Information Technology Reference
In-Depth Information
P
must now be adjusted to include
the loading. By definit
io
n vector
P
∗
contains prescribed nodal loading. Inclusion of an
element loading vector
p
i
0
to account for effects distributed between the nodes will involve
the special assembly of a new global loading vector. The global stiffness matrix is still
assembled in terms of stiffness coefficients, or submatrices as appropriate, using
The assembly of the global stiffn
es
s equations
KV
=
M
k
i
jk
K
jk
=
(5.99a)
i
=
1
where
M
is the number of elements in the structural model, while the global vector at node
j
is formed as
M
P
0
j
p
i
0
j
=
(5.99b)
i
=
1
Fo
r the whole system, the
P
0
j
due to applie
d
distributed loads forms a global nodal vector
P
0
. This can be incorporated directly into
P
of the glob
a
l stiffness equations
KV
=
P
.Itis
advisable to distinguish between the direct nodal loads
P
∗
and the influence of the elements
on the nodes due to distributed loading, represented by
P
0
. Then we may write the system
equilibrium equations as
P
∗
P
0
KV
=
+
=
P
(5.100)
Nodal vector
containing
direct nodal
loads
Nodal vector
due to applied
distributed
loading
Upon solution of the system equations for
V
, the usual procedure is followed to compute
the local displacements and forces. Use the incidence conditions
to relate the
member end displacements to the global node displacements. After transformation of
v
to
local coordinates using
(
v
=
aV
)
v
i
T
i
v
i
, Eq. (5.97) can be employed to calculate member forces.
=
EXAMPLE 5.6 Beam with Linearly Varying Loading
Return to the fixed-simply supported beam of Fig. 5.5. This beam has been treated exten-
sively in Chapters 3 and 4 as well as in the present chapter.
The solution procedure for this structure with loading distributed between the nodes
follows the outline of Section 5.3.9. For this straight beam, the local and global coordinate
systems coincide; consequently, no transformation of variables from local to global coordi-
nates is required. Hence, the stiffness matrix of Chapter 4, Eq. (4.13) is assembled directly
into the global stiffness matrix
K
.
The element loading vectors
p
i
0
can be taken from Chapter 4, Table 4.2. For the first
element, the distributed load begins with a magnitude of
p
0
and ends with
p
0
/
2. The second
element begins with a magnitude of
p
0
2 and ends with a zero. With this information, the
loading vectors are provided directly by Chapter 4, Table 4.2. If desired, the loading vectors
can be calculated using C
ha
pter 4, Eq. (4.58). We choose to treat it he
re
for the quite general
case of a distributed load
p
z
(ξ )
/
varying linearly as in Fig. 5.24. Then
p
z
can be described as
p
z
(ξ )
=
p
a
+
(
p
b
−
p
a
)ξ
(1)
Rewrite this in the form
p
z
(ξ )
=
N
p
G
p
p
p
(2)
Search WWH ::
Custom Search