{"id":9271,"date":"2021-08-15T13:13:16","date_gmt":"2021-08-15T17:13:16","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me323\/?page_id=9271"},"modified":"2022-08-09T14:29:28","modified_gmt":"2022-08-09T18:29:28","slug":"simple-buckling-model","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me323\/animations-and-demonstrations\/simple-buckling-model\/","title":{"rendered":"Simple buckling model"},"content":{"rendered":"<p>A model of a simple system that exhibits buckling is the rigid bar system presented in the lecture book with the bar pinned to ground at O, with a \u201crotational spring\u201d of rotational stiffness <em>K<\/em> connecting the bar to ground. (Note: a rotational spring is a device that resists rotation with a generated moment of\u00a0 <em>K\u03b8<\/em>.) With the spring being unstressed at <em>\u03b8<\/em> = 0, an axial load <em>P<\/em> is applied to the free end. The strength of the load is measured inversely by the parameter <em>\u03b1 = K\/PL<\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9272 aligncenter\" src=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/Untitled-2-1-159x300.jpg\" alt=\"\" width=\"129\" height=\"243\" srcset=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/Untitled-2-1-159x300.jpg 159w, https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/Untitled-2-1.jpg 186w\" sizes=\"auto, (max-width: 129px) 100vw, 129px\" \/><\/p>\n<p><span style=\"font-size: 14pt\"><em><strong>Observations<\/strong><\/em><\/span><\/p>\n<ul>\n<li>The system can ALWAYS be in equilibrium at <em>\u03b8<\/em> = 0 for any value of the loading parameter <em>\u03b1<\/em>.<\/li>\n<li>Analysis in the lecture book shows that for values of loading for which <em>\u03b1<\/em> &lt; 1, a second equilibrium state (the \u201cbuckled\u201d state) can exist.<\/li>\n<li>When this second equilibrium exists, the <em>\u03b8<\/em> = 0 equilibrium state is UNSTABLE.<\/li>\n<li>An \u201cunstable\u201d equilibrium state is one for which the system can be remain at rest; however, if this state is given a small \u201cbump\u201d, the system moves from the undeformed state to the buckled state.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9279 aligncenter\" src=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/Untitled-2-2-300x188.jpg\" alt=\"\" width=\"373\" height=\"234\" srcset=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/Untitled-2-2-300x188.jpg 300w, https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/Untitled-2-2.jpg 441w\" sizes=\"auto, (max-width: 373px) 100vw, 373px\" \/><\/p>\n<p>The following simulations show how \u201cbumps\u201d will result in different final equilibrium states for three different loadings.\u00a0Recall that for <em>\u03b1<\/em> &lt; 1, the system is unstable. The loadings on the left two structures correspond to stable loadings and allows the bar to return to the upright state, whereas the loading on the right (unstable) produces buckling where the structure collapses.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/buckling01d.gif\" width=\"652\" height=\"370\" \/><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A model of a simple system that exhibits buckling is the rigid bar system presented in the lecture book with the bar pinned to ground at O, with a \u201crotational spring\u201d of rotational stiffness K connecting the bar to ground. (Note: a rotational spring is a device that resists rotation with a generated moment of\u00a0 &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me323\/animations-and-demonstrations\/simple-buckling-model\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Simple buckling model<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":10,"featured_media":0,"parent":5074,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-9271","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/9271","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/comments?post=9271"}],"version-history":[{"count":5,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/9271\/revisions"}],"predecessor-version":[{"id":9280,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/9271\/revisions\/9280"}],"up":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/5074"}],"wp:attachment":[{"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/media?parent=9271"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}