494999246-Matrix-structural-analysis-Truss.pdf

Matrix analysis

1. MATRIX STRUCTURAL
ANALYSIS

2. STRUCTURAL ANALYSIS
Itistheprocessofpredictingthe
performanceofagivenstructureundera
prescribedloadingcondition.
(a) stresses or stress resultants (i.e., axial forces,
shears, and bending moments);
(b) deflections; and
(c) support reactions

3. SIGNIFICANCE OF
LEARNING THE COURSE
1.ETABS
2.MIDAS GEN
3.STAAD
4.GRASP
Itisthereforeessentialthatstructural
engineersunderstandthebasicprinciplesof
matrixanalysis,sothattheycandeveloptheir
owncomputerprogramsand/orproperlyuse
commercially availablesoftware—and
appreciatethephysicalsignificanceofthe
analyticalresults.

4. Methods of
Structural Analysis
1.ClassicalMethod
2.MatrixMethod
3.FiniteElementMethod

5. Classical versus
Matrix Methods
Classical:
•Intended forhand
calculations
•ofteninvolvecertain
assumptionstoreduce
the amount of
computational effort
requiredforanalysis.
•Getscomplicated
Matrix:
•Systematic,sothatthey
canbeconveniently
programmed.
•General,inthesense
thatthesameoverall
formatoftheanalytical
procedure canbe
appliedtothevarious
types of framed
structures.

6. MATRIX VERSUS:
FiniteElementMethod
i.MatrixanalyzeframedstructuresonlywhereasFEM
hasbeendevelopedtotheextentthatitcanbe
appliedtostructuresandsolidsofpracticallyany
shapeorform.
ii.inmatrixmethods,thememberforce–displacement
relationshipsarebasedontheexactsolutionsofthe
underlyingdifferentialequations,whereasinfiniteel
ementmethods,suchrelationsaregenerally
derivedbywork-energyprinciplesfromassumed
displacementorstressfunctions.

7. MATRIX METHODS|
FLEXIBILITY AND STIFFNESS METHODS
1.Flexibilitymethod(forceorcompatibility
method)
Inthisapproach,theprimaryunknownsarethe
redundantforces,whicharecalculatedfirstby
solvingthestructure’scompatibilityequations.
2.Stiffnessmethod(displacementorequilibrium
method)
Inthisapproach,theprimaryunknownsarethejoint
displacements,whicharedeterminedfirstbysolving
thestructure’sequationsofequilibrium.
*thiscoursewillfocusonstiffnessmethod(directstiffness
method)

8. CLASSIFICATION OF
FRAMED STRUCTURES
•Framedstructuresarecomposedofstraightmembers
whoselengthsaresignificantlylargerthantheircross-
sectionaldimensions.
•Common framedstructurescanbeclassifiedintosix
basiccategoriesbasedonthearrangementoftheir
members,andthetypesofprimarystressesthatmay
developintheirmembersundermajordesignloads.

9. CLASSIFICATION OF
FRAMED STRUCTURES
1.Plane Trusses
2.Beams
3.Plane Frames (Rigid Frames)
4.Space Trusses
5.Grids
6.Space Frames

11. ANALYTICAL MODELS
•Ananalyticalmodelisanidealizedrepresentationofa
realstructureforthepurposeofanalysisrepresentedby
LINEDIAGRAMS
•Inmatrixmethodsofanalysis,astructureismodeledas
anassemblageofstraightmembersconnectedattheir
endstojoints.
1.Amemberisdefinedasapartofthestructureforwhichthe
memberforce-displacementrelationshipstobeusedintheanal
ysisarevalid.
2.Ajointisdefinedasastructuralpartofinfinitesimalsizetowhich
theendsofthemembersareconnected.
*(Infinite-elementterminology,themembersandjointsofstructuresare
generallyreferredtoaselementsandnodes,respectively.)

12. ANALYTICAL MODELS

13. ANALYTICAL MODELS|
LINE DIAGRAMS
•eachmemberisdepictedbya
linecoinciding withits
centroidalaxis
•thememberdimensionsand
thesizeofconnectionsarenot
shown
•rigidjointsareusually
representedbypoints,and
hingedjointsbysmallcircles,at
theintersectionsofmembers.
•thejointsandmembers
ofthestructureare
identifiedbynumbersin
whichthejointnumbers
areenclosedwithincircles
todistinguishthemfrom
themember numbers
e n c l o s e dw i t h i n
rectangles.

14. FUNDAMENTAL
RELATIONSHIPS FOR
STRUCTURAL ANALYSIS
Structuralanalysis,ingeneral,involvestheuseof
threetypesofrelationships:
•Equilibrium equations,
•compatibility conditions, and
•constitutive relations.

15. Equilibrium Equations
Astructureisconsideredtobeinequilibriumif,initiallyat
rest,itremainsatrestwhensubjectedtoasystemof
forcesandcouples.
•Plane (two-dimensional) structure
•Space (three-dimensional) structure

16. Compatibility Conditions
Alsorefferedto
as continuity
conditions,relate
thedeformationsof
astructuresothat
itsvariousparts
(members, joints,
andsupports)fit
togetherwithout
a n yg a p sor
overlaps.

17. Constitutive Relations
•Alsoknownasstress-strainrelations,describethe
relationshipsbetweenthestressesandstrainsofa
structureinaccordancewiththestress-strainproperties
ofthestructuralmaterial
•Itprovideslinkbetweentheequilibriumequationsand
compatibilityconditionsthatisnecessarytoestablish
theload-deformationrelationshipsforastructureora
member

18. MATRIX ALGEBRA

19. DEFINITION OF A MATRIX
Amatrixisdefinedasarectangulararrayofquantities
arrangedinrowsandcolumns.Amatrixwithmrowsandn
columnscanbeexpressedasfollows.

20. TYPES OF MATRICES
1.ColumnMatrix(Vector)
2.RowMatrix
3.SquareMatrix

21. TYPES OF MATRICES
1.ColumnMatrix(Vector)
2.RowMatrix
3.SquareMatrix
4.SymmetricMatrix
5.LowerTriangularMatrix
6.UpperTriangularMatrix

22. TYPES OF MATRICES
1.ColumnMatrix(Vector)
2.RowMatrix
3.SquareMatrix
4.SymmetricMatrix
5.LowerTriangularMatrix
6.UpperTriangularMatrix
7.DiagonalMatrix
8.UnitorIdentityMatrix
9.NullMatrix

23. MATRIX OPERATIONS
1.Equality
2.AdditionandSubtraction
3.MultiplicationbyaScalar
4.MultiplicationofMatrices
5.TransposeofaMatrix
6.InverseofaSquareMatrix
7.OrthogonalMatrix

24. MATRIX OPERATIONS||
Equality
SincebothAandBareoforder3×2,andsinceeach
elementofAisequaltothecorrespondingelementofB,the
matricesAandBareequaltoeachother;thatis,A=B.

25. MATRIX OPERATIONS||
Addition and Subtraction
CalculatethematricesC=A+BandD=A−Bif

26. MATRIX OPERATIONS||
Multiplication by a Scalar
CalculatethematrixB=cAifc=−6and

27. MATRIX OPERATIONS||
Multiplication of Matrices
Twomatricescanbemultipliedonlyifthenumberofcolumns
ofthefirstmatrixequalsthenumberofrowsofthesecond
matrix.
Ex.CalculatetheproductC=ABofthematricesAandB.

28. MATRIX OPERATIONS||
Multiplication of Matrices
Animportantapplicationofmatrixmultiplicationistoexpress
simultaneousequationsincompactmatrixform.Considerthe
followingsystemoflinearsimultaneousequations.
or symbolically
in matrix form:
NOTE:
•Matrixmultiplicationisgenerally
notcommutative.
•Matrix multiplicationis
associativeanddistributive,
providedthatthesequential
orderinwhichthematricesare
tobemultipliedismaintained.

29. CalculatetheproductsABandBAif
AretheproductsABandBAequal?
MATRIX OPERATIONS||
Multiplication of Matrices

30. MATRIX OPERATIONS||
Transpose of a Matrix
Thetransposeofamatrixisobtainedbyinterchangingits
correspondingrowsandcolumns.Thetransposedmatrixis
commonlyidentifiedbyplacingasuperscriptTonthesymbol
oftheoriginalmatrix.
Example:
1.
2.
Thetransposeofa
productofmatrices
equalstheproductofthe
transposedmatricesin
reverseorder

31. MATRIX OPERATIONS||
Inverse of a Square Matrix
TheinverseofasquarematrixAisdefinedasamatrixA
−1
with
elementsofsuchmagnitudesthattheproductoftheoriginal
matrixAanditsinverseA
-1
equalsaunitmatrixI;thatis,
Theoperationofinversionisdefinedonlyforsquare
matrices,withtheinverseofsuchamatrixalsobeinga
squarematrixofthesameorderastheoriginalmatrix.

32. MATRIX OPERATIONS||
Orthogonal Matrix
Iftheinverseofamatrixisequaltoitstranspose,thematrixis
referredtoasanorthogonalmatrix.Inotherwords,amatrixA
isorthogonalif
Example.
DeterminewhethermatrixAgivenbelowisanorthogonal
matrix.

33. EXERCISES
1.Showthat(ABC)
T
=C
T
B
T
A
T
byusingthefollowingmatrices
2.DeterminewhethermatrixBgivenbelowisanorthogonal
matrix.

34. PLANE TRUSSES

35. OUTLINE:
1.Global and Local Coordinate Systems
2.Degrees of Freedom
3.Stiffness method of analysis
1.Member Stiffness Relations (Local Coordinate System)
2.Coordinate Transformations
3.Member Stiffness Relations (Global Coordinate System)
4.Structure Stiffness Relations
5.Procedure for Analysis
Summary
Problems

36. 1. GLOBAL AND LOCAL
COORDINATE SYSTEMS
Global Coordinate System
Theoverallgeometryandtheload–deformation
relationshipsforanentirestructurearedescribedwith
referencetoaCartesianorrectangularglobal
coordinatesystem.
LocalCoordinateSystem
Definedforeachmemberofthestructuretoderivethe
basicmemberforce–displacementrelationshipsinterms
oftheforcesanddisplacementsinthedirectionsalong
andperpendiculartomembers.

37. Theoriginofthelocalxyzcoordinatesystemforamembermaybearbitrarily
locatedatoneoftheendsofthememberinitsundeformedstate,withthexaxis
directedalongthemember’scentroidalaxisintheundeformedstate.The
positivedirectionoftheyaxisisdefinedsothatthecoordinatesystemisright-
handed,withthelocalzaxispointinginthepositivedirectionoftheglobalZaxis.
1. GLOBAL AND LOCAL
COORDINATE SYSTEMS

38. OUTLINE:
1.Global and Local Coordinate Systems
2.Degrees of Freedom
3.Stiffness method of analysis
1.Member Stiffness Relations (Local Coordinate System)
2.Coordinate Transformations
3.Member Stiffness Relations (Global Coordinate System)
4.Structure Stiffness Relations
5.Procedure for Analysis
Summary
Problems

39. 2. DEGREES OF FREEDOM
Thedegreesoffreedomofastructure,ingeneral,are
definedastheindependent jointdisplacements
(translationsandrotations)thatarenecessarytospecify
thedeformedshapeofthestructurewhensubjectedtoan
arbitraryloading.

40. 2. DEGREES OF FREEDOM

41. 2. DEGREES OF FREEDOM
NDOF = NSC (NJ) –NR
Where:
•NDOF=numberofdegreesoffreedomofthestructure(someti
mesreferredtoasthedegreeofkinematicindeterminacyofthe
structure);
•NSC=numberofdegreesoffreedomofafreejoint(alsocalled
thenumberofstructurecoordinatesperjoint);
•NJ=numberofjoints;
•NR=numberofjointdisplacementsrestrainedbysupports.
Forplanetrusses:
NDOF = 2 NJ–NR
NumberofDegreesofFreedom(NDOF)

42. 2. DEGREES OF FREEDOM

43. DOF–representedby
RestrainedCoordinate
1.DOFisnumberedstartingatthelowestnumberedjointthat
hasaDOF,andproceeding sequentiallytothe
highest-numberedjoint.
InthecaseofmorethanoneDOFatajoint,thetranslationintheXdirection
isnumberedfirst,followedbythetranslationintheYdirection.ThefirstDOFis
assignedas“1”,andthelastDOFisassignedanumberequaltoNDOF.
2.OncealltheDOFofthestructurehavebeennumbered,we
numbertherestrainedcoordinatesinasimilarmanner,but
beginwithanumberequaltoNDOF+1.Westartatthelowest-
numberedjointthatisattachedtoasupport,andproceed
sequentiallytothehighest-numberedjoint.
Inthecaseofmorethanonerestrainedcoordinateatajoint,thecoordinate
intheXdirectionisnumberedfirst,followedbythecoordinateintheY
direction.
2. DEGREES OF FREEDOM|
Numbering of DOF & Restrained Coordinates
/

44. Externalloadsappliedtothejointsoftrussesarespecifiedasforce
componentsintheglobalXandYdirections.Anyloadsinitiallygivenin
inclineddirectionsareresolvedintotheirXandYcomponents,before
proceedingwithananalysis.
2. DEGREES OF FREEDOM|
JOINT LOAD VECTOR
inwhichPiscalled
thejointload
vectorofthetruss.

45. 2. DEGREES OF FREEDOM|
REACTIONVECTOR
Asindicated there,the
numbers assignedtothe
restrainedcoordinatesare
usedtoidentifythesupport
reactions.Inotherwords,a
reactioncorrespondingtoan
ithrestrainedcoordinateis
denotedbythesymbolRi.The
threesupportreactionsofthe
trusscanbecollectively
expressedinmatrixformas
inwhichRiscalledthe
reactionvectorofthestructure

46. ACTIVITY 2
CreateanAnalyticModelofthefollowingstructures
shownandidentifynumericallythedegreesoffreedom
andrestrainedcoordinatesofthestructureshown.Also,
formthejointloadvectorPandreactionvectorRforthe
structure.

47. ACTIVITY 2
CreateanAnalyticModelofthe
followingstructuresshownandidentify
numericallythedegreesoffreedomand
restrainedcoordinatesofthestructure
shown.Also,formthejointloadvectorP
andreactionvectorRforthestructure.
1.
2.

48. OUTLINE:
1.Global and Local Coordinate Systems
2.Degrees of Freedom
3.Stiffness method of analysis
1.Member Stiffness Relations (Local Coordinate System)
2.Coordinate Transformations
3.Member Stiffness Relations (Global Coordinate System)
4.Structure Stiffness Relations
5.Procedure for Analysis
Summary
Problems

49. 3. STIFFNESS METHOD OF
ANALYSIS
(A.K.A. DIRECT STIFFNESS METHOD)

50. 3.STIFFNESS METHOD OF ANALYSIS
Inthestiffnessmethodofanalysis,thejointdisplacements,
d,ofastructureduetoanexternalloading,P,are
determinedbysolvingasystemofsimultaneousequations,
expressedintheform
P = Sd
In which S is called the structure stiffness matrix.
Structurestiffnessmatrixisformedbyassemblingthe
stiffnessmatricesforitsindividualmember inglobal
coordinatesystem.

51. 3.STIFFNESS METHOD OF ANALYSIS
GENERALPROCEDURE
Local Member Stiffness Matrix, k
Global Member Stiffness Matrix, K
Structure Member Stiffness Matrix, S
P = Sd
Tansformation Matrix Q = k u
F = K v
Where:
k = member stiffness matrix in the LCS
u = member end displacement vector in the LCS
Q = member end force vector in the LCS (Axial Force)
K = member stiffness matrix in the GCS
v = member end displacement vector in the GCS
F = member end force vector in the GCS
S = Structure stiffness matrix
d = joint displacements
P = external loading

52. GENERALPROCEDURE
Local Member Stiffness Matrix, k
Global Member Stiffness Matrix, K
Structure Member Stiffness Matrix, S
P = Sd
Tansformation Matrix Q = k u
F = K v

53. OUTLINE:
1.Global and Local Coordinate Systems
2.Degrees of Freedom
3.Stiffness method of analysis
1.Member Stiffness Relations (Local Coordinate System)
2.Coordinate Transformations
3.Member Stiffness Relations (Global Coordinate System)
4.Structure Stiffness Relations
5.Procedure for Analysis
Summary
Problems
GENERALPROCEDURE
Local Member Stiffness Matrix, k
Global Member Stiffness Matrix, K
Structure Member Stiffness Matrix, S
P = Sd
Tansformation Matrix Q = k u
F = K v

54. 3.1 MEMBER STIFFNESS RELATIONS
(LOCAL COORDINATE SYSTEM)
Objective:
Derive the stiffness matrix for the members of plane
trusses in the local coordinate system.

55. 3.1 MEMBER STIFFNESS RELATIONS
(LOCAL COORDINATE SYSTEM)

56. 3.1 MEMBER STIFFNESS RELATIONS
(LOCAL COORDINATE SYSTEM)

57. 3.1 MEMBER STIFFNESS RELATIONS
(LOCAL COORDINATE SYSTEM)
From Figs. b through f, we can see that
Q
1= k
11u
1+ k
12u
2+ k
13u
3+ k
14u
4
Q
2= k
21u
1+ k
22u
2+ k
23u
3+ k
24u
4
Q
3= k
31u
1+ k
32u
2+ k
33u
3+ k
34u
4
Q
4= k
41u
1+ k
42u
2+ k
43u
3+ k
44u
4
inwhichk
ijrepresentstheFORCEatthelocationandinthedirectionofk
i
required,alongwithotherendforces,tocauseaunitvalueof
displacementu
j,whileallotherenddisplacementsarezero.
Thesek
ijarecalledstiffnesscoefficientsexpressedinforcesperunit
displacement(ex.N/m,lb/in)
Doublesubscript,k
ij:i=forceandj=displacement)

58. 3.1 MEMBER STIFFNESS RELATIONS
(LOCAL COORDINATE SYSTEM)
Byusingthedefinitionofmatrixmultiplication,the4equationsof
Qcanbeexpressedinmatrixformas
orsymbolicallyas Q=ku

59. 3.1 MEMBER STIFFNESS RELATIONS
(LOCAL COORDINATE SYSTEM)
Relatetheaxialforcek
ijtoaxialdeformationu
jusingthestress–strain
relationshipforlinearlyelasticmaterialsgivenbyHooke’slawas
σ = E ε
Where:
Thus,
1
L


60. 3.1 MEMBER STIFFNESS RELATIONS
(LOCAL COORDINATE SYSTEM)
Determinethevaluesofthestiffnesscoefficients.
k
11=EA/L k
12=0 k
13=-EA/L k
14=0
k
21=0 k
22=0 k
23=0 k
24=0
k
31=-EA/L k
32=0 k
33=EA/L k
34=0
k
41=0 k
42=0 k
43=0 k
44=0
OR
Note:
Member stiffness matrix is symmetric

63. ACTIVITY 3
(FOR 20 MINUTES)
Thedisplacedpositionofmember8ofthetrussinFig.(a)is
giveninFig.(b).Calculatetheaxialforce,Q,inthis
member.Isitundertensionorcompression?

64. ACTIVITY 3
Ifenddisplacementsin
thelocalcoordinate
systemformember9of
thetrussshownare
Calculatetheaxialforce
inthemember.
What istheglobal
memberendforce?

65. OUTLINE:
1.Global and Local Coordinate Systems
2.Degrees of Freedom
3.Stiffness method of analysis
1.Member Stiffness Relations (Local Coordinate System)
2.Coordinate Transformations
3.Member Stiffness Relations (Global Coordinate System)
4.Structure Stiffness Relations
5.Procedure for Analysis
Summary
Problems
GENERALPROCEDURE
Local Member Stiffness Matrix, k
Global Member Stiffness Matrix, K
Structure Member Stiffness Matrix, S
P = Sd
Tansformation Matrix Q = k u
F = K v

66. 3.2 Coordinate Transformations
Whenmembersofastructureareorientedin
differentdirections,itbecomesnecessarytotransformthe
stiffnessrelationsforeachmember fromitslocal
coordinatesystemtoasingleglobalcoordinatesystem
selectedfortheentirestructure.Thememberstiffness
relationsasexpressedintheglobalcoordinatesystemare
thencombinedtoestablishthestiffnessrelationsforthe
wholestructure.

67. 3.2 Coordinate Transformations||
Transformation from Global to Local Coordinate Systems

68. 3.2 Coordinate Transformations||
Transformation from Global to Local Coordinate Systems||
Member End Forces
BycomparingFigs.(b)and(c),weobservethatatend
bofm,thelocalforceQ
1mustbeequaltothe
algebraicsumofthecomponentsoftheglobalforces
F
1andF
2inthedirectionofthelocalxaxis;thatis,
Q
1= F
1cos θ + F
2sin θ
Similarly,thelocalforceQ2equalsthealgebraicsumof
thecomponentsofF1andF2inthedirectionofthe
localyaxis.Thus,
Q
2 = −F
1sin θ + F
2cos θ
Byusingasimilarreasoningatende,weexpressthe
localforcesintermsoftheglobalforcesas
Q
3 = F
3cos θ + F
4sin θ
Q
4= −F
3sin θ + F
4cos θ

69. Where:
T = TRANSFORMATION MATRIX
3.2 Coordinate Transformations||
Transformation from Global to Local Coordinate Systems||
Member End Forces
Q
1= F
1cos θ + F
2sin θ
Q
2 = −F
1sin θ + F
2cos θ
Q
3 = F
3cos θ + F
4sin θ
Q
4= −F
3sin θ + F
4cos θ
These equations can be written as:
In symbols:
Q = TF
Where:
Q = member end forces in LCS
F = member end forces in GCS

70. 3.2 Coordinate Transformations||
Transformation from Global to Local Coordinate Systems||
Member End Forces
Thedirectioncosinesofthemember,necessaryfortheevaluationofT,
canbeconvenientlydeterminedbyusingthefollowingrelationships:
inwhichXbandYbdenotetheglobalcoordinatesofthebeginningjoint
bforthemember,andXeandYerepresenttheglobalcoordinatesofthe
endjointe.
Lettingλ
x=cos(θ)andλ
y=sin(θ)representthedirectioncosinesforthe
member,wehave T

x

y
0
0

y

x
0
0
0
0

x

y

0
0

y

x
















71. 3.2 Coordinate Transformations||
Transformation from Global to Local Coordinate Systems||
Member End Displacement
ThetransformationmatrixT,developedfortransformingendforces,can
alsobeusedtotransformmemberenddisplacementsfromtheglobal
tolocalcoordinatesystem;thatis,
From Q = T F :
u = Tv
where:
u = displacement in the LCS
v = displacement in the GCS
T = transformation matrix

72. 3.2 Coordinate Transformations||
Transformation from Local to Global Coordinate Systems

73. 3.2 Coordinate Transformations||
Transformation from Local to Global Coordinate Systems ||
Member End Forces
A comparison of Figs. (b) and (c) indicates that at end b of m,
F1 = Q1 cos θ −Q2 sin θ
F2 = Q1 sin θ + Q2 cos θ
F3 = Q3 cos θ −Q4 sin θ
F4 = Q3 sin θ + Q4 cos θ
These equations can be written as:
Where:
In symbols:
F = T
T
Q

74. 3.2 Coordinate Transformations||
Transformation from Local to Global Coordinate Systems ||
Member End Displacement
Asdiscussedpreviously,because themember end
displacementsarealsovectors,whicharedefinedinthesame
directionsasthecorrespondingforces,thematrixTalsodefines
thetransformationofmemberenddisplacementsfromthelocal
totheglobalcoordinatesystem;thatis,
v = T
T
u
Where:
v = displacement in the GCS
u = displacement in the LCS

75. ACTIVITY 4
(FOR 10 MINUTES)
Forthetrussshown,theenddisplacementsofmember2(in
inches)intheglobalcoordinatesystemare
Calculatetheendforcesforthismember intheglobal
coordinatesystem.Isthememberinequilibriumunderthese
forces.v
2
0 . 7 5
0
1 . 5
2
















76. OUTLINE:
1.Global and Local Coordinate Systems
2.Degrees of Freedom
3.Stiffness method of analysis
1.Member Stiffness Relations (Local Coordinate System)
2.Coordinate Transformations
3.Member Stiffness Relations (Global Coordinate System)
4.Structure Stiffness Relations
5.Procedure for Analysis
Summary
Problems
GENERALPROCEDURE
Local Member Stiffness Matrix, k
Global Member Stiffness Matrix, K
Structure Member Stiffness Matrix, S
P = Sd
Tansformation Matrix Q = k u
F = K v

77. 3.3 Member Stiffness Relations
(Global Coordinate System)
First,wesubstitutethelocalstiffnessrelationsQ=kuinto
theforcetransformationrelationsF=T
T
Qtoobtain
F = T
T
(ku)
Then,bysubstitutingthedisplacementtransformation
relations,u=Tvintoit,wedeterminethatthedesired
relationshipbetweenthememberendforcesFandend
displacementsv,intheglobalcoordinatesystem,is
F = T
T
[k(Tv)]
F = Kv Where:
K = T
T
k T
K = global member stiffness matrix

78. 3.3 Member Stiffness Relations
(Global Coordinate System)
Performing thematrix
multiplications,weobtain:
Note:Likethememberlocalstiffness
matrixk,thememberglobalstiffness
matrixK,issymmetricT

x

y
0
0

y

x
0
0
0
0

x

y

0
0

y

x














 K
EA
L

x
2

x
y

x
2


x
y

x

y

y
2

x

y

y
2


x
2


x
y

x
2

x
y

x

y

y
2


x

y

y
2





















F = Kv ; K = T
T
k T

79. ACTIVITY 5
(FOR 10 MINUTES)
Forthetrussshown,theenddisplacementsofmember2(in
inches)intheglobalcoordinatesystemare
Calculatetheendforcesforthismember intheglobal
coordinatesystem.Isthememberinequilibriumunderthese
forces.(Thistime,useGlobalMemberstiffnesmatrix)v
2
0 . 7 5
0
1 . 5
2




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80. OUTLINE:
1.Global and Local Coordinate Systems
2.Degrees of Freedom
3.Stiffness method of analysis
1.Member Stiffness Relations (Local Coordinate System)
2.Coordinate Transformations
3.Member Stiffness Relations (Global Coordinate System)
4.Structure Stiffness Relations
5.Procedure for Analysis
Summary
Problems
GENERALPROCEDURE
Local Member Stiffness Matrix, k
Global Member Stiffness Matrix, K
Structure Member Stiffness Matrix, S
P = Sd
Tansformation Matrix Q = k u
F = K v

81. 3.4 STRUCTURE STIFFNESS
RELATIONS

82. 3.4 Structure Stiffness Relations||
•Havingdetermined themember force–displacement
relationshipsintheglobalcoordinatesystem,wearenow
readytoestablishthestiffnessrelationsfortheentirestructure.
ThestructurestiffnessrelationsexpresstheexternalloadsP
actingatthejointsofthestructure,asfunctionsofthejoint
displacementsd.Suchrelationshipscanbeestablishedas
follows:
1.ThejointloadsParefirstexpressedintermsofthemember
endforcesintheglobalcoordinatesystem,F,byapplyingthe
equationsofequilibriumforthejointsofthestructure.
2.Thejointdisplacementsdarethenrelatedtothemember
enddisplacementsintheglobalcoordinatesystem,v,by
usingthecompatibilityconditionsthatthedisplacementsof
thememberendsmustbethesameasthecorresponding
jointdisplacements.

83. 3.4 Structure Stiffness Relations||
3.Next,thecompatibilityequationsaresubstitutedintothe
memberforce–displacementrelations,F=Kv,toexpressthe
memberglobalendforcesintermsofthejointdisplacementsd.
TheF–drelationsthusobtainedarethensubstitutedintothejoint
equilibriumequationstoestablishthedesiredstructurestiffness
relationshipsbetween thejointloadsPandthejoint
displacementsd.

84. OUTLINE FOR MATRIX STRUCTURAL ANALYSIS
ANALYTICAL MODEL
LINE DIAGRAM
which each joint and member is
identified by a number
Establish the global and Local
coordinate system
Identify the degrees of freedom
EVALUATE THE STRUCTURE
STIFFNESS MATRIX, S
Calculate its length and direction
cosines
Compute the global member
stiffness matrix, K
Identify the degrees of freedom
Identify its code numbers, and store
the pertinent elements of K in their
proper positions in S
(complete structure stiffness matrix
must be a symmetric matrix)
Procedure for Analysis

85. Find the Member end forces (LCS and GCS) and
displacements (GCS) for member 5 and
STRUCTURE STIFFNESS MATRIX, S

86. Assignment 1
D e t e r m i n et h ej o i n t
displacements,memberaxial
forces,andsupportreactions
forthetrussesshowninthe
figureusingthematrixstiffness
method.Checkthehand-
calculatedresultsbyusing
GRASP.

87. Assignment 2
D e t e r m i n et h ej o i n t
displacements,memberaxial
forces,andsupportreactions
forthetrussesshowninthe
figureusingthematrixstiffness
method.Checkthehand-
calculatedresultsbyusing
GRASP.

88. 1. 2.