Consider an internally pressurized thick-walled spherical vessel,

(a). Develop an equation for the maximum shear stress at any radial position in the vessel, expressing this as a function of the radii r ₁ , r ₂ , and R. and the pressure p. Also show that the overall maximum shear stress in the vessel occurs at the inner wall.

(b). Assume that the vessel has an inner diameter of 100 mm and an outer diameter of 150 mm and contains an internal pressure of 300 MPa. Then determine the principal normal and shear stresses at the inner wall.

(c). For the same case as in (b). plot the variations of σ _r , σ _t , and T _max versus R.