Performance-based plastic design of steel plate shear walls

Swapnil B. Kharmale1
Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India
Corresponding author. Tel.: + 91 22 2576 7309; fax: + 91 22 2576 7302.
1Former doctoral research scholar.
Abstract
The existing codes and design guidelines for steel plate shear walls (SPSWs) fail to utilise the excellent ductility capacity of SPSW systems to its fullest extent, because these methods do not consider the inelastic
displacement demand or ductility demand as their design objective. A performance-based plastic design method for SPSW systems with rigid beam-to-column connections is proposed in this work, which sets a specific ductility
demand and a preferred yield mechanism as its performance targets. The effectiveness of the proposed method in achieving these targets is illustrated through sample case studies of four- and eight-storey SPSW systems for
varied design scenarios. A comparison with the existing AISC method for the same design scenario shows that the proposed method consistently performs better, in achieving these performance-based targets. The proposed
method is modified to account for P-Delta effects, wherever necessary. This modified method is found to be more effective than the original proposal, whenever P-Delta effects are significant.
Keywords: Steel plate shear wall; Performance-based design; Inelastic deformation capacity; Ductility-based design; P-Delta effects
1 IntroductionExisting design provisions for steel plate shear walls
During the 1980s and 1990s, a significant amount of research works, both ‘analytical’ and ‘experimental’, was conducted on the post-buckling behaviour of thin unstiffened steel plate shear wall (SPSW) systems [1]. These research
works, conducted primarily in Canadian and U.S. universities (for example, [2–6]), resulted in the incorporation of design specifications for SPSW systems in design standards/codes. In 1994, the Canadian steel design standard [7] included
design provisions for unstiffened thin SPSW, although only as an appendix to the main design code. The 2001 Canadian standard [8] incorporated mandatory clauses on the design of steel plate shear walls. This standard had provisions for
both ‘limited ductility’ and ‘ductile’ steel plate shear walls. For the limited ductility SPSW, no special requirements were made for the beam-to-column connections and a response modification factor (R) of 2.0 was assigned for these systems.
For the ductile SPSW, however, the beam-to-column connections had to be moment resisting and the response modification factor was higher (R = 5.0). In order to ensure a ductile failure mode for SPSW structures, this code recommended
an indirect capacity design approach. In this approach, a factor B (ratio of the probable shear resistance at the base of the wall for a given plate thickness to the factored lateral force at the base of the wall, obtained from the calculated
seismic load) was used to magnify the moments and axial forces in columns obtained from an elastic analysis. This magnification was not required if column forces and moments were obtained from a nonlinear pushover analysis. Further
research on SPSW systems in the last decade, particularly the plastic analysis and design methods for SPSW [9], resulted in newer design provisions, for example, as in the AISC Seismic Provisions [10,11] and the Canadian standard
CAN/CSA S16 [8,12].
The AISC SPSW specifications followed the load and resistance factor design (LRFD) format based on the limit state of collapse. The concept of capacity design was incorporated in this standard. For example, all edge/boundary
elements (‘horizontal boundary elements/HBE’ and ‘vertical boundary elements/VBE’) were designed to resist the maximum forces that could be generated by fully yielded steel ‘infill panels’. These provisions also indicated to a preferred
mechanism of failure through specifications, such as that the boundary elements were required to be proportioned in order to meet the ‘strong-column-weak-beam’ criterion, and that in boundary elements plastic hinging was permitted only at
HBE ends. The recently published AISC Design Guide 20 for SPSW [13] developed the 2005 AISC Seismic Provisions into a complete design methodology. It included step-by-step design procedures as well as design examples for two
types of steel plate shear walls: high-ductility SPSW (with R = 7.0) for high-seismic regions and low-ductility SPSW (with R = 3.0) for low seismic regions. This design guide was developed in accordance with the then existing relevant
standards ASCE7-05 for minimum design loads in buildings [14], ANSI/AISC 360-05 for structural steel [15], and 2005 AISC Seismic Provisions [10].
standards ASCE7-05 for minimum design loads in buildings [14], ANSI/AISC 360-05 for structural steel [15], and 2005 AISC Seismic Provisions [10].
Although elements of capacity design concepts were incorporated in the latest Canadian and U.S. steel design standards, there are a few limitations when assessed from a performance-based seismic design (PBSD) perspective
1. Significant inelastic deformation capacity (ductility) of SPSW systems cannot be fully utilised by these codes, as the design is primarily based on an elastic force/strength-based approach where the inelastic behaviour is implicitly accounted for through a response modification factor, R.
2. These guides specify a desirable yield mechanism, however they do not provide specific design equations to attain this yield mechanism [16], especially for the VBE and HBE in the SPSW system.
3. These standards do not provide the designer any option to choose a specific yielding hierarchy or failure mechanism for the SPSW structure.
In more recent times, Berman and Bruneau [17] proposed a reasonably accurate and relatively effective capacity design method for SPSW columns (VBE). Their procedure combined a linear elastic model of SPSW and plastic
analysis concepts. Research works by [18–21] provided capacity design provisions for boundary beams (HBE) in SPSW systems. These design equations, especially those for ‘anchor beams’ (beams at roof and ground level with infill panels
only at one side) were derived considering local collapse mechanism (‘beam mechanism’) with plastic hinges forming at the ends of the HBE and close to the mid-span of the HBE. Vian et al. [19] also recommended the use of ‘reduced beam
section/RBS’ at the ends of the HBE to ensure the preferred failure mechanism of the AISC Seismic Provisions.
Over the last decade, the performance-based seismic design philosophy has emerged as a promising and efficient seismic design approach. PBSD explicitly accounts for the inelastic behaviour of a structural system in the design
process itself. PBSD approaches based on plastic analysis and design concepts called as performanceperformanc-based plastic design (PBPD) methods were recently developed for different lateral load resisting systems (such as steel
moment resisting frames, steel braced frames, etc.) in the University of Michigan [22,23]. In these design methods a pre-selected yield/failure mechanism and a uniform target drift (based on inelastic behaviour) were considered as
performance objectives. The analytical validation of these methods showed that structures designed using these methods were very effective in achieving the pre-selected performance objectives. Details of these methods and step-by-step
procedures were later compiled in a book by Goel and Chao [24]. Considering a gradual shift towards PBSD for seismic design methods in general, Ghosh et al. [25] proposed a displacement/ductility-based design methodology for steel plate
shear wall systems with pin-connected boundary beams. Similar to the methods developed in the University of Michigan, they also considered the target displacement ductility ratio and a pre-selected yield mechanism as the design criterion;
and an inelastic energy balance concept was used in the formulation of the design method. Ghosh et al. validated this method by designing a four-storey SPSW with pin-connected beams subjected to various ground motion scenarios and for
different target ductility ratios. Gupta et al. [26] successfully applied the inelastic displacement ductility-based method proposed by Ghosh et al. using standard hot rolled-sections (for boundary elements) available in the U.S. [15] and in India
[27]. More recently, while investigating for a suitable (height-wise) distribution of the design base shear for this method, [28] applied this method effectively to SPSW with pin-connected beams of various heights.
2 Objective
Considering that the existing U.S. and Canadian design standards/guides for ‘ductile’ SPSW recommend the use of only rigid beam-to-column connections, an inelastic displacement-based seismic design method similar to that
proposed by Ghosh et al. [25] needs to be formulated for SPSW systems with rigid beam-to-column connections. The primary objective of the work presented here, thus, is to develop a PBPD method for SPSW systems with rigid beam-to-
column connections, with the following performance goals:
1. achieving a target displacement ductility ratio demand considering the inelastic behaviour of the SPSW system, and
2. achieving a pre-selected yield/failure mechanism for this inelastic behaviour.
It must be mentioned here that in order to develop a full-fledged PBSD framework for any structural system, the first important task is to define acceptable performanceperfromance levels in a specific quantitative manner in terms of
structural, non-structural and component behaviours. The focus of this work, however, is on the structural design calculations once a performance level is selected and limits are defined in terms of displacement-based quantities. Before we
begin with the proposal of a PBPD method for SPSW, the existing design method (based on AISC Design Guide 20) is reviewed through a sample design case and it is checked if this sample design meets the stated performance objectives
(Section 3). Section 4 provides the fundamentals and the framework of the proposed PBPD method for SPSW with rigid beam-to-column connections. This method is validated in Section 5 through sample designs of low-rise (four-storey) and
medium-rise (eight-storey) SPSW buildings, for different target ductility ratios, and subjected to various earthquake scenarios. Results of this validation are discussed in detail, along with a comparison with the sample design based on
existing AISC guidelines. A modification of the proposed PBPD method – to account for P-Delta effects (which are predominant for medium- and high-rise SPSW systems with large displacement ductility demands) – is provided in Section 6.
Section 7 presents the significant conclusions of this work and also discusses the limitations thereof. It should however, be noted that the work presented here does not address the issue of formulating the design method in a probabilistic
framework, which is the most significant feature for a PBSD methodology, other than the explicit consideration of inelastic behaviour and damage in a structure. We are currently engaged in developing a reliability-based framework for the
performance-based plastic design method, which will be reported in future.
3 Design of a SPSW system following AISC Design Guide 20, and its performance assessment
Kharmale and Ghosh
3 Design of a SPSW system following AISC Design Guide 20, and its performance assessment
To assess the seismic performance of a steel plate shear wall structure, which is designed following provisions of AISC Design Guide 20 [13], a four-storey steel plate shear wall building is considered. The configuration of this four-
storey building is illustrated in detail in Fig. 1. The building has a five bay by six bay plan, with one SPSW bay along each outer frame. All beams, except those in the SPSW bays, are pin-connected (shear-connected) to the frame, and
therefore only the SPSW frames form the lateral load resisting system. The building is assumed to have seismic weights of 4690 kN per floor, except for the roof, where it is 5090 kN. For seismic force calculations, this study building is
assumed to be located in downtown San Francisco, CA, USA. The building site is categorised as Site Class D for ‘stiff soil’ and the its occupancy category is adopted as ‘I’, based on its use as an office building. The site location and the soil
characteristics considered here are the same as those that of the design example II (‘high-seismic’ design) provided in AISC Design Guide 20 [13].
As mentioned earlier, the seismic force calculations in AISC Design Guide 20 are based on ASCE7 [14], where the inelastic behaviour of a structure is only implicitly accounted for through a response modification/reduction factor, R.
For the design of SPSW systems subjected to a ‘high seismic’ scenario, ASCE7 specifies a response modification factor, R = 7.0 and a system overstrength factor, Ω0 = 2.0. These values were later supported by a detailed numerical study
on 44 SPSW designs [29]. Assuming that the structure under consideration does not have any supplemental damping device and also adopting a minimum reliability factor, one can conclude that SPSW systems are designed for a target
displacement ductility ratio, µ = 3.5. It should be noted here that ASCE7 and AISC Design Guide 20 do not directly include the displacement ductility ratio of a system in the calculation of the design base shear. The MCE spectral
accelerations for this location are, SS = 1.70 g and S1 = 0.850 g. Following ASCE7, the design spectral acceleration parameters are calculated: SDS = 1.14 g and SD1 = 0.850 g. The design base shear is calculated as 3120 kN following ASCE7
guidelines. The different components of the SPSW system are designed as per the AISC Design Guide 20 and the AISC Seismic Provisions. Here onwards, this design is referred to as the ‘AISC method’ design. The final dimensions of
various components and other features of this design are
• Infill panel thickness (from the first storey upwards) = 5.50, 5.00, 4.00, and 2.30 mm.
• HBE (same section for all floors) = W 27 × 94
• VBE (same section for all storeys) = W 14X × 398
• Fundamental time period (T1), estimated as per ASCE7 = 0.585 s
• Spectral acceleration for T1 (Sa) = 1.14 g.
As per the design specifications, the displacement ductility (ratio) demand for this structure subjected to a design level earthquake should ideally be 3.50, which will exploit the capacity to the fullest without overshooting it. The
seismic performance assessment for this structure is based on this perspective. This ductility demand (µd) is evaluated using nonlinear response history analyses (NLRHA) subjected to real earthquake records. Three strong motion records
(Table 1), scaled to the design Sa of this typical design, are used in these NLRHA. The ductility ratio demand is calculated as
Fig. 1 Configuration of the hypothetical study building(s).
(1)
where, Dm is the maximum roof displacement obtained from an NLRHA and Dy is the yield roof displacement. Dy for a SPSW structure is obtained from the conventional nonlinear static pushover analysis (NSPA) using the lateral load
distribution recommended in the IBC [30]. The base shear (Vb) vs. roof displacement (D) pushover plot is bilinearized using an elastic-perfectly plastic force-deformation behaviour so that the areas under the pushover curve and its bilinear
approximation are equal (Fig. 2). FromForm the NSPA, the yield displacement (Dy) is obtained as 0.123 m, with a yield base shear of 3892 kN. The three earthquakes result in µd values of 2.36, 2.18 and 1.76 (giving an average of 2.10). These
values show that the existing design method do not (always) utilise the excellent displacement capacity of SPSW systems. The three different earthquakes result in very different yielding patterns. The lack of an effective energy dissipation
through inelastic activity is also evident from the fact that beams do not form plastic hinges at both ends. Besides, the plasticity is observed to be concentrated more in the second and third storeys, resulting in higher interstorey drift (ratio)
demands in these two than the other storeys. Fig. 3 provides the displacement profile for this design at the instant of peak roof displacement for each of the three records, along with an ‘ideal’ response based on the ductility factor.
Table 1 Details of earthquake records used for designs of four-storey and eight-storey SPSWs.
Earthquake Date Station Component PGA (g) Name
Northridge Jan 17,1994 Sylmar converter Horizontal -052 0.612 SYL
Northridge Jan 17, 1994 Newhall fire station Horizontal - 360 0.589 NH
Kobe Jan 16, 1995 KJMA Horizontal - 000 0.821 KJM
Landers Jun 28, 1992 SCE station 24 Horizontal -000 0.785 LAN
Imperial Valley Oct 15, 1979 USGS station 5054 Horizontal -140 0.775 IMV
Cape Mendocino Apr 25,1992 CDMG station 89005 Horizontal- 000 1.497 CM
(1)
Fig. 2 Obtaining the yield base shear (Vy) and yield displacement (Dy) from the bilinearized pushover plot.
4 Proposed performance-based design methodology
The proposed performance-based design method broadly follows the performance-based plastic design methodology recommended by [24] for various other lateral load resisting systems in steel. As mentioned earlier, the proposed
design method considers a uniform interstorey drift ratio and a pre-selected yield mechanism as performance targets. Fig. 4 shows a typical one-bay SPSW configuration with rigid beam-to-column connections, along with the selected
unidirectional and uniform yield mechanism. This preferred yield mechanism consists of the yielding of all steel infill panels, formation of plastic hinges at the column bases, and formation of plastic hinges at the two ends of each beam
(HBE). Capacity design approaches proposed earlier for SPSW systems also specify this as the desired yield mechanism.
Fig. 3 Displacement profiles for the ‘AISC method’ design SPSW system, at the instant of maximum roof displacement.
The proposed design method adopts the concept of energy balance, in which the inelastic energy demand on a structural system is equated with the inelastic work done, internally, through the plastic deformations. The total strain
energy demand to an inelastic single-degree-of-freedom (SDOF) system is estimated as
where, Ee = elastic strain energy demand, Ep = plastic strain energy demand, γ = energy modification factor, M = total seismic mass of the structure, Sv = spectral velocity corresponding to T1, and Ce = elastic force coefficient. Based
on the study by Akiyama [31], the elastic vibrational energy can be written, assuming that the entire structure is reduced into a SDOF system:
W is the total seismic weight of the structure and Vb is the base shear. The energy modification factor (γ) comes from the equal displacement law, and is calculated based on the target displacement ductility ratio of the system (µt)
and the ductility-based reduction factor (Rµ):
Any suitable ‘R–µ–T’ relation can be used to estimate Rµ. Here, we use the relationships suggested originally by Newmark and Hall [32]. Based on recent findings [33], the effects of material strain-hardening are neglected in this work,
although the proposed method can be easily modified to incorporate this aspect. The elastic force coefficient is expressed in terms of the design pseudo- acceleration (Sa) or the design yield base shear (Vby), as
The structure is idealised as an elastic-perfectly plastic (EPP) equivalent single degree system by selecting the preferred yield mechanism up to the peak monotonic drift demand (Fig. 4). The elastic part of the total strain energy
demand is calculated by replacing Vb with the yield base shear (Vby) in Eq. (3). Substituting this in Eq. (2), we get
This plastic energy demand is the same as the work done by the equivalent lateral forces Fi (Fig. 4) going through the inelastic drift (θp):
Fig. 4 (a) Schematic of a SPSW system with rigid beam-to-column connections; and (b) the selected uniform and unidirectional yield mechanism.
(2)
(3)
(4)
(5)
(6)
(7)
where, hi is the height of ith floor measured from the ground, and Cvi is the lateral force distribution factor for this floor. Recent research by Kharmale and Ghosh [28] recommended that any commonly followed lateral force distribution
(that is, Cvi) can be adopted in the inelastic displacement-based design method for SPSW systems. In this work, the distribution recommended by ASCE7 is adopted. Eq. (7) can be rearranged in the form of a quadratic equation and its
solution provides an expression for the required yield base shear for this performance-based design:
Once Vby is obtained, a preliminary design of the steel infill panels is considered first. Assuming that each infill panel takes the full storey shear (Vi), the initial thickness of the panel at the ith storey (t′) is obtained, similarly as by
Ghosh et al. [25]:
where, Fy is the material yield stress and L is the infill panel width. The selection of the column section is primarily based on the minimum (strength and) stiffness requirement prescribed in the Canadian…