A solid circular bar is surrounded by a layer of copper rigidly attached to it, the assembly is subjected to a torsional moment if the copper supports 1.5 times the torque supported by the material of the circular bar. Find the relationship between the outer and inner diameters of the copper tube which has a young's modulus of 4x10^3 Kg/mm2 and the modulus of rigidity is 8x10^3 Kg/mm2

An idealized model of a thin-walled convertible car body cross-section is shown in figure Q16. The effect of doors and windows has been ignored. The car has vertical side walls of thickness t1 and length 1 m and a horizontal base of thickness t2 and length 1.9 m. It is constructed from aluminium with a shear modulus of 24 GPa. t₁ = 15 mm t2 = 15 mm a. For this open section determine the torsional stiffness? Give your answer to 3 significant figures in Pa m4 and non-scientific notation e.g. 10000 rather than 10e05. Do not include units in your answer Note that for an open section, the total area moment JT is the sum over all plates i as given by equation (1) below. Ir = = = Ei bit? Torsional stiffness = GJT t₂ 1.9 m 1m Figure Q16 - idealised model of a thin-walled convertible car body

Consider a two-component rod fixed at the left end. The first component (on the left, positioned from z=0 to z=L/2) is a solid rod with a shear modulus of G and radius c1. The second component (on the right, positioned from z=L/2 to z=L) is a hollow pipe fixed to component 1 at z=L/2 with the same shear modulus and outer radius c1, but with an inner radius of c2. A torque of 6To is applied at z=L/2, and a torque of -2To is applied at z=L. What value of c2 is required to make the maximum shear stress equivalent in each component? What is the angle of twist at the free end at z=L?

A solid steel bar 250 mm long is 10 mm square over part of its length, the remainder being circular with a diameter of 25 mm. A gradually applied tensile load P kN induces maximum stress in the bar of 200 MPa. The total strain energy in the bar when this stress of 200 MPa  is acting is 1,3 Joule. Take E = 200 GPa   Calculate : The load The length of the square portion of the bar The load that falls from a height of 8 mm and induces 1,3 Joules in the bar

A thin tube of uniform thickness 5 mm and square cross section, shown below, is subject to a torque of 1500 N·m. The shear modulus of the material is 80 GPa. What will be the angle of twist per length (in unit: rad/m)? 50 mm 50 mm

Q3

The figure illustrates a torsion-bar spring OA having a diameter d = 12 mm. The actuating cantilever AB also has d = 12 mm. Both parts are of carbon steel. Use Castigliano’s theorem and find the spring rake k corresponding to a force F acting at B.

4. A ball in a cylindrical inner race has a radius of 5mm. The radius from the center of the race to the bottom of the groove is 20mm. If a load of 1,500N is applied: A. Find the dimensions of the contact ellipse B. Find the maximum deformation *Esteel = 207MPA, Vsteel = 0.3

Problem 15 The compound bar shown below is made of a light, strong material with E= 70,000 N/mm². Two forces are applied to the rigid plates connecting the bars. The yield strength is o, = 200 N/mm². The bar has a rectangular cross section with height H = 12.5 mm and thickness t = 1.0 mm. You may neglect the weight of the bars. (a) Calculate the x-direction internal force in segments AB and BC of the compound bar. Show all work. (b) Calculate the total length change of the compound bar. (c) The plane 1-2 represents a lap joint. Calculate the normal stress on the lap joint. L₂= 625 mm A L₁ = 775 mm H = 12.5 mm X 1800 N B 1 H = 12.5 mm =35° 2 F₂ = 1000 N Cross section H-12 H = 12.5 mm t = 1.0 mm