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Seismic Performance Factors for Low- to Midrise Steel Diagrid Structural Systems Esmaeel Asadi1 and Hojjat Adeli2 1. Department of Civil Engineering, Case Western Reserve University, 10900 Euclid Ave, Cleveland, OH 44106, Email:
[email protected]
2. Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, 470 Hitchcock Hall, 2070 Neil Avenue, Columbus, OH 43220, USA, Email:
[email protected]
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ABSTRACT
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Diagrids are known as an aesthetically pleasing and structurally efficient system. The
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current design codes and provisions, however, provide no specific guidelines for their
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design under extreme events such as earthquakes. This paper presents a comprehensive
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investigation of the performance of steel diagrid structures to evaluate their key seismic
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performance factors. Nonlinear static, time-history dynamic, and incremental dynamic
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analyses are used to assess diagrid performance and collapse mechanisms in a high
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seismic region. Seismic performance factors including response modification factor,
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ductility factor, overstrength factor, and deflection amplification factor are quantified
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using four different methodologies. Four archetype groups of diagrid buildings ranging
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in height from 4 to 30 stories have been investigated. An R factor in the range of 4 to
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5 is recommended for steel diagrid frames in the range of 8 to 30 stories unless
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supplementary analyses are conducted to find the optimal diagonal angle. For low-rise
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steel diagrids (under 8 stories) an R factor in the range of 3.5 to 4 is recommended.
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Further, an overstrength and ductility of 2.5 and 2 is recommended. This paper lays the
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groundwork for including steel diagrids in design provisions.
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Keywords: Seismic Performance, Response Modification Factor, Steel structure,
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Diagrid
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1. INTRODUCTION
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Two current trends in design of tall and special buildings are creating freeform signature
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buildings and using sustainable and efficient structural systems (Wang and Adeli 2014, Rafiei and
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Adeli 2016). Diagrid, a variation of the tubular structures, enjoys both of those trendy attributes.
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Diagrids are known as an aesthetically pleasing and structurally efficient system and for their
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flexibility to create free-form structures. Their inclined diagonal members can carry both gravity
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and lateral loads (Moon 2007, Mele et al., 2014). Fig. 1 shows the main components of a diagrid
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frame along with its basic triangular element. Diagrids have been used in a number of signature
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and free-form high-rise buildings across the world such as the 103-story Guangzhou International
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Finance Centre, Guangzhou, China, 57-story The Bow, Calgary, Canada, the 51-story Tornado
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Tower, Doha, Qatar, and the 595.7-m Canton Tower (Moon 2008, Niu et al. 2015). Diagrids have
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also been used for iconic mid-rise buildings such as the 11-story Seattle Central Library, Seattle,
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U.S and the 11-story Macquarie Bank, Sydney, Australia (Boake 2014). Further, diagrids have
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been used for free-form steel space-frame roof structures (Kociecki and Adeli 2013, Asadi and
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Adeli 2017). However, current codes of practice such as ASCE7-10 (2010) provide no specific
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design specifications and seismic performance factors for diagrids. Such specifications and factors
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will make this structural system more widespread for low- to high-rise buildings.
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The design codes such as ASCE7-10 (2010) permit elastic analysis for the design of different
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structural systems by providing a response modification factor, R, (ASCE7-10 2010, FEMA 2009
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and 2012) to account for the nonlinear response of the structure during extreme events (Adeli et
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al. 1978). Applied Technology Council (ATC 1978) introduced a base shear reduction factor for
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the computation of the design base shear of a structure using an elastic response spectrum by taking
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the ductility into account. Early values of the R factor were mostly based on engineering judgment
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and previous behavior of each structural system during earthquakes (Whittaker et al. 1999).
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Other design codes also follow this approach and employ performance/behavior factors to
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account for the nonlinear performance of the structure during earthquakes. The methods to
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calculate these factors and the effective parameters, however, vary. For instance, Eurocode-8
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(2004) uses a behavior factor (q) to account for energy dissipation during earthquakes, which is
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based mostly on ductility and plan and elevation regularity/irregularity of the frame. The Chinese
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code (GB 50011, 2010) uses a seismic influence coefficient (α1) which is tabulated based on the
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seismic class and intensity of the site and natural period and damping ratio of the structure.
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Ductility affects the performance of the structure to dynamic loads and generally reduces the
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effective base shear (Newmark and Hall 1982). Several architectural and structural factors may
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cause an amount of overstrength in the building including expected-to-nominal strength ratio of
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members, detailing of the components and connections in the design process, and non-structural
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components such as partition walls (Whittaker et al. 1999, FEMA 2009).
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FEMA (2009) published a comprehensive report on the assessment of building performance
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and response parameters under seismic loads. The main goal of that report is to set reliable
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minimum design criteria and provide a consistent approach applicable to building codes when a
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linear design method is used. The report introduces a procedure to quantify the key structural
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factors of seismic building structural systems such as period-dependent ductility and overstrength
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factors. The recommended approach is compatible with previous FEMA reports (FEMA 2000,
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2005) and ASCE7-10 (2010).
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The objective of the current research is to evaluate the seismic performance of diagrid
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structures and propose seismic performance factors (SPFs) including overstrength, ductility,
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deflection amplification and response modification factors for steel diagrid structures. Nonlinear
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static, time-history dynamic, and incremental dynamic analyses have been performed to assess the
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performance of archetype diagrid structures. Key factors affecting diagrid performance are
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specified. Four different methodologies have been explored to evaluate and quantify SPFs of
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diagrid archetypes with different heights and diagonal angles: FEMA P-695 (2009), Miranda and
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Bertero (1994), Vidic et al. (1994), and Newmark and Hall (1982).
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2.
METHODOLOGY FOR QUANTIFYING SEISMIC PERFORMANCE
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FACTORS
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This section describes briefly the four methodologies employed in this research. First, FEMA
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P-695 (2009) method, the more recent approach, is described in detail. Then, the procedures used
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to quantify each SPF value is discussed.
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2.1. FEMA P-695 METHODOLOGY
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FEMA P-695 report, referred as FEMA (2009) in this article, introduces a comprehensive
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method to estimate SPFs for new seismic-resisting structural systems. This method integrates
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uncertainty in demand and performance and employs advanced nonlinear analysis techniques
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including incremental dynamic analysis (IDA). Fig. 2 shows the main steps of the FEMA (2009)
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method and the evaluation procedure.
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The first step is to define a series of criteria for the desired structural system including the
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structural material, configuration, inelastic and nonlinear properties, and intended scope of
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application. In the case of diagrids, configuration and particularly the diagonal angle is a key
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property dictating the performance of the structure. Next, a set of archetypes reflecting a wide
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range of possible archetype structures should be developed given the scope and criteria defined
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initially. Archetypes are assembled into a number of performance groups. Given that it is
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impractical to consider or find the SPFs for all possible applications, the archetypes should
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represent the typical applications of the system (FEMA, 2009). Using nonlinear static analyses,
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statistical data on the overstrength, ductility, and median collapse capacities of the structural
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system should be evaluated. The expressions for quantifying each factor are discussed in the
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section.
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2.2. QUANTIFICATION OF SEISMIC RESPONSE FACTORS
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2.2.1. Response Modification Factor
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ATC (1995a, b) present a formula for determination of the R factor as a product of three
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important properties of the structure: redundancy factor (Rr) and period-dependent ductility (Rµ)
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and overstrength factors (Ro).
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𝑅𝑅 = 𝑅𝑅𝑜𝑜 𝑅𝑅𝜇𝜇 𝑅𝑅𝑟𝑟
(1)
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The Ro factor represents the capacity-to-demand strength ratio in a building which is generally
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larger than 1.0 as there is always some overstrength in the structure. The Rµ factor is associated
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with the global nonlinear response of the structural system. The Rr factor represents the degree of
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indeterminacy of the structure or the number of available load carrying lines to transfer seismic
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loads to the foundation (Whittaker et al. 1999).
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Eq. 1 has been used for assessment of the R factor for various structural systems such as
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chevron-braced frames (Kim and Choi 2005), Buckling-Restrained Braced Frames (BRBF)
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(Asgarian and Shokrgozar 2009), tubular structures in tall buildings (Kim et al. 2009), moment
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resisting frames with TADAS (triangular-plate added damping and stiffness) dampers (Mahmoudi
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and Abdi 2012), and steel frames utilizing shape memory alloys (Ghassemieh and Kargarmoakhar
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2013). Such a study has not been reported for diagrids, however.
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The redundancy represents an enhanced reliability of the system when multiple vertical lines
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are used to carry the lateral load. Whittaker et al. (1999) recommend a redundancy factor of
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approximately 1.0 for structures with at least 4 lines of vertical seismic framing in each
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perpendicular direction. Since there are at least 4 lateral load-carrying lines (at least 4 diagonals)
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in each direction in the examples studied in this research, the redundancy factor is assumed to be
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equal to 1.0.
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2.2.2. Overstrength Factor
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The overstrength factor for the structural system is the largest average of all archetype
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performance groups. FEMA (2009) defines the overstrength factor, Ro, as the ratio of the maximum
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base shear resistance, Vmax, to the design base shear of the structure, Vd.: 𝑅𝑅𝑜𝑜 =
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𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 𝑉𝑉𝑑𝑑
(2)
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where Vmax is based on the nonlinear static pushover curve (Fig. 3) and Vd is the required design
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base shear defined by ASCE7 (2010).
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In the literature, other slightly different definitions of this factor have been suggested based on
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FEMA (2000) idealized pushover curve. FEMA (2000) states that the pushover curve shall be
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idealized with a bilinear curve using an “iterative graphical procedure” so that the areas above
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and below the ideal curve are approximately equal as shown in Fig. 3. The secant stiffness (Ke) is
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taken equal to the slope of the ideal curve where it crosses the primary curve at 0.6 times the
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effective yield strength, Vy, (Fig. 3). Kim and Choi (2005) use the overstrength factor based on the
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idealized force-displacement curve introduced in FEMA (2000) as follows: 𝑅𝑅𝑜𝑜 =
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𝑉𝑉𝑦𝑦
𝑉𝑉𝑑𝑑
(3)
The effective yield strength (Vy) shall be equal to or less than the maximum base shear (Vmax
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in Fig. 3). Eq. 2 results in a larger than or equal value for the overstrength factor than Eq. 3.
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2.2.3. Ductility Factor
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The relationship between the period-dependent ductility reduction factor (Rµ) and the ductility
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ratio (µ) has been studied by several researchers in the past decades (FEMA 2009, Whittaker et al.
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1999, Miranda and Bertero 1994, Vidic et al. 1994, Newmark and Hall 1982). The relationship
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depends mostly on the fundamental period of the structure. It changes for displacement, velocity,
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and acceleration segments of a linear response spectrum (Whittaker et al. 1999).
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Newmark and Hall (1982) proposed a method to construct the inelastic response spectrum of
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a structural system from an elastic one which can be used to find the Rµ factor as well. Chopra and
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Goel (1999) refined the Newmark˗Hall method as follows:
𝑅𝑅𝜇𝜇 =
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where
1 𝑇𝑇 < 𝑇𝑇𝑎𝑎 ⎧ 𝛽𝛽⁄2 𝑇𝑇𝑎𝑎 < 𝑇𝑇 < 𝑇𝑇𝑏𝑏 ⎪(2𝜇𝜇 − 1) 𝑇𝑇
⎨ 𝑇𝑇𝑐𝑐 𝜇𝜇 ⎪ ⎩ 𝜇𝜇
𝑇𝑇𝑏𝑏 < 𝑇𝑇 < 𝑇𝑇𝑐𝑐 𝑇𝑇 > 𝑇𝑇𝑐𝑐
𝛽𝛽 = ln(𝑇𝑇/𝑇𝑇𝑎𝑎 )⁄ln(𝑇𝑇𝑏𝑏 /𝑇𝑇𝑎𝑎 )
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(4a)
(4b)
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and T is the natural period of the structure, µ is the ratio of the maximum permissible lateral
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displacement (δu) to the effective elastic displacement (δy) of the structure, and the period limits
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are defined based on the elastic response spectra with the following recommended values:
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Ta=0.030 s, Tb=0.125 s and Tc is the period where the constant acceleration and the constant
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velocity segments of the inelastic design spectrum meet.
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Miranda and Bertero (1994) observed that the Rµ˗µ relationship also depends on the site
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soil condition. Based on a study of 124 ground motions collected on different site conditions they
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introduced a Φ coefficient to take into account the soil condition as follows: 𝑅𝑅𝜇𝜇 =
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𝜇𝜇−1 Φ
+ 1 ≥ 1.0
(5)
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They proposed three different equations (for rock, alluvium, and soft soil sites) for the Φ
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coefficient based on ductility ratio and period of the structure.
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Vidic et al. (1994) propose the following equation for the Rµ˗µ relationship: 𝑇𝑇
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where
𝑐𝑐1 (𝜇𝜇 − 1)𝑐𝑐𝑅𝑅 + 1 𝑇𝑇 ≤ 𝑇𝑇0 𝑇𝑇0 𝑅𝑅𝜇𝜇 = � 𝑐𝑐𝑅𝑅 𝑐𝑐1 (𝜇𝜇 − 1) + 1 𝑇𝑇 > 𝑇𝑇0 𝑇𝑇0 = 𝑐𝑐2 𝜇𝜇 𝑐𝑐𝑇𝑇 𝑇𝑇𝑐𝑐
(6)
(7)
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and the coefficients c1, c2, cR, and cT depend on the damping ratio and hysteresis response of the
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structure. The recommended values for these coefficients used in this research for a bilinear
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hysteretic behavior (similar to the dashed line in Fig. 3) and 5 percent damping are 1.35, 0.75,
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0.95, and 0.2, respectively (Chopra and Goel 1999).
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FEMA (2009), on the other hand, recommends a different approach to find the archetype
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period-dependent ductility. It requires the lateral load to be applied monotonically until the base
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shear capacity of 0.8Vmax. Then, the period-dependent ductility factor is obtained by dividing the
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corresponding lateral roof displacement (δu) by the effective yield lateral roof displacement (δy,eff)
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(Fig. 3). The effective yield roof displacement is found using the following equation:
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where
𝛿𝛿𝑦𝑦,𝑒𝑒𝑒𝑒𝑒𝑒 = 𝐶𝐶0 𝐶𝐶0 = 𝜑𝜑1,𝑟𝑟
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𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 𝑊𝑊
�
𝑔𝑔
4𝜋𝜋2
∑𝑁𝑁 1 𝑚𝑚𝑛𝑛 𝜑𝜑1,𝑛𝑛
2 ∑𝑁𝑁 1 𝑚𝑚𝑛𝑛 𝜑𝜑1,𝑛𝑛
� (𝑇𝑇𝑝𝑝 ) 2
(8)
(9)
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and W, g, and Tp are building weight, gravity acceleration, and the maximum permissible
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fundamental period of the structure according to ASCE7-10 (2010). The fundamental period of the
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structure can be computed based on an eigenvalue analysis but shall not be greater than CuTa (the
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product of the coefficient for the upper limit and the approximate fundamental period). The
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modification factor C0 converts the spectral displacement of the equivalent SDOF system to the
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roof displacement of the multi-degree of freedom (MDOF) structure and is found by Eq. 9 which
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is Eq. C3-4 of ASCE/SEI 41-06 (2006) or alternatively using the values from Table 3-2 of FEMA
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(2000). In Eq. 9, mn, φ1,n, φ1,r are the mass at level n, the ordinate of fundamental mode shape at
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level n, and the roof, respectively.
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2.2.4. Deflection Amplification Factor
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The ASCE7-10 (2010) uses the deflection amplification factor (Cd) to determine the amplified
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story deflection at level x (δx) from an elastic analysis. The Cd factor can be calculated using the
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structure ductility ratio (µ), the effective yield strength (Vy), and the design base shear (Vd) as
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follows (Uang and Maarouf 1994):
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𝐶𝐶𝑑𝑑 =
𝑉𝑉𝑦𝑦
𝑉𝑉𝑑𝑑
𝜇𝜇
(10)
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FEMA (2009) considers the Cd factor as a reduced R factor and recommends the following equation to calculate the Cd: 𝐶𝐶𝑑𝑑 =
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𝑅𝑅
𝐵𝐵𝐼𝐼
(11)
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where the BI is a numerical coefficient provided in Table 18.6-1 of the ASCE7-10 (2010) in terms
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of the inherent damping ratio of the structure. The BI value for 5 percent effective damping ratio
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is 1.0 where the period of the structure is larger than T0 = 0.2SD1/SDS. This means that in this case
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the Cd factor is equal to the R factor.
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2.2.5. Relation Between Ductility Factor (Rµ) and Ductility Ratio (µ)
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The ductility factor (Rµ) is calculated using three methods: Newmark˗Hall (NH), Miranda-
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Bertero (MB), and Vidic-Fajfar-Fischinger (VFF). These methods are selected to cover a
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reasonably wide range of methodologies proposed for quantifying Rµ. The Newmark and Hall
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(1982) methodology paved the way for ATC method (1995a&b) and is used by a number of
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previous researchers (Kim and Choi 2005, Kim et al. 2009). The MB method includes the effect
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of soil property and the VFF method considers hysteresis response and damping ratio of the
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system. By considering these three methods in addition to FEMA (2009) method, the authors aim
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to obtain more reliable values for the Rµ and R factors.
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Fig. 4 shows the variation of Rµ versus the period of the structure for three different ductility
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ratios of 2, 4, and 6 using the three aforementioned approaches. In general, the VFF approach
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offers an upper bound for the Rµ factor in periods larger than Tb whereas the refined NH approach
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is the lower bound in that range. The MB approach for rock soils generally yields Rµ values
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between the other two approaches. Also, this last approach results in larger Rµ values than the NH
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approach for periods between 1 to 3 seconds but not as large as the VFF approach. For periods
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larger than 3 seconds, all three methods result in virtually a constant value of Rµ (equal to µ) where
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the VFF approach yields a relatively larger value than the others. In case of diagrids, the
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fundamental period of the structure is relatively low due to their substantial lateral stiffness (Kim
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and Lee 2012, Asadi and Adeli 2017). Therefore, low- to mid-rise diagrids are expected to fall in
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the range of small periods (lower than 2.0s). More information on Rµ-µ relationship is presented
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in Chopra and Goel (1999), Whittaker et al. (1999), and ATC (1995a&b).
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3. ARCHETYPE DIAGRID STRUCTURES 3.1. ARCHETYPE BUILDINGS
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Four groups of archetype steel diagrids with 4, 8, 15, and 30 stories are considered for
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assessment of SPFs to cover a wide range of possible applications of diagrids. As is common in
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diagrid structures, the diagrid frames form the perimeter of the building (Ali and Moon, 2007).
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The archetypes are office buildings located in Los Angeles, CA with Ss (spectral response
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acceleration at 0.2 sec) and S1 (spectral response acceleration at 1 sec.) of 1.803g and 0.649g,
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respectively. Dead and Live loads are 4 and 2.4 kN/m2, respectively. The floor plan for the 4-, 8-
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and 15-story groups is similar as presented in Fig. 5a. Fig. 5b presents the typical floor plan for
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the 30-story group. For each group, three diagrid patterns with diagonal angles of approximately
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45°, 63°, and 72° with the horizon are studied to account for various possible diagonal
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configurations. Fig. 6 shows the diagrid patterns used in this research. The structures are labeled
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by using the number of stories and their diagonal angle. For instance, archetype 15-63 refers to a
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15-story diagrid structure with a diagonal angle of 63°.
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3.2. DESIGN AND ANALYSIS CRITERIA
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AISC Load and Resistance Factor Design (LRFD) (AISC 2011), AISC 360-10 (2010), and
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ASCE7-10 (2010) are used for the structural design of all archetypes. Note that ASCE7-10 (2010)
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does not provide any specific performance factor for the seismic design of diagrids. For
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preliminary design, following previous studies (Kim and Lee 2010&2012; Kwon and Kim 2014),
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all three seismic coefficients including response modification, overstrength, and deflection
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amplification factors are assumed to be equal to 3.0. The A992 grade 50 steel with yield and
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ultimate strength of 345 and 450 MPa, respectively, is used for all steel members. The standard W
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section is used for diagonals, beams, and interior columns. Diagonals are designed to carry both
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lateral and gravity loads. For wind loads, Exposure Category C and wind speed of 110 mph are
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used based on the location of the structures. Floors are reinforced concrete slabs with a thickness
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of 6 in. The ASCE7 (2010) requirements for story drift are satisfied.
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Three types of analyses are performed for SPF evaluation including three-dimensional
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nonlinear static analysis using SAP2000 (CSI, 2011) per FEMA (2005; 2009) and NTHA and IDA
250
using OpenSees planar models (Mazzoni et al., 2006) models per FEMA (2009; 2012). SAP2000
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has been used by a number of other diagrid system researchers in recent years in a similar fashion
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(Moon et al., 2007; Kim et al., 2010&2012; Mele et al., 2014). The plastic performance and
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modeling criteria of FEMA (2005) and ASCE/SEI 41-13 (2014) for braced frames are adapted for
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diagonals. Geometric nonlinearity (P-∆ effect) is considered directly in the computation of the
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stiffness matrix. Fig. 7 depicts the force-deformation curves used for modeling plastic hinges in
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SAP2000 models. The modeling parameters a, b, c in Fig. 7 are calculated per FEMA (2005)
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Tables 5-6 and 5-7. The parameter ∆c represents the axial deformation at expected buckling load
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(Pcr).
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For OpenSees models, the hysteretic model of Menegotto-Pinto with fiber element and 0.02
260
hardening is used for both diagonals and beams (Mazzoni et al., 2006). The method developed by
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Uriz (2005) for modeling Concentrically Braced Frames (CBFs) in OpenSees which considers low
262
cycle fatigue and global brace buckling is adapted to model nonlinear behavior of diagonals. To
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simulate P-∆ effect, a leaning column linked to the main frame is considered (Moghaddasi and
264
Zhang, 2013). Fig. 7 also depicts the key parameters of the OpenSees models. The nonlinear
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dynamic analyses are conducted for 4- and 8-story archetypes.
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The few experimental studies published on diagrid connections (Kim et al., 2010 and 2011)
267
provide little information on design and modeling of diagrid connection. Hence, connections are
268
not explicitly modeled in this research. Different possible approaches were tested and assuming
269
moment-resisting connections seem to provide the most accurate results. Note that regardless of
270
the connection type, i.e. hinged or moment-resisting, the diagonal design and behavior are
271
governed by axial strength not flexural strength (Moon et al., 2007; Kim and Lee, 2012; Mele et
272
al., 2014).
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4. NONLINEAR PERFORMANCE OF STEEL DIAGRIDS
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Nonlinear static analysis, NTHA, and IDA are performed per FEMA (2005, 2009, 2012a,
275
2012b) on archetype diagrid structures to assess their seismic performance and evaluate the SPFs.
276
This section summarizes key findings of the static and dynamic performance of diagrids and
277
clarifies details of the analyses used for SPF assessment.
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4.1. NONLINEAR STATIC ANALYSIS
279
In static nonlinear analyses, three lateral load distributions are considered for each model: 1)
280
uniform distribution, 2) the distribution provided by the Equivalent Lateral Force (ELF) method
281
of ASCE7-10 (2010), and 3) the modal shape distribution, and the critical one, showing the least
282
lateral stiffness and load capacity, is chosen for quantification of the SPFs.
283
Figs. 8a-d show the lateral force versus roof displacement, the pushover curve, obtained by a
284
static nonlinear analysis for the 4-, 8-, 15-, and 30-story models, respectively. Diagrids show large
285
initial lateral stiffness and collapse capacity which is consistent with previous studies (Moon et al.,
286
2007; Kim and Lee, 2012; Milana et al., 2015). Plastic hinges are well spread across the diagrid
287
frames for most archetypes except the 4-72 and 8-72. These two archetypes have incomplete
288
uppermost diagrid modules (shown in Fig. 9) which adversely impact their behavior. The optimal
289
archetype in terms of nonlinear behavior and lateral stiffness is found to be the 4-63, 8-63, 15-72,
290
and 30-72 models among the 4-, 8-, 15-, 30-story structures, respectively. In these optimal models,
291
plastic hinges are spread both horizontally and vertically across the width and height of the
292
structure more broadly than the other case; these models have a larger load carrying capacity
293
compared with other models (Fig. 8).
294
4.2. NONLINEAR TIME-HISTORY ANALYSIS
295
NTHA is conducted on OpenSees models of 4- and 8-story archetype buildings considering
296
the gravity loads and a set of 22 far-field ground motion records recommended for SPF assessment
297
in Appendix A of FEMA (2009).
298
4.2.1. Earthquake Ground Motions
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A representative set of ground motions is a critical component of a reliable seismic
300
performance assessment. FEMA (2012a) recommends a minimum of 11 ground motion records
301
for collapse analysis of building structures. In this research, a set of 22 far-field records
302
recommended by FEMA (2009) to appropriately represent record-to-record variability is used in
303
NTHAs and IDAs. The records obtained from 14 different events include site class C and D and
304
magnitudes in the range of 6.5 to 7.6. Based on FEMA (2009) instructions, they are normalized
305
with respect to peak ground velocity (PGV) and scaled to match the design response spectrum of
306
ASCE7-10 (2010) at the fundamental period of the structure. Fig. 10 depicts the scaled response
307
spectrum alongside the design response spectrum for the 8-45 archetype.
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4.2.2. Diagrid Performance under Nonlinear Dynamic Analysis
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The nonlinear dynamic performance of diagrid archetypes is studied under two earthquake
310
hazard intensities: DBE (10% probability of exceedance in 50 years) and MCE (2% probability of
311
exceedance in 50 years) per United States Geological Survey (USGS) hazard maps. The NTHAs
312
are performed in OpenSees for 4- and 8-story archetypes. Table 1 summarizes the key engineering
313
demand parameters (EDPs) of the archetypes studied, including the maximum and mean inter-
314
story drift (IDRmax and IDRavg, respectively) under DBE and MCE.
315
Diagrids shows relatively small IDRmax and IDRavg under both DBE and MCE compared to
316
other structural systems (on average, 1.26% and 0.64% for DBE and 2.02% and 0.96% for MCE).
317
For comparison, Chen et al. (2008) report an IDRmax of 1.46% and 2.78% under DBE and MCE
318
for a 3-story steel CBFs located in downtown Los Angeles, CA. As noted under static analysis,
319
the incomplete uppermost module in 4-72 and 8-72 archetypes causes a substantially larger IDR
320
in these cases. Table 1 also shows the IDRmax to IDRavg ratio which is an indicator of the soft-story
321
vulnerability of the structure. Large IDRmax/ IDRavg values indicate concentration of damage in a
322
specific story and likely soft-story formation in the structure. The IDRmax/IDRavg of diagrids is
323
close to CBFs, 1.84 and 2.00 for diagrids compared to 1.76 and 2.08 for the 3-story CBFs under
324
DBE and MCE, respectively (Chen et al., 2008). Moreover, if cases with 72° diagonal angle are
325
excluded, diagrids are much less likely to form soft-story than CBFs. These cases have an
326
incomplete uppermost module. Fig. 11 and 12 depict the variation and the logarithmic trendline of
327
IDRmax, IDRavg, and their ratio for different models and hazard levels. The vertical axis shows the
328
pseudo spectral displacement based on 5% damped design spectra of the site at the fundamental
329
period of the structure, Sd (T1,5%). Accordingly, the IDRmax values and IDRmax to IDRavg ratio for
330
diagrid with 72° diagonal angle has much more dispersion (i.e. standard deviation) than other cases
331
in addition to larger values indicating more uncertainty in the behavior of these models. For
332
instance, the dispersion for IDRmax to IDRavg ratio for 4-72 archetype under DBE is 0.39 compared
333
to 0.18 and 0.05 for 4-45 and 4-63 archetypes, respectively. Further, the trendlines for the 72°
334
archetype in Figs. 11 and 12 has a much larger slope than that of 63° and 45° archetypes for all
335
EDPs. This indicates that the IDRmax and IDRavg increase at a higher rate in 72° models compared
336
to others with the increase of spectral displacement. On the other hand, the trendlines for 63° and
337
45° models are close to each other. The 4-63 archetype shows a slightly higher rate than 4-45
338
archetype (Fig. 11). While the 8-45 archetype shows a considerably higher rate than 8-63 archetype
339
(Fig. 12). Generally, diagrid archetypes absorb large spectral acceleration under both DBE and
340
MCE earthquakes which implies a large capacity and stiffness for diagrid frames. This will
341
manifest in larger overstrength factor for diagrids compared to similar structural systems namely
342
CBFs.
343
4.3. INCREMENTAL DYNAMIC ANALYSIS
344
The IDA is used to explicitly consider record-to-record uncertainty in collapse evaluation
345
(FEMA, 2012a; Vamvatsikos and Cornell, 2002). The IDA includes hundreds of time-history
346
analyses where the ground motion intensity is increased gradually to achieve collapse in the
347
structure. Fig. 13 shows IDA curves for the 4-45 diagrid archetype. The vertical axis shows the
348
spectral acceleration based on 5% damped design spectra of the site at the fundamental period of
349
the structure, Sa (T1,5%). For each archetype, 748 NTHAs are performed to develop the
350
corresponding collapse fragility curve. The IDA is conducted for 4- and 8-story archetypes.
351
Collapse is achieved if an insignificant increase in ground motion intensity (spectral acceleration)
352
result in a significant increase in the governing EDP (maximum IDR) or a dynamic instability
353
happens (FEMA, 2009; Yamin et al, 2017).
354
4.4. COLLAPSE FRAGILITY ASSESSMENT
355
For 4- and 8-story archetypes, collapse fragility functions for each archetype were developed
356
in terms of cumulative distribution function (CDF) of ground motion intensities leading to
357
collapse. According to FEMA (2009), adjusted collapse margin ratio (ACMR) is defined as the
358
ratio of median collapse capacity (ŜCT) to the MCE intensity (SMT) multiplied by Spectral Shape
359
Factor (SSF):
360
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 𝑆𝑆𝑆𝑆𝑆𝑆 × Ŝ𝐶𝐶𝐶𝐶 /𝑆𝑆𝑀𝑀𝑀𝑀
(12)
361
SSF values are provided in Table 7-1 of FEMA (2009) based on ductility ratio and fundamental
362
period of the structure. Also, the acceptable values of ACMR are provided in Table 7-3 of FEMA
363
(2009) based on total system collapse uncertainty.
364
The collapse capacity, ŜCT, obtained from IDA for each ground motion record is used to
365
evaluate the probability of collapse at a certain Sa (T1,5%). Then, a lognormal distribution function
366
is used to estimate the collapse fragility function for each archetype. The empirical CDF and the
367
fitted lognormal collapse curve are illustrated in Fig. 14 for 4- and 8-story archetypes. The Sa
368
(T1,5%) associated with DBE and MCE are shown with dashed lines in Fig. 14.
369
Table 2 lists ŜCT, collapse IDRmax, and their corresponding lognormal dispersion (𝛽𝛽𝑆𝑆𝐶𝐶𝐶𝐶 and 𝛽𝛽𝐼𝐼𝐼𝐼𝐼𝐼
370
respectively), as well as ACMR and probability of collapse for 4- and 8-story archetypes. The mean
371
ACMR is 2.05 which is higher than the acceptable value, i.e. 1.96, and individual cases has a higher
372
ACMR than individual ACMR limit, i.e. 1.56. The acceptable ACMR margins are evaluated for a
373
good model quality conditions based on Table 7-3 of FEMA (2009). The probability of collapse
374
under MCE for each individual archetype is less than 20%, the acceptance limit suggested by
375
FEMA (2009). The mean probability of collapse under MCE, however, is 13.6% which is higher
376
than the suggested acceptance limit, i.e. 10%. To achieve the suggested acceptance limit,
377
archetypes were redesigned several times with different approaches. The design options which
378
passes the later acceptance limit has an extremely small demand to capacity ratio (Fu /Fn) for all or
379
many structural elements and as a result, are economically inefficient and impractical. Note that
380
per ASCE and AISC design codes, the archetypes need to be designed for DBE not MCE. Thus,
381
given that the archetypes listed in Table 2 meet ACMR requirements, they are accepted for collapse
382
fragility assessment.
383
5. SEISMIC PERFORMANCE FACTORS FOR STEEL DIAGRIDS
384
The effective yield strength (Vy) and effective elastic displacement as well as the maximum
385
permissible lateral displacement (δu) are found using the idealized bilinear pushover curve
386
introduced in FEMA (2000) (Fig. 3). Table 3 presents the seismic response parameters including
387
the effective yield strength (Vy), ductility ratio (µ), deflection amplification factor (Cd), and
388
overstrength factor (Ro), period-dependent ductility factor (Rµ), and response modification factor
389
(R) found using four different methods discussed in the previous section. Table 4 presents the
390
mean, standard deviation (SD), maximum (Max), and minimum (Min) values for the seismic
391
response parameters for models investigated in this research.
392
5.1. DUCTILITY RATIO
393
Fig. 15 presents a bar chart for the ductility ratio (µ) of archetypes studied defined as the ratio
394
of the ultimate lateral displacement (δu) to the effective elastic displacement (δy) based on the
395
idealized bilinear nonlinear static pushover curve (Fig. 3). The mean and the SD values of ductility
396
ratio, as reported in Table 4, are 2.65 and 0.79, respectively. The largest ductility ratio in the 4-
397
and 8-story archetypes belongs to 4-63 and 8-63, respectively while in 15- and 30-story archetypes
398
the 72° archetypes show a considerably larger ductility than others. This result indicates that the
399
diagonal angle has a major effect on the ductility of the diagrid structures and by choosing an
400
optimal diagonal angel, the ductility of diagrids can be improved noticeably.
401
5.2. OVERSTRENGTH FACTOR
402
The overstrength factor (Ro) is evaluated using Eqs. 2 and 3. The results are presented in Fig.
403
15. As expected, the FEMA (2009) equation yields a larger value for the Ro factor than Eq. 3 used
404
by Kim and Choi (2005) by an average of 27%. In 4-, 8- and 30-story archetypes, the 63° case has
405
the largest overstrength whereas the overstrength of 15-story cases are relatively similar. This
406
difference is mostly due to the section grouping in the design process and the capacity-to-demand
407
ratio of diagonal sections in the 63° archetypes. Note that the architectural and material factors
408
causing overstrength are similar in all cases.
409
The average value of the overstrength factor for all diagrid archetypes is 2.89 and 2.28 based
410
on Eqs. 2 and 3, respectively. This is relatively close to that of steel moment-resisting frames
411
(MRFs) which is equal to 3.0 per ASCE7-10 (2010), and noticeably larger than the overstrength
412
factors of CBFs and eccentrically braced frames (EBFs) which are both equal to 2.0 per ASCE7-
413
10 (2010). This points to a notably large reserve strength for steel diagrid frames.
414
5.3. DUCTILITY FACTOR
415
Four different approaches, NH, MB, VFF, and FEMA (2009), were used to calculate the
416
ductility factor (Rµ). Fig. 16 shows the variation of the ductility factor for four groups of diagrid
417
structures and three diagonal angles. In general, the ductility factor calculated based on the VFF
418
and MB approaches are larger than the others with the VFF approach being the largest. The result
419
from the NH and FEMA (2009) approaches are relatively close except in the 30-63 archetype,
420
even though the corresponding equations are completely different. The mean ductility factor based
421
on the FEMA (2009) approach is 11, 21, and 28 percent smaller than that of the NH, MB, and VFF
422
approaches, respectively.
423
The mean ductility factors for 4-, 8-, 15-, and 30-story archetypes are relatively close indicating
424
that the number of stories does not have a significant effect on the ductility factor. The diagonal
425
angle, however, has a major effect on the ductility factor similar to the ductility ratio. The
426
archetypes with the largest lateral stiffness in each group, 4-63, 8-63, 15-72, and 30-72, show a
427
noticeably larger ductility factor than others in all approaches studied. The 72° archetypes are
428
found to have a larger ductility factor than others unless the uppermost diagrid module is
429
incomplete. Clearly, the uppermost incomplete diagrid module in 4-72 and 8-72 archetypes
430
adversely affects their ductility factors.
431
The bar chart in Fig. 17 compares the mean SPFs (Ro, Rµ, and R) obtained using various
432
approaches. The SD is also shown in the middle of each bar. In case of Rµ, the SD value for VFF
433
approach is the largest showing undesirable large variations of calculated values using this method.
434
At the same time, the Rµ values calculated based on the NH approach are less disperse showing
435
the smallest SD among all four approaches.
436
5.4. RESPONSE MODIFICATION FACTOR
437
The response modification factor is found for all archetypes using the four approaches
438
discussed earlier. The results are presented in Fig. 18 and Tables 3 and 4. In most cases, the NH
439
approach yields the smallest and most conservative R factor than others except for the 30-63 and
440
all 4-story archetypes. For 30-63 and 4-story archetypes, the FEMA (2009) approach yields the
441
smallest R factor because of the small Rµ factor of these archetypes. Note that the FEMA (2009)
442
approach for finding the period-dependent Rµ factor is different from other approaches. Its
443
equation depends on the period and weight of the structure. In the case of the 30-63 archetype, the
444
period of the structure is relatively large (1.26 seconds, a 105-meter high diagrid) but the weight
445
of this archetype is the smallest among the 30-story cases. This leads to a large effective yield roof
446
displacement (δy,eff) and consequently smaller Rµ (Eq. 8). On average, NH approach gives an R
447
factor of 5.06 with the SD of 1.39 which are both the smallest values among all approaches (Fig.
448
17). This shows that the NH method yields the most conservative and the least disperse results.
449
The MB approach for rock soil yields a mean R factor of 5.67 which is close to the FEMA (2009)
450
mean value of 5.60 but smaller than that of 6.21 obtained using the VFF. Generally, the VFF
451
approach shows the largest R factor (up to 10.76) which can be considered as an upper bound for
452
the R factor of steel diagrid structures (Table 3).
453
In general, four parameters affect the SPFs and in particular, the R factor of diagrid structures.
454
Diagonal angle has the largest influence. As mentioned earlier, the 4-63, 8-63, 15-72, and 30-72
455
archetypes are the optimal archetypes corresponding to each group in terms of nonlinear behavior
456
and lateral stiffness. They have the largest value of the R factor in their corresponding group as
457
well. This indicates that an optimal diagonal angle can increase the R factor of the diagrid steel
458
structure substantially, up to 61 percent in the case of the FEMA (2009) approach. In order to
459
benefit from a large R factor in the range 6-8 (based on the mean value of optimal cases for four
460
approaches), the designer needs to find the optimal diagonal angle based on aspect ratio, structural
461
configuration, effective loads, etc. This can be achieved by checking multiple diagonal
462
configurations or more effectively through a formal optimization method (). In the absence of an
463
extensive parametric study or a formal optimization approach, a smaller conservative R factor
464
between 4-5 (based on the mean value of non-optimal cases for four approaches) is recommended.
465
The number of stories or the height of the structure also affects the diagrid R factor. Low-rise
466
cases (4-story archetypes) show a noticeably smaller R factor than other cases; an average of 3.8
467
compared to overall average of 5.64. Since diagrids are most effective against large lateral loads,
468
a smaller R factor between 3.5-4 is recommended for low-rise buildings (under 8-story). This value
469
is based on mean R factor of 4-story archetypes for four approaches. The mean value of the R
470
factor increases with an increase in the number of stories except for the FEMA (2009) approach
471
where the mean value for the 30-story archetypes is smaller than those of 8- and 15-story
472
archetypes. Similarly, the aspect ratio of the building also influences the R value. The low-rise
473
buildings with smaller aspect ratio have a noticeably smaller R factor.
474
Another key influencing parameter is the configuration and in particular the incomplete
475
uppermost diagrid module. As shown in Fig. 9, the 4-72 and 8-72 have an incomplete uppermost
476
diagrid module which adversely affects their performance and causes a reduction in the R factor.
477
The structural deformation increases in these incomplete modules initiating failure of diagonals.
478
This adverse impact is clearer in the 8-72 archetype where its R factor is noticeably lower than 8-
479
63 archetype. Consequently, if the designer wants to have an incomplete upper module in the
480
diagrid structure, the R factor needs to be reduced accordingly. The magnitude of this reduction
481
should be evaluated by supplementary analyses per FEMA (2009).
482
5.5. DEFLECTION AMPLIFICATION FACTOR
483
The results for the deflection amplification factor (Cd) using Eqs. 10 and 11 are presented in
484
Tables 3 and 4 and as a bar chart in Fig. 15. The result from each equation is noticeably different
485
for different archetypes but the mean values are relatively close. The mean value for the Cd factor
486
using Eq. 10 and Eq. 11 is 6.03 and 5.60, respectively. Note that the average of results from Eq.
487
13 (6.03) is larger than the overall mean R factor obtained (5.64). The mean values for the Cd factor
488
are larger than those of steel MRFs which are in the range 3.0-5.5 per ASCE7-10 (2010), and CBFs
489
which are in the range 3.25-5.00 per ASCE7-10 (2010).
490
6. CONCLUSIONS
491
This paper presented a comprehensive study of the key seismic performance factors needed
492
for the seismic design of diagrids including ductility ratio, deflection amplification factor, ductility
493
factor, overstrength factor, and the response modification factor. Four different methodologies
494
were employed: Newmark˗Hall, Miranda-Bertero, Vidic-Fajfar-Fischinger, and FEMA P-695.
495
Four archetype groups of diagrids ranging in height from 4 to 30 stories were studied in a high
496
seismic region.
497
The mean value of R factor based on four methods and twelve structural archetypes (a total of
498
48 cases) is 5.64. This value is relatively large given that no special seismic design consideration
499
is included in the modeling. Experimental studies aiming to develop special seismic considerations
500
for design of diagrid connections and diagonal members will improve the performance of steel
501
diagrids. For comparison, steel concentrically braced frames (CBFs) has an R factor of 3.25 for
502
ordinary CBFs to 6.00 for special CBFs per ASCE7-10. The optimal archetypes in terms of
503
nonlinear behavior and lateral stiffness in each group (4-63, 8-63, 15-72, and 30-72 archetypes)
504
have a disproportionately large effect on the overall mean seismic response factor of steel diagrid
505
structures. The mean R factor for all archetypes excluding those four is 4.76.
506
In general, four parameters affect the seismic performance factors of diagrid structures: 1)
507
diagonal angle, 2) number of story, 3) height to width ratio (aspect ratio) of the building, and 4)
508
having an incomplete upper diagrid module. Diagonal angle is the most effective parameter and
509
an optimal diagonal angle can significantly improve the ductility and seismic performance of the
510
diagrid structure. An R factor in the range of 4 to 5 is recommended for steel diagrid frames in the
511
range of 8 to 30 stories unless supplementary analyses are conducted to find the optimal diagonal
512
angle. For low-rise steel diagrids (under 8 stories) an R factor in the range of 3.5 to 4 is
513
recommended. Further, an overstrength and ductility of 2.5 and 2 is recommended based on the
514
mean values for all archetypes excluding the optimal ones. The deflection amplification factor
515
needs to be calculated based on the R factor and in most cases, it can be taken as equal to the R
516
factor. This paper lays the groundwork for including steel diagrids design provisions in ASCE and
517
AISC standards. The seismic overstrength and response modification factors of steel diagrids are
518
found to be larger than the concentrically braced frames, particularly when an optimal diagonal
519
angle is used, thus making them a superior alternative structural system.
520
FUTURE RESEARCH
521
The fundamental period of diagrids is notably smaller than conventional tubular systems (Kim
522
and Lee 2012), Moment-Resisting Frames (MRFs), and Concentrically Braced Frames (CBFs).
523
The current ASCE7-10 (2010) equation for approximate fundamental period is Ta = Ct hnx where
524
of Ct and x depend on the type of structure and h is the structure height. For diagrids, using the Ct
525
and x values of 0.0488 and 0.75 (for “other structural systems”) the equation generally yields a
526
larger period than direct modal analysis which may lead to a non-conservative design. Therefore,
527
new equations for the fundamental period similar to those developed for MRF (Adeli 1985, Young
528
and Adeli 2014), CBF (Young and Adeli 2014), and EBF (Young and Adeli 2016) should be
529
developed for diagrids to reflect the diagrid characteristics more accurately.
530
A conclusion of this research is the diagonal angle plays a key role in the structural
531
performance of diagrids. The optimal diagonal angle can change based on the aspect ratio,
532
structural configuration, effective loads, etc. As such, finding the optimal angle is a challenging
533
problem. Additional research is needed on the application of optimization techniques for the most
534
efficient design of diagrid systems. Authors advocate the use of nature-inspired computing
535
techniques (Siddique and Adeli, 2017) such as evolutionary computing (Wright and Jordanov,
536
2017; Pillon et al., 2016; Siddique and Adeli, 2013) or neural dynamics model of Adeli and Park
537
(Park and Adeli, 1995; Tashakori and Adeli, 2002) that have been used effectively for both
538
minimum weight and cost optimization of highrise and superhighrise building structures with
539
thousands of members (Adeli and Park 1998; Aldwaik and Adeli, 2014).
540
In addition to introduction of innovative structural systems for highrise building structures,
541
two key technologies at the frontiers of structural engineering research have been health
542
monitoring of structures (Shan et al., 2016; Tsogka et al., 2017), and active, semi-active, and hybrid
543
vibration control of structures under dynamic earthquake and wind loading where significant
544
advances have been made in recent years (Kim and Adeli, 2005a&b; Karami and Akbarabadi,
545
2016). Application of these technologies can lead to development of smart/adaptive diagrid
546
systems where data collected by sensors are processed using advanced signal processing
547
techniques (Amezquita-Sanchez and Adeli, 2016) and machine learning approaches (Palomo and
548
Lopez-Rubio, 2016; Lin et al., 2017; Zhang et al., 2017) to monitor their health in real-time, and
549
actuators or tuned liquid column dampers are used to modify their behavior, reduce their
550
vibrations, and lessen the impact of extreme dynamic loading.
551 552 553 554 555 556 557 558 559 560 561
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BIOSKETCHES OF AUTHORS
706
Esmaeel Asadi is a Ph.D. Candidate in Department of Civil Engineering at Case Western Reserve
707
University and former research assistant at The Ohio State University. His research interests
708
include seismic performance and resilience assessment of innovative structural systems including
709
steel diagrids and steel shear walls. He has recently published two journal papers on these topics
710
including a review paper on diagrid structures.
711
Hojjat Adeli received his Ph.D. from Stanford University in 1976 at the age of 26. He has authored
712
over 600 research and scientific publications in various fields of computer science, engineering,
713
applied mathematics, and medicine, including 16 high-technology books, and holds a United States
714
patent in the area of design optimization. He is the recipient of 55 awards and honors. In 1998 he
715
received the Distinguished Scholar Award, from The Ohio State University’s highest research
716
award “in recognition of extraordinary accomplishment in research and scholarship”. In 2010, he
717
was profiled as an Engineering Legend in the ASCE journal of Leadership and Management in
718
Engineering. He is a corresponding member of the Spanish Royal Academy of Engineering, a
719
foreign member of Lithuanian Academy of Sciences and Polish Academy of Science, a
720
Distinguished Member of ASCE, and a Fellow of AAAS, IEEE, AIMBE, and American
721
Neurological Association.
722
723 724
TABLES Table 1. Median expected engineering demand parameters DBE
725 726
MCE
Archetype
IDRmax (%)
IDRavg (%)
IDRmax/ IDRavg
IDRmax (%)
IDRavg (%)
4-45
0.50
0.44
1.16
0.71
0.62
1.17
4-63
0.55
0.47
1.17
0.93
0.69
1.41
4-72
3.04
0.86
3.49
5.03
1.41
3.54
8-45
1.24
0.96
1.25
1.77
1.48
1.19
8-63
0.65
0.54
1.26
0.95
0.75
1.31
8-72
1.61
0.58
2.76
2.70
0.86
3.39
IDRmax/ IDRavg
727 728
Table 2. Expected collapse capacity and IDR, the corresponding lognormal dispersion, the adjusted collapse margin and the probability of collapse
729 730
ŜCT (g)
Collapse IDRmax (%)
Prob. of collapse (%) under
4-45
3.75
𝛽𝛽𝑆𝑆𝐶𝐶𝐶𝐶
0.46
1.16
𝛽𝛽𝐼𝐼𝐼𝐼𝐼𝐼
0.53
2.4
0.6
5.6
4-63
3.07
0.52
1.5
0.62
2.0
3.6
15.3
4-72
3.35
0.47
5.91
0.42
2.1
1.5
9.5
8-45
3.15
0.65
1.37
0.41
2.0
6.9
19.5
8-63
2.93
0.52
1.25
0.57
2.0
4.4
17.6
8-72
2.86
0.43
2.21
0.46
1.8
2.2
14.1
Model
ACMR DBE
MCE
731
Table 3. Seismic performance parameters evaluated using four different methods Parameter
4-45
4-63
4-72
8-45
8-63
8-72
1545
1563
1572
3045
3063
3072
Vy
20.6
26.8
26.3
61.6
71.5
66.0
87.3
92.0
125
495
484
520
Ro= Vy /Vd
1.60
2.13
2.10
2.34
2.68
2.10
2.06
2.14
2.18
2.29
3.49
2.21
2.29 µ= δu/ δy Cd=µ Vy /Vd 3.66 Newmark-Hall:
2.69 5.73
2.33 4.89
2.29 5.36
3.93 10.5
2.42 5.08
2.31 4.77
2.09 4.47
4.41 9.60
1.70 3.88
1.99 6.96
3.39 7.50
1.89
2.09
1.91
1.89
2.62
1.96
2.31
2.09
2.80
1.70
1.99
3.39
R 3.02 Miranda-Bertero:
4.46
4.01
4.42
7.01
4.12
4.77
4.47
6.09
3.88
6.96
7.50
Rµ
Rµ
1.83
2.16
2.01
2.01
3.08
2.10
2.37
2.11
3.84
1.91
2.28
4.01
R
2.93
4.60
4.23
4.70
8.24
4.41
4.89
4.53
8.36
4.38
7.96
8.87
1.74
2.15
2.22
2.28
3.30
2.36
2.75
2.46
4.95
1.96
2.34
4.09
R 2.78 FEMA (2009):
4.58
4.67
5.33
8.84
4.96
5.67
5.28
10.7
4.48
8.17
9.04
Ro= Vmax /Vd Rµ R& Cd=R/BI
2.30 1.19
2.87 1.54
2.74 1.34
2.92 1.81
3.54 2.54
2.63 2.11
2.58 2.26
2.56 1.87
2.70 3.38
2.77 1.78
4.27 1.03
2.76 2.72
2.75
4.41
3.68
5.28
9.00
5.56
5.81
4.78
9.13
4.94
4.42
7.50
Vidic et al.: Rµ
732 733
734
Table 4. Statistical measures for seismic performance parameters Parameter
Mean
SD
Max
Min
Ro= Vy /Vd
2.28
0.43
3.49
1.60
µ= δu/ δy
2.65
0.79
4.41
1.70
Cd=µ Vy /Vd
6.03
2.10
10.51
3.66
Rµ
2.22
0.47
3.39
1.70
R Miranda-Bertero:
5.06
1.39
7.50
3.02
Rµ
2.48
0.72
4.01
1.83
5.67
1.96
8.87
2.93
2.72
0.90
4.95
1.74
R FEMA (2009):
6.21
2.29
10.76
2.78
Ro= Vmax /Vd
2.89
0.50
4.27
2.30
Rµ
1.96
0.65
3.38
1.03
R and Cd=R/BI
5.60
1.90
9.13
2.75
Newmark-Hall:
R Vidic et al: Rµ
735 736
737
738 739 740
FIGURES
Fig. 1. Main components of a diagrid frame and its basic triangular element
741 742 743
Fig. 2. Main components of FEMA (2009) method for SPF evaluation
744 745 746 747
Fig. 3. Idealized force-displacement curve based on pushover analysis adapted from FEMA (2000; 2009)
748 749 750 751
Fig. 4. Variation of ductility factor (Rµ) versus period of the structure for three different ductility ratios of 2, 4, and 6 using three approaches NH, MB and VFF.
752 753 754
Fig. 5. Typical floor plan for (a) 4-, 8-, and 15-story (b) 30-story diagrid archetypes
755 756
Fig. 6. Diagrid patterns used in this research
757 758 759
(b) (a) Fig. 7. General member force-deformation relationship and modeling parameters adapted from ASCE/SEI 41-13 (2014) (a) flexural elements (b) diagonals
(a)
760 761
(b)
(c) (d) Fig. 8. Pushover curves for (a) 4- (b) 8- (c) 15- (d) 30-story archetypes
762 763 764
Fig. 9. Elevation of 4-72 model with uppermost incomplete module
765 766 767 768
Fig. 10. Response spectrum of scaled ground motion records and the design response spectrum for the 8-45 archetype
DBE (10%/50-yr)
MCE (2%/50-yr)
(a) IDRmax
(b) IDRavg
769 770 771
(c) IDRmax/IDRavg Fig. 11. Engineering demand parameters for 4-story archetypes (a) IDRmax (b) IDRavg (c) IDRmax to IDRavg ratio under DBE and MCE using NTHA
DBE (10%/50-yr)
MCE (2%/50-yr)
(a) IDRmax
(b) IDRavg
772 773 774
(c) IDRmax/IDRavg Fig. 12. Engineering demand parameters for 8-story archetypes (a) IDRmax (b) IDRavg (c) IDRmax to IDRavg ratio under DBE and MCE using NTHA
775 776 777
Fig. 13. Incremental dynamic analysis curves for 4-45 diagrid archetype
778 779 780
(b) (a) Fig. 14. Empirical CDF of Sa (T1,5%) and fitted lognormal fragility functions for (a) 4story and (b) 8-story archetypes
781 782 783 784
Fig. 15. Seismic ductility ratio (µ), overstrength factor (Ro), and deflection amplification factor (Cd)
785 786 787
Fig. 16. Seismic ductility factor (Rµ) using four different approaches
788 789 790 791
Fig. 17. Mean seismic performance factors, Ro, Rµ, and R, obtained using four different approaches (SD is shown in the middle of each bar)
792 793 794 795 796
Fig. 18. Seismic modification factor (R) using four different approaches
