 Review Article
 Open access
 Published:
A rolling shear analysisbased method for determining the apparent stiffness and bending capacity of CLT panel under outofplane load
Journal of Wood Science volumeÂ 70, ArticleÂ number:Â 8 (2024)
Abstract
Most of the currently employed methods, such as Gamma method and shear analogy method, to estimate the bending stiffness and bending capacity of crosslaminated timber (CLT) beams, are computationally extensive. In our previous study, a rolling shear analysis (RSA)based method, which simplifies the calculation, was developed to determine the shearing capacity of CLT beams. In the present study, the authors expand upon the RSA method to determine the apparent stiffness and bending capacity of 3 and 5layer CLT beams. By considering the shear deformation of cross layers, simplified formulas to determine the apparent bending stiffness of CLT beam was derived. Two schemes to determine the CLT bending capacity were proposed. One is based on the shear stress analysis, and the other is based on the formula specified in Canadian standard, CSA O86, by replacing the effective stiffness with the apparent stiffness. Test results from the authors and the other researchers were adopted to validate the method. The findings showed that the RSA method, using the apparent stiffness obtained from the proposed method along with the bending capacity formula in CSA O86, can provide a simpler and more reliable estimation of the apparent bending stiffness and bending capacity of CLT beams as compared to the Gamma method and shear analogy method.
Introduction
The heretical structure of crosslaminated timber (CLT) determines that shear weakness must be an inherent property of CLT panels when subjected to outofplane bending [1]. As specified in Canadian standard, CSA O86 [2], rolling shear modulus of a CLT board is estimated as 1/160 of the modulus parallel to grain. Hence, the shear deformation in cross layers could be significant, consequently the beam theory based on the Eulerâ€“Bernoulli assumption, which stated that the strain on any arbitrary position of a cross section is linearly proportional to the distances from the position to the neutral axis of global bending, cannot be directly applied to the bending problems of CLT beam to predict their bending stiffness and bending capacity. Currently, the most accepted methods for predicting the bending properties of CLT beams [3, 4] are modified from beam theory, such as shear analogy method and Gamma method, etc. These methods assume the cross layer as mechanical joints that carry all the shear deformation of the beam, while the applied load was carried by the parallel layers. The apparent stiffness of the beam was estimated by introducing a connection efficiency parameter (Gamma method) or using Timoshenko beam theory (shear analogy method). The assessment of analytical methods, i.e., shear analogy method and Gamma method for predicting the bending performances of CLT panels was investigated by Christovasilis et al. [5]. They studied the accuracy and the reliability of these methods by comparing the prediction results with the test results, and concluded that all the methods are applicable for determining the bending strength and stiffness whereas have lower reliability for determining the shear performances because of the complicated mechanism of rolling shear.
Extensive studies confirmed that the layup, thickness, and even the saw pattern of CLT boards can significantly influence the performances of rolling shear. Flores et al. [6] studied the rolling shear properties by homogenization and cohesive model on multiscale beams. The constitutive properties of the model included the information from cellwall in nanometerscale and the growing ring in millimeterscale, whereas the governing equation was obtained by using homogenization technique and was solved by using finite element method. Obviously, this multiscale model is not applicable for design calculation because it is impossible for the manufactures to provide the material properties on micro scale. Franzonia et al. [7] presented an analytical solution for twolayered beam with interlayer slip and provided an analytical solution for deflection, slip, crosssectional rotation, and internal forces. In order to assess the CLT panels with regular gaps in cross layers, Ecsedi et al. [8] modeled the CLT panels with gaps as a space frame of beams connected with wooden blocks. A closedform formulas for predicting the panelâ€™s stiffnesses and maximum longitudinal and rolling shear stresses were developed. In general, the methods mentioned above have taken comprehensive effects of rolling shear into consideration, however, they are complicated for design use.
Numerical methods are considered as the most powerful way to deal with the material anisotropy of CLT beams. The literatures on this field are abundant [9,10,11,12,13,14,15]. However, because of the difficulties to simulate the complicated damage process, extensive calculations are also need [16]. For this reason, analytical model is still an attractive method for the analysis of CLT beams.
More recently, Huang et al. [17] developed an analytical model to predict the shear capacity of shortspan CLT beams. The model takes the cross layer as a continuum joint medium between the parallel layers. CLT bending member is treated as a composite beam that consist of the parallel layers coupled by the joint medium. The simplified formulas to predict the shear capacity of CLT beams were obtained based on the analysis of rolling shear stress. Hence, the model was named as rolling stress analysisbased model (RSA) model. To deal with CLT cross layers as continuum joint medium was firstly reported by Huang et al. [18] when they developed a model to predict the loadcarrying capacity of CLT panels loaded with combined bending and compression.
By using RSA method, the present study aims to develop an analytical method to determine the apparent stiffness and the bending capacity of CLT beams. Simplified formulas applicable for design calculations were proposed. The formulas are much simpler than the currently used approaches but without losing accuracy. This paper is arranged as follows. After introduction, the most popular methods accepted by the design standard / manus, i.e., Gamma method, shear analogy method, and the simplified method in CSA O86, were summarized in â€˜Summary of currently used methodsâ€™. The establishment of RSA model used for design purpose to predict the apparent bending stiffness and bending capacity of CLT beams was presented in â€˜Methodsâ€™. Validation of the proposed RSA model was given in â€˜Validationâ€™; and finally, discussions and conclusions on the use of RSA method were provided in â€˜Discussions and conclusionsâ€™.
Summary of currently used methods
Apparent bending stiffness
Gamma method and shear analogy method are based on the mechanical joint theory which was originally developed for the composite beams connected with fasteners to calculate their apparent stiffness [19]. It was modified for CLT bending problem by imaging the cross layers as continuously distributed fasteners with slip modulus being equal to the modulus of rolling shear [20]. It assumes the load on CLT beam are only carried by the boards oriented in the longitudinal direction, while shear deformations are only occurred in cross layers. A coefficient, \(\gamma\), was introduced to measure the connection efficiency of cross layer, with \(\gamma = 1\) representing completely glued members, and \(\gamma = 0\) representing no connection at all. The apparent bending stiffness of CLT beam can be determined by
The connection coefficient, \(\gamma\), can be determined as follow
Shear analogy method models CLT panel as two virtual beams, beams A and B, coupled by infinitely rigid web. They are used to obtain the sum of the moment of inertia and the second moment of area for all the individual layers, respectively. The effective bending stiffness of the CLT beam equals to the summation of the stiffness of beam A and beam B, and is denoted as
The effective shear stiffness of the beam is determined by
The deflection of the beam is calculated referring to Timoshenko beam theory by replacing the bending stiffness, EI, and shear stiffness, GA, with the effective bending stiffness, \((\text{EI})_{{\text{eff}}}\), and effective shear stiffness, \((\text{GA})_{{\text{eff}}}\), respectively. For the simply supported CLT beam, the apparent bending stiffness can be shown as Eq.Â (5).
Shear analogy method has been employed in Canadian standard CSA O86 [2] to calculate the bending stiffness of CLT beams.
Moment resistance
Gamma method assumes the failure happens if the summation of normal stress on the outmost parallel layer caused by local and global bending reaches its bending strength. The moment of resistance of CLT beam, according to Gamma method, can be given as
where, \(h_1\) is the thickness of the outer parallel layer.
The method to calculate the moment resistance for CLT beam used in CSA O86 [2] is a simplified method. It was modified from Eulerâ€™s beam theory for calculating normal stress over the crosssection of the isotropic beams by replacing the bending stiffness with the effective stiffness. The moment resistance is denoted as
where, H is the depth of CLT beam; \(\phi = 0.9\) is the resistance factor. This formula is also adopted by shear analogy method to calculated the moment resistance by replacing the effective stiffness with the associated apparent bending stiffness.
Obviously, the above methods ignored the mechanism of rolling shear. Furthermore, Eqs. (1), (2), and (5) show that the apparent bending stiffness calculated form Gamma method and shear analogy method are reduced with the decrease of squared length of the span of CLT beam. As shown in the following text, the apparent bending stiffness can be significantly lower than the effective stiffness if the spantodepth ratio is less than 15. This could lead to unacceptable errors for calculating the moment resistance and deformation for CLT beams of small spantodepth ratio. To overcome the above shortcomings, RSA method was adopted in the present study. The method firstly derived the apparent bending stiffness on the bases of rolling shear analysis, and then provide a simplified formula to determine the bending capacity of CLT beams. The details are presented as follows.
Methods
Rolling shear stress
For simplicity but without losing generality, consider a simply supported 3layer CLT beam with a concentrated load between the two supports, shown as Fig.Â 1a. According to the RSA method proposed by Huang et al. [17], the cross layer is deemed as a continuum joint medium while the external load is carried by the parallel layers. Take the segment from the left support to the arbitrary position \(x \le l\), which subjected to combined bending and shearing actions (BS segment), into consideration. The forces on the BS segment of a length x can be illustrated as Fig.Â 1(b), where, x is the distance from the left support to the loading position. Supposing the moduli of elasticity (MOE) of parallel layers and cross layers in longitudinal direction are \(E_0\) and \(E_{90}\), respectively, and the modulus of rolling shear is \(G_{\text{R}}\). The rolling shear stress varies against the location, x, has been derived by previous study [17] and can be noted as Eq.Â (8).
where, \(V_2\) is the shear force carried by the lower parallel layer; \(\alpha\), \(\beta\), and \(\psi\) are parameters independent of the variable x. For 3layer panel, \(\alpha\) is given as
while for 5layer CLT, \(\alpha\) is given as
The other two parameters are given as Eqs. (11) and (12).
where, h and t are the thickness of parallel and cross layers, respectively; I is the section moment of inertia of parallel layers; A is the section area of parallel layers. For simplicity, supposing the beam has unit width, hence there must have \(I = h^3 /12\) and \(A = h\). The axial force and the moment carried by the outmost parallel layer can be obtained by the equilibrium conditions, shown as Fig.Â 1b, which can be expressed as [17]
Because the parallel layers have the same thickness, there must have that \(V_1 = V_2\), \(M_1 = M_2\), and \(N_1 =  N_2\). Thus, the axial force and moment on the upper parallel can be obtained.
Apparent bending stiffness
For the simply supported beam shown in Fig.Â 1, the effective bending stiffness, \(\left( {\text{EI}} \right)_{{\text{eff}}}\), can be noted as Eq.Â (15) if the shear deformation is ignored.
where, \(E_i\) is the MOE of each layer. It equals to \(E_{0}\) for parallel layer while equals to \(E_{{90}}\) for cross layer; \(I_i\) and \(A_i\) are the moment of inertia and area of each layer, respectively; \(Z_i\) is the distance from the center of each layer to the neutral axial of global bending. Assuming the momentinduced deflection is \(y_{\text{m}}\), the deflection of the beam at any position must fulfill Eq.Â (16).
where, M is the moment on the beam. On the other hand, if the deflection caused by shear deformation is considered, the overall deflection of the beam must be the sum of the deflections caused by bending and shearing, i.e., \(y = y_s + y_m\). Thus, the deformation of the beam can be expressed as
where, \((\text{EI})_{{\text{app}}}\) is the apparent stiffness of which shear deformation is considered; \(y_{\text{s}}\) is the deflection caused by shear deformation. Obviously, Eqs. (16) and (17) give the relationship between the effective stiffness and apparent stiffness, shown as Eq.Â (18).
where, \(y^{\prime \prime} = \frac{d^2 y}{{dx^2 }}\). According to the beam theory [21], the contribution of the axial deformation to the vertical deflection can be ignored, thus, the shear strain of the cross layer can be expressed as Eq.Â (19).
where, \(\gamma_{\text{R}}\) is the strain of rolling shear. For the rolling shear stress, \(\tau\), the constitutive law yields \(\gamma_{\text{R}} = \tau /G_{\text{R}}\), hence we further have
Substituting Eq.Â (8) into Eq.Â (20) leads to
Substituting Eq.Â (12) into Eq.Â (21), and considering \(V = 2V_2\) for 3layer CLT panel, Eq.Â (21) can be further written as
Using Eqs. (16), (18), (20), and, (22), and considering \(M = Vx\), and \({{(e^{\alpha x} + e^{  \alpha x} )} / {\left( {e^{\alpha x}  e^{  \alpha x} } \right) \to 1}}\) as x increasing, the relationship between the apparent stiffness and the effective stiffness can be obtained, shown as Eq.Â (23).
EquationÂ (23) indicates that the apparent stiffness is the function of position, x. It varies along the longitudinal direction from one section to another. Since the amount of the variation is not significant, the apparent stiffness at the loading position, where the maximum deflection occurs, is taken as the apparent stiffness of the beam. Let \(x = l\), and substituting Eqs. (9) or (10), and (11) into Eq.Â (22), yield the apparent stiffness of 3layer CLT beam and 5layer CLT beam, shown as Eqs.Â (24) and Â (25), respectively.
Usually, the thickness of parallel layers and cross layers are the same. The modulus of rolling shear \(G_{\text{R}}\) can be estimated as \(E_{0} /160\) according CSA O86. Therefore, Eqs. (24) and (25) can be further simplify as Eqs. (26) and (27), respectively.
where, H is the depth of CLT panel. For conservation, it can be taken as 3Â h for 3layer CLT panel and 5Â h for 5layer CLT panel; L is the span of CLT beam. EquationsÂ (26) and (27) show that the apparent stiffness depends on the spantodepth ratio, section sizes, and the MOE of material. This finding differs from that of Gamma method and shear analogy method because the apparent stiffness calculated from to these two methods depends on L^{2}.
In order to illustrate the influence of spantodepth ratio on the reduction of apparent stiffness of CLT beams, we define the quantity
as the coefficient of stiffness reduction due to rolling shear. For 3layer and 5layer, the coefficients can be noted as Eq.Â (29) and (30), respectively.
FigureÂ 2 compares the coefficient of stiffness reduction versus the spantodepth ratio calculated by RSA method, Gamma method, and shear analogy method. The results of all three methods indicated that the stiffness reduction for 3layer CLT panels is smaller than that for 5layer CLT panels. Based on the RSA method, the coefficient of stiffness reduction for the RSA method ranged from 0.77 to 0.94 for 3layer panels and from 0.87 to 0.97 for 5layer panels, with a change in the spantodepth ratio between 6 and 30. On the other hand, the Gamma method showed a coefficient of stiffness reduction ranging from 0.23 to 0.83 for 3layer panels and from 0.38 to 0.92 for 5layer panels. Similarly, the results for shear analogy method are consistent with RSA and Gamma methods. FigureÂ 2 shows that the apparent stiffness of CLT beams, calculated by both Gamma method and shear analogy method, is significantly lower than the effective stiffness for beams with small spantodepth ratios. For example, for 5layer panels with a spantodepth ratio less than 12, the apparent stiffness decreases by 25% while for 3layer panels, the decrease is 50%. However, the RSA method shows the apparent stiffness only about 10% lower for 5layer panels and up to 25% lower for 3layer panels. For larger spantodepth ratios, like those greater than 30, the apparent stiffness calculated using all three methods becomes identical for 5layer panels. In contrast, for 3layer panels, the RSA method produces results that are approximately 10% higher than those obtained from Gamma method and shear analogy method. These results are similar to those reported earlier by Christovasilis in their research [5], which stated that the shear analogy method underestimates the bending stiffness of softwood and hardwood CLT panels by less than 5% and 25%, respectively. These findings suggest that the RSA method is more accurate in predicting the real bending stiffness of CLT beams compared to Gamma method and shear analogy method.
Moment resistance
It has been confirmed by extensive studies [22,23,24,25] that the bending failure of CLT beams are usually caused by the rupture of the outmost parallel layer. Therefore, it is reasonable to establish the failure criterion of CLT beam on the bases of analyzing outmost parallel layer. We firstly take the 3layer CLT beam into consideration. Assuming parallel layers have the same thickness, the internal forces on the interested section are shown in Fig.Â 3a. It shows that the moment resistance at the cross section can be noted as
where, t is the thickness of cross layer; h is the thickness of parallel layers. Let \(M_2 (l)\), \(N_2 (l)\), and \(V_2 (l)\), denote the moment, axial force, and the shear force at the failure section, respectively, the failure condition of the layer can be expressed as
where, \(f_{\text{b}}\) is the bending strength of the lower parallel layer; \(A_{2}\) and \(W_{2}\) are the area and the section modulus of the failure cross section, respectively. Using Eqs. (13), (14) and Eq.Â (32), and considering that the value of item \(e^{  \alpha l}\) is much smaller than that of \(e^{\alpha l}\) because the bending member usually has large span, the vertical force on the failure cross section can be obtained, as shown in Eq.Â (33).
Therefore, the axial force and the moment at failure section can be obtained by using Eqs. (33), (13), and (14), shown as Eqs. (34) and (35), respectively.
The moment resistance of the 3layer CLT beam can be obtained by using Eqs. (31), (33), and (35), as shown in Eqs. (36).
For 5layer CLT beam, the local moment carried by the middle parallel layer is \(M_3 = h\int_0^l {\tau (x)_{\,} } dx = N_2 h = 2M_2\), as shown in Fig.Â 3 (b). Thus, the moment resistance of the 5layer CLT beam can be calculated as
EquationsÂ (36) and (37) can be further expressed as Eqs. (38) and (39) by using Eqs. (36), (37), and (32).
where, the coefficients are denoted as
EquationsÂ (38) and (39) are complicated and are not user friendly for design calculation. It is perhaps more convenient to use the method specified in CSA O86 by replacing the effective stiffness with the apparent stiffness shown in Eqs.Â (42) and (43) for 3layer CLT beam and 5layer CLT beam, respectively. Thus, the formula to determine the bending resistances of 3layer and 5layer CLT beams can be noted as Eqs. (42) and (43), respectively.
If we let \(\phi = \left( {1 + \frac{{\sqrt {30} H}}{3L}} \right)^{  1}\) for 3layer CLT beam and \(\phi = \left( {1 + \frac{4H}{{5L}}} \right)^{  1}\) for 5layer CLT beam, Eqs. (42) and (43) have the same form as Eq.Â (7) specified on Canadian standard CSA O86 to determine the bending capacity of CLT beams. However, it is important to note that the definition of \(\phi\) in CSA O86 pertains to reliability considerations, which differs from the concept of stiffness reduction coefficient proposed in this study. As demonstrated in â€˜Apparent bending stiffnessâ€™ that coefficient \(\phi\) varies from 0.85 to 0.94 for 3layer CLT panel, while from 0.94 to 0.97 for 5layer CLT panel, as the spantodepth ratio varies from 12 to 30. This happens to be largely consistent with the value specified in CSA O86.
Validation
In order to validate the proposed model, specimens tested by the authors and the other researchers [26,27,28] were adopted as the samples for predicting the apparent stiffness and bending capacity. The material properties and the specimen dimensions are presented in TableÂ 1. The specimens named as SPF3 and SPF5 were test by the authors. The laminations that were used to make the CLT panels came from Canada and glued by a Chinese Company. The MOE and the strength of bending strength of the laminations were obtained from 4point bending test in accordance with the ASTM standard D 198 [29]. In total 9 specimens were tested to determine the MOE and the bending strength, respectively. The mean values of the tested results are shown in TableÂ 1. The mean value of the MOE of bending is 8360Â MPa with the variable coefficient of 11.71, and the bending strength is 31.02Â MPa with the variable coefficient of 5.49. The specimens tested by the other researchers were originally used to determine the bending strength of the CLT laminations by using Gamma method or shear analogy method. In the following validation calculations, the bending strength is used as the input to calculate the maximum load of CLT beams.
Table 2 presents the apparent bending stiffnesses obtained from the test and the prediction by using RSA method, Gamma method, and shear analogy method. The effective stiffnesses are also included. It is evident that the RSA method gives the most accurate prediction for all test specimens. Except for specimen CL3/105/2100, differences between the prediction and the test results fall withinâ€‰Â±â€‰20%. Gamma method and Shear analogy method yield similar prediction on apparent bending stiffness, however, the differences between the prediction and test results are larger than RSA method.
It should be noted from TableÂ 2 that for the first seven specimens, the apparent bending stiffness from the tests are greater than the effective stiffness. As the apparent stiffness must be smaller than the effective stiffness due to the shear deformation. Therefore, these apparent stiffnesses must be overestimated in tests.
Table 3 compares bending capacities of the specimens obtained by testing with those determined by RSA method, Gamma method, shear analogy method, and the method in CSA O86. The values in RSA1 column were calculated by using Eq.Â (36) or Eq.Â (37), and those in RSA2 column were calculated by using Eq.Â (42) or Eq.Â (43). The discrepancies between the prediction and the test results are also presented in TableÂ 3. Discrepancies that are greater thanâ€‰Â±â€‰20% are highlighted in grey. It was found that for the specimens CLTS/4825 and CLT3/105/2100, the discrepancies are greater thanâ€‰Â±â€‰20%. For specimen CL3/105/2940, only the proposed method, i.e., RSA2, gives the closest prediction with a discrepancy of 10.96%, whereas the predictions from the other methods significantly deviated from test results.
Furthermore, it is evident, according to TableÂ 3, that the predictions obtained using Eq.Â (31) and CSA O86 are highly comparable. With the exception of unreliable data highlighted in gray, both RSA method and CSA O86, i.e., Eqs. (31) and (7), provide reasonable estimations for the bending capacity of CLT beams. The accuracy of the Gamma method surpasses that of the shear analogy method. Therefore, it can be inferred that the RSA and CSA O86 methods are both straightforward and dependable in determining the bending capacity of CLT beams.
Discussions and conclusions
In general, the RSA method yields larger predictions of apparent stiffness for CLT beams compared to those predicted by the Gamma method and shear analogy method. However, as the spantodepth ratio increases, the predicted bending stiffness from these three methods tend to convergence. Comparing to the effective bending stiffness which ignores the effect of rolling shear, when the spantodepth ratio of CLT beam less than 12, the prediction apparent stiffness is up to 80% lower for 5layer beam and 60% lower for 3layer beam. Test results indicated that RSA method gives more reliable prediction for bending stiffness. Therefore, it is recommended that RSA method can be used to determine the apparent stiffness.
RSA method takes the cross layer of CLT beam as the continuum joint media that couples the parallels as a composite beam. By replacing the constrains of the cross layers with the rolling shear stress for the parallel layers, RSA method converts the analysis of the parallel layer of CLT beam to the problem of elasticfoundation beam. Bending capacity was obtained by treating the rolling shear stress as an applied load on parallel layer. The limitation of this method is that the vertical forces at the end of cross layers are ignored, which could lead to the predicted rolling shear stress deviating from the real case, especially for more than 3layer CLT beams. Therefore, RSA method may lead to great errors in predicting the bending capacity of the CLT beams with more than five layers. However, because RSA method yields good prediction of apparent bending stiffness, Eqs. (42) and (43) are recommended to determine the bending capacity of CLT beams.
Availability of data and materials
Not applicable.
Abbreviations
 \(A_i\) :

Section area of ith layer
 a :

The distance between the centers of outmost layers
 b :

Width of CLT beam
 \(E_i\) :

MOE of ith layer in longitudinal direction
 \(E_{\text{L}}\) :

MOE in longitudinal direction of CLT beam
 \(E_{0}\) :

MOE parallel to grain of a panel
 \(E_{{90}}\) :

MOE perpendicular to grain of a panel
 \(f_{\text{b}}\) :

The bending strength of outer layer
 \(G_{\text{R}}\) :

Modulus of rolling shear
 \(G_i\) :

Shear modulus of ith layer; for cross layer, it equals to \(G_{\text{R}}\)
 \(h_i\) :

The thickness of ith layer
 H :

The depth of CLT beam
 \(I_i\) :

The section moment of inertia of ith layer
 L :

Span of CLT beam
 l :

Length of bendingâˆ’shearing segment
 \(M_i\) :

Bending moment on ith layer
 \(N_i\) :

Axial force on ith layer
 \(V_i\) :

Shear force on ith layer
 t :

Thickness of cross layer
 \(Z_i\) :

The distance from the center if ith layer to the neutral axis of global bending
 \(\alpha\),\(\beta\) :

Distribution parameters of the rolling shear stress
 \(\gamma_i\) :

Coefficient of connection efficiency for ith layer
 \(\gamma_{\text{R}}\) :

Rolling shear strain
 \(\psi\) :

Magnitude of rolling shear stress distribution
 \(\phi\) :

Factor of moment resistance
 \((\text{EI})_{{\text{eff}}}\) :

Effective bending stiffness
 \((\text{EI})_{{\text{app}}}\) :

Apparent bending stiffness
 \((\text{GA})_{{\text{eff}}}\) :

Effective shear modulus
 CLT:

Crosslaminated timber
 RSA:

Rolling Shear Analysis
 BS:

Bending and Shearing
 MOE:

Modulus of Elasticity
 SPF:

SprucePineFir
 CSA:

Canadian Standard Association
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Acknowledgements
The authors would like to thank Prof. Dongsheng Huang of Nanjing Forestry University, China, for his gently guidance during the whole program; Prof. Zhongfan Chen from Southeast University and Dr. Yurong Shen from Nanjing Forestry University, China, for supporting of the related data collection and analysis provided for some of the work reported herein; Zhongjia Low Carbon Co. Ltd. at Ningbo, China, for their contributions of manufacturing the crosslaminated timber used in the tests.
Funding
Funding for the research was provided by the National Natural Science Foundation of China (Grant No. 52208234), through the young scholar scientific research program implemented in Southeast University.
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ZH accomplished all the theoretical analysis and the manuscript writing work. YB supported some of the data analysis of the tests. CN provided theoretical and technical support of the whole research work presented in the manuscript. All the authors have read and approved the final manuscript.
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Huang, Z., Bian, Y. & Ni, C. A rolling shear analysisbased method for determining the apparent stiffness and bending capacity of CLT panel under outofplane load. J Wood Sci 70, 8 (2024). https://doi.org/10.1186/s10086024021219
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DOI: https://doi.org/10.1186/s10086024021219