FINITE ELEMENT METHOD 311
Writing Eq. (11.1) in the matrix form, we get
kk
kk
u
u
f
f
pp
pp
i
j
i
j
=
⎡
⎣
⎢
⎤
⎦
⎥
⎧
⎨
⎩
⎫
⎬
⎭
⎧
⎨
⎩
⎫
⎬
⎭
(11.2a)
or in compact notation
kpup = fp (11.2b)
In the finite element nomenclature, matrix kp =
kk
kk
pp
pp
⎡
⎣
⎢
⎤
⎦
⎥ is called the element stiffness matrix,
up =
u
u
i
j
⎧
⎨
⎩
⎫
⎬
⎭
the element displacement vector and fp =
f
f
i
j
⎧
⎨
⎩
⎫
⎬
⎭
the element force vector. Note from
Eq. (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
j
T
pp
pp
i
j
i
j
T
pi j
pj i
u
u
kk
kk
u
u
u
u
ku 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
kk
u
u
f
f
qq
qq
l
k
l
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
kk
u
u
u
f
f
pp
pp
i
j
k
i
j
–0
–0
000
=
0
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(11.5a)
and
00 0
0–
0–
=
0
kk
kk
u
u
u
f
f
qq
qq
i
j
k
l
k
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(11.5b)