6.8 Finite Element Method for Continuum Structures 197Thetotalpotentialenergyofthebeamhasastationaryvaluewithrespecttothenodaldisplacements
{δe}T;hence,fromEq.(6.71),
∂(U+V)
∂{δe}T=
∫
vol[A−^1 ]T[C]T[D][C][A−^1 ]{δe}d(vol)−{Fe}= 0 (6.72)fromwhich
{Fe}=⎡
⎣
∫
vol[C]T[A−^1 ]T[D][C][A−^1 ]d(vol)⎤
⎦{δe} (6.73)orwriting[C][A−^1 ]as[B]weobtain
{Fe}=⎡
⎣
∫
vol[B]T[D][B]d(vol)⎤
⎦{δe} (6.74)fromwhichtheelementstiffnessmatrixisclearly
[Ke]=⎡
⎣
∫
vol[B]T[D][B]d(vol)⎤
⎦ (6.75)
FromEqs.(6.62)and(6.64),wehave
[B]=[C][A−^1 ]=[0026x]⎡
⎢
⎢
⎢
⎣
1000
0100
− 3 /L^2 − 2 /L 3 /L^2 − 1 /L
2 /L^31 /L^2 − 2 /L^31 /L^2
⎤
⎥
⎥
⎥
⎦
or
[B]T=
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
−
6
L^2
+
12 x
L^3−4
L
+
6 x
L^2
6
L^2−
12 x
L^3−2
L
+
6 x
L^2