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Buckling, frame element

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ExampleReferenceRecordsLargest deviation
Column, pinned-pinnedPublished source10.0033%
Column, cantileverPublished source1< 0.001%
Column, fixed-fixedPublished source10.0512%

Problem. The buckling load is the load factor at which the stiffness matrix becomes singular under compression. Four classical support conditions of a column are tested.

Reference document. The Euler critical loads from Timoshenko, Theory of Elastic Stability, are used.

The critical load factor λ\lambda is found from the buckling eigenvalue problem.

K ϕ=λ (−Kg) ϕK\,\phi = \lambda\,(-K_g)\,\phi

The critical loads of pinned-pinned, cantilever and fixed-fixed columns are given by these relations in turn.

Pcr=π2EIL2;π2EI4L2;4π2EIL2P_{cr} = \dfrac{\pi^2 EI}{L^2};\quad \dfrac{\pi^2 EI}{4L^2};\quad \dfrac{4\pi^2 EI}{L^2}

Numerical model. The column is modelled with eight elements. The computed critical load stays above the closed form and approaches it as the mesh is refined.

Column, pinned-pinned

Parameters. mesh = 8 elements · L = 4 · EI = 6,000

Published source. The Euler critical load (Timoshenko, Theory of Elastic Stability)

Pcr=π2EI/L2P_{cr} = \pi^2 EI / L^2
QuantityReferenceComputedDeviation
Critical load3,701.13,701.220.0033%

Column, cantilever

Parameters. mesh = 8 elements · L = 4 · EI = 6,000

Published source. The Euler critical load (Timoshenko, Theory of Elastic Stability)

Pcr=0.25 π2EI/L2P_{cr} = 0.25\,\pi^2 EI / L^2
QuantityReferenceComputedDeviation
Critical load925.275925.277< 0.001%

Column, fixed-fixed

Parameters. mesh = 8 elements · L = 4 · EI = 6,000

Published source. The Euler critical load (Timoshenko, Theory of Elastic Stability)

Pcr=4 π2EI/L2P_{cr} = 4\,\pi^2 EI / L^2
QuantityReferenceComputedDeviation
Critical load14,804.414,8120.0512%