7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
1/14
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.
You are encouraged to submit comments, suggestions, and questions to [email protected].
It is unnecessary to print these notes because they will remain available online.
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
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
2/14
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.
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
3/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 3
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:
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
4/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 4
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
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
5/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 5
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.
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
6/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 6
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.
!
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
7/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 7
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)
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
8/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 8
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
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
9/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 9
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:
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
10/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 10
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)
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
11/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 11
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
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
12/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 12
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
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
13/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 13
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
7/30/2019 STRUCTURAL ANALYSIS Matrix Structural Analysis
14/14
Terje Haukaas University of British Columbia www.inrisk.ubc.ca
Matrix Structural Analysis Page 14
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.)