A laminated wood beam is built up by gluing together three 2- in.-by-4-in. members to form a solid beam 4-in.-by-6-in. in cross section, as shown. The allowable shear stress in the glued joints is 60 psi. The beam is a 3-ft-long cantilever with a load P applied at its free end. Neglecting the weight of the beam:
<\/p>\n\n\n\n\n\n
(a) Calculate the maximum allowable load P that can be applied at the free end.<\/p>\n\n\n\n
(b) Calculate the maximum bending stress in the beam.<\/p>\n\n\n\n
The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n To solve this problem, we’ll determine the maximum allowable load PP based on the shear stress at the glued joints and then calculate the maximum bending stress.<\/p>\n\n\n\n The critical shear force will act at the glued joints. The maximum horizontal shear force VV in the beam occurs at the fixed support due to the applied load PP. For a cantilever beam: V=PV = P<\/p>\n\n\n\n The glued joints split the beam into two horizontal laminations:<\/p>\n\n\n\n The first moment of area QQ for the section above the joint is: Q=Area above joint\u00d7distance to centroid of area above jointQ = \\text{Area above joint} \\times \\text{distance to centroid of area above joint} Q=(2\u2009in.\u00d74\u2009in.)\u00d7(4\u2009in.+2\u2009in.2)Q = (2 \\, \\text{in.} \\times 4 \\, \\text{in.}) \\times \\left( \\frac{4 \\, \\text{in.} + 2 \\, \\text{in.}}{2} \\right) Q=8\u2009in.2\u00d73\u2009in.=24\u2009in.3Q = 8 \\, \\text{in.}^2 \\times 3 \\, \\text{in.} = 24 \\, \\text{in.}^3<\/p>\n\n\n\n The moment of inertia of the rectangular beam ( 4\u2009in.\u00d76\u2009in.4 \\, \\text{in.} \\times 6 \\, \\text{in.}) about its neutral axis is: I=112bh3=112(4)(63)I = \\frac{1}{12} b h^3 = \\frac{1}{12} (4) (6^3) I=112\u22c54\u22c5216=72\u2009in.4I = \\frac{1}{12} \\cdot 4 \\cdot 216 = 72 \\, \\text{in.}^4<\/p>\n\n\n\n The horizontal shear stress in the glued joint is given by: \u03c4=VQIb\\tau = \\frac{V Q}{I b}<\/p>\n\n\n\n Where:<\/p>\n\n\n\n Setting \u03c4=\u03c4allow\\tau = \\tau_{\\text{allow}}: 60=P(24)72(4)60 = \\frac{P (24)}{72 (4)} P=60\u22c572\u22c5424=720\u2009lbP = \\frac{60 \\cdot 72 \\cdot 4}{24} = 720 \\, \\text{lb}<\/p>\n\n\n\n The bending stress in the beam is given by: \u03c3=McI\\sigma = \\frac{M c}{I}<\/p>\n\n\n\n Where:<\/p>\n\n\n\n Substitute P=720\u2009lbP = 720 \\, \\text{lb} and L=36\u2009in.L = 36 \\, \\text{in.}: M=PL=720\u22c536=25,920\u2009lb-in.M = P L = 720 \\cdot 36 = 25,920 \\, \\text{lb-in.} \u03c3=McI=25,920\u22c5372\\sigma = \\frac{M c}{I} = \\frac{25,920 \\cdot 3}{72} \u03c3=1,080\u2009psi\\sigma = 1,080 \\, \\text{psi}<\/p>\n\n\n\n (a) The maximum allowable load PP is: P=720\u2009lbP = 720 \\, \\text{lb}<\/p>\n\n\n\n (b) The maximum bending stress in the beam is: \u03c3=1,080\u2009psi\\sigma = 1,080 \\, \\text{psi}<\/p>\n","protected":false},"excerpt":{"rendered":" A laminated wood beam is built up by gluing together three 2- in.-by-4-in. members to form a solid beam 4-in.-by-6-in. in cross section, as shown. The allowable shear stress in the glued joints is 60 psi. The beam is a 3-ft-long cantilever with a load P applied at its free end. Neglecting the weight of […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-184179","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184179","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=184179"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184179\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=184179"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=184179"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=184179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\n(a) Maximum Allowable Load PP Based on Shear Stress<\/strong><\/h3>\n\n\n\n
Key Dimensions and Properties:<\/strong><\/h4>\n\n\n\n
\n
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Shear Force at the Joint:<\/strong><\/h4>\n\n\n\n
First Moment of Area QQ:<\/strong><\/h4>\n\n\n\n
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Moment of Inertia II:<\/strong><\/h4>\n\n\n\n
Shear Stress Formula:<\/strong><\/h4>\n\n\n\n
\n
\n\n\n\n(b) Maximum Bending Stress<\/strong><\/h3>\n\n\n\n
Bending Stress Formula:<\/strong><\/h4>\n\n\n\n
\n
\n\n\n\nFinal Answers:<\/strong><\/h3>\n\n\n\n