Computer and Structures SAP

8/20/2019 Computer and Structures SAP

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CSI Analysis Reference Manual


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For SAP2000®, ETABS®, SAFE® and CSiBridge®

ISO# GEN062708M1 Rev.11

Berkeley, California, USA January 2014


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COPYRIGHT

Copyright © Com puters & Structures, Inc., 1978-2014All rights reserved.

The CSI Logo®, SAP2000®, ETABS®, SAFE®, CSiBridge®, and SAPFire® are

registered trademarks of Com puters & Structures, Inc. Model-AliveTM

and Watch

& LearnTM

are trademarks of Com puters & Structures, Inc. Windows® is a reg is-

tered trademark of the Microsoft Cor poration. Adobe® and Acro bat® are reg is-

tered trademarks of Adobe Systems Incor porated.

The com puter programs SAP2000®, ETABS®, SAFE®, and CSiBridge® and all

associated documentation are pro prietary and copyrighted products. Worldwiderights of ownership rest with Com puters & Structures, Inc. Unlicensed use of these

programs or re production of documentation in any form, without prior written au-

thorization from Com puters & Structures, Inc., is ex plicitly prohibited. No part of

this publication may be re produced or distributed in any form or by any means, or

stored in a data base or retrieval sys tem, with out the prior ex plicit written permis-

sion of the publisher.

Further information and copies of this documentation may be obtained from:

Com puters & Structures, Inc.

www.csiamerica.com

[email protected] (for general information)

sup [email protected] (for technical sup port)


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DISCLAIMER

CONSID ER ABLE TIME, EF FORT AND EX PENSE HAVE GONEINTO THE DE VEL OP MENT AND TEST ING OF THIS SOFTWARE.

HOWEVER, THE USER ACCEPTS AND UN DERSTANDS THAT

NO WARRANTY IS EX PRESSED OR IMPLIED BY THE DE VEL-

OPERS OR THE DISTRIBUTORS ON THE AC CURACY OR THE

RELIABILITY OF THE PROGRAMS THESE PRODUCTS.

THESE PROD UCTS ARE PRAC TI CAL AND POW ER FUL TOOLS

FOR STRUC TURAL DE SIGN. HOWEVER, THE USER MUST EX -

PLICITLY UNDERSTAND THE BASIC ASSUMPTIONS OF THE

SOFTWARE MODELING, ANALYSIS, AND DESIGN ALGO-

RITHMS AND COMPENSATE FOR THE AS PECTS THAT ARE

NOT ADDRESSED.

THE INFOR MATION PRODUCED BY THE SOFTWARE MUST BE

CHECKED BY A QUALIFIED AND EXPERIENCED ENGI NEER.

THE ENGI NEER MUST INDEPENDENTLY VERIFY THE RE-

SULTS AND TAKE PROFESSIONAL RESPONSIBILITY FOR THE

INFORMATION THAT IS USED.


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ACKNOWLEDGMENT

Thanks are due to all of the numerous structural engineers, who over theyears have given valuable feed back that has contributed toward the en-

hancement of this product to its current state.

Special recognition is due Dr. Edward L. Wilson, Professor Emeritus,

University of California at Berkeley, who was responsi ble for the con-

ception and development of the original SAP series of programs and

whose continued originality has produced many unique concepts that

have been im plemented in this version.


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Table of Contents

Chapter I Introduction 1Analysis Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Structural Analysis and Design . . . . . . . . . . . . . . . . . . . . . . 3

About This Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Ty pographical Conventions . . . . . . . . . . . . . . . . . . . . . . . 4

Bold for Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 4

Bold for Variable Data. . . . . . . . . . . . . . . . . . . . . . . . 4

Italics for Mathematical Variables . . . . . . . . . . . . . . . . . . 4

Italics for Em phasis . . . . . . . . . . . . . . . . . . . . . . . . . 5Capitalized Names . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Bibliographic References . . . . . . . . . . . . . . . . . . . . . . . . . 5

Chapter II Ob jects and Elements 7

Ob jects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Ob jects and Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter III Coordinate Systems 11

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Global Coordinate System . . . . . . . . . . . . . . . . . . . . . . . 12

Upward and Horizontal Directions . . . . . . . . . . . . . . . . . . . 13

Defining Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . 13

Vector Cross Product . . . . . . . . . . . . . . . . . . . . . . . . 13

Defining the Three Axes Using Two Vectors . . . . . . . . . . . 14

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Local Coordinate Systems. . . . . . . . . . . . . . . . . . . . . . . . 14

Alternate Coordinate Systems. . . . . . . . . . . . . . . . . . . . . . 16

Cylindrical and Spherical Coordinates . . . . . . . . . . . . . . . . . 17

Chapter IV Joints and Degrees of Free dom 21

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Modeling Considerations . . . . . . . . . . . . . . . . . . . . . . . . 23

Local Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . 24

Advanced Local Coordinate System . . . . . . . . . . . . . . . . . . 24

Reference Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 25

Defining the Axis Reference Vector . . . . . . . . . . . . . . . . 26

Defining the Plane Reference Vector. . . . . . . . . . . . . . . . 26

Determin ing the Local Axes from the Reference Vec tors . . . . . 27

Joint Coordinate Angles . . . . . . . . . . . . . . . . . . . . . . 28

Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Available and Unavailable Degrees of Free dom . . . . . . . . . . 31

Restrained Degrees of Free dom . . . . . . . . . . . . . . . . . . 32

Constrained Degrees of Freedom. . . . . . . . . . . . . . . . . . 32

Mix ing Restraints and Constraints Not Recommended . . . . . . 32

Active Degrees of Freedom . . . . . . . . . . . . . . . . . . . . 33

Null Degrees of Freedom. . . . . . . . . . . . . . . . . . . . . . 34

Restraint Sup ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Spring Sup ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Nonlinear Sup ports . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Distributed Sup ports . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Joint Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Base Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Force Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Ground Dis placement Load . . . . . . . . . . . . . . . . . . . . . . . 42

Restraint Dis placements . . . . . . . . . . . . . . . . . . . . . . 43

Spring Dis placements . . . . . . . . . . . . . . . . . . . . . . . 44

Link/Sup port Dis placements . . . . . . . . . . . . . . . . . . . . 45

Generalized Dis placements . . . . . . . . . . . . . . . . . . . . . . . 45

Degree of Freedom Out put . . . . . . . . . . . . . . . . . . . . . . . 46

Assem bled Joint Mass Out put. . . . . . . . . . . . . . . . . . . . . . 47

Dis placement Out put . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Force Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Element Joint Force Out put . . . . . . . . . . . . . . . . . . . . . . . 48

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Chapter V Constraints and Welds 49

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Body Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Joint Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . 51

Local Coordinate System. . . . . . . . . . . . . . . . . . . . . . 51Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . 51

Plane Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Dia phragm Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 53


Local Coordinate System. . . . . . . . . . . . . . . . . . . . . . 53

Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . 54

Plate Con straint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55



Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . 55Axis Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Rod Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56




Beam Constraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58




Equal Constraint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59



Selected Degrees of Free dom . . . . . . . . . . . . . . . . . . . 60


Local Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61


No Local Coordinate System . . . . . . . . . . . . . . . . . . . . 62

Selected Degrees of Free dom . . . . . . . . . . . . . . . . . . . 62

Constraint Equations . . . . . . . . . . . . . . . . . . . . . . . . 62Welds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Automatic Master Joints. . . . . . . . . . . . . . . . . . . . . . . . . 66

Stiffness, Mass, and Loads . . . . . . . . . . . . . . . . . . . . . 66

Local Coordinate Systems . . . . . . . . . . . . . . . . . . . . . 67

Constraint Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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Chapter VI Material Properties 69

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70


Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Isotro pic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Uniaxial Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Orthotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Anisotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Tem perature-De pendent Properties . . . . . . . . . . . . . . . . . . . 76

Element Material Tem perature . . . . . . . . . . . . . . . . . . . . . 77

Mass Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Weight Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Material Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Modal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 79Viscous Pro portional Damping. . . . . . . . . . . . . . . . . . . 80

Hysteretic Pro portional Damping . . . . . . . . . . . . . . . . . 80

Nonlinear Material Behavior . . . . . . . . . . . . . . . . . . . . . . 80

Tension and Com pression . . . . . . . . . . . . . . . . . . . . . 81

Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Ap plication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Fric tion and Dilitational An gles . . . . . . . . . . . . . . . . . . 84

Time-de pendent Properties . . . . . . . . . . . . . . . . . . . . . . . 85

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Time-Integration Control . . . . . . . . . . . . . . . . . . . . . . 86

Design-Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Chapter VII The Frame Element 89

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Joint Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Insertion Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


Longitudinal Axis 1 . . . . . . . . . . . . . . . . . . . . . . . . 93

Default Orientation . . . . . . . . . . . . . . . . . . . . . . . . . 93

Coordinate Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Advanced Local Coordinate System . . . . . . . . . . . . . . . . . . 94

Reference Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Determining Transverse Axes 2 and 3 . . . . . . . . . . . . . . . 97

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Section Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98


Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . 99

Geometric Properties and Section Stiffnesses. . . . . . . . . . . 100

Shape Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Automatic Section Property Calculation . . . . . . . . . . . . . 102Section Property Data base Files. . . . . . . . . . . . . . . . . . 102

Section-Designer Sections . . . . . . . . . . . . . . . . . . . . 102

Additional Mass and Weight . . . . . . . . . . . . . . . . . . . 104

Non-prismatic Sections . . . . . . . . . . . . . . . . . . . . . . 104

Property Modifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Named Prop erty Sets . . . . . . . . . . . . . . . . . . . . . . . 108

Insertion Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Local Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

End Offsets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Clear Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Rigid-end Factor . . . . . . . . . . . . . . . . . . . . . . . . . 113

Effect upon Non-prismatic Elements . . . . . . . . . . . . . . . 114

Effect upon Internal Force Out put . . . . . . . . . . . . . . . . 114

Effect upon End Releases . . . . . . . . . . . . . . . . . . . . . 114

End Releases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Unsta ble End Releases . . . . . . . . . . . . . . . . . . . . . . 116

Effect of End Offsets . . . . . . . . . . . . . . . . . . . . . . . 116


Nonlinear Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Tension/Com pression Limits . . . . . . . . . . . . . . . . . . . 117

Plastic Hinge . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Concentrated Span Load . . . . . . . . . . . . . . . . . . . . . . . . 119

Distributed Span Load . . . . . . . . . . . . . . . . . . . . . . . . . 121

Loaded Length . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Load Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Pro jected Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 121Tem perature Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Strain Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Deformation Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Target-Force Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Internal Force Out put . . . . . . . . . . . . . . . . . . . . . . . . . 126

Effect of End Offsets . . . . . . . . . . . . . . . . . . . . . . . 128

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Stress Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Chapter VIII Frame Hinge Proper ties 131

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Hinge Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Hinge Length . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Plastic Deformation Curve . . . . . . . . . . . . . . . . . . . . 134

Scaling the Curve . . . . . . . . . . . . . . . . . . . . . . . . . 135

Strength Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Cou pled P-M2-M3 Hinge . . . . . . . . . . . . . . . . . . . . . 136

Fi ber P-M2-M3 Hinge . . . . . . . . . . . . . . . . . . . . . . 139

Automatic, User-Defined, and Generated Properties . . . . . . . . . 139

Automatic Hinge Properties . . . . . . . . . . . . . . . . . . . . . . 141

Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Chapter IX The Cable Element 145

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Joint Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Undeformed Length . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Shape Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Ca ble vs. Frame Elements. . . . . . . . . . . . . . . . . . . . . 149

Num ber of Segments . . . . . . . . . . . . . . . . . . . . . . . 150

Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Local Coordinate System . . . . . . . . . . . . . . . . . . . . . . . 150

Section Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Material Properties . . . . . . . . . . . . . . . . . . . . . . . . 151


Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153


Tem perature Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Strain and Deformation Load . . . . . . . . . . . . . . . . . . . . . 154


Nonlinear Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Element Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Chapter X The Shell Element 157

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

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Homogeneous . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Layered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159


Shape Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . 160

Edge Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

De grees of Free dom . . . . . . . . . . . . . . . . . . . . . . . . . . 164


Normal Axis 3. . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Default Orientation . . . . . . . . . . . . . . . . . . . . . . . . 165

Element Coordinate Angle . . . . . . . . . . . . . . . . . . . . 167

Advanced Local Coordinate System. . . . . . . . . . . . . . . . . . 167

Reference Vector . . . . . . . . . . . . . . . . . . . . . . . . . 167

Determining Tangential Axes 1 and 2 . . . . . . . . . . . . . . 169


Area Sec tion Type. . . . . . . . . . . . . . . . . . . . . . . . . 170Shell Sec tion Type . . . . . . . . . . . . . . . . . . . . . . . . 170

Homogeneous Section Properties . . . . . . . . . . . . . . . . . 171

Layered Section Property . . . . . . . . . . . . . . . . . . . . . 174

Property Modifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . 181


Joint Offsets and Thickness Overwrites . . . . . . . . . . . . . . . . 183

Joint Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Effect of Joint Offsets on the Local Axes . . . . . . . . . . . . . 184

Thickness Overwrites . . . . . . . . . . . . . . . . . . . . . . . 185

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Uniform Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Surface Pressure Load . . . . . . . . . . . . . . . . . . . . . . . . . 188


Strain Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Internal Force and Stress Out put. . . . . . . . . . . . . . . . . . . . 190

Chapter XI The Plane Element 195

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196




Stresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . 197


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Section Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 198


Material Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Incom pati ble Bending Modes . . . . . . . . . . . . . . . . . . . 200

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201


Pore Pressure Load. . . . . . . . . . . . . . . . . . . . . . . . . . . 202


Stress Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Chapter XII The Asolid Ele ment 205

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206






Section Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 208


Material Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Axis of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 210Arc and Thickness. . . . . . . . . . . . . . . . . . . . . . . . . 211


Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213




Rotate Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Stress Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Chapter XIII The Solid Element 217

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218


Degenerate Solids . . . . . . . . . . . . . . . . . . . . . . . . . 219


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Reference Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 221

Defining the Axis Reference Vector . . . . . . . . . . . . . . . 221

Defining the Plane Reference Vector . . . . . . . . . . . . . . . 222

De termining the Local Axes from the Reference Vec tors . . . . 223Element Coordinate Angles . . . . . . . . . . . . . . . . . . . . 223


Solid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226


Material Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 226


Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229


Pore Pres sure Load. . . . . . . . . . . . . . . . . . . . . . . . . . . 229


Stress Out put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

Chapter XIV The Link/Support Element—Basic 231

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232


Conversion from One-Joint Ob jects to Two-Joint Elements . . . 233Zero-Length Elements . . . . . . . . . . . . . . . . . . . . . . . . . 233



Longitudinal Axis 1 . . . . . . . . . . . . . . . . . . . . . . . . 235

Default Orientation . . . . . . . . . . . . . . . . . . . . . . . . 235

Coordinate Angle . . . . . . . . . . . . . . . . . . . . . . . . . 236


Axis Reference Vector . . . . . . . . . . . . . . . . . . . . . . 237

Plane Reference Vector . . . . . . . . . . . . . . . . . . . . . . 238

Determining Transverse Axes 2 and 3 . . . . . . . . . . . . . . 239

Internal Deformations . . . . . . . . . . . . . . . . . . . . . . . . . 240

Link/Sup port Properties . . . . . . . . . . . . . . . . . . . . . . . . 243

Local Coordinate System . . . . . . . . . . . . . . . . . . . . . 244

Internal Spring Hinges . . . . . . . . . . . . . . . . . . . . . . 244

Spring Force-Deformation Relationships . . . . . . . . . . . . . 246

Element Internal Forces . . . . . . . . . . . . . . . . . . . . . . 247

Uncou pled Linear Force-Deformation Relationships . . . . . . . 247

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Types of Linear/Nonlinear Properties. . . . . . . . . . . . . . . 249

Cou pled Linear Property . . . . . . . . . . . . . . . . . . . . . . . . 250

Fixed Degrees of Free dom. . . . . . . . . . . . . . . . . . . . . . . 250

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Internal Force and Deformation Out put . . . . . . . . . . . . . . . . 253

Chapter XV The Link/Support Element—Ad vanced 255

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

Nonlinear Link/Sup port Properties . . . . . . . . . . . . . . . . . . 256

Linear Effective Stiffness . . . . . . . . . . . . . . . . . . . . . . . 257

Special Considerations for Modal Analyses . . . . . . . . . . . 257

Linear Effective Damping . . . . . . . . . . . . . . . . . . . . . . . 258Ex ponential Maxwell Damper Property . . . . . . . . . . . . . . . . 259

Bilinear Maxwell Damper Property . . . . . . . . . . . . . . . . . . 261

Friction-Spring Damper Property . . . . . . . . . . . . . . . . . . . 262

Gap Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Hook Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Multi-Linear Elasticity Property . . . . . . . . . . . . . . . . . . . . 267

Wen Plasticity Property . . . . . . . . . . . . . . . . . . . . . . . . 268

Multi-Linear Kinematic Plasticity Property . . . . . . . . . . . . . . 269

Multi-Linear Takeda Plasticity Property. . . . . . . . . . . . . . . . 272Multi-Lin ear Pivot Hysteretic Plasticity Property . . . . . . . . . . . 272

Hysteretic (Rub ber) Isolator Property . . . . . . . . . . . . . . . . . 274

Friction-Pendulum Isolator Property. . . . . . . . . . . . . . . . . . 276

Axial Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 276

Shear Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 278

Linear Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Dou ble-Acting Friction-Pendulum Isolator Property . . . . . . . . . 280

Axial Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Shear Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Linear Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 282

Tri ple-Pendulum Isolator Property. . . . . . . . . . . . . . . . . . . 282

Axial Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Shear Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 284

Linear Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Nonlinear Deformation Loads . . . . . . . . . . . . . . . . . . . . . 287

Frequency-De pendent Link/Sup port Properties . . . . . . . . . . . . 289

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Chapter XVI The Tendon Ob ject 291

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Tendons Modeled as Loads or Elements. . . . . . . . . . . . . . . . 293Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293


Local Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . 295

Base-line Local Coordinate System . . . . . . . . . . . . . . . . 295

Natural Local Coordinate System . . . . . . . . . . . . . . . . . 295




Tension/Com pression Limits . . . . . . . . . . . . . . . . . . . . . 297Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

Prestress Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300


Strain Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300



Internal Force Out put . . . . . . . . . . . . . . . . . . . . . . . . . 302

Chapter XVII Load Patterns 303

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Load Patterns, Load Cases, and Load Com binations . . . . . . . . . 305

Defining Load Patterns . . . . . . . . . . . . . . . . . . . . . . . . 305

Coordinate Systems and Load Com ponents . . . . . . . . . . . . . . 306

Effect upon Large-Dis placements Analysis. . . . . . . . . . . . 306

Force Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Ground Dis placement Load . . . . . . . . . . . . . . . . . . . . . . 307Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Gravity Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

Concentrated Span Load . . . . . . . . . . . . . . . . . . . . . . . . 309


Tendon Prestress Load . . . . . . . . . . . . . . . . . . . . . . . . . 309

Uniform Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

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Strain Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313



Rotate Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Joint Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Mass Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

Mass from Specified Load Patterns . . . . . . . . . . . . . . . . 317

Negative Mass. . . . . . . . . . . . . . . . . . . . . . . . . . . 318

Multi ple Mass Sources . . . . . . . . . . . . . . . . . . . . . . 318

Automated Lateral Loads . . . . . . . . . . . . . . . . . . . . . 320

Acceleration Loads. . . . . . . . . . . . . . . . . . . . . . . . . . . 320

Chapter XVIII Load Cases 323

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Types of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Sequence of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 326

Running Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Linear and Nonlinear Load Cases . . . . . . . . . . . . . . . . . . . 328

Linear Static Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 329

Multi-Step Static Analysis . . . . . . . . . . . . . . . . . . . . . . . 330

Linear Buckling Analysis . . . . . . . . . . . . . . . . . . . . . . . 331

Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

Load Com binations (Com bos) . . . . . . . . . . . . . . . . . . . . . 333

Contributing Cases . . . . . . . . . . . . . . . . . . . . . . . . 333

Types of Com bos . . . . . . . . . . . . . . . . . . . . . . . . . 334

Exam ples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . 336

Additional Considerations. . . . . . . . . . . . . . . . . . . . . 339

Equation Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339Accessing the Assem bled Stiffness and Mass Matrices . . . . . . . . 340

Chapter XIX Modal Anal ysis 341

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Eigenvector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 342

Num ber of Modes . . . . . . . . . . . . . . . . . . . . . . . . . 343

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Frequency Range . . . . . . . . . . . . . . . . . . . . . . . . . 344

Automatic Shifting . . . . . . . . . . . . . . . . . . . . . . . . 345

Convergence Tolerance . . . . . . . . . . . . . . . . . . . . . . 345

Static-Correction Modes . . . . . . . . . . . . . . . . . . . . . 346

Ritz-Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 348

Num ber of Modes . . . . . . . . . . . . . . . . . . . . . . . . . 349Starting Load Vectors . . . . . . . . . . . . . . . . . . . . . . . 349

Num ber of Generation Cycles. . . . . . . . . . . . . . . . . . . 351

Modal Analysis Out put . . . . . . . . . . . . . . . . . . . . . . . . 351

Periods and Frequencies . . . . . . . . . . . . . . . . . . . . . 352

Partici pation Factors . . . . . . . . . . . . . . . . . . . . . . . 352

Partici pating Mass Ratios . . . . . . . . . . . . . . . . . . . . . 353

Static and Dynamic Load Partici pation Ratios . . . . . . . . . . 354

Chapter XX Response-Spectrum Anal ysis 359

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359


Response-Spectrum Function . . . . . . . . . . . . . . . . . . . . . 361

Damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

Modal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Modal Com bination . . . . . . . . . . . . . . . . . . . . . . . . . . 364

Periodic and Rigid Response . . . . . . . . . . . . . . . . . . . 364

CQC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

GMC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

SRSS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 366Absolute Sum Method . . . . . . . . . . . . . . . . . . . . . . 367

NRC Ten-Percent Method . . . . . . . . . . . . . . . . . . . . 367

NRC Dou ble-Sum Method . . . . . . . . . . . . . . . . . . . . 367

Directional Com bination . . . . . . . . . . . . . . . . . . . . . . . . 367

SRSS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

CQC3 Method. . . . . . . . . . . . . . . . . . . . . . . . . . . 368

Absolute Sum Method . . . . . . . . . . . . . . . . . . . . . . 369

Response-Spectrum Analysis Out put . . . . . . . . . . . . . . . . . 370

Damping and Accelerations . . . . . . . . . . . . . . . . . . . . 370

Modal Am plitudes. . . . . . . . . . . . . . . . . . . . . . . . . 370

Base Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Chapter XXI Linear Time-History Anal ysis 373

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

Defining the Spatial Load Vectors . . . . . . . . . . . . . . . . 375

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Defining the Time Functions . . . . . . . . . . . . . . . . . . . 376

Initial Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

Time Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

Modal Time-History Analysis . . . . . . . . . . . . . . . . . . . . . 379

Modal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 380

Direct-Integration Time-History Analysis . . . . . . . . . . . . . . . 381

Time Integration Parameters . . . . . . . . . . . . . . . . . . . 382

Damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

Chapter XXII Geometric Nonlinearity 385

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

Nonlinear Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . 387

The P-Delta Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

P-Delta Forces in the Frame Element . . . . . . . . . . . . . . . 391

P-Delta Forces in the Link/Sup port Element . . . . . . . . . . . 394

Other Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 395

Initial P-Delta Analysis . . . . . . . . . . . . . . . . . . . . . . . . 395

Building Structures . . . . . . . . . . . . . . . . . . . . . . . . 396

Ca ble Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 398

Guyed Towers. . . . . . . . . . . . . . . . . . . . . . . . . . . 398

Large Dis placements . . . . . . . . . . . . . . . . . . . . . . . . . . 398

Ap plications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

Initial Large-Dis placement Analysis . . . . . . . . . . . . . . . 399

Chapter XXIII Nonlinear Static Anal ysis 401

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

Im portant Considerations . . . . . . . . . . . . . . . . . . . . . . . 403

Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

Load Ap plication Control . . . . . . . . . . . . . . . . . . . . . . . 404

Load Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Dis placement Control . . . . . . . . . . . . . . . . . . . . . . . 405


Out put Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

Saving Multi ple Steps . . . . . . . . . . . . . . . . . . . . . . . 407

Nonlinear Solution Control . . . . . . . . . . . . . . . . . . . . . . 409

Maximum Total Steps . . . . . . . . . . . . . . . . . . . . . . . 410

Max imum Null (Zero) Steps . . . . . . . . . . . . . . . . . . . 410

Maximum Iterations Per Step . . . . . . . . . . . . . . . . . . . 410

Iteration Convergence Tolerance . . . . . . . . . . . . . . . . . 411

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Event-to-Event Iteration Control . . . . . . . . . . . . . . . . . 411

Hinge Unloading Method . . . . . . . . . . . . . . . . . . . . . . . 411

Unload Entire Structure . . . . . . . . . . . . . . . . . . . . . . 412

Ap ply Local Redistri bution . . . . . . . . . . . . . . . . . . . . 413

Restart Using Secant Stiffness . . . . . . . . . . . . . . . . . . 413

Static Pushover Analysis. . . . . . . . . . . . . . . . . . . . . . . . 414Staged Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 416

Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

Changing Section Properties . . . . . . . . . . . . . . . . . . . 419

Out put Steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Exam ple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Target-Force Iteration . . . . . . . . . . . . . . . . . . . . . . . . . 421

Chapter XXIV Nonlinear Time-History Anal ysis 425

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427


Time Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

Nonlinear Modal Time-History Analysis (FNA) . . . . . . . . . . . 429

Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 429

Link/Sup port Effective Stiffness . . . . . . . . . . . . . . . . . 430

Mode Su per position . . . . . . . . . . . . . . . . . . . . . . . . 430

Modal Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 432

Iterative Solution . . . . . . . . . . . . . . . . . . . . . . . . . 433

Static Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

Nonlinear Direct-Integration Time-History Analysis . . . . . . . . . 436

Time Integration Parameters . . . . . . . . . . . . . . . . . . . 436

Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 437

Damping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Iterative Solution . . . . . . . . . . . . . . . . . . . . . . . . . 438

Chapter XXV Frequency-Domain Anal yses 441Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

Sources of Damping. . . . . . . . . . . . . . . . . . . . . . . . 444

Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

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Defining the Spatial Load Vectors . . . . . . . . . . . . . . . . 446

Frequency Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

Steady-State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 447

Exam ple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

Power-Spectral-Density Analysis . . . . . . . . . . . . . . . . . . . 449

Exam ple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

Chapter XXVI Moving-Load Anal ysis 453

Overview for CSiBridge . . . . . . . . . . . . . . . . . . . . . . . . 454

Moving-Load Analysis in SAP2000 . . . . . . . . . . . . . . . . . . 455

Bridge Modeler . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

Moving-Load Analysis Procedure . . . . . . . . . . . . . . . . . . . 456

Lanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

Centerline and Direction . . . . . . . . . . . . . . . . . . . . . 458Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Interior and Exterior Edges . . . . . . . . . . . . . . . . . . . . 459

Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

In fluence Lines and Surfaces . . . . . . . . . . . . . . . . . . . . . 460

Vehicle Live Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 462

Direction of Loading . . . . . . . . . . . . . . . . . . . . . . . 462

Distri bution of Loads . . . . . . . . . . . . . . . . . . . . . . . 462

Axle Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

Uniform Loads . . . . . . . . . . . . . . . . . . . . . . . . . . 463Minimum Edge Distances . . . . . . . . . . . . . . . . . . . . . 463

Restricting a Ve hicle to the Lane Length . . . . . . . . . . . . . 463

Ap plication of Loads to the Influence Surface . . . . . . . . . . 463

Length Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . 465

Ap pli cation of Loads in Multi-Step Analysis . . . . . . . . . . . 466

General Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Moving the Vehicle . . . . . . . . . . . . . . . . . . . . . . . . 468

Vehicle Response Com ponents . . . . . . . . . . . . . . . . . . . . 469

Su perstructure (Span) Moment . . . . . . . . . . . . . . . . . . 469 Negative Su perstructure (Span) Moment . . . . . . . . . . . . . 470

Reactions at Interior Sup ports . . . . . . . . . . . . . . . . . . 471

Standard Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

Vehicle Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Moving-Load Load Cases . . . . . . . . . . . . . . . . . . . . . . . 479

Exam ple 1 — AASHTO HS Loading. . . . . . . . . . . . . . . 480

Exam ple 2 — AASHTO HL Loading. . . . . . . . . . . . . . . 482

xvi



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Ex am ple 3 — Caltrans Permit Loading . . . . . . . . . . . . . . 483

Ex am ple 4 — Re stricted Caltrans Per mit Load ing . . . . . . . . 485

Exam ple 5 — Eurocode Characteristic Load Model 1 . . . . . . 486

Moving Load Response Control . . . . . . . . . . . . . . . . . . . . 488

Bridge Response Groups . . . . . . . . . . . . . . . . . . . . . 488

Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . 489Influence Line Tolerance . . . . . . . . . . . . . . . . . . . . . 489

Exact and Quick Response Calculation . . . . . . . . . . . . . . 489

Step-By-Step Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 490

Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Static Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . 491

Time-History Analysis . . . . . . . . . . . . . . . . . . . . . . 492

Enveloping and Load Com binations . . . . . . . . . . . . . . . 492

Com putational Considerations . . . . . . . . . . . . . . . . . . . . . 493

Chapter XXVII References 495

xvii

Table of Contents


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C h a p t e r I

Introduction

SAP2000, ETABS, SAFE, and CSiBridge are software packages from Com puters

and Structures, Inc. for structural analy sis and design. Each package is a fully inte-

grated system for modeling, analyzing, designing, and optimizing structures of a

particular type:

• SAP2000 for general structures, including stadiums, towers, industrial plants,

offshore structures, piping systems, buildings, dams, soils, machine parts and

many others

• ETABS for building structures

• SAFE for floor slabs and base mats

• CSiBridge for bridge structures

At the heart of each of these soft ware packages is a common anal ysis en gine, re -

ferred to throughout this manual as SAPfire. This engine is the latest and most pow-erful version of the well-known SAP series of structural analy sis programs. The

pur pose of this manual is to describe the features of the SAPfire analy sis engine.

Throughout this manual reference may be made to the program SAP2000, although

it often ap plies equally to ETABS, SAFE, and CSiBridge. Not all features de-

scribed will ac tu ally be available in ev ery level of each pro gram.

1


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Analysis Features

The SAPfire analysis engine offers the following features:

• Static and dynamic analysis

• Linear and nonlinear analysis

• Dynamic seismic analysis and static push over analysis

• Vehicle live-load analysis for bridges

• Geometric nonlinearity, including P-delta and large-dis placement effects

• Staged (incremental) construction

• Creep, shrinkage, and aging effects

• Buckling analysis

• Steady-state and power-spec tral-density analysis

• Frame and shell structural elements, including beam-column, truss, mem brane,

and plate behavior

• Ca ble and Tendon elements

• Two-dimensional plane and axisymmetric solid elements

• Three-dimensional solid elements

• Nonlinear link and sup port elements

•

Frequency-de pendent link and sup port properties• Multi ple coordinate systems

• Many types of constraints

• A wide variety of loading options

• Alpha-numeric la bels

• Large ca pacity

• Highly efficient and sta ble solution algorithms

These fea tures, and many more, make CSI product the state-of-the-art for structuralanal ysis. Note that not all of these features may be available in every level of

SAP2000, ETABS, SAFE, and CSiBridge.

2 Analysis Features



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Structural Analysis and Design

The following general steps are required to analyze and design a structure using

SAP2000, ETABS, SAFE, and CSiBridge:

1. Create or modify a model that numerically defines the geometry, properties,loading, and analysis parameters for the structure

2. Perform an analysis of the model

3. Review the re sults of the analysis

4. Check and optimize the design of the structure

This is usu ally an iterative pro cess that may involve sev eral cycles of the above se-

quence of steps. All of these steps can be performed seamlessly using the SAP2000,

ETABS, SAFE, and CSiBridge graph ical user inter faces.

About This Manual

This manual describes the theoretical concepts behind the modeling and analysis

features offered by the SAPfire analysis engine that underlies the various structural

analy sis and design software packages from Com puters and Structures, Inc. The

graphi cal user in ter face and the design features are described in separate manu als

for each program.

It is im perative that you read this manual and understand the assumptions and pro-

cedures used by these software packages be fore attempting to use the analysis fea-

tures.

Throughout this manual reference may be made to the program SAP2000, although

it often ap plies equally to ETABS, SAFE, and CSiBridge. Not all features de-

scribed will ac tu ally be available in ev ery level of each pro gram.

Topics

Each Chapter of this manual is divided into topics and subtopics. All Chap ters be-

gin with a list of topics covered. These are divided into two groups:

• Basic topics — recommended reading for all users

Structural Analysis and Design 3

Chapter I Introduction


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• Advanced topics — for users with specialized needs, and for all users as they

become more familiar with the program.

Following the list of topics is an Overview which provides a summary of the Chap-

ter. Reading the Overview for every Chapter will acquaint you with the full scope

of the program.

Typographical Conventions

Throughout this manual the following ty pographic conventions are used.

Bold for Definitions

Bold roman type (e.g., example) is used whenever a new term or concept is de-

fined. For exam ple:

The global coordinate system is a three-dimensional, right-handed, rectangu-

lar coordinate system.

This sentence begins the definition of the global coordinate system.

Bold for Variable Data

Bold roman type (e.g., example) is used to represent variable data items for which

you must specify values when defining a structural model and its analysis. For ex-

am ple:

The Frame element coordinate angle, ang, is used to define element orienta-

tions that are different from the default orientation.

Thus you will need to sup ply a numeric value for the variable ang if it is different

from its default value of zero.

Italics for Mathematical Variables Normal italic type (e.g., exam ple) is used for scalar mathematical variables, and

bold italic type (e.g., exam ple) is used for vectors and matrices. If a vari able data

item is used in an equation, bold roman type is used as discussed above. For exam-

ple:

0 da < db L

4 Typographical Conventions



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Here da and db are variables that you specify, and L is a length cal culated by the

program.

Italics for Emphasis

Normal italic type (e.g., exam ple) is used to em phasize an im portant point, or for the title of a book, manual, or journal.

Capitalized Names

Capi talized names (e.g., Exam ple) are used for cer tain parts of the model and its

analysis which have special meaning to SAP2000. Some exam ples:

Frame element

Dia phragm Constraint

Frame Section

Load Pat tern

Common en ti ties, such as “joint” or “element” are not capi talized.

Bibliographic References

References are indicated throughout this manual by giving the name of theauthor(s) and the date of publication, using parentheses. For exam ple:

See Wilson and Tetsuji (1983).

It has been demonstrated (Wilson, Yuan, and Dickens, 1982) that …

All biblio graphic references are listed in al pha beti cal or der in Chapter “Refer-

ences” (page 495).

Bibliographic References 5

Chapter I Introduction


28/5236 Bibliographic References



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C h a p t e r II

Objects and Elements

The physical structural mem bers in a structural model are represented by ob jects.

Using the graphical user in terface, you “draw” the geometry of an ob ject, then “as-

sign” properties and loads to the ob ject to com pletely define the model of the physi-

cal mem ber. For analy sis pur poses, SAP2000 converts each ob ject into one or more

elements.

Basic Topics for All Users

• Objects

• Ob jects and Elements

• Groups

ObjectsThe following ob ject types are available, listed in order of geometrical dimension:

• Point ob jects, of two types:

– Joint ob jects: These are automati cally created at the corners or ends of all

other types of ob jects below, and they can be ex plicitly added to represent

sup ports or to capture other localized behavior.

Objects 7


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– Grounded (one-joint) link/support ob jects: Used to model special sup-

port behavior such as isolators, dampers, gaps, multi-linear springs, and

more.

• Line ob jects, of four types

–

Frame ob jects: Used to model beams, columns, braces, and trusses – Cable ob jects: Used to model slender ca bles under self weight and tension

– Tendon ob jects: Used to prestressing tendons within other ob jects

– Connecting (two-joint) link/support ob jects: Used to model special

mem ber behavior such as isolators, dampers, gaps, multi-linear springs,

and more. Unlike frame, ca ble, and tendon ob jects, connecting link ob jects

can have zero length.

• Area ob jects: Shell elements (plate, mem brane, and full-shell) used to model

walls, floors, and other thin-walled mem bers; as well as two-dimensional sol-ids (plane-stress, plane-strain, and axisymmetric solids).

• Solid ob jects: Used to model three-dimensional solids.

As a general rule, the geometry of the ob ject should correspond to that of the physi-

cal mem ber. This sim plifies the visualization of the model and helps with the de-

sign process.

Ob jects and Elements

If you have ex perience using traditional finite element programs, including earlier

versions of SAP2000, ETABS, and SAFE, you are proba bly used to meshing phys-

ical models into smaller finite elements for analysis pur poses. Ob ject-based model-

ing largely eliminates the need for doing this.

For users who are new to finite-element modeling, the ob ject-based concept should

seem perfectly natural.

When you run an analy sis, SAP2000 automatically converts your ob ject-based

model into an element-based model that is used for analysis. This element-basedmodel is called the analy sis model, and it con sists of tradi tional finite elements and

joints (nodes). Results of the analysis are re ported back on the ob ject-based model.

You have control over how the meshing is performed, such as the degree of refine-

ment, and how to handle the connections between intersecting ob jects. You also

have the option to manually mesh the model, resulting in a one-to-one correspon-

dence between ob jects and elements.


8 Ob jects and Elements


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In this manual, the term “element” will be used more often than “ob ject”, since

what is described herein is the finite-element analy sis portion of the program that

operates on the element-based analysis model. However, it should be clear that the

prop erties de scribed here for elements are ac tually assigned in the interface to the

ob jects, and the conversion to analysis elements is automatic.

One specific case to be aware of is that both one-joint (grounded) link/sup port ob-

jects and two-joint (connecting) link/support ob jects are always converted into

two-joint link/support elements. For the two-joint ob jects, the conversion to ele-

ments is direct. For the one-joint ob jects, a new joint is created at the same location

and is fully restrained. The generated two-joint link/support element is of zero

length, with its original joint connected to the structure and the new joint connected

to ground by restraints.

GroupsA group is a named collection of ob jects that you define. For each group, you must

provide a unique name, then select the ob jects that are to be part of the group. You

can include ob jects of any type or types in a group. Each ob ject may be part of one

of more groups. All ob jects are always part of the built-in group called “ALL”.

Groups are used for many pur poses in the graphical user interface, including selec-

tion, design optimization, defining section cuts, controlling out put, and more. In

this manual, we are primarily interested in the use of groups for defining staged

construction. See Topic “Staged Construction” (page 79) in Chapter “Nonlinear Static Analysis” for more information.

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 ferent coordinate systems to de scribe the location

of points and the directions of loads, dis placement, internal forces, and stresses.

Understanding these different coordinate systems is crucial to being able to prop-

erly define the model and inter pret the results.


• Overview

• Global Coordinate System

• Upward and Horizontal Directions

• Defining Coordinate Systems

• Local Coordinate Systems

Advanced Topics

• Alternate Coordinate Systems

• Cylindrical and Spherical Coordinates

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Overview

Coordinate systems are used to locate different parts of the structural model and to

define the directions of loads, dis placements, internal forces, and stresses.

All coordinate systems in the model are defined with respect to a single global coor-dinate system. Each part of the model (joint, element, or constraint) has its own lo-

cal coordinate system. In addition, you may create alternate coordinate systems that

are used to define locations and directions.

All coordinate systems are three-dimensional, right-handed, rectangular (Carte-

sian) systems. Vector cross products are used to define the local and alternate coor-

dinate systems with respect to the global system.

SAP2000 always assumes that Z is the vertical axis, with +Z being upward. The up-

ward direction is used to help define local coordinate systems, although local coor-dinate systems themselves do not have an upward direc tion.

The locations of points in a coordinate system may be specified using rectangular

or cylindrical coordinates. Likewise, directions in a coordinate system may be

specified using rectangular, cylindrical, or spherical coordinate directions at a

point.

Global Coordinate System

The global coordinate system is a three-dimensional, right-handed, rectangular

coordinate system. The three axes, denoted X, Y, and Z, are mutually per pendicular

and satisfy the right-hand rule.

Locations in the global coordinate system can be specified using the variables x, y,

and z. A vector in the global coordinate system can be specified by giving the loca-

tions of two points, a pair of angles, or by specifying a coordinate direction. Coor-

dinate directions are indicated using the values X, Y, and Z. For exam ple, +Xdefines a vector parallel to and directed along the positive X axis. The sign is re-

quired.

All other coordinate systems in the model are ultimately de fined with respect to the

global coordinate system, either directly or indirectly. Likewise, all joint coordi-

nates are ultimately converted to global X, Y, and Z coordinates, regardless of how

they were speci fied.

12 Overview



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Upward and Horizontal Directions

SAP2000 always assumes that Z is the vertical axis, with +Z being upward. Local

coordinate systems for joints, elements, and ground-acceleration loading are de-

fined with respect to this upward direction. Self-weight loading always acts down-

ward, in the –Z direction.

The X-Y plane is horizontal. The primary horizontal direction is +X. Angles in the

horizontal plane are measured from the positive half of the X axis, with positive an-

gles ap pearing counterclockwise when you are looking down at the X-Y plane.

If you prefer to work with a different upward direction, you can define an alternate

coordinate system for that pur pose.

Defining Coordinate SystemsEach coordinate system to be defined must have an origin and a set of three,

mutually- perpendicular axes that satisfy the right-hand rule.

The origin is defined by sim ply specifying three coordinates in the global coordi-

nate system.

The axes are de fined as vectors using the concepts of vector alge bra. A fundamental

knowledge of the vec tor cross product operation is very helpful in clearly under-

standing how co ordinate system axes are defined.

Vector Cross Product

A vector may be defined by two points. It has length, direction, and location in

space. For the pur poses of defining coordinate axes, only the direction is im portant.

Hence any two vectors that are parallel and have the same sense (i.e., pointing the

same way) may be consid ered to be the same vector.

Any two vectors, V i and V

j , that are not par allel to each other define a plane that is

parallel to them both. The location of this plane is not im portant here, only its orien-tation. The cross product of V

i and V

j defines a third vector, V

k , that is per pendicular

to them both, and hence normal to the plane. The cross product is written 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 portant here. The side of the V

i -V

j plane to which V

k points

is determined by the right-hand rule: The vector V k points toward you if the acute

angle (less than 180°) from V i to V

j ap pears counterclockwise.

Thus the sign of the cross product de pends upon the order of the operands:

V j V i = – V i V j

Defining the Three Axes Using Two Vectors

A right-handed coordinate system R-S-T can be represented by the three mutually-

perpendicular vectors V r , V

s, and V

t , respectively, that satisfy the relationship:

V t = V r V s

This coordinate system can be defined by specifying two non- parallel vectors:

• An axis ref erence vec tor, V a, that is parallel to axis R

• A plane ref erence vec tor, V p, that is parallel to plane R-S, and points toward 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 convenient vector parallel to the R-S plane; it does not have

to be parallel to the S axis. This is illustrated in Figure 1 (page 15).

Local Coordinate Systems

Each part (joint, element, or constraint) of the structural model has its own local co-

ordinate system used to define the properties, loads, and response for that part. Theaxes of the local coordinate systems are denoted 1, 2, and 3. In general, the local co-

ordinate systems may vary from joint to joint, element to element, and constraint to

constraint.

There is no preferred upward direction for a local coordinate system. However, the

upward +Z direction is used to define the default joint and element local coordinate

systems with respect to the global or any alter nate coor di nate system.

14 Local Coordinate Systems



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The joint local 1-2-3 coordinate system is normally the same as the global X-Y-Z

coordinate system. However, you may define any ar bitrary orientation for a joint

local coordinate system by specifying two reference vectors and/or three angles of

rotation.

For the Frame, Area (Shell, Plane, and Asolid), and Link/Sup port elements, one of

the element lo cal axes is deter mined by the geometry of the individual ele ment.

You may define the orientation of the remaining two axes by specify ing a single

reference vector and/or a single angle of ro tation. The exception to this is one-joint

or zero-length Link/Sup port elements, which require that you first specify the lo-

cal-1 (ax ial) axis.

The Solid element local 1-2-3 coordinate system is normally the same as the global

X-Y-Z coordinate system. However, you may define any ar bitrary orientation for a

solid local coordinate system by specifying two reference vectors and/or three an-

gles of rotation.

The local coordinate system for a Body, Dia phragm, Plate, Beam, or Rod Con-

straint is normally determined automatically from the geometry or mass distri bu-

tion of the constraint. Optionally, you may specify one local 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 Constraint (but not for the Body Constraint); the re-

maining two axes are determined auto matically.

The local co or di nate system for an Equal Constraint may be ar bi trarily speci fied;

by default it is the global coordinate system. The Local Constraint does not have its

own local coordinate system.

For more information:

• See Topic “Local Coordinate System” (page 24) in Chapter “Joints and De-

grees of Freedom.”

• See Topic “Local Coordinate System” (page 92) in Chap ter “The Frame Ele-

ment.”

• See Topic “Local Coordinate System” (page 164) in Chapter “The Shell Ele-

ment.”

• See Topic “Local Coordinate System” (page 197) in Chapter “The Plane Ele-

ment.”

• See Topic “Local Coordinate System” (page 207) in Chapter “The Asolid Ele-

ment.”

• See Topic “Local Coordinate System” (page 220) in Chapter “The Solid Ele-

ment.”

• See Topic “Local Coordinate System” (page 233) in Chapter “The Link/Sup-

port Element—Basic.”

• See Chapter “Constraints and Welds (page 49).”

Alternate Coordinate Systems

You may define alternate coordinate systems that can be used for locating the

joints; for defining local coordinate systems for joints, elements, and constraints;

and as a reference for defining other properties and loads. The axes of the alternate

coordinate systems are denoted X, Y, and Z.

The global co or di nate system and all alter nate systems are called fixed coordinate

systems, since they ap ply to the whole structural model, not just to individual parts

as do the local coor di nate systems. Each fixed coor di nate system may be used in

rectangular, cylindrical or spherical form.

Asso ci ated with each fixed coor dinate system is a grid system used to locate ob jects

in the graphical user interface. Grids have no meaning in the analy sis model.

16 Alternate Coordinate Systems



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Each alternate coordinate system is defined by specifying the location of the origin

and the orientation of the axes with respect to the global coordinate system. You

need:

• The global X, Y, and Z coordinates of the new origin

• The three angles (in degrees) used to rotate from the global coordinate systemto the new system

Cylindrical and Spherical Coordinates

The location of points in the global or an alternate coordinate system may be speci-

fied using polar coordinates instead of rectangular X-Y-Z coordinates. Polar coor-

dinates include cylindrical CR-CA-CZ coordinates and spherical SB-SA-SR coor-

dinates. See Figure 2 (page 19) for the definition of the polar coordinate systems.

Polar co ordinate systems are always defined with respect to a rectangular X-Y-Z

system.

The coordinates CR, CZ, and SR are lineal and are specified in length units. The co-

or di nates CA, SB, and SA are angular and are speci fied in de grees.

Locations are specified in cylindrical coordinates using the variables cr, ca, and cz.

These are related to the rectangular coordinates as:

cr x y= +2 2

cay

x= tan

-1

cz z=

Locations are specified in spherical coordinates using the variables sb, sa, and sr.

These are related to the rectangular coordinates as:

sb

x y

z= tan

+-12 2

say

x= tan

-1

sr x y z= + +2 2 2

Cylindrical and Spherical Coordinates 17



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A vector in a fixed coordinate system can be specified by giving the locations of

two points or by specifying a coordinate direction at a single point P . Coordinate

directions are tangential to the coordinate curves at point P . A positive coordinate

direction indicates the direction of increasing coordinate value at that point.

Cylindrical coordinate directions are indicated using the values CR, CA, andCZ. Spherical coordinate directions are indicated using the values SB, SA, andSR. The sign is required. See Figure 2 (page 19).

The cylindrical and spherical coordinate directions are not constant but vary with

angular position. The coordinate directions do not change with the lineal coordi-

nates. For exam ple, +SR defines a vector directed from the origin to point P .

Note that the coordinates Z and CZ are identical, as are the corresponding coordi-

nate directions. Similarly, the coordinates CA and SA and their corresponding co-

ordinate directions are identical.

18 Cylindrical and Spherical Coordinates



41/523Cylindrical and Spherical Coordinates 19


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 2

Cylindrical and Spherical Coordinates and Coordinate Directions


42/52320 Cylindrical and Spherical Coordinates



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C h a p t e r IV

Joints and Degrees of Freedom

The joints play a fundamental role in the analysis of any structure. Joints are the

points of connection between the elements, and they are the primary locations in

the structure at which the dis placements are known or are to be determined. The

dis placement com ponents (translations and rotations) at the joints are called the de-

grees of freedom.

This Chapter describes joint properties, degrees of freedom, loads, and out put. Ad-

ditional information about joints and degrees of freedom is given in Chapter “Con-

straints and Welds” (page 49).


• Overview

• Modeling Considerations

• Local Coordinate System

• Degrees of Freedom

• Restraint Supports

• Spring Sup ports

• Joint Reactions

• Base Reactions

21


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• Masses

• Force Load

• Degree of Freedom Out put

• Assem bled Joint Mass Out put

• Dis placement Out put

• Force Out put

Advanced Topics

• Advanced Local Coordinate System

• Nonlinear Sup ports

• Distributed Supports

• Ground Dis placement Load

• Generalized Displacements

• Element Joint Force Output

Overview

Joints, also known as nodal points or nodes, are a fun da mental part of every struc-

tural model. Joints perform a variety of functions:

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

• The structure is sup ported at the joints using Restraints and/or Springs

• Rigid- body behavior and symmetry conditions can be specified using Con-

straints that ap ply to the joints

• Concentrated loads may be ap plied at the joints

• Lumped (con centrated) masses and rotational inertia may be placed at the

joints

• All loads and masses ap plied to the elements are ac tu ally trans ferred to the

joints

• Joints are the primary locations in the structure at which the dis placements are

known (the sup ports) or are to be determined

All of these functions are discussed in this Chapter except for the Constraints,

which are described in Chapter “Constraints and Welds” (page 49).

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Joints in the analysis model correspond to point ob jects in the structural-ob ject

model. Using the SAP2000, ETABS, SAFE, or CSiBridge graphical user interface,

joints (points) are au to matically created at the ends of each Line ob ject and at the

corners of each Area and Solid ob ject. Joints may also be defined inde pendently of

any ob ject.

Automatic meshing of ob jects will create additional joints corresponding to any el-

ements that are cre ated.

Joints may themselves be considered as elements. Each joint may have its own lo-

cal coordinate system for defining the degrees of freedom, restraints, joint proper-

ties, and loads; and for inter preting joint out put. In most cases, however, the global

X-Y-Z coordinate system is used as the local coordinate system for all joints in the

model. Joints act inde pendently of each other unless connected by other elements.

There are six dis placement degrees of free dom at every joint — three transla tions

and three rotations. These dis placement com ponents are aligned along the local co-

ordinate system of each joint.

Joints may be loaded directly by concentrated loads or indirectly by ground dis-

placements acting though Restraints, spring sup ports, or one-joint (grounded)

Link/Sup port objects.

Dis placements (translations and rotations) are produced at every joint. Reaction

forces and moments acting at each sup ported joint are also produced.

For more information, see Chapter “Constraints and Welds” (page 49).

Modeling Considerations

The location of the joints and elements is critical in determining the accuracy of the

structural model. Some of the factors that you need to consider when defining the

elements, and hence the joints, for the structure are:

• The number of elements should be sufficient to describe the geometry of the

structure. For straight lines and edges, one element is adequate. For curves andcurved surfaces, one element should be used for every arc of 15° or less.

• Element boundaries, and hence joints, should be located at points, lines, and

surfaces of discontinuity:

– Structural boundaries, e.g., corners and edges

– Changes in material properties

Modeling Considerations 23

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– Changes in thickness and other geometric properties

– Sup port points (Restraints and Springs)

– Points of ap pli cation of concentrated loads, ex cept that Frame elements

may have concentrated loads ap plied within their spans

• In regions having large stress gradients, i.e., where the stresses are chang ingrapidly, an Area- or Solid-element mesh should be refined using small ele-

ments and closely-spaced joints. This may require changing the mesh after one

or more preliminary analyses.

• More that one element should be used to model the length of any span for

which dy namic be havior is im portant. This is required because the mass is al-

ways lumped at the joints, even if it is contributed by the elements.

Local Coordinate SystemEach joint has its own joint local coordinate system used to define the degrees of

freedom, Restraints, properties, and loads at the joint; and for inter preting joint out-

put. The axes of the joint local coordinate system are denoted 1, 2, and 3. By default

these axes are identical to the global X, Y, and Z axes, respec tively. Both systems

are right-handed coordinate systems.

The default local coordinate system is adequate for most situations. However, for

certain modeling pur poses it may be useful to use different local coordinate sys-

tems at some or all of the joints. This is described in the next topic.

For more information:

• See Topic “Upward and Horizontal Directions” (page 13) in Chapter “Coordi-

nate Systems.”

• See Topic “Advanced Local Coordinate System” (page 24) in this Chapter.

Advanced Local Coordinate System

By default, the joint local 1-2-3 coordinate system is identical to the global X-Y-Z

coordinate system, as described in the previous topic. However, it may be neces-

sary to use different local coordi nate sys tems at some or all joints in the following

cases:

• Skewed Restraints (sup ports) are present

• Constraints are used to im pose rotational symmetry

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• Constraints are used to im pose symmetry about a plane that is not parallel to

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