Stress Distribution of a thick Tube, subjecting to an Internal Pressure, which has a Cross Section of Regular Polygon and a Circular Hole in its Center

Abstract

(This Paper was read before the Meeting of the Japan Society of Mechanical Engineers and Applied Mechanics held in Nagoya on the 26th. Nov. 1945) The solution for an internal pressure acting at a circular hole in the center of a regular polygonal tube can be obtained from the following complex function z=C∫ (1-tⁿ)ⁿ^^2 dt, z=x+iy, t=ρ (cos θ+i sin θ) using the orthogonal curvilinear coordinate. The differential equation for Airy's stress function F in this case becomes [numerical formula] where h²=c^-2 (1-2ρⁿcosnθ+ρ²ⁿ) n^^2 The boundary conditions for this problem are σ_ρ=0, τ_ρθ=0 for ρ=1 σ_ρ=-p, τ_ρθ=0 for ρ=ρ₀, where σ_ρ, σ_θ, represent a normal stress acting a curve of ρ=const. and θ=const. on the x, y plane, and τ_ρθ a tangential stress acting along the both curves. Then σ_ρ, σ_θ, τ_ρθ and F can be write in the following form : [numerical formula] where A_o, A_s, B_s, C_o, C_s and D_s are constants and they are determined by the boundary conditions. The details are descibed in the original paper.