Issue 74

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

C OMPREHENSIVE VALIDATION OF THE PROPOSED EQUATION AGAINST FEM RESULTS AND ESTABLISHED FORMULAS FROM LITERATURE he accuracy of the suggested formula for estimating the fundamental vibration period of reinforced concrete moment-resisting frame (RC MRF) structures is confirmed in Fig. 14 by comparing it with existing formulations from Goel and Chopra [5], Salama [6], Aninthaneni and Dhakal [23], as well as seismic design codes such as ASCE 7-05 [32] and UBC [1], using Finite Element Method (FEM) results as reference. Two structural layouts are considered: frames with a 5 m span (Fig. 14a) and frames with a 7 m span (Fig. 14b). The comparison is based on two important statistical indicators: the coefficient of determination (R²), which measures the degree of correlation between predicted and reference values, and the root mean square error (RMSE), which quantifies the average prediction error. For the 5 m span illustration case (Fig. 14a), the new equation is well correlated with the FEM solution with R² = 0.998 and RMSE = 0.039s. Both indicators confirm the high accuracy and reliability of the new model in estimating RC MRF frames' dynamic behavior. For comparison, Aninthaneni and Dhakal's equation [23] gives fair agreement (R² = 0.81, RMSE = 0.50 s), whereas Goel and Chopra's equation [5] is poor (R² = 0.14, RMSE = 1.09 s) with considerable departure from FEM based periods. Likewise, Salama's equation [6] exhibits poor predictive strength (R² = 0.62, RMSE = 0.71 s) due to overestimation of stiffness. Of the seismic code equations, ASCE 7-05 [32] is moderately consistent with FEM solutions (R² = 0.80) but tends to underestimate the fundamental period for higher structures, which may result in overly conservative seismic force estimates. The most conservative estimates overall are given by UBC [1], with lower predicted periods at every height and relatively high R² =0.86, indicating trend conformity more than actual accuracy. In the case of 7 m span (Fig. 14b), the proposed equation maintains the predicting proficiency with R² = 0.999 and RMSE = 0.027 s, and makes it robust and span-insensitive. Aninthaneni and Dhakal's model [23] in the present case shows enhanced performance (R² = 0.93, RMSE = 0.29 s), depicting higher applicability for wide spans. Goel and Chopra's equation [5] still performs poorly (R² = 0.016, RMSE = 1.12 s), whereas Salama's equation [6] still has moderate accuracy (R² = 0.56, RMSE = 0.75 s). The ASCE 7-05 [32] model once more has moderate correlation (R² = 0.80) but still underestimates periods for high-rise structures. The UBC [1] formula has R² = 0.84 but is still too conservative, severely underestimating the fundamental period for all heights. Overall, for both span cases, the new equation outperforms any other empirical and code-based formula in terms of prediction accuracy (lowest RMSE) and magnitude of correlation (highest R²). Its excellent agreement with FEM solutions, regardless of span length, demonstrates its utility in the seismic analysis and design of RC MRF buildings in practice. These outcomes validate its usage as a rational alternative to conventional empirical period estimation formulas. T

Figure 1 4 : Comparison of fundamental vibration periods estimated using the proposed equation with those predicted by FEM analysis and existing formulas for reinforced concrete moment-resisting frame (RC MRF) buildings : (a) frames with a 5 m span ; (b) frames with a 7 m span.

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