CIVIL ENGINEERING FORMULAS

Chapter 3. Column Formulas

GENERAL CONSIDERATIONS

Columns are structural members subjected to direct compression. All columns
can be grouped into the following three classes:

1.Compression blocksare so short (with a slenderness ratio — that is, unsup-
ported length divided by the least radius of gyration of the member — below 30)
that bending is not potentially occurring.

2.Columns so slender that bending under load is given are termedlong columns
and are defined by Euler’s theory.

3.Intermediate-length columns, often used in structural practice, are called
short columns.

Long and short columns usually fail by buckling when their critical loadis
reached. Long columns are analyzed using Euler’s column formula, namely,

(3.1)

In this formula, the coefficient naccounts for end conditions. When the column
is pivoted at both ends, n1; when one end is fixed and the other end is
rounded,n2; when both ends are fixed, n4; and when one end is fixed
and the other is free, n0.25. The slenderness ratio separating long columns
from short columns depends on the modulus of elasticity and the yield strength
of the column material. When Euler’s formula results in (Pcr/A)>Sy, strength
instead of buckling causes failure, and the column ceases to be long. In quick
estimating numbers, this critical slenderness ratiofalls between 120 and 150.
Table 3.1 gives additional column data based on Euler’s formula.

SHORT COLUMNS

Stress in short columns can be considered to be partly due to compression and
partly due to bending. Empirical, rational expressions for column stress are, in

Pcr

n 

2 EI

l^2



n 

2 EA

(l/r)^2

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