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2.13 (B). A 10 mm long column is constructed from two 375 mm x 100 mm channels placed back to back with a distance h between their centroids and connected together by means of narrow batten plates, the effects of which may be ignored. Determine the value of h at which the section develops its maximum resistance to buckling.

Estimate the safe axial load on the column using the Perry-Robertson formula (a) with a load factor of 2, (b) with a factor of safety of 2. For each channel= 175 x 10"6 m4,/vy = 7 x 10~6 m4,/\ = 6.25 x 10"3 m2, E = 210 GN/m2 and yield stress = 300 MN/m2. Assume i] = 0.003 L/k and that the ends of the column are effectively pinned. [328 mm; 1.46, 1.55 MN.]

2.14 (B). (a) Compare the buckling loads that would be obtained from the Rankine-Gordon formula for two identical steel columns, one having both ends fixed, the other having pin-jointed ends, if the slenderness ratio is 100.

(b) A steel column, 6 m high, of square section 120 mm x 120 mm, is designed using the Rankine-Gordon expression to be used as a strut with both ends pin-jointed.

The values of the constants used were a = 1/7500, and ac = 300 MN/m2. If, in service, the load is applied axially but parallel to and a distance x from the vertical centroidal axis, calculate the maximum permissible value of x. Take E = 200 GN/m2. [7.4; 0.756 m.]

2.15 (B). Determine the maximum compressive stress set up in a 200 mm x 60 mm I-section girder carrying a load of 100 kN with an eccentricity of 6 mm. Assume that the ends of the strut are pin-jointed and that the overall length is 4 m.

Take / = 3 x 10"6 m4; A = 6 x 10"3 m2 and E = 207 GN/m2. [25.4 MN/m2.]

2.16 (B). A slender strut, initially straight, is pinned at each end. It is to be subjected to an eccentric compressive load whose line of action is parallel to the original centre-line of the strut.

(a) Prove that the central deflection y of the strut, relative to its initial centre-line, is given by the expression y = e

Secl2\j(lt)

where P is the applied load, L is the effective length of the strut, e is the eccentricity of the line of action of the load from the initially straight strut axis and EI is the flexural rigidity of the strut cross-section, (b) Using the above formula, and assuming that the strut is made of a ductile material, show that, for a maximum compressive stress, a, the value of P is given by the expression oA

the symbols A, h and k having their usual meanings, (c) Such a strut, of constant tubular cross-section throughout, has an outside diameter of 64 mm, a principal second moment of area of 52 x 10~8m4 and a cross-sectional area of 12.56 x 10_4m2. The effective length of the strut is 2.5 m. If P = 120 kN and a = 300 MN/m2, determine the permissible value of e. Take E = 200 GN/m2.

2.17 (C). A strut of length L has each end fixed in an elastic material which can exert a restraining moment p-per radian. Prove that the critical load P is given by the equation

The designed buckling load of a 1 m long strut, assuming the ends to be rigidly fixed, was 2.5 kN. If, during service, the ends were found to rotate with each mounting exerting a restraining moment of 1 kN m per radian, show that the buckling load decreases by 20%. [C.E.I.]

2.18 (C). A uniform elastic bar of circular cross-section and of length L, free at one end and rigidly built-in at the other end, is subjected to a single concentrated load P at the free end. In general the line of action of P may be at an angle 9 to the axis of the bar (0 < 9 < n/2) so that the bar is simultaneously compressed and bent. For this general case:

(a) Show that the deflection at the free end is given by tan mL

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