Buckling of Plates and Columns Questions and Answers

Answer: A

Correct answer: EI over effective length squared

\[ P_{cr} = \frac{\pi^2 EI}{L_e^2}. \] Two things follow at once. The critical load depends on the length squared, so doubling an unsupported length quarters the capacity; and the material enters only through \( E \), so the yield stress plays no part - a high-strength alloy buckles at the same load as a mild one of the same section.

Answer: B

At the critical load the straight configuration ceases to be the only equilibrium state, and a bent form becomes possible at the same load. Nothing about the material has failed, which is why the yield stress plays no part in the classical result.

Answer: C

The pinned-pinned case is the reference against which the others are quoted. Fixing both ends halves the effective length and so quadruples the critical load, while a cantilevered column doubles it and cuts the load to a quarter.

Answer: D

The cantilevered column is the weakest of the standard cases, with an effective length twice its actual length and a critical load one quarter of the pinned-pinned value. An undercarriage leg approximates this condition.

Answer: A

Clamping both ends halves the effective length, so the critical load is four times the pinned-pinned value. Real end fittings never achieve full fixity, so a factor between the theoretical value and one is normally assumed in design.

Answer: B

Slenderness combines the column's length with the shape of its section into a single parameter that governs the buckling stress. The least radius of gyration is used because the column will buckle about whichever axis is weakest.

Answer: C

It expresses how far the material effectively lies from the neutral axis and has units of length. A section with a large radius of gyration for its area is efficient in compression, which is why tubes and wide-flanged sections are favoured for struts.

Answer: D

Dividing the critical load by the area gives pi squared E over slenderness squared. The curve falls away steeply as the column becomes more slender, and it rises without limit for very short columns, which is where the formula stops being physically sensible.

Answer: A

Below a transition slenderness the formula returns a stress the material cannot reach elastically, so the column yields or crushes before it buckles. The inelastic range is covered by empirical or tangent modulus treatments instead.

Answer: B

It fits a parabola tangent to the Euler curve at the transition slenderness and passing through the yield stress at zero slenderness. The result is a continuous curve covering the practical range in which most real struts fall.