[{"data":1,"prerenderedAt":39},["ShallowReactive",2],{"topic-en-strength-of-materials\u002Fgeometric-properties-of-plane-areas\u002Fparallel-axis-theorem-and-composite-sections":3},{"topic":4,"trail":24,"children":34,"tasks":35,"alternates":36},{"id":5,"name":6,"locale":7,"path":8,"seo_title":6,"seo_description":9,"seo_text":10,"content_html":11,"content_chunks":12},121,"Parallel-Axis Theorem and Composite Sections","en","strength-of-materials\u002Fgeometric-properties-of-plane-areas\u002Fparallel-axis-theorem-and-composite-sections","Transfer second moments to parallel axes and calculate composite-section centroids and moments of inertia, including holes and simple components.","This topic explains the parallel-axis theorem and its application to composite cross-sections. It covers decomposition into simple shapes, centroid determination, transfer of component second moments to common centroidal axes, algebraic summation, and treatment of holes.","\u003Cp>For a composite section, tabulated properties of individual simple shapes are not sufficient because their second moments must first be referred to a common axis. The \u003Cstrong>parallel-axis theorem\u003C\u002Fstrong> provides this transfer.\u003C\u002Fp>\u003Cdiv class=\"formula-chunk\">\u003Cp>The second moment of area about an axis parallel to a centroidal axis is found from the parallel-axis theorem:\u003C\u002Fp>\u003Cp>$$I_x=I_{x_c}+Aa^2,$$\u003C\u002Fp>\u003Cp>where \u003Cstrong>Ixc\u003C\u002Fstrong> is the second moment about the parallel centroidal axis, \u003Cstrong>A\u003C\u002Fstrong> is the area, and \u003Cstrong>a\u003C\u002Fstrong> is the distance between the axes.\u003C\u002Fp>\u003Cp>Similarly:\u003C\u002Fp>\u003Cp>$$I_y=I_{y_c}+Ab^2.$$\u003C\u002Fp>\u003Cp>For a composite section, the transferred contributions of all parts are summed algebraically; holes are treated with negative area.\u003C\u002Fp>\u003Cdiv data-formula-calculator-config=\"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\">\u003C\u002Fdiv>\u003C\u002Fdiv>\u003Ch2>Why the centroid comes first\u003C\u002Fh2>\u003Cp>Centroidal second moments of a composite section are calculated about axes through the centroid of the complete area. Therefore, the first step is to determine that centroid.\u003C\u002Fp>\u003Cdiv class=\"formula-chunk\">\u003Cp>The first moments of area about the x and y axes are:\u003C\u002Fp>\u003Cp>$$S_x=\\int_A y\\,dA,\\qquad S_y=\\int_A x\\,dA.$$\u003C\u002Fp>\u003Cp>The centroid coordinates are:\u003C\u002Fp>\u003Cp>$$x_c=\\frac{S_y}{A},\\qquad y_c=\\frac{S_x}{A}.$$\u003C\u002Fp>\u003Cp>For a composite section divided into simple parts:\u003C\u002Fp>\u003Cp>$$x_c=\\frac{\\sum_i A_i x_i}{\\sum_i A_i},\\qquad y_c=\\frac{\\sum_i A_i y_i}{\\sum_i A_i}.$$\u003C\u002Fp>\u003Cp>Holes can conveniently be treated as negative areas in algebraic summation.\u003C\u002Fp>\u003C\u002Fdiv>\u003Ch2>Procedure\u003C\u002Fh2>\u003Cdiv class=\"algorithm-chunk\">\u003Col>\u003Cli>Divide the composite section into simple shapes with known geometric properties. Treat holes as negative areas.\u003C\u002Fli>\u003Cli>Choose a convenient reference coordinate system and determine the areas Ai and centroid coordinates xi, yi of all parts.\u003C\u002Fli>\u003Cli>Find the centroid of the complete section from the algebraic sums Ai xi and Ai yi.\u003C\u002Fli>\u003Cli>Draw centroidal axes through the calculated centroid.\u003C\u002Fli>\u003Cli>Determine the centroidal second moments of each part and transfer them to the common centroidal axes using the parallel-axis theorem.\u003C\u002Fli>\u003Cli>Sum the contributions algebraically to obtain Ix, Iy, and, if required, Ixy.\u003C\u002Fli>\u003Cli>If principal axes are required, determine their rotation angle and the principal second moments of area.\u003C\u002Fli>\u003Cli>For bending calculations, determine the section moduli for the required extreme fibers and verify units and geometric symmetry.\u003C\u002Fli>\u003C\u002Fol>\u003C\u002Fdiv>\u003Ch2>Holes\u003C\u002Fh2>\u003Cp>A hole can be treated as a negative component: its area, first moments, and transferred second moment are subtracted from the corresponding sums for solid parts.\u003C\u002Fp>\u003Ch2>Example structure\u003C\u002Fh2>\u003Cp>For a T-section, represent the flange and web as two rectangles. First determine the centroid coordinate of the complete section. Then calculate the centroidal $I_x$ of each rectangle and add $A_i a_i^2$, where $a_i$ is the distance between the component centroid and the common centroidal axis.\u003C\u002Fp>\u003Ch2>Check\u003C\u002Fh2>\u003Cp>Verify that all distances are measured to the same reference axis, units are consistent, and the $Aa^2$ term has not been added to a second moment that is already taken about the required common axis.\u003C\u002Fp>",[13,17,20],{"id":14,"code":15,"type":16,"locale":7},89,"parallel-axis-theorem","formula",{"id":18,"code":19,"type":16,"locale":7},87,"area-static-moments-centroid",{"id":21,"code":22,"type":23,"locale":7},91,"composite-section-properties-algorithm","algorithm",[25,29,33],{"id":26,"name":27,"path":28},45,"Strength of Materials","strength-of-materials",{"id":30,"name":31,"path":32},59,"Geometric Properties of Plane Areas","strength-of-materials\u002Fgeometric-properties-of-plane-areas",{"id":5,"name":6,"path":8},[],[],{"en":37,"uk":38},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Fstrength-of-materials\u002Fgeometric-properties-of-plane-areas\u002Fparallel-axis-theorem-and-composite-sections","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fopir-materialiv\u002Fheometrychni-kharakterystyky-pereriziv\u002Fteorema-shteinera-ta-skladeni-pererizy",1787712539938]