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Partial Answer:
σ
55 mm at both ends of the beam.
5.9 Use the transfer matrix method to find the displacements and forces along the plane
frame of Fig. 5.22.
5.10 Compute, with transfer matrices, the displacement and force responses of the plane
frame of Fig. 5.28.
=
5
.
34 MPa,
w
=
2
.
max
max
Displacement method
5.11 Find the displacements, reaction forces, and spring forces in the one-dimensional
system of springs connected to rigid bars shown in Fig. P5.11. For numerical results,
let the spring constants be given by k 1
2 ,k 2
1 ,k 3
1, and k 4
=
=
=
=
1
.
FIGURE P5.11
A spring system.
Hint:
For a direct stiffness, nonvariational formulation, use the following procedure. Take
the element stiffness matrix for a spring from Chapter 3, i.e.,
k i
k i v i
k i
p i
=
k i
Equilibrium at the nodes:
Node a : P a =
p a +
p a =
reaction force
p b +
p b +
p b =
Node b : P b =
0
=
0
p c +
p c +
p c
Node c : P c =
1
=
Compatibility at the nodes:
Node a : u a =
u a =
U a =
0
Node b : u b =
u b =
u b =
U b
Node c : u c =
u c =
u c =
U c
Global Stiffness Matrix:
?
0
1
K 11
K 12
K 13
U a
U b
U c
P a
P b
P c
=
=
K 21
K 22
K 23
K 31
K 32
K 33
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