Consider the two pressure vessel geometries shown below. Both are thin-walled vessels made of titanium 6Al-4V;...

  

Transcribed Image Text:

Consider the two pressure vessel geometries shown below. Both are thin-walled vessels made of titanium 6Al-4V; both are to be pressurized from 0 to the same pressure, p; and both have the same internal volume, Vo (the volume of the internal cavity, not the volume of wall material). The first is a spherical vessel with radius r. The second is a cylindrical vessel with hemispherical end caps of radius Tes and in between, a cylindrical portion with length L = 8re. The vessel initial volumes, pressures, and titanium mechanical properties (Young's modulus, Poisson's ratio, yield stress and density) are: Vo= 0.639 m², p = 30 MPa, E = 114 GPa, v= 0.34, oy= 880 MPa, p= 4.43 g/cm? a) Find the respective constant wall thicknesses, t, or te, needed for each vessel type to support pressure, p, with a factor of safety against yielding of 1. b) Find the resulting weight of each vessel, W, or We, when empty. c) When the pressure is increased from 0 to p, calculate the thickness change, At, or Ate, observed in each vessel type and where it occurs. d) Find the volume change, AV, or AVe, in each vessel when pressurized from 0 to p. e) In comparison to the result obtained in part (b), how much weight can be saved in the cylindrical vessel supporting pressure p, by allowing a reduction in the material thickness in locations where the maximum stress (whether hoop or axial) happens to be significantly less than the yields stress, oy= 880 MPa. Note: We ignore any effects of the small transition region in the cylindrical vessel where the end caps join the cylindrical middle section. -87- 2re Consider the two pressure vessel geometries shown below. Both are thin-walled vessels made of titanium 6Al-4V; both are to be pressurized from 0 to the same pressure, p; and both have the same internal volume, Vo (the volume of the internal cavity, not the volume of wall material). The first is a spherical vessel with radius r. The second is a cylindrical vessel with hemispherical end caps of radius Tes and in between, a cylindrical portion with length L = 8re. The vessel initial volumes, pressures, and titanium mechanical properties (Young's modulus, Poisson's ratio, yield stress and density) are: Vo= 0.639 m², p = 30 MPa, E = 114 GPa, v= 0.34, oy= 880 MPa, p= 4.43 g/cm? a) Find the respective constant wall thicknesses, t, or te, needed for each vessel type to support pressure, p, with a factor of safety against yielding of 1. b) Find the resulting weight of each vessel, W, or We, when empty. c) When the pressure is increased from 0 to p, calculate the thickness change, At, or Ate, observed in each vessel type and where it occurs. d) Find the volume change, AV, or AVe, in each vessel when pressurized from 0 to p. e) In comparison to the result obtained in part (b), how much weight can be saved in the cylindrical vessel supporting pressure p, by allowing a reduction in the material thickness in locations where the maximum stress (whether hoop or axial) happens to be significantly less than the yields stress, oy= 880 MPa. Note: We ignore any effects of the small transition region in the cylindrical vessel where the end caps join the cylindrical middle section. -87- 2re