Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

6.8 Finite Element Method for Continuum Structures 203

asinEq.(6.74),fromwhich

[Ke]=

⎡

⎣

∫

vol

[B]T[D][B]d(vol)

⎤

⎦

Inthisexpression[B]=[C][A−^1 ],where[A]isdefinedinEq.(6.85)and[C]inEq.(6.89).Theelasticity
matrix[D]isdefinedinEq.(6.92)forplanestressproblemsorinEq.(6.93)forplanestrainproblems.
Wenotethatthe[C],A,and[D]matricescontainonlyconstanttermsandmaytherefore
betakenoutsidetheintegrationintheexpressionfor[Ke],leavingonly

∫

d(vol),whichissimplythe
areaA,ofthetriangletimesitsthicknesst.Thus,

[Ke]=[[B]T[D][B]At] (6.94)

Finally,theelementstressesfollowfromEq.(6.79),thatis,

{σ}=[H]{δe}

where[H]=[D][B]and[D]and[B]havepreviouslybeendefined.Itisusuallyfoundconvenienttoplot
thestressesatthecentroidoftheelement.
Of all the finite elements in use, the triangular element is probably the most versatile. It may
be used to solve a variety of problems ranging from two-dimensional flat plate structures to three-
dimensionalfoldedplatesandshells.Forthree-dimensionalapplications,theelementstiffnessmatrix
[Ke] is transformed from an in-planexycoordinate system to a three-dimensional system of global
coordinatesbytheuseofatransformationmatrixsimilartothosedevelopedforthematrixanalysisof
skeletal structures. In addition to the preceding, triangular elements may be adapted for use in plate
flexureproblemsandfortheanalysisofbodiesofrevolution.

Example 6.3
Aconstantstraintriangularelementhascorners1(0,0),2(4,0),and3(2,2)referredtoaCartesianOxy
axessystemandis1unitthick.Iftheelasticitymatrix[D]haselementsD 11 =D 22 =a,D 12 =D 21 =b,
D 13 =D 23 =D 31 =D 32 =0,andD 33 =c,derivethestiffnessmatrixfortheelement.

FromEq.(6.82),

u 1 =α 1 +α 2 ( 0 )+α 3 ( 0 )

thatis,

u 1 =α 1 (i)

u 2 =α 1 +α 2 ( 4 )+α 3 ( 0 )

thatis,

u 2 =α 1 + 4 α 2 (ii)

u 3 =α 1 +α 2 ( 2 )+α 3 ( 2 )