If the maximum working stress in compression a for this strut is given by a = 135[1 — 0.005 L/k] MN/m2, what factor of safety must be used with the Rankine formula to give the same result? For each R.SJ. A = 6250 mm2, kx = 85 mm, ky = 35 mm. [180.6 mm, 6.23 MN; 2.32.]

2.6 (B). A stanchion is made from two 200 mm x 75 mm channels placed back to back, as shown in Fig. 2.19, with suitable diagonal bracing across the flanges. For each channel Ixx = 20 x 10-6m4, /vv = 1.5 x 10-6 m4, the cross-sectional area is 3.5 x 10~3 m2 and the centroid is 21 mm from the back of the web.

At what value of p will the radius of gyration of the whole cross-section be the same about the X and Y axes? The strut is 6 m long and is pin-ended. Find the Euler load for the strut and discuss briefly the factors which cause the actual failure load of such a strut to be less than the Euler load. £ = 210 GN/m2. [163.6 mm; 2.3 MN.]

2.7 (B). In tests it was found that a tube 2 m long, 50 mm outside diameter and 2 mm thick when used as a pin-jointed strut failed at a load of 43 kN. In a compression test on a short length of this tube failure occurred at a load of 115 kN.

(a) Determine whether the value of the critical load obtained agrees with that given by the Euler theory.

(b) Find from the test results the value of the constant a in the Rankine-Gordon formula. Assume E = 200 GN/m2.

2.8 (B). Plot, on the same axes, graphs of the crippling stresses for pin-ended struts as given by the Euler and Rankine-Gordon formulae, showing the variation of stress with slenderness ratio

21 mm

21 mm

For the Euler formula use L/k values from 80 to 150, and for the Rankine formula L/k from 0 to 150, with <TV = 315 MN/m2 and a = 1/7500.

From the graphs determine the values of the stresses given by the two formulae when L/k = 130 and the slenderness ratio required by both formulae for a crippling stress of 135 MN/m2. E = 210 GN/m2.

2.9 (B/C). A timber strut is 75 mm x 75 mm square-section and is 3 m high. The base is rigidly built-in and the top is unrestrained. A bracket at the top of the strut carries a vertical load of 1 kN which is offset 150 mm from the centre-line of the strut in the direction of one of the principal axes of the cross-section. Find the maximum stress in the strut at its base cross-section if E = 9 GN/m2. [I.Mech.E.] [2.3 MN/m2.]

2.10 (B/C). A slender column is built-in at one end and an eccentric load is applied at the free end. Working from first principles find the expression for the maximum length of column such that the deflection of the free end does not exceed the eccentricity of loading. [I.Mech.E.] [sec-1 2/^/(P/EI).]

2.11 (B/C). A slender column is built-in one end and an eccentric load of 600 kN is applied at the other (free) end. The column is made from a steel tube of 150 mm o.d. and 125 mm i.d. and it is 3 m long. Deduce the equation for the deflection of the free end of the beam and calculate the maximum permissible eccentricity of load if the maximum stress is not to exceed 225 MN/m2. E = 200 GN/m2. [I.Mech.E.] [4 mm.]

2.12 (B). A compound column is built up of two 300 mm x 125 mm R.S.J.s arranged as shown in Fig. 2.20. The joists are braced together; the effects of this bracing on the stiffness may, however, be neglected. Determine the safe height of the column if it is to carry an axial load of 1 MN. Properties of joists: A = 6 x I0~3 m2; kyy = 27 mm; kxx = 125 mm.

The allowable stresses given by BS449: 1964 may be found from the graph of Fig. 2.9. [8.65 m.]

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