FINITE ELEMENT METHOD 311Writing Eq. (11.1) in the matrix form, we get
kk
kku
uf
fpp
ppi
ji
j=⎡
⎣⎢⎤
⎦⎥⎧
⎨
⎩⎫
⎬
⎭⎧
⎨
⎩⎫
⎬
⎭(11.2a)or in compact notation
kpup = fp (11.2b)
In the finite element nomenclature, matrix kp =
kk
kkpp
pp⎡
⎣⎢⎤
⎦⎥ is called the element stiffness matrix,up =
u
ui
j⎧
⎨
⎩⎫
⎬
⎭the element displacement vector and fp =f
fi
j⎧
⎨
⎩⎫
⎬
⎭the element force vector. Note fromEq. (11.1) that fj = –fi which suggests spring equilibrium. It is for this reason that the matrix kp is
singular (its determinant is zero) since Eq. (11.1) is, in a way, a single equation. It may further be
noted that kp is symmetric and is positive semi-definite as one of the eigen-values is zero and the
other is positive. Consider
ukuTp pp
i
jT
pp
ppi
ji
jT
pi j
pj iu
ukk
kku
uu
uku u
ku u==( – )
( – )⎧
⎨
⎩⎫
⎬
⎭⎡
⎣⎢⎤
⎦⎥⎧
⎨
⎩⎫
⎬
⎭⎧
⎨
⎩⎫
⎬
⎭⎡
⎣⎢⎤
⎦⎥ = kp(ui – uj)ui + kp(uj – ui)uj=kp(ui – uj)(ui – uj) (11.2c)which is twice the strain energy stored in the spring. Thus, ukuTp ppis related to the strain energy
which can never be negative. The singularity, symmetry and positive semi-definiteness are inherent
properties of finite element stiffnesses.
We prefer, however, the elaborate form in Eq. (11.2a) for convenience in matrix assembly. Consider
now another spring of stiffness kq as shown in Figure 11.1(b). The element equations in matrix form
can be written by inspection from Eq. (11.2a), that is
kk
kku
uf
fqq
qql
kl
k=⎡
⎣⎢⎤
⎦⎥⎧
⎨
⎩⎫
⎬
⎭⎧
⎨
⎩⎫
⎬
⎭(11.3)If nodes j and l are to coincide such that the two springs are in series with 3 degrees of freedom
as in Figure 11.1(c), then
uj = ul (11.4)
Expressing Eqs. (11.2a) and (11.3) in all three degrees of freedom, we have for springs p and q,
respectively
kk
kku
u
uf
fpp
ppi
j
ki
j–0
–0
000=
0⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥(11.5a)and
00 0
0–
0–=0
kk
kku
u
uf
fqq
qqi
j
kl
k⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥(11.5b)