What is the longitudinal stress in a thin cylindrical shell under internal pressure?

• A. PR/t
• B. PR/(2t)
• C. R t / P
• D. PR t

Answer: (A)

In a thin-walled cylinder under internal pressure, the wall carries membrane stresses in two perpendicular directions: around the circumference (hoop stress) and along the length (longitudinal stress). To balance the pressure on the ends, the end-cap force P times the end area π r^2 must be supported by the wall's axial tension. The axial tension is the longitudinal stress σ_longitudinal times the cross-sectional area of the wall resisting that load, which is the circumference times thickness, 2π r t. Setting the forces equal: σ_longitudinal · (2π r t) = P · π r^2. Solving gives σ_longitudinal = P r / (2 t). The hoop stress would be σ_hoop = P r / t, which is twice the longitudinal stress. So, the longitudinal stress is P r / (2 t). The other expressions either correspond to the hoop stress or are not dimensionally appropriate for stress.