Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

204 CHAPTER 6 Matrix Methods

thatis,

u 3 =α 1 + 2 α 2 + 2 α 3 (iii)

FromEq.(i),

α 1 =u 1 (iv)

andfromEqs.(ii)and(iv),

α 2 =

u 2 −u 1

4

(v)

Then,fromEqs.(iii)to(v),

α 3 =

2 u 3 −u 1 −u 2

4

(vi)

Substitutingforα 1 ,α 2 ,andα 3 inthefirstofEqs.(6.82)gives

u=u 1 +

(

u 2 −u 1

4

)

x+

(

2 u 3 −u 1 −u 2

4

)

y

or

u=

(

1 −

x

4

−

y

4

)

u 1 +

(x

4

−

y

4

)

u 2 +

y

2

u 3 (vii)

Similarly,

v=

(

1 −

x

4

−

y

4

)

v 1 +

(x

4

−

y

4

)

v 2 +

y

2

v 3 (viii)

NowfromEq.(6.88),

εx=

∂u

∂x

=−

u 1

4

+

u 2

4

εy=

∂v

∂y

=−

v 1

4

−

v 2

4

+

v 3

2

and

γxy=

∂u

∂y

+

∂v

∂x

=−

u 1

4

−

u 2

4

−

v 1

4

+

v 2

4

Hence,

[B]{δe}=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

∂u

∂x

∂v

∂y

∂u

∂y

+

∂v

∂x

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

=

1

4

⎡

⎣

− 101000

0 − 10 − 102

− 1 − 1 − 1120

⎤

⎦

⎧

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎩

u 1

v 1

u 2

v 2

u 3

v 3

⎫

⎪⎪

⎪⎪

⎪⎪

⎬

⎪⎪

⎪⎪

⎪⎪

⎭

(ix)