{"id":9200,"date":"2021-08-15T08:27:02","date_gmt":"2021-08-15T12:27:02","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me323\/?page_id=9200"},"modified":"2022-08-09T14:29:13","modified_gmt":"2022-08-09T18:29:13","slug":"influence-of-shear-deformations-in-beams","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me323\/animations-and-demonstrations\/influence-of-shear-deformations-in-beams\/","title":{"rendered":"Shear stress in beams"},"content":{"rendered":"

Bending beam<\/strong><\/em><\/span><\/p>\n

\"\"<\/p>\n

The transverse load P<\/em> will produce a shear force V in the positive y<\/em>-direction on a mathematical cut through the beam at point \u201ca\u201d. This shear force is directly responsible for the shear stress \u03c4<\/em> on the x<\/em>-face of a stress element at \u201ca\u201d in the positive y-direction. In class, we have used the Euler-Bernoulli theory of beams to calculate this shear stress. Recall that the E-B theory of beam assumes that beam cross sections always remain perpendicular to the neutral plane of the beam.<\/p>\n

Beam without shear stiffness
\n<\/strong><\/em>Say we consider a beam made up of thin layers (laminae) aligned with the longitudinal axis of the beam, and for which the laminae can slide with respect to each other along the longitudinal axis as the beam deflects. (Think a deck of playing cards.)<\/span><\/span><\/p>\n

Under the action of a transverse load we see below how the beam deflects in the absence of shear stiffness.<\/p>\n

<\/p>\n

Focus, in particular, on the deflection at the right end of the beam, as seen below. Here, you can easily see how the laminae slide with respect to each other in the absence of shear stress between the laminae.<\/p>\n

<\/p>\n

 <\/p>\n

Beam with shear stiffness
\n<\/strong><\/em><\/span>Now let’s consider \u00a0reconsider our laminated beam where shear stresses are produced as the laminae slide with respect to each other under loading. For example, these stresses could result from applying a sticky glue between the playing cards in the deck.\u00a0<\/span><\/span><\/p>\n

Under the action of a transverse load we see below how the beam deflects in the absence of shear stiffness.<\/p>\n

<\/p>\n

Focus, in particular, on the deflection at the right end of the beam, as seen below. Here, you can easily see that now the beam cross sections remain perpendicular to the neutral plane of the beam as the beam deflects.<\/p>\n

<\/p>\n

 <\/p>\n

CONCLUSION<\/span><\/em>: Shear stresses play an important role in the Euler-Bernoulli theory for the bending of beams when shear force resultants exist at a given cross section.<\/p>\n","protected":false},"excerpt":{"rendered":"

Bending beam The transverse load P will produce a shear force V in the positive y-direction on a mathematical cut through the beam at point \u201ca\u201d. This shear force is directly responsible for the shear stress \u03c4 on the x-face of a stress element at \u201ca\u201d in the positive y-direction. In class, we have used … Continue reading Shear stress in beams<\/span> →<\/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-9200","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/9200","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=9200"}],"version-history":[{"count":11,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/9200\/revisions"}],"predecessor-version":[{"id":9354,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/9200\/revisions\/9354"}],"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=9200"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}