General Design Principles for DuPont Engineering Polymers - Module

Form of bar;
manner of loading and support
Uniform straight bar under end load
One end free, other end fixed
Uniform straight bar under end load
Both ends hinged
l
l
P
P
P
Table 4.04. Buckling of Columns, Rings and Arches
E = modulus of elasticity, I = moment of inertia of cross section about central axis perpendicular to plane of buckling.
All dimensions are in inches, all forces in pounds, all angles in radians.
Uniform straight bar under end load
One end fixed, other end hinged and
horizontally constrained over fixed end
0.7 l
0.3 l
Uniform circular ring under uniform radial
pressure p lb/in. Mean radius of ring r.
Uniform circular arch under uniform radial
pressure p lb/in. Mean radius r.
Ends hinged
Uniform circular arch under uniform radial
pressure p lb/in. Mean radius r.
Ends fixed
r
P
2�
P
2�
P
P' = π 2 El
4 l 2
P' = π 2 El
l 2
Formulas for critical load P', critical unit load p', critical torque T', critical bending moment M',
or critical combination of loads at which elastic buckling occurs
P' = π 2 El
(0.7 l ) 2
p' =
p' =
3 El
r 2
( – 1 )
2 El π
r 3 �2 (For symmetrical arch of any form under central concentrated loading see Ref. 40)
p' = El (k 2 – 1)
r 3
Where k depends on α and is found by trial from the equation: k tan α cot kα = 1 or from the following
table:
α = 15° 30° 45° 60° 75° 90° 120° 180°
k = 17.2 8.62 5.80 4.37 3.50 3.00 2.36 2.00
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