Introducing Damage Index of Moment-Resisting Frames with Setback Using Gene Expression Programming

Architectural requirements and specific functional demands often lead to vertical irregularities in buildings, such as height setbacks. Structural damage depends on several parameters, including plastic displacement, dynamic characteristics, structural type, and design variables. A well-defined damage index is a normalized parameter that equals zero when the structure remains elastic (i.e., no significant damage occurs) and approaches unity when failure is imminent. In this study, 28 two-dimensional reinforced concrete frames with 3, 6, 9, and 12 stories and eight different setback configurations were analyzed using nonlinear dynamic time-history analysis in IDARC. The input motions used in this study were selected from the Pacific Earthquake Engineering Research (PEER) Center strong ground motion database and were compatible with soil type II conditions and located far from the causative fault. All records were normalized to the intensity of the design spectrum of Iranian Code 2800 for a peak ground acceleration of 0.35g. Three categories of normalization factors were presented to increase the accuracy of the calculations. The total structural damage was quantified using the Park–Ang damage index, selected for its consistency with experimental observations and its simplicity in correlating with observed damage levels. This index accounts for the combined effects of maximum lateral deformation and cumulative plastic energy dissipation. It is calculated based on maximum and yield rotations, ultimate rotation capacity, yield moment, a structural parameter, and the history of inelastic response. Using Gene Expression Programming (GEP), an explicit relationship was developed to predict the damage index as a function of three parameters: irregularity index (Φ), fundamental period (T), and overall drift (OD), defined as the ratio of maximum roof displacement to the total frame height. Ten models with one to four genes were examined. A set of 26 mathematical functions was used to generate candidate relationships. For multi-gene models, gene linking was performed using addition, multiplication, and division operators. The results demonstrate that the proposed method can estimate the damage index with satisfactory accuracy, eliminating the need for nonlinear dynamic analysis. The best-performing model achieved an R² of 0.95 and a relative absolute error of 0.22. The correlation between predicted and calculated damage indices yielded an R² of 0.88. The minimum, maximum, and average absolute error percentages were 1.03%, 21.71%, and 8.3%, respectively. For 17 out of 28 frames, the prediction error was less than 10%. Moreover, the proposed model outperformed both power and quadratic regression models in terms of maximum and average errors. Model validation was conducted using three additional two-dimensional frames with 5, 8, and 11 stories. The average error percentage for these validation models was 9.64%, and the predictions were generally conservative and reliable. Sensitivity analysis was performed using both the Taguchi method and the Range method. In the Taguchi method, the relative sensitivity of the fundamental period parameter was greater than that of the overall drift parameter, whereas in the Range method this order was reversed. Overall, the results of the two methods show reasonable agreement. Sensitivity analysis indicated that the irregularity index has a more significant influence on the predicted damage index than the fundamental period and overall drift.