## 4 Numerical Modeling and Methodology
### Loading Regime  
The evaluation of seismic performance outlined here operates on two primary levels. Initially, a dynamic implicit simulation is executed using a representative segment of the Mw 7.7 Kahramanmararas earthquake record. Using the implicit dynamic solver included in ABAQUS [33], which is based on the Newmark-beta integration approach as described by [PERSON] [42], nonlinear time-history analyses were conducted in this work. By solving the linked system of equilibrium equations, this formulation naturally includes the consequences of inertia:  
\[\mathbf{M}i(t)+\mathbf{C}i(t)+\mathbf{K}u(t)=F(t)\]  
where \(\mathbf{M}\) represents the consistent mass matrix, \(\mathbf{C}\) is the damping matrix, and \(\mathbf{K}\) is the developing tangent stiffness matrix. The time-varying effective force applied at the base is denoted by the word \(F(t)\). Inertial forces, resulting from mass acceleration, are calculated at every time increment and immediately affect the worldwide response when the structural system reacts to seismic stimulation. Dynamic amplification events are recorded by the simulation framework as a natural result of modal development and stiffness deterioration. The stiffness matrix \(\mathbf{K}\) shows incremental softening as inelastic deformations accumulate, especially inside the ACPH  
Figure 8: FEM representations of the 1B1S, 1B2S, 2B1S, and 2B2S PRC frames both with and without ACPHs. Figure 9: Detailed FEM of the ACPH subassembly in the beam–column joint region. subassemblies, causing changes in the fundamental frequency of the structure and increasing the demands in certain modes. These consequences result from the interplay of material nonlinearity and inertial reaction, not artificial imposition. Rayleigh damping was included to simulate actual energy dissipation during shaking; it was set to reach around 5% critical damping in the first two vibration modes of the undamaged frames. This degree of damping corresponds to standard levels used in performance-based earthquake engineering for RC structures. Although the implicit approach has its drawbacks, it provides strong convergence and consistent temporal integration even in very nonlinear settings. When damage starts or advances quickly, the approach calls for rather tiny time increments to preserve accuracy, which might raise computing time. The internal stabilization methods employed to improve numerical convergence may also marginally reduce high-frequency material, so the approach is more appropriate for obtaining global response metrics (e.g., drift, base shear, and damage localization) than for resolving extremely short-duration transients. The analysis specifically isolates a critical 10 s interval surrounding the record's peak acceleration, concentrating on the phase of peak acceleration to capture essential inelastic responses. The ground motion considered is illustrated in Figure 10, which also highlights the isolated interval on the graph. In the dynamic implicit analysis, the most intense base accelerations corresponding to the earthquake are applied to the supports of each frame. The computational solver iteratively processes each small time step, continuously adjusting system stiffness and damping in reflection of the changes in material states. During increased shaking, localized concrete cracking occurs, reinforcing bars may undergo yielding, and plastic deformations are noted particularly around beam-column joints or at specialized ACPH devices. The analytical outputs, including time histories of base shear, inter-story drift, and localized damage, reveal the responses of each frame, whether equipped with an ACPH or not, under real-time seismic forces characterized by inertia and dynamic amplification effects. Only a single nonlinear dynamic analysis was conducted for each frame configuration, using the 10 s interval extracted from the 2023 Kahramanmaras earthquake record. No additional static or quasi-static loading procedures were applied.