{"id":959,"date":"2020-12-16T16:08:41","date_gmt":"2020-12-16T16:08:41","guid":{"rendered":"https:\/\/pickedshares.com\/?p=959"},"modified":"2021-05-08T12:25:44","modified_gmt":"2021-05-08T12:25:44","slug":"engineering-mechanics-1-exercise-34-determination-of-the-center-of-area","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-34-determination-of-the-center-of-area\/","title":{"rendered":"Center of area, with bore"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>The common center of area has to be calculated for the shown circular cross-section with a bore.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"834\" height=\"1024\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225108_Sketch-834x1024.jpg\" alt=\"Circular cross-section with bore\" class=\"wp-image-952\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225108_Sketch-834x1024.jpg 834w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225108_Sketch-244x300.jpg 244w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225108_Sketch-768x943.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225108_Sketch-400x491.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225108_Sketch-800x982.jpg 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225108_Sketch.jpg 1105w\" sizes=\"auto, (max-width: 834px) 100vw, 834px\" \/><figcaption>Circular cross-section with bore<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<p>The individual areas are numbered in the first step.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225145_Sketch-1024x1024.jpg\" alt=\"Numbering of the individual areas and assumed x-coordinate of the centre of area\" class=\"wp-image-954\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225145_Sketch-1024x1024.jpg 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225145_Sketch-300x300.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225145_Sketch-150x150.jpg 150w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225145_Sketch-768x767.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225145_Sketch-400x400.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225145_Sketch-800x799.jpg 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201215-225145_Sketch.jpg 1095w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Numbering of the individual areas and assumed x-coordinate of the centre of area<\/figcaption><\/figure>\n\n\n\n<script src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-AMS_HTML\" async=\"async\">  \/\/ A comment that hinders wxWidgets from optimizing this tag too much.\n<\/script>\n\\[ \\require{cancel} \\]\n\n<p>Since the component is mirror-symmetrical about the x-axis, it is sufficient to consider the x-coordinate. The general equation for the x-coordinate of the center of area is:<\/p>\n<p>\\[ \\tag{1} x_S = \\frac{\\sum A_i \\cdot x_i}{\\sum A_i} \\]<\/p>\n\n<p>And for the area of \u200b\u200bthe circle<\/p>\n<p>\\[ \\tag{2} A = \\frac{\\pi}{4} \\cdot d^2 \\]<\/p>\n\n<p>The individual areas and their centre point coordinates are listed in a table. Empty areas, such as e.g. the bore with the diameter of 22mm, are negative.<\/p>\n\n<p>\n<\/p>\n\n<div style=\"overflow:auto;\">\n<table>\n<thead>\n<tr>\n<th>\\[ i \\]<\/th><th>\\[ A_i \\]<\/th><th>\\[ x_i \\, in \\, mm \\]<\/th><th>\\[ A_i \\cdot x_i\\, in \\,mm^3 \\]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<\/tbody>\n<tbody><tr>\n<td>1<\/td><td>3.848<\/td><td>0<\/td><td>0<\/td>\n<\/tr>\n<tr>\n<td>2<\/td><td>-380<\/td><td>11<\/td><td>-4.180<\/td>\n<\/tr>\n<tr>\n<td>\\[ \\sum \\]<\/td><td>3.468<\/td><td><\/td><td>-4.180<\/td>\n<\/tr>\n<\/tbody><\/table>\n<\/div>\n<p><\/p>\n\n<p>\\[ \\tag{3} x_S = \\frac{-4180mm^3}{3468mm^2} \\]<\/p>\n<p>\\[ \\tag{4} x_S = -1,21 mm \\]<\/p>\n<p>And for the sake of completeness:<\/p>\n<p>\\[ \\tag{5} y_S = 0 mm \\]<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Center of area, with bore\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-34-determination-of-the-center-of-area\/\" aria-label=\"Read more about Center of area, with bore\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":956,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,60],"tags":[50],"class_list":["post-959","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-i","category-exercises","tag-centre-points","infinite-scroll-item","masonry-post","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-33"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Center of area, with bore &#8226; 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