{"id":3106,"date":"2021-02-17T12:48:02","date_gmt":"2021-02-17T12:48:02","guid":{"rendered":"https:\/\/pickedshares.com\/?p=3106"},"modified":"2021-05-08T13:16:29","modified_gmt":"2021-05-08T13:16:29","slug":"deflection-line-of-a-cantilever-beam","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/deflection-line-of-a-cantilever-beam\/","title":{"rendered":"Deflection line of a cantilever beam"},"content":{"rendered":"\n<p>In this exercise the bending line for a cantilevered beam with two opposite loads is calculated according to the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Euler%E2%80%93Bernoulli_beam_theory#Static_beam_equation\" target=\"_blank\" rel=\"noreferrer noopener\">Bernoulli beam theory<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>A beam fixed on one side is loaded by the opposing forces F. The bending line for the beam has to be determined!<\/p>\n\n\n\n<figure class=\"wp-block-image size-large custom-flex-image75\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"715\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-1-1024x715.png\" alt=\"Beam clamped on one side with two opposing forces\" class=\"wp-image-3087\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-1-1024x715.png 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-1-300x209.png 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-1-768x536.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-1-400x279.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-1-800x559.png 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-1.png 1047w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Beam clamped on one side with two opposing forces<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\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 equilibria are established. The beam is divided into two sections.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large custom-flex-image75\"><img loading=\"lazy\" decoding=\"async\" width=\"1021\" height=\"783\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-2.png\" alt=\"Two sections of the beam\" class=\"wp-image-3093\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-2.png 1021w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-2-300x230.png 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-2-768x589.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-2-400x307.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-2-800x614.png 800w\" sizes=\"auto, (max-width: 1021px) 100vw, 1021px\" \/><figcaption>Two sections of the beam<\/figcaption><\/figure>\n\n\n\n<p>In this case, it is not necessary to determine the <a href=\"https:\/\/pickedshares.com\/en\/reaction-forces-in-bearings-joints-and-guidances\/\">support reactions<\/a> of the fixed restraint, so that the internal forces for section I can be started immediately.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Determination of the bending moments<\/h3>\n\n\n\n<p>Left-turning moments are positive in the following. The internal forces are applied negatively on a negative cutting edge.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Section I<\/h4>\n\n\n\n<figure class=\"wp-block-image size-large custom-flex-image75\"><img loading=\"lazy\" decoding=\"async\" width=\"1023\" height=\"747\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-4.png\" alt=\"Internal forces and moments in section I\" class=\"wp-image-3097\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-4.png 1023w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-4-300x219.png 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-4-768x561.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-4-400x292.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-4-800x584.png 800w\" sizes=\"auto, (max-width: 1023px) 100vw, 1023px\" \/><figcaption>Internal forces and moments in section I<\/figcaption><\/figure>\n\n\n\n<p>The moment balance in section I 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\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{1} \\sum M(x) = 0 = -M_{bI} + F \\cdot (l-x) - F \\cdot (2 \\cdot l -x) \\]<\/p>\n<\/div>\n<p>\\[ \\tag{2} M_{bI} = - F \\cdot l \\]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Section II<\/h4>\n\n\n\n<figure class=\"wp-block-image size-large custom-flex-image75\"><img loading=\"lazy\" decoding=\"async\" width=\"953\" height=\"615\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-3.png\" alt=\"Internal forces and moments in section II\" class=\"wp-image-3095\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-3.png 953w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-3-300x194.png 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-3-768x496.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-3-400x258.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-18-3-800x516.png 800w\" sizes=\"auto, (max-width: 953px) 100vw, 953px\" \/><figcaption>Internal forces and moments in section II<\/figcaption><\/figure>\n\n\n\n<p>The moment balance in section II 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\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{3} \\sum M(x) = 0 = -M_{bII} - F \\cdot (2 \\cdot l -x) \\]<\/p>\n<\/div>\n<p>\\[ \\tag{4} M_{bI} = F \\cdot x - 2 \\cdot F \\cdot l \\]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Bending lines<\/h3>\n\n\n\n<p>The bending lines are determined on the basis of the Bernoulli beam bending. The basic relation 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>\\[ w'' = \\frac{-M_b}{E \\cdot I} \\]<\/p>\n\n\n\n<p>Here, w'' is the second derivative of the bending line, E is the Young's modulus and I is the area moment of inertia. The bending line w is obtained by double integration. It follows from this for the present case<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Section I<\/h4>\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{5}  E \\cdot I \\cdot w''_I = F \\cdot l \\]<\/p>\n<p>\\[ \\tag{6} E \\cdot I \\cdot w'_I = F \\cdot l \\cdot x + c_1\\]<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{7} E \\cdot I \\cdot w_I = \\frac{1}{2} F \\cdot l \\cdot x^2 + c_1 \\cdot x + c_2 \\]<\/p>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Section II<\/h4>\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{8} E \\cdot I \\cdot w''_{II} = - F \\cdot x + 2 \\cdot F \\cdot l \\]<\/p>\n<p>\\[ \\tag{9} E \\cdot I \\cdot w'_{II} = - \\frac{1}{2} F \\cdot x^2 + 2 \\cdot F \\cdot l \\cdot x + c_3 \\]<\/p>\n<p>\\[ \\tag{10} E \\cdot I \\cdot w_{II} = - \\frac{1}{6} F \\cdot x^3 + F \\cdot l \\cdot x^2 + c_3 \\cdot x + c_4 \\]<\/p>\n<\/div>\n\n\n\n<p>In order to be able to determine the constants of integration c<sub>1<\/sub> to c<sub>4<\/sub>, the boundary and transition conditions must be established.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Boundary and transition conditions<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">w<sub>I<\/sub>(x=0) = 0<\/h4>\n\n\n\n<p>The deflection at the point x = 0 is zero.<\/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\\[ \\require{cancel} \\]\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{11}  0 = \\frac{1}{E \\cdot I} \\left( \\bcancel{\\frac{1}{2}F \\cdot l \\cdot 0^2} + \\bcancel{c_1 \\cdot 0} + c_2 \\right) \\]<\/p>\n<\/div>\n<p>\\[ \\tag{12}  c_2 = 0 \\]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">w'<sub>I<\/sub>(x=0) = 0<\/h4>\n\n\n\n<p>The angle at the point x = 0 is equal to zero.<\/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\\[ \\require{cancel} \\]\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{13}  0 = \\frac{1}{E \\cdot I} \\left(\\bcancel{ F \\cdot l \\cdot 0} + c_1 \\right) \\]<\/p>\n<\/div>\n<p>\\[ \\tag{14}  c_1 = 0 \\]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">w<sub>I<\/sub>(x=l) = w<sub>II<\/sub>(x=l)<\/h4>\n\n\n\n<p>The deflection of the two bending lines at the point x = l is equal.<\/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\\[ \\require{cancel} \\]\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{15}  \\frac{1}{2} F \\cdot l \\cdot l^2 + \\bcancel{c_1 \\cdot l} + \\bcancel{c_2} = - \\frac{1}{6} F \\cdot l^3 + F \\cdot l \\cdot l^2 + c_3 \\cdot l + c_4 \\]<\/p>\n<p>\\[ \\tag{16}  \\frac{1}{2} F \\cdot l^3 = \\frac{5}{6} F \\cdot l^3 + c_3 \\cdot l + c_4 \\]<\/p>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\">w'<sub>I<\/sub>(x=l) = w'<sub>II<\/sub>(x=l)<\/h4>\n\n\n\n<p>The angle of the two bending lines at the point x = l is the same.<\/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\\[ \\require{cancel} \\]\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{17} F \\cdot l \\cdot l + \\bcancel{c_1} = - \\frac{1}{2} F \\cdot l^2 + 2 \\cdot F \\cdot l \\cdot l + c_3 \\]<\/p>\n<\/div>\n<p>From this context the integration constant c<sub>3<\/sub> can be resolved.<\/p>\n<p>\\[ \\tag{18} c_3 = -\\frac{1}{2}F \\cdot l^2  \\]<\/p>\n\n<p>And finally c<sub>4<\/sub>:<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{19}  \\frac{1}{2} F \\cdot l^3 = \\frac{5}{6} F \\cdot l^3 - \\frac{1}{2}F \\cdot l^3 + c_4 \\]<\/p>\n<\/div>\n<p>\\[ \\tag{20}  c_4 = \\frac{1}{6}F \\cdot l^3  \\]<\/p>\n<p>The equations of the bending lines are thus<\/p>\n<p>\\[ \\tag{21}  w_I = \\frac{1}{2 E \\cdot I} F \\cdot l \\cdot x^2 \\]<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{22}  w_{II} = \\frac{F}{E \\cdot I} \\left( - \\frac{1}{6} x^3 + l \\cdot x^2 -\\frac{1}{2} l^2 \\cdot x + \\frac{1}{6} l^3 \\right) \\]<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Deflection line of a cantilever beam\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/deflection-line-of-a-cantilever-beam\/\" aria-label=\"Read more about Deflection line 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