{"id":28686,"date":"2013-05-21T17:01:50","date_gmt":"2013-05-21T11:31:50","guid":{"rendered":"http:\/\/www.kopykitab.com\/blog\/?p=28686"},"modified":"2013-05-21T17:01:50","modified_gmt":"2013-05-21T11:31:50","slug":"classical-plate-equation-notes","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/classical-plate-equation-notes\/","title":{"rendered":"Classical Plate Equation Notes"},"content":{"rendered":"<h1 style=\"text-align: center;\">Classical Plate Equation Notes<\/h1>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #000000;\">The\u00a0<em>small<\/em>\u00a0transverse (out-of-plane) displacement\u00a0<i>w<\/i>\u00a0of a\u00a0<em>thin<\/em>\u00a0plate is governed by the\u00a0Classical Plate Equation,<\/span><\/p>\n<p>&nbsp;<\/p>\n<div align=\"center\"><span style=\"color: #000000;\"><img alt=\"\" src=\"http:\/\/www.samconsult.biz\/Science\/Plates\/Images\/ClassicalPlate.gif\" width=\"98\" height=\"28\" align=\"absMiddle\" border=\"0\"><\/span><\/div>\n<p><span style=\"color: #000000;\">where\u00a0<i>p<\/i>\u00a0is the distributed load (force per unit area) acting in the same direction as\u00a0<i>z<\/i>\u00a0(and\u00a0<i>w<\/i>), and D is the bending\/flexual rigidity of the plate defined as follows,<\/span><\/p>\n<p>&nbsp;<\/p>\n<div align=\"center\"><span style=\"color: #000000;\"><img alt=\"\" src=\"http:\/\/www.samconsult.biz\/Science\/Plates\/Images\/DDef.gif\" width=\"102\" height=\"64\" align=\"absMiddle\" border=\"0\"><\/span><\/div>\n<p><span style=\"color: #000000;\">in which\u00a0<i>E<\/i>\u00a0is the Young&#8217;s modulus,\u00a0<img alt=\"nu\" src=\"http:\/\/www.samconsult.biz\/Science\/Plates\/Images\/nu.gif\" width=\"12\" height=\"14\" align=\"absMiddle\" border=\"0\" \/>\u00a0is the Poisson&#8217;s ratio of the plate material, and\u00a0<i>t<\/i>\u00a0is the thickness of the plate.<\/span><\/p>\n<p><span style=\"color: #000000;\">Furthermore, the differential operator\u00a0<img alt=\"\" src=\"http:\/\/www.samconsult.biz\/Science\/Plates\/Images\/Del2.gif\" width=\"25\" height=\"25\" align=\"absMiddle\" border=\"0\">\u00a0is called the Laplacian differential operator\u00a0<img alt=\"\" src=\"http:\/\/www.samconsult.biz\/Science\/Plates\/Images\/LaplaceOp.gif\" width=\"17\" height=\"17\" align=\"absMiddle\" border=\"0\">,<\/span><\/p>\n<p>&nbsp;<\/p>\n<div align=\"center\"><span style=\"color: #000000;\"><img alt=\"\" src=\"http:\/\/www.samconsult.biz\/Science\/Plates\/Images\/Del2B.gif\" width=\"378\" height=\"112\" align=\"absMiddle\" border=\"0\"><\/span><\/div>\n<p><span style=\"color: #000000;\">If the bending rigidity\u00a0<i>D<\/i>\u00a0is constant throughout the plate, the plate equation can be simplified to,<\/span><\/p>\n<p>&nbsp;<\/p>\n<div align=\"center\"><span style=\"color: #000000;\"><img alt=\"\" src=\"http:\/\/www.samconsult.biz\/Science\/Plates\/Images\/ClassicalPlateB.gif\" width=\"70\" height=\"41\" align=\"absMiddle\" border=\"0\"><\/span><\/div>\n<p><span style=\"color: #000000;\">where\u00a0<img alt=\"\" src=\"http:\/\/www.samconsult.biz\/Science\/Plates\/Images\/BiharmonicOp.gif\" width=\"132\" height=\"25\" align=\"absMiddle\" border=\"0\">\u00a0is called the bi-harmonic differential operator.<\/span><\/p>\n<p><span style=\"color: #000000;\">This small deflection theory assumes that\u00a0<i>w<\/i>\u00a0is small in comparison to the thickness of the plate\u00a0<i>t<\/i>, and the strains and the mid-plane slopes are much smaller than 1.<br \/>\n<\/span><\/p>\n<ul>\n<li type=\"disc\"><span style=\"color: #000000;\">A plate is called thin when its thickness\u00a0<i>t<\/i>\u00a0is at least one order of magnitude smaller than the span or diameter of the plate.<\/span><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Classical Plate Equation Notes &nbsp; The\u00a0small\u00a0transverse (out-of-plane) displacement\u00a0w\u00a0of a\u00a0thin\u00a0plate is governed by the\u00a0Classical Plate Equation, &nbsp; where\u00a0p\u00a0is the distributed load (force per unit area) acting in the same direction as\u00a0z\u00a0(and\u00a0w), and D is the bending\/flexual rigidity of the plate defined as follows, &nbsp; in which\u00a0E\u00a0is the Young&#8217;s modulus,\u00a0\u00a0is the Poisson&#8217;s ratio of the plate material, &#8230; <a title=\"Classical Plate Equation Notes\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/classical-plate-equation-notes\/\" aria-label=\"More on Classical Plate Equation Notes\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[4773],"tags":[],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/28686"}],"collection":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=28686"}],"version-history":[{"count":0,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/28686\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=28686"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=28686"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=28686"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}