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For the linearly varying load of Fig. 6.13, the distribution of
p
is a known function of
ξ
,
i.e.,
η
−
ξ
p
3
p
4
N
p
p
3
p
4
p
4
)ξ
=
ξ
.
p
η
(ξ
,
η
=
1
)
=
p
4
+
(
p
3
−
1
=
(6.45)
Then the loading vector integral of Eq. (6.38) becomes
0
0
00
00
00
00
00
00
0
0
ξ
0
1
00
ξ
d
a
p
3
p
4
a
p
3
p
4
1
−
ξ
0
p
i
0
=
ξ
·
=
(6.46)
0
0
1
−
ξ
0
0
0
ξ
01
/
/
0
1
31
6
−
ξ
/
/
1
61
3
Assembly of the System Stiffness Matrix and Loading Vector
The procedure for assembling the system stiffness matrix and system loading vector will
be demonstrated with a specific numerical example.
EXAMPLE 6.2 Planar structure
To illustrate the fundamentals of applying the displacement finite element method to a
multidimensional structure, consider the problem of determining the in-plane stresses and
displacements in a flat structure of constant thickness
t
lying in the
xy
plane as shown
in Fig. 6.14a. A linearly varying line load is applied to the top edge. Suppose this two-
dimensional planar structure is in a state of plane stress.
Begin the solution by taking advantage of the vertical axis of symmetry and choosing
the model of Fig. 6.14b to replace the actual structure of Fig. 6.14a. Note that supports have
been added to the model. Support is necessary in order to assure that, at least from a rigid
body motion viewpoint, the model is a reasonable idealization of the structure of Fig. 6.14a.
Without support, a singular stiffness matrix would be expected, which mathematically
corresponds to rigid body motion of the structure.
The selection of the finite element model, along with appropriate constraints, is the most
important step in modern structural analyses. The model must be established by the engi-
neer even when readily available general purpose finite element programs are used. The
supports (constraints) of Fig. 6.14b were chosen to achieve a symmetric deformation pattern
like that of the original system of Fig. 6.14a. Of course, if we do not take advantage of the
symmetry, the larger model of the complete original system could be used. However, the
desire for computational economy usually prevails, and symmetry is utilized whenever
appropriate.
The model of Fig. 6.14b is discretized into four rectangular elements. The use of only
four elements can lead to rather inaccurate results. Each of these elements can be illustrated
as in Fig. 6.10, which shows the local, natural coordinates
for this particular element.
The nodes, which in this case are the four corners of the element, are numbered counter-
clockwise. The elements are attached to each other only at the nodes. For this elementary
example, the local coordinates used as a reference for displacements and forces for the
elements are the same as the global coordinates used for the system displacements and
loading. Hence it is not necessary to perform a coordinate transformation for the element
matrices and loading vectors. Furthermore,
x
and
y
will designate the global coordinate
system, rather than
X
and
Y
of Chapter 5.
ξ
,
η
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