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methods)</span> </div> </a> <button aria-controls="toc-Strength_of_materials_methods_(classical_methods)-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Strength of materials methods (classical methods) subsection</span> </button> <ul id="toc-Strength_of_materials_methods_(classical_methods)-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> <li id="toc-Method_of_joints" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Method_of_joints"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Method of joints</span> </div> </a> <ul id="toc-Method_of_joints-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Method_of_sections" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Method_of_sections"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Method of sections</span> </div> </a> <ul id="toc-Method_of_sections-sublist" class="vector-toc-list"> <li id="toc-Method_1:_Ignore_the_right_side" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Method_1:_Ignore_the_right_side"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2.1</span> <span>Method 1: Ignore the right side</span> </div> </a> <ul id="toc-Method_1:_Ignore_the_right_side-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Method_2:_Ignore_the_left_side" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Method_2:_Ignore_the_left_side"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2.2</span> <span>Method 2: Ignore the left side</span> </div> </a> <ul id="toc-Method_2:_Ignore_the_left_side-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Elasticity_methods" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elasticity_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Elasticity methods</span> </div> </a> <ul id="toc-Elasticity_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Methods_using_numerical_approximation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Methods_using_numerical_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Methods using numerical approximation</span> </div> </a> <ul id="toc-Methods_using_numerical_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Timeline" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Timeline"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Timeline</span> </div> </a> <ul id="toc-Timeline-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> 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class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Structural analysis</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%88%98%E1%8B%8B%E1%89%85%E1%88%AD_%E1%89%B5%E1%8A%95%E1%89%B3%E1%8A%94" title="የመዋቅር ትንታኔ – Amharic" lang="am" hreflang="am" data-title="የመዋቅር ትንታኔ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%A5%D9%86%D8%B4%D8%A7%D8%A6%D9%8A" title="تحليل إنشائي – Arabic" lang="ar" hreflang="ar" data-title="تحليل إنشائي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%82%D1%80%D0%BE%D0%B8%D1%82%D0%B5%D0%BB%D0%BD%D0%B0_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Строителна статика – Bulgarian" lang="bg" hreflang="bg" data-title="Строителна статика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_estructural" title="Anàlisi estructural – Catalan" lang="ca" hreflang="ca" data-title="Anàlisi estructural" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Baustatik" title="Baustatik – German" lang="de" hreflang="de" data-title="Baustatik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_estructural" title="Análisis estructural – Spanish" lang="es" hreflang="es" data-title="Análisis estructural" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%DB%8C%D9%84_%D8%B3%D8%A7%D8%B2%D9%87%E2%80%8C%D9%87%D8%A7" title="تحلیل سازه‌ها – Persian" lang="fa" hreflang="fa" data-title="تحلیل سازه‌ها" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Analyse_structurelle" title="Analyse structurelle – French" lang="fr" hreflang="fr" data-title="Analyse structurelle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%B0%E0%A4%9A%E0%A4%A8%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%AE%E0%A4%95_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E2%80%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="संरचनात्मक विश्‍लेषण – Hindi" lang="hi" hreflang="hi" data-title="संरचनात्मक विश्‍लेषण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_struktur" title="Analisis struktur – Indonesian" lang="id" hreflang="id" data-title="Analisis struktur" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Analisi_strutturale" title="Analisi strutturale – Italian" lang="it" hreflang="it" data-title="Analisi strutturale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%98%E0%B4%9F%E0%B4%A8%E0%B4%BE_%E0%B4%B5%E0%B4%BF%E0%B4%B6%E0%B4%95%E0%B4%B2%E0%B4%A8%E0%B4%82" title="ഘടനാ വിശകലനം – Malayalam" lang="ml" hreflang="ml" data-title="ഘടനാ വിശകലനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D0%B0%D1%80%D0%B8%D0%BB%D0%B3%D1%8B%D0%BD_%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA" title="Барилгын статик – Mongolian" lang="mn" hreflang="mn" data-title="Барилгын статик" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Sterkteberekeningen" title="Sterkteberekeningen – Dutch" lang="nl" hreflang="nl" data-title="Sterkteberekeningen" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_dynamiczna" title="Analiza dynamiczna – Polish" lang="pl" hreflang="pl" data-title="Analiza dynamiczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/An%C3%A1lise_estrutural" title="Análise estrutural – Portuguese" lang="pt" hreflang="pt" data-title="Análise estrutural" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D1%80%D1%83%D0%BA%D1%82%D1%83%D1%80%D0%BD%D1%8B%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7" title="Структурный анализ – Russian" lang="ru" hreflang="ru" data-title="Структурный анализ" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Byggnadsstatik" title="Byggnadsstatik – Swedish" lang="sv" hreflang="sv" data-title="Byggnadsstatik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Yap%C4%B1_stati%C4%9Fi" title="Yapı statiği – Turkish" lang="tr" hreflang="tr" data-title="Yapı statiği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D1%82%D1%80%D1%83%D0%BA%D1%82%D1%83%D1%80%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7" title="Структурний аналіз – Ukrainian" lang="uk" hreflang="uk" data-title="Структурний аналіз" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/K%E1%BA%BFt_c%E1%BA%A5u_x%C3%A2y_d%E1%BB%B1ng" title="Kết cấu xây dựng – Vietnamese" lang="vi" hreflang="vi" data-title="Kết cấu xây dựng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a 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For social-science usage, see <a href="/wiki/Structuralism" title="Structuralism">Structuralism</a>. For other uses, see <a href="/wiki/Structure_(disambiguation)" class="mw-disambig" title="Structure (disambiguation)">Structure (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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.sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title">Mechanical failure modes</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Buckling" title="Buckling">Buckling</a></li> <li><a href="/wiki/Corrosion" title="Corrosion">Corrosion</a></li> <li><a href="/wiki/Corrosion_fatigue" title="Corrosion fatigue">Corrosion fatigue</a></li> <li><a href="/wiki/Creep_(deformation)" title="Creep (deformation)">Creep</a></li> <li><a href="/wiki/Fatigue_(material)" title="Fatigue (material)">Fatigue</a></li> <li><a href="/wiki/Fouling" title="Fouling">Fouling</a></li> <li><a href="/wiki/Fracture" title="Fracture">Fracture</a></li> <li><a href="/wiki/Hydrogen_embrittlement" title="Hydrogen embrittlement">Hydrogen embrittlement</a></li> <li><a href="/wiki/Impact_(mechanics)" title="Impact (mechanics)">Impact</a></li> <li><a href="/wiki/Liquid_metal_embrittlement" title="Liquid metal embrittlement">Liquid metal embrittlement</a></li> <li><a href="/wiki/Mechanical_overload" title="Mechanical overload">Mechanical overload</a></li> <li><a href="/wiki/Metal-induced_embrittlement" title="Metal-induced embrittlement">Metal-induced embrittlement</a></li> <li><a href="/wiki/Stress_corrosion_cracking" title="Stress corrosion cracking">Stress corrosion cracking</a></li> <li><a href="/wiki/Sulfide_stress_cracking" title="Sulfide stress cracking">Sulfide stress cracking</a></li> <li><a href="/wiki/Thermal_shock" title="Thermal shock">Thermal shock</a></li> <li><a href="/wiki/Wear" title="Wear">Wear</a></li> <li><a href="/wiki/Yield_(engineering)" title="Yield (engineering)">Yielding</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar 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.mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Structural_analysis" title="Special:EditPage/Structural analysis">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Structural+analysis%22">"Structural analysis"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Structural+analysis%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Structural+analysis%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Structural+analysis%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Structural+analysis%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Structural+analysis%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2018</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><b>Structural analysis</b> is a branch of <a href="/wiki/Solid_mechanics" title="Solid mechanics">solid mechanics</a> which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of <a href="/wiki/Structural_load" title="Structural load">loads</a> on physical <a href="/wiki/Structure#Load-bearing" title="Structure">structures</a> and their <a href="/wiki/Structural_engineering#Structural_elements" title="Structural engineering">components</a>. In contrast to theory of elasticity, the models used in structural analysis are often differential equations in one spatial variable. Structures subject to this type of <a href="/wiki/Analysis" title="Analysis">analysis</a> include all that must withstand loads, such as buildings, bridges, aircraft and ships. Structural analysis uses ideas from <a href="/wiki/Applied_mechanics" title="Applied mechanics">applied mechanics</a>, <a href="/wiki/Materials_science" title="Materials science">materials science</a> and <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> to compute a structure's <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deformations</a>, internal <a href="/wiki/Force" title="Force">forces</a>, <a href="/wiki/Stress_analysis" class="mw-redirect" title="Stress analysis">stresses</a>, support reactions, velocity, accelerations, and <a href="/wiki/Structural_stability" title="Structural stability">stability</a>. The results of the analysis are used to verify a structure's fitness for use, often precluding <a href="/wiki/Physical_test" title="Physical test">physical tests</a>. Structural analysis is thus a key part of the <a href="/wiki/Structural_engineering" title="Structural engineering">engineering design of structures</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Structures_and_loads">Structures and loads</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=1" title="Edit section: Structures and loads"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the context to structural analysis, a <a href="/wiki/Structure" title="Structure">structure</a> refers to a body or system of connected parts used to support a load. Important examples related to <a href="/wiki/Civil_Engineering" class="mw-redirect" title="Civil Engineering">Civil Engineering</a> include buildings, bridges, and towers; and in other branches of engineering, ship and aircraft frames, tanks, pressure vessels, mechanical systems, and electrical supporting structures are important. To design a structure, an engineer must account for its safety, aesthetics, and serviceability, while considering economic and environmental constraints. Other branches of <a href="/wiki/Engineering" title="Engineering">engineering</a> work on a wide variety of <a href="/wiki/Non-building_structure" class="mw-redirect" title="Non-building structure">non-building structures</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Classification_of_structures">Classification of structures</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=2" title="Edit section: Classification of structures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>structural system</i> is the combination of structural elements and their materials. It is important for a structural engineer to be able to classify a structure by either its form or its function, by recognizing the various <a href="/wiki/Structural_engineering#Structural_elements" title="Structural engineering">elements</a> composing that structure. The structural elements guiding the systemic forces through the materials are not only such as a connecting rod, a truss, a beam, or a column, but also a cable, an arch, a cavity or channel, and even an angle, a surface structure, or a frame. </p> <div class="mw-heading mw-heading3"><h3 id="Loads">Loads</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=3" title="Edit section: Loads"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Structural_load" title="Structural load">Structural load</a></div> <p>Once the dimensional requirement for a structure have been defined, it becomes necessary to determine the loads the structure must support. Structural design, therefore begins with specifying loads that act on the structure. The design loading for a structure is often specified in <a href="/wiki/Building_code" title="Building code">building codes</a>. There are two types of codes: general building codes and design codes, engineers must satisfy all of the code's requirements in order for the structure to remain reliable. </p><p>There are two types of loads that structure engineering must encounter in the design. The first type of loads are dead loads that consist of the weights of the various structural members and the weights of any objects that are permanently attached to the structure. For example, columns, beams, girders, the floor slab, roofing, walls, windows, plumbing, electrical fixtures, and other miscellaneous attachments. The second type of loads are live loads which vary in their magnitude and location. There are many different types of live loads like building loads, highway bridge loads, railroad bridge loads, impact loads, wind loads, snow loads, earthquake loads, and other natural loads. </p> <div class="mw-heading mw-heading2"><h2 id="Analytical_methods">Analytical methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=4" title="Edit section: Analytical methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To perform an accurate analysis a structural engineer must determine information such as <a href="/wiki/Structural_load" title="Structural load">structural loads</a>, <a href="/wiki/List_of_structural_elements" class="mw-redirect" title="List of structural elements">geometry</a>, support conditions, and material properties. The results of such an analysis typically include support reactions, <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stresses</a> and <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacements</a>. This information is then compared to criteria that indicate the conditions of failure. Advanced structural analysis may examine <a href="/wiki/Dynamic_response" class="mw-redirect" title="Dynamic response">dynamic response</a>, <a href="/wiki/Buckling" title="Buckling">stability</a> and <a href="/wiki/Non-linear" class="mw-redirect" title="Non-linear">non-linear</a> behavior. There are three approaches to the analysis: the <a href="/wiki/Strength_of_materials" title="Strength of materials">mechanics of materials</a> approach (also known as strength of materials), the <a href="/wiki/3-D_elasticity" class="mw-redirect" title="3-D elasticity">elasticity theory</a> approach (which is actually a special case of the more general field of <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a>), and the <a href="/wiki/Finite_element" class="mw-redirect" title="Finite element">finite element</a> approach. The first two make use of analytical formulations which apply mostly simple linear elastic models, leading to closed-form solutions, and can often be solved by hand. The finite element approach is actually a numerical method for solving differential equations generated by theories of mechanics such as elasticity theory and strength of materials. However, the finite-element method depends heavily on the processing power of computers and is more applicable to structures of arbitrary size and complexity. </p><p>Regardless of approach, the formulation is based on the same three fundamental relations: <a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">equilibrium</a>, <a href="/wiki/Constitutive_equation" title="Constitutive equation">constitutive</a>, and <a href="/wiki/Compatibility_(mechanics)" title="Compatibility (mechanics)">compatibility</a>. The solutions are approximate when any of these relations are only approximately satisfied, or only an approximation of reality. </p> <div class="mw-heading mw-heading3"><h3 id="Limitations">Limitations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=5" title="Edit section: Limitations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each method has noteworthy limitations. The method of mechanics of materials is limited to very simple structural elements under relatively simple loading conditions. The structural elements and loading conditions allowed, however, are sufficient to solve many useful engineering problems. The theory of elasticity allows the solution of structural elements of general geometry under general loading conditions, in principle. Analytical solution, however, is limited to relatively simple cases. The solution of elasticity problems also requires the solution of a system of partial differential equations, which is considerably more mathematically demanding than the solution of mechanics of materials problems, which require at most the solution of an ordinary differential equation. The finite element method is perhaps the most restrictive and most useful at the same time. This method itself relies upon other structural theories (such as the other two discussed here) for equations to solve. It does, however, make it generally possible to solve these equations, even with highly complex geometry and loading conditions, with the restriction that there is always some numerical error. Effective and reliable use of this method requires a solid understanding of its limitations. </p> <div class="mw-heading mw-heading2"><h2 id="Strength_of_materials_methods_(classical_methods)"><span id="Strength_of_materials_methods_.28classical_methods.29"></span>Strength of materials methods (classical methods)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=6" title="Edit section: Strength of materials methods (classical methods)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The simplest of the three methods here discussed, the mechanics of materials method is available for simple structural members subject to specific loadings such as axially loaded bars, prismatic <a href="/wiki/Beam_(structure)" title="Beam (structure)">beams</a> in a state of <a href="/wiki/Pure_bending" title="Pure bending">pure bending</a>, and circular shafts subject to torsion. The solutions can under certain conditions be superimposed using the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</a> to analyze a member undergoing combined loading. Solutions for special cases exist for common structures such as thin-walled pressure vessels. </p><p>For the analysis of entire systems, this approach can be used in conjunction with statics, giving rise to the <i>method of sections</i> and <i>method of joints</i> for <a href="/wiki/Truss" title="Truss">truss</a> analysis, <a href="/wiki/Moment_distribution_method" title="Moment distribution method">moment distribution method</a> for small rigid frames, and <i>portal frame</i> and <i>cantilever method</i> for large rigid frames. Except for moment distribution, which came into use in the 1930s, these methods were developed in their current forms in the second half of the nineteenth century. They are still used for small structures and for preliminary design of large structures. </p><p>The solutions are based on linear isotropic infinitesimal elasticity and Euler–Bernoulli beam theory. In other words, they contain the assumptions (among others) that the materials in question are elastic, that stress is related linearly to strain, that the material (but not the structure) behaves identically regardless of direction of the applied load, that all <a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">deformations</a> are small, and that beams are long relative to their depth. As with any simplifying assumption in engineering, the more the model strays from reality, the less useful (and more dangerous) the result. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=7" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are 2 commonly used methods to find the truss element forces, namely the method of joints and the method of sections. Below is an example that is solved using both of these methods. The first diagram below is the presented problem for which the truss element forces have to be found. The second diagram is the loading diagram and contains the reaction forces from the joints. </p> <dl><dd><figure class="mw-halign-center mw-image-border" typeof="mw:File"><a href="/wiki/File:Truss_Structure_Analysis,_Full_Figure2.jpg" class="mw-file-description" title="A simple triangular truss with loads imposed ."><img alt="A simple triangular truss with loads imposed ." src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Truss_Structure_Analysis%2C_Full_Figure2.jpg/638px-Truss_Structure_Analysis%2C_Full_Figure2.jpg" decoding="async" width="638" height="450" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/3c/Truss_Structure_Analysis%2C_Full_Figure2.jpg 1.5x" data-file-width="652" data-file-height="460" /></a><figcaption>A simple triangular truss with loads imposed .</figcaption></figure></dd></dl> <p>Since there is a pin joint at A, it will have 2 reaction forces. One in the x direction and the other in the y direction. At point B, there is a roller joint and hence only 1 reaction force in the y direction. Assuming these forces to be in their respective positive directions (if they are not in the positive directions, the value will be negative). </p> <dl><dd><figure class="mw-halign-center mw-image-border" typeof="mw:File"><a href="/wiki/File:Truss_Structure_Analysis,_FBD2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/12/Truss_Structure_Analysis%2C_FBD2.jpg" decoding="async" width="467" height="450" class="mw-file-element" data-file-width="434" data-file-height="418" /></a><figcaption></figcaption></figure></dd></dl> <p>Since the system is in static equilibrium, the sum of forces in any direction is zero and the sum of moments about any point is zero. Therefore, the magnitude and direction of the reaction forces can be calculated. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum M_{A}=0=-10*1+2*R_{B}\Rightarrow R_{B}=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mo>−</mo> <mn>10</mn> <mo>∗</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>∗</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum M_{A}=0=-10*1+2*R_{B}\Rightarrow R_{B}=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9a6b1a1bac4d5da7564bfda8cd764a284dec043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:42.87ex; height:3.843ex;" alt="{\displaystyle \sum M_{A}=0=-10*1+2*R_{B}\Rightarrow R_{B}=5}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{y}=0=R_{Ay}+R_{B}-10\Rightarrow R_{Ay}=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>−</mo> <mn>10</mn> <mo stretchy="false">⇒</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{y}=0=R_{Ay}+R_{B}-10\Rightarrow R_{Ay}=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83989dc58a84678c90c9d518e3b47350a582b56b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:40.861ex; height:3.843ex;" alt="{\displaystyle \sum F_{y}=0=R_{Ay}+R_{B}-10\Rightarrow R_{Ay}=5}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{x}=0=R_{Ax}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{x}=0=R_{Ax}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b843188ec6205176532da94c3a6f771e2b9b50aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.938ex; height:3.843ex;" alt="{\displaystyle \sum F_{x}=0=R_{Ax}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Method_of_joints">Method of joints</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=8" title="Edit section: Method of joints"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This type of method uses the force balance in the x and y directions at each of the joints in the truss structure. </p> <dl><dd><span class="mw-image-border" typeof="mw:File"><a href="/wiki/File:Truss_Structure_Analysis,_Method_of_Joints2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Truss_Structure_Analysis%2C_Method_of_Joints2.png/350px-Truss_Structure_Analysis%2C_Method_of_Joints2.png" decoding="async" width="350" height="358" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/af/Truss_Structure_Analysis%2C_Method_of_Joints2.png 1.5x" data-file-width="418" data-file-height="427" /></a></span></dd></dl> <p>At A, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{y}=0=R_{Ay}+F_{AD}\sin(60)=5+F_{AD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{AD}=-{\frac {10}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>D</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{y}=0=R_{Ay}+F_{AD}\sin(60)=5+F_{AD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{AD}=-{\frac {10}{\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aef1b754ffbee9b6c1ac32aaa667989521be10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:64.36ex; height:6.843ex;" alt="{\displaystyle \sum F_{y}=0=R_{Ay}+F_{AD}\sin(60)=5+F_{AD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{AD}=-{\frac {10}{\sqrt {3}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{x}=0=R_{Ax}+F_{AD}\cos(60)+F_{AB}=0-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>D</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{x}=0=R_{Ax}+F_{AD}\cos(60)+F_{AB}=0-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ba7cea95e2ef40c429fa6a7415ae64ef2316d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:74.713ex; height:6.176ex;" alt="{\displaystyle \sum F_{x}=0=R_{Ax}+F_{AD}\cos(60)+F_{AB}=0-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}"></span></dd></dl> <p>At D, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{y}=0=-10-F_{AD}\sin(60)-F_{BD}\sin(60)=-10-\left(-{\frac {10}{\sqrt {3}}}\right){\frac {\sqrt {3}}{2}}-F_{BD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mo>−</mo> <mn>10</mn> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>D</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>10</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{y}=0=-10-F_{AD}\sin(60)-F_{BD}\sin(60)=-10-\left(-{\frac {10}{\sqrt {3}}}\right){\frac {\sqrt {3}}{2}}-F_{BD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f305c2a064fa86f6130c2868cc12f99bef771ac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:98.325ex; height:6.843ex;" alt="{\displaystyle \sum F_{y}=0=-10-F_{AD}\sin(60)-F_{BD}\sin(60)=-10-\left(-{\frac {10}{\sqrt {3}}}\right){\frac {\sqrt {3}}{2}}-F_{BD}{\frac {\sqrt {3}}{2}}\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{x}=0=-F_{AD}\cos(60)+F_{BD}\cos(60)+F_{CD}=-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{CD}\Rightarrow F_{CD}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>D</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{x}=0=-F_{AD}\cos(60)+F_{BD}\cos(60)+F_{CD}=-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{CD}\Rightarrow F_{CD}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/239c461ab6d797656d1fe9d68bd7f628f662e7be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:88.515ex; height:6.176ex;" alt="{\displaystyle \sum F_{x}=0=-F_{AD}\cos(60)+F_{BD}\cos(60)+F_{CD}=-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{CD}\Rightarrow F_{CD}=0}"></span></dd></dl> <p>At C, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{y}=0=-F_{BC}\Rightarrow F_{BC}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>C</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{y}=0=-F_{BC}\Rightarrow F_{BC}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d850817d66cb01a338321d69f951982cdb28556c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.775ex; height:3.843ex;" alt="{\displaystyle \sum F_{y}=0=-F_{BC}\Rightarrow F_{BC}=0}"></span></dd></dl> <p>Although the forces in each of the truss elements are found, it is a good practice to verify the results by completing the remaining force balances. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{x}=-F_{CD}=-0=0\Rightarrow verified}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>v</mi> <mi>e</mi> <mi>r</mi> <mi>i</mi> <mi>f</mi> <mi>i</mi> <mi>e</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{x}=-F_{CD}=-0=0\Rightarrow verified}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/239e497bb13cc82ec8f0e00d1d62cb1c87a18e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:38.039ex; height:3.843ex;" alt="{\displaystyle \sum F_{x}=-F_{CD}=-0=0\Rightarrow verified}"></span></dd></dl> <p>At B, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{y}=R_{B}+F_{BD}\sin(60)+F_{BC}=5+\left(-{\frac {10}{\sqrt {3}}}\right){\frac {\sqrt {3}}{2}}+0=0\Rightarrow verified}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>v</mi> <mi>e</mi> <mi>r</mi> <mi>i</mi> <mi>f</mi> <mi>i</mi> <mi>e</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{y}=R_{B}+F_{BD}\sin(60)+F_{BC}=5+\left(-{\frac {10}{\sqrt {3}}}\right){\frac {\sqrt {3}}{2}}+0=0\Rightarrow verified}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67658a576bc487489b6154d1ebcab583315626ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:75.151ex; height:6.843ex;" alt="{\displaystyle \sum F_{y}=R_{B}+F_{BD}\sin(60)+F_{BC}=5+\left(-{\frac {10}{\sqrt {3}}}\right){\frac {\sqrt {3}}{2}}+0=0\Rightarrow verified}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{x}=-F_{AB}-F_{BD}\cos(60)={\frac {5}{\sqrt {3}}}-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}=0\Rightarrow verified}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>v</mi> <mi>e</mi> <mi>r</mi> <mi>i</mi> <mi>f</mi> <mi>i</mi> <mi>e</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{x}=-F_{AB}-F_{BD}\cos(60)={\frac {5}{\sqrt {3}}}-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}=0\Rightarrow verified}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f598e001a6a6943f0a12b0188cc7969c42f922d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:62.454ex; height:6.176ex;" alt="{\displaystyle \sum F_{x}=-F_{AB}-F_{BD}\cos(60)={\frac {5}{\sqrt {3}}}-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}=0\Rightarrow verified}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Method_of_sections">Method of sections</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=9" title="Edit section: Method of sections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This method can be used when the truss element forces of only a few members are to be found. This method is used by introducing a single straight line cutting through the member whose force has to be calculated. However this method has a limit in that the cutting line can pass through a maximum of only 3 members of the truss structure. This restriction is because this method uses the force balances in the x and y direction and the moment balance, which gives a maximum of 3 equations to find a maximum of 3 unknown truss element forces through which this cut is made. Find the forces FAB, FBD and FCD in the above example </p> <div class="mw-heading mw-heading5"><h5 id="Method_1:_Ignore_the_right_side">Method 1: Ignore the right side</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=10" title="Edit section: Method 1: Ignore the right side"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mw-image-border" typeof="mw:File"><a href="/wiki/File:Truss_Structure_Analysis,_Method_of_Sections_Left2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/62/Truss_Structure_Analysis%2C_Method_of_Sections_Left2.jpg" decoding="async" width="385" height="450" class="mw-file-element" data-file-width="359" data-file-height="419" /></a></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum M_{D}=0=-5*1+{\sqrt {3}}*F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mo>−</mo> <mn>5</mn> <mo>∗</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>∗</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum M_{D}=0=-5*1+{\sqrt {3}}*F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47ae32688ca60353f08f5c646800af1a7e99a88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:48.47ex; height:6.176ex;" alt="{\displaystyle \sum M_{D}=0=-5*1+{\sqrt {3}}*F_{AB}\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{y}=0=R_{Ay}-F_{BD}\sin(60)-10=5-F_{BD}{\frac {\sqrt {3}}{2}}-10\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>10</mn> <mo>=</mo> <mn>5</mn> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>10</mn> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{y}=0=R_{Ay}-F_{BD}\sin(60)-10=5-F_{BD}{\frac {\sqrt {3}}{2}}-10\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80030b4e0b91003c1c6b4b2580fceb40fca8442d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:74.735ex; height:6.843ex;" alt="{\displaystyle \sum F_{y}=0=R_{Ay}-F_{BD}\sin(60)-10=5-F_{BD}{\frac {\sqrt {3}}{2}}-10\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{x}=0=F_{AB}+F_{BD}\cos(60)+F_{CD}={\frac {5}{\sqrt {3}}}-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{CD}\Rightarrow F_{CD}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{x}=0=F_{AB}+F_{BD}\cos(60)+F_{CD}={\frac {5}{\sqrt {3}}}-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{CD}\Rightarrow F_{CD}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a045870f4898b9bb64ad634b3fad50bbf01f170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:75.155ex; height:6.176ex;" alt="{\displaystyle \sum F_{x}=0=F_{AB}+F_{BD}\cos(60)+F_{CD}={\frac {5}{\sqrt {3}}}-{\frac {10}{\sqrt {3}}}{\frac {1}{2}}+F_{CD}\Rightarrow F_{CD}=0}"></span></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Method_2:_Ignore_the_left_side">Method 2: Ignore the left side</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=11" title="Edit section: Method 2: Ignore the left side"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span class="mw-image-border" typeof="mw:File"><a href="/wiki/File:Truss_Structure_Analysis,_Method_of_Sections_Right2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6c/Truss_Structure_Analysis%2C_Method_of_Sections_Right2.jpg" decoding="async" width="351" height="450" class="mw-file-element" data-file-width="290" data-file-height="372" /></a></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum M_{B}=0={\sqrt {3}}*F_{CD}\Rightarrow F_{CD}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>∗</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> </msub> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum M_{B}=0={\sqrt {3}}*F_{CD}\Rightarrow F_{CD}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd96ec1221cfc69cbb3eeabb42eddf584cddbae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:36.676ex; height:3.843ex;" alt="{\displaystyle \sum M_{B}=0={\sqrt {3}}*F_{CD}\Rightarrow F_{CD}=0}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{y}=0=F_{BD}\sin(60)+R_{B}=F_{BD}{\frac {\sqrt {3}}{2}}+5\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mn>5</mn> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{y}=0=F_{BD}\sin(60)+R_{B}=F_{BD}{\frac {\sqrt {3}}{2}}+5\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d47dc1848f7f2ce876ea64945ec8537fcfc534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:63.602ex; height:6.843ex;" alt="{\displaystyle \sum F_{y}=0=F_{BD}\sin(60)+R_{B}=F_{BD}{\frac {\sqrt {3}}{2}}+5\Rightarrow F_{BD}=-{\frac {10}{\sqrt {3}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum F_{x}=0=-F_{AB}-F_{BD}\cos(60)-F_{CD}=-F_{AB}-\left(-{\frac {10}{\sqrt {3}}}\right){\frac {1}{2}}-0\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∑</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mi>D</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>60</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> <mi>D</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum F_{x}=0=-F_{AB}-F_{BD}\cos(60)-F_{CD}=-F_{AB}-\left(-{\frac {10}{\sqrt {3}}}\right){\frac {1}{2}}-0\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cad06df2c2f60f5675af6ce815d7a9f696f0265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:84.128ex; height:6.509ex;" alt="{\displaystyle \sum F_{x}=0=-F_{AB}-F_{BD}\cos(60)-F_{CD}=-F_{AB}-\left(-{\frac {10}{\sqrt {3}}}\right){\frac {1}{2}}-0\Rightarrow F_{AB}={\frac {5}{\sqrt {3}}}}"></span></dd></dl> <p>The truss elements forces in the remaining members can be found by using the above method with a section passing through the remaining members. </p> <div class="mw-heading mw-heading2"><h2 id="Elasticity_methods">Elasticity methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=12" title="Edit section: Elasticity methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Elasticity methods are available generally for an elastic solid of any shape. Individual members such as beams, columns, shafts, plates and shells may be modeled. The solutions are derived from the equations of <a href="/wiki/Linear_elasticity" title="Linear elasticity">linear elasticity</a>. The equations of elasticity are a system of 15 partial differential equations. Due to the nature of the mathematics involved, analytical solutions may only be produced for relatively simple geometries. For complex geometries, a numerical solution method such as the finite element method is necessary. </p> <div class="mw-heading mw-heading2"><h2 id="Methods_using_numerical_approximation">Methods using numerical approximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=13" title="Edit section: Methods using numerical approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is common practice to use approximate solutions of differential equations as the basis for structural analysis. This is usually done using numerical approximation techniques. The most commonly used numerical approximation in structural analysis is the <a href="/wiki/Finite_Element_Method" class="mw-redirect" title="Finite Element Method">Finite Element Method</a>. </p><p>The finite element method approximates a structure as an assembly of elements or components with various forms of connection between them and each element of which has an associated stiffness. Thus, a continuous system such as a plate or shell is modeled as a discrete system with a finite number of elements interconnected at finite number of nodes and the overall stiffness is the result of the addition of the stiffness of the various elements. The behaviour of individual elements is characterized by the element's stiffness (or flexibility) relation. The assemblage of the various stiffness's into a master stiffness matrix that represents the entire structure leads to the system's stiffness or flexibility relation. To establish the stiffness (or flexibility) of a particular element, we can use the <i>mechanics of materials</i> approach for simple one-dimensional bar elements, and the <i>elasticity approach</i> for more complex two- and three-dimensional elements. The analytical and computational development are best effected throughout by means of <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix algebra</a>, solving <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. </p><p>Early applications of matrix methods were applied to articulated frameworks with truss, beam and column elements; later and more advanced matrix methods, referred to as "<a href="/wiki/Finite_element_method_in_structural_mechanics" title="Finite element method in structural mechanics">finite element analysis</a>", model an entire structure with one-, two-, and three-dimensional elements and can be used for articulated systems together with continuous systems such as a <a href="/wiki/Pressure_vessel" title="Pressure vessel">pressure vessel</a>, plates, shells, and three-dimensional solids. Commercial computer software for structural analysis typically uses matrix finite-element analysis, which can be further classified into two main approaches: the displacement or <a href="/wiki/Stiffness_method" class="mw-redirect" title="Stiffness method">stiffness method</a> and the force or <a href="/wiki/Flexibility_method" title="Flexibility method">flexibility method</a>. The stiffness method is the most popular by far thanks to its ease of implementation as well as of formulation for advanced applications. The finite-element technology is now sophisticated enough to handle just about any system as long as sufficient computing power is available. Its applicability includes, but is not limited to, linear and non-linear analysis, solid and fluid interactions, materials that are isotropic, orthotropic, or anisotropic, and external effects that are static, dynamic, and environmental factors. This, however, does not imply that the computed solution will automatically be reliable because much depends on the model and the reliability of the data input. </p> <div class="mw-heading mw-heading2"><h2 id="Timeline">Timeline</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=14" title="Edit section: Timeline"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>1452–1519 <a href="/wiki/Leonardo_da_Vinci" title="Leonardo da Vinci">Leonardo da Vinci</a> made many contributions</li> <li>1638: <a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo Galilei</a> published the book "<a href="/wiki/Two_New_Sciences" title="Two New Sciences">Two New Sciences</a>" in which he examined the failure of simple structures</li> <li>1660: <a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a> by <a href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</a></li> <li>1687: <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> published "<i><a href="/wiki/Philosophiae_Naturalis_Principia_Mathematica" class="mw-redirect" title="Philosophiae Naturalis Principia Mathematica">Philosophiae Naturalis Principia Mathematica</a></i>" which contains the <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a></li> <li>1750: <a href="/wiki/Euler%E2%80%93Bernoulli_beam_equation" class="mw-redirect" title="Euler–Bernoulli beam equation">Euler–Bernoulli beam equation</a></li> <li>1700–1782: <a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a> introduced the principle of <a href="/wiki/Virtual_work" title="Virtual work">virtual work</a></li> <li>1707–1783: <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> developed the theory of <a href="/wiki/Buckling" title="Buckling">buckling</a> of columns</li> <li>1826: <a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Claude-Louis Navier</a> published a treatise on the elastic behaviors of structures</li> <li>1873: <a href="/wiki/Carlo_Alberto_Castigliano" title="Carlo Alberto Castigliano">Carlo Alberto Castigliano</a> presented his dissertation "<i>Intorno ai sistemi elastici</i>", which contains <a href="/wiki/Castigliano%27s_method" title="Castigliano's method">his theorem</a> for computing displacement as partial derivative of the strain energy. This theorem includes the method of 'least work' as a special case</li> <li>1878-1972 <a href="/wiki/Stephen_Timoshenko" title="Stephen Timoshenko">Stephen Timoshenko</a> father of modern <a href="/wiki/Applied_mechanics" title="Applied mechanics">Applied mechanics</a> including the <a href="/wiki/Timoshenko%E2%80%93Ehrenfest_beam_theory" title="Timoshenko–Ehrenfest beam theory">Timoshenko–Ehrenfest beam theory</a></li> <li>1936: <a href="/wiki/Hardy_Cross" title="Hardy Cross">Hardy Cross</a>' publication of the moment distribution method which was later recognized as a form of the relaxation method applicable to the problem of flow in pipe-network</li> <li>1941: <a href="/wiki/Alexander_Hrennikoff" title="Alexander Hrennikoff">Alexander Hrennikoff</a> submitted his D.Sc. thesis in <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">MIT</a> on the discretization of plane elasticity problems using a lattice framework</li> <li>1942: <a href="/wiki/Richard_Courant" title="Richard Courant">R. Courant</a> divided a domain into finite subregions</li> <li>1956: J. Turner, <a href="/wiki/Ray_W._Clough" class="mw-redirect" title="Ray W. Clough">R. W. Clough</a>, H. C. Martin, and L. J. Topp's paper on the "Stiffness and Deflection of Complex Structures" introduces the name "finite-element method" and is widely recognized as the first comprehensive treatment of the method as it is known today</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Geometrically_and_materially_nonlinear_analysis_with_imperfections_included" title="Geometrically and materially nonlinear analysis with imperfections included">Geometrically and materially nonlinear analysis with imperfections included</a></li> <li><a href="/wiki/Limit_state_design" title="Limit state design">Limit state design</a></li> <li><a href="/wiki/Structural_engineering_theory" title="Structural engineering theory">Structural engineering theory</a></li> <li><a href="/wiki/Structural_integrity_and_failure" title="Structural integrity and failure">Structural integrity and failure</a></li> <li><a href="/wiki/Stress%E2%80%93strain_analysis" title="Stress–strain analysis">Stress–strain analysis</a></li> <li><a href="/wiki/Von_Mises_yield_criterion" title="Von Mises yield criterion">von Mises yield criterion</a></li> <li><a href="https://en.wikiversity.org/wiki/Probabilistic_Assessment_of_Structures" class="extiw" title="v:Probabilistic Assessment of Structures">Probabilistic Assessment of Structures</a></li> <li><a href="/wiki/Structural_testing" title="Structural testing">Structural testing</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Structural_analysis&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span 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