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Buckling of Columns

1) Columns can buckle under compressive loads if they are too slender, with the critical buckling load determined by the Euler equation that depends on the column effective length and moment of inertia. 2) The column effective length depends on end conditions and may be less than the actual length, with pinned ends having the lowest effective length. 3) The AISC buckling criterion is used to determine the allowable stress in a column based on its slenderness ratio and accounts for both elastic and inelastic buckling.

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239 views10 pages

Buckling of Columns

1) Columns can buckle under compressive loads if they are too slender, with the critical buckling load determined by the Euler equation that depends on the column effective length and moment of inertia. 2) The column effective length depends on end conditions and may be less than the actual length, with pinned ends having the lowest effective length. 3) The AISC buckling criterion is used to determine the allowable stress in a column based on its slenderness ratio and accounts for both elastic and inelastic buckling.

Uploaded by

M036
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Chapter 9: Columns

And as imagination bodies forth the forms of things unknown, The poets pen turns them to shapes And gives to airy nothingness a local habitation and a name. William Shakespeare A Midsummer Nights Dream Hamrock Fundamentals of Machine Elements

Equilibrium Regimes

P P 0 Re 0 We (a) (b) 0 P Re

We (c)

We

Figure 9.1 Depiction of equilibrium regimes. (a) Stable; (b) neutral; (c) unstable.

Hamrock Fundamentals of Machine Elements

Example 9.1
l

P mag

Figure 9.2 Pendulum used in Example 9.1.

Hamrock Fundamentals of Machine Elements

Column with Pinned Ends


P x P

y l dx y dy ds P M V

y P P (a) (b) (c) P

Figure 9.3 Column with pinned ends. (a) Assembly; (b) deformation shape; (c) load acting.

Hamrock Fundamentals of Machine Elements

P y x

Buckling of Columns

Euler Equation:

n2!2EI Pcr = l2

Figure 9.4 Buckling of rectangular section. Hamrock Fundamentals of Machine Elements

Column Effective Lengths


End condition description Both ends pinned One end pinned and one end fixed Both ends fixed One end fixed and one end free P

le = 0.7l Illustration of end condition l = le l l le = 0.5l le = 2l

Theoretical effective column length AISC (1989) recommended effective column length

le = l le = l

le = 0.7l le = 0.8l

le = 0.5l le = 0.65l

le = 2l le = 2.1l

Table 9.1 Effective length for four end conditions. Hamrock Fundamentals of Machine Elements

Buckling for Different Slenderness Ratio


Critical Slenderness Ratio
Yielding Yield strength A Buckling Normal stress, Johnson equation T Safe AISC Equations (le /rg) T Slenderness ratio, le /rg Euler equation

Cc =

le rg

=
E

2E ! 2 Sy

Euler Equation
(Pcr )E "2 E (!cr )E = = A (le/rg)2

Johnson Parabola
2 Sy (Pcr )J (!cr )J = = Sy 2 A 4" E

Figure 9.5 Normal stress as a function of slenderness ratio.

le rg

Hamrock Fundamentals of Machine Elements

AISC Buckling Criterion


12"2E 23 (le/rg)2
(le /rg )2 2 2Cc

Elastic Buckling Inelastic Buckling

!all =

!all =

Sy

n!

Allowable Stress Reduction

5 3 (le/rg) (le/rg)3 n! = + 3 3 8Cc 8Cc

Hamrock Fundamentals of Machine Elements

50 mm 27.6 mm

Example 9.3

41.7 mm

Pcr = 738.6 N Pcr = 4094 N (a) 50 mm 24.5 mm (b)

43.6 mm

Pcr = 773.5 N Pcr = 5672 N (c) (d)

Figure 9.6 Cross-sectional areas, drawn to scale, from results of Exampe 9.3, aas well as critical buckling load for each cross-sectional area.

Hamrock Fundamentals of Machine Elements

P e A x y l l x

P M = Pe y x y

P e

Eccentrically Loaded Column

M y P (c)

Secant Equation:
le ymax = e sec 2 P EI 1

B e P x (a) x (b)

M = Pe P

Figure 9.7 Eccentrically loaded column. (a) Eccentricity; (b) statically equivalent bending moment; (c) free-body diagram through arbitrary section.

Hamrock Fundamentals of Machine Elements

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