Civil Engineering Reference
In-Depth Information
M
21
=
M
45
=−
Pl
/16 (hogging)
M
23
=
M
43
=−
Pl
/12 (hogging)
Twisting moment in 62, 82, 74 and 94 is
Pl
/
96.
1.6
l
7
5
1.6
l
6
4
P
3
l
2
l
2
l
9
1
2
l
8
F
IGURE
P.17.7
P.17.8
It is required to formulate the stiffness of a triangular element 123 with
coordinates (0, 0) (
a
, 0) and (0,
a
) respectively, to be used for 'plane stress' problems.
(a) Form the [
B
] matrix.
(b) Obtain the stiffness matrix [
K
e
].
Why, in general, is a finite element solution not an exact solution?
P.17.9
It is required to form the stiffness matrix of a triangular element 123 for use in
stress analysis problems. The coordinates of the element are (1, 1), (2, 1) and (2, 2)
respectively.
(a) Assume a suitable displacement field explaining the reasons for your choice.
(b) Form the [
B
] matrix.
(c) Form the matrix which gives, whenmultiplied by the element nodal displacements,
the stresses in the element. Assume a general [
D
] matrix.
P.17.10
It is required to form the stiffness matrix for a rectangular element of side
2
a
×
2
b
and thickness
t
for use in 'plane stress' problems.
(a) Assume a suitable displacement field.
(b) Form the [
C
] matrix.
(c) Obtain
vol
[
C
]
T
[
D
][
C
]d
V
.
Note that the stiffness matrix may be expressed as
[
A
−
1
]
T
[
C
]
T
[
D
][
C
]d
V
[
A
−
1
]
[
K
e
]
=
vol
