{"id":160127,"date":"2022-09-26T10:00:37","date_gmt":"2022-09-26T10:00:37","guid":{"rendered":"\/knowledge\/forums\/topic\/discovery-aim-tutorial-structural-analysis-of-plate-with-hole\/"},"modified":"2023-08-16T06:33:34","modified_gmt":"2023-08-16T06:33:34","slug":"discovery-aim-tutorial-structural-analysis-of-plate-with-hole","status":"publish","type":"topic","link":"https:\/\/innovationspace.ansys.com\/knowledge\/forums\/topic\/discovery-aim-tutorial-structural-analysis-of-plate-with-hole\/","title":{"rendered":"Discovery AIM tutorial &#8211; Structural Analysis of Plate with Hole"},"content":{"rendered":"<p><strong>This example is taken from\u00a0<u><a href=\"https:\/\/confluence.cornell.edu\/display\/SIMULATION\/ANSYS+AIM+-+Plate+with+Hole\" target=\"_blank\" rel=\"nofollow noopener noreferrer\">Cornell University&#8217;s ANSYS AIM Learning Modules<\/a><\/u><\/strong><\/p>\n<hr \/>\n<nav class=\"toc -selected\">Contents<\/p>\n<ol class=\"toc__section -lev0\">\n<li class=\"toc__item -lev0\">Problem Specification<\/li>\n<li class=\"toc__item -lev0\">Pre-Analysis<\/li>\n<li class=\"toc__item -lev0\">Geometry<\/li>\n<li class=\"toc__item -lev0\">Mesh<\/li>\n<li class=\"toc__item -lev0\">Physics Setup<\/li>\n<li class=\"toc__item -lev0\">Results Evaluation<\/li>\n<li class=\"toc__item -lev0\">Verification<\/li>\n<\/ol>\n<\/nav>\n<h4 content_id=\"problem-specification\" class=\"toc__permalink\" content_id=\"problem-specification\" class=\"toc__permalink\"  id=\"PROBLEM-SPECIFICATION\">Problem Specification<\/h4>\n<p>Consider the classic example of a circular hole in a rectangular plate of constant thickness. The plate is A514 steel with a modulus of elasticity of 29e6 psi and a Poisson ratio of 0.3. The thickness of the plate is 0.2 in., the diameter of the hole is 0.5 in., the length of the plate is 10 in. and the width of the plate 5 in., as the figure below indicates.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-160424\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-10-300x196.png\" alt=\"\" width=\"300\" height=\"196\" srcset=\"https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-10-300x196.png 300w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-10-50x33.png 50w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-10-100x65.png 100w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-10-24x16.png 24w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-10-36x24.png 36w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-10-48x31.png 48w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-10.png 350w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>This tutorial will show you how to use ANSYS AIM to find the displacement and the stresses in the plate.<\/p>\n<hr \/>\n<h4 content_id=\"pre-analysis\" class=\"toc__permalink\" content_id=\"pre-analysis\" class=\"toc__permalink\"  id=\"PRE-ANALYSIS\">Pre-Analysis<\/h4>\n<p><strong>Analytical vs. Numerical Approaches<\/strong><\/p>\n<p>We can either assume the geometry as an infinite plate and solve the problem analytically, or approximate the geometry as a collection of &#8220;finite elements&#8221;, and solve the problem numerically. The following flow chart compares the two approaches.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-160425\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-11-279x300.png\" alt=\"\" width=\"279\" height=\"300\" srcset=\"https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-11-279x300.png 279w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-11-46x50.png 46w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-11-93x100.png 93w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-11-22x24.png 22w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-11-33x36.png 33w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-11-45x48.png 45w, https:\/\/innovationspace.ansys.com\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/09\/HM-11.png 601w\" sizes=\"auto, (max-width: 279px) 100vw, 279px\" \/><\/p>\n<p>Let&#8217;s first review the analytical results for the infinite plate. We&#8217;ll then use these results to check the numerical solution from ANSYS.<\/p>\n<p><strong>Analytical Results<\/strong><\/p>\n<p><strong>Displacement<\/strong><\/p>\n<p>Let&#8217;s estimate the expected displacement of the right edge relative to the center of the hole. We can get a reasonable estimate by neglecting the hole and approximating the entire plate as being in uniaxial tension. Dividing the applied tensile stress by the Young&#8217;s modulus gives the uniform strain in the x direction.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157054\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-12.png\" alt=\" width=\" height=\"93\" \/><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157055\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-13.png\" alt=\" width=\" height=\"38\" \/><\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157057\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-14.png\" alt=\" width=\" height=\"63\" \/><\/p>\n<p>Multiplying this by the half-width (5 in) gives the expected displacement of the right edge as ~ 0.1724 in. We&#8217;ll check this against ANSYS AIM.<\/p>\n<p><strong>\u03c3r<\/strong><\/p>\n<p>Let&#8217;s consider the expected trends for \u03c3r, the radial stress, in the vicinity of the hole and far from the hole. The analytical solution for \u03c3r in an infinite plate is:<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157058\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-15.png\" alt=\" width=\" height=\"58\" \/><\/p>\n<p>where a is the hole radius and \u03c3o is the applied uniform stress (denoted P in the problem specification). At the hole (r=a), this reduces to<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157059\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-16.png\" alt=\" width=\" height=\"27\" \/><\/p>\n<p>This result can be understood by looking at a vanishingly small element at the hole as shown schematically below.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157060\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-17.png\" alt=\" width=\" height=\"241\" \/><\/p>\n<p>We see that \u03c3r\u00a0at the hole is the normal stress at the hole. Since the hole is a free surface, this has to be zero.<\/p>\n<p>For r &gt;&gt; a,<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157061\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-18.png\" alt=\" width=\" height=\"33\" \/><\/p>\n<p>Far from the hole, \u03c3r\u00a0is a function of only. At = 0, \u03c3r\u00a0~ \u03c3o. This makes sense since r is aligned with x when = 0. At = 90 deg., \u03c3r\u00a0~ 0 which also makes sense since r is now aligned with y. We&#8217;ll check these trends in the ANSYS AIM results.<\/p>\n<p><strong>\u03c3 <\/strong><\/p>\n<p>Let&#8217;s next consider the expected trends for \u03c3 , the circumferential stress, in the vicinity of the hole and far from the hole. The analytical solution for \u03c3 \u00a0in an infinite plate is:<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157062\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-19.png\" alt=\" width=\" height=\"54\" \/><\/p>\n<p>At r = a, this reduces to<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157064\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-20.png\" alt=\" width=\" height=\"53\" \/><\/p>\n<p>At\u00a0 \u00a0= 90 deg., \u03c3 \u00a0= 3*\u03c3o\u00a0for an infinite plate. This leads to a stress concentration factor of 3 for an infinite plate.<\/p>\n<p>For r &gt;&gt; a,<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157065\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-21.png\" alt=\" width=\" height=\"36\" \/><\/p>\n<p>At\u00a0 \u00a0= 0 and\u00a0 \u00a0= 90 deg., we get<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157068\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-22.png\" alt=\" width=\" height=\"104\" \/><\/p>\n<p>Far from the hole, \u03c3 \u00a0is a function of\u00a0 \u00a0only but its variation is the opposite of \u03c3r\u00a0(which is not surprising since r and\u00a0 \u00a0are orthogonal coordinates; when r is aligned with x,\u00a0 \u00a0is aligned with y and vice-versa). As one goes around the hole from\u00a0 \u00a0= 0 to\u00a0 \u00a0= 90 deg., \u03c3 \u00a0increases from 0 to \u03c3o. More trends to check in the ANSYS AIM results!<\/p>\n<p><strong> r <\/strong><\/p>\n<p>The analytical solution for the shear stress,\u00a0 r , in an infinite plate is:<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157069\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-23.png\" alt=\" width=\" height=\"51\" \/><\/p>\n<p>At r = a,<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157072\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-24.png\" alt=\" width=\" height=\"39\" \/><\/p>\n<p>By looking at a vanishingly small element at the hole, we see that\u00a0 r \u00a0is the shear stress on a stress surface, so it has to be zero.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157073\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-25.png\" alt=\" width=\" height=\"225\" \/><\/p>\n<p>For r &gt;&gt; a,<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157074\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-26.png\" alt=\" width=\" height=\"47\" \/><\/p>\n<p>We can deduce that, far from the hole, r \u00a0= 0 both at\u00a0 \u00a0= 0 and\u00a0 \u00a0= 90 deg. Even more trends to check in ANSYS AIM!<\/p>\n<p><strong>\u03c3x<\/strong><\/p>\n<p>First, let&#8217;s begin by finding the average stress, the nominal area stress, and the maximum stress with a concentration factor.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157076\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-27.png\" alt=\" width=\" height=\"116\" \/><\/p>\n<p>The concentration factor for an infinite plate with a hole is K = 3. The maximum stress for an infinite plate with a hole is:<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157078\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-28.png\" alt=\" width=\" height=\"90\" \/><\/p>\n<p>Although there is no analytical solution for a finite plate with a hole, there is empirical data available to find a concentration factor. Using a Concentration Factor Chart (Cornell 3250 Students: See Figure 4.22 on page 158 in Deformable Bodies and Their Material Behavior), we find that d\/w = 1 and thus K ~ 2.73. Now, we can find the maximum stress using the nominal stress and the concentration factor:<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157080\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-29.png\" alt=\" width=\" height=\"32\" \/><\/p>\n<hr \/>\n<h4 content_id=\"geometry\" class=\"toc__permalink\" content_id=\"geometry\" class=\"toc__permalink\"  id=\"GEOMETRY\"><strong>Geometry<\/strong><\/h4>\n<p>Since the problem given to us represents a 2D state of stress, there are several simplifications that must be made due to the fact that Discovery\u00a0AIM model will be 3D. Since the geometry is symmetric, when we draw the shape we want to create a one-quarter model. By taking advantage of symmetry, we can simplify the problem and also ensure that the model is not over constrained. Below video shows how to create the geometry.<\/p>\n<p><iframe loading=\"lazy\" class=\"vidyard_iframe\" src=\"\/\/play.vidyard.com\/d4XCbuyzXRHkbUFJpkT42Z.html?\" width=\"700\" height=\"400\" frameborder=\"0\" scrolling=\"no\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<hr \/>\n<h4 content_id=\"mesh\" class=\"toc__permalink\" content_id=\"mesh\" class=\"toc__permalink\"  id=\"MESH\">Mesh<\/h4>\n<p>In this tutorial, we will be using\u00a0<strong>Physics-Aware Meshing<\/strong>. Physics-aware\u00a0meshing helps automate and simplify your problem setup.\u00a0With physics-aware meshing, the computational mesh is generated automatically based on the solution fidelity setting and the physics inputs.<\/p>\n<hr \/>\n<h4 content_id=\"physics-setup\" class=\"toc__permalink\" content_id=\"physics-setup\" class=\"toc__permalink\"  id=\"PHYSICS-SETUP\">Physics Setup<\/h4>\n<p>In this video, you will learn how to define symmetric boundary conditions and pressure to the plate<\/p>\n<p><iframe loading=\"lazy\" class=\"vidyard_iframe\" src=\"\/\/play.vidyard.com\/SrvjyXs3hyVrBzkazDTtMy.html?\" width=\"700\" height=\"400\" frameborder=\"0\" scrolling=\"no\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<hr \/>\n<h4 content_id=\"results-evaluation\" class=\"toc__permalink\" content_id=\"results-evaluation\" class=\"toc__permalink\"  id=\"RESULTS-EVALUATION\">Results Evaluation<\/h4>\n<p>In this video, you will evaluate stresses generated in the plate and its deformation<\/p>\n<p><iframe loading=\"lazy\" class=\"vidyard_iframe\" src=\"\/\/play.vidyard.com\/u8t4nc1UVPkT4exiy3nxaX.html?\" width=\"700\" height=\"400\" frameborder=\"0\" scrolling=\"no\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<hr \/>\n<h4 content_id=\"verification\" class=\"toc__permalink\" content_id=\"verification\" class=\"toc__permalink\"  id=\"VERIFICATION\">Verification<\/h4>\n<p>In the pre-analysis, the maximum stress was calculated. \u00a0In order to verify that our simulation was accurate, a comparison must be made. From\u00a0 the Results Evaluation video, the maximum stress value can be determined.<\/p>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-157085\" src=\"\/knowledge\/wp-content\/uploads\/sites\/4\/2022\/08\/HM-30.png\" alt=\" width=\" height=\"648\" \/><\/p>\n<p>The table below compares the calculated and simulated values for maximum stress in the plate with a hole. With a difference of less than 5%, we can consider our simulation to be accurate. This small difference in results may be caused by not using enough mesh refinement.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Simulated Value<\/td>\n<td>Calculated Value<\/td>\n<td>Difference<\/td>\n<\/tr>\n<tr>\n<td>3.1157E6 psi<\/td>\n<td>3.033E6 psi<\/td>\n<td>2.71%<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"template":"","class_list":["post-160127","topic","type-topic","status-publish","hentry","topic-tag-aim-tutorials","topic-tag-discovery-aim","topic-tag-structures"],"aioseo_notices":[],"acf":[],"custom_fields":[{"0":{"_wp_page_template":["default"],"_bbp_last_active_time":["09-13-2022  20:20:04"],"_bbp_forum_id":["159552"],"_btv_view_count":["7496"],"_edit_lock":["1665575980:77457"],"_edit_last":["77457"],"_bbp_topic_id":["160127"],"_yoast_wpseo_content_score":["30"],"_yoast_wpseo_estimated-reading-time-minutes":["9"],"_yoast_wpseo_wordproof_timestamp":[""],"family":[""],"application_name":[""],"product_version":[""]},"test":"articlesansys-com"}],"_links":{"self":[{"href":"https:\/\/innovationspace.ansys.com\/knowledge\/wp-json\/wp\/v2\/topics\/160127","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/innovationspace.ansys.com\/knowledge\/wp-json\/wp\/v2\/topics"}],"about":[{"href":"https:\/\/innovationspace.ansys.com\/knowledge\/wp-json\/wp\/v2\/types\/topic"}],"version-history":[{"count":0,"href":"https:\/\/innovationspace.ansys.com\/knowledge\/wp-json\/wp\/v2\/topics\/160127\/revisions"}],"wp:attachment":[{"href":"https:\/\/innovationspace.ansys.com\/knowledge\/wp-json\/wp\/v2\/media?parent=160127"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}