- Email: [email protected]
Procedure for parameter identification and mechanical properties assessment of CLT connections
Engineering Structures xxx (xxxx) xxxx
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Procedure for parameter identification and mechanical properties assessment of CLT connections Jixing Cao, Haibei Xiong , Lin Chen ⁎
Department of Disaster Mitigation for Structures, Tongji University, Shanghai 200092, China
ARTICLE INFO
ABSTRACT
Keywords: CLT connections Mechanical properties Unscented Kalman filtering Hysteretic model Assessment methods
Connections in cross-laminated timber (CLT) structures are crucial components that can affect the behaviour of the whole structure. A novel framework that integrates the unscented Kalman filter (UKF) as an estimation tool with a hysteretic model is developed to identify the model parameters of a nonlinear system. The UKF estimates the mean and covariance of the model parameters using unscented transformation (UT) by a set of deterministically chosen sample points. The proposed framework is applied to identify the unknown model parameters of CLT connections using available experimental data, where the cycle behaviour of CLT connections is simulated using a spring element assigned to the hysteretic model. The comparison of hysteretic curves between the test and model shows that the results identified with UKF are precise. Two different approaches are proposed: an assessment according to the EN 12512 standard, and a damage accumulation assessment. The EN 12512 standard assessment is evaluated from the elastic stiffness, ductility ratio, and energy dissipation, whereas the damage accumulation assessment considers the effects of low amplitude and accumulated damage. Together, these two methods fully evaluate the identified result and the mechanical characteristics of CLT connections. The proposed procedure of parameter identification using the UKF (together with the mechanical properties assessment) can be applied to other connections in timber engineering.
1. Introduction Cross-laminated timber (CLT) is a widely-used innovative building material because it has advantages such as structural strength, fire resistance, favourable environmental attributes, and the ability to facilitate modularisation [1,2]. CLT is frequently applied to multi-storey buildings [3], such as the Växjö condominium in Sweden, the Murray Grove Stadthaus in London, and the Forte building in Australia [4]. Extensive multi-storey CLT buildings have prompted some researchers to gain an understanding of the seismic performance of CLT, including full-scale shake table tests and numerical analyses [5–7], and the behaviour of shear wall components [8,9]. The results showed that the CLT panels are relatively stiff, and the CLT connections are the source of energy dissipation [10,11], which influences static and seismic performance [12]. Therefore, it is essential to assess the mechanical properties of CLT connections. Reports on CLT connections are primarily focused on experimental [13,14] and numerical analyses [15,16]. Hossain investigated the performance of CLT panels connected with self-tapping screws under vertical shear loading and concluded that the screw layout had a significant influence on the joint ductility ratio [17]. Sullivan [18] tested ⁎
six different CLT connections, and compared the mechanical characteristics between test and design values calculated from the technical specifications, such as the building codes in the U.S. and Europe. Apart from the mechanical characteristics of CLT connections, seismic performance is also widely reported [19–21]. Fragiacomo et al. [22] discussed the seismic design of a multi-storey CLT building and suggested that the energy dissipation resulted from the connections as well as the friction between timber panels. Pozza et al. [23] characterised four massive wooden shear walls in terms of strength, stiffness, ductility, and hysteresis behaviour using test results, evaluated the dissipative capacity, and estimated the suitable intrinsic q-factor via numerical simulations. Based on the test results, numerical analysis was also performed. Shen et al. [24] calibrated two hysteretic models using test data of CLT connections and discussed the estimated result. Schneider et al. [25] utilised six CLT connections to assess and compare their mechanical characteristics under cyclic loading in experimental and numerical studies. Although these hysteretic models were demonstrated to simulate the nonlinear response accurately, the real behaviour of connections under different loading situations is so complicated that the value of model parameters is not unique; this complexity means that the stability of the estimated parameters is not guaranteed.
Corresponding author. E-mail address: [email protected] (H. Xiong).
https://doi.org/10.1016/j.engstruct.2019.109867 Received 9 June 2019; Received in revised form 13 October 2019; Accepted 28 October 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Jixing Cao, Haibei Xiong and Lin Chen, Engineering Structures, https://doi.org/10.1016/j.engstruct.2019.109867
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Recently, the Bayesian statistical method has attracted attention because it incorporates test information from a probabilistic point of view, which serves as an effective solution to the parameter identification problem. Liu and Au [26] utilised the Bayesian probabilistic approach to identify the parameters of a hysteretic model for multi-grid composite walls and discussed the effects of stiffness degradation, strength degradation, and pinching. Green [27] developed a novel variant of simulated annealing for Bayesian system identification of a nonlinear dynamic system. Some researchers have also applied the Bayesian filtering algorithm to identify model parameters [28,29]. Ching compared the performance of the extended Kalman filter and the particle filter using three numerical examples and reported that the particle filter provided consistent state and parameter estimates for highly nonlinear models [30]. Astroza et al. [31] employed the extended Kalman filter, unscented Kalman filter (UKF), and iterated extended Kalman filter to update the nonlinear finite element model and compared the performance of these models in terms of convergence, accuracy, robustness, and computational cost. Ebrahimian et al. [32] took the extended Kalman filter as a parameter estimation tool to estimate time-invariant parameters associated with the nonlinear models used in the finite element model of the structural system of interest. The Bayesian filtering algorithm described above performs parameter identification in a very efficient manner; however, there are few reports about parameter identification using the Bayesian filtering algorithm in timber engineering. In this paper, the UKF is introduced to identify the hysteretic model parameters of CLT connections using test data. The objective is to estimate the model parameters of CLT connections under cyclic loading and assess their mechanical properties. The identified parameters can be regarded as a valuable database if such types of CLT connections are used as crucial connector elements for an entire structure.
Gaussian distribution. This method employs the UT to generate a set of deterministically chosen sigma points (SPs). When the SPs are propagated through a nonlinear state equation, the mean vector and covariance matrix of the parameters can be estimated. Since the UKF does not require linearisation of the nonlinear state-space model or computation of the response sensitivities concerning model parameters, it is very efficient for determining parameters, especially in nonlinear estimation problems. The UT propagates through the nonlinear system with the following procedure. Suppose a random vector x with mean x and covariance matrix P transfers to a nonlinear function y = h (x ) . To estimate the mean and covariance matrix of y , a set of (2n + 1) SPs, denoted by (i) , are defined as follows:
+ wk
1
(i )
P^ =
+ )
(3b)
(4) is a scaling (5)
n
(i) ),
(6)
i = 0, , 2n ,
2n
Wm(i)
(i )
W c(i ) [
(i )
i=0 2n
, y ][
(i )
y]
(7)
where the weight coefficients Wm(i) and W c(i) are given as
Wm(i) =
n+
Wm(i)
2(n + )
=
, i=0 , i = 1, , 2n
Wc(i) =
n+
Wc(i) =
1 , 2(n + )
+ (1
2
,
+ ), i = 0
i = 1, , 2n
(8a)
, (8b)
in which is the scaling factor. A two-dimensional example is considered to explain the above procedure. The mean vector x and covariance matrix P of the state vector shown in Eq. (9) is propagated through the nonlinear system in Eq. (10).
(2)
yk = hk ( k ) + vk
+ wk
= h(
y^ =
(1)
(3a)
k 1
2 (n
i=0
1
=
[( n +
,
The mean and covariance matrix of the transformed SPs can be estimated as
n × 1 is zero-mean Gaussian white noise. in which wk 1 The goal of a nonlinear filtering problem is to recursively estimate the posterior probability density function (PDF) of the state and parameters at the given data. If the measurement equation contains all the time step of output response, then the state estimation problem converts to a parameter estimation problem, which means that only the parameters k are included in the state equations. The new state-space equation can be formulated as
k
=x
P^ )i], i = n + 1, , 2n.
in which the parameter affects the spread of SPs around the mean x , and denotes the secondary scaling parameter. The SPs are then propagated through the nonlinear function y = h (x ) that obtains the transformed (i) for the SPs.
where k denotes the time-invariant parameters vector to be identified; hk (·) is the nonlinear function representing the parameterised model; the subscript k indicates the time step of input-output ny × 1 are the measurement vector and the ny × 1 and v data; yk k corresponding noise vector, respectively. In the Bayesian filtering algorithm, the parameters to be identified are typically considered as random variables, the evolution of which is assumed to follow the Gaussian Markov process modelled as k 1
P )i], i = 1, , n
=
n ×1
=
= x + [( n +
where [·]i represents the ith column of the matrix, and parameter calculated by
Suppose a nonlinear hysteretic system can be constructed as
k
= x, i = 0
(i ) (i )
2. Two-step procedure for evaluating the mechanical properties of CLT connections
yk = hk ( k ) + v k ,
(i )
x=
0 85 , P= , 0 5 20
y1 = 2x1 + 3x2 y2 = 0.1x12 + x 22
,
(9)
(10)
The SPs for Eq. (9) are presented in Fig. 1, where the covariance ellipse depicts the first and second standard deviations, and the points are scaled based on the mean weights. All the five SPs lie between the first and second standard deviation, and different values of influence the spread of sample points. As the value of increases, the distance between the SPs and the centre point becomes larger, and the weight of the centre point increases, whereas the rest of the SPs have less weight. The process of the UT in Eq. (10) is illustrated in Fig. 2. In Fig. 2(a), both the black points and red points are generated according to Eq. (9),
where Eq. (3a) is the state equation that governs the evolution of system parameters over time, and the measurement equation (Eq. (3b)) corrects the prediction by comparing the measured and predicted responses. To solve the nonlinear parameter estimation problem in Eq. (3), the UKF [33] approach is used to approximate the posterior PDF by a 2
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 1. Two-dimensional SPs with different values of .
where the number of black points and red points is 50 000 and 5, respectively. When these points are propagated through the nonlinear Eq. (9) to capture the transfer estimation, as shown in Fig. 2, the estimated mean by UT (red points in Fig. 2(b)) consists of the calculated result from 50,000 random points (green point). This demonstrates that the estimated result using UT has high precision. When applying the UT to solve the recursive estimation problem defined by Eq. (3), time and measurement updates are involved. The time update generates SPs using UT, and the generated SPs feed into the numerical model. When the measured data is received, the predicted response is corrected by computing the Kalman gain. As shown in Eq. (3a), the predicted response to characterise the model parameters is not included explicitly in the state equation; this simplification allows the model parameters to be identified in different software platforms. For example, the UKF method is programmed in the Python environment [34], and the numerical model is developed in OpenSees [35]. The OpenSees is interfaced with Python to interact recursively during the nonlinear parameter identification. The recursive procedure of parameter identification using UKF is shown in Fig. 3, where k | k 1 and Pk | k 1 denote the prior estimates of the mean and covariance matrix of the parameters at time step (k ), respectively, and k | k and Pk | k denote the posterior estimates of the mean and covariance matrix of the parameters at time step (k ); Rk is the diagonal matrix corresponding to the measurement error; and (ki) denotes the predicted response corresponding to the SPs ( k(i) ).
Fig. 3. Parameter estimation using UKF.
When the parameters of a hysteretic model are identified, the identified result can be utilised to assess the mechanical properties of CLT connections. The performance evaluation between test and model involves the assessment according to the EN 12512 standard and the damage accumulation assessment. These two methods are evaluated from different aspects; the former compares mechanical properties including the maximum load capacity, ductility ratio, and energy dissipation, and the latter considers the cumulative damage. A two-step procedure is proposed in Fig. 4, where the first step is to identify the parameters using the UKF, and the second step is to assess the identified result using the two different approaches. The proposed procedure will be applied in the CLT connections in the following section.
Fig. 2. Example of a nonlinear transformation using UKF; (a) original and (b) transformed. 3
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
3.2. Test results The backbone curves of CLT connections in shear loading are presented in Fig. 8, where the legend “M” denotes one sample under a monotonous load, and “CSSJ-S-1” represents the specimen ID 1 of CSSJ in shear loading. For each type of connection, the performance of the samples under cyclic loading are consistent with the one under monotonous loading in the elastic stage. When the specimens suffer from yielding, their properties exhibit differences due to cracking of the CLT panels and yielding of the connectors. The ABJ samples have the highest maximum loading capacity among the three types of connections. Although the maximum loading capacity of SBJSS samples is low, they reach a large displacement at the failure point. For the CSSJ samples, both the failure displacement and the maximum loading capacity remain relatively low. Fig. 9 displays the backbone curves of three types of connections in tension loading, where the legend “M” represents one sample under monotonous loading, and “ABJ-T-1” denotes the specimen ID 1 of ABJ in the tension loading. For each connection, the specimens under both monotonous and cyclic loading have a similar trend before reaching the maximum loading capacity, beyond which there are steep decreases and irregular drops at the failure point.
Fig. 4. Two-step assessment procedure between the test and model.
3. Experimental and numerical study of CLT connections 3.1. Test introduction To study the mechanical properties of CLT connections, a variety of specimens were tested at the state key laboratory of disaster reduction in civil engineering, Tongji University, China. Three types of CLT connections, including angle bracket joint (ABJ), crossed self-tapping screws joint (CSSJ), and simple butt joint with self-tapping screws (SBJSS) were conducted under both monotonic and cycle tests, as summarised in Fig. 5. Fig. 6 presents the test photos of specimens under six different loading cases. More detailed information can be found in [36]. The monotonic test was conducted under displacement control at a constant slip rate of 0.05 mm/s, and it was continued until there was a 20% reduction from the maximum load. The specimens under cyclic loading were tested in both shear and tension directions. The shear loading protocol followed by the EN 12512 standard is shown in Fig. 7(a) [37], in which the specimens in the tension direction were subjected to a non-reversed modification of the procedure outlined in EN 12512 due to restrained movement on the compression side, as shown in Fig. 7(b). The predefined yield value (d y ) was obtained from the monotonic test.
3.3. Numerical model of CLT connections According to the test results, the cyclic loading response of CLT connections is highly nonlinear and exhibits pinching behaviour with strength and stiffness degradation. To simulate the behaviour of CLT connections under cyclic loading, a spring element assigned to the SAWS [38] model is created in the OpenSees [35], as shown in Fig. 10. The hysteretic rule of the SAWS model consists of an exponential function and a linear descent [39].
sgn(d )·(F0 + r1 S0 |d|)· 1 F=
sgn(d )· Fu + r2 S0 (d 0, |d| > |dF |
exp
(
S0 | d| F0
sgn(d )· du ), |du|
)
, |du| |d |
|d|
|dF |
|dF | (11)
where F and d are the force and displacement variables, respectively; F1 denotes the intercept strength for an asymptotic line to the envelope
Fig. 5. Test information for three CLT connections.
4
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 6. Photos of six different specimens.
Fig. 7. Procedure for the cyclic loading test in two directions; (a) in shear loading and (b) in tension loading.
Fig. 8. Backbone curves of CLT connections in shear loading.
curve; F0 is the intercept strength for the pinching part; S0 is the initial stiffness of the hysteresis curve; r1 is the stiffness ratio of an asymptotic line to the envelope curve; r2 is the stiffness ratio for the descending segment of the envelope curve; r3 is the stiffness ratio of the unloading segment of the envelope curve; and r4 is the stiffness ratio of the pinching part where it is simplified to a parallelogram. The parameters and control stiffness and energy degradation, respectively.
4. Parameter identification using UKF and the mechanical properties assessment 4.1. Parameter identification results Ten parameters should be identified in the SAWS model: = [F0, F1, S0, du , , , r1, r2, r3, r4 ]. During the estimation process, the loading displacement and the recorded force (F ) are regarded as input and output data, respectively. Because the experimental data inevitably 5
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 9. Backbone curves of CLT connections in tension loading.
Fig. 10. Numerical model of CLT connections; (a) spring element in shear loading, (b) hysteresis rule of the SAWS model, and (c) spring element in tension loading. Table 1 The identified parameters of SAWS model and its normalized relative error. Specimen ID
S0
r1
r2
r3
r4
F1
F0
du
ABJ-S-1 ABJ-S-2 ABJ-S-3 CSSJ-S-1 CSSJ-S-2 CSSJ-S-3 SBJSS-S-1 SBJSS-S-2 SBJSS-S-3 ABJ-T-1 ABJ-T-2 ABJ-T-3 CSSJ-T-1 CSSJ-T-2 CSSJ-T-3 SBJSS-T-1 SBJSS-T-2 SBJSS-T-3
7.38 6.84 5.58 4.03 1.92 2.06 2.52 2.63 2.48 7.17 7.23 6.13 8.15 6.92 10.54 5.65 7.45 7.82
0.08 0.09 0.15 0.056 0.053 0.041 0.057 0.054 0.045 0.07 0.16 0.11 0.036 0.052 0.042 0.052 0.043 0.035
−0.12 −0.25 −0.15 −0.17 −0.15 −0.18 −0.083 −0.045 −0.062 −0.13 −0.16 −0.165 −0.18 −0.14 −0.076 −0.09 −0.21 −0.16
1.08 1.38 1.24 3.26 4.09 4.21 2.18 2.57 1.83 1.23 1.18 1.32 0.79 0.93 0.72 0.93 0.69 0.82
0.025 0.05 0.04 0.015 0.034 0.021 0.001 0.006 0.004 0.015 0.026 0.032 0.013 0.024 0.027 0.032 0.027 0.024
32.06 43.75 33.73 10.52 9.48 9.19 9.13 8.52 9.35 26.67 15.32 15.42 14.78 20.21 14.65 12.86 12.28 13.21
1.94 3.22 3.58 0.84 0.92 0.55 0.46 0.55 0.63 1.31 1.06 1.76 0.42 0.49 0.38 0.32 0.31 0.34
22.16 28.31 31.99 12.95 18.09 15.39 41.9 30.32 36.25 40.02 32.66 40.23 4.13 2.83 2.43 2.83 5.61 2.78
contain noise, a zero mean, and a standard deviation of 0.5%F is assumed to construct the covariance matrix R for the measurement noise. The process noise wk is assumed to be time-invariant. The first-order statistics of wk are the zero-mean, and the second-order statistics are the diagonal covariance matrix Q = (1 × 10 3 × 0 |0 ) 2 , where 0 |0 is the prior distribution of the parameters according to the experience of engineers. The parameters of the scaled UT are chosen as = 0.1, = 0 , and = 2 [40]. The algorithm presented in Fig. 3 is employed to estimate the
NRE 0.35 0.26 0.42 0.73 0.43 0.39 0.56 0.75 0.68 0.62 0.33 0.56 0.76 0.72 0.45 0.41 0.46 0.55
0.92 1.17 1.08 1.09 0.86 0.97 1.08 1.04 0.93 0.98 0.95 1.06 1.13 1.07 1.01 1.07 1.03 1.05
0.0530 0.1079 0.0925 0.0224 0.0203 0.0121 0.0096 0.0101 0.0119 0.0332 0.0207 0.0433 0.0251 0.0238 0.0230 0.0205 0.0191 0.0242
unknown model parameters recursively, and the identified parameters of all connections are summarised in Table 1. Using the identified parameters, the generated hysteresis curves from the SAWS model are compared with the test results, as shown in Fig. 11. Due to the space limitations, only one specimen in the same loading case is presented in Fig. 11. In Fig. 11, the identified hysteresis curves of CLT connections in shear loading are consistent with the test results, whereas there are some differences between test and model in tension loading. There are 6
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 11. Six typical hysteresis curves that compare test and model.
negative forces for the tests when specimens are not subjected to negative displacement in Fig. 11(d), (e), and (f). This phenomenon is due to the plastic deformation of connectors, as shown in Fig. 12. When the connections are in the plastic stage, even if the force is unloading to zero, the displacement cannot be restored to the original point (i.e., displacement is zero) due to the residual deformation. The displacement continues unloading until zero, and the force then becomes negative. This test phenomenon cannot be considered in the SAWS model, leading to a significant difference between test and model. To quantify the estimation results, the normalised relative error (NRE) [41] is defined as follows:
n
NRE =
1 n k=1
zk
zk zk
2
, 2
(12)
where k is the time step, n is the total number of time steps, · 2 is the 2norm, and z k and zk are the forces obtained from the test and identified results, respectively. The NRE are also exhibited in the last column of Table 1, ranging from 0.0096 to 0.1079. If the evaluation of the identified results only depends on the values of NRE, then ABJ-S-1, CSSJ-S-3, SBJSS-S-1, ABJT-2, CSSJ-T-3, and SBJSS-T-2 are the best-identified specimens for each loading case. However, these evaluation results are from the mathematics aspect; it would be meaningful if the identified results are 7
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 13 compares the total energy dissipation between test and model for all the specimens, in which the total energy dissipation is calculated by the sum of the enclosed area of each loop. All three connections exhibit greater energy dissipation in shear loading than that in tension loading. The ABJ connections show the highest energy dissipation in both shear and tension directions, whereas the CSSJ connections exhibit the lowest value of energy dissipation. For the SBJSS connections, a large amount of energy is dissipated in shear loading, but in tension loading, a much smaller amount of energy dissipation occurs. Comparing the estimated and test results, the RE ranges from 0.4% (e.g., case CSSJ-S-3) to 24.7% (e.g., case CSSJ-T-2), demonstrating that the identified parameters have relatively high precision. 4.3. Identified results using the accumulative damage assessment In addition to the mechanical properties, damage evaluation is a critical index in the study of CLT connection behaviours. In recent decades, several damage models have been proposed, including the index of curvature, displacement, and energy [42]. Notably, Kraetzing et al. [43] developed a cumulative damage model based on the energy dissipation in a half cycle; i.e.,
Fig. 12. Plastic deformation of connectors.
assessed from the mechanical characteristics. The following sections will discuss the identified results from the mechanical properties. 4.2. Identified results using the EN 12512 standard assessment
D+ =
In this section, the identified results are compared with the test results in terms of strength, stiffness, ductility, and energy dissipation according to the EN 12512 standard. These mechanical properties of specimens under shear loading are computed by taking both sides of the hysteresis loops into account. In the case of specimens under tension loading, only the positive side is considered. Appendix Tables A and B compare the mechanical characteristics between the test and model in shear loading and tension loading, respectively, where the Fmax and dmax are the maximum load capacity and the corresponding displacement, respectively; Fy and d y denote the yielding load and the corresponding displacement, respectively; K 0 denotes the initial stiffness; and is the ductility ratio between failure displacement (du ) and yield displacement
(i. e. , RE =
=
du dy
Vt Vt
× 100%
E+ f
+
Ei+ Ei+
,
(14)
where D+ is the positive damage index, Ep+, i denotes the energy dissipation in a primary half cycle (PHC), Ei+ represents the energy dissipation in a follower half cycle (FHC), and E + f is the energy dissipation
in a monotonic test to failure. Fig. 14 depicts the protocol of PHC and FHC, where the PHC represents any half cycle whose amplitude exceeds those in all previous cycles, and the FHC denotes all subsequent cycles of smaller or equal amplitude. The damage index for the negative part is calculated using the same formula as Eq. (14), with a negative superscript inserted (i.e., Ep, i , Ei , E f ). Therefore, the overall damage index (D ) can be computed as
). The relative error (RE) is calculated as
Vm
Ep+, i +
D = D+ + D
D+D
(15)
In Eq. (14), both the numerator and denominator include the FHC energy, meaning that the FHC makes less of a contribution to the damage index than the PHC. By applying Eqs. (14) and (15) to the hysteresis response, the cumulative damage of CLT connections at each time step is computed and will be discussed in the following subsections.
(13)
where Vt is the value obtained from the test, Vm is the value calculated from the identified result, and |·| is the absolute operation. According to Table A1 in the Appendix, the maximum load shows good agreement between test and model, and the value of RE is less than 10% except for the specimen CSSJ-S-2. The RE of the corresponding displacement ranges from 0.76% to 27.35%. The initial stiffness difference between test and model is no more than 20%, and the RE of ductility ratio ranges from 1.78% (e.g., SBJSS-S-2) to 43.48% (e.g., CSSJ-S-2). Such large differences may be caused by model error. In some specimens, the measured force shows an irregular fluctuation, which is not well captured by the SAWS model, resulting in a variation of yielding displacement. The difference in yielding displacement between test and model aggravates the RE of the ductility ratio. The comparison of the mechanical properties of CLT connections in tension loading between test and model are shown in Appendix Table B1. The difference in yielding force between test and model is less than 17%, whereas the RE of ductility ratio ranges from 5.49% (e.g., SBJSS-T-1) to 62.44% (e.g., ABJ-T-3). The negative forces for the test data provide clues to the reasons for such large differences. Although the results of three specimens for different types of connections show great dispersion, this paper focuses on developing the procedure of parameter identification using UKF, which is capable of illustrating the suitable of UKF method for the parameter identification of a hysteretic model.
4.3.1. Cumulative damage of CLT connections in shear loading A comparison of the cumulative damage curves from the test and model is shown in Fig. 15; the legends ‘T’ and ‘M’ designate the curves generated from the test and model, respectively; the numbers ‘1, 2, 3’ denote the specimen ID; the legend ‘Aver’ represents the average value calculated from the three specimens. The overall trend of the model is similar to that of the test results, but differences appear over time. Several factors influence the characteristics of the damage accumulation curve. In Fig. 11, the test hysteresis curves show asymmetry between the positive and negative sides. However, the SAWS model can only consider situations where there is symmetry between the two sides. Another factor that causes variation is seen in pinching behaviour. A visible pinching effect is observed in the test results, whereas two parallel lines are adopted to describe the pinching in the SAWS model. According to the calculated results shown in Fig. 15, the failure point of CLT connections is defined at approximately D = 0.8 [44]. The t time to failure point (i.e., tD = 0.8) and the failure time ratio = Dt=0.8 are w summarised in Table 2, where tw denotes the total time of the loading protocol. The last column in Table 2 presents the relative error of the
8
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 13. Comparison of total energy dissipation between test and model.
failure time ratio ( ) between test ( t ) and model ( m ); i.e.,
=
t
m t
the variation is the subsequent cycles of the loading procedure. The SAWS model defines a repetitive cycle with a slope of 1 (r4 S0) (shown in Fig. 10b), which is a linear approximation relative to the test observations.
,
where |·| is the absolute operation. The value of ranges from 2.98% to 31.73%. The reasons for such a significant variation can be attributed to the hysteresis response. The test envelope curve is often irregular following the maximum load, whereas the envelope curve of the SAWS model follows a linear degradation. Another factor that contributes to
4.3.2. The cumulative damage of CLT connections in tension loading A comparison of cumulative damage curves between test and model 9
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Table 2 Comparison of time at D = 0.8 and the failure time ratio in the shear loading. Specimen ID
ABJ-S-1 ABJ-S-2 ABJ-S-3 CSSJ-S-1 CSSJ-S-2 CSSJ-S-3 SBJSS-S-1 SBJSS-S-2 SBJSS-S-3
tD = 0.8
(%)
tw
Test
Model
355.01 423.65 496.16 329.52 473.43 602.56 584.84 678.58 613.54
395.72 436.28 454.13 379.24 623.63 528.20 511.38 614.64 462.44
659.72 527.60 571.76 908.40 1201.08 1198.04 1236.16 1329.88 1235.60
t (%)
m (%)
53.81 80.30 86.78 36.27 39.42 50.30 47.31 51.03 49.66
59.98 82.69 79.43 41.75 51.92 44.09 41.37 46.22 37.43
11.47 2.98 8.47 15.09 31.73 12.34 12.56 9.42 24.63
model increases because the CLT connections suffer yielding of the screws in the test. However, only two parameters are directly defined for cyclic degradation in the SAWS model, which leads to the SAWS model having the same degradation factor over the entire protocol. Table 3 compares the time of D = 0.8 between the test and model. The relative error for all specimens does not exceed 11%, except for specimen ABJ-T-1.
Fig. 14. Protocol of PHC and FHC.
in tension loading is given in Fig. 16. The shapes of the cumulative damage curves from the model have trends similar to the test curves. Before the curves reach D = 0.2, the variation between them is negligible. When D ranges from 0.2 to 0.8, the difference between test and
Fig. 15. Comparison of the cumulative damage in shear loading. 10
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 16. Comparison of the cumulative damage in tension loading.
load capacity, elastic stiffness, and ductility ratio are compared and evaluated. This is a straightforward method, and its results can be summarised and compared easily in a table. In addition, the mechanical properties obtained from the test also provide reliable design values to engineers and a better understanding of the mechanical characteristics of CLT connections. However, this method is focused on the envelope curve, which means that the influences of lower amplitude and cumulative damage are not considered. The second method is a damage accumulation assessment that examines the relationship between force, displacement, and time. This approach assesses the damage that accounts for a single high-amplitude cycle and repeated cycles at a low amplitude. However, this cumulative damage is considered from an initial stage, and small variations will be accumulated during the whole loading process, which means that the cumulative damage is focused on the overall performance and is not sensitive to the detailed damage of components. These two assessment methods have advantages and disadvantages. When assessing the mechanical characteristic of CLT connections using these two methods together, they fully evaluate the identified results.
Table 3 Comparison of time at D = 0.8 and the failure time ratio in the tension loading. Specimen ID
ABJ-T-1 ABJ-T-2 ABJ-T-3 CSSJ-T-1 CSSJ-T-2 CSSJ-T-3 SBJSS-T-1 SBJSS-T-2 SBJSS-T-3
tD = 0.8
(%)
tw
Test
Model
322.33 434.64 458.17 202.90 395.25 383.08 424.86 438.59 446.33
402.76 440.47 463.53 197.36 384.81 366.53 467.95 423.03 416.34
748.52 881.40 902.80 499.40 647.04 647.08 605.96 607.84 606.44
t (%)
m (%)
43.06 49.31 50.75 40.63 61.09 59.20 70.11 72.16 73.60
53.81 49.97 51.34 39.52 59.47 56.64 77.22 69.60 68.65
24.95 1.34 1.17 2.73 2.64 4.32 10.14 3.55 6.72
4.4. Comparison of the two assessment methods Two different methods are adopted to evaluate and compare the identified results. The first approach is assessed from the mechanical properties according to the EN 12512 standard, where the maximum
11
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
5. Conclusions
sults in terms of the mechanical properties: the assessment according to the EN 12512 standard, and the damage accumulation assessment. The former is evaluated in term of elastic stiffness, ductility ratio, and energy dissipation, whereas the latter considers the accumulated damage. Together, these two methods fully evaluate the identified result and the mechanical characteristics of CLT connections.
In this paper, a novel procedure that combines a parameter identification technique and two assessment methods is presented to compare and evaluate the mechanical performance of CLT connections. A numerical model is created, for which the nonlinear behaviour is governed by the SAWS model, and the unscented Kalman filter is employed to identify the model parameters. Once parameters are identified, two different approaches are used to evaluate the identified results. The following conclusions can be drawn from the results:
Declaration of Competing Interest The author declare that there is no conflict of interest.
(1) A novel framework that integrates the UKF as an estimation tool with a hysteretic model provides parameter identification for the nonlinear system. The novelty is in coupling the nonlinear model in OpenSees, which allows the complex finite element model to be analysed. (2) The developed framework is applied to identify the hysteretic model parameters of CLT connections using the available experimental data. The identified results have good agreement with the test data, demonstrating that the UKF is suitable for the nonlinear system identification. (3) Two different methods are proposed to evaluate the identified re-
Acknowledgments This paper is funded by the National Natural Science Foundation (51978502), Key project of international cooperation in science and technology innovation (2016YFE0105600), Fundamental Research Funds for the Central University (22120180315, 22120170521). The financial support is greatly appreciated. Special thanks are given to the China Scholarship Council, which supports the first author for the research work in UCLA. The authors also thank the anonymous reviewers for their constructive comments.
Appendix A See Tables A1 and B1.
Table A1 Comparison of mechanical properties for CLT connections in shear direction between test and model. Connections types
Fmax [kN]
dmax [mm]
K0
kN mm
Fy [kN]
dy [mm]
[ ]
ABJ-S-1
Test Model RE/%
49.92 45.17 9.51
28.08 20.93 25.48
4.28 5.10 19.23
33.59 33.48 0.33
8.71 6.34 27.19
3.94 5.28 33.97
ABJ-S-2
Test Model RE/%
59.98 60.31 0.55
24.25 28.01 15.51
5.14 4.85 5.48
38.86 45.36 16.72
9.39 9.05 3.57
3.51 3.87 10.02
ABJ-S-3
Test Model RE/%
59.29 60.14 1.44
27.20 31.93 17.39
4.29 3.79 11.65
44.98 47.68 5.99
12.15 12.18 0.21
3.44 3.61 4.96
CSSJ-S-1
Test Model RE/%
12.32 12.76 3.58
12.95 11.98 7.45
2.01 2.11 5.35
8.47 8.38 1.00
4.05 3.61 10.99
4.08 4.46 9.27
CSSJ-S-2
Test Model RE/%
13.22 8.36 36.71
18.09 15.55 14.07
1.35 1.43 6.03
6.90 6.62 3.99
5.06 4.48 11.56
4.27 6.13 43.48
CSSJ-S-3
Test Model RE/%
10.24 9.83 3.92
15.39 14.99 2.60
1.64 1.48 9.69
6.95 7.52 8.24
4.07 4.93 21.06
4.72 4.15 11.95
SBJSS-S-1
Test Model RE/%
13.71 13.94 1.74
41.90 41.07 1.98
1.56 1.55 0.59
9.40 10.45 11.23
16.61 13.58 18.22
3.31 3.72 12.46
SBJSS-S-2
Test Model RE/%
11.62 12.40 6.68
41.32 30.02 27.35
1.63 1.66 2.19
9.49 10.13 6.69
15.48 13.57 12.31
3.78 3.85 1.78
SBJSS-S-3
Test Model RE/%
13.12 13.82 5.34
36.25 35.98 0.76
1.82 1.58 12.76
10.53 10.88 3.33
15.54 14.11 9.21
3.78 3.91 3.48
12
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Table B1 Comparison of mechanical properties for CLT connections in tension direction between test and model. Connections types
Fmax [kN]
dmax [mm]
K0
kN mm
Fy [kN]
dy [mm]
[ ]
ABJ-T-1
Test Model RE
34.22 38.19 11.61
23.02 23.04 0.09
5.61 4.80 14.36
23.64 26.85 13.58
7.23 9.94 37.48
6.49 5.62 13.48
ABJ-T-2
Test Model RE
48.06 44.17 8.10
32.66 33.14 1.47
3.83 3.96 3.43
30.56 25.66 16.03
7.42 8.29 11.80
6.31 5.53 12.28
ABJ-T-3
Test Model RE
33.66 42.27 25.59
46.00 40.50 11.96
3.31 2.83 14.44
22.08 23.08 4.51
12.39 7.33 40.84
4.45 7.23 62.44
CSSJ-T-1
Test Model RE
13.62 14.37 5.48
3.60 4.14 15.00
5.97 6.16 3.22
11.35 12.09 6.48
1.96 1.92 2.04
3.16 3.50 10.75
CSSJ-T-2
Test Model RE
12.78 12.58 1.56
2.95 2.25 23.73
6.13 6.32 3.06
11.98 10.16 15.25
2.11 1.61 23.70
2.44 3.81 56.09
CSSJ-T-3
Test Model RE
12.09 13.01 7.59
2.28 2.51 10.09
7.11 7.64 7.49
11.38 10.97 3.59
1.62 1.96 21.36
4.08 3.56 12.78
SBJSS-T-1
Test Model RE
9.45 10.42 10.30
2.83 3.51 24.03
4.23 4.31 1.93
8.94 9.43 5.47
2.13 2.17 1.88
2.87 3.03 5.49
SBJSS-T-2
Test Model RE
13.06 13.58 3.97
5.36 5.74 7.09
5.94 5.85 1.54
9.68 9.92 2.49
1.59 1.65 3.79
5.40 4.77 11.81
SBJSS-T-3
Test Model RE
11.11 11.33 1.95
2.78 2.75 1.08
6.01 6.41 6.76
10.03 10.04 0.16
1.66 1.54 6.95
2.77 3.27 18.28
Appendix B. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engstruct.2019.109867.
[13] D’Arenzo G, Rinaldin G, Fossetti M, et al. Tensile and shear behaviour of an innovative angle bracket for CLT structures. In: 2018 World Conference on Timber Engineering, Seoul, Korea; 2018. [14] Liu J, Lam F. Experimental test of coupling effect on CLT angle bracket connections. Eng Struct 2018;171:862–73. [15] Pozza L, Saetta A, Savoia M, Talledo D. Coupled axial-shear numerical model for CLT connections. Constr Build Mater 2017;150:568–82. [16] Bita HM, Tannert T. Numerical optimisation of novel connection for cross-laminated timber buildings. Eng Struct 2018;175:273–83. [17] Hossain A, Lakshman R, Tannert T. Shear connections with self-tapping screws for cross-laminated timber panels. In: Structures Congress, Oregon, Portland; 2015. [18] Sullivan K, Miller TH, Gupta R. Behavior of cross-laminated timber diaphragm connections with self-tapping screws. Eng Struct 2018;168:505–24. [19] Pei S, Van De Lindt JW, Popovski M, Berman JW, Dolan JD, Ricles J, et al. Crosslaminated timber for seismic regions: progress and challenges for research and implementation. J Struct Eng 2014;142(4):E2514001. [20] Reynolds T, Casagrande D, Tomasi R. Comparison of multi-storey cross-laminated timber and timber frame buildings by in situ modal analysis. Constr Build Mater 2016;102:1009–17. [21] Pozza L, Savoia M, Franco L, Saetta A, Talledo D. Effect of different modelling approaches on the prediction of the seismic response of multi-storey CLT buildings. Timber Struct Eng 2018::163. [22] Fragiacomo M, Dujić B, Šušteršič cI. Elastic and ductile design of multi-storey crosslam massive wooden buildings under seismic actions. Eng Struct 2011;33(11):3043–53. [23] Pozza L, Scotta R, Trutalli D, Polastri A. Behaviour factor for innovative massive timber shear walls. B Earthq Eng 2015;13(11):3449–69. [24] Shen YL, Schneider J, Tesfamariam S, Stiemer SF, Mu ZG. Hysteresis behavior of bracket connection in cross-laminated-timber shear walls. Constr Build Mater 2013;48:980–91. [25] Schneider J, Shen Y, Stiemer SF, Tesfamariam S. Assessment and comparison of experimental and numerical model studies of cross-laminated timber mechanical connections under cyclic loading. Constr Build Mater 2015;77:197–212. [26] Liu P, Au SK. Bayesian parameter identification of hysteretic behavior of composite walls. Probabilist Eng Mech 2013;34:101–9. [27] Green PL. Bayesian system identification of a nonlinear dynamical system using a novel variant of simulated annealing. Mech Syst Signal Pr 2015;52:133–46.
References [1] Brandner R, Flatscher G, Ringhofer A, Schickhofer G, Thiel A. Cross laminated timber (CLT): overview and development. Eur J Wood Prod 2016;74(3):331–51. [2] Yates M, Linegar M, Dujić B. Design of an 8 storey Residential Tower from KLH Cross Laminated Solid Timber Panels. In: Proceedings of the 10th World Conference on Timber Engineering, Miyazaki, Japan; 2008. [3] Ramage MH, Burridge H, Busse-Wicher M, Fereday G, Reynolds T, Shah DU. The wood from the trees: the use of timber in construction. Renew Sust Energ Rev 2017;68:333–59. [4] Šušteršič I, Fragiacomo M, Dujić B. Seismic analysis of cross-laminated multistory timber buildings using code-prescribed methods: influence of panel size, connection ductility, and schematization. J Struct Eng 2015;142(4):E4015012. [5] Ceccotti A, Sandhaas C, Okabe M, Yasumura M, Minowa C, Kawai N. SOFIE project—3D shaking table test on a seven-storey full-scale cross-laminated building. Earthq Eng Struct Dyn 2013;42(13):2003–21. [6] Popovski M, Gavrić I, Schneider J. Performance of two-storey CLT house subjected to lateral loads. In: 12th World Conference of Timber Engineering 2014, Auckland, New Zealand; 2014. [7] Pozza L, Scotta R, Vitaliani,R. A non-linear numerical model for the assessment of the seismic behavior and ductility factor of X-Lam timber structures. In: International symposium on timber structures, Istanbul, Turkey; 2009. [8] Gavrić I, Fragiacomo M, Ceccotti A. Cyclic behavior of CLT wall systems: Experimental tests and analytical prediction models. J Struct Eng 2015;141(11):04015034. [9] Akbas T, Sause R, Ricles JM, Ganey R, Berman J, Loftus S. Analytical and experimental lateral-load response of self-centering posttensioned CLT walls. J Struct Eng 2017;143(6):04017019. [10] Schneider J. Connections in cross-laminated-timber shear walls considering the behaviour under monotonic and cyclic lateral loading. MS thesis. Univ. Stuttgart, Germany; 2009. [11] Fragiacomo M, Dujić B, Šušteršič I. Elastic and ductile design of multi-storey crosslam massive wooden buildings under seismic actions. Eng Struct 2011;33(11):3043–53. [12] Gavrić I, Fragiacomo M, Ceccotti A. Cyclic behaviour of typical metal connectors for cross-laminated (CLT) structures. Mater Struct 2015;48(6):1841–57.
13
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al. [28] Corigliano A, Mariani S. Parameter identification in explicit structural dynamics: performance of the extended Kalman filter. Comput Method Appl M 2004;193(36–38):3807–35. [29] Wu M, Smyth AW. Application of the unscented Kalman filter for real-time nonlinear structural system identification. Struct Control Hlth 2007;14(7):971–90. [30] Ching Jianye, Beck James L, Porter Keith A. Bayesian state and parameter estimation of uncertain dynamical systems. Probabilist Eng Mech 2006;21(1):81–96. [31] Astroza R, Ebrahimian H, Conte JP. Performance comparison of Kalman−based filters for nonlinear structural finite element model updating. J Sound Vib 2019;438:520–42. [32] Ebrahimian H, Astroza R, Conte JP. Extended Kalman filter for material parameter estimation in nonlinear structural finite element models using direct differentiation method. Earthq Eng Struct D 2015;44(10):1495–522. [33] Wan, Eric A, Rudolph Van Der Merwe. The unscented Kalman filter for nonlinear estimation. In: Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373), Alberta, Canada; 2000. [34] https://www.python.org/. [35] Mazzoni S, McKenna F, Scott MH, Fenves GL. OpenSees command language manual [J]. Pacific Earthq Eng Res (PEER) Center 2006:264. [36] Cao Jixing, Xiong Haibei, Chen Jiawei, Huynh Alexandre. Bayesian parameter
[37] [38] [39] [40] [41] [42] [43] [44]
14
identification for empirical model of CLT connections. Constr Build Mater 2019;218:254–69. EN 12512 (2001). Timber structures—test methods—cyclic testing of joints made with mechanical fasteners. CEN, Brussels, Belgium. https://opensees.berkeley.edu/wiki/index.php/SAWS_Material. Folz Bryan, Filiatrault Andre. Seismic analysis of woodframe structures. I: Model formulation. J Struct Eng 2004;130(9):1353–60. Julier S, Uhlmann J, Durrant-Whyte HF. A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Automat Contr 2000;45(3):477–82. Erazo K, Hernandez EM. Uncertainty quantification of state estimation in nonlinear structural systems with application to seismic response in buildings. ASCE-ASME J Risk Uncertainty Eng Syst A: Civil Eng 2015;2(3):B5015001. Williams Martin S, Sexsmith Robert G. Seismic damage indices for concrete structures: a state-of-the-art review. Earthq Spectra 1995;11(2):319–49. Kraetzig W, Meyer I, Meskouris K. Damage evolution in reinforced concrete members under cyclic loading. In: 5th international conference on structural safety and reliability, San Francisco, USA; 1989. Schneider J, Karacabeyli E, Popovski M, Stiemer SF, Tesfamariam S. Damage assessment of connections used in cross-laminated timber subject to cyclic loads. J Perform Constr Fac 2013;28(6):A4014008.
Contents lists available at ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Procedure for parameter identification and mechanical properties assessment of CLT connections Jixing Cao, Haibei Xiong , Lin Chen ⁎
Department of Disaster Mitigation for Structures, Tongji University, Shanghai 200092, China
ARTICLE INFO
ABSTRACT
Keywords: CLT connections Mechanical properties Unscented Kalman filtering Hysteretic model Assessment methods
Connections in cross-laminated timber (CLT) structures are crucial components that can affect the behaviour of the whole structure. A novel framework that integrates the unscented Kalman filter (UKF) as an estimation tool with a hysteretic model is developed to identify the model parameters of a nonlinear system. The UKF estimates the mean and covariance of the model parameters using unscented transformation (UT) by a set of deterministically chosen sample points. The proposed framework is applied to identify the unknown model parameters of CLT connections using available experimental data, where the cycle behaviour of CLT connections is simulated using a spring element assigned to the hysteretic model. The comparison of hysteretic curves between the test and model shows that the results identified with UKF are precise. Two different approaches are proposed: an assessment according to the EN 12512 standard, and a damage accumulation assessment. The EN 12512 standard assessment is evaluated from the elastic stiffness, ductility ratio, and energy dissipation, whereas the damage accumulation assessment considers the effects of low amplitude and accumulated damage. Together, these two methods fully evaluate the identified result and the mechanical characteristics of CLT connections. The proposed procedure of parameter identification using the UKF (together with the mechanical properties assessment) can be applied to other connections in timber engineering.
1. Introduction Cross-laminated timber (CLT) is a widely-used innovative building material because it has advantages such as structural strength, fire resistance, favourable environmental attributes, and the ability to facilitate modularisation [1,2]. CLT is frequently applied to multi-storey buildings [3], such as the Växjö condominium in Sweden, the Murray Grove Stadthaus in London, and the Forte building in Australia [4]. Extensive multi-storey CLT buildings have prompted some researchers to gain an understanding of the seismic performance of CLT, including full-scale shake table tests and numerical analyses [5–7], and the behaviour of shear wall components [8,9]. The results showed that the CLT panels are relatively stiff, and the CLT connections are the source of energy dissipation [10,11], which influences static and seismic performance [12]. Therefore, it is essential to assess the mechanical properties of CLT connections. Reports on CLT connections are primarily focused on experimental [13,14] and numerical analyses [15,16]. Hossain investigated the performance of CLT panels connected with self-tapping screws under vertical shear loading and concluded that the screw layout had a significant influence on the joint ductility ratio [17]. Sullivan [18] tested ⁎
six different CLT connections, and compared the mechanical characteristics between test and design values calculated from the technical specifications, such as the building codes in the U.S. and Europe. Apart from the mechanical characteristics of CLT connections, seismic performance is also widely reported [19–21]. Fragiacomo et al. [22] discussed the seismic design of a multi-storey CLT building and suggested that the energy dissipation resulted from the connections as well as the friction between timber panels. Pozza et al. [23] characterised four massive wooden shear walls in terms of strength, stiffness, ductility, and hysteresis behaviour using test results, evaluated the dissipative capacity, and estimated the suitable intrinsic q-factor via numerical simulations. Based on the test results, numerical analysis was also performed. Shen et al. [24] calibrated two hysteretic models using test data of CLT connections and discussed the estimated result. Schneider et al. [25] utilised six CLT connections to assess and compare their mechanical characteristics under cyclic loading in experimental and numerical studies. Although these hysteretic models were demonstrated to simulate the nonlinear response accurately, the real behaviour of connections under different loading situations is so complicated that the value of model parameters is not unique; this complexity means that the stability of the estimated parameters is not guaranteed.
Corresponding author. E-mail address: [email protected] (H. Xiong).
https://doi.org/10.1016/j.engstruct.2019.109867 Received 9 June 2019; Received in revised form 13 October 2019; Accepted 28 October 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Jixing Cao, Haibei Xiong and Lin Chen, Engineering Structures, https://doi.org/10.1016/j.engstruct.2019.109867
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Recently, the Bayesian statistical method has attracted attention because it incorporates test information from a probabilistic point of view, which serves as an effective solution to the parameter identification problem. Liu and Au [26] utilised the Bayesian probabilistic approach to identify the parameters of a hysteretic model for multi-grid composite walls and discussed the effects of stiffness degradation, strength degradation, and pinching. Green [27] developed a novel variant of simulated annealing for Bayesian system identification of a nonlinear dynamic system. Some researchers have also applied the Bayesian filtering algorithm to identify model parameters [28,29]. Ching compared the performance of the extended Kalman filter and the particle filter using three numerical examples and reported that the particle filter provided consistent state and parameter estimates for highly nonlinear models [30]. Astroza et al. [31] employed the extended Kalman filter, unscented Kalman filter (UKF), and iterated extended Kalman filter to update the nonlinear finite element model and compared the performance of these models in terms of convergence, accuracy, robustness, and computational cost. Ebrahimian et al. [32] took the extended Kalman filter as a parameter estimation tool to estimate time-invariant parameters associated with the nonlinear models used in the finite element model of the structural system of interest. The Bayesian filtering algorithm described above performs parameter identification in a very efficient manner; however, there are few reports about parameter identification using the Bayesian filtering algorithm in timber engineering. In this paper, the UKF is introduced to identify the hysteretic model parameters of CLT connections using test data. The objective is to estimate the model parameters of CLT connections under cyclic loading and assess their mechanical properties. The identified parameters can be regarded as a valuable database if such types of CLT connections are used as crucial connector elements for an entire structure.
Gaussian distribution. This method employs the UT to generate a set of deterministically chosen sigma points (SPs). When the SPs are propagated through a nonlinear state equation, the mean vector and covariance matrix of the parameters can be estimated. Since the UKF does not require linearisation of the nonlinear state-space model or computation of the response sensitivities concerning model parameters, it is very efficient for determining parameters, especially in nonlinear estimation problems. The UT propagates through the nonlinear system with the following procedure. Suppose a random vector x with mean x and covariance matrix P transfers to a nonlinear function y = h (x ) . To estimate the mean and covariance matrix of y , a set of (2n + 1) SPs, denoted by (i) , are defined as follows:
+ wk
1
(i )
P^ =
+ )
(3b)
(4) is a scaling (5)
n
(i) ),
(6)
i = 0, , 2n ,
2n
Wm(i)
(i )
W c(i ) [
(i )
i=0 2n
, y ][
(i )
y]
(7)
where the weight coefficients Wm(i) and W c(i) are given as
Wm(i) =
n+
Wm(i)
2(n + )
=
, i=0 , i = 1, , 2n
Wc(i) =
n+
Wc(i) =
1 , 2(n + )
+ (1
2
,
+ ), i = 0
i = 1, , 2n
(8a)
, (8b)
in which is the scaling factor. A two-dimensional example is considered to explain the above procedure. The mean vector x and covariance matrix P of the state vector shown in Eq. (9) is propagated through the nonlinear system in Eq. (10).
(2)
yk = hk ( k ) + vk
+ wk
= h(
y^ =
(1)
(3a)
k 1
2 (n
i=0
1
=
[( n +
,
The mean and covariance matrix of the transformed SPs can be estimated as
n × 1 is zero-mean Gaussian white noise. in which wk 1 The goal of a nonlinear filtering problem is to recursively estimate the posterior probability density function (PDF) of the state and parameters at the given data. If the measurement equation contains all the time step of output response, then the state estimation problem converts to a parameter estimation problem, which means that only the parameters k are included in the state equations. The new state-space equation can be formulated as
k
=x
P^ )i], i = n + 1, , 2n.
in which the parameter affects the spread of SPs around the mean x , and denotes the secondary scaling parameter. The SPs are then propagated through the nonlinear function y = h (x ) that obtains the transformed (i) for the SPs.
where k denotes the time-invariant parameters vector to be identified; hk (·) is the nonlinear function representing the parameterised model; the subscript k indicates the time step of input-output ny × 1 are the measurement vector and the ny × 1 and v data; yk k corresponding noise vector, respectively. In the Bayesian filtering algorithm, the parameters to be identified are typically considered as random variables, the evolution of which is assumed to follow the Gaussian Markov process modelled as k 1
P )i], i = 1, , n
=
n ×1
=
= x + [( n +
where [·]i represents the ith column of the matrix, and parameter calculated by
Suppose a nonlinear hysteretic system can be constructed as
k
= x, i = 0
(i ) (i )
2. Two-step procedure for evaluating the mechanical properties of CLT connections
yk = hk ( k ) + v k ,
(i )
x=
0 85 , P= , 0 5 20
y1 = 2x1 + 3x2 y2 = 0.1x12 + x 22
,
(9)
(10)
The SPs for Eq. (9) are presented in Fig. 1, where the covariance ellipse depicts the first and second standard deviations, and the points are scaled based on the mean weights. All the five SPs lie between the first and second standard deviation, and different values of influence the spread of sample points. As the value of increases, the distance between the SPs and the centre point becomes larger, and the weight of the centre point increases, whereas the rest of the SPs have less weight. The process of the UT in Eq. (10) is illustrated in Fig. 2. In Fig. 2(a), both the black points and red points are generated according to Eq. (9),
where Eq. (3a) is the state equation that governs the evolution of system parameters over time, and the measurement equation (Eq. (3b)) corrects the prediction by comparing the measured and predicted responses. To solve the nonlinear parameter estimation problem in Eq. (3), the UKF [33] approach is used to approximate the posterior PDF by a 2
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 1. Two-dimensional SPs with different values of .
where the number of black points and red points is 50 000 and 5, respectively. When these points are propagated through the nonlinear Eq. (9) to capture the transfer estimation, as shown in Fig. 2, the estimated mean by UT (red points in Fig. 2(b)) consists of the calculated result from 50,000 random points (green point). This demonstrates that the estimated result using UT has high precision. When applying the UT to solve the recursive estimation problem defined by Eq. (3), time and measurement updates are involved. The time update generates SPs using UT, and the generated SPs feed into the numerical model. When the measured data is received, the predicted response is corrected by computing the Kalman gain. As shown in Eq. (3a), the predicted response to characterise the model parameters is not included explicitly in the state equation; this simplification allows the model parameters to be identified in different software platforms. For example, the UKF method is programmed in the Python environment [34], and the numerical model is developed in OpenSees [35]. The OpenSees is interfaced with Python to interact recursively during the nonlinear parameter identification. The recursive procedure of parameter identification using UKF is shown in Fig. 3, where k | k 1 and Pk | k 1 denote the prior estimates of the mean and covariance matrix of the parameters at time step (k ), respectively, and k | k and Pk | k denote the posterior estimates of the mean and covariance matrix of the parameters at time step (k ); Rk is the diagonal matrix corresponding to the measurement error; and (ki) denotes the predicted response corresponding to the SPs ( k(i) ).
Fig. 3. Parameter estimation using UKF.
When the parameters of a hysteretic model are identified, the identified result can be utilised to assess the mechanical properties of CLT connections. The performance evaluation between test and model involves the assessment according to the EN 12512 standard and the damage accumulation assessment. These two methods are evaluated from different aspects; the former compares mechanical properties including the maximum load capacity, ductility ratio, and energy dissipation, and the latter considers the cumulative damage. A two-step procedure is proposed in Fig. 4, where the first step is to identify the parameters using the UKF, and the second step is to assess the identified result using the two different approaches. The proposed procedure will be applied in the CLT connections in the following section.
Fig. 2. Example of a nonlinear transformation using UKF; (a) original and (b) transformed. 3
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
3.2. Test results The backbone curves of CLT connections in shear loading are presented in Fig. 8, where the legend “M” denotes one sample under a monotonous load, and “CSSJ-S-1” represents the specimen ID 1 of CSSJ in shear loading. For each type of connection, the performance of the samples under cyclic loading are consistent with the one under monotonous loading in the elastic stage. When the specimens suffer from yielding, their properties exhibit differences due to cracking of the CLT panels and yielding of the connectors. The ABJ samples have the highest maximum loading capacity among the three types of connections. Although the maximum loading capacity of SBJSS samples is low, they reach a large displacement at the failure point. For the CSSJ samples, both the failure displacement and the maximum loading capacity remain relatively low. Fig. 9 displays the backbone curves of three types of connections in tension loading, where the legend “M” represents one sample under monotonous loading, and “ABJ-T-1” denotes the specimen ID 1 of ABJ in the tension loading. For each connection, the specimens under both monotonous and cyclic loading have a similar trend before reaching the maximum loading capacity, beyond which there are steep decreases and irregular drops at the failure point.
Fig. 4. Two-step assessment procedure between the test and model.
3. Experimental and numerical study of CLT connections 3.1. Test introduction To study the mechanical properties of CLT connections, a variety of specimens were tested at the state key laboratory of disaster reduction in civil engineering, Tongji University, China. Three types of CLT connections, including angle bracket joint (ABJ), crossed self-tapping screws joint (CSSJ), and simple butt joint with self-tapping screws (SBJSS) were conducted under both monotonic and cycle tests, as summarised in Fig. 5. Fig. 6 presents the test photos of specimens under six different loading cases. More detailed information can be found in [36]. The monotonic test was conducted under displacement control at a constant slip rate of 0.05 mm/s, and it was continued until there was a 20% reduction from the maximum load. The specimens under cyclic loading were tested in both shear and tension directions. The shear loading protocol followed by the EN 12512 standard is shown in Fig. 7(a) [37], in which the specimens in the tension direction were subjected to a non-reversed modification of the procedure outlined in EN 12512 due to restrained movement on the compression side, as shown in Fig. 7(b). The predefined yield value (d y ) was obtained from the monotonic test.
3.3. Numerical model of CLT connections According to the test results, the cyclic loading response of CLT connections is highly nonlinear and exhibits pinching behaviour with strength and stiffness degradation. To simulate the behaviour of CLT connections under cyclic loading, a spring element assigned to the SAWS [38] model is created in the OpenSees [35], as shown in Fig. 10. The hysteretic rule of the SAWS model consists of an exponential function and a linear descent [39].
sgn(d )·(F0 + r1 S0 |d|)· 1 F=
sgn(d )· Fu + r2 S0 (d 0, |d| > |dF |
exp
(
S0 | d| F0
sgn(d )· du ), |du|
)
, |du| |d |
|d|
|dF |
|dF | (11)
where F and d are the force and displacement variables, respectively; F1 denotes the intercept strength for an asymptotic line to the envelope
Fig. 5. Test information for three CLT connections.
4
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 6. Photos of six different specimens.
Fig. 7. Procedure for the cyclic loading test in two directions; (a) in shear loading and (b) in tension loading.
Fig. 8. Backbone curves of CLT connections in shear loading.
curve; F0 is the intercept strength for the pinching part; S0 is the initial stiffness of the hysteresis curve; r1 is the stiffness ratio of an asymptotic line to the envelope curve; r2 is the stiffness ratio for the descending segment of the envelope curve; r3 is the stiffness ratio of the unloading segment of the envelope curve; and r4 is the stiffness ratio of the pinching part where it is simplified to a parallelogram. The parameters and control stiffness and energy degradation, respectively.
4. Parameter identification using UKF and the mechanical properties assessment 4.1. Parameter identification results Ten parameters should be identified in the SAWS model: = [F0, F1, S0, du , , , r1, r2, r3, r4 ]. During the estimation process, the loading displacement and the recorded force (F ) are regarded as input and output data, respectively. Because the experimental data inevitably 5
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 9. Backbone curves of CLT connections in tension loading.
Fig. 10. Numerical model of CLT connections; (a) spring element in shear loading, (b) hysteresis rule of the SAWS model, and (c) spring element in tension loading. Table 1 The identified parameters of SAWS model and its normalized relative error. Specimen ID
S0
r1
r2
r3
r4
F1
F0
du
ABJ-S-1 ABJ-S-2 ABJ-S-3 CSSJ-S-1 CSSJ-S-2 CSSJ-S-3 SBJSS-S-1 SBJSS-S-2 SBJSS-S-3 ABJ-T-1 ABJ-T-2 ABJ-T-3 CSSJ-T-1 CSSJ-T-2 CSSJ-T-3 SBJSS-T-1 SBJSS-T-2 SBJSS-T-3
7.38 6.84 5.58 4.03 1.92 2.06 2.52 2.63 2.48 7.17 7.23 6.13 8.15 6.92 10.54 5.65 7.45 7.82
0.08 0.09 0.15 0.056 0.053 0.041 0.057 0.054 0.045 0.07 0.16 0.11 0.036 0.052 0.042 0.052 0.043 0.035
−0.12 −0.25 −0.15 −0.17 −0.15 −0.18 −0.083 −0.045 −0.062 −0.13 −0.16 −0.165 −0.18 −0.14 −0.076 −0.09 −0.21 −0.16
1.08 1.38 1.24 3.26 4.09 4.21 2.18 2.57 1.83 1.23 1.18 1.32 0.79 0.93 0.72 0.93 0.69 0.82
0.025 0.05 0.04 0.015 0.034 0.021 0.001 0.006 0.004 0.015 0.026 0.032 0.013 0.024 0.027 0.032 0.027 0.024
32.06 43.75 33.73 10.52 9.48 9.19 9.13 8.52 9.35 26.67 15.32 15.42 14.78 20.21 14.65 12.86 12.28 13.21
1.94 3.22 3.58 0.84 0.92 0.55 0.46 0.55 0.63 1.31 1.06 1.76 0.42 0.49 0.38 0.32 0.31 0.34
22.16 28.31 31.99 12.95 18.09 15.39 41.9 30.32 36.25 40.02 32.66 40.23 4.13 2.83 2.43 2.83 5.61 2.78
contain noise, a zero mean, and a standard deviation of 0.5%F is assumed to construct the covariance matrix R for the measurement noise. The process noise wk is assumed to be time-invariant. The first-order statistics of wk are the zero-mean, and the second-order statistics are the diagonal covariance matrix Q = (1 × 10 3 × 0 |0 ) 2 , where 0 |0 is the prior distribution of the parameters according to the experience of engineers. The parameters of the scaled UT are chosen as = 0.1, = 0 , and = 2 [40]. The algorithm presented in Fig. 3 is employed to estimate the
NRE 0.35 0.26 0.42 0.73 0.43 0.39 0.56 0.75 0.68 0.62 0.33 0.56 0.76 0.72 0.45 0.41 0.46 0.55
0.92 1.17 1.08 1.09 0.86 0.97 1.08 1.04 0.93 0.98 0.95 1.06 1.13 1.07 1.01 1.07 1.03 1.05
0.0530 0.1079 0.0925 0.0224 0.0203 0.0121 0.0096 0.0101 0.0119 0.0332 0.0207 0.0433 0.0251 0.0238 0.0230 0.0205 0.0191 0.0242
unknown model parameters recursively, and the identified parameters of all connections are summarised in Table 1. Using the identified parameters, the generated hysteresis curves from the SAWS model are compared with the test results, as shown in Fig. 11. Due to the space limitations, only one specimen in the same loading case is presented in Fig. 11. In Fig. 11, the identified hysteresis curves of CLT connections in shear loading are consistent with the test results, whereas there are some differences between test and model in tension loading. There are 6
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 11. Six typical hysteresis curves that compare test and model.
negative forces for the tests when specimens are not subjected to negative displacement in Fig. 11(d), (e), and (f). This phenomenon is due to the plastic deformation of connectors, as shown in Fig. 12. When the connections are in the plastic stage, even if the force is unloading to zero, the displacement cannot be restored to the original point (i.e., displacement is zero) due to the residual deformation. The displacement continues unloading until zero, and the force then becomes negative. This test phenomenon cannot be considered in the SAWS model, leading to a significant difference between test and model. To quantify the estimation results, the normalised relative error (NRE) [41] is defined as follows:
n
NRE =
1 n k=1
zk
zk zk
2
, 2
(12)
where k is the time step, n is the total number of time steps, · 2 is the 2norm, and z k and zk are the forces obtained from the test and identified results, respectively. The NRE are also exhibited in the last column of Table 1, ranging from 0.0096 to 0.1079. If the evaluation of the identified results only depends on the values of NRE, then ABJ-S-1, CSSJ-S-3, SBJSS-S-1, ABJT-2, CSSJ-T-3, and SBJSS-T-2 are the best-identified specimens for each loading case. However, these evaluation results are from the mathematics aspect; it would be meaningful if the identified results are 7
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 13 compares the total energy dissipation between test and model for all the specimens, in which the total energy dissipation is calculated by the sum of the enclosed area of each loop. All three connections exhibit greater energy dissipation in shear loading than that in tension loading. The ABJ connections show the highest energy dissipation in both shear and tension directions, whereas the CSSJ connections exhibit the lowest value of energy dissipation. For the SBJSS connections, a large amount of energy is dissipated in shear loading, but in tension loading, a much smaller amount of energy dissipation occurs. Comparing the estimated and test results, the RE ranges from 0.4% (e.g., case CSSJ-S-3) to 24.7% (e.g., case CSSJ-T-2), demonstrating that the identified parameters have relatively high precision. 4.3. Identified results using the accumulative damage assessment In addition to the mechanical properties, damage evaluation is a critical index in the study of CLT connection behaviours. In recent decades, several damage models have been proposed, including the index of curvature, displacement, and energy [42]. Notably, Kraetzing et al. [43] developed a cumulative damage model based on the energy dissipation in a half cycle; i.e.,
Fig. 12. Plastic deformation of connectors.
assessed from the mechanical characteristics. The following sections will discuss the identified results from the mechanical properties. 4.2. Identified results using the EN 12512 standard assessment
D+ =
In this section, the identified results are compared with the test results in terms of strength, stiffness, ductility, and energy dissipation according to the EN 12512 standard. These mechanical properties of specimens under shear loading are computed by taking both sides of the hysteresis loops into account. In the case of specimens under tension loading, only the positive side is considered. Appendix Tables A and B compare the mechanical characteristics between the test and model in shear loading and tension loading, respectively, where the Fmax and dmax are the maximum load capacity and the corresponding displacement, respectively; Fy and d y denote the yielding load and the corresponding displacement, respectively; K 0 denotes the initial stiffness; and is the ductility ratio between failure displacement (du ) and yield displacement
(i. e. , RE =
=
du dy
Vt Vt
× 100%
E+ f
+
Ei+ Ei+
,
(14)
where D+ is the positive damage index, Ep+, i denotes the energy dissipation in a primary half cycle (PHC), Ei+ represents the energy dissipation in a follower half cycle (FHC), and E + f is the energy dissipation
in a monotonic test to failure. Fig. 14 depicts the protocol of PHC and FHC, where the PHC represents any half cycle whose amplitude exceeds those in all previous cycles, and the FHC denotes all subsequent cycles of smaller or equal amplitude. The damage index for the negative part is calculated using the same formula as Eq. (14), with a negative superscript inserted (i.e., Ep, i , Ei , E f ). Therefore, the overall damage index (D ) can be computed as
). The relative error (RE) is calculated as
Vm
Ep+, i +
D = D+ + D
D+D
(15)
In Eq. (14), both the numerator and denominator include the FHC energy, meaning that the FHC makes less of a contribution to the damage index than the PHC. By applying Eqs. (14) and (15) to the hysteresis response, the cumulative damage of CLT connections at each time step is computed and will be discussed in the following subsections.
(13)
where Vt is the value obtained from the test, Vm is the value calculated from the identified result, and |·| is the absolute operation. According to Table A1 in the Appendix, the maximum load shows good agreement between test and model, and the value of RE is less than 10% except for the specimen CSSJ-S-2. The RE of the corresponding displacement ranges from 0.76% to 27.35%. The initial stiffness difference between test and model is no more than 20%, and the RE of ductility ratio ranges from 1.78% (e.g., SBJSS-S-2) to 43.48% (e.g., CSSJ-S-2). Such large differences may be caused by model error. In some specimens, the measured force shows an irregular fluctuation, which is not well captured by the SAWS model, resulting in a variation of yielding displacement. The difference in yielding displacement between test and model aggravates the RE of the ductility ratio. The comparison of the mechanical properties of CLT connections in tension loading between test and model are shown in Appendix Table B1. The difference in yielding force between test and model is less than 17%, whereas the RE of ductility ratio ranges from 5.49% (e.g., SBJSS-T-1) to 62.44% (e.g., ABJ-T-3). The negative forces for the test data provide clues to the reasons for such large differences. Although the results of three specimens for different types of connections show great dispersion, this paper focuses on developing the procedure of parameter identification using UKF, which is capable of illustrating the suitable of UKF method for the parameter identification of a hysteretic model.
4.3.1. Cumulative damage of CLT connections in shear loading A comparison of the cumulative damage curves from the test and model is shown in Fig. 15; the legends ‘T’ and ‘M’ designate the curves generated from the test and model, respectively; the numbers ‘1, 2, 3’ denote the specimen ID; the legend ‘Aver’ represents the average value calculated from the three specimens. The overall trend of the model is similar to that of the test results, but differences appear over time. Several factors influence the characteristics of the damage accumulation curve. In Fig. 11, the test hysteresis curves show asymmetry between the positive and negative sides. However, the SAWS model can only consider situations where there is symmetry between the two sides. Another factor that causes variation is seen in pinching behaviour. A visible pinching effect is observed in the test results, whereas two parallel lines are adopted to describe the pinching in the SAWS model. According to the calculated results shown in Fig. 15, the failure point of CLT connections is defined at approximately D = 0.8 [44]. The t time to failure point (i.e., tD = 0.8) and the failure time ratio = Dt=0.8 are w summarised in Table 2, where tw denotes the total time of the loading protocol. The last column in Table 2 presents the relative error of the
8
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 13. Comparison of total energy dissipation between test and model.
failure time ratio ( ) between test ( t ) and model ( m ); i.e.,
=
t
m t
the variation is the subsequent cycles of the loading procedure. The SAWS model defines a repetitive cycle with a slope of 1 (r4 S0) (shown in Fig. 10b), which is a linear approximation relative to the test observations.
,
where |·| is the absolute operation. The value of ranges from 2.98% to 31.73%. The reasons for such a significant variation can be attributed to the hysteresis response. The test envelope curve is often irregular following the maximum load, whereas the envelope curve of the SAWS model follows a linear degradation. Another factor that contributes to
4.3.2. The cumulative damage of CLT connections in tension loading A comparison of cumulative damage curves between test and model 9
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Table 2 Comparison of time at D = 0.8 and the failure time ratio in the shear loading. Specimen ID
ABJ-S-1 ABJ-S-2 ABJ-S-3 CSSJ-S-1 CSSJ-S-2 CSSJ-S-3 SBJSS-S-1 SBJSS-S-2 SBJSS-S-3
tD = 0.8
(%)
tw
Test
Model
355.01 423.65 496.16 329.52 473.43 602.56 584.84 678.58 613.54
395.72 436.28 454.13 379.24 623.63 528.20 511.38 614.64 462.44
659.72 527.60 571.76 908.40 1201.08 1198.04 1236.16 1329.88 1235.60
t (%)
m (%)
53.81 80.30 86.78 36.27 39.42 50.30 47.31 51.03 49.66
59.98 82.69 79.43 41.75 51.92 44.09 41.37 46.22 37.43
11.47 2.98 8.47 15.09 31.73 12.34 12.56 9.42 24.63
model increases because the CLT connections suffer yielding of the screws in the test. However, only two parameters are directly defined for cyclic degradation in the SAWS model, which leads to the SAWS model having the same degradation factor over the entire protocol. Table 3 compares the time of D = 0.8 between the test and model. The relative error for all specimens does not exceed 11%, except for specimen ABJ-T-1.
Fig. 14. Protocol of PHC and FHC.
in tension loading is given in Fig. 16. The shapes of the cumulative damage curves from the model have trends similar to the test curves. Before the curves reach D = 0.2, the variation between them is negligible. When D ranges from 0.2 to 0.8, the difference between test and
Fig. 15. Comparison of the cumulative damage in shear loading. 10
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Fig. 16. Comparison of the cumulative damage in tension loading.
load capacity, elastic stiffness, and ductility ratio are compared and evaluated. This is a straightforward method, and its results can be summarised and compared easily in a table. In addition, the mechanical properties obtained from the test also provide reliable design values to engineers and a better understanding of the mechanical characteristics of CLT connections. However, this method is focused on the envelope curve, which means that the influences of lower amplitude and cumulative damage are not considered. The second method is a damage accumulation assessment that examines the relationship between force, displacement, and time. This approach assesses the damage that accounts for a single high-amplitude cycle and repeated cycles at a low amplitude. However, this cumulative damage is considered from an initial stage, and small variations will be accumulated during the whole loading process, which means that the cumulative damage is focused on the overall performance and is not sensitive to the detailed damage of components. These two assessment methods have advantages and disadvantages. When assessing the mechanical characteristic of CLT connections using these two methods together, they fully evaluate the identified results.
Table 3 Comparison of time at D = 0.8 and the failure time ratio in the tension loading. Specimen ID
ABJ-T-1 ABJ-T-2 ABJ-T-3 CSSJ-T-1 CSSJ-T-2 CSSJ-T-3 SBJSS-T-1 SBJSS-T-2 SBJSS-T-3
tD = 0.8
(%)
tw
Test
Model
322.33 434.64 458.17 202.90 395.25 383.08 424.86 438.59 446.33
402.76 440.47 463.53 197.36 384.81 366.53 467.95 423.03 416.34
748.52 881.40 902.80 499.40 647.04 647.08 605.96 607.84 606.44
t (%)
m (%)
43.06 49.31 50.75 40.63 61.09 59.20 70.11 72.16 73.60
53.81 49.97 51.34 39.52 59.47 56.64 77.22 69.60 68.65
24.95 1.34 1.17 2.73 2.64 4.32 10.14 3.55 6.72
4.4. Comparison of the two assessment methods Two different methods are adopted to evaluate and compare the identified results. The first approach is assessed from the mechanical properties according to the EN 12512 standard, where the maximum
11
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
5. Conclusions
sults in terms of the mechanical properties: the assessment according to the EN 12512 standard, and the damage accumulation assessment. The former is evaluated in term of elastic stiffness, ductility ratio, and energy dissipation, whereas the latter considers the accumulated damage. Together, these two methods fully evaluate the identified result and the mechanical characteristics of CLT connections.
In this paper, a novel procedure that combines a parameter identification technique and two assessment methods is presented to compare and evaluate the mechanical performance of CLT connections. A numerical model is created, for which the nonlinear behaviour is governed by the SAWS model, and the unscented Kalman filter is employed to identify the model parameters. Once parameters are identified, two different approaches are used to evaluate the identified results. The following conclusions can be drawn from the results:
Declaration of Competing Interest The author declare that there is no conflict of interest.
(1) A novel framework that integrates the UKF as an estimation tool with a hysteretic model provides parameter identification for the nonlinear system. The novelty is in coupling the nonlinear model in OpenSees, which allows the complex finite element model to be analysed. (2) The developed framework is applied to identify the hysteretic model parameters of CLT connections using the available experimental data. The identified results have good agreement with the test data, demonstrating that the UKF is suitable for the nonlinear system identification. (3) Two different methods are proposed to evaluate the identified re-
Acknowledgments This paper is funded by the National Natural Science Foundation (51978502), Key project of international cooperation in science and technology innovation (2016YFE0105600), Fundamental Research Funds for the Central University (22120180315, 22120170521). The financial support is greatly appreciated. Special thanks are given to the China Scholarship Council, which supports the first author for the research work in UCLA. The authors also thank the anonymous reviewers for their constructive comments.
Appendix A See Tables A1 and B1.
Table A1 Comparison of mechanical properties for CLT connections in shear direction between test and model. Connections types
Fmax [kN]
dmax [mm]
K0
kN mm
Fy [kN]
dy [mm]
[ ]
ABJ-S-1
Test Model RE/%
49.92 45.17 9.51
28.08 20.93 25.48
4.28 5.10 19.23
33.59 33.48 0.33
8.71 6.34 27.19
3.94 5.28 33.97
ABJ-S-2
Test Model RE/%
59.98 60.31 0.55
24.25 28.01 15.51
5.14 4.85 5.48
38.86 45.36 16.72
9.39 9.05 3.57
3.51 3.87 10.02
ABJ-S-3
Test Model RE/%
59.29 60.14 1.44
27.20 31.93 17.39
4.29 3.79 11.65
44.98 47.68 5.99
12.15 12.18 0.21
3.44 3.61 4.96
CSSJ-S-1
Test Model RE/%
12.32 12.76 3.58
12.95 11.98 7.45
2.01 2.11 5.35
8.47 8.38 1.00
4.05 3.61 10.99
4.08 4.46 9.27
CSSJ-S-2
Test Model RE/%
13.22 8.36 36.71
18.09 15.55 14.07
1.35 1.43 6.03
6.90 6.62 3.99
5.06 4.48 11.56
4.27 6.13 43.48
CSSJ-S-3
Test Model RE/%
10.24 9.83 3.92
15.39 14.99 2.60
1.64 1.48 9.69
6.95 7.52 8.24
4.07 4.93 21.06
4.72 4.15 11.95
SBJSS-S-1
Test Model RE/%
13.71 13.94 1.74
41.90 41.07 1.98
1.56 1.55 0.59
9.40 10.45 11.23
16.61 13.58 18.22
3.31 3.72 12.46
SBJSS-S-2
Test Model RE/%
11.62 12.40 6.68
41.32 30.02 27.35
1.63 1.66 2.19
9.49 10.13 6.69
15.48 13.57 12.31
3.78 3.85 1.78
SBJSS-S-3
Test Model RE/%
13.12 13.82 5.34
36.25 35.98 0.76
1.82 1.58 12.76
10.53 10.88 3.33
15.54 14.11 9.21
3.78 3.91 3.48
12
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al.
Table B1 Comparison of mechanical properties for CLT connections in tension direction between test and model. Connections types
Fmax [kN]
dmax [mm]
K0
kN mm
Fy [kN]
dy [mm]
[ ]
ABJ-T-1
Test Model RE
34.22 38.19 11.61
23.02 23.04 0.09
5.61 4.80 14.36
23.64 26.85 13.58
7.23 9.94 37.48
6.49 5.62 13.48
ABJ-T-2
Test Model RE
48.06 44.17 8.10
32.66 33.14 1.47
3.83 3.96 3.43
30.56 25.66 16.03
7.42 8.29 11.80
6.31 5.53 12.28
ABJ-T-3
Test Model RE
33.66 42.27 25.59
46.00 40.50 11.96
3.31 2.83 14.44
22.08 23.08 4.51
12.39 7.33 40.84
4.45 7.23 62.44
CSSJ-T-1
Test Model RE
13.62 14.37 5.48
3.60 4.14 15.00
5.97 6.16 3.22
11.35 12.09 6.48
1.96 1.92 2.04
3.16 3.50 10.75
CSSJ-T-2
Test Model RE
12.78 12.58 1.56
2.95 2.25 23.73
6.13 6.32 3.06
11.98 10.16 15.25
2.11 1.61 23.70
2.44 3.81 56.09
CSSJ-T-3
Test Model RE
12.09 13.01 7.59
2.28 2.51 10.09
7.11 7.64 7.49
11.38 10.97 3.59
1.62 1.96 21.36
4.08 3.56 12.78
SBJSS-T-1
Test Model RE
9.45 10.42 10.30
2.83 3.51 24.03
4.23 4.31 1.93
8.94 9.43 5.47
2.13 2.17 1.88
2.87 3.03 5.49
SBJSS-T-2
Test Model RE
13.06 13.58 3.97
5.36 5.74 7.09
5.94 5.85 1.54
9.68 9.92 2.49
1.59 1.65 3.79
5.40 4.77 11.81
SBJSS-T-3
Test Model RE
11.11 11.33 1.95
2.78 2.75 1.08
6.01 6.41 6.76
10.03 10.04 0.16
1.66 1.54 6.95
2.77 3.27 18.28
Appendix B. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engstruct.2019.109867.
[13] D’Arenzo G, Rinaldin G, Fossetti M, et al. Tensile and shear behaviour of an innovative angle bracket for CLT structures. In: 2018 World Conference on Timber Engineering, Seoul, Korea; 2018. [14] Liu J, Lam F. Experimental test of coupling effect on CLT angle bracket connections. Eng Struct 2018;171:862–73. [15] Pozza L, Saetta A, Savoia M, Talledo D. Coupled axial-shear numerical model for CLT connections. Constr Build Mater 2017;150:568–82. [16] Bita HM, Tannert T. Numerical optimisation of novel connection for cross-laminated timber buildings. Eng Struct 2018;175:273–83. [17] Hossain A, Lakshman R, Tannert T. Shear connections with self-tapping screws for cross-laminated timber panels. In: Structures Congress, Oregon, Portland; 2015. [18] Sullivan K, Miller TH, Gupta R. Behavior of cross-laminated timber diaphragm connections with self-tapping screws. Eng Struct 2018;168:505–24. [19] Pei S, Van De Lindt JW, Popovski M, Berman JW, Dolan JD, Ricles J, et al. Crosslaminated timber for seismic regions: progress and challenges for research and implementation. J Struct Eng 2014;142(4):E2514001. [20] Reynolds T, Casagrande D, Tomasi R. Comparison of multi-storey cross-laminated timber and timber frame buildings by in situ modal analysis. Constr Build Mater 2016;102:1009–17. [21] Pozza L, Savoia M, Franco L, Saetta A, Talledo D. Effect of different modelling approaches on the prediction of the seismic response of multi-storey CLT buildings. Timber Struct Eng 2018::163. [22] Fragiacomo M, Dujić B, Šušteršič cI. Elastic and ductile design of multi-storey crosslam massive wooden buildings under seismic actions. Eng Struct 2011;33(11):3043–53. [23] Pozza L, Scotta R, Trutalli D, Polastri A. Behaviour factor for innovative massive timber shear walls. B Earthq Eng 2015;13(11):3449–69. [24] Shen YL, Schneider J, Tesfamariam S, Stiemer SF, Mu ZG. Hysteresis behavior of bracket connection in cross-laminated-timber shear walls. Constr Build Mater 2013;48:980–91. [25] Schneider J, Shen Y, Stiemer SF, Tesfamariam S. Assessment and comparison of experimental and numerical model studies of cross-laminated timber mechanical connections under cyclic loading. Constr Build Mater 2015;77:197–212. [26] Liu P, Au SK. Bayesian parameter identification of hysteretic behavior of composite walls. Probabilist Eng Mech 2013;34:101–9. [27] Green PL. Bayesian system identification of a nonlinear dynamical system using a novel variant of simulated annealing. Mech Syst Signal Pr 2015;52:133–46.
References [1] Brandner R, Flatscher G, Ringhofer A, Schickhofer G, Thiel A. Cross laminated timber (CLT): overview and development. Eur J Wood Prod 2016;74(3):331–51. [2] Yates M, Linegar M, Dujić B. Design of an 8 storey Residential Tower from KLH Cross Laminated Solid Timber Panels. In: Proceedings of the 10th World Conference on Timber Engineering, Miyazaki, Japan; 2008. [3] Ramage MH, Burridge H, Busse-Wicher M, Fereday G, Reynolds T, Shah DU. The wood from the trees: the use of timber in construction. Renew Sust Energ Rev 2017;68:333–59. [4] Šušteršič I, Fragiacomo M, Dujić B. Seismic analysis of cross-laminated multistory timber buildings using code-prescribed methods: influence of panel size, connection ductility, and schematization. J Struct Eng 2015;142(4):E4015012. [5] Ceccotti A, Sandhaas C, Okabe M, Yasumura M, Minowa C, Kawai N. SOFIE project—3D shaking table test on a seven-storey full-scale cross-laminated building. Earthq Eng Struct Dyn 2013;42(13):2003–21. [6] Popovski M, Gavrić I, Schneider J. Performance of two-storey CLT house subjected to lateral loads. In: 12th World Conference of Timber Engineering 2014, Auckland, New Zealand; 2014. [7] Pozza L, Scotta R, Vitaliani,R. A non-linear numerical model for the assessment of the seismic behavior and ductility factor of X-Lam timber structures. In: International symposium on timber structures, Istanbul, Turkey; 2009. [8] Gavrić I, Fragiacomo M, Ceccotti A. Cyclic behavior of CLT wall systems: Experimental tests and analytical prediction models. J Struct Eng 2015;141(11):04015034. [9] Akbas T, Sause R, Ricles JM, Ganey R, Berman J, Loftus S. Analytical and experimental lateral-load response of self-centering posttensioned CLT walls. J Struct Eng 2017;143(6):04017019. [10] Schneider J. Connections in cross-laminated-timber shear walls considering the behaviour under monotonic and cyclic lateral loading. MS thesis. Univ. Stuttgart, Germany; 2009. [11] Fragiacomo M, Dujić B, Šušteršič I. Elastic and ductile design of multi-storey crosslam massive wooden buildings under seismic actions. Eng Struct 2011;33(11):3043–53. [12] Gavrić I, Fragiacomo M, Ceccotti A. Cyclic behaviour of typical metal connectors for cross-laminated (CLT) structures. Mater Struct 2015;48(6):1841–57.
13
Engineering Structures xxx (xxxx) xxxx
J. Cao, et al. [28] Corigliano A, Mariani S. Parameter identification in explicit structural dynamics: performance of the extended Kalman filter. Comput Method Appl M 2004;193(36–38):3807–35. [29] Wu M, Smyth AW. Application of the unscented Kalman filter for real-time nonlinear structural system identification. Struct Control Hlth 2007;14(7):971–90. [30] Ching Jianye, Beck James L, Porter Keith A. Bayesian state and parameter estimation of uncertain dynamical systems. Probabilist Eng Mech 2006;21(1):81–96. [31] Astroza R, Ebrahimian H, Conte JP. Performance comparison of Kalman−based filters for nonlinear structural finite element model updating. J Sound Vib 2019;438:520–42. [32] Ebrahimian H, Astroza R, Conte JP. Extended Kalman filter for material parameter estimation in nonlinear structural finite element models using direct differentiation method. Earthq Eng Struct D 2015;44(10):1495–522. [33] Wan, Eric A, Rudolph Van Der Merwe. The unscented Kalman filter for nonlinear estimation. In: Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No. 00EX373), Alberta, Canada; 2000. [34] https://www.python.org/. [35] Mazzoni S, McKenna F, Scott MH, Fenves GL. OpenSees command language manual [J]. Pacific Earthq Eng Res (PEER) Center 2006:264. [36] Cao Jixing, Xiong Haibei, Chen Jiawei, Huynh Alexandre. Bayesian parameter
[37] [38] [39] [40] [41] [42] [43] [44]
14
identification for empirical model of CLT connections. Constr Build Mater 2019;218:254–69. EN 12512 (2001). Timber structures—test methods—cyclic testing of joints made with mechanical fasteners. CEN, Brussels, Belgium. https://opensees.berkeley.edu/wiki/index.php/SAWS_Material. Folz Bryan, Filiatrault Andre. Seismic analysis of woodframe structures. I: Model formulation. J Struct Eng 2004;130(9):1353–60. Julier S, Uhlmann J, Durrant-Whyte HF. A new method for the nonlinear transformation of means and covariances in filters and estimators. IEEE Automat Contr 2000;45(3):477–82. Erazo K, Hernandez EM. Uncertainty quantification of state estimation in nonlinear structural systems with application to seismic response in buildings. ASCE-ASME J Risk Uncertainty Eng Syst A: Civil Eng 2015;2(3):B5015001. Williams Martin S, Sexsmith Robert G. Seismic damage indices for concrete structures: a state-of-the-art review. Earthq Spectra 1995;11(2):319–49. Kraetzig W, Meyer I, Meskouris K. Damage evolution in reinforced concrete members under cyclic loading. In: 5th international conference on structural safety and reliability, San Francisco, USA; 1989. Schneider J, Karacabeyli E, Popovski M, Stiemer SF, Tesfamariam S. Damage assessment of connections used in cross-laminated timber subject to cyclic loads. J Perform Constr Fac 2013;28(6):A4014008.