6.8 Finite Element Method for Continuum Structures 195FromEqs.(6.54)and(6.56),wecanwritedownexpressionsforthenodaldisplacementsvi,θiandvj,θj
atx=0andx=L,respectively.Hence,
vi=α 1
θi=α 2
vj=α 1 +α 2 L+α 3 L^2 +α 4 L^3
θj=α 2 + 2 α 3 L+ 3 α 4 L^2⎫
⎪⎪
⎬
⎪⎪
⎭
(6.57)
WritingEqs.(6.57)inmatrixformgives
⎧
⎪⎪
⎨
⎪⎪
⎩
vi
θi
vj
θj⎫
⎪⎪
⎬
⎪⎪
⎭
=
⎡
⎢
⎢
⎣
100 0
010 0
1 LL^2 L^3
012 L 3 L^2
⎤
⎥
⎥
⎦
⎧
⎪⎪
⎨
⎪⎪
⎩
α 1
α 2
α 3
α 4⎫
⎪⎪
⎬
⎪⎪
⎭
(6.58)
or
{δe}=[A]{α} (6.59)ThethirdstepfollowsdirectlyfromEqs.(6.58)and(6.55)inthatweexpressthedisplacementat
anypointinthebeamelementintermsofthenodaldisplacements.UsingEq.(6.59),weobtain
{α}=[A−^1 ]{δe} (6.60)SubstitutinginEq.(6.55)gives
{v(x)}=[f(x)][A−^1 ]{δe} (6.61)where[A−^1 ]isobtainedbyinverting[A]inEq.(6.58)andmaybeshowntobegivenby
[A−^1 ]=
⎡
⎢
⎢
⎣
1000
0100
− 3 /L^2 − 2 /L 3 /L^2 − 1 /L
2 /L^31 /L^2 − 2 /L^31 /L^2
⎤
⎥
⎥
⎦ (6.62)
Instepfour,werelatethestrain{ε(x)}atanypointxintheelementtothedisplacement{v(x)}and
hencetothenodaldisplacements{δe}.Sinceweareconcernedherewithbendingdeformationsonly,
wemayrepresentthestrainbythecurvature∂^2 v/∂x^2 .Hence,fromEq.(6.54),
∂^2 v
∂x^2= 2 α 3 + 6 α 4 x (6.63)orinmatrixform
{ε}=[0026x]⎧
⎪⎪
⎨
⎪⎪
⎩
α 1
α 2
α 3
α 4