Long Columns with Central Loading (Buckling)


A column failure is always sudden, total, and unexpected, and hence dangerous. There is no advance warning. A beam will bend and give visual warning that it is overloaded; but not so for a column.

The relation between the critical load and the column material and geometry is developed with reference to the figure shown. We assume a bar of length l loaded by a force p acting along the centroidal axis with different end conditions.

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Equation (1) is called the Euler column formula,


Pcr = critical load (N)

          C = end condition constant (see Table)

          E = modulus of elasticity (N/m2)

          I = second moment of area (m4)

          l = shaft length (m)

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Table (Theoretical End Restraint Coefficients)


End Conditions



Both ends pinned



Both ends built in



One end pinned, one end built in



One end built in, one end free



One end built in , one end fixed against rotation but free



One end pinned, one end fixed against rotation but free

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