Civil Engineering Reference
In-Depth Information
M
1
=−
60
.
00
+
60
.
00
=
0
.
00
F
x
1
=
30
.
38
+
0
.
00
=
30
.
38
F
y
1
=
19
.
75
+
60
.
00
=
79
.
75
M
2
=−
58
.
49
−
60
.
00
=−
118
.
49
Element 2
F
x
1
=−
23
.
25
+
0
.
00
=−
23
.
25
F
y
1
=
8
.
24
+
120
.
00
=
128
.
24
M
1
=−
2
.
52
+
140
.
00
=
137
.
48
=
+
=
F
x
1
23
.
25
0
.
00
23
.
25
=−
+
=
F
y
1
8
.
24
120
.
00
111
.
76
M
2
=
−
=−
51
.
95
140
.
00
88
.
05
Element 3
F
x
1
=
0
.
00
+
0
.
00
=
0
.
00
F
y
1
=
20
.
00
+
20
.
00
=
40
.
00
M
1
=
33
.
33
+
6
.
67
=
40
.
00
F
x
1
=
0
.
00
+
0
.
00
=
0
.
00
F
y
1
=−
20
.
00
+
20
.
00
=
0
.
00
M
2
=
6
.
67
−
6
.
67
=
0
.
00
Moment equilibrium is established by adding the moments from the appropriate end of
all elements coming into the joint.
The second example to be analysed by Program 4.4 is shown in Figure 4.22 and rep-
resents a 3D rigid-jointed frame subjected to a vertical point load of
100.0. In 3D, the
elements have 12 degrees of freedom as shown in Figure 4.23. At each node there are three
translational freedoms in
−
-, and three rotations about each of the global axes.
The extension to 3D is conceptually simple, but considerably more care is required in the
preparation of data and attention to sign conventions. The data organisation is virtually the
same as in the previous example, except
ndim
is set to 3 in the data.
In addition to the axial stiffness (
EA
), 3D involves the flexural stiffness about the
element's local
x
-,
y
-, and
z
y
z
and
axes (
EI
and
EI
respectively) and a torsional stiffness (
GJ
),
y
z
x
thus
nprops
is 4. The local coordinate
defines the long axis of the element. The
z
) must be considered
for 3D space frames because in addition to the six coordinates that define the position of
each node of the element in space, a seventh rotational “coordinate”
x
,
y
,
relationship between the global axes (
x
,
y
,
z
) and local axes (
must be read in as
data. The additional real vector
gamma
is provided to hold this information (in degrees)
for each element.
For the purposes of data preparation, a “vertical” element is defined as one which
lies parallel to the global
γ
y
axis. For non-vertical elements the angle
γ
is defined as
x
the rotation of the element about its local
axis as shown in Figure 4.24. For “ver-
tical” elements however,
γ
is defined as the angle between the global
z
axis and the
z
local
axis, measured towards the global
x
axis as shown in Figure 4.25. For “vertical”
x
elements it is essential that the local
axis points in the same direction as the global
y
axis.
