{"id":1219,"date":"2020-11-30T10:39:11","date_gmt":"2020-11-30T10:39:11","guid":{"rendered":"https:\/\/pickedshares.com\/?p=1219"},"modified":"2021-05-08T11:25:52","modified_gmt":"2021-05-08T11:25:52","slug":"engineering-mechanics-ii-exercise-1-normal-forces-in-members","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-ii-exercise-1-normal-forces-in-members\/","title":{"rendered":"Normal forces in members"},"content":{"rendered":"\n<p>This exercise is our introductory exercise in strength theory and deals with the following questions:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>How to calculate the normal force in a member?<\/li><li>How to establish an equilibrium of forces with internal forces?<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>A weight G<sub>2<\/sub> is attached to a rod. The weight G<sub>1<\/sub> is applied at connection point C. The members 1 and 2 are fastened with fixed bearings. The members are without mass.<\/p>\n\n\n\n<p>The normal forces that occur in members 1 to 3 have to be calculated.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"916\" height=\"1024\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1-916x1024.png\" alt=\"Members with loads G1 and G2\" class=\"wp-image-1197\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1-916x1024.png 916w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1-268x300.png 268w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1-768x859.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1-400x447.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1-800x895.png 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1.png 947w\" sizes=\"auto, (max-width: 916px) 100vw, 916px\" \/><figcaption>Members with loads G1 and G2<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<p>The normal forces are referred to as N and the index of their member number. The normal force in member 1 is the easiest to calculate.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"671\" height=\"1009\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_2.png\" alt=\"Calculation of the normal force in member 1\" class=\"wp-image-1199\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_2.png 671w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_2-200x300.png 200w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_2-400x601.png 400w\" sizes=\"auto, (max-width: 671px) 100vw, 671px\" \/><figcaption>Calculation of the normal force in member 1<\/figcaption><\/figure>\n\n\n\n<p>The equilibrium of forces in the y-direction yields<\/p>\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<p>\\[ \\tag{1} \\sum F_y = 0 = N_1 - G_2\\]<\/p>\n<p>\\[ \\tag{2} N_1 = G_2\\]<\/p>\n\n\n\n<p>Now, starting from point C, mentally cut through the members and apply their respective normal forces. A small trap lurks here in this task, because the force G<sub>1<\/sub> must of course not be missing in this free cut.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"871\" height=\"1024\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_3-871x1024.png\" alt=\"Calculation of the normal forces in members 2 and 3\" class=\"wp-image-1201\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_3-871x1024.png 871w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_3-255x300.png 255w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_3-768x903.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_3-400x470.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_3-800x941.png 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm2-1_3.png 911w\" sizes=\"auto, (max-width: 871px) 100vw, 871px\" \/><figcaption>Calculation of the normal forces in members 2 and 3<\/figcaption><\/figure>\n\n\n\n<p>The two normal forces in members 2 and 3 can now be calculated using the force equilibria in the x and y directions.<\/p>\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\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{3} \\sum F_x = 0 = - N_3 \\cdot sin \\alpha + N_2\\]<\/p>\n<p>\\[ \\tag{4} \\sum F_y = 0 = N_3 \\cdot cos \\alpha - G_1 - N_1\\]<\/p>\n<\/div>\n<p>or converted to N<sub>3<\/sub> and the normal force N<sub>1<\/sub> replaced by its value:<\/p>\n<p>\\[ \\tag{5} N_3 = \\frac{G_1 + G_2}{cos \\alpha} \\]<\/p>\n<p>Finally, N<sub>2<\/sub> follows from equation 3<\/p>\n<p>\\[ \\tag{6} N_2 = \\frac{G_1 + G_2}{cos \\alpha} \\cdot sin \\alpha\\]<\/p>\n<p>or in other words:<\/p>\n<p>\\[ \\tag{7} N_2 = (G_1+G_2) \\cdot tan \\alpha\\]<\/p>\n\n\n\n<p>The member lengths given in the sketch were not relevant for the calculation of the normal forces.<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Normal forces in members\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-ii-exercise-1-normal-forces-in-members\/\" aria-label=\"Read more about Normal forces in members\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":1203,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[52,60],"tags":[],"class_list":["post-1219","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-ii","category-exercises","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>Normal forces in members &#8226; pickedshares<\/title>\n<meta name=\"description\" content=\"How to calculate the normal forces in the members for the shown arrangement with the two weight loads? 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