198 CHAPTER 6 Matrix Methods
Hence,
[Ke]=∫L
0⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
−
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⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
[EI]
[
−
6
L^2
+
12 x
L^3−
4
L
+
6 x
L^26
L^2
−
12 x
L^3−
2
L
+
6 x
L^2]
dxwhichgives
[Ke]=EI
L^3
⎡
⎢
⎢
⎣
12 − 6 L − 12 − 6 L
− 6 L 4 L^26 L 2 L^2
−12 6L 12 6L
− 6 L 2 L^26 L 4 L^2
⎤
⎥
⎥
⎦ (6.77)
Equation(6.77)isidenticaltothestiffnessmatrix(seeEq.(6.44))fortheuniformbeamofFig.6.6.
Finally,instepseven,werelatetheinternal“stresses,”{σ},intheelementtothenodaldisplacements
{δe}.ThishasinfactbeenachievedtosomeextentinEq.(6.69),namely
{σ}=[D][C][A−^1 ]{δe}or,fromthepreceding,
{σ}=[D][B]{δe} (6.78)Equation(6.78)isusuallywrittenas
{σ}=[H]{δe} (6.79)inwhich[H]=[D][B]isthestress–displacementmatrix.Forthisparticularbeamelement,[D]=EI
and[B]isdefinedinEq.(6.76).Thus,
[H]=EI
[
−
6
L^2
+
12 x
L^3−
4
L
+
6 x
L^26
L^2
−
12 x
L^3−
2
L
+
6 x
L^2]
(6.80)
6.8.2 Stiffness Matrix for a Triangular Finite Element
Triangular finite elements are used in the solution of plane stress and plane strain problems. Their
advantageoverothershapedelementsliesintheirabilitytorepresentirregularshapesandboundaries
withrelativesimplicity.
In the derivation of the stiffness matrix, we shall adopt the step-by-step procedure of the previ-
ousexample.Initially,therefore,wechooseasuitablecoordinateandnodenumberingsystemforthe
elementanddefineitsnodaldisplacementandnodalforcevectors.Figure6.13showsatriangularele-
ment referred to axes Oxyand having nodesi,j,andklettered counterclockwise. It may be shown