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4.15 Convert the stiffness matrix
k
i
for an Euler-Bernoulli beam into a transfer matrix
U
i
.
Hint:
Use Eq. (4.79).
4.16 Derive the element stiffness matrix for a bar in extension on an elastic foundation
(modulus
k
x
).
Hint:
Your answer can be checked by comparison with
k
i
of Problem 4.26, which
was obtained by converting the transfer matrix into a stiffness matrix.
Answer:
EA
β
cosh
β/(
sinh
β)
−
EA
β/(
sinh
β)
k
i
2
=
β
=
k
x
/
EA
−
EA
β/(
sinh
β)
EA
β
cosh
β/(
sinh
β)
4.17 Use the transfer matrix for an Euler-Bernoulli beam element [Eq. (4.8c)] to derive the
shape function
N
of
w(
x
)
=
Nv
.
Hint:
Condense
V
a
and
M
a
out of the standard transfer matrix.
Answer:
Your
N
should be the same as
N
of Eq. (4.47a).
4.18 For a rod subject to extension, the kinematic relation is
=
d
x
u
0
, the material law
0
x
is
N
=
EA
0
x
, and the condition of equilibrium is
d
x
N
+
p
x
=
0
.
See Chapter 2,
Problem 2.21.
(a) Derive
A
of the first-order governing equations
d
x
z
=
Az
+
P
.
(b) Use
A
to compute the transfer matrix
U
i
.
(c) Convert the transfer matrix
U
i
into the element stiffness matrix
k
i
.
Hint:
Use Eq. (4.16).
(d) Derive
k
D
of the principle of virtual work.
(e) Derive the stiffness matrix
k
i
using the differential stiffness operator
k
D
.
(f) Convert the stiffness matrix
k
i
into the transfer matrix
U
i
.
Hint:
Use Eq. (4.79).
Transfer Matrices
4.19 Calculate the
e
i
functions of Eqs. (4.112) and (4.113). Use can be made of the roots of
the denominator. The four roots of
s
4
s
2
+
(ζ
−
η)
+
λ
−
ζη
=
0
are
2
s
1
,
2
,
3
,
4
=±
−
(ζ
−
η) /
2
±
(ζ
−
η)
/
4
−
(λ
−
ηζ )
Find
e
4
()
for
(a)
ζ
=
η
=
λ
=
0
(b)
η
=
λ
=
0
,
ζ<
0
(c)
η
=
λ
=
0
,
ζ>
0
(d)
η
=
ζ
=
0
,
λ<
0
(e)
η
=
ζ
=
0
,
λ>
0
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