Cylinder problems

1. Derive the thin cylinder design equations ( ii).
    
2. It is required to choose a pipe from the following availability list, from AS 1835-1976, for a pipeline pressurised to 20 MPa. Use a design stress of 110 MPa in conjunction with the thin cylinder equations to select a suitable size.
   

   OD	Wall thickness
   	3.6	4.0	4.5	4.9	5.4	5.9	6.3	8.0	9.5
   60.3	X		X	X	X			X
   76.1	X		X		X	X	X		X
   88.9		X		X	X	X	X		X
   101.6					X		X		X
   114.3			X		X		X	X	X
   139.7				X	X		X	X	X
   165.1				X	X		X	X	X

    
3. It has been noted that the thin cylinder equations ( 1), which use the internal diameter D_i on the numerator of the stress equations, underestimate the stresses. The equations are sometimes modified by substituting, instead of D_i, either :
   • the mean diameter D_m = ( D_i + D_o )/2, as used in AS 1210, or
   • the outside diameter D_o (giving the so-called 'Barlow's formula')
   1. Calculate the error using D_i, D_m and D_o in turn for a closed steel cylinder with D_i/t = 25.
   2. Derive the basic AS 1210 design formula for thin cylinders subjected to internal pressure p_i, namely :
      t = D_i /( 2 σ_d / p_i - 1 )     where σ_d is the allowable (design) stress.
   3. A pressure vessel to contain steam at 2 MPa is designed to AS 1210 with a design stress of 80 MPa. The vessel is 780 mm bore, 2.5 m long and simply supported over a 2 m span. Is bending significant when the vessel is filled with water for an hydraulic test?
    
4. A tube is made from an aluminium alloy with properties E = 70 GPa, ν = 0.33. The tube OD and wall thickness are 90 and 2 mm. Determine the principal strains on the exterior surface when internally pressurised to 12 MPa. By how much does the OD increase under load?       [ 0.33 mm open; 0.28 mm closed ]
    
5. A thin-walled steel pipeline of 84 mm OD contains pressurised hot oil. Since it is feared that expansion is giving rise to bending, three strain gauges are mounted at the critical location and indicate the strains shown (in 10^-6 units). What is the maximum equivalent stress at the cross-section?       [ 208 MPa ]
    
6. A closed steel cylinder having bore and OD of 80 and 240 mm respectively, is pressurised to 60 MPa internally and 20 MPa externally. Plot the stress variation through the wall and calculate the maximum equivalent stress.       [ σ* = 90 MPa]
    
7. Verify the design equations ( viii).
    
8. Derive design equations, analogous to ( ix), for a brittle closed cylinder. Plot the failure locus for γ_o = 4 and m = 3.5 (a typical value for cast iron).
    
9. The table's boldface italic figures are known for each steel cylinder. Verify the remaining tabulated values. F_a is an axial compressive force applied externally to a cylinder's ends.
   

   		========== Inside ==========								===== Outside =====
   Cylinder		Fa	D	p	σt	σr	εD	σ*		D	p	σt	σr	εD
   		kN	mm	MPa	MPa	MPa	x104	MPa		mm	MPa	MPa	MPa	x104
   a	open	0	40	120	60	-120	4.6	180		120	40	-20	-40	-0.4
   b	open	0	20	190	125	-190	8.8	315		60	50	-15	-50	0
   c	open	0	15	120	40	-120	3.7	160		60	45	-35	-45	-1.0
   d	closed	0	60	75	125	-75	6.76	200		120	0	50	0	2.1
   e	closed	600	60	0	-59	0	-1.4	100		120	22	-37	-22	0

    
10. An open cylinder of 50 mm bore is required to withstand internal and external pressures of 55 and 45 MPa respectively. Determine the necessary OD if the safety factor is 4 and :
    1. the material's yield is 200 MPa; use the distortion energy theory       [ 69 mm ]
    2. ditto - but using the maximum shear stress criterion
    3. the brittle material has tensile and compressive ultimates of 200 and 700 MPa.       [ 56 mm ]
     
11. The surface diameters of an open, steel compound cylinder are 100, 180 and 270 mm. The components are assembled with an interference of 0.05 mm prior to loading with an internal pressure of 50 MPa. Plot the tangential, radial and equivalent stresses across the wall of each component after initial assembly and after loading.       [ σ*= 82 MPa ]
    If a single cylinder had been used in this application
    1. what pressure could it withstand if it were the same overall size and the equivalent stress was limited to 82 MPa?       [ 35 MPa ]
    2. what outside diameter would it need to be to contain the pressure of 50 MPa with a maximum equivalent stress of 82 MPa?
     
12. The wall of an air compressor's cylinder liner is 5 mm thick. When cold, the liner sits in the block with a diametral clearance of 0.05 mm. The temperatures of the liner and the block rise by 150 and 80 ^oC respectively when on load. If the mean pressure in the cylinder is then 1 MPa, what is the contact pressure between liner and block - which are both of steel?       [ 6.2 MPa ]
     
13. Two coaxial hollow shafts are to be coupled together by means of an interference fit, and are required to transmit a torque of 7.3 kNm. Both shafts are made from a 250 MPa yield steel. The inner shaft is 100 x 130 mm (bore x OD), the outer shaft is 130 x 155, and the effective axial length of the interference is 125 mm. Assuming a friction coefficient of 1/3, and neglecting axial non-uniformity :
    1. determine the necessary diametral interference       [ 0.04 mm ]
    2. plot the variation of equivalent stress through both walls
    3. ascertain the safety factor.       [ 4.9 ]
     
14. The overall diameter ratio of an internally pressurised open steel compound cylinder is Γ = (D_o/D_i)². Show that :
    1. the optimum choice of interface diameter D_c is such that :
             γ_o^inner   = γ_o^outer   =   Γ^1/2  =   1 /( 1 - n p_i / S_y )
    2. the corresponding optimum interference is :     Δ_opt = p_i D_c /E