{"id":139,"date":"2018-02-14T13:57:05","date_gmt":"2018-02-14T18:57:05","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/powr4406\/?post_type=chapter&#038;p=139"},"modified":"2018-06-25T11:55:12","modified_gmt":"2018-06-25T15:55:12","slug":"properties-of-areas","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/powr4406\/chapter\/properties-of-areas\/","title":{"raw":"Properties of Areas","rendered":"Properties of Areas"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3 itemprop=\"educationalUse\"><strong>Learning Objectives<\/strong><\/h3>\r\nUpon completion of this chapter you should be able to\r\n<ul>\r\n \t<li>Determine the centroid location of a given cross-section<\/li>\r\n \t<li>Calculate the moment of inertia for a given cross-section, with both SI and US Customary units<\/li>\r\n<\/ul>\r\n<\/div>\r\nFinding the location of the centroid is needed when calculating the moment of inertia (or second moment of areas) of beams subjected to bending.\u00a0 For convenience, you may used the table provided in <a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/back-matter\/appendix\/\">Appendix 1<\/a>.\r\n\r\nThe geometric properties of areas for common shapes are given in textbook Appendix C.\u00a0 Common industrial shapes like W-beams and pipes are listed in Appendix D.\r\n<div class=\"textbox shaded\" style=\"text-align: center\"><strong>Centroids of composite areas<\/strong><\/div>\r\nDetermining the location of the centroid of a composite area uses the concept of moment of an area; this is why textbooks may refer to this as \"first moments of areas\".\u00a0 Mathematically this principle is expressed as:\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig1.jpg\" alt=\"\" class=\"alignnone wp-image-149\" width=\"204\" height=\"91\" \/>\r\n\r\nwhere:\r\n<ul>\r\n \t<li>Y is the distance to the centroid from some reference axis.\u00a0 Commonly, the reference axis is the base of the figure.<\/li>\r\n \t<li>A<sub>i<\/sub> is the area of one part of the composite area.\u00a0 Typically, the composite areas are split into common shapes of known geometric properties, summarized in the textbook Appendix C.<\/li>\r\n \t<li>y<sub>i<\/sub> is the distance from the reference axis (commonly the base of the figure) to the centroid of each part of the composite figure.<\/li>\r\n \t<li>\u03a3(A<sub>i<\/sub>) is the entire area of the composite area.<\/li>\r\n<\/ul>\r\nWhen determining the location of a centroid please observe the following rules:\r\n<ul>\r\n \t<li>If the cross-section has one axis of symmetry then the centroid will be located on this axis.<\/li>\r\n \t<li>if the cross-section has two axes of symmetry then the centroid will be located at the intersection of the two axes.<\/li>\r\n \t<li>If the cross-section is not symmetric about any axis then two calculations are required:\r\n<ul>\r\n \t<li>one for determining the centroid location Y<\/li>\r\n \t<li>one for determining the centroid location X, commonly measured from the extreme left end.\u00a0 For this second calculation imagine that you rotate the figure 90\u00ba counter-clockwise and repeat the first calculation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>If the composite area has a part that is removed from the figure (a void), this missing part can be treated as a negative area.<\/li>\r\n<\/ul>\r\n<div class=\"textbox shaded\" style=\"text-align: center\"><strong>Moments of inertia of composite areas<\/strong><\/div>\r\nIn rotational kinetics we learned that the \"rotational\" moment of inertia of a flywheel (function of its mass, size and shape) represents a resistance to change in its motion. This moment of inertia multiplied by the angular acceleration \u03b1, gives an inertia-moment reaction that attempts to balance the accelerating moment action (accelerating torque).\u00a0 In general, a moment of inertia is a resistance to change.\r\n\r\nBeams are subject to bending and as a result they tend to deform (deflect).\u00a0 The moment of inertia of a beam cross-section can be related to the stiffness of the beam.\u00a0 The deflection of the beam is inverse proportional to the moment of inertia.\r\n\r\nFormulas for moments of inertia of simple shapes are given in Textbook Appendix C.\u00a0 They will also be provided in the exam.\r\n\r\nWhen dealing with a composite area, divide the shape into basic parts for which the moment of inertia can be easily calculated.\u00a0 The combined moment of inertia of the entire shape is the sum of moments of inertia of constituent parts plus their corresponding transfer term.\u00a0 The transfer term is calculated as the area of the part multiplied by the squared distance between the centroid of the part and the common centroid of the entire area.\u00a0 This transfer term represents the additional stiffness of each part due to its relative distance from the common centroid.\u00a0 The table given in Supplement Appendix 1 can be used for calculations; it is useful when the shape is more complex.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3 itemprop=\"educationalUse\"><strong>Assigned Problems<\/strong><\/h3>\r\nWhen completing these exercises please make sure that you clearly identify and number the parts of your composite area.\r\n\r\n<\/div>\r\n<strong>Problem 1: <\/strong>Determine the moment of inertia about the vertical and horizontal centroidal axes for the following figure.\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig2-1.jpg\" alt=\"\" class=\"alignnone wp-image-159\" width=\"302\" height=\"450\" \/>\r\n\r\n<strong>Problem 2: <\/strong>For the following cross-section determine the location (elevation) of the centroid and the moments of inertia with respect to the horizontal and vertical centroidal axes.\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig3-1.jpg\" alt=\"\" class=\"alignnone wp-image-160\" width=\"266\" height=\"370\" \/>\r\n\r\n<strong>Problem 3: <\/strong>For the following figure<span style=\"text-decoration: underline\"><\/span> determine Y, the vertical location of the centroid, and calculate the moment of inertia with respect to the <span style=\"text-decoration: underline\">horizontal<\/span> centroidal axis..\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig4-1.jpg\" alt=\"\" class=\"alignnone wp-image-161\" width=\"272\" height=\"274\" \/>\r\n\r\n<strong>Problem 4: <\/strong>Suggest one improvement to this chapter; this may include an original cross-section.\r\n\r\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3 itemprop=\"educationalUse\"><strong>Learning Objectives<\/strong><\/h3>\n<p>Upon completion of this chapter you should be able to<\/p>\n<ul>\n<li>Determine the centroid location of a given cross-section<\/li>\n<li>Calculate the moment of inertia for a given cross-section, with both SI and US Customary units<\/li>\n<\/ul>\n<\/div>\n<p>Finding the location of the centroid is needed when calculating the moment of inertia (or second moment of areas) of beams subjected to bending.\u00a0 For convenience, you may used the table provided in <a href=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/back-matter\/appendix\/\">Appendix 1<\/a>.<\/p>\n<p>The geometric properties of areas for common shapes are given in textbook Appendix C.\u00a0 Common industrial shapes like W-beams and pipes are listed in Appendix D.<\/p>\n<div class=\"textbox shaded\" style=\"text-align: center\"><strong>Centroids of composite areas<\/strong><\/div>\n<p>Determining the location of the centroid of a composite area uses the concept of moment of an area; this is why textbooks may refer to this as &#8220;first moments of areas&#8221;.\u00a0 Mathematically this principle is expressed as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig1.jpg\" alt=\"\" class=\"alignnone wp-image-149\" width=\"204\" height=\"91\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig1.jpg 307w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig1-300x134.jpg 300w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig1-65x29.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig1-225x100.jpg 225w\" sizes=\"auto, (max-width: 204px) 100vw, 204px\" \/><\/p>\n<p>where:<\/p>\n<ul>\n<li>Y is the distance to the centroid from some reference axis.\u00a0 Commonly, the reference axis is the base of the figure.<\/li>\n<li>A<sub>i<\/sub> is the area of one part of the composite area.\u00a0 Typically, the composite areas are split into common shapes of known geometric properties, summarized in the textbook Appendix C.<\/li>\n<li>y<sub>i<\/sub> is the distance from the reference axis (commonly the base of the figure) to the centroid of each part of the composite figure.<\/li>\n<li>\u03a3(A<sub>i<\/sub>) is the entire area of the composite area.<\/li>\n<\/ul>\n<p>When determining the location of a centroid please observe the following rules:<\/p>\n<ul>\n<li>If the cross-section has one axis of symmetry then the centroid will be located on this axis.<\/li>\n<li>if the cross-section has two axes of symmetry then the centroid will be located at the intersection of the two axes.<\/li>\n<li>If the cross-section is not symmetric about any axis then two calculations are required:\n<ul>\n<li>one for determining the centroid location Y<\/li>\n<li>one for determining the centroid location X, commonly measured from the extreme left end.\u00a0 For this second calculation imagine that you rotate the figure 90\u00ba counter-clockwise and repeat the first calculation<\/li>\n<\/ul>\n<\/li>\n<li>If the composite area has a part that is removed from the figure (a void), this missing part can be treated as a negative area.<\/li>\n<\/ul>\n<div class=\"textbox shaded\" style=\"text-align: center\"><strong>Moments of inertia of composite areas<\/strong><\/div>\n<p>In rotational kinetics we learned that the &#8220;rotational&#8221; moment of inertia of a flywheel (function of its mass, size and shape) represents a resistance to change in its motion. This moment of inertia multiplied by the angular acceleration \u03b1, gives an inertia-moment reaction that attempts to balance the accelerating moment action (accelerating torque).\u00a0 In general, a moment of inertia is a resistance to change.<\/p>\n<p>Beams are subject to bending and as a result they tend to deform (deflect).\u00a0 The moment of inertia of a beam cross-section can be related to the stiffness of the beam.\u00a0 The deflection of the beam is inverse proportional to the moment of inertia.<\/p>\n<p>Formulas for moments of inertia of simple shapes are given in Textbook Appendix C.\u00a0 They will also be provided in the exam.<\/p>\n<p>When dealing with a composite area, divide the shape into basic parts for which the moment of inertia can be easily calculated.\u00a0 The combined moment of inertia of the entire shape is the sum of moments of inertia of constituent parts plus their corresponding transfer term.\u00a0 The transfer term is calculated as the area of the part multiplied by the squared distance between the centroid of the part and the common centroid of the entire area.\u00a0 This transfer term represents the additional stiffness of each part due to its relative distance from the common centroid.\u00a0 The table given in Supplement Appendix 1 can be used for calculations; it is useful when the shape is more complex.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3 itemprop=\"educationalUse\"><strong>Assigned Problems<\/strong><\/h3>\n<p>When completing these exercises please make sure that you clearly identify and number the parts of your composite area.<\/p>\n<\/div>\n<p><strong>Problem 1: <\/strong>Determine the moment of inertia about the vertical and horizontal centroidal axes for the following figure.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig2-1.jpg\" alt=\"\" class=\"alignnone wp-image-159\" width=\"302\" height=\"450\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig2-1.jpg 497w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig2-1-201x300.jpg 201w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig2-1-65x97.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig2-1-225x335.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig2-1-350x522.jpg 350w\" sizes=\"auto, (max-width: 302px) 100vw, 302px\" \/><\/p>\n<p><strong>Problem 2: <\/strong>For the following cross-section determine the location (elevation) of the centroid and the moments of inertia with respect to the horizontal and vertical centroidal axes.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig3-1.jpg\" alt=\"\" class=\"alignnone wp-image-160\" width=\"266\" height=\"370\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig3-1.jpg 582w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig3-1-216x300.jpg 216w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig3-1-65x90.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig3-1-225x312.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig3-1-350x486.jpg 350w\" sizes=\"auto, (max-width: 266px) 100vw, 266px\" \/><\/p>\n<p><strong>Problem 3: <\/strong>For the following figure<span style=\"text-decoration: underline\"><\/span> determine Y, the vertical location of the centroid, and calculate the moment of inertia with respect to the <span style=\"text-decoration: underline\">horizontal<\/span> centroidal axis..<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig4-1.jpg\" alt=\"\" class=\"alignnone wp-image-161\" width=\"272\" height=\"274\" srcset=\"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig4-1.jpg 730w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig4-1-150x150.jpg 150w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig4-1-298x300.jpg 298w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig4-1-65x66.jpg 65w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig4-1-225x227.jpg 225w, https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-content\/uploads\/sites\/290\/2018\/02\/Ch5Fig4-1-350x353.jpg 350w\" sizes=\"auto, (max-width: 272px) 100vw, 272px\" \/><\/p>\n<p><strong>Problem 4: <\/strong>Suggest one improvement to this chapter; this may include an original cross-section.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"author":239,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"Inertia Moment","pb_subtitle":"Inertia Moment","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[57],"license":[],"class_list":["post-139","chapter","type-chapter","status-publish","hentry","contributor-alex-podut"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapters\/139","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/wp\/v2\/users\/239"}],"version-history":[{"count":21,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapters\/139\/revisions"}],"predecessor-version":[{"id":565,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapters\/139\/revisions\/565"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapters\/139\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/wp\/v2\/media?parent=139"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/pressbooks\/v2\/chapter-type?post=139"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/wp\/v2\/contributor?post=139"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/powr4406\/wp-json\/wp\/v2\/license?post=139"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}