{"id":933,"date":"2020-12-15T15:59:32","date_gmt":"2020-12-15T15:59:32","guid":{"rendered":"https:\/\/pickedshares.com\/?p=933"},"modified":"2021-05-08T12:24:56","modified_gmt":"2021-05-08T12:24:56","slug":"engineering-mechanics-1-exercise-33-determine-the-center-of-area","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-33-determine-the-center-of-area\/","title":{"rendered":"Center of area, combined profiles"},"content":{"rendered":"\n<p>This exercise is about the following questions:<\/p>\n\n\n\n<ul class=\"wp-block-list\" id=\"block-02cc2f27-0db1-439d-8de7-c6bd2166ddc1\"><li>How to calculate the common centroid of composited areas?<\/li><li>How to determine the centroid of a rectangle?<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>For the cross-section of two adjacent profiles, the common center of area is to be determined.<\/p>\n\n\n\n<p>Given: a = 50 mm, b = 22 mm, c = 70 mm, d = 20 mm.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"964\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33big-1024x964.jpg\" alt=\"Compound faces\" class=\"wp-image-926\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33big-1024x964.jpg 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33big-300x282.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33big-768x723.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33big-400x377.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33big-800x753.jpg 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33big.jpg 1094w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Compound faces<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<p>Since the adjacent profiles are purely rectangular cross-sections, the individual centroids can be easily determined.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"744\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33solved-1024x744.jpg\" alt=\"Individual centres of area\" class=\"wp-image-930\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33solved-1024x744.jpg 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33solved-300x218.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33solved-768x558.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33solved-400x291.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33solved-800x581.jpg 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-33solved.jpg 1480w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Individual centres 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>Both surfaces are mirror-symmetrical about the x-axis, so it is sufficient to consider the x-coordinate. The general equation for the x-coordinate of the centre of area is:<\/p>\n<p>\\[ \\tag{1} x_S = \\frac{\\sum A_i \\cdot x_i}{\\sum A_i} \\]<\/p>\n\n<p>If there are several areas, it is advisable to list the individual areas and the relevant coordinates of the individual centre points in a table:<\/p>\n\n<!--a = 50 mm, b = 22 mm, c = 70 mm, d = 20 mm -->\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>1.100<\/td><td>11<\/td><td>12.100<\/td>\n<\/tr>\n<tr>\n<td>2<\/td><td>960<\/td><td>46<\/td><td>44.160<\/td>\n<\/tr>\n<tr>\n<td>\\[ \\sum \\]<\/td><td>2.060<\/td><td><\/td><td>56.260<\/td>\n<\/tr>\n<\/tbody><\/table>\n<\/div>\n<p><\/p>\n\n<p>\\[ \\tag{2} x_S = \\frac{56260mm^3}{2060mm^2} \\]<\/p>\n<p>\\[ \\tag{3} x_S = 27,3 mm \\]<\/p>\n<p>And for the sake of completeness:<\/p>\n<p>\\[ \\tag{4} y_S = 0 mm \\]<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Center of area, combined profiles\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-33-determine-the-center-of-area\/\" aria-label=\"Read more about Center of area, combined profiles\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":928,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,60],"tags":[50],"class_list":["post-933","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, combined profiles &#8226; pickedshares<\/title>\n<meta name=\"description\" content=\"Two profiles are attached together. 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