Civil Engineering Reference
In-Depth Information
Introducing the discretised approximation
w
, from (2.21) and (2.40) this becomes
1
2
(
{
T
[
D
]
δU
=
A
}
[
N
]
{
w
}
)
{
A
}
[
N
]
{
w
}
δx
1
2
{
T
T
[
D
]
{
}
{
}
δx
=
w
}
(
{
A
}
[
N
]
)
A
[
N
]
w
(2.43)
The total strain energy of the element is thus,
L
1
2
T
T
[
D
]
U
=
{
w
}
(
{
A
}
[
N
]
)
{
A
}
[
N
]
{
w
}
d
x
(2.44)
0
The product
{
A
}
[
N
] is usually written as [
B
], and since
{
w
}
are nodal values and
therefore, constants
T
1
2
{
w
}
[
B
]
T
[
D
][
B
] d
x
{
w
}
U
=
(2.45)
Similar operations on (2.39) lead to the total external work done,
W
, and hence the
stored potential energy of the beam is given by
=
U
−
W
T
L
0
L
1
2
{
1
2
{
[
B
]
T
[
D
][
B
] d
x
{
T
[
N
]
T
d
x
=
w
}
w
} −
w
}
q
(2.46)
0
A state of stable equilibrium is achieved when
is a minimum with respect to all
{
w
}
.
That is,
L
L
∂
∂
{
[
B
]
T
[
D
][
B
] d
x
{
[
N
]
T
d
x
=
=
w
} −
q
0
(2.47)
T
w
}
0
0
or
L
L
[
B
]
T
[
D
][
B
] d
x
{
[
N
]
T
d
x
w
} =
q
(2.48)
0
0
which is simply another way of writing (2.25).
Thus we see from (2.28) that the elastic element stiffness matrix [
k
m
] can be written
in the form,
L
[
B
]
T
[
D
][
B
] d
x
[
k
m
]
=
(2.49)
0
which will prove to be a useful general matrix form for expressing stiffnesses of all elastic
solid elements. The computer programs for analysis of solids developed in the next chapter
use this notation and method of stiffness formation.
The “energy” formulation described above is clearly valid only for “conservative” sys-
tems. Galerkin's method is more generally applicable.
