Issue 74

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

 The UBC [1] predictions are relatively aligned with FEM results for low- to mid-rise structures (about 15–40 m), but diverge significantly as building height increases. It is obvious that the UBC formula is relatively adequate up to a height of 40 m, but is not reliable in the case of high-rise buildings.  Salama [6] presents the most conservative of the three estimates, significantly underestimating the fundamental period for all heights of buildings. This shows that the model is probably missing the structural flexibility and cracking influences, and hence it overestimates the lateral stiffness. The discrepancy between empirical predictions and FEM highlights the restrictions of generalized code equations, especially for irregular geometries or taller structures. This warrants calibration of empirical models with newer numerical data and consideration of FEM-based strategies in performance-based seismic design. eismic design typically begins with estimating a trustworthy seismic force, usually based on the periods of the building's vibration. These periods are primarily influenced by the mass and stiffness of the building. In this case, seismic design often becomes a process of trial and error, since the relationship between seismic capacity, mass, and stiffness must be well recognized to ensure effective seismic performance. To avoid unnecessary trial and error, most standards include starting points for element size, construction orientation, number of levels, and material properties. These provide natural period calculations, and many formulas from codes and literature are used to calculate the building's seismic load. The fundamental period of the building, during which a significant portion of the mass participates in the associated mode of vibration, is by definition the natural period of vibration. While more complex formulas are available in building codes, simpler equations based on height or the number of stories are often recommended to streamline the calculation process. Several studies have examined the effect of both structural and non-structural factors on the fundamental period of vibration in reinforced concrete buildings. Consequently, the empirical equations have revealed that the natural period is mostly determined by the building's height or number of stories. More specifically, the taller or higher a building is, the longer the period of vibration will be [34, 35]. These two properties, height and number of stories, are found to be the two most influential ones that determine a building's fundamental period. Consequently, most design codes rely on one of these two properties, preferably height or number of floors, in developing their empirical equations for the vibration period. Notable examples of such codes include the “building standard law of Japan” (BSLJ) [36], the National Building Code of Canada (NBCC, 1995) [37] the Uniform UBC, [1], and the ATC [29], all of which incorporate height or number of stories as key parameters in their period formulas. A simplified technique for determining the fundamental period The most practical approach to developing an analytical model of a real-frame structure is to model it as a skeleton with masses lumped at each floor level. In this section, a simplified technique is outlined for calculating the fundamental period of vibration of such structures. As previously mentioned, and as widely recognized, the period of vibration of a building depends on its mass and stiffness. For an idealized Single-Degree-of-Freedom (SDOF) system, the fundamental period is classically expressed by Equation (7), which defines the dynamic relationship between mass and stiffness. S E STIMATION OF THE F UNDAMENTAL P ERIOD OF V IBRATION Equation (7) forms the base of this study. An RC building can be modeled by a single-degree-of-freedom (SDOF) system using an equivalent lumped mass m eq and equivalent stiffness k eq as shown in Fig. 2. This idealization is concerned with the fundamental period of vibration and the interaction between the mass and stiffness in the system, making it much easier to analyze the dynamic behavior of the building. Procedure for estimating the equivalent lateral stiffness of the frame (K eq ) The following explanation may help clarify the development of the current formula. Initially, the building is simplified to achieve a more accurate estimate of lateral stiffness. The proposed method begins by calculating the equivalent stiffness for a two-storey building, as shown in Fig. 3. Using the Rayleigh approach, the process for estimating the fundamental period of the building is formulated in this section. 2   m K T (7)

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