Accuracy of a Simplified Analysis Model for Modern Skyscrapers

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1 Brgham Young Unversty BYU ScholarsArchve All Theses and Dssertatons Accuracy of a Smplfed Analyss Model for Modern Syscrapers Jacob Scott Lee Brgham Young Unversty - Provo Follow ths and addtonal wors at: Part of the Cvl and Envronmental Engneerng Commons BYU ScholarsArchve Ctaton Lee, Jacob Scott, "Accuracy of a Smplfed Analyss Model for Modern Syscrapers" (013). All Theses and Dssertatons Ths Thess s brought to you for free and open access by BYU ScholarsArchve. It has been accepted for ncluson n All Theses and Dssertatons by an authorzed admnstrator of BYU ScholarsArchve. For more nformaton, please contact scholarsarchve@byu.edu, ellen_amatangelo@byu.edu.

2 Accuracy of a Smplfed Analyss Model for Modern Syscrapers Jacob S. Lee A thess submtted to the faculty of Brgham Young Unversty n partal fulfllment of the requrements for the degree of Master of Scence Rchard J. Ballng, Char Paul W. Rchards Fernando S. Fonseca Department of Cvl and Envronmental Engneerng Brgham Young Unversty June 013 Copyrght 013 Jacob S. Lee All Rghts Reserved

3 ABSTRACT Accuracy of a Smplfed Analyss Model for Modern Syscrapers Jacob S. Lee Department of Cvl and Envronmental Engneerng, BYU Master of Scence A new smplfed syscraper analyss model (SSAM) was developed and mplemented n a spreadsheet to be used for prelmnary syscraper desgn and teachng purposes. The SSAM predcts lnear and nonlnear response to gravty, wnd, and sesmc loadng of "modern" syscrapers whch nvolve a core, megacolumns, outrgger trusses, belt trusses, and dagonals. The SSAM may be classfed as a dscrete method that constructs a reduced system stffness matrx nvolvng selected degrees of freedom (DOF's). The steps n the SSAM consst of: 1) determnaton of megacolumn areas, ) constructon of stffness matrx, 3) calculaton of lateral forces and dsplacements, and 4) calculaton of stresses. Seven confguratons of a generc syscraper were used to compare the accuracy of the SSAM aganst a space frame fnte element model. The SSAM was able to predct the exstence of ponts of contraflexure n the deflected shape whch are nown to exst n modern syscrapers. The accuracy of the SSAM was found to be very good for dsplacements (translatons and rotatons), and reasonably good for stress n confguratons that exclude dagonals. The speed of executon, data preparaton, data extracton, and optmzaton were found to be much faster wth the SSAM than wth general space frame fnte element programs. Keywords: Jacob S. Lee, syscraper, structural analyss, optmzaton, prelmnary desgn

4 ACKNOWLEDGEMENTS I wsh to express my sncere apprecaton and thans to my advsor, Dr. Rchard J. Ballng, for hs help and dedcaton that went nto our research. I am grateful for hs motvaton and encouragement n stretchng my mnd n ways I never thought possble. I was blessed wth the opportunty he gave me to explore an area of structural engneerng that nspres me and share that experence alongsde hm. I am ndeed ndebted to hm for mang t possble to travel to Chna to study the syscrapers there and collaborate wth the engneers behnd ther desgns as part of hs Megastructures class. That unforgettable experence has gven me a wder perspectve of what structural engneerng has to offer and exctes me about my future career aspratons n engneerng. I would also le to than my commttee members, Dr. Paul W. Rchards and Dr. Fernando S. Fonseca, who have truly been nspratonal professors to me n my academc lfe at BYU. I would le to than my famly and frends who have supported me throughout ths endeavor and my graduate degree, and who have nstlled n me the values of one who strves to emulate the pure love of Chrst. I would also le to than my fellow colleagues for ther hard wor and nsght durng ths project.

5 TABLE OF CONTENTS LIST OF TABLES... v LIST OF FIGURES... x 1 Introducton... 1 Lterature revew Contnuum Methods for Low/Medum-Rse Buldngs Contnuum Method for Framed Tubes Inherent Problems wth Contnuum Methods Dscrete Outrgger Methods Dscrete Substructurng Methods... 3 Smplfed Syscraper Analyss Model Determnaton of Megacolumn Areas Constructon of the Stffness Matrx Calculaton of Lateral Forces/Dsplacements Calculaton of Stresses Rapd Tral-and-Error Optmzaton Space frame model Nodes Members Supports Loads Output Results Confguraton #1 core+megacolumns v

6 5. Confguraton # core+megacolumns+outrggers Confguraton #3 core+megacolumns+belts Confguraton #4 core+megacolumns+dagonals Confguraton #5 core+megacolumns+outrggers+belts Confguraton #6 core+megacolumns+outrggers+belts+dagonals Confguraton #7 core+megacolumns+outrgger at one level only Conclusons REFERENCES APPENDIX A. SSAM Excel Spreadsheet (Confguraton #6) v

7 LIST OF TABLES Table 3-1: Stffness matrx - contrbuton of the core and megacolumns...9 Table 3-: Stffness matrx - contrbuton of outrggers...3 Table 3-3: Stffness matrx - contrbuton of the belt trusses...35 Table 3-4: Stffness matrx - contrbuton of dagonals...37 Table 5-1: Confguraton #1 - desgn varables and calculated megacolumn areas...61 Table 5-: Confguraton #1 - lateral core translaton (m)...6 Table 5-3: Confguraton #1 - core rotaton (rad)...6 Table 5-4: Confguraton #1 - vertcal megacolumn translaton mnus vertcal core translaton...63 Table 5-5: Confguraton #1 - core stress (KPa)...63 Table 5-6: Confguraton #1 - megacolumn stress (KPa)...64 Table 5-7: Confguraton # - desgn varables and calculated megacolumn areas...65 Table 5-8: Confguraton # - lateral core translaton (m)...65 Table 5-9: Confguraton # - core rotaton (rad)...65 Table 5-10: Confguraton # - vertcal megacolumn translaton mnus vertcal core translaton...66 Table 5-11: Confguraton # - core stress (KPa)...66 Table 5-1: Confguraton # - megacolumn stress (KPa)...67 Table 5-13: Confguraton # - outrgger stress under lateral load only (KPa)...67 Table 5-14: Confguraton #3 desgn varables and calculated megacolumn areas...68 Table 5-15: Confguraton #3 - lateral core translaton (m)...69 Table 5-16: Confguraton #3 - core rotaton (rad)...69 Table 5-17: Confguraton #3 - vertcal megacolumn translaton mnus vertcal core translaton...70 Table 5-18: Confguraton #3 - core stress (KPa)...70 v

8 Table 5-19: Confguraton #3 - megacolumn stress (KPa)...71 Table 5-0: Confguraton #3 - belt truss stress under lateral load only (KPa)...71 Table 5-1: Confguraton #4 desgn varables and calculated megacolumn areas...7 Table 5-: Confguraton #4 - lateral core translaton (m)...73 Table 5-3: Confguraton #4 - core rotaton (rad)...73 Table 5-4: Confguraton #4 - vertcal megacolumn translaton mnus core vertcal translaton...73 Table 5-5: Confguraton #4 - core stress (KPa)...74 Table 5-6: Confguraton #4 - megacolumn stress (KPa)...74 Table 5-7: Confguraton #4 - dagonal stress (KPa)...75 Table 5-8: Confguraton #5 desgn varables and calculated megacolumn areas...76 Table 5-9 Confguraton #5: - lateral core translaton (m)...77 Table 5-30: Confguraton #5 - core rotaton (rad)...77 Table 5-31: Confguraton #5 - vertcal megacolumn translaton mnus core vertcal translaton...78 Table 5-3: Confguraton #5 - core stress (KPa)...78 Table 5-33: Confguraton #5 - megacolumn stress (KPa)...79 Table 5-34: Confguraton #5 - outrgger stress under lateral load only (KPa)...79 Table 5-35: Confguraton #5 - belt truss stress under lateral load only (KPa)...80 Table 5-36: Confguraton #6 desgn varables and calculated megacolumn areas...81 Table 5-37: Confguraton #6 - lateral core translaton (m)...81 Table 5-38: Confguraton #6 - core rotaton (rad)...81 Table 5-39: Confguraton #6 - vertcal megacolumn translaton mnus vertcal core translaton...8 Table 5-40: Confguraton #6 - core stress (KPa)...8 Table 5-41: Confguraton #6 - megacolumn stress (KPa)...83 v

9 Table 5-4: Confguraton #6 - outrgger stress under lateral load only (KPa)...83 Table 5-43: Confguraton #6 - belt truss stress under lateral load only (KPa)...84 Table 5-44: Confguraton #6 - dagonal stress (KPa)...84 Table 5-45: Confguraton #7 - desgn varables and calculated megacolumn areas...85 Table 5-46: Confguraton #7 - lateral core translaton (m)...86 Table 5-47: Confguraton #7 - core rotaton (rad)...86 Table 5-48: Confguraton #7 - vertcal megacolumn translaton mnus vertcal core translaton...87 x

10 LIST OF FIGURES Fgure 1-1: Jn Mao Tower Shangha, Chna... Fgure 1-: Jn Mao Tower - structural system elevaton vew (Cho et al. 01)...3 Fgure 1-3: Jn Mao Tower - typcal framng plan (Cho et al. 01)...3 Fgure 1-4: Two Internatonal Fnance Centre (IFC) - Hong Kong, Chna...4 Fgure 1-5: IFC - typcal floor plan...5 Fgure 1-6: IFC - typcal layout of outrgger and belt trusses (Empors.com)...5 Fgure 1-7: Shangha Tower...6 Fgure 1-8: Shangha Tower - sometrc of core, megacolumns, outrgger, and belt trusses...7 Fgure 1-9: Guangzhou Internatonal Fnance Center...8 Fgure 1-10: Shangha World Fnancal Center (WFC)...9 Fgure 1-11: Shangha WFC structural system - core, megacolumns, outrgger truss, belt truss, and megadagonal (Katz et al. 008)...9 Fgure 1-1: Shangha WFC structural system elevaton vews ( Fgure 1-13: Generc syscraper elevaton and plan vews...11 Fgure 1-14: Generc syscraper - outrgger truss, belt truss, and dagonal systems...1 Fgure -1: One Lberty Place deflected shape...17 Fgure -: Contraflexure n core created by cap truss (Taranath 005)...19 Fgure -3: Behavor of system wth outrgger located at z = L (Taranath 005)...19 Fgure -4: Behavor of system wth outrgger located at z = 0.75L (Taranath 005)...19 Fgure -5: Behavor of system wth outrgger located at z = 0.5L (Taranath 005)...0 Fgure -6: Behavor of system wth outrgger located at z = 0.5L (Taranath 005)...0 Fgure 3-1: Dsplaced core wth locaton of DOF's...8 Fgure 3-: Typcal outrgger truss subject to unt load...30 x

11 Fgure 3-3: Two-member outrgger truss subject to unt load...31 Fgure 3-4: Unt upward vertcal dsplacement (top) and unt clocwse core rotaton (bottom)...33 Fgure 3-5: Eght-member belt truss subject to unt load...33 Fgure 3-6: Belt truss n generc syscraper subject to unt load...34 Fgure 3-7: Unt dsplacement at megacolumns A (top) and D (mddle), and a unt core rotaton (bottom)...36 Fgure 3-8: Dagonal n generc syscraper subject to unt load...37 Fgure 3-9: Unt vertcal dsplacements at megacolumns A (left), D (mddle), and E (rght)...38 Fgure 3-10: Interval wth a wnd/sesmc force at a partcular story...41 Fgure 3-11: Interval and a P-delta moment at a partcular story...44 Fgure 3-1: Typcal outrgger member subject to unt load...47 Fgure 3-13: Two-member outrgger truss subject to unt load...47 Fgure 3-14: Eght-member belt truss subject to unt load...49 Fgure 3-15: Belt truss n generc syscraper subject to unt load...49 Fgure 3-16: Dagonal n generc syscraper subject to unt load...51 Fgure 4-1: Space frame model all members wthout floors...57 Fgure 4-: Space frame model sngle floor confguratons between ntervals and at ntervals...58 Fgure 5-1: Confguraton #1 - lateral dsplacement and nterstory drft...64 Fgure 5-: Confguraton # - lateral dsplacement and nterstory drft...68 Fgure 5-3: Confguraton #3 - lateral dsplacement and nterstory drft...7 Fgure 5-4: Confguraton #4 - lateral dsplacement and nterstory drft...76 Fgure 5-5: Confguraton #5 - lateral dsplacement and nterstory drft...80 Fgure 5-6: Confguraton #6 - lateral dsplacement and nterstory drft...85 Fgure 5-7: Confguraton #7 - lateral dsplacement and nterstory drft...88 x

12 1 INTRODUCTION A new smplfed syscraper analyss model (SSAM) s descrbed heren. The model can be mplemented on a spreadsheet. The accuracy of the SSAM has been compared to results from sophstcated space frame and fnte element analyss models. Those results wll be presented and dscussed n ths thess. The SSAM s ntended to be used n the prelmnary desgn phase of syscrapers where a fast, reasonably-accurate model s needed n desgn teratons. The model can also be used n an educatonal settng where senor/graduate students are ntroduced to the behavor and desgn of syscrapers. The SSAM predcts the lnear and nonlnear response of "modern" syscrapers subject to gravty, wnd, and sesmc loads. Modern syscrapers are defned heren to be thrd generaton syscrapers. Frst generaton syscrapers such as the Empre State Buldng n New Yor Cty conssted of steel braced and unbraced frames. Such syscrapers had many nteror columns obstructng the space. Fazlur Khan s regarded as the father of second generaton syscrapers characterzed as framed tubes or tube-n-tube syscrapers such as the former World Trade Center n New Yor Cty. These syscrapers possess an nteror core tube that encloses elevator shafts, and a permeter tube wth many columns. There are no columns n between the core tube and permeter tube, thus provdng unobstructed space. By movng the columns to the permeter, a system wth maxmum moment of nerta s created to resst lateral loads. The thrd generaton of syscrapers coalesces permeter columns nto a few megacolumns to provde an unobstructed vew to the outsde. Such megacolumns are usually composte members made from steel 1

13 sectons encased n hgh-stffness, hgh-strength concrete. To provde the necessary moment of nerta to resst lateral loads, the megacolumns and core are perodcally connected wth outrgger trusses, belt trusses, and dagonals. The SSAM s used to analyze core-megacolumn-outrggerbelt-dagonal systems. Some examples of core-megacolumn-outrgger-belt-dagonal syscrapers wll now be gven. The 88-story Jn Mao Tower (see Fgure 1-1), completed n 1999 n Shangha, Chna, conssts of an octagonal concrete core and eght composte steel/concrete megacolumns. Steel outrgger trusses connect the core and megacolumns at stores 5, 54, and 86 (see Fgure 1- and Fgure 1-3). A belt truss s located at the top of the tower, whch s typcally called a cap truss. Fgure 1-1: Jn Mao Tower Shangha, Chna SOM

14 Fgure 1-: Jn Mao Tower - structural system elevaton vew (Cho et al. 01) Fgure 1-3: Jn Mao Tower - typcal framng plan (Cho et al. 01) 3

15 Hong Kong s Internatonal Fnance Centre s an excellent example of a core-mega column-outrgger-belt system (see Fgure 1-4). Ths 88-story syscraper completed n 004 has a core wth eght megacolumns shown n red n Fgure 1-5. Outrgger and belt trusses are located at stores 33, 55, and 67. The 4m spacng between megacolumns provdes unobstructed vew to the outsde for offces on the permeter. A typcal outrgger-belt truss confguraton s shown n Fgure 1-6. Belt trusses transfer loads from secondary corner columns to the megacolumns (Cho et al. 01). Fgure 1-4: Two Internatonal Fnance Centre (IFC) - Hong Kong, Chna Antony Wood/CTBUH 4

16 Fgure 1-5: IFC - typcal floor plan Arup Fgure 1-6: IFC - typcal layout of outrgger and belt trusses (Empors.com) 5

17 The Shangha Tower s a 16-story 63m tall syscraper scheduled for completon n 014 (see Fgure 1-7). Even though the exteror facade has a twstng rregular shape that sgnfcantly reduces wnd load, the core s square and the composte megacolumns are arranged n a regular crcular pattern whose dameter decreases wth heght (see Fgure 1-8). The outrgger trusses and crcular belt trusses occur at nne levels separated by 1 to 15 stores (Mass et al. 010). Radal trusses extend outward from the megacolumns to support the rregular twstng facade. The space between the permeter megacolumns and the exteror facade wll be used as atra open to the publc. Fgure 1-7: Shangha Tower Gensler 6

18 Fgure 1-8: Shangha Tower - sometrc of core, megacolumns, outrgger, and belt trusses Thornton Tomasett The Guangzhou Internatonal Fnance Center s a 440m hgh syscraper wth 73 stores of offce space and 30 stores of hotel space (see Fgure 1-9). The structure conssts of a central core and permeter dagonals arranged n what s nown as a dagrd system. There are no vertcal megacolumns, outrggers, or belt trusses n the dagrd system. The dagonals are concrete-flled steel tubes. 7

19 Fgure 1-9: Guangzhou Internatonal Fnance Center Chrstan Rchters The Shangha World Fnancal Center ncludes core, megacolumns, outrgger trusses, belt trusses, and megadagonals (see Fgures 1-10, 1-11, and 1-1). Ths mxed lateral load-resstng system was motvated by the need to reduce weght n the structure (Katz et al. 008). Note that two of the megacolumns splt part way up so that there are four megacolumns at the base and sx megacolumns after the splt. 8

20 Fgure 1-10: Shangha World Fnancal Center (WFC) Kohn Pederson Fox Assocates/CTBUH Fgure 1-11: Shangha WFC structural system - core, megacolumns, outrgger truss, belt truss, and megadagonal (Katz et al. 008) 9

21 Fgure 1-1: Shangha WFC structural system elevaton vews ( The generc syscraper n Fgures 1-13 and 1-14 wll be used throughout ths thess for analyss comparson. The concrete core s shown n yellow, the 16 concrete megacolumns are shown n red, the two-member steel outrgger trusses are shown n green, the 8-member steel belt trusses are shown n blue, and the steel dagonals are shown n blac. Multple confguratons of ths generc syscraper wll be consdered n the thess: 1) core+megacolumns ) core+megacolumns+outrggers 3) core+megacolumns+belts 4) core+megacolumns+dagonals 5) core+megacolumns+outrggers+belts 6) core+megacolumns+outrggers+belts+dagonals 10

22 The remander of the thess s dvded nto fve chapters. Chapter revews the lterature on related approxmate analyss methods. Chapter 3 descrbes the SSAM. Chapter 4 descrbes the sophstcated lnear space frame and nonlnear ADINA models. Chapter 5 presents results from the SSAM, the space frame model, and the ADINA model for the sx confguratons of the generc syscraper. Chapter 6 submts conclusons based on the results. The appendx ncludes a copy of the spreadsheet mplementaton of the SSAM for the generc syscraper. Fgure 1-13: Generc syscraper elevaton and plan vews 11

23 Fgure 1-14: Generc syscraper - outrgger truss, belt truss, and dagonal systems 1

24 LITERATURE REVIEW The lterature on approxmate analyss methods for tall buldngs can be subdvded nto contnuum methods and dscrete methods. Contnuum methods model tall buldngs as vertcal cantlevers, and approxmate dsplacements as contnuous functons of vertcal poston usng flexure/shear beam theory. Dscrete methods construct stffness or flexblty matrces for the system. The fnte element method s an example of a dscrete method. Some of the approxmate dscrete methods surveyed enforce compatblty condtons at the dscrete locatons of outrgger and belt trusses. Other approxmate dscrete methods construct reduced system stffness matrces through the use of substructurng or super-elements. The SSAM s a dscrete method that constructs a reduced system stffness matrx..1 Contnuum Methods for Low/Medum-Rse Buldngs Bozdogan and Oztur (009) proposed an approxmate method based on the contnuum method dealzng low-rse wall-frame and tube-n-tube structures of 11 stores and 15 stores, respectvely, as sandwch beams. Ther sandwch beam conssts of two vertcal Tmosheno cantlever beams attached by horzontal connectng beams n parallel. One beam conssts of the sum of the flexural and shear rgdtes of shear walls and columns. The second beam conssts of the sum of shear rgdtes of frames and connectng beams. By solvng a set of dfferental equatons for the shear force equlbrum n both beams, contnuous equatons for dsplacement 13

25 and rotaton wth respect to vertcal poston are obtaned. Bozdogan (009) also appled ths method to dynamc analyses on the same example wall-frame structure. Potzta and Kollar (003) dscussed the development of replacement beams as sandwch beams n smplfyng the analyss of low-rse buldngs wth combnatons of shear walls, coupled shear walls, frames, and trusses. Agan, the sandwch beam apples the contnuum method by representng the system as a Tmosheno beam that s supported laterally by a beam wth bendng stffness. Each lateral load-resstng system s replaced by a contnuous cantlever beam wth connectng beams between them. The stran energy of the sandwch beam s presented as the stran energes of a Tmosheno beam and of a beam wth bendng deformaton only. An example 7-story buldng wth two coupled shear walls and a frame s used to demonstrate ths method. Ths same procedure s used by Kavan et al. (008) who extends the method to structures of varable cross-secton. An approxmate hand calculated method for asymmetrc wall-frame structures was proposed by Rutenberg and Hedebrecht (1975). Lateral loads from wnd or earthquaes produce both lateral deflectons and twstng n asymmetrc confguratons. The flexural walls and frames are modeled as vertcal flexural and shear cantlevers where torsonal behavor s treated n addton. Coupled torson-bendng dfferental equatons governng the statc equlbrum of the structure are solved to obtan contnuous functons for story dsplacements and rotatons wth heght. Ther method s appled to a 16-story wall-frame structure. A new concept to ncrease the lateral stffness of wall-frame tall buldng structures by stffenng a story of the frame system was proposed by Nollet and Smth (1997). The wall-frame structure s modeled usng the contnuum theory by representng the system as two cantlever beams n parallel by connectng beams. The shear wall, wth a modfed flexural rgdty, s 14

26 connected to and constraned to have the same deflected shape as the frame by axally rgd connectng lns. The rgd lns that provde horzontal rgdty represent a contnuum between the wall and frame. A contnuous dsplacement functon wth respect to heght was then obtaned by modfyng and solvng the dfferental equaton for bendng moment of a cantlever wth the added stffness parameter. An example 0-story wall-frame structure wth shear walls and four moment resstng frames was analyzed to verfy the method. Abergel and Smth (1983) developed an approxmate method of analyss for non-twstng medum-rse structures composed of shear walls, cores, and dentcal coupled walls. An alternatve to prevous approxmatons s made based on the dfferental equatons of deflecton of a cantlever beam. By replacng the coupled wall wth a comparable structure where the connectng beams are treated as a contnuous medum wth equvalent bendng and shear propertes, a dfferental equaton relatng the horzontal loadng s derved. The dfferental equaton relatng horzontal loadng was developed from two prevously derved equatons for shear walls. A 0-story buldng wth four coupled walls, two shear walls, and a core s used as an example. Hedebrecht and Smth (1973) present a smple hand method for the statc and dynamc analyss of unform low to medum-rse structures consstng of nteractng shear walls and frames. The mathematcal model conssts of a combnaton of flexural and shear vertcal cantlever beams deformng ether n shear or bendng and s very smlar to other methods where the governng dfferental equatons for flexural and shear beams subject to lateral load are solved. Dfferently from other approxmatons, ther method has applcaton to nonunform shear wall-frame structures. The method s appled to a 1-story wall-frame buldng. Smlarly, Hoenderamp et al. (1984) and Toutanj (1997) use varatons of ths method by modelng 15

27 medum-rse buldngs wth coupled walls and shear walls wth frames as flexural and shear vertcal cantlevers.. Contnuum Method for Framed Tubes Kwan (1994) developed a smple hand calculaton method for the analyss of framed tube structures accountng for shear lag effects. Ths method assumes that framed tube structures can prmarly behave le cantlevered box beams. The framed tube structure s modeled as two web panels and two flange panels. It s assumed that there s unform stffness throughout the structure and the dfferental equaton of moment equlbrum n a cantlever beam was solved to obtan contnuous functons of dsplacement and rotaton wth respect to heght. Two examples of a 40-story hgh-rse and a 15-story low-rse composed of framed tubes are presented. Rahgozar and Sharf (009) appled a varaton of Kwan s (1994) method on 30, 40, and 50- story framed tube buldngs wth shear cores and belt trusses. Taabatae (01) refers to the one-dmensonal rod theory as a method n the prelmnary desgn stage that s most sutable when replacng a hgh rse structure as a contnuous member. Ths extended rod theory ncludes the Tmosheno beam theory effects along wth longtudnal deformaton and shear-lag effects by replacng the structure wth an equvalent stffness dstrbuton. The theory s extended to two-dmensonal extended rod theory by consderng structural components wth dfferent stffness and mass dstrbutons that are contnuously connected. They are modeled as several parallel beams. Governng equatons are solved for shear and flexure n a cantlever beam where a contnuous dsplacement functon s obtaned that satsfes contnuty condtons between the parallel beams. A 30-story framed tube s used as an example. Kobayash et al. (1995) appled Taabatae s (01) method to a 30- story tube-n-tube example buldng. 16

28 .3 Inherent Problems wth Contnuum Methods Note that none of the contnuum models surveyed thus far have been appled to buldngs wth outrggers. Ths s because contnuum models based on cantlever beam theory cannot reproduce the ponts of contraflexure exhbted n the deflected shapes of tall buldngs wth outrggers as shown n Fgure -1 taen from Cho et al. (01). The bendng moment n a cantlever beam loaded laterally n one drecton does not change sgn, and therefore, ponts of contraflexure do not exst. Many studes have recognzed the possblty that ponts of contraflexure exst n tall buldngs wth outrggers. Fgure -1: One Lberty Place deflected shape Thornton Tomasett 17

29 Cho et al. (01) explan the outrgger-core couplng as follows: "When laterally loaded the outrggers resst core rotaton by usng permeter columns to push and pull n opposton, ntroducng a change n slope of the vertcal deflecton curve, a porton of the core overturnng moment s transferred to the outrggers and, n turn, tenson n wndward columns and compresson n leeward columns... Analyss and desgn of a complete core-and-outrgger system s not that smple: dstrbuton of forces between the core and the outrgger system depends on the relatve stffness of each element. One cannot arbtrarly assgn overturnng forces to the core and the outrgger columns. However, t s certan that brngng permeter structural elements together wth the core as one lateral load resstng system wll reduce core overturnng moment." Kowalczy et al. (1995) explan the functon of outrggers wth the followng: outrggers serve to reduce the overturnng moment n the core that would otherwse act as a pure cantlever, and to transfer the reduced moment to columns outsde the core by way of a tenson-compresson couple, whch taes advantage of the ncreased moment arm between these columns. Stafford Smth and Coull (1991) state that, "the outrgger-braced structure, wth at most four outrggers, s not strctly amenable to a contnuum analyss and has to be consdered n ts dscrete arrangement." Fgure - through Fgure -6 were taen from a study by (Taranath 005) about the relatonshp between outrgger locaton and the exstence of ponts of contraflexure. In Fgure - 4, the te-down acton of the cap truss generates a restorng couple at the buldng top, resultng n a pont of contraflexure n ts deflecton curve. Fgure -3, Fgure -4, Fgure -5, and Fgure -6 show deflected shape and bendng moment dagrams for dfferent vertcal locatons of a sngle outrgger truss. 18

30 Fgure -: Contraflexure n core created by cap truss (Taranath 005) Fgure -3: Behavor of system wth outrgger located at z = L (Taranath 005) Fgure -4: Behavor of system wth outrgger located at z = 0.75L (Taranath 005) 19

31 Fgure -5: Behavor of system wth outrgger located at z = 0.5L (Taranath 005) Fgure -6: Behavor of system wth outrgger located at z = 0.5L (Taranath 005).4 Dscrete Outrgger Methods Hoenderamp and Baer (003) wrote about analyzng hgh-rse braced frames wth outrggers. Three stffness parameters are consdered whch represent the frame wall, outrggers and columns at the sngle story where the outrgger s present. Two degrees of freedom for the braced frame are taen as a rotaton and a translaton about the vertcal axs. The rotaton equaton assumes the rotaton of a free cantlever wth respect to heght subject to a unformly dstrbuted load. A thrd degree of freedom comes from the rotaton of the outrgger that produces a restranng moment n the frame. The total rotaton of the braced frame at the outrgger level becomes a product of the cantlever rotaton reduced by the moment rotaton 0

32 created by the outrggers. The horzontal deflecton at the top of the structure s then determned by a compatblty equaton for the rotaton at the nterface of the braced frame and outrgger. The method was tested on three braced-frame-outrgger hgh-rse buldngs of 57.5m, 7m, and 93.6m n heght. Hoenderamp (008) apples the method to hgh-rses wth outrggers at two levels, and Hoenderamp (004) apples the method to hgh-rses wth outrggers and flexble foundatons. Taranath (005) conceptualzes outrggers as restranng sprngs located on the cantlever. The rato of the outrgger moment to the outrgger stffness s equated to the rotaton of a unformly loaded cantlever beam wth constant stffness. The resultng deflecton s obtaned by superposng the deflecton of the cantlever and the moment nduced by the sprng. Rahgozar et al. (010) apply a smlar method to 45-story and 55-story buldngs composed of framed tube, shear core, belt truss, and an outrgger where the belt truss, outrgger, and shear core are consdered as a bendng sprng wth constant rotatonal stffness actng as a concentrated moment where the belt truss and outrgger are located. Stafford Smth and Coull (1991) created compatblty equatons for each outrgger level to equate the rotaton of the core to the rotaton of the outrgger. The rotaton of the core s expressed n terms of ts bendng deformaton and that of the outrgger n terms of the axal deformatons of the columns and the bendng of the outrgger. The top drft of the structure may then be determned from the resultng bendng moment dagram for the core by usng the moment-area method. Furthermore, ths same method of analyss can be appled to structures wth more than two outrggers by expressng them as restranng moments n the equaton of horzontal deflecton for a cantlever beam. These multple restranng moments can be expressed n matrx form for smultaneous soluton of multple equatons. Ths method of 1

33 compatblty was publshed earler by Smth and Salm (1981) whch was then mproved upon by Stafford Smth and Coull (1991). Wu and L (003) tae ths compatblty approach as well for mult-outrgger-braced tall buldngs wth an addtonal applcaton to ther dynamc characterstcs. Rutenberg (1987) made a parametrc study for ths method nvestgatng the effect of outrgger locaton, rato of permeter column to core stffness, and stffness varaton along the heght on the horzontal dsplacement at roof level..5 Dscrete Substructurng Methods Ln et al. (1994) presented an approxmate approach called the fnte story method (FSM) to analyze the dsplacement and natural frequences of tall framed tube buldngs. The method reduces the system stffness matrx to nvolve horzontal dsplacements and rotatons about the vertcal axs. It s based on the dsplacements of two-story substructures to approxmate shear, bendng, and torson components of global deformatons. A 30-story framed tube buldng s used as an example. De Llera and Chopra (1995) developed a new smplfed model for analyss and desgn of multstory buldngs. The model s based on a sngle super-element per buldng story that s capable of representng the elastc and nelastc propertes of the story. Ths s done by matchng the stffness matrces and ultmate yeld surface of the story wth that of the element. The analyss conssts of multstory buldngs wth rgd daphragms where the masses are lumped together and where lateral resstance s provded by resstng planes n both horzontal drectons composed of elasto-plastc elements. A sngle fcttous structural super-element per story has three degrees of freedom, two horzontal translatons and the rotaton of the floor connected by

34 the element, where a reduced stffness matrx s created. Ths method was appled to a small buldng wth 4 stores. 3

35 3 SIMPLIFIED SKYSCRAPER ANALYSIS MODEL The steps of the smplfed syscraper analyss model (SSAM) consst of: 1) determnaton of megacolumn areas, ) constructon of stffness matrx, 3) calculaton of lateral forces and dsplacements, and 4) calculaton of stresses. The SSAM was mplemented on a spreadsheet. The spreadsheet can be used for rapd tral-and-error optmzaton of the syscraper. Such usage wll be addressed at the end of ths chapter. 3.1 Determnaton of Megacolumn Areas The SSAM subdvdes the syscraper vertcally nto ntervals. Outrgger and belt trusses are located at nterval boundares. It wll be assumed that the cross-sectonal areas of the core, megacolumns, and dagonals reman constant n each nterval. It wll also be assumed that the cross-sectonal areas of composte steel/concrete cores and megacolumns are the cross-sectonal areas of the transformed all-concrete sectons where steel area has been multpled by the rato of steel elastc modulus to concrete elastc modulus. Defne the followng terms: n = number of stores n nterval (0 for generc syscraper) h = vertcal heght of nterval (80m for generc syscraper) A core = cross-sectonal area of the core n nterval A colj = cross-sectonal area of megacolumn j n nterval A dagj = cross-sectonal area of dagonal j n nterval V dag = volume of all dagonal members n nterval S dag = sne of angle from horzontal for dagonals n nterval L dag = length of dagonals n nterval 4

36 dag = vertcal stffness of all dagonals n nterval F core = axal force n core at base of nterval excludng nterval self weght F colj = axal force n megacolumn j at base of nterval excludng nterval self weght γ = concrete unt weght (core and megacolumns) ε = axal stran at bottom of nterval E = concrete modulus of elastcty (core and megacolumns) E s = steel modulus of elastcty (dagonals, outrggers, belts) A T core = core trbutary area A T colj = trbutary area for megacolumn j P T colj = trbutary permeter for megacolumn j L dead = floor dead load per area L lve = floor lve load per area L clad = claddng load per area T core = outrgger truss weght n nterval supported by the core T colj = outrgger-belt-dagonal truss weght n nterval supported by megacolumn j Assume that ntervals are numbered wth =1 beng the top nterval and ncreasng downward. Assume that h 0 = A 0 core = A 0 colj = 0 n the followng formulas. Assume that the weght of any pnnacle or cap on top of the syscraper s dstrbuted approprately among the core and megacolumns to get values for F 0 core and F 0 colj. The core and megacolumn axal forces excludng nterval self weght are calculated from Equatons 3-1 and 3-: core core core core dead lve F = F 1 + γh 1A 1 + nat (L + L ) + T core (3-1) colj colj colj colj dead lve colj clad F = F 1 + γh 1A 1 + nat (L + L ) + hpt L + T colj (3-) Gven the cross-sectonal area of the core, the cross-sectonal areas of the megacolumns are determned from the prncple that the axal stran n the megacolumns must be the same as the axal stran n the core under gravty loads n order to prevent unacceptably large dfferental vertcal dsplacements from accumulatng n the upper floors of the syscraper. If there are no dagonals, then at the base of nterval, the axal stran n the core s equated to the axal stran n each megacolumn j n Equaton 3-3, 5

37 F ε = core + γh A EA core core F = colj + γh A EA colj colj (3-3) Ths can be solved for the area of megacolumn j n nterval n Equaton 3-4: A = A colj core F F colj core (3-4) The above formula must be modfed f dagonals are present because dagonals contrbute to the support of gravty loads. The vertcal stffness of all dagonals n nterval s calculated n Equaton 3-5: dag s E = j A dagj L dag dag s dagj dag ( S ) E A ( S ) = j h 3 (3-5) The sum of dagonal areas n nterval can be calculated from the volume of dagonal members n nterval from Equaton 3-6: dagj dag A = V j S h dag (3-6) At the base of nterval, the axal stran n the core s equated to the axal stran n all the megacolumns and dagonals together by Equaton 3-7: ε F = core + γh A EA core core colj colj F h A + γ j j = colj s dagj E A + E A j j ( S dag ) 3 (3-7) Ths can be solved for the sum of megacolumn areas n nterval n Equaton 3-8: 6

38 j A colj colj F core j = A core A F j dagj s core dag 3 E γh ( ) A S 1+ core E F (3-8) = A core F colj j core F V h s core dag 4 E γh ( ) A S 1+ core E F The area of megacolumn j n nterval s solved for n Equaton 3-9: dag A colj = F colj colj F j j A colj (3-9) = A core F F colj core V h dag colj s core dag 4 F γ ( ) E ha S 1+ colj core F E F j The above formula s used n the spreadsheet. Note that f the area of the dagonals s bg enough, the megacolumn areas may drop to zero resultng n a dagrd syscraper. 3. Constructon of the Stffness Matrx Lateral load analyss n the SSAM s performed by constructng a stffness matrx n the spreadsheet for the syscraper. The degrees of freedom (DOF's) consst of the horzontal dsplacement of the core at the top of each nterval, the rotaton of the core at the top of each nterval, and the vertcal dsplacements of each of the megacolumns at the top of each nterval. Fgure 3-1 below shows a laterally dsplaced core (thc lne), a sngle megacolumn B (thn lne), and outrgger trusses at the top of each nterval (dotted lnes). The dashed lnes show the undsplaced poston of the structure. The DOF's are dentfed n Fgure 3-1 where subscrpts correspond to story numbers. Symmetry s exploted f possble. The generc syscraper s doubly symmetrc so that only one quarter of the syscraper s ncluded n the model. The model conssts of one-fourth of the core, one-half of megacolumns C and E, and a full porton of 7

39 megacolumns A, B, and D. Assume that the lateral load s perpendcular to the wall contanng megacolumns A, B, and C. Snce the vertcal dsplacement n megacolumn E s zero under lateral loadng, only the vertcal dsplacements for megacolumns A, B, C, and D wll be counted as DOF's. Thus, there are 6 DOF's at the top of each nterval for a total of 30 DOF's. 10 B θ 100 B 80 B θ 60 θ 80 B 40 0 θ 40 B 0 θ 0 Fgure 3-1: Dsplaced core wth locaton of DOF's The moment of nerta of the core must be calculated for each nterval. Ths s done by dvdng the core nto thn rectangles where t s assumed that all rectangles have the same thcness n Equaton 3-10: I core = moment of nerta of the core n nterval t = core wall thcness n nterval d j = length of rectangle j n nterval y j = dstance from centrod of rectangle j to neutral axs n nterval α j = angle from neutral axs to axs parallel to length of rectangle j n nterval I core = t j d j j 3 j j ( ) ( d ) ( sn α ) y + 1 (3-10) 8

40 The local moments of nerta of the megacolumns are much less than the core moment of nerta, and may be calculated from the megacolumn areas n Equaton 3-11: I colj = local moment of nerta of megacolumn j n nterval A colj = cross-sectonal area of megacolumn j n nterval η = 1 for sold square and 4π for sold crcle colj colj ( A ) I = η (3-11) For the generc syscraper, the contrbuton of the core and megacolumns to the frst 1 rows and columns of the stffness matrx s shown n Table 3-1 wth Equatons 3-1 to 3-0: I I = 4 I + core colc cola colb cold + I + I + I + I cole (3-1) 1EI 6EI = cor1 cor = 3 h h = cor3 4EI h = cor4 EI h (3-13, 3-14, 3-15, 3-16) cola EA = h cola = colb EA h colb = colc EA h colc = cold EA h cold (3-17, 3-18, 3-19, 3-0) Table 3-1: Stffness matrx - contrbuton of the core and megacolumns 100 θ 100 A 100 B 100 C 100 D θ 80 A 80 B 80 C 80 D θ 100 A 100 B 100 C 100 D θ 80 A 80 B 80 C 80 D 80 cor1 1 cor cor1-1 cor cor - 1 cor3 1 1 cor 1 cor4 1 cola - 1 cola 1 colb - 1 colb 1 colc - 1 colc 1 cold - 1 cold cor1-1 cor 1 cor1 1 cor1 + 1 cor - cor cor - 1 cor4 1 cor 1 cor - 1 cor3 + cor3-1 cola 1 cola + cola - 1 colb 1 colb + colb - 1 colc 1 colc + colc - 1 cold 1 cold + cold 9

41 The shear stffness of a typcal outrgger truss as shown n Fgure 3- s the recprocal of the vertcal tp dsplacement due to a unt load. L + 1 = w h L + = w (h / ) Fgure 3-: Typcal outrgger truss subject to unt load Assume that the cross-sectonal area of each member of the outrgger truss s proportonal to the magntude of the axal force F ndcated n the fgure above as n Equaton 3-1. Let C be the constant of proportonalty: A = C F (3-1) The total volume of N outrgger trusses at the top of an nterval s calculated n Equaton 3-: V = N AL = NC F L (3-) The stffness of any outrgger truss s the recprocal of the tp dsplacement as determned by the prncple of vrtual forces n Equaton 3-3: out = 1 F L EA = CE F L = N EV ( F L) (3-3) For the outrgger truss n Fgure 3- ts stffness s calculated from Equatons 3-4 and 3-5: 30

42 F L = w h w + h L1 + h L + h + h 6w = + h h (3-4) out = ( + h ) N 6w Eh V (3-5) The shear stffness of each of the 8 two-member outrgger trusses per nterval n the generc syscraper s shown n Fgure 3-3 and calculated n Equaton 3-6: L = w + (h / ) S = h L Fgure 3-3: Two-member outrgger truss subject to unt load out = shear stffness of an outrgger truss at top of nterval V out = volume of all outrgger trusses at top of nterval S out = sne of angle from horzontal of members of outrgger truss at top of nterval h out = heght of outrgger truss at top of nterval (16m for generc syscraper) E s = steel modulus of elastcty out EV EV E = = = N( F L) 1 h 8 S S s out 4 ( S ) out ( h ) V out (3-6) For the generc syscraper, the contrbuton of the outrggers to the frst 1 rows and columns of the stffness matrx s shown n Table 3-. Snce there are no outrggers at story 100 n the generc syscraper, 1 out = 0, but t s retaned n the table to llustrate the pattern. 31

43 Table 3-: Stffness matrx - contrbuton of outrggers 100 θ 100 A 100 B 100 C 100 D θ 80 A 80 B 80 C 80 D θ out +5 1 out -5 1 out out A 100 B out 1 out C 100 D out 1 out 80 θ out +5 out -5 out -1.5 out A 80 B 80-5 out out C 80 D out out Note that there s couplng between the vertcal dsplacements of megacolumns B and D and the rotaton of the core. To understand ths couplng, Fgure 3-4 shows the left half of the core n sold blac and a two-member outrgger truss extendng from the core to megacolumn B. The top part of the fgure shows a unt upward vertcal dsplacement at megacolumn B and the bottom part of the fgure shows a unt clocwse core rotaton. 3

44 out half core 1 megacolumn out (5m) out 5m 1 5m (5m) out (5m) out (5m) out Fgure 3-4: Unt upward vertcal dsplacement (top) and unt clocwse core rotaton (bottom) The shear stffness of each of the 16 eght-member belt trusses per nterval n the generc syscraper s shown n Fgure 3-5 and calculated n Equaton 3-7: L = w + S = h L h Fgure 3-5: Eght-member belt truss subject to unt load 33

45 belt = shear stffness of a belt truss at top of nterval V belt = volume of all belt trusses at top of nterval S belt = sne of angle from horzontal of members of belt truss at top of nterval h belt = heght of belt truss at top of nterval (8m for generc syscraper) E s = steel modulus of elastcty belt = N EV ( F L) EV = 4w 4L 16 + h S = 64 s belt 4 belt E ( S ) V belt belt ( h ) ( S ) ( ) (3-7) If t s assumed that the horzontal members of the belt truss consst of nfntely stff floor daphragms, then the shear stffness of each of the belt trusses n the generc syscraper s ncreased as shown n Fgure 3-6 and calculated n Equaton 3-8: L = w + S = h L h Fgure 3-6: Belt truss n generc syscraper subject to unt load belt EV EV E = = = N( F L) 1 h 16 4 S S belt 4 ( S ) V belt 64( h ) s belt (3-8) For the generc syscraper, the contrbuton of the belts to the frst 1 rows and columns of the stffness matrx s shown n Table 3-3. Snce there are no belts at story 100 n the generc syscraper, 1 belt = 0, but t s retaned n the table to llustrate the pattern. 34

46 Table 3-3: Stffness matrx - contrbuton of the belt trusses 100 θ 100 A 100 B 100 C 100 D θ 80 A 80 B 80 C 80 D θ 100 (1.5 ) A 100 B 100 C 100 D belt belt belt belt 1 belt belt belt - 1 belt 1 belt - 1 belt - 1 belt 1-1 belt 1 belt 80 θ 80 (1.5 ) A 80 B 80 C 80 D 80 belt -1.5 belt -1.5 belt belt belt - belt - belt - belt belt - belt - belt - belt belt Note that there s couplng between the vertcal dsplacement of megacolumn A and the rotaton of the core. To understand ths couplng, Fgure 3-7 shows belt trusses spannng between megacolumn A on the left, megacolumn D n the mddle, and megacolumn E on the rght. The top part of the fgure shows a unt vertcal dsplacement at megacolumn A, the mddle part of the fgure shows a unt vertcal dsplacement at megacolumn D, and the bottom part of the fgure shows a unt core rotaton. It s assumed that the rotaton of all megacolumns s the same as the rotaton of the core because the core and megacolumns are connected wth axally rgd floor daphragms at every story. 35

47 belt belt 1 1.5m 1.5m (1.5m) belt belt belt belt belt m 1.5m 1 (1.5m) belt (1.5m) belt (1.5m) belt (1.5m) belt (1.5m) belt Fgure 3-7: Unt dsplacement at megacolumns A (top) and D (mddle), and a unt core rotaton (bottom) The vertcal stffness of each of the 3 dagonals per nterval n the generc syscraper s gven n Fgure 3-8 and calculated n Equaton 3-9: 36

48 L = w + S = h L h Fgure 3-8: Dagonal n generc syscraper subject to unt load dag = vertcal stffness of a dagonal member n nterval V dag = volume of all dagonal members n nterval S dag = sne of angle from horzontal for dagonals n nterval h dag = heght of dagonal between adjacent megacolumns (0m for generc syscraper) E s = steel modulus of elastcty s dag 4 dag dag EV EV E ( S ) V = = = (3-9) dag N( F L) 1 h 3( h ) 3 S S For the generc syscraper, the contrbuton of the dagonals to the frst 1 rows and columns of the stffness matrx s shown n Table 3-4. Table 3-4: Stffness matrx - contrbuton of dagonals 100 θ 100 A 100 B 100 C 100 D θ 80 A 80 B 80 C 80 D dag /6.4-1 dag / dag /6.4-1 dag /6.4 θ 100 A dag /6.4 1 dag B 100 C 100 D dag - 1 dag dag - 1 dag 1 dag - 1 dag - 1 dag 1-1 dag 1 dag 1 dag / dag 1 /6.4 dag 1 /6.4 4 dag 1 /6.4 dag 1 / dag /6.4 - dag /6.4 θ 80 A 80 - dag 1 /6.4 dag 1 /6.4 - dag /6.4 B 80 C 80 D 80 dag 1 + dag dag - 1 dag dag - dag - 1 dag dag - dag 1 + dag dag - 1 dag dag dag - dag 1 dag dag - dag 1 dag + dag 37

49 Note that there s couplng between the vertcal dsplacement of megacolumn A and the horzontal dsplacement of the core. To understand ths couplng, Fgure 3-9 shows dagonals and megacolumns A, D, and E n the top two ntervals. The left part of the fgure shows a unt vertcal dsplacement at megacolumn A at the top of nterval (bottom of nterval 1), the mddle part of the fgure shows a unt vertcal dsplacement at megacolumn D at the top of nterval, and the rght part of the fgure shows a unt horzontal dsplacement at the top of nterval. It s assumed that the horzontal dsplacement of all megacolumns s the same as the horzontal dsplacement of the core because the core and megacolumns are connected wth axally rgd floor daphragms at every story. 1 dag /6.4 80m/1.5m = dag /(6.4) 1 dag /6.4 1 dag /6.4 dag 1 /6.4 dag /6.4 1 dag 1 1 dag 80m dag 1 dag dag dag /(6.4) dag 1 /6.4 1 dag dag dag dag 4 dag /(6.4) 80m dag dag /6.4 dag /6.4 4 dag /(6.4) dag / m 1.5m dag /6.4 Fgure 3-9: Unt vertcal dsplacements at megacolumns A (left), D (mddle), and E (rght) 38

50 3.3 Calculaton of Lateral Forces/Dsplacements The lateral force vectors have zero values for the DOF's correspondng to vertcal dsplacements n the megacolumns at the top of each nterval. To get the values for the DOF's correspondng to the horzontal dsplacements and rotatons n the core at the top of each nterval, the lateral forces for wnd and sesmc loadng are determned at every story and then aggregated over the ntervals. The spreadsheet ncludes a sheet wth a row for each story n the buldng startng at the bottom and ncreasng upward (100 stores for the generc syscraper). For the lateral wnd pressure, a formula such as Equaton (3-30 taen from ASCE 7-05 could be used: P wnd = wnd pressure at story n psf v = desgn wnd speed n mles per hour (13mph for the generc syscraper) H = heght of story above the ground H g = reference heght parameter reflectng exposure (74m for the generc syscraper) α = another parameter reflectng the exposure (9.5 for the generc syscraper) / α wnd H P =.01 v (3-30) H g After gettng the wnd pressure at each story and convertng t to the approprate unts, the wnd force at each story s obtaned from Equaton 3-31: F wnd = lateral wnd force at story s = story heght for story (4m for the generc syscraper) w = buldng wdth at story (50m for the generc syscraper) F = P wnd wnd s w (3-31) For lateral sesmc forces, the dead weght of each story must be obtaned from Equaton 3-3: 39

51 W = weght of story (excludng lve load) A = floor area of story P = buldng permeter at story s = story heght for story (4m for the generc syscraper) L dead = floor dead load per area L clad = claddng load per area γ = concrete unt weght A core-col = cross-sectonal area of core and all megacolumns at story γ s = steel unt weght V out-belt-dag = volume of all outrggers, belts, and dagonals at story W = A L dead + s P L clad + γs A core col s + γ V out belt dag (3-3) The sesmc force at each story s obtaned wth a formula such as Equaton 3-33 taen from ASCE 7-05: F sesmc = lateral sesmc force at story H = heght of story above the ground S a = spectral acceleraton n g (0. for generc syscraper) R = ductlty factor (3 for generc syscraper) β = sesmc exponent ( for generc syscraper) F sesmc = β ( H ) ( H ) a ( ) W β R W S W (3-33) The wnd and sesmc forces at each story must be aggregated over ntervals to get the forces and moments at the DOF's correspondng to the horzontal dsplacements and rotatons n the core at the top of each nterval. Fgure 3-10 shows a partcular nterval of heght h and a wnd or sesmc lateral force F at a partcular story. In the generc syscraper there are 0 stores n each nterval. Formulas for the fxed end force and moment support reactons at the top and bottom of the nterval are gven n the fgure. In the spreadsheet, these formulas are evaluated for every lateral force n every nterval. The negatve of these support reactons are the 40

52 equvalent forces and moments appled at the DOF's. The rghtward force at a partcular DOF correspondng to a core horzontal dsplacement s equal to the sum of F bot for all lateral forces n the nterval above the DOF plus the sum of F top for all lateral forces n the nterval below the DOF. The clocwse moment at a partcular DOF correspondng to a core rotaton s equal to the sum of M bot for all lateral forces n the nterval above the DOF mnus the sum of M top for all lateral forces n the nterval below the DOF. F top F = ( a ) ( 3h a ) ( h ) 3 M top F (h = a ) ( a ) ( h ) h -a F a F bot F = ( h a ) ( a + h ) ( h ) 3 M bot F a = ( h a ) ( h ) Fgure 3-10: Interval wth a wnd/sesmc force at a partcular story The stffness matrx s nverted and multpled by the lateral force vector for wnd loadng to get the core horzontal dsplacements, the core rotatons, and the megacolumn vertcal dsplacements at the top of each nterval. The nverted stffness matrx s multpled by the lateral force vector for sesmc loadng to get these same dsplacements for sesmc loadng. The prncple of superposton s used to get the lateral dsplacement at a partcular story wthn an nterval. Superposton begns wth a cubc polynomal for the dsplacement due to dsplacements and rotatons at the top and bottom of the nterval, and then adds the 41

53 4 dsplacements of the fxed-fxed beam n Fgure 3-11 due to all of the pont loads F n the nterval and calculated n Equaton 3-34: = lateral dsplacement at story h = heght of nterval a = heght from bottom of nterval to story = core lateral dsplacement at top of nterval +1 = core lateral dsplacement at bottom of nterval θ = core rotaton at top of nterval θ +1 = core rotaton at bottom of nterval F m bot = fxed end force at bottom of nterval due to pont force m F m top = fxed end force at top of nterval due to pont force m M m bot = fxed end moment at bottom of nterval due to pont force m M m bot = fxed end moment at top of nterval due to pont force m E = concrete modulus of elastcty (core and megacolumns) I = moment of nerta of core and local moment of nerta of megacolumns (3-34) Interstory drfts can be calculated and compared to allowable values (e.g. 1/360 for wnd and 1/50 for sesmc) n Equaton 3-35: D = nterstory drft at story = lateral dsplacement at story s = story heght for story (4m for the generc syscraper) (3-35) As the syscraper dsplaces laterally under wnd and sesmc loads, the weght of the structure creates an addtonal overturnng moment equal to the weght tmes the lateral ( ) ( ) a a h h 3 a h h θ θ + θ + θ + θ + = ( ) ( ) + > > m top m m top m 3 m bot m m bot m 3 EI 1 M a h 6 F a h M a 6 F a 1 s D =

54 dsplacement. Ths moment, called the P effect, ncreases the lateral dsplacement, and thus, nonlnear teraton s necessary to converge to the fnal equlbrum poston when the P moments no longer change: W = weght of story (ncludng lve load) F = F +1 +W = total axal force at story = lateral dsplacement at story M = F ( - -1 ) moment at story due to P effect The moments at each story must be aggregated over ntervals to get the forces and moments at the DOF's correspondng to the horzontal dsplacements and rotatons n the core at the top of each nterval. Fgure 3-11 shows a partcular nterval of heght h and a P moment M at a partcular story. Formulas for the fxed end force and moment support reactons at the top and bottom of the nterval are gven n the fgure. In the spreadsheet, these formulas are evaluated for every P moment n every nterval. The negatve of these support reactons are the equvalent forces and moments appled at the DOF's. The rghtward force at a partcular DOF correspondng to a core horzontal dsplacement s equal to the sum of F bot for all lateral forces n the nterval above the DOF plus the sum of F top for all lateral forces n the nterval below the DOF. The clocwse moment at a partcular DOF correspondng to a core rotaton s equal to the sum of M bot for all lateral forces n the nterval above the DOF mnus the sum of M top for all lateral forces n the nterval below the DOF. 43

55 F top 6Ma = ( h a ) ( h ) 3 M top Ma (h 3a ) = ( h ) h -a M a F bot 6Ma = ( h a ) ( h ) 3 M bot M = ( h a )( h 3a ) ( h ) Fgure 3-11: Interval and a P-delta moment at a partcular story Nonlnear teraton s accomplshed n the spreadsheet by creatng two columns for the P lateral force vector -- a startng column and an endng column. The startng column s ntalzed to zero and s added to the wnd lateral force vector. The endng column calculates the new P lateral force vector by the procedure descrbed above. The values from the endng column are repeatedly pasted nto the startng column untl the two columns are the same. Startng and endng columns for the P lateral force vector are lewse created for sesmc loadng. 3.4 Calculaton of Stresses Gravty load stresses are greatest at the bottom of each nterval for the core, megacolumns, and dagonals. The gravty load stress s the same for the core and the megacolumns snce megacolumn areas were determned earler by equatng ther gravty load 44

56 strans to that of the core (see Equatons 3-36 and 3-37). The gravty load stress n dagonals s also determned by equatng the respectve gravty load stran to that of the core n Equaton The gravty load stress n nteror dagonal members s decreased by a fracton of the relatve ncrement n axal force for the nterval n Equaton 3-39: σ core_grav = gravty load stress n core at bottom of nterval σ colj_grav = gravty load stress n megacolumn j at bottom of nterval σ outb_grav = gravty load stress n outrgger B at top of nterval σ outd_grav = gravty load stress n outrgger D at top of nterval σ dagab_grav = gravty load stress n bottom dagonal AB n nterval σ dagad_grav = gravty load stress n bottom dagonal AD n nterval σ dagbc_grav = gravty load stress n bottom dagonal BC n nterval σ dagde_grav = gravty load stress n bottom dagonal DE n nterval h = vertcal heght of nterval (80m for generc syscraper) A core = cross-sectonal area of the core n nterval F core = axal force n core at base of nterval excludng nterval self weght γ = concrete unt weght (core and megacolumns) E = concrete modulus of elastcty E s = steel modulus of elastcty S out = sne of angle from horzontal of members of outrgger truss at top of nterval S dag = sne of angle from horzontal for dagonals n nterval core core _ grav F σ = + γh core (3-36) A σ σ σ colj_ grav = σ dagab _ grav dagbc _ grav core _ grav = σ = σ dagad _ grav dagde _ grav E = s s E = dag ( S ) dag ( S ) σ E σ E core _ grav core _ grav F 1.5 core F core F core 1 (3-37) (3-38) (3-39) The lateral load stress n the core and megacolumns at the bottom of each nterval s equal to the modulus of elastcty tmes axal stran plus the modulus of elastcty tmes flexural curvature tmes dstance from local neutral axs to outermost fber n Equatons 3-40 and The flexural curvature s the same for core and megacolumns and s obtaned by dfferentatng 45

57 the lateral dsplacement formulas twce and evaluatng at a = 0 (bottom of the nterval). Under lateral loadng, the axal stran s zero n the core. The megacolumn axal stran s equal to the dfference between vertcal dsplacements n the megacolumn at the top and bottom of the nterval dvded by the nterval heght: σ core_lat = lateral load stress n core at bottom of nterval σ colj_lat = lateral load stress n megacolumn j at bottom of nterval h = heght of nterval E = concrete modulus of elastcty (core and megacolumns) I = moment of nerta of core and local moment of nerta of megacolumns c core = dstance to outermost fber n core n nterval (1.5m for generc syscraper) c colj = dstance to outermost fber n megacolumn j n nterval A colj = cross-sectonal area of megacolumn j n nterval µ = 4 for sold square and π for sold crcle = core lateral dsplacement at top of nterval +1 = core lateral dsplacement at bottom of nterval θ = core rotaton at top of nterval θ +1 = core rotaton at bottom of nterval colj = vertcal dsplacement n megacolumn j at top of nterval +1 colj = vertcal dsplacement n megacolumn j at bottom of nterval M m bot = moment at bottom of nterval due to lateral force at story m wthn nterval Ec 6 ( ) θ + 4θ core_ lat core σ = h h c + core m I M bot m (3-40) E colj colj ( ) colj_ lat + 1 σ = h + σ core _ lat c c colj core c colj = A µ colj (3-41) For the typcal outrgger truss n Fgure 3-1, recall the formulas developed earler when the stffness of ths truss was consdered: 46

58 A = C F V = NC F L out = N EV ( F L) 6w F L = + h h L + 1 = w h L + = w (h / ) Fgure 3-1: Typcal outrgger member subject to unt load To get the lateral load stress for the members of ths outrgger truss, the axal forces due to a unt load must be multpled by the stffness out tmes the shear dsplacement out. These axal forces must then be dvded by the cross-sectonal area to get stress as n Equaton 3-4: σ out _ lat = F A out out = E F L out Eh = 6w + h out (3-4) The lateral load stress n the members of each of the 8 two-member outrgger trusses per nterval n the generc syscraper s shown n Fgure 3-13 and calculated n Equaton 3-43: L = w + (h / ) S = h L Fgure 3-13: Two-member outrgger truss subject to unt load 47

59 σ out_lat = lateral load stress n outrgger at top of nterval out = shear dsplacement n outrgger at top of nterval S out = sne of angle from horzontal of members of outrgger truss at top of nterval h out = heght of outrgger truss at top of nterval (16m for generc syscraper) E s = steel modulus of elastcty σ out _ lat = E F L out = E 1 S h S out = E out ( S ) out s out h (3-43) The shear dsplacement n each outrgger s the dfference between megacolumn vertcal dsplacement and the product of core rotaton and dstance from core centerlne to megacolumn as calculated n Equatons 3-44 and The shear dsplacement s greater for the upper member of the outrgger than for the lower member. The core rotaton at the top of the upper member must be approprately nterpolated from the core rotaton at the top of the nterval and the core rotaton at the top nterval above: outb = shear dsplacement n outrgger B at top of nterval outd = shear dsplacement n outrgger D at top of nterval θ = core rotaton at top of nterval θ -1 = core rotaton at top of nterval -1 colb = vertcal dsplacement n megacolumn B at top of nterval cold = vertcal dsplacement n megacolumn D at top of nterval colb ( θ ) outb = 5m (3-44) cold ( θ ) outd = 1.5m (3-45) The lateral load stress n the members of each of the 16 eght-member belt trusses per nterval n the generc syscraper s shown n Fgure 3-14 and calculated n Equaton 3-46: 48

60 L = w + S = h L h Fgure 3-14: Eght-member belt truss subject to unt load σ belt = lateral load stress n belt truss at top of nterval belt = shear dsplacement n belt truss at top of nterval S belt = sne of angle from horzontal of members of belt truss at top of nterval h belt = heght of belt truss at top of nterval (8m for generc syscraper) E s = steel modulus of elastcty σ belt _ lat = E F L belt E = 4w 4L + h S belt E = h belt belt ( S ) belt ( S ) s ( ) belt (3-46) If t s assumed that the horzontal members of the belt truss consst of nfntely stff floor daphragms, then the lateral load stress n the members of each of the belt trusses n the generc syscraper s ncreased n Fgure 3-15 and calculated n Equaton 3-47: L = w + S = h L h Fgure 3-15: Belt truss n generc syscraper subject to unt load 49

61 σ belt _ lat = E F L belt = E 1 h 4 S S belt = belt ( S ) belt s E h belt (3-47) The shear dsplacement n belts AB and BC s the dfference between the two ends of the belt of the megacolumn vertcal dsplacements as calculated n Equatons 3-48 and The shear dsplacement n belt DE s the dfference between megacolumn D vertcal dsplacement and the product of core rotaton and dstance from core centerlne to megacolumn D as calculated n Equaton The shear dsplacement n belt AD s the dfference between the two ends of the belt of the dfference between megacolumn vertcal dsplacement and the product of core rotaton and dstance from core centerlne to megacolumn as calculated n Equaton 3-51: beltab = shear dsplacement n belt AB at top of nterval beltbc = shear dsplacement n belt BC at top of nterval beltad = shear dsplacement n belt AD at top of nterval beltde = shear dsplacement n belt DE at top of nterval cola = vertcal dsplacement n megacolumn A at top of nterval colb = vertcal dsplacement n megacolumn B at top of nterval colc = vertcal dsplacement n megacolumn C at top of nterval cold = vertcal dsplacement n megacolumn D at top of nterval θ = core rotaton at top of nterval beltab = cola colb beltbc = colb colc (3-48, 3-49) beltad cola = m( θ) 1.5m( θ) 5 + cold beltde =.5m( θ) 1 cold (3-50, 3-51) The lateral load stress n each of the 3 dagonals per nterval n the generc syscraper s shown n Fgure 3-16 and calculated n Equaton 3-5: 50

62 L = w + S = h L h Fgure 3-16: Dagonal n generc syscraper subject to unt load σ dag_lat = lateral load stress n a dagonal member n nterval dag = vertcal dsplacement n dagonal members n nterval S dag = sne of angle from horzontal for dagonals n nterval h dag = heght of dagonal between adjacent megacolumns (0m for generc syscraper) E s = steel modulus of elastcty σ dag _ lat = E F L dag E = 1 h S S dag s E = h dag ( S ) dag dag (3-5) The vertcal dsplacement n dagonals AB and BC s the dfference between the two ends of the dagonal of the megacolumn vertcal dsplacements as calculated n Equatons 3-53 and The vertcal dsplacement n dagonal DE s the dfference between megacolumn D vertcal dsplacement and the product of lateral drft and dstance from core centerlne to megacolumn D as calculated n Equaton The vertcal dsplacement n dagonal AD s the dfference between the two ends of the dagonal of the dfference between megacolumn vertcal dsplacement and the product of lateral drft and dstance from core centerlne to megacolumn as calculated n Equaton In these formulas, megacolumn vertcal dsplacements must be approprately nterpolated between the top and bottom of the nterval to get values at the ends of the dagonal: 51

63 dagab = stress n bottom dagonal AB n nterval dagad = stress n bottom dagonal AD n nterval dagbc = stress n bottom dagonal BC n nterval dagde = stress n bottom dagonal DE n nterval cola = vertcal dsplacement n megacolumn A at top of nterval colb = vertcal dsplacement n megacolumn B at top of nterval colc = vertcal dsplacement n megacolumn C at top of nterval cold = vertcal dsplacement n megacolumn D at top of nterval +1 cola = vertcal dsplacement n megacolumn A at bottom of nterval +1 colb = vertcal dsplacement n megacolumn B at bottom of nterval +1 colc = vertcal dsplacement n megacolumn C at bottom of nterval +1 cold = vertcal dsplacement n megacolumn D at bottom of nterval D dagad = lateral drft at the center of dagonal AD n nterval D dagde = lateral drft at the center of dagonal DE n nterval dagab cola colb = colb (3-53) dagbc colb colb colc = colc (3-54) dagad dagad cola cold = D (5m 1.5m) cold (3-55) dagde dagde cold = D (1.5m) cold (3-56) 3.5 Rapd Tral-and-Error Optmzaton Now that the descrpton of the SSAM s complete, the spreadsheet can be used to optmze the syscraper desgn. The desgn varables are the core thcness at each nterval, the outrgger truss volume at each nterval, the belt truss volume at each nterval, and the dagonal volume at each nterval. The objectve s the mnmzaton of structural cost whch s the total volume of concrete n the core and megacolumns multpled by the specfed concrete cost per unt volume plus the total volume of steel n the outrgger trusses, belt trusses, and dagonals 5

64 multpled by the specfed steel cost per unt volume. The constrants to be satsfed nclude lateral drft n every story under wnd loadng, lateral drft n every story under sesmc loadng, stress n every member under combned gravty and wnd loadng, and stress n every member under combned gravty and sesmc loadng. For each of these types of constrants, the spreadsheet calculates a constrant rato of actual value to allowable value. For example, the constrant rato for wnd lateral drft s equal to the maxmum drft over the 100 stores dvded by the specfed allowable such as 1/360 or 1/400. The constrant rato for wnd+gravty belt stress s equal to the maxmum wnd+gravty stress over all belt truss members n all ntervals dvded by the allowable stress for steel. Desgn constrants are satsfed when the constrant ratos are less than or equal to one. The desgn varables, desgn objectve, and desgn constrants are located together on the spreadsheet to facltate rapd tral-and-error optmzaton. Ths process was carred out for all sx confguratons of the generc syscraper. 53

65 4 SPACE FRAME MODEL A 3D, seletal, lnear, statc, small-dsplacement, space frame model was developed to compare the accuracy of the SSAM. The space frame model was executed on a program wrtten by Ballng (1991) as well as on the commercal program, ADINA. Both programs gave the same results for lnear analyss. The ADINA program was also executed to get nonlnear (largedsplacement) results for one confguraton of the space frame model. The space frame model wll be descrbed n fve sectons: nodes, members, supports, loads, and output. 4.1 Nodes There were a total of 1877 nodes n the space frame model. The y-axs was taen as the vertcal axs of the buldng located n the center of the core. There were 101 "core-center" nodes wth x=z=0 equally spaced every 4m n the y-drecton correspondng to the 100 stores of the generc syscraper. Lewse, there were 101 "megacolumn" nodes for each of the 16 megacolumns. For a partcular megacolumn, the x and z-coordnates were constant and depended on the locaton of the megacolumn n the plan, and the y-coordnates were equally spaced every 4m. Four "core-corner" nodes were located at each of stores 18,, 38, 4, 58, 6, 78, and 8 wth horzontal coordnates x=±1.5m and z=±1.5m. Outrgger members connected core-corner nodes wth megacolumn nodes. Sxteen "belt" nodes were located mdway between 54

66 megacolumn nodes at each of levels 19, 1, 39, 41, 59, 61, 79, and 81. Belt truss members connected belt nodes to megacolumn nodes. 4. Members There were a total of 5668 members n the space frame model (see Fgure 4-1), ncludng 100 core members (yellow), 1600 megacolumn members (red), 64 outrgger members (green), 56 belt truss members (lght blue), 160 dagonal members (blac), 3 rgd ln members (dar blue), and 3456 floor members (not shown n Fgure 4-1, but shown n Fgure 4-). Shear deformaton was neglected n all members, and the Posson's rato was assumed to be 0.5. Core members connect core-center nodes, and megacolumn members connect megacolumn nodes. Core and megacolumn members possess axal, flexural, and torsonal stffness. The modulus of elastcty and cross-sectonal areas were set equal to the values used n the SSAM. Both the strong and wea core moments of nerta were set equal to the values used n the SSAM. The torson constant was arbtrarly set to 1000m 4, and t was verfed that ths dd not mpact the results because of the symmetry of the structure and loadng, and the axal rgdty of the floor daphragms. Both ends of core and megacolumn members were connected rgdly. Outrgger, belt truss, and dagonal members were modeled as truss members that only possess axal stffness. The outrgger members connect between core-corner nodes and adjacent megacolumn nodes. The belt truss members connect between megacolumn nodes and belt nodes. The dagonal members connect between megacolumn nodes. Snce these members possess axal stffness only, ther moments of nerta and torson constants were set to zero, and both ends were hnge-connected. The modulus of elastcty was set equal to the value used n the SSAM. The cross-sectonal areas were calculated by dvdng the volumes used n the SSAM by the number of members and member length. 55

67 Rgd ln members connect core-center nodes located at the ntersecton of ntervals to core-corner nodes. They model the fnte sze of the core. Rgd ln members possess nfnte axal, flexural, and torsonal stffness. Infnte stffness was obtaned by settng the modulus of elastcty to 10 1 KPa. The moments of nerta and torson constant were arbtrarly set to 1000m 4. Both ends of the rgd ln members were connected rgdly. Floor members extend radally from core-center nodes to megacolumn and belt nodes n the same horzontal plane. Addtonal floor members connect crcumferentally between megacolumn and belt nodes n the same horzontal plane (see Fgure 4-). These members model the axally rgd floor daphragms. They were modeled as truss members where ther moments of nerta and torson constants were set to zero, and both ends were hnge-connected. Axal rgdty was obtaned by settng the modulus of elastcty to 10 1 KPa. The cross-sectonal areas were arbtrarly set to 1000m. Cho et al. (01) mentoned that f a belt truss s used, a stff floor daphragm s requred at the top and bottom chord of each belt truss n order to transfer the core bendng moment, n the form of floor shear and axal forces, to the belt wall and eventually to the columns. Also, mproperly modeled daphragms wll result n msleadng behavors and load paths, and ncorrect member desgn forces. 56

68 Fgure 4-1: Space frame model all members wthout floors 57

69 Fgure 4-: Space frame model sngle floor confguratons between ntervals and at ntervals 4.3 Supports Supports restraned some of the DOF's. Each of the 1877 nodes n the space frame model had sx dsplacement DOF s: three translatons (Δ x, Δ y, and Δ z ) and three rotatons (θ x, θ y, and θ z ). A total of 486 restrants were needed n ths model. The core-center node and the sxteen megacolumn nodes at the base of the structure were fxed-supported to create 6x17=10 restrants. The rotatonal DOF's of the 18 belt nodes had to be supported for stablty snce the belt members were hnge-connected. Ths created 3x18=384 restrants. The number of unrestraned DOF's n the space frame model was 6 x = 10,776. Note that ths s far greater than the 30 DOF's of the SSAM. 58

70 4.4 Loads Both pont loads and dstrbuted loads were ncluded n ths model. A downward pont load was appled at each of the core-center and megacolumn nodes representng the external dead, lve, and claddng loads (1700 pont loads). Horzontal pont loads n the postve x drecton were appled to each core-center node representng the lateral loads (100 pont loads). Downward dstrbuted loads were appled to core, megacolumn, outrgger, belt truss, and dagonal members representng member self-weght (180 dstrbuted loads). The magntudes for all of these loads were obtaned from the SSAM. 4.5 Output Output from the space frame program conssted of nodal dsplacements and member end forces. These output fles were mported nto a macro-enabled Excel spreadsheet. The macro n the spreadsheet extracted approprate data and calculated the followng for comparson wth the results from the SSAM: core lateral translatons core rotatons core stresses megacolumn stresses outrgger stresses belt truss stresses dagonal stresses 59

71 5 RESULTS Results from the SSAM and the space frame model (Sframe) are compared n the followng tables for the sx confguratons of the generc syscraper. For each confguraton, the frst table gves values of the desgn varables and calculated megacolumn areas for each nterval. In the remanng tables for each confguraton, the term "rato" s the rato of the SSAM value over the Sframe value. The "max error" gven as a percentage below each table s equal to 100 tmes the maxmum absolute value of the rato mnus one. The values n all tables are for lnear analyss only wth the excepton of Table 5- and Table 5-3 (Confguraton #1) where values are gven for both lnear and nonlnear analyss. All tables gve results for combned gravty and lateral loadng wth the excepton of tables for outrgger and belt stresses where the results are for lateral loadng only. It was observed n the space frame model, for confguratons nvolvng belt trusses, that the stress n the megacolumn located nsde belt trusses was much greater than the stress n the megacolumn located outsde belt trusses. It was assumed that the megacolumn cross-sectonal area for an nterval refers to the megacolumn outsde belt trusses. The Sframe megacolumn stress reported n the followng tables s the value for the member just above the belt truss at the bottom of the nterval. The lateral dsplacement and nterstory drft are also plotted after the comparson tables for each confguraton. Recall that nterstory drft s defned as the dfference n lateral dsplacement 60

72 between the top and bottom of the story dvded by the story heght. Snce rotaton s the dervatve of lateral dsplacement, nterstory drft s effectvely a fnte dfference approxmaton of rotaton. The nterstory drft that s plotted n the fgures that follow has been normalzed by the allowable value so that when the normalzed value s less than one, the drft constrant s satsfed. Ponts of contraflexure are also ndcated on the plots. They are located at ponts where the drft s vertcal because that s the pont where the rotaton changes from ncreasng to decreasng wth heght. Ponts of contraflexure also correspond to ponts where curvature changes n the plot of lateral dsplacement. However, the curvature changes are too subtle to observe n the lateral dsplacement plots for the sx confguratons. A seventh confguraton was added wth outrggers located only at nterval. Here the ponts of contraflexure are observable n both the plot of drft and the plot of lateral dsplacement. 5.1 Confguraton #1 core+megacolumns Table 5-1: Confguraton #1 - desgn varables and calculated megacolumn areas Interval Core t Outrg V Belt V Dag V Megacolumn A Megacolumns B/D Megacolumns C/E (m) (m 3 ) (m 3 ) (m 3 ) Area (m ) Area (m ) Area (m )

73 Table 5-: Confguraton #1 - lateral core translaton (m) Top of Lnear Lnear Lnear Nonlnear Nonlnear Nonlnear Interval SSAM Sframe Rato SSAM Sframe Rato max lnear error = 0.05 max nonlnear error = 0.1 Table 5-3: Confguraton #1 - core rotaton (rad) Top of Lnear Lnear Lnear Nonlnear Nonlnear Nonlnear Interval SSAM Sframe Rato SSAM Sframe Rato max lnear error = 0.07% max nonlnear error = 0.15% 6

74 Table 5-4: Confguraton #1 - vertcal megacolumn translaton mnus vertcal core translaton Megacolumn A Megacolumn B Top of SSAM Sframe Rato SSAM Sframe Rato Interval Megacolumn C Megacolumn D Top of SSAM Sframe Rato SSAM Sframe Rato Interval max error = 0.00% Table 5-5: Confguraton #1 - core stress (KPa) Bottom of Interval SSAM Sframe Rato max error = 0.06% 63

75 Table 5-6: Confguraton #1 - megacolumn stress (KPa) Megacolumn A Megacolumn B Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato Megacolumn C Megacolumn D Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 1.05% Heght (m) SSAM Wnd Dsp SSAM Wnd Drft Sframe Wnd Dsp Sframe Wnd Drft Lateral dsplacement (m) and normalzed nterstory drft Fgure 5-1: Confguraton #1 - lateral dsplacement and nterstory drft 64

76 5. Confguraton # core+megacolumns+outrggers Table 5-7: Confguraton # - desgn varables and calculated megacolumn areas Interval Core t Outrg V Belt V Dag V Megaolumn A Megaolumns B/D Megaolumns C/E (m) (m 3 ) (m 3 ) (m 3 ) Area (m ) Area (m ) Area (m ) Table 5-8: Confguraton # - lateral core translaton (m) Top of Interval SSAM Sframe Rato max error = 0.07% Table 5-9: Confguraton # - core rotaton (rad) Top of Interval SSAM Sframe Rato max error = 0.11% 65

77 Table 5-10: Confguraton # - vertcal megacolumn translaton mnus vertcal core translaton Megaolumn A Megaolumn B Top of SSAM Sframe Interval Rato SSAM Sframe Rato Megaolumn C Megaolumn D Top of SSAM Sframe Interval Rato SSAM Sframe Rato max error = 0.07% Table 5-11: Confguraton # - core stress (KPa) Bottom of Interval SSAM Sframe Rato max error = 0.0% 66

78 Table 5-1: Confguraton # - megacolumn stress (KPa) Megaolumn A Megaolumn B Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato Megaolumn C Megaolumn D Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 0.65% Table 5-13: Confguraton # - outrgger stress under lateral load only (KPa) B D Top of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 0.08% 67

79 Heght (m) SSAM Wnd Dsp SSAM Wnd Drft Sframe Wnd Dsp Sframe Wnd Drft Ponts of Contraflexure Lateral dsplacement (m) and normalzed nterstory drft Fgure 5-: Confguraton # - lateral dsplacement and nterstory drft 5.3 Confguraton #3 core+megacolumns+belts Table 5-14: Confguraton #3 desgn varables and calculated megacolumn areas Interval Core t (m) Outrg V (m 3 ) Belt V (m3 ) Dag V (m 3 ) Megacolumn A Megacolumns B/D Megacolumns C/E Area (m ) Area (m ) Area (m )

80 Table 5-15: Confguraton #3 - lateral core translaton (m) Top of Interval SSAM Sframe Rato max error = 0.68% Table 5-16: Confguraton #3 - core rotaton (rad) Top of Interval SSAM Sframe Rato max error = 0.84% 69

81 Table 5-17: Confguraton #3 - vertcal megacolumn translaton mnus vertcal core translaton Megacolumn A Megacolumn B Top of SSAM Sframe Interval Rato SSAM Sframe Rato Megacolumn C Megacolumn D Top of SSAM Sframe Interval Rato SSAM Sframe Rato max error = 0.56% Table 5-18: Confguraton #3 - core stress (KPa) Bottom of Interval SSAM Sframe Rato max error = 3.17% 70

82 Table 5-19: Confguraton #3 - megacolumn stress (KPa) Megacolumn A Megacolumn B Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato Megacolumn C Megacolumn D Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 5.3% Table 5-0: Confguraton #3 - belt truss stress under lateral load only (KPa) AB BC Top of Interval SSAM Sframe Rato SSAM Sframe Rato AD DE Top of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 5.54% 71

83 Heght (m) SSAM Wnd Dsp SSAM Wnd Drft Sframe Wnd Dsp Sframe Wnd Drft Ponts of Contraflexure Lateral dsplacement (m) and normalzed nterstory drft Fgure 5-3: Confguraton #3 - lateral dsplacement and nterstory drft 5.4 Confguraton #4 core+megacolumns+dagonals Table 5-1: Confguraton #4 desgn varables and calculated megacolumn areas Interval Core t Outrg Belt V Dag V Megacolumn A Megacolumns B/D Megacolumns C/E (m) V (m 3 ) (m 3 ) (m 3 ) Area (m ) Area (m ) Area (m )

84 Table 5-: Confguraton #4 - lateral core translaton (m) Top of Interval SSAM Sframe Rato max error = 0.61% Table 5-3: Confguraton #4 - core rotaton (rad) Top of Interval SSAM Sframe Rato max error = 0.96% Table 5-4: Confguraton #4 - vertcal megacolumn translaton mnus core vertcal translaton Megacolumn A Megacolumn B Top of SSAM Sframe Interval Rato SSAM Sframe Rato Megacolumn C Megacolumn D Top of SSAM Sframe Interval Rato SSAM Sframe Rato max error = 7.35% 73

85 Table 5-5: Confguraton #4 - core stress (KPa) Bottom of Interval SSAM Sframe Rato max error = 3.65% Table 5-6: Confguraton #4 - megacolumn stress (KPa) Megaolumn A Megaolumn B Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato Megaolumn C Megaolumn D Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 17% 74

86 Table 5-7: Confguraton #4 - dagonal stress (KPa) AB BC Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato AD DE Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 76% 75

87 Heght (m) SSAM Wnd Dsp SSAM Wnd Drft Sframe Wnd Dsp Sframe Wnd Drft Ponts of Contraflexure Lateral dsplacement (m) and normalzed nterstory drft Fgure 5-4: Confguraton #4 - lateral dsplacement and nterstory drft 5.5 Confguraton #5 core+megacolumns+outrggers+belts Table 5-8: Confguraton #5 desgn varables and calculated megacolumn areas Interval Core t Outrg V Belt V Dag V Megacolumn A Megacolumns B/D Megacolumns C/E (m) (m 3 ) (m 3 ) (m 3 ) Area (m ) Area (m ) Area (m )

88 Table 5-9 Confguraton #5: - lateral core translaton (m) Top of Interval SSAM Sframe Rato max error = 0.09% Table 5-30: Confguraton #5 - core rotaton (rad) Top of Interval SSAM Sframe Rato max error = 0.19% 77

89 Table 5-31: Confguraton #5 - vertcal megacolumn translaton mnus core vertcal translaton Megacolumn A Megacolumn B Top of SSAM Sframe Interval Rato SSAM Sframe Rato Megacolumn C Megacolumn D Top of SSAM Sframe Interval Rato SSAM Sframe Rato Max error = 0.79% Table 5-3: Confguraton #5 - core stress (KPa) Bottom of Interval SSAM Sframe Rato max error = 3.17% 78

90 Table 5-33: Confguraton #5 - megacolumn stress (KPa) Megacolumn A Megacolumn B Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato Megacolumn C Megacolumn D Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 5.04% Table 5-34: Confguraton #5 - outrgger stress under lateral load only (KPa) B D Top of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 1.37% 79

91 Table 5-35: Confguraton #5 - belt truss stress under lateral load only (KPa) AB BC Top of Interval SSAM Sframe Rato SSAM Sframe Rato AD DE Top of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 11.7% Heght (m) SSAM Wnd Dsp SSAM Wnd Drft Sframe Wnd Dsp Sframe Wnd Drft Ponts of Contraflexure Lateral dsplacement (m) and normalzed nterstory drft Fgure 5-5: Confguraton #5 - lateral dsplacement and nterstory drft 80

92 5.6 Confguraton #6 core+megacolumns+outrggers+belts+dagonals Table 5-36: Confguraton #6 desgn varables and calculated megacolumn areas Interval Core t Outrg V Belt V Dag V Megacolumn A Megacolumns B/D Megacolumns C/E (m) (m 3 ) (m 3 ) (m 3 ) Area (m ) Area (m ) Area (m ) Table 5-37: Confguraton #6 - lateral core translaton (m) Top of Interval SSAM Sframe Rato max error = 0.09% Table 5-38: Confguraton #6 - core rotaton (rad) Top of Interval SSAM Sframe Rato max error = 0.% 81

93 Table 5-39: Confguraton #6 - vertcal megacolumn translaton mnus vertcal core translaton Megacolumn A Megacolumn B Top of SSAM Sframe Interval Rato SSAM Sframe Rato Megacolumn C Megacolumn D Top of SSAM Sframe Interval Rato SSAM Sframe Rato max error = 1.07% Table 5-40: Confguraton #6 - core stress (KPa) Bottom of Interval SSAM Sframe Rato max error =.89% 8

94 Table 5-41: Confguraton #6 - megacolumn stress (KPa) Megacolumn A Megacolumn B Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato Megacolumn C Megacolumn D Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 4.97% Table 5-4: Confguraton #6 - outrgger stress under lateral load only (KPa) B D Top of Interval SSAM Sframe Rato SSAM Sframe Rato max error =.85% 83

95 Table 5-43: Confguraton #6 - belt truss stress under lateral load only (KPa) AB BC Top of Interval SSAM Sframe Rato SSAM Sframe Rato AD DE Top of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 91.7% Table 5-44: Confguraton #6 - dagonal stress (KPa) AB BC Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato AD DE Bottom of Interval SSAM Sframe Rato SSAM Sframe Rato max error = 5.5% 84

96 Heght (m) SSAM Wnd Dsp SSAM Wnd Drft Sframe Wnd Dsp Sframe Wnd Drft Ponts of Contraflexure Lateral dsplacement (m) and normalzed nterstory drft Fgure 5-6: Confguraton #6 - lateral dsplacement and nterstory drft 5.7 Confguraton #7 core+megacolumns+outrgger at one level only Table 5-45: Confguraton #7 - desgn varables and calculated megacolumn areas Interval Core t Outrg V Belt V Dag V Megacolumn A Megacolumns B/D Megacolumns C/E (m) (m 3 ) (m 3 ) (m 3 ) Area (m ) Area (m ) Area (m )

97 Table 5-46: Confguraton #7 - lateral core translaton (m) Top of Interval SSAM Sframe Rato max error = 0.10% Table 5-47: Confguraton #7 - core rotaton (rad) Top of Interval SSAM Sframe Rato max error = 0.11% 86

98 Table 5-48: Confguraton #7 - vertcal megacolumn translaton mnus vertcal core translaton Megacolumn A Megacolumn B Top of SSAM Sframe Interval Rato SSAM Sframe Rato Megacolumn C Megacolumn D Top of SSAM Sframe Interval Rato SSAM Sframe Rato max error = 0.11% 87

99 Heght (m) SSAM Wnd Dsp SSAM Wnd Drft Sframe Wnd Dsp Sframe Wnd Drft Ponts of Contraflexure Lateral dsplacement (m) and normalzed nterstory drft Fgure 5-7: Confguraton #7 - lateral dsplacement and nterstory drft 88

100 6 CONCLUSIONS The SSAM was developed, mplemented, and tested. The SSAM was able to predct the exstence of ponts of contraflexure n the deflected shape of confguratons nvolvng outrggers, belts, and dagonals, as verfed by the space frame model. Such ponts of contraflexure cannot be predcted wth contnuum models. The accuracy of the SSAM was compared aganst the space frame model. For all confguratons that exclude dagonals, the maxmum error was 1% for lnear and nonlnear lateral translatons, 1% for lnear and nonlnear rotatons, and 1% for vertcal translatons. Furthermore, the maxmum error n stress was 3% for the core, 3% for the megacolumns, 1% for outrggers, and 1% for belts. For confguratons that ncluded dagonals, the maxmum error was 1% for lnear lateral translatons, 1% for lnear rotatons, and 7% for vertcal translatons. Addtonally, the maxmum error n stress was 4% for the core, 17% for the megacolumns, 3% for the outrggers, 9% for the belts, and 76% for the dagonals. Thus, the accuracy of the SSAM s very good for translatons and rotatons, and reasonably good for stress n confguratons that exclude dagonals. Stress formulas for confguratons that nclude dagonals need further development. The speed of executon, data preparaton, data extracton, and optmzaton s much faster wth the SSAM than wth general space frame programs, both that of Ballng (1991) and ADINA. Executon of the SSAM s nstantaneous snce t only nvolves 30 DOF's for the generc syscraper. Executon of the space frame model of the generc syscraper wth 10,776 89

101 DOF's on the space frame program from Ballng (1991) requred about 5 mnutes on a computer. Preparaton of data for the SSAM spreadsheet on a new syscraper wll tae some tme. But preparaton/extracton of data for general space frame and fnte element programs for a syscraper nvolvng 5668 members and 1877 nodes wll tae more tme. Rapd tral-and-error optmzaton s possble wth the SSAM spreadsheet, but not possble wth general space frame and fnte element programs. The SSAM appears to be deal for prelmnary syscraper desgn and educatonal purposes for students learnng about the behavor and desgn of modern syscrapers. 90

102 REFERENCES Abergel, D. P., and Smth, B. S. (1983). "Approxmate analyss of hgh-rse structures comprsng coupled walls and shear walls." Buldng and Envronment, 18(1), Amercan Socety of Cvl Engneers (ASCE). (006). Mnmum desgn loads for buldngs and other structures. ASCE 7-05, New Yor. Baer, M. C. M., Hoenderamp, J. C. D., and Snjder, H. H. (003). "Prelmnary desgn of hgh-rse outrgger braced shear wall structures on flexble foundatons." HERON, 48(), Ballng, R. J. (1991). Computer Structural Analyss, BYU Academc Publshng, Provo, UT. Bozdogan, K. B. (009). "An approxmate method for statc and dynamc analyses of symmetrc wall-frame buldngs." The Structural Desgn of Tall and Specal Buldngs, 18(3), Bozdogan, K. B., and Oztur, D. (009). "An approxmate method for lateral stablty analyss of wall-frame buldngs nculdng shear deformatons of walls." Indan Academy of Scences, 35(3), Cho, H. S., Ho, G., Joseph, L., and Mathas, N. (01). "Outrgger Desgn for Hgh-Rse Buldngs: An output of the CTBUH Outrgger Worng Group." De Llera, J. C. L., and Chopra, A. K. (1995). "A smplfed model for analyss and desgn of asymmetrc-plan buldngs." Earthquae engneerng & structural dynamcs, 4, 1. Hedebrecht, A. C., and Smth, B. S. (1973). "Approxmate analyss of tall wall-frame structures." Journal of the Structural Dvson, 99(),

103 Hoenderamp, J. C. D. (004). "Shear wall wth outrgger trusses on wall and column foundatons." The Structural Desgn of Tall and Specal Buldngs, 13(1), Hoenderamp, J. C. D. (008). "Second outrgger at optmum locaton on hgh-rse shear wall." The Structural Desgn of Tall and Specal Buldngs, 17(3), Hoenderamp, J. C. D., and Baer, M. C. M. (003). "Analyss of hgh-rse braced frames wth outrggers." The Structural Desgn of Tall and Specal Buldngs, 1(4), Hoenderamp, J. C. D., Kuster, M., and Smth, B. S. (1984). "Generalzed method for estmatng drft n hgh-rse structures." Journal of Structural Engneerng, 110(7), Inc., K. K. E. "Structural Desgn for Hgh-rse Buldng." < Katz, P., Robertson, L. E., and See, S. (008). "Case Study: Shanga World Fnancal Center." CTBUH Journal, 1(), Kavan, P., Rahgozar, R., and Saffar, H. (008). "Approxmate analyss of tall buldngs usng sandwch beam models wth varable cross secton." The Structural Desgn of Tall and Specal Buldngs, 17(), Kan, P. S., and Sahaan, F. T. (001). "The Use of Outrgger and Belt Truss System for Hgh- Rse Concrete Buldngs." Dmensons of Cvl Engneerng, 3(1), 6. Kobayash, M., Taabatae, H., and Taesao, R. (1995). "A smlfed analyss of doubly symmetrc tube structures." The Structural Desgn of Tall Buldngs, 4, Kwan, A. K. H. (1994). "Smple method for approxmate analyss of framed tube structures." Journal of Structural Engneerng, 10(4), Ln, L., Peau, O. A., and Zelns, Z. A. (1994). "Dsplacement and natural frequences of tall buldng structures by fnte story method." Computers & Structures, 54(1), Mass, D. C., Poon, D., and Xa, J. (010). "Case Study: Shangha Tower." CTBUH Journal, 1(),

104 Nollet, M.-J., and Smth, B. S. (1997). "Stffened-story wall-frame tall buldng structure." Computers & Structures, 66(-3), Potzta, G., and Kollar, L. P. (003). "Analyss of buldng structures by replacement sandwch beams." Internatonal Journal of Solds and Structures, 40, Rahgozar, R., and Sharf, Y. (009). "An approxmate analyss of combned system of framed tube, shear core and belt truss n hgh-rse buldngs." The Structural Desgn of Tall and Specal Buldngs, 18(6), Rahgozar, R., Ahmad, A. R., and Sharf, Y. (010). "A smple mathematcal model for approxmate analyss of tall buldngs." Appled Mathematcal Modellng, 34(9), Rutenberg, A. (1987). "Lateral load response of belted tall buldng structures." Engneerng Structures, 9(1), Rutenberg, A., and Hedebrecht, A. C. (1975). "Approxmate analyss of asymmetrc wall-frame structures." Buldng Scence, 10(1), Stafford Smth, B., and Coull, A. (1991). Tall buldng structures: analyss and desgn, John Wley & Sons, Inc., New Yor. Smth, B. S., and Salm, I. (1981). "Parameter Study of Outrgger-Braced Tall Buldng Structures." Journal of the Structural Dvson, 107(ST10), Taabatae, H. (01). "A Smplfed Analytcal Method for Hgh-Rse Buldngs." Taranath, B. S. (005). Wnd and earthquae resstant buldngs: sturctural analyss and desgn, Marcel Deer, New Yor. Toutanj, H. A. (1997). "The effect of foundaton flexblty on the nteracton between shear walls and frames." Engneerng Structures, 19(1), Wong, V. (June 00). "Concrete core detal." < 93

105 Wu, J. R., and L, Q. S. (003). "Structural performance of mult-outrgger-braced tall buldngs." The Structural Desgn of Tall and Specal Buldngs, 1(),

106 APPENDIX A. SSAM EXCEL SPREADSHEET (CONFIGURATION #6) The SSAM was executed on an Excel spreadsheet. A typcal spreadsheet has fve sheets: 1) Propertes sheet, ) Desgn sheet, 3) Matrces sheet, 4) Lateral sheet, and 6) Stress sheet. An example wll follow wth Confguraton #6. Propertes Sheet Concrete allowable stress (KPa) modulus (KPa) densty (KN/m^3) 1.7 cost ($/m^3) 157 Steel allowable stress (KPa) modulus (KPa) densty (KN/m^3) 77 cost ($/KN) 70 Weght Data floor dead load (KPa) 4.34 floor lve load (KPa).4 claddng weght (KPa) 1.3 Wnd Data speed (mph) 13 reference heght (m) 74 exponent alpha 9.5 drft allowable 360 Sesmc Data spectral acceleraton (g) 0. ductlty factor 3 exponent drft allowable 50 95

107 Desgn Sheet 96

108 Matrces Sheet 97

109 98

110 99

111 Lateral Sheet (Only 35 of 100 stores) 100

112 101

113 10

114 103

115 Stress Sheet 104