An Analytical Method to Reproduce Seismic Behavior of a Two-Story Cross-Laminated Timber Building at Large Deformation

A full-scale two-story CLT building with narrow panels comprised a gravity frame, CLT wall, CLT floor, and CLT roof. The specimen and coordinate directions of the shaking table test (Fig. 2), the floor plan of the second story (Fig. 3), and its elevation in the $Y2$ and $X1$ planes in which the CLT shear walls were located (Fig. 4) are shown. The specimen size was 6.0 m long in the $X$-direction, 2.5 m wide in the $Y$-direction, and 6.0 m high. Two arrangements were made for the floor—one with the outermost lamina parallel to the $X$-direction, and the other with the outermost lamina perpendicular to the $X$-direction—to verify various conditions. As the main seismic structure, two CLT shear walls were at the center along the $X$-direction on the first and second stories. The strength grading of glued laminated timber (glulam), which was E95-F315, is specified in the Japan Agricultural Standard (JAS). The meaning of E95 is that the laminas in all layers have an average Young’s modulus of $9.5\mathrm{kN}/{\mathrm{mm}}^2$ or greater, whereas F315 indicates that the bending strength of the glulam is 31.5 MPa. The material of columns and beams was glued laminated timber made of Scotch pine (Pinus sylvestris), and had two different strength gradings, E95–F315 and E105–F300. The cross-section area was $120\times 120\mathrm{mm}$ and $120\times 300\mathrm{mm}$ (width × height), respectively. The horizontal diaphragm was made of seven-layer 210-mm-thick CLT panels of Japanese cedar (Cryptomeria japonica) grade Mx60 according to the JAS. The average Young’s modulus of the lamina in the outer layers was equal to or greater than $6.0\mathrm{kN}/{\mathrm{mm}}^2$, whereas it was $3.0\mathrm{kN}/{\mathrm{mm}}^2$ in the inner layers. The vertical diaphragm, e.g., shear wall, hanging wall, and orthogonal wall, comprised three-layer 90-mm-thick CLT panels of Japanese cedar (Cryptomeria japonica) grade S60, meaning that the average Young’s modulus of the lamina in every lamina was equal to or greater than ${6.0\mathrm{kN}/\mathrm{mm}}^2$. Fig. 5 shows details of the connectors and the springs to which they correspond, and Fig. 6 shows images of the joints. Tensile bolts (ABR490 and M16) and U-shaped connectors (TB-90) with holes for screws were used in the wall–foundation joints. It is not common in Japan to install a stopper to resist shear force, but the purpose of this experiment was to excite the structure to a large-deformation domain. Thus, stoppers were introduced to prevent the experiment from stopping early due to the shear failure of the CLT shear wall foot of the first story [Fig. 6(a)]. The metal protectors were attached between the wall panel and stoppers to prevent the shear wall from being embedded in the stoppers due to drifting during excitation. At the wall–wall and wall–roof joints, bolts (ABR490 and M20) and U-shaped metal connectors (TC-90) with holes for screws were used as tensile connectors [Fig. 6(b)], and angle brackets (LST) with holes for screws were used as shear connectors [Fig. 6(c)]. Wall–hanging wall and floor–floor connections were made with steel plates secured with screws [Fig. 6(d)]. STS6.5F screws were used for plywood and long steel plates in the floor–floor shear joint, and STSC65 screws were used for other joints. The specimen was designed according to Route 1 provided in CLT manual (Japan Housing and Wood Technology Center 2016). Route 1 is the simple method of allowable stress design against 0.2 base shear capacity for moderate earthquake and ultimate strength design against 1.0 base shear capacity for a maximum considered earthquake. The total seismic weight of the specimen was set to 175.95 kN based on the specimen specifications of the Japanese building standard law [Notification 611 of the Ministry of Land, Infrastructure, Transport and Tourism (2016)]. The specimen was subjected to a sine wave with a frequency that was constant in Sequence 0, and to the north–south (N–S) component of JMA Kobe waves, which was recorded during the Osaka–Kobe Earthquake in 1995, at 100% and 140% intensity in Sequences 1 and 2, respectively (Table 1). The acceleration response spectrum of the three seismic waves is presented in Fig. 7. The exciting direction was the $X$-direction of the specimen.

Fourier transformation was performed on the acceleration response time history on the shaking table and on the roof of the specimen in Sequence 0, Sequence 1, and Sequence 2; then the Fourier spectrum was calculated. The Fourier spectrum of the roof was divided by that of the shaking table to obtain a spectral ratio (transfer function), and the lowest frequency among the peaks of the spectral ratio was defined as the natural frequency. In addition, the base shear capacity was derived by dividing the maximum shear force by the total seismic weight, in which the maximum shear force was determined from the acceleration observed during excitation. Table 2 shows the natural frequency and base shear capacity. The specimen’s maximum overall drift and interstory drift in Sequences 1 and 2 are listed in Table 3. During Sequence 1, the maximum roof displacement was 200 mm, corresponding to 3.33% overall drift. A split was seen at the edge of the CLT shear wall due to pulling by the hanging wall, but there was no significant damage other than the split [Fig. 8(a)]. During Sequence 2, the maximum roof displacement was 397 mm, corresponding to 6.62% overall drift, and the interstory drift of the first story was 8.77%. In addition, wall head embedment into the floor panel was observed [Fig. 8(b)]. No repair work was performed after Sequence 1. Some damage and deterioration in the specimen from Sequence 1 was considered in the analysis because the numerical models were subjected to Sequences 1 and 2 sequentially.

In this study, the structural analysis software wallstat was used for pushover and time-history analysis. The analysis model shown in Fig. 9 was built based on the specimen’s joint specifications. The number of nodes was 382, and the number of springs was 668. This model was fixed in the $Y$-direction of translation because twisting of the specimen was not observed in the shaking table test. Fig. 10 shows the spring arrangement in the $Y2$ plane. The analysis model included tensile, shear, and compression springs. Two tensile springs were used for the two types of tensile joints, wall–foundation (T1) and wall–wall or wall–roof (T2). Three types of shear springs were inserted corresponding to the stoppers in wall–foundation joints (S1) and to the shear joints with steel-plated screws in wall–floor or wall–roof joints (S2) and wall–hanging wall joints (S3). Three differently defined compression springs were inserted to express the embedment ability of CLT under loads applied to wall–foundation joints (C1), wall–floor and wall–roof joints (C2), and wall–hanging wall and orthogonal wall–floor joints (C3). Other planes, such as $Y1$ and $X1$, were modeled similarly.

The CLT shear wall modeling method is depicted in Fig. 11, in which the CLT shear wall is modeled as a beam element with rigid beams at its top and bottom ends with lengths corresponding to the wall width. The joints at the wall head and foot were modeled with tensile springs, shear springs, and 11 compression springs. The 11 compression springs at equal intervals are called a multiple-spring; this modeling method has been used in some numerical analyses of specimens with CLT to reproduce the rocking behavior of a CLT shear wall (Sato et al. 2017; Azumi et al. 2019). The orthogonal walls, hanging walls, and floor panels were all modeled with beam elements and rigid beams, similar to the CLT shear walls. The beam elements were given only one Young’s modulus, which was applicable for both bending and compression. The Young’s modulus was set to $4.0\mathrm{kN}/{\mathrm{mm}}^2$ for CLT shear walls and hanging walls, because the major direction layer was $0\mathrm{kN}/{\mathrm{mm}}^2$ according to the CLT manual. The Young’s modulus was set to $2.571\mathrm{kN}/{\mathrm{mm}}^2$ for the floor panels along the major direction, and to $1.285\mathrm{kN}/{\mathrm{mm}}^2$ along the minor direction. When establishing the floor panel parameters, the cooperation width was considered. The gravity frame was modeled as a beam element with elastoplastic rotational springs at the element ends. The columns and beams’ Young’s moduli were set to 9.5 and $10.5\mathrm{kN}/{\mathrm{mm}}^2$, respectively, and the bending strength was set to 31.5 and 30.0 MPa, respectively. This Young’s modulus was applied for both bending and compression.

Nonlinear tensile shear springs were used to simulate each joint, and nonlinear compression springs were used to model the CLT’s embedment properties. Fig. 12 shows hysteretic rules of tensile–compressional and shear springs. The tensile–compressional spring was modeled as a slip type in tension and as elastoplastic in compression. Shear springs were set to bilinear for blind prediction, and subsequent wall–foundation shear springs were reset to the slip type considering the load–displacement relationship of the experimental results. The ultimate capacity of the tensile springs at the wall–foundation joint (T1) were determined to be 59.3 and 93 kN, and the Young’s modulus and effective area of the tensile bolts (ABR490 and M16) were $205\mathrm{kN}/{\mathrm{mm}}^2$ and $157{\mathrm{mm}}^2$, respectively, according to JIS B 1220 (JISC 2010). In addition, the length between the nuts was 400 mm; therefore, the first stiffness and the ultimate displacement for the T1 spring were set to $26.0\mathrm{kN}/\mathrm{mm}$ and 41 mm, respectively, based on the CLT manual. Similarly, for the wall–wall joint (T2), the Young’s modulus and effective area of tensile bolts (ABR490 and M20) were ${205\mathrm{kN}/\mathrm{mm}}^2$ and ${245\mathrm{mm}}^2$, respectively, according to JIS B 1220, and the length between the nuts was 200 mm. Therefore, the first stiffness and the ultimate displacement for the T2 spring were set to $30.0\mathrm{kN}/\mathrm{mm}$ and 20 mm, respectively, according to the CLT manual. For the shear springs at the wall–floor joints (S2), the yielding and ultimate capacity of the two angle brackets were determined to 54 and 90 kN, respectively, based on the CLT manual. In addition, considering friction, the first stiffness was assumed to be rigid, and the value obtained by multiplying the allowable (79.6 kN) and ultimate capacities (93.0 kN) of the tensile joint from JIS B 1220 by a friction coefficient of 0.3 were added to 54 and 90 kN, giving 77.9 and 118 kN. Because the four shear springs were distributed at the wall–floor joints (Fig. 11), the calculated yielding and ultimate capacities were divided by 2. Finally, yielding and ultimate capacities for the shear spring were set to 38.9 and 59.0 kN.

The ultimate displacement was set to 23.86 mm based on the CLT manual. Skeleton curves of the wall–hanging wall shear spring (S3) and wall–foundation compression spring (C1) were determined from the element test results [Figs. 13(c and d)]. The critical tensile–compressional and shear spring properties in this analysis model are presented in Tables 4 and 5. All analytical results were generated using a time-integration step of ${10}^{-6}\mathrm{s}$. The numerical analysis used the measured acceleration at the center of the shaking table as ground motions. The model weights were equal to the seismic weights of the full-scale specimen, including members and loaded weights, and the weights of the first and second stories were set to 100.54 and 75.41 kN, respectively.

Pushover analysis was performed on the analysis model based on the element test results and reference value, i.e, blind prediction. Fig. 14 depicts the blind prediction and hysteresis of the experiment on the story shear force–interstory drift for the first and second stories. The hysteresis of the experimental results was drawn using the acceleration and displacement measured in the shaking table tests. The first stiffness agreed with the experiments for both the first and second stories, but the maximum capacity of the second story and the ductility of the first and second stories were insufficient, indicating that the analysis could not reproduce the full-scale shaking table test results as described by Yasumura et al. (2016) when the parameters of the spring were determined by element test results. Thus, reproductive analysis was performed to better match the experimental results.

As previously stated, pushover analysis based on elemental tests and references could not reproduce the behavior of full-scale shaking table test at large deformation. The shaking table test results were used to calibrate the spring parameters for the wall–foundation tensile and shear joints (Fig. 15). For the tensile spring, the axial force–displacement relationship of the tensile bolts at the wall–foundation was traced, and the skeleton curve was defined by the slip-type hysteresis characteristics. In addition, a pre-tension load of 20 kN was added according to the experimental condition. Half of the shear force for the first story against wall–foundation relative displacement was traced for the shear spring skeleton curve. Half of that was used because there were two CLT shear walls on the first story. The skeleton curve was defined by the slip-type hysteresis characteristics. Table 6 presents the two spring properties after calibration. Comparing the skeleton curve in blind prediction and after calibration showed that the tensile spring parameters after calibration matched well with those in blind prediction. For the shear spring, the difference in the first stiffness was large, indicating that the shear resistance ability of the stoppers was not rigid due to the embedment of the CLT shear wall even if the metal protectors were installed.

Furthermore, to enhance reproducibility, the analytical method of Namba et al. (2021), data assimilation, was performed for the reproductive analysis. The 24 critical parameters of springs and elements in Table 7 were the target of data assimilation, and were multiplied by the correction factors to create various skeleton curves, and the parameter conbinations when fit to the experimental results were explored. The correction factors had the range to account for variations in materials and resistance factors which were not considered in the analytical model. Assuming that the variation due to these uncertainties was expressed as the correction factors’ coefficient of variation $\mathrm{COV}=0.2$ and that the mean value of the correction factors $\mu$ was 1, the standard deviation $\sigma =0.2$ was derived using Eq. (1). Assuming that the correction factor $x$ was normally distributed, standardization was performed using the random variable $z$ according to Eq. (2). In this case, $0.5<x<2.0$ accounted for 99.38% of the total, which covered almost all patterns. Therefore, the correction factors were varied in the range 0.5–2.0 in intervals of 0.15 for the Young’s modulus, first stiffness, and first yield point. For the second stiffness, the correction factors were varied in the range 0.0001–0.8 in intervals of 0.08, considering that the most possible values of the stiffness after yielding were covered

With this method, various skeleton curves for the springs were created. Then the experimental results and many analytical results were compared in terms of only the shear force–interstory drift of the first story through Sequences 1 and 2, not in terms of the time history of interstory drift and uplift displacement. Then the analytical results with the smallest error from the experimental results were extracted. By comparing the skeleton curves before assimilation and after assimilation, five analysis results with the five smallest errors from the experimental results were extracted. Thus, assimilation was performed to match both Sequences 1 and 2. Therefore, the damage from Sequence 1 also was considered in the analytical results after data assimilation. In addition, because all springs and elements were free in all directions except the $Y$-direction of translation, if multiple factors are included in the springs and elements, data assimilation of the parameters will result in an accurate analysis model.

The interstory drift time history for the first and second stories in Sequences 1 and 2 from data assimilation is illustrated in Fig. 16, together with the blind prediction and experimental results. The blind prediction did not agree with the experiment results except for the initial drift response to the excitation. After assimilation, both the interstory drift and phase agreed well with the experimental results through Sequence 1. In Sequence 2, there was a slight error in interstory drift with the experiment results after the maximum drift, but the trends of the interstory drift and phase agreed well, implying a good reproduction result.

Fig. 17 illustrates the shear force–interstory drift relationships for the first story in Sequences 1 and 2 in the experiment, in blind prediction, and after assimilation. For the blind prediction, the maximum load was nearly reproduced in Sequence 1, but not in Sequence 2. Nevertheless, the interstory drift and stiffness were not consistent with the experiment results. However, the analytical results after assimilation were almost identical to the experimental result, indicating that the overall behavior of the specimen could be tracked by reproductive analysis.

Fig. 18 shows the uplift displacement time history at the CLT shear wall foot of the first story in Sequence 2 in the experiment and the analysis results after assimilation. The analytical result after assimilation agreed with the experimental results in drift and phase, demonstrating that the detailed behavior of the two-story CLT building was reproduced.

Fig. 19 shows the skeleton curves for the four springs before and after assimilation. The solid lines in Figs. 19(a–d) are skeleton curves in the analysis results with the five smallest errors from the experimental results. The five analytical results (labeled 1–5 in order from the smallest to the fifth smallest) are shown to indicate the trend of how the skeleton curves changed.

For the wall–wall tensile springs [Fig. 19(a)], a clear trend of the first stiffness and yielding capacity was not seen. For the wall–floor shear spring [Fig. 19(b)] considering friction when determining the parameters, the skeleton curves tended to be the same before and after assimilation. For the wall–hanging wall shear spring [Fig. 19(c)] without considering friction, both stiffness and yield capacity tended to increase after assimilation. This implies that the friction between the shear connectors under suppressed force contributed to an increase in both stiffness and yielding capacity. However, it is recommended to reconsider the friction coefficient, which was 0.3 in this paper. In addition, the difference in the loading speed between the dynamic loading during the full-scale shaking table test and the static loading in the element tests can be another reason for increase of the stiffness and yielding capacity. In additioin, for Analytical result 5, the yielding capacity of the wall–floor shear spring was the highest, although the first stiffness and yielding capacity of the wall–hanging wall shear spring were the smallest, implying that the wall–floor and wall–hanging wall shear springs had an inverse relationship.

An increasing trend of the first stiffness and yielding capacity also was seen in the case of the compression springs [Fig. 19(d)]. Similar to the shear springs, the dynamic effects of loading can be a contributing factor to the increase in stiffness and yielding capacity. Another possible factor is that the dead weights were fixed to the floor with screws in the experiment, which increased the floor rigidity significantly and suppressed the vertical deformation caused by the rocking of the CLT shear walls. These inverse interactions and factors for increase of stiffness and capacity, such as friction, dynamic effects, and increase of floor rigidity, will be verified by comparing the detailed behavior and performing static loading tests in the future.