Comparative Study of Steel Buildings with Eccentrically Inverted V-Brace Frames and Concentrically Inverted V-Brace Frames

Brace frames are structural systems designed to reduce the lateral deflection of steel structures, making them a common choice for resisting the forces of an earthquake. The brace frames enhance the stiffness of a building, enabling it withstand lateral loads like those from wind or seismic activity. While they are designed to perform well under normal conditions, brace frames are expected to undergo inelastic deformation (permanent deformation) when subjected to severe seismic actions (Malley & Popov, 1983; Kasai & Popov, 1986; Yiğitsoy, 2010; Sabouri-Ghomi & Payandehjoo, 2017; Salmasi & Sheidaii, 2017; Mirjalali, Ghasemi, & Labbafzadeh, 2019; Prinz, 2010). This controlled yielding is a primary part of their design, permitting them to absorb and dissipate energy from an earthquake, which prevents an abrupt collapse. The braces can be configured in various ways within a building's design to optimize their effectiveness.

There are two types of brace frames, which are concentrically brace frames (CBFs) and eccentrically brace frames (EBFs). The prime difference in the bracing members connect to the beams and columns.

Researches on concentrically brace frames (CBFs) have focused on improving their performance, specifically in high-seismicity regions (Banihashemi, Mirzagoltabar, & Tavakoli, 2015; Mahmoudi, Shirpour, & Zarezadeh, 2019; Yang, Sheikh, & Tobber, 2019; Zeng, Zhang, & Ding, 2019; Tajmir Riahi, Zeynalian, Rabiei, & Ferdosi, 2020; Hammad & Moustafa, 2021). The main deficiency with conventional CBFs is their behavior under cyclic loading, such as during an earthquake. While the brace in tension is yields and dissipates energy, the brace in compression is susceptible to buckling, this can cause to a fast loss of strength and stiffness (Astaneh-Asl, Topkaya, & Kazemzadeh Azad, 2017; Tamboli, n.d.; Issa, Stephen, & Mwafy, 2024; Al-Safi, Alameri, Ezzedine, & Alwalidi, 2022; Souri & Mofid, 2023; Gottem et al., 2023; Sugihardjo, Habieb, & Karuniawan, 2022).

The concept of eccentrically brace frames and the research into their seismic performance is documented in academic literature and design codes. Much of the foundational research on the EBFs was conducted at the University of California, Berkeley, in the late 1970s and 1980s (Roeder & Popov, 1978). The principles established by this early research were later incorporated into seismic design codes around the world, e.g., AISC 341-22 (American Institute for Steel Construction, 2022), Eurocode 8 (European Committee for Standardization, 2004), CSA S16-94 (Canadian Standards Association, 1994).

EBFs successfully combine the high level of ductility of moment resisting frames (MRFs) and the high level of stiffness of CBFs by inserting eccentricity between a frames braces and columns (Canadian Standards Association, 1994). The braces of the EBFs supply the elastic stiffness of the frames and the eccentricity of the braces forms a link that provides the ductility, consequently, energy dissipation capacity of the MRFs, where all other elements of the EBFs, including the beams, columns, and braces outside the link, are designed to remain elastic.

The link's behavior of the EBFs is highly impacted by its length, consequently effects on the type of hinges and the type of mechanism. Links are typically classified into two main types based on their length relative to their depth (American Institute for Steel Construction, 2022): short links: these links are designed to yield essentially in shear, and long links: these are designed to yield essentially in flexure (bending).

The fragility curves are a lognormal functions used to seismic risk assessment, they are probabilistic curves that show the likelihood of a structure experiencing a certain level of damage (or “damage state”) at a given intensity of earthquake (Popov & Engelhardt, 1988; Federal Emergency Management Agency, 2003).

By using fragility curves, engineers can make informed decisions about structural design and risk management of different bracing systems by comparing the fragility curves of their to determine which one is less vulnerable to a specific hazard. (Kianmehr, 2021) has evaluated how different types of bracing systems influence the seismic performance and collapse probability of steel structures during a maximum reliable earthquake, fragility curves show the probability of a structure reaching or exceeding a certain damage state (like complete damage) for a given earthquake intensity.

According to earlier versions of the Iraqi Seismic Code, Baghdad was placed in seismic zone II. Modern seismic codes as the ISC 2017 (C.O.S.Q.C., 2017) is used spectral response acceleration parameters (S₁ and S_s) for seismic design instead of a zone number.

The purpose of this study is to assess the seismic vulnerability for two systems, eccentrically inverted V-brace frames (EBFs) and concentrically inverted V-brace frames (CBFs) in Iraq -Baghdad were designed according to codes (American Institute for Steel Construction, 2022; American Society of Civil Engineers, 2022; C.O.S.Q.C., 2017). This study is the first to develop fragility curves for CBF and EBF models designed per (American Institute for Steel Construction, 2022) under Iraqi seismic conditions. The fragility curves developed based on nonlinear static (pushover) analysis (NSP) in order to estimate seismic damage probability in terms of spectrum displacements. By comparing the seismic response of concentric vs. eccentric bracing systems, the results of story drift, formed plastic hinges, and fragility curves shows the eccentric system is more ductile and has a lower probability of extensive damage or collapse.

The dynamic response of the structures is evaluated by nonlinear static (pushover) analysis (NSP) in ETABS - 19 (Computers and Structures, Inc., 2017). A 3-dimention structural model was created. The models that were designed for seismic zones of Baghdad according to codes (American Society of Civil Engineers, 2022; C.O.S.Q.C., 2017). The seismic parameters for structures are shown in Table 2. According to (American Society of Civil Engineers, 2022; C.O.S.Q.C., 2017), For NSP analysis, the modified coefficient method was utilized according to (Federal Emergency Management Agency, 2005) and recommendation in (American Society of Civil Engineers, 2017). To determine the target displacement, the pushover curves were developed by adopted the first mode distribution. The structural elements (braces, beams, and columns) was modelled as elastic elements and concentrated plastic hinges at each element end was employed, a lumped plasticity model is employed for frame elements (columns, brace, link and beams) as per (American Society of Civil Engineers, 2017). For beams and link, the moment rotation relationship was entered to ETAB utilizing steel beam – flexure hinge property type M3, also for columns utilizing steel columns – flexure hinge property type P-M2-M3 and for brace utilizing steel braces-axial. The lengths of plastic hinges was evaluated according to (Federal Emergency Management Agency, 2000), Lp = 0.5 of the section depth.

In order to modal the link element was used a technique employed by (Ramadan & Ghobarah, 1995), then modified by (Prinz, 2010). In this technique, the links was modeled as a linear elastic member with concentrated plastic hinges at each end.

The design load combinations that include earthquake effects, which should be used according to (American Concrete Institute, 2022; American Society of Civil Engineers, 2022), and the Iraqi seismic code (C.O.S.Q.C., 2017), are: (1) U1=1.2 DL+1.6 LL U1 = 1.2\,DL + 1.6\,LL (2) U2=1.2 DL+0.5 LL±1.0 E U2 = 1.2\,DL + 0.5\,LL \pm 1.0\,E (3) U3=0.9 DL±1.0 E U3 = 0.9\,DL \pm 1.0\,E Where:

• DL = Dead load,

• LL = Live load,

• E = Effect of horizontal and vertical earthquake induced forces.

Nonlinear static (pushover) analysis (NSP) was performed using ETABS-19 (Computers and Structures, Inc., 2017) software program to calculate the target displacement (δt) then will be determined the capacity curve which represented the relationship between base shear force and displacement. The pushover curve adopting the modified coefficient method according to (Federal Emergency Management Agency, 2005), that was recommendation in (American Society of Civil Engineers, 2017). The NSP was performed in the both (±X and ±Y) directions, the analyses was including P-effects. According on (Federal Emergency Management Agency, 2005; American Society of Civil Engineers, 2017), the first mode distribution was utilized as a first distribution. In addition according on (Federal Emergency Management Agency, 2000), “the pushover curve is developed for at least two vertical distributions of lateral loads”. Therefore, the second distribution was uniform pattern of lateral force and third distribution was the equivalent lateral force (ELF), then the worst case will be the one governing. The design inter-story drifts was computed according to (American Society of Civil Engineers, 2022). The office buildings, as those in the current study, are classified as occupancy category type II according to (American Society of Civil Engineers, 2022). The life safety of performance level was employed according to (American Society of Civil Engineers, 2022), for which the maximum values of drift must not be exceed the 2%.

Seismic actions influences can be expressed in the fragility curves to assess the vulnerability of different steel bracing structures based on their probability of damage. In this study the fragility curves of the buildings under consideration were expressed the probability of the spectral displacement exceeds a specific damage state P (d ≥ d_s).

Where the spectral displacement Sd take into account as a function to determine the strength of seismic action. The fragility curves characterized by the standard deviation βd_s and mean displacement Sd_ds. Thus, for a specified of damage state d_si, the fragility curves are expressed by the lognormal functions explained in Equation 4 (Federal Emergency Management Agency, 2003): (4) P[ds/Sd]=ϕ1βdslnsdsdds {\rm{P}}[{d_s}/Sd] = \phi \left( {{1 \over {\beta {d_s}}}ln\left( {{{sd} \over {{sd}_{ds}}}} \right)} \right) Where:

• ϕ - Standard normal cumulative distribution function,

• βd_s - Standard deviation of the natural logarithm of spectral displacement of damage state, d_si,

• Sd_ds - Median value of spectral displacement at which the building reaches the threshold of the damage state, d_s.

The thresholds Sd_dsi, are expressed of yield and ultimate displacement of the structure obtained from the bilinear representation of the capacity curves as explained in Table 3.

To estimate the variability of fragility curves for the damage states, the standard deviation (βdx_si) were used determined from values existed in (Federal Emergency Management Agency, 2003) for mid-rise buildings. Some propositions was made to fulfil this purpose as following:

• 1)

  The values of degradation (k) factor was determined accordance to (Canadian Standards Association, 1994) with assumptions that the seismic design level designation was high-code (HC) and for construction quality was Superior (S), the moderate of post-yield duration = 0.9

• 2)

  The structural systems of buildings with large capacity curves variability, β_c = 0.4

• 3)

  The damage variability was small (0.2), β_T,ds = 0.7 for slight damage.

• 4)

  The damage variability was moderate (0.4), β_T,ds = 0.75 for moderate damage.

• 5)

  The damage variability is large (0.6), β_T,ds = 0.9 for severe and complete damage, then was computed the lognormal standard deviation from Equation 5 (Federal Emergency Management Agency, 2003): (5) βds=(βc)2+(βT,ds)2 {\beta _{ds}} = \sqrt {{{({\beta _c})}^2} + {{({\beta _{T,ds}})}^2}} Where:

  • β_ds - The lognormal standard deviation, which expresses the total variability of damage state (d_s),

  • β_c - The lognormal standard deviation, which expresses the variability of the capacity curve,

  • β_T,ds - The lognormal standard deviation, which expresses the variability of the threshold of damage state.

The damage states classification are described in in Table 4.

As mentioned in previous section, the NSP analysis was performed in (X and Y) directions for all models, the first load distribution in Y direction was adopted. Figure 2 shows the development of plastic hinges in models.

5.1.

Development of Plastic Hinges

From Figure 3, it can be noticed that the forming of plastic hinges for concentrically inverted V-brace frames (CBFs) models was in braces, because the (CBFs) essentially resist lateral loads through axial forces (tension and compression) in the braces. When braces are exposes to a lateral load, one diagonal brace is put in tension, while the other is in compression. In an intense seismic activity, energy is dissipated primarily through the buckling and yielding of the braces. While the forming of plastic hinges for eccentrically inverted V-brace frames (EBFs) models was in links because the eccentrically brace frame is characterized by eccentricity through its possession of the link that is designed to yield and dissipate energy during an earthquake. This ductile behavior prevent premature buckling of braces, enabling a structure to absorb more seismic energy without an abrupt loss of strength.

Figure 3:

Plastic hinges in models with 12 and 9 storyes

5.3.

The Fragility Curves

Figure 4 present the fragility curves of models under consideration, these curves were developed to quantify of the influence eccentrically inverted V-brace frames and concentrically inverted V-brace frames on the model's vulnerability. The displacement corresponding to the damage states for 50% and 90% probabilities is illustrated in Table 7.

Figure 4:

Probabilities for models

Table 7:

Models	Probability of slight state		Probability of moderate state		Probability of extensive state		Probability of complete state
	at 50%	at 90%	at 50%	at 90%	at 50%	at 90%	at 50%	at 90%
EBFs-12 story	97	100	133	142	181	204	438	-
CBFs-12 story	66	95	123	137	157	167	198	-
EBFs-9 story	115	134	182	226	293	334	640	-
CBFs-9 story	90	92	122	132	203	219	392	-

It can be observed from the comparison of the results for models with 9 stories that the spectral displacement identical to the probability of slight damage state 50% and 90% of the EBFs is higher about 28% and 46% ,respectively as compared with the CBFs. For the probability of moderate damage state 50% and 90%, the spectral displacement of the EBFs is higher by 38 % and 71%, respectively as compared with the CBFs. For the probability of the extensive damage state 50% and 90%, the spectral displacement of the EBFs is higher about 44 % and 53%, respectively as compared with the CBFs. The probability of the complete damage state 50%, the spectral displacement of the EBFs is higher about 63 % as compared with the CBFs.

For models with 12 storyes that the spectral displacement identical to the probability of slight damage state 50% and 90% for the EBFs is higher about 21 % and 8%, respectively as compared with the CBFs. For probability of moderate damage state 50% and 90%, the spectral displacement of the EBFs is higher by 4 % and 13%, respectively as compared with the CBFs. For the probability of the extensive damage state 50% and 90%, the spectral displacement of the EBFs is higher about 24 % and 21%, respectively as compared with the CBFs. For probability of the complete damage state 50%, the spectral displacement of the EBFs is higher about 16% as compared with the CBFs.

In this study, the seismic response of the eccentrically inverted V-brace frames (EBFs) models and the concentrically inverted V-brace frames (CBFs) models with 12 and 9 story which are designed based on the codes (American Institute for Steel Construction, 2022; American Society of Civil Engineers, 2022; C.O.S.Q.C., 2017) locating on Baghdad were studied; the vulnerability was investigated by using fragility curves. The following observations are obtained:

• ○

  Story drifts for the NSP analysis in Y-direction are less for the EBFs about (78% to 7%) along the height when compared to the CBFs models. The current models analysis lacks a dynamic time-history validation, although is more accurate. However, it can still proceed with a preliminary assessment using a simpler method as NSP.

• ○

  For NSP analysis, it can be observed that the fragility curves for the CBFs models present poorer seismic performance and an increased damage hazard compared to the EBFs models. This is because the two systems have fundamental differences in how they dissipate seismic energy.

• ○

  Fragility curves for the EBFs presents their controlled energy dissipation and higher ductility, significance they have a lower probability of significant damage or collapse even at higher seismic intensity levels. The damage is limited to the link beams, which can be reconditioned or replaced, thereby reducing the overall damage hazard.

• ○

  Fragility curves for the CBFs indicates a higher probability of exposing a severe damage state or collapse at lower seismic intensity levels. The sudden loss of strength due to brace buckling and fracture contributes to this increased fragility.

From the above, by comparing the story drift, formed plastic hinges, and fragility curves of concentric vs. eccentric to determine which one is less vulnerable to a specific seismic hazard, it can be concluded that the eccentric bracing system is more ductile for a building in a seismic region, because its fragility curve shows a lower probability of extensive damage or collapse. For practice in Iraq, the EBFs are recommended for zones with high spectral acceleration; however, cost and detailing complexity should be considered.