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

1.15
e = 0
693
THE SECANT FORMuLA
P
P cr
e decreasing
0.00
deflected column is no longer sufficiently small to be neglected in the expression for the
curvature. As a result, accurate deflections can be obtained only by using the nonlinear
form of the differential equation of the elastic curve.
Plots of Equation (d) for various values of eccentricity e are shown in Figure 16.9.
These curves reveal that the maximum column deflection v max is extremely small as
e approaches zero until the load P approaches the Euler critical load P cr . As P nears P cr ,
v max increases rapidly. In the limit as e→0, the curve degenerates into two lines that represent
the straight unbuckled column (P < P cr ) and the buckled configuration (P = P cr )—in
other words, nothing more than Euler column buckling.
Secant Formula
In writing the elastic curve equation, it was assumed that stresses do not exceed the proportional
limit. On the basis of this assumption, the maximum compressive stress can be obtained
by superposition of the axial stress and the maximum bending stress. The maximum
bending stress occurs on a section at the midspan of the column, where the bending moment
attains its largest value, M max = P(e + v max ). Thus, the magnitude of the maximum
compressive stress in the column can be expressed as
P Mmaxc
P Pe ( + vmax
) c
σ max = + = +
(e)
2
A I A Ar
in which r is the radius of gyration of the column cross section about the axis of bending.
If we now start with Equation (c),
⎡ ⎛ L P ⎞ ⎤
vmax = e⎢sec⎜
⎟ − 1⎥
⎣ ⎝ 2 EI ⎠ ⎦
add e to both sides, and perform some algebra on the right-hand side, we obtain the following
expression for e + v max :
⎛ L P ⎞
e + vmax
= esec⎜
⎟
⎝ 2 EI ⎠
v max
FIGURE 16.9 Load–deflection diagram for an eccentrically
loaded column.
Using this expression allows Equation (e) to be written as
P ⎡ ec ⎛ L
σ max = ⎢1+
sec⎜
2
A⎣
r ⎝ 2
P
EI
⎞⎤
⎟⎥
⎠⎦