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

graph shows a scattered range of values that transition from the yield stress for the very
shortest columns to the Euler buckling stress for the very longest columns. In the broad
range of slenderness ratios between these two extremes, neither the yield stress nor the
Euler buckling stress is a good predictor of the strength of the column. Furthermore,
most practical columns fall within this intermediate range of slenderness ratios. Consequently,
practical column design is based primarily on empirical formulas that have been
developed to represent the best fit of test results for a range of realistic full-size columns.
These empirical formulas incorporate appropriate factors of safety, effective-length factors,
and other modifying factors.
The strength of a column and the manner in which it fails are greatly dependent on its
effective length. For example, consider the behavior of columns made of steel:
Short steel columns: A very short steel column may be loaded until the steel: reaches the
yield stress; consequently, very short columns do not buckle. The strength of these
members is the same in both compression and tension; however, the columns are so
short that they have no practical value.
intermediate-length steel columns: Most practical steel columns fall into this category. As
the effective length (or slenderness ratio) increases, the cause of failure becomes more
complicated. In steel columns—in particular, hot-rolled steel columns—the applied
load may cause compression stresses that exceed the proportional limit in portions
of the cross section; thus, the column will fail both by yielding and by buckling.
These columns are said to buckle inelastically. The buckling strength of hot-rolled
steel columns is particularly influenced by the presence of residual stresses—stresses
that are “locked into” the steel shape during the manufacturing process because the
steel flanges and webs cool faster than the fillet regions that connect them. Because of
residual stress and other factors, the analysis and design of intermediate-length steel
columns are based on empirical formulas developed from test results.
long steel columns: Long, slender steel columns buckle elastically, since the Euler
buckling stress is well below the proportional limit (even taking into account the
presence of residual stress). Consequently, the Euler buckling equations are reliable
predictors for long columns. Long, slender columns, however, are not very efficient,
since the Euler buckling stress for these columns is much less than the proportional
limit for the steel.
Several representative empirical design formulas for centrically loaded steel, aluminum,
and wood columns will be presented to introduce basic aspects of column design.
697
EMPIRICAL COLuMN
FORMuLAS—
CENTRIC LOAdINg
Structural Steel columns
Structural steel columns are designed in accordance with specifications published by the
American Institute of Steel Construction (AISC). The AISC Allowable Stress Design 1
(ASD) procedure differentiates between short and intermediate-length columns and long
columns. The transition point between these two categories is defined by an effectiveslenderness
ratio
KL
r
= 4.71
E
σ Y
This effective-slenderness ratio corresponds to an Euler buckling stress of 0.44σ Y .
1
Specification for Structural Steel Buildings, ANSI/AISC 360-10, American Institute of Steel Construction,
Chicago, 2010.