Currently, the reduction factor is widely used in most of the seismic design codes, trying to compensate the effects of significant overstrength and ductility of the structure to withstand seismic load. In other words, design codes allow simplified elastic analysis to be performed, calculating the design forces acting on the structure from spectra based on linear behavior and properly scaled down by a suitable reduction factor that accounts for the global nonlinear effects. The reduction factor is called behavior factor (
Previous studies were conducted in order to define the parameters affecting the
Concentric X-braced system (XBF) is known to be a very effective structural element that can provide lateral stiffness and strength in steel building structures. This system is used as lateral (i.e., seismic and wind) force resisting system through the vertical concentric truss system [13]. In XBF structures, according to EC8, braces are the dissipative members, while all the other members (i.e., beams and columns) should remain elastic. The braces act as fuses which dissipate the input seismic energy through axial deformations in tension and compression loading cycles. Hereafter, the main design rules of the EC8 for seismic resistant systems with X-braces are summarized.
Firstly, the slenderness ratio of braces members should be limited to 1.3 ≤ λ ≤ 2.0, where the upper limit has the aim to avoid excessive distortions due to buckling of braces in compression, which could cause damage to connections, while the lower limit ensures the validity of the structural model with only active tensile braces as well as restricts the design internal forces in the columns. Moreover, a special rule is presented in EC8 to ensure homogeneous dissipative behavior of braces along the height of a structure. According to EC8, the overstrength of a brace member Ω_{i} is calculated as follows:
Thus, it can be seen clearly that the EC8 does not give enough precision about the effect of structural characteristics (stories number, brace slenderness ratio, local response of structural members, and support type) on the
Nowadays, the linear elastic design method is commonly used in most seismic codes. In this method, the most important parameter is the
The effect of vibration period on the nonlinear response of frame is of major interest, in particular, on the components of
These studies did not consider the effect of “brace slenderness ratio” and local response of the braces on the
Clear definitions of the
Observations during earthquakes have shown that building structures could take the forces considerably larger than those that they were designed for. This is explained by the presence in such structures of significant overstrength not accounted for in design. The main possible sources of overstrength that have been reviewed by Rahgozar and Humar [15] include the difference between the actual and the design material strength, the effect of minimum requirements on member sections in order to meet the stability and serviceability limits, and the redistribution of internal forces. The presence of the overstrength
The displacement ductility
In the EC8 [1], the
The approach presented above is commonly used to obtain the
Based on the existing studies [3, 5, 27, 28], the structural response curve (capacity curve) is obtained by two different methods of analysis: the nonlinear static pushover analysis (NSPA) and the nonlinear dynamic analysis. The first method is simple to use and gives accurate results. It is chosen in the present study.
In the current work, a number of XBFs having three, six, and nine stories and three bays of 6 m each with a height of 3 m for each floor (Fig. 2) are analyzed to evaluate the impact of various parameters on the
XBF steel structures considered in parametric studies.
Three stories | λ_{1} = 1.93 | 220 (1–3) | 127X4 (1) + 108X3.6 (2) + 101.6X3.6 (3) |
λ_{2} = 1.56 | 240 (1–3) | 152.4X4 (1) + 133X4 (2) + 127X4 (3) | |
λ_{3} = 1.30 | 260 (1–3) | 193.7X4.5 (1) + 159X4 (2) + 139.7X4 (3) | |
Six stories | λ_{1} = 1.93 | 240 (1–2) + 220 (3–4) + 200 (5–6) | 127X4 (1–3) + 108X3.6 (4) + 101.6X3.6 (5) + 82.5X3.2 (6) |
λ_{2} = 1.56 | 260 (1–2) + 240 (3–4) + 220 (5–6) | 152.4X4 (1–2) + 139.7X4 (3) + 133X4 (4) + 127X4(5) + 101.6X3.6 (6) | |
λ_{3} = 1.30 | 280 (1–2) + 260 (3–4) + 240 (5–6) | 193.7X4.5 (1–2) + 168.3X4 (3) + 159X4 (4) + 139.7X4(5) + 127X4 (6) | |
Nine stories | λ_{1} = 1.93 | 260 (1–3) + 240 (4–6) + 220 (7–9) | 127X4 (1–4) + 108X3.6 (5–6) + 101.6X3.6 (7) + 88.9X3.2 (8) + 76.1X3.2 (9) |
λ_{2} = 1.56 | 280 (1–3) + 260 (4–6) + 240 (7–9) | 152.4X4 (1–3) + 139.7X4 (4) + 133X4 (5) + 127X4 (6–7) + 108X3.6 (8) + 88.9X3.2 (9) | |
λ_{3} = 1.30 | 320 (1–3) + 300 (4–6) + 280 (7–9) | 193.7X4.5 (1–4) + 159X4 (5) + 152.4X4 (6) + 139.7X4 (7) + 127X4 (8) + 108X3.6 (9) |
For each frame corresponding to a stories number (three, six, or nine stories), three design processes were performed in order to obtain three distinct values (1.93, 1.56, and 1.30) of the brace slenderness ratio, λ_{i}, defined as
Table 2 presents the natural periods and the modal mass ratios M^{*1}, M^{*2}, M^{*3} of the first three mode shapes of the dynamic modal analysis, where it can be observed that the total mass participating in the fundamental mode of the studied frames is higher than 75%. This allows using the NSPA, which is mainly based on the fundamental mode.
Modal periods and mass ratios of the analyzed XBF steel structures.
Three stories | λ_{1} = 1.93 | 0.41 | 0.14 | 0.09 | 0.86 | 0.11 | 0.02 | 0.22 |
λ_{2} = 1.56 | 0.36 | 0.13 | 0.08 | 0.87 | 0.10 | 0.02 | 0.23 | |
λ_{3} = 1.30 | 0.32 | 0.11 | 0.07 | 0.85 | 0.12 | 0.02 | 0.23 | |
Six stories | λ_{1} = 1.93 | 0.83 | 0.28 | 0.16 | 0.78 | 0.15 | 0.03 | 0.13 |
λ_{2} = 1.56 | 0.75 | 0.25 | 0.14 | 0.79 | 0.15 | 0.03 | 0.14 | |
λ_{3} = 1.30 | 0.68 | 0.22 | 0.12 | 0.77 | 0.16 | 0.04 | 0.16 | |
Nine stories | λ_{1} = 1.93 | 1.37 | 0.44 | 0.24 | 0.75 | 0.17 | 0.04 | 0.08 |
λ_{2} = 1.56 | 1.24 | 0.40 | 0.22 | 0.76 | 0.17 | 0.04 | 0.09 | |
λ_{3} = 1.30 | 1.10 | 0.34 | 0.19 | 0.75 | 0.19 | 0.04 | 0.10 |
Analyses have been performed using the finite element software SAP2000 [30], which is a general-purpose structural analysis program for static and dynamic analyses of structures. In this study, SAP2000 Nonlinear Version 14 has been used. A description of the modeling details is provided in the following.
A two-dimensional finite element model of each frame structure is created in SAP2000 to perform NSPA. Structural members are modeled as beam elements with lumped plasticity. Plastic hinges of columns and beams are assigned to the end of the members, while for modeling the nonlinear behavior of braces, the plastic hinges are defined at the midpoint of each brace. Moreover, for columns, the effect of the axial load is considered using a model of P–M interaction diagram (axial force P and bending moment M). The plastic hinge properties in SAP2000 are determined according to the provisions of FEMA 356 [31] (Fig. 3).
In this analysis, the geometric and mechanical characteristics of steel members are considered. Also, all sources of geometrical nonlinearity have been included, namely
The frames are subjected to two horizontal load patterns, a uniform pattern UD, whereby lateral forces are proportional to the total mass at each floor level, and an inverted triangular pattern TD, in which seismic forces are proportional to the product of floor mass and storey height [32, 33, 34]. The horizontal loads are distributed along the frame height with the intensity increased incrementally until a mechanism is reached. It leads to construct the capacity curve, which is used to obtain the
In this investigation, the capacity curve plotted using DRAIN-2DX computer program of six-storey steel frame studied by Karavasilis et al. [35] was used to validate the capacity curve obtained by SAP2000 [30]. The considered frame has three bays of 6 m each and storey height equal to 3 m. The first three stories have columns with HEB340 sections and beams with IPE450 sections, whereas the last three (upper) stories have columns with HEB280 sections and beams with IPE360 sections. Steel members are made of grade S235. The comparison between the two curves represented in Fig. 4 shows that the two curves are very close.
The numerical results of the studied XBFs are presented and discussed in this section. NSPA using inverted TD and UD load pattern distribution was carried out to compute the
In Fig. 5, the capacity curves of the studied XBF structures with the three values of brace slenderness ratio (λ_{1}, λ_{2}, and λ_{3}) for the two types of supports (pinned and fixed supports) are depicted. The section damage levels of the first-storey brace (FS-B) are marked on the capacity curve (FS-B). It can be seen that the brace slenderness ratio, λ_{i}, does not make any major difference in terms of lateral displacement. However, this parameter has a significant influence on the base shear force. The base shear force obtained for the frames with λ_{3} = 1.30 is greater than that of those with λ_{1} = 1.93 and λ_{2} = 1.56. This is due to the increase in the dimensions of braces’ cross sections (hence, an increase in their axial capacity), which will absorb more forces. For such frames (especially for nine-storey frame), greater differences have been found between the results obtained from NSPA under UD and TD. The outcome of the present study is in good agreement with the numerical work reported by Ferraioli et al. [5]. Besides, Fig. 5 indicates that the risk of local brace instability, due to the loss of rigidity after reaching the ultimate capacity (point C in Fig. 3), increases as the number of stories increases. Moreover, for such frames, the FS-B section is the first and the most damaged section, which leads to a premature failure of the frame. The appearance of such premature failure (soft-storey mechanism) confirms the importance of design methods focusing on the performance-based plastic design (PBPD) of steel frame structures [36, 37].
Figs 6 and 7 show the distribution of plastic hinges for the studied frames with λ_{1} = 1.93 (the maximum value of brace slenderness ratio). It appears that there is a good distribution of energy dissipation along the height and across the length of low-rise frames (three stories). However, for medium (six stories) and high-rise frames (nine stories), the ultimate capacity is reached in the plastic hinge of FS-Bs, while those on the top of the frame remain in the elastic domain without plastic dissipation. This is due to the concentration of high axial force at the FS-Bs’ sections. Furthermore, in the case of the frames with fixed supports (λ_{1}’), it is observed that the plastic hinges are formed also at the first-storey column sections in all the frames (three, six, and nine stories). This distribution could be explained by the sensitivity of the frames to the
In order to compute the
Fig. 8 shows the variation of the
In Fig. 8, it can be noticed that the number of stories influences the
The
Fig. 8 also shows the variation of the calculated
The effect of support type on the
In Fig. 8, it can be observed that the
In order to study the effects of brace slenderness ratio, λ_{i}, on the
The variation of
In Fig. 8, the
Fig. 9 shows the effect of the axial force ratio (N/N_{pl}) and the weight ratio (W_{1}/W_{t}) on the variation of
On the basis of the above results, the studied XBF structures present a weak point in their lateral strength. In fact, this structural typology is not able to provide sufficient resistance as far as the height of the structures or the axial force increases. As it has already been pointed out, the
In Fig. 9, the calculated
A detailed study concerning the XBFs designed according to the European codes (EC3 and EC8) has been conducted. In this context, the effects of structural characteristics, brace slenderness ratio, and type of support are considered. The partial components (overstrength
Overstrength
When the number of stories increases, the axial force at the FS-B sections increases. This leads to premature failure of these braces. Thus, the values of the
The triangular and the uniform lateral load distributions give a
The
The
The brace slenderness ratio λ_{i} makes a major difference in terms of
For the studied structures, EC8 gives a constant value of the
Based on the ultimate limit capacity (using FEM-356 limits on member rotation capacity), the EC8 overestimates the
A local strength criterion based on the control of axial force level and related to the local response of brace sections has been proposed to avoid the overestimation of
The results of this study confirm the importance of the special rule given in EC8, which requires using overstrength factor in the design of brace to ensure homogeneous dissipative behavior of braces along the height of a structure.
As a final note, it is worth emphasizing that the results presented in this study are based on the NSPA performed on XBFs with three, six, and nine stories. Therefore, further research considering other parameters (taller structures) by means of nonlinear dynamic analysis is necessary for generalizing the presented conclusions.
Modal periods and mass ratios of the analyzed XBF steel structures.
Three stories | λ_{1} = 1.93 | 0.41 | 0.14 | 0.09 | 0.86 | 0.11 | 0.02 | 0.22 |
λ_{2} = 1.56 | 0.36 | 0.13 | 0.08 | 0.87 | 0.10 | 0.02 | 0.23 | |
λ_{3} = 1.30 | 0.32 | 0.11 | 0.07 | 0.85 | 0.12 | 0.02 | 0.23 | |
Six stories | λ_{1} = 1.93 | 0.83 | 0.28 | 0.16 | 0.78 | 0.15 | 0.03 | 0.13 |
λ_{2} = 1.56 | 0.75 | 0.25 | 0.14 | 0.79 | 0.15 | 0.03 | 0.14 | |
λ_{3} = 1.30 | 0.68 | 0.22 | 0.12 | 0.77 | 0.16 | 0.04 | 0.16 | |
Nine stories | λ_{1} = 1.93 | 1.37 | 0.44 | 0.24 | 0.75 | 0.17 | 0.04 | 0.08 |
λ_{2} = 1.56 | 1.24 | 0.40 | 0.22 | 0.76 | 0.17 | 0.04 | 0.09 | |
λ_{3} = 1.30 | 1.10 | 0.34 | 0.19 | 0.75 | 0.19 | 0.04 | 0.10 |
XBF steel structures considered in parametric studies.
Three stories | λ_{1} = 1.93 | 220 (1–3) | 127X4 (1) + 108X3.6 (2) + 101.6X3.6 (3) |
λ_{2} = 1.56 | 240 (1–3) | 152.4X4 (1) + 133X4 (2) + 127X4 (3) | |
λ_{3} = 1.30 | 260 (1–3) | 193.7X4.5 (1) + 159X4 (2) + 139.7X4 (3) | |
Six stories | λ_{1} = 1.93 | 240 (1–2) + 220 (3–4) + 200 (5–6) | 127X4 (1–3) + 108X3.6 (4) + 101.6X3.6 (5) + 82.5X3.2 (6) |
λ_{2} = 1.56 | 260 (1–2) + 240 (3–4) + 220 (5–6) | 152.4X4 (1–2) + 139.7X4 (3) + 133X4 (4) + 127X4(5) + 101.6X3.6 (6) | |
λ_{3} = 1.30 | 280 (1–2) + 260 (3–4) + 240 (5–6) | 193.7X4.5 (1–2) + 168.3X4 (3) + 159X4 (4) + 139.7X4(5) + 127X4 (6) | |
Nine stories | λ_{1} = 1.93 | 260 (1–3) + 240 (4–6) + 220 (7–9) | 127X4 (1–4) + 108X3.6 (5–6) + 101.6X3.6 (7) + 88.9X3.2 (8) + 76.1X3.2 (9) |
λ_{2} = 1.56 | 280 (1–3) + 260 (4–6) + 240 (7–9) | 152.4X4 (1–3) + 139.7X4 (4) + 133X4 (5) + 127X4 (6–7) + 108X3.6 (8) + 88.9X3.2 (9) | |
λ_{3} = 1.30 | 320 (1–3) + 300 (4–6) + 280 (7–9) | 193.7X4.5 (1–4) + 159X4 (5) + 152.4X4 (6) + 139.7X4 (7) + 127X4 (8) + 108X3.6 (9) |
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