{"id":3238,"date":"2021-02-22T13:28:57","date_gmt":"2021-02-22T13:28:57","guid":{"rendered":"https:\/\/pickedshares.com\/?p=3238"},"modified":"2021-05-08T13:18:11","modified_gmt":"2021-05-08T13:18:11","slug":"cantilever-beam-under-ascending-triangle-load","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/cantilever-beam-under-ascending-triangle-load\/","title":{"rendered":"Cantilever beam under ascending triangle load"},"content":{"rendered":"\n<p>In this exercise the bending line is calculated for a cantilever beam with ascending triangular load. <a href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-2-exercise-10-beam-with-triangular-line-load\/\">How to determine the internal forces for a beam with descending triangular load is described here<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>A beam with a fixed restraint on one side is loaded by the linearly increasing line load (triangular load) q<sub>0<\/sub>. The bending line for the beam has to be determined!<\/p>\n\n\n\n<figure class=\"wp-block-image size-large custom-flex-image\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"630\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-1-1024x630.png\" alt=\"Cantilever beam under ascending triangle load\" class=\"wp-image-3226\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-1-1024x630.png 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-1-300x185.png 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-1-768x473.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-1-400x246.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-1-800x492.png 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-1.png 1136w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Cantilever beam under ascending triangle load<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<p>The triangular load must be described as a function. In the present case it is<\/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} q(x) = q_0 \\frac{x}{l} \\]<\/p>\n\n\n\n<p>To determine the bending line, the internal forces of the beam must be determined. Only the bending moment is relevant for the bending, i.e. in the following equations only the moment equilibrium is considered.<\/p>\n\n\n\n<p>In this case, it is not necessary to determine the support reactions of the fixed restraint, as the section is chosen in such a way that the support reactions do not occur. The internal forces can be determined directly.<\/p>\n\n\n\n<p><a href=\"https:\/\/pickedshares.com\/en\/right-hand-rules\/\" target=\"_blank\" rel=\"noreferrer noopener\">Left-turning moments are positive in the following.<\/a> The internal forces are applied negatively on a negative cutting edge. The coordinate direction x is supplemented by the auxiliary coordinate \u03be.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Determination of the internal forces<\/h3>\n\n\n\n<figure class=\"wp-block-image size-large custom-flex-image\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"667\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-2-1024x667.png\" alt=\"Internal forces of the cantilever beam under ascending triangle load\" class=\"wp-image-3232\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-2-1024x667.png 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-2-300x196.png 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-2-768x501.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-2-400x261.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-2-800x521.png 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-19-2.png 1054w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Internal forces of the cantilever beam under ascending triangle load<\/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<p>The moment balance delivers<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{2} \\sum M(x) = 0 = -M_b - \\int_x^l{q(\\xi)\\cdot \\left(\\xi - x \\right) d \\xi} \\]<\/p>\n<p>\\[ \\tag{3} M_b = - \\left( \\frac{q_0 x^3 - 3l^2q_0x+2l^3q_0}{6l}  \\right)  \\]<\/p>\n<\/div>\n\n<p>The bending line is determined on the basis of the Bernoulli beam bending. The basic relation is<\/p>\n<p>\\[ w'' = \\frac{-M_b}{E \\cdot I} \\]<\/p>\n<p>Here, w'' is the second derivative of the bending line, E is the modulus of elasticity and I is the area moment of inertia. The bending line w is obtained by integrating twice over x. It follows from this for the present case<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{4} w'' = \\frac{q_0x^3-3l^2q_0x+2l^3q_0}{6EIl}  \\]<\/p>\n<p>\\[ \\tag{5} w' = \\frac{\\frac{q_0x^4}{4}-\\frac{3l^2q_0x^2}{2}+2l^3q_0x}{6EIl} + c_1 \\]<\/p>\n<p>\\[ \\tag{6} w = \\frac{\\frac{q_0x^5}{20}-\\frac{l^2q_0x^3}{2}+l^3q_0x^2}{6EIl} +c_1x + c_2 \\]<\/p>\n<\/div>\n\n<p>In order to be able to determine the constants of integration c<sub>1<\/sub> and c<sub>2<\/sub>, the boundary conditions must be established.<\/p>\n<h3>Boundary conditions<\/h3>\n<p>The first condition is: the deflection at the point x = 0 is equal to 0.<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{7} w(x=0) = 0 = \\frac{\\frac{q_00^5}{20}-\\frac{l^2q_00^3}{2}+l^3q_00^2}{6EIl} +c_10 + c_2 \\]<\/p>\n<\/div>\n<p>\\[ \\tag{8} c_2 = 0\\]<\/p>\n<p>The second condition is: the angle at the point x = 0 is equal to 0.<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{9} w'(x=0) = 0 = \\frac{\\frac{q_00^4}{4}-\\frac{3l^2q_00^2}{2}+2l^3q_00}{6EIl} + c_1 \\]<\/p>\n<\/div>\n<p>\\[ \\tag{10} c_1 = 0 \\]<\/p>\n<p>The bending line for the cantilever beam with ascending triangular load is thus completely defined. It is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{11} w = \\frac{\\frac{q_0x^5}{20}-\\frac{l^2q_0x^3}{2}+l^3q_0x^2}{6EIl} \\]<\/p>\n<\/div>\n\n\n\n<p>Don't miss the other <a href=\"https:\/\/pickedshares.com\/en\/category\/exercises\/engineering-mechanics-ii\/\">exercises regarding Engineering Mechanics II<\/a>!<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Cantilever beam under ascending triangle load\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/cantilever-beam-under-ascending-triangle-load\/\" aria-label=\"Read more about Cantilever beam under ascending triangle load\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":3228,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[52,60],"tags":[66,70],"class_list":["post-3238","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-ii","category-exercises","tag-bending-line","tag-line-load","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>Cantilever beam under ascending triangle load &#8226; 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