Civil Engineering Reference
In-Depth Information
rod axial stiffness
EA
1
distributed
load
nodal
displacement
2
u
1
u
2
F
a)
f
x
1
f
x
2
x
d
x
nodal
force
L
F
d
x
P+
dP
d
x
dx
P
A
b)
stress =
s
= P/A
strain =
e
=
s
/E
2
u
d
x
∂
t
2
ρ
A
∂
P
P+ P
d
x
x
∂
∂
A
c)
Figure 2.1 Equilibrium of a rod element
assuming “small” strain, and for equilibrium from Figure 2.1(b),
d
d
x
+
F
=
0
(2.2)
hence the differential equation to be solved is
EA
d
2
u
d
x
2
+
F
=
0
(2.3)
u
in terms
of its nodal values,
u
1
and
u
2
, through simple functions of the space variable called
shape
functions
.Thatis
In the finite element technique, the continuous variable
u
is approximated by
u
=
N
1
u
1
+
N
2
u
2
or
N
2
]
u
1
u
2
{
}
u
=
[
N
1
=
[
N
]
u
(2.4)
where
x
L
,N
2
=
x
L
N
1
=
1
−
(2.5)
