[{"data":1,"prerenderedAt":37},["ShallowReactive",2],{"topic-en-strength-of-materials\u002Fgeometric-properties-of-plane-areas\u002Fsecond-and-polar-moments-of-area":3},{"topic":4,"trail":22,"children":32,"tasks":33,"alternates":34},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},120,"Second and Polar Moments of Area","en","strength-of-materials\u002Fgeometric-properties-of-plane-areas\u002Fsecond-and-polar-moments-of-area","Second and Polar Moments of Area — Ix, Iy and Jp","Definitions of Ix, Iy, Ixy and Jp, geometric meaning, units, and standard formulas for rectangular, circular, and annular sections.","This topic introduces the second moments of area Ix and Iy, product of inertia Ixy, and polar moment Jp as geometric measures of area distribution. It explains their integral definitions, units, geometric meaning, and standard formulas for rectangles, circles, and annuli.","\u003Cp>A \u003Cstrong>second moment of area\u003C\u002Fstrong> characterizes how cross-sectional area is distributed relative to a selected axis. It is a geometric property and must not be confused with a mass moment of inertia.\u003C\u002Fp>\u003Cdiv class=\"formula-chunk\">\u003Cp>The second moments of area about the x and y axes are:\u003C\u002Fp>\u003Cp>$$I_x=\\int_A y^2\\,dA,\\qquad I_y=\\int_A x^2\\,dA.$$\u003C\u002Fp>\u003Cp>The product moment of area is:\u003C\u002Fp>\u003Cp>$$I_{xy}=\\int_A xy\\,dA.$$\u003C\u002Fp>\u003Cp>The polar second moment of area about the intersection of orthogonal axes is:\u003C\u002Fp>\u003Cp>$$J_p=\\int_A(x^2+y^2)dA=I_x+I_y.$$\u003C\u002Fp>\u003Cp>Second and polar moments of area have dimensions of length to the fourth power, for example mm⁴.\u003C\u002Fp>\u003C\u002Fdiv>\u003Ch2>Geometric meaning\u003C\u002Fh2>\u003Cp>Area elements located farther from an axis contribute much more strongly because the distance enters quadratically. This is why sections that place material far from the centroidal axis can achieve high bending stiffness with relatively modest area.\u003C\u002Fp>\u003Ch2>Standard shapes\u003C\u002Fh2>\u003Ctable>\u003Cthead>\u003Ctr>\u003Cth>Shape\u003C\u002Fth>\u003Cth>Area A\u003C\u002Fth>\u003Cth>Centroidal second moment\u003C\u002Fth>\u003Cth>Section modulus\u003C\u002Fth>\u003C\u002Ftr>\u003C\u002Fthead>\u003Ctbody>\u003Ctr>\u003Ctd>Rectangle b × h, x-axis parallel to side b\u003C\u002Ftd>\u003Ctd>$bh$\u003C\u002Ftd>\u003Ctd>$I_x=bh^3\u002F12$\u003C\u002Ftd>\u003Ctd>$W_x=bh^2\u002F6$\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>Rectangle b × h, y-axis parallel to side h\u003C\u002Ftd>\u003Ctd>$bh$\u003C\u002Ftd>\u003Ctd>$I_y=hb^3\u002F12$\u003C\u002Ftd>\u003Ctd>$W_y=hb^2\u002F6$\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>Circle of diameter d\u003C\u002Ftd>\u003Ctd>$\\pi d^2\u002F4$\u003C\u002Ftd>\u003Ctd>$I_x=I_y=\\pi d^4\u002F64$\u003C\u002Ftd>\u003Ctd>$W_x=W_y=\\pi d^3\u002F32$\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>Circle of diameter d, polar property\u003C\u002Ftd>\u003Ctd>$\\pi d^2\u002F4$\u003C\u002Ftd>\u003Ctd>$J_p=\\pi d^4\u002F32$\u003C\u002Ftd>\u003Ctd>$W_p=\\pi d^3\u002F16$\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Ctd>Annulus D, d\u003C\u002Ftd>\u003Ctd>$\\pi(D^2-d^2)\u002F4$\u003C\u002Ftd>\u003Ctd>$I_x=I_y=\\pi(D^4-d^4)\u002F64$\u003C\u002Ftd>\u003Ctd>$W_x=2I_x\u002FD$\u003C\u002Ftd>\u003C\u002Ftr>\u003C\u002Ftbody>\u003C\u002Ftable>\u003Ch2>Polar moment\u003C\u002Fh2>\u003Cp>For two mutually perpendicular axes $x$ and $y$ passing through the same point, $J_p=I_x+I_y$. For a circular shaft, this property enters directly into the classical torsion formulas.\u003C\u002Fp>\u003Ch2>Units\u003C\u002Fh2>\u003Cp>If geometric dimensions are given in millimetres, second moments of area are expressed in mm⁴. Because characteristic dimensions enter to the fourth power, relatively small dimensional changes can strongly affect $I$.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>For a rectangle with $b=40\\ \\text{mm}$ and $h=80\\ \\text{mm}$, about its centroidal $x$-axis parallel to $b$:\u003C\u002Fp>\u003Cp>$$I_x=\\frac{40\\cdot80^3}{12}\\approx1.707\\cdot10^6\\ \\text{mm}^4.$$\u003C\u002Fp>",[14,18],{"id":15,"code":16,"type":17,"locale":7},88,"area-second-moments","formula",{"id":19,"code":20,"type":21,"locale":7},90,"rectangle-circle-area-properties","table",[23,27,31],{"id":24,"name":25,"path":26},45,"Strength of Materials","strength-of-materials",{"id":28,"name":29,"path":30},59,"Geometric Properties of Plane Areas","strength-of-materials\u002Fgeometric-properties-of-plane-areas",{"id":5,"name":6,"path":8},[],[],{"en":35,"uk":36},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Fstrength-of-materials\u002Fgeometric-properties-of-plane-areas\u002Fsecond-and-polar-moments-of-area","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fopir-materialiv\u002Fheometrychni-kharakterystyky-pereriziv\u002Fosiovi-ta-poliarnyi-momenty-inertsii-ploshchi",1787712539864]