Issue 74

E. Sharaf et alii, Fracture and Structural Integrity, 74 (2025) 262-293; DOI: 10.3221/IGF-ESIS.74.17

Fig. 15c, relative to a 7 m span, still confirms the goodness of the new model, which provides R² = 0.999 and RMSE = 0.018 s. Aninthaneni and Dhakal's formulation [23] has some improvement (R² = 0.71, RMSE = 0.48 s) but is still less accurate than that of the new model. Goel and Chopra's method [5] remains the least accurate (R² = 0.37, RMSE = 0.72 s), whereas Salama's equation [6] provides the most accurate estimate of the available models (R² = 0.84, RMSE = 0.36 s). In general, the statistical analysis for all three cases exhibits the continual outperformance of the new equation in predicting the fundamental time period of moment-resisting frames. Its R² values close to unity and substantially lower RMSE values than those of the existing models show both high precision and low error of prediction. This implies that the new model provides a more precise and generalized tool for the dynamic analysis of frame structures for a broad spectrum of changing span lengths. Effect of span variation The sensitivity study in Fig. 16a and Fig. 16b presents a comprehensive evaluation of the effect of building span length on the fundamental time period of reinforced concrete (RC) moment-resisting frames of two various building heights, 60 m and 54 m, respectively. The performance of the suggested analytical formula is compared to the established formulations of [5, 6, 23] with the benchmark being the finite element method (FEM). As can be seen from the two figures, the FEM results show a reducing trend for the period over the increasing span length. This is due to the fact that there is more lateral stiffness in buildings with larger widths, and therefore, the natural period decreases. The equation we have developed predicts this trend accurately and agrees very well with the FEM results for the range of spans considered whereas the other models have large differences. To determine the predictive capability of both models, statistical measures in the form of the coefficient of determination (R 2 ) and the Root Mean Square Error (RMSE) are provided. These measures criticize the correlation as well as predictive precision of both methods when compared to FEM results. Statistical comparison firmly confirms the higher performance of the new formula with the highest R 2 values (0.91 and 0.94) and the lowest RMSE values (0.025 and 0.016) for both building heights. These are indicators of high correlation with FEM results and low error in prediction and suggest the ability of the formula to capture the influence of variable span length on dynamic response. In stark contrast, the other existing formulas all have negative R 2 values, meaning that their predictions deviate so significantly from the FEM results. This is most evident for [5, 23], who have RMSE values above 0.4 and R 2 values below -40. Salama’s formula [6] has a slightly better fit, but once more, it performs badly in comparison to the suggested method, having low correlation and moderate error. In conclusion, the comparison indicates the inability of the equations from the literature to effectively account for the influence of span length, especially in mid- to high-rise buildings. The suggested method is, however, invariably correct irrespective of height and span.

Figure 16: Variation of fundamental time period with span length for different building heights: (a) 60 m height; (b) 54 m height.

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