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INPUT   DATA

♥ EXAMPLE Of Input/Output

Title

Length of column, L		m
Diameter of areal cross-section, d		m
Modulus of elasticity, E		109N/m2
Yield strength of material, Sy		106N/m2

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OUTPUT   VARIABLES   &   GRAPHS

VARIABLES	Pin-Pin	Fixed-Pin	Fixed-Fixed	Fixed-Free	Units
♦ Radius of gyration, k					m
♦ Slenderness ratio, L/k
♦ Critical slenderness ratio, L/k*
♦ Applicable Equation
♦ Critical stress, σcr					106N/m2
♦ Critical force, Pcr					N

Buckling of Slender Columns

A straight sufficiently slender column subjected to compressive axial force at it ends can fail(buckle) due to elastic instability. The onset of buckling will depend on the coupling conditions of the two ends of the column, eg pin,fixed, free. The critical buckling stress can be estimated by the Euler Equation for long columns and by Johnson (parabolic) Equation for moderate length ones. The critical slenderness ratio determines which of these two equations is applicable. If the slenderness ratio is above the critical ratio the Euler equation is used, otherwise Johnson's is used.

where
     σ_cr = critical stress to cause buckling
     C = end condition dependent constant: pin-pin=1, fixed-pin=2, fixed-fixed=4, fixed-free=1/4
     L = length of column
     r = radius of cross-sectional area
     d = diameter of cross-sectional area = 2r
     E = modulus of elasticity of column material
     S_y = yield point of column material
     k = radius of gyration
     L/k = slenderness ratio of column
     (L/k)^* = critical slenderness ratio of column
     P = Compressive force
     P_cr = Critical compressive force = σ_crπr²

Tips

    ◊ Use link EXAMPLE Of Input/Output  to demo data entry expectations and results; you may edit & use it as starting point

BIBLIOGRAPHY

• Brown TH; Marks' Calculations for Machine Design; McGraw Hill, New York, 2005, Chapter 6.
• Hicks TG; Mechanical Engineering Formulas, Pocket Guide; McGraw Hill, New York, 2003, Section 5.
• Young WC & Budynas RG; Roark's Formulas for Stress and Strain; 7th Edition, McGraw Hill, New York, 2002, Chapter 12.
• Moulton A, Grosjean J & Owen G; The Moulton Formulae and Methods - directly usable for calculations in mechanical engineering; Professional Engineering Publishing Ltd, London, UK, 2005.