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FIGURE P2.11
Statically indeterminate truss.
2.14 Consider the quadratic function in the
n
variables
x
1
,x
2
,
...
,x
n
:
n
x
T
Ax
=
a
ik
x
i
x
k
i,k
=
1
where
a
11
···
a
1
n
x
1
x
n
.
.
A
=
x
=
a
n
1
···
a
nn
Assume that
A
is symmetic. Show that this quadratic function is positive definitive
if and only if
>
>
...
>
D
1
0
,
D
2
0
,
,
D
n
0
where
D
1
,D
2
,
...
,D
n
are the principal minors of
A
a
11
···
a
1
n
a
11
a
12
.
.
D
1
=
a
11
,
D
2
=
,
D
n
=
a
21
a
22
a
n
1
···
a
nn
This theorem is useful in studying inequality relations for material constants using
the positive definitive strain energy density written in terms of the strain components.
Hint:
See F. R. Gantmacher,
The Theory of Matrices
, Vol. I, Chelsea, 1959, p. 306.
2.15 Use the principle of virtual work to find the forces in the bars of the system shown
in Fig. P2.15 if the material is linearly elastic. Find the horizontal displacement of
point 0.
303
AE
Answer:
N
a
=−
303 kN,
N
b
=
175 kN,
N
c
=
289 kN, Horizontal disp. =
to
the right
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