Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

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202 CHAPTER 6 Matrix Methods


Thus,inmatrixform,


{ε}=




εx
εy
γxy




=

1

E



1 −ν 0
−ν 10
002 ( 1 +ν)






σx
σy
τxy




(6.91)

Itmaybeshownthat(seeChapter1)


{σ}=




σx
σy
τxy




=

E

1 −ν^2



1 ν 0
ν 10
0012 ( 1 −ν)






εx
εy
γxy




(6.92)

whichhastheformofEq.(6.68),thatis,


{σ}=[D]{ε}

Substitutingfor{ε}intermsofthenodaldisplacements{δe},weobtain


{σ}=[D][B]{δe} (seeEq.(6.69))

Inthecaseofplanestrain,theelasticitymatrix[D]takesadifferentformtothatdefinedinEq.(6.92).
Forthistypeofproblem,


εx=

σx
E


νσy
E


νσz
E
εy=

σy
E


νσx
E


νσz
E
εz=

σz
E


νσx
E


νσy
E

= 0

γxy=

τxy
G

=

2 ( 1 +ν)
E

τxy

Eliminatingσzandsolvingforσx,σy,andτxygive


{σ}=




σx
σy
τxy




=

E( 1 −ν)
( 1 +ν)( 1 − 2 ν)


⎢⎢





1

ν
1 −ν

0

ν
1 −ν

10

00

( 1 − 2 ν)
2 ( 1 −ν)


⎥⎥








εx
εy
γxy




(6.93)

whichagaintakestheform


{σ}=[D]{ε}

Step six, in which the internal stresses{σ}are replaced by the statically equivalent nodal forces
{Fe},proceedsinanidenticalmannertothatdescribedforthebeamelement.Thus,


{Fe}=




vol

[B]T[D][B]d(vol)


⎦{δe}
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