Problem #2 (25 points): Consider a thin-walled cylindrical pressure vessel (R/t >> 1) with closed ends, and internal gage pressure Pint as shown in the schematic below. A stress element with faces parallel and perpendicular to the axis of the cylinder is also shown in the figure. The normal stresses d, and d₂ are acting on the side faces of this element as shown. No shear stresses act on the faces of this element because of the symmetry of the vessel. Therefore, stresses σ, and o₂ are also the principal stresses. Stress 01 is called the circumferential stress, or hoop stress. Stress d₂ is called the longitudinal stress, or axial stress. Each of the stresses can be calculated from force-equilibrium equations using appropriate free body diagrams as shown below. 771 P b n 9 Free body diagrams for calculating circumferential stress σ, and longitudinal stress d₂ in terms of internal pressure Pint, radius of the cylinder R, and thickness t, along with their expressions are shown below: m R Pint Pint 12 Circumferential stress or Hoop Longitudinal b Stress Stress σ => Pint R t Pint R 0₂ = 2t 02