Fundamental Period Prediction of Steel Plate Shear Wall Structure

Benjamin Kean1 and Cheng Chen2

1Graduate student, School of Engineering, San Francisco State University, USA, keanben@mail.sfsu.edu 2Associate Professor of civil engineering, School of Engineering, San Francisco State University, USA, chcsfsu@sfsu.edu

ABSTRACT: The Steel Plate Shear Wall (SPSW) is a structural system used today in design as the primary Lateral Force Resisting System (LFRS) of a building. In the initial design stage the fundamental period of the structure is used to calculate the seismic forces using the Equivalent Lateral Force Procedure (ELFP). The American Society of Civil Engineers (ASCE) allows for the fundamental period to be approximated using a general formula using two coefficients whose values are dependent on the type of LFRS. It has been shown that the current procedure produces overly conservative estimations of the fundamental period for SPSWs. This study evaluates the fundamental period of a large population of code-designed SPSW prototypes as well as SPSWs that are designed with consideration of plate-frame interaction. A formula for better fundamental period approximation coefficients is proposed and verified.

INTRODUCTION

The Steel Plate Shear Wall (SPSW) is a structural system which can be used as the primary lateral force resisting system (LFRS) of a building (Sabelli and Bruneau 2007). Developed in the 1970s, the SPSW has seen an increase in use over the past few decades due to increased efficiency in the design process from continuing research efforts (Seilie and Hooper 2005). Typically, a SPSW consists of a vertical steel infill plate connected to surrounding beams and columns (Sabelli and Bruneau 2007). The beams and columns are often referred to as horizontal boundary elements (HBEs) and vertical boundary elements (VBEs), respectively, as shown in Fig. 1. SPSWs have demonstrated high initial stiffness, ductile behavior, and good energy

dissipation under cyclical loading thus making them a viable alternative to moment resisting frames and braced frames (Sabelli and Bruneau 2007).

Fig. 1. Configuration of typical SPSW (Sabelli and Bruneau 2007)

The SPSWs designed for this study were based on Bermans “9M” SPSW structure (Berman 2010) which was modeled after a modified prototype of the Los Angeles structure used in the SAC building research project (Gupta and Krawinkler 1999). The structures are assumed to be constructed on Site D soil. Provisions and specifications for the design of a SPSW are taken from AISC Design Guide 20 (2007) and AISC 341-10 (2010). Loading used in the design was determined using ASCE 7-10 (2010). The design of a SPSW is determined utilizing the post-buckling strength of the web-plate developed by the tension-field action when subjected to lateral loading. Web-plates are assumed to be connected to rigid beams and columns making a moment resisting frame (Sabelli and Bruneau 2007). The story shear is assumed to be resisted completely by the web-plate for prototypes designed for 100 percent base-shear participation. The amount of story shear resisted can also be reduced to account for the resistive capacity of the horizontal and vertical boundary elements (Enright 2014; Barghi 2015), which is often referred to as plate-frame interaction.

The HBEs and VBEs of a SPSW are designed based on a capacity design approach to ensure that the post-buckling limit state in the web-plate is achieved. More specifically, the HBEs are designed to resist the demands resulting from yielding of the tension field in the web-plates, and the VBEs are designed to resist the tension field yielding as well the flexural yielding in the HBEs (Sabelli and Bruneau 2007; AISC 2010). The SPSW prototypes in this study were optimized in the design process by matching the shear capacity of the web-plate to the shear demand and measured the demand of the system to the capacity of the member. A total of 54 SPSWs are designed for different bay widths (10, 15, and 20 feet denoted as N, M, and W respectively), different stories (3, 5, 6, 8, 9, and 10), and different base shear

ratio (65%, 85%, and 100%). More details for the design of these SPSWs can be found in Kean (2016).

FINITE ELEMENT MODELING OF SPSW PROTOTYPES

For this study SPSW prototypes were modeled and analyzed using the software platform Open System for Earthquake Engineering Simulation (OpenSees) (McKenna et al. 2000). OpenSees is open source software produced by the Pacific Earthquake Engineering Research (PEER) center for modeling and analyzing the nonlinear response of structural systems. The finite element models used in this study for SPSW prototypes were based on models developed in studies at San Francisco State University (Favreto 2013; Enright 2014; Barghi 2015). Modeling of the web-plate was achieved using the strip method outlined in AISC Design Guide 20 (Sabelli and Bruneau 2007), where the web-plate is modeled through a series of pined tension only truss members placed diagonally along the HBE. These truss members are defined to have a significant tension capacity but negligible compression strength. To achieve the effects of the distributed loading that the web-plate acts onto the HBE member AISC recommends at least ten strips be used in each direction when modeling the web-plate. VBEs and HBEs were modeled so that plasticity was distributed along the length of the member so that the tension effects of the truss pulling at different locations on the members would be accurately modeled.

EXISTING METHODS FOR FUNDAMENTAL PERIOD APPROXIMATION

The overly conservative approximation of an SPSW’s fundamental period has been studied by researchers (Berman 2010; Enright 2014). Along with the current prescribed method from ASCE 7-10, two methods that have been developed by previous researchers will considered in this study.

Approximation Method in ASCE 7-10

Current code provisions allow the fundamental period of a structure to be approximated using the expression

𝑇! = 𝐶!𝐻! (1) where in Eq. (1), Ta is the fundamental period of the structure in seconds; H is the total height of the structure in feet; and the parameters Ct and x are empirical coefficients to be determined from Table 12.8-2 in ASCE 7-10 (2010). Values for Ct and x are only defined for steel and concrete moment frames, and both steel buckling-restrained and eccentrically braced frames. Other LFRS systems including SPSW must use a generalized value when approximating the fundamental period. The approximation method prescribed by ASCE 7-10 has been shown to lead top overly conservative results when compared with that from finite element analysis. Shown in

Fig. 2 is a comparison of fundamental periods from ASCE 7-10 and those from finite element analysis for the SPSW prototypes used in this study.

Fig 2. Comparison of SPSW fundamental period from finite element and ASCE

7-10 methods.

As observed in Fig. 2, the ASCE 7-10 method is overly conservative in the approximation of the fundamental period for SPSWs when compared to results from finite element analysis. As the height of the structure is increased ASCE 7-10 cannot accurately approximate the period of the structure which will lead to higher base shear approximations which ultimately leads to larger requirements for strength and size of structural members.

Approximation Method by Liu et al. (2013)

Liu et al. (2013) proposed a simplified method to approximate the fundamental period of a SPSW system. The method can be expressed as follows

1𝜔!!

≈1𝜔!"!

+1𝜔!"!

(2)

where in Eq. (2), ωi, ωsi, and ωfi, represent the combined shear-flexure, shear, and flexure frequencies of the ith mode of vibration, respectively. The fundamental period of the SPSW is then calculated as

𝑇! =2𝜋𝜔!

(3)

In Eq. (3), Ti is the natural period of a SPSW structure for the ith mode of vibration. The shear frequency of the system is found through an eigenvalue analysis using a

lumped mass assumption and the summation of lateral stiffness from the VBEs and web-plate. The flexural frequency of the system is determined assuming the structure acts as a vertical cantilever which has solutions available for analysis.

Approximation Method by Topkaya and Kurban (2008)

Topkaya and Kurban (2008) proposed a simplified method for estimating the fundamental period of a SPSW structure, where the SPSW is assumed to behave as a vertical cantilever similar to Liu et al. (2013). In this method, the fundamental period of a SPSW is calculated as

𝑇! ≈1𝑓!!+1𝑓!!

(4a)

where, Tw is the natural period of the SPSW structure; fb and fs are the natural frequencies of two cantilever beams, subject to bending and shear, respectively.

The natural frequency of the beam subject to bending is calculated as

𝑓! = 𝑟!0.5595𝐻!

𝐸𝐼!𝑚 (4b)

where H is the height of the structure; m is the mass of the SPSW structure; and Iw is the moment of inertia of the web-plate. The parameter rf is a factor proposed by Zalka (2000) to account for lumped masses at story levels.

The natural frequency of the SPSW deforming in shear is calculated as

𝑓! = 𝑟!14𝐻

𝐾𝐺𝐴!𝑚 (4c)

where G is the shear modulus of the steel plate; and KAw is the effective shear area of the web plate defined as

𝐾𝐴! =𝐼!!

𝛽 (4d)

where the parameter β is defined as

𝛽 =𝑄!

𝑏!!!𝑑𝐴 (4e)

where Q is the static moment of area; and b is the width of the web-plate. Determining the exact value of β could be quite time consuming due to the possibility of integration of fourth-order polynomials (Topkaya and Kurban 2008). An approximation proposed by Atasoy (2008) assuming the ratio Q/b varies linearly is used in this study to simplify the calculation process. More details can be found in Topkaya and Kurban (2008) regarding this approximation. This method assumes that the SPSW will be designed with uniform boundary element sizes and web-plate thicknesses along the height of the structure. This is often not consistent with

common practice as the designer will take advantage of the reduction in strength requirements along the height of the structure. For this reason, an upper bound limit using the most ductile elements and a lower bound limit using the most rigid elements is computed so it can be said that the fundamental period of the SPSW will fall within a known range.

Evaluation of SPSW Fundamental Period Approximation Methods

Both methods proposed by Liu et al. (2013) and Topkaya and Kurban (2008) were applied to the fifty-four SPSW prototypes used in this study for a comparison. The results from the 10-foot (N) bay width SPSW prototypes are presented in Fig. 3.

Fig. 3. Comparison of fundamental period using approximation methods by Liu

et al. and Topkaya and Kurban

As can be observed in Fig. 3 the method proposed by Liu et al. (2013) is able to approximate the fundamental period of the SPSW prototypes that closely match the results from finite element analysis. While this method produces attractive results, it is not practical for application in the preliminary design stage since it requires the designer to have sized the boundary elements and web-plate prior to applying the method. In addition, the method proposed by Liu et al. (2013) requires an eigenvalue analysis to determine the shear frequency of the SPSW. This requirement involves complicated linear algebra analysis as the total number of stories increases in the structure.

It can also be observed in Fig. 3 that the method proposed by Topkaya and Kurban (2008) is able to approximate the fundamental period of a SPSW prototype within the upper and lower bound limits. Similar to Liu et al. (2013) this method is not practical for application in the preliminary design stage. The assumption that the SPSW is designed with uniform section properties limits the methods ability to

approximate the fundamental period of the entire structure. For these reasons both methods discussed in this section are not practical for use in the preliminary design stage. A simple expression which is not dependent on the section properties of the SPSW would be beneficial to the designer in the preliminary stage as fundamental periods would be quickly approximated to values close to the results from finite element analysis. PROPOSED METHOD FOR SPSW FUNDMANETAL PERIOD

This study develops a simplified expression for approximating the fundamental period of SPSW structures. The method proposed in this section is developed so that the height of the structure, bay-width, and base-shear participation level are the only parameters needed which are typically known to the engineer at the preliminary design stage. The proposed method will adjust the current generalized value of the empirical coefficients so that it will more accurately predict the fundamental periods obtained from finite element analysis.

The method is developed considering first only the SPSW prototypes designed for 100 percent base-shear participation with a varying bay-width. Additional adjustment expressions are then added to account for possible variation in base-shear participation levels.

Bay-Width Adjustment

Bay-width adjustment expressions for the empirical fundamental period coefficients in Eq. (1) can be expressed as

𝐶!! = 𝐶!𝑓!(𝐿) (5a)

𝑥! = 𝑥𝑓!(𝐿) (5b)

where in Eq. (5a) 𝐶!! is the corrected fundamental period coefficient; Ct is the code prescribed value for “All Other Structures”; and fc(L) is the bay-width adjustment function for 𝐶!!. Similarly, in Eq. (5b) 𝑥!is the corrected fundamental period coefficient; xt is the code prescribed value for “All Other Structures”; and fx(L) is the adjustment function of 𝑥!. It is assumed that bay-width adjustment functions will take on a power series regression similar to Eq. (1). The forms that the adjustment function will take on are expressed as

𝑓! 𝐿 = 𝛼!𝐿!! (5c)

𝑓! 𝐿 = 𝛼!𝐿!! (5d)

in Eqs. (5c) and (5d) the values of α and β are determined from a power series regression analysis. Eqs. (5a) through (5d) can be used to approximate the fundamental period of an SPSW which has been designed for 100 percent base-shear

participation. The following section introduces adjustment expressions which consider a reduction in the base-shear participation of an SPSW.

Base-Shear Participation Level Adjustment

To adjust the fundamental period coefficient values based on the base-shear participation level parameter adjustment functions were derived for the parameters α and β in Eqs. (5c) and (5d) and are expressed as

𝛼! = 𝛼!!""𝑓!!(𝑉) (6a) 𝛽! = 𝛽!!""𝑓!!(𝑉) (6b) 𝛼! = 𝛼!!""𝑓!!(𝑉) (6c) 𝛽! = 𝛽!!""𝑓!!(𝑉) (6d)

A linear model is used to describe the behavior of the α and β coefficients over the base-shear participation levels. The adjustment functions for α and β are expressed as

𝑓!!(𝑉) = 𝜃!!𝑉 + 𝛾!! (7a) 𝑓!!(𝑉) = 𝜃!!𝑉 + 𝛾!! (7b) 𝑓!!(𝑉) = 𝜃!!𝑉 + 𝛾!! (7c) 𝑓!!(𝑉) = 𝜃!!𝑉 + 𝛾!! (7d)

where in Eqs. (7a) through (7d) the parameter V is the base-shear participation level in decimal form; the parameters θα, θβ, γα, and γβ are coefficients determined from a linear regression analysis.

Table 1. Parameter values for proposed method for fundamental period approximation

Parameter Value1 Parameter Value1

Ct 0.02(0.0488) βC100 0.5239(0.2979)

x 0.75(0.75) αx100 2.1368(1.6652)

αC100 0.1897(0.6016) βx100 -0.2099(-0.2099) Parameter Value1 Parameter Value1

θαc -1.856(-0.9681) θαx 0.2894(0.1442)

γαc 3.187(2.0747) γαx 0.6777(0.8355)

θβc 0.7671(0.9767) θβx 0.6331(0.6331)

γβc 0.0937(-0.1457) γβx 0.3095(0.3095) 1S.I. equivalents given in parenthesis

Tabulated values for the parameters shown in Eqs. (5) through (7) are given in Table 1 for both U.S. customary units and Standard International (S.I) units. This

method was applied to the fifty-four SPSW prototypes used in this study and the results are shown in Fig. 4.

Fig 4. Comparison of proposed method of predicting SPSW fundamental period to finite element analysis

As can be observed in Fig. 4 the fundamental periods from the proposed method agree well with those from the finite element analysis. This implies that the proposed method provides an accurate and practical method for fundamental period approximation of the SPSW prototypes.

Table 2. Comparison of fundamental periods for three validation cases

SPSW Prototype Finite Element (s) Proposed Method (s) Error

3K100 0.486 0.503 3.50% 4N80 0.738 0.731 0.95% 7W60 1.003 1.017 1.40%

To further validate this proposed method, three more SPSW prototypes are designed with story heights, bay-width, and base-shear participation levels not used in the initial set of fifty-four SPSW prototypes. These include a 3-story twelve-foot bay-width SPSW designed for 100 percent base-shear participation (denoted 3K100); a 4-story ten-foot (N) bay-width SPSW designed for 80 percent base-shear participation (denoted 4N80); and a 7-story twenty-foot (W) bay-width SPSW designed for 60 percent base-shear participation (denoted 7W60). Results from the three SPSW prototypes are presented in Fig. 5 as well as tabulated in Table 2.

Fig 5. Comparison of fundamental periods from proposed method, finite element analysis, and ASCE 7-10

It can be observed in both Fig. 5 and Table 2 that the proposed method provides comparable fundamental periods of SPSWs to those from finite element analysis. Table 2 also shows that the error for the method is as small as 0.95 percent

SUMMARY AND CONCLUSIONS

This study explored the fundamental period of an SPSW structure and developed a simple and practical method for better predicting the fundamental period. The method was developed so that only the height, bay-width, and base-shear participation level in the web-plate be variables making this method applicable in the preliminary design stage when determining seismic loads. It was found that this method could accurately predict the fundamental periods when compared with finite element analysis. The proposed method is limited by the configuration of the SPSW prototypes used in this study which were assumed to span over the entire bay-width length with no discontinuities in the plate. SPSWs can take advantage of a perforated web-plate to provide openings over the span. This consideration would affect the systems stiffness and therefore affect the fundamental period of the SPSW. Optimization of the method proposed in this study by considering this parameter would increase the range of applicability for approximating the fundamental period of SPSW structures.

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