106
E. Chircan et al.
where P tot represents the axial force in the beam cross section at distance x. The force
components acting at the right beam end considered in the local coordinate system
are represented by P x , P y = 0, P z = 0. Beside these components, the value of P and
the components of the inertia forces acting upon the portion of the beam between x
and L are being determined.
The total internal energy is:
E p =
1
2
{δ e }
T
[k eb ] + [k ea ] + [k et ] +
k
G
e
{δ e }
=
1
2
{δ e }
T [k e ]{δ e }
(20)
The external work of distributed loads is:
W =
L
0
p x u + p y v + p z w + m x α + m y β + m z γ
dx
=
L
0
p x p y p z m x m y m z
N
N
∗
{δ e }dx
=
q
∗
eL
T {δ e },
(21)
here the vector {q
*
eL } contains the three components of the distributed loads and the
three components of the distributed moments.
The external work of concentrated loads {q eL } in the nodes is:
W
c
= {q eL }
T
{δ e }
(22)
After deformation, the position vector of point M becomes M
and it is expressed
by:
r M ,L
=
r M,L
+ {δ} =
r M,L
+
⎧
⎨
⎩
u
v
w
⎫
⎬
⎭
=
r o,L
+
⎧
⎨
⎩
x + u
v
w
⎫
⎬
⎭
,
(23)
or, with respect to the global coordinate system:
r M ,G
=
r M,G
+ [R]
⎧
⎨
⎩
u
v
w
⎫
⎬
⎭
=
r o,G
+ [R]
⎧
⎨
⎩
x
0
0
⎫
⎬
⎭
+ [R]
⎧
⎨
⎩
u
v
w
⎫
⎬
⎭
=
r o,G
+ [R]
⎧
⎨
⎩
x
0
0
⎫
⎬
⎭
+ [R][N ]{δ e }
(24)