5.1 Backbone curve analysis
The backbone curves obtained from the tests were modelled using numerical fitting [10]. The dimensionless backbone curves are shown in Fig. 5.
By analysing the trend of the dimensionless backbone curves, the load-carrying and deformation characteristics of the PBCJ are roughly divided into three stages:
(1) The elastic stage: Before the PBCJ entered the elastoplastic state, the load-displacement curve was basically linear, and only a few tiny cracks were generated. At this stage, the slope of the curve was the initial stiffness of the PBCJ, and the prefabricated beam-to-column subassemblage was basically in an elastic state.
(2) The strengthening stage: After the PBCJ entered the elastoplastic stage, as the displacement increased, the load growth rate decreased and the stiffness degraded severely. This is mainly because as the PBCJ yielded, the cracks at the beam ends propagated continuously, causing the accumulation of damage to concrete.
(3) The degradation stage: After the PBCJ entered the plastic stage, the load and displacement were negatively correlated. The concrete in the plastic hinge region of the panel zone of the PBCJ was crushed. When the load degraded to 85% of the ultimate load, the loading was stopped.
By summarising the three stages that reflect the basic change trend of the backbone curves, the characteristics of the load-carrying and deformation process of the specimens can be visually observed.
5.2 Backbone curve model
Based on the analysis of the test results, an asymmetric trilinear model with descending sections in both positive and reverse directions was used to fit the backbone curves of the PBCJ specimens. A three-segment polyline was obtained by fitting the dimensionless data of the specimens using the yield point, the ultimate load point, and the maximum deformation point as the boundary points [9]. Three points in each direction were determined from the fitted three-segment polyline: the yield point (0.38, 0.88), the ultimate load point (1.0, 1.0) and the maximum deformation point (1.41, 0.9) in the positive direction; and the yield point (-0.375, -0.84), ultimate load point (-1.0, -1.0) and maximum deformation point (-1.30, -0.88) in the negative direction. It is noted that the maximum displacement was the average of the ultimate displacements of different specimens. The fitting equations are provided in Table 2.
Table 2
Backbone curve fitting equations
Line segment
|
Fitting equation
|
OA
|
P/Pu+=2.314Δ/Δu+
|
AB
|
P/Pu+=0.205Δ/Δu+ + 0.802
|
BC
|
P/Pu+=-0.056Δ/Δu+ + 1.028
|
OD
|
P/Pu−=2.236Δ/Δu−
|
DE
|
P/Pu−=0.267Δ/Δu−-0.745
|
EF
|
P/Pu−=-0.234Δ/Δu−-1.215
|
5.3 Backbone curve verification
Using the regression equation of each stage of the backbone curve model and the coordinates of the characteristic points given in Table 2, the theoretical backbone curve of the PBCJ was calculated and then compared with the test result of each PBCJ specimen, as shown in Fig. 6. The data in the figure show that the proposed trilinear backbone curve model is consistent with the test values of the joints and thus well reflects the load-displacement relationship of this type of joint under the quasi-static loading test.
5.4 Analysis of hysteresis curve test results
The red dashed lines in Fig. 9 are the hysteresis curves of the six specimens obtained from the tests. These curves show that:
(1) The hysteresis curves obtained from the new type of bolted PBCJ are full, basically without pinching, and roughly in a parallelogram shape. The positive and reverse stiffnesses of the PBCJs are similar.
(2) With the increase of the horizontal load at the end of the column, minor vertical cracks appeared in the concrete at both upper and lower flanges of the beam in the plastic hinge region, and the area enclosed by the hysteresis curve is very small. After the longitudinal regular reinforcement yielded, the hysteresis curve of the PBCJ is full and spindle-shaped.
(3) At the ultimate load, the concrete cover in the plastic hinge region spalled severely and some stirrups were exposed; however, the load-carrying capacity of the PBCJ decreased slowly after the peak load was reached. It is thus concluded that the PBCJ can still carry the corresponding load after the concrete cover spalls, demonstrating good seismic performance.
5.5 Stiffness degradation law
(1) Degeneration law of the positive unloading stiffness K1
The relation between
and
is calculated by regression analysis, as shown in Fig. 7.
After the yielding of the PBCJ, the slope of the section of the curve from the starting point of positive unloading to the point at which the force is zero on the hysteresis loop, as obtained from the regression analysis, is the positive unloading stiffness [8] K1. The regression equation for this section of the stiffness degradation curve is as follows:
![](https://myfiles.space/user_files/83064_0857a92044b57365/83064_custom_files/img1624285201.png)
(2) Degeneration law of the reverse unloading stiffness K2
The slope of the section of the curve from the point at which the positive unloading becomes zero to the point at which the reverse loading reaches the peak under the same cyclic load, as obtained from the regression analysis, is the reverse loading stiffness [8] K2. The regression equation for this section of the stiffness degradation curve is as follows:
![](https://myfiles.space/user_files/83064_0857a92044b57365/83064_custom_files/img1624285229.png)
(3) Degeneration law of the negative unloading stiffness K3
Under the same cyclic loading, the slope of the section of the curve from the starting point of reverse unloading to the point at which the force is unloaded to zero, as obtained from the regression analysis, is the reverse loading stiffness [8] K3. The regression equation for this section of the stiffness degradation curve is as follows :
![](https://myfiles.space/user_files/83064_0857a92044b57365/83064_custom_files/img1624285291.png)
(4) Degeneration law of the positive loading stiffness K4
Under the same cyclic loading, the slope of the section of the curve from the point at which the reverse unloading reaches zero to the point at which the positive loading reaches the peak, as obtained from the regression analysis, is the positive loading stiffness [8] K4. The regression equation for this section of the stiffness degradation curve is as follows:
![](https://myfiles.space/user_files/83064_0857a92044b57365/83064_custom_files/img1624285314.png)
where
and
are the initial stiffnesses corresponding to positive and reverse loading, respectively;
and
are the displacements corresponding to the maximum positive and reverse loading, respectively; and Δ1, Δ2, Δ3, and Δ4 are the displacements under positive unloading, displacement after positive unloading, displacement under reverse unloading, and displacement after reverse unloading, respectively.
5.6 Hysteresis rule
The fitted trilinear backbone curve agrees well with the stiffness degradation law, and thus the hysteresis rule applicable to this type of joint is proposed, as shown in Fig. 8.
(1) In the OA and OD sections, the structural member is in the elastic state, where there is a linear load-displacement relationship. The positive and reverse loading and unloading stiffnesses are the initial stiffnesses.
(2) In the AB and DE sections, the structural member enters the yielding stage, and its loading and unloading stiffnesses at all levels gradually decrease. The unloading [8] starts at point a towards point b, with an unloading stiffness Kab; then, the reverse loading is directed towards point c, with a loading stiffness Kbc; finally, the reverse unloading is directed towards point d, with an unloading stiffness Kcd.
(3) The positive loading is directed towards point a, with a loading stiffness Kda; the positive loading continues along the backbone curve described by the equations in Table 2 and starts the next cycle of displacement-controlled loading.
5.7 Verification of the restoring force model for the new type of PBCJ
The seismic performance of the new type of bolted PBCJ is affected by various parameters that defy the establishment of a unified equation to accurately describe the load-displacement relationship of the joint. High regularity can be attained from the backbone curve and the stiffness degradation curve, thereby proving the reasonableness of the restoring force model of the new type of bolted PBCJ. Figure 9 compares the hysteresis curve constructed according to the above stiffness degradation and hysteresis rules with the hysteresis curves obtained from the tests. Evidently, the proposed restoring force model has a high goodness of fit, and the discrepancies in the calculation results for some individual members were mainly due to the dispersion of the backbone curves after nondimensionalisation. The overall results were very good, proving that the proposed restoring force model can accurately simulate the seismic performance of this type of joint.