STRUCTURAL ANALYSIS Matrix Structural Analysis

7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis

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This document is part of the notes written by Terje Haukaas and posted at www.inrisk.ubc.ca.The notes are revised without notice and they are provided as is without warranty of any kind.

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Matrix StructuralAnalysis

This document presents the computational stiffness method. In the classicalstiffnessmethod,handcalculationsarecentral.Inthatapproach,thestiffnessmatrix

isestablishedcolumn-by-columnbysettingthestructuraldegreesoffreedomequal

toone,oneatatime.Theloadvectorisalsoestablishedbyhand.Incontrast,the

ultimateobjectiveinthisdocumentiscomputerimplementation.Thus,genericand

programmable procedures are emphasized. In this document, the concept of

transformation matrices between element configurations is central. In another

documentonthefiniteelementmethodthestiffnessmethodisfurthergeneralized

tootherelementtypes.

ConfigurationsThewordconfigurationisusedthroughoutthisdocument.Itreferstoanelementsdegreesoffreedomandtheirdirectionsrelativetoacoordinatesystem.Itmayalso


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Terje Haukaas University of British Columbia www.inrisk.ubc.ca

Matrix Structural Analysis Page 2

describetheentirestructuresdegreesoffreedom.Fiveelementconfigurationsare

identifiedinthisdocument:

Basic:InthisconfigurationtheelementhastheminimumnumberofDOFstodescribeanyelementdeformationbutnotrigid-bodymotion

Local:ThisconfigurationhasenoughDOFstofullydescribedeformationand

rigid-bodymotion,buttheDOFsareinthelocalcoordinatesystem

Global:ThiselementconfigurationisthesameastheLocal,buttheDOFsarenowalignedwiththeglobalcoordinatesystem

All: This is a structural configuration, in which absolutely all DOFsof thestructureareincluded,eventhoseassociatedwithboundaryconditions

Final: In this structural configuration the boundary conditions, aswell asotherrestraintsanddependenciesareintroduced

AnalysisProcedure

Theanalysisprocedureofthecomputationalstiffnessmethodis:1. kb:Setupthestiffnessmatrixinthebasicconfiguration2. Kf:Assemblethefinalstiffnessmatrixforthestructurebythetransformation

matricesthataredescribedshortly

3. Fl:Iftherearedistributedelementloadsthensetuptheloadvectorinthebasicorlocalconfiguration,whicheverismoreconvenient.Asintheclassical

stiffnessmethod,thefixed-endforcesarefirstcomputedandthenmultiplied

by(-1)toobtaintheloadvectortobeusedintheanalysis.

4. Ff:Iftherearedistributedelementloadsthenassemblethefinalloadvectorfor the structure from Fb or Fl by the transformation matrices that are

describedshortly.ForconcentratednodalloadsinthedirectionoftheDOFs,

Ff is established by inserting the nodal loads in the correct position in Ffwithoutmodificationofthesign.

5. uf:SolvethesystemofequationsKfuf=Ff6. ub: Calculate the deformations in the basic element configuration by

employingthetransformationmatricesthataredescribedshortly7. Fb: Calculate the forces in the basic element configuration by the basic

stiffnessrelationship,whichyieldsthehomogeneoussolutionofthesection

forcediagrams

8. Add the particular solution to the section force diagrams, i.e., thecontribution from distributed element loads. Contrary to the moment

distributionmethod and the slope-deflectionmethod, the structure is still

considered fully clamped after the stiffness method analysis is completed.

Hence, the particular solution is the section forcediagrams for fixed-fixed

beams.


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TransformationMatricesA transformation matrix defines the relationship between the DOFs in two

configurations.DenotingbyuthevectorofDOFstherelationshipsarewritten

ub= T

blu

l (1)

ul = Tlgug (2)

ug= T

gaua (3)

ua = Tafuf (4)

The subscripts are selected by the first letter of the name of the configuration it

refersto.Inthisway,Tblreferstothetransformationmatrixbetweenthebasicandlocalconfigurations.Noticethatthebasic-mostconfigurationappearson theleft-

handsideinthedisplacementrelationshipsinEqs.(1)to(4).Theprincipleofvirtual

workisinvokedtodeterminetheassociatedforcerelationships.Whenanelement

orthestructuredeformsthenthevirtualworkinanyoftheconfigurationsmustbe

equal.Considerthebasicandlocalconfigurationsasanexample.Bytheprincipleof

virtualdisplacements,equalityofvirtualworkinthetwoconfigurationsrequires:

FgT!ug " Fl

T!ul = 0 (5)

Introducingthetransformationmatrixbetweenthetwoconfigurationsyields:

FgT!ug " Fl

TTlg!ug = Fg

T" Fl

TTlg( )!ug = 0 (6)

Since ug is an arbitrary virtual displacementpattern, the parenthesis in Eq. (6)

mustbezeroforEq.(6)toholdtrue.Consequently,becausetheparenthesisinEq.

(6)isarowvector:

FgT! Fl

TTlg = 0 " Fg !Tlg

TFl = 0 " Fg = Tlg

TFl (7)

Itisthusobservedthattheforcestransformaccordingtothetransposedversionoftheearliertransformationmatrices,withthefinal-mostconfigurationontheleft-

handside.

Next,therelationshipbetweentheforcesanddisplacementsineachconfigurationis

investigated.Initially,thisrelationshipmayseemobvious:inthestiffnessmethod

the forces are related to the displacement by the stiffness matrix. However, the

question is how the stiffness matrix in each configuration is established. First,

assumethatthestiffnessmatrixinthebasicconfigurationisknown.Thisrepresents

themateriallawatthebasiclevel.

CombinedwiththeknownequilibriumandkinematicsrelationshipsinFigure1the

stiffnessmatrixinthelocalconfigurationis:


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Fl= T

bl

TF

b

= Tbl

Tk

bu

b

= Tbl

Tk

bT

bl

kl

!"# $#

ul

(8)

Thatis,thestiffnessmatrixinanaboveconfigurationisobtainedbypre-andpost-

multiplyingthestiffnessmatrixinthebelowconfigurationbythetransformationmatrixbetweenthetwoconfigurations,schematicallywritten:

K = TTkT (9)

Thisissometimesreferredtoasacontragradienttransformation.Noticealsothat

the load vector also transforms according to the transformation matrix. With

reference to the left-hand side of Figure 1, the load vector in an above

configuration is obtained by pre-multiplying the load vector from the below

configuration by the transpose of the transformation matrix between the twoconfigurations.

Figure1:EquilibriumandkinematicsbetweentheBasicandLocalconfigurations.

Figure2providesanoverviewofalltheconfigurationsandmatrixrelationships.In

thefollowingsectionsthetransformationmatricesareexplicitlyestablished,for2D

trussandframeelements.Itturnsoutthatthetransformationmatricesaresetupinamanner similar to the creation of the stiffnessmatrix in the classical stiffness

method;DOFsatsetequaltounity,oneatatime.Forsomeofthetransformationsit

ispossibletoestablishcomputeralgorithmsthatarefarmoreefficientthanusing

transformationmatricesforestablishingthestiffnessmatrixandloadvector.Thisisparticularly the case with the transformation from Global to All, and the

introduction of boundary conditions when going from the All to the Finalconfiguration.While the use of transformation matrices is a pedagogical way to

ubFb Fb=kbub

ub=T

blulFl=T

TFb

ul

Fl

Equilibrium Kinematics

bl

Local

Basic


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establish a consistent methodology, the more efficient algorithms are also

mentioned.

StiffnessMatrixandLoadVectorintheBasicConfiguration

Beforeinvokingthetransformationmatrices,thestartingpointoftheanalysisistoestablishthefundamentalstiffnessmatrixandloadvectorfortheelement.Forthe

frameelementinFigure2thebasicstiffnessmatrixis

kb=

EA

L0 0

04EI

L

2EI

L

02EI

L

4EI

L

!

"

#######

$

%

&&&&&&&

(10)

Theentriesinthismatrixareobtainedbysolvingthedifferentialequationforthe

element, or equivalent methods to determine element deformations. If there aredistributedelementloadsthentheloadvectorisestablishedbyenteringthefixed-

end forces into the basic or local force vector, whichever is more convenient.

Thereafter,thesignofthevectorisflippedandthefinalloadvectorisestablishedbyutilizingthetransformationmatricesthataredescribedinthefollowing.


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Figure2:Overviewofconfigurationsandmatrixrelationships.

Basic-LocalTransformationTheobjectiveinthissectionistoestablishthetransformationmatrixTblinEq.(1).This is accomplished by setting the local DOFs equal to one, one at a time. By

observingtheresultingdeformationsinthebasicconfigurationthisestablishesthe

correspondingrowofTbl.Asanexample,considerthe2DframeelementinFigure2.SettingthefirstlocalDOFequaltounity,andallothersequaltozero,impliesaunit

shorteningoftheelement.ThusthevalueofthefirstDOFinthebasicconfiguration

is-1,whilethebendingDOFsarezero.ThisformsthefirstrowofTbl,whichinits

entiretyreads

ubFbkb

BASIC

ub=TblulFl=TblFb

ulFlkl=TblkbTbl

ul=TlgugFg=TlgFl

ugFg kg=TlgklTlg

ug=TgauaFa=TgaFg

uaFa

ua=TafufFf=TafFa

ufFfKf=TafKaTaf

LOCAL

GLOBAL

ALL

FINAL

T

T

T

T

T

T

T

2

1

3

4

6

5

31

2

3

1

2

4 6

5

1

2

3

4

5

6

7

8

910

11

12

1

2

3

4

5

6

Ka =

Tga

Tk

gT

ga

Num. el.

!


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Tbl=

!1 0 0 1 0 0

0 !1 / L 1 0 1 / L 0

0 !1 / L 0 0 1 / L 1

"

#

$$$

%

&

'''

(11)

Thesecondandfifthcolumnofthismatrixareconfusingatfirst;ithelpstodrawthe

deformed shape of the element in the local configuration and identify the endrotationscomparedwiththestraightlinefromoneelementendtotheother.

Local-GlobalTransformationThe transformationinto the globalcoordinatesystem isa functionoftheelement

orientation.For2Delements,theorientationisrepresentedbytheanglebetweenthe localelement axisandthehorizontal axis.The columnsofthe transformation

matrixareestablishedbysettingtheglobalDOFsequaltounity,oneatatime.Forthe2DframeelementinFigure2theresultis

Tlg =

cos(!) sin(!) 0 0 0 0

"sin(!) cos(!) 0 0 0 0

0 0 1 0 0 0

0 0 0 cos(!) sin(!) 0

0 0 0 "sin(!) cos(!) 0

0 0 0 0 0 1

#

$

%%%%%%%%

&

'

((((((((

(12)

Tlgcanberegardedasamatrixofdirectioncosines,whichisaconceptfromthe

moregeneral fieldofcoordinate transformations,described inthemath-notes on

geometry.Thedirectioncosinesaredefinedas

cx !dx

L, cy !

dy

L, cz !

dz

L (13)

Hence,Tlgcanalsobewritten

Tlg =

cx cy 0 0 0 0

!cy cx 0 0 0 0

0 0 1 0 0 0

0 0 0 cx cy 0

0 0 0 !cy cx

0

0 0 0 0 0 1

"

#

$$$$$$

$$

%

&

''''''

''

(14)

Fora3DframeelementwithsixDOFsateachendthetransformationmatrixfrom

thelocaltotheglobalcoordinatesystemreads

Tlg =R 0

0 R

!

"#

$

%& (15)


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where0isathree-by-threesub-matrixofzerosand Ristherotationmatrixthathas

threeorthogonalvectorsasrows:

R =

x

y

z

!

"

##

#

$

%

&&

&

(16)

wherexisthethree-dimensionalrowvectorthatcontainsthedirectioncosinesof

Eq.(13),whiletherowvectorsyandzareexpressedasthecrossproducts

y = x ! v (17)

z = x ! y (18)

wherev isa user-definedvector thatorientsthe localz-axisoftheelementinthe

globalcoordinatesystem.Asanexample,consideraframeelementthathasthelocalx-axis along the element length. Suppose the end nodes of this element are

positionedintheglobalcoordinatesystemsothattheelementisorientedparallelto

theglobalz-axis.Practically,thismeansthattheelementisaverticalcolumnifthe

globalz-axisdefinestheupwarddirection.Toorientthelocalz-axisoftheelement

alongtheglobalx-directionthevectorvisgivenas{100}.Conversely,toorientthe

localz-axisalongtheglobaly-directionthevector visgivenas{010}.Giventhe

element orientation it is impossible to orient the localz-axis along the globalz-

direction.Whenprovidingthevectorvitissufficienttogiveadirectionintheglobal

systemthatliesinthex-z-planeofthelocalelement.Thatis,thevectors{100}and

{100}wouldworkfineeveniftheelementisnotorientedcompletelyverticalintheglobal system. Importantly, however, all the vectors v, y, and z must have unit

length;xisalreadynormalizedbyitsdefinitionintermsofthedirectioncosines.

Finally,asapracticalinterpretationofR,noticethatthevectorsx,y,andzrepresentorthogonalvectors intheglobalcoordinatesystem.For trussand frameelements

the vector x is aligned with the longitudinal element axis,which is given by the

direction cosines of the element. The y and z vectors are rotated around the

longitudinalelementaxisbymeansoftheauxiliaryvectorvtoorienttheelements

cross-sectionaxesintheglobalcoordinatesystem.

Global-AllTransformationTheobjectiveofthistransformationistolinktheelementDOFswiththestructuralDOFs.The size of the transformationmatrix Tga is (numberof elementDOFs) by

(numberofstructuralDOFs).Clearly,whenthenumberofstructuralDOFsislargethenthecontragradienttransformationinEq.(9)toobtaintheAllstiffnessmatrixis

computationally expensive. Therefore, in addition to establishing Tga, it is an

objective in this section to look for algorithms that are more efficient for

establishingthestiffnessmatrixandloadvectorintheAllconfiguration.

Tgaisestablishedbythesameprocedureasallothertransformationmatrices:one

DOF at a time in the All configuration is set equal to unity. For the structure in


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Figure2,setDOFnumberfourequaltoone.ThisDOFcorrespondstoDOFnumber

onefortheelementbelow.Hence,inthiscasethefourthcolumnofthematrixTga

hasvalueoneinthefirstrow:

Tga=

0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0

!

"

#

######

$

%

&

&&&&&&

(19)

Thepresenceofonlyzerosandonesisageneralcharacteristicofthe Tgamatrix.Itis

relatively straightforward to implement an algorithm that places the ones in the

correctpositioninthematrix.Thisisdoneinthetransformationmatrixassembler

in St. However, the computational cost increases rapidly with the number of

structural DOFs. In the transformation matrix approach, one Tga matrix is

established for each element followedby the followingmatrixmultiplication andsummation:

Ka = Tga

Tk

gT

ga( )Element 1

+ TgaTk

gT

ga( )Element 2

+ TgaTk

gT

ga( )Element 3

+! (20)

Similarly,theloadvectoris

Fa = Tga

TF

g( )Element 1

+ TgaTF

g( )Element 2

+ TgaTF

g( )Element 3

+! (21)

Algorithmsthatarefarmoreefficient,albeitperhapslesspedagogical,exist.After

all,theobjectiveissimplytoentercontributionsfromtheelementstiffnessmatrix

andloadvectorintotheirstructuralcounterparts.Thisisachievedbyenteringsub-

matricesfromkgintoKa,andbyenteringsub-vectorsfromFgintoFa:

Ka=

[ ] [ ]

[ ] [ ]

!

"

######

$

%

&&&&&&

, Fa=

{ }

{ }

'

(

)))

*

)))

+

,

)))

-

)))

(22)

All-FinalTransformation

Fixed boundary conditions are introduced by means of the Taf transformationmatrix. Thedimensions of thismatrix is given by the number of DOFs in theAllconfigurationandthenumberofDOFsinthefinalconfiguration.Theprocedureto

establishTafisnotnew:eachDOFinthefinalconfigurationissetequaltounity,oneatatime.ForthestructureinFigure2,settingthefirstDOFinthefinalconfiguration

equaltoonewiththeotherszeroimpliesthatthefourthDOFintheAllconfiguration

isequaltooneandtheotherzero.Thus,thefirstrowin Tafhasaoneinthefourthposition:


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Taf =

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

!

"

#####

##########

$

%

&&&&&

&&&&&&&&&&

(23)

Thefinalstiffnessmatrixandloadvectorare

Kf = TafT

KaTaf,

Ff = TafT

Fa (24)

Depending on the number of DOFs in the structure, these matrix multiplicationoperations are unnecessarily computationally expensive, albeit pedagogically

appealing.Therefore,insteadofcarryingouttheoperationsinEq.(24),considerthe

effect of these operations. For the stiffness matrix, the row and column that

correspondstothefixedDOFisremoved.Similarlyfortheloadvector,therowthat

correspondstothefixedDOFisremoved.ThisistheeffectofEq.(24),andthiscanbeaccomplishedwithefficientalgorithms.

ShearWallTransformationOne useful application ofmatrix structural analysis is shear wall analysis. Theobjectiveofthistypeofanalysisistodeterminetheforcesonindividualshearwalls

duetoaglobalforce,F,onalateralforceresistingsystemconsistingofcolumnsand

shearwalls. The analysisprocedure isdescribedearlier inthisdocumentand the

stiffnessmatrix for a shearwall element is discussed in the document on frame

elements. Figure 3 shows the plan view of a generic building and one of the

supportingshearwalls.The relationshipbetween theDOFsofthe floor,ufloor,and

theDOFsoftheshearwall,uwall,issoughtfortheanalysis.TheDOFsofthefloorarearbitrarilyselectedtooriginateinthelowerleftcorner.Settingthecomponentsof

ufloor equal to one, one at a time, establishes the columns of the transformation

matrix:

Twf =

1 0 b

0 1 !a

0 0 1

"

#

$$$

%

&

'''

(25)


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Figure3:Rigidfloorwithshearwall.

Theloadvectorfortheshownflooris:

F =

0

F

!c "F

#

$%

&%

'

(%

)% (26)

SpecialTopics

StaticCondensation

Static condensation isa technique that ispossible in linear structural analysis toremoveDOFsfromthesystemofequationswithoutlockingthem.TheDOFsthatare

removed remain free to translate or rotate, but the value of the translation orrotation remains unknown after the system of equations in solved. Static

condensation is particularly useful in the development of super-elements. Asuper-element isone that isinitiallydevelopedwithmanyDOFs,someofwhich

areremovedbystaticcondensation.Usually,onlytheDOFsattheboundaryofthe

element are retained. To understand static condensation, consider the followingsortedsystemofequationsforanelementorastructure

Kii

Kio

Koi

Koo

!

"##

$

%&&

ui

uo

'

(

)

*)

+

,

)

-)=

Fi

Fo

'

(

)

*)

+

,

)

-) (27)

wheresubscriptiidentifiestheDOFsthatarekeptin,whilesubscriptoidentifiesthe

DOFsthataretobetossedout.Next,writeoutthetwosub-systemsofequations

K

iiu

i+K

iou

o= F

i

Koiu

i+K

oou

o= F

o

(28)

b

c

u1,floor

u2,floor

u3,floor

u1,wall

u2,wall

u3,wall u

1,wall

u2,wall

u3,wall

F


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Solveforuointhesecondsub-systemtoobtain:

uo= K

oo

!1F

o!K

oiu

i( ) (29)

Substitutionintothefirstsub-systemyields:

K

ii

ui

+Kio

Koo

!1F

o

!Koi

ui( )( )

= Fi (30)

Re-arrangetoobtainthenewsystemofequationsintheunknownsui:

Kii!K

ioK

oo

!1K

oi( )K

! "### $###

ui= F

i!K

ioK

oo

!1F

o

F

! "## $##

(31)

where the new stiffness matrix, K, and load vector, F, for the super-element or

super-structurewithouttheuoDOFsareidentified.

SettlementsandImposedDeformations

DOFs that experience settlementsand imposeddisplacements are not unknowns.

Rather, it is the forces along those DOFs that are unknown. Consequently, thesystemofequationsisreducedbytheintroductionoftheseeffects.Tounderstand

howthisishandled,considerthesortedsystemofequationsforthestructure

K

iiK

is

Ksi

Kss

!

"

##

$

%

&&

ui

us

'()

*)

+,)

-)=

Fi

Fs

'()

*)

+,)

-) (32)

wheresubscriptiidentifiesthefreeDOFs,whilesubscriptsidentifiestheDOFsthat

aresubjecttosettlementorimposeddisplacementorrotation.Next,writeoutthe

twosub-systemsofequations

Kiiu

i+K

isu

s= F

i

Ksiu

i+K

ssu

s= F

s (33)

The firstof thesetwosub-systemsisreadilysolvedbymovingthesecond termintheleft-handsideovertotheright-handside:

Kiiu

i= F

i!K

isu

s (34)

becauseusisavectorofknowndisplacements.Uponsolvingforui,thesecondsub-

systemimmediatelyprovidesthevalueoftheforcesFsalongtheDOFswithimposed

deformations.

Springs

Springs are concentrated stiffness values along selected DOFs. They typicallyrepresentflexiblefoundations,i.e.,theyaremeanttobeconnectedtoground.There

aretwowaystoaddressconcentratedsprings.Oneapproachistointroduceextra

axialbars,i.e.,trusselementsthatareattachedtothestructureatoneendandtoa

fixednode at the otherend. This isa straightforward approach, inwhichthe bar

stiffnessEA/L istuned tothe desired spring stiffness.Thisapproachalsohas the

advantage that the force in the spring is read from the axial force in the truss

member.Anotherapproachistointroduceaspecialoptiontoallowscalarstiffness


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values to be inserted into the diagonal of the structural stiffness matrix. This is

perhapssimplerfortheanalystbecauseiteliminatestheneedtocreateanauxiliary

nodeandtrusselement.

DOFDependencies

ADOF dependencymeans that oneDOF isequal toor linearlydependentonanother DOF. One example is when the axial deformation of a horizontal frame

element is neglected; then both horizontal end displacements are the same, i.e.,dependent.Anotherexampleisaninclinedrollersupport,inwhichcasethevertical

displacementis linearlyrelated tothehorizontal displacement. It isdangerousto

addressthisproblembysettingamemberstiffness,e.g.,axialstiffness,tosomevery

highnumber.Thismayleadtopoorconditioningofthestiffnessmatrixand,asa

result,inaccuraciesinthesolutionofthesystemofequations.Abetterapproachis

tointroduceDOFdependenciesbyaspecial-purposetransformationmatrix.Forthis

purpose,considerthefollowingrelationshipbetweenDOFsintheAllconfiguration:

u1

u2

u3

!

!

"##

$

##

%

#

'

##

ua

"#$ %$

=

0 ( 00 1 0 &

0 0 1

! '

)

*

++++

,

-

.

.

.

.

Td

" #$$$ %$$$

u1

u2

u3

!

!

"##

$

##

%

#

'

##

ua

"#$ %$

(35)

where the transformation matrix Td (d stands for dependency) states that DOF

numberoneisequaltotimesDOFnumberto,i.e.,

u1=! "u

2 (36)

Inotherwords,u2istheindependentDOFandu1isthedependentDOF.Asusual,

thenewstiffnessmatrixandloadvectorareobtainedby

Ka= T

d

TK

aT

d, F

a= T

d

TF

a (37)

wherethemodifiedstiffnessmatrixhasretaineditsdimensionbuthaszerosinthe

rowandcolumnthatcorrespondtothedependentDOF.Similarly,themodifiedload

vectorhasazeroentryinthecomponentthatcorrespondstothedependentDOF.

Thus, the dependentDOFmust be removed from the system ofequationsby the

transformationfromtheAllconfigurationtotheFinalconfiguration,i.e.,bythe Taftransformationmatrix.

BucklingLoadsandModes

Whenthestiffnessmatrixisamendedwithgeometricstiffnesstermsitispossibletocomputethebucklingloads,andassociateddisplacedshapes,i.e.,modes,ofthe

structure.Therewill be as many buckling loads as there are DOFs, but only the

smallest is relevant in practice because it is the governing buckling load. Thegeometric stiffness matrix for each element type is established in separate

documents, while the procedure to compute the buckling loads and modes is


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presented here. Upon assembling the final structural stiffness matrix, including

geometricstiffnesscontributions,thesystemofequilibriumequationsiswritten

K ! P "KG( )u = F (38)

ThefactorizationoftheaxialforcelevelPasamultiplierofthegeometricstiffness

matrixofthestructure isnoted.This ispossiblewhentheconsideredaxialforcesare fromonesource, such asgravity. It isalso necessary that the formulationbe

basedonthesecond-orderlinearizedtheory.TheuseofexactLivesleyfunctions,

i.e., the exact solutionof the differential equation, prevents the form inEq. (38).

However,whentheforminEq.(38)ispossible,thebucklingloads, Pcr,ofthesystem

canbecomputed.First,removetheotherloadsthatarerepresentedbytheexternal

loadvector,i.e.,setF=0.Next,recognizethattheremainingsystemofequationsisaneigenvalueproblem;itishomogeneouswithnon-trivialsolutionsonlywhenthe

determinantofthecoefficientmatrixiszero.Hence,theequation

det K ! P "KG( ) = 0 (39)

issolvedtoobtainthecriticalvaluesoftheaxialloadlevel,whicharethebuckling

loads,Pcr.Therewillbeasmanybucklingloadsastherearedegreesoffreedom,but

usually only the lowest value is relevant in design. Each buckling load has a

corresponding bucklingmode.While the buckling loads are the eigenvalues, the

bucklingmodesaretheeigenvectors.Themodesrepresentthedisplacedshapeof

thestructurewhenitbucklesatthecorrespondingbucklingload.Theamplitudeof

the deformation isnot uniquelydetermined,butthe shape isobtainedbysetting

onecomponentofuequaltounityandsolvingfortheothers.Softwareapplications

thatsolveeigenvalueproblemsdothisautomatically.

ResponseSensitivityAnalysis

(Notwrittenyet.)

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STRUCTURAL ANALYSIS Matrix Structural Analysis