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TABLE 11.8
Buckling Load Convergence as the Number of Elements Increases
with a lowest root of
1
by 7%. Note that, as
more terms are retained, the solution improves, always approaching the exact value from
above.
Improved solutions are obtained by increasing the number of elements used to model
the beam. Table 11.8 illustrates this type of convergence, where for each element terms up
to and including
ε
=
4
.
80, which differs from the exact value of
ε
1
2
are retained. The four so-called
Euler cases
of boundary conditions are
shown in order in the table. For cases 2 and 4, the exact values were given in Examples 11.7
and 11.6, respectively. Note that in all cases the solutions approach the exact buckling load
from above.
ε
EXAMPLE 11.11 Critical Loads and Buckling Shapes of a Framework
Return to the framework of the Example 11.5 to evaluate the critical loads for which bi-
furcations of the equilibrium state occurs. The fundamental state is taken either from the
second order analysis (for instance, after the first iteration) or approximately from the linear
analysis. It is also of interest to learn whether the latter possibility is applicable for this type
of structure.
The geometry and load combinations are shown in Fig. 11.34, in which the loads are
multiplied by the common factor
λ
, the value of which is to be determined by the conditions
of Eq. (11.66), det
K
0.
In Fig. 11.35, the load level for bifurcation of equilibrium is shown in the general load
displacement diagram. The relation between the load factor
=
λ
and the deflection, chosen
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