The critical loads of thin columns depend on the end conditions of the column. The value of the Euler load P₁ in Example 4 was derived under the assumption that the column was hinged at both ends. Suppose that a thin vertical homogeneous column is embedded at its base (x = 0) and free at its top (x = L) and that a constant axial load P is applied to its free end. This load either causes a small deflection 8 as shown in FIGURE 3.9.9 or does not cause such a deflection. In either case the differential equation for the deflection y(x) is
The critical loads of thin columns depend on the end conditions of the column. The value of the Euler load P₁ in Example 4 was derived under the assumption that the column was hinged at both ends. Suppose that a thin vertical homogeneous column is embedded at its base (x = 0) and free at its top (x = L) and that a constant axial load P is applied to its free end. This load either causes a small deflection 8 as shown in FIGURE 3.9.9 or does not cause such a deflection. In either case the differential equation for the deflection y(x) is
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:The critical loads of thin columns depend on the end conditions
of the column. The value of the Euler load P₁ in Example 4 was
derived under the assumption that the column was hinged at
both ends. Suppose that a thin vertical homogeneous column
is embedded at its base (x = 0) and free at its top (x = L) and
that a constant axial load P is applied to its free end. This load
either causes a small deflection 8 as shown in FIGURE 3.9.9 or
does not cause such a deflection. In either case the differential
equation for the deflection y(x) is
x = Lt
EI
x=0
d²y
dx²
+ Py
=
(a) What is the predicted deflection when 8 = 0?
(b) When 8 ‡ 0, show that the Euler load for this column is
one-fourth of the Euler load for the hinged column in
Example 4.
↓P
PS.
FIGURE 3.9.9 Deflection of vertical column in Problem 24
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