6.8 Finite Element Method for Continuum Structures 205Also,
[D]=
⎡
⎣
ab 0
ba 0
00 c⎤
⎦
Hence,
[D][B]=
1
4
⎡
⎣
−a −ba−b 02 b
−b −ab−a 02 a
−c −c −cc 2 c 0⎤
⎦
and
[B]T[D][B]=
1
16
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
a+cb+c −a+cb−c − 2 c − 2 b
b+ca+c −b+ca−c − 2 c − 2 a
−a+c −b+ca+c −b−c − 2 c 2 b
b−ca−c −b−ca+c 2 c − 2 a
− 2 c − 2 c − 2 c 2 c 4 c 0
− 2 b − 2 a 2 b − 2 a 04 a⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Then,fromEq.(6.94),
[Ke]=1
4
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
a+cb+c −a+cb−c − 2 c − 2 b
b+ca+c −b+ca−c − 2 c − 2 a
−a+c −b+ca+c −b−c − 2 c 2 b
b−ca−c −b−ca+c 2 c − 2 a
− 2 c − 2 c − 2 c 2 c 4 c 0
− 2 b − 2 a 2 b − 2 a 04 a⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
6.8.3 Stiffness Matrix for a Quadrilateral Element
Quadrilateralelementsarefrequentlyusedincombinationwithtriangularelementstobuildupparticular
geometricalshapes.Figure6.14showsaquadrilateralelementreferredtoaxesOxyandhavingcorner
nodes,i,j,k,andl;thenodalforcesanddisplacementsarealsoshown,andthedisplacementandforce
vectorsare
{δe}=⎧
⎪⎪
⎪⎪
⎪⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩
ui
vi
uj
vj
uk
vk
ul
vl⎫
⎪⎪
⎪⎪
⎪⎪⎪
⎪⎪
⎪⎬
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎭
{Fe}=⎧
⎪⎪
⎪⎪
⎪⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩
Fx,i
Fy,i
Fx,j
Fy,j
Fx,k
Fy,k
Fx,l
Fy,l