SAP2000 Basic

SAP2000® Ba sic

Anal y sis Ref er ence Manual

Lin ear and Non lin ear Static and Dy namicAnal y sis and De sign of Three-Di men sional

Struc tures

ISO# SAP041709M2 Rev. 0 Ver sion 14Berke ley, Cal i for nia, USA April 2009

COPYRIGHT

Copy right ã Com put ers and Struc tures, Inc., 1978-2009.All rights re served.

The CSI Logo® and SAP2000® are reg is tered trade marks of Com put ersand Struc tures, Inc. Model-AliveTM is a trade mark of Com put ers andStruc tures, Inc. Win dows is a reg is tered trade mark of Microsoft Cor po -ra tion.

The com puter pro gram SAP2000® and all as so ci ated doc u men ta tion are pro pri etary and copy righted prod ucts. World wide rights of own er shiprest with Com put ers & Struc tures, Inc. Unlicensed use of this pro gram or re pro duc tion of doc u men ta tion in any form, with out prior writ ten au tho -ri za tion from Com put ers & Struc tures, Inc., is ex plic itly pro hib ited.

No part of this pub li ca tion may be re pro duced or dis trib uted in any formor by any means, or stored in a da ta base or re trieval sys tem, with out theprior ex plicit writ ten per mis sion of the pub lisher.

Fur ther in for ma tion and cop ies of this doc u men ta tion may be ob tainedfrom:

Com put ers & Struc tures, Inc.1995 Uni ver sity Av e nueBerke ley, Cal i for nia 94704 USAPhone: (510) 649-2200FAX: (510) 649-2299

e-mail: [email protected] (for gen eral ques tions)e-mail: sup [email protected] (for tech ni cal sup port ques tions)web: www.csiberkeley.com

DISCLAIMER

CON SID ER ABLE TIME, EF FORT AND EX PENSE HAVE GONEINTO THE DE VEL OP MENT AND TEST ING OF THIS SOFT WARE.HOW EVER, THE USER AC CEPTS AND UN DER STANDS THATNO WAR RANTY IS EX PRESSED OR IM PLIED BY THE DE VEL -OP ERS OR THE DIS TRIB U TORS ON THE AC CU RACY OR THERE LI ABIL ITY OF THIS PRODUCT.

THIS PROD UCT IS A PRAC TI CAL AND POW ER FUL TOOL FORSTRUC TURAL DE SIGN. HOW EVER, THE USER MUST EX PLIC -ITLY UN DER STAND THE BA SIC AS SUMP TIONS OF THE SOFT -WARE MOD EL ING, ANAL Y SIS, AND DE SIGN AL GO RITHMSAND COM PEN SATE FOR THE AS PECTS THAT ARE NOT AD -DRESSED.

THE IN FOR MA TION PRO DUCED BY THE SOFT WARE MUST BECHECKED BY A QUAL I FIED AND EX PE RI ENCED EN GI NEER.THE EN GI NEER MUST IN DE PEND ENTLY VER IFY THE RE -SULTS AND TAKE PRO FES SIONAL RE SPON SI BIL ITY FOR THEIN FOR MA TION THAT IS USED.

ACKNOWLEDGMENT

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

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

Ta ble of Con tents

Chap ter I In tro duc tion 1

About This Man ual . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Im por tant No tice — No men cla ture Change for Load ing. . . . . . . . . 2

Top ics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Ty po graphic Con ven tions . . . . . . . . . . . . . . . . . . . . . . . . 2

Bib lio graphic Ref er ences . . . . . . . . . . . . . . . . . . . . . . . . . 4

Chap ter II Ob jects and El e ments 5

Chap ter III Co or di nate Sys tems 7

Over view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Global Co or di nate Sys tem . . . . . . . . . . . . . . . . . . . . . . . . 8

Up ward and Hor i zon tal Di rec tions . . . . . . . . . . . . . . . . . . . . 8

Lo cal Co or di nate Sys tems . . . . . . . . . . . . . . . . . . . . . . . . 9

Chap ter IV The Frame El e ment 11

Over view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Joint Con nec tiv ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Joint Off sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

De grees of Free dom . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Lo cal Co or di nate Sys tem . . . . . . . . . . . . . . . . . . . . . . . . 14

Lon gi tu di nal Axis 1 . . . . . . . . . . . . . . . . . . . . . . . . 15De fault Ori en ta tion . . . . . . . . . . . . . . . . . . . . . . . . . 15Co or di nate An gle . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Sec tion Prop er ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

i

Lo cal Co or di nate Sys tem. . . . . . . . . . . . . . . . . . . . . . 17Ma te rial Prop er ties . . . . . . . . . . . . . . . . . . . . . . . . . 17Geo met ric Prop er ties and Sec tion Stiffnesses . . . . . . . . . . . 17Shape Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Au to matic Sec tion Prop erty Cal cu la tion . . . . . . . . . . . . . . 20Sec tion Prop erty Da ta base Files . . . . . . . . . . . . . . . . . . 20

In ser tion Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

End Off sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Clear Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Ef fect upon In ter nal Force Out put . . . . . . . . . . . . . . . . . 25Ef fect upon End Re leases . . . . . . . . . . . . . . . . . . . . . 25

End Re leases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Un sta ble End Re leases . . . . . . . . . . . . . . . . . . . . . . . 27Ef fect of End Off sets . . . . . . . . . . . . . . . . . . . . . . . . 27

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Con cen trated Span Load . . . . . . . . . . . . . . . . . . . . . . . . 28

Dis trib uted Span Load. . . . . . . . . . . . . . . . . . . . . . . . . . 29

Loaded Length . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Load In ten sity . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

In ter nal Force Out put . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Ef fect of End Off sets . . . . . . . . . . . . . . . . . . . . . . . . 34

Chap ter V The Shell El e ment 35

Over view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Joint Con nec tiv ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37



Nor mal Axis 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 41De fault Ori en ta tion . . . . . . . . . . . . . . . . . . . . . . . . . 41Co or di nate An gle . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Sec tion Prop er ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Sec tion Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Thick ness For mu la tion . . . . . . . . . . . . . . . . . . . . . . . 43Ma te rial Prop er ties . . . . . . . . . . . . . . . . . . . . . . . . . 44Thick ness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Self-Weight Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Uni form Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

In ter nal Force and Stress Out put . . . . . . . . . . . . . . . . . . . . 46

ii

SAP2000 Basic Analysis Reference

Chap ter VI Joints and De grees of Free dom 49

Over view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Mod el ing Con sid er ations . . . . . . . . . . . . . . . . . . . . . . . . 51



Avail able and Un avail able De grees of Free dom . . . . . . . . . . 53Re strained De grees of Free dom . . . . . . . . . . . . . . . . . . 54Con strained De grees of Free dom. . . . . . . . . . . . . . . . . . 54Ac tive De grees of Free dom . . . . . . . . . . . . . . . . . . . . 54Null De grees of Free dom. . . . . . . . . . . . . . . . . . . . . . 55

Re straints and Re ac tions . . . . . . . . . . . . . . . . . . . . . . . . 55

Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Force Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Ground Dis place ment Load . . . . . . . . . . . . . . . . . . . . . . . 59

Re straint Dis place ments . . . . . . . . . . . . . . . . . . . . . . 61Spring Dis place ments . . . . . . . . . . . . . . . . . . . . . . . 61

Chap ter VII Joint Con straints 63

Over view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Di a phragm Con straint . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Joint Con nec tiv ity . . . . . . . . . . . . . . . . . . . . . . . . . 64Plane Def i ni tion . . . . . . . . . . . . . . . . . . . . . . . . . . 65Lo cal Co or di nate Sys tem. . . . . . . . . . . . . . . . . . . . . . 66Con straint Equa tions . . . . . . . . . . . . . . . . . . . . . . . . 66

Chap ter VIII Static and Dy namic Anal y sis 67

Over view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Load Pat terns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Ac cel er a tion Loads . . . . . . . . . . . . . . . . . . . . . . . . . 69

Load Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Lin ear Static Anal y sis . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Modal Anal y sis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Eigenvector Anal y sis . . . . . . . . . . . . . . . . . . . . . . . . 71Ritz-vec tor Anal y sis . . . . . . . . . . . . . . . . . . . . . . . . 72Modal Anal y sis Re sults . . . . . . . . . . . . . . . . . . . . . . 73

Re sponse-Spec trum Anal y sis . . . . . . . . . . . . . . . . . . . . . . 75

Lo cal Co or di nate Sys tem. . . . . . . . . . . . . . . . . . . . . . 76

iii

Ta ble of Con tents

Re sponse-Spec trum Func tions . . . . . . . . . . . . . . . . . . . 77Re sponse-Spec trum Curve . . . . . . . . . . . . . . . . . . . . . 77Modal Com bi na tion . . . . . . . . . . . . . . . . . . . . . . . . 79Di rec tional Com bi na tion . . . . . . . . . . . . . . . . . . . . . . 82Re sponse-Spec trum Anal y sis Re sults . . . . . . . . . . . . . . . 83

Chap ter IX Bib li og ra phy 85

iv


C h a p t e r I

Introduction

SAP2000 is the lat est and most pow er ful ver sion of the well- known SAP se ries ofstruc tural analy sis pro grams.

About This Manual

This man ual de scribes the ba sic and most com monly used mod el ing and analy sisfea tures of fered by the SAP2000 struc tural analy sis pro gram. It is im pera tive thatyou read this man ual and un der stand the as sump tions and pro ce dures used by thepro gram be fore at tempt ing to cre ate a model or per form an analy sis.

The com plete set of mod el ing and analy sis fea tures is de scribed in the CSI Anal y sisRef er ence Manual.

As back ground ma te rial, you should first read chap ter “The Struc tural Model” inthe SAP2000 Getting Started man ual ear lier in this vol ume. It de scribes the over allfea tures of a SAP2000 model. The pres ent man ual (Ba sic Anal y sis Ref er ence) willpro vide more de tail on some of the el e ments, prop er ties, loads, and anal y sis types.

About This Manual 1

Important Notice — Nomenclature Change for Loading

If you are fa mil iar with SAP2000 Version 11 or ear lier, you should be aware thatthere has been a sim ple but sig nif i cant change made start ing with Ver sion 12 in theno men cla ture used for load ing:

• The term Load Case has been changed to Load Pat tern

• The term Anal y sis Case has been changed to Load Case

• The term Load Com bi na tion is un changed.

Al though the ter mi nol ogy has changed, the fun da men tal con cepts and be hav ior ofthe pro gram are still the same with re gard to load ing and analysis. Mod els cre atedin pre vi ous ver sions of the pro gram are com pat i ble with Ver sion 12 and 14.

This new no men cla ture will be used in all fu ture prod ucts of Computers and Struc -tures, Inc.

Topics

Each chap ter of this man ual is di vided into top ics and subtopics. Most chap ters be -gin with a list of top ics cov ered. Fol low ing the list of top ics is an Over view whichpro vides a sum mary of the chap ter.

Typographic Conventions

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

Bold for Definitions

Bold ro man type (e.g., ex am 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 tan gu -lar co or di nate sys tem.

This sen tence be gins the defi ni tion of the global co or di nate sys tem.

2 Important Notice — Nomenclature Change for Loading


Bold for Variable Data

Bold ro man type (e.g., ex am ple) is used to rep re sent 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 el e ment co or di nate an gle, ang, is used to de fine el e ment ori en ta -tions that are dif fer ent from the de fault ori en ta tion.

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

Italics for Mathematical Variables

Nor mal italic type (e.g., ex am ple) is used for sca lar mathe mati cal vari ables, andbold italic type (e.g., ex am 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 cu lated by thepro gram.

Italics for Emphasis

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

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 el e ment

Dia phragm Con straint

Frame Sec tion

Load Pattern

Com mon en ti ties, such as “joint” or “el e ment” are not cap i tal ized.

Typographic Conventions 3

Chapter I Introduction

Bibliographic References

Ref er ences are in di cated through out this man ual by giv ing the name of theauthor(s) and the date of pub li ca tion, us ing pa ren the ses. For ex am ple:

See Wil son and Tet suji (1983).

It has been dem on strated (Wil son, Yuan, and Dick ens, 1982) that …

All bib lio graphic ref er ences are listed in al pha beti cal or der in Chap ter “Bib li og ra -phy” (page 85).

4 Bibliographic References


C h a p t e r II

Objects and Elements

The phys i cal struc tural mem bers in a SAP2000 model are rep re sented by ob jects.Using the graph i cal user in ter face, you “draw” the ge om e try of an ob ject, then “as -sign” prop er ties and loads to the ob ject to com pletely de fine the model of the phys i -cal mem ber.

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

• Point ob jects, of two types:

– Joint ob jects: These are au to mat i cally cre ated at the cor ners or ends of allother types of ob jects be low, and they can be ex plic itly added to modelsup ports or other lo cal ized be hav ior.

– Grounded (one-joint) sup port ob jects: Used to model spe cial sup portbe hav ior such as iso la tors, damp ers, gaps, multilinear springs, and more.These are not cov ered in this manual

• Line ob jects, of sev eral types

– Frame ob jects: Used to model beams, col umns, braces, and trusses; theymay be straight or curved

– Ca ble ob jects: Used to model flex i ble ca bles

– Ten don ob jects: Used to prestressing ten dons in side other ob jects

5

– Con necting (two-joint) link ob jects: Used to model spe cial mem ber be -hav ior such as iso la tors, damp ers, gaps, multilinear springs, and more. Un -like frame, ca ble, and tendon objects, con nect ing link ob jects can have zero length. These are not cov ered in this man ual.

• Area ob jects: Used to model walls, floors, and other thin-walled mem bers, aswell as two-di men sional sol ids (plane stress, plane strain, and axisymmetricsol ids). Only shell-type area ob jects are cov ered in this man ual.

• Solid ob jects: Used to model three-di men sional sol ids. These are not cov eredin this manual.

As a gen eral rule, the ge om e try of the ob ject should cor re spond to that of the phys i -cal mem ber. This sim pli fies the vi su al iza tion of the model and helps with the de -sign pro cess.

If you have ex pe ri ence us ing tra di tional fi nite el e ment pro grams, in clud ing ear lierver sions of SAP2000, you are prob a bly used to mesh ing phys i cal mod els intosmaller fi nite el e ments for anal y sis pur poses. Ob ject-based mod el ing largely elim i -nates the need for do ing this.

For us ers who are new to fi nite-el e ment mod el ing, the ob ject-based con cept shouldseem per fectly nat u ral.

When you run an anal y sis, SAP2000 au to mat i cally con verts your ob ject-basedmodel into an el e ment-based model that is used for anal y sis. This el e ment-basedmodel is called the anal y sis model, and it con sists of tra di tional fi nite el e ments andjoints (nodes). Re sults of the anal y sis 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 ter sect ing ob jects. You alsohave the op tion to man u ally sub di vide the model, re sult ing in a one-to-one cor re -spon dence be tween ob jects and el e ments.

In this man ual, the term “el e ment” will be used more of ten than “ob ject”, sincewhat is de scribed herein is the fi nite-el e ment anal y sis por tion of the pro gram thatop er ates on the el e ment-based anal y sis model. However, it should be clear that theprop er ties de scribed here for el e ments are ac tu ally as signed in the in ter face to theob jects, and the con ver sion to anal y sis el e ments is au to matic.


6

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 ca tionof points and the di rec tions of loads, dis place ment, in ter nal forces, and stresses.Un der stand ing these dif fer ent co or di nate sys tems is cru cial to be ing able to prop -erly de fine the model and in ter pret the re sults.

Topics

• Over view

• Global Co or di nate Sys tem

• Up ward and Hori zon tal Di rec tions

• Lo cal Co or di nate Sys tems

Overview

Co or di nate sys tems are used to lo cate dif fer ent parts of the struc tural model and tode fine the di rec tions of loads, dis place ments, in ter nal forces, and stresses.

All co or di nate sys tems in the model are de fined with re spect to a sin gle, globalX-Y-Z co or di nate sys tem. Each part of the model (joint, el e ment, or con straint) hasits own lo cal 1-2-3 co or di nate sys tem. In ad di tion, you may cre ate al ter nate co or di -

Overview 7

nate sys tems that are used to de fine lo ca tions and di rec tions. All co or di nate sys tems are three-di men sional, right-handed, rect an gu lar (Car te sian) sys tems.

SAP2000 al ways as sumes that Z is the ver ti cal axis, with +Z be ing up ward. The up -ward di rec tion is used to help de fine lo cal co or di nate sys tems, al though lo cal co or -di nate sys tems them selves do not have an up ward di rec tion.

For more in for ma tion and ad di tional fea tures, see Chap ter “Co or di nate Sys tems” in the CSI Anal y sis Ref er ence Manual and the Help Menu in the SAP2000 graphi caluser in ter face.

Global Coordinate System

The global co or di nate sys tem is a three- dimensional, right- handed, rec tan gu larco or di nate sys tem. The three axes, de noted X, Y, and Z, are mu tu ally per pen dicu lar and sat isfy the right- hand rule. The lo ca tion and ori en ta tion of the global sys tem are ar bi trary.

Lo ca tions in the global co or di nate sys tem can be speci fied us ing the vari ables x, y,and z. A vec tor in the global co or di nate sys tem can be speci fied by giv ing the lo ca -tions of two points, a pair of an gles, or by speci fy ing a co or di nate di rec tion. Co or -

di nate di rec tions are in di cated us ing the val ues ±X, ±Y, and ±Z. For ex am ple, +Xde fines a vec tor par al lel to and di rected along the posi tive X axis. The sign is re -quired.

All other co or di nate sys tems in the model are de fined with re spect to the global co -or di nate sys tem.

Upward and Horizontal Directions

SAP2000 al ways as sumes that Z is the ver ti cal axis, with +Z be ing up ward. Lo calco or di nate sys tems for joints, el e ments, and ground-ac cel er a tion 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 counter- clockwise when you are look ing down at the X-Y plane.

8 Global Coordinate System


Local Coordinate Systems

Each part (joint, el e ment, or con straint) of the struc tural model has its own lo cal co -or di nate sys tem used to de fine the prop er ties, loads, and re sponse for that part. Theaxes of the lo cal co or di nate sys tems are de noted 1, 2, and 3. In gen eral, the lo cal co -or di nate sys tems may vary from joint to joint, el e ment to el e ment, and con straint tocon straint.

There is no pre ferred up ward di rec tion for a lo cal co or di nate sys tem. How ever, thejoint and el e ment lo cal co or di nate sys tems are de fined with re spect to the globalup ward di rec tion, +Z.

The joint lo cal 1- 2-3 co or di nate sys tem is nor mally the same as the global X- Y-Zco or di nate sys tem.

For the Frame and Shell el e ments, one of the el e ment lo cal axes is de ter mined bythe ge om e try of the in di vid ual el e ment. You may de fine the ori en ta tion of the re -main ing two axes by spec i fy ing a sin gle an gle of ro ta tion.

The lo cal co or di nate sys tem for a Dia phragm Con straint is nor mally de ter minedauto mati cally from the ge ome try or mass dis tri bu tion of the con straint. Op tion ally,you may spec ify one global axis that de ter mines the plane of a Dia phragm Con -straint; the re main ing two axes are de ter mined auto mati cally.

For more in for ma tion:

• See Topic “Lo cal Co or di nate Sys tem” (page 14) in Chap ter “The Frame El e -ment.”

• See Topic “Lo cal Co or di nate Sys tem” (page 40) in Chap ter “The Shell El e -ment.”

• See Topic “Lo cal Co or di nate Sys tem” (page 52) in Chap ter “Joints and De -grees of Free dom.”

• See Topic “Dia phragm Con straint” (page 64) in Chap ter “Joint Con straints.”

Local Coordinate Systems 9

Chapter III Coordinate Systems

10 Local Coordinate Systems


C h a p t e r IV

The Frame Element

The Frame el e ment is used to model beam, col umn, truss, and brace be hav ior inpla nar and three-di men sional struc tures. Al though not dis cussed in this man ual,some of the top ics in this chap ter ap ply to other line ob jects: the curved frame, theca ble, and the ten don.

Topics

• Over view

• Joint Con nec tivity

• De grees of Free dom

• Lo cal Co or di nate Sys tem

• Sec tion Prop erties

• In ser tion Point

• End Off sets

• End Re leases

• Mass

• Self- Weight Load

• Con cen trated Span Load

11

• Dis trib uted Span Load

• In ter nal Force Out put

Overview

The Frame el e ment uses a gen eral, three-di men sional, beam-col umn for mu la tionwhich in cludes the ef fects of bi axial bend ing, tor sion, ax ial de for ma tion, and bi -axial shear de for ma tions. See Bathe and Wil son (1976).

Struc tures that can be mod eled with this el e ment in clude:

• Three- dimensional frames

• Three- dimensional trusses

• Pla nar frames

• Pla nar gril lages

• Pla nar trusses

A Frame el e ment is mod eled as a straight line con nect ing two points. In the graph i -cal user in ter face, you can di vide curved ob jects into mul ti ple straight ob jects,subject to your spec i fi ca tion.

Each el e ment has its own lo cal co or di nate sys tem for de fin ing sec tion prop er tiesand loads, and for in ter pret ing out put.

Each Frame el e ment may be loaded by self-weight, mul ti ple con cen trated loads,and mul ti ple dis trib uted loads.

In ser tion points and end off sets are avail able to ac count for the fi nite size of beamand col umn in ter sec tions. End re leases are also avail able to model dif fer ent fix itycon di tions at the ends of the el e ment.

El e ment in ter nal forces are pro duced at the ends of each el e ment and at a user-spec -i fied num ber of equally-spaced out put sta tions along the length of the el e ment.

For more in for ma tion and ad di tional fea tures, see Chap ter “The Frame El e ment” inthe CSI Anal y sis Ref er ence Man ual.

12 Overview


Joint Connectivity

A Frame el e ment is rep re sented by a straight line con nect ing two joints, I and j, un -less mod i fied by joint off sets as de scribed be low. The two joints must not share thesame lo ca tion in space. The two ends of the el e ment are de noted end I and end J, re -spec tively.

By de fault, the neu tral axis of the el e ment runs along the line con nect ing the twojoints. How ever, you can change this us ing the in ser tion point, as de scribed inTopic “In ser tion Point” (page 22).

Joint Off sets

Some times the axis of the el e ment can not be con ve niently spec i fied by joints thatcon nect to other el e ments in the struc ture. You have the op tion to spec ify joint off -sets in de pend ently at each end of the el e ment. These are given as the three dis tancecom po nents (X, Y, and Z) par al lel to the global axes, mea sured from the joint to theend of the el e ment (at the in ser tion point.)

The two lo ca tions given by the co or di nates of joints I and j, plus the cor re spond ingjoint off sets, de fine the axis of the el e ment. These two lo ca tions must not be co in ci -dent. It is gen er ally rec om mended that the off sets be per pen dic u lar to the axis of the el e ment, al though this is not re quired.

Off sets along the axis of the el e ment are usu ally spec i fied us ing end off sets ratherthan joint off sets. See topic “End Off sets” (page 24). End off sets are part of thelength of the el e ment, have el e ment prop er ties and loads, and may or may not berigid. Joint off sets are ex ter nal to the el e ment, and do not have any mass or loads.In ternally the pro gram cre ates a fully rigid con straint along the joints off sets.

Joint off sets are spec i fied along with the car di nal point as part of the in ser tion pointas sign ment, even though they are in de pend ent features.


• See Topic “In ser tion Point” (page 22) in this chap ter.

• See Topic “End Off sets” (page 24) in this chap ter.

Joint Connectivity 13

Chapter IV The Frame Element

Degrees of Freedom

The Frame el e ment ac ti vates all six de grees of free dom at both of its con nectedjoints. If you want to model truss el e ments that do not trans mit mo ments at the ends, you may ei ther:

• Set the geo met ric Sec tion prop er ties j, i33, and i22 all to zero (a is non- zero;as2 and as3 are ar bi trary), or

• Re lease both bend ing ro ta tions, R2 and R3, at both ends and re lease the tor -sional ro ta tion, R1, at ei ther end


• See Topic “De grees of Free dom” (page 52) in Chap ter “Joints and De grees ofFree dom.”

• See Topic “Sec tion Prop erties” (page 17) in this chap ter.

• See Topic “End Re leases” (page 26) in this chap ter.

Local Coordinate System

Each Frame el e ment has its own el e ment lo cal co or di nate sys tem used to de finesec tion prop er ties, loads and out put. The axes of this lo cal sys tem are de noted 1, 2and 3. The first axis is di rected along the length of the el e ment; the re main ing twoaxes lie in the plane per pen dic u lar to the el e ment with an ori en ta tion that you spec -ify.

It is im por tant that you clearly un der stand the def i ni tion of the el e ment lo cal 1-2-3co or di nate sys tem and its re la tion ship to the global X-Y-Z co or di nate sys tem. Bothsys tems are right-handed co or di nate sys tems. It is up to you to de fine lo cal sys temswhich sim plify data in put and in ter pre ta tion of re sults.

In most struc tures the def i ni tion of the el e ment lo cal co or di nate sys tem is ex -tremely sim ple us ing the de fault ori en ta tion and the Frame el e ment co or di natean gle. Ad di tional meth ods are avail able.


• See Chap ter “Co or di nate Sys tems” (page 7) for a de scrip tion of the con ceptsand ter mi nol ogy used in this topic.

• See Topic “Ad vanced Lo cal Co or di nate Sys tem” in Chap ter “The Frame El e -ment” in the CSI Anal y sis Ref er ence Man ual.

14 Degrees of Freedom


• See Topic “Joint Offsets” (page 13) in this chap ter.

Longitudinal Axis 1

Lo cal axis 1 is al ways the lon gi tu di nal axis of the el e ment, the pos i tive di rec tion be -ing di rected from end I to end J.

Spe cifically, end I is joint I plus its joint off sets (if any), and end J is joint j plus itsjoint off sets (if any.) The axis is de ter mined in de pend ently of the cardinal point; see Topic “In ser tion Point” (page 22.)

Default Orientation

The de fault ori en ta tion of the lo cal 2 and 3 axes is de ter mined by the re la tion shipbe tween the lo cal 1 axis and the global Z axis:

• The lo cal 1-2 plane is taken to be ver ti cal, i.e., par al lel to the Z axis

• The lo cal 2 axis is taken to have an up ward (+Z) sense un less the el e ment is ver -ti cal, in which case the lo cal 2 axis is taken to be hor i zon tal along the global +Xdi rec tion

• The lo cal 3 axis is al ways hori zon tal, i.e., it lies in the X-Y plane

An el e ment is con sid ered to be ver ti cal if the sine of the an gle be tween the lo cal 1axis and the Z axis is less than 10-3.

The lo cal 2 axis makes the same an gle with the ver ti cal axis as the lo cal 1 axismakes with the hor i zon tal plane. This means that the lo cal 2 axis points ver ti callyup ward for hor i zon tal el e ments.

Coordinate Angle

The Frame el e ment co or di nate an gle, ang, is used to de fine el e ment ori en ta tionsthat are dif fer ent from the de fault ori en ta tion. It is the an gle through which the lo cal 2 and 3 axes are ro tated about the pos i tive lo cal 1 axis from the de fault ori en ta tion.The ro ta tion for a pos i tive value of ang ap pears coun ter-clock wise when the lo cal+1 axis is point ing to ward you.

For ver ti cal el e ments, ang is the an gle be tween the lo cal 2 axis and the hor i zon tal+X axis. Oth er wise, ang is the an gle be tween the lo cal 2 axis and the ver ti cal planecon tain ing the lo cal 1 axis. See Figure 1 (page 16) for ex am ples.

Local Coordinate System 15


16 Local Coordinate System


Y

Y

Y

Y

Z

Z

Z

Z

X

X

X

X

ang=90°

ang=90°

ang=30°

ang=30°

2

2

2

2

3

3

3

3

1

1

1

1

i

i

i

i

j

j

j

j

Local 1 Axis is Parallel to +Y AxisLocal 2 Axis is Rotated 90° from Z-1 Plane

Local 1 Axis is Parallel to +Z AxisLocal 2 Axis is Rotated 90° from X-1 Plane

Local 1 Axis is Parallel to –Z AxisLocal 2 Axis is Rotated 30° from X-1 Plane

Local 1 Axis is Not Parallel to X, Y, or Z AxesLocal 2 Axis is Rotated 30° from Z-1 Plane

Figure 1The Frame Element Coordinate Angle with Respect to the Default Orientation

Section Properties

A Frame Sec tion is a set of ma te rial and geo met ric prop er ties that de scribe thecross-sec tion of one or more Frame el e ments. Sec tions are de fined in de pend entlyof the Frame el e ments, and are as signed to the el e ments.


Sec tion prop er ties are de fined with re spect to the lo cal co or di nate sys tem of aFrame el e ment as fol lows:

• The 1 di rec tion is along the axis of the el e ment. It is nor mal to the Sec tion andgoes through the in ter sec tion of the two neu tral axes of the Sec tion.

• The 2 and 3 di rec tions de fine the plane of the Sec tion. Usu ally the 2 di rec tion istaken along the ma jor di men sion (depth) of the Sec tion, and the 3 di rec tionalong its mi nor di men sion (width), but this is not re quired.

See Topic “Lo cal Co or di nate Sys tem” (page 14) in this chap ter for more in for ma -tion.

Material Properties

The ma te rial prop er ties for the Sec tion are speci fied by ref er ence to a previously- defined Ma te rial. The ma te rial prop er ties used by the Sec tion are:

• The modu lus of elas tic ity, e1, for ax ial stiff ness and bend ing stiff ness;

• The shear modu lus, g12, for tor sional stiff ness and trans verse shear stiff ness;this is com puted from e1 and the Pois son's ra tio, u12

• The mass den sity (per unit of vol ume), m, for com put ing el e ment mass;

• The weight den sity (per unit of vol ume), w, for com put ing Self- Weight Load.

• The de sign-type in di ca tor, ides, that in di cates whether el e ments us ing this Sec -tion should be de signed as steel, con crete, alu mi num, cold-formed steel, ornone (no de sign).

Geometric Properties and Section Stiffnesses

Six ba sic geo met ric prop er ties are used, to gether with the ma te rial prop er ties, togen er ate the stiff nesses of the Sec tion. These are:

• The cross- sectional area, a. The ax ial stiff ness of the Sec tion is given by a e1× ;

Section Properties 17


• The mo ment of in er tia, i33, about the 3 axis for bend ing in the 1-2 plane, andthe mo ment of in er tia, i22, about the 2 axis for bend ing in the 1-3 plane. Thecor re spond ing bend ing stiff nesses of the Sec tion are given by i33 e1× and i22 e1× ;

• The tor sional con stant, j. The tor sional stiff ness of the Sec tion is given by j g12× . Note that the tor sional con stant is not the same as the po lar mo ment ofin er tia, ex cept for cir cu lar shapes. See Roark and Young (1975) or Cook andYoung (1985) for more in for ma tion.

• The shear ar eas, as2 and as3, for trans verse shear in the 1-2 and 1-3 planes, re -spec tively. The cor re spond ing trans verse shear stiff nesses of the Sec tion aregiven by as2 g12× and as3 g12× . For mu lae for cal cu lat ing the shear ar eas oftypi cal sec tions are given in Figure 2 (page 19).

Set ting a, j, i33, or i22 to zero causes the cor re spond ing sec tion stiff ness to be zero.For ex am ple, a truss mem ber can be mod eled by set ting j = i33 = i22 = 0, and a pla -nar frame mem ber in the 1-2 plane can be mod eled by set ting j = i22 = 0.

Set ting as2 or as3 to zero causes the cor re spond ing trans verse shear de for ma tion tobe zero. In ef fect, a zero shear area is in ter preted as be ing in fi nite. The trans verseshear stiff ness is ig nored if the cor re spond ing bend ing stiff ness is zero.

Shape Type

For each Sec tion, the six geo met ric prop er ties (a, j, i33, i22, as2 and as3) may bespeci fied di rectly, com puted from speci fied Sec tion di men sions, or read from aspeci fied prop erty da ta base file. This is de ter mined by the shape type, sh, speci fiedby the user:

• If sh=G (gen eral sec tion), the six geo met ric prop er ties must be ex plic itly speci -fied

• If sh=R, P, B, I, C, T, L, or 2L, the six geo met ric prop er ties are auto mati callycal cu lated from speci fied Sec tion di men sions as de scribed in “Auto matic Sec -tion Prop erty Cal cu la tion” be low.

• If sh is any other value (e.g., W27X94 or 2L4X3X1/4), the six geo met ric prop -er ties are ob tained from a speci fied prop erty da ta base file. See “Sec tion Prop -erty Da ta base Files” be low.

18 Section Properties




tf

tf

bf

bf

Section Description EffectiveShear Area

d

b

Rectangular Section

Shear Forces parallel to the b or d

directions

bd5/6

Wide Flange Section

Shear Forces parallel to flanget f bf

5/3

w

Wide Flange Section

Shear Forces parallel to webtw d

d

t

r

t

Thin Walled

Circular Tube Section

Shear Forces from any direction

r t

r Solid Circular Section

Shear Forces from any direction 0.9 r 2

d

t

2 t d

Q(Y) = n b(n) dny

General SectionShear Forces parallel to

I = moment of inertia of

I x2

(y)dy

Q2

b(y)yb

Y

X

n

dn

y

yb

b(y)

n.a.

Thin WalledRectangular Tube Section

Shear Forces parallel tod-direction

Y-direction

section about X-Xx

yt

yt

yt

Figure 2Shear Area Formulae

Automatic Section Property Calculation

The six geo met ric Sec tion prop er ties can be auto mati cally cal cu lated from speci -fied di men sions for the sim ple shapes shown in Figure 3 (page 21). The re quired di -men sions for each shape are shown in the fig ure.

Note that the di men sion t3 is the depth of the Sec tion in the 2 di rec tion and con trib -utes pri mar ily to i33.

Auto matic Sec tion prop erty cal cu la tion is avail able for the fol low ing shape types:

• sh=R: Rec tan gu lar Sec tion

• sh=P: Pipe Sec tion, or Solid Cir cu lar Sec tion if tw=0 (or not speci fied)

• sh=B: Box Sec tion

• sh=I: I- Section

• sh=C: Chan nel Sec tion

• sh=T: T- Section

• sh=L: An gle Sec tion

• sh=2L: Double- angle Sec tion

Section Property Database Files

Geo met ric Sec tion prop er ties may be ob tained from one or more Sec tion prop ertyda ta base files. Sev eral da ta base files are sup plied with SAP2000, including:

• AISC.pro and AISC3.pro: Amer i can In sti tute of Steel Con struc tion shapes

• AA6061-T6.pro: Alu mi num As so ci a tion shapes

• CISC.pro: Ca na dian In sti tute of Steel Con struc tion shapes

• SEC TIONS8.pro: This is just a copy of AISC3.pro.

Ad di tional files are pro vided con tain ing stan dard shapes for other coun tries.

You may cre ate your own prop erty da ta base files us ing the pro gram PROPER,which is avail able upon re quest from Com put ers and Struc tures, Inc.

The geo met ric prop er ties are stored in the length units speci fied when the da ta basefile was cre ated. These are auto mati cally con verted to the units be ing used bySAP2000.





SH = TSH = I SH = C

SH = L SH = 2L

SH = R SH = P SH = B

3

2

t3

tw

t3

t2

tf

tf

tw tw

2

3

t2

t33

2

t2t

tfb

tft

t2b

tw

3

22

3 t3

tf

tf

tf

t2

tw

t2

tw

dis

tf

2

3

22

3 3

t2

t2

tf

t3 t3

tw

tw

Figure 3Automatic Section Property Calculation

Each shape type stored in a da ta base file may be ref er enced by one or two dif fer entla bels. For ex am ple, the W 36x300 shape type in file AISC.PRO may be ref er enced ei ther by la bel “W36X300” or by la bel “W920X446”. Shape types stored inCISC.PRO may only be ref er enced by a sin gle la bel.

You may se lect one da ta base file to be used when de fin ing a given Frame Sec tion.The da ta base file in use can be changed at any time when de fin ing Sec tions. If noda ta base file name is speci fied, the de fault file SEC TIONS8.PRO is used. You maycopy any prop erty da ta base file to SEC TIONS8.PRO.

All Sec tion prop erty da ta base files, in clud ing file SEC TIONS8.PRO, must be lo -cated ei ther in the di rec tory that con tains the data file, or in the di rec tory that con -tains the SAP2000 pro gram files. If a speci fied da ta base file is pres ent in both di -rec to ries, the pro gram will use the file in the data- file di rec tory.

In ser tion Point

By de fault the lo cal 1 axis of the el e ment runs along the neu tral axis of the sec tion,i.e., at the cen troid of the sec tion. It is of ten con ve nient to spec ify an other lo ca tionon the sec tion, such as the top of a beam or an out side cor ner of a col umn. This lo ca -tion is called the car di nal point of the sec tion.

22 In ser tion Point


Note: For doubly symmetric members such asthis one, cardinal points 5, 10, and 11 arethe same.

1 2 3

4

51011

6

7 8 9

1. Bottom left2. Bottom center3. Bottom right4. Middle left5. Middle center6. Middle right7. Top left8. Top center9. Top right10. Centroid11. Shear center

2 axis

3 axis

Figure 4Frame Cardinal Points

The avail able car di nal point choices are shown in Figure 4 (page 22). The de faultlo ca tion is point 10.

Joint off sets are spec i fied along with the car di nal point as part of the in ser tion pointas sign ment, even though they are in de pend ent fea tures. Joint off sets are used first

In ser tion Point 23


Elevation

Cardinal Point B2

B2

Cardinal Point C1

B1

C1

Cardinal Point B1

X

Z

X

Y

Plan

B2C1

B1

2"

2"

Figure 5Example Showing Joint Offsets and Cardinal Points

to cal cu late the el e ment axis and there fore the lo cal co or di nate system, then the car -di nal point is lo cated in the re sult ing lo cal 2-3 plane.

This fea ture is use ful, as an ex am ple, for mod el ing beams and col umns when thebeams do not frame into the cen ter of the col umn. Figure 5 (page 22) shows an el e -va tion and plan view of a com mon fram ing ar range ment where the ex te rior beamsare off set from the col umn cen ter lines to be flush with the ex te rior of the build ing.Also shown in this fig ure are the cardinal points for each mem ber and the joint off -set di men sions.

End Offsets

Frame ele ments are mod eled as line ele ments con nected at points (joints). How -ever, ac tual struc tural mem bers have fi nite cross- sectional di men sions. When twoele ments, such as a beam and col umn, are con nected at a joint there is some over lapof the cross sec tions. In many struc tures the di men sions of the mem bers are largeand the length of the over lap can be a sig nifi cant frac tion of the to tal length of acon nect ing ele ment.

You may spec ify two end off sets for each ele ment us ing pa rame ters ioff and joffcor re spond ing to ends I and J, re spec tively. End off set ioff is the length of over lapfor a given ele ment with other con nect ing ele ments at joint I. It is the dis tance fromthe joint to the face of the con nec tion for the given ele ment. A simi lar defi ni tion ap -plies to end off set joff at joint j. See Figure 6 (page 25).

End off sets can be auto mati cally cal cu lated by the SAP2000 graphi cal user in ter -face for se lected ele ments based on the maxi mum Sec tion di men sions of all otherele ments that con nect to that ele ment at a com mon joint.

Clear Length

The clear length, de noted Lc , is de fined to be the length be tween the end off sets

(sup port faces) as:

L Lc = - +( )ioff joff

where L is the to tal ele ment length. See Figure 6 (page 25).

If end off sets are speci fied such that the clear length is less than 1% of the to tal ele -ment length, the pro gram will is sue a warn ing and re duce the end off sets pro por -tion ately so that the clear length is equal to 1% of the to tal length. Nor mally the endoff sets should be a much smaller pro por tion of the to tal length.

24 End Offsets


Effect upon Internal Force Output

All in ter nal forces and mo ments are out put at the faces of the sup ports and at otherequally- spaced points within the clear length of the ele ment. No out put is pro ducedwithin the end off set, which in cludes the joint.

See Topic “In ter nal Force Out put” (page 32) in this chap ter for more in for ma tion.

Effect upon End Releases

End re leases are al ways as sumed to be at the sup port faces, i.e., at the ends of theclear length of the ele ment. If a mo ment or shear re lease is speci fied in ei ther bend -ing plane at ei ther end of the ele ment, the end off set is as sumed to be rigid for bend -ing and shear in that plane at that end.

See Topic “End Re leases” (page 26) in this chap ter for more in for ma tion.

End Offsets 25


End Offsets

HorizontalMember

Support Face

Total Length L

JI

CL

CLCL

ioff joff

Clear Length Lc

Figure 6Frame Element End Offsets

End Releases

Nor mally, the three trans la tional and three ro ta tional de grees of free dom at eachend of the Frame ele ment are con tinu ous with those of the joint, and hence withthose of all other ele ments con nected to that joint. How ever, it is pos si ble to re lease(dis con nect) one or more of the ele ment de grees of free dom from the joint when itis known that the cor re spond ing ele ment force or mo ment is zero. The re leases areal ways speci fied in the ele ment lo cal co or di nate sys tem, and do not af fect any other ele ment con nected to the joint.

In the ex am ple shown in Figure 7 (page 26), the di ago nal ele ment has a mo mentcon nec tion at End I and a pin con nec tion at End J. The other two ele ments con nect -ing to the joint at End J are con tinu ous. There fore, in or der to model the pin con di -tion the ro ta tion R3 at End J of the di ago nal ele ment should be re leased. This as -sures that the mo ment is zero at the pin in the di ago nal ele ment.

26 End Releases


Axis 3

Axis 2

Axis 1ContinousJoint

Z

Global

ContinousJoint

Pin Joint

I

J

X

For diagonal element: R3 is released at end J

Figure 7Frame Element End Releases

Unstable End Releases

Any com bi na tion of end re leases may be speci fied for a Frame ele ment pro videdthat the ele ment re mains sta ble; this as sures that all load ap plied to the ele ment istrans ferred to the rest of the struc ture. The fol low ing sets of re leases are un sta ble,ei ther alone or in com bi na tion, and are not per mit ted:

• Re leas ing U1 at both ends



• Re leas ing R1 at both ends

• Re leas ing R2 at both ends and U3 at ei ther end

• Re leas ing R3 at both ends and U2 at ei ther end

Effect of End Offsets

End re leases are al ways ap plied at the sup port faces, i.e., at the ends of the ele mentclear length. The pres ence of a mo ment or shear re lease will cause the end off set tobe rigid in the cor re spond ing bend ing plane at the cor re spond ing end of the ele -ment.

See Topic “End Off sets” (page 24) in this chap ter for more in for ma tion.

Mass

In a dy namic analy sis, the mass of the struc ture is used to com pute in er tial forces.The mass con trib uted by the Frame ele ment is lumped at the joints I and j. No in er -tial ef fects are con sid ered within the ele ment it self.

The to tal mass of the ele ment is equal to the in te gral along the length of the massden sity, m, mul ti plied by the cross- sectional area, a.

The to tal mass is ap por tioned to the two joints in the same way a similarly- distributed trans verse load would cause re ac tions at the ends of a simply- supportedbeam. The ef fects of end re leases are ig nored when ap por tion ing mass. The to talmass is ap plied to each of the three trans la tional de grees of free dom: UX, UY, andUZ. No mass mo ments of in er tia are com puted for the ro ta tional de grees of free -dom.


Mass 27


• See Topic “Sec tion Prop er ties” (page 17) in this chap ter for the defi ni tions ofm and a.

• See Chap ter “Static and Dy namic Analy sis” (page 67).

Self-Weight Load

Self- Weight Load can be ap plied in any Load Pattern to ac ti vate the self- weight ofall ele ments in the model. For a Frame ele ment, the self- weight is a force that is dis -trib uted along the length of the ele ment. The mag ni tude of the self- weight is equalto the weight den sity, w, mul ti plied by the cross- sectional area, a.

Self- weight al ways acts down ward, in the global –Z di rec tion. The self- weight may be scaled by a sin gle fac tor that ap plies to the whole struc ture.


• See Topic “Sec tion Prop er ties” (page 17) in this chap ter for the defi ni tions of wand a.


Concentrated Span Load

The Con cen trated Span Load is used to ap ply con cen trated forces and mo ments atar bi trary lo ca tions on Frame ele ments. The di rec tion of load ing may be speci fied in the global co or di nate sys tem or in the ele ment lo cal co or di nate sys tem.

The lo ca tion of the load may be speci fied in one of the fol low ing ways:

• Speci fy ing a rela tive dis tance, rd, meas ured from joint I. This must sat isfy 0 1£ £rd . The rela tive dis tance is the frac tion of ele ment length;

• Speci fy ing an ab so lute dis tance, d, meas ured from joint I. This must sat isfy 0 £ £d L, where L is the ele ment length.

Any number of con cen trated loads may be ap plied to each ele ment. Loads given inglobal co or di nates are trans formed to the ele ment lo cal co or di nate sys tem. SeeFigure 8 (page 29). Mul ti ple loads that are ap plied at the same lo ca tion are addedto gether.

See Chap ter “Static and Dy namic Analy sis” (page 67) for more in for ma tion.

28 Self-Weight Load


Distributed Span Load

The Dis trib uted Span Load is used to ap ply dis trib uted forces and mo ments onFrame ele ments. The load in ten sity may be uni form or trape zoi dal. The di rec tion of load ing may be speci fied in the global co or di nate sys tem or in the ele ment lo cal co -or di nate sys tem.


Loaded Length

Loads may ap ply to full or par tial ele ment lengths. Mul ti ple loads may be ap plied to a sin gle ele ment. The loaded lengths may over lap, in which case the ap plied loadsare ad di tive.

Distributed Span Load 29


X Y

Z

Global

1

2

3

uz

1

2

3

u2

1

2

3

rz

1

2

3

r2

Global Z Force Global Z Moment

Local 2 Force Local 2 Moment

All loads appliedat rd=0.5

Figure 8Examples of the Definition of Concentrated Span Loads

A loaded length may be speci fied in one of the fol low ing ways:

• Speci fy ing two rela tive dis tances, rda and rdb, meas ured from joint I. Theymust sat isfy 0 1£ < £rda rdb . The rela tive dis tance is the frac tion of ele mentlength;

• Speci fy ing two ab so lute dis tances, da and db, meas ured from joint I. Theymust sat isfy 0 £ < £da db L, where L is the ele ment length;

• Speci fy ing no dis tances, which in di cates the full length of the ele ment.

30 Distributed Span Load


X

Z

Global

All loads applied from rda=0.25 to rdb=0.75

1

2

1

2

1

2

1

2

uz rz

u2 r2

Y

Global Z Force Global Z Moment

Local 2 Force Local 2 Moment

Figure 9Examples of the Definition of Distributed Span Loads

Distributed Span Load 31


AXIS 2

AXIS 1

5

1020

AXIS 3

AXIS 1

55

416

20

AXIS 2

AXIS 14

1016

20

5

10

rda=0.0rdb=0.5u2a=–5u2b=–5

da=4db=16u3a=5u3b=5

da=4db=10u2a=5u2b=5

da=10db=16u2a=10u2b=10

da=0db=4u3a=0u3b=5

da=16db=20u3a=5u3b=0

Figure 10Examples of Distributed Span Loads

Load Intensity

The load in ten sity is a force or mo ment per unit of length. For each force or mo ment com po nent to be ap plied, a sin gle load value may be given if the load is uni formlydis trib uted. Two load val ues are needed if the load in ten sity var ies line arly over itsrange of ap pli ca tion (a trape zoi dal load).

See Figure 9 (page 30) and Figure 10 (page 31).

Internal Force Output

The Frame ele ment in ter nal forces are the forces and mo ments that re sult from in -te grat ing the stresses over an ele ment cross sec tion. These in ter nal forces are:

• P, the ax ial force

• V2, the shear force in the 1-2 plane

• V3, the shear force in the 1-3 plane

• T, the ax ial torque

• M2, the bend ing mo ment in the 1-3 plane (about the 2 axis)

• M3, the bend ing mo ment in the 1-2 plane (about the 3 axis)

These in ter nal forces and mo ments are pres ent at every cross sec tion along thelength of the ele ment.

The sign con ven tion is il lus trated in Figure 11 (page 33). Posi tive in ter nal forcesand ax ial torque act ing on a posi tive 1 face are ori ented in the posi tive di rec tion ofthe ele ment lo cal co or di nate axes. Posi tive in ter nal forces and ax ial torque act ingon a nega tive face are ori ented in the nega tive di rec tion of the ele ment lo cal co or di -nate axes. A posi tive 1 face is one whose out ward nor mal (point ing away from ele -ment) is in the posi tive lo cal 1 di rec tion.

Posi tive bend ing mo ments cause com pres sion at the posi tive 2 and 3 faces and ten -sion at the nega tive 2 and 3 faces. The posi tive 2 and 3 faces are those faces in theposi tive lo cal 2 and 3 di rec tions, re spec tively, from the neu tral axis.

The in ter nal forces and mo ments are com puted at equally- spaced out put pointsalong the length of the ele ment. The nseg pa rame ter speci fies the number of equalseg ments (or spaces) along the length of the ele ment be tween the out put points. Forthe de fault value of “2”, out put is pro duced at the two ends and at the mid point ofthe ele ment. See “Ef fect of End Off sets” be low.

32 Internal Force Output


Internal Force Output 33


Positive Axial Force and Torque

Positive Moment and Shearin the 1-2 Plane

Axis 1

Axis 3

Compression Face

Axis 2

M2

Tension Face

Axis 3

M3

V2

Axis 2Axis 1

Axis 2

PT

Axis 1

Axis 3T

P

Positive Moment and Shearin the 1-3 Plane

Tension Face

Compression Face

V2

V3

V3

M3

M2

Figure 11Frame Element Internal Forces and Moments

The Frame ele ment in ter nal forces are com puted for all Load Cases: Lin ear andnon lin ear static, mode shapes, response-spec trum, time-his tory, etc.

It is im por tant to note that the response spectrum re sults are al ways posi tive, andthat the cor re spon dence be tween dif fer ent val ues has been lost.


Effect of End Offsets

When end off sets are pres ent, in ter nal forces and mo ments are out put at the faces of the sup ports and at nseg - 1 equally- spaced points within the clear length of the ele -ment. No out put is pro duced within the length of the end off set, which in cludes thejoint. Out put will only be pro duced at joints I or j when the cor re spond ing end off -set is zero.

See Topic “End Off sets” (page 24) in this chap ter for more in for ma tion.

34 Internal Force Output


C h a p t e r V

The Shell Element

The Shell el e ment is used to model shell, mem brane, and plate be hav ior in pla narand three-di men sional struc tures. The shell el e ment/ob ject is one type of area ob -ject. De pending on the type of sec tion prop er ties you as sign to an area, the ob jectcould also be used to model plane stress/strain and axisymmetric solid be hav ior;these types of be hav ior are not con sid ered in this man ual.

Topics

• Over view

• Joint Con nec tivity



• Sec tion Prop er ties

• Mass

• Self- Weight Load

• Uni form Load

• In ter nal Force and Stress Out put

35

Overview

The Shell el e ment is a three- or four-node for mu la tion that com bines sep a ratemem brane and plate-bend ing be hav ior. The four-joint el e ment does not have to bepla nar.

The mem brane be hav ior uses an isoparametric for mu la tion that in cludestranslational in-plane stiff ness com po nents and a ro ta tional stiff ness com po nent inthe di rec tion nor mal to the plane of the el e ment. See Tay lor and Simo (1985) andIbrahimbegovic and Wil son (1991).

The plate bend ing be hav ior in cludes two-way, out-of-plane, plate ro ta tional stiff -ness com po nents and a translational stiff ness com po nent in the di rec tion nor mal tothe plane of the el e ment. By de fault, a thin-plate (Kirchhoff) for mu la tion is usedthat ne glects trans verse shear ing de for ma tion. Op tionally, you may choose athick-plate (Mindlin/Reissner) for mu la tion which in cludes the ef fects of trans verse shear ing de for ma tion.

Struc tures that can be mod eled with this el e ment in clude:

• Three- dimensional shells, such as tanks and domes

• Plate struc tures, such as floor slabs

• Mem brane struc tures, such as shear walls

For each Shell el e ment in the struc ture, you can choose to model pure mem brane,pure plate, or full shell be hav ior. It is gen er ally rec om mended that you use the fullshell be hav ior un less the en tire struc ture is pla nar and is ad e quately re strained.

Each Shell el e ment has its own lo cal co or di nate sys tem for de fin ing Ma te rial prop -er ties and loads, and for in ter pret ing out put. Each el e ment may be loaded by grav -ity or uni form load in any di rec tion.

A vari able, four-to-eight-point nu mer i cal in te gra tion for mu la tion is used for theShell stiff ness. Stresses and in ter nal forces and mo ments, in the el e ment lo cal co or -di nate sys tem, are eval u ated at the 2-by-2 Gauss in te gra tion points and ex trap o -lated to the joints of the el e ment. An ap prox i mate er ror in the el e ment stresses or in -ter nal forces can be es ti mated from the dif fer ence in val ues cal cu lated from dif fer -ent el e ments at tached to a com mon joint. This will give an in di ca tion of the ac cu -racy of a given fi nite-el e ment ap prox i ma tion and can then be used as the ba sis forthe se lec tion of a new and more ac cu rate fi nite el e ment mesh.

For more in for ma tion and ad di tional fea tures, see Chap ter “The Shell El e ment” inthe CSI Anal y sis Ref er ence Man ual.

36 Overview


Joint Connectivity

Each Shell el e ment may have ei ther of the fol low ing shapes, as shown in Figure 12(page 38):

• Quad ri lat eral, de fined by the four joints j1, j2, j3, and j4.

• Tri an gu lar, de fined by the three joints j1, j2, and j3.

The quad ri lat eral for mu la tion is the more ac cu rate of the two. The tri an gu lar el e -ment is rec om mended for tran si tions only. The stiff ness for mu la tion of thethree-node el e ment is rea son able; how ever, its stress re cov ery is poor. The use ofthe quad ri lat eral el e ment for mesh ing var i ous ge om e tries and tran si tions is il lus -trated in Figure 13 (page 39).

The lo ca tions of the joints should be cho sen to meet the fol low ing geo met ric con di -tions:

• The in side an gle at each cor ner must be less than 180°. Best re sults for thequad ri lat eral will be ob tained when these an gles are near 90°, or at least in therange of 45° to 135°.

• The as pect ra tio of an el e ment should not be too large. For the tri an gle, this isthe ra tio of the lon gest side to the short est side. For the quad ri lat eral, this is thera tio of the lon ger dis tance be tween the mid points of op po site sides to theshorter such dis tance. Best re sults are ob tained for as pect ra tios near unity, or at least less than four. The as pect ra tio should not ex ceed ten.

• For the quad ri lat eral, the four joints need not be coplanar. A small amount oftwist in the el e ment is ac counted for by the pro gram. The an gle be tween thenor mals at the cor ners gives a mea sure of the de gree of twist. The nor mal at acor ner is per pen dic u lar to the two sides that meet at the cor ner. Best re sults areob tained if the larg est an gle be tween any pair of cor ners is less than 30°. Thisan gle should not ex ceed 45°.

These con di tions can usu ally be met with ade quate mesh re fine ment.


Chapter V The Shell Element

38 Joint Connectivity


Axis 1

Axis 1

Axis 3

Axis 3

Axis 2

Axis 2

j1

j1

j2

j2

j3

j3

j4

Face 1

Face 1

Face 3

Face 3

Face 4

Face 2

Face 2

Face 6: Top (+3 face)

Face 5: Bottom (–3 face)

Face 6: Top (+3 face)

Face 5: Bottom (–3 face)

Four-node Quadrilateral Shell Element

Three-node Triangular Shell Element

Figure 12Shell Element Joint Connectivity and Face Definitions



Triangular Region Circular Region

Infinite Region Mesh Transition

Figure 13Mesh Examples Using the Quadrilateral Shell Element

Degrees of Freedom

The Shell el e ment al ways ac ti vates all six de grees of free dom at each of its con -nected joints. When the el e ment is used as a pure mem brane, you must en sure thatre straints or other sup ports are pro vided to the de grees of free dom for nor mal trans -la tion and bend ing ro ta tions. When the el e ment is used as a pure plate, you must en -sure that re straints or other sup ports are pro vided to the de grees of free dom forin-plane trans la tions and the ro ta tion about the nor mal.

The use of the full shell be hav ior (mem brane plus plate) is rec om mended for allthree- dimensional struc tures.

See Topic “De grees of Free dom” (page 52) in Chap ter “Joints and De grees of Free -dom” for more in for ma tion.


Each Shell el e ment has its own el e ment lo cal co or di nate sys tem used to de fineMa te rial prop er ties, loads and out put. The axes of this lo cal sys tem are de noted 1, 2and 3. The first two axes lie in the plane of the el e ment with an ori en ta tion that youspec ify; the third axis is nor mal.

It is im por tant that you clearly un der stand the def i ni tion of the el e ment lo cal 1-2-3co or di nate sys tem and its re la tion ship to the global X-Y-Z co or di nate sys tem. Bothsys tems are right-handed co or di nate sys tems. It is up to you to de fine lo cal sys temswhich sim plify data in put and in ter pre ta tion of re sults.

In most struc tures the def i ni tion of the el e ment lo cal co or di nate sys tem is ex -tremely sim ple us ing the de fault ori en ta tion and the Shell el e ment co or di natean gle. Ad di tional meth ods are avail able.


• See Chap ter “Co or di nate Sys tems” (page 7) for a de scrip tion of the con ceptsand ter mi nol ogy used in this topic.

• See Topic “Ad vanced Lo cal Co or di nate Sys tem” in Chap ter “The Shell El e -ment” in the CSI Anal y sis Ref er ence Man ual.



Normal Axis 3

Lo cal axis 3 is al ways nor mal to the plane of the Shell el e ment. This axis is di rectedto ward you when the path j1-j2-j3 ap pears coun ter-clock wise. For quad ri lat eral el -e ments, the el e ment plane is de fined by the vec tors that con nect the mid points ofthe two pairs of op po site sides.

Default Orientation

The de fault ori en ta tion of the lo cal 1 and 2 axes is de ter mined by the re la tion shipbe tween the lo cal 3 axis and the global Z axis:

• The lo cal 3-2 plane is taken to be ver ti cal, i.e., par al lel to the Z axis

• The lo cal 2 axis is taken to have an up ward (+Z) sense un less the el e ment ishor i zon tal, in which case the lo cal 2 axis is taken to be hor i zon tal along theglobal +Y di rec tion

• The lo cal 1 axis is al ways hori zon tal, i.e., it lies in the X-Y plane

The el e ment is con sid ered to be hor i zon tal if the sine of the an gle be tween the lo cal3 axis and the Z axis is less than 10-3.

The lo cal 2 axis makes the same an gle with the ver ti cal axis as the lo cal 3 axismakes with the hor i zon tal plane. This means that the lo cal 2 axis points ver ti callyup ward for ver ti cal el e ments.

Coordinate Angle

The Shell el e ment co or di nate an gle, ang, is used to de fine el e ment ori en ta tions that are dif fer ent from the de fault ori en ta tion. It is the an gle through which the lo cal 1and 2 axes are ro tated about the pos i tive lo cal 3 axis from the de fault ori en ta tion.The ro ta tion for a pos i tive value of ang ap pears coun ter-clock wise when the lo cal+3 axis is point ing to ward you.

For hori zon tal ele ments, ang is the an gle be tween the lo cal 2 axis and the hori zon tal +Y axis. Oth er wise, ang is the an gle be tween the lo cal 2 axis and the ver ti cal planecon tain ing the lo cal 3 axis. See Figure 14 (page 42) for ex am ples.

Local Coordinate System 41


Section Properties

A Shell Sec tion is a set of ma te rial and geo met ric prop er ties that de scribe thecross- section of one or more Shell ele ments. Sec tions are de fined in de pend ently ofthe Shell ele ments, and are as signed to the area objects.



Z

X

Y

45°

90°

–90°

3

For all elements,Axis 3 points outward,

toward viewer

1

2

1

2

1

2

1

2

3

3

3

Top row: ang = 45°2nd row: ang = 90°3rd row: ang = 0°4th row: ang = –90°

Figure 14The Shell Element Coordinate Angle with Respect to the Default Orientation

Section Type

When de fin ing an area sec tion, you have a choice of three ba sic el e ment types:

• Shell– the sub ject of this chap ter, with translational and ro ta tional de grees offree dom, ca pa ble of sup port ing forces and moments

• Plane (stress or strain) – a two-di men sional solid, with translational de grees offree dom, ca pa ble of sup port ing forces but not mo ments. This el e ment is notcov ered in this manual.

• Asolid – axisymmetric solid, with translational de grees of free dom, ca pa ble ofsup port ing forces but not mo ments. This el e ment is not cov ered in this manual.

For shell sections, you may choose one of the fol low ing sub-types of be hav ior:

• Mem brane – pure mem brane be hav ior; only the in- plane forces and the nor mal(drill ing) mo ment can be sup ported

• Plate – pure plate be hav ior; only the bend ing mo ments and the trans verse forcecan be sup ported

• Shell – full shell be hav ior, a com bi na tion of mem brane and plate be hav ior; allforces and mo ments can be sup ported

It is gen er ally rec om mended that you use the full shell be hav ior un less the en tirestruc ture is pla nar and is ade quately re strained.

Thickness Formulation

Two thick ness for mu la tions are avail able, which de ter mine whether or not trans -verse shearing de for ma tions are in cluded in the plate- bending be hav ior of a plate or shell element:

• The thick-plate (Mindlin/Re iss ner) for mu la tion, which in cludes the ef fects oftrans verse shear de for ma tion

• The thin-plate (Kirch hoff) for mu la tion, which ne glects tran sverse shearingdeformation

Shear ing de for ma tions tend to be im por tant when the thick ness is greater thanabout one- tenth to one- fifth of the span. They can also be quite sig nifi cant in the vi -cin ity of bending- stress con cen tra tions, such as near sud den changes in thick nessor sup port con di tions, and near holes or re- entrant corners.

Even for thin- plate bend ing prob lems where shear ing de for ma tions are truly neg li -gi ble, the thick-plate formulation tends to be more ac cu rate, al though somewhat



stiffer, than the thin- plate for mu la tion. How ever, the ac cu racy of the thick- platefor mu la tion is more sen si tive to large as pect ra tios and mesh dis tor tion than is thethin- plate for mu la tion.

It is gen er ally rec om mended that you use the thick-plate for mu la tion un less you are us ing a dis torted mesh and you know that shear ing de for ma tions will be small, orun less you are try ing to match a theo reti cal thin- plate so lu tion.

The thick ness for mu la tion has no ef fect upon mem brane be hav ior, only uponplate- bending be hav ior.

Material Properties

The ma te rial prop er ties for each Sec tion are speci fied by ref er ence to a previously- defined Ma te rial. The ma te rial prop er ties used by the Shell Sec tion are:

• The modu lus of elas tic ity, e1, and Pois son’s ra tio, u12, to com pute the mem -brane and plate- bending stiff ness;

• The mass den sity (per unit vol ume), m, for com put ing ele ment mass;

• The weight den sity (per unit vol ume), w, for com put ing Self- Weight Load.

Orthotropic prop er ties are also avail able, as dis cussed in the com plete CSI Anal y sisRef er ence Man ual.

Thickness

Each Shell Sec tion has a con stant mem brane thick ness and a con stant bend ingthick ness. The mem brane thick ness, th, is used for cal cu lat ing:

• The mem brane stiff ness for fullshell and pure- membrane Sec tions

• The ele ment vol ume for the ele ment self- weight and mass cal cu la tions

The bend ing thick ness, thb, is use for cal cu lat ing:

• The plate- bending stiff ness for fullshell and pure- plate Sec tions

Nor mally these two thick nesses are the same. How ever, for some ap pli ca tions,such as mod el ing cor ru gated sur faces, the mem brane and plate- bending be hav iorcan not be ade quately rep re sented by a ho mo ge ne ous ma te rial of a sin gle thick ness.For this pur pose, you may spec ify a value of thb that is dif fer ent from th.



Mass

In a dy namic analy sis, the mass of the struc ture is used to com pute in er tial forces.The mass con trib uted by the Shell ele ment is lumped at the ele ment joints. No in er -tial ef fects are con sid ered within the ele ment it self.

The to tal mass of the ele ment is equal to the in te gral over the plane of the ele ment of the mass den sity, m, mul ti plied by the thick ness, th. The to tal mass is ap por tionedto the joints in a man ner that is pro por tional to the di ago nal terms of the con sis tentmass ma trix. See Cook, Malkus, and Ple sha (1989) for more in for ma tion. The to talmass is ap plied to each of the three trans la tional de grees of free dom: UX, UY, andUZ. No mass mo ments of in er tia are com puted for the ro ta tional de grees of free -dom.


• See Sub topic “Thick ness” (page 44) in this chap ter for the defi ni tion of th.


Self-Weight Load

Self- Weight Load can be ap plied in any Load Pat tern to ac ti vate the self- weight ofall ele ments in the model. For a Shell ele ment, the self- weight is a force that is uni -formly dis trib uted over the plane of the ele ment. The mag ni tude of the self- weightis equal to the weight den sity, w, mul ti plied by the thick ness, th.

Self- weight al ways acts down ward, in the global –Z di rec tion. The self- weight may be scaled by a sin gle fac tor that ap plies to the whole struc ture.


• See Topic “Sec tion Prop er ties” (page 42) in this chap ter for the defi ni tions of wand th.


Uniform Load

Uni form Load is used to ap ply uni formly dis trib uted forces to the mid sur faces ofthe Shell ele ments. The di rec tion of the load ing may be speci fied in the global co or -di nate sys tem or in the ele ment lo cal co or di nate sys tem.

Mass 45


Load in ten si ties are given as forces per unit area. Load in ten si ties speci fied in dif -fer ent co or di nate sys tems are con verted to the ele ment lo cal co or di nate sys tem andadded to gether. The to tal force act ing on the ele ment in each lo cal di rec tion is given by the to tal load in ten sity in that di rec tion mul ti plied by the area of the mid sur face.This force is ap por tioned to the joints of the ele ment.


Internal Force and Stress Output

The Shell ele ment stresses are the forces- per- unit- area that act within the vol umeof the ele ment to re sist the load ing. These stresses are:

• In- plane di rect stresses: S11 and S22

• In- plane shear stress: S12

• Trans verse shear stresses: S13 and S23

• Trans verse di rect stress: S33 (al ways as sumed to be zero)

The three in- plane stresses are as sumed to be con stant or to vary line arly throughthe ele ment thick ness.

The two trans verse shear stresses are as sumed to be con stant through the thick ness.The ac tual shear stress dis tri bu tion is para bolic, be ing zero at the top and bot tomsur faces and tak ing a maxi mum or mini mum value at the mid sur face of the ele ment.

The Shell ele ment in ter nal forces (also called stress re sul tants) are the forces and mo ments that re sult from in te grat ing the stresses over the ele ment thick ness. Thesein ter nal forces are:

• Mem brane di rect forces: F11 and F22

• Mem brane shear force: F12

• Plate bend ing mo ments: M11 and M22

• Plate twist ing mo ment: M12

• Plate trans verse shear forces: V13 and V23

It is very im por tant to note that these stress re sul tants are forces and mo ments perunit of in- plane length. They are pres ent at every point on the mid sur face of the ele -ment.

The sign con ven tions for the stresses and in ter nal forces are il lus trated in Figure 15(page 47). Stresses act ing on a posi tive face are ori ented in the posi tive di rec tion of

46 Internal Force and Stress Output


the ele ment lo cal co or di nate axes. Stresses act ing on a nega tive face are ori ented inthe nega tive di rec tion of the ele ment lo cal co or di nate axes. A posi tive face is one

Internal Force and Stress Output 47


j1

j1

j2

j2

j3

j3

j4

j4

F11

F12F22

F-MIN

F-MAX

Axis 1

Axis 1

Axis 2

Axis 2

Forces are per unitof in-plane length

Moments are per unitof in-plane length

M-MAXM-MIN

M12

M11

M22

PLATE BENDING AND TWISTING MOMENTS

Transverse Shear (not shown)

Positive transverse shear forces andstresses acting on positive faces

point toward the viewer

STRESSES AND MEMBRANE FORCES

Stress Sij Has Same Definition as Force Fij

ANGLE

ANGLE

M12

Figure 15Shell Element Stresses and Internal Forces

whose out ward nor mal (point ing away from ele ment) is in the posi tive lo cal 1 or 2di rec tion.

Posi tive in ter nal forces cor re spond to a state of posi tive stress that is con stantthrough the thick ness. Posi tive in ter nal mo ments cor re spond to a state of stress thatvar ies line arly through the thick ness and is posi tive at the bot tom.

The stresses and in ter nal forces are evalu ated at the stan dard 2- by-2 Gauss in te gra -tion points of the ele ment and ex trapo lated to the joints. Al though they are re portedat the joints, the stresses and in ter nal forces ex ist through out the ele ment. SeeCook, Malkus, and Ple sha (1989) for more in for ma tion.

The Shell ele ment stresses and in ter nal forces are com puted for all Load Cases:Lin ear and non lin ear static, mode shapes, re sponse-spec trum, time-his tory, etc.

Prin ci pal val ues and the as so ci ated prin ci pal di rec tions are also com puted for thestatic load cases and mode shapes. The an gle given is meas ured counter- clockwise(when viewed from the top) from the lo cal 1 axis to the di rec tion of the maxi mumprin ci pal value.

It is im por tant to note that the Re sponse Spec trum re sults are al ways posi tive, andthat the cor re spon dence be tween dif fer ent val ues has been lost. For this rea son,prin ci pal val ues are not avail able.


48 Internal Force and Stress Output


C h a p t e r VI

Joints and Degrees of Freedom

The joints play a fun da men tal role in the analy sis of any struc ture. Joints are thepoints of con nec tion be tween the ele ments, and they are the pri mary lo ca tions inthe struc ture at which the dis place ments are known or are to be de ter mined. Thedis place ment com po nents (trans la tions and ro ta tions) at the joints are called the de -grees of free dom.

Topics

• Over view

• Mod el ing Con sid era tions



• Re straints and Re ac tions

• Springs

• Masses

• Force Load

• Ground Dis place ment Load

49

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 ri ety of func tions:

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

• The struc ture is sup ported at the joints us ing restraints and/or springs

• Rigid- body be hav ior and sym me try con di tions 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 ta tional in er tia may be placed at thejoints

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

• Joints are the pri mary lo ca tions in the struc ture at which the dis place ments areknown (the sup ports) or are to be de ter mined

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 “Joint Con straints” (page 63).

Joints in the anal y sis model cor re spond to point ob jects in the struc tural-ob jectmodel. Us ing the SAP2000 graphi cal in ter face, joints (points) are auto mati callycre ated at the ends of each frame ob ject and at the cor ners of each shell object.Joints may also be de fined in de pend ently of any ele ment.

Au to matic mesh ing of frame and shell ob jects will cre ate ad di tional joints cor re -spond ing to any frame and shell el e ments that are created.

Joints may them selves be con sid ered as el e ments. Each joint may have its own lo -cal co or di nate sys tem for de fin ing the de grees of free dom, restraints, 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 or di nate sys tem is used as the lo cal co or di nate sys tem for all joints in themodel.

There are six dis place ment de grees of free dom at every joint — three trans la tionsand three ro ta tions. These dis place ment com po nents are aligned along the lo cal co -or di nate sys tem of each joint.

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

50 Overview


Dis place ments (trans la tions and ro ta tions) are pro duced at every joint. The ex ter nal and in ter nal forces and mo ments act ing on each joint are also pro duced.

For more in for ma tion and ad di tional fea tures:

• See Chap ter “Joint Co or di nates” in the CSI Anal y sis Ref er ence Man ual.

• See Chap ter “Joints and De grees of Free dom” in the CSI Anal y sis Ref er enceMan ual.

• See Chap ter “Con straints and Welds” in the CSI Anal y sis Ref er ence Man ual.

Modeling Considerations

The lo ca tion of the joints and ele ments is criti cal in de ter min ing the ac cu racy of thestruc tural model. Some of the fac tors that you need to con sider when de fin ing theele ments (and hence joints) for the struc ture are:

• The number of ele ments should be suf fi cient 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 ti nu ity:

– Struc tural bounda ries, e.g., cor ners and edges

– Changes in ma te rial prop er ties

– Changes in thick ness and other geo met ric prop er ties

– Sup port points (re straints and springs)

– Points of ap pli ca tion of con cen trated loads, ex cept that Frame ele mentsmay have con cen trated loads ap plied within their spans

• In re gions hav ing large stress gra di ents, i.e., where the stresses are chang ingrap idly, a shell ele ment mesh should be re fined us ing small ele ments andclosely- spaced joints. This may re quire chang ing the mesh af ter one or morepre limi nary anal y ses, which can be done by mod i fy ing the au to mated-meshpa ram e ters for an area ob ject.

• More that one ele ment should be used to model the length of any span forwhich 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 trib uted by the ele ments.

Modeling Considerations 51

Chapter VI Joints and Degrees of Freedom


Each joint has its own joint lo cal co or di nate sys tem used to de fine the de grees offree dom, restraints, prop er ties, and loads at the joint; and for in ter pret ing joint out -put. The axes of the joint lo cal co or di nate sys tem are de noted 1, 2, and 3. By de fault these axes are iden ti cal to the global X, Y, and Z axes, re spec tively. Both sys temsare right- handed co or di nate sys tems.

The de fault lo cal co or di nate sys tem is ad e quate for most sit u a tions. How ever, forcer tain mod el ing pur poses it may be use ful to use dif fer ent lo cal co or di nate sys -tems at some or all of the joints. This is de scribed in Chap ter “Joint De grees ofFree dom” in the CSI Anal y sis Ref er ence Man ual.

For more in for ma tion see Chap ter “Co or di nate Sys tems” (page 7).

Degrees of Freedom

The de flec tion of the struc tural model is gov erned by the dis place ments of thejoints. 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 la tions are de -noted U1, U2, and U3.

• The joint may ro tate about its three lo cal axes. These ro ta tions 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 thejoint. The joint lo cal de grees of free dom are il lus trated in Figure 16 (page 53).

In ad di tion to the regu lar joints de fined as part of your struc tural model, the pro -gram auto mati cally cre ates mas ter joints that gov ern the be hav ior of any Con -straints that you may have de fined. Each mas ter joint has the same six de grees offree dom as do the regu lar joints. See Chap ter “Joint Con straints” (page 63) formore in for ma tion.

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

• Ac tive — the dis place ment is com puted dur ing the analy sis

• Re strained — the dis place ment is speci fied, and the cor re spond ing re ac tion iscom puted dur ing the analy sis

52 Local Coordinate System


• Con strained — the dis place ment is de ter mined from the dis place ments at otherde grees of free dom

• Null — the dis place ment does not af fect the struc ture and is ig nored by theanaly sis

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

These dif fer ent types of de grees of free dom are de scribed in the fol low ing sub top -ics.

Available and Unavailable Degrees of Freedom

You may ex plic itly spec ify the set of 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 to every joint. This de fault should gen er ally be used for all three- dimensionalstruc 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-Z plane: a pla nar truss needs only UX andUZ; a pla nar frame needs only UX, UZ, and RY; and a pla nar grid or flat plateneeds only UY, RX, and RZ.

Degrees of Freedom 53


Joint

U2

U3

U1

R2

R3

R1

Figure 16The Six Displacement Degrees of Freedom in the Joint Local Coordinate System

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 constraints thatare ap plied to the un avail able de grees of free dom are ig nored by the analy sis.

Avail able de grees of free dom may be re strained, con strained, ac tive, or null.

Restrained Degrees of Freedom

If the dis place ment of a joint along any one of its avail able de grees of free dom isknown, such as at a sup port point, that de gree of free dom is re strained. The knownvalue of the dis place ment may be zero or non- zero, and may be dif fer ent in dif fer -ent Load Pat terns. The force along the re strained de gree of free dom that is re quiredto im pose the speci fied re straint dis place ment is called the re ac tion, and is de ter -mined by the analy sis.

Un avail able de grees of free dom are es sen tially re strained. How ever, they are ex -cluded from the analy sis and no re ac tions are com puted, even if they are non- zero.

See Topic “Re straints and Re ac tions” (page 55) in this chap ter for more in for ma -tion.

Constrained Degrees of Freedom

Any joint that is part of a con straint may have one or more of its avail able de greesof free dom con strained. The pro gram auto mati cally cre ates a mas ter joint to gov -ern the be hav ior of each con straint. The dis place ment of a con strained de gree offree dom is then com puted as a lin ear com bi na tion of the dis place ments along thede grees of free dom at the cor re spond ing mas ter joint.

If a con strained de gree of free dom is also re strained, the re straint will ap ply to thewhole set of con strained joints.

See Chap ter “Joint Con straints” (page 63) for more in for ma tion.

Active Degrees of Freedom

All avail able de grees of free dom that are nei ther con strained nor re strained must beei ther ac tive or null. The pro gram will auto mati cally de ter mine the ac tive de greesof free dom as fol lows:

• If any load or stiff ness is ap plied along any trans la tional de gree of free dom at ajoint, then all avail able trans la tional de grees of free dom at that joint are madeac tive un less they are con strained or re strained.



• If any load or stiff ness is ap plied along any ro ta tional de gree of free dom at ajoint, then all avail able ro ta tional de grees of free dom at that joint are made ac -tive un less they are con strained or re strained.

• All de grees of free dom at a mas ter joint that gov ern con strained de grees offree dom are made ac tive.

A joint that is con nected to any frame or shell ele ment will have all of its avail ablede grees of free dom ac ti vated. An ex cep tion is a Frame ele ment with only truss- type stiff ness, which will not ac ti vate ro ta tional de grees of free dom.

Every ac tive de gree of free dom has an as so ci ated equa tion to be solved. If there areN ac tive de grees of free dom in the struc ture, there are N equa tions in the sys tem,and the struc tural stiff ness ma trix is said to be of or der N. The amount of com pu ta -tional ef fort re quired to per form the analy sis in creases with N.

The load act ing along each ac tive de gree of free dom is known (it may be zero). Thecor re spond ing dis place ment will be de ter mined by the analy sis.

If there are ac tive de grees of free dom in the sys tem at which the stiff ness is knownto be zero, such as the out- of- plane trans la tion in a planar- frame, these must ei therbe re strained or made un avail able. Oth er wise, the struc ture is un sta ble and the so lu -tion of the static equa tions will complain.


• See Topic “De grees of Free dom” (page 14) in Chap ter “The Frame Ele ment.”

• See Topic “De grees of Free dom” (page 40) in Chap ter “The Shell Ele ment.”

Null Degrees of Freedom

The avail able de grees of free dom that are not re strained, con strained, or ac tive, arecalled the null de grees of free dom. Be cause they have no load or stiff ness, their dis -place ments and re ac tions are zero, and they have no ef fect on the rest of the struc -ture. The pro gram auto mati cally ex cludes them from the analy sis.

Restraints and Reactions

If the dis place ment of a joint along any of its de grees of free dom has a known value, ei ther zero (e.g., at sup port points) or non-zero (e.g., due to sup port set tle ment), are straint must be ap plied to that de gree of free dom. The known value of the dis -place ment may dif fer from one Load Pat tern to the next, but the de gree of free domis re strained for all Load Pat terns. In other words, it is not pos si ble to have the dis -

Restraints and Reactions 55


place ment known in one Load Pat tern and un known (un re strained) in an other LoadPat tern.

56 Restraints and Reactions


2-D Frame Structure, X-Z plane

3-D Frame Structure

Notes: Joints are indicated with dots: Solid dots indicate moment continuity Open dots indicate hinges All joint local 1-2-3 coordinate systems are identical to the global X-Y-Z coordinate system

Z

X

Global

Global

Z

XYRollers

Hinge

Fixed

SpringSupport

1

2

3

4

5

6

7

8

HingeFixedRoller1 2 3

4 5 6

Joint Restraints 1 U1, U2, U3 2 U3 3 U1, U2, U3, R1, R2, R3 4 None

Joint Restraints All U3, R1, R2 1 U2 2 U1, U2, R3 3 U1, U2

Figure 17Examples of Restraints

Re straints should also be ap plied to avail able de grees of free dom in the sys tem atwhich the stiff ness is known to be zero, such as the out- of- plane trans la tion and in- plane ro ta tions of a planar- frame. Oth er wise, the struc ture is un sta ble and the so lu -tion of the static equa tions will complain.

The force or mo ment along the de gree of free dom that is re quired to en force the re -straint is called the re ac tion, and it is de ter mined by the anal y sis. The re ac tion maydif fer from one Load Pat tern to the next. The re ac tion in cludes the forces (or mo -ments) from all el e ments con nected to the re strained de gree of free dom, as well asall loads ap plied to the de gree of free dom.

Ex am ples of Re straints are shown in Figure 17 (page 56).


• See Topic “De grees of Free dom” (page 52) in this chap ter.

• See Topic “Ground Dis place ment Load” (page 59) in this chap ter.

Springs

Any of the six de grees of free dom at any of the joints in the struc ture can have trans -la tional or ro ta tional spring sup port con di tions. These springs elas ti cally con nectthe joint to the ground. Spring sup ports along re strained de grees of free dom do notcon trib ute to the stiff ness of the struc ture.

The spring forces that act on a joint are re lated to the dis place ments of that joint by a 6x6 sym met ric ma trix of spring stiff ness co ef fi cients. These forces tend to op posethe dis place ments. Spring stiff ness co ef fi cients are speci fied in the joint lo cal co or -di nate sys tem. The spring forces and mo ments F1, F2, F3, M1, M2 and M3 at a jointare given by:

(Eqn. 1)F

F

F

M

M

M

1

2

3

1

2

3

ì

í

ïïï

î

ïïï

ü

ý

ïïï

þ

ïïï

= -

u1 0 0 0 0 0

u2 0 0 0 0

u3 0 0 0

r1 0 0

sym. r2 0

r3

é

ë

êêêêêêê

ù

û

úúúúúúú

ì

í

ïïï

î

ï

u

u

u

r

r

r

1

2

3

1

2

3

ïï

ü

ý

ïïï

þ

ïïï

where u1, u2, u3, r1, r2 and r3 are the joint dis place ments and ro ta tions, and theterms u1, u2, u3, r1, r2, and r3 are the speci fied spring stiff ness co ef fi cients.

Springs 57


The dis place ment of the grounded end of the spring may be spec i fied to be zero ornon-zero (e.g., due to sup port set tle ment). This ground dis place ment may varyfrom one Load Pat tern to the next.



• See Topic “Ground Dis place ment Load” (page 59) in this chap ter.

Masses

In a dy namic analy sis, the mass of the struc ture is used to com pute in er tial forces.Nor mally, the mass is ob tained from the ele ments us ing the mass den sity of the ma -te rial and the vol ume of the ele ment. This auto mati cally pro duces lumped (un cou -pled) masses at the joints. The ele ment mass val ues are equal for each of the threetrans la tional de grees of free dom. No mass mo ments of in er tia are pro duced for thero ta tional de grees of free dom. This ap proach is ade quate for most analy ses.

It is of ten nec es sary to place ad di tional con cen trated masses and/or mass mo mentsof in er tia at the joints. These can be ap plied to any of the six de grees of free dom atany of the joints in the struc ture.

For com pu ta tional ef fi ciency and so lu tion ac cu racy, SAP2000 al ways uses lumpedmasses. This means that there is no mass cou pling be tween de grees of free dom at ajoint or be tween dif fer ent joints. These un cou pled masses are al ways re ferred to the lo cal co or di nate sys tem of each joint. Mass val ues along re strained de grees of free -dom are ig nored.

In er tial forces act ing on the joints are re lated to the ac cel era tions at the joints by a6x6 ma trix of mass val ues. These forces tend to op pose the ac cel era tions. In a jointlo cal co or di nate sys tem, the in er tia forces and mo ments F1, F2, F3, M1, M2 and M3

at a joint are given by:

F

F

F

M

M

M

1

2

3

1

2

3

ì

í

ïïï

î

ïïï

ü

ý

ïïï

þ

ïïï

= -

u1 0 0 0 0 0

u2 0 0 0 0

u3 0 0 0

r1 0 0

sym. r2 0

r3

é

ë

êêêêêêê

ù

û

úúúúúúú

&&

&&

&&

&&

&&

u

u

u

r

r

1

2

3

1

2

3&&r

ì

í

ïïï

î

ïïï

ü

ý

ïïï

þ

ïïï

58 Masses


where &&u1, &&u2 , &&u3 , &&r1, &&r2 and &&r3 are the trans la tional and ro ta tional ac cel era tions at thejoint, and the terms u1, u2, u3, r1, r2, and r3 are the speci fied mass val ues.

Mass val ues must be given in con sis tent mass units (W/g) and mass mo ments of in -er tia must be in WL2/g units. Here W is weight, L is length, and g is the ac cel era tiondue to grav ity. The net mass val ues at each joint in the struc ture should be zero orposi tive.

See Figure 18 (page 60) for mass mo ment of in er tia for mu la tions for vari ous pla narcon figu ra tions.




Force Load

Force load is used to ap ply con cen trated forces and mo ments at the joints. Valuesare spec i fied in global co or di nates as shown in Figure 19 (page 61). The force loadmay vary from one Load Pat tern to the next.

Forces and mo ments ap plied along re strained de grees of free dom add to the cor re -spond ing re ac tion, but do not oth er wise af fect the struc ture.

For more in for ma tion, see Topic “De grees of Free dom” (page 52) in this chap ter.

Ground Displacement Load

The ground displacement load is used to ap ply speci fied dis place ments (trans la -tions and ro ta tions) at the grounded end of joint restraints and spring sup ports. Dis -place ment val ues are speci fied in global co or di nates as shown in Figure 19 (page61). These val ues are con verted to joint lo cal co or di nates be fore be ing ap plied tothe joint through the re straints and springs.

Re straints may be con sid ered as rigid con nec tions be tween the joint de grees offree dom and the ground. Springs may be con sid ered as flexi ble con nec tions be -tween the joint de grees of free dom and the ground.

It is very im por tant to un der stand that ground dis place ment load ap plies to theground, and does not af fect the struc ture un less the struc ture is sup ported by re -straints or springs in the di rec tion of loading!

Force Load 59


60 Ground Displacement Load


Mass Moment of Inertia about vertical axis(normal to paper) through center of mass

Line mass:Uniformly distributed mass per unit length

Total mass of line = M (or w/g)

Formula

d

b

c.m.

c.m.

d

c.m.

Y

XX

Y

Triangular diaphragm:Uniformly distributed mass per unit areaTotal mass of diaphragm = M (or w/g)

General diaphragm:

Total mass of diaphragm = M (or w/g)Area of diaphragm = A

Moment of inertia of area about X-X = IX

Circular diaphragm:Uniformly distributed mass per unit areaTotal mass of diaphragm = M (or w/g)

Rectangular diaphragm:Uniformly distributed mass per unit area

Total mass of diaphragm = M (or w/g)

Moment of inertia of area about Y-Y = IY

Shape inplan

Use generaldiaphragm formula

c.m.

XX

Y

Y

If mass is a point mass, MMI = 0o

Axis transformation for a mass:

Uniformly distributed mass per unit area

MMI = cm

MMI = cm

8

12

2Md

2Md

2MMI = MMI + MDcm o

d

c.m.

oD

c.m.

MMIcm = M 2 2b +d( )

12

MMIcm =M( )

A

I +IX Y

Figure 18Formulae for Mass Moments of Inertia

Restraint Displacements

If a par ticu lar joint de gree of free dom is re strained, the dis place ment of the joint isequal to the ground dis place ment along that lo cal de gree of free dom. This ap pliesre gard less of whether or not springs are pres ent.

The ground dis place ment, and hence the joint dis place ment, may vary from oneLoad Pat tern to the next. If no ground displacement load is speci fied for a re strained de gree of free dom, the joint dis place ment is zero for that Load Pat tern.

Com po nents of ground dis place ment that are not along re strained de grees of free -dom do not load the struc ture (ex cept pos si bly through springs). An ex am ple of this is il lus trated in Figure 20 (page 62).

Spring Displacements

The ground dis place ments at a joint are mul ti plied by the spring stiff ness co ef fi -cients to ob tain ef fec tive forces and mo ments that are ap plied to the joint. Springdis place ments ap plied in a di rec tion with no spring stiff ness re sult in zero ap plied

Ground Displacement Load 61


Joint Load Components

Joint

uy

uz

ux

ry

rz

rx

GlobalOrigin

Y

Z

X

Figure 19Specified Values for Force Load and Ground Displacement Load

load. The ground dis place ment, and hence the ap plied forces and mo ments, mayvary from one Load Pat tern to the next.

In a joint lo cal co or di nate sys tem, the ap plied forces and mo ments F1, F2, F3, M1,M2 and M3 at a joint due to ground dis place ments are given by:

(Eqn. 2)F

F

F

M

M

M

1

2

3

1

2

3

ì

í

ïïï

î

ïïï

ü

ý

ïïï

þ

ïïï

= -

u1 0 0 0 0 0

u2 0 0 0 0

u3 0 0 0

r1 0 0

sym. r2 0

r3

é

ë

êêêêêêê

ù

û

úúúúúúú

ì u

u

u

r

r

r

g

g

g

g

g

g

1

2

3

1

2

3

í

ïïï

î

ïïï

ü

ý

ïïï

þ

ïïï

where ug1, ug 2 , ug 3 , rg1, rg 2 and rg 3 are the ground dis place ments and ro ta tions, andthe terms u1, u2, u3, r1, r2, and r3 are the speci fied spring stiff ness co ef fi cients.

The net spring forces and mo ments act ing on the joint are the sum of the forces andmo ments given in Equa tions (1) and (2); note that these are of op po site sign. At are strained de gree of free dom, the joint dis place ment is equal to the ground dis -place ment, and hence the net spring force is zero.


• See Topic “Re straints and Re ac tions” (page 55) in this chap ter.

• See Topic “Springs” (page 57) in this chap ter.

62 Ground Displacement Load


Z

X

1

Global

Local

At the roller support, the vertical component of ground displacement, UZ = –0.8, is imposed upon the joint due to the restraint in the vertical (U3) direction.

The horizontal component of ground displacement, UX = 0.6, has no effect on the joint because there is no restraint in the horizontal (U1) direction.

The U1 displacement will be determined by the analysis.

3

UX = +0.6UZ = –0.8 Ground Displacement

Figure 20Ground Displacement at Restrained and Unrestrained Degrees of Freedom

C h a p t e r VII

Joint Constraints

Con straints are used to en force cer tain types of rigid- body be hav ior, to con nect to -gether dif fer ent parts of the model, and to im pose cer tain types of sym me try con di -tions.

Topics

• Over view

• Dia phragm Con straint

Overview

A con straint con sists of a set of two or more con strained joints. The dis place ments of each pair of joints in the con straint are re lated by con straint equa tions. The typesof be hav ior that can be en forced by con straints are:

• Rigid- body be hav ior, in which the con strained joints trans late and ro tate to -gether as if con nected by rigid links. The types of rigid be hav ior that can bemod eled are:

– Rigid Body: fully rigid for all dis place ments

– Rigid Dia phragm: rigid for mem brane be hav ior in a plane

Overview 63

– Rigid Plate: rigid for plate bend ing in a plane

– Rigid Rod: rigid for ex ten sion along an axis

– Rigid Beam: rigid for beam bend ing on an axis

• Equal- displacement be hav ior, in which the trans la tions and ro ta tions are equalat the con strained joints

• Sym me try and anti- symmetry con di tions

The use of con straints re duces the number of equa tions in the sys tem to be solvedand will usu ally re sult in in creased com pu ta tional ef fi ciency.

Only the diaphragm con straint is de scribed in the chap ter, since it is the most com -monly used type of con straint.

For more in for ma tion and ad di tional fea tures see Chap ter “Con straints and Welds”in the CSI Anal y sis Ref er ence Man ual.

Diaphragm Constraint

A diaphragm constraint causes all of its con strained joints to move to gether as apla nar dia phragm that is rigid against mem brane de for ma tion. Ef fec tively, all con -strained joints are con nected to each other by links that are rigid in the plane, but donot af fect out- of- plane (plate) de for ma tion.

This constraint can be used to:

• Model con crete floors (or concrete- filled decks) in build ing struc tures, whichtypi cally have very high in- plane stiff ness

• Model dia phragms in bridge su per struc tures

The use of the diaphragm constraint for build ing struc tures elimi nates thenumerical- accuracy prob lems cre ated when the large in- plane stiff ness of a floordia phragm is mod eled with mem brane ele ments. It is also very use ful in the lat eral(hori zon tal) dy namic analy sis of build ings, as it re sults in a sig nifi cant re duc tion inthe size of the ei gen value prob lem to be solved. See Figure 21 (page 65) for an il lus -tra tion of a floor dia phragm.

Joint Connectivity

Each Dia phragm Con straint con nects a set of two or more joints to gether. Thejoints may have any ar bi trary lo ca tion in space, but for best re sults all joints should

64 Diaphragm Constraint


lie in the plane of the con straint. Oth er wise, bend ing mo ments may be gen er atedthat are re strained by the Con straint, which un re al is ti cally stiff ens the struc ture.

Plane Definition

The con straint equa tions for each diaphragm constraint are writ ten with re spect to a par ticu lar plane. The lo ca tion of the plane is not im por tant, only its ori en ta tion.

By de fault, the plane is de ter mined auto mati cally by the pro gram from the spa tialdis tri bu tion of the con strained joints. If no unique di rec tion can be found, the hori -

Diaphragm Constraint 65

Chapter VII Joint Constraints

Z

X Y

ConstrainedJoint

ConstrainedJoint

ConstrainedJoint

ConstrainedJoint

Global

Column

Beam

AutomaticMaster Joint

Rigid Floor Slab

EffectiveRigid Links

Figure 21Use of the Diaphragm Constraint to Model a Rigid Floor Slab

zon tal (X-Y) plane is as sumed; this can oc cur if the joints are co in ci dent or co -linear, or if the spa tial dis tri bu tion is more nearly three- dimensional than pla nar.

You may over ride auto matic plane se lec tion by speci fy ing the global axis (X, Y, orZ) that is nor mal to the plane of the con straint. This may be use ful, for ex am ple, tospec ify a hori zon tal plane for a floor with a small step in it.


Each diaphragm constraint has its own lo cal co or di nate sys tem, the axes of whichare de noted 1, 2, and 3. Lo cal axis 3 is al ways nor mal to the plane of the con straint.The pro gram auto mati cally ar bi trar ily chooses the ori en ta tion of axes 1 and 2 in the plane. The ac tual ori en ta tion of the pla nar axes is not im por tant since only the nor -mal di rec tion af fects the con straint equa tions.

Constraint Equations

The con straint equa tions re late the dis place ments at any two con strained joints(sub scripts I and j) in a diaphragm constraint. These equa tions are ex pressed interms of in- plane trans la tions (u1 and u2), the ro ta tion (r3) about the nor mal, and the in- plane co or di nates (x1 and x2), all taken in the constraint lo cal co or di nate sys tem:

u1j = u1i – r3i Dx2

u2j = u2i + r3i Dx1

r3i = r3j

where Dx1 = x1j - x1i and Dx2 = x2j - x2i.

66 Diaphragm Constraint


C h a p t e r VIII

Static and Dynamic Analysis

Static and dy namic analy ses are used to de ter mine the re sponse of the struc ture tovari ous types of load ing. This chap ter de scribes the ba sic types of analy sis avail -able for SAP2000.

Topics

• Over view

• Loads

• Load Cases

• Lin ear Static Anal y sis

• Ei gen vec tor Analy sis

• Ritz- vector Analy sis

• Mo dal Analy sis Re sults

• Response- Spectrum Analy sis

• Response- Spectrum Analy sis Re sults

67

Overview

Many dif fer ent types of analy sis are avail able us ing pro gram SAP2000. These in -clude:

• Lin ear static analy sis

• Mo dal analy sis for vi bra tion modes, us ing ei gen vec tors or Ritz vec tors

• Response- spectrum analy sis for seis mic re sponse

• Other types of lin ear and non lin ear, static and dynamic anal y sis that are be yond the scope of this in tro duc tory man ual

These dif fer ent types of analy ses can all be de fined in the same model, and the re -sults com bined for out put.

For more in for ma tion and ad di tional fea tures see Chap ter “Load Cases” in the CSIAnal y sis Ref er ence Man ual.

Loads

Loads rep re sent ac tions upon the struc ture, such as force, pres sure, sup port dis -place ment, ther mal ef fects, ground ac cel er a tion, and oth ers. You may de finenamed Load Pat terns con tain ing any mix ture of loads on the ob jects. The pro gramau to mat i cally com putes built-in ground ac cel er a tion loads.

In or der to cal cu late any re sponse of the struc ture due to the Load Pat terns, youmust de fine and run Load Cases which spec ify how the Load Pat terns are to be ap -plied (e.g., stat i cally, dy nam i cally, etc.) and how the struc ture is to be an a lyzed(e.g., lin early, nonlinearly, etc.) The same Load Pat tern can be ap plied dif fer entlyin dif fer ent Load Cases.

By de fault, the pro gram cre ates a lin ear static Load Case cor re spond ing to eachLoad Pattern that you de fine.

Load Patterns

You can de fine as many named Load Pat terns as you like. Typically you wouldhave sep a rate Load Pat terns for dead load, live load, wind load, snow load, ther malload, and so on. Loads that need to vary in de pend ently, ei ther for de sign pur posesor be cause of how they are ap plied to the struc ture, should be de fined as sep a rateLoad Pat terns.

68 Overview


Af ter de fin ing a Load Pat tern name, you must as sign spe cific load val ues to the ob -jects as part of that Load Pat tern. Each Load Pat tern may in clude:

• Self-Weight Loads on Frame and/or Shell el e ments

• Con cen trated and Dis trib uted Span Loads on Frame el e ments

• Uni form Loads on Shell el e ments

• Force and/or Ground Dis place ment Loads on Joints

• Other types of loads de scribed in the CSI Anal y sis Ref er ence Man ual

Each ob ject can be sub jected to mul ti ple Load Pat terns.

Acceleration Loads

The pro gram au to mat i cally com putes three Ac cel er a tion Loads that act on thestruc ture due to unit translational ac cel er a tions in each of the three global di rec -tions. They are de ter mined by d’Alembert’s prin ci pal, and are de noted mx, my, andmz. These loads are used for ap ply ing ground ac cel er a tions in re sponse-spec trumanal y ses, and are used as start ing load vec tors for Ritz-vec tor anal y sis.

These loads are com puted for each joint and el e ment and summed over the wholestruc ture. The Ac cel er a tion Loads for the joints are sim ply equal to the neg a tive ofthe joint translational masses in the joint lo cal co or di nate sys tem. These loads aretrans formed to the global co or di nate sys tem.

The Ac cel er a tion Loads for all el e ments are the same in each di rec tion and areequal to the neg a tive of the el e ment mass. No co or di nate trans for ma tions are nec es -sary.

The Ac cel er a tion Loads can be trans formed into any co or di nate sys tem. In theglobal co or di nate sys tem, the Ac cel er a tion Loads along the pos i tive X, Y, and Zaxes are de noted UX, UY, and UZ, re spec tively. In a lo cal co or di nate sys tem de -fined for a re sponse-spec trum anal y sis, the Ac cel er a tion Loads along the pos i tivelo cal 1, 2, and 3 axes are de noted U1, U2, and U3, re spec tively.

Load Cases

Each dif fer ent analy sis per formed is called an Load Case. You as sign a la bel toeach Load Case as part of its defi ni tion. These la bels can be used to cre ate ad di -tional com bi na tions and to con trol out put.

The ba sic types of Load Cases are:

Load Cases 69

Chapter VIII Static and Dynamic Analysis

• Lin ear static anal y sis

• Modal anal y sis

• Re sponse-spec trum anal y sis

You may de fine any number of each dif fer ent type of Load Case for a sin gle model.Other types of Load Cases are also avail able.

By de fault, the pro gram cre ates a lin ear static Load Case for each Load Pat tern thatyou de fine, as well as a sin gle modal Load Case for the first few eigen-modes of thestruc ture.

Lin ear com bi na tions and en ve lopes of the vari ous Load Cases are avail ablethrough the SAP2000 graphi cal in ter face.


• See Topic “Ei gen vec tor Analy sis” (page 71) in this chap ter.

• See Topic “Ritz- vector Analy sis” (page 72) in this chap ter.

• See Topic “Response- Spectrum Analy sis” (page 75) in this chap ter.

Linear Static Analysis

The static analy sis of a struc ture in volves the so lu tion of the sys tem of lin ear equa -tions rep re sented by:

K u r=

where K is the stiff ness ma trix, r is the vec tor of ap plied loads, and u is the vec -tor of re sult ing dis place ments. See Bathe and Wil son (1976).

For each lin ear static Load Case that you de fine, you may spec ify a lin ear com bi na -tion of one or more Load Pat terns and/or Ac cel er a tion Loads to be ap plied in vec tor r .Most com monly, however, you will want to solve a sin gle Load Pat tern in eachlin ear static Load Case, and com bine the re sults later.

Modal Analysis

You may de fine as many modal Load Cases as you wish, al though for most pur -poses one case is enough. For each modal Load Case, you may choose ei thereigenvector or Ritz-vec tor anal y sis.

70 Linear Static Analysis


Eigenvector Analysis

Eigenvector analy sis de ter mines the un damped free- vibration mode shapes and fre -quen cies of the sys tem. These natu ral Modes pro vide an ex cel lent in sight into thebe hav ior of the struc ture. They can also be used as the ba sis for response- spectrumanaly ses, al though Ritz vec tors are rec om mended for this pur pose.

Ei gen vec tor analy sis in volves the so lu tion of the gen er al ized ei gen value prob lem:

[ ]K M 0- =W F2

where K is the stiff ness ma trix, M is the di ago nal mass ma trix, W2 is the di ago nalma trix of ei gen val ues, and F is the ma trix of cor re spond ing ei gen vec tors (modeshapes).

Each eigenvalue- eigenvector pair is called a natu ral Vi bra tion Mode of the struc -ture. The Modes are iden ti fied by num bers from 1 to n in the or der in which themodes are found by the pro gram.

The ei gen value is the square of the cir cu lar fre quency, w, for that Mode. The cy clicfre quency, f, and pe riod, T, of the Mode are re lated to w by:

Tf

f= =1

2and

w

p

You may spec ify the number of Modes, n, to be found. The pro gram will seek the nlowest- frequency (longest- period) Modes.

The number of Modes ac tu ally found, n, is lim ited by:

• The number of Modes re quested, n

• The number of mass de grees of free dom in the model

A mass de gree of free dom is any ac tive de gree of free dom that pos sesses trans la -tional mass or ro ta tional mass mo ment of in er tia. The mass may have been as signed di rectly to the joint or may come from con nected ele ments.

Only the Modes that are ac tu ally found will be avail able for any sub se quentresponse- spectrum analy sis proc ess ing.

See Topic “De grees of Free dom” (page 52) in Chap ter “Joints and De grees of Free -dom.”

Modal Analysis 71


Ritz-vector Analysis

Re search has in di cated that the nat u ral free-vi bra tion mode shapes are not the bestba sis for a mode-su per po si tion anal y sis of struc tures sub jected to dy namic loads. Ithas been dem on strated (Wil son, Yuan, and Dick ens, 1982) that dy namic anal y sesbased on a spe cial set of load-de pend ent Ritz vec tors yield more ac cu rate re sultsthan the use of the same num ber of nat u ral mode shapes.

The rea son the Ritz vec tors yield ex cel lent re sults is that they are gen er ated by tak -ing into ac count the spa tial dis tri bu tion of the dy namic load ing, whereas the di rectuse of the natu ral mode shapes ne glects this very im por tant in for ma tion.

The spa tial dis tri bu tion of the dy namic load vec tor serves as a start ing load vec torto ini ti ate the pro ce dure. The first Ritz vec tor is the static dis place ment vec tor cor -re spond ing to the start ing load vec tor. The re main ing vec tors are gen er ated from are cur rence re la tion ship in which the mass ma trix is mul ti plied by the pre vi ously ob -tained Ritz vec tor and used as the load vec tor for the next static so lu tion. Each static so lu tion is called a gen era tion cy cle.

When the dy namic load is made up of sev eral in de pend ent spa tial dis tri bu tions,each of these may serve as a start ing load vec tor to gen er ate a set of Ritz vec tors.Each gen era tion cy cle cre ates as many Ritz vec tors as there are start ing load vec -tors. If a gen er ated Ritz vec tor is re dun dant or does not ex cite any mass de grees offree dom, it is dis carded and the cor re spond ing start ing load vec tor is re moved fromall sub se quent gen era tion cy cles.

For seis mic anal y sis, in clud ing re sponse-spec trum anal y sis, you should use thethree ac cel era tion loads as the start ing load vec tors. This pro duces bet ter response- spectrum re sults than us ing the same number of ei gen Modes.

Stan dard ei gen so lu tion tech niques are used to or thogonal ize the set of gen er atedRitz vec tors, re sult ing in a fi nal set of Ritz- vector Modes. Each Ritz- vector Modecon sists of a mode shape and fre quency. The full set of Ritz- vector Modes can beused as a ba sis to rep re sent the dy namic dis place ment of the struc ture.

Once the stiff ness ma trix is tri an gu lar ized it is only nec es sary to stati cally solve forone load vec tor for each Ritz vec tor re quired. This re sults in an ex tremely ef fi ciental go rithm. Also, the method auto mati cally in cludes the ad van tages of the provennu meri cal tech niques of static con den sa tion, Guyan re duc tion, and static cor rec tion due to higher- mode trun ca tion.

The al go rithm is de tailed in Wil son (1985).

72 Modal Analysis


When a suf fi cient number of Ritz- vector Modes have been found, some of themmay closely ap proxi mate natu ral mode shapes and fre quen cies. In gen eral, how -ever, Ritz- vector Modes do not rep re sent the in trin sic char ac ter is tics of the struc -ture in the same way the natu ral eigen- Modes do. The Ritz- vector Modes are bi ased by the start ing load vec tors.

You may spec ify the to tal number of Modes, n, to be found. The to tal number ofModes ac tu ally found, n, is lim ited by:

• The number of Modes re quested, n

• The number of mass de grees of free dom pres ent in the model

• The number of natu ral free- vibration modes that are ex cited by the start ing load vec tors (some ad di tional natu ral modes may creep in due to nu meri cal noise)

A mass de gree of free dom is any ac tive de gree of free dom that pos sesses trans la -tional mass or ro ta tional mass mo ment of in er tia. The mass may have been as signed di rectly to the joint or may come from con nected ele ments.

Only the Modes that are ac tu ally found will be avail able for any sub se quentresponse- spectrum analy sis proc ess ing.

See Topic “De grees of Free dom” (page 52) in Chap ter “Joint De grees of Free dom.”

Modal Analysis Results

Each modal Load Case re sults in a set of modes. Each mode con sists of a modeshape (nor mal ized de flected shape) and a set of modal prop er ties. These are avail -able for dis play and print ing from the SAP2000 graphi cal in ter face. This in for ma -tion is the same re gard less of whether you use ei gen vec tor or Ritz- vector analy sis,and is de scribed in the fol low ing sub top ics.

Periods and Frequencies

The fol low ing time- properties are given for each Mode:

• Pe riod, T, in units of time

• Cy clic fre quency, f, in units of cy cles per time; this is the in verse of T

• Cir cu lar fre quency, w, in units of ra di ans per time; w = 2 p f

• Ei gen value, w2, in units of radians- per- time squared

Modal Analysis 73


Modal Stiffness and Mass

For each mode shape, only the rel a tive de flec tion val ues are im por tant. The over allscal ing is ar bi trary. In SAP2000, the modes shapes are each nor mal ized, or scaled,with re spect to the mass ma trix such that:

M n n n= =j jT

M 1

This quan tity is called the modal mass. Sim i larly, the modal stiff ness is de fined as:

Kn n n= j jT

K

Re gard less of how the modes are scaled, the ra tio of modal stiff ness to modal massal ways gives the modal eigenvalue:

K

Mn

n

n= w2

Participation Factors

The mo dal par tici pa tion fac tors are the dot prod ucts of the three Ac cel era tionLoads with the modes shapes. The par tici pa tion fac tors for Mode n cor re spond ingto Ac cel era tion Loads in the global X, Y, and Z di rec tions are given by:

f xn n x= jT

m

f yn n y= jT

m

f zn n z= jT

m

where jn is the mode shape and mx, my, and, mz are the unit Ac cel era tion Loads.

These fac tors are the gen er al ized loads act ing on the Mode due to each of the Ac cel -era tion Loads. They are re ferred to the global co or di nate sys tem.

These participation fac tors in di cate how strongly each mode is ex cited by the re -spec tive ac cel er a tion loads.

Participating Mass Ratios

The par tici pat ing mass ra tio for a Mode pro vides a rel a tive mea sure of how im por -tant the Mode is for com put ing the re sponse to the Ac cel era tion Loads in each ofthe three global di rec tions. Thus it is use ful for de ter min ing the ac cu racy ofresponse- spectrum analy ses.

74 Modal Analysis


The par tici pat ing mass ra tios for Mode n cor re spond ing to Ac cel era tion Loads inthe global X, Y, and Z di rec tions are given by:

pf

Mxn

xn

x

=( )2

pf

Myn

yn

y

=( )2

pf

Mzn

zn

z

=( )2

where fxn, fyn, and fzn are the par tici pa tion fac tors de fined in the pre vi ous sub topic;and Mx, My, and Mz are the to tal un re strained masses act ing in the X, Y, and Z di rec -tions. The par tici pat ing mass ra tios are ex pressed as per cent ages.

The cu mu la tive sums of the par tici pat ing mass ra tios for all Modes up to Mode nare printed with the in di vid ual val ues for Mode n. This pro vides a sim ple meas ureof how many Modes are re quired to achieve a given level of ac cu racy for ground- acceleration load ing.

If all eigen- Modes of the struc ture are pres ent, the par tici pat ing mass ra tio for eachof the three Ac cel era tion Loads should gen er ally be 100%. How ever, this may notbe the case in the pres ence of cer tain types of Con straints where sym me try con di -tions pre vent some of the mass from re spond ing to trans la tional ac cel era tions.

Response-Spectrum Analysis

The dy namic equi lib rium equa tions as so ci ated with the re sponse of a struc ture toground mo tion are given by:

K u C u M u m m m( ) & ( ) &&( ) && ( ) && ( ) &&t t t u t u t ux gx y gy z gz+ + = + + ( )t

where K is the stiff ness ma trix; C is the pro por tional damp ing ma trix; M is the di -ago nal mass ma trix; u, &u, and &&u are the rela tive dis place ments, ve loci ties, and ac cel -era tions with re spect to the ground; mx, my, and mz are the unit Ac cel era tion Loads;and &&ugx , &&ugy , and &&ugz are the com po nents of uni form ground ac cel era tion.

Re sponse-spec trum anal y sis seeks the likely max i mum re sponse to these equa tionsrather than the full time his tory. The earth quake ground ac cel er a tion in each di rec -tion is given as a dig i tized re sponse-spec trum curve of pseudo-spec tral ac cel er a tion re sponse ver sus pe riod of the struc ture.

Response-Spectrum Analysis 75


Even though ac cel er a tions may be spec i fied in three di rec tions, only a sin gle, pos i -tive re sult is pro duced for each re sponse quan tity. The re sponse quan ti ties in cludedis place ments, forces, and stresses. Each com puted re sult rep re sents a sta tis ti calmea sure of the likely max i mum mag ni tude for that re sponse quan tity. The ac tualre sponse can be ex pected to vary within a range from this pos i tive value to its neg a -tive.

No cor re spon dence be tween two dif fer ent re sponse quan ti ties is avail able. No in -for ma tion is avail able as to when this ex treme value oc curs dur ing the seis mic load -ing, or as to what the val ues of other re sponse quan ti ties are at that time.

Re sponse-spec trum anal y sis is per formed us ing mode su per po si tion (Wil son andBut ton, 1982). Modes may have been com puted us ing eigenvector anal y sis orRitz-vec tor anal y sis. Ritz vec tors are rec om mended since they give more ac cu ratere sults for the same num ber of Modes. You must de fine a Modal Load Case thatcom putes the modes, and then re fer to that Modal Load Case in the def i ni tion of theRe sponse-Spec trum Case.

Re sponse-spec trum can con sider high-fre quency rigid re sponse if re quested and ifap pro pri ate modes have been com puted. When eigen modes are used, you shouldre quest that static cor rec tion vec tors be com puted. This in for ma tion is au to mat i -cally avail able in Ritz modes gen er ated for ground ac cel er a tion. In ei ther case, youmust be sure to have suf fi cient dy nam i cal modes be low the rigid fre quency of theground mo tion.

Any number of response- spectrum analy ses can be de fined in a sin gle model. Youas sign a unique la bel to each re sponse-spec trum Load Case. Each case can dif fer inthe ac cel era tion spec tra ap plied, the modal Load Case used to gen er ate the modes,and in the way that re sults are com bined.

The fol low ing sub top ics de scribe in more de tail the pa rame ters that you use to de -fine each case.


Each re sponse-spec trum case has its own response- spectrum lo cal co or di natesys tem used to de fine the di rec tions of ground ac cel era tion load ing. The axes ofthis lo cal sys tem are de noted 1, 2, and 3. By de fault these cor re spond to the globalX, Y, and Z di rec tions, re spec tively.

You may change the ori en ta tion of the lo cal co or di nate sys tem by speci fy ing a co -or di nate an gle, ang (the de fault is zero). The lo cal 3 axis is al ways the same as thever ti cal global Z axis. The lo cal 1 and 2 axes co in cide with the X and Y axes if an gle

76 Response-Spectrum Analysis


ang is zero. Oth er wise, ang is the an gle in the hori zon tal plane from the global Xaxis to the lo cal 1 axis, meas ured coun ter clock wise when viewed from above. Thisis il lus trated in Figure 22 (page 77).

Response-Spectrum Functions

A Response- spectrum Func tion is a se ries of dig it ized pairs of structural- period and cor re spond ing pseudo- spectral ac cel era tion val ues. You may de fine any number ofFunc tions, as sign ing each one a unique la bel. You may scale the ac cel era tion val -ues when ever the Func tion is used.

Spec ify the pairs of pe riod and ac cel era tion val ues as:

t0, f0, t1, f1, t2, f2, ..., tn, fn

where n + 1 is the number of pairs of val ues given. All val ues for the pe riod and ac -cel era tion must be zero or posi tive. These pairs must be speci fied in or der of in -creas ing pe riod.

Response-Spectrum Curve

The response- spectrum curve for a given di rec tion is de fined by dig it ized points ofpseudo- spectral ac cel era tion re sponse ver sus pe riod of the struc ture. The shape ofthe curve is given by speci fy ing the name of a Response- spectrum Func tion.



ang

ang

ang

Z, 3

X

1

2

Y

Figure 22Definition of Response Spectrum Local Coordinate System

If no Func tion is speci fied, a con stant func tion of unit ac cel era tion value for allstruc tural pe ri ods is as sumed.

The pseudo spec tral ac cel era tion re sponse of the Func tion may be scaled by the fac -tor sf. Af ter scal ing, the ac cel era tion val ues must be in con sis tent units. See Figure23 (page 78).

The response- spectrum curve cho sen should re flect the damp ing that is pres ent inthe struc ture be ing mod eled. Note that the damp ing is in her ent in the response- spectrum curve it self. It is not af fected by the damp ing ra tio, damp, used for theCQC or GMC method of mo dal com bi na tion, al though nor mally these two damp -ing val ues should be the same.

If the response- spectrum curve is not de fined over a pe riod range large enough tocover the modes cal cu lated in the modal Load Case, the curve is ex tended to largerand smaller pe ri ods us ing a con stant ac cel era tion equal to the value at the near estde fined pe riod.



Pseudo-SpectralAccelerationResponse

Period (time)

10 2 3 4

10

0

20

30

40

Figure 23Digitized Response-Spectrum Curve

Modal Combination

For a given di rec tion of ac cel er a tion, the max i mum dis place ments, forces, andstresses are com puted through out the struc ture for each of the Vi bra tion Modes.These modal val ues for a given re sponse quan tity are com bined to pro duce a sin gle, pos i tive re sult for the given di rec tion of ac cel er a tion us ing one of the fol low ingmeth ods. The method of modal com bi na tion may be af fected by the modal damp -ing ra tio, damp, spec i fied for the re sponse spec trum load case. Given re -sponse-spec trum curves will be ad justed, if nec es sary, to re flect this damp ingvalue.

Pe riodic and Rigid Response

For all modal com bi na tion meth ods ex cept Ab so lute Sum, there are two parts to the modal re sponse for a given di rec tion of load ing: pe ri odic and rigid. The dis tinc tionhere is a prop erty of the load ing, not of the struc ture. Two fre quen cies are de fined,

f1 and f2, which de fine the rigid-re sponse con tent of the ground mo tion, where f1 £f2.

For struc tural modes with fre quen cies less than f1 (lon ger pe ri ods), the re sponse isfully pe ri odic. For struc tural modes with fre quen cies above f2 (shorter pe ri ods), the re sponse is fully rigid. Be tween fre quen cies f1 and f2, the amount of pe ri odic andrigid re sponse is in ter po lated, as de scribed by Gupta (1990).

Fre quen cies f1 and f2 are prop er ties of the seis mic in put, not of the struc ture. Gupta de fines f1 as:

f1 =S

SA

V

max

max2p

where S Amax is the max i mum spec tral ac cel er a tion and SVmax is the max i mum spec -tral ve loc ity for the ground mo tion con sid ered. The de fault value for f1 is unity.

Gupta de fines f2 as:

f2 f1= +1

3

2

3f r

where f r is the rigid fre quency of the seis mic in put, i.e., that fre quency abovewhich the spec tral ac cel er a tion is es sen tially con stant and equal to the value at zerope riod (in fi nite fre quency). Oth ers have de fined f2 as:

f2 = f r



The fol low ing rules ap ply when spec i fy ing f1 and f2:

• If f2 = 0, no rigid re sponse is cal cu lated and all re sponse is pe ri odic, re gard lessof the value spec i fied for f1.

• Oth er wise, the fol low ing con di tion must be sat is fied: 0 £ f1 £ f2.

• Spec i fy ing f1 = 0 is the same as spec i fy ing f1 = f2.

For any given re sponse quan tity (dis place ment, stress, forces, etc.), the pe ri odic re -sponse, R p , is com puted by one of the modal com bi na tion meth ods de scribed be -low. The rigid re sponse, Rr , is al ways com puted as an al ge braic (fully cor re lated)sum of the re sponse from each mode with fre quency above f2, and an in ter po latedpor tion of the re sponse from each mode be tween f1 and f2. The to tal re sponse, R, iscom puted by one of the fol low ing two meth ods:

• SRSS, as rec om mended by Gupta (1990) and NRC (2006), which as sumes thatthese two parts are sta tis ti cally in de pend ent:

R R Rp r= +2 2

• Ab so lute Sum, for com pat i bil ity with older meth ods:

R R Rp r= +

Please note that the choice of us ing the SRSS or Ab so lute Sum for com bin ing pe ri -odic and rigid re sponse is in de pend ent of the modal com bi na tion or the di rec tionalcom bi na tion meth ods de scribed be low.

CQC Method

The Com plete Qua dratic Com bi na tion tech nique for cal cu lat ing the pe ri odic re -sponse is de scribed by Wil son, Der Kiureghian, and Bayo (1981). This is the de -fault method of modal com bi na tion.

The CQC method takes into ac count the sta tis ti cal cou pling be tweenclosely-spaced Modes caused by modal damp ing. In creas ing the modal damp ingin creases the cou pling be tween closely-spaced modes. If the damp ing is zero for allModes, this method de gen er ates to the SRSS method.

GMC Method

The Gen eral Modal Com bi na tion tech nique for cal cu lat ing the pe ri odic re sponse isthe com plete modal com bi na tion pro ce dure de scribed by Equa tion 3.31 in Gupta(1990). The GMC method takes into ac count the sta tis ti cal cou pling be tween



closely-spaced Modes sim i larly to the CQC method, but uses the Rosenblueth cor -re la tion co ef fi cient with the time du ra tion of the strong earth quake mo tion set to in -fin ity. The re sult is es sen tially iden ti cal to the CQC method.

In creas ing the modal damp ing in creases the cou pling be tween closely-spacedmodes. If the damp ing is zero for all Modes, this method de gen er ates to the SRSSmethod.

SRSS Method

This method for cal cu lat ing the pe ri odic re sponse com bines the modal re sults bytak ing the square root of the sum of their squares. This method does not take intoac count any cou pling of the modes, but rather as sumes that the re sponse of themodes are all sta tis ti cally in de pend ent.

Absolute Sum Method

This method com bines the modal re sults by tak ing the sum of their ab so lute val ues.Es sen tially all modes are as sumed to be fully cor re lated. This method is usu allyover-con ser va tive. The dis tinc tion be tween pe ri odic and rigid re sponse is not con -sid ered for this method. All modes are treated equally.

NRC Ten-Percent Method

This tech nique for cal cu lat ing the pe ri odic re sponse is the Ten-Per cent method ofthe U.S. Nu clear Reg u la tory Com mis sion Reg u la tory Guide 1.92.

The Ten-Per cent method as sumes full, pos i tive cou pling be tween all modes whosefre quen cies dif fer from each other by 10% or less of the smaller of the two fre quen -cies. Modal damp ing does not af fect the cou pling.

NRC Dou ble-Sum Method

This tech nique for cal cu lat ing the pe ri odic re sponse is the Dou ble-Sum method ofthe U.S. Nu clear Reg u la tory Com mis sion Reg u la tory Guide 1.92.

The Dou ble-Sum method as sumes a pos i tive cou pling be tween all modes, with cor -re la tion co ef fi cients that de pend upon damp ing in a fash ion sim i lar to the CQC andGMC meth ods, and that also de pend upon the du ra tion of the earth quake. Youspec ify this du ra tion as pa ram e ter td as part of the Load Cases def i ni tion.



Directional Combination

For each dis place ment, force, or stress quan tity in the struc ture, mo dal com bi na tion pro duces a sin gle, posi tive re sult for each di rec tion of ac cel era tion. These di rec -tional val ues for a given re sponse quan tity are com bined to pro duce a sin gle, posi -tive re sult. Two meth ods are avail able for com bin ing the di rec tional re sponse,SRSS and Ab so lute Sum.

SRSS Method

This method com bines the re sponse for dif fer ent di rec tions of loading by tak ing the square root of the sum of their squares:

R R R R= + +12

22

32

where R1, R2 , and R3 are the modal-com bi na tion val ues for each di rec tion. Thismethod is in vari ant with re spect to co or di nate sys tem, i.e., the re sults do not de pend upon your choice of co or di nate sys tem when the given response- spectrum curvesare the same in each direction. This is the rec om mended method for di rec tionalcom bi na tion, and is the de fault.

Absolute Sum Method

This method com bines the re sponse for dif fer ent di rec tions of load ing by tak ing the sum of their ab so lute val ues. A scale fac tor, dirf, is avail able for re duc ing the in ter -ac tion be tween the dif fer ent di rec tions.

Spec ify dirf = 1 for a sim ple ab so lute sum:

R R R R= + +1 2 3

This method is usu ally over-con ser va tive.

Spec ify 0 < dirf < 1 to com bine the di rec tional re sults by the scaled ab so lute summethod. Here, the di rec tional re sults are com bined by tak ing the maxi mum, over all di rec tions, of the sum of the ab so lute val ues of the re sponse in one di rec tion plusdirf times the re sponse in the other di rec tions.

For ex am ple, if dirf = 0.3, the spec tral re sponse, R, for a given dis place ment, force,or stress would be:

R R R R= max ( , , )1 2 3

where:



R R R R1 1 2 30 3= + +. ( )

R R R R2 2 1 30 3= + +. ( )

R R R R3 3 1 20 3= + +. ( )

and R1, R2 , and R3 are the modal- combination val ues for each di rec tion.

The re sults ob tained by this method will vary de pend ing upon the co or di nate sys -tem you choose. Re sults ob tained us ing dirf = 0.3 are com pa ra ble to the SRSSmethod (for equal in put spec tra in each di rec tion), but may be as much as 8% un -con ser va tive or 4% over- conservative, de pend ing upon the co or di nate sys tem.Larger val ues of dirf tend to pro duce more con ser va tive re sults.

Response-Spectrum Analysis Results

Cer tain in for ma tion about each response- spectrum Load Case is avail able for print -ing from the SAP2000 graphi cal in ter face. This in for ma tion is de scribed in the fol -low ing sub top ics.

Damping and Accelerations

The mo dal damp ing and the ground ac cel era tions act ing in each di rec tion are givenfor every Mode. The damp ing value printed for each Mode is just the speci fieddamp ing ra tio, damp, spec i fied for the load case (un less ad vanced damp ing is de -fined for the the model, in which case it will be larger.)

The ac cel era tions printed for each Mode are the ac tual val ues as in ter po lated at themo dal pe riod from the response- spectrum curves af ter scal ing by the speci fiedvalue of sf. The ac cel era tions are al ways re ferred to the lo cal axes of the response- spectrum analy sis. They are iden ti fied in the out put as U1, U2, and U3.

Modal Amplitudes

The re sponse-spec trum modal am pli tudes give the mul ti pli ers of the mode shapesthat con trib ute to the dis placed shape of the struc ture for each di rec tion of Ac cel er -a tion Load. For a given Mode and a given di rec tion of ac cel er a tion, this is the prod -uct of the modal par tic i pa tion fac tor and the re sponse-spec trum ac cel er a tion, di -

vided by the eigenvalue, w2, of the Mode.

This am pli tude, mul ti plied by any modal re sponse quan tity (dis place ment, force,stress, etc.), gives the con tri bu tion of that mode to the value of the same re sponsequan tity re ported for the re sponse-spec trum load case.



The ac cel era tion di rec tions are al ways re ferred to the lo cal axes of the response- spectrum analy sis. They are iden ti fied in the out put as U1, U2, and U3.


• See the pre vi ous Sub topic “Damp ing and Ac cel era tion” for the defi ni tion ofthe response- spectrum ac cel era tions.

• See Topic “Mo dal Analy sis Re sults” (page 73) in this chap ter for the defi ni tionof the mo dal par tici pa tion fac tors and the ei gen val ues.

Base Reactions

The base re ac tions are the to tal forces and mo ments about the global ori gin re quired of the sup ports (re straints and springs) to re sist the in er tia forces due to response- spectrum load ing. These are printed for each in di vid ual Mode af ter per form ingonly di rec tional com bi na tion, and then summed for all Modes af ter per form ing mo -dal com bi na tion and di rec tional com bi na tion.

The re ac tion forces and mo ments are al ways re ferred to the lo cal axes of theresponse- spectrum analy sis. They are iden ti fied in the out put as F1, F2, F3, M1,M2, and M3.



C h a p t e r IX

Bibliography

AASHTO, 2002

Stan dard Spec i fi ca tions for High ways Bridges, 17th Edi tion, The Amer i canAs so ci a tion of State High way and Trans por ta tion Of fi cials, Inc., Wash ing ton,D.C.

AASHTO, 2008

AASHTO LRFD Bridge De sign Spec i fi ca tions, 4th Edi tion 2007, with 2008 In -terim Re vi sions, The Amer i can As so ci a tion of State High way and Trans por ta -tion Of fi cials, Inc., Wash ing ton, D.C.

ACI, 2005

Build ing Code Re quire ments for Struc tural Con crete (ACI 318-05) and Com -men tary (ACI 318R-05), Amer i can Con crete In sti tute, Farmington Hills, Mich.

AISC, 2005

ANSI/AISC 360-05: An Amer i can Na tional Stan dard ¾ Spec i fi ca tion forStruc tural Steel Build ings, Amer i can In sti tute of Steel Con struc tion, Chi cago,Ill.

85

K. J. Bathe, 1982

Fi nite El e ment Pro ce dures in En gi neer ing Anal y sis, Prentice-Hall, Englewood Cliffs, N.J.

K. J. Bathe and E. L. Wil son, 1976

Nu mer i cal Meth ods in Fi nite El e ment Anal y sis, Prentice-Hall, EnglewoodCliffs, N.J.

K. J. Bathe, E. L. Wil son, and F. E. Pe ter son, 1974

SAP IV — A Struc tural Anal y sis Pro gram for Static and Dy namic Re sponse ofLin ear Sys tems, Re port No. EERC 73-11, Earth quake En gi neer ing Re searchCen ter, Uni ver sity of Cal i for nia, Berke ley.

J. L. Batoz and M. B. Tahar, 1982

“Eval u a tion of a New Quad ri lat eral Thin Plate Bend ing El e ment,” In ter na -tional Jour nal for Nu mer i cal Meth ods in En gi neer ing, Vol. 18, pp.1655–1677.

Caltrans, 1995

Bridge De sign Spec i fi ca tions Man ual, as amended to De cem ber 31, 1995,State of Cal i for nia, De part ment of Trans por ta tion, Sac ra mento, Ca lif.

Comite Euro-In ter na tional Du Beton, 1993

CEB-FIP Modal Code, Thomas Tel ford, Lon don

P. C. Roussis and M. C. Constantinou, 2005

Ex per i men tal and An a lyt i cal Stud ies of Struc tures Seis mi cally Iso lated withand Up lift-Re straint Iso la tion Sys tem, Re port No. MCEER-05-0001, MCEER, State Uni ver sity of New York, Buf falo

R. D. Cook, D. S. Malkus, and M. E. Plesha, 1989

Con cepts and Ap pli ca tions of Fi nite El e ment Anal y sis, 3rd Edi tion, John Wiley & Sons, New York, N.Y.

R. D. Cook and W. C. Young, 1985

Ad vanced Me chan ics of Ma te ri als, Macmillan, New York, N.Y.

86


R. K. Dowell, F. S. Seible, and E. L. Wil son, 1998

“Pivot Hysteretic Model for Re in forced Con crete Mem bers,” ACI Struc turalJour nal, Vol. 95, pp. 607–617.

FEMA, 2000

Prestandard and Com men tary for Seis mic Re ha bil i ta tion of Build ings, Pre -pared by the Amer i can So ci ety of Civil En gi neers for the Fed eral Emer gencyMan age ment Agency (Re port No. FEMA-356), Wash ing ton, D.C.

A. K. Gupta, 1990

Re sponse Spec trum Method in Seis mic Anal y sis and De sign of Struc tures,Blackwell Sci en tific Pub li ca tions, Cam bridge, Mass.

J. P. Hollings and E. L. Wil son, 1977

3–9 Node Isoparametric Pla nar or Axisymmetric Fi nite El e ment, Re port No.UC SESM 78-3, Di vi sion of Struc tural En gi neer ing and Struc tural Me chan ics,Uni ver sity of Cal i for nia, Berke ley.

T. J. R. Hughes, 2000

The Fi nite El e ment Method: Lin ear Static and Dy namic Fi nite El e ment Anal y -sis, Do ver.

A. Ibrahimbegovic and E. L. Wil son, 1989

“Sim ple Nu mer i cal Al go rithms for the Mode Su per po si tion Anal y sis of Lin earStruc tural Sys tems with Non-pro por tional Damp ing,” Com put ers and Struc -tures, Vol. 33, No. 2, pp. 523–531.

A. Ibrahimbegovic and E. L. Wil son, 1991

“A Uni fied For mu la tion for Tri an gu lar and Quad ri lat eral Flat Shell Fi nite El e -ments with Six Nodal De grees of Free dom,” Com mu ni ca tions in Ap plied Nu -mer i cal Meth ods, Vol. 7, pp. 1–9.

M. A. Ketchum, 1986

Re dis tri bu tion of Stresses in Seg men tally Erected Pre stressed Con creteBridges, Re port No. UCB/SESM-86/07, De part ment of Civil En gi neer ing,Uni ver sity of Cal i for nia, Berke ley.

87

Chapter IX Bibliography

N. Makris and J. Zhang, 2000

“Time-do main Viscoelastic Anal y sis of Earth Struc tures,” Earth quake En gi -neer ing and Struc tural Dy nam ics, Vol. 29, pp. 745–768.

L. E. Malvern, 1969

In tro duc tion to the Me chan ics of a Con tin u ous Me dium, Prentice-Hall,Englewood Cliffs, N.J.

S. Nagarajaiah, A. M. Reinhorn, and M. C. Constantinou, 1991

3D-Ba sis: Non lin ear Dy namic Anal y sis of Three-Di men sional Base Iso latedStruc tures: Part II, Tech ni cal Re port NCEER-91-0005, Na tional Cen ter forEarth quake En gi neer ing Re search, State Uni ver sity of New York at Buf falo,Buf falo, N. Y.

NRC, 2006

“Com bin ing Modal Re sponses and Spa tial Com po nents in Seis mic Re sponse Anal y sis,” Reg u la tory Guide 1.92, Re vi sion 2, U.S. Nu clear Reg u la toryCom mis sion.

Y. J. Park, Y. K. Wen, and A. H-S. Ang, 1986

“Ran dom Vi bra tion of Hysteretic Sys tems un der Bi-Di rec tional Ground Mo -tions,” Earth quake En gi neer ing and Struc tural Dy nam ics, Vol. 14.

R. J. Roark and W. C. Young, 1975

For mu las for Stress and Strain. 5th Edi tion, McGraw-Hill, New York, N.Y.

T. Takeda, M. A. Sozen, and N. N. Niel sen, 1970

“Re in forced Con crete Re sponse to Sim u lated Earth quakes,” J. Struct. Engrg.Div., ASCE, Vol. 96, No. 12, pp. 2257–2273.

R. L. Tay lor and J. C. Simo, 1985

“Bend ing and Mem brane El e ments for Anal y sis of Thick and Thin Shells,”Pro ceed ings of the NUMEETA 1985 Con fer ence, Swansea, Wales.

K. Terzaghi and R. B. Peck, 1967

Soil Me chan ics in En gi neer ing Prac tice, 2nd Edi tion, John Wiley & Sons,New York, N.Y.

88


S. Timoshenko and S. Woinowsky-Krieger, 1959

The ory of Plates and Shells, 2nd Edi tion, McGraw-Hill, New York, N.Y.

Y. K. Wen, 1976

“Method for Ran dom Vi bra tion of Hysteretic Sys tems,” Jour nal of the En gi -neer ing Me chan ics Di vi sion, ASCE, Vol. 102, No. EM2.

D. W. White and J. F. Hajjar, 1991

“Ap pli ca tion of Sec ond-Or der Elas tic Anal y sis in LRFD: Re search to Prac -tice,” En gi neer ing Jour nal, AISC, Vol. 28, No. 4, pp. 133–148.

E. L. Wil son, 1970

SAP — A Gen eral Struc tural Anal y sis Pro gram, Re port No. UC SESM 70-20,Struc tural En gi neer ing Lab o ra tory, Uni ver sity of Cal i for nia, Berke ley.

E. L. Wil son, 1972

SOLID SAP — A Static Anal y sis Pro gram for Three Di men sional Solid Struc -tures, Re port No. UC SESM 71-19, Struc tural En gi neer ing Lab o ra tory, Uni -ver sity of Cal i for nia, Berke ley.

E. L. Wil son, 1985

“A New Method of Dy namic Anal y sis for Lin ear and Non-Lin ear Sys tems,”Fi nite El e ments in Anal y sis and De sign, Vol. 1, pp. 21–23.

E. L. Wil son, 1993

“An Ef fi cient Com pu ta tional Method for the Base Iso la tion and En ergy Dis si -pa tion Anal y sis of Struc tural Sys tems,” ATC17-1, Pro ceed ings of the Sem i naron Seis mic Iso la tion, Pas sive En ergy Dis si pa tion, and Ac tive Con trol, Ap pliedTech nol ogy Coun cil, Red wood City, Ca lif.

E. L. Wil son, 1997

Three Di men sional Dy namic Anal y sis of Struc tures with Em pha sis on Earth -quake En gi neer ing, Com put ers and Struc tures, Inc., Berke ley, Ca lif.

E. L. Wil son and M. R. But ton, 1982

“Three Di men sional Dy namic Anal y sis for Multicom ponent Earth quake Spec -tra,” Earth quake En gi neer ing and Struc tural Dy nam ics, Vol. 10.

89

Chapter IX Bibliography

E. L. Wil son, A. Der Kiureghian, and E. P. Bayo, 1981

“A Re place ment for the SRSS Method in Seis mic Anal y sis,” Earth quake En gi -neer ing and Struc tural Dy nam ics, Vol. 9.

E. L. Wil son and I. J. Tetsuji, 1983

“An Eigensolution Strat egy for Large Sys tems,” Com put ers and Struc tures,Vol. 16.

E. L. Wil son, M. W. Yuan, and J. M. Dick ens, 1982

“Dy namic Anal y sis by Di rect Su per po si tion of Ritz Vec tors,” Earth quake En -gi neer ing and Struc tural Dy nam ics, Vol. 10, pp. 813–823.

V. Zayas and S. Low, 1990

“A Sim ple Pen du lum Tech nique for Achiev ing Seis mic Iso la tion,” Earth quake Spec tra, Vol. 6, No. 2.

O. C. Zienkiewicz and R. L. Tay lor, 1989

The Fi nite El e ment Method, 4th Edi tion, Vol. 1, McGraw-Hill, Lon don.

O. C. Zienkiewicz and R. L. Tay lor, 1991

The Fi nite El e ment Method, 4th Edi tion, Vol. 2, McGraw-Hill, Lon don.

90


Index

B

Beamsee Frame el e ment

C

Col umnsee Frame el e ment

Co or di nate di rec tions, 8

Co or di nate sys tems, 7Di a phragm con straint, 9Frame el e ment, 9,14global, 8hor i zon tal di rec tions, 8lo cal, 9over view, 7Shell el e ment, 9up ward di rec tion, 8ver ti cal di rec tion, 8X-Y-Z, 8

F

Frame el e ment, 11clear length, 24con nec tions, 24con nec tiv ity, 13

co or di nate an gle, 15co or di nate sys tem, 14de fault ori en ta tion, 15de grees of free dom, 14end I, 13end J, 13end off sets, 24,27end re leases, 25 - 26in ter nal forces, 25joints, 13lon gi tu di nal axis, 15mass, 27over view, 12sec tion prop er ties

see Frame sec tionsup port faces, 24truss be hav ior, 14ver ti cal, 15

Frame sec tion, 17an gle sec tion, 20area, 17box sec tion, 20chan nel sec tion, 20da ta base file, 20da ta base sec tion, 20dou ble-an gle sec tion, 20gen eral sec tion, 18

91

I sec tion, 20lo cal co or di nate sys tem, 17ma te rial prop er ties, 17mo ment of in er tia, 17pipe sec tion, 20rect an gu lar sec tion, 20shape, 18shear area, 17shear de for ma tion, 17solid cir cu lar sec tion, 20T sec tion, 20tor sional con stant, 17

T

Trusssee Frame el e ment

Ty po graph i cal con ven tions, 2

92


Documents

SAP2000 Basic