[{"data":1,"prerenderedAt":40},["ShallowReactive",2],{"topic-en-strength-of-materials\u002Fshell-analysis":3},{"topic":4,"trail":25,"children":31,"tasks":36,"alternates":37},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},124,"Shell Analysis","en","strength-of-materials\u002Fshell-analysis","Shell Analysis — Membrane Forces and Thin-Walled Structures","Thin-walled shell analysis: midsurface geometry, membrane force resultants, stresses, Laplace equilibrium, pressure vessels, and strength checks.","This section introduces the strength analysis of thin-walled shells. It covers midsurface geometry, principal radii of curvature, membrane force resultants and stresses, Laplace equilibrium, cylindrical and spherical pressure vessels, strength assessment, and the limitations of membrane theory near edges and local disturbances.","\u003Cp>A \u003Cstrong>shell\u003C\u002Fstrong> is a thin-walled spatial structural element whose thickness $t$ is much smaller than the characteristic dimensions of its curved midsurface. Examples include tanks, large-diameter pipes, domes, pressure-vessel walls, and other curved thin structures.\u003C\u002Fp>\u003Ch2>Midsurface\u003C\u002Fh2>\u003Cp>The geometry of a thin shell is conveniently described by its midsurface, located approximately midway through the thickness. At each point, two principal directions of curvature can be identified with radii $R_1$ and $R_2$. For a shell of revolution, these commonly correspond to meridional and circumferential directions.\u003C\u002Fp>\u003Ch2>Internal force resultants\u003C\u002Fh2>\u003Cp>General shell theory may include membrane forces, transverse shear forces, bending moments, and twisting moments per unit length. In membrane theory, the dominant resultants lie in the tangent plane of the midsurface.\u003C\u002Fp>\u003Cp>For normal membrane resultants $N_1$ and $N_2$ and shell thickness $t$, the corresponding average stresses are:\u003C\u002Fp>\u003Cp>$$\\sigma_1=\\frac{N_1}{t},\\qquad \\sigma_2=\\frac{N_2}{t}.$$\u003C\u002Fp>\u003Ch2>Equilibrium of a curved element\u003C\u002Fh2>\u003Cdiv class=\"formula-chunk\">\u003Cp>For an element of a thin shell in a membrane state under normal pressure $p$, local equilibrium in the surface-normal direction gives:\u003C\u002Fp>\u003Cp>$$\\frac{N_1}{R_1}+\\frac{N_2}{R_2}=p.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$N_1$, $N_2$\u003C\u002Fstrong> — membrane normal force resultants per unit length in the two principal directions of the midsurface;\u003C\u002Fli>\u003Cli>\u003Cstrong>$R_1$, $R_2$\u003C\u002Fstrong> — corresponding principal radii of curvature;\u003C\u002Fli>\u003Cli>\u003Cstrong>$p$\u003C\u002Fstrong> — resultant normal load per unit area of the midsurface.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>For a shell of constant thickness $t$, the corresponding average normal stresses are $\\sigma_1=N_1\u002Ft$ and $\\sigma_2=N_2\u002Ft$. Signs must follow the adopted convention for curvature, pressure, and membrane forces.\u003C\u002Fp>\u003Cdiv data-formula-calculator-config=\"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\">\u003C\u002Fdiv>\u003C\u002Fdiv>\u003Cp>Curvature allows in-plane membrane forces to balance loading normal to the surface. This is why shells can carry pressure efficiently with relatively small wall thickness.\u003C\u002Fp>\u003Ch2>Cylindrical and spherical shells\u003C\u002Fh2>\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>\u003Cp>These relations are basic examples of membrane action under internal pressure and illustrate how geometry affects stress distribution.\u003C\u002Fp>\u003Ch2>Strength assessment\u003C\u002Fh2>\u003Cp>After membrane stresses $\\sigma_1$ and $\\sigma_2$ are determined, the critical stress state is assessed. For ductile isotropic materials, an appropriate multiaxial failure criterion may be used; the selected criterion and allowable values depend on the material and design method.\u003C\u002Fp>\u003Ch2>When membrane theory is insufficient\u003C\u002Fh2>\u003Cp>Near rigid restraints, flanges, supports, openings, nozzles, joints, concentrated forces, and abrupt changes in thickness or curvature, edge and local effects can generate significant bending moments. Membrane theory should therefore not be used automatically for local verification in these regions.\u003C\u002Fp>\u003Ch2>Calculation procedure\u003C\u002Fh2>\u003Cdiv class=\"algorithm-chunk\">\u003Col>\u003Cli>Check that the shell is thin-walled and that bending moments may reasonably be neglected in the region being analyzed.\u003C\u002Fli>\u003Cli>Define the midsurface, thickness $t$, principal directions, and radii of curvature $R_1$ and $R_2$.\u003C\u002Fli>\u003Cli>Determine the external loading: normal pressure $p$, self-weight, or other distributed actions.\u003C\u002Fli>\u003Cli>Write the local shell equilibrium equations and, where necessary, equilibrium of a cut-off portion of the shell.\u003C\u002Fli>\u003Cli>Determine membrane force resultants $N_1$ and $N_2$ per unit length.\u003C\u002Fli>\u003Cli>Calculate average membrane stresses $\\sigma_1=N_1\u002Ft$ and $\\sigma_2=N_2\u002Ft$.\u003C\u002Fli>\u003Cli>Identify critical regions and perform the required strength check using the adopted criterion.\u003C\u002Fli>\u003Cli>Assess edges, openings, supports, joints, concentrated loads, and abrupt changes in geometry separately because membrane theory may be insufficient there.\u003C\u002Fli>\u003C\u002Fol>\u003C\u002Fdiv>\u003Cp>The child topic develops membrane theory in more detail, including assumptions, equilibrium relations, and typical pressure-shell calculations.\u003C\u002Fp>",[14,18,21],{"id":15,"code":16,"type":17,"locale":7},121,"shell-laplace-equilibrium","formula",{"id":19,"code":20,"type":17,"locale":7},122,"thin-pressure-vessel-membrane-stress",{"id":22,"code":23,"type":24,"locale":7},123,"membrane-shell-calculation-algorithm","algorithm",[26,30],{"id":27,"name":28,"path":29},45,"Strength of Materials","strength-of-materials",{"id":5,"name":6,"path":8},[32],{"id":33,"name":34,"path":35},125,"Membrane Theory","strength-of-materials\u002Fshell-analysis\u002Fmembrane-theory-of-shells",[],{"en":38,"uk":39},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Fstrength-of-materials\u002Fshell-analysis","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fopir-materialiv\u002Frozrakhunok-obolonok",1787712540994]