Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

6.8 Finite Element Method for Continuum Structures 207

or
{
u(x,y)
v(x,y)

}

=[f(x,y)]{α} (6.98)

Now,substitutingthecoordinatesandvaluesofdisplacementateachnode,weobtain

⎧

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

ui

vi

uj

vj

uk

vk

ul

vl

⎫

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎭

=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1 xi yi xiyi 0000

0000 1xi yi xiyi

1 xj yj xjyj 0000

0000 1xj yj xjyj

1 xk yk xkyk 0000

0000 1xk yk xkyk

1 xl yl xlyl 0000

0000 1xl yl xlyl

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎧

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

α 1

α 2

α 3

α 4

α 5

α 6

α 7

α 8

⎫

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎭

(6.99)

whichisoftheform

{δe}=[A]{α}

Then,

{α}=[A−^1 ]{δe} (6.100)

Theinversionof[A]isillustratedinExample6.4but,asinthecaseofthetriangularelement,ismost
easilycarriedoutbymeansofacomputer.Theremaininganalysisisidenticaltothatforthetriangular
elementexceptthatthe{ε}–{α}relationship(seeEq.(6.89))becomes

{ε}=

⎡

⎣

010 y 0000

0000001 x

001 x 010 y

⎤

⎦

⎧

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

α 1

α 2

α 3

α 4

α 5

α 6

α 7

α 8

⎫

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎭

(6.101)

Example 6.4
Arectangularelementusedinaplanestressanalysishascornerswhosecoordinates(inmeters),referred
toanOxyaxessystem,are1(−2,−1),2(2,−1),3(2,1),and4(−2,1);thedisplacements(alsoinmeters)
ofthecornerswere

u 1 =0.001, u 2 =0.003, u 3 =−0.003, u 4 = 0

v 1 =−0.004, v 2 =−0.002, v 3 =0.001, v 4 =0.001