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

708
COLuMNS
where the compressive stresses are treated as positive quantities. In Equation (16.30), σ allow
is the allowable stress calculated from one of the empirical design formulas presented in
Section 16.5. The formula uses the largest value of the effective-slenderness ratio for the
cross section, irrespective of the axis about which bending occurs. Values of c and I used in
calculating the bending stress, however, do depend on the axis of bending. Thus, the allowable
stress method and Equation (16.30) generally produce a conservative design.
Interaction method
In an eccentrically loaded column, much of the total stress may be caused by the bending
moment. However, the allowable bending stress is generally larger than the allowable compression
stress. How, then, can some balance be attained between the two allowable stresses?
Consider the axial stress σ a = P/A. If the allowable axial stress for a member acting as a column
is denoted by (σ allow ) a , then the area required for a given axial force P can be expressed as
A
a
P
=
( )
σ allow
Next, consider the bending stress given by σ b = Mc/I. The moment of inertia I can be expressed
in terms of the area and the radius of gyration as I = Ar 2 , where r is the radius of
gyration in the plane of bending. Let the allowable bending stress be designated (σ allow ) b .
Then the area required for a given bending moment M can be expressed as
A
b
a
Mc
=
2
r ( )
σ allow
Therefore, the total area required for a column subjected to an axial force and a bending
moment can be expressed as the sum of the expressions for A a and A b :
P Mc
A = Aa
+ Ab
= +
2
( σallow) a r ( σ allow )
Dividing this equation by the total area A and letting Ar 2 = I gives
b
b
PA /
( σ )
allow
a
Mc/
I
+ = 1
(16.31)
( σ )
allow
b
If the column has an axial load, but no bending moment (i.e., if it is a centrically loaded
column), then Equation (16.31) indicates that the column is analyzed in accordance with
the allowable axial stress. If the column has a bending moment, but no axial load (in other
words, if it is truly a beam), then the normal stresses must satisfy the allowable bending
stress. Between these two extremes, Equation (16.31) accounts for the relative importance
of each normal stress component in relation to the combined effect. Equation (16.31) is
known as an interaction formula, and the approach that it presents is a common method for
considering the combined effect of an axial load and a bending moment in columns.
In Equation (16.31), (σ allow ) a is the allowable axial stress given by one of the empirical
column design formulas in Section 16.5 and (σ allow ) b is the allowable bending stress. The
AISC specifications use the general form of Equation (16.31) to analyze combined axial
compression and bending; however, additional modification factors are added to that equation,
depending on whether (P/A)/(σ allow ) a is less than or greater than 0.2. Since the purpose
of this discussion is to introduce the concept of interaction equations, rather than to teach the
specific details of AISC steel column design, Equation (16.31) without additional factors will
be used here to analyze columns subjected to both axial compression and bending moments.