Issue 74

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

The heatmap shows in Fig. 20b how the level of prediction error for the fundamental period varies with frame span and building height (number of stories). Shorter spans (6–7 m) at lower error levels (1–2) are clustered, as well as at moderate heights ( ≤ 10 stories), whereas taller buildings at moderate spans (9–12 m) perform erratically. The most significant error levels (4–5) occur mainly at long spans ( ≥ 12 m) with minimal stories or very tall structures, implying that both the extreme span–height ratios lead to greater deviation in period estimation.

a)

1

1

2

3

2

2

2

5

b)

15

5

4

1

3

2

5

Beam moment of inertia/column moment of inertia ( α ) 0.1296 3 4 2 0.0337 4 3 3 0.0123 5 4 3 2 0.008 5 3 3

2

3

2

4

12

4

4

1

4

3

4

4

2

2

3

9

4

3

2

5

4

3

Error %

1

3

2

2

7

2

1

5

5

4

2

Frame spam (m)

Error %

2

2

3

1

6

1

1

5

5

5

1

5

10 15 20 25 30

5

8

10 15 20

Number of stories

Number of stories

Figure 20: Heat maps showing the sensitivity of fundamental period prediction error to variations in (a) α versus number of stories and (b) span length versus number of stories. Statistical indicators of agreement with FEM To verify the efficacy of the provided analytical formula in estimating the fundamental time period, a comparison was also made using FEM results. Tab. 11 illustrates the comparison for a number of structural configurations that involve various span lengths, column and beam cross-sections, and building heights. The result shows a very good agreement between the values computed by the new approach and those computed using numerical analysis. FEM/proposed ranges from 0.96 to 1.04, with an average value of approximately 1.00. This reveals that the proposed model is not biased toward overestimation or underestimation of the fundamental period. Besides, the coefficient of variation (COV) is just 2%, and the standard deviation of ratios is 0.02, reflecting the uniformity and consistency of the proposed formulation. For the different structural configurations, the close agreement between numerical and analytical solutions verifies the capability of the proposed equation in simulating the dynamic properties of the framed buildings. This accuracy renders the proposed method an effective tool in early design and seismic analysis

Fundamental time period

Col cross section (m)

Beam cross section (m)

Span (m)

Building height (m)

Proposed Equation (sec)

FEM (sec)

FEM/proposed

4 4 4 5 5 6 6 7 7

0.4*0.4 0.4*0.4 0.5*0.5 0.4*0.4 0.5*0.5 0.6*0.6 0.6*0.6 0.7*0.7 0.7*0.7

0.3*0.3 0.4*0.4 0.4*0.4 0.4*0.4 0.5*0.5 0.5*0.5 0.6*0.6 0.5*0.5 0.6*0.6

30 30 30 30 45 45 45 60 60

0.99 0.775 0.756 0.822 1.12 1.181 1.023 1.833 1.536

0.95 0.77 0.78 0.82 1.22 0.99 1.82 1.55 1.1

1.04 1.006 0.969 1.002 1.018 0.96 1.03 1.007 0.99

Mean

1

St.deviation

0.02

COV% 2 Table 12: Comparison of the predicted fundamental periods using the proposed model and numerical modal analysis.

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