{"id":990,"date":"2016-09-28T21:59:58","date_gmt":"2016-09-28T21:59:58","guid":{"rendered":"http:\/\/blogs.napier.ac.uk\/cwst\/?p=990"},"modified":"2016-09-29T12:43:30","modified_gmt":"2016-09-29T12:43:30","slug":"5th-percentile-shear-modulus","status":"publish","type":"post","link":"https:\/\/blogs.napier.ac.uk\/cwst\/5th-percentile-shear-modulus\/","title":{"rendered":"5th percentile shear modulus?"},"content":{"rendered":"<p>In the current version of Eurocode 5\u00a0\u00a0(EN 1995-1-1:2004+A2:2014) you can find equations that relate to the lateral torsional buckling of beams.<\/p>\n<p>Equation 6.31 gives the critical bending stress<\/p>\n<p><a href=\"http:\/\/blogs.napier.ac.uk\/cwst\/wp-content\/uploads\/sites\/23\/2016\/09\/Capture.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-991 alignleft\" src=\"http:\/\/blogs.napier.ac.uk\/cwst\/wp-content\/uploads\/sites\/23\/2016\/09\/Capture-300x81.png\" alt=\"Eqn 6.31\" width=\"300\" height=\"81\" srcset=\"https:\/\/blogs.napier.ac.uk\/cwst\/wp-content\/uploads\/sites\/23\/2016\/09\/Capture-300x81.png 300w, https:\/\/blogs.napier.ac.uk\/cwst\/wp-content\/uploads\/sites\/23\/2016\/09\/Capture.png 409w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>Where:<br \/>\n<em>E<sub>0,05<\/sub><\/em> is the fifth percentile value of modulus of elasticity parallel to grain<br \/>\n<em>G<sub>0,05<\/sub><\/em> is the fifth percentile value of modulus of elasticity parallel to grain<br \/>\n<em>I<sub>z<\/sub><\/em> is the second moment of area about the weak axis<br \/>\n<em>I<sub>tor<\/sub><\/em> is the torsional moment of inertia<br \/>\n<em>l<sub>ef<\/sub><\/em> is the effective length of the beam (depending on support conditions and loading configuration)<br \/>\n<em>W<sub>y<\/sub><\/em> is the section modulus about the strong axis<\/p>\n<p>For graded timber, you can find a value of\u00a0fifth percentile value of modulus of elasticity parallel to grain (<em>E<sub>0,05<\/sub><\/em>) by looking at the strength class table in EN338 (or the equations for secondary properties (which have been moved to\u00a0the 2016 version of EN384). \u00a0But you won&#8217;t find a value for fifth percentile value of modulus of elasticity parallel to grain. \u00a0So what do you do?<\/p>\n<p>Well, if you have rectangular cross-section softwood you have a solution in the form of equation 6.32:<\/p>\n<p><a href=\"http:\/\/blogs.napier.ac.uk\/cwst\/wp-content\/uploads\/sites\/23\/2016\/09\/Capture2-e1475096927396.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-992 alignleft\" src=\"http:\/\/blogs.napier.ac.uk\/cwst\/wp-content\/uploads\/sites\/23\/2016\/09\/Capture2-e1475096927396-300x97.png\" alt=\"Eqn 6.32\" width=\"300\" height=\"97\" srcset=\"https:\/\/blogs.napier.ac.uk\/cwst\/wp-content\/uploads\/sites\/23\/2016\/09\/Capture2-e1475096927396-300x97.png 300w, https:\/\/blogs.napier.ac.uk\/cwst\/wp-content\/uploads\/sites\/23\/2016\/09\/Capture2-e1475096927396.png 313w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>So what value of\u00a0<em>G<sub>0,05<\/sub> <\/em>does this equation imply? \u00a0We might use the EN338\/EN384 calculation\u00a0of\u00a0<em>G<sub>mean<\/sub><\/em> from<em> E<sub>mean<\/sub><\/em> and make an educated guess that\u00a0<em>G<sub>0,05<\/sub><\/em> =\u00a0<em>E<sub>0,05<\/sub><\/em> \/ 16<br \/>\nDoes this guess check out? \u00a0The answer is interesting: it depends.<\/p>\n<p>We have a rectangular cross-section in which:<br \/>\n<em>b<\/em> = the width of the beam (smaller cross-section dimension)<br \/>\n<em>h<\/em> = the depth of the beam\u00a0(larger cross-section dimension)<br \/>\n<em>I<sub>z<\/sub><\/em> = <em>bh<\/em><sup>3<\/sup>\/12<br \/>\n<em>W<sub>y<\/sub><\/em> = <em>bh<\/em><sup>2<\/sup>\/6<br \/>\n<em>I<sub>tor<\/sub><\/em> = <em>\u0392hb<\/em><sup>3<\/sup> in which <em>\u0392<\/em> is a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Torsion_constant#Rectangle\" target=\"_blank\">constant that depends on the aspect ratio of the cross-section<\/a><\/p>\n<p>If we set that\u00a0<em>G<sub>0,05<\/sub><\/em> =\u00a0<em>E<sub>0,05<\/sub><\/em> \/ <em>A<\/em><\/p>\n<p>And rearrange and simply Equation 6.31 = Equation 6.32<\/p>\n<p>We end up with<\/p>\n<p><em>A<\/em> = 48.7<em>B<\/em><\/p>\n<p>The value of <em>B<\/em> for a cross-section of infinite aspect ratio is 1\/3 so our guess that <em>A<\/em> is 16 checks out at that end of the scale<\/p>\n<p>But for non-infinite aspect ratios\u00a0we have a rather strange situation &#8211; the ratio of <em>G<\/em> and <em>E<\/em> assumed varies, depending on the ratio of <em>b<\/em> and <em>h<\/em>.<\/p>\n<p>&nbsp;<\/p>\n<table>\n<tbody>\n<tr>\n<td width=\"200\"><em>h\/b<\/em><\/td>\n<td width=\"200\"><i>B <\/i>(a textbook value)<\/td>\n<td width=\"200\"><em>A <\/em>(which is\u00a0<em>E<sub>0,05<\/sub><\/em> \/ <em>G<sub>0,05<\/sub><\/em>)<\/td>\n<\/tr>\n<tr>\n<td width=\"200\">1<\/td>\n<td width=\"200\">0.141<\/td>\n<td width=\"200\">6.8<\/td>\n<\/tr>\n<tr>\n<td width=\"200\">1.5<\/td>\n<td width=\"200\">0.196<\/td>\n<td width=\"200\">9.5<\/td>\n<\/tr>\n<tr>\n<td width=\"200\">2<\/td>\n<td width=\"200\">0.229<\/td>\n<td width=\"200\">11.1<\/td>\n<\/tr>\n<tr>\n<td width=\"200\">2.5<\/td>\n<td width=\"200\">0.249<\/td>\n<td width=\"200\">12.1<\/td>\n<\/tr>\n<tr>\n<td width=\"200\">3<\/td>\n<td width=\"200\">0.263<\/td>\n<td width=\"200\">12.8<\/td>\n<\/tr>\n<tr>\n<td width=\"200\">4<\/td>\n<td width=\"200\">0.281<\/td>\n<td width=\"200\">13.7<\/td>\n<\/tr>\n<tr>\n<td width=\"200\">5<\/td>\n<td width=\"200\">0.291<\/td>\n<td width=\"200\">14.2<\/td>\n<\/tr>\n<tr>\n<td width=\"200\">6<\/td>\n<td width=\"200\">0.299<\/td>\n<td width=\"200\">14.6<\/td>\n<\/tr>\n<tr>\n<td width=\"200\">10<\/td>\n<td width=\"200\">0.312<\/td>\n<td width=\"200\">15.2<\/td>\n<\/tr>\n<tr>\n<td width=\"200\">Infinity<\/td>\n<td width=\"200\">0.333<\/td>\n<td width=\"200\">16.2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now, we know that <em>E<\/em> and <em>G<\/em> are not well correlated for pieces within a species (e.g. <a href=\"http:\/\/researchrepository.napier.ac.uk\/4406\/\" target=\"_blank\">ref<\/a>) so actually it is not so likely that a piece will, simultaneously, have both low <em>E<\/em> and low <em>G. \u00a0<\/em>Lateral torsional buckling is also less critical for the smaller aspect ratios &#8230;and so perhaps it is reasonable that we have an equation that gets closer to the more conservative value as buckling becomes more of an issue.<\/p>\n<p>This leaves an open question &#8211; what value of 5th percentile shear modulus should you take for hardwoods? EN338\/EN384 imply the same ratio <em>G<sub>0,05<\/sub><\/em> =\u00a0<em>E<sub>0,05<\/sub><\/em> \/ 16\u00a0 for hardwoods so why is equation 6.32 specifically limited to softwoods? \u00a0Not sure &#8211; perhaps it is thought that the lower amount of knowledge we have of hardwood properties means that you should not be allowed that extra headroom for the beams with lower aspect ratios.<\/p>\n<p>In summary &#8211; it is not very helpful that EN1995 requires you to use a property that is not listed among the strength class values.<\/p>\n<p>By the way, for softwood glulam, EN14080:2013 gives a conversion in clause 5.1.3<\/p>\n<p><em>G<sub>g,k<\/sub><\/em>\/<em>G<sub>g,mean<\/sub><\/em> = 5\/6<\/p>\n<p>Since, for softwood, EN338\/EN384 tells us that\u00a0<em>E<sub>0,05<\/sub><\/em> is 2\/3 *\u00a0<em>E<sub>mean<\/sub><\/em>\u00a0and \u00a0<em>G<sub>mean<\/sub><\/em>\u00a0is <em>E<sub>mean<\/sub><\/em>\/16 this is equivalent to<\/p>\n<p><em>G<sub>0,05\u00a0<\/sub><\/em>= <em>G<sub>mean<\/sub><\/em>\u00a0* 5\/6 =\u00a0<em>E<sub>mean<\/sub><\/em>\/16 * 5\/6 = 3\/2 *\u00a0<em>E<sub>0,05<\/sub><\/em>\/16 * 5\/6<\/p>\n<p><em>G<sub>0,05\u00a0<\/sub><\/em>= <em>E<sub>0,05<\/sub><\/em>\/12.8<\/p>\n<p><em>E<sub>0,05<\/sub><\/em>\u00a0= 2\/3 *\u00a0<em>E<sub>mean<\/sub><\/em>\u00a0is equivalent to an assumption that <em>E<\/em> is normally distributed with a coefficient of variation of about 20%<\/p>\n<p><em>G<sub>0,05\u00a0<\/sub><\/em>= 5\/6 *\u00a0<em>G<sub>mean<\/sub><\/em> is equivalent to an assumption that <i>G<\/i>\u00a0is normally distributed with a coefficient of variation of about 10%<\/p>\n","protected":false},"excerpt":{"rendered":"<div class=\"mh-excerpt\"><p>In the current version of Eurocode 5\u00a0\u00a0(EN 1995-1-1:2004+A2:2014) you can find equations that relate to the lateral torsional buckling of beams. Equation 6.31 gives the <a class=\"mh-excerpt-more\" href=\"https:\/\/blogs.napier.ac.uk\/cwst\/5th-percentile-shear-modulus\/\" title=\"5th percentile shear modulus?\">[&#8230;]<\/a><\/p>\n<\/div>","protected":false},"author":57,"featured_media":995,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_crdt_document":"","jetpack_post_was_ever_published":false,"footnotes":""},"categories":[5,37],"tags":[108,68,10],"class_list":["post-990","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-standards","category-thoughts","tag-en1995-1-1","tag-en338","tag-en384"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>5th percentile shear modulus? - Centre for Wood Science &amp; Technology<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.napier.ac.uk\/cwst\/5th-percentile-shear-modulus\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"5th percentile shear modulus? - Centre for Wood Science &amp; Technology\" \/>\n<meta property=\"og:description\" content=\"In the current version of Eurocode 5\u00a0\u00a0(EN 1995-1-1:2004+A2:2014) you can find equations that relate to the lateral torsional buckling of beams. 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