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FIGURE 11.27
Column with hinged-hinged boundaries, Euler Case 4.
Equation (2), with the definitions of A
and B
of Eq. (11.51), becomes
ε
sin
ε =
0
or
sin
ε =
0
(3)
The roots of this transcendental equation are displayed in Fig. 11.27b. With
ε 1 = π
and
ε 2 =
as the first two eigenvalues. The lowest bifurcation load (first eigenvalue), which
corresponds to instability (buckling) of the beam, is from
2
π
ε 1 = π
,
P cr = ε
2 EI
L 2
= π
2 EI
L 2
(4)
The corresponding eigenshapes (mode shapes or buckling shapes) are depicted in Fig.
11.27b.
EXAMPLE 11.7 Buckling of Fixed-Hinged Beam
Find the first two bifurcation loads (the lowest eigenvalues) of the fixed-hinged column of
Fig. 11.28a. This corresponds to Euler case 2.
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