Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

6.8 Finite Element Method for Continuum Structures 201

FromEqs.(1.18)and(1.20),weseethat

εx=

∂u

∂x

εy=

∂v

∂y

γxy=

∂u

∂y

+

∂v

∂x

(6.88)

SubstitutingforuandvinEqs.(6.88)fromEqs.(6.82)gives

εx=α 2

εy=α 6

γxy=α 3 +α 5

orinmatrixform

{ε}=

⎡

⎣

010000

000001

001010

⎤

⎦

⎧

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪⎪

⎪⎩

α 1

α 2

α 3

α 4

α 5

α 6

⎫

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪⎪

⎪⎭

(6.89)

whichisoftheform

{ε}=[C]{α} (seeEqs.(6.64)and(6.65))

Substitutingfor{α}(=[A−^1 ]{δe})weobtain

{ε}=[C][A−^1 ]{δe} (comparewithEq.(6.66))

or

{ε}=[B]{δe} (seeEq.(6.76))

where[C]isdefinedinEq.(6.89).
In step five, we relate the internal stresses{σ}to the strain{ε}and hence, using step four, to the
nodaldisplacements{δe}.Forplanestressproblems,

{σ}=

⎧

⎨

⎩

σx

σy

τxy

⎫

⎬

⎭

(6.90)

and

εx =

σx

E

−

νσy

E

εy =

σy

E

−

νσx

E

γxy=

τxy

G

=

2 ( 1 +ν)

E

τxy

⎫

⎪⎪

⎪⎪

⎪⎬

⎪⎪

⎪⎪

⎪⎭

(seeChapter1)