period $j$. In other words, it quantifies how each of the $n$ oscillators respond to all $n$ individual wavelets. The summation of these responses must satisfy the adjustments required by vector $\mathbf{R}$ across all $n$ periods simultaneously. Hence, the magnitudes of the wavelets $\mathbf{b}$ are found by solving this system of simultaneous equations.

(5)$\mathbf{C}\mathbf{b}=∆\mathbf{R}$

(6)$\left[\begin{array}{ccc}{\mathrm{c}}_{11}& \cdots & {\mathrm{c}}_{1\mathrm{n}}\\ ⋮& \ddots & ⋮\\ {\mathrm{c}}_{\mathrm{n}1}& \cdots & {\mathrm{c}}_{\mathrm{n}\mathrm{n}}\end{array}\right]\left[\begin{array}{c}{\mathrm{b}}_1\\ ⋮\\ {\mathrm{b}}_{\mathrm{n}}\end{array}\right]=\left[\begin{array}{c}∆{\mathrm{R}}_1\\ ⋮\\ ∆{\mathrm{R}}_{\mathrm{n}}\end{array}\right]$

One argument against spectral matching is that the loss of record-to-record variability can be unconservative. Because of this, ASCE7-22 penalizes spectral matching by requiring a target spectrum that is 10% higher. Meanwhile, mean spectrum matching attempts to strike a middle ground between the two extremes of amplitude scaling and spectral matching. The goal is to ensure that the mean spectrum (averaged across all records) matches the target spectrum while preserving the variability between records, as shown in Figure 1b [16,17]. Thus, the records are only lightly modified by the addition of wavelets. This requires a linear scaling of the set of records, period-by-period, by the individual differences between the mean spectrum and target spectrum, as shown in Figure 1b. To achieve this, the procedure begins by amplitude scaling all records to give an approximate fit and avoid excessive alteration from matching. The mean spectrum is calculated as the average of all the records’ spectra. Next, find the set of $n$ scale factors needed to match the mean spectrum to the target spectrum at each of the n periods. Now, spectral matching is performed for each record, except that the adjusted spectrum (and hence, the input $∆\mathrm{R}$) is determined by the individual record’s spectrum multiplied by the set of $n$ scale factors at $n$ period points. By performing this for all records, the mean spectrum is shifted by the scale factor corresponding to each of the n period points. Thus, the records are only lightly modified by spectral matching.

2.2. Ground Motion Selection

Traditionally, the ground motions selected for structural analyses would possess similar seismological (tectonic source and site) characteristics as the seismic hazard expected for the structure [18]. However, engineering parameters, like the response spectrum, are known to correlate better with structural response quantities [19]. This is also acknowledged by design codes like ASCE7-22, which allow a looser match of the seismological parameters if a better match of spectral shapes is possible. Some regions like southwest British Columbia can be exposed to earthquakes arising from multiple types of source mechanisms (tectonic source) simultaneously, namely shallow crustal, subduction interface, and subduction intraslab mechanisms [20]. Similarly, New Zealand has faults that exhibit these main tectonic mechanisms and characteristics. In New Zealand, these faults have all been identified, and zones have been defined [21,22]. Large-magnitude subduction earthquakes produce longer shaking duration that induce a larger number of inelastic cycles on the structure [23]. Thus, it may not be sufficient to select ground motions based on the fit of the response spectrum alone, since it indicates the peak response only and does not offer any information on the number of cycles expected.

2.3. Friction Hold-Downs

Shear walls have performed well in past earthquakes, with only a few structures sustaining catastrophic damage or collapse [24]. The walls are, however, often severely damaged and endure plastic deformation (particularly at the bottom of the wall) or large residual deformations due to a ratcheting effect. In many cases, these structures are rendered non-operational and must be demolished due to the high cost of repairs and replacements. These damage observations have led to efforts to reduce structural damage during major earthquakes through the implementation of energy dissipation systems or low-damage structural systems. By maintaining the structure’s elastic capacity (via capacity protection and overstrength) and incorporating energy dissipators such as friction or viscous dampers, structures can achieve low-damage performance. Friction connections have been extensively developed since the study by [25]. A variety of symmetrical and asymmetrical friction connections have been developed to meet different application needs and achieve optimal energy dissipation (Figure 2). While friction solutions provide highly effective and noticeable damping [26], they may suffer residual displacement following an earthquake, requiring adjustments to the connections and the structure as a whole. The Resilient Slip Friction Joint (RSFJ) was developed at the University of Auckland [27] to provide a complete system that provides self-centering and damping while offering a resilient and low-damage solution (Figure 2). In a variety of fields ranging from concrete to steel and mass timber, it has been the subject of numerous numerical and experimental studies [3,8,28,29]. The RSFJ delivers repeatable and multi-cycle flag-shaped energy dissipation without damage or yielding to the elements.

2.4. Numerical Modeling Assumptions and Variables

In numerical modeling, particularly in dynamic nonlinear analyses, variations in assumptions and considerations regarding modeling techniques and methods selected can influence the outcomes of the analysis and the determination of seismic demand. It is likely that the majority of these selections will be constrained or predefined by the commercial software employed. Selections and assumptions will also vary depending on the type of nonlinearity and archetype being numerically analyzed. The damping model and the integration technique are two major considerations that could influence the numerical results. The definition and selection of the damping model have been extensively studied and discussed [30,31,32,33], mainly in the form of modal damping, Rayleigh damping, proportional damping, and material damping models. There has also been extensive study and discussion on the definition and selection of time integration methods [30,31,34,35], primarily in the form of the Newmark method, Hilber–Hughes–Taylor method, Wilson Theta method, and Runge–Kutta method, amongst others. Section 3.4 provides a brief description of the damping models and time integration methods available in ETABS that are suitable for the numerical modeling of rocking walls with friction dampers. Subsequently, Section 4.3 examines their influence on performance and seismic demand determination.

3. Methodology

3.1. Selected Archetype and Numerical Mmodelling

This paper presents a two-dimensional model of a ten-story mass timber structure with two rocking walls in ETABS [36]. As of now, engineers have pushed the limits of engineering and successfully constructed mass timber buildings up to eleven stories in earthquake-prone regions. In a study on rocking Cross Laminated Timber (CLT) walls with friction hold-downs, higher mode effects were shown to become evident after six stories [37]. Therefore, a ten-story building would be ideal for capturing these higher mode effects and highlighting the purpose of this study. The structure consists of a ten-story gravity frame made of Glulam columns and Laminated Veneer Lumber (LVL) beams, measuring 41 m wide and consisting of seven bays (Figure 3). Two balloon-type CLT rocking walls are offset (de-coupled) from the gravity frame in bays number two and five. The five-layered CLT rocking walls are constructed from Machine Stress-Graded (MSG) sawn timber with a modulus of elasticity of 8 GPa in the longitudinal direction and 6 GPa in the transverse direction. All subsequent floors measure an equal height of 3.8 m, except the first floor, which measures 4.8 m, making the total height of the structure and rocking walls 39 m. The building is assumed to be located in Wellington, New Zealand, with hazard factor Z = 0.4 and considering 1/500 earthquake event. Given the resilient low-damage design approach applied to the structure’s design and modeling, a conservative Sp value of 0.85 has been adopted for this study. Loading considerations are based on a mix of residential and commercial structures. There is a dead load of 2.75 kPa on all floors and a live load of 3 kPa on all floors except for the roof, which carries a dead load of 1.7 kPa and a live load of 1 kPa. Self-weight of the members and the structure is calculated automatically by ETABS 22.4. As the mass timber gravity frame completely supports the gravity load, rocking CLT walls are decoupled from the gravity load-resisting system in this study. The rocking walls utilize flag-shaped friction hold-downs (RSFJ) to provide energy dissipation and self-centering, allowing safe rocking movement of the wall. It has been demonstrated through tests with rocking CLT walls and flag-shaped friction hold-downs that the combined rocking and friction joints provide self-centering and damage-free deformation and ductility [3,38]. To ensure adequate load transfer to the walls, shear key links have been modeled between the frame structure and the rocking walls at every level. These shear key links transfer lateral loads to the walls while simultaneously allowing uplift and upward movement of the wall, preventing displacement incompatibility with the floor or beam [37]. It is important to emphasize that the gravity frame only transfers gravitational demand and does not bear lateral demand except for transferring this demand to the walls, as all beams are released at both ends and column bases are pinned. System’s entire nonlinearity is contained within rocking of the CLT walls, i.e., the flag-shaped friction hold-downs (RSFJ). In the ETABS model, effective section properties were implemented based on the FPI Innovation Modelling Guide [39] and the Canadian CLT handbook [40], and stress checks were conducted to ensure stresses were within elastic limits. Flag-shaped friction wall hold-downs are modeled using “Damper-friction spring” and rocking toes are modeled using nonlinear “Gap” link elements in ETABS [38]. In practice, to facilitate the rotation of the hold-downs in line with the rocking wall, a spherical swivel bearing will be used at the base of the friction hold-downs. Therefore, in the numerical model, a pinned boundary condition is assigned at the bottom of the damper-friction spring links. Mass is lumped at each story by default via ETABS (CSI Analysis Reference Manual). Through ETABS, the key time periods of the structure and their mass participations have been calculated based on the effective stiffness of the flag-shaped friction hold-downs. A mode 1 time period of 1.99 s, a mode 2 time period of 0.335 s, and a mode 3 time period of 0.144 s were determined with mass participation ratios of 69%, 21%, and 6%, respectively (Figure 4). Due to the fact that the case study structure is considered a tall structure, nonlinear pushover analysis is not appropriate for this study, as the first three modes of vibration have a significant effect on the structure’s response, and the fundamental time period exceeds one second [41].

The investigation of this paper can be divided into three parts. In the first part, the impact of scaling method of ground motion is investigated, and in the second part, the impact of ground motion source and characteristics is investigated. The third part discusses and examines the assumptions and considerations required for numerical modeling of such models and the variables that may be involved. The information presented herein holds significant relevance and utility, particularly in the context of New Zealand and regions with similar seismic characteristics, which encompasses fault characteristics of the three principal tectonic sources. This paper offers valuable insights for practitioners and researchers considering the implementation of friction connections in the low-damage design of rocking wall structures.

3.2. Ground Motion Scaling

The first suite of records is obtained by amplitude scaling to the target spectrum, as determined by the best fit (minimum SSE) from 50 points spread across the period range for scaling. This set of pre-scaled records is then used as the seed for subsequent wavelet modifications to produce the second and third suite of records. For the suite of spectrally matched records, the accelerograms were individually matched to the target spectrum. The steps are as follows:

1. The pre-scaled record (from amplitude scaling) is used as the seed record. This helps to avoid excessive alteration to the record due to the addition of wavelets.

2. Calculate the response spectrum and the timing of the peak responses.

3. Pad the record at the beginning and/or end if necessary. This avoids the need for baseline correction due to the wavelet.

4. Calculate the adjustment required at each period. This is simply the difference between the target spectrum and the response spectrum.

5. Calculate the amplitude of the wavelets required to achieve these adjustments.

6. The record is adjusted by inserting the sum of the 50 wavelets, and the responses are re-computed by returning to Step 2.

7. The iterations are repeated until convergence is achieved. The tolerances are satisfied when the average misfit and maximum misfit are within 1% and 5% of the target spectrum.

With mean spectrum matching, the goal is to match the mean spectrum of all records rather than the spectrum of individual records. Hence, a similar process was used to obtain the suite of mean spectrum-matched records. Only a minor difference is needed to Step 4, which is now as follows:

• 4.. Calculate the mean spectrum of the pre-scaled records (from amplitude scaling). To match the mean spectrum, the ratio between the target spectrum and the mean spectrum is calculated for each period point. This gives a set of 50 ratios that must be applied to all records. Thus, the adjustment required for each record is simply the record’s spectrum multiplied by the set of 50 ratios.

DTHAs are carried out for each scaling method suit. As a means of highlighting the output, results are presented as the average of each suit together with the standard deviation in order to examine the impact of each scaling method on the response of structures. Standard deviations are added to the average primarily to account for variability and enhance interpretation, and in the case of NBCC2020, the addition is required as part of the compliance process. It should be noted that, to date, no dedicated studies have been conducted to investigate the dynamic response of rocking walls in relation to different ground motion scaling methods. Figure 5 represents the scaled ground motions with their average spectra against the target spectra for each scaling method.

As presented in Figure 6, the amplitude scaling method preserves the shape and characteristics of the records since it entails a linear scaling process. In contrast, the other two methods alter the shape, characteristics, and peaks of the record. This alteration is more apparent for spectral matching as it requires more adjustments to fit the record closely to the target spectra. This is also more evident in Figure 7, where the spectral acceleration of RSN1495_ChiChi record scaled for the three-scaling method is plotted against the target spectra.

3.3. Ground Motion Selection

In practice, it is best to carry out site-specific investigation, taking into account the soil–structure interaction and the characteristics of seismic waves as they impact the given location. Often, this step is not carried out or skipped due to cost, design, or construction time constraints. Each tectonic source produces ground motion excitation that differs from one another in terms of their characteristics, frequency, duration, and how they commence and end, all of which have been well documented [42,43]. Large-magnitude subduction earthquakes produce longer shaking duration that induce a larger number of inelastic cycles on the structure [23]. As a result, it may not be sufficient to select ground motions solely based on the fit on the response spectrum, as it indicates only the peak response and does not offer any information on the number of cycles expected, nor does it indicate any information about the structure’s response to large cycles of high- or low-frequency shaking observed from subduction sources. It has never been studied how different ground motions originating from different tectonic sources affect rocking walls with friction hold-down responses. The results of this study can assist engineers in selecting ground motions based on the rocking wall system selected and the anticipated seismic zonation [21,22] and allocating proportionate sources of ground motion for the structure for the purpose of DTHA. For this purpose, ground motion records were selected from a bank of records sourced from the PEER NGA-West2 database [44] for shallow crustal earthquakes (RSN1—RSN10000) and the PEER NGA-Sub database [45] for the subduction earthquakes (using only a subset of records with S_a (T = 1) > 0.05 g) (Figure 8). A set of 14 records for each tectonic ground motion source is selected based on which fits the best within the scaling criteria prescribed by NZS1170.5. The time period scaled ranges from the minimum period during which combined modes of vibration include 90% of mass participation to the maximum period of 1.5 times the first mode of vibration T₁. In order to conduct comparable comparisons, similar criteria have been selected for all the records. All records have been selected with similar conditions, including all within site soil classification D (180_m/s < V_s30 < 360_m/s), magnitude between 6 and 8.5 Richter, and eruption depth less than 20km (for shallow crustal records). These conditions are selected according to recommendations by [21] for Wellington, New Zealand. Ground motions are then amplitude scaled (NZS1170.5 method) to preserve the characteristics of the records. DTHAs are carried out for each tectonic source suit, and results are presented as the average of each suit together with the standard deviation in order to compare them with one another and investigate the impact of each tectonic source on the response of structure.

3.4. Numerical Modeling Considerations

Currently, there is no consensus on the most appropriate approach for modeling viscous damping in nonlinear time history analyses of buildings with friction dampers and other energy dissipation systems, primarily due to uncertainties in accurately quantifying energy dissipation [46,47]. Extensive research has been conducted over the years on various damping methods, exploring their advantages, disadvantages, and limitations for different structural types, archetypes, and lateral load-resisting systems [48,49,50,51,52,53]. There are primarily three types of damping that can be introduced into a numerical model, namely material damping, modal damping, and proportional damping. Material damping represents the inherent energy dissipation properties of materials, such as internal friction and other mechanisms. It must be incorporated within the definitions of materials or structural members, including frames, shells, cables, and similar components. Given that the archetype under study adheres to a low-damage resilient design, all nonlinearities are confined to the friction hold-downs, with all structural members remaining fully elastic. As a result, material damping is not applicable in this context and should not be considered as it offers no utility within the low-damage design framework. In modal damping, each mode of vibration is dampened separately. Complex structures with significant damping variations across modes may benefit from this approach. Modal damping, also known as constant damping, is applied in the modal space if all modes are dampened at a fixed percentage, where each mode of vibration is considered separately, and damping is applied independently to each mode regardless of the dynamic state of the structure. In highly nonlinear systems, this can lead to inconsistencies when modes evolve, interact, or change due to large displacements and nonlinear behavior. The fixed nature of damping may not adapt well to localized nonlinearities, leading to energy imbalances and poor convergence. Constant damping ratios tied to pre-selected modes (or all modes) may no longer align with the actual modal characteristics of the deformed system. This misalignment can cause over- or under-damping of key modes, leading to convergence issues. If nonlinear elements (like dampers or hinges) engage or disengage, constant modal damping may fail to balance the forces correctly. Modal damping is frequently employed as it is required for conducting nonlinear modal time history analysis, also known as Fast Nonlinear Analysis (FNA). FNA method often appeals to engineers due to its fast and efficient runtime; however, it is commonly associated with convergence challenges.

In proportional damping or viscous proportional damping (commonly referred to as Rayleigh damping), the damping is directly proportional to the stiffness matrix. This damping model is particularly suitable for structures where damping is primarily associated with stiffness properties, and discrete nonlinearities are explicitly modeled. It provides a continuous distribution of damping across frequencies and dynamically adjusts as stiffness changes due to nonlinearity. Rayleigh damping inherently accommodates evolving modal characteristics in a nonlinear system, ensuring stable energy dissipation throughout the analysis. According to Rayleigh’s damping model, the damping force is proportional to both mass and stiffness of the structure, as shown in the following equation:

(7)$\mathrm{C}=\mathsf{\alpha }\mathrm{M}+\mathsf{\beta }\mathrm{K}$

where C is the damping matrix, α and β are the Rayleigh damping coefficients, M is the mass matrix, and K is the stiffness matrix. The mass and stiffness proportional components ensure that damping is distributed smoothly and remains numerically stable. Even when stiffness changes dramatically (like damper activation or hinge formation), Rayleigh damping recalculates based on the updated stiffness matrix, aiding convergence. Nevertheless, Rayleigh damping applies different levels of damping to different frequency ranges. This can lead to under-damping or over-damping of certain modes, especially outside the selected frequency range. There is a risk that important modes, particularly higher modes, which are present on tall wall structures, may receive excessive or insufficient damping, leading to inaccurate dynamic responses, especially for short-duration (high-frequency) modes. Therefore, it is crucial to carefully select the α (mass-proportional) and β (stiffness-proportional) coefficients to target the dominant frequency range of interest. This is typically achieved by specifying the target frequencies (or primary mode time periods) in the numerical software, which then computes the corresponding coefficients.

In this study, the two damping models, modal and Rayleigh, are implemented separately to evaluate the differences in seismic demands derived by each method and to determine whether these differences are significant enough to warrant consideration. The analysis involves three approaches: Fast Nonlinear Analysis (FNA) with constant damping, Direct Integration (DI) nonlinear time history analysis with constant damping, and DI nonlinear time history analysis with Rayleigh damping. A conservative 2% inherent system (elastic) damping is assumed for this structure. This consideration aligns with the resilient wall structure under study, which adheres to a low-damage resilient design philosophy and complies with current codes, including NZS1170.5, ASCE7, and NBCC.

In this archetype, the rocking wall with friction hold-downs serves as the sole lateral load-resisting system, with all nonlinearities confined to the friction hold-downs while all structural members remain fully elastic. Accordingly, a 2% constant damping is applied across all modes for the constant damping models, while a 2% Rayleigh damping is assigned, pivoting at the first mode (fundamental mode) and the second mode, which collectively contribute to 90% of the mass participation. NZS1170.5 requires that the lower bound ensures at least 90% mass participation across the collective modes. Similarly, ASCE7 mandates a minimum of 90% mass participation, with the lower bound period not exceeding 20% of the smallest first-mode period for the two principal horizontal directions of response. NBCC also specifies a 90% mass participation requirement but permits a smaller lower bound period of 0.15 times the first-mode period. To evaluate whether the 90% mass participation requirement is sufficient and to determine if it may lead to over-damping of higher modes, which are prevalent in tall wall structures and observed in the structure under study, a second Rayleigh damping curve is defined. This curve pivots at the first and third modes, collectively covering 96% of mass participation. The differences in the resulting seismic demands are analyzed and presented in the following section. Figure 9 presents the constant modal versus Rayleigh damping curve for the structure under numerical investigation. By default, for all DTHAs, the time step is set to match the time step of each individual ground motion record. Thus, the total number of time steps for each record is determined by dividing the duration of the ground motion by its corresponding time step.

To represent nonlinearity and damping from the dampers (links), ETABS allows both nonlinear viscous damping (velocity-based damper) and nonlinear hysteretic damping (displacement-based damper) to be modeled discretely in the numerical model. In this archetype, where RSFJ hold-downs are utilized, RSFJs are modeled using “damper friction spring”, and ETABS is able to reproduce the same flag-shaped hysteresis that RSFJs produce in practice and during component testing.

Time integration methods are numerical techniques used to solve the equations of motion for dynamic systems over time. According to the CSI analysis manual, the Hilber–Hughes–Taylor alpha method should be used by default regardless of the type of structural numerical model used. This study examines the influence and impact of alpha value selection on the results and outcomes of DTHA. Hilber–Hughes–Taylor is an implicit time integration method, is a modification of the Newmark method, and includes an additional parameter, alpha (α), to control the numerical behavior of the solution. The alpha value can range from 0 to −0.33. With an alpha value of zero, it practically represents the Newmark method [29]. According to CSI reference manual, “Using α = 0 offers the highest accuracy but may permit excessive vibrations in the higher frequency modes… for more negative values of alpha, the higher frequency modes are more severely damped”. While the use of negative values of alpha helps significantly with the computational speed of the numerical model, it has been observed to introduce unbalanced energies and energy errors in the system, which result in unconservative seismic demands predicted. A sensitivity analysis is performed in this study with alpha values of −0.33, −0.3, −0.2, −0.1, −0.05, 0.0, and the results are discussed.

4. Results and Discussion

The results presented in this section aim to assist practitioners and engineers in conducting numerical models for the design of resilient rocking wall systems with friction connections, providing additional insights and guidance for this process. Based on the observational data and the compiled results, it is found that following the initial oscillation characterized by mode shape one, the structure predominantly oscillated in modes two and three, in line with analytical studies on rocking structures [54,55]. In fact, this trend was much more pronounced with the records that had high-frequency excitations. This observation highlights the significance of dynamic analysis, as evidenced by the notable influence of these subsequent modes on response parameters, such as base shear, story shear demands, and peak story accelerations. This reiterates the susceptibility of wall systems to the higher mode effects, and with no exception, those employing rocking walls with friction damper hold-downs. It is observed that the structure’s vibrational response is coupled with the rocking mechanism and influenced by the rocking behavior [56]; however, the rocking response behavior does not influence the performance or efficiency of the hold-downs [57]. This emphasizes the necessity for careful consideration and attention during both the design and numerical modeling processes for such structures. DTHA is the only method for observing and evaluating such trends and responses.

4.1. Ground Motion Scaling

The DTHA results indicate that the spectral matching results in the highest base shear and drifts demands compared to the other scaling methods (Figure 10). Although this is not entirely unexpected, it was anticipated that the acceleration hikes associated with amplitude scaling and mean spectrum matching would result in large spikes in story shear and displacement demands. Nevertheless, a structure sensitive and susceptible to higher mode effects can be affected more and put under more stress if elevated accelerations are consistently present within a scaled period range (which correspond to the structures’ main modes of participation) close to the target spectrum, instead of pulses or abrupt increases and decreases in excitation, as seen in amplitude scaling. This is further magnified in rocking walls featuring friction hold-downs, primarily due to the elongated period following the slip or activation of friction hold-downs. The consistent elevation of accelerations resulting from spectral-matched records has significantly triggered both mode 1 and mode 2, which govern the displacement behavior and the story shear behavior of the structure, respectively. The spectral matching method is therefore more conservative than the other two scaling methods; however, it may be overly conservative if the design is penalized by the code requirements, such as the requirement to achieve 10% above the spectrum as prescribed in ASCE7-22. It is more forgiving to select records using spectral matching, but it requires greater computational effort. Some records may require up to 100 iterations to meet the criteria for misfit limits set by the user, resulting in an entirely new excitation record.

While mean spectrum matching demands fall behind spectral matching, it has predicted the least displacement demands amongst other scaling methods. This method requires considerable yet less intense computational effort and is an iterative process that allows the user to determine how much of the record to modify and set misfit limits while still retaining a majority of the original characteristics. It is also possible to examine the structure’s response to pulses and peak input accelerations by means of mean spectrum matching. Moreover, it should not be penalized by codal requirements. The results indicate that the standard deviations for spectral matching and mean spectrum matching are quite similar, and both are noticeably higher than the amplitude scaling method. Although it may be anticipated that spectral matching would lead to minimal standard deviation, given the close alignment of all records with the target spectrum, empirical observation reveals that spectral matching predominantly results in consistently elevated intensities across all records throughout all periods of vibration; therefore, each record is likely to generate peak responses throughout the duration of structural excitation. An alternative perspective emerges from the derivation of the target spectrum for a single degree of freedom (SDOF) system, which may not accurately reflect the behavior of the multi-degree of freedom (MDOF) numerical model under examination. This is particularly relevant given the model’s demonstrated sensitivity to higher mode effects.

The amplitude scaling method has resulted in the lowest base shear in comparison to all three methods. However, it predicts higher displacement demands than mean spectrum matching, despite being lower than spectral matching. DHTA, with amplitude scaling records, often results in one or two records being discarded as having unreasonable outputs. It requires rigorous selection and criteria to find “good fit” records, whether to meet codal requirements or to achieve design objectives, such as limiting the k₁ scale factor or reaching target peak ground acceleration. Overall, while the three scaling methods offer varying predictions of demands, their outcomes remain closely clustered. However, penalizing spectrum-matched records by 10% would be overly conservative. As such, all three methods can be confidently employed, provided adequate care is exercised throughout the scaling procedures. However, given the computational effort involved and the sensitivity of the structure to the characteristics of ground motion records, as demonstrated in the next section, it is recommended to prioritize amplitude scaling. Using the mean plus standard deviation approach provides a more efficient and practical method for analysis.

Figure 11 illustrates the performance comparison between the flag-shaped friction hold-down and the top displacements across three scaling methods. It demonstrates that amplitude scaling reflects the realistic displacement characteristics of the structure and hold-down for the particular ground motion record. For mean spectrum matching, the behavior and characteristics remain broadly consistent, with only an increase in demand observed. Conversely, spectral matching reveals noticeable alterations in the displacement profiles of both the structure and hold-down, with a notably greater increase in demand. Nonetheless, the flag-shaped friction hold-down has provided complete self-centering for the rocking wall structure.

4.2. Ground Motion Selection

Compared with subduction interface and shallow crustal records, subduction intraslab ground motions exhibit significantly higher base shear demands (Figure 12). This is due to the long duration of high-frequency vibration, and it is observed that within this long vibration, the structure mainly transitions and oscillates between modes 2 and 3. This is further confirmed with the roof and story drift outputs from intraslab ground motions, which have resulted in the least among others. This is evident from modes 2 and 3, which are the story shear dominant modes, prevailing over mode 1, which is the displacement dominant mode. The peak ground floor accelerations (Figure 13) also reveal the same pattern. Ground motions originating from subduction zones are generally associated with long-duration, high-energy earthquakes. It is evident that tall rocking wall structures with friction hold-downs exhibit greater sensitivity and more pronounced responses to long-duration, high-frequency excitations generated from intraslab sources, as opposed to short-duration, intense, low-frequency excitations from shallow crustal sources. For wall systems dependent solely on friction or yielding of hold-downs, prolonged exposure to large-cycle subduction records can offer insights into the potential occurrence of ratcheting behavior. However, resilient friction hold-downs, such as RSFJ hold-down, which provide restoring force and self-centering capabilities, have been demonstrated to mitigate any concerns regarding ratcheting behavior.

Additionally, it is evident that short, peak-pulsed records generated from shallow crustal sources distinctly challenge the displacement demands of structures, pushing them to their maximum displacement capacity. These records can offer valuable insights into the potential displacement demands that may be required, prompting engineers or designers to re-evaluate the system for necessary adjustments or refinements. By selecting ground motion records that fit well with the spectrum and the codal criteria and by using good engineering judgment and scaling procedures, all three scaling methods should be valid and provide adequate results to be used for design. On the other hand, if a subduction intraslab source exists within the zone of the structure, special consideration should be given, and the authors recommend adopting the NBCC approach, wherein one-third of the records selected for DTHA should be selected from subduction intraslab ground motion source.

It may not be sufficient to choose ground motions based solely on how well they match the response spectrum since this approach only indicates the highest response and does not indicate the anticipated number of cycles that may affect residual drifts. This detail is crucial, especially when utilizing friction connections. This is of less concern when flag-shaped friction hold-downs such as RSFJ are implemented, and as shown in Figure 10 and Figure 12, the structure for all scaling methods and ground motions is fully self-centered. However, in cases where subduction intraslab or interface faults are present within the structure zone, it is recommended to diversify ground motion selection. It is recommended to adopt the NBCC approach by allocating one-third of the ground motions from subduction interface sources, another third from subduction intraslab sources, and the remaining third from shallow crustal sources based on site-specific considerations. NBCC prescribes the use of three suites of records for each source, with each suite containing 11 records, resulting in a total of 33 records. In New Zealand, the common practice is to either select three records (as suggested by the allocated geotechnical engineer) and consider the critical record demands as the design demands or to use a suite of seven records and take the average of the demands as the design demands [58]. In the context of New Zealand, the authors recommend extending the suite of seven records to nine records, selecting three records from each tectonic source to align effectively with the NBCC approach. Further statistical and mathematical studies could refine and enhance this methodology.

Figure 14 presents a distinct example of each ground motion tectonic source, along with their respective structural displacement responses and flag-shaped friction hold-down performance. The variations in ground motion inputs, including duration of vibration, pulses, and characteristics, are evident. These differences markedly influence the structural displacement response and the performance of the flag-shaped friction hold-down, as clearly depicted in the figure. Despite the varied input characteristics, the flag-shaped friction hold-down has consistently provided self-centering behavior for the structure. This outcome is achievable only when the friction hold-down exhibits repeatable and pinching-free hysteresis behavior. The findings also highlight the necessity of subjecting such hold-downs to rigorous testing regimes, like those outlined in ASCE7-22, to verify their ability to withstand intense, prolonged, and highly cyclic pulsed ground excitations typical of subduction tectonic sources.

4.3. Modeling Considerations

There is currently no direct method to determine which damping model can more accurately predict the seismic demand of the archetype structure due to the lack of benchmark data from monitored structures during actual earthquakes or shake table tests. None of the existing archetype structures have been exposed to real earthquakes, nor do they have the instrumentation required to capture response data. Furthermore, large-scale shake table tests of such structures are rare, extremely costly, and typically conducted only once a decade. Given these limitations, engineers designing and constructing resilient structures have relied on numerical models and engineering judgment, supported by the proven reliability of current commercial software in terms of computational accuracy and solver precision [6,8]. These designs are also subjected to thorough peer reviews to ensure adequacy and safety.

It is worth noting that the differences in demands and behavior observed are likely as significant as they can be for such an archetype structure. This is because the ten-story structure under investigation represents the current maximum height achieved by engineers for mass timber wall structures and appears to be the upper limit of viable design options. The FNA analyses were completed in a fraction of the time required for the DI analyses, and it is demonstrated that the FNA results provide a higher estimation of seismic demands (Figure 15). The analyses demonstrate that, irrespective of the modeling considerations and variables, the selected archetype consistently achieves complete self-centering.

It can be argued that for such an archetype structure, FNA is a viable, reliable, and conservative analysis method for use in the design process. Considering that designs incorporating supplementary dampers often require numerous time history iterations to achieve target seismic performance factors (such as the ductility factor μ for NZS 1170.5 or the R factor for ASCE and NBCC) and target drifts, FNA offers a significantly faster and more efficient alternative compared to DI methods. The analyses reveal that both FNA and DI methods using constant damping result in higher base shear compared to those using Rayleigh damping, primarily due to the over-damping of higher modes inherent in the Rayleigh damping approach. This trend is consistent with the higher roof displacements observed in the DI Rayleigh damping analysis (T1 to T2). While the maximum difference in base shear is noticeable at 14%, the roof displacement difference is minimal at only 2%, and the maximum interstory drift is nearly identical across all cases. When Rayleigh damping coefficients are more carefully selected (T1 to T3) to account for all three major modes, rather than simply meeting the 90% mass participation requirement, the differences in base shear become negligible. The analyses demonstrate that, instead of merely meeting the codal 90% mass participation requirement, it is essential to include all major modes, defined here as any mode contributing more than 3% mass participation, within the Rayleigh damping range. For resilient archetype structures of this kind, the design is typically governed by target drifts, as the lateral load-resisting system (LLRS) is consistently capacity-protected using appropriate overstrength factors. As evidenced by the results, all cases produce similar seismic drift demands. Therefore, the authors conclude that all methods of modal consideration, provided they are selected and defined carefully, should produce comparable results and are acceptable for this resilient archetype. For this archetype, it is recommended to utilize the FNA method during the design process, as it offers a fast, efficient, and reliable approach to time history analysis, which is particularly beneficial during iterative design phases. If additional confidence is required or as part of a peer review process, a final DI analysis with Rayleigh damping can be conducted to verify the results.

However, modeling damping using the constant damping model often encounters convergence issues. To mitigate these issues, the following recommendations should be considered: 1—Reducing the time step size. By default, the time step size should match the time steps of the ground motion record. If convergence issues persist, further reducing the step size can help stabilize the analysis. 2—Defining an adequate number of modes to ensure that the minimum and maximum number of modes specified for the Ritz modal load case are sufficient to capture all relevant modes and achieve 100% mass participation. Comparison of the Ritz modes with the Eigen modes will ensure the accuracy of the Ritz modes. There should be a close match between the modes, providing confidence in the validity of the modal analysis. 3—The ramp function defined for FNA gravity loads must be sufficiently long and gradual to prevent unintended excitation of the structure.

It is observed that the Hilber–Hughes–Taylor alpha method of integration, though it noticeably helps with the speed of DTHA, can produce significant energy errors. This method employs a single parameter, denoted as alpha (α), to incorporate numerical damping into the system. It serves to regulate the extent of high-frequency response in the analysis and can vary within the range of zero to −0.33. It is observed, however, that for any alpha value smaller than −0.1, there would be a significant increase in the energy error in the system, which would lead to an invalidation of the results. These energy errors are observed to lead to mispredictions and an unconservative estimation of demands. For instance, base shear may be up to 20% lower, and displacement demands up to 10% lower than what would be predicted with an alpha value of zero. Figure 16 illustrates the progression of error for various values of alpha utilized in the sensitivity analysis. In cases where the structure is sensitive to higher mode effects, such as tall rocking walls, it is strongly recommended that the alpha value be set to zero. The alpha value of 0 effectively transforms the integration system into the Newmark integration system, which is particularly stable and does not produce any energy imbalance. An alternative approach involves reducing the time history step size sufficiently to address discretization errors and minimize overall error while avoiding convergence issues. In the case of this numerical model, it was found that a time step size of 0.0005 effectively reduced the error to negligible levels. Base shear and displacement demands were within a 5% difference compared to predictions from the Newmark method. However, this very small step size notably increased computational time.

5. Conclusions

Selecting ground motion records that fit well with the target spectrum and code requirements, along with applying sound engineering judgment and scaling techniques, ensures that all three scaling methods are reliable and produce results suitable for design purposes; however, it would be overly conservative to penalize spectrum-matched records by 10%. It is crucial to exercise additional caution if the structure is situated in a region containing subduction tectonic sources. It is recommended to diversify ground motion selection. It is recommended to adopt the NBCC approach by allocating one-third of the ground motions from subduction interface sources, another third from subduction intraslab sources, and the remaining third from shallow crustal sources based on site-specific considerations. For the New Zealand context, selecting a suite of nine records and three records from each tectonic source effectively aligns with the NBCC approach. Further statistical and mathematical studies could refine and enhance this methodology. This is to account for the long duration of vibration resulting in a high number of short duration cycles (high frequency). As observed, tall rocking wall structures with friction hold-downs exhibit greater sensitivity and more pronounced responses to long-duration, high-frequency excitations generated from intraslab sources. Both Rayleigh and modal damping models, when carefully selected and properly defined, should yield comparable results and are deemed acceptable for this resilient archetype. For this specific archetype, it is recommended to adopt the FNA method during the design process, as it provides a fast, efficient, and reliable approach to time history analysis, making it especially advantageous for iterative design phases. It is observed that while the Hilber–Hughes–Taylor alpha method of integration noticeably aids in the speed of DTHA, it can lead to significant energy errors and unconservative demand predictions. Therefore, for this archetype, it is strongly advised to set the alpha value to zero.

The combined conclusions from all analyses conducted demonstrate that the rocking wall system with friction hold-downs is a resilient and low-damage solution. Across all ground motion records, modeling approaches, and selection methods, no scenario resulted in catastrophic displacement or force demands. This indicates that the LLRS can reliably withstand and perform as expected under a wide range of earthquake conditions and characteristics, maintaining its resilient performance. Furthermore, the results consistently show that, irrespective of analysis considerations and variables, the selected archetype exhibits complete self-centering behavior.

The presence and influence of higher mode effects have recently become a focal point of attention and are shown here to be both crucial and significant, highlighting the need for careful consideration. It is recommended that future research should focus on addressing these effects, ideally incorporating low-damage friction joints as part of the solution.

Author Contributions

Conceptualization, S.A. and A.H.; methodology, S.A. and A.H.; software, S.A. and N.C.; validation, S.A., A.H. and N.C.; formal analysis, S.A.; investigation, S.A.; resources, S.A. and N.C.; data curation, S.A; writing—original draft preparation, S.A.; writing—review and editing, S.A. and A.H.; visualization, S.A.; supervision, A.H. and P.Q.; project administration, A.H. and P.Q.; funding acquisition, A.H. and P.Q. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures

Figure 1. (a) Amplitude scaling. (b) Mean spectrum matching. (c) Spectral matching.

Figure 2. (a) Slip friction joint, (b) idealized load deformation of slip friction joint, (c) RSFJ wall hold-down, (d) flag-shaped load deformation of RSFJ.

Figure 3. Selected archetype and numerical modeling of ten-story mass timber structure with balloon-type CLT rocking walls, utilizing flag-shaped friction damper (RSFJ) hold-downs.

Figure 4. Exaggerated mode shapes of the numerical model structure.

Figure 5. Scaled ground motions via method: (a) amplitude scaling, (b) spectral matching, (c) mean spectrum matching.

Figure 5. Scaled ground motions via method: (a) amplitude scaling, (b) spectral matching, (c) mean spectrum matching.

Figure 6. RSN1495_ChiChi ground motion scaled records plotted as a comparison between the different scaling methods.

Figure 7. Scaled spectral acceleration of RSN1495_ChiChi record versus target spectra for three scaling methods.

Figure 8. Scaled ground motion samples from each tectonic source: shallow crustal, subduction interface, subduction intraslab.

Figure 9. Rayleigh damping curves vs constant damping for the structure under numerical study.

Figure 10. DTHA output as mean plus the standard deviation for base shear, maximum roof displacement, maximum interstory drift, and maximum interstory residual drift.

Figure 11. Performance variations between the flag-shaped friction hold-down and top displacements under three different ground motion scaling methods for RSN4458: (a) amplitude scaling, (b) spectral matching. (c) mean spectrum matching.

Figure 12. DTHA output as mean plus the standard deviation for base shear, maximum roof displacement, maximum interstory drift, and maximum interstory residual drift.

Figure 13. Peak floor acceleration: (left) ground motion records scaled for three main scaling methods and (right) records from three main tectonic sources.

Figure 14. A distinct example of each ground motion tectonic source, along with their respective structural displacement responses and flag-shaped friction hold-down performance: (a) shallow crustal ground motion RSN803, (b) subduction interface ground motion NGAsubRSN4024857, (c) subduction intraslab ground motion NGAsubRSN4032649.

Figure 15. DTHA output as mean plus the standard deviation for base shear, maximum roof displacement, maximum interstory drift, and maximum interstory residual drift.

Figure 16. The alpha values and the progression of error in numerical analysis.

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