Issue 74

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

Most current design codes specify that the design base shear should be calculated using the following equation:

V CW  (6) where V is the design base shear, W is the total seismic dead load, and C is the seismic coefficient. The seismic coefficient C depends on several parameters: the soil profile, the seismic zone factor, the importance factor, the building's fundamental period T, and a numerical coefficient representing overstrength and global ductility capacity inherent in the lateral load resisting system. The fundamental period Ta calculated by the empirical formulas from equations (1) and (2) is typically designed to be shorter than the true period in order to provide a conservative estimate of the base shear, as structures are required to be designed for larger seismic forces. Thus, code formulas are intentionally calibrated to underestimate the fundamental period by about 10–20%. Furthermore, while building codes allow the determination of the fundamental period using rational methods, such as Rayleigh's method [33], they also specify that the period computed based on these methods shall not exceed the period obtained from the empirical formula by more than a certain factor. In this way, consistency in seismic design is achieved, while there is provision for making more accurate calculations when required.

Figure 1: Comparaison of FEM and code periods for RC MRF buildings.

EFFECTIVENESS OF FORMULAS FROM CODES AND LITERATURE

T

o evaluate the validity of frequently used empirical formulas in identifying the fundamental period of structures, comparative study was performed between code expressions and FEM. The comparison given in Figure 1 shows a detailed assessment of accuracy of different empirical expressions to comprehensive finite element method (FEM) analysis for calculation of the fundamental period of structures for a broad range of heights. Specifically, the comparison incorporates empirical period formulae from UBC [1], ASCE 7-05 [32], and also Salama's equation [6] as written in Equations (1), (5) and (3) respectively. Figure 1 presents several important observations concerning the accuracy of these formulae relative to FEM results:  FEM calculations consistently forecast longer fundamental periods than all empirical formulas, particularly for structures exceeding 30 meters in height. This reflects the enhanced flexibility represented in numerical models, which is typically underestimated in empirical formulations.  ASCE 7-05 [32] calculates the period using the formula T a =0.0466h 0.9 , with an upper limit adjusted by C u =1.4. While the upper bound curve somewhat aligns with the FEM results for mid-heights; nevertheless, it significantly underestimates periods for high-rise buildings, indicating inadequate applicability for tall structures.  The ASCE 7-05 [32] code formula produces significantly shorter period estimates than those obtained from FEM analysis, particularly for buildings exceeding 45 m in height. At 90 m, the percentage error is approximately − 45%, reflecting a rather large underestimation of the fundamental period and that the code may not capture accurately the dynamic behaviour of high-rise buildings.

266