Information Technology Reference
In-Depth Information
In transfer matrix form
4
24 EI
3
6 EI
2
2 EI
w
1
w
0
3
6 EI
V
2
2 EI
EI
V
01
=
+
p 0
00
1
0
M
M
2
2
00
1
4.24 Find the deflection, slope, moment, and shear along a uniform beam fixed on both
ends.
U i z a +
z i , with U i
Answer:
z b =
taken from Eq. (4.8c) and
w 0 = θ 0 =
0 ,
12
L 3
o
b
6
L 2
0
b
2
0
b
6
L 2
0
b
V 0 =−
EI
(
w
+
θ
)
, and M 0 =
EI
(
L θ
+
w
)
4.25 Use the Laplace transform to derive the transfer matrix for a bar in extension.
4.26 The governing differential equations for a bar in extension on an elastic foundation
(modulus k x )are du
EA , d 2 u
dx 2
/
dx
=
N
/
EA, dN
/
dx
=
k x u or du
/
dx
=
N
/
/
=
k x u
EA . Do not include the effects of applied loading. Derive the element transfer
matrix, using
(a) An exponential series expansion
(b) The Cayley-Hamilton theorem
(c) The Laplace transform
Use this transfer matrix to derive the element stiffness matrix.
/
Answer:
cosh
β
sinh
β/(
EA
β)
U i
2
=
β
=
k x /
EA
EA
β
sinh
β
cosh
β
cosh
β
1
EA
β
k i
=
sinh
β
1
cosh
β
Flexibility Matrices
4.27 Show that the flexibility matrix for a beam element that is hinged at the left end ( x
=
a )
and guided at the right end ( x
=
b ) is given by
w b
θ a
2
V b
M a
6 EI
2
3
=
3
6
4.28 Derive the complete (not reduced) element stiffness matrix of a beam beginning with
any of the flexibility matrices discussed in this chapter.
4.29 For an extension bar on an elastic foundation (modulus k x ) with the left end ( x
a )
fixed, find the nonsingular stiffness matrix and then find the flexibility matrix for an
element of length
=
.
Hint:
Remove a dependent DOF from the stiffness matrix of Problem 4.16.
Answer:
Reduced Stiffness Matrix
Flexibility Matrix
β
β
cosh
l
sinh
l
N b =
EA
β
l
u
u
=
l N b
β
β
β
sinh
EA
cosh
2
u
=
u b
u a =
u b
β
=
k x /
EA
Search WWH ::




Custom Search