Page 158 - Civil Engineering Formulas

96 CHAPTER THREE

where G shear modulus of elasticity, psi (MPa)
J torsional constant
2
2
A cross-sectional area, in (mm )
4
4
I p polar moment of inertia I x I y , in (mm )
If the section possesses a significant amount of warping rigidity, the axial buck-
ling load is increased to
2
A
EC w
P GJ 2 (3.28)
I p L
where C w is the warping constant, a function of cross-sectional shape and
dimensions.

ALLOWABLE DESIGN LOADS FOR ALUMINUM COLUMNS

Euler’s equation is used for long aluminum columns, and depending on the
material, either Johnson’s parabolic or straight-line equation is used for short
columns. These equations for aluminum follow
Euler’s equation
2
c
E
F e (3.29)
(L/ ) 2
Johnson’s generalized equation

(3.30)
n
(L/ )
F c F ce 1 K cE
B F ce
The value of n, which determines whether the short column formula is the
straight-line or parabolic type, is selected from Table 3.5. The transition from the
long to the short column range is given by
cr
kcE (3.31)
L
B F ce

where F e allowable column compressive stress, psi (MPa)
F ce column yield stress and is given as a function of F cy (compres-
sive yield stress), psi (MPa)
L length of column, ft (m)
radius of gyration of column, in (mm)
E modulus of elasticity—noted on nomograms, psi (MPa)
c column-end fixity from Fig. 3.9
n, K, k constants from Table 3.5