78
5 Finite Element Formulations
Fig. 5.1 Physical meaning
of the resultant internal
forces and moments
0
L
11
0
L
22
0
L
21
0
L
12
1
L
11
1
L
22
1
L
12
1
L
21
0
L
33
0
L
13
0
L
23
Transverse shear forces
Longitudinal forces
In-plane shear forces
Bending moments
Transverse normal force
Torsional moments
a 3
a 2
a 1
a 3
a 2
a 1
a 3
a 2
a 1
a 3
a 2
a 1
a 3
a 2
a 1
a 3
a 2
a 1
moments (
1
L
11 ,
1
L
22 ), the torsional moments (
1
L
12 ,
1
L
21 ), the transverse shear forces
(
0
L
13 ,
0
L
23 ), and the transverse normal force (
0
L
33 ), as shown in Fig. 5.1.
The resultant stress vector L and the corresponding resultant strain vector S are
defined as
L =
0
L
11
,
0
L
22
,
0
L
12
,
1
L
11
,
1
L
22
,
1
L
12
,
2
L
11
,
2
L
22
,
2
L
12
,
0
L
23
,
0
L
13
,
1
L
23
,
1
L
13
T , (5.4)
S =
0
ε 11 ,
0
ε 22 , 2
0
ε 12 ,
1
ε 11 ,
1
ε 22 , 2
1
ε 12 ,
2
ε 11 ,
2
ε 22 , 2
2
ε 12 , 2
0
ε 23 , 2
0
ε 13 , 2
1
ε 23 , 2
1
ε 13
T . (5.5)
Therefore, the strain components given in (3.68)–(3.70) can be expressed in terms
of the resultant strain vector S as
ε = H s S ,
(5.6)
with