Tutorial 8

For the following problems, assume ideal, slender, prismatic, linearly elastic columns (Euler buckling). Buckling occurs in the plane of the figure unless otherwise stated.

Problems

1. Calculate the critical load, Pcr, for a USB 305×127×48 kg steel column having length, L = 7 m, and E = 200 GPa under the following conditions: (a) the column buckles by bending about its strong axis (axis 1-1), and (b) the column buckles by bending about its weak axis (axis 2-2) 1.

In both cases, assume that the column has pinned ends and use the following standard structural steel property data for moment of inertia, I, and radius of gyration, r:

I (cm4) r (cm) Axis 1-1 9485 12.5 Axis 2-2 438 2.68

Answer: (a) 3.821 MN, and (b) 176.44 kN.

1 J. M. Gere, p. 798 in Mechanics of materials, 5th SI edition (Nelson Thornes Ltd., Cheltenham) (2002).

Materials 337 (Advanced Strength of Materials)

2

2. A horizontal beam, AB, is supported by a pinned-end column, CD. The column is a solid steel bar (E = 200 GPa) of square cross-section having length, L = 1.8 m, and side dimensions, b = 50 mm. Based upon the critical load of the column, determine the allowable load, Q, if the factor of safety with respect to buckling is n = 2.0 2.

Answer: 52.88 kN.

3. Three pinned-end columns of the same material have the same length and the same cross-sectional area. The columns are free to buckle in any direction. The columns have cross-sections as follows: (a) a circle, (b) a square, and (c) an equilateral triangle. Determine the ratios P1 : P2 : P3 of the critical loads for these columns3.

Answer: P1 : P2 : P3 = 1 : 1.047 : 1.209.

4. The hoisting arrangement for lifting a large pipe is shown in the figure below. The spreader is a steel tubular section with outer diameter 70 mm and inner diameter 55 mm. Its length is 2.5 m and its modulus of elasticity is 200 GPa. Based upon a factor of safety of 2.0 with respect to Euler buckling of the spreader, what is the maximum weight of pipe that can be lifted? (Assume pinned conditions at the end of the spreader.)4

Answer: 161.26 kN.

2 J. M. Gere, p. 799 in Mechanics of materials, 5th SI edition (Nelson Thornes Ltd., Cheltenham) (2002). 3 J. M. Gere, p. 800 in Mechanics of materials, 5th SI edition (Nelson Thornes Ltd., Cheltenham) (2002). 4 J. M. Gere, p. 801 in Mechanics of materials, 5th SI edition (Nelson Thornes Ltd., Cheltenham) (2002).