Computer (matrix) version of the stiffness method 1. The computer ...

Since
we get
or
T
e
T T = I
~
R +
e
T
T
e = Te
KeTeqe
Te
R0e
R = K q + R
e e e 0e
with the element stiffness matrix in the global co-ordinates
and the vector of nodal reactions due to the span loading in the global co-ordinates
T ~
R = T R
K
e
= T
T
e
~
K
e
T
e
0e e 0e
R e is the corresponding vector of reactions expressed in the global co-ordinates.
In the case of the plane beam element the transformation matrix has the form
T e
⎡C
0⎤
= ⎢ ⎥
⎣0
C⎦
where the component matrix C is related to the transformation of the nodal displacement vector
q
n,
e
⎡ui
⎤
⎢ ⎥
=
⎢
v i ⎥
⎢⎣
ϕ ⎥
i ⎦
In this situation the axis z and z ~ coincide and the matrix C has the form
⎡cos
⎢
C = ⎢cos
⎢
⎣cos
( x
~ , x) cos( x
~ , y ) cos( x
~ , z)
( y
~ , x) cos( y
~ , y ) cos( y
~ , z)
( z
~ , x) cos( z
~ , x) cos( z
~ , z)
⎤ ⎡ cosα
⎥ ⎢
⎥ =
⎢
cos
⎥
⎦
⎢⎣
0
( π 2 + α )
cos
( π 2 − α )
cosα
0
0⎤
⎡ cosα
⎥ ⎢
0
⎥
=
⎢
− sinα
1⎥⎦
⎢⎣
0
sinα
cosα
0
0⎤
⎥
0
⎥
1⎥⎦
α
y ~
z ~ =
z
y
α
x ~
x
The transformation of the element stiffness matrix can be given in terms of submatrices. If we
represent the stiffness matrix in the local co-ordinates for an element e with nodes i and j as
then the transformation follows as
K
e
⎡C
= ⎢
⎣0
0⎤
⎥
C⎦
T
⎡~
K
⎢~
⎢⎣
K
ii,
e
ji,
e
~
K
e
~
K
~
K
⎡~
K
= ⎢~
⎢⎣
K
ij,
e
jj,
e
ii,
e
ji,
e
⎤⎡C
⎥⎢
⎥⎦
⎣0
Similarly for the vector of reactions we can write down
~
K
~
K
ij,
e
jj,
e
⎤
⎥
⎥⎦
⎡ T ~
0⎤
C K
⎥ = ⎢
T ~
C⎦
⎢⎣
C K
ii,
e
ji,
e
C
C
T ~
C Kij
T ~
C K
, e
jj,
e
C⎤
⎥
C⎥⎦