Page 143 - Civil Engineering Formulas

COLUMN FORMULAS 83

S
Euler column
Critical L/r
Compression blocks Straight line
Parabolic type

type

L/r
Short Long
FIGURE 3.1 L/r plot for columns.

general, based on the assumption that the permissible stress must be reduced
below that which could be permitted were it due to compression only. The manner
in which this reduction is made determines the type of equation and the slender-
ness ratio beyond which the equation does not apply. Figure 3.1 shows the curves
for this situation. Typical column formulas are given in Table 3.2.

ECCENTRIC LOADS ON COLUMNS

When short blocks are loaded eccentrically in compression or in tension, that is,
not through the center of gravity (cg), a combination of axial and bending stress
results. The maximum unit stress S M is the algebraic sum of these two unit stresses.
In Fig. 3.2, a load, P, acts in a line of symmetry at the distance e from cg; r
radius of gyration. The unit stresses are (1) S c , due to P, as if it acted through cg,
and (2) S b , due to the bending moment of P acting with a leverage of e about cg.
Thus, unit stress, S, at any point y is

S S c S b (3.2)
(P/A) Pey/I
2
S c (1 ey/r )
y is positive for points on the same side of cg as P, and negative on the opposite
side. For a rectangular cross section of width b, the maximum stress, S M
S c (1 6e/b). When P is outside the middle third of width b and is a compres-
sive load, tensile stresses occur.
For a circular cross section of diameter d, S M S c (1 8e/d). The stress due to
the weight of the solid modifies these relations.
Note that in these formulas e is measured from the gravity axis and gives
tension when e is greater than one-sixth the width (measured in the same direction
as e), for rectangular sections, and when greater than one-eighth the diameter,
for solid circular sections.