{"id":3043,"date":"2021-02-11T19:32:37","date_gmt":"2021-02-11T19:32:37","guid":{"rendered":"https:\/\/pickedshares.com\/?p=3043"},"modified":"2021-05-08T13:10:46","modified_gmt":"2021-05-08T13:10:46","slug":"mechanics-ii-safety-against-plastic-flow-acc-tresca-shear-stress-hypothesis","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/mechanics-ii-safety-against-plastic-flow-acc-tresca-shear-stress-hypothesis\/","title":{"rendered":"Safety against plastic flow acc. TRESCA (shear stress hypothesis)"},"content":{"rendered":"\n<p>This exercise shows how the safety against flow is calculated according to the <a href=\"https:\/\/de.wikipedia.org\/wiki\/Henri_Tresca\" target=\"_blank\" rel=\"noreferrer noopener\">TRESCA<\/a> shear stress hypothesis. The equivalent stress is calculated from the principal stresses and then compared with the yield stress of the material.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>A load case with the stress tensor is given<\/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\\[ \\newcommand{\\myvec}[1]{{\\begin{bmatrix}#1\\end{bmatrix}}} \\]\n<p>\\[ S = \\myvec{-160 &amp; 120 &amp; 0\\\\120 &amp; 80 &amp; 0\\\\0 &amp; 0 &amp; 0}  \\, MPa \\]<\/p>\n\n\n\n<p>The yield point for the material is \u03c3<sub>F<\/sub> = 650 MPa. Is there a double safety against flowing according to TRESCA?<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\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 yield condition according to TRESCA (shear stress hypothesis) is<\/p>\n<p>\\[ \\tag{1} |\\tau_{max}| \\overset{!}{=} \\frac{\\sigma_0}{2} \\]<\/p>\n<p>with<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{2} \\frac{1}{2}\\,Max \\left[ |\\sigma_1 - \\sigma_2|,|\\sigma_2 - \\sigma_3|,|\\sigma_1 - \\sigma_3| \\right] = \\frac{\\sigma_0}{2}  \\]<\/p>\n<\/div>\n<p>Since it is a plane stress state, only \u03c3<sub>1<\/sub> and \u03c3<sub>2<\/sub> have to be considered.<\/p>\n\n<p>The dimensions of the tensions are not noted in the following equations to improve readability. The angle of the principal stresses is<\/p>\n<p>\\[ \\tag{3} \\phi_0 = \\frac{arctan\\left( \\frac{2\\tau_{xy}}{\\sigma_x -\\sigma_y} \\right)}{2}\\]<\/p>\n<p>\\[ \\tag{4} \\phi_0 = \\frac{arctan\\left( \\frac{2 \\cdot 120}{-160 -80} \\right)}{2}\\]<\/p>\n<p>\\[ \\tag{5} \\phi_0 = -22.5\u00b0 \\]<\/p>\n<p>The principal stresses are<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{6} \\sigma_1 = \\frac{\\sigma_x + \\sigma_y}{2} + \\frac{\\sigma_x-\\sigma_y}{2} \\cdot cos 2 \\phi_0 + \\tau_{xy} \\cdot sin 2 \\phi_0 \\]<\/p>\n<p>\\[ \\tag{7} \\sigma_1 = \\frac{-160 + 80}{2} + \\frac{-160-80}{2} \\cdot cos (-45\u00b0) + 120 \\cdot sin (-45\u00b0) \\]<\/p>\n<\/div>\n<p>\\[ \\tag{8} \\sigma_1 = -210 \\, MPa \\]<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{9} \\sigma_2 = \\frac{\\sigma_x + \\sigma_y}{2} - \\frac{\\sigma_x-\\sigma_y}{2} \\cdot cos 2 \\phi_0 - \\tau_{xy} \\cdot sin 2 \\phi_0 \\]<\/p>\n<p>\\[ \\tag{10} \\sigma_2 = \\frac{-160 + 80}{2} - \\frac{-160-80}{2} \\cdot cos (-45\u00b0) - 120 \\cdot sin (-45\u00b0) \\]<\/p>\n<\/div>\n<p>\\[ \\tag{11} \\sigma_2 = 130 \\, MPa \\]<\/p>\n<p>This results in the decisive tension<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{12} \\sigma_0 = |-210 - 130| \\, MPa = 340 \\, MPa \\]<\/p>\n<\/div>\n\n<p>The safety against flow S<sub>F<\/sub> is<\/p> \n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{13} S_F = \\frac{\\sigma_F}{\\sigma_0} = \\frac{650\\,MPa}{340\\,MPa} = 1.91 \\]<\/p>\n<\/div>\n\n\n\n<p>This shows that there is no double safety against flow according to the shear stress hypothesis according to TRESCA.<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Safety against plastic flow acc. TRESCA (shear stress hypothesis)\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/mechanics-ii-safety-against-plastic-flow-acc-tresca-shear-stress-hypothesis\/\" aria-label=\"Read more about Safety against plastic flow acc. TRESCA (shear stress hypothesis)\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":3034,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[52,60],"tags":[],"class_list":["post-3043","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>Safety against plastic flow acc. 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