Page 208 - Shigley's Mechanical Engineering Design

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Deflection and Stiffness 183
Table 4–2
End-Condition Constant C
End-Condition Constants Column End Theoretical Conservative Recommended
Conditions Value Value Value*
for Euler Columns [to Be
Used with Eq. (4–43)] Fixed-free 1 1 1
4 4 4
Rounded-rounded 1 1 1
Fixed-rounded 2 1 1.2
Fixed-fixed 4 1 1.2

*To be used only with liberal factors of safety when the column load is accurately known.

Figure 4–19
P
Euler curve plotted using
Eq. (4–43) with C = 1.
S Q
y
P cr A Parabolic
Unit load T
curve
Euler curve
R
l l
k Q k 1
l
Slenderness ratio
k

compression member. Thus it would appear that any compression member having an
l/k value less than (l/k) Q should be treated as a pure compression member while all
others are to be treated as Euler columns. Unfortunately, this is not true.
In the actual design of a member that functions as a column, the designer will be
aware of the end conditions shown in Fig. 4–18, and will endeavor to configure the ends,
using bolts, welds, or pins, for example, so as to achieve the required ideal end condi-
tions. In spite of these precautions, the result, following manufacture, is likely to contain
defects such as initial crookedness or load eccentricities. The existence of such defects
and the methods of accounting for them will usually involve a factor-of-safety approach
or a stochastic analysis. These methods work well for long columns and for simple
compression members. However, tests show numerous failures for columns with
slenderness ratios below and in the vicinity of point Q, as shown in the shaded area in
Fig. 4–19. These have been reported as occurring even when near-perfect geometric
specimens were used in the testing procedure.
A column failure is always sudden, total, unexpected, and hence dangerous. There
is no advance warning. A beam will bend and give visual warning that it is over-
loaded, but not so for a column. For this reason neither simple compression methods
nor the Euler column equation should be used when the slenderness ratio is near
(l/k) Q . Then what should we do? The usual approach is to choose some point T on
the Euler curve of Fig. 4–19. If the slenderness ratio is specified as (l/k) 1 correspond-
ing to point T, then use the Euler equation only when the actual slenderness ratio is