[{"data":1,"prerenderedAt":24},["ShallowReactive",2],{"formula-en-thin-pressure-vessel-membrane-stress":3},{"formula":4,"related_topics":12,"alternates":21},{"id":5,"code":6,"slug":6,"title":7,"formula":8,"locale":9,"seo_title":10,"seo_description":10,"content_html":11},122,"thin-pressure-vessel-membrane-stress","Membrane Stresses in Thin-Walled Cylinders and Spheres","σθ = pr \u002F t","en",null,"\u003Cdiv class=\"formula-chunk\">\u003Cp>For a thin-walled closed cylindrical shell of radius $r$ and thickness $t$ under uniform internal gauge pressure $p$, sufficiently far from edges and local disturbances:\u003C\u002Fp>\u003Cp>$$N_\\theta=pr,\\qquad \\sigma_\\theta=\\frac{pr}{t},$$\u003C\u002Fp>\u003Cp>$$N_z=\\frac{pr}{2},\\qquad \\sigma_z=\\frac{pr}{2t}.$$\u003C\u002Fp>\u003Cp>The hoop stress $\\sigma_\\theta$ is twice the longitudinal stress $\\sigma_z$.\u003C\u002Fp>\u003Cp>For a thin-walled closed spherical shell:\u003C\u002Fp>\u003Cp>$$N=\\frac{pr}{2},\\qquad \\sigma=\\frac{pr}{2t}.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$p$\u003C\u002Fstrong> — internal gauge pressure;\u003C\u002Fli>\u003Cli>\u003Cstrong>$r$\u003C\u002Fstrong> — shell radius;\u003C\u002Fli>\u003Cli>\u003Cstrong>$t$\u003C\u002Fstrong> — wall thickness;\u003C\u002Fli>\u003Cli>\u003Cstrong>$N_\\theta$, $N_z$\u003C\u002Fstrong> — hoop and longitudinal membrane force resultants per unit length;\u003C\u002Fli>\u003Cli>\u003Cstrong>$\\sigma_\\theta$, $\\sigma_z$\u003C\u002Fstrong> — hoop and longitudinal membrane stresses;\u003C\u002Fli>\u003Cli>\u003Cstrong>$N$, $\\sigma$\u003C\u002Fstrong> — membrane resultant and stress in a spherical shell.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>These are membrane approximations for $t\\ll r$ and uniform pressure. Additional bending and local stresses may occur near edges, openings, supports, and abrupt geometric changes.\u003C\u002Fp>\u003Cp>The calculator below uses the hoop-stress relation for a thin-walled cylinder.\u003C\u002Fp>\u003Cdiv data-formula-calculator-config=\"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\">\u003C\u002Fdiv>\u003C\u002Fdiv>",[13,17],{"id":14,"name":15,"path":16},125,"Membrane Theory","strength-of-materials\u002Fshell-analysis\u002Fmembrane-theory-of-shells",{"id":18,"name":19,"path":20},124,"Shell Analysis","strength-of-materials\u002Fshell-analysis",{"en":22,"uk":23},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Fformulas\u002Fthin-pressure-vessel-membrane-stress","https:\u002F\u002Fmechclassroom.com\u002Fformulas\u002Fthin-pressure-vessel-membrane-stress",1787712543143]