200 CHAPTER 6 Matrix Methods
compatibilityofdisplacementalongtheedgesofadjacentelements.WritingEqs.(6.82)inmatrixform
gives
{
u(x,y)
v(x,y)}
=
[
1 xy 000
0001 xy]
⎧
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪⎩
α 1
α 2
α 3
α 4
α 5
α 6⎫
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎪⎭
(6.83)
ComparingEq.(6.83)withEq.(6.55),weseethatitisoftheform
{
u(x,y)
v(x,y)
}
=[f(x,y)]{α} (6.84)SubstitutingvaluesofdisplacementandcoordinatesateachnodeinEq.(6.84),wehavefornodei
{
ui
vi
}
=
[
1 xi yi 000
0001 xi yi]
{α}Similarexpressionsareobtainedfornodesjandksothatforthecompleteelementweobtain
⎧
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎩
ui
vi
uj
vj
uk
vk⎫
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎪
⎭
=
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
1 xi yi 00 0
0001 xi yi
1 xj yj 00 0
0001 xj yj
1 xk yk 00 0
0001 xk yk⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎧
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎩
α 1
α 2
α 3
α 4
α 5
α 6⎫
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪
⎪⎪
⎭
(6.85)
FromEq.(6.81)andbycomparingwithEqs.(6.58)and(6.59),weseethatEq.(6.85)takestheform
{δe}=[A]{α}Hence(step3)weobtain
{α}=[A−^1 ]{δe} (comparewithEq.(6.60))The inversion of [A], defined in Eq. (6.85), may be achieved algebraically as illustrated in Example
6.3.Alternatively,theinversionmaybecarriedoutnumericallyforaparticularelementbycomputer.
Substitutingfor{α}fromtheprecedingintoEq.(6.84)gives
{
u(x,y)
v(x,y)
}
=[f(x,y)][A−^1 ]{δe} (6.86)(comparewithEq.(6.61)).
Thestrainsintheelementare
{ε}=⎧
⎨
⎩
εx
εy
γxy