fundamental period prediction of steel plate shear wall

FUNDAMENTAL PERIOD PREDICTION OF STEEL PLATE SHEAR WALLSTRUCTURES

A Thesis submitted to the faculty of San Francisco State University

In partial fulfillment of The requirements for

The Degree

. V: 4-'2- Master of Science

In

Engineering: Structural/Earthquake Engineering

By

Benjamin Aaron Kean

San Francisco, California

May 2017

Copyright by Benjamin Aaron Kean

2017

CERTIFICATION OF APPROVAL

I certify that I have read Fundamental Period Prediction o f Steel Plate Shear Wall

Structures by Benjamin Aaron Kean, and that in my opinion this work meets the

criteria for approving a thesis submitted in partial fulfillment of the requirement for

the degree Master of Science in Engineering: Structural/Earthquake Engineering at

San Francisco State University.

Cheng Chen, Ph.D.Professor of Civil Engineering

Professor of Civil Engineering

Fundamental Period Prediction of Steel Plate Shear Wall Structures

Benjamin Aaron Kean San Francisco, California

2017

The Steel Plate Shear Wall (SPSW) is a structural system used today in design for

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 approximation of the fundamental period

to be performed using a general formula using two coefficients whose values are

dependent on the type of LFRS. The SPSW uses a general value for the

fundamental period coefficients and it has been shown that the current procedure

for approximating the fundamental period of an SPSW produces overly

conservative estimations. This study evaluates the fundamental period of a large

population of SPSW prototypes. A formula for better approximation of the

fundamental period coefficients is derived and then verified on code-designed

SPSW prototypes.

I certify that the Abstract is a correct representation of the content of this thesis.

Date

PREFACE AND/OR ACKNOWLEDGEMENTS

I would like to thank the efforts of Dr. Cheng Chen, Amado Flores Renteria, Jolani

Chun-Moy, Daniel Salmeron, and David Flores for making this research possible.

Amado Flores Renteria, Jolani Chun-Moy, Daniel Salmeron, and David Flores

contributed as undergraduate students in the process of design and analysis of the

5-,8-, and 10-story SPSWs. Acknowledgements are also given to Ramin Bahargi

and Shihang Guo who contributed on the 3-, 6-, and 9-Story SPSWs. The author

would also like to acknowledge the constant support of his family and friends. The

results and findings in this thesis would not be possible without the help from the

individuals listed above.

TABLE OF CONTENTS

List of Table.................................................................................................................. viii

List of Figures...................................................................................................................x

List of Appendices.......................................................................................................... ix

CHAPTER 1. INTRODUCTION..................................................................................1

1.1 The Steel Plate Shear W all..........................................................................1

1.1.1 Previous SPSW research............................................................. 3

1.2 Current Code for Fundamental Period Estimation for SPSW ................. 5

1.3 Previous Research on Fundamental Period Estimation for S P S W 6

1.3.1 Liu etal. Approximation M ethod.............................................. 6

1.3.2 Topkaya and Kurban Approximation M ethod..........................8

CHAPTER 2. DESIGN OF SPSW PROTOTYPES.................................................10

2.1 Building Information..................................................................................10

2.2 Design of SPSW Prototypes......................................................................12

2.3 SPSW Prototype Design Summary...........................................................13

2.3.1 3-Story SPSW Design Summary..............................................13





2.3.6 10-Story SPSW Design Summary........................................... 17

CHAPTER 3. FUNDAMENTAL PERIOD APPROXIMATION METHODS ...19

3.1 Equivalent Lateral Force Procedure..........................................................19

3.1.1 Approximation of Fundamental Period.................................. 20

3.1.2 Seismic Response Coefficient...................................................24

3.2 Proposed Simplified Fundamental Period Approximation Methods...27

3.2.1 Liu etal. Approximation Method ........................................... 28

3.2.1.1 Application of Liu et al. method..............................30

3.2.2 Topkaya and Kurban Approximation M ethod....................... 32

3.2.2.1 Application of Topkaya and Kurbans method.......34

3.3 Comparison of Period Approximations...................................................37

CHAPTER 4. FINITE ELEMENT MODELING.....................................................44

4.1 Open System for Earthquake Engineering Simulation (OpenSees) ....44

4.2 Modeling of SPSW prototypes.................................................................44

4.3 Panel Zone Modeling.................................................................................46

CHAPTER 5. STATISTICAL ANALYSIS..............................................................50

5.1 Statistical Analysis of the SPSW Prototypes..........................................50

5.2 Bay-Width Adjustment..............................................................................52

5.2.1 Coefficient C t..............................................................................52

5.2.2 Coefficient x................................................................................ 54

5.3 Derivation of Base-Shear Participation Adjustment Function..............56

5.3.1 Ct Coefficient Adjustment........................................................ 57

5.3.2 x Coefficient Adjustment.......................................................... 61

5.3.3 Error Analysis.............................................................................66

5.4 Proposed Method for Approximating SPSW Fundamental Period.....67

CHAPTER 6. VALIDATION OF PROPOSED METHOD................................... 70

6.1 SPSW Validation Prototypes.................................................................... 70

6.2 Results from Approximation M ethod...................................................... 71

6.3 Summary......................................................................................................75

CHAPTER 7. CONCLUSIONS.................................................................................. 82

REFERENCES...............................................................................................................83

vii

LIST OF TABLES

Table 1-1. Values of Parameters Ct and x .................................................................... 6

Table 2-1. Seismic mass and weights for SPSW prototypes...................................10

Table 2-2. Design summary SPSW prototype 3M100.............................................14

Table 2-3. Design summary SPSW prototype 5M100............................................ 14

Table 2-4. Design summary SPSW 6M100................................................................ 15



Table 2-7. Design summary SPSW 10M100.............................................................. 18

Table 3-2. Values for factor r f ..................................................................................... 21

Table 3-3. 10-foot bay width fundamental period results.........................................33



Table 5-1. Fundamental period coefficients from regression analysis...................42

Table 5-2. /(L ) values for fundamental period coefficient fc(L ) ............................51

Table 5-3. Values for ^c100ari^ c 100 ............................................................................^3

Table 5-4. Coefficient values for the adjustment function.......................................55

Table 5-5. Coefficient values for the adjustment function/*(L)..............................55

Table 5-6. fac(V) at varying levels of base-shear participation..............................58

Table 5-7. fpc(V) at varying levels of base-shear participation............................. 58

Table 5-8. Values for aUc and bac assuming a power series model....................... 60

Table 5-9. Values for apc and bpc assuming a power series m odel...................... 60

Table 5-10. Values for aUc and bac assuming a linear variation.............................61

viii

Table 5-11. Values for a.pc and bpc assuming a linear variation.............................. 61

Table 5-12. Values of fax(Y) at varying base-shear participation levels...............62

Table 5-13. Values of fpx(V) values at each base-shear participation value 62

Table 5-14. Values for aUxand /^assum ing a power series variation.................65

Table 5-15. Values for a^x and bpx assuming a power series variation................65

Table 5-16. Values for aUx and bUx assuming a linear variation...........................65

Table 5-17. Values for a^x and bpx assuming a linear variation.............................66

Table 5-18. Root Mean Squared Analysis for the power series and linear model66

Table 5-19. Values of all coefficients used in proposed method............................68

Table 6-1. 3K100 SPSW design sum m ary.................................................................70

Table 6-2. 4N80 SPSW design summary................................................................... 71

Table 6-3. 7W60 SPSW Design Summary.................................................................71

Table 6-4. Comparison of fundamental periods from finite element analysis and the proposed method for the three validation prototypes.......................................... 74

Table 6-5. Proposed method coefficient values........................................................ 76

Table 6-6. 10-foot bay-width SPSW fundamental period results............................79



LIST OF FIGURES

Figure 1-1 Typical SPSW configuration [1],

Figure 1-2 Hour-glass shape of web-plate due to pull-in effect (courtesy of Carlos Ventura, University of Columbia, Vancouver, Canada) [1]....................................... 4

Figure 2-1. Floor plan for 3-story SPSW prototypes................................................. 11

Figure 2-2. Floor plan for 5-, 6-, 8-, 9-, and 10-story SPSW prototypes..................11

Figure 3-1. Comparison of fundamental period for SPSW prototypes designed with 100 percent base shear..........................................................................................22

Figure 3-2. Comparison of fundamental period for SPSW prototypes designed with 85 percent base shear.........................................................................................23

Figure 3-3. Comparison of fundamental period for SPSW prototypes designed with 65 percent base shear..........................................................................................24

Figure 3-4. Comparison of Cs max values of SPSW N prototypes........................26

Figure 3-5. Comparison of Cs max values of SPSW M prototypes........................26

Figure 3-6. Comparison of Cs max values o f SPSW W prototypes........................27

Figure 3-7. Application of Liu et al. method to 10-foot bay-width SPSW prototypes........................................................................................................................ 30



Figure 3-10. 10-foot bay width SPSW results from Topkayas SPSW fundamental period m ethod.................................................................................................................35



Figure 3-13. 10-foot SPSW period approximations from Liu and Topkaya..... 38

Figure 3-14. 15-foot SPSW period approximations from Liu and Topkaya.....38

Figure 3-15. 20-foot SPSW period approximations from Liu and Topkaya.....39

Figure 4-1. Visualization o f the strip method [1 ] ......................................................45

Figure 4-2. 6-Story OpenSees SPSW m odel............................................................ 46

Figure 4-3. Visualization of a typical panel zone configuration at beam-to-column connection....................................................................................................................... 47

Figure 4-4. HBE node conventions for typical SPSW web-plate............................48

Figure 4-5. Configuration of the modified panel zones............................................ 49

Figure 5-1. Comparison between periods from regression and finite element analysis.............................................................................................................................51

Figure 5-2. Comparison of adjusted Ct values and finite element values.............54

Figure 5-3. Validation o f power series assumption for “x” coefficient adjustment function............................................................................................................................56

Figure 5-4. aC over base-shear participation levels..................................................59

Figure 5-5. PC over base-shear participation levels..................................................59

Figure 5-6. Variation o f ax over base-shear participation levels.............................63

Figure 5-7. Variation of Px over base-shear participation levels.............................64

Figure 5-8. Results from application of proposed method on SPSW prototypes ..69

Figure 6-1. Comparing fundamental period analysis methods for the first validation SPSW prototype, SPSW 3K 100................................................................72

Figure 6-2. Comparing fundamental period analysis methods for the second validation SPSW prototype, SPSW 4N 80..................................................................73

Figure 6-3. Comparing fundamental period analysis methods for the third validation SPSW prototype, SPSW 7W60..................................................................74

Figure 6-4. 10-foot bay-width SPSW fundamental period results..........................77

LIST OF APPENDICES

A-2-1. 3-STORY SPSW DESIGN SUMMARIES.................................................... 87

A-2-2. 5-STORY SPSW DESIGN SUMMARIES.................................................... 88

A-2-3. 6-STORY SPSW DESIGN SUMMARIES....................................................89



A-2-6. lO-STORY SPSW DESIGN SUMMARIES..................................................95

A-3-1. LIU ETAL. SPSW FUNDAMENTAL PERIOD APPROXIMATION M ETHOD....................................................................................................................... 97

A-3-2. TOPKAYA AND KURBAN SPSW FUNDAMENTAL PERIOD APPROXIMATION M ETHOD................................................................................ I l l

A-4-1. lO-STORY SPSW PANEL ZONE OPENSEES CODE............................ 124

A-6-1. VERIFICATION OF PROPOSED METHOD WITH VALIDATION PROTOTYPE...............................................................................................................127

Chapter 1. INTRODUCTION

In today’s design industry, there are many different Lateral Force Resisting Systems

(LFRS) for the structural engineer to choose from. The Steel Plate Shear Wall (SPSW) is

a structural system that can be used as the primary LFRS in a building [1], The SPSW has

been utilized in LFRS design since the 1970s with their use steadily increasing over the

past few decades [2]. The use of SPSWs can be advantageous when compared with other

LFRSs. Performance of the SPSW has been shown to be highly ductile when subjected to

seismic loading. In addition, the cost and time to construct the SPSW system makes it

competitive to other LFRSs [1].

While advances in research on the behavior of the SPSW continue, some aspects of the

systems behavior remain unknown. Approximating the fundamental period of a SPSW is

one area of research where more work is required. The fundamental period is used to

calculate the design base-shear of a structure using provisions from ASCE 7-10 [3].

Current code and provision has been shown to produce overly conservative approximations

of the fundamental period for a SPSW when compared to that from a finite element

analysis. This error in estimation leads to overly conservative approximations of design

base-shear and costly designs. The focus of this study is to improve estimation of the

fundamental period of steel plate shear wall structures. A total of fifty-four SPSWs are

designed and analyzed using finite element models developed for its fundamental period.

Statistical analysis is then performed to explore an optimal estimation for the fundamental

period of a SPSW system.

1.1 The Steel Plate Shear Wall

Typically, a SPSW consists of a vertical steel infill plate connected to surrounding beams

and columns [1], The beams and columns are often referred to as horizontal boundary

elements (HBEs) and vertical boundary elements (VBEs), respectively. The steel web-

plates are installed in one or more bays along the full height of the structure to form a

cantilever wall, a typical SPSW configuration is presented in Figure 1-1. The behavior of

2

the SPSW is similar to a plate-girder where forces are resisted by flexure and axial capacity

in the VBEs and by in-plane shear resistance of the web-plate bounded by the HBEs.

Figure 1-1. Typical SPSW configuration [1],

SPSWs have been shown to have high initial stiffness, behave ductile, and dissipate

a good amount of energy under cyclical loading, which makes them a viable alternative for

moment resisting and braced frames in LFRS design. Compared with the concrete shear

wall system SPSW offers advantages due average 18 percent decrease in building weight

and average 2 percent increase in available floor space [2]. The SPSWs in this study are

designed using the requirements outlined in the “American Institute of Steel Construction

(AISC) Design Guide 20: Steel Plate Shear Walls” (Design Guide 20) [1] and “AISC

Seismic Provisions for Structural Steel Buildings” (AISC 341-10) [10], Lateral loading

for the SPSW structures are calculated using the ELFP outlined in the “American Society

o f Civil Engineers (ASCE) Minimum Design Loads for Buildings and Other Structures”

3

(ASCE 7-10) [3]. A detailed discussion on the design of a SPSW prototype is presented in

Chapter 2.

1.1.1 Previous SPSW research

Prior to research performed in the 1980s, the assumed limit state for the SPSW was out-of-

plane buckling of the web-plate [4]. Satisfying this limit state required using web-plates

with high stiffness which typically made the system uncompetitive economically. Through

a number of studies, it was shown that the post-buckling strength of the web-plate

contributes significantly to the strength and ductility of the system. This observation allows

the designer to select thinner web-plates and ultimately select smaller boundary elements

making the system more economically competitive. This has also led to an increase in

research in both analytical and experimental analysis of the performance of SPSWs.

A study performed in 1993 by Elgaaly et al. [5] used the strip method to analytically

model three SPSW prototypes developed for a previous study by Caccese et al. [6] to verify

the accuracy of the model. The results indicated that a large number of tension strips were

required for accurate approximations of the deformations and other properties o f the web-

plate. For the time at which this study was conducted this resulted in large computational

power requirements, the computer analysis would also require a large amount o f time.

Finite element analysis consisted of applying an increasing load until a loss o f stability due

to plastic hinges forming in the base of the column. It was found that using plates much

thicker than required did not substantially add strength to the system as the column yielding

was the failure mode for each analysis and the web-plates were not able to achieve the

capacity of their post-buckling strength.

From the analytical results a method was developed to predict the hysteretic

behavior of the SPSW. A modified version of the strip method which included truss

elements in each direction to account for the earthquake action in both direction was used

4

for the derivation of the method. The results from this method were able to closely match

the hysteretic behavior of the SPSWs tested experimentally.

A study performed in 2000 by Lubell et al. [7] designed, constructed, and tested

SPSW prototypes under cyclical quasi-static loading to observe their performance. Under

the loading the first prototype experienced failure in the lateral brace due to out-of-plane

deflection of the top HBE. To avoid this failure mode, the top HBE was stiffened for the

remaining prototypes. Failure modes for the remaining SPSW prototypes were plastic

hinging in the column. One thing observed from the failure of the SPSWs was significant

pull-in at the column to web-plate interface. The pull-in effect o f the cyclical deformations

resulted in an hour-glass shape forming in the web-plate as shown in Figure 1-2. The main

conclusions drawn from this study was the importance of proper anchorage of the top level

HBE and the necessity of capacity design of the VBEs to ensure the web-plate reaches its

full post-buckling strength and to avoid the hour-glass effect from the VBE pull-in.

Figure 1-2. Hour-glass shape of web-plate due to pull-in effect (courtesy of Carlos

Ventura, University of Columbia, Vancouver, Canada) [1].

5

Both experimental and analytical analyses however showed that the post-buckling

strength of thin steel plates can be quite substantial. Current code provisions require that

the web-plate of a SPSW should resist the design lateral load calculated from the

Equivalent Lateral Force Procedure (ELFP) [3], This requirement neglects any lateral

resistance in the boundary elements that could potentially add to the strength of the SPSW

ultimately leading to overly conservative designs.

Previous studies at San Francisco State University from Enright [8] and Barghi [9]

have explored the performance of SPSWs whose web-plates were designed to resist less

than one-hundred percent o f the design base-shear. These studies suggest that the boundary

elements lateral resistance could provide a substantial increase in strength and stiffness for

a SPSW. This could be advantageous in the design due to the reduction in size and strength

requirements for the boundary elements and web-plates. For this study the contribution of

lateral resistance effects of reducing the lateral load resisted in the web-plate was

considered in a similar way to that by Enright [8] and Barhgi [9].

1.2 Current code for fundamental period estimation

The natural period of a structure is the duration of time it takes for the structure to complete

one full cycle of vibratory motion. In structural design using the ELFP, the natural period

is a used in calculating the design base-shear on the structure accordingly to chapter 12 o f

ASCE 7-10 [3]. Current code allows for the engineer to approximate a structures

fundamental period using the following equation:

[ASCE 7-10 12.8-7] Ta = Ct Hx (1-1)

where in Eq. (1-1), G and x are empirical coefficients from Table 12.8-2 in ASCE 7-10 [3]

as presented in Table 1-1; and H is the total height of the structure in feet. ASCE 7-10

provides specific values for the coefficients of G and x for only four distinct LFRSs

including steel and concrete moment frames, steel eccentrically braced frames, and steel

buckling-restrained braced frames. All other LFRSs, including the SPSW, are categorized

6

under “All other structural systems” with generalized values for G and x. This

generalization contributes to the inaccurate and overly conservative approximations of the

fundamental period during the initial design of SPSW structures.

Table 1-1. Values of Parameters Ct and x

ASCE 7-10 Table 12.8-2 Values of Approximate Period ________________ Parameters Ct and x

Structure Type C, X

Steel moment-resisting frames 0.28 0.8Concrete moment-resisting frames 0.016 0.9Steel eccentrically braced frames 0.03 0.75

Steel buckling-restrained braced frames 0.03 0.75All other structural systems 0.02 0.75

1.3 Previous research on fundamental period estimation for SPSWs

Research on more accurate approximation methods for the fundamental period of SPSWs

have been conducted by researchers. While the provision of ASCE 7-10 allows the

engineer to approximate the fundamental period using finite element analysis, it is often

time consuming. Moreover, member sizes of the LFRS are not available at this stage for a

finite element analysis. There is a need for a simplified method to provide better

approximations of the fundamental period for a SPSW structure. Two approximation

methods developed by researchers are presented and reviewed in this section. A more

detailed discussion on both approximation methods is presented in Chapter 3.

1.3.1 Liu et al. [11] approximation method

Liu et al. [11] proposed a simplified method to approximate the fundamental period of a

SPSW system. The SPSW prototypes designed by Berman [12] are used to evaluate the

7

proposed method through a comparison with finite element analysis. The method proposed

by Liu et al. can be expressed as follows

where in Eq. (1-2), on, coSi, and co/i, represent the combined shear-flexure, shear, and flexure

frequencies o f the ith mode of vibration, respectively. The fundamental period o f the SPSW

is then calculated as

In Eq. (1-3), Ti is the natural period of a SPSW structure for the ith mode of vibration. The

calculation of the shear frequency is performed using the assumption that the SPSW

behaves as a lumped mass system [11], which allows the shear frequency to be calculating

using the expression

In Eq. (1-4), K and M represent the global stiffness and mass matrices of the SPSW

structure, respectively. The global mass matrix is a diagonal matrix composed of the

individual story masses for the assumed lumped mass system. The global stiffness matrix

is composed of the individual story stiffness, which is considered to be a summation of the

column and web-plate stiffness as following

[Liu et al. Eq. (3)]1 1 12 ~ 2* 2~ < oj2fi ( 1-2)

2n[Liu et al. Eq. (4)] Ti = (1-3)

[Liu et al. Eq. (5)] d e t(K - <o2 M) = 0 (1-4)

[Liu et al. Eq. (8)] (1-5)

[Liu et al. Eq. (9)]ELt

sin2 a cos2 a ( 1-6)k p i~ h

8

where in Eq. (1-5), fc/the lateral stiffness of VBEs at the ith story; E is the elasticity modulus

for steel; Ic is the moment of inertia for the VBE member; and h is the typical story height.

In Eq (1-6), kPi is the stiffness o f the web-plate; a is the tension field angle of the web-plate;

t is the thickness of the web-plate; L is the center to center distance between VBE members.

The flexural frequency in Eq. (1-2) is calculated assuming the SPSW frame as a

cantilever wall with uniformly distributed mass and stiffness

[Liu et al. Eq. (10a)] W/1 = 1.8752\

El(1-7)

mHA

In Eq. (1-7), co/i is the flexural frequency of the first mode of vibration; H is the height of

the structure; m is as the distributed mass of the structure; E is the elasticity modulus of

the steel; and / is the moment of inertia of the VBE assuming the entire structure acts as a

cantilever beam. To account for varying VBE member sizing the mean value of the VBEs

moment of inertia is used in the calculation [11].

1.3.2 Topkaya and Kurban [13] approximation method

Topkaya and Kurban [13] proposed a simplified method for estimating the fundamental

period of a SPSW. For this method the structure is assumed to behave as a vertical

cantilever similar to the assumption made by Liu et al. [11]. In this method, the

fundamental period of a SPSW is approximated as

[Topkaya Eq. (3)] TxW1 1

7 2 + 7 1 ( 1-8)Jb Js

where in Eq. (1-8), Tw is the natural period o f the SPSW structure; f i and f s are the natural

frequencies of two cantilever beams, experiencing bending and shear, respectively.

The natural frequency of the vertical cantilever subject to bending is calculated as

9

[Topkaya Eq. (4)] f b = VfM

El(1-9)

m

where H is the height of the structure; m is the mass; and Iw is the moment of inertia of the

web-plate. The parameter r/ is a factor proposed by Zalka [14] accounting for lumped

masses at story levels [13].

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

KGAw ( 1- 10)[Topkaya Eq. (5)] f ' = r —7 4 m

where KAW is the effective shear area of the web plate; and G is the shear modulus of the

steel plate. The effective shear area of the web-plate KAw can be expressed as

[Topkaya Eq. (6)] KAW = (1-11)

where the parameter fi is defined as

f Q2[Topkaya Eq. (6)] /? = I ~^dA (1-12)

Aw

where Q is the static moment of area; and b is the width of the web-plate. Determining the

exact value of /? could be quite time consuming due to the possibility of integration of

fourth-order polynomials [13]. An approximation proposed by Atasoy [15] could be used

to simplify the calculation which assumes the ratio Q/b varies linearly over the region of

continuity.

10

CHAPTER 2. DESIGN OF SPSW PROTOTYPES

For this study, a total of fifty-four SPSW structures were designed to create a population

of fundamental period data with varying story height, bay width, and base-shear

participation ratios.

2.1 Building information

The SPSW prototypes in this study are modeled after structures described in the SAC joint

venture [16] using a Los Angeles structure with a seismic design category D. The SPSW

structures are designed in accordance to the procedures outlined in AISC Design Guide 20

[1], and requirements from AISC 341-10 [10]. The design base shear was calculated using

the equivalent lateral force procedure [3] with seismic masses listed in Table 2-1. Building

weights were calculated using 86 psf for roof dead loading and 96 psf for dead loading on

a typical floor [16]. From ASCE 7-10 Table 12.2-1 the response modification factor, R,

overstrength factor, Qo, and deflection amplification factor, Cd are given as 7, 2.5, and 6.5

respectively.

Table 2-1. Seismic mass and weights for SPSW prototypes

Level

3-Story 5,6,8,9,& 10-Story

Seismic Mass Seismic Weight SeismicMass

SeismicWeight

kip sec / ft kips kip sec/ft kipsRoof 55.7 1793.5 58 1867.6

TypicalFloor 64.4 2073.7 67.1 2160.6

Buildings are constructed at three-, five-, six-, eight-, nine-, and ten-story with a

typical story height o f thirteen feet. Three-story SPSWs are designed with the floor plan

presented in Figure 2-1, the remaining SPSWs are designed with the floor plan presented

in Figure 2-2. All SPSW prototypes are designed with varying bay widths at ten, fifteen,

11

and twenty feet denoted N, M, and W respectively. Base-shear participation ratios in the

web-plate varied at 100%, 85%, and 65% to account for plate-frame interaction based on

previous works by Enright [8] and Barghi [9].

/7T

§>

&r.

~T~

111

“ “ “ ***•“ -* — *■«

c 1 1

I B B a* m m

I11

* ________L . _______111

_________ %_______

--------------

— ; —

i i

i i i i

1 I

I 1 I 1

1I

r i ---------------1i i i i i i i i i

1 - - « .

ii

i

L ..............................................11 4 bays (§ 30' 1

Figure 2-1. Floor plan for 3-story SPSW prototypes.

i f -

om

</aC3JD

■f-

i i i i

i i. - — f ----------- - i

i i i i i i

i ii i

i t

i i i i

I t

i

i

i

i i i 1i i i i ii i i i i

i I ! 1 1i I 1 1 11 i I 1 1

5 bays @ 30’Figure 2-2. Floor plan for 5-, 6-, 8-, 9-, and 10-story SPSW prototypes.

Two material grades were considered for the selection of web-plates. Both ASTM

A36 steel plate and ASTM A 1001 SS Gr. 36 Type 2 structural steel gauge plate are selected

based on the optimal thickness needed to resist the shear loading at each story. All HBEs

and VBEs are assumed to be ASTM 992 wide flange sections.

12

2.2 Design of SPSW prototypes

The design of a SPSW is determined utilizing the post-buckling strength which is

developed by the tension-field action of the web-plate when subjected to lateral loading.

Web-plates are assumed to be of the unstiffened slender type. It is assumed in design that

the web-plate experiences only shear deformations and does not carry any lateral loading.

Web-plates are assumed to be connected to rigid beams and columns making a moment

resisting frame (MRF) [1]. The story shear is assumed to be resisted completely by the

web-plate in the preliminary design for prototypes designed for 100 percent base-shear

participation. The amount of story shear resisted can be reduced to account for the resistive

capacity of the horizontal and vertical boundary elements.

To determine the required thickness of the web-plate, equation 3-21 shown in Eq.

(2-1), from AISC Design Guide 20 is used.

Vn[AISC 3-21] tw > -— — (2-1)

QAlFyRy sin 2a

In Eq. (2-1), U is the web-plate thickness; Vn is the individual story shear for the

nth story; Fy is the nominal yield stress; Ry is the modification factor for expected yield

strength; and a is the developed tension field angle in the web-plate. The value of a is

determined calculated using equation 17-2 in AISC 341-10 [10] and presented in Eq. (2-

2).

i + hvL2 A

[AISC 17-2] a = ta n _4(------------— — c - ) (2-2)1 + twh - T - + -Ab 360ICL

where in Eq. (2-2), L is the bay-width of the structure; h is the typical story height; Ac is

the cross-sectional area of the VBE; Ab is the cross-sectional area of the HBE; and Ic is the

moment of inertia for the VBE.

13

It can be seen in Eq. (2-2) that the buildings geometry and properties of the

boundary elements are used for calculating a. The boundary elements are not typically

known when the ELFP is performed making the value of tension field angle difficult to

initially approximate. For this reason, a conservative assumption of 30 degrees is typically

taken as a for initial web-plate sizing.

The HBEs and VBEs of an SPSW are designed based on a capacity design approach

to ensure 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 [1, 7]. 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 o f the member. The SPSW

prototype is considered optimized when the web-plate DCR is approximately one.

Variation of the HBEs and VBEs were also limited to create a realistic design. VBE

selections were limited to one-member size spliced at every second floor. HBE selections

were limited to one or two sizes to create a typical beam section for each floor. While this

process does result in the overdesign of VBE and HBE elements it creates a more realistic

design prototype.

2.3 Design summary of SPSW prototypes

2.3.1 3-story SPSW design summary

The 3-story SPSW prototypes were designed with a rectangular floor plan of 180 feet

longitudinal and 120 transverse as shown in Figure 2-1. The W prototypes were designed

with 2 SPSW frames in each direction. The M and N prototypes were designed with 4

SPSW frames in each direction. SPSW prototypes were named based on their number of

stories, bay width, and web-plate shear participation. A sample design summary is

presented in Table 2-2 for the 3-story SPSW structure with 15-foot bay width designed for

14

100% web-plate shear participation (3M100). The remaining 3-story SPSW designs are

presented in Appendix A-2-1.

Table 2-2. Design summary SPSW prototype 3M100

SPSW Design Summary

Prototype: 3M100

Story HBE VBE Web-Plate (in.)

Roof W 18X65 - -

3 W18X71 W14X132 18 gauge

2 W18X65 W14X193 13 gauge

1 W18X119 W14X193 12 gauge


The 5-story SPSW prototypes were designed with a square floor plan of 150 feet in both

longitudinal and transverse as seen in Figure 2-2. The W prototypes were designed with 4

SPSW frames in each direction. M and N prototypes were designed with 6 SPSW frames

in each direction. Naming convention follows the procedure outlined for 3-story

prototypes. A sample design summary is presented in Table 2-3 for the 5M100 prototype.

The remaining design summaries are presented in Appendix A-2-2.

Table 2-3. Design summary SPSW prototype 5M100

SPSW Design Summary

Prototype: 5M100


Roof W 18X65 - -

5 W 18X65 W14X132 19 gauge

4 W 18X65 W14X132 14 gauge

3 W 14X74 W14X193 12 gauge

2 W 14X74 W14X193 11 gauge

1 W24X192 W 14X233 10 gauge

15


The 6-story SPSW prototypes were designed using the same floor plan as the 5-story SPSW

prototypes in Figure 2-2. The number of SPSW frames used in each direction remains the

same as the 5-story SPSW prototypes for the W, M, and N 6-story prototypes. Similar

naming conventions are used as the 3- and 5-story prototypes. A sample design summary

is presented in Table 2-4 for the 6M100 prototype. The remaining design summaries for

the 6-story SPSWs are presented in Appendix A-2-3

Table 2-4. Design summary SPSW 6M100

SPSW Design Summary

Prototype: 6M100


Roof W 18X65 - -

6 W 18X65 W14X176 19 gauge

5 W18X65 W14X176 14 gauge

4 W 18X65 W14X176 12 gauge

3 W 18X65 W 14X283 1/8

2 W 18X65 W 14X283 9 gauge

1 W21X132 W 14X283 9 gauge


The 8-story SPSW prototypes were designed using the same floor plan as the 5- and 6-

story SPSW prototypes in Figure 2-2. The same number of SPSW frames in each direction

is used as the 5- and 6- story frames for the W and M prototypes. 8-story N prototypes use

8 SPSW frames in each direction. Naming convention remains the same as the 3-, 5-, and

6-story prototypes. A sample design summary is presented in Table 2-5 for the 8M100

prototype. The remaining 8-story designs are presented in Appendix A-2-4.

16


SPSW Design Summary

Prototype: 8M100


Roof W 16X77 - -

8 W 16X77 W14X145 19 gauge

7 W 14X68 W14X145 13 gauge

6 W 14X74 W 14X283 11 gauge

5 W 16X77 W 14X283 10 gauge

4 W 14X68 W 14X283 7 gauge

3 W 14X68 W 14X426 6 gauge

2 W 14X68 W 14X426 5 gauge

1 W24X146 W 14X426 5 gauge


The 9-story SPSW prototypes were designed using the same floor plan as the 5-, 6-, and 8-

story SPSW prototypes in Figure 2-2. The number o f SPSW frames used in each direction

remains the same as the 8-story prototypes. Similar naming conventions are used. A sample

design summary is presented in Table 2-6 for the 9M100 prototype. The remaining design

summaries are presented in Appendix A-2-5.

17


SPSW Design Summary

Prototype: 9M100


Roof W 16X77 - -

9 W 16X77 W14X211 19 gauge

8 W 16X77 W14X211 13 gauge

7 W16X77 W14X211 1/8

6 W16X77 W 14X426 8 gauge

5 W 16X77 W 14X426 6 gauge

4 W 16X77 W 14X426 5 gauge

3 W 16X77 W 14X665 4 gauge

2 W 16X77 W 14X665 3 gauge

1 W24X192 W 14X665 1/4


The 10-story SPSW prototypes were designed using the same floor plan in Figure 2-2. The

number of SPSW frames used in each direction remains the same as the 8- and 9-story

prototypes. Same naming convention is used. A sample design summary is presented in

Table 2-7 for the 10M100 prototype. The remaining design summaries are presented in

Appendix A-2-6.

18


SPSW Design Summary

Prototype: 10M100


Roof W 16X77 - -

10 W 16X77 W14X257 19 gauge

9 W 14X68 W14X257 13 gauge

8 W 14X68 W14X257 11 gauge

7 W16X77 W 14X257 8 gauge

6 W 14X82 W 14X500 6 gauge

5 W 14X74 W 14X500 5 gauge

4 W 14X74 W 14X500 4 gauge

3 W 18X86 W14X500 3 gauge

2 W14X132 W 14X665 1/4

1 W40X264 W 14X665 1/4

19

Chapter 3. EXISTING METHODS FOR FUNDAMENTAL PERIOD

APPROXIMATION OF SPSW STRUCTURES

The code prescribed methodology for approximating the fundamental period of a structure

is outlined in chapter 12 of ASCE 7-10 [3] as a part of the Equivalent Lateral Force

Procedure (ELFP), the estimated fundamental period is used to calculate the structures

design base-shear. The current code method has been shown to produce overly

conservative estimations of the period for SPSW structures. While AISC 7-10 does allow

the use of a fundamental period obtained through computer analysis this is an impractical

solution. Periods obtained from these types of analysis require the engineer to already have

member sizes selected for the HBE, VBE, and web-plates which are typically unknown in

the preliminary design.

A simplified method to accurately estimate the fundamental period of an SPSW

structure would be a useful tool for a SPSW design. Previous studies have focused on the

development of a simple and accurate method to estimate the fundamental period of SPSW

structures. Two of these methods are considered in this study to serve as a comparison.

This section discusses the current accepted method of approximating a SPSWs

fundamental period. As well the two proposed simplified methods are discussed and

applied to the SPSW prototypes developed for this study.

3.1 Equivalent lateral force procedure

The ELFP, outlined in section 12.8 of ASCE 7-10, is used to calculated the design base-

shear of a structure expressed as

[ASCE 12.8-1] V = CSW (3-1)

where in Eq. (3-1), V is the base shear of the structure; W is the seismic weight of the

structure; and Cs is the seismic response coefficient. The seismic response coefficient is a

parameter which depends on several factors including the location o f the structure,

20

structural system being used, and the fundamental period. The base shear is then

distributed along the height of the structure. Point loading at each level is calculated as

[ASCE 12.8-11] Fx = CvxV (3-2)

In Eq. (3-2), Fx is the point load found at each level, x; CW is the vertical distribution

coefficient. The vertical distribution coefficient determines what percentage of the base

shear will be applied at each level and is calculated for each story as

[ASCE 12.8-12] Cvx = vnW*fe* . fc (3-3)I"= iW iK

In Eq. (3-3), h is the height from the base of the structure to the level of interest; w is the

portion of seismic weight applied to the story of interest; and k is an exponent obtained

from section 12.8.3 of ASCE 7-10. ASCE 7-10 defines k as an exponent related to the

period of the structure [3]. The exponent k is taken as 1 for structures with a period less

than 0.5 seconds and 2 for structures with a period greater than 2.5 seconds. For

intermediate periods between 0.5 and 2.5 seconds the value of k is determined through

linear interpolation. Seismic forces at each level are calculated as

n

[ASCE 12.8-13] Vx = 1 Ft (3-4)i-x

In Eq. (3-4) the seismic force at a particular story is found through the summation of the

story loads found in Equation (3-2) above the level of interest. Applying Eqs. (3-1) through

(3-4) allows the designer to calculate the seismic loads that the structure will need to resist

in order to be code compliant.

3.1.1 Approximation of the fundamental period

The ELFP outlined in Section 12.8 of ASCE 7-10 requires the approximation of the

fundamental period of the structure for the calculation of the base shear described in Eq.

(3-1). It is stated in Section 12.8.2 of ASCE 7-10 that the period of the structure may be

determined from a sufficient analysis using the structural properties and deformation

21

characteristics. This method of analysis requires sizes of the structural elements to be

known, which is generally not the case in the preliminary design stage when the ELFP is

applied. ASCE 7-10 does allow for the use of an approximated formula for the

approximation of a structures fundamental period calculated as

[ASCE 12.8-7] Ta = Cth* (3-5)

In Eq. (3-5), Ta is the approximate fundamental period of the structure; hnis the full height

of the structure; and both G and x are fundamental period coefficients from Table 12.8-2

of ASCE 7-10 [3]. Values for these two coefficients are dependent on the type of LFRS

and are only defined for four LFRSs including steel moment-resisting frames, concrete

moment-resisting frames, steel eccentrically braced frames, and steel buckling-restrained

braced frames. LFRSs not described in Table 12.8-2 are allowed to use the generalized

values for the two coefficients. Values o f G and x are shown below in Table 3-1 for U.S.

customary units. Values of the coefficients are available for the S.I. unit system in ASCE

7-10 Table 12.8-2.

Table 3-1. Values of Period Coefficients Ct and x.

ASCE 7-10 Table 12.8-2 Values of Approximate Period ________________ Parameters Ctand x

Structure Type C, X

Steel moment-resisting frames 0.28 0.8Concrete moment-resisting frames 0.016 0.9Steel eccentrically braced frames 0.03 0.75

Steel buckling-restrained braced frames 0.03 0.75All other structural systems 0.02 0.75

As indicated above, the generalized values of the two coefficients are often used to

approximate the fundamental period of the SPSWs for the use of the ELFP. For the

prototype SPSWs in the previous chapter, Figures 3-1 through 3-3 present the comparison

between the fundamental period using ASCE 7-10 and those from finite element analysis.

22

It can be observed that the use of the generalized values results in overly conservative

estimations.

N100

15r1h

05-

%

15r1to

■aoq3 0 5 “ cl

40

40

40

50

50

50

60

60

60

70 80

M10090 100 110 120 130

70 80

W10090 100

..X.70

Height (ft)

110 120 130

i l . i l 90 100 110 120 130

Finite Element “ — ASCE 7-10

Figure 3-1. Comparison of fundamental period for SPSW prototypes designed with 100

percent base shear.

Perio

d (s

)

23

N85

M85

W85

Finite Element ASCE7-10


percent base shear.

24

N65

96“ 40 50 60 70 80

M6590 100 110 120 130

9 r 40 50 60 70 80

W65

—1_______I_______!_______I_______I90 100 110 120 130


percent base shear.

3.1.2 Seismic response coefficient

The fundamental period of a LFRS is used in the calculation of the seismic response

coefficient, Cs. The coefficient is used to determine what percentage of the seismic weight

will be distributed to the structure as base shear. The seismic response coefficient is

calculated using equation 12.8-2 in section 12.8.1.f of ASCE 7-10. The coefficient is

calculated as

_ Sds[ASCE 12.8-2] ~ ~r ~ (3-6)

VIn Eq. (3-6), G is the seismic response coefficient; S d s is the short period range

design spectral acceleration; Ie is the importance factor; and R is the response modification

factor. Ie is taken from section 11.5 o f ASCE 7-10 and is determined from the inherent risk

25

to human life the structure would have if it were to fail, for this study /<? is taken as equal to

1 for all SPSW prototypes suggesting the structures carry a low risk if it were to fail. R is

a factor which considers the overall ductility of the system; values for R can be found in

Table 12.14-1 in ASCE 7-10, for this study R is taken to equal 7.5 for all SPSW prototypes

suggesting the system will behave in a ductile manner. Sas is a parameter determined from

the geographic location o f the structure; for this study S d s is taken to equal 1.07.

ASCE 7-10 requires checking the seismic response factor calculated in Eq. (3-6)

with a maximum and a minimum value. The minimum value allowed by ASCE 7-10 is

calculated as

[ASCE 12.8-5] Cs = 0.044 S D S I e > 0.01 (3-7)

The maximum allowable value for the seismic response coefficient is calculated as

Sdi[ASCE 12.8-3] Cs ~ ~ R ~ (3-8)

l e

In Eq. (3-8), Sdi is the one second period design spectral response acceleration; and

T is the fundamental period of the structure calculated in Eq. (3-5). As seen in Eq. (3-8) the

maximum value the seismic response coefficient can take is dependent on the period of the

structure calculated using Eq. (3-5). When the overly conservative approximation of the

fundamental period is used the values Cs takes will become larger. A comparison of the

maximum allowed Cs using the fundamental periods from Eq. (3-5) and finite element

analysis are presented in Figures 3-4, 3-5, and 3-6 for the N, M, and W SPSW prototypes

respectfully.

C m

ax

26

Q 4r~

0 2r n*—

0 4j

0

0 4p 0 2b

0*—

3N100

5N100

6N100

10N100

3N85

5N85

6N85

3N65

5N65

6N65

10N85 10N65

lASCE 7-10 ■ F in ite Element

Figure 3-4. Comparison of Cs max values for SPSW N prototypes.

10M100 10M85 10MW65

I ASCE 7-10B|Ftntte Element]

Figure 3-5. Comparison of Cs max values of SPSW M prototypes.

27

OAr0 2 -

oL-0 4r-0 . 2 “

0-0.4«-02-

0-0.2- 0 1 -

0- 0.2- 0 1 -

0-X 0.2rE 0.11-o* 0L

Figure 3-6. Comparison of Cs max values of SPSW W prototypes.

As seen in Figures 3-4, 3-5, and 3-6 the maximum values of Cs calculated using the

fundamental period from finite element analysis are smaller in magnitude when compared

to the values computed using fundamental periods from Eq. (3-5). Using the fundamental

periods from the finite element data would reduce the magnitude of the maximum

allowable Cs potentially reducing the design base-shear of the structure.

3.2 Simplified fundamental period approximation methods

A simple and accurate method for approximating the fundamental period of a SPSW

structure would reduce the iteration during the design process and increase efficiency.

Research has been conducted in the attempt of proposing a simplified method of

approximating SPSW fundamental periods. Two methods are considered in this study,

both of which treat the SPSW as a vertical cantilever and analyze the system as a vibrating

body. These methods are applied to the SPSW and compared with a new method proposed

in this study.

3W100 3W85 3W65

5W100 5W85 5W65

6W100

8W100

6W85

8W85

6W65

8W65

9W100 9W85 9W65

10W1GQ 10W85

|ASCE 7 -1 0 lF in ite Element]

1QW65

28

3.2.1 Approximation method by Liu et al. [11]

A study by Liu et al. [11] proposes a method to estimate the fundamental period of SPSW

structures. In this method, the fundamental period of the structure is approximated by

analyzing both the flexure and shear frequencies using Dunkerley’s equation [17].

Considering the fact that many designers take advantage of the reduction in strength

requirements along the height of the structure and design a SPSW with varying properties

at each story, this method takes into consideration the possibility that the material

properties of the system may not be uniform along the height of the structure.

The shear frequency of the system is calculated from an eigenvalue analysis with

the frame modeled as a lumped mass system [11]. The eigenvalue analysis is calculated

by taking the determinant of the characteristic equation of the system expressed as

[Liu et al. (5)] d e t(K - (OgM) = 0 (3-9)

In Eq. (3-9), M is the mass matrix; K is the stiffness matrix; and tos is the shear frequency.

For a lumped mass system, the mass matrix can be calculated as

M =

m 1000

0m 2

00

00

0 0

... 00 m n

(3-10)

In Eq. (3-10), mn is the story mass at the nth level. The stiffness matrix, K, is similarly

defined as

[Liu et al. (6)] K =

kt + k2 - k 2 k2 k2 + k3

00

0 0 ivn 1V.JJ J

(3-11)

- k n krIn Eq. (3-11) the value kn represents the stiffness of the nth story. It is worth noting that Eq.

(6) in Liu et al. states that a value of -k i should be used for the non-diagonal terms in the

29

first row and first column of the stiffness matrix. It is assumed that this is a simple

typographical error and the correct value of -k2 is therefore used in Eq. (3-11) [18].

The story stiffness used in Eq. (3-11) is a summation of contributions from the

frame and the web-plate. Frame stiffness is determined using the assumption that the

columns are connected to beams with high rigidity as

In Eq. (3-12), kd is the lateral frame stiffness at the ith story; E is the elasticity modulus of

steel; Ic is the moment of inertia of the VBEs; and h is the height of a typical column. The

web-plate stiffness is calculated as

In Eq. (3-13), k pi is the web-plate stiffness at the ith story; L is the bay-width of the SPSW

frame; t is the thickness of the web-plate; and a is the tension field angle. With the stiffness

kci and kpi defined for each story in Eqs. (3-12) and (3-13), the natural frequency of the

SPSW system under shear can be found using Eq. (3-9).

To determine the flexural frequency, the SPSW is modeled as a vertical cantilever

with mass and stiffness uniformly distributed along the height of the structure [11]. The

first mode natural frequency for a cantilever with uniformly distributed mass can be

calculated as [ 19]

In Eq. (3-14), cof is the flexure frequency; H is the full height of the structure; and m is the

distributed mass of the structure taken as the seismic mass divided by the height of the

structure. Using both the natural frequencies of shear and flexure the combined shear-

[Liu et al. (8)] (3-12)

[Liu et al. (9)]E L t

sin2 a cos2 a (3-13)k p i ~ h

[Liu et al. (10 a)] (3-14)

30

flexure frequency can be determined using Dunkerly’s equation [17], which can be

expressed as

0)f[Liu et al. (3)] (i) UUjr UUS

The natural period of the SPSW is then found using the expression

(3-15)

[Liu et al. (4)] T =2nco

(3-16)

3.2.1.1 Application of method by Liu et al.

The method proposed by Liu et al. is applied to the fifty-four SPSW prototypes developed

for this study. The fundamental periods for these SPSW prototypes are shown in Figures

3-4 through 3-6 in comparison with those from ASCE 7-10 and from finite element

analysis. An example o f the method proposed by Lie et a l being used to approximate an

SPSWs fundamental period is presented in Appendix A-3-1.

0L3N100 3N85 3N65

-E Z L T v

%5N100 5N85 5N65

o n_ ; l_[ [_[ l l I

ft

6N100 6N85 6N65

?F8N100 8N85

'zs M l . ' ■■ i

8N65

X ~ 1

o l |0Cl

9N100 9N85 9N65

p - i . ■ •- r - r "

■r / :/ \ ........

; ______________________ P v ^ L 1 ...................

10N100 10N85 10N65F33ASCE 7-10 I [OpenSees Analysis [73 Liu

Figure 3-7. Application of Liu et al. method to 10-foot bay-width SPSW prototypes.

31

?FO'— M -Lfl Z L

?(3M100

l tmm

3M85 3M65

L5M100

E5M85

s a i l ‘ LESS

5M65

6M100

' Li

6M85

* d n s z

6M65

r n P t8M100

f r e t i i ' \ 'EiSi

8M85 8M65

m lyZ 7/

9M100 9M85 9M65

5 i f■c nL_■g °LCLJ 1 l O .

F~/i5, -I u' ;

10M100 10M85 10MW65


Pll_______ 1 « i _ J Y lw M 'x.A _______________

3W100 3W85

E X 11

J LI IL3W65

5W100 5W85 5W65

E6W100 6W85

i s i r ± i i 2 a _

6W65

-T ^ n J ' U / I8W100 8W85

E jnijfelJ LEZZI

8W65

j j j ig i] 5 L f ^ v l9W100 9W85 9W65

1 iF r r f t T~ Lf ~ l : i x t . -TSXL10W100 10W85 10W65


32

3.2.2 Approximation method by Topkaya and Kurban [13]

A study by Topkaya and Kurban [13] proposed another method for approximating the

natural period of SPSWs. In this method, the fundamental period is determined through

the cyclical natural frequencies of the system under shear and flexure as

[Topkaya (3)] TxW

In Eq. (3-17), Tw is the fundamental period of an SPSW structure; fs is the natural frequency

of the SPSW under shear; and ft is the natural frequency of the SPSW subject to bending.

The natural frequency under bending can be calculated as

r r i rAM r 0.5595[Topkaya (4)] f b = Tf —ElZ^L (3-18)m

In Eq. (3-18), H is the total height of the SPSW structure; E is the elasticity

modulus of steel for the VBE; I is the moment o f inertia of the VBE; m is the distributed

seismic mass of the structure; and rf is a factor proposed by Zalka [14] to take into account

for the lumped mass assumption at each story level. The factor rf is dependent on the total

number of stories of the SPSW structure and is defined for a structure up to 50 stories.

Selected values of rf are shown below in Table 3-2 which encompasses all prototypes of

SPSWs considered in this study.

33

Table 3-2. Values for factor rf

Number of Stories rf1 0.4932 0.6533 0.7704 0.8125 0.8426 0.8637 0.8798 0.8929 0.90210 0.911

The natural frequency of a structure under shear is calculated as

[Topkaya (5)] £ = r / _ L G K A W (3. 19)mN

In Eq. (3-19), G is the shear modulus of steel; and KAW is the effective shear area calculated

as

[Topkaya (6a)] / J (3-20)a A

where the parameter p is calculated as

'Aw Q2[Topkaya (6b)] p = [ j j d A (3-21)

J o b

In Eq. (3-21), Q is the first moment of area of the web-plate; and b is the bay-width of the

SPSW frame. Eq. (3-21) requires integrating polynomials of the fourth degree which is not

practical in a preliminary design stage. An approximation of the value P can be determined

with the assumption that a linear variation exists between Q/b over regions of continuity

[13, 20], where the parameter p is then calculated as

34

[Topkaya (7a)] P = P i + P 2 (3-22)

where Pi is the contribution from the VBE element; P2 is the contribution from the web-

plate. Pi is calculated as

[Topkaya (7b)] A = ~ + ~ d VBE (3-23)

The parameter Qi is calculated as

[Topkaya (7d)] Q i = A f i ( 0 . S p l w + d V B E ) (3-24)

The parameter Q2 is calculated as

[Topkaya (7e)] Q2 = Qi + A w e b 0 . S ( p l w + d V B E ) (3-25)

The contribution from the web-plate is calculated as

[Topkaya (7c)] fe = p l w (3-26)2 p t k


[Topkaya (7f)] Q 3 = A V B E 0 . S ( p l w + d V B E ) (3-27)


[Topkaya (7g)] Q4 = Q3 + ^ f ^ p t k (3-28)O

3.2.2.1 Application of method by Topkaya and Kurban

The method proposed by Topkaya and Kurban was applied to the SPSW prototypes

developed for this study. It is worth noting that this method was developed for SPSW

frames utilizing the same boundary elements and web-plate along the full height of the

structure. This however is not practical in the design of an optimized SPSW structure as

the strength requirements reduce at each story level allowing reduction in the size of

boundary elements and web-plates [11].

35

To apply this method by Topkaya and Kurban to the SPSW prototypes in this study

a lower bound and upper bound estimation are calculated. The lower bound estimations

are calculated using the VBE and web-plate elements from the bottom level of each

prototype representing the period of the SPSW using the most rigid elements. The upper

bound estimations are calculated using VBE and web-plate elements from the top level of

each prototype representing the period of the SPSW using the least rigid elements. The

results from this method are presented in Figures 3-7 through 3-9 in comparison with those

using ASCE 7-10 and those from finite element method. An example of the method

proposed by Topkaya and Kurban being used to approximate the fundamental period of an

SPSW is presented in Appendix A-3-2.

2r1 -

-iX3N100

ZLl Li

3N85

( J

3N65

5N100

l I L 1..j l

5N85 5N65

6N100

c z l

6N85

JU L

6N65

I rZ L8N100 8N85 8N65

n n:2r

i i£ o

9N100

1ON100

9N85

10N85

9N65

i10N65

2 3 L

| | ASCE 7-10 j | Open Sees Analysis j | Topkaya Upper Bound C M Topkaya Lower Bound

Figure 3-10. 10-foot bay width SPSW results from Topkayas SPSW Fundamental Period

Method.

Perio

d (s

)

36

r m i---- »,— i______ cmU___k ,.K>1 ...........jtttl___li3M100 3M85 3M65

2 r1 -qI----- i—_lJ-----bJ J ® .............. J - l i li , s m ________r— 1..I L M M ......2 -

1 -

5M100 5M85

e h

5M65

6M100 6M85 6M65

8M10G

f ZZL

8M85 8M65

[£E L9M100

EZZL

9M85

fTTl

9M65

10M100 10M85 10MW65

11 [ASCE 7-10 | |openSees Analysis j [Topkaya Upper Bound |' * ] Topkaya Lower Bound


Method.

37

3W100 3W85

1 - ...— ................0 F 'T v l I li m m ............ r ’ - i i — i . ’ E H r ^ r i 1 li BBS

5W12 r

0 I - I f li

00 5W

r— 1 1 1.

85 5W

r ~ i f li

65

m6W1

2r*.......00 6W

j s s i ______i

85

u s r ~ i

6W

i

65

j - ...j___2

10

? 1

lo

8W85

H H i

8W65

9W100 9W85

10W100 10W85 10W85

ASCE 7-10 [ [OpenSees Analysis |Topkaya Upper Bound f v j Topkaya Lower Bound


Method.

3.3 Discussion of the existing two methods

The two methods explored in this study for approximating the fundamental period o f a

SPSW structure were shown to agree well through visual inspection with the periods

obtained through finite element analysis. The results from both methods are shown below

in Figures 3-10 through 3-12 along with the ASCE 7-10 approximation and finite element

analysis. A tabulated version of the values obtained through the two approximation

methods is shown in Tables 3-3 through 3-5.

Perio

d (s)

Pe

tiod(

s)

38

2p1 -

3N100lCZL

3N85 3N65

5N100 5N85I I I n n

5N65J__ L

6N100xm J .. ... L

6N85 6N65JZZU

8N100iZILI

8N85i Li

8N65

j | j o9N100 9N85 9N65

A m

10N100jzzu

10N85J— L

10N65f lA S C E 7-10 I ; Open Sees Analysis I iTopkaya Upper Bound HlTopkaya Lower Bound E lL iu

Figure 3-13. 10-foot SPSW period approximations from Liu and Topkaya.

J, Ml ,...L3M100

f ' j...;;,,;,.., ..i3M85

lE h J lE 23M65

2-i ... ........ .0---- t l L l Z I I i lM . .. .... .j— U I n P

I5M100 SM85

J .715M65

J Z l i6M100

-T Li6M85

J L I : : j ~ ~ i □6M65

E U l2f1 -

8M100J---- 18M85

m J8M65

C-...... iVd'tt ' I BO1 “* I

o__ I....L...... j ^ a M l ___ i i n t s i ___ m i i ted9M100 9M85 9M65

X U 10M100 10M85X U

10MW65In A S C E 7-10 ( jQpenSees Analysis I iTopkaya Upper Bound I iTopkaya Lower Bound PSlLiu]


Perio

d (s

)

39

02p1 -0 —

3W100

5W1D0

8W100

9W100

10W100

............ 1

3W85 3W65

[ b £ L.....I I. 1 J— iM L ---------[ m l 1. i_ iJiX l....... ...m l 1,5W85 5W65

I i— i f - ! □n r a r - , r r m m r ~ i c n ® _

2r

1 — r ~ i ____

>W1CX

□0 (5W8e

□awee

r>

8W85 8W65

9W85 9W65

10W85EZL i n i n E - M

10W65[I 1ASCE7-10I OpenSees Analysis I iTopkaya Upper Bound I iTopkava Lower Bound F 'lL iu


40

Table 3-3. 10-foot bay width fundamental period results

Fundamental Period Analysis Method

SPSWPrototype

ASCE 7-10 (s)

Finite Element

Analysis (s)

Upper Bound: Topkaya's Method

(s)

Lower Bound: Topkaya's Method

(s)

Liu'sMethod

(s)3N100 0.499 0.700 0.669 0.512

3N85 0.312 0.566 0.883 0.768 0.571

3N65 0.618 0.953 0.788 0.661

5N100 0.807 0.880 0.495 0.861

5N85 0.458 0.854 0.940 0.532 0.907

5N65 0.952 1.080 0.608 1.017

6N100 0.860 0.870 0.437 0.993

6N85 0.525 0.993 0.884 0.477 1.133

6N65 1.045 0.997 0.533 1.237

8N100 1.245 1.085 0.522 1.358

8N85 0.651 1.303 1.183 0.563 1.420

8N65 1.491 1.293 0.673 1.657

9N100 1.334 1.192 0.510 1.515

9N85 0.711 1.521 1.269 0.580 1.734

9N65 1.543 1.269 0.580 1.745

10N100 1.583 1.203 0.555 1.799

10N85 0.770 1.722 1.405 0.563 1.972

10N65 1.940 1.475 0.686 2.176

41



SPSWPrototype

ASCE 7- 10 (s)

Finite Element

Analysis (s)Upper Bound:

Topkaya's Method (s)Lower Bound:

Topkaya's Method (s)Liu's

Method (s)

3M100 0.469 0.953 0.944 0.518

3M85 0.312 0.512 1.018 0.782 0.573

3M65 0.564 1.100 0.806 0.667

5M100 0.710 1.053 0.611 0.832

5M85 0.458 0.753 1.200 0.685 0.887

5M65 0.872 1.391 0.744 1.023

6M100 0.812 1.141 0.644 0.960

6M85 0.525 0.895 1.281 0.704 1.065

6M65 0.957 1.408 0.763 1.1398M100 1.124 1.366 0.664 1.278

8M85 0.651 1.171 1.440 0.701 1.332

8M65 1.332 1.647 0.808 1.530

9M100 1.146 1.423 0.643 1.296

9M85 0.711 1.200 1.514 0.712 1.331

9M65 1.429 1.772 0.816 1.641

10M100 1.378 1.295 0.604 1.555

10M85 0.770 1.473 1.645 0.715 1.664

10M65 1.675 1.597 0.734 1.904

42



SPSWPrototype

ASCE 7- I 0 ( s )

Finite Element

Analysis (s)Upper Bound: Topkaya's

Method (s)


(s)

Liu'sMethod

(s)3W100 0.442 1.083 0.739 0.4993W85 0.312 0.490 1.207 0.813 0.5543W65 0.536 1.347 0.919 0.602

5W100 0.645 1.285 0.739 0.7905W85 0.458 0.688 1.440 0.774 0.822

5W65 0.771 1.582 0.839 1.108

6W100 0.737 1.307 0.748 0.922

6W85 0.525 0.788 1.493 0.783 0.979

6W65 0.911 1.725 0.927 1.151

8W100 0.895 1.236 0.611 1.081

8W85 0.651 1.016 1.316 0.681 1.237

8W65 1.085 1.529 0.739 1.315

9W100 1.028 1.625 0.780 1.169

9W85 0.711 1.127 1.729 0.819 1.253

9W65 1.228 2.027 0.928 1.355

10W100 1.214 1.478 0.671 1.402

10W85 0.770 1.344 1.496 0.685 1.562

10W65 1.521 1.547 0.714 1.759

While the two methods proposed by Liu et al. [11] and Topkaya [13] were accurate

in predicting the fundamental period of an SPSW frame both contain some limitation from

being a practical and simplified method. The method proposed by Liu et al. [11] requires

taking the determinant o f a matrix which grows in complexity with an increase in total

story height. Not only does limitation introduce a possibility for error if done quickly by

hand, it also will require the engineer to spend some time to set up and analyze their

matrices when done by hand.

The method proposed by Topkaya [13] carries the limitation that the SPSW be

designed using uniform sizes for HBE, VBE, and web-plate elements. This assumption is

43

not practical especially when designing an SPSW greater than 3 stories. While the upper

and lower bound estimations provide a range where the engineer can approximate the

period it does not give a definite value to use in the design.

44

CHAPTER 4. FINITE ELEMENT MODELING OF SPSW

PROTOTYPES

This chapter discusses the finite element modeling and analysis for the fundamental period

o f the SPSW prototypes.

4.1 Open System for Earthquake Engineering Simulation (OpenSees)

For this study SPSW prototypes were modeled and analyzed using the software platform

Open System for Earthquake Engineering Simulation (OpenSees). OpenSees is open

source software produced by the Pacific Earthquake Engineering Research (PEER) Center

for modeling and analyzing the nonlinear response of structural systems. For this study

OpenSees version 2.4.6 was used for all modeling and analysis of the SPSW prototypes

[21]. OpenSees analysis was performed to obtain the fundamental period for each SPSW

prototype using the Eigen algorithm [22].

4.2 Modeling of SPSW Prototypes

The finite element models used in this study were based on models previously used in

studies at San Francisco State University performed by Enright [8] and Barghi [9]. For

specific modeling questions the OpenSees Wiki [23] was utilized.

Web-plate modeling was achieved using the strip method outlined in AISC Design

Guide 20 [1]. The strip method models the web-plate as a series of pined tension only truss

members which are placed diagonally along the HBE. The truss members are modeled to

have a significant tension capacity and negligible compression strength. To achieve the

effects o f 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. A

visualization of the strip method is provided below in Figure 4-1.

45

Th1

y y y y / ^

:Y / .Y vf> ...................... A i

■------------L-------------

Figure 4-1. Visualization of the strip method [1].

The web-plate tension strips are modeled so their angle matches closely to the

tension-field angle, a, shown in Figure 4-1. Tension strips are assumed to be distributed

along the HBE using similar node locations for each story level. This is a simplifying

assumption as the tension-field angle will slightly vary at each story level. An average of

the tension field angles at each level are taken when determining how many tension strips

to use for web-plate modeling. This technique is allowed by AISC Design Guide 20 but is

only recommended when bay-width and story heights are similar, which is the case for the

prototypes used in this study [1].

Determining the number of tension strips to use for the web-plate modeling was

done based on work by Barghi [8]. Tension strips were selected so that diagonal of each

strip would lie within a 5% tolerance o f the average tension field angle used for each SPSW

prototype. Using this requirement, the number of strips in each direction fell between 15

46

and 16 for each prototype. Figure 4-2 shows a 6-story OpenSees model whose setup is

typical for all prototypes.

VBE and HBE elements were modeled so that plasticity was distributed along the

length of the member. This was done so that the tension effects o f the truss pulling at

different locations on the members would be accurately modeled.

4.3 Panel zone modeling

Panel zones at beam-to-column connections were modeled using an algorithm provided by

Lingos [24] which is based on definitions provided by Gupta et al. Modification of Lingos

algorithm was required for the 10-story SPSW prototypes due to node conventions used by

OpenSees. A visualization of a typical OpenSees Panel Zone configuration is shown in

Figure 4-3.

47

Rotational Spring (element 4xy00)500xyl x

[xy02] _ A |xy03][xy04]- 500xy3

[xyOl]500xy8 — - - — 500xy3

« [Xy'° \x y05] -500xy7 - — 500xy4

[xy091 I[xy08] |

500xy6 —

Figure 4-3. Visualization of a typical panel zone configuration at beam-to-column

connection.

The panel zones at beam-to-column connections are modeled similarly to the

configuration shown in Figure 4-3. Coordinates xyOl through xylO represent the node

locations defined by the user to create the location for each panel zone. The variable x

represents the pier each panel zone is being defined on. The variable y represents the story

level where each panel zone is being defined. Panel zone elements are modeled using eight

beam-column elements which carry a very high stiffness represented by elements 500xyl

through 500xy8 in Figure 4-3 [25].

The configuration shown in Figure 4-3 was used for all SPSW models excluding

the 10 story SPSW prototypes. An error in modeling the 10 story SPSW prototypes

presented itself when the panel zones and tension strips were defined at the roof level. The

source of error was found to be from the node conventions used by OpenSees to define the

locations of both the tension strips and panel zones. To remove this error, the node

conventions used for defining the panel zone locations were modified.

48

To visualize the source of this error, Figure 4-4 shows the typical convention for

defining nodes on the SPSW HBE elements. The variable n represents the story being

modeled. When the top level o f the 10 story SPSW prototypes were being modeled in

OpenSees the node coordinates of nOOl with n equal to 10 was required resulting in a node

coordinate of 10001. Based on the panel zone node definitions shown if Figure 4-3 and

the beam node definitions shown in Figure 4-4 an identical node of 11005 is generated

when modeling the first pier, tenth story panel zone and the roof HBE. OpenSees is not

capable to distinguish the difference between the two different nodes and was unable to

run any analysis using the algorithms for the previous models.

To account for this error, the panel zone source code was modified at the top two

HBE elements. Modifying the code was accomplished by inserting an additional zero at

the node coordinate to distinguish it from the node defined on the HBE. This required

modifying the OpenSees source code [24] to account for the additional zero. The modified

code used for the 10 story SPSW models is provided in Appendix A-4-1. A visualization

of the modified panel zone configuration is shown below in Figure 4-5.

49

Figure 4-5. Configuration of the modified panel zones.

The configuration presented in Figure 4-5 was adopted into the 10-story SPSW

finite element models. Once adopted the Eigen analysis was able to converge and the

fundamental period was computed.

50

Chapter 5. STATISTICAL ANALYSIS

To develop a more efficient method for approximating the fundamental period of SPSWs,

a statistical analysis is performed with MATLAB [25] using fundamental period data

obtained from the SPSW prototypes presented in Chapter 2. The objective of this analysis

is to develop adjustment expressions to better approximate the fundamental period

coefficients, Ct and x, used in Eq. (3-5). This method is developed so that the bay-width

and the base-shear participation level of the structure are the only parameters required.

These parameters are typically known to the designer when calculating seismic loads

making it ideal for application in the initial design stage of SPSWs. In this section, a method

for approximating the fundamental period coefficients of SPSWs is developed and

proposed. Adjustment expressions are first developed considering only the bay-width of

the prototypes; expressions are then developed to adjust the bay-width expressions

according to the structures base-shear participation levels.

5.1 Statistical analysis of the SPSW prototypes

ASCE 7-10 Eq. (12.8-7) approximates the fundamental period as a power function of the

structure height as shown in Eq. (3-5). It was assumed for the development o f the proposed

method that the periods obtained from the finite element analysis would follow a similar

trend to be behavior of Eq. (3-5). Using MATLAB [27] a first-degree power regression

was performed assuming the data to follow the following form:

y = a x b (5-1)

where in Eq. (5-1) the coefficients a and b represent the fundamental period coefficients Ct

and x, respectfully; the variable y represents the fundamental period of the system in

seconds; and x represents the height of the structure in feet. Shown in Figure 5-1 is a

comparison of the regression analysis and the finite element analysis. Table 5-1 gives

tabulated values of the coefficients a and b obtained from the regression analysis.

51

N100 N85 N652 2

1.5 . / . 1 5

1 1

0 5 _______ 0.5150 150M100 M85

50 100M65

150

151

05

Q 50 100W100

150

151

05

0,50 100

W85150

2

151

0 5,50 100

W65150

1.5

' 1a> 0 5 a.

15 2

£ 1 3q> 0 5Q-

n

Perio

d (s

.........

Height (ft)150

Height (ft)150

Height (ft)150

| — Power Regression Finite Element

Figure 5-1. Comparison between periods from regression and finite element analysis.

Table 5-1. Fundamental period coefficients from regression analysis.

SPSW Prototype C, X

N100 0.0125 0.9877

N85 0.0144 0.9774

N65 0.0153 0.9839

M100 0.0159 0.9097

M85 0.0192 0.8828

M65 0.0175 0.9312

W100 0.0181 0.8534

W85 0.0174 0.8833

W65 0.0197 0.8786

As seen in Figure 5-1 the power regression models the behavior of the finite

element analysis closely. This suggests that the assumed behavior described in Eq. (5-1)

appropriately models the behavior of the fundamental period of a SPSW. The G and x

coefficient values presented in Table 5-1 are derived using only the height of the structure

52

as a parameter. To adjust the fundamental period coefficients to their assumed correct

value, adjustment functions for Ct and jc are derived first only considering the bay-width of

the structure as a parameter. For this only SPSW prototypes designed for 100 percent base-

shear participation were considered. An additional expression adjusting the coefficients

considering the base-shear participation of the structure using the remaining SPSW

prototypes is then presented.

5.2 Bay-width adjustment function

5.2.1 Ct bay-width adjustment function

The Ct adjustment function for bay-width is assumed to take the form

where, Ct' is the adjusted fundamental period coefficient; G is the code prescribed value

taken from ASCE 7-10t; and f c {L) is the bay-width adjustment function.

To determine the form of the Ct adjustment function Eq. (5-2) can be modified as

Using Eq. (5-3), values for f c (L) can be calculated for the bay widths considered in this

study using SPSWs designed for 100 percent base-shear participation. Table 5-2 gives

values o f / c (L)at the considered bay-widths.

c l = Ctfc(U) (5-2)

(5-3)l a s c e

Table 5-2. /(L ) values for fundamental period coefficient fc(L).

53

Ctfo il)

l a s c e

/c(10 fe e t ) 0.7646

fc( 15 f e e t ) 0.8751

A (20 f e e t ) 0.9840

To model the behavior of the adjustment function,/C(L) it is assumed that a power

series similar to Eq. (5-1) will be an appropriate model to use for calculating the value of

the bay-width adjustment function. With this assumption, the bay-width adjustment

function is expressed as

U L ) = cccLPc (5-4)

In Eq. (5-4), L is the bay-width of the SPSW; a cand /?care coefficients determined through

regression [27] and tabulated in Table 5-3.

Table 5-3. Values for a cand/?c .

Coefficient Value

«c 0.1897

Pc 0.5239

To determine the accuracy of Eq. (5-4) the values of a cand /?cwere applied and

the value of the adjustment function was calculated for each bay-width considered. The

results from Eq. (5-4) were then compared to the values of G obtained from finite elment

analysis. The results from this comparison is presented in Figure 5-2.

54

0,054

0 052

0 05

0 048

O0 046

0.044

0 042

0 04 35j ’ —Finite Element Coefficient Adjustment Function Coefficient t

45 5Bay Width (ft)

55 65

Figure 5-2. Comparison of adjusted G values and finite element values.

As seen in Figure 5-2 the Ct values using Eq. (5-4) can acceptably model the

behavior of Ct values taken from the finite element analysis. From a visual inspection, it

can be assumed that behavior o f the adjustment function described in Eq. (5-4) is an

accurate model to use.

5.2.2 x bay-width adjustment function

The derivation of the adjustment function for the coefficient x used in Eq. (3-5) is

performed in a similar fashion to the Ct adjustment function derived in the previous section.

The adjustment of x to the regression values presented in Table 5-1 was performed

assuming the following expression

x' = xfx(L) (5-5)

where in Eq. (5-5), x' is the adjusted value of the fundamental period coefficient; x is the

code prescribed value obtained from ASCE 7-10; and fx(L) is the bay-width adjustment

function. To determine the appropriate model to use for describing the behavior o ffx(L),

55

values were computed at the SPSW bay-widths similar to the procedure presented in Eq.

(5-3) and are presented in Table 5-4.

Table 5-4. Values of fx{L) at considered SPSW bay-widths.

/*(£) X

XASCE

fx( 10 fe e t) 1.3118

fx{ 15 fe e t) 1.2416

A (20 fe e t ) 1.1714

The behavior of fx(L) is assumed to behave similar to f c {L) in Eq. (5-4) and can

be expressed as

/ , ( i ) = axLV* (5-6)

where in Eq. (5-6), L is the bay-width of the SPSW in feet; a* and /?xare coefficients found

through a power regression analysis in MATLAB [27] and are presented in Table 5-5.

Table 5-5. Coefficient values for the adjustment functionfx(L).

Coefficient Value

2.1368

Px -0.2099

To determine if Eq. (5-6) is an appropriate model to use adjusted values o f x using

Eq. (5-6) were computed and compared with the x values obtained from the finite element

analysis. This comparison is presented in Figure 5-3.

1 — Finite Element CoefHcient — Adjustment Function Coefficient ! . , . .Q = ^ ; v : .= == = = === i-------- 1-------------------------------- 1-------------------------------- 1--------------------------------- 1

3 3 5 4 4 5 5 5 5 6 6 5Bay Width (ft)

Figure 5-3. Validation of power series assumption for jc coefficient adjustment function.

As seen in Figure 5-3, the model presented in Eq. (5-6) is able to closely

approximate the adjusted values of x to those computed from the finite element analysis.

Therefore, Eq. (5-6) is used for adjusting the coefficient x according to the bay-width of

the structure.

5.3 Derivation of base-shear participation adjustment function

Current code provisions require the web-plate of the SPSW to be designed to resist 100

percent of the design-base shear [1], Previous studies conducted by Berman, Enright, and

Bhargi [9,5,6] suggest that the boundary elements in the SPSW can add substantial strength

to the system which is currently ignored. If this additional strength is considered in the

design a reduction in base-shear resisted in the web-plate will be conducted to optimize the

system. The reduction in base-shear resisted in the web-plate due to the plate-frame

interactions were considered in the study with SPSW prototypes being designed for 85 and

57

65 percent of the design base shear in addition to the code prescribed 100 percent

prototypes.

The adjustment functions derived in the previous sections adjust the fundamental

period coefficients, G and x, used in Eq. (3-5) with the bay-width of the SPSW as the only

parameter. To continue exploring the effects of reducing the design base-shear in the web-

plate additional adjustment functions are developed considering a possible variation in the

base-shear participation ratio. These expressions are developed adjusting the a and P

coefficients derived in Eqs. (5-4) and (5-6) based on the design base-shear resisted in the

web-plate.

5.3.1 G base-shear adjustment function.

The adjusted fundamental period coefficient Ct' is expressed as:

where in Eq. (5-7), a cand /?care the bay-width adjustment coefficients whose values are

taken from Table 5-3. To account for a variation in the design base-shear resisted in the

web-plate, adjustment functions for both a cand /?c are derived and presented in this

section.

The behavior of the adjustment functions for a cand /?c are assumed to be modeled

similar to Eq. (5-2) and are expressed as:

C' = Cta cLPc (5-7)

a'c = a cfac ( y ) (5-8)

Pc = PcfficQO (5-9)

The expressions fa (V) and fpc (V) in Eqs. (5-8) and (5-9) are found in a similar

way to / c ( i ) in Eq. (5-3). Values o f fac(V) and fpc (V) at each base-shear participation

ratio considered in this study are presented in Tables 5-6 and 5-7, respectively.

58

Table 5-6. fac(Y) at base-shear participation ratios.

/«c0 0Cioo

/«c(ioo% ) 1

fac (85%) 2.2146

foci 65%) 1.7313

Table 5-7. fpc(V) at base-shear participation ratios.

fficOO «c

ac100

fpc( 100%) 1

fpci 85%) 0.5019

fpci 65%) 0.6967

To determine the appropriate model to use for describing the behavior o f the

adjustment functions over the considered base-shear participation ratios a visual inspection

was performed for both fac(V) and fpc(V). For this inspection values of a cand /?c were

computed for each considered base-shear participation level and plotted over the base-shear

participation ratios. The results from this inspection are presented in Figures 5-4 and

Figures 5-5, respectively.

alph

^.

59

Base-Shear Participation

Figure 5-4. ac over base-shear participation levels.

Figure 5-5. Pc over base-shear participation levels

60

As seen in Figures 5-4 and 5-5 a visual inspection of the adjusted coefficients over

the base-shear participation levels gives no explicit model to accurately describe the

behavior of the data. Therefore, two models are presented and explored to determine the

appropriate form of the base-shear adjustment function. Both models are presented in this

section and later analyzed to determine the appropriate model to use for describing the

behavior of the adjustment functions. The first model assumes the adjustment functions

fac (V) and fp (V) to be modeled as a power series similar to Eqs. (5-6) and (5-7),

expressed as:

fa cOO = (5-1°)ftcQO = e/cv ri,c (5-ii)

In Eqs. (5-10) and (5-11), V is the base shear participation ratio;0ac, yac, 6pc, and

Ypc are coefficients whose values are determined through power series regression in

MATLAB [27]. Values for 6ac, yac, 0pc, and ypc assuming the form presented in Eqs. (5-

10) and (5-11) are provided in Tables 5-8 and 5-9 for fac(V) and fpc(V), respectively.

Table 5-8. Values for 6ac and yac assuming a power series model.

fa c (Y y coefficient value Value

1.493

Vac -0.691

Table 5-9. Values for °Pc and Ypc assuming a power series model.

fpc (K ) coefficient value Value

dPc 0.870

YPc 0.948

61

The second model considered assumes the adjustment functions fac(V) and

fpc(V) to vary linearly over the base-shear participation ratios. With this assumption fa (V)

and fpc (K) are expressed as

fac(V) = eacV + Yac (5- 12)

fpc (v ) = dPcV + Ypc (5_13)

In Eqs. (5-12) and (5-13), V is the base-shear participation ratio; 9ac, yac, 9pc, and

Ypc are coefficients whose values are determined through a linear regression in MATLAB

[28]. Values for 6ac, yac, 6pc, and ypc assuming the form presented in Eqs. (5-12) and (5-

13) are provided in Tables 5-10 and 5-11 for fac(V) and fpc(V), respectively.

Table 5-10. Values for Gac and yac assuming a linear variation.

fa (V) coefficient value Value

0 «c -1.846

Yac 3.187

Table 5-11. Values for 6pc and ypc assuming a linear variation.

fpc ( I /) coefficient value Value

0.7671

YPc 0.0937

5.3.2 x base-shear adjustment function.

A similar procedure is used for the derivation of a base-shear participation adjustment

function for x to that of Ct. The adjusted fundamental period coefficient x' is expressed as:

x' = xaxL^x (5-14)

62

where in Eq. (5-14), a xand /?x are coefficients whose values are obtained through

regression analysis and are presented in Table 5-5. To account for a possible variation in

base-shear resisted in the web-plate, additional adjustment functions are derived for a xand

(3X considering the variation of base-shear participation ratios considered in this study. The

adjusted values of the coefficients a^and j3x are assumed to take the form:

a'x = ocxfax(V) (5-15)

P x = P x f p x GO (5-16)

To determine the behavior o f fUx(V) and fpx(V), values of a xand /3X are calculated

at each base-shear participation level considered in this study, tabulated values are

presented in Tables 5-12 and 5-13, respectively.

Table 5-12. Values of fa (V) at varying base-shear participation levels.

fax 0 0 ax

aXioo

/ax(100%) 1

fax (85%) 0.8663

faxi 65%) 0.8905

Table 5-13. Values of fpx(V) values at each base-shear participation value.

/fe(v0“ *100

/ fe (100%) 1

/& ( 85%) 0.7471

/fc ( 65%) 0.7641

alph

^

63

To determine the appropriate model to use for describing the behavior of a x and

(3X ,the values of ax and (3X at each base-shear participation level are computed and plotted

over the varying base-shear participation levels and presented in Figures 5-6 and 6-7.

Figure 5-6. Variation of axover base-shear participation levels.

64

Figure 5-7. Variation of |3x over base-shear participation levels.

As seen in Figures 5-6 and 5-7, the appropriate model to use for describing the

behavior of axand (3X is not explicitly given. Similar to a c and /3C, two models are

presented and later analyzed to determine the appropriate to use.

The first model assumes the adjustment function to behave as a power series. Using

this assumption, the adjustment functions fa (V) and fpx(V) are expressed as

fax0 0 = 0 « V Ya* (5-17)

fpx(V ) = dpVVPx (5-18)

In Eqs. (5-17) and (5-18), Vis the base-shear participation level; and 0Ux, yUx, Qpx,

and y ^are coefficients obtained through a power regression analysis in MATLAB [27].

Tabulated values of 6Ux, yUx, 0px, and y ^are provided in Tables 5-14 and 5-15 for fUx(V)

and fp QT), respectively.

65

Table 5-14. Values for 9Uxand /^assum ing a power series variation.

fax (V) coefficient value Value

0.9642

Yax 0.2484

Table 5-15. Values for 9px and yp assuming a power series variation.

fpx ( 7 ) coefficient value Value

dPx 0.9420

Ypx 0.6304

The second model assumes a linear variation over base-shear participation ratios.

Under this assumption f a (V) and fp (V) are expressed as

/ o ,0 0 = K V + fax (5-20)

f e W = ek v + Yh (5-2i)

where in Eqs. (5-20) and (5-21), V is the base-shear participation ratio; and the

coefficients 9Ux, yUx, 9px, and yp are obtained through a linear regression analysis in

MATLAB [28]. The values of 9Ux, yUx, 9px, and y^are shown in Tables 5-16 and 5-17 for

fax(V) and fpx(V), respectively.

Table 5-16. Values for 9Ux and ya% assuming a linear variation.

/^^(F)coefficient value Value

0.2894

Yax 0.6777

66

Table 5-17. Values for 0px and Ypx assuming a linear variation.

fp ( V ) coefficient value Value

0.6331

ypx 0.3095

5.3.3 Error analysis of base-shear adjustment function models

The model to describe the base-shear adjustment function was not explicitly given from

the visual inspections performed in Figures 5-4 through 5-7. Both a power series model

and a linear series modeled were considered. To determine which of these two models are

more appropriate to use an error analysis in the form of the root mean squared method was

performed using MATLAB [29]. Results from the error analysis are tabulated in Table 5-

18.

Table 5-18. Root Mean Squared Analysis results for the power series and linear model.

Root Mean Squared Error Analysis

Structure

Class

Power Series

model

Linear

Model

N100 0.2191 0.1414

N85 0.0864 0.0939

N65 0.1007 0.0751

Ml 00 0.1448 0.0783

M85 0.0775 0.0844

M65 0.1113 0.069

W100 0.1605 0.0971

W85 0.0676 0.0746

W65 0.0868 0.0642

As seen in Table 5-18, the linear model for the base shear adjustment function is

more appropriate for modeling the behavior of the SPSWs fundamental period

67

coefficient. From this observation the linear models presented in Eqs. (5-12), (5-13), (5-

20), and (5-21) will be adopted for the proposed method.

5.4 Proposed method for approximating the fundamental period

Based on the derivations outlined in this chapter the proposed method of approximating

the fundamental period of an SPSW prototype can be summarized as

T = Ctf c (L)Hxf* V (5-22)

fc(L) = a cfacm L M >cm (5-23)

fx(L) = axfXc(V )L^I>,m (5-24)

fac(.V) = e„cV + Yac (5-12)

ff,c{V) = V + Yt>c (5-13)

faxm = eaxv + (5-20)

+OSII (5-21)

Tabulated values for all variables used in the method are presented in Table 5-19.

The method proposed in this section was applied to the SPSWs developed for this study.

The fundamental period of the SPSWs using the proposed method are compared with the

fundamental periods obtained from the finite element analysis. This comparison is

presented in Figure 5-8.

68

Table 5-19. Parameter values for proposed methods.

Parameter Value1

Q 0.02(0.0488)

X 0.75(0.75)

«c 0.1897(0.6016)

Pc 0.5239(0.2979)

2.1368(1.6652)

Px -0.2099(-0.2099)

-1.8S6<-0.9681>

Vac 3.187(2.0747)

0.7671(0.9767)

ypc 0.0937(-0.1457)

6 a ax 0.2894(0.1442)

Yax 0.6777(0.8355)

0.6331(0.6331)

y p x 0.3095(0.3095)

'S.I. equivalents are given in parenthesis

69

N100

0

1 5f

1

05

0

1 5f

3 1ES 0.5Q,

qo

50 100M100

150

50 100W100

150

1 5-

1 ■

1 5

1

05

q

1 5

H 1 E§ 0.5

50 100Height (ft)

150

N85

50 100M85

N65

50 100W85

50 100Height (ft)

150

150

2'15

1

2

1 5

10 5,

150

E 1 S

CL.

0,

50 100M65

50 100W65

150

150

50 100 150 ________ Height (ft) .........- —Proposed Function Finite Element Period;

Figure 5-8. Results from application of proposed method on SPSW prototypes.

As seen in Figure 5-8, the proposed method can closely approximate the

fundamental period of the SPSWs designed for this study when compared to the finite

element analysis. From this comparison, it can be concluded that the proposed method

can be used to approximate the fundamental period of an SPSW if designed similar to the

prototypes developed for this study.

70

CHAPTER 6. VALIDATION OF PROPOSED METHOD

To determine the validity of the proposed method, SPSWs with story heights, bay-widths,

and base-shear participation ratios not considered in the initial population. The SPSW

validation prototypes were designed using the specification from AISC Design Guide 20

[1]. Finite element modeling and analysis of the fundamental period was performed using

OpenSees [23]. The fundamental period of the prototypes were then obtained using the

proposed method and compared with the finite element analysis.

6.1 SPSW validation prototypes

Three SPSW prototypes were designed for the validation of the method proposed in this

study. The validation prototypes include a three story, twelve-foot bay-width SPSW with

a 100 percent base-shear participation ratio denoted 3K100; a four story, ten-foot bay width

SPSW with an 80 percent base-shear participation ratio denoted 4N80; and a seven story,

twenty-foot bay width SPSW with a 60 percent base-shear participation ratio denoted

7W60. Design summaries for each validation prototype are presented below in Tables 6-1

through 6-3.

Table 6-1. 3K 100 SPSW design summary

Design Summary: 3K100

Story HBE VBE Web-Plate

Roof W 18X65 - -

3 W 18X65 W 14X233 15 gauge

2 W16X57 W 14X233 11 gauge

1 W24X250 W 14X233 9 gauge

71

Table 6-2. 4N80 SPSW design summary

Design Summary: 4N80

Story HBE VBE Web-Plate

Roof W 14X68 - -

4 W 14X68 W14X145 18 gauge

3 W 14X68 W14X145 13 gauge

2 W14X68 W14X233 11 gauge

1 W 18X65 W 14X233 10 gauge

Table 6-3. 7W60 SPSW Design Summary

Design Summary: 7W60

Story HBE VBE Web-plate

Roof W18X119 - -

7 W18X119 W14X132 23 gauge

6 W18X119 W 14X233 17 gauge

5 W18X119 W 14X23 3 14 gauge

4 W 18X65 W 14X233 12 gauge

3 W 18X65 W 14X45 5 12 gauge

2 W 14X53 W 14X45 5 11 gauge

1 W24X306 W 14X45 5 11 gauge

6.2 Results from approximation method.

The method proposed in this study was applied to the three validation prototypes and then

compared with the results from a finite element analysis results. Tabulated values from the

proposed method and finite element analysis comparison are given in Table 6-5. The

comparison is presented in Figures 6-1 through 6-3.

Perio

d (s

)

72

3K1000.7

0.6

0.5

0.4

0.3

0.2

0 1

p.A. * ■

- .

H i Finite Element mProposed Method □ A S C E 7-10

Figure 6-1. Comparing fundamental period analysis methods for the first validation

SPSW prototype, SPSW 3K100

Perio

d (s

)

73

4N800.8r

0,7-

0 6 -

0.5-

0 4 -

0.3-

02-

0.1

' " "* “ . M*' *" S''1*'*VVvvVV/y-v

' V V X % <V\ *V*/,v%'VvS.\

> x x v

XV«LiV %%:*

F I Finite Element ( I Proposed Method □ A S C E 7-10

Figure 6-2. Comparing fundamental period analysis methods for the second validation

SPSW prototype, SPSW 4N80

74

7W60

i fkmW

■ ■ /■/-> .. '

.■ ■ . ■ ■ ■ ,

* A \■ >

□ Finite Element □Proposed Method □ A S C E 7-10

Figure 6-3. Comparing fundamental period analysis methods for the third validation

SPSW prototype, SPSW 7W60.

Table 6-4. Comparison of fundamental periods from finite element analysis and the

proposed method for the three validation prototypes.

Comparison of Fundamental Period Analysis for Validation SPSW PrototypesSPSW

Prototype Finite Element Period (s) Proposed Method Period (s) Error3K100 0.486 0.503 3.50%4N80 0.738 0.731 0.95%7W60 1.003 1.017 1.40%

As seen in Figures 6-1 through 6-3 the proposed method is able to closely

approximate the fundamental period of an SPSW when compared to a finite element

analysis.

75

6.3 Summary

In this study the fundamental period of fifty-four SPSW structures was used to develop a

simplified method for approximating the fundamental period. The proposed method

approximates by adjusting the fundamental period coefficients to use in ASCE 7-10

equation 12.8-7 for a SPSW structure based on the bay-width and base-shear participation

level. The method proposed in this study is summarized as follows in Eqs (6-1) through

(6-7).

T = Ctf c (L)Hxf*M (6-1)

fc ( 0 = a cfac(V)LPcfec(v) (6-2)

fx ( 0 = axfXc( V ) L ^ f ^ (6-3)

fac (V) = dacV + yac (6-4)

fpcW = ePcV + Ypc (6-5)

fax(V) = da V + Yax (6-6)

fpx(y ) = ep v + Ypx (6-7)

The values for each variable used in Eqs (6-1) through (6-7) are presented in Table

6-5.

76

Table 6-5. Proposed method coefficient values

Parameter Value1

Ct 0.02(0.0488)

X 0.75(0.75)

«c 0.1897(0.6016)

Pc 0.5239(0.2979)

2.1368(1.6652)

Px -0.2099(-0.2099)

-1.856(-0.9681)

Yac 3.187(2.0747)

0.7671(0.9767)

Ypc 0.0937(-0.1457)

a X0.2894(0.1442)

Yax 0.6777(0.8355)

0.6331(0.6331)

Ypx 0.3095(0.3095)

'S.I. equivalents are given in parenthesis

In addition to deriving the proposed method, two methods proposed in previous

research were considered as a method of comparison. These methods approximate the

fundamental period of the SPSW through analysis of its dynamic and material properties

[8, 10]. The two methods were applied to the fifty-four SPSW prototypes to approximate

the fundamental period. The results from each method used in this study are presented in

Figures 6-4 through 6-6 and in Tables 6-6 through 6-8.

Perio

d(s)

77

DM m l1GN100 10N85 10N65

|ASCE 7~10MQpenSees Analysis I iTopkaya Upper Bound | iTopkaya Lower Bound 1HLiu H i Proposed Method]

Figure 6-4. 10-foot bay-width SPSW fundamental period results

Perio

d (s)

Pe

riod

(s)

78

5M100

. n M I6M100

l.i—i M l8M100

l£=L9M100

HI

l£ n10M100

LM

.no3M85

□ M l5M85

£ n M J6M85

8M85

n9M85

□ i

j it—ii3M65

n i l5M65

L t m6M65

8M65

9M65

£ H10M85

I «■10MW65

|ASCE 7-10 WM OpenSees Analysis! iTopkaya Upper Bound I iTopkaya Lower Bound M L iu M i Proposed Method]

Figure 6-5. 15-foot bay-width SPSW fundamental period results

10W100

1f -.1 I h m b

u

J —

3 W 1 0 0 3VV85

n n i5V\

J — -

m o o

, r n n « — ■

5VV85

■ r n n *

t m m

6V\MOO

n n i - ■

6VV85

a l AO' ' i

; f ■ ■

8V\im o o

n W M m M

8VV85

, n ( ! ■U *-------........... ........

L m *

9WMOO

n i l ■ !

9VV85

, n l ■10W85

a3W65

□ j5W65

o M j6W65

8W65

9W65n

Q10W65

0lASCE 7-10 B | OpenSees Analysis I ITopkaya Upper Bound □Topkaya Lower Bound B B t iu M Proposed Method]

Figure 6-6. 20-foot bay-width SPSW fundamental period results.

79

Table 6-6. 10-foot bay-width SPSW fundamental period results.


SPSWPrototype

ASCE 7- 10 (s)

Finite Element

Analysis (s)


‘ (s)


’ IS)

Liu'sMethod

(s)

Kean'sMethod

(s)3N100 0.499 0.700 0.669 0.512 0.526

3N85 0.312 0.566 0.883 0.768 0.571 0.553

3N65 0.618 0.953 0.788 0.661 0.557

5N100 0.807 0.880 0.495 0.861 0.870

5N85 0.458 0.854 0.940 0.532 0.907 0.913

5N65 0.952 1.080 0.608 1.017 0.919

6N100 0.860 0.870 0.437 0.993 1.040

6N85 0.525 0.993 0.884 0.477 1.133 1.092

6N65 1.045 0.997 0.533 1.237 1.098

8N100 1.245 1.085 0.522 1.358 1.380

8N85 0.651 1.303 1.183 0.563 1.420 1.449

8N65 1.491 1.293 0.673 1.657 1.456

9N100 1.334 1.192 0.510 1.515 1.550

9N85 0.711 1.521 1.269 0.580 1.734 1.627

9N65 1.543 1.269 0.580 1.745 1.634

10N100 1.583 1.203 0.555 1.799 1.719

10N85 0.770 1.722 1.405 0.563 1.972 1.804

10N65 1.940 1.475 0.686 2.176 1.811

80



SPSWPrototype

ASCE 7-10 (s)

FiniteElementAnalysis

(s)


(s)


' ( s )

Liu'sMethod

(s)

Kean'sMethod

(s)3M100 0.469 0.953 0.944 0.518 0.479

3M85 0.312 0.512 1.018 0.782 0.573 0.504

3M65 0.564 1.100 0.806 0.667 0.510

5M100 0.710 1.053 0.611 0.832 0.761

5M85 0.458 0.753 1.200 0.685 0.887 0.804

5M65 0.872 1.391 0.744 1.023 0.817

6M100 0.812 1.141 0.644 0.960 0.898

6M85 0.525 0.895 1.281 0.704 1.065 0.950

6M65 0.957 1.408 0.763 1.139 0.966

8M100 1.124 1.366 0.664 1.278 1.166

8M85 0.651 1.171 1.440 0.701 1.332 1.236

8M65 1.332 1.647 0.808 1.530 1.259

9M100 1.146 1.423 0.643 1.296 1.297

9M85 0.711 1.200 1.514 0.712 1.331 1.376

9M65 1.429 1.772 0.816 1.641 1.404

10M100 1.378 1.295 0.604 1.555 1.427

10M85 0.770 1.473 1.645 0.715 1.664 1.516

10M65 1.675 1.597 0.734 1.904 1.547

81



SPSWPrototype

ASCE 7-10 (s)

FiniteElementAnalysis

(s)Upper Bound:

Topkaya's Method (s)

Lower Bound: Topkaya's Method (s)

Liu'sMethod

(s)

Kean'sMethod

(s)3W100 0.442 1.083 0.739 0.499 0.454

3W85 0.312 0.490 1.207 0.813 0.554 0.477

3W65 0.536 1.347 0.919 0.602 0.483

5W100 0.645 1.285 0.739 0.790 0.703

5W85 0.458 0.688 1.440 0.774 0.822 0.744

5W65 0.771 1.582 0.839 1.108 0.758

6W100 0.737 1.307 0.748 0.922 0.821

6W85 0.525 0.788 1.493 0.783 0.979 0.871

6W65 0.911 1.725 0.927 1.151 0.890

8W100 0.895 1.236 0.611 1.081 1.051

8W85 0.651 1.016 1.316 0.681 1.237 1.119

8W65 1.085 1.529 0.739 1.315 1.147

9W100 1.028 1.625 0.780 1.169 1.163

9W85 0.711 1.127 1.729 0.819 1.253 1.239

9W65 1.228 2.027 0.928 1.355 1.273

10W100 1.214 1.478 0.671 1.402 1.273

10W85 0.770 1.344 1.496 0.685 1.562 1.358

10W65 1.521 1.547 0.714 1.759 1.397

82

CHAPTER 7. CONCLUSIONS

This study proposes a simplified method for approximating the fundamental period of a

SPSW structure. Fifty-four SPSW were designed in accordance to specifications outlined

in AISC Design Guide 20 [1], ASCE 7-10 [3], and AISC 341-10 [10], Using the finite

element analysis software OpenSees [21] finite element models were created for each

SPSW prototype and their fundamental period was calculated using an Eigen value

approach. A statistical analysis of the SPSW fundamental period data was performed using

MATLAB [26]. Using the results from the statistical analysis a method was developed

which adjusted the fundamental period coefficients used in ASCE 7-10 equation 12.8-7

based on the structures bay-width and base-shear participation ratios.

The proposed method was applied to the SPSW prototypes developed for this study

and could closely approximate the fundamental period when compared to a finite element

analysis. Three validation prototypes were then developed with story heights, bay-widths,

and base-shear participation ratios not considered in the initial SPSW population. The

proposed method was then used to approximate the fundamental period o f the validation

prototypes and compared with the finite element analysis. From the comparison, it was

found that the proposed method could closely approximate the fundamental period of the

validation SPSWs.

Based on the results from this study it can be stated that the proposed method can

approximate the fundamental period of a SPSW if designed similarly to the prototypes in

this method. The method proposed in this study is limited to the type of web-plate used in

the SPSW design. The prototypes developed for this study were all unstiffened and

unperforated steel web-plates. Further on the effects o f using web-plates with eccentric

configurations could expand the range where this method is valid.

83

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http://opensees.berkeley.edu/wiki/index.php/Main_Page

http://opensees.berkeley.edu/wiki/index.php/ElemPanelZone2D.tcl

http://opensees.berkeley.edu/wiki/index.php/Pushover_and_Dynamie_Analyses_of_2-

http://www.mathworks.com/help/matlab/

http://www.mathworks.com/help/curvefit/power.html

86

[28] "Documentation." Linear Regression. Mathworks, n.d. Web. 19 Dec. 2015.

<http://www.mathworks.com/help/matlab/data_analysis/linear-regression.html>.

[29] "Documentation." Root-mean-square Level. Web. 19 Mar. 2016.

<http://www.mathworks.com/help/signal/ref/rms.html?s_tid=gn_loc_drop>.

http://www.mathworks.com/help/matlab/data_analysis/linear-regression.html

http://www.mathworks.com/help/signal/ref/rms.html?s_tid=gn_loc_drop

87

APPENDIX A-2-1. 3-STORY SPSW DESIGN SUMMARIES

SPSW Design Summary: Building Frame 3N - 10ft Bay-Width3N65 3N85 3N100

Story HBE VBE

Web-Plate(in.) HBE VBE


Web-Plate(in.)

R W 18x65 - - W 16x57 - - W18x71 - -

3 W 14x48 W14xl32 18 ga W16x57 W14xl59 1/16 W18x71 W14x211 13 ga

2 W 14x48 W14xl59 13 ga W 16x57 W14x211 11 ga W 18x71 W 14x211 9 ga

1 W14xl58 W14xl59 12 ga W 18x86 W14x211 9 ga W 18x86 W 14x283 7 ga

SPSW Design Summary: Building Frame 3M - 15ft Bay-Width

3M65 3M85 3M100

Story HBE VBE



Web-Plate(in.)

R W16x57 - - W 16x77 - - W 18x65 - -

3 W 16x57 W 14x74 20 ga W 14x68 W14xl32 19 ga W 18x71 W14xl32 18 ga

2 W 12x45 W14xl32 16 ga W 14x48 W14xl45 14 ga W 18x65 W14xl93 13 ga

1 W 18x86 W14xl32 15 ga W18xl 19 W14xl45 13 ga W18xl 19 W14xl93 12 ga

SPSW Design Summary: Building Frame 3W - 20ft Bay-Width

3W65 3W85 3W100

Story HBE VBE



Web-Plate(in.)

R W21xl32 - - W24xl31 - - W24xl62 - -

3 W21xl32 W14xl59 18 ga W21xl22 W14xl76 16 ga W24xl31 W14x211 14 ga

2 W21xl32 W 14x257 13 ga W21xl22 W 14x283 12 ga W24xl31 W 14x342 1/81 W24x229 W 14x257 12 ga W27x217 W 14x283 10 ga W27x281 W 14x342 3/16

NotesR denotes Roof Level ga deontoes gauge thickness

88

APPENDIX 2-2-2. 5-STORY SPSW DESIGN SUMMARIES


Story HBE VBE



Web-Plate(in.)

R W 14x68 - W 14x68 - - W 12x50 - -

5 W 14x68 W 14x82 22 ga W 14x68 W14xl32 20 ga W12x50 W14xl59 18 ga

4 W 14x68 W14xl32 17 ga W 14x68 W14xl32 14 ga W12x50 W14xl59 13 ga

3 W 14x68 W14xl32 14 ga W 14x68 W14xl59 12 ga W12x50 W14xl59 11 ga

2 W 14x68 W14xl93 13 ga W 14x68 W 14x233 11 ga W 12x50 W14x211 9 ga


SPSW Design Summary: Building Frame 5M - 15ft Bay-Width5M65 5M85 5M100

Story HBE VBE



Web-Plate(in.)

R W 14x53 - - W 14x68 - - W 18x65 - -

5 W 14x53 W 14x74 24 ga W 14x68 W 14x82 21 ga W 18x65 W14xl32 19 ga



2 W 14x53 W14xl32 14 ga W 14x68 W14x211 12 ga W 14x74 W14xl93 11 ga

1 W14x257 W14xl59 13 ga W30x211 W14x211 12 ga W24xl92 W 14x233 10 ga

SPSW Design Summary: Building Frame 5W - 20ft Bay-Width5W65 5W85 5W100

Story HBE VBE



Web-Plate(in.)

R W 18x97 - - W21xl 11 - - W21xl22 - -

5 W 18x86 W14xl32 23 ga W14xl59 W14xl32 21 ga W21xl32 W14xl45 19 ga

4 W16x77 W14xl32 17 ga W14xl93 W14xl76 15 ga W 16x77 W14x211 13 ga


2 W 14x48 W14xl76 13 ga W 16x77 W14xl93 12 ga W21xl 11 W14x257 10 ga

1 W30x211 W14xl76 13 ga W27x217 W 14x370 11 ga W33x236 W14x311 10 ga


89



Story HBE VBE

Web-Plate( in .) HBE VBE


Web-Plate(in.)

R W 18x65 - W21x93 - - W21x93 - -

6 W16x57 W14xl45 19 ga W16x57 W14xl76 17 ga W 18x65 W14x257 16 ga

5 W 14x48 W14xl45 13 ga W16x57 W14xl76 12 ga W 16x57 W14x257 11 ga

4 W 14x48 W 14x233 11 ga W16x57 W 14x257 9 ga W16x57 W 14x342 7 ga

3 W 14x48 W14x233 10 ga W 16x57 W 14x257 7 ga W 18x65 W 14x342 5 ga

2 W 14x48 W14x311 9 ga W18x65 W14x370 5 ga W 18x65 W14x455 3 ga

1 W 14x86 W14x311 8 ga W21x93 W14x370 5 ga W21xl 11 W 14x455 1/4


Story HBE VBE



Web-Plate(in.)

R W16x57 - W16x57 - - W 18x65 - -

6 W16x57 W14xl32 23 ga W16x57 W14xl45 21 ga W 18x65 W14xl76 19 ga

5 W16x57 W14xl32 18 ga W 16x57 W14xl45 1/16 W 18x65 W14xl76 14 ga


3 W 16x57 W 14x211 13 ga W16x57 W 14x233 12 ga W 18x65 W 14x283 1/82 W16x57 W14x211 12 ga W16x57 W14x233 11 ga W 18x65 W 14x283 9 ga

1 W18xl 19 W14x211 12 ga W21x211 W 14x233 1/8 W21xl32 W 14x283 9 ga

90


Story HBE VBE



Web-Plate(in.)

R W18xl06 - - W24xl31 - - W21xl32 - -

6 W18xl06 W14xl32 23 ga W21xl 11 W14xl93 20 ga W21xl32 W 14x211 19 ga

5 W18xl06 W14xl32 17 ga W21x93 W14xl93 14 ga W21xl32 W14x211 13 ga

4 W 18x65 W14xl76 14 ga W 16x77 W14x233 12 ga W16x77 W 14x257 11 ga3 W 18x65 W14xl76 13 ga W 16x77 W14x233 11 ga W 16x77 W 14x257 10 ga

2 W18x65 W 14x233 12 ga W 16x77 W14x311 10 ga W 16x77 W14x342 9 ga

1 W24x229 W 14x233 12 ga W30x211 W14x311 9 ga W30x235 W 14x342 8 ga


91



Story HBE VBE



Web-Plate(in.)

R W 14x68 - W 14x82 - - W 14x68 - -

8 W 14x68 W14xl59 21 ga W 14x82 W14xl45 19 ga W 14x82 W14x211 18 ga


6 W 14x68 W14xl59 13 ga W 14x68 W14x370 11 ga W 14x68 W14x211 9 ga

5 W 14x68 W 14x283 11 ga W 14x68 W 14x370 9 ga W 14x68 W14x455 7 ga

4 W 14x68 W 14x283 10 ga W14xl32 W14x370 7 ga W 14x68 W 14x455 5 ga

3 W 16x89 W 14x289 9 ga W14xl32 W14x370 6 ga W 14x68 W 14x455 3 ga

2 W 16x89 W 14x370 9 ga W14xl32 W 14x500 5 ga W14x68 W14x455 1/4

1 W 16x89 W14x370 8 ga W16xl00 W 14x500 5 ga W14xl32 W14x550 1/4


Story HBE VBE



Web-Plate(in.)

R W 14x68 - - W 14x74 - - W 14x82 - -






3 W 14x68 W14xl93 13 ga W 14x68 W14x342 8 ga W 14x48 W14x283 10 ga

2 W 14x68 W14x257 12 ga W21xl 11 W 14x342 7 ga W 14x48 W14x283 9 ga

1 W 18x86 W 14x257 12 ga W30x235 W14x426 7 ga W21xl22 W14x342 9 ga

92


Story HBE VBE

Web- Plate

....(in.).... HBE VBE


Web-Plate(in.)

R W18xl 19 - W21xl22 - W18xl92 -

8 W 18xl 19 W14x211 22 ga W21xl 11 W 14x233 19 ga W18xl92 W l 4x257 18 ga

7 W18xl 19 W14x211 16 ga W24xl31 W 14x233 14 ga W18xl58 W14x257 12 ga

6 W14xl32 W14x211 13 ga W24xl31 W14x283 11 ga W 18xl 19 W 14x426 9 ga

5 W14xl32 W 14x342 12 ga W 18x86 W14x283 9 ga W18xl 19 W 14x426 8 ga

4 W 18x86 W14x342 11 ga W18x86 W 14x426 8 ga W18xl75 W 14x426 6 ga

3 W14xl32 W 14x342 10 ga W18x86 W 14x426 7 ga W18x97 W 14x426 5 ga

2 W24x306 W 14x342 9 ga W 18x86 W 14x426 3/16 W27x368 W 14x500 4 ga

1 W24x306 W14x500 9 ga W33x221 W14x426 6 ga W27x668 W l 4x730 4 ga

NotesR denotes Roof Levelga deontoes gauge thickness

93



Story HBE VBE

Web-Plate( in .) HBE VBE


Web-Plate(in.)

R W21x93 - W 12x50 - - W16x77 -

9 W 14x48 W14xl59 21 ga W12x50 W14xl59 19 ga W16x77 W14xl76 18 ga

8 W 14x48 W14xl59 15 ga W12x50 W14xl59 13 ga W 16x77 W14xl76 12 ga

7 W21x93 W14xl59 13 ga W16x57 W14xl59 11 ga W 16x77 W14x311 9 ga

6 W21x93 W 14x342 11 ga W16x57 W 14x342 9 ga W16x77 W14x311 6 ga

5 W 14x48 W 14x342 9 ga W16x57 W 14x342 7 ga W16x77 W 14x455 3 ga

4 W 14x48 W 14x342 8 ga W16x57 W 14x342 5 ga W16x77 W14x455 1/4

3 W21x93 W14x550 7 ga W16x57 W14x550 4 ga W 16x77 W14x730 5/16

2 W21x93 W14x550 3/16 W16x57 W14x550 4 ga W 16x77 W14x730 5/16

1 W 14x90 W14x550 3/16 W 18x97 W14x550 3 ga W18xl30 W14x730 5/16


Story HBE VBE



Web-Plate(in.)

R W16x57 - - W21x93 - - W 16x77 - -

9 W16x57 W14xl32 23 ga W 18x65 W14x211 20 ga W 16x77 W14x211 19 ga

8 W16x57 W14xl32 17 ga W 18x65 W14x211 15 ga W16x77 W 14x211 13 ga

7 W16x57 W14xl32 14 ga W21x93 W14x211 12 ga W 16x77 W 14x211 1/86 W16x57 W14x283 12 ga W 18x65 W 14x455 10 ga W16x77 W 14x426 8 ga

5 W16x57 W14x283 11 ga W 18x65 W 14x455 9 ga W 16x77 W 14x426 6 ga

4 W 16x57 W14x283 10 ga W21x93 W14x455 8 ga W16x77 W 14x426 5 ga

3 W 16x57 W 14x426 10 ga W 18x65 W 14x605 7 ga W 16x77 W 14x665 4 ga

2 W16x57 W 14x426 9 ga W 18x65 W 14x605 6 ga W16x77 W 14x665 3 ga

1 W18xl75 W 14x426 9 ga W33x241 W 14x605 6 ga W24xl92 W 14x665 1/4

94


Story HBE VBE



Web-Plate(in.)

R W21x93 - W21xl32 - - W21xl32 - -

9 W21x93 W14xl76 22 ga W21xl32 W 14x233 19 ga W21xl22 W14x257 18 ga

8 W21x93 W14xl76 16 ga W21x93 W 14x233 13 ga W21xl22 W 14x257 13 ga

7 W21x93 W14xl76 13 ga W21x93 W l 4x233 11 ga W21xl22 W 14x257 10 ga

6 W 18x97 W 14x455 11 ga W21x93 W 14x500 9 ga W21xl22 W l 4x605 7 ga

5 W 18x97 W l 4x455 1/8 W21x93 W 14x500 7 ga W21x93 W 14x605 5 ga

4 W18x86 W14x550 9 ga W21x93 W l 4x500 3/16 W21x93 W 14x605 4 ga

3 W 18x86 W 14x605 9 ga W21x93 W 14x665 5 ga W21x93 W14x730 3 ga

2 W18x86 W 14x605 8 ga W21x93 W 14x665 4 ga W21x93 W 14x730 1/4

1 W24x335 W 14x605 ....8 £a.... W33x263 W 14x665 4 ga W30x326 W l 4x730 1/4


95

APPENDIX A-2-6. lO-STORY SPSW DESIGN SUMMARIES

SPSW Design Summary: Building Frame ION - 10ft Bay-Width10N65 10N85 10N100

Story HBE VBE


Web-Plate

......<>"•) ....._ HBE VBE

Web-Plate(in.)

R W 14x48 - - W 14x48 - W 16x77 - -

10 W 14x48 W14xl76 21 ga W14x53 W14xl32 19 ga W 16x77 W14x211 17 ga


8 W14x48 W14xl76 12 ga W 14x48 W 14x283 10 ga W16x77 W 14x211 9 ga

7 W 14x48 W14xl76 11 ga W14x53 W 14x283 8 ga W 16x77 W14x398 3/16

6 W14xl32 W 14x426 9 ga W 14x48 W14x550 6 ga W16x77 W l 4x398 4 ga

5 W14xl32 W 14x426 8 ga W14x68 W14x550 5 ga W 16x77 W14x398 1/4

4 W14xl32 W 14x426 7 ga W 14x68 W14x550 3 ga W 16x77 W14x730 5/16

3 W 14x82 W 14x426 7 ga W 14x48 W14x550 1/4 W 16x77 W l 4x730 5/16

2 W14x82 W 14x426 6 ga W 14x48 W l 4x550 5/16 W 16x77 W 14x730 5/16

1 W14xl32 W l 4x500 6 ga W14xl59 W 14x665 5/16 W14xl59 W14x730 5/16


Story HBE VBE



Web-Plate(in.)

R W 14x74 - - W 16x57 - - W16x77 - -

10 W 14x74 W14xl76 23 ga W 16x77 W14xl32 20 ga W16x77 W 14x257 19 ga

9 W 14x74 W14xl76 17 ga W 16x77 W14xl32 14 ga W 14x68 W l 4x257 13 ga

8 W14x53 W14xl76 13 ga W16x77 W 14x342 12 ga W 18x86 W14x257 11 ga

7 W14x53 W14xl76 12 ga W 16x77 W 14x342 10 ga W 16x77 W l 4x257 8 ga

6 W16x57 W 14x342 11 ga W 16x77 W 14x342 8 ga W 14x82 W 14x500 6 ga

5 W 14x68 W 14x342 10 ga W 16x77 W 14x342 7 ga W 14x74 W 14x500 5 ga

4 W 14x48 W 14x342 9 ga W 16x77 W14x550 6 ga W 14x74 W l 4x500 4 ga

3 W14xl32 W 14x342 9 ga W 14x82 W14x550 5 ga W18x86 W 14x500 3 ga

2 W14xl32 W 14x455 8 ga W 14x82 W14x550 5 ga W14xl32 W 14x665 1/4

1 W l 4x370 W l 4x455 8 ga W24xl46 W14x550 5 ga W40x264 W 14x665 1/4

96


Story HBE VBE

Web-Plate

HBE VBE


Web-Plate(in.)

R W18xl 19 -

.....W21xl22 - - W l 8x234 - -

10 W18xl 19 W14xl59 22 ga W24xl31 W 14x233 19 ga W18xl58 W 14x283 18 ga

9 W18xl 19 W14xl59 16 ga W21xl 11 W 14x233 13 ga W18xl58 W 14x283 13 ga

8 W18xl 19 W14x257 13 ga W21xl 11 W l 4x233 11 ga W18xl58 W 14x283 10 ga

7 W18xl 19 W 14x283 11 ga W 18x86 W l 4x283 9 ga W18xl58 W 14x455 8 ga

6 W18xl 19 W l 4x283 10 ga W21xl 11 W 14x500 8 ga W18xl58 W l 4x455 6 ga

5 W 16x89 W l 4x455 9 ga W21xl22 W 14x500 6 ga W18xl58 W 14x455 4 ga

4 W 16x77 W14x455 8 ga W 18x97 W 14x500 5 ga W 18x234 W l 4x730 3 ga

3 W 16x89 W 14x455 8 ga W21xl22 W 14x500 4 ga W18xl58 W 14x730 5/16

2 W 16x77 W l 4x455 7 ga W21xl22 W 14x605 4 ga W18xl58 W14x730 5/16

1 W24x335 W 14x455 7 ga W40x278 W 14x605 4ga W27x537 W14x730 5/16


97

APPENDIX A-3-1. LIU E T A L . SPSW FUNDAMENTAL PERIOD

APPROXIMATION METHOD

Liu et al. SPSW Fundamental Period Approximation Method

SPSW Prototype: 5W85

Input:Percentage of Shear in Pate

Building Information

Number of Stoires:

Levels:

Frame Width:

k := 0 .8 f

n := 5

i := 0 .. n - 1

Length := 24( L* := Length-in

LT = (240 240 240 240 2 4 0 )-in

Frame Height:

Unbraced Length:

Height := 15f

T

h := Height in-(1 1 1 1 1 1 )

h * = (13 13 13 13 13 13) ft

H :=AW

^6- 156-in >

oor-

5-156-in 65

4-156-in 52

3-156-in 39

2-156-in 26

v 156-in J ,13 J

• f t

Lb := — 3

Lb =(6.667 6.667 6.667 6.667 6.667)ft

Number of Shear Walls on a Story: Nsh := 4

Dead Load: Po l ^ 96 96 96 % % > V f

Live Load: pLL:= lOOpsf

98

Seismic Design sDS := 1.0'Acceleration:

Sds := 1.0'

Sdl := 0.7S

Material Properties

Web Plate

Steel Web Yield Strength:

Ratio of expected yield stress to the minimum yield stress:

Boundary ElementsSteel Yield Strength:

Ration of expected yield stress to the minimum yield stress:

Steel Modulus of Elasticity:

Fyweb := 36'ksi

Ryw :=l-2

Fy := 50ksi

:= 1.1

E := 29000ksi

Effective Length factor:

CodeBending phi factor

Tension phi factor

Shear phi factor

Compression phi factor

K := l.CAAAA

* b := 0.S

<j,t := O.S

<t>v := 1C

d»c := O.S

Braced member phi factor := 0.7f

ASTM A36

99

Connection

Weld Yield Strength: FyWeld := 36 ksi

Weld ELectrode Strength: FEXX:= 70 ksi

Constant Variables

P I n := 3.1415926

Equivalent Lateral Force Procedure: Steel special plate shear wall

r := 7 ASCE 7-10 Table 12.2-1AAA/

Q := 2 .fAAAA

Cd := 6.5

I := i Importance Factor for Typical Buildings w/outlargeimpact to community see Table 1.5-1

Building Weight:

Lbuilding := 150ft

Wbuilding := 150ft

4 2FloorArea := Lbuilding Wbuilding = 2.25 x 10 ft

100

Wh. := p ^ L ’FloorArea

Wh =

1.867x 10

2.16x 10

2.16x 10

2.16x 10

2.16x 10

kip

FlrW := stack(w h0 ,W hr Wh2 ,W h3 ,W h4j

FlrW

1.867x 10

2.16x 10

2.16x 10

2.16x 10

kip

2.16x 10 J

W := Y FlrW. = 1.051 x 104-kip/WW / , 1 ri

WEB PLATE DESIGN PROPERTIES

Standard Steel Plate Thicknesses

wSTD '1 3 1 5 3 7 1 9 5 Y 1

- ins j _1 16 8 16 4 16 8 16 2 16

101

Gauge Steel Defined

wg •

0

0

0

0.2391

0.2242

0.2092

0.1943

0.1793

0.1644

0.1495

0.1345

0.1198

0.1046

0.0897

0.0747

0.0673

0.0598

0.0538

0.0478

0.0418

0.0359

0.0329

0.0299

0.0269

v0.0239y

in

Steel Gauge begins at 3

Web Plate Thickness Per Story

V :=

•wg21

^0.0027^'wSl5 0.0056

wSi3 = 0.0075

0.0087w®12 v 0.01 j

W lly

fit

HORIZONTAL BEAM ELEMENT DESIGN PROPERTIES

ROOF HBE:

FIFTH FLOOR HBE:

FOURTH FLOOR HBE:

THIRD FLOOR HBE:

SECOND FLOOR HBE:

FIRST FLOOR HBE:

Whbe6 Q = "W21X111"

Whbe5Q = "W21X111"

Whbe4Q = "W14X193"

Whbe3Q = "W16X77"

Whbe2 Q = "W16X77"

Whbel Q = "W27X217"

W Section Properties Table 1-1 [AISC 316-10]Area:

p := 2

. 2' Whbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel in I P P P P P P

Plastic Section Modulus:

Lxb '=. 3Whbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel in

I P P P P P P

Zyb :_. 3

Whbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel in I p P P P P P

103

Inertial Properties:

v := 1CHAKi

4Iu := ( Whbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel )mb LA P P P P P P /

J L := 3

du:=f('w hbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel jinj1b LA P P P P P P / J

S U = 17

r , ;= [YWhbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel W"IT y t > L v P P P P P P / J

13r , := [YWhbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel )in“|TxD LA P P P P P P / J

£L'-=4bfl, := |7Whbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel )in“|T

™ Lv P P P P P Pt J

&,:= 6

tfh '■-[(Whbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel )in1Ttb LI p p p p p p / J

JL--8htwu := (Ywhbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel 'l'iT

b U P P P P P pJ J

£J= 5

t , := 7 Whbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel l in fw b Lv p p p p p p / J

Plastic Development Length:

JL:= 9

L. := \ ( Whbe6 Whbe5 Whbe4 Whbe3 Whbe2 Whbel Vftli3 LV P P P P P P / J

104

10.24 > ^0.227 > f 1.208 N f 0.129 N

10.24 0.227 1.208 0.129

14.31 0.394 2 1.537 0.116ft A b = Z xb = gaI Jb =8.72 u 0.157 0.649 u 0.054

8.72 0.157 0.649 0.054

,11.73 , v 0.444 , v 3.078 j v 0.43 ,

f t

d b =

1.792 A

1.792

1.292

1.375

1.375

2.367 )

VERTICAL BOUNDARY ELEMENT DESIGN PROPERTIES

FIFTH FLOOR VBE: Wvbe50 = "W14X132’

FOURTH FLOOR VBE: Wvbe40 = "W14X176'

THIRD FLOOR VBE: Wvbe3Q = "W14X2111

SECOND FLOOR VBE: Wvbe2Q = "W14X193'

FIRST FLOOR VBE: Wvbel = "W14X370'

W Section Properties Table 1-1 [AISC 316-10]

Area:

Ac .

SJ-=2

(Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel P P P P PKf

105

Modulus of Plasticity:

&„■= 1 1

z •=xc •. 3f Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel W

{ P P P P P/

JL:=

zyc • Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel I P P P P


d := 1C

!C := (Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel )in \ P P P P P/

£ .:= 3

dp := r(Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel )in1 c Ll P P P P P / J

i T

r :=f(Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel V I 1 yc LI P P P P P / J

13

■ iTr ;= [Y Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel \in~l xc LV P P P P P / J£L-=4

- [ (b^c := Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel • -|T

• nTtfP :=|7Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel W~|tc L\ p p p p p / J

htw. := ( Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel )T c \ p p p p p)

t_ := rfWvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel Vnlwc LI P P P P P / J


S c == [(Wvbe5 Wvbe4 Wvbe3 Wvbe2 Wvbel P P P P P

S c T =

r 13.28^

14.2

14.38

14.31

v 15.08y

).«]

V

0.269^

0.36

0.431

0.394

0.757y

f t 7 -xc

r 1.013n

1.385

1.688

1.537

v3.186y

gal L =

r 0.074^

0.103

0.128

0.116

v0.262y

dc =

r 1.225^

1.267

1.308

1.292

0 .4 9 2 ,

f t

Angle of Inclination

a . := atan i

1 +tw L.Wj i

2* A c-

1 + tw -h .■Wj i 360 L L. A h c. , b.

( 0.764 > 43.791 ^

0.75180

42.963

0.757 a deg a — - 43.352

0.723 41.401

v 0.716 j v41.016 j

alpha := reverse (a)

Length of beam from centerline: Lcf := L - d

Approximation MethodShear Frequency Calculation

Column Stiffness

1 2 E L

(hif

r 280.496^

392.328

487.66

439.994

v 997.32

kipin

/

Web Plate Stiffness

EL. t .i i:= “ ( h y (sin(alpbai)) -(cos(alphai))

Total Stiffness

1.311 x 10

1.148x 103

997.192

746.864

v 366.309 j

kipin

k. := k„ +. + kp. = - 1 F1

/

Define Global Stiffness Matrix

1.591 x 10

1.541 x 10

1.485x 10

1.187 x 10

1.364 x 10 J

108

% + k3 " k3

-k3 k3 + k2

“ tot :=-k 2 k2 + kj

0

0 0

k l + k0 “ko

Define Mass Matrix

a := 32.2g

m. := 1

FlrW.

Nsh 'gm :

4.669 x 10

5.4 x 10

5.4 x 10

5.4 x 10

5.4 x 10

•lb

M := diag ( m4 m3 m2 m i m0 f

cos := J sort eigenvals M ' • JJ =

f 9.043 ^

27.595

42.628

53.528

v 62.66 j

Shear Natural Frequency of System

©c = 9.043— s0 s

2-nT •=----- = 0.695s

°°ss0

109

Flexure Frequency Calcualtion

Average mass

Z mi

Have • n-13ft

Height of structure

Not := n ' 13-ft = 780in

Moment of Inertia of system

:= 2-L +cantilever j • Cj 2 = ...-ft

.6 . 4^average • mean(Wntilever) l-84x 10 -in

cantilever

r 54.036^

72.151

86.368

79.12

151.914;

• f t

Flexural Frequency

p := 1.8752

cof := P-El,

2=11 . ,4.29,1

110

Natural Frequency of system

1 1 2 invco := ----------+ ------ = 0.017s

co := I—!— = 7.642- \j invco s

Natural Period of System

T := — = 0.82220%/wv co

I l l

APPENDIX A-3-2. TOPKAYA AND KURBAN SPSW FUNDAMENTAL PERIOD

APPROXIMATION METHOD

Topkaya and Kurban Fundamental Period Approximation Method

SPSW Prototype: 3N85

Input:Percentage of Shear in Pate K := i

Building Information

Number of Stoires: n := 3

Levels: i : = 0 . . n - l

Frame Width: Length := 12C L. := Length-in

LT =(120 120 120)-in

Frame Height: Height := 15( h := Height in ( 1 1 1)

hT =(13 13 13) ft

,T

( 3-156-in^ ( 39 1

H := 2156in = 26 -ftWVv 156in J Vl3y

Unbraced Length: LbT =(3.333 3.333 3.333) ft

Number of Shear Walls on a Story:

Dead Load: pDL := (83 96 96)T psf

Live Load: PLl := lOOpsf

112

Seismic Design Acceleration:

SDS := 1.0 /

Sds := 1.0 /

Sdl := 0.7S

Material Properties

Web Plate

Steel Web Yield Strength:

Ratio of expected yield stress to the minimum yield stress:

Boundary ElementsSteel Yield Strength:

Ration of expected yield stress to the minimum yield stress:

Steel Modulus of Elasticity:

Steel Modulus of Rigidity:

Effective Length factor:

CodeBending phi factor

Tension phi factor

Shear phi factor

Compression phi factor

Fyweb := 36'ksi

^yw • 1.3

Fy := 50ksi

Ry := L]

E := 29000 ksi

G := 11.5106 psiAAA/ r

K := l.CAAAA

* b := 0.S

♦ t := 0.S

<t.v :=LC

♦ c : = 0 5

Braced member phi factor ^)br := 0.7f

ASTM A36

113

Connection

Weld Yield Strength: FyWeld := 36ksi

Weld ELectrode Strength: FEXX:= 7(>ksi

Constant Variables

pj n := 3.1415926

Equivalent Lateral Force Procedure: Steel special plate shear wall

R ;=7 ASCE 7-10 Table 12.2-1AAAt

Q := 2 .fAAAA

Cd := 6.f

I := i Importance Factor for Typical Buildings w/outlargeimpact to community see Table 1.5-1

Building Weight:

Lbuilding := 180ft

Wbuilding := 120ft

FloorArea := Lbuilding-Wbuilding = 2.16x 104-ft2

Wh. := Pql.- Floor Area

Wh =

1.793 x 10'

2.074 x 10'

2.074 x 10'

kip

FlrW := s tack |w hQ,Wh j , Wh 2j

FlrW

1.793x 10

2.074 x 10

V2.074x 10

kip

W - V FlrW. = 5.94 x 10 -kit/WW / J 1 r

WEB PLATE DESIGN PROPERTIES

Standard Steel Plate Thicknesses

t,wSTD •_ L 1 _ L J . _ L 2 J L J . J L 116 8 16 4 16 8 16 2 16 8

115

Gauge Steel Defined

Wg ■

\

V

0

0

0

0.2391

0.2242

0.2092

0.1943

0.1793

0.1644

0.1495

0.1345

0.1198

0.1046

0.0897

0.0747

0.0673

0.0598

0.0538

0.0478

0.0418

0.0359

0.0329

0.0299

0.0269

0.0239^

in

Steel Gauge begins at 3

Web Plate Thickness Per Story

wSTD0( 0.0625^

tw§ n = 0.1198 •in

twg9 ,

v0.1495y

116

HORIZONTAL BEAM ELEMENT DESIGN PROPERTIES

ROOF HBE: Whbe30 = "W16X57"

Third Floor HBE: Whbe2() = "W16X57"

Second Floor HBE: Whbel Q = "W16X57"

First Floor HBE: Whbe0() = "W18X86"

W Section Properties Table 1-1 [AISC 316-10] Area:

. 2~|TAb := Whbe3„ Whbe2„ Whbel „ WhbeO„ -in'Y

Modulus of Plasticity:

,)z x b :" Whbe3 j j Whbe2 ] Whbel x ] WhbeO ] )in3

Zyb := [ Whbe3 j 5 Whbe2J5 Whbel |5 WhbeO ]5 )-in3

l bWhbe3 jQ Whbe2 ]Q Whbel 1() WhbeO ]Q |in 4

■ iT

• nT

Inertial Properties

.1db :=[7whbe33 Whbe23 Whbel 3 WhbeO3 jinjT

ryb := g w h b e3 17 Whbe2]? Whbel ]? WhbeO 1?)inj

rxb := [ (Whbe3 i3 w hb e2 13 Whbel ]3 WhbeO )3)inJ

bf b :=^W hbe34 Whbe24 Whbel 4 Whbe04 j-inj

tfb ;= [ ( whbe3 6 Whbe26 Whbel 6 WhbeO6jinJ

htwb := ^Whbe3g Whbe2g Whbel g WhbeO J

twb := r(w hbe35 Whbe2$ Whbel 5 WhbeO^-inj1

L p:= ^W hbe39 Whbe29 Whbel g Whbe09 )-ftj

• iT

iT

\T

Vertical Boundary Element Design Properties

Third Floor VBE: Wvbe3 Q =

Second Floor VBE: Wvbe2 Q =

First Floor VBE: Wvbel Q =

W Section Properties Table 1-1 [AISC 316-10]

Area:

Ac : [ Wvbe32 Wvbe22 Wvbel A in 2

Modulus of Plastcity:

3Z •= xc *

^yc

|W vbe311 Wvbe211 Wvbel ^ j i n

Wvbe3 Wvbe215 Wvbel 15


Ic :=[(W vbe3 |0 Wvbe2|0 Wvbel ,„ )

dc := ^W vbe33 Wvbe23 Wvbel 3 jinj

ryC := [(W vbe317 Wvbe217 Wvbel jinj

rxe := [( Wvbe3 J3 Wvbe2 J3 Wvbel ]3 jinj

iT

m]

• 1T m]iTbfc :=[(W vbe34 Wvbe2^ Wvbel^j inj

tfc ^W vbe36 Wvbe2^ Wvbel 6 j-inJT

htwc := ( Wvbe3 g Wvbe2 g Wvbe 1 g jT

twc ^ Wvbe3 Wvbe2^ Wvbel 5 jinJT


LpC :=^W vbe39 Wvbe29 Wvbel

kdet := [(Wvbe322 Wvbe222 Wvbel 22 j-inj7

"W14X159

"W14X211

MW14X211

118

Angle of Inclination

a. := atan 1

1 +tw LWj l

2-A

1 + tw *h .■Wj iw3

360 L L. Al e. , b.

^0.707^180

a deg a ' n

( 40.496^

a = 0.674 38.646

v 0.66 y v37.807y

Length of beam from centerline: Lcf := L - dc

Application of Topkaya's Method: Upper Bound

Define Necessary Parameters

Thickness of Web-Plate: ptk„ := t,„ =0.063inu wo

Height of Web-Plate: piwu := h Q = 156in

Column Moment of Inertia: IyBEu:= = 19x lo3-ir|4

2Column Cross-Sectional Area: AVBEu: =Ac =46.7 in

Column Depth: dVBEu:= dc = 15-in

119

Cross-Sectional Area of Colunm Flange:

. 2Aflu := tfco bfcQ = 18-564in

Cross-Sectional Area of Column Flange:

Mass per story:

^webu * ( Cq ‘ detQjVcQ 7.45in. 2

^tory *W

32.2-g- 156-in3 lb

= 1.183 x 10 —in

Zalkas Factor: iv :=

r 0.493^

0.653

0.770

0.812

0.842

0.863

0.879

0.892

0.902

0.911)V

Calculate Necessary Q values

Q]upper := Aflu(°-5-Plwu + dVBEi) = L726x ^ ' i "3

S upper := Q]upper + Aw ebu°-5'(Plwu + dVBEi) = 2-363x 10'3 . 3 in

Supper A VBEu(p1wu + dVBEi) " 7‘986x 1()3' in3

(Plwu )2 3 3Q4upper := Q3upper + ---- Ptku = 8176x 10 in

120

Calculate P 1 and p 2

2 2 Ql upper + S upper , 8 . 6

Pj - dVBEu_ l 725x 10 inwc0

2 2:= ^ u p p e , + S upper _ |()n

2 lp tk u

Calculate Total p

p := pj + p2 = 1.632x 10U -in6

Determine Moment of Inertia for the Wall with respect to the Neutral Axis

Centroid of Web-VBE Interface:

0 '5' tfcj' twi-tfci + (°-5-h i)-(h i-tWj) + (°-5'tfcj + h ; -d. := ----------- 7— = ...-in

2-twi '( tfci) + *Wj

Averaged Centroid: dsys := mean(d) = 6.5ft

Moment of Inertia for 1 3 2 5 .4Web-VBE Interface: *w := ^ 'P tku 'P lwu + 2 AVBEiidsys + 2 I VBEu = 5-918x 10 in

121

Calculate Approximate Fundamental Period of SPSW

Application of Topkaya's Method: Lower Bound

Define Necessary Parameters

Thickness of Web-Plate: ptk, := t = 0.15-inn-1

Height of Web-Plate: piw, := h = 156in

Column Moment of Inertia: Iy B£i:= Ic = 2.66x l 03-in4n-1

2Column Cross-Sectional Area: A VBEj := A c = 62-in

Column Depth: d VBE1:= dc = 15.7-in

Cross-Sectional Areaof Colunm Flange: Af l l := tfcn_ , 'b fcn_ 1 = 24-648in

Fundamental Period: T,upper ' + ------- = 0.883s2 2 f fbu su

n-1

Cross-Sectional Area ,of Column Flange: A webl := e •twc'n-1

122

Calculate Necessary Q values

Qllower := Afir(0-5-plwj + dVBE|) = 2.31 x 103-in3

.3 . 3S lo w er Qllower + Awebl'°-5'(Plwl + dVBEj) “ 3-147x 10 in

.4 . 3S lo w er AVBEl(Plwl + dVBEj) ~ 1 065x 10 -in

(Plwl)2 4 3Q41ower := Q31ower + ~ J -----Ptkl = 1-Hx 10 -in

Calculate p 1 and (3 2

2 2 ^1 lower + S low er 8 . 6

P |j VBE1- 2*441 x 10 -in

twCn -l

2 2^3 lower + S low er ll . 6

B91 := ----------------------------- plwi = 1.234x 10 -in21 2-ptk, 1

Calculate Total (3

^ : = P n + P 2I = 1.237x 1011 -in6

Determine Moment of Inertia for the Wall with respect to the Neutral Axis

Centroid of Web-VBE Interface:

°-5' tfci' twi'tfci + (°-5'h i)'(^h i‘twij + + h i _ ‘fC j^ fC j 'Vd. := ----------- 7— = ...-in

1 2 ' S f tfci ) + h *’twi

123

Averaged Centroid: d mean(d) = 6 .5ft

Moment of Inertia for 1 3 2Web-VBE Interface: !w l := ~ Ptkl Plwl + 2 AVBEIdsys + 2 IVBE1:

Calculate Effective Shear Area

i 2'w 2" r>"2-inEffective Shear Area: KA . := — = 2.832-Wl p

Calculate Bending and Shearing Natural Frequencies

0 5595 I ^ ^wlBending Natural Fequency: f , . := rf •—1 I-----------= 5 .437-H2n~' |H Qj 2 J mstory

Shearing Natural Frequency: f , := rf ------------= 1.341-H2n -l 4-Hq J ^ to ry

Calculate Approximate Fundamental Period of SPSW

T *=Fundamental Period: lower • — + — = 0.768s 2 2

fbl fsl

8.07 x 105-in4

124

APPENDIX A-4-1. lO-STORY SPSW PANEL ZONE OPENSEES CODE

# SPSW PROTOTYPE: 10W10 0# PANELZONE MODELING WITH MO DIF ICAT ION AT ROOF LEVEL# d e f i n e e l a s t i c p a n e l z o n e e l e m e n t s ( a s s u m e r i g i d )# e l e m P a n e l Z o n e 2 D c r e a t e s 8 e l a s t i c e l e m e n t s t h a t f o r m a r e c t a n g u l a r p a n e l z o n e# r e f e r e n c e s p r o v i d e d i n e l e m P a n e l Z o n e 2 D . t e l# n o t e : t h e n o d e l D a n d e l e l D o f t h e u p p e r l e f t c o r n e r o f t h e PZ m u s t b e i m p o r t e d# e l e l D c o n v e n t i o n : 5 0 0 x y a , 5 0 0 = p a n e l z o n e e l e m e n t , x = P i e r # , y = F l o o r ## " a " c o n v e n t i o n : d e f i n e d i n e l e m P a n e l Z o n e 2 D . t e l , b u t 1 = t o p l e f t e l e m e n tset Apz 1100.0; # a r e a o f p a n e l z o n e e l e m e n t ( m a k e m u c h l a r g e r t h a n A o f f r a m e e l e m e n t s )set E 29000 . 0;set Ipz 1 . 0 e 5 ; # m o m e n t o f i n t e r t i a o f p a n e l z o n e e l e m e n t ( m a k e m u c h l a r g e r t h a n I o f f r a m e e l e m e n t s )# e l e m P a n e l Z o n e 2 D e l e l D n o d e R E A_P Z I _ P Z t r a n s f T a ge 1 e m P a n e 1 Z o n e 2 D 500121 1201 $E $Apz $Ipz $PDeltaTransf; # P i e r 1 , F l o o rZe l e m P a n e l Z o n e 2 Do

500221 2201 $E $Apz $ Ipz $PDeltaTransf; # P i e r 2 , F l o o rze l e m P a n e l Z o n e 2 DQ

500131 1301 $E $Apz $Ipz $PDeltaTransf; # P i e r 1, F l o o rO

e l e m P a n e 1 Z o n e 2 D3e l e m P a n e l Z o n e 2 D

500231 2301 $E $Apz $ Ipz $PDeltaTransf; # P i e r 2 , F l o o r

500141 1401 $E $Apz $Ipz $PDeltaTransf ; # P i e r 1 , F l o o r*±

# [ J T 3 0 1 - 2 7 ] c h a n g e F l o o r 4 t o R o o f 2 / 2 5 / 2 0 1 5 R o o f C h a n g e d t o F l o o r 4e 1 e m P a n e l Z o n e 2 DA

500241 2401 $E $Apz $Ipz $PDeltaTransf ; # P i e r 2 , F l o o r

# [J T 3 0 1 - 2 8 ] c h a n g e F l o o r 4 F l o o r 4

t o R o o f CHANGE; 2 / 2 5 / 2 0 1 5 R o o f <C h a n g e d t o

e l e m P a n e l Z o n e 2 Dcr

500151 1501 $E $Apz $ Ipz $PDeltaTransf; # P i e r 1 , F l o o rbe l e m P a n e l Z o n e 2 Dc

500251 2501 $E $Apz $Ipz $PDeltaTransf; # P i e r 2 , F l o o rDe l e m P a n e l Z o n e 2 D 500161 1601 $E $Apz $Ipz $PDeltaTransf; # P i e r 1 , F l o o rbe l e m P a n e l Z o n e 2 D 500261 2601 $E $Apz $Ipz $PDeltaTransf; # P i e r 2 , F l o o rbe l e m P a n e l Z o n e 2 D7e l e m P a n e l Z o n e 2 D7e l e m P a n e l Z o n e 2 D8

500171 1701 $E $Apz $Ipz $PDeltaTransf; # P i e r 1 , F l o o r



125

elemPanelZone2D 500281 2801 $E $Apz $Ipz $PDeltaTransf; # P i e r 2 , F l o o r

elemPanelZone2D 500191 1901 $E $Apz $Ipz $PDeltaTransf; # P i e r 1 , F l o o r 9elemPanelZone2D 500291 2901 $E $Apz $Ipz $PDeltaTransf; # P i e r 2 , F l o o r 9# e l e I D { 5 0 0 1 1 0 1 5 0 0 2 1 0 1 5 0 0 1 1 1 1 5 0 0 2 1 1 1 } n o d e R { 1 1 0 0 0 1 2 1 0 0 0 1 1 1 1 0 0 1 2 1 1 0 0 1 } $E $ A p z $ I p z $ P D e l t a T r a n s f# p r o c e l e m P a n e l Z o n e 2 D { e l e l D n o d e R E A _P Z I__PZ t r a n s f T a g }foreach elelD {5001101 5002101 5001111 5002111} nodeR {llQOOl 210001 111001 211002} {set A_PZ 1100.0; # a r e a o f p a n e l z o n e e l e m e n t ( m a k e m u c h l a r g e r t h a n Ao f f r a m ee l e m e n t s )set E 29000 . 0;set I_PZ 1. 0 e 5 ;set transfTag 1;# d e f i n e p a n e l z o n e n o d e sset node_xy01 $nodeR; # t o p l e f t o f j o i n t

[expr $node__xy01 + 1][expr $node__xy01 + 2][expr $node_xy01 + ][expr $node_xy01 + 4]

set node_xy02 set node_xy03 set node_xy04 set node_xy05m i d d l e ,h o r i z o n t a l r i g h t ) set node_xy06 [expr $node_xy01

[expr $node__xy01 [expr $node_xy0i [expr $node_xy01 [expr ($node__xy01

# t o p l e f t o f j o i n t# t o p r i g h t o f j o i n t# t o p r i g h t o f j o i n t# m i d d l e r i g h t o f j o i n t ( v e r t i c a l

5] ;6] ;

] ;S] ; )/K

# b t m r i g h t o f j o i n t# b t m r i g h t o f j o i n t# b t m l e f t o f j o i n t# b t m l e f t o f j o i n t+ 10]; # m i d d l e l e f t o f j o i n t

set node_xy07 set node_xy08 set node_xy0 9 set node_xyl0( v e r t i c a l

m i d d l e , h o r i z o n t a l l e f t ) set node_xy6 [expr ( $ n o d e _ _ x y 0 1 - 1 ) / set node_xy7 [expr ($ n o d e _ _ x y 0 1 - 1 ) /# c r e a t e e l e m e n t I D s a s a f u n c t i o n o f f i r s t i n p u t e l e l D (8 p e r p a n e l z o n e )set xl $eleID; # l e f t e l e m e n t o n t o p o f p a n e l z o n e

+ 6 ] ; # b t m c e n t e r o f j o i n t + 3 ; # t o p c e n t e r o f j o i n t

1 ] ; # r i g h t e l e m e n t o n t o p o f p a n e l z o n e2 ] ; # t o p e l e m e n t o n r i g h t s i d e o f p a n e l z o n e3 ] ; # b t m e l e m e n t o n r i g h t s i d e o f p a n e l z o n e4 ] ; # r i g h t e l e m e n t o n b t m o f p a n e l z o n e

set x2 [expr $xl +set x3 [expr $xl +set x4 [expr $xl +set x5 [expr $xl +- 1-

C:\Users\Benjamin\Desktop\A-4-1.tcl Monday, March 21, 2016 7:56 PMset x6 [expr $xl + 5 ] ; # l e f t e l e m e n t o n b t m o f p a n e l z o n eset x7 [expr $xl + 6 ] ; # b t m e l e m e n t o n l e f t s i d e o f p a n e l z o n eset x8 [expr $xl + 7 ] ; # t o p e l e m e n t o n l e f t s i d e o f p a n e l z o n e# c r e a t e p a n e l z o n e e l e m e n t s# t a g n d l n d J A__PZ E I _ P Z t r a n s f T a gelement elasticBeamColumn $xl $node_xy02 $node_xy7 $A_PZ $E $1 PZ $transfTag;

126

element elasticBeamColumn $x2 $node_xy7 $node_xy03 $A__PZ $E $I_PZ $transfTag;element elasticBeamColumn $x3 $node__xy05 $node_xy04 $A__PZ $E $I_PZ $transfTag;element elasticBeamColumn $x4 $node_xy0 6 $node_xy05 $A_PZ $E $I_PZ $transfTag;element elasticBeamColumn $x5 $node_xy6 $node_xy07 $A_PZ $E $I__PZ $transfTag;element elasticBeamColumn $x6 $node_xy08 $node_xy6 $A__PZ $E $I__PZ $transfTag;element elasticBeamColumn $x7 $node_xy0 9 $node__xylO $A_PZ $E $I_PZ $transfTag;element elasticBeamColumn $x8 $node_xylO $node_xy01 $A_PZ $E $I_PZ $transfTag;}

127

APPENDIX A-6-1. VERIFICATION OF PROPOSED METHOD WITH

VALIDATION PROTOTYPE

Validation of Kean's Fundamental Period Approxiamtion Method

SPSW PROTOTYPE: 3K100

Building Parameters

Building Height (feet)

H := 3SAAA/

Base-Shear Participation Level

V := 1AA A /

Bay-Width (feet)

ASCE 7-10 Equation 12-8-2 Parameters

c t := 0.02

x := 0.75

128

Proposed Method Coefficients

a c := 0.189'

Pc := 0.523‘

Px := - 0.209‘

epc

?pc

a x := 2 -136f e a x :

ax *

0a C := -l.B 46( 0p x :

Y a C := 3.187C Ypx

Proposed Method

faC := 0a C V+ YaC = L341

fpc := 0pc v + y p c = 0 861

W -= ®ax ^ + Y ax = 0-967

fp x := 0p xV+ Ypx = 0-943

P c fpcfC := a C fa C L = 0.78

f, := CL,-f„v-L^X = 1.264x ax

xf

T = 0.503 seconds

TpE *— 0.48(

Errortf e - t

lFE3.466%

= 0.767

= 0.093'

- 0.289^

= 0.677'

= 0.633:

= 0.3095

fundamental period prediction of steel plate shear wall

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