200912311 Csi Analysis Reference Manual for Sap2000 Etabs and Safe PDF

8/11/2019 200912311 Csi Analysis Reference Manual for Sap2000 Etabs and Safe PDF

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CSI Anal ysis ReferenceManual

For SAP2000, ETABS, and SAFE

COMPUTERS &STRUCTURES

INC.

R

Com put ers and Struc tures, Inc.Ber keley, Cali fornia, USA Octo ber 2005


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COPYRIGHT

The com puter pro grams SAP2000, ETABS, and SAFE and all as soci-ated docu men tation are pro prie tary and copy righted prod ucts. World -wide rights of own ership rest with Com put ers and Struc tures, Inc. Un li-censed use of the pro gram or re pro duc tion of the docu men tation in anyform, with out prior writ ten authori zation from Com put ers and Struc -tures, Inc., is ex plic itly pro hib ited.

Fur ther in formation and cop ies of this docu men tation may be ob tainedfrom:

Com put ers and Struc tures, Inc.1995 Uni versity Av enue

Berke ley, Cal ifornia 94704 USA

tel: (510) 845-2177fax: (510) 845-4096

e-mail: [email protected]: www.computersandstructures.com

Copy right Com put ers and Struc tures, Inc., 19782005.The CSI Logo is a reg istered trade mark of Com put ers and Struc tures, Inc.SAP2000 is a reg istered trade mark of Com put ers and Struc tures, Inc.ETABS is a reg istered trade mark of Com put ers and Struc tures, Inc.SAFE is a trade mark of Com put ers and Struc tures, Inc.Win dows is a reg istered trade mark of Microsoft Cor poration.


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DISCLAIMER

CON SID ER ABLE TIME, EF FORT AND EX PENSE HAVE GONEINTO THE DE VEL OP MENT AND DOCU MEN TA TION OFSAP2000, ETABS AND SAFE. THE PRO GRAMS HAVE BEENTHOR OUGHLY TESTED AND USED. IN US ING THE PRO -GRAMS, HOW EVER, THE USER AC CEPTS AND UN DER STANDSTHAT NO WAR RANTY IS EX PRESSED OR IM PLIED BY THE DE -VEL OPERS OR THE DIS TRIBU TORS ON THE AC CU RACY OR THE RE LIABIL ITY OF THE PRO GRAMS.

THE USER MUST EX PLIC ITLY UN DER STAND THE AS SUMP -TIONS OF THE PRO GRAMS AND MUST IN DE PEND ENTLY VER -IFY THE RE SULTS.


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ACKNOWLEDGMENT

Thanks are due to all of the nu mer ous struc tural en gineers, who over theyears have given valu able feed back that has con tributed to ward the en -hance ment of this prod uct to its cur rent state.

Spe cial rec ognition is due Dr. Ed ward L. Wil son, Pro fessor Emeri tus,Uni ver sity of Cali fornia at Ber keley, who was re spon si ble for the con -cep tion and de velopment of the origi nal SAP se ries of pro grams andwhose con tinued origi nal ity has pro duced many unique con cepts thathave been im ple mented in this ver sion.


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Table of Contents

Chapter I In tro duction 1Anal ysis Fea tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Struc tural Anal ysis and De sign . . . . . . . . . . . . . . . . . . . . . . 2About This Man ual . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Top ics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Ty po graph ical Con ven tions . . . . . . . . . . . . . . . . . . . . . . . 4

Bold for Def initions . . . . . . . . . . . . . . . . . . . . . . . . . 4Bold for Vari able Data. . . . . . . . . . . . . . . . . . . . . . . . 4Ital ics for Math emat ical Vari ables . . . . . . . . . . . . . . . . . . 4

Ital ics for Em pha sis . . . . . . . . . . . . . . . . . . . . . . . . . 4All Cap itals for Lit eral Data . . . . . . . . . . . . . . . . . . . . . 5Cap italized Names . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Bib liographic Ref erences . . . . . . . . . . . . . . . . . . . . . . . . . 5

Chapter II Ob jects and Elements 7

Ob jects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Ob jects and El ements . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter III Coordinate Systems 11

Over view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Global Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . . 12Upward and Hor izon tal Di rec tions . . . . . . . . . . . . . . . . . . . 13Defining Co ordinate Sys tems . . . . . . . . . . . . . . . . . . . . . . 13

Vec tor Cross Prod uct . . . . . . . . . . . . . . . . . . . . . . . . 13

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Con straint Equa tions . . . . . . . . . . . . . . . . . . . . . . . . 49Plane Def inition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Dia phragm Con straint . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . 51Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . 51

Con straint Equa tions . . . . . . . . . . . . . . . . . . . . . . . . 52Plate Con straint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . 53Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . 53Con straint Equa tions . . . . . . . . . . . . . . . . . . . . . . . . 53

Axis Def inition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Rod Con straint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . 55Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . 56Con straint Equa tions . . . . . . . . . . . . . . . . . . . . . . . . 56

Beam Con straint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . 56Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . 57Con straint Equa tions . . . . . . . . . . . . . . . . . . . . . . . . 57

Equal Con straint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . 58Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . 58Selected De grees of Free dom . . . . . . . . . . . . . . . . . . . 58Con straint Equa tions . . . . . . . . . . . . . . . . . . . . . . . . 58

Local Con straint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . 59

No Lo cal Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . 59Selected De grees of Free dom . . . . . . . . . . . . . . . . . . . 60Con straint Equa tions . . . . . . . . . . . . . . . . . . . . . . . . 60

Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Automatic Mas ter Joints. . . . . . . . . . . . . . . . . . . . . . . . . 64

Stiff ness, Mass, and Loads . . . . . . . . . . . . . . . . . . . . . 64Local Co ordinate Sys tems . . . . . . . . . . . . . . . . . . . . . 64

Con straint Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Chapter VI Material Properties 67

Over view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . . . 68Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Iso tro pic Ma terials . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Orthotropic Ma terials . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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Anisotropic Ma terials . . . . . . . . . . . . . . . . . . . . . . . . . . 72Tem per ature-De pend ent Prop erties . . . . . . . . . . . . . . . . . . . 73Element Ma terial Tem per ature . . . . . . . . . . . . . . . . . . . . . 74Mass Den sity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Weight Den sity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Material Damp ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Modal Damp ing . . . . . . . . . . . . . . . . . . . . . . . . . . 76Vis cous Pro por tional Damp ing . . . . . . . . . . . . . . . . . . . 76Hysteretic Pro por tional Damp ing . . . . . . . . . . . . . . . . . 76

Design-Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Time-de pend ent Prop erties . . . . . . . . . . . . . . . . . . . . . . . 77

Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Time-In tegra tion Con trol . . . . . . . . . . . . . . . . . . . . . . 78

Stress-Strain Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Chapter VII The Frame/Cable Element 79

Over view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Joint Off sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Degrees of Free dom . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . . . 82

Lon gitudinal Axis 1 . . . . . . . . . . . . . . . . . . . . . . . . 83Default Ori entation . . . . . . . . . . . . . . . . . . . . . . . . . 83

Coordinate An gle . . . . . . . . . . . . . . . . . . . . . . . . . . 85Advanced Lo cal Co ordinate Sys tem . . . . . . . . . . . . . . . . . . 85

Ref erence Vec tor . . . . . . . . . . . . . . . . . . . . . . . . . . 86Determin ing Trans verse Axes 2 and 3 . . . . . . . . . . . . . . . 87

Sec tion Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Local Co ordinate Sys tem. . . . . . . . . . . . . . . . . . . . . . 89Material Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . 89Geo met ric Prop erties and Sec tion Stiffnesses . . . . . . . . . . . 89Shape Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Automatic Sec tion Prop erty Cal culation . . . . . . . . . . . . . . 92Sec tion Prop erty Da ta base Files . . . . . . . . . . . . . . . . . . 92Sec tion-De signer Sec tions . . . . . . . . . . . . . . . . . . . . . 94Additional Mass and Weight . . . . . . . . . . . . . . . . . . . . 94

Non-pris matic Sec tions . . . . . . . . . . . . . . . . . . . . . . . 94Prop erty Mod ifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Inser tion Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98End Off sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Clear Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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Rigid-end Fac tor . . . . . . . . . . . . . . . . . . . . . . . . . 101Effect upon Non-pris matic El ements . . . . . . . . . . . . . . . 102Effect upon In ternal Force Out put . . . . . . . . . . . . . . . . 102Effect upon End Re leases . . . . . . . . . . . . . . . . . . . . . 102

End Re leases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Unsta ble End Re leases . . . . . . . . . . . . . . . . . . . . . . 104Effect of End Off sets . . . . . . . . . . . . . . . . . . . . . . . 104

Non linear Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . 104Ten sion/Com pres sion Lim its . . . . . . . . . . . . . . . . . . . 104Plas tic Hinge . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Grav ity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Con cen trated Span Load . . . . . . . . . . . . . . . . . . . . . . . . 107

Dis tributed Span Load . . . . . . . . . . . . . . . . . . . . . . . . . 107Loaded Length . . . . . . . . . . . . . . . . . . . . . . . . . . 107Load In tensity . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Pro jected Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Tem per ature Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Internal Force Out put . . . . . . . . . . . . . . . . . . . . . . . . . 112

Effect of End Off sets . . . . . . . . . . . . . . . . . . . . . . . 114

Chapter VIII Frame Hinge Prop erties 115

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Hinge Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Hinge Length . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Plas tic De formation Curve . . . . . . . . . . . . . . . . . . . . 117Scal ing the Curve . . . . . . . . . . . . . . . . . . . . . . . . . 118Cou pled P-M2-M3 Hinge . . . . . . . . . . . . . . . . . . . . . 119Fi ber P-M2-M3 Hinge . . . . . . . . . . . . . . . . . . . . . . 121

Default, User-De fined, and Gen erated Prop erties . . . . . . . . . . . 121Default Hinge Prop erties. . . . . . . . . . . . . . . . . . . . . . . . 123

Default Con crete Hinge Prop erties . . . . . . . . . . . . . . . . 124

De fault Steel Hinge Prop er ties . . . . . . . . . . . . . . . . . . 125Anal ysis Re sults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Chapter IX The Shell Element 127

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 129De grees of Free dom . . . . . . . . . . . . . . . . . . . . . . . . . . 131

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Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . . 132 Nor mal Axis 3. . . . . . . . . . . . . . . . . . . . . . . . . . . 132Default Ori entation . . . . . . . . . . . . . . . . . . . . . . . . 133Element Co ordinate An gle . . . . . . . . . . . . . . . . . . . . 133

Advanced Lo cal Co ordinate Sys tem. . . . . . . . . . . . . . . . . . 133

Ref erence Vec tor . . . . . . . . . . . . . . . . . . . . . . . . . 135Determin ing Tan gen tial Axes 1 and 2 . . . . . . . . . . . . . . 136

Sec tion Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Sec tion Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Thick ness For mulation . . . . . . . . . . . . . . . . . . . . . . 138Material Prop erties . . . . . . . . . . . . . . . . . . . . . . . . 138Material An gle . . . . . . . . . . . . . . . . . . . . . . . . . . 139Thick ness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Grav ity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Uni form Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Sur face Pres sure Load . . . . . . . . . . . . . . . . . . . . . . . . . 143Tem per ature Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Internal Force and Stress Out put. . . . . . . . . . . . . . . . . . . . 144

Chapter X The Plane Element 149

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Degrees of Free dom . . . . . . . . . . . . . . . . . . . . . . . . . . 151Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . . 151Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . 151Sec tion Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Sec tion Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Material Prop erties . . . . . . . . . . . . . . . . . . . . . . . . 153Material An gle . . . . . . . . . . . . . . . . . . . . . . . . . . 153Thick ness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Incom pat i ble Bend ing Modes . . . . . . . . . . . . . . . . . . . 154

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Grav ity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Sur face Pres sure Load . . . . . . . . . . . . . . . . . . . . . . . . . 156Pore Pres sure Load. . . . . . . . . . . . . . . . . . . . . . . . . . . 156Tem per ature Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Stress Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

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Chapter XI The Asolid El ement 159

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 160De grees of Free dom . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . . 161Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . 162Sec tion Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Sec tion Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 162Material Prop erties . . . . . . . . . . . . . . . . . . . . . . . . 163Material An gle . . . . . . . . . . . . . . . . . . . . . . . . . . 163Axis of Sym metry . . . . . . . . . . . . . . . . . . . . . . . . . 164Arc and Thick ness. . . . . . . . . . . . . . . . . . . . . . . . . 165Incom pat i ble Bend ing Modes . . . . . . . . . . . . . . . . . . . 166

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 166Grav ity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167Sur face Pres sure Load . . . . . . . . . . . . . . . . . . . . . . . . . 167Pore Pres sure Load. . . . . . . . . . . . . . . . . . . . . . . . . . . 168Tem per ature Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 168Rotate Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168Stress Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Chapter XII The Solid Element 171

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 172De grees of Free dom . . . . . . . . . . . . . . . . . . . . . . . . . . 173Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . . 174Advanced Lo cal Co ordinate Sys tem. . . . . . . . . . . . . . . . . . 174

Ref erence Vec tors . . . . . . . . . . . . . . . . . . . . . . . . . 175Defin ing the Axis Ref erence Vec tor . . . . . . . . . . . . . . . 175Defin ing the Plane Ref erence Vec tor . . . . . . . . . . . . . . . 176De ter min ing the Lo cal Axes from the Ref erence Vec tors . . . . 177

Element Co ordinate An gles . . . . . . . . . . . . . . . . . . . . 178Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . 178Solid Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Material Prop erties . . . . . . . . . . . . . . . . . . . . . . . . 180Material An gles . . . . . . . . . . . . . . . . . . . . . . . . . . 180Incom pat i ble Bend ing Modes . . . . . . . . . . . . . . . . . . . 180

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

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Grav ity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Sur face Pres sure Load . . . . . . . . . . . . . . . . . . . . . . . . . 182Pore Pres sure Load. . . . . . . . . . . . . . . . . . . . . . . . . . . 183Tem per ature Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 183Stress Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Chapter XIII The Link/Support ElementBasic 185

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186Joint Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 187Zero-Length El ements . . . . . . . . . . . . . . . . . . . . . . . . . 187Degrees of Free dom . . . . . . . . . . . . . . . . . . . . . . . . . . 187Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . . 188

Lon gitudinal Axis 1 . . . . . . . . . . . . . . . . . . . . . . . . 188Default Ori entation . . . . . . . . . . . . . . . . . . . . . . . . 189Coordinate An gle . . . . . . . . . . . . . . . . . . . . . . . . . 189

Advanced Lo cal Co ordinate Sys tem. . . . . . . . . . . . . . . . . . 190Axis Ref erence Vec tor . . . . . . . . . . . . . . . . . . . . . . 191Plane Ref erence Vec tor . . . . . . . . . . . . . . . . . . . . . . 192Determin ing Trans verse Axes 2 and 3 . . . . . . . . . . . . . . 193

Internal De formations . . . . . . . . . . . . . . . . . . . . . . . . . 194Link/Sup port Prop erties . . . . . . . . . . . . . . . . . . . . . . . . 196

Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . 197Internal Spring Hinges . . . . . . . . . . . . . . . . . . . . . . 197

Spring Force-De formation Re lation ships . . . . . . . . . . . . . 199Element In ternal Forces . . . . . . . . . . . . . . . . . . . . . . 200Uncou pled Lin ear Force-De formation Re lation ships . . . . . . . 201Types of Lin ear/Non linear Prop erties. . . . . . . . . . . . . . . 203

Cou pled Lin ear Prop erty . . . . . . . . . . . . . . . . . . . . . . . . 203Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Grav ity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Internal Force and De for mation Out put . . . . . . . . . . . . . . . . 206

Chapter XIV The Link/Support ElementAd vanced 207

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 Non linear Link/Sup port Prop erties . . . . . . . . . . . . . . . . . . 208Linear Ef fec tive Stiff ness . . . . . . . . . . . . . . . . . . . . . . . 209

Spe cial Con siderations for Modal Anal yses . . . . . . . . . . . 209Linear Ef fec tive Damp ing . . . . . . . . . . . . . . . . . . . . . . . 210Vis cous Damper Prop erty . . . . . . . . . . . . . . . . . . . . . . . 211

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Gap Prop erty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212Hook Prop erty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212Multi-Lin ear Elas ticity Prop erty . . . . . . . . . . . . . . . . . . . . 213Wen Plas ticity Prop erty . . . . . . . . . . . . . . . . . . . . . . . . 214Multi-Lin ear Ki nematic Plas tic ity Prop erty . . . . . . . . . . . . . . 215Multi-Lin ear Takeda Plas ticity Prop erty . . . . . . . . . . . . . . . . 218Multi-Lin ear Pivot Hysteretic Plas tic ity Prop erty . . . . . . . . . . . 218Hysteretic (Rub ber) Iso lator Prop erty . . . . . . . . . . . . . . . . . 220Fric tion-Pen dulum Iso lator Prop erty. . . . . . . . . . . . . . . . . . 222

Non linear De formation Loads . . . . . . . . . . . . . . . . . . . . . 225Fre quency-De pend ent Link/Sup port Prop erties . . . . . . . . . . . . 227

Chapter XV The Tendon Ob ject 229

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231Ten dons Mod eled as Loads or El ements. . . . . . . . . . . . . . . . 231Con nec tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231De grees of Free dom . . . . . . . . . . . . . . . . . . . . . . . . . . 232Local Co ordinate Sys tems . . . . . . . . . . . . . . . . . . . . . . . 232

Base-line Lo cal Co ordinate Sys tem . . . . . . . . . . . . . . . . 233 Nat ural Lo cal Co ordinate Sys tem . . . . . . . . . . . . . . . . . 233

Sec tion Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . 234Material Prop erties . . . . . . . . . . . . . . . . . . . . . . . . 234Geo met ric Prop erties and Sec tion Stiffnesses. . . . . . . . . . . 234Prop erty Mod ifiers . . . . . . . . . . . . . . . . . . . . . . . . 235

Non linear Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . 235Ten sion/Com pres sion Lim its . . . . . . . . . . . . . . . . . . . 236Plas tic Hinge . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236Prestress Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Grav ity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238Tem per ature Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 238Internal Force Out put . . . . . . . . . . . . . . . . . . . . . . . . . 239

Chapter XVI Load Cases 241

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242Load Cases, Anal ysis Cases, and Com binations . . . . . . . . . . . . 243

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Defin ing Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 243Coordinate Sys tems and Load Com po nents . . . . . . . . . . . . . . 244

Effect upon Large-Dis place ments Anal ysis . . . . . . . . . . . . 244Force Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245Restraint Dis place ment Load . . . . . . . . . . . . . . . . . . . . . 245Spring Dis place ment Load. . . . . . . . . . . . . . . . . . . . . . . 245Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 245Grav ity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246Con cen trated Span Load . . . . . . . . . . . . . . . . . . . . . . . . 247Dis tributed Span Load . . . . . . . . . . . . . . . . . . . . . . . . . 247Ten don Pre stress Load . . . . . . . . . . . . . . . . . . . . . . . . . 247Uni form Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248Sur face Pres sure Load . . . . . . . . . . . . . . . . . . . . . . . . . 248Pore Pres sure Load. . . . . . . . . . . . . . . . . . . . . . . . . . . 249Tem per ature Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 250Ref erence Tem per ature . . . . . . . . . . . . . . . . . . . . . . . . 251Rotate Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251Joint Pat terns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252Acceleration Loads. . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Chapter XVII Anal ysis Cases 255

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Anal ysis Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257Types of Anal ysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 257Sequence of Anal ysis . . . . . . . . . . . . . . . . . . . . . . . . . 258Run ning Anal ysis Cases . . . . . . . . . . . . . . . . . . . . . . . . 259Linear and Non linear Anal ysis Cases . . . . . . . . . . . . . . . . . 260Lin ear Static Anal ysis . . . . . . . . . . . . . . . . . . . . . . . . . 261Lin ear Buck ling Anal ysis . . . . . . . . . . . . . . . . . . . . . . . 262Func tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263Com binations (Com bos) . . . . . . . . . . . . . . . . . . . . . . . . 264

Equa tion Solv ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267Access ing the As sem bled Stiff ness and Mass Ma trices . . . . . . . . 267

Chapter XVIII Modal Anal ysis 269

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Eigenvector Anal ysis . . . . . . . . . . . . . . . . . . . . . . . . . 270

Num ber of Modes . . . . . . . . . . . . . . . . . . . . . . . . . 271

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Fre quency Range . . . . . . . . . . . . . . . . . . . . . . . . . 271Automatic Shift ing . . . . . . . . . . . . . . . . . . . . . . . . 273Con vergence Tol erance . . . . . . . . . . . . . . . . . . . . . . 273Static-Cor rec tion Modes . . . . . . . . . . . . . . . . . . . . . 274

Ritz-Vec tor Anal ysis . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Num ber of Modes . . . . . . . . . . . . . . . . . . . . . . . . . 276Start ing Load Vec tors . . . . . . . . . . . . . . . . . . . . . . . 277

Num ber of Gen eration Cy cles. . . . . . . . . . . . . . . . . . . 278Modal Anal ysis Out put . . . . . . . . . . . . . . . . . . . . . . . . 279

Periods and Fre quen cies . . . . . . . . . . . . . . . . . . . . . 279Par tici pa tion Fac tors . . . . . . . . . . . . . . . . . . . . . . . 279Par tici pat ing Mass Ra tios . . . . . . . . . . . . . . . . . . . . . 280Static and Dy namic Load Par tic i pa tion Ra tios . . . . . . . . . . 281

Chapter XIX Response-Spectrum Anal ysis 285

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285Local Co ordinate Sys tem . . . . . . . . . . . . . . . . . . . . . . . 286Response-Spec trum Curve . . . . . . . . . . . . . . . . . . . . . . . 287

Damp ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288Modal Damp ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289Modal Com bination . . . . . . . . . . . . . . . . . . . . . . . . . . 290

CQC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 290GMC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 290SRSS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Absolute Sum Method . . . . . . . . . . . . . . . . . . . . . . 291

NRC Ten-Per cent Method . . . . . . . . . . . . . . . . . . . . 292 NRC Dou ble-Sum Method . . . . . . . . . . . . . . . . . . . . 292

Direc tional Com bination . . . . . . . . . . . . . . . . . . . . . . . . 292SRSS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 292Absolute Sum Method . . . . . . . . . . . . . . . . . . . . . . 292Scaled Ab solute Sum Method. . . . . . . . . . . . . . . . . . . 293

Response-Spec trum Anal ysis Out put . . . . . . . . . . . . . . . . . 293Damp ing and Ac celerations . . . . . . . . . . . . . . . . . . . . 293Modal Am pli tudes. . . . . . . . . . . . . . . . . . . . . . . . . 294Modal Cor relation Fac tors . . . . . . . . . . . . . . . . . . . . 294Base Re actions . . . . . . . . . . . . . . . . . . . . . . . . . . 294

Chapter XX Linear Time-History Anal ysis 295

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296Load ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Defining the Spa tial Load Vec tors . . . . . . . . . . . . . . . . 297

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Defin ing the Time Func tions . . . . . . . . . . . . . . . . . . . 298Ini tial Con ditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 300Time Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300Modal Time-His tory Anal ysis . . . . . . . . . . . . . . . . . . . . . 301

Modal Damp ing . . . . . . . . . . . . . . . . . . . . . . . . . . 302Direct-In tegra tion Time-His tory Anal ysis . . . . . . . . . . . . . . . 303

Time In tegra tion Pa ram eters . . . . . . . . . . . . . . . . . . . 303Damp ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Chapter XXI Geometric Nonlinearity 307

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Non linear Anal ysis Cases . . . . . . . . . . . . . . . . . . . . . . . 309The P-Delta Ef fect . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

P-Delta Forces in the Frame El ement . . . . . . . . . . . . . . . 313P-Delta Forces in the Link/Sup port El ement . . . . . . . . . . . 316Other El ements . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Ini tial P-Delta Anal ysis . . . . . . . . . . . . . . . . . . . . . . . . 317Build ing Struc tures . . . . . . . . . . . . . . . . . . . . . . . . 318Ca ble Struc tures . . . . . . . . . . . . . . . . . . . . . . . . . . 319Guyed Tow ers. . . . . . . . . . . . . . . . . . . . . . . . . . . 320

Large Dis place ments . . . . . . . . . . . . . . . . . . . . . . . . . . 321Ap plications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321Ini tial Large-Dis place ment Anal ysis . . . . . . . . . . . . . . . 322

Chapter XXII Nonlinear Static Anal ysis 323

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324Im por tant Con siderations . . . . . . . . . . . . . . . . . . . . . . . 325Load ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326Load Ap pli cation Con trol . . . . . . . . . . . . . . . . . . . . . . . 326

Load Con trol . . . . . . . . . . . . . . . . . . . . . . . . . . . 327Dis place ment Con trol . . . . . . . . . . . . . . . . . . . . . . . 327

Ini tial Con ditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 328Out put Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

Sav ing Mul ti ple Steps . . . . . . . . . . . . . . . . . . . . . . . 329 Non linear So lution Con trol . . . . . . . . . . . . . . . . . . . . . . 331

Max imum To tal Steps . . . . . . . . . . . . . . . . . . . . . . . 331Max i mum Null (Zero) Steps . . . . . . . . . . . . . . . . . . . 331Max imum It erations Per Step . . . . . . . . . . . . . . . . . . . 332Iteration Con ver gence Tol erance . . . . . . . . . . . . . . . . . 332

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Event Lump ing Tol erance. . . . . . . . . . . . . . . . . . . . . 332Hinge Un load ing Method . . . . . . . . . . . . . . . . . . . . . . . 332

Unload En tire Struc ture . . . . . . . . . . . . . . . . . . . . . . 333Ap ply Lo cal Re dis tri bu tion . . . . . . . . . . . . . . . . . . . . 334Restart Us ing Se cant Stiff ness . . . . . . . . . . . . . . . . . . 334

Static Push over Anal ysis. . . . . . . . . . . . . . . . . . . . . . . . 335Staged Con struc tion . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338Out put Steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . 339Exam ple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

Chapter XXIII Nonlinear Time-History Anal ysis 343

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

Load ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345Ini tial Con ditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 345Time Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

Non linear Modal Time-His tory Anal ysis (FNA) . . . . . . . . . . . 347Ini tial Con ditions . . . . . . . . . . . . . . . . . . . . . . . . . 347Link/Sup port Ef fec tive Stiff ness . . . . . . . . . . . . . . . . . 348Mode Su per po sition . . . . . . . . . . . . . . . . . . . . . . . . 348Modal Damp ing . . . . . . . . . . . . . . . . . . . . . . . . . . 350Iterative So lution . . . . . . . . . . . . . . . . . . . . . . . . . 351Static Pe riod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

Non linear Di rect-In tegra tion Time-His tory Anal ysis . . . . . . . . . 354Time In tegra tion Pa ram eters . . . . . . . . . . . . . . . . . . . 354

Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354Ini tial Con ditions . . . . . . . . . . . . . . . . . . . . . . . . . 355Damp ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355Iterative So lution . . . . . . . . . . . . . . . . . . . . . . . . . 356

Chapter XXIV Frequency-Domain Anal yses 359

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360Har monic Mo tion . . . . . . . . . . . . . . . . . . . . . . . . . . . 360Fre quency Do main . . . . . . . . . . . . . . . . . . . . . . . . . . . 361Damp ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

Sources of Damp ing. . . . . . . . . . . . . . . . . . . . . . . . 362Load ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Defining the Spa tial Load Vec tors . . . . . . . . . . . . . . . . 364Frequency Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

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Steady-State Anal ysis . . . . . . . . . . . . . . . . . . . . . . . . . 366Exam ple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

Power-Spec tral-Den sity Anal ysis . . . . . . . . . . . . . . . . . . . 367Exam ple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

Chapter XXV Bridge Anal ysis 371

Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372Mod eling the Bridge Struc ture. . . . . . . . . . . . . . . . . . . . . 373

Frame El ements . . . . . . . . . . . . . . . . . . . . . . . . . . 373Sup ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374Bear ings and Ex pan sion Joints . . . . . . . . . . . . . . . . . . 375Other El ement Types . . . . . . . . . . . . . . . . . . . . . . . 375

Road ways and Lanes. . . . . . . . . . . . . . . . . . . . . . . . . . 377Road ways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377Lanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377Eccen tric ities . . . . . . . . . . . . . . . . . . . . . . . . . . . 378Mod eling Guide lines . . . . . . . . . . . . . . . . . . . . . . . 378Exam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

Spa tial Res olution . . . . . . . . . . . . . . . . . . . . . . . . . . . 381Load and Out put Points . . . . . . . . . . . . . . . . . . . . . . 381Res olution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382Mod eling Guide lines . . . . . . . . . . . . . . . . . . . . . . . 383

Influence Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383Ve hicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

Direc tion of Loads . . . . . . . . . . . . . . . . . . . . . . . . 384Ap pli cation of Loads . . . . . . . . . . . . . . . . . . . . . . . 385Option to Al low Re duced Re sponse Se ver ity . . . . . . . . . . . 386Gen eral Ve hicle . . . . . . . . . . . . . . . . . . . . . . . . . . 387Stan dard Ve hicles . . . . . . . . . . . . . . . . . . . . . . . . . 390

Ve hicle Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394Mov ing Load Anal ysis Cases . . . . . . . . . . . . . . . . . . . . . 397

Exam ple 1 AASHTO HS Load ing. . . . . . . . . . . . . . . 398Exam ple 2 AASHTO HL Load ing. . . . . . . . . . . . . . . 399

Exam ple 3 Caltrans Per mit Load ing . . . . . . . . . . . . . . 400Exam ple 4 Re stricted Caltrans Per mit Load ing . . . . . . . . 402Influence Line Tol erance . . . . . . . . . . . . . . . . . . . . . . . 404Exact and Quick Re sponse Cal culation . . . . . . . . . . . . . . . . 404Mov ing Load Re sponse Con trol . . . . . . . . . . . . . . . . . . . . 405Cor respon dence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405Com pu tational Con siderations . . . . . . . . . . . . . . . . . . . . . 406

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Chapter XXVI References 407

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C h a p t e r I

Introduction

SAP2000, ETABS and SAFE are soft ware pack ages from Com put ers and Struc -tures, Inc. for struc tural anal ysis and de sign. Each package is a fully in tegrated sys -tem for mod eling, an alyz ing, de sign ing, and op timiz ing struc tures of a par ticular type:

SAP2000 for gen eral struc tures, in clud ing bridges, sta diums, tow ers, in dus trial plants, off shore struc tures, pip ing sys tems, build ings, dams, soils, ma chine parts and many oth ers

ETABS for build ing struc tures SAFE for floor slabs and base mats

At the heart of each of these soft ware pack ages is a com mon anal y sis en gine, re -ferred to through out this man ual as SAP2000. This en gine is the lat est and most

pow erful ver sion of the well-known SAP se ries of struc tural anal y sis pro grams.The pur pose of this man ual is to de scribe the fea tures of the SAP2000 anal y sis en -gine.

Through out this man ual the anal ysis en gine will be re ferred to as SAP2000, al -though it ap plies also to ETABS and SAFE. Not all fea tures de scribed will ac tually

be avail able in ev ery level of each pro gram.

1


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Analysis FeaturesThe CSI anal y sis en gine of fers the fol low ing fea tures:

Static and dy namic analy sis Lin ear and non linear analy sis Dynamic seis mic analy sis and static push over analysis Ve hicle live- load analy sis for bridges Geo met ric nonlinearity, in clud ing P-delta and large-dis place ment ef fects Staged (in cremen tal) con struc tion Creep, shrink age, and ag ing effects Buckling anal ysis Steady-state and power-spec tral-den sity analysis Frame and shell struc tural ele ments, in clud ing beam- column, truss, mem brane,

and plate be hav ior Two-di men sional plane and axi sym met ric solid el ements Three-di men sional solid el ements Non linear link and sup port el ements Fre quency-de pend ent link and sup port prop erties

Mul ti ple co ordinate sys tems Many types of con straints A wide va riety of load ing op tions Alpha- numeric la bels Large ca pac ity Highly ef ficient and sta ble so lution al gorithms

These fea tures, and many more, make CSI pro grams the state-of-the-art for struc -tural anal y sis. Note that not all of these fea tures may be avail able in ev ery level of SAP2000, ETABS and SAFE.

Structural Analysis and DesignThe fol low ing gen eral steps are re quired to ana lyze and de sign a struc ture us ingSAP2000, ETABS and SAFE:

2 Analysis Features



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1. Cre ate or mod ify a model that nu meri cally de fines the ge ome try, prop erties,load ing, and analy sis pa rame ters for the struc ture

2. Per form an analy sis of the model

3. Re view the re sults of the analy sis

4. Check and op timize the de sign of the struc ture

This is usu ally an it era tive pro cess that may in volve sev eral cy cles of the above se -quence of steps. All of these steps can be per formed seam lessly us ing the SAP2000,ETABS, and SAFE graph ical user in terfaces.

About This ManualThis man ual de scribes the theo reti cal con cepts be hind the mod eling and analy sisfea tures of fered by the SAP2000 anal ysis en gine that un der lies the SAP2000,ETABS and SAFE struc tural anal y sis and de sign soft ware pack ages. The graphi caluser in terface and the de sign fea tures are de scribed in sepa rate man u als for each

program.

It is im per ative that you read this man ual and un der stand the as sump tions and pro -cedures used by these soft ware packages be fore at tempt ing to use the anal ysis fea -tures.

Through out this man ual the anal y sis en gine may be re ferred to as SAP2000, al -though it ap plies also to ETABS and SAFE. Not all fea tures de scribed will ac tually

be avail able in ev ery level of each pro gram.

TopicsEach Chap ter of this man ual is di vided into top ics and sub top ics. All Chap ters be -gin with a list of top ics cov ered. These are di vided into two groups:

Basic top ics rec ommended read ing for all us ers Advanced top ics for us ers with spe cial ized needs, and for all us ers as they

be come more fa mil iar with the pro gram.

Fol lowing the list of top ics is an Over view which pro vides a sum mary of the Chap -ter. Read ing the Over view for every Chap ter will ac quaint you with the full scopeof the pro gram.

About This Manual 3

Chapter I Introduction


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Typographical ConventionsThrough out this man ual the fol low ing ty po graphic con ven tions are used.

Bold for DefinitionsBold ro man type (e.g., exam ple ) is used when ever a new term or con cept is de -fined. For ex am ple:

The global co or di nate sys tem is a three- dimensional, right- handed, rec tangu-lar co ordinate sys tem.

This sen tence be gins the defi nition of the global co ordinate sys tem.

Bold for Variable DataBold ro man type (e.g., exam ple ) is used to rep resent vari able data items for whichyou must spec ify val ues when de fin ing a struc tural model and its analy sis. For ex -am ple:

The Frame ele ment co ordinate an gle, ang , is used to de fine ele ment ori enta-tions that are dif ferent from the de fault ori entation.

Thus you will need to sup ply a nu meric value for the vari able ang if it is dif ferentfrom its de fault value of zero.

Italics for Mathematical Variables

Nor mal italic type (e.g., exam ple ) is used for sca lar mathe mati cal vari ables, and bold italic type (e.g., exam ple ) is used for vec tors and ma tri ces. If a vari able dataitem is used in an equa tion, bold ro man type is used as dis cussed above. For ex am-

ple:

0 da < db L

Here da and db are vari ables that you spec ify, and L is a length cal culated by the pro gram.

Italics for Emphasis

Nor mal italic type (e.g., exam ple ) is used to em pha size an im por tant point, or for the ti tle of a book, man ual, or jour nal.

4 Typographical Conventions



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All Capitals for Literal Data

All capi tal type (e.g., EX AM PLE) is used to rep resent data that you type at the key - board ex actly as it is shown, ex cept that you may ac tu ally type lower- case if you pre fer. For ex am ple:

DIAPHRAGM

indicates that you type DIAPHRAGM or di a phragm at the key board.

Capitalized Names

Capi tal ized names (e.g., Ex am ple) are used for cer tain parts of the model and itsanaly sis which have spe cial mean ing to SAP2000. Some ex am ples:

Frame ele mentDia phragm Con straint

Frame Sec tion

Load Case

Com mon en ti ties, such as joint or ele ment are not capi tal ized.

Bibliographic ReferencesRef erences are in dicated through out this man ual by giv ing the name of theauthor(s) and the date of pub lication, us ing pa ren theses. For ex am ple:

See Wil son and Tet suji (1983).

It has been dem onstrated (Wil son, Yuan, and Dick ens, 1982) that

All bib lio graphic ref erences are listed in al pha beti cal or der in Chap ter Ref er -ences (page 407 ).

Bibliographic References 5

Chapter I Introduction


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C h a p t e r II

Objects and Elements

The phys ical struc tural mem bers in a structural model are rep re sented by ob jects.Using the graph ical user in ter face, you draw the ge ometry of an ob ject, then as -sign prop erties and loads to the ob ject to com pletely de fine the model of the phys i-cal mem ber. For anal y sis pur poses, SAP2000 con verts each ob ject into one or more

elements.

Basic Topics for All Users Objects Ob jects and Elements Groups

ObjectsThe fol low ing ob ject types are avail able, listed in or der of geo met rical di men sion:

Point ob jects, of two types: Joint ob jects: These are au tomat i cally cre ated at the cor ners or ends of all

other types of ob jects be low, and they can be ex plic itly added to rep resentsup ports or to cap ture other lo cal ized be hav ior.

Objects 7


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Grounded (one-joint) sup port ob jects: Used to model spe cial sup port behav ior such as iso lators, damp ers, gaps, multi-lin ear springs, and more.

Line ob jects, of two types Frame/ca ble ob jects: Used to model beams, col umns, braces, trusses,

and/or ca ble mem bers Con necting (two-joint) link ob jects: Used to model spe cial mem ber be -

hav ior such as iso lators, damp ers, gaps, multi-lin ear springs, and more.Unlike frame/ca ble ob jects, con nect ing link ob jects can have zero length.

Area ob jects: Shell el ements (plate, mem brane, and full-shell) used to modelwalls, floors, and other thin-walled mem bers; as well as two-di men sional sol -ids (plane-stress, plane-strain, and axisymmetric sol ids).

Solid ob jects: Used to model three-di men sional sol ids.

As a gen eral rule, the ge ometry of the ob ject should cor respond to that of the phys i-cal mem ber. This sim pli fies the vi sualiza tion of the model and helps with the de -sign pro cess.

Ob jects and ElementsIf you have ex pe rience us ing tra ditional fi nite el ement pro grams, in clud ing ear lier ver sions of SAP2000, ETABS or SAFE, you are prob a bly used to mesh ing phys i-cal mod els into smaller fi nite el ements for anal y sis pur poses. Ob ject-based mod el-ing largely elim inates the need for do ing this.

For us ers who are new to fi nite-el ement mod eling, the ob ject-based con cept shouldseem per fectly nat ural.

When you run an anal y sis, SAP2000 au tomat ically con verts your ob ject-basedmodel into an el ement-based model that is used for anal ysis. This el ement-basedmodel is called the anal y sis model, and it con sists of tra di tional fi nite el ements and

joints (nodes). Re sults of the anal ysis are re ported back on the ob ject-based model.

You have con trol over how the mesh ing is per formed, such as the de gree of re fine -ment, and how to han dle the con nec tions be tween in tersect ing ob jects. You alsohave the op tion to man ually mesh the model, re sult ing in a one-to-one cor respon -dence be tween ob jects and el ements.

In this man ual, the term el ement will be used more of ten than ob ject, sincewhat is de scribed herein is the fi nite-el ement anal y sis por tion of the pro gram thatoperates on the el ement-based anal ysis model. However, it should be clear that the


8 Ob jects and Elements


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prop er ties de scribed here for el ements are ac tually as signed in the in terface to theob jects, and the con version to anal ysis el ements is au tomatic.

GroupsA group is a named col lection of ob jects that you de fine. For each group, you must

pro vide a unique name, then se lect the ob jects that are to be part of the group. Youcan in clude ob jects of any type or types in a group. Each ob ject may be part of oneof more groups. All ob jects are al ways part of the built-in group called ALL.

Groups are used for many pur poses in the graph ical user in terface, in clud ing se lec-tion, de sign op timization, de fining sec tion cuts, con trol ling out put, and more. Inthis man ual, we are pri mar ily in terested in the use of groups for de fin ing stagedcon struc tion. See Topic Staged Con struc tion (page 77) in Chap ter Non linear Static Anal ysis for more in forma tion.

Groups 9

Chapter II Objects and Elements


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C h a p t e r III

Coordinate Systems

Each struc ture may use many dif fer ent co or di nate sys tems to de scribe the lo cationof points and the di rec tions of loads, dis place ment, in ternal forces, and stresses.Under stand ing these dif ferent co ordinate sys tems is cru cial to be ing able to prop -erly de fine the model and in ter pret the re sults.

Basic Topics for All Users Over view Global Co ordinate Sys tem Upward and Hori zon tal Di rec tions Defining Co ordinate Sys tems Local Co ordinate Sys tems

Advanced Topics Alternate Co ordinate Sys tems Cylindrical and Spheri cal Co ordinates

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Overview Coordinate sys tems are used to lo cate dif ferent parts of the struc tural model and todefine the di rec tions of loads, dis place ments, in ternal forces, and stresses.

All co ordinate sys tems in the model are de fined with re spect to a sin gle global co or-dinate sys tem. Each part of the model (joint, ele ment, or con straint) has its own lo -cal co ordinate sys tem. In ad dition, you may cre ate al ternate co ordinate sys tems thatare used to de fine lo cations and di rec tions.

All co ordinate sys tems are three- dimensional, right- handed, rec tangular (Car te-sian) sys tems. Vec tor cross prod ucts are used to de fine the lo cal and al ternate co or-dinate sys tems with re spect to the global sys tem.

SAP2000 al ways as sumes that Z is the ver tical axis, with +Z be ing up ward. The up -

ward di rec tion is used to help de fine lo cal co ordinate sys tems, al though lo cal co or-dinate sys tems them selves do not have an up ward di rec tion.

The lo cations of points in a co ordinate sys tem may be speci fied us ing rect angular or cy lindrical co ordinates. Like wise, di rec tions in a co ordinate sys tem may bespeci fied us ing rec tangular, cy lindrical, or spheri cal co ordinate di rec tions at a

point.

Global Coordinate SystemThe global co or di nate sys tem is a three- dimensional, right- handed, rec tangular coordinate sys tem. The three axes, de noted X, Y, and Z, are mu tually per pen dicu larand sat isfy the right- hand rule.

Locations in the global co ordinate sys tem can be speci fied us ing the vari ables x, y,and z. A vec tor in the global co ordinate sys tem can be speci fied by giv ing the lo ca-tions of two points, a pair of an gles, or by speci fying a co ordinate di rec tion. Co or-dinate di rec tions are in dicated us ing the val ues X, Y, and Z. For ex am ple, +Xdefines a vec tor par allel to and di rected along the posi tive X axis. The sign is re -quired.

All other co ordinate sys tems in the model are ul timately de fined with re spect to theglobal co ordinate sys tem, ei ther di rectly or in directly. Like wise, all joint co ordi-nates are ul timately con verted to global X, Y, and Z co ordinates, re gard less of howthey were speci fied.

12 Overview



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Upward and Horizontal DirectionsSAP2000 al ways as sumes that Z is the ver tical axis, with +Z be ing up ward. Lo calcoordinate sys tems for joints, ele ments, and ground- acceleration load ing are de -fined with re spect to this up ward di rec tion. Self- weight load ing al ways acts down -ward, in the Z di rec tion.

The X-Y plane is hori zon tal. The pri mary hori zon tal di rec tion is +X. An gles in thehori zon tal plane are meas ured from the posi tive half of the X axis, with posi tive an -gles ap pear ing coun terclock wise when you are look ing down at the X-Y plane.

If you pre fer to work with a dif ferent up ward di rec tion, you can de fine an al ternatecoordinate sys tem for that pur pose.

Defining Coordinate SystemsEach co ordinate sys tem to be de fined must have an ori gin and a set of three,mutually- perpendicular axes that sat isfy the right- hand rule.

The ori gin is de fined by sim ply speci fying three co ordinates in the global co ordi-nate sys tem.

The axes are de fined as vec tors us ing the con cepts of vec tor al ge bra. A fun damen talknowl edge of the vec tor cross prod uct opera tion is very help ful in clearly un der -

stand ing how co ordinate sys tem axes are de fined.

Vector Cross Product

A vec tor may be de fined by two points. It has length, di rec tion, and lo cation inspace. For the pur poses of de fin ing co ordinate axes, only the di rec tion is im por tant.Hence any two vec tors that are par al lel and have the same sense (i.e., point ing thesame way) may be con sid ered to be the same vec tor.

Any two vec tors, V i and V j , that are not par al lel to each other de fine a plane that is par allel to them both. The lo cation of this plane is not im por tant here, only its ori en-tation. The cross prod uct of V i and V j de fines a third vec tor, V k , that is per pen dicu larto them both, and hence nor mal to the plane. The cross prod uct is writ ten as:

V k = V i V j

Upward and Horizontal Directions 13

Chapter III Coordinate Systems


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The length of V k is not im por tant here. The side of the V i -V j plane to which V k pointsis de termined by the right- hand rule: The vec tor V k points to ward you if the acuteangle (less than 180) from V i to V j ap pears coun terclock wise.

Thus the sign of the cross prod uct de pends upon the or der of the op erands:

V j V i = V i V j

Defining the Three Axes Using Two Vectors

A right- handed co ordinate sys tem R- S-T can be rep resented by the three mutually- perpendicular vec tors V r , V s, and V t , re spec tively, that sat isfy the re lationship:

V t = V r V s

This co ordinate sys tem can be de fined by speci fying two non- parallel vec tors: An axis ref er ence vec tor, V a , that is par allel to axis R A plane ref er ence vec tor, V p, that is par allel to plane R-S, and points to ward the

positive-S side of the R axis

The axes are then de fined as:

V r = V a

V t = V r V p

V s = V t V r

Note that V p can be any con ven ient vec tor par allel to the R-S plane; it does not haveto be par allel to the S axis. This is il lus trated in Figure 1 (page 15).

Local Coordinate SystemsEach part (joint, ele ment, or con straint) of the struc tural model has its own lo cal co -

ordinate sys tem used to de fine the prop erties, loads, and re sponse for that part. Theaxes of the lo cal co ordinate sys tems are de noted 1, 2, and 3. In gen eral, the lo cal co -ordinate sys tems may vary from joint to joint, ele ment to ele ment, and con straint tocon straint.

There is no pre ferred up ward di rec tion for a lo cal co ordinate sys tem. How ever, theupward +Z di rec tion is used to de fine the de fault joint and ele ment lo cal co ordinatesys tems with re spect to the global or any al ter nate co or di nate sys tem.

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The joint lo cal 1- 2-3 co ordinate sys tem is nor mally the same as the global X- Y-Zcoordinate sys tem. How ever, you may de fine any ar bi trary ori entation for a jointlocal co ordinate sys tem by speci fying two ref erence vec tors and/or three an gles of rotation.

For the Frame, Area (Shell, Plane, and Asolid), and Link/Sup port ele ments, one of the ele ment lo cal axes is de ter mined by the ge ome try of the in dividual ele ment.You may de fine the ori entation of the re main ing two axes by speci fying a sin glereference vec tor and/or a sin gle an gle of ro tation. The ex cep tion to this is one-jointor zero-length Link/Sup port el ements, which re quire that you first spec ify the lo -cal-1 (ax ial) axis.

The Solid el ement lo cal 1-2-3 co ordinate sys tem is nor mally the same as the globalX-Y-Z co ordinate sys tem. How ever, you may de fine any ar bi trary ori entation for asolid lo cal co ordinate sys tem by spec ifying two ref erence vec tors and/or three an -gles of ro tation.

The lo cal co ordinate sys tem for a Body, Dia phragm, Plate, Beam, or Rod Con -straint is nor mally de termined auto mati cally from the ge ome try or mass dis tri bu -tion of the con straint. Op tionally, you may spec ify one lo cal axis for any Dia -

Local Coordinate Systems 15


V is parallel to R axisaV is parallel to R-S plane p

V = V r aV = V x V t r p V = V x V s t r

Y

Z

Global

Plane R-S

V r

V t

V s

V a

V p

Cube is shown for visualization purposes

Figure 1 Determining an R-S-T Coordinate System from Reference Vectors V a and V p


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phragm, Plate, Beam, or Rod Con straint (but not for the Body Con straint); the re -main ing two axes are de termined auto mati cally.

The lo cal co or di nate sys tem for an Equal Con straint may be ar bi trar ily speci fied; by de fault it is the global co ordinate sys tem. The Lo cal Con straint does not have its

own lo cal co ordinate sys tem.For more in formation:

See Topic Lo cal Co ordinate Sys tem (page 24) in Chap ter Joints and De -grees of Free dom.

See Topic Lo cal Co ordinate Sys tem (page 82) in Chap ter The Frame Ele -ment.

See Topic Lo cal Co ordinate Sys tem (page 132 ) in Chap ter The Shell Ele -ment.

See Topic Lo cal Co ordinate Sys tem (page 151 ) in Chap ter The Plane Ele -ment.

See Topic Lo cal Co ordinate Sys tem (page 161 ) in Chap ter The Aso lid Ele -ment.

See Topic Lo cal Co ordinate Sys tem (page 174 ) in Chap ter The Solid Ele -ment.

See Topic Lo cal Co ordinate Sys tem (page 187 ) in Chap ter The Link/Sup - port El ementBasic.

See Chap ter Con straints and Welds (page 47).

Alternate Coordinate SystemsYou may de fine al ter nate co or di nate sys tems that can be used for lo cat ing the

joints; for de fin ing lo cal co ordinate sys tems for joints, ele ments, and con straints;and as a ref erence for de fin ing other prop erties and loads. The axes of the al ternatecoordinate sys tems are de noted X, Y, and Z.

The global co or di nate sys tem and all al ter nate sys tems are called fixed co or di natesys tems , since they ap ply to the whole struc tural model, not just to in dividual partsas do the lo cal co or di nate sys tems. Each fixed co or di nate sys tem may be used inrec tangular, cy lindrical or spheri cal form.

Asso ci ated with each fixed co or di nate sys tem is a grid sys tem used to lo cate ob jectsin the graph ical user in terface. Grids have no mean ing in the anal y sis model.

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Each al ternate co ordinate sys tem is de fined by spec ifying the lo cation of the or iginand the ori entation of the axes with re spect to the global co ordinate sys tem. Youneed:

The global X, Y, and Z co ordinates of the new or igin The three an gles (in de grees) used to ro tate from the global co ordinate sys tem

to the new sys tem

Cylindrical and Spherical CoordinatesThe lo cation of points in the global or an al ternate co ordinate sys tem may be speci -fied us ing po lar co ordinates in stead of rec tangular X- Y-Z co ordinates. Po lar co or-dinates in clude cy lindrical CR- CA- CZ co ordinates and spheri cal SB- SA- SR co or-dinates. See Figure 2 (page 19) for the defi nition of the po lar co ordinate sys tems.Polar co ordinate sys tems are al ways de fined with re spect to a rec tangular X- Y-Zsys tem.

The co ordinates CR, CZ, and SR are lin eal and are speci fied in length units. The co -or di nates CA, SB, and SA are an gu lar and are speci fied in de grees.

Locations are speci fied in cylindri cal co ordinates us ing the vari ables cr , ca , and cz .These are re lated to the rec tangular co ordinates as:

cr x y= +2 2

cayx

= tan -1

cz z=

Locations are speci fied in spheri cal coordinates us ing the vari ables sb , sa , and sr .These are re lated to the rec tangular co ordinates as:

sb

x y

z= tan

+-12 2

sayx

= tan -1

sr x y z= + +2 2 2

Cylindrical and Spherical Coordinates 17



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A vec tor in a fixed co ordinate sys tem can be speci fied by giv ing the lo cations of two points or by speci fying a co ordinate di rec tion at a sin gle point P . Co ordinatedirec tions are tan gen tial to the co ordinate curves at point P . A posi tive co ordinatedirec tion in dicates the di rec tion of in creas ing co ordinate value at that point.

Cylindrical co ordinate di rec tions are in dicated us ing the val ues CR, CA, andCZ. Spheri cal co ordinate di rec tions are in dicated us ing the val ues SB, SA, andSR. The sign is re quired. See Figure 2 (page 19).

The cy lindrical and spheri cal co ordinate di rec tions are not con stant but vary withangular po sition. The co ordinate di rec tions do not change with the lin eal co ordi-nates. For ex am ple, +SR de fines a vec tor di rected from the ori gin to point P .

Note that the co ordinates Z and CZ are iden tical, as are the cor respond ing co ordi-nate di rec tions. Simi larly, the co ordinates CA and SA and their cor respond ing co -

ordinate di rec tions are iden tical.

18 Cylindrical and Spherical Coordinates



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CylindricalCoordinates

SphericalCoordinates

X

Y

Z, CZ

ca

cr

cz

P

X

Y

Z

sa

sb

sr

P

+CR

+CA

+CZ

+SB

+SA

+SR

Cubes are shown for visualization purposes

Figure 2Cylindrical and Spherical Coordinates and Coordinate Directions


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C h a p t e r IV

Joints and Degrees of Freedom

The joints play a fun damen tal role in the analy sis of any struc ture. Joints are the points of con nec tion be tween the ele ments, and they are the pri mary lo cations inthe struc ture at which the dis place ments are known or are to be de termined. Thedis place ment com po nents (trans lations and ro tations) at the joints are called the de -

grees of free dom .

This Chap ter de scribes joint prop erties, de grees of free dom, loads, and out put. Ad -ditional in formation about joints and de grees of free dom is given in Chap ter Con -straints and Welds (page 47).

Basic Topics for All Users Over view Mod eling Con sidera tions Local Co ordinate Sys tem Degrees of Free dom Restraints and Re actions Springs Masses Force Load

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Ground Dis place ment Load Degree of Free dom Out put Assem bled Joint Mass Out put Dis place ment Out put Force Out put

Advanced Topics Advanced Lo cal Co ordinate Sys tem Gen eralized Displacements Element Joint Force Output

Overview Joints , also known as nodal points or nodes , are a fun da men tal part of every struc -tural model. Joints per form a va riety of func tions:

All ele ments are con nected to the struc ture (and hence to each other) at the joints

The struc ture is sup ported at the joints us ing Re straints and/or Springs Rigid- body be hav ior and sym metry con ditions can be speci fied us ing Con -

straints that ap ply to the joints Con cen trated loads may be ap plied at the joints Lumped (con cen trated) masses and ro tational in ertia may be placed at the

joints All loads and masses ap plied to the ele ments are ac tu ally trans ferred to the

joints Joints are the pri mary lo cations in the struc ture at which the dis place ments are

known (the sup ports) or are to be de termined

All of these func tions are dis cussed in this Chap ter ex cept for the Con straints,which are de scribed in Chap ter Con straints and Welds (page 47).

Joints in the anal y sis model cor respond to point ob jects in the struc tural-ob jectmodel. Using the SAP2000, ETABS or SAFE graph ical user in terface, joints(points) are au to mat ically cre ated at the ends of each Line ob ject and at the cor nersof each Area and Solid ob ject. Joints may also be de fined in de pend ently of any ob -

ject.

22 Overview



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Automatic mesh ing of ob jects will cre ate ad ditional joints cor respond ing to any el -ements that are cre ated.

Joints may them selves be con sidered as el ements. Each joint may have its own lo -cal co ordinate sys tem for de fin ing the de grees of free dom, re straints, joint prop er-

ties, and loads; and for in ter pret ing joint out put. In most cases, how ever, the globalX-Y-Z co ordinate sys tem is used as the lo cal co ordinate sys tem for all joints in themodel. Joints act in de pend ently of each other un less con nected by other el ements.

There are six dis place ment de grees of free dom at ev ery joint three trans lationsand three ro tations. These dis place ment com po nents are aligned along the lo cal co -ordinate sys tem of each joint.

Joints may be loaded di rectly by con cen trated loads or in directly by ground dis - place ments act ing though Re straints or spring sup ports.

Dis place ments (trans lations and ro tations) are pro duced at every joint. The ex ternaland in ternal forces and mo ments act ing on each joint are also pro duced.

For more in formation:

See Chap ter Con straints and Welds (page 47).

Modeling ConsiderationsThe lo cation of the joints and ele ments is criti cal in de termin ing the ac curacy of thestruc tural model. Some of the fac tors that you need to con sider when de fin ing theelements, and hence the joints, for the struc ture are:

The number of ele ments should be suf ficient to de scribe the ge ome try of thestruc ture. For straight lines and edges, one ele ment is ade quate. For curves andcurved sur faces, one ele ment should be used for every arc of 15 or less.

Ele ment bounda ries, and hence joints, should be lo cated at points, lines, andsur faces of dis con tinuity:

Struc tural bounda ries, e.g., cor ners and edges Changes in ma terial prop erties Changes in thick ness and other geo met ric prop erties Sup port points (Re straints and Springs) Points of ap pli cation of con cen trated loads, ex cept that Frame/Cable el e-

ments may have con cen trated loads ap plied within their spans

Modeling Considerations 23

Chapter IV Joints and Degrees of Freedom


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In re gions hav ing large stress gra dients, i.e., where the stresses are chang ingrap idly, an Area- or Solid-el ement mesh should be re fined us ing small ele -ments and closely- spaced joints. This may re quire chang ing the mesh af ter oneor more pre limi nary analy ses.

More that one ele ment should be used to model the length of any span for which dy namic be hav ior is im por tant. This is re quired be cause the mass is al -ways lumped at the joints, even if it is con tributed by the ele ments.

Local Coordinate SystemEach joint has its own joint lo cal co or di nate sys tem used to de fine the de grees of free dom, Re straints, prop erties, and loads at the joint; and for in ter pret ing joint out -

put. The axes of the joint lo cal co ordinate sys tem are de noted 1, 2, and 3. By de fault

these axes are iden tical to the global X, Y, and Z axes, re spec tively. Both sys temsare right- handed co ordinate sys tems.

The de fault lo cal co ordinate sys tem is ade quate for most situa tions. How ever, for cer tain mod eling pur poses it may be use ful to use dif ferent lo cal co ordinate sys -tems at some or all of the joints. This is de scribed in the next topic.


See Topic Up ward and Hori zon tal Di rec tions (page 13) in Chap ter Co ordi-nate Sys tems.

See Topic Ad vanced Lo cal Co ordinate Sys tem (page 24) in this Chap ter.

Advanced Local Coordinate SystemBy de fault, the joint lo cal 1- 2-3 co ordinate sys tem is iden tical to the global X- Y-Zcoordinate sys tem, as de scribed in the pre vious topic. How ever, it may be nec es-sary to use dif ferent lo cal co ordinate sys tems at some or all joints in the fol low ingcases:

Skewed Re straints (sup ports) are pres ent Con straints are used to im pose ro tational sym metry Con straints are used to im pose sym metry about a plane that is not par allel to a

global co ordinate plane The prin ci pal axes for the joint mass (trans lational or ro tational) are not aligned

with the global axes

24 Local Coordinate System



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Joint dis place ment and force out put is de sired in an other co ordinate sys tem

Joint lo cal co ordinate sys tems need only be de fined for the af fected joints. Theglobal sys tem is used for all joints for which no lo cal co ordinate sys tem is ex plic itlyspeci fied.

A va riety of meth ods are avail able to de fine a joint lo cal co or dinate sys tem. Thesemay be used sepa rately or to gether. Lo cal co or di nate axes may be de fined to be par -allel to ar bi trary co ordinate di rec tions in an ar bi trary co ordinate sys tem or to vec -tors be tween pairs of joints. In ad dition, the joint lo cal co ordinate sys tem may bespeci fied by a set of three joint co or di nate an gles. These meth ods are de scribed inthe sub topics that fol low.


See Chap ter Co ordinate Sys tems (page 11 ). See Topic Lo cal Co ordinate Sys tem (page 24) in this Chap ter.

Reference Vectors

To de fine a joint lo cal co ordinate sys tem you must spec ify two ref er ence vec torsthat are par allel to one of the joint lo cal co ordinate planes. The axis ref er ence vec -tor , Va , must be par allel to one of the lo cal axes ( I = 1, 2, or 3) in this plane and

have a posi tive pro jec tion upon that axis. The plane ref er ence vec tor , V p , must

have a posi tive pro jec tion upon the other lo cal axis ( j = 1, 2, or 3, but I j) in this plane, but need not be par allel to that axis. Hav ing a posi tive pro jec tion means thatthe posi tive di rec tion of the ref erence vec tor must make an an gle of less than 90with the posi tive di rec tion of the lo cal axis.

To gether, the two ref erence vec tors de fine a lo cal axis, I , and a lo cal plane, i- j.From this, the pro gram can de termine the third lo cal axis, k , us ing vec tor al ge bra.

For ex am ple, you could choose the axis ref er ence vec tor par allel to lo cal axis 1 andthe plane ref erence vec tor par allel to the lo cal 1-2 plane ( I = 1, j = 2). Al ternatively,

you could choose the axis ref erence vec tor par allel to lo cal axis 3 and the plane ref -erence vec tor par allel to the lo cal 3-2 plane ( I = 3, j = 2). You may choose the planethat is most con ven ient to de fine us ing the pa rame ter local , which may take on thevalues 12, 13, 21, 23, 31, or 32. The two dig its cor respond to I and j, re spec tively.The de fault is value is 31.

Advanced Local Coordinate System 25



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Defining the Axis Reference Vector

To de fine the axis ref erence vec tor for joint j , you must first spec ify or use the de -fault val ues for:

A co ordinate di rec tion ax dir (the de fault is +Z) A fixed co ordinate sys tem csys (the de fault is zero, in dicat ing the global co or-

dinate sys tem)

You may op tionally spec ify:

A pair of joints, ax veca and ax vecb (the de fault for each is zero, in dicat ing joint j itself). If both are zero, this op tion is not used.

For each joint, the axis ref erence vec tor is de termined as fol lows:

1. A vec tor is found from joint ax veca to joint ax vecb . If this vec tor is of fi nitelength, it is used as the ref erence vec tor Va

2. Oth erwise, the co ordinate di rec tion ax dir is evalu ated at joint j in fixed co ordi-nate sys tem csys , and is used as the ref erence vec tor Va

Defining the Plane Reference Vector

To de fine the plane ref er ence vec tor for joint j , you must first spec ify or use the de -

fault val ues for: A pri mary co ordinate di rec tion pldirp (the de fault is +X) A sec ondary co ordinate di rec tion pldirs (the de fault is +Y). Di rec tions pldirs

and pldirp should not be par allel to each other un less you are sure that they arenot par allel to lo cal axis 1

A fixed co ordinate sys tem csys (the de fault is zero, in dicat ing the global co or-di nate sys tem). This will be the same co or di nate sys tem that was used to de finethe axis ref er ence vec tor, as de scribed above

You may op tionally spec ify:

A pair of joints, plveca and plvecb (the de fault for each is zero, in dicat ing joint j itself). If both are zero, this op tion is not used.

For each joint, the plane ref er ence vec tor is de ter mined as fol lows:

26 Advanced Local Coordinate System



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1. A vec tor is found from joint plveca to joint plvecb . If this vec tor is of fi nitelength and is not par allel to lo cal axis I , it is used as the ref er ence vec tor V p

2. Oth erwise, the pri mary co ordinate di rec tion pldirp is evalu ated at joint j infixed co ordinate sys tem csys . If this di rec tion is not par allel to lo cal axis I , it is

used as the ref er ence vec tor V p

3. Oth erwise, the sec ondary co ordinate di rec tion pldirs is evalu ated at joint j infixed co ordinate sys tem csys . If this di rec tion is not par allel to lo cal axis I , it isused as the ref er ence vec tor V p

4. Oth erwise, the method fails and the analy sis ter minates. This will never hap penif pldirp is not par al lel to pldirs

A vec tor is con sid ered to be par allel to lo cal axis I if the sine of the an gle be tweenthem is less than 10 -3.

Determining the Local Axes from the Reference Vectors

The pro gram uses vec tor cross prod ucts to de termine the lo cal axes from the ref er-ence vec tors. The three axes are rep resented by the three unit vec tors V1, V2 andV3 , re spec tively. The vec tors sat isfy the cross- product re lationship:

V V V1 2 3

The lo cal axis Vi is given by the vec tor Va af ter it has been nor mal ized to unitlength.

The re main ing two axes, V j and Vk , are de fined as fol lows:

If I and j per mute in a posi tive sense, i.e., local = 12, 23, or 31, then:

V V Vk i p andV V V j k i

If I and j per mute in a nega tive sense, i.e., local = 21, 32, or 13, then:

V V Vk p i

andV V V j i k

An ex am ple show ing the de termination of the joint lo cal co ordinate sys tem us ingref er ence vec tors is given in Figure 3 (page 28).

Advanced Local Coordinate System 27



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Joint Coordinate Angles

The joint lo cal co or di nate axes de ter mined from the ref er ence vec tors may be fur -ther modi fied by the use of three joint co or di nate an gles , de noted a , b , and c. Inthe case where the de fault ref erence vec tors are used, the joint co or di nate an gles de -fine the ori entation of the joint lo cal co ordinate sys tem with re spect to the globalaxes.

The joint co ordinate an gles spec ify ro tations of the lo cal co ordinate sys tem aboutits own cur rent axes. The re sult ing ori entation of the joint lo cal co ordinate sys temis ob tained ac cord ing to the fol low ing pro cedure:

1. The lo cal sys tem is first ro tated about its +3 axis by an gle a

2. The lo cal sys tem is next ro tated about its re sult ing +2 axis by an gle b

3. The lo cal sys tem is lastly ro tated about its re sult ing +1 axis by an gle c

The or der in which the ro tations are per formed is im por tant. The use of co ordinateangles to ori ent the joint lo cal co ordinate sys tem with re spect to the global sys tem isshown in Figure 4 (page 30).

28 Advanced Local Coordinate System


V is parallel to axveca -axvecbaV is parallel to plveca -plvecb p

V = V 3 aV = V x V All vectors normalized to unit length.2 3 p V = V x V 1 2 3

Y

Z

Global

axveca

axvecb

plvecaplvecb

Plane 3-1

j

V 3

V 2

V 1

V a

V p

Figure 3 Example of the Determination of the Joint Local Coordinate System

Using Reference Vectors for local =31


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Degrees of FreedomThe de flec tion of the struc tural model is gov erned by the dis place ments of the

joints. Every joint of the struc tural model may have up to six dis place ment com po -nents:

The joint may trans late along its three lo cal axes. These trans lations are de -noted U1, U2, and U3.

The joint may ro tate about its three lo cal axes. These ro tations are de noted R1,R2, and R3.

These six dis place ment com po nents are known as the de grees of free dom of the joint. In the usual case where the joint lo cal co ordinate sys tem is par allel to theglobal sys tem, the de grees of free dom may also be iden tified as UX, UY, UZ, RX,RY and RZ, ac cord ing to which global axes are par allel to which lo cal axes. The

joint lo cal de grees of free dom are il lus trated in Figure 5 (page 31).

In ad dition to the regu lar joints that you ex plic itly de fine as part of your struc turalmodel, the pro gram auto mati cally cre ates mas ter joints that gov ern the be hav ior of any Con straints and Welds that you may have de fined. Each mas ter joint has thesame six de grees of free dom as do the regu lar joints. See Chap ter Con straints andWelds (page 47) for more in formation.

Each de gree of free dom in the struc tural model must be one of the fol low ing types:

Active the dis place ment is com puted dur ing the analy sis Restrained the dis place ment is speci fied, and the cor respond ing re action is

com puted dur ing the analy sis Con strained the dis place ment is de termined from the dis place ments at other

de grees of free dom Null the dis place ment does not af fect the struc ture and is ig nored by the

analy sis Un avail able the dis place ment has been ex plic itly ex cluded from the analy -

sisThese dif ferent types of de grees of free dom are de scribed in the fol low ing sub top-ics.

Degrees of Freedom 29



49/43230 Degrees of Freedom


a

a

a

b

b

b

c

c

c

Z, 3

Z

Z

X

X

X

1

2

2

2

3

3

1

1

Y

Y

Y

Step 1: Rotation about

local 3 axis by angle a

Step 2: Rotation about newlocal 2 axis by angle b

Step 3: Rotation about newlocal 1 axis by angle c

Figure 4Use of Joint Coordinate Angles to Orient the Joint Local Coordinate System


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Available and Unavailable Degrees of Freedom

You may ex plic itly spec ify the global de grees of free dom that are avail able to every joint in the struc tural model. By de fault, all six de grees of free dom are avail able toevery joint. This de fault should gen erally be used for all three- dimensional struc -tures.

For cer tain pla nar struc tures, how ever, you may wish to re strict the avail able de -grees of free dom . For ex am ple, in the X-Y plane: a pla nar truss needs only UX andUY; a pla nar frame needs only UX, UY, and RZ; and a pla nar grid or flat plateneeds only UZ, RX, and RY.

The de grees of free dom that are not speci fied as be ing avail able are called un avail -able de grees of free dom . Any stiff ness, loads, mass, Re straints, or Con straints thatare ap plied to the un avail able de grees of free dom are ig nored by the analy sis.

The avail able de grees of free dom are al ways re ferred to the global co or di nate sys -tem, and they are the same for every joint in the model. If any joint lo cal co ordinatesys tems are used, they must not cou ple avail able de grees of free dom with the un -avail able de grees of free dom at any joint. For ex am ple, if the avail able de grees of free dom are UX, UY, and RZ, then all joint lo cal co ordinate sys tems must have onelocal axis par allel to the global Z axis.

Degrees of Freedom 31


Joint

U2U1

R2

R3

R1

Figure 5The Six Displacement Degrees of Freedom in the Joint Local Coordinate System


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Restrained Degrees of Freedom

If the dis place ment of a joint along any one of its avail able

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