Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

692
COLuMNS
In this case, a relationship can be found between the pinned–pinned column’s maximum
deflection v max , which occurs at x = L/2, and the load P. First,
v
max
⎡ kL kL kL ⎤
= e tan sin + cos − 1
⎣⎢ 2 2 2 ⎦⎥
⎡ 2 2
sin ( kL/2)
cos ( kL/2)
⎤
= e⎢
+ − 1⎥
⎣ cos( kL/2)
cos( kL/2)
⎦
⎡ 1 ⎤ ⎡ kL ⎤
= e⎢
−1⎥ = e sec −1
⎣cos( kL/2)
⎦ ⎣⎢ 2 ⎦⎥
(b)
Then, since k 2 = P/EI, Equation (b) can be restated in terms of the load P and the flexural
rigidity EI as
v
max =
⎡ ⎛ L P ⎞ ⎤
e⎢sec⎜
⎟ − 1⎥
⎣ ⎝ 2 EI ⎠ ⎦
(c)
Equation (c) indicates that, for a given column in which E, I, and L are fixed and e > 0, the
column exhibits lateral deflection for even small values of the load P. For any value of e,
the quantity
⎛ L P ⎞
sec⎜
⎟ − 1
⎝ 2 EI ⎠
approaches positive or negative infinity as the argument of the function approaches π/2, 3π/2,
5π/2,…, and the deflection v increases without bound, indicating that the critical load corresponds
to one of these angles. If π/2 (the angle that yields the smallest load) is chosen, then
or
from which it follows that
L P π
=
2 EI 2
P
cr
P
EI
π
=
L
2
π EI
= (16.19)
2
L
which is the Euler formula discussed in Section 16.2.
Unlike an Euler column, which deflects laterally only if P equals or exceeds the Euler
buckling load, an eccentrically loaded column deflects laterally for any value of P. To illustrate
this property, the quantities E, I, and L can be eliminated from Equation (b) by
using Equation (16.19) to produce an expression for the maximum lateral column deflection
in terms of P and P cr . From Equation (16.19), let EI = P cr L 2 /π 2 . Substituting this expression
into Equation (b) then gives
v
max
⎡ ⎛ L P ⎞ ⎤ ⎡ ⎛
2
L P π ⎞ ⎤
π P
= e⎢sec⎜
⎟ − 1⎥ = e⎢sec 1 e sec 1
2 EI
⎜
2
2
⎣ ⎝ ⎠ ⎦ ⎝ Pcr
L
⎟ − ⎥
⎣⎢
⎠ ⎦⎥ =
⎡ ⎛ ⎞ ⎤
⎢ ⎜ ⎟ − ⎥
⎣⎢
⎝ 2 Pcr
⎠ ⎦⎥
(d)
From this equation, it would appear that the maximum deflection becomes infinite as
P approaches the Euler buckling load P cr ; however, under these conditions, the slope of the