Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

178 CHAPTER 6 Matrix Methods

where[T]isknownasthetransformationmatrix.Asimilarrelationshipexistsbetweenthesetsofnodal
displacements.Thus,againusingourshorthandnotation,

{δ ̄}=[T]{δ} (6.26)

Substitutingnowfor{F ̄}and{ ̄δ}inEq.(6.23)fromEqs.(6.25)and(6.26),wehave

[T]{F}=[Kij][T]{δ}

Hence,

{F}=[T−^1 ][Kij][T]{δ} (6.27)

Itmaybeshownthattheinverseofthetransformationmatrixisitstranspose:

[T−^1 ]=[T]T

Thus,werewriteEq.(6.27)as

{F}=[T]T[Kij][T]{δ} (6.28)

The nodal force system referred to global coordinates{F}is related to the corresponding nodal
displacementsby

{F}=[Kij]{δ} (6.29)

where[Kij]isthememberstiffnessmatrixreferredtoglobalcoordinates.ComparisonofEqs.(6.28)
and(6.29)showsthat

[Kij]=[T]T[Kij][T]

Substitutingfor[T]fromEq.(6.24)and[Kij]fromEq.(6.23),weobtain

[Kij]=

AE

L

⎡

⎢

⎢

⎣

λ^2 λμ −λ^2 −λμ

λμ μ^2 −λμ −μ^2

−λ^2 −λμ λ^2 λμ

−λμ −μ^2 λμ μ^2

⎤

⎥

⎥

⎦ (6.30)

Byevaluatingλ(=cosθ)andμ(=sinθ)foreachmemberandsubstitutinginEq.(6.30),weobtainthe
stiffnessmatrix,referredtoglobalcoordinates,foreachmemberoftheframework.
InSection6.3,wedeterminedtheinternalforceinaspringfromthenodaldisplacements.Applying
similarreasoningtotheframeworkmember,wemaywritedownanexpressionfortheinternalforce
Sijintermsofthelocalcoordinates.Thus,

Sij=

AE

L

(uj−ui) (6.31)

Now,

uj=λuj+μvj

ui=λui+μvi