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Second order (P-Delta)

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ExampleReferenceRecordsLargest deviation
Cantilever column, axial load at 20% of the critical loadPublished source10.266%
Cantilever column, axial load at 40% of the critical loadPublished source10.543%
Cantilever column, axial load at 60% of the critical loadPublished source10.832%

Problem. In second-order analysis the geometric stiffness is updated with the displacement. The amplification of the lateral drift of a cantilever under lateral and axial load is tested at axial load ratios of one fifth, two fifths and three fifths of the critical load.

Reference document. The classical amplification factor is used. The relation is approximate, and the deviation grows with the load ratio.

In second-order equilibrium the geometric stiffness depends on the displacement and is updated throughout the solution.

(K+Kg(u))u=f\left(K + K_g(u)\right)u = f

The critical load of a cantilever and the classical magnification factor that follows from it are given by these relations.

δδ0≈11−P/Pcr,Pcr=π2EI4L2\dfrac{\delta}{\delta_0} \approx \dfrac{1}{1 - P/P_{cr}},\qquad P_{cr} = \dfrac{\pi^2 EI}{4L^2}

Numerical model. A separate test checks that beyond the critical load the solution does not converge and that this is reported explicitly.

Cantilever column, axial load at 20% of the critical load

Parameters. P/PcrP/P_{cr} = 0.2 · mesh = 8 elements

Published source. The classical magnification factor

11−P/Pcr,Pcr=π2EI/(4L2)\dfrac{1}{1 - P/P_{cr}},\qquad P_{cr} = \pi^2 EI/(4L^2)
QuantityReferenceComputedDeviation
Magnification factor δ/δ₀1.251.246670.266%

Cantilever column, axial load at 40% of the critical load

Parameters. P/PcrP/P_{cr} = 0.4 · mesh = 8 elements

Published source. The classical magnification factor

11−P/Pcr,Pcr=π2EI/(4L2)\dfrac{1}{1 - P/P_{cr}},\qquad P_{cr} = \pi^2 EI/(4L^2)
QuantityReferenceComputedDeviation
Magnification factor δ/δ₀1.666671.657620.543%

Cantilever column, axial load at 60% of the critical load

Parameters. P/PcrP/P_{cr} = 0.6 · mesh = 8 elements

Published source. The classical magnification factor

11−P/Pcr,Pcr=π2EI/(4L2)\dfrac{1}{1 - P/P_{cr}},\qquad P_{cr} = \pi^2 EI/(4L^2)
QuantityReferenceComputedDeviation
Magnification factor δ/δ₀2.52.479210.832%