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

694
COLuMNS
which can be further simplified with the use of I = Ar 2 to give an expression for the maximum
compressive stress in the deflected column:
P ⎡ ec ⎛ L
σ max = ⎢1+
sec⎜
2
A⎣
r ⎝ 2 r
P ⎞⎤
⎟⎥ (16.20)
EA ⎠⎦
The maximum compressive
stress σ max occurs at midheight
of the column on the inner
(concave) side.
Equation (16.20) is known as the secant formula, and it relates the average force per unit
area, P/A, that causes a specified maximum stress σ max in a column to the dimensions of the
column, the properties of the material the column is made of, and the eccentricity e from
the centerline of the undeformed column. The term L/r is the same slenderness ratio found
in the Euler buckling stress formula [Equation (16.8)]; thus, for columns with different end
conditions (see Section 16.3), the secant formula can be restated as
P ⎡ ec ⎛ KL
σ max = ⎢1+
sec⎜
2
A⎣
r ⎝ 2 r
P ⎞⎤
⎟⎥ (16.21)
EA ⎠⎦
The quantity ec/r 2 is called the eccentricity ratio and is seen to depend on the eccentricity of
the load and the dimensions of the column. If the column is loaded precisely at its centroid then,
e = 0 and σ max = P/A. It is virtually impossible, however, to eliminate all eccentricity that might
result from various factors, such as initial crookedness of the column, minute flaws in the material,
and a lack of uniformity of the cross section, as well as accidental eccentricity of the load.
To determine the maximum compressive load that can be applied at a given eccentricity
to a particular column, the maximum compressive stress can be set equal to the yield
stress in compression and Equation (16.20) can then be solved numerically for P/A.
Figure 16.10 is a plot of the force per unit area, P/A, versus the slenderness ratio L/r for
300
40
36
ec
r
2 = 0
0.1
σY
= 36 ksi
E = 29,000 ksi
250
ec
r
2 = 0
0.1
σY = 250 MPa
E = 200 GPa
(ksi)
P
A
30
25
20
0.2
0.4
0.6
1.0
Euler’s curve
(MPa)
P
A
200
150
0.2
0.4
0.6
1.0
Euler’s curve
15
10
2
4
5
ec
r
2 = 8
0
0 50 100 150 200
100
50
2
4
ec
r
2 = 8
0
0 50 100 150 200
KL
r
(a)
FIGURE 16.10 Average compression stress versus slenderness ratio, based on the secant formula.
KL
r
(b)