[{"data":1,"prerenderedAt":24},["ShallowReactive",2],{"formula-en-shell-laplace-equilibrium":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},121,"shell-laplace-equilibrium","Laplace Equilibrium Equation for a Membrane Shell","p = N₁\u002FR₁ + N₂\u002FR₂","en",null,"\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>",[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\u002Fshell-laplace-equilibrium","https:\u002F\u002Fmechclassroom.com\u002Fformulas\u002Fshell-laplace-equilibrium",1787712543127]