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4.7 Derive a stiffness matrix for a beam element if the moment of interia is given by
I
Use an approximate series along with the principle of virtual
work in this derivation.
4.8 A stiffness matrix which provides a relationship between forces
p
i
(
x
)
=
I
0
(
1
+
2
x
/).
and displacements
v
i
on both ends of a member, is usually considered to have the following properties
•
It is singular and, hence, cannot be inverted.
•
It is symmetric.
•
All diagonal elements are positive.
•
After elimination of rigid body motion, it is not singular.
Suppose the stiffness matrix for a beam is
12
−
6
12
−
6
EI
2
2
−
6
4
6
2
k
i
=
12
−
6
12
−
6
−
6
4
6
2
and for a bar is
−
EA
2
−
1
k
i
=
−
12
Are these matrices proper stiffness matrices? If not, what is wrong with them?
Answer:
The first matrix is not symmetic. Some elements do not have correct
units. It is singular. The second matrix does not have positive diagonal units. It is
not singular.
4.9 Can
1
EA
−
2
/
−
11
be a valid stiffness matrix for a bar in extension?
4.10 Consider the stiffness matrix
12
−
6
−
12
−
6
2
2
6
−
4
6
2
3
EI
/
−
11
.
99
6
12
6
2
2
−
6
2
6
4
Which elements may not be correct? Why? What is the meaning of the
k
22
element?
4.11 For a simple beam, i.e., beam theory with no consideration taken for shear deforma-
tion, show that the stiffness
m
atrix loading vector of Eq. (4.16) is the same as obtained
using [Eq. (4.66)]
G
T
a
N
u
.
4.12 Derive an element stiffness matrix for a beam with axial compressive force
N
, shear
deformation, and elastic foundations
k
,
k
∗
. To do so, convert the transfer matrix of Eq.
(4.102) into a stiffness matrix
k
i
p
z
dx
.
4.13 Convert the 6
6 transfer matrix of a beam with axial extension [Eq. (4.123)] into a
stiffness matrix
k
i
.
×
Hint:
Use Eq. (4.16).
4.14 Derive the element stiffness matrix
k
i
for a rod of circular cross-section subject to
torsional loading.
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