6.8 Finite Element Method for Continuum Structures 199Fig.6.13
Triangular element for plane elasticity problems.
thattheinverseofthe[A]matrixforatriangularelementcontainstermsgivingtheactualareaofthe
element;thisareaispositiveiftheprecedingnodeletteringornumberingsystemisadopted.Theele-
ment is to be used for plane elasticity problems and has therefore two degrees of freedom per node,
giving a total of six degrees of freedom for the element, which results in a 6×6 element stiffness
matrix[Ke].Thenodalforcesanddisplacementsareshown,andthecompletedisplacementandforce
vectorsare
{δe}=⎧
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪⎪
⎪⎪
⎪⎩
ui
vi
uj
vj
uk
vk⎫
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪⎪
⎪⎪
⎪⎭
{Fe}=⎧
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪⎪
⎪⎪
⎪⎩
Fx,i
Fy,i
Fx,j
Fy,j
Fx,k
Fy,k⎫
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎬
⎪⎪
⎪⎪⎪
⎪⎪
⎪⎭
(6.81)
Wenowselectadisplacementfunctionwhichmustsatisfytheboundaryconditionsoftheelement—
that is, the condition that each node possesses two degrees of freedom. Generally, for computational
purposes, a polynomial is preferable to, say, a trigonometric series, since the terms in a polynomial
canbecalculatedmuchmorerapidlybyadigitalcomputer.Furthermore,thetotalnumberofdegrees
of freedom is six so that only six coefficients in the polynomial can be obtained. Suppose that the
displacementfunctionis
u(x,y)=α 1 +α 2 x+α 3 y
v(x,y)=α 4 +α 5 x+α 6 y}
(6.82)
Theconstantterms,α 1 andα 4 ,arerequiredtorepresentanyin-planerigidbodymotion—thatis,motion
withoutstrain—whilethelineartermsenablestatesofconstantstraintobespecified;Eqs.(6.82)ensure