{"id":3061,"date":"2021-02-15T14:05:22","date_gmt":"2021-02-15T14:05:22","guid":{"rendered":"https:\/\/pickedshares.com\/?p=3061"},"modified":"2021-05-08T13:13:05","modified_gmt":"2021-05-08T13:13:05","slug":"comparison-of-the-shear-stress-hypothesis-and-the-v-mises-criterion","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/comparison-of-the-shear-stress-hypothesis-and-the-v-mises-criterion\/","title":{"rendered":"Comparison of the shear stress hypothesis and the v.MISES criterion"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">This exercise compares the shear stress hypothesis and the maximum&nbsp;shear&nbsp;strain&nbsp;energy&nbsp;criterion for a plane stress state.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A thin-walled pipe is is under load of tension and torsion at the same time. The corresponding stress tensor 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\\[ \\newcommand{\\myvec}[1]{{\\begin{bmatrix}#1\\end{bmatrix}}} \\]\n<p>\\[ S = \\myvec{\\sigma^* &amp; \\tau &amp; 0\\\\ \\tau &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 0} \\]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Set up the flow conditions according to TRESCA and HUBER-v.MISES. We are looking for the respective value for \u03c3 * that flow begins.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The yield point for the material is \u03c3<sub>F<\/sub> = 300 MPa, the shear stress \u03c4 = 145 MPa.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Yield condition according to TRESCA, shear stress hypothesis<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">In the following equations, the unit MPa is omitted for better readability. The equivalent stress according to TRESCA is for the plane stress state<\/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>\\[ \\tag{1} \\sigma_v = \\sqrt{\\left( \\sigma_x - \\sigma_y \\right)^2 + 4 \\tau^2} \\]<\/p>\n<p>The equivalent stress is set equal to the yield point, so that the following results:<\/p>\n<p>\\[ \\tag{2} 300 = \\sqrt{\\left( \\sigma^* \\right)^2 + 4 \\left( 145 \\right)^2} \\]<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{3} \\sigma^* = \\sqrt(5900) = 76.8 \\, MPa \\]<\/p>\n<\/div>\n<p>The stress state at the moment of the start of flow is according to TRESCA<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{4} S = \\myvec{76.8 &amp; 145 &amp; 0\\\\ 145 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 0} \\,MPa \\]<\/p>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Yield condition according to HUBER-v.MISES, maximum&nbsp;shear&nbsp;strain&nbsp;energy&nbsp;criterion<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The equivalent stress according to HUBER-v.MISES is for the plane stress state<\/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\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{5} \\sigma_v = \\sqrt{\\sigma_x^2+\\sigma_y^2-\\sigma_x \\sigma_y + 3 \\tau^2} \\]<\/p>\n<\/div>\n<p>The equivalent stress is set equal to the yield point, so that the following results:<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{6} 300 = \\sqrt{(\\sigma^*)^2-\\sigma^* \\cdot 0 + 63075} \\]<\/p>\n<p>\\[ \\tag{7} \\sigma^* = \\sqrt{26925} = 164.1 \\,MPa \\]<\/p>\n<\/div>\n<p>According to HUBER-v.MISES, the state of stress at the moment the flow begins<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{8} S = \\myvec{164.1 &amp; 145 &amp; 0\\\\ 145 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 0} \\,MPa \\]<\/p>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The shear stress hypothesis calculates a significantly lower failure stress and is therefore the conservative approach.<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Comparison of the shear stress hypothesis and the v.MISES criterion\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/comparison-of-the-shear-stress-hypothesis-and-the-v-mises-criterion\/\" aria-label=\"Read more about Comparison of the shear stress hypothesis and the v.MISES criterion\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":3056,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[52,60],"tags":[],"class_list":["post-3061","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 v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Comparison of the shear stress hypothesis and the v.MISES criterion<\/title>\n<meta name=\"description\" content=\"This exercise shows a comparison between the calculation of the equivalent tensions according to TRESCA and v.MISES for a plane stress state.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, 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