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

Plane frame
x
i
v i
u i
q
n
⎡ui
⎤
⎢ ⎥
=
⎢
v i ⎥
⎢⎣
ϕ ⎥
i ⎦
y
x
z
3D frame
y
u i
ϕ x
ϕ z
ϕ i
i
w i
v i
ϕ x
ϕ y
Plane yz
q
n
⎡ui
⎤
⎢ ⎥
⎢
v i ⎥
⎢w
⎥
i
= ⎢ ⎥
⎢ϕ
x ⎥
⎢ϕ
⎥
y
⎢ ⎥
⎢⎣
ϕz
⎥⎦
Interpretation
of the rotation angle
The displacements presented in the above figures are referred to the global sets of co-ordinates xy
or xyz. In practice it is much easier to form the basic equations on the level of elements, hence, the
local systems of co-ordinates are introduced. The values corresponding to the local co-ordinates
are additionally denoted with tildes.
The slope-defection formulae from the classical stiffness method are replaced by the element
stiffness matrix, which relates the element displacements in local co-ordinates to the reactions at
the element supports in local co-ordinates:
~ K
~ ~
Re
= Keq
~
e e
[ k eij
]
n×
n
where n is the number of element degrees of freedom. The elements of the stiffness matrix
represent reactions at the element supports created by unit displacements. For instance, the
element k ~ eij represents the reaction number i created by the action of the displacement q
~ = j 1
First, let us consider the plane truss element. In this case the vectors of element displacements
and reactions have four components
= ~
~ q
~
i
R 1,
1
2
~ R 3,
q
~
3
x ~
k
e
~ ~
l
R 4,
q 4
~ R
~
2,
q
y ~
q
~
e
⎡u
~ ⎤ ⎡ ~
i q1
⎤
⎢ ⎥ ⎢ ⎥
⎢v
~ ~
i ⎥ = ⎢q2
= ⎥
⎢u
~ ⎥ ⎢~
⎥
k q3
⎢ ⎥ ⎢~
⎥
~
⎣v
k
⎦ ⎢⎣
q4
⎥⎦
~
R
e
⎡ ~
R
⎢ ~
⎢R
=
⎢ ~
R
⎢ ~
⎢⎣
R
1
2
3
4
⎤ ⎡−
Ni
⎤
⎥ ⎢ ⎥
⎥ ⎢
−Ti
= ⎥
⎥ ⎢ N ⎥
k
⎥ ⎢ ⎥
⎥⎦
⎣ Tk
⎦