Information Technology Reference
In-Depth Information
These conditions applied to (1) appear as
.
.
U
U
w =
0
w =
0
w
V
.
.
V
V
···
···
···
···
=
(4)
.
.
···
···
···
···
M
=
0
M
=
0
.
.
L
a
U M θ
U MV
U i
z b
=
z a
or
0
= θ
a U +
V a U w V
(5)
= θ
a U M θ +
0
V a U MV
The determinant (set equal to zero) of these homogeneous equations constitutes the char-
acteristic equation. That is,
∇=
U
U MV
U M θ
U
=
0
(6)
w
V
leads to the buckling load. The relation of (6) becomes
2
1
2
2
1
2
2
1 ε
1 LEI 1
/
P b
+
I 1
/
2 ε
I 2
2 LEI 2
/
P b
2 I 1
/
I 2
1
2
2
2
=
0
(7)
ε
ε 1 tan
ε 1
ε
ε 2 tan
ε 2
where
P a
EI 1
P a +
P b
2
1
2
1
2
2
2
2
ε
=
ε
=
EI 2
0d efine the conditions of instabilit y. Nor-
m ally, P a and P b are not independent. Typically, P b is known to be proportional to P a , i.e.,
P b = β
Com b inatio ns of P a and P b that satisfy
∇=
is a known constant. Then (6) is the characteristic equation for a single
unknown, the lowest value of which is the buckling load.
The buckling loads for complicated stepped columns must be found numerically. In
such cases, a numerical search for the lowest root of
P a where
β
∇=
0 is performed. Typically, this
process begins by evaluating
for an estimated buckling load that is believed to be below
the actual buckling load. Increase the estimate, and repeat the evaluation of
. Continue
the process until
changes sign. The desired buckling load, which lies between the two
previous estimates, is then found to a prescribed accuracy by utilizing a nonlinear equation
solver such as Newton-Raphson. 1 See the references in Chapter 10 for a review of various
techniques for conducting numerical determinant searches.
11.5.3 Determination of the Buckling Loads Using Geometric Stiffness Matrices
In those cases where a closed form solution of the condition det K
0 is not available or is
too complex, the linear and geometric stiffness matrices can be used to obtain the first root
of equations developed using the displacement method. The assembled stiffness matrices
of the system can then be written as Eq. (11.67), KV
=
= (
K lin + λ
K geo )
V
=
0 in which
λ
is
1 Joseph D. Raphson (1648-1715) was a British disciple of Newton. Indeed, it is said that his devotion to Newton
clouded his assessment of the notation used for the calculus in that he favored Newton's notation over Leibniz's.
This viewpoint hampered the adoption of the Leibniz calculus notation in England for over a century. In 1690 he
published a method for approximating the real roots of a numerical equation. It supplanted a technique proposed
by Newton and is now referred to as the Newton-Raphson method.
Search WWH ::




Custom Search