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Figure 6. (a) Sections of the 1:6 scaled RC circular piers whose materials (CLS C30, HRB400E and R235) come from the Chinese Code JTG D62-2004 (JTG D62-2004, 2004) the pier R16-1A is obtained through repair and retrofit of the pier P16-1A, after being tested. (b) View of the cyclic test setup; a constant vertical load and an horizontal displacement are applied on RC pier specimens at the Fuzhou University Lab.

Figure 6 (a) Sections of the 1:6 scaled RC circular piers whose materials (CLS C30, HRB400E and R235) come from the Chinese Code JTG D62-2004 (JTG D62-2004, 2004) the pier R16-1A is obtained through repair and retrofit of the pier P16-1A, after being tested. (b) View of the cyclic test setup; a constant vertical load and an horizontal displacement are applied on RC pier specimens at the Fuzhou University Lab.

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Table 1. Summary of parameters for Bouc-Wen (BW) and Bouc-Wen-Baber-Noori (BWBN) models and their roles for hysteresis shape definition. Finally, the term h(z(f)) in Equation (4) is the pinching function and it is expressed by the relation:
Table 1. Summary of parameters for Bouc-Wen (BW) and Bouc-Wen-Baber-Noori (BWBN) models and their roles for hysteresis shape definition. Finally, the term h(z(f)) in Equation (4) is the pinching function and it is expressed by the relation:
Figure 1. Effect of Bouc-Wen model parameters variation on the simulated hysteretic behaviour of the system, in terms of force (F) and displacement (x).
Figure 1. Effect of Bouc-Wen model parameters variation on the simulated hysteretic behaviour of the system, in terms of force (F) and displacement (x).
(b) Regulation of the second term of 7(e) in order to limit the increasing of the stiffness degradation rate.
(b) Regulation of the second term of 7(e) in order to limit the increasing of the stiffness degradation rate.
Figure 3. Normalised Gaussian function fp (x) used to simulate the pinching effect as a reduction of the stiffness confined in a central range of displacemen Figure 2. Stiffness degradation control in the proposed modified BW model.
Figure 3. Normalised Gaussian function fp (x) used to simulate the pinching effect as a reduction of the stiffness confined in a central range of displacemen Figure 2. Stiffness degradation control in the proposed modified BW model.
Figure 4. Degradation of the stiffness (k) in the central range of displacement (x); o controls the width of the displacement interval in which the stiffness reduction occurs, p defines the amount of reduction with respect to the initial stiffness value kj, and u determines the slope with which the stiffness decrease occurs.
Figure 4. Degradation of the stiffness (k) in the central range of displacement (x); o controls the width of the displacement interval in which the stiffness reduction occurs, p defines the amount of reduction with respect to the initial stiffness value kj, and u determines the slope with which the stiffness decrease occurs.
Table 2. Summary of the parameters of the proposed modified BW model and their roles for hysteresis shape definition.
Table 2. Summary of the parameters of the proposed modified BW model and their roles for hysteresis shape definition.
Table 3. Optimal values of the modified BW model parameters obtained through the GA and relative errors in terms of force (OF) for the analysed piers P16-1A and R16-1A.
Table 3. Optimal values of the modified BW model parameters obtained through the GA and relative errors in terms of force (OF) for the analysed piers P16-1A and R16-1A.
Table 4. Optimal values of the BWBN model parameters obtained through the GA and relative errors in terms of force (OF) for the analysed piers P16-1A and R16-1A. 5.4. Comparison between the results obtained with the two models
Table 4. Optimal values of the BWBN model parameters obtained through the GA and relative errors in terms of force (OF) for the analysed piers P16-1A and R16-1A. 5.4. Comparison between the results obtained with the two models
Table 5. Comparison of the relative errors er in terms of dissipated energy committed with the proposed modified BW model (BWM) and the BWBN model. limit the already enormous computational burden due to the complexity of the differential equation and the large number of unknown parameters. Figure 10 shows the decrease of the OF value as the generations progress for both piers P16-1A and R16-1A, in the case of modified BW model (BWM) and BWBN model. This gives an idea of the rapidity with which the models reach the final simulated results.
Table 5. Comparison of the relative errors er in terms of dissipated energy committed with the proposed modified BW model (BWM) and the BWBN model. limit the already enormous computational burden due to the complexity of the differential equation and the large number of unknown parameters. Figure 10 shows the decrease of the OF value as the generations progress for both piers P16-1A and R16-1A, in the case of modified BW model (BWM) and BWBN model. This gives an idea of the rapidity with which the models reach the final simulated results.
Figure 7. Comparison between experimental force data (Fey) and force simulated through the proposed modified BW model (Fgyyjy) in terms of force—displacement curves for the piers P16-1A and R16-1A.
Figure 7. Comparison between experimental force data (Fey) and force simulated through the proposed modified BW model (Fgyyjy) in terms of force—displacement curves for the piers P16-1A and R16-1A.
Figure 8. Comparison between experimental force data (Fey) and force simulated through the BWBN model (Fgygy) in terms of force-displacement curves for the piers P16-1A and R16-1A.
Figure 8. Comparison between experimental force data (Fey) and force simulated through the BWBN model (Fgygy) in terms of force-displacement curves for the piers P16-1A and R16-1A.
Figure 10. OF value with increasing the GA iterations (generations); comparison of the trend for the piers P16-1A and R16-1A by using the proposed modified BW model (BWM) and the BWBN model. Figure 9. Comparison between experimental dissipated energy (€ex), dissipated energy simulated through the proposed modified BW model (egy), and dissipated energy simulated through the BWBN model (egy) for the piers P16-1A and R16-1A.
Figure 10. OF value with increasing the GA iterations (generations); comparison of the trend for the piers P16-1A and R16-1A by using the proposed modified BW model (BWM) and the BWBN model. Figure 9. Comparison between experimental dissipated energy (€ex), dissipated energy simulated through the proposed modified BW model (egy), and dissipated energy simulated through the BWBN model (egy) for the piers P16-1A and R16-1A.
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