# Development of a New Rubber Buckling-Restrained Brace System for Structures
## 10 Cumulative Plastic Deformation (CPD)  
Cumulative Plastic Deformation (CPD) capacity is a critical parameter for Buckling-Restrained Braces (BRBs), indicating the total plastic deformation a BRB can endure before fracturing. Accurately predicting CPD capacity is essential for ensuring the reliability and safety of structures employing BRBs. Factors affecting Cumulative Plastic Deformation include material properties (e.g., elastic modulus, yield strength), structural geometry (e.g., slenderness ratio, cross-sectional shape), loading conditions (e.g., intensity, frequency, and type of loading), and environmental factors (e.g., temperature, corrosion conditions), all of which can either amplify or limit the accumulation of plastic deformation. To determine the Cumulative Plastic Deformation (CPD), the hysteretic response of the considered Buckling-Restrained Brace (BRB) (or RBRB in this study) can be used. The CPD is then calculated as the cumulative sum of the plastic deformations across all load cycles. Plastic deformation is defined as the residual deformation that remains unrecovered after the load is removed. In Equation (5) For the i(Pull) half-load cycle, the plastic deformation (\(\Delta_{\text{P}}\)) is calculated as the difference in deformation at zero force at the start (\(\Delta_{\text{i(Push)}}\)) and end (\(\Delta_{\text{i(Pull)}}\)) of the half cycle:  
\[\Delta_{\text{P}}=|\Delta_{\text{i(Pull)}}-\Delta_{\text{i(Push)}}| \tag{5}\]  
Equation (6) shows that the total CPD is the summation of plastic deformations across all half cycles:  
\[\text{CPD}=\sum_{i=1}^{n}|\ \Delta_{\text{i(Pull)}}-\Delta_{\text{i(Push)}}| \tag{6}\]  
Figure 61: Plastic hinge status in MCE-hazard-level Y-direction. (**a**) Percentage of plastic hinges formed in A-IO. (**b**) Percentage of plastic hinges formed in IO-LS. (**c**) Percentage of plastic hinges formed in LS-CP. (**d**) Percentage of plastic hinges formed in >CP. To facilitate comparisons with seismic code requirements, Equation (7) shows that the CPD is normalized by the yield deformation of the BRB (\(\Delta_{\rm y,BRB}\)) to obtain the CPD ratio (CPDR):  
\[\text{CPDR}=\frac{\sum_{i=1}^{\text{\tiny{$\mathrm{\mathrm{\pi}}$}}}\mid\Delta_{ \rm i(Full)}-\Delta_{\rm i(Push)}\mid}{\Delta_{y,BRB}} \tag{7}\]  
The plastic deformation of BRB is shown in Figure 62, wherein \(\Delta_{\rm t}\) is tension displacement, \(\Delta_{\rm c}\) is compression displacement, and \(\Delta_{\rm i}\) is iteration displacement. The CPDR calculated for the considered BRBs and RBRBs in this study was 470 and 780, respectively, and we compared these results with previous research available in the literature, as reported in Table 24.  
The CPDR results in this study (470 for BRB and 780 for RBRB) indicate a significant plastic deformation capacity for both brace configurations, with the RBRB demonstrating a substantially higher ability to withstand cumulative inelastic deformations compared to the standard BRB. This superior performance suggests that the RBRB is better suited for applications requiring enhanced energy dissipation and higher deformation tolerance under cyclic loading, such as in seismic conditions. Compared to other studies, the CPDR values for the BRB and RBRB exceed those reported by [PERSON] et al. (2019) (\(\sim\)600), [PERSON] et al. (2007) (17.64), and [PERSON] et al. (2020) (16.6), while aligning with [PERSON] et al. (2024) (200-600). The displacement range in this study (\(-\)25 to +25 mm for BRB and \(-\)40 to +40 mm for RBRB) surpasses those in prior research, such as [PERSON] et al. and [PERSON] et al. (\(-\)10 to +10 mm), [PERSON] et al. (\(-\)6 to +6 mm), and [PERSON] et al. (\(-\)3 to +3 mm), reflecting the higher deformation capacity of the tested braces. Furthermore, despite having a smaller cross-sectional area (2000 mm\({}^{2}\)) compared to [PERSON] et al. (3731 mm\({}^{2}\)) and [PERSON] et al. (6361 mm\({}^{2}\)), the BRB and