Parametric Analysis of Moment-Resisting Timber Frames Combined with Cross Laminated Timber Walls and Prediction Models Using Nonlinear Regression and Artificial Neural Networks

Abstract

The light weight and moderate stiffness of multistorey timber buildings make them susceptible to increased lateral displacements and accelerations under service-level wind loading. Therefore, the fulfilment of serviceability requirements is a major challenge. In this study, linear elastic finite element analysis was used to perform a parametric study of moment-resisting timber frames combined with cross laminated timber walls. In the parametric study, various mechanical and geometrical parameters were varied within practical ranges. The results of the parametric study were used to derive simplified analytical expressions and to train artificial neural networks which can be used to estimate fundamental frequency, mode shape, top floor displacement, maximum inter-storey drift, and wind-induced acceleration. The analytical expressions and the artificial neural networks can be used for the preliminary assessment of serviceability performance of moment-resisting timber frames with and without cross laminated timber walls, under service-level wind loading.

1. Introduction

Timber has an excellent weight-to-strength ratio, primarily due to its light weight compared to other building materials. Nevertheless, the lightweight nature and moderate stiffness of timber buildings make them susceptible to serviceability problems, such as excessive lateral displacements and accelerations under service-level wind loading [1,2,3,4,5,6,7]. Large lateral displacements can cause damage to non-structural elements and may impact the building’s functionality, and large accelerations may cause discomfort to the occupants. Therefore, it is important to ensure that lateral displacements and accelerations are within acceptable limits.

Lateral displacements can be reduced by enhancing the lateral stiffness of the building, e.g., by use of stiffer connections [1,2,3,4]. Wind-induced accelerations can be reduced by increasing the lateral stiffness [1,2,3,4], the mass [1,2,4], or the damping [8] of the building, or a combination.

Several Lateral Load-Resisting Systems (LLRSs) can be used to provide the lateral stability for multistorey timber buildings. A common LLRS is based on Cross Laminated Timber (CLT) walls. Examples for buildings that employ CLT walls as LLRS are Stadthaus in London [9], and Moholt 50/50 [10] in Trondheim. Multistorey timber buildings employing lateral bracing as LLRS are also prevalent, such as Treet [5] and Mjøstårnet [11] in Norway. Nevertheless, these LLRSs impose architectural restrictions and space limitations.

Moment-Resisting Timber Frames (MRTFs) can be used as an alternative structural system that can provide more open space with fewer architectural restrictions. In MRTFs, the lateral stiffness is achieved by use of moment-resisting beam-to-column connections. Such connections can be based on screwed-in threaded rods [3,12,13,14,15], glued-in rods [16,17,18], self-tapping screws [19], or dowel-type connections [20,21]. The use of moment-resisting connections reduces the need for lateral bracing and improves the performance of floors against human-induced vibrations [1,14].

A parametric study by Vilguts et al. [2] highlighted the feasibility of MRTFs made with Glued Laminated Timber (glulam) for multistorey timber buildings. However, under high wind velocities, the system can hardly be used for more than eight storeys with limited out-of-plane spacing of 2.4 m between adjacent frames, due to increased wind-induced accelerations and lateral displacements [2].

The use of MRTFs in combination with CLT walls (i.e., dual frame-wall) can allow for buildings up to 12 storeys with out-of-plane spacing of up to 5 m under moderate to high wind velocities [1]. Moreover, stairs and elevators usually exist in multistorey buildings and walls often surround the staircase and elevator shaft; hence, employing CLT walls in combination with MRTFs presents a practical consideration.

In this study, a parametric study of planar (i.e., 2D) MRTFs in combination with CLT walls was performed using linear elastic Finite Element Analysis (FEA). The results of the parametric study were used to derive simplified analytical expressions for fundamental frequency, mode shape, top floor displacement, maximum inter-storey drift, and wind-induced acceleration by use of nonlinear regression. Additionally, the results were utilized to train Artificial Neural Networks (ANNs) that can be used to predict the aforementioned response parameters with improved accuracy.

For non-seismic design, the design of multistorey timber buildings is typically governed by Serviceability Limit State (SLS) [1,2,3,5]. A parametric study of timber dual frame-wall system highlighted that SLS requirements are more critical than the ultimate limit state [1]. The derived analytical expressions and ANNs can be used to facilitate the preliminary assessment of multistorey timber buildings, specifically addressing the SLS requirements.

Although the parametric study was performed using 2D FEA and resulted in analytical expressions and ANNs specifically designed for 2D analysis, a discussion on the behaviour of the structural system in a 3D context is also included.

2. Methods

2.1. Structural System

Figure 1 shows an example of a multistorey timber building. The LLRS in X direction is a dual frame-wall system. In Y direction, the building is stabilized using CLT walls around the staircase or elevator shaft, and lateral bracing. In this example, CLT rib panel floors were assumed. The focus in this paper is on the system in X direction (i.e., the dual frame-wall system). The dual system consists of continuous CLT walls and glulam columns, and glulam beams. The connections between beams and columns or walls are moment-resisting. In X direction, all beams, columns, and walls were assumed of a double cross section; consult Figure 1.

2.2. Parametric Study

Figure 2 shows an example of a dual frame-wall system. The present study considers the response of the dual system in 2D. The cases of two and three bays were considered, and the location and number of CLT walls were varied; see Figure 2b,c. The case of no CLT walls was also considered; see variation 3 and variation 7 in Figure 2b,c.

Table 1. Parameters of the parametric study.

Parameter	Value	Parameter	Value	Parameter	Value
$n$	4/6/8/10/12/14	$q$ (kN/m2)	1.5/3.0	$h_{\mathrm{b}}$ (mm)	450/540/630
$h$ (m)	3.0/3.5	$w$ (kN/m)	1.0	$h_{\mathrm{c}}$ (mm)	420/585/765
$n_{\mathrm{b}}$	2/3	$C/C$ (m)	3.0/4.5/6.0	$h_{\mathrm{w}}$ (m)	1.0/2.0/3.0
$L_{\mathrm{b}}$ (m)	6.0/7.5/9.0	$b$ (mm)	140/215/430	${\beta }_{\mathrm{b}}$, ${\beta }_{\mathrm{c}}$, ${\beta }_{\mathrm{w}}$	1/2/5/10

summarizes the parameters of the considered frames; the relevant notations are shown in Figure 1 and Figure 2. The values of the varied parameters were selected based on practical ranges commonly used in practice.

The ratios ${\beta }_{\mathrm{b}},{\beta }_{\mathrm{c}}$, and ${\beta }_{\mathrm{w}}$ were defined as the ratio of the connection stiffness to the flexural stiffness of beams, columns, and walls, respectively:

$${\beta }_{\mathrm{b}}={\displaystyle \frac{K_{\mathrm{\theta },\mathrm{b}}·L_{\mathrm{b}}}{E{I}_{\mathrm{b}}}};{\beta }_{\mathrm{c}}={\displaystyle \frac{K_{\mathrm{\theta },\mathrm{c}}·H}{E{I}_{\mathrm{c}}}};{\beta }_{\mathrm{w}}={\displaystyle \frac{K_{\mathrm{\theta },\mathrm{w}}·H}{E{I}_{\mathrm{w}}}}$$

(1)

where $K_{\mathrm{\theta },\mathrm{b}}$ is the rotational stiffness of beam-to-column/wall connections, $K_{\mathrm{\theta },\mathrm{c}}$ is the rotational stiffness of column-to-foundation connections, and $K_{\mathrm{\theta },\mathrm{w}}$ is the rotational stiffness of wall-to-foundation connections. $L_{\mathrm{b}}$ and $H$ are the bay length and the total height, respectively. $E{I}_{\mathrm{b}}$, $E{I}_{\mathrm{c}}$, and $E{I}_{\mathrm{w}}$ are the beam, the column, and the wall bending stiffness, respectively. All relevant notations are shown in Figure 1 and Figure 2.

For each FEA simulation, the location and the number of the CLT walls were randomly selected from the variations shown in Figure 2b,c. The preceding variation of parameters in Table 1 results in a total of 746,496 analyses.

2.3. Finite Element Analysis

Figure 2a shows the FEA model considered in this paper. OpenSeesPy [22] Python library was used to perform linear elastic 2D FEA. Beams, columns, and walls were modelled using Timoshenko beam elements. The beam elements representing the CLT walls were verified against layered shell elements under in-plane loading. The verification was performed on both stresses and deformations, and the difference was of the order of 5%. Hence, beam elements were deemed to have sufficient accuracy for the purpose of this study.

The glulam beams and columns were assumed of strength class GL30c according to EN 14080 [23], with a mean elastic modulus E_0,mean of 13,000 N/mm² and a mean shear modulus G_mean of 650 N/mm². The connections between the beams and the columns/walls were modelled using rotational springs with stiffness $K_{\mathrm{\theta },\mathrm{b}}$; consult Figure 2a. Similarly, the connections of the columns and the walls to the foundation were modelled using rotational springs with stiffness $K_{\mathrm{\theta },\mathrm{c}}$ and $K_{\mathrm{\theta },\mathrm{w}}$, respectively.

The lamellae constituting the CLT panels were assumed of strength class C24 according to EN 338 [24]. It was assumed that 2/3 of the lamellae are parallel to the longitudinal (main) direction of the CLT walls (main direction as indicated in Figure 2a). The remaining 1/3 is orthogonal to the main direction. The CLT walls were modelled assuming a homogeneous cross section with equivalent material properties as proposed by [25]. This simplified modelling approach was deemed to have reasonable accuracy for CLT walls loaded in-plane [26]. Based on the preceding assumptions, an equivalent elastic modulus E_eq,0,mean and an equivalent shear modulus G_eq,mean of 8000 and 518 N/mm² were considered in this study, respectively.

Modelling columns and walls using beam elements (i.e., 1D) results in an increased beam span. Rigid elements with high stiffness were added at both ends of the beam to avoid this increase in beam span; see Figure 2a. The rigid elements allow no shear or bending deformations.

To estimate the modal properties (frequency and mode shape), modal analysis was performed. For lateral displacements, the second-order effects due to nonlinear geometry were not included. The exclusion of the second-order effects was implemented to reduce computational time and enable extrapolation to loads beyond those presented in Table 1. Nevertheless, an additional dataset of approximately 130,000 analyses was created to quantify the magnitude of including the second-order effects on lateral displacements.

2.4. Calculation of Wind-Induced Acceleration

Several methods exist in building design codes for the estimation of wind-induced acceleration; a comparison between different methods can be found in [27,28]. In this study, the peak wind-induced acceleration was calculated using the method described in EN 1991-1-4 Annex B [29]. Relevant parameters that were kept constant in the parametric study (see Section 2.2 in this paper) are summarized in

Table 2. Assumed constant parameters for the calculation of wind-induced acceleration.

Parameter	Value
Directional factor $C_{\mathrm{d}\mathrm{i}\mathrm{r}}$	1.0
Seasonal factor $C_{\mathrm{s}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{o}\mathrm{n}}$	$1$.00
Probability factor $C_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}}$	$0.73$
Orography factor $C_{0\left(\mathrm{z}\right)}$	$1$.00
Turbulence factor $k_{\mathrm{l}}$	$1$.00
Terrain category	Urban environment ($\mathrm{I}\mathrm{V}$)
Reference height $Z_{\mathrm{t}}$	200
Reference length $L_{\mathrm{t}}$	300

.

Additional to the parameters in Table 2, fundamental frequency, mode shape, basic wind velocity, dimensions of the building (L, B, H; consult Figure 1), and floors masses are required for the calculation of wind-induced acceleration. EN 1998-1 [30] and EN 1991-1-4 Annex F [29] provide the simplified Equations (2) and (3) for the fundamental frequency, respectively:

$$f={\displaystyle \frac{1}{C_{\mathrm{t}}}}·H^{-0.75}\mathrm{for}H\le 40\mathrm{m}$$

(2)

$$f={\displaystyle \frac{46}{H}}\mathrm{for}H>50\mathrm{m}$$

(3)

where $f$ is the fundamental frequency in Hz, $H$ is the total height of the building in m, and $C_{\mathrm{t}}$ is 0.085 for moment-resisting steel frames, 0.075 for moment-resisting concrete frames and eccentrically braced steel frames, and 0.05 for other structures.

According to EN 1991-1-4 [29], the mode shape can be approximated using Equation (4):

$$\Phi (z)={\left({\displaystyle \frac{z}{H}}\right)}^{\zeta }$$

(4)

where $z$ is the height above ground in m, $H$ is in m, and $\zeta$ is the mode shape exponent. The mode shape exponent depends on the type of the building and varies from 0.6 to 2.5 [29].

In combination with the 746,496 analyses generated based on Table 1, three additional parameters are required for acceleration calculation. The values of these parameters were defined as below:

• Fundamental basic wind velocity $v_{\mathrm{b},0}$: 10, 15, 20, 25, 30, 35, and 40 m/s.
• Damping ratio ξ: 0.01, 0.015, 0.02, 0.025, and 0.03.
• Cross wind dimension of the building $B$ (confer Figure 1): 10, 15, 20, 25, and 30 m.

For the calculation of wind-induced acceleration, $v_{\mathrm{b},0}$, ξ, and $B$ were randomly selected from the defined set of values for each frame (i.e., dual frame-wall).

Little research has been done on the damping of timber structures. In this study, the values of the damping ratio were reasonably assumed based on [10,31].

3. Results and Discussion

3.1. Nonlinear Regression

The FEA results generated from the parametric study were used to derive analytical expressions for fundamental frequency, mode shape exponent, top floor displacement, and maximum inter-storey drift. Prior to performing the nonlinear regression, some frames were deemed unrealistic:

• Frames with fundamental frequency $f$, where $f>5.0$ Hz or $f<0.20$ Hz, were excluded.
• For frames with a number of storeys $n\ge 10$, ${\beta }_{\mathrm{b}}$, ${\beta }_{\mathrm{c}}$, and ${\beta }_{\mathrm{w}}$ $\ge 2.0$ were considered.
• For frames with a number of storeys $n\ge 10$, $b\ge 215$ mm was considered.
• For frames with number of storeys $n\le 6$, $h_{\mathrm{w}}\le 1.0$ m was considered.

3.1.1. Fundamental Frequency

Figure 3 shows the parity plot for the fundamental frequencies estimated from the FEA results and those calculated using Equations (2) and (3). None of the equations provide accurate predictions of the fundamental frequency compared to the FEA. The equations rely solely on the total height of the building as an input. To provide better predictions, factors such as quasi-permanent load, stiffness of connections, and cross-sectional dimensions of structural elements should be considered.

The FEA results were used to derive Equation (5):

$$f=C_1·{\displaystyle \frac{{n_{\mathrm{b}}}^{0.7}·{h_{\mathrm{b}}}^{0.8}·b^{0.5}·{{\beta }_{\mathrm{b}}}^{0.15}}{L^{0.85}·H^{1.1}·Q^{0.5}}}$$

(5)

$$C_1=1100·{h_{\mathrm{c}}}^{0.5}·{{\beta }_{\mathrm{c}}}^{0.07}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{C}\mathrm{L}\mathrm{T}\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}$$

$$C_1=930·{n_{\mathrm{w}}}^{0.15}·{h_{\mathrm{c}}}^{0.15}·{h_{\mathrm{w}}}^{0.45}·{{\beta }_{\mathrm{w}}}^{0.1}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{C}\mathrm{L}\mathrm{T}\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}$$

where $f$ is the fundamental frequency in Hz, $n_{\mathrm{b}}$ is the number of bays, $L$ is the length of the frame in m, $H$ is the total height of the frame in m, $Q$ is the quasi-permanent load in kN/m, $h_{\mathrm{b}}$, $h_{\mathrm{c}}$, and $h_{\mathrm{w}}$ are the heights of beam, column, and wall cross sections in m, respectively, $b$ is the width of beam, column, and wall cross sections in m, and $n_{\mathrm{w}}$ is the number of CLT walls. The stiffness ratios ${\beta }_{\mathrm{b}}$, ${\beta }_{\mathrm{c}}$, and ${\beta }_{\mathrm{w}}$ are defined in Equation (1). The parameters are also depicted in Figure 1 and Figure 2.

Figure 4 shows the parity plots for the fundamental frequency obtained from FEA and Equation (5). The goodness of fit was evaluated using the coefficient of determination $R^2$:

$$R^2=1-{\displaystyle \frac{{\Sigma }_{i=1}^n{\left(y_{\mathrm{i}}-{\widehat{y}}_{\mathrm{i}}\right)}^2}{{\Sigma }_{i=1}^n{\left(y_{\mathrm{i}}-{\overline{y}}_{\mathrm{i}}\right)}^2}}$$

(6)

where $y_{\mathrm{i}}$ is the value from FEA, ${\widehat{y}}_{\mathrm{i}}$ is the predicted value, and ${\overline{y}}_{\mathrm{i}}$ is the mean value of FEA results. Hereafter, Equation (6) is also used for the estimation of $R^2$ for all other response parameters.

As shown in Figure 4, Equation (5) can provide better predictions of the fundamental frequency than Equations (2) and (3). The predictions for frames without CLT walls are slightly better than those for frames with CLT walls.

3.1.2. Mode Shape

According to EN 1991-1-4 [29], a mode shape exponent $\zeta$ [consult Equation (4)] of 0.60 is recommended for slender frame structures with non-load-sharing walls, and a value of 1.0 is recommended for buildings with a central core plus peripheral columns or larger columns plus shear bracing. Figure 5 shows the mode shapes obtained from FEA and those obtained using Equation (4) with the values of $\zeta$ recommended in EN 1991-1-4 [29].

As shown in Figure 5, the mode shapes obtained from FEA vary among different frames and cannot be approximated by a single value of $\zeta$.

A mode shape exponent based on the structural parameters of individual frames can provide a better approximation of the mode shape. For each individual frame, the best-fitting mode shape exponent was obtained using the method of least squares. The results were then used to derive Equation (7) for frames without CLT walls and Equation (8) for frames with CLT walls:

$$\zeta ={\displaystyle \frac{0.85}{{n_{\mathrm{b}}}^{0.2}}}·{\left({\displaystyle \frac{L}{H}}\right)}^{0.15}·{\left({\displaystyle \frac{h_{\mathrm{c}}}{h_{\mathrm{b}}}}\right)}^{0.4}·{\left({\displaystyle \frac{{\beta }_{\mathrm{c}}}{{\beta }_{\mathrm{b}}}}\right)}^{0.1}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{C}\mathrm{L}\mathrm{T}\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}$$

(7)

$$\zeta ={\displaystyle \frac{0.85·{n_{\mathrm{w}}}^{0.1}}{{n_{\mathrm{b}}}^{0.3}}}·{\left({\displaystyle \frac{L}{H}}\right)}^{0.2}·{\left({\displaystyle \frac{h_{\mathrm{w}}}{h_{\mathrm{b}}}}\right)}^{0.3}·{\left({\displaystyle \frac{{\beta }_{\mathrm{w}}}{{\beta }_{\mathrm{b}}}}\right)}^{0.08}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{C}\mathrm{L}\mathrm{T}\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}$$

(8)

where $\zeta$ is the mode shape exponent to be used in Equation (4) and $L$, $H$, $h_{\mathrm{b}}$, $h_{\mathrm{c}}$, $h_{\mathrm{w}}$ are in m.

Figure 6 shows the parity plots for the mode shape exponent obtained based on FEA and using Equations (7) and (8). As shown in the figure, both equations provide a fairly good accuracy with a coefficient of determination exceeding 0.90.

3.1.3. Top Floor Displacement

Using the FEA results, Equation (9) has been derived for the top floor displacement (i.e., roof displacement) due to a uniform lateral load $w$:

$$\Delta =C_2·{\displaystyle \frac{L^{0.8}·h^{3.1}·n^{2.2}}{{n_{\mathrm{b}}}^{1.6}·{h_{\mathrm{b}}}^{2.1}·{{\beta }_{\mathrm{b}}}^{0.45}·b}}·w$$

(9)

$$C_2={\displaystyle \frac{2.1·{10}^{-4}}{{h_{\mathrm{c}}}^{0.7}·{{\beta }_{\mathrm{c}}}^{0.1}}}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{C}\mathrm{L}\mathrm{T}\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}$$

$$C_2={\displaystyle \frac{2.35·{10}^{-4}}{{h_{\mathrm{c}}}^{0.3}·{h_{\mathrm{w}}}^{0.85}·{{\beta }_{\mathrm{w}}}^{0.1}·{n_{\mathrm{w}}}^{0.1}}}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{C}\mathrm{L}\mathrm{T}\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}$$

where $\Delta$ is the top floor displacement in mm, $h$ is the storey height in m, $n$ is the number of storeys, $w$ is a uniform lateral load in kN/m, and $L$, $b$, $h_{\mathrm{b}}$, $h_{\mathrm{c}}$, and $h_{\mathrm{w}}$ are in m.

Figure 7 shows the parity plots for the top floor displacement obtained from FEA and Equation (9). As shown in the figure, Equation (9) shows a good correlation with FEA results. The displacement calculated using Equation (9) does not include second-order effects. To quantify the influence of these effects, a smaller parametric study ($\approx$130,000 analyses) was performed with and without the inclusion of second-order effects. The results show that the inclusion of the second-order effects results in an approximately 5% increase in top floor displacements for stiff frames (${\beta }_{\mathrm{b}}$, ${\beta }_{\mathrm{c}}$, and ${\beta }_{\mathrm{w}}$ $\ge 3.0$) and a 9% increase for softer frames (${\beta }_{\mathrm{b}}$, ${\beta }_{\mathrm{c}}$, and ${\beta }_{\mathrm{w}}$ $\le 2.0$).

3.1.4. Inter-Storey Drift

Inter-storey drift ($\delta$) is the relative displacement between two successive floors. The inter-storey drift from FEA for a representative set of 10,000 frames (randomly selected) is plotted in Figure 8. As shown in the figure, the maximum inter-storey drift occurs at lower storeys. The maximum inter-storey drift is, on average, located at $0.30H$ and $0.18H$ (measured from ground level) for frames with and without walls, respectively. It is therefore possible to optimize the design of connections by employing stiffer connections at lower storeys with a larger inter-storey drift. The concept of varying the stiffness of connections across different storeys has been explored in [1].

Equation (10) has been derived for the maximum inter-storey drift ${\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ using the FEA results:

$${\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}=C_3·{\displaystyle \frac{L^{0.6}·h^{3.3}·n^{1.4}}{{n_{\mathrm{b}}}^{1.4}·{h_{\mathrm{b}}}^{1.6}·{{\beta }_{\mathrm{b}}}^{0.35}·b}}·w$$

(10)

$$C_3={\displaystyle \frac{3·{10}^{-4}}{{h_{\mathrm{c}}}^{1.3}·{{\beta }_{\mathrm{c}}}^{0.2}}}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{C}\mathrm{L}\mathrm{T}\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}$$

$$C_3={\displaystyle \frac{3.5·{10}^{-4}}{{h_{\mathrm{c}}}^{0.4}·h_{\mathrm{w}}·{{\beta }_{\mathrm{w}}}^{0.15}·{n_{\mathrm{w}}}^{0.2}}}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{C}\mathrm{L}\mathrm{T}\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{s}$$

where ${\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum inter-storey drift in mm and $L$, $h$, $b$, $h_{\mathrm{b}}$, $h_{\mathrm{c}}$, and $h_{\mathrm{w}}$ are in m.

Figure 9 shows the parity plots for the maximum inter-storey drift obtained from FEA and Equation (10). As shown in the figure, Equation (10) provides good predictions. Similar to the case with top floor displacement, Equation (10) does not include second-order effects. The inclusion of the second-order effects results in an approximately 6% increase in maximum inter-storey drift for stiff frames (${\beta }_{\mathrm{b}}$, ${\beta }_{\mathrm{c}}$, and ${\beta }_{\mathrm{w}}$ $\ge 3.0$) and a 10% increase for softer frames (${\beta }_{\mathrm{b}}$, ${\beta }_{\mathrm{c}}$, and ${\beta }_{\mathrm{w}}$ $\le 2.0$).

3.1.5. Wind-Induced Acceleration

Wind-induced accelerations at the top storey, calculated using the FEA results in combination with EN 1991-1-4 [29] (consult Section 2.4 in this paper), were used to derive Equation (11). For the derivation of Equation (11), additional to the excluded frames listed in Section 3.1, frames with wind-induced acceleration $a$ at the top storey, where $a>0.50$ m/s², were also deemed unrealistic and hence were excluded.

$$a={\displaystyle \frac{0.0044·{v_{\mathrm{b},0}}^3}{f^{0.9}·{q_{\mathrm{m}}}^{0.9}·L^{1.1}·{B^{0.2}·\xi }^{0.4}}}$$

(11)

$a$ is the peak wind-induced acceleration at the top storey in m/s², $v_{\mathrm{b},0}$ is the fundamental basic wind velocity in m/s, $f$ is the fundamental frequency in Hz, $q_{\mathrm{m}}$ is the quasi-permanent mass per unit area of the floor in kg/m², $L$ and $B$ are in m (consult Figure 1), and $\xi$ is the damping ratio.

The parity plot for the peak acceleration calculated using EN 1991-1-4 [29] and using Equation (11) is depicted in Figure 10. Despite the simplicity of Equation (11) compared the procedure described in EN 1991-1-4 [29], it provides good estimates for the acceleration and hence can be used for the preliminary design of multistorey timber buildings in urban environments.

3.2. Artificial Neural Networks (ANNs)

The research community has observed a remarkable increase in the applications of machine learning in structural engineering. A comprehensive review on applications of machine learning techniques in structural engineering can be found in [32]. ANNs are a subset of various machine learning techniques which mimic how biological neurons work. Rosenblatt [33] pioneered the first ANN in 1958, which was employed for pattern recognition purposes.

Five ANNs have been trained for prediction of fundamental frequency $f$, mode shape exponent $\zeta$, top floor displacement $\Delta$, maximum inter-storey drift ${\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}$, and peak wind-induced acceleration $a$. For the training, 70% of the FEA results was used; the remaining 30% was used for testing the accuracy of the ANNs (referred to as the testing dataset). Figure 11 depicts the architecture of all ANNs. All neurons employ a sigmoid activation function.

Figure 12 depicts the parity plots for the FEA results (and acceleration according to EN 1991-1-4 [29]) and the values predicted using the ANNs for the testing dataset. Each ANN can predict frames with and without CLT walls. As shown in Figure 12, the accuracy of the developed ANNs slightly outperforms the analytical expressions derived using nonlinear regression, with a coefficient of determination of 0.99 for all ANNs.

3.3. Solved Example

The derived analytical expressions and the ANNs were used to estimate lateral displacements and wind-induced acceleration for a dual frame-wall structural system. The results were compared with FEA results. Figure 13 shows the considered example building. In X direction, the building was stabilized using a dual frame-wall system. In Y direction, lateral bracing was assumed.

Table 3. Parameters of the dual frame-wall systems in X.

Parameter	Frames 1 and 6	Frames 2 and 5	Frames 3 and 4
$n$	8
$h$ (m)	3.0
$n_{\mathrm{b}}$	3
$L_{\mathrm{b}}$ (m)	9.0
$n_{\mathrm{w}}$	2	2	1
$h_{\mathrm{w}}$ (m)	2.5	2.5	2.5
$h_{\mathrm{b}}$ (m)	0.585	0.585	0.585
$h_{\mathrm{c}}$ (m)	0.54	0.54	0.54
$b$ (m) a	0.43	0.43	0.43
${\beta }_{\mathrm{w}}$	1.5	1.5	1.5
${\beta }_{\mathrm{b}}$	2.5	2.5	2.5
${\beta }_{\mathrm{c}}$	2.0	2.0	2.0
$K_{\mathrm{\theta },\mathrm{w}}$ (kNm/rad) b	279,948	279,948	279,948
$K_{\mathrm{\theta },\mathrm{b}}$ (kNm/rad) b	25,906	25,906	25,906
$K_{\mathrm{\theta },\mathrm{c}}$ (kNm/rad) b	6113	6113	6113
$q$ (kN/m2)	2.0	2.0	2.0
$Q$ (kN/m)	5.0	10.0	10.0
$w$ (kN/m) c	2.125	4.25	4.25

^a Double cross section was assumed, ^b Substituting $\beta$ in Equation (1), ^c Corresponds to a fundamental basic wind velocity of 26 m/s and urban environment (IV) as defined in EN 1991-1-4 [29].

summarizes the properties of the dual systems in X.

3.3.1. Solving the Individual Frames in 2D

Substituting in Equations (5) and (8)–(10) for frames 1 and 6:

$$C_1=930·2^{0.15}·{0.54}^{0.15}·{2.5}^{0.45}·{1.5}^{0.1}=1479.7$$

$$f=1479.7·{\displaystyle \frac{3^{0.7}·{0.585}^{0.8}·{0.43}^{0.5}·{2.5}^{0.15}}{{27}^{0.85}·{24}^{1.1}·5^{0.5}}}=1.29\mathrm{H}\mathrm{z}$$

(12)

$$\zeta ={\displaystyle \frac{0.85·2^{0.1}}{3^{0.3}}}·{\left({\displaystyle \frac{27}{24}}\right)}^{0.2}·{\left({\displaystyle \frac{2.5}{0.585}}\right)}^{0.3}·{\left({\displaystyle \frac{1.5}{2.5}}\right)}^{0.08}=0.99$$

(13)

$$C_2={\displaystyle \frac{2.35·{10}^{-4}}{{0.54}^{0.3}·{2.5}^{0.85}·{1.5}^{0.1}·2^{0.1}}}=1.162·{10}^{-4}$$

$$\Delta =1.162·{10}^{-4}·{\displaystyle \frac{{27}^{0.8}·3^{3.1}·8^{2.2}}{3^{1.6}·{0.585}^{2.1}·{2.5}^{0.45}·0.43}}·2.125=8.26\mathrm{m}\mathrm{m}$$

(14)

$$C_3={\displaystyle \frac{3.5·{10}^{-4}}{{0.54}^{0.4}·2.5·{1.5}^{0.15}·2^{0.2}}}=1.47·{10}^{-4}$$

$${\delta }_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.47·{10}^{-4}·{\displaystyle \frac{{27}^{0.6}·3^{3.3}·8^{1.4}}{3^{1.4}·{0.585}^{1.6}·{2.5}^{0.35}·0.43}}·2.125=1.33\mathrm{m}\mathrm{m}$$

(15)

The ANNs were used to estimate frequency, mode shape exponent, and lateral displacements. Frequencies and lateral displacements calculated using the derived analytical expressions, the ANNs, and the FEA are summarized in

Table 4. Summary of the results calculated using the derived expressions, ANNs, and FEA in 2D.

Method of Calculation	Frames 1 and 6			Frames 2 and 5			Frames 3 and 4
	$\mathit{f}$ (Hz)	$\mathit{\Delta }$ (mm)	${\mathit{\delta }}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ (mm)	$\mathit{f}$ (Hz)	$\mathit{\Delta }$ (mm)	${\mathit{\delta }}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ (mm)	$\mathit{f}$ (Hz)	$\mathit{\Delta }$ (mm)	${\mathit{\delta }}_{\mathbf{m}\mathbf{a}\mathbf{x}}$ (mm)
Analytical expressions	1.29	8.26	1.33	0.91	16.51	2.66	0.82	17.70	3.05
ANNs	1.24	8.81	1.34	0.88	17.62	2.69	0.77	23.22	3.58
FEA	1.27	8.69	1.27	0.89	17.37	2.55	0.76	24.10	3.85

; the mode shapes are plotted in Figure 14.

3.3.2. Estimating the 3D Properties Based on the 2D Results

Assuming symmetry about X (i.e., no eccentricity/torsional modes; consult Figure 13), and diaphragm action at all floor levels, the frequency of the building in Figure 13 in X direction can be estimated based on the frequencies of the 2D frames in Table 4 using Equation (16):

$$f_{3\mathrm{D}}=\sqrt{{\displaystyle \frac{\sum _{\mathrm{i}=1}^N{n}_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},\mathrm{i}}·{f_{\mathrm{i}}}^2}{\sum _{\mathrm{i}=1}^N{n}_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},\mathrm{i}}}}}$$

(16)

where $f_{3\mathrm{D}}$ is the frequency of the 3D building in Hz, $f_{\mathrm{i}}$ is the frequency of frame (i) in Hz, $N$ is the number of frames (here, 6), and $n_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},\mathrm{i}}$ is the mass ratio (i.e., $Q$ ratio) of frame (i) to a selected reference frame. The derivation of Equation (16) is shown in Appendix A.

Assuming frame 1 to be the reference frame, the mass ratios can be calculated:

$$\begin{gathered} n_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},1}=n_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},6}={\displaystyle \frac{5.0}{5.0}}=1.0 \\ {n}_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},2}=n_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},3}=n_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},4}=n_{\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s},5}={\displaystyle \frac{10.0}{5.0}}=2.0 \end{gathered}$$

(17)

Using the frequencies obtained with the derived analytical expressions:

$$f_{3\mathrm{D}}=\sqrt{{\displaystyle \frac{2·1·{1.29}^2+2·2·{0.91}^2+2·2·{0.82}^2}{2·1+2·2+2·2}}}=0.97\mathrm{H}\mathrm{z}$$

(18)

Using the frequencies obtained with the ANNs:

$$f_{3\mathrm{D}}=\sqrt{{\displaystyle \frac{2·1·{1.24}^2+2·2·{0.88}^2+2·2·{0.77}^2}{2·1+2·2+2·2}}}=0.92\mathrm{H}\mathrm{z}$$

(19)

The frequency estimated from 3D FEA is 0.96 Hz.

Setting the wind load $w$ in Equations (9) and (10) to unity yields the frame compliance. Based on the assumption of diaphragm action, the parallel frames (frames 1–6) can be considered a system of springs in parallel. The compliance of the 3D building can therefore be calculated:

$${C_{3\mathrm{D}}}^{-1}=\sum _{i=1}^N{C_{\mathrm{i}}}^{-1}$$

(20)

where $C_{3\mathrm{D}}$ is the compliance of the 3D building in mm/(kN/m), $C_i$ is the compliance of frame (i) in mm/(kN/m), and $N$ is the number of frames (here, 6).

The displacements (${\Delta }_{3\mathrm{D}}$ and ${\delta }_{\mathrm{m}\mathrm{a}\mathrm{x},3\mathrm{D}}$) can be calculated by multiplying the compliance term $C_{3\mathrm{D}}$ by the total load $(P=2.125·2+4.25·4=21.25\mathrm{k}\mathrm{N}/\mathrm{m})$.

Using the results of the analytical expressions, the compliance terms for top floor displacement and inter-storey drift can be calculated:

$$C_{3\mathrm{D},\mathsf{\Delta }}=0.662\mathrm{m}\mathrm{m}/(\mathrm{k}\mathrm{N}/\mathrm{m}),C_{3\mathrm{D},\mathsf{\delta }}=0.109\mathrm{m}\mathrm{m}/(\mathrm{k}\mathrm{N}/\mathrm{m})$$

(21)

Multiplying by the total load $P$:

$${\Delta }_{3\mathrm{D}}=14.07\mathrm{m}\mathrm{m},{\delta }_{\mathrm{m}\mathrm{a}\mathrm{x},3\mathrm{D}}=2.31\mathrm{m}\mathrm{m}$$

Using the results of the ANNs:

$$C_{3\mathrm{D},\mathsf{\Delta }}=0.751\mathrm{m}\mathrm{m}/(\mathrm{k}\mathrm{N}/\mathrm{m}),C_{3\mathrm{D},\mathsf{\delta }}=0.115\mathrm{m}\mathrm{m}/(\mathrm{k}\mathrm{N}/\mathrm{m})$$

(22)

Multiplying by the total load $P$:

$${\Delta }_{3\mathrm{D}}=15.97\mathrm{m}\mathrm{m},{\delta }_{\mathrm{m}\mathrm{a}\mathrm{x},3\mathrm{D}}=2.44\mathrm{m}\mathrm{m}$$

The top floor displacement and inter-storey drift obtained from 3D FEA are 15.20 mm and 2.24 mm, respectively. The results of the 3D analysis are summarized in

Table 5. Summary of results using the derived expressions, ANNs, and FEA in 3D.

Method of Calculation	${\mathit{f}}_{3\mathbf{D}}$ (Hz)	${\mathit{\Delta }}_{3\mathbf{D}}$ (mm)	${\mathit{\delta }}_{\mathbf{m}\mathbf{a}\mathbf{x},3\mathbf{D}}$ (mm)
Analytical expressions	0.97	14.07	2.31
ANNs	0.92	15.97	2.44
FEA	0.96	15.20	2.24

.

Using the frequency from Equation (18), and for a basic wind velocity of 26 m/s and a damping ratio of 0.02, the peak wind-induced acceleration can be calculated:

$$a={\displaystyle \frac{0.0044·{26}^3}{{0.97}^{0.9}·{\left(2·\frac{1000}{9.81}\right)}^{0.9}·{27}^{1.1}·{{25}^{0.2}·0.02}^{0.4}}}=0.044\mathrm{m}/{\mathrm{s}}^2$$

(23)

The peak wind-induced acceleration estimated using the developed ANN and the procedure described in EN 1991-1-4 [29] are 0.043 m/s² and 0.037 m/s², respectively.

The frequency, top floor displacement, and maximum inter-storey drift obtained using the derived analytical expressions and ANNs show good accuracy compared with the results obtained from the FEA. The accelerations calculated using Equation (11) and the ANN are approximately 20% higher than the acceleration calculated using EN 1991-1-4 [29].

4. Conclusions

The fulfilment of serviceability requirements is challenging for multistorey timber buildings, due to their light weight and moderate stiffness. A parametric study using 2D linear elastic Finite Element Analysis (FEA) was performed for Moment-Resisting Timber Frames (MRTFs) with and without Cross Laminated Timber (CLT) walls. Various mechanical and geometrical parameters were varied within practical ranges, resulting in a total of 746,496 analyses.

The FEA results were used to derive simplified analytical expressions for the fundamental frequency, mode shape, top floor displacement, maximum inter-storey drift, and wind-induced acceleration by use of nonlinear regression. The derived analytical expressions show a coefficient of determination (R²) with the FEA results in the range of 0.92–0.99, indicating excellent goodness of fit. The FEA results were also used to train Artificial Neural Networks (ANNs) that can predict the same response parameters. All the developed ANNs show an R² of 0.99, indicating better accuracy than the analytical expressions.

EN 1998-1 [30] and EN 1991-1-4 [29] provide simplified expressions for the estimation of the fundamental frequency. Comparison between the frequency calculated using FEA and using the expressions provided in EN 1998-1 [30] and EN 1991-1-4 [29] shows that the provided expressions cannot be used and may result in significant underestimation or overestimation of the frequency. Similarly, the mode shape cannot be represented by the exponent values given in EN 1991-1-4 [29]. Therefore, the derived expressions and the ANNs can be used to provide better estimation of the frequency and the mode shape.

Although the derived analytical expressions and the ANNs are designed for 2D analysis, analytical expressions are proposed to allow for extrapolation to a 3D building. A 3D building was solved using FEA, and the results were compared with those obtained using the derived analytical expressions and the ANNs. The comparison showed good agreement, indicating the possibility of using the analytical expressions or the ANNs for preliminary assessment of multistorey timber buildings employing MRTFs, with and without CLT walls, specifically addressing serviceability requirements.

The data used for the development of the analytical expressions and the ANNs were generated based on an assumption of a simplified building layout. This simplification may not accurately capture the complexities and variations found in practical buildings. Future research addressing this limitation is necessary.